Spintronics: Materials, Applications, and Devices

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SPINTRONICS: MATERIALS, APPLICATIONS, AND DEVICES No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

SPINTRONICS: MATERIALS, APPLICATIONS, AND DEVICES

GIULIA C. LOMBARDI AND

GINEVRA E. BIANCHI EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Lombardi, Giulia C. Spintronics : materials, applications, and devices / Giulia C. Lombardi and Ginevra E. Bianchi. p. cm. ISBN 978-1-61668-279-8 (E-Book) I. Bianchi, Ginevra E. II. Title. TK7874.887.L66 2008 621.381--dc22 2008019258

Published by Nova Science Publishers, Inc. Ô  New York

CONTENTS Preface

vii

Chapter 1

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides Yoshihide Kimishima

Chapter 2

Weak High-Temperature Bulk Ferromagnetism in Mn-Doped CdGeAs2 Semiconductors Vyacheslav G. Storchak, Jess H. Brewer, Dmitri G. Eshchenko, Scott L. Stubbs, Stephen P. Cottrell, Andrey A. Nikonov, Oleg E. Parfenov and Sergey F. Marenkin

49

Chapter 3

Ferromagnetism in Semiconductors Doped with Non-magnetic Elements Y.P. Feng, H. Pan, R.Q. Wu, L. Shen, J. Ding, J.B. Yi and Y.H. Wu

59

Chapter 4

Magnetic Resonance and Spin-Wave (Magnon) Excitations in Ferromagnetic Semiconductor (Ga,Mn)As Xinyu Liu, Y.-Y. Zhou, J.K. Furdyna, Eric C.T. Harley and L.E. McNeil

79

Chapter 5

Spin Transfer Torque Effect and Its Applications Haiwen Xi, Xiaobin Wang, Yuankai Zheng and Xiaohua Lou

97

Chapter 6

Bloch Sphere, Spin Surfaces and Spin Precession in Quantum Wells A. Dargys

141

Chapter 7

Electrical and Magnetic Properties of Nano-scale Samanta Piano

Chapter 8

Fundamentals of Half-Metallic Full-Heusler Alloys K. Özdoğan, E. Şaşıoğlu and I. Galanakis

1

π -junctions

185 213

vi

Contents

Chapter 9

Spin Drift–Diffusion and Its Crossover in Semiconductors M. Idrish Miah

227

Chapter 10

Electronics, Spintronics and Orbitronics M. Idrish Miah

241

Index

245

PREFACE “Spintronics” is based on the control and manipulation of electron spin instead of, or in addition to, its charge, leading to novel electronic devices (spintronic devices) such as spin field effect transistors (SFETs), spin storage/ memory devices or spin quantum computers, which hold the promise of reduced power consumption, faster operation and smaller size. This new book provides leading-edge research from around the globe on the materials, applications and devices associated with spintronics. Perfect spin polarization of half metal suggests that there is a large magneto-resistance (MR) effect, making these oxides applicable to spintronics devices such as spin valve and the new magnetic storage device of magnetic random access memory (MRAM). The intergranular room temperature (RT)-MR effects in the Cr2O5/CrO2, MoO2/CrO2, MoxCr1-xOy and Fe3O4/CrO2 system are described, and the intergranular RT-MR effects in the Fe3O4/Ag system are also discussed in Chapter 1. Granular MR-ratio (MRR) were observed as -0.23 to -0.45% at room-temperature for some [Cr2O5/CrO2] and [(Cr2O3+Cr2O5)/CrO2] samples. These results indicate that Cr2O5 also forms efficient tunnel barriers between CrO2 grains, probably due to the paramagnetic property. Mixtures of metallic Pauli paramagnet MoO2 and half-metallic ferromagnet CrO2 were prepared by planetary ball milling. Magnetization M and |MRR| of (MoO2)x(CrO2)1-x steeply decreased with increasing MoO2 content and disappeared at x~0.7. At values of x above 0.6, the remanent magnetization Mr nearly disappeared but the saturation magnetization Ms kept a finite value, where the CrO2 nano-particles or small crystalline parts contributed to weak ferromagnetism. At x=0.05, anomalous rpm-dependence was observed for the coercive force Hc in this system. Next, MoO3 was mixed with CrO2, and the effects for the magnetism and conductivity were investigated. X ray diffraction, magnetization M, resistivity ρ and magneto-resistance ratio MRR were measured for MoxCr1-xOy. Tetragonal phase of CrO2-type transformed to monoclinic phase of MoO3(II)-type at x above 0.7. M and |MRR| rapidly decreased with increasing Mo content, and disappeared at x above 0.3. Above x = 0.3, trivalent Cr ions were assumed to be dominant for the magnetic property of present system. For the (Fe3O4)1-x(CrO2)x granular system, the expected inverse TMR effects were observed near or just below the percolation threshold of xc ~0.3. In the range 0.4 ≤ x ≤ 1, MR behaviors were characterized by low-resistive CrO2/I/CrO2 junctions. For x ≤ 0.2, highresistive Fe3O4/I/Fe3O4 junctions were dominant in the magnetic conduction behaviors. The

viii

Giulia C. Lombardi and Ginevra E. Bianchi

oxygen reduction of commercial Fe3O4 was useful for obtaining samples with visible and large inverse TMR effects. RT-MR were measured and discussed for some kinds of Fe3O4/Ag granular systems. Most of them commonly showed weak peaks of |MRR| and 0-field resistivity ρ 0 near the percolation threshold xc of about 0.2. The maximum |MRR| of about 5%, which is fairly high value for the granular MR of Fe3O4, was observed for nano-Fe3O4/Ag system prepared by sintering the mixture of Fe3O4 and Ag2O (samples II). It was also revealed that the site percolation problem for the mixed system, including two kinds of conductive particles with different diameters, is important to explain the MRR and ρ 0 behaviors in the granular systems. The remainder of AgCl was confirmed in the production process of nano-Fe3O4. In the present system, AgCl impurities contributed to large negative MR-ratio of about -5.5 % at 300 K. It was suggested that Fe3O4/AgCl/Fe3O4 paths were effective for the large MRR in nano-Fe3O4 system. Modern information technology utilizes electrons’ charge degrees of freedom to process information in semiconductors and their spin degrees of freedom to store information in magnetic materials. Materials in which the two degrees of freedom interact strongly offer applications known as “spintronics”. The chalcopyrite semiconductor CdGeAs2 is a promising candidate because it exhibits ferromagnetism up to ~350 K when doped with manganese. In Chapter 2, the authors have studied CdGeAs2:Mn with 18 and 36 mole% Mn using high transverse magnetic field muon spin relaxation, X-ray diffraction, magnetization, AC susceptibility, Auger spectroscopy and electronic transport measurements. The authors observe a muon spin precession frequency shift characteristic of a ferromagnetic medium, indicating that the ferromagnetism in CdGeAs:Mn is a bulk phenomenon. However, the net contribution to the magnetic field from dipole and contact hyperfine interactions is found to be at least 3-4 orders of magnitude weaker than in ferromagnetic d-metals like Fe, Co and Ni. Materials with specific magnetic properties are required for spin generation, manipulation, and detection for spintronics applications. Transition metal (TM) doped semiconductors and oxides, which were expected to meet this requirement, are facing problems due to formation of secondary phases and clusters. To overcome this problem, researchers have gone beyond traditional magnetic TM doped semiconductors and oxides and have started looking for alternative dopants. Some breakthroughs have been made recently. In Chapter 3, we review progress made along this direction and discuss possible origins of the unexpected ferromagnetism observed in materials without magnetic elements. They focus on dilute magnetic semiconductors and oxides obtained by doping with elements with a full dshell or 2p light elements. It has been demonstrated that room temperature ferromagnetism can be achieved in such systems. In Chapter 4 the authors review recent studies of spin (or magnetization) dynamics in thin layers of the ferromagnetic semiconductor (Ga,Mn)As, including Brillouin light scattering, ferromagnetic resonance, and time-resolved magneto-optical measurements carried out to investigate the various spin dynamic phenomena in this material. In most experiments the analysis of spin wave (magnon) modes in terms of the Landau-Lifshitz-Gilbert equation and specific boundary conditions allows one to determine the value of the exchange stiffness constant, along with the bulk and surface anisotropy in this material. The relationships between the results obtained by nonlinear optics and magnetic resonance, and between frequency and time-domain experiments are also discussed.

Preface

ix

In Chapter 5, the authors give a review of spin transfer torque effect in magnetic devices. Spin transfer torques arise from the interaction between the spins of conduction electrons and the local magnetic moments. Phenomena of spin transfer torques include current induced magnetization reversal and microwave excitation in magnetic nanostructures and current driven domain wall motion in magnetic wires, which will be discussed in detail. They will also discuss the implications of the effect in microelectronic applications and associated problems. The Bloch sphere represents all allowed spin states. It is often used to visualize dynamical spin properties since the points on the Bloch sphere, in contrast to Hilbert space, are directly related with the experiment. In semiconductors, where spin-orbit interaction may rearrange energy bands at critical points of the Brillouin zone, the spin is not a good quantum number and a more general form of the spin surface is required in this case to describe free electron spin properties in bulk semiconductors and quantum wells (QWs). In the first part of Chapter 6 the main properties of the Bloch sphere are summarized, with a particular emphasis on the free charge carrier spins in bulk semiconductors. Starting from the Kramers theorem for spin split bands it is shown that the spin surface, an analogue of the Bloch sphere, remains a useful concept in the analysis of dynamical properties of free carrier spins in crystalline solids as well. In the second part, to illustrate typical shapes of the spin surfaces and how they may be calculated, the properties of electron and hole spin surfaces in HgTe/HgCdTe quantum wells are considered in detail. The most interesting case, when the energy bands are inverted and the energy gap is negative, is analyzed using the eight-band k p Hamiltonian. Characteristic shapes of the spin surfaces are presented in a form of graphs. It is shown that, depending on the free electron (hole) wave vector magnitude and symmetry of the considered 2D energy subband, the spin surface may be an ellipsoid, disk, line, or reduced to the Bloch sphere. In the third part, the spin properties of a wider class of nanostructures (n- and p type GaAs/AlAs QWs, hollow cylindrical QWs) are briefly reviewed. The role of the spin surface as well as of natural quantization axis in the design of spintronics devices is discussed. The physics of the π phase shift in ferromagnetic Josephson junctions enables a range of applications for spin-electronic devices and quantum computing. In this respect the authors research is devoted to the evaluation of the best materials for the development and the realization of the quantum devices based on superconductors and at the same point towards the reduction of the size of the employed heterostructures towards and below nano-scale. In Chapter 7 the authors report our investigation of transitions from 0 to π states in Nb Josephson junctions with strongly ferromagnetic barriers of Co, Ni, Ni80Fe20 (Py) and Fe. They show that it is possible to fabricate nanostructured Nb/ Ni(Co, Py, Fe)/Nb π -junctions with a nano-scale magnetic dead layer and with a high level of control over the ferromagnetic barrier thickness variation. In agreement with the theoretical model they estimate, from the oscillations of the critical current as function of the ferromagnetic barrier thickness, the exchange energy of the ferromagnetic material and we obtain that it is close to bulk ferromagnetic materials implying that the ferromagnet is clean and S/F roughness is minimal. The authors conclude that S/F/S Josephson junctions are viable structures in the development of superconductor-based quantum electronic devices; in particular Nb/Co/Nb and Nb/Fe/Nb multilayers with their low value of the magnetic dead layer and high value of the exchange energy can readily be used in controllable two-level quantum information systems. In this

x

Giulia C. Lombardi and Ginevra E. Bianchi

respect, we discuss applications of our nano-junctions to engineering magnetoresistive devices such as programmable pseudo-spin-valve Josephson structures. Intermetallic Heusler alloys are amongst the most attractive half-metallic systems due to the high Curie temperatures and the structural similarity to the binary semiconductors. In Chapter 8 we present an overview of the basic electronic and magnetic properties of the halfmetallic full-Heusler alloys like Co2MnGe. Ab-initio results suggest that the electronic and magnetic properties in these compounds are intrinsically related to the appearance of the minority-spin gap. The total spin magnetic moment in the unit cell, Mt, scales linearly with the number of the valence electrons, Zt, such that Mt = Zt - 24 for the full-Heusler alloys opening the way to engineer new half-metallic alloys with the desired magnetic properties. Moreover, the authors present analytical results on the disorder in Co2Cr(Mn)Al(Si) alloys, which is susceptible to destroy the perfect half-metallicity of the bulk compounds and thus degrade the performance of devices. Finally we discuss the appearance of the half-metallic ferrimagnetism due to the creation of Cr(Mn) antisites in these compounds and the Co-doping in Mn2VAl(Si) alloys which leads to the fully-compensated half-metallic ferrimagnetism. In Chapter 9 a two-component drift-diffusion equation for spin density is derived and is used to model spin transport in nonmagnetic semiconductors in different transport regimes. The authors study spin current (js) and find that drift (j) and diffusion (jdi) currents contribute to js in the down-stream and up-stream directions and that jdi decreases with the electric field (E) while j increases and there is a spin drift-diffusion crossover field (Ec) in the down-stream direction. It is also found that js increases in the degenerate regime, suggesting a possible way for the enhancement of js in semiconductors. However, js in the up-stream direction is found to be vanished when E is very large. The authors derive the expressions for the intrinsic spin diffusion length (δs) of a semiconductor in different electron statistical regimes, namely nondegenerate, degenerate and highly degenerate and show that δs can be obtained directly from Ec. The results of the investigation are also shown to be useful in identifying whether the process for a given E would be in the spin drift ( j jdi ), spin drift-diffusion crossover ( j ≈ jdi ) or spin diffusive ( j jdi ) regime and in obtaining transport properties of the electron spin in semiconductors, the crucial requirements for practical spintronic devices. As explained in Chapter 10, the control and manipulation of electronic charge by electric fields is well established for electronics and the progress in conventional electronics has proven that the spin of the electron was ignored in past in the mainstream of electronics, because the electron has spin as well as charge. Electron spin (self-rotation - rotation of the electron on its axis) is a quantum mechanical property associated with its intrinsic angular momentum, and as an intrinsic magnetic moment is associated with spin, spin is closely related to magnetic phenomena.

In: Spintronics: Materials, Applications and Devices Editors: G.C. Lombardi and G.E. Bianchi

ISBN: 978-1-60456-734-2 © 2009 Nova Science Publishers, Inc.

Chapter 1

ROOM TEMPERATURE MAGNETO-RESISTANCE EFFECTS OF HALF-METALLIC OXIDES Yoshihide Kimishima Department of Physics, Division of Physics, Electrical and Computer Engineering, Graduate School of Engineering, Yokohama National University Tokiwadai 19-5, Hodogaya-ku, Yokohama 240-8501, JAPAN

Abstract Perfect spin polarization of half metal suggests that there is a large magneto-resistance (MR) effect, making these oxides applicable to spintronics devices such as spin valve and the new magnetic storage device of magnetic random access memory (MRAM). The intergranular room temperature (RT)-MR effects in the Cr2O5/CrO2, MoO2/CrO2, MoxCr1-xOy and Fe3O4/CrO2 system are described, and the intergranular RT-MR effects in the Fe3O4/Ag system are also discussed in this chapter. Granular MR-ratio (MRR) were observed as -0.23 to -0.45% at room-temperature for some [Cr2O5/CrO2] and [(Cr2O3+Cr2O5)/CrO2] samples. These results indicate that Cr2O5 also forms efficient tunnel barriers between CrO2 grains, probably due to the paramagnetic property. Mixtures of metallic Pauli paramagnet MoO2 and half-metallic ferromagnet CrO2 were prepared by planetary ball milling. Magnetization M and |MRR| of (MoO2)x(CrO2)1-x steeply decreased with increasing MoO2 content and disappeared at x~0.7. At values of x above 0.6, the remanent magnetization Mr nearly disappeared but the saturation magnetization Ms kept a finite value, where the CrO2 nano-particles or small crystalline parts contributed to weak ferromagnetism. At x=0.05, anomalous rpm-dependence was observed for the coercive force Hc in this system. Next, MoO3 was mixed with CrO2, and the effects for the magnetism and conductivity were investigated. X ray diffraction, magnetization M, resistivity ρ and magneto-resistance ratio MRR were measured for MoxCr1-xOy. Tetragonal phase of CrO2-type transformed to monoclinic phase of MoO3(II)-type at x above 0.7. M and |MRR| rapidly decreased with increasing Mo content, and disappeared at x above 0.3. Above x = 0.3, trivalent Cr ions were assumed to be dominant for the magnetic property of present system. For the (Fe3O4)1-x(CrO2)x granular system, the expected inverse TMR effects were observed near or just below the percolation threshold of xc ~0.3. In the range 0.4 ≤ x ≤ 1, MR behaviors were characterized by low-resistive CrO2/I/CrO2 junctions. For x ≤ 0.2, highresistive Fe3O4/I/Fe3O4 junctions were dominant in the magnetic conduction behaviors. The

Yoshihide Kimishima

2

oxygen reduction of commercial Fe3O4 was useful for obtaining samples with visible and large inverse TMR effects. RT-MR were measured and discussed for some kinds of Fe3O4/Ag granular systems. Most of them commonly showed weak peaks of |MRR| and 0-field resistivity ρ 0 near the percolation threshold xc of about 0.2. The maximum |MRR| of about 5%, which is fairly high value for the granular MR of Fe3O4, was observed for nano-Fe3O4/Ag system prepared by sintering the mixture of Fe3O4 and Ag2O (samples II). It was also revealed that the site percolation problem for the mixed system, including two kinds of conductive particles with different diameters, is important to explain the MRR and ρ 0 behaviors in the granular systems. The remainder of AgCl was confirmed in the production process of nano-Fe3O4. In the present system, AgCl impurities contributed to large negative MR-ratio of about -5.5 % at 300 K. It was suggested that Fe3O4/AgCl/Fe3O4 paths were effective for the large MRR in nano-Fe3O4 system.

1. Introduction Here the author reviews his studies on the magneto-resistance (MR) effects in the granular systems at room temperature, based on the half metallic oxides of CrO2 and Fe3O4. CrO2 is known as a half-metallic and ferromagnetic oxide with 100% spin polarization of conductive electrons. Band calculations for CrO2 have shown that the majority spin electrons have a metallic band structure, while the minority ones have a semiconductive energy gap at the Fermi level [1-5]. The half-metallic property of this compound has been confirmed by several experiments [6-11]. Many MR results have been reported in earlier papers [12-23]. At room temperature (RT), 0.1 to 1 μ m thick films showed metallic resistivity and MR ratios of -0.5% to -2% at 10 kOe [12,13,18,20,22], which were interpreted as indicating a double exchange mechanism [5], while cold pressed powder compacts with 0.1 to 2 μ m-long acicular grains showed intergranular resistivity and negative MR ratio (MRR) lower than 0.1% [15,17]. The MRR was defined here as [ ρ (H)-

ρ max] / ρ max, where ρ was the resistivity and ρ max was the maximum one in a low magnetic field region. The small RT-MRR was considered to be due to the spin-flip scattering of polarized electrons by the production of antiferromagnetic Cr2O3 in the intergrains between CrO2 grains [24,25]. Magnetite (Fe3O4) has been known as the other half metal with the spin polarized conduction electrons in the 3d minority spin bands [26,27]. In the recent years, the large negative MR effect in low field was expected, and many experimental results were reported for the Fe3O4 thin films [28-35]. However, small negative RT-MRR of about -1% have been observed for polycrystalline and single crystal thin films under the magnetic field of 1 T. Relatively high RT-|MRR|-results of about 5 % at 1 T have been reported for TMR in Fe3O4 films with oxide barriers such as MgO [36], γ -Fe2O3 [37-39] and ZnFe2O4 [40]. In the bulk granular system, the Zn0.41Fe2.59O4 with α -Fe2O3 grain boundaries [41] showed a largely negative RT-MRR of -61 % at 1 T, although it disappeared above 320 K. For nano-contacted Fe3O4 particles [42], the RT-MRR of -75 % at 8mT was reported, but the nano-contacting parts were unstable against a small external turbulence. The perfect spin polarization of half metal suggests that there is a large MR effect, making these oxides applicable to spintronics devices such as spin valve and the new

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

3

magnetic storage device of magnetic random access memory (MRAM). In Sections 2-5, the intergranular RT-MR effects in the Cr2O5/CrO2, MoO2/CrO2, MoxCr1-xOy and Fe3O4/CrO2 system are described. In Sections 6 and 7, the intergranular RT-MR effects in the Fe3O4/Ag system are discussed.

2. Intergranular Magneto-resistance of Half-Metallic Ferromagnet CrO2 with Cr2O5 Barriers It should be noted that chromium oxide has another stage of Cr2O5 adjacent to the CrO2phase in the oxygen pressure versus temperature (p-T) phase diagram [13,14,43]. Since Cr2O5 shows paramagnetism above 80 K [44], the suppression of spin-flip scattering and large RT-MR effects were expected in the Cr2O5 tunneling barrier between CrO2 grains. In this study, the efficiency of Cr2O5 tunneling barriers was confirmed first by the relatively large MR in the CrO2 granular system at 300 K. The method of sample preparation and experimental results of electronic and magnetic measurements are presented as following.

2.1. Sample Preparation Sintered granular samples of chromium oxide were prepared from the precursor of CrO3 in the micro-reactor. Referring the p-T phase diagram [43], twelve kinds of samples were obtained between 598 K and 723 K under an oxygen gas pressure of 0.5 MPa, and several samples were subjected to second sintering under an O2 gas pressure of 10 MPa. The sintering conditions of samples 1-12 are given in Table 2.1. Table 2.1. Sintering conditions of samples 1-12. No. 1 2 3 4 5 6 7 8 9 10 11 12

T1st/0.5Mpa (K) 598 623 610 623 648 623 610 648 673 673 698 723

T2nd/10Mpa (K) 623 598 610 648 673 -

The results of powder x-ray diffraction measurements are shown in Fig. 2.1 for samples 4 and 9, where the diffraction peaks of tetragonal CrO2 [45], monoclinic Cr2O5 [44], and

Yoshihide Kimishima

4

rhombohedral Cr2O3 [46] are indicated by the symbols *, +, and x, respectively. In Fig.2.2,

* CrO2 + Cr2O5 x Cr2O3

1.0

(110)

No. * 4 9 +

0.6

x

0.4 0.2

++

0.0 20

x

(104)

0.8

(-221)

Intensity (arb. units)

the intensity ratios of the ( 2 21)-peak of Cr2O5, the (110) peak of CrO2, and the (104) peak of Cr2O5 are depicted together with the room-temperature magneto-resistance ratios. The left longitudinal axis shows the relative intensity of I(CrnOm) normalized by Itot=[I(Cr2O5)+I(CrO2)+I(Cr2O3)]. Samples 1-3 were sintered once at temperatures between 598 K and 623 K under 0.5MPa O2. They were Cr2O5-rich samples with [I(Cr2O5)/Itot]>0.8 and [I(CrO2)/Itot]<0.2 Samples 10-12 were sintered at temperatures above 673 K with [I(Cr2O3)/Itot]>0.4 and [I(CrO2)/Itot]<0.6. All of these samples showed extreme high resistance above 10M Ω at 300 K, and R-T curves and MR could not be measured between 77 K and 300 K. Therefore, magnetization measurements were not performed for samples 1-3 and 10-12.

+

x

*

+ 30 2θ (deg)

40

Figure 2.1. Powder x-ray patterns of samples 4 and 9.

1 CrO

0.5 Cr O 2

5

(%)

Intensity ratio

0.4 0.2 0

-MRR

1.5

2

Cr O 2

3

0 1 2 3 4 5 6 7 8 9 10 11 12 Sample number Figure 2.2. Intensity ratios of three kinds of chromium oxide. Room temperature MRR values at 10 kOe are also given.

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

5

Samples 4-9 were magneto-resisitive ones which were sintered twice between 610K and 648K under 0.5MPa and 10MPa O2, except for samples 5 and 9, which were sintered once at 648K and 673K, respectively, under 0.5MPa O2. The ratios of I(Cr2O5):I(CrO2) were 0.34:0.66,0.32:0.68, 0.10:0.90, and 0.09:0.91 for samples 4,5,6, and 7, respectively. In sample 8, I(Cr2O5):I(CrO2): I(Cr2O3) was 0.03:0.94:0.03, while I(CrO2):I(Cr2O3) was 0.66:0.34 in sample 9. The averaged diameters of CrO2 particles are important for understanding the magneto-resistive behaviors in these samples. From the half-widths of the main peak of (110) in the rocking curves, the particle diameters d of CrO2 were estimated as about 15,50,10,14,50 and 50nm for samples 4,5,6,7,8 and 9, respectively. In the following sections, the electronic and magnetic properties of samples 4-9 are presented.

2.2. Experimental Results 2.2.1. Electrical Resistivity Electrical resistivities ρ were measured by the usual four-terminals method between 77 K and 300 K. In Fig.2.3, log ρ (T) of samples 4-9 is depicted for 1/ T , because the tunneling magneto-resistance (TMR) theory for the granular system [47] gives the resistivity as

ρ=

⎛ Δ⎞ ⎟ exp⎜⎜ ⎟ T 1+ P m ⎝ ⎠ ρ0

2

2

(1),

where ρ0 and Δ are constant values, P is the spin polarization coefficient of conduction electrons, and m is defined by M/Ms. The parameter Δ is defined by 8 κ C, where

κ = 2m * (V − E F ) / h 2 and C = sEc = const. The parameter m* is the effective mass of electrons, V is the barrier potential, EF is the Fermi energy, s is the barrier thickness, and Ec is the charging energy. Equation (1) is derived from the condition that s gives the maximum transmission coefficient of conduction electrons [48]. In this figure, the magnitude of ρ (T) is not of the same order as the content ratio of CrO2 in Fig.2.2. The resistivity ratio of ρ 77K/ ρ 300K is affected by the grain size d of CrO2. Samples 4-9 can be classified into two groups as regards d. One is the 10nm class of samples 4,6, and 7, and the other is the 50nm class of samples 5, 8 and 9. The resistivity ratios of 50nm-class samples were lower than those of 10-nm-class samples. In each class, ρ 77K/ ρ 300K became lower as the CrO2 content increased. The 10-nm-class samples showed granular tunneling behaviors of Equation (1) of ρ (T) ∝ exp

Δ T . The Δ -values were obtained as 92K, 29K, and 43K for the samples 4, 6 and

7, respectively, which reflecting the barrier thickness of Cr2O5 between CrO2 grains with temperature independent Coulomb energy, effective mass of electrons, and barrier height. These are consistent with previous results for CrO2 powder compacts with Cr2O3-barriers

Yoshihide Kimishima

6

[21]. Therefore we expected the granular tunnel MR especially for sample 4, 6, and 7 of the 10-nm -class between 77 K and 300 K.

10

log [ρ (Ωcm)]

No. 5 9

1

4 7 8 6

0.1 300 K

77 K

0.01 0.06

Figure 2.3.

0.08 0.1 1/2 -1/2 1/T (K )

0.12

ρ (T) curves of samples 4-9.

On the other hand, the ρ (T) of 50-nm-class samples showed metallic temperature dependences above 200K and granular ones between 77K and 150K. These behaviors are similar to those of CrO2 powder compact or granular films [49-51], where the conducting paths were formed at the areas of direct contact between CrO2 particles. If the contact resistance was dominant in samples 5, 8, and 9, both the relatively high resistivity and the metallic temperature dependence could be explained by the direct contacts of the large CrO2 particles above 200 K. At low temperatures below about 150 K, the contact resistance increased as a result of the shrinking of contact areas, and the tunnel resistance through the Cr2O5 or Cr2O3 barriers should be predominant. We were thus able to determine the Δ -values between 77 K and 150 K as shown in Table 2.3.

2.2.2. Magnetization The temperature and field dependences of the magnetization were measured by using a Quantum Design MPMS SQUID magnetometer. The M(T) and M(H) behaviors are shown in Figs.2.4 and 2.5, respectively. As shown in Fig.2.4, all of samples 4-9 showed ferromagnetic behavior at 300 K, and the saturation magnetization Ms depends on the CrO2 content. The ferromagnetic transition temperature Tc was near 400 K, and had no sample dependence, as shown in Fig.2.5. These results mean that crystallographic phases of CrO2, Cr2O5, and Cr2O3 are isolated from each other in these samples, and only the ferromagnetism of CrO2 was apparent. Cr2O5 is a Curie-Weiss paramagnet above 80 K, with an effective moment of 4.3 μ B and a Weiss temperature of -204 K [44]. On the other hand, Cr2O3 is known to be an antiferromagnet with TN near 310 K [52], and therefore the magnetic contribution of Cr2O5

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

7

and Cr2O3 are negligible. The coercive forces Hc of samples 4, 5, 6, 7, 8 and 9 were 220, 62, 152, 152, 117 and 85 Oe at 77 K, and 91, 42, 81, 82, 64 and 53 Oe at 300 K, respectively. The relatively large Hc of 10-nm-class samples 4, 6 and 7 indicate the formation of single ferromagnetic domains in CrO2 nano-grains.

No. 8 7 6 9 5 4

100

M (emu/g)

50 0 -50 -100

-40

-20

0 20 H (kOe)

40

Figure 2.4. M(H) curves of samples 4-9.

120

No.8

M (emu/g)

7 80

40

6 9 5 4

0

100

Figure 2.5. M(T) curves of samples 4-9.

200 300 T (K)

400

Yoshihide Kimishima

8

2.2.3. Magneto-resistance The field-dependent resistivity ρ (H) was also measured by the four terminals method at 77 K and 300 K under dc fields between -10 kOe and 10 kOe for samples 4-9. The magnetoresistance ratio MRR was determined from following definition of MRR=[ ρ (H) - ρ (Hp)] / ρ (Hp)

(2).

The MRR(%) at 77 K and 300 K is shown for eliminated samples in Figs.2.6 and 2.7, respectively, to avoid complication. Here ρ (Hp) is the peak value of ρ under a low field. At 77 K, all of the samples showed hysteretic ρ (H) with two maximums at about ± Hc of the coercive force. They exhibited tunnel magneto-resistive behaviors at 77 K, where the MRR rapidly decreased in the low field region and gradually decreased near 10kOe, as shown in Fig.2.6. The MRR of -2 to -3 % are comparable to the results obtained in earlier studies for Cr2O3/CrO2 powder compacts or polycrystalline films [14,15,17,21]. The TMR effects in samples 4, 5, 6 and 7 are the first reported in a Cr2O5/CrO2 granular system. As regards the room temperature (RT) MR, the 10-nm -class samples 4, 6 and 7 showed TMR-like behaviors, except for the 50-nm-class samples 5, 8 and 9. The RT-MRR of 50-nmclass samples with a metallic temperature dependence of ρ were decreased almost linearly with increasing magnetic field H at 300 K. These features of 50-nm-class samples are the same as the earlier results for CrO2 thin films on substrates of TiO2 or Al2O3 [13, 22]. In these films, the CrO2 grains have diameters of about 100 nm. Therefore, the MR of the 50-nm-class can be considered to be part of its intrinsic nature due to the double exchange mechanism [5] in CrO2 granular systems with diameters larger than about 50 nm.

MRR (%)

0.0 No. 5

-0.1 -0.2 300 K -0.3 -10

-5

Figure 2.6. Magneto-resistance ratio at 77 K.

0 5 H (kOe)

9 6 10

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

9

MRR (%)

0 -1 -2 77 K -3 -10

-5

0 5 H (kOe)

No. 5 6 4 9 10

Figure 2.7. Magneto-resistance ratio at 300 K.

In the theory of TMR in granular systems [47,48], the MR ratio was given by the relation of MRR=P2m2/(1+P2m2)

(3)

at constant temperature, where P is the spin polarization and m is the magnetization normalized by the saturation magnetization Ms. We can therefore estimate the effective polarization P from the experimental results of MRR and magnetization at constant temperature. The fitting curves for sample 6 are given in Figs.2.6 and 2.7. P values of 17% and 5% were obtained for 77 K and 300 K, respectively. Unexpectedly small effective polarization of electron spins may be due to the destruction of the boundary order between CrO2 and Cr2O5 grains. Nearly the same P-results were obtained for samples 4 and 7, as shown in Tables 2.2 and 2.3. The realization of granular MR behaviors at room temperature in the 10 nm class samples 4, 6, and 7 is probably due to the paramagnetic property of Cr2O5 grains. In contrast to the antiferromagnetically ordered Cr2O3 grains, Cr2O5 barriers may support elastic tunneling of polarized conduction electrons of CrO2. Table 2.2. Characteristic parameters at 77 K.

Δ

No.

d (nm)

MRR1T (%)

Hc (Oe)

Ms (emu/g)

(K)

4 5 6 7 8 9

15 50 10 14 50 50

-2.9 -2.3 -2.7 -3.4 -3.2 -3.2

220 62 152 152 117 85

35 52 73 95 113 56

92 29 29 43 14 14

P 0.192 0.155 0.172 0.188 0.180 0.185

Yoshihide Kimishima

10

Table 2.3. Characteristic parameters at 300K.

Δ

No.

d (nm)

MRR1T (%)

Hc (Oe)

Ms (emu/g)

(K)

4 5 6 7 8 9

15 50 10 14 50 50

-0.23 -0.13 -0.29 -0.24 -0.45 -0.25

91 42 81 82 64 53

25 42 53 73 87 44

92 – 29 43 – –

P 0.048 – 0.049 0.044 – –

2.3. Summary Magneto-resistances were measured at 77 K and 300 K for 100% spin-polarized half-metallic ferromagnet CrO2 grains with Cr2O5 or Cr2O3 insulating barriers. The samples prepared from the precursor of CrO3 in the micro-reactor showed granular MRR of -2.3 to -3.4 % at 77 K. Granular MRR were also observed as -0.23 to -0.45% at room-temperature for some [Cr2O5/CrO2] and [(Cr2O3+Cr2O5)/CrO2] samples. These results indicate that Cr2O5 also forms efficient tunnel barriers between CrO2 grains, probably due to the paramagnetic property at 77 K and 300 K. The spin flip scattering of spin-polarized electrons from CrO2 may be large in the antiferromagnetically ordered Cr2O3 barriers below 300 K. A larger RTMRR will be realized in [Cr2O5/CrO2] crystalline thin films by the use of the CVD method, among others.

3. Magnetization and Magneto-resistance in an MoO2/CrO2 Mixed System We found that a paramagnetic Cr2O5 barrier can enhance the |MRR| of a CrO2 system as described in Section 2. Impurity doping effects calculated for CrO2 by the DV-X α method have shown that Mo4+-doping causes a slight increase in the magnetic moment and Curie temperature Tc [53]. In this study, metallic dioxide of MoO2 was mixed with CrO2, and the effects on the magnetism and conductivity of the latter were investigated. MoO2 has a monoclinic crystal structure with lattice parameters of a = 0.56096 nm, b = 0.4857 nm, c = 0.56259 nm, and β = 120.912 deg [54]. CrO2 has a tetragonal crystal structure with a = 0.4419 nm and b = 0.29154 nm [45]. Both MoO2 and CrO2 belong to the Rutile structure family, but they have different energy band diagrams [55]. In metallic MoO2, one of the 4d 2 electrons is used for the metalmetal σ -bonding and the other 4d 2 electron partially fills the metal-oxygen π * band. In halfmetallic CrO2, all of the 3d 2 electrons exist in the majority up spin band and behave like as polarized conduction electrons. A previous study of the effect of doping CrO2 with a few percentage points of Mo6+(4d 0 ) revealed a steep disappearance of ferromagnetism along with a reduction in magnetization

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

11

and a lowering of the Curie temperature Tc [56]. We also expected to observe a so-called ferromagnetic quantum critical point (FQCP) [57] as a result of slight doping of CrO2-phase in an (MoO2)x(CrO2)1-x system with Mo4+. At the FQCP, the ferromagnetism of CrO2 disappears and, for example, a possibility of p-type superconductivity may occur. Here we will report the effects of mechanical milling on the conductivity and magnetism of MoO2 /CrO2 mixtures.

3.1. Sample Preparation and Experiment Commercial 99.9% CrO2 (Aldrich Chemical Company, Inc.) and 99.9% MoO2 (Soekawa Chemicals) were used as the precursor for (MoO2)x(CrO2)1-x mixed powder, where x is the molar concentration of MoO2. The x-values are between 0 and 1 with step of 0.1. The powder mixture of CrO2 and MoO2 was milled with a planetary ball mill (Fritch Pulverisette-7, Germany) using Cr-steel vials. The inner diameter and volume of each vial were 40 mm and 45cm3, respectively, and Cr-steel balls with a diameter of 15 mm were used as the grinding media. About 2 g of powder mixture was the starting material. The volume ratio of balls and powder was about 30:1. The rotation speed was controlled between 100 and 700rpm, and the rotation time was 30min. The ground products were characterized by x-ray powder diffraction (XRD), and were then mixed with an aqueous solution of polyvinyl alcohol (PVA) and pressed into a 1mm-thick pellets with a diameter of 5 mm. Measurements of the resistivity and magneto- resistance were performed for dried samples by the usual 4-terminals method in dc magnetic fields between -1 T and 1 T. A vibrating sample magnetometer (VSM) and a superconducting quantum interference device (SQUID) were used for the magnetization measurements.

3.2. Results and Discussion Figure 3.1 shows the CuK α x-ray powder diffraction patterns of (MoO2)0.05(CrO2)0.95 prepared by mechanical milling at 100,400 and 700rpm for 30min. The (110)-peak of CrO2 was greatly broadened at 400 and 700rpm. The sharpness of the (110)-peak of MoO2- phase was kept at 400rpm, but was followed by broadening at 700rpm. If we use Sherrer's formula for the half-width of XRD peaks, the mean crystallographic correlation length d of CrO2 can be estimated as 5-10nm as a result of milling at 400rpm. This shows that either CrO2 nano-particles or amorphous phase is produced in this sample. The d value of MoO2 particles did not decrease for rotation speeds up to 700 rpm, indicating that MoO2 particles are much harder than CrO2 particles. X-ray powder diffraction patterns of x=0-0.8, which were milled at 290 rpm for 30 min., are depicted in Fig.3.2. The (110) diffraction peak of CrO2 rapidly broadened at x=0.4 and disappeared at x=0.8. On the other hand, the (110) peak intensity of MoO2 rose as x increased, without broadening. This means that d in CrO2 was steeply reduced by milling with the harder MoO2 particles. At x=0.8, the CrO2 crystal phase becomes unstable, and may be completely transformed into amorphous phase.

Yoshihide Kimishima

12

600

Intensity

(MoO2)0.05(CrO2)0.95 100 rpm, 30 min 400 rpm 700 rpm

400

MoO2 (110)

200

0

CrO2 (110)

26

27

28 29 2θ (deg)

30

Figure 3.1. X-ray powder diffraction patterns of the x=0.05 sample after milling at 100-700 rpm for 30 min.

2000 MoO2 (110)

Intensity

1500

(MoO2)x(CrO2)1-x 290 rpm, 30 min x=0 0.1 0.4 0.8 CrO2 (110)

1000 500 0

26

27 28 2θ (deg)

29

Figure 3.2. X-ray powder diffraction patterns of x=0, 0.1, 0.4, and 0.8 samples after 290 rpm milling for 30 min.

In Fig.3.3, the field dependences of magnetization M at 300 K are shown for the samples of x = 0, 0.2, 0.4, and 0.6 prepared after 290 rpm milling for 30 min. According to the x-ray results of Fig.3.2, the ferromagnetic magnetization of CrO2 rapidly decreased as the MoO2

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

13

content increased. At x=0.6, the sample showed very weak ferromagnetism or superparamagnetism, which may be due to the production of CrO2 nano-particles or amorphous CrO2-phase by mechanical milling.

Magnetization (emu/g)

60

x=0 300 K

30

0.2

0

0.4 0.6

-30 -60 -2000 -1000 0 1000 Applied field (Oe)

2000

Figure 3.3. Field dependence of the magnetization M at 300 K for x=0, 0.2, 0.4, and 0.6 samples after 290 rpm milling for 30 min.

Magnetization (emu/g)

120

Ms 77 K 300 K Mr 77 K 300 K

80

40

0 0.0

0.2

0.4

0.6

0.8

1.0

x Figure 3.4. x-dependence of the saturated magnetization Ms and remanent magnetization Mr at 77 K and 300 K for the x=0 -1 samples after 290 rpm milling for 30 min.

Yoshihide Kimishima

14

The x-dependences of the saturation magnetization Ms and remanent magnetization Mr at 77 K and 300 K are shown in Fig.3.4 for x=0-1.0 samples, prepared by 290 rpm milling for 30 min. The Ms and Mr decreased non-linearly with x. At values of x above 0.6, the Mr nearly disappeared but the Ms kept a finite value. This may be due to the production of CrO2 nanoparticles or small crystalline CrO2-phase by mechanical milling with MoO2.

Magnetization (emu/g)

100

10 kOe

x=0 80 60

0.2

40 0.4 20 0

0.6 100 0.8 200 300 400 Temperature (K)

Figure 3.5. Temperature dependence of the magnetization M for x=0, 0.2, 0.4, 0.6, and 0.8 samples between 77 K and 300 K.

x 0.5 4

Resistivity (Ωcm)

10

0.3 2

10

0.1 0

0.7 0

10

0.9 -2

10

0.08

0.10 1/2

1/2

1/T (1/K ) Figure 3.6. T-1/2-dependence of the electrical resistivity

ρ

for x=0, 0.1, 0.3, 0.5, 0.7, and 0.9 samples.

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

15

The same situation was observed in the temperature dependences of M, as shown in Fig.3.5. For the x=0, 0.2, and 0.4 samples, the Curie temperature Tc could be identified as about 400 K from the inflection point of the M-T curve, but the Tc's were very ambiguous for x=0.6 and 0.8 samples. We can therefore assume that the ferromagnetism of half-metallic CrO2 disappeared at x~0.7. The temperature dependences of the electrical resistivity ρ are shown in Fig.3.6. The transverse axis was taken as T-1/2 and the vertical axis as log10 ρ , considering Equation (1) of

ρ (T) ∝ exp Δ T . The linearity of log ρ vs. T-1/2 was found for x=0 and 0.1 samples, but non-linear hopping-like behavior appeared in x=0.3 and 0.5 samples. The Δ values of x=0 and 0.1 samples between 80 K and 200 K were 0.012eV and 0.035eV, respectively. The monotonic increase in ρ in the range 0 ≤ x ≤ 0.5 shows that the dominant conduction paths of polarized electrons are grain boundaries between CrO2 particles or amorphous boundaries between small CrO2 crystalline phase. Since the number of grain boundaries or phase boundaries increases with a reduction in CrO2 particle size or crystalline volume, the resistivity increases in the range 0 ≤ x ≤ 0.5. However, most of the CrO2 crystalline parts disappear at x~0.7, and the behavior of ρ changes to metallic at value of x above 0.7, mainly due to the use of MoO2.

x 0.7

Magneto-resistance ratio (%)

0

0.5

0.3

-2

-4

0.1 77 K

-6 -10000 -5000 0 5000 Applied field (Oe)

0 10000

Figure 3.7. x-dependence of the magneto-resistance ratio (MRR) at 77 K.

The field dependences of the MRR of the x = 0, 0.1, 0.3, 0.5, and 0.7 samples at 77 K are shown in Fig.3.7. The MRR is defined here as [ ρ (H) - ρ (Hp)] / ρ (Hp) of Equation (2), where Hp is the peak field at which ρ (H) becomes maximum. The |MRR| at 77 K was about 6% for pure CrO2, and it decreased steeply as the amount of MoO2 was increased. At x=0.7,

Yoshihide Kimishima

16

|MRR| became negligibly small. By the theory of TMR in a granular system [47], MRR is derived from Equation (3) of P2m2/[1+P2m2] at constant temperature. For an external field of 1T, the magnetization nearly saturated for all of the samples in the range 0 ≤ x ≤ 0.9. Therefore the decrease in |MRR| indicates a reduction of the electron spin polarization coefficient P in the grain boundary between CrO2 nano-particles or in the amorphous phase boundary between small crystalline parts.

Magnetic field (Oe)

1200 77 K Hc Hp

100 rpm

800 200 rpm

290 rpm

400

0 0.0

0.2

0.4

x

0.6

0.8

1.0

Figure 3.8. x-dependence of the coercive force Hc at 77 K and peak field Hp of the MRR-curve at 77 K.

In Fig.3.8, the coercive force Hc is plotted for each sample milled at 290 rpm for 30 min, in the range 0 ≤ x ≤ 0.9, together with the peak field Hp at which ρ (H) became maximum. Both of Hc and Hp of the samples agreed well with each other in 0 ≤ x ≤ 0.6 at 290 rpm for 30 min. This represents one of the characteristics of TMR in half-metallic CrO2. Nearly constant Hc-values above x>0.6 show that the magnetism of CrO2 nano-particles or small crystalline parts is not super-paramagnetic but weakly ferromagnetic. Here it should be noted that an enhancement of Hc was observed for x=0.05 samples milled at 100 and 200 rpm for 30 min. Though there is no reasonable explanation for this result at the present time, it might be due to the slight doping of the CrO2-phase with Mo4+.

3.3. Summary Mixtures of metallic Pauli paramagnet MoO2 and half-metallic ferromagnet CrO2 were prepared by planetary ball milling. The magnetization M, resistivity ρ , and magnetoresistance ratio MRR were measured for (MoO2)x(CrO2)1-x from x=0 to 1. M and |MRR| steeply decreased with increasing MoO2 content and disappeared at x~0.7. At values of x

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

17

above 0.6, the Mr nearly disappeared but the Ms kept a finite value, where the CrO2 nanoparticles or small crystalline parts contributed to weak ferromagnetism. At x=0.05, anomalous rpm-dependence was observed for the coercive force Hc in this system.

4. Magnetism and Magneto-resistance of MoxCr1-xOy System In this work, tri-oxide of MoO3 was mixed with CrO2, and the effects for the magnetism and conductivity were investigated. In the present experiments, MoO3 has a monoclinic MoO3(II) crystal structure with the lattice parameters of a=0.3954 nm, b=0.3687 nm, c=0.7095 nm and β = 103.75 deg [58]. CrO2 has a tetragonal crystal structure with a of 0.4419 nm and b of 0.29154 nm [45]. In the half-metallic CrO2, all of 3d2 electrons exist in the majority up spin band and behave as the polarized conduction electrons. Previous study on the effect of a few % doping of Mo6+(4d0) into CrO2 revealed steep disappearance of ferromagnetism with reduction of magnetization and Curie point Tc [56]. We also expected so-called ferromagnetic quantum critical point (FQCP) [57] for the doping of Mo6+ into the CrO2-phase in the (MoO3)x(CrO2)1-x mixed system. At FQCP, the ferromagnetism of CrO2 disappears and, for example, a possibility of ptype superconductivity may occur. Here we will report the effects of mechanical milling on the conductivity and magnetism of MoO3 /CrO2 mixtures.

4.1. Sample Preparation and Experimental Commercial CrO3, Cr(OH)3 • nH2O and MoO3 were used as the precursor for MoxCr1-xOy samples, where the x-values are between 0 and 1. First, we obtained CrOOH • 0.5H2O by sintering Cr(OH)3 • nH2O at 523 K for 1 hour in air. Then the powder mixture of CrO3, CrOOH • 0.5H2O and MoO3 with the mole ratio of (1-x)/3 : 2(1-x)/3 : 1 was milled for 1 hour by the planetary ball mill (Fritch Pulverisette-7, Germany) with Cr-steel vials. The inner diameter and volume of each vial were 40 mm and 45 cm3, respectively, and the Cr-steel balls with 15 mm diameter were used as the grinding media. About 2 g of powder mixture was the starting material. The volume ratio of balls and powder was about 30 : 1. Rotation speed was kept as 700 rpm. The ground products were followed by annealing at 573 K for 4 hours in the flow of oxygen gas, and they were characterized as the solid solutions of MoxCr1-xOy by x-ray powder diffraction (XRD) as mentioned below. Then they were mixed with aqueous solution of Polyvinyl Alcohol (PVA) and pressed to be a 1 mm thick pellet with 5 mm diameter. Vibrating sample magnetometer (VSM) and the super- conducting quantum interference devise (SQUID) were used for the magnetization measurements. Measurements of resistivity and magneto-resistance were performed for dried samples by usual 4-terminals method in DC magnetic field between -1 T and 1 T.

Yoshihide Kimishima

18

4.2. Experimental Results 4.2.1. X-ray Diffractions Powder CuK α -XRD were measured for the milled samples of nominal (MoO3)x(CrO2)1-x. We obtained the XRD pattern like as that of CrO2 in 0 ≤ x ≤ 0.7 accompanied by the decreasing of diffraction angle of 2 θ as x increased. Meanwhile, in 0.7 ≤ x ≤ 1.0, the XRD pattern was similar to that of MoO3(II). At x = 0.7, two XRD patterns like as CrO2 and MoO3 coexisted. As depicted in Fig. 1, the diffraction angle of (110) reflection of CrO2 shifts to lower value, according to the increasing of Mo content. Above x = 0.7, diffraction angle 2 θ of (110)-reflection becomes nearly constant as shown in Fig.4.1.

Intensity (arb. units)

2000 1500

MoO3 (110)

1000

0.4

x=0.2 CrO2 (110)

0.8 500 0.7 0 27

28 2θ (deg)

29

Lattice parameter (nm)

Figure 4.1. (110) main peaks of CrO2 and MoO3 in 0 ≤ x ≤ 1.0.

0.7

c

0.6

CrO2-type

0.5

a

MoO3 -type a

0.4

c

b

0.3 0.0

0.2

0.4

0.6

0.8

1.0

x Figure 4.2. Lattice parameters of tetragonal phase in 0 ≤ x ≤ 0.7, and monoclinic phase in 0.7 ≤ x ≤ 1.0.

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

19

In Fig.4.2, calculated lattice parameters from XRD results are shown. They show that the extension of a-axis in the tetragonal phase below x=0.7, and the shrinking of c-axis in the monoclinic phase above x=0.7 as x increases. The β of monoclinic phase was almost constant at about 104 deg in 0.7 ≤ x ≤ 1.0. From the above results, we convinced of the successful doping of Mo6+ to the CrO2 phase in 1 ≤ x ≤ 0.6, considering the larger ion radius of Mo6+ than that of Cr4+. However, above x=0.8, the stable crystalline phase changed to the MoO3(II)-type. In 0.8 ≤ x ≤ 1, we can assume that the Cr3+-ions are mainly doped into MoO3 from the electric and magnetic properties as will be discussed below. In anyway, we can express the prepared samples as MoxCr1-xOy, where x = 0~1 and y=2~3. From the half widths of XRD peaks, the grain sizes of MoxCr1-xOy were estimated as 10~20 nm, which means that the present system is composed of the MoxCr1-xOy nano-particles.

4.2.2. Magnetization Magnetization curves at 77 K are shown in Fig.4.3. The saturation magnetization Ms rapidly decreased at x=0.2, and kept low values above x=0.3.

100

x=0 0.05 0.1

77K M (emu/g)

50

0.2 0.3 0.7

0 -50

-100 -10000 -5000

0 5000 H (Oe)

10000

Figure 4.3. Magnetization curves of MoxCr1-xOy at 77 K.

Temperature dependences of magnetization M under the field of 5 kOe are shown in Fig.4.4. The M were nearly saturated at 77 K in 0 ≤ x ≤ 0.3, but the M-value at 77 K steeply decreased at x=0.2. The vertical arrows are indicating the inflection points of the M(T)-curves which roughly give the ferromagnetic Curie point Tc. As shown in Fig. 4, rapid decreasing of Tc was observed at x=0.2~0.3.

Yoshihide Kimishima

20

100 x=0

5 kOe

M (emu/g)

0.05

50

0.1

0.2 0.3

0

100

200 300 T (K)

400

Figure 4.4. Temperature dependence of magnetization at 5 kOe.

Ms (emu/g)

100

300 K 77 K

50

0 0.0

0.2

0.4

0.6

0.8

x Figure 4.5. x-dependence of saturation magnetization Ms at 77 K and 300 K.

1.0

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

21

Mr (emu/g)

15

10 300 K 77 K

5

0 0.0

0.1

0.2

x 0.3

0.4

0.5

Figure 4.6. x-dependence of remanent magnetization Mr at 77 K and 300 K.

Hc (Oe)

300

200 300 K 77 K

100

0 0.0

0.1

0.2

0.3

0.4

0.5

x Figure 4.7. x-dependence of coercive force Hc at 77 K and 300 K.

The above results mean that the ferromagnetism, which is induced from large electron spin polarization, smeared out by the Mo6+-doping above x=0.3. The x-dependence of saturation magnetization Ms, remanent magnetization Mr and coercive force Hc at 77 K and 300 K are shown in Fig.4.5, 4.6 and 4.7, respectively. All of the magnetic quantities become very small at x=0.2~0.3. Therefore we can identify the vanishing point of half metallic ferromagnetism in MoxCr1-xOy (y~2) as x~0.3.

Yoshihide Kimishima

22

4.2.3. Resistivity and Magneto-resistance Effect In Fig.4.8, temperature dependence of electrical resistivity ρ is shown for x=0~0.5 samples. Since pure CrO2 showed the tunneling magneto-resistance (TMR) at 77K and 300K by the spin polarized 3d electrons, we plotted log ρ for 1/T1/2 following the TMR theory [47]. As shown in Fig. 4.8, log ρ is linear with 1/T1/2 up to x=0.5 between about 150 K and 350 K. This result looks like as that the half metallic property does not disappear completely at x=0.5. However the magnitude of ρ , which is also shown in Fig.4.9, shows steeply increases above x = 0.3 and becomes one thousand time larger than that of CrO2 (x=0) at x = 0.5. The increasing of ρ corresponds to the disappearance of ferromagnetism in this system. 5

10

0.5

3

ρ (Ωcm)

10

0.4

1

10

0.3 0.2 0.1 x=0

-1

10

0.06

0.07 -1/2 1/T (K )

0.08

1/2

Figure 4.8. Temperature dependence of electrical resistivity.

5

ρ (Ω cm)

10

300 K 77 K

3

10

1

10

-1

10

0.0

0.1

0.2

x

0.3

0.4

Figure 4.9. x-dependence of electrical resistivity at 77 K and 300 K.

0.5

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

23

In Fig.4.10, magneto-resistance ratio (MRR) of MoxCr1-xOy are shown, where MRR is defined here as [ ρ (Hp)- ρ (H)]/ ρ (Hp) of Equation (2), where Hp is the peak field at which

ρ (H) becomes maximum. The |MRR| at 77 K was about 4 % for pure CrO2, but it steeply decreased as the Mo-content increased. Above x=0.3, |MRR| becomes negligibly small, which also shows the disappearance of ferromagnetism.

|MRR|1T (%)

4 3 300 K 77 K

2 1 0 0.0

0.1

0.2

x

0.3

0.4

0.5

Figure 4.10. x-dependence of magneto-resistance ratio at 77 K and 300 K.

4.3. Discussion Now we discuss the ionic states of Cr based on the above experimental results. From the XRD results, we confirmed that the successful doping of Mo6+ to the CrO2 phase in 1 ≤ x ≤ 0.6. Meanwhile, the MoO3-type crystalline phase was observed in 0.8 ≤ x ≤ 1. Here we think that the Cr3+-ions play the important role in the present MoxCr1-xOy system as following. For example, the MoxCr1-xOy can be assumed as (Mo6+)x(Cr4+)1-3x(Cr3+)2xO2 in 0 ≤ x ≤ 1/3. In that case, no Cr4+ ion exists at x =1/3 (~ 0.3), where the half- metallic ferromagnetism disappears. In the region of 1/3 ≤ x ≤ 2/3, (Mo6+)4x/[3(1+x)](Cr3+)4(1-x)/[3(1+x)]□(3x1)/[3(1+x)] O2 is possible considering □ as the vacancy in CrO2- type crystal structure. In 2/3 ≤ x ≤ 1, we can assume (Mo6+)x(Cr3+)1-x O3- δ where δ =3(1-x)/2 is the oxygen deficiency. Then the coexistence of two phases of Mo6+0.53Cr3+0.27□0.2O2 (CrO2-type) and Mo6+0.67Cr3+0.33 O3-0.5 (MoO3-type) is capable at x =2/3 (~0.7). The above example shows one of the possibilities, but the importance of Cr3+, which composes an antiferro- magnetic oxide of Cr2O3, can be sufficiently elucidated for the present system. If there is the ferromagnetic quantum critical point (FQCP) of MoxCr1-xOy near x = 0.3, p-type superconductivity can be expected at very low temperature [57].

24

Yoshihide Kimishima

Therefore, electronic and magnetic properties below 77 K is very interesting, and they shall be reported in the near future.

4.4. Summary MoO3 was mixed with CrO2, and the effects for the magnetism and conductivity were investigated. Mixed samples of MoxCr1-xOy were prepared by planetary ball mill. X ray diffraction, magnetization M, resistivity ρ and magneto-resistance ratio MRR were measured for x = 0 to 1 with 0.1 step. Tetragonal phase of CrO2-type transformed to monoclinic phase of MoO3(II)-type at x above 0.7. M and |MRR| rapidly decreased with increasing Mo content, and disappeared at x above 0.3. Above x = 0.3, trivalent Cr ions were assumed to be dominant for the magnetic property of present system.

5. Inverse TMR Effect in a Granular Fe3O4/CrO2 System near the Percolation Threshold Spin-polarized electron transport is expected to be used to develop the magnetic storage devices such as magnetic random access memories with high magneto-resistance ratios at room temperature. Half-metallic (HFM) oxides with insulating barriers have attracted particular attention on account of their 100% spin polarization of conduction electrons and large tunneling magneto-resistance (TMR) effects. In recent years, inverse, or positive, TMR effects have been studied [59-66] for ferromagnetic layers whose spin polarization coefficients P have opposite signs. If P=1, the electrons in conduction d band have only majority spins parallel to the magnetic field. On the other hand, all of the conductive d electrons have minority spins anti-parallel to the magnetic field for P=-1. For tunneling junctions composed of two different HFM oxides with opposite signs of P, such as HFM (P=-1)/I/HFM (P=1), the tunneling resistivity ρ has a low value under a low magnetic field, on account of the anti-parallel alignment of the magnetic moments. However, ρ becomes large under a high field due to the parallel alignment of magnetic moments in the junctions. Inverse TMR was observed for Fe3O4(P=-1)/SrTiO3 (STO)/La0.7Sr0.3MnO3 (LSMO, P=1) epitaxial hetero-structures [59,60]. At 10 kOe, the inverse TMR ratios of Fe3O4/STO/LSMO junctions were 19%, 7%, and 3% at 80 K, 150 K, and 200 K, respectively. However, the TMR ratio changed to a negative value under fields larger than 10 kOe at 150 K and 80 K, and the inverse (positive) TMR ratio disappeared above 200 K. One of the problems of the above study was the absence of correlation between the MR data and the coercive forces Hc of the ferromagnetic layers. Hu et al. [61,62] studied the inverse TMR of epitaxial trilayer junctions of Fe3O4/CoCr2O4(CCO) /LSMO, and reported a maximum inverse MR ratio at 4 kOe of 25% at 60 K, and 0.5% at room temperature. The magnetic fields in which the MR curve showed jumping and step like behavior nearly corresponded to the Hc of Fe3O4 and LSMO, respectively, due to the smaller lattice mismatch of CCO than that of STO with Fe3O4. Inverse TMR has also been used as a sensor for negative spin polarization of conduction electrons. For example, the negative spin polarization of Co was confirmed by the inverse

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

25

TMR of Co/I/LSMO junctions with insulating barriers I of STO, Ce0.69La0.31 O1.845 (CLO) [63,64] and anatase TiO2 [65]. Instead of LSMO, another HFM of CrO2 with positive spin polarization has been used to examine the negative P of Co and permalloy Ni81Fe19 [66]. In these Co/I/HFM (P = 1) junctions, inverse TMR ratios of 40%-50 % were observed at 5 K and 1 kOe, but the room temperature TMR ratio was less than 0.1%. In the present work, a new inverse MR junction of Fe3O4 (P = -1)/I/CrO2 (P = 1) was investigated using granular Fe3O4/CrO2 samples, where we defined x as VCrO2/Vsample, which is the volume fraction of CrO2 in the measured samples. In the following sections, the experimental results fot the magnetization and MR will be presented for values of x between 0 and 1.

5.1. Sample Preparation Granular Fe3O4/CrO2 samples were prepared by mixing commercial powder specimens of Fe3O4 and CrO2 supplied by Wako Pure Chemical Industries, Ltd. and Aldrich Chemical Company, Inc., respectively. The diameters of powder particles were confirmed to be about 100-500 nm for both Fe3O4 and CrO2 by scanning electron microscopy (SEM). Powder x-ray diffraction patterns suggested the presence of very small amounts of Cr2O3 impurity phase in CrO2 and γ -Fe2O3 or Fe3O4+ δ in Fe3O4. We think that these impurities are insulating barriers in this system. The mixed powders were glued by application of adequate amounts of polyvinyl alcohol (PVA), and the resulting material was suitably cut for magnetic and electrical measurements. Eleven kinds of samples were prepared, with volume fractions x of CrO2 between 0 and 1 with 0.1 intervals. Magnetization measurements were performed with an MPMS SQUID magnetometer from 5 K to 300 K between ± 50 kOe. The temperature and field dependence of the electrical resistivity ρ were measured by the four-terminal method between 77 K and 300 K. The Au leading wires were pasted to the sample surface with silver paste. The size of the measured samples was 3 × 2 × 1 mm, and the current direction was parallel to H. In this (Fe3O4)1-x(CrO2)x system, ρ values decreased with increasing x from the order of 106 to 10-1 Ω cm.

5.2. Results and Discussion In Fig.5.1, the temperature dependence of the magnetization M under a magnetic field H of 5 kOe is shown for the sample where x = 0.5, with the ferromagnetic transition of CrO2 near 400 K clearly indicated. Since the Curie temperature Tc of Fe3O4 is known to be 860 K, far above the Tc of CrO2, the M value of about 50 emu/g at 400 K is mainly due to saturation magnetization of Fe3O4. The H dependences of M are shown in Fig.5.2 for x = 0.5 at 300 K and 77 K. The magnetization M was almost saturated at 77 K and the coercive forces Hc were obtained as 234 Oe and 307 Oe at 300 K and 77 K, respectively. The unique Hc without a step-like M-H curve shows the characteristic of the present mixed granular system. In Fig.5.3, the x-dependences of Hc are plotted for 300 K and 77 K. The linear change in the unique Hc with x shows the averaging effect in this granular system of the random magnetic anisotropy.

Yoshihide Kimishima

26

100 x = 0.5 5 kOe

M (emu/g)

90 80 70 60 50 100

200

300

400

T (K) Figure 5.1. Temperature dependence of the magnetization between 77 K and 400 K under 5 kOe for a CrO2 volume fraction of x = 0.5.

120

M (emu/g)

80 40

x = 0.5 300 K 77 K

0 80 40

-40

0 -40

-80

-80 -1.0

-120 -50

-25

-0.5

0

0.0

25

0.5

1.0

50

H (kOe) Figure 5.2. Magnetic field dependence of the magnetization between -50 kOe and 50 kOe at 300 K and 77 K for x = 0.5. The inset shows the M-H curve in the low-field region.

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

27

In Fig.5.4, the magneto-resistance ratio MRR, defined by [ ρ (H)- ρ max]/ ρ max, is shown for the sample in which x = 0.5, where ρ is the resistivity and ρ max is the maximum ρ in the low-field region. The ρ max values at 300 K and 77 K were about 2.0 and 3.4 Ω cm, respectively. The magnetic fields Hmax when ρ = ρ max were ± 670 Oe and ± 1 kOe at 300 K and 77 K, respectively. These |Hmax|-values in the sample where x = 0.5 were a few times larger than the coercive forces Hc given above for x=0.5. The physical meaning of these discrepancies between Hmax and Hc is an unresolved question in the present magnetically coupled granular system.

600

Hc (Oe)

300 K 77 K

400

200

0

0.0

0.2

0.4

0.6

0.8

1.0

x Figure 5.3. x-dependence of the coercive force Hc of all samples at 300 K and 77 K.

Whatever the answer may be, the negative MRR should arise from the tunnel junction between two ferromagnetic electrodes whose polarization coefficients P have the same sign, so the |MRR| at 10 kOe, which is less than 0.5% at 300 K and about 8% at 77 K, may be mainly due to the CrO2/I(Cr2O3)/CrO2 and Fe3O4/I( γ -Fe2O3 or Fe3O4+δ )/Fe3O4 junctions. The tunneling gap parameter Δ , obtained from the relation of ρ (T) ∝ exp

ΔT

between 77 K

and 300 K, was 40 -80 K in the range 0.4 ≤ x ≤ 1, but Δ abruptly increased to about 700 K at x=0.2. The value of ρ at room temperature under a 0-field was between 0.4 and 4 Ω cm in the range 0.4 ≤ x ≤ 1, but it became higher than 10k Ω cm below x=0.2. These results also mean that the main conduction paths were CrO2/I/CrO2 junctions in the range 0.4 ≤ x ≤ 1, and Fe3O4 /I/Fe3O4 junctions for x ≤ 0.2, because the Δ - and ρ values in the range 0.4 ≤ x ≤ 1 were of the same order as those of the other CrO2/I/CrO2 granular junctions as shown in Section 2. Therefore the percolation threshold xc for the current through the CrO2/I/CrO2 junctions should be about 0.3 for the present system. The value of xc ~0.3 coincides with the value predicted by percolation theory for bcc site model [67,68]. The

Yoshihide Kimishima

28

values of Δ and ρ at x = 0.3 were intermediate between those at x=0.2 and 0.4. However, as shown in Fig.5.5, MRR at 10 kOe was much small as at this specific volume fraction of x=0.3.

0 MRR (%)

300 K -2 x = 0.5 -4 -6

77 K

-8 -10

-5

0 5 H (kOe)

10

Figure 5.4. Field dependence of the magneto-resistance ratio (MRR) at 300 K and 77 K for x = 0.5.

0

MRR (%)

-2

300 K 77 K

-4 -6 -8 -10 0.2

0.4

0.6

0.8

1.0

x Figure 5.5. x-dependence of MRR at 10 kOe for all samples at 300 K and 77 K.

The MRR curve for x=0.3 showed complicated behavior at 77 K, as shown in Fig.5.6. For this curve, not only an Hmax of about ± 800 Oe but also an Hmin at which ρ became minimum was seen at ± 1.7kOe. Above 1.7kOe, MRR increased with the magnetic field up to

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

29

about 3 kOe. Under fields larger than 3kOe, the MRR curve saturated at about -1.3%. This inverse TMR-like behavior was only observed for the sample in which x = 0.3 in the present granular system. Since the coercive field Hc of this sample was about 200 Oe, as given in Fig.5.3, the above behavior of the MRR curve had no correlation with the Hc value. Just below the percolation threshold xc, it can be assumed that the number of isolated CrO2 (P=1) grains or clusters, which are enclosed completely by the Fe3O4 (P=-1) grains, becomes maximum in the present system [67]. The SEM picture of the x=0.3 sample in Fig.5.7 also shows this situation, where the darkened grains of CrO2 were confirmed by energy-dispersive x-ray (EDX) analysis. Therefore the above inverse-TMR-like result is considered to be mainly due to the passage of polarized conduction electrons through the Fe3O4 (P=-1)/I/CrO2 (P=1) junctions.

MRR (%)

0.0 x = 0.3 77 K

-0.5 -1.0 -1.5 -2.0 -10

-5

0

5

10

H (kOe) Figure 5.6.

Field dependence of MRR at 77 K for x = 0.3.

Figure 5.7. SEM picture of the x=0.3 sample. CrO2 grains seem to be completely enclosed by Fe3O4 grains.

Yoshihide Kimishima

30

The very small value of MRR at x=0.3 indicates a compensation effect caused by positive and negative changes in the MR of this sample. It should be noted that the switching MRR behavior at x =0.3 was very sensitive to the magnetic field at about 1kOe, with about 2% variation in the MRR over a field interval of about 0.5kOe.

8

18.3

6

18.0

x = 0.2 4 77 K

17.7

2

17.4 -10000 -5000

0

5000 10000

MRR (%)

ρ (Ωcm)

18.6

0

H (Oe) Figure 5.8.

ρ -H and MRR results for a mixed Fe3O4-II/CrO2 system at x=0.2.

For the magnetite-rich samples where x ≤ 0.2, resistivity values were relatively high at 105 -10 Ω cm, probably due to the insulating barriers of γ -Fe2O3 or Fe3O4+δ . We therefore 6

prepared low-resistive magnetite (Fe3O4-II) by sintering the commercial one in an Ar atmosphere with 2.5% H2 at 623 K for 3 hours. The resistivity of granular Fe3O4-II was of the order of 104 Ω cm between 77 K and 300 K. In the mixed Fe3O4-II/CrO2 system, ρ -values were of the order of 1 to 104 Ω cm. The inverse TMR effect was observed at x = 0.2, as shown in Fig.5.8. This result means that reduction of the proportion of γ -Fe2O3 or Fe3O4+δ allows clearer observation of inverse TMR in an Fe3O4/CrO2 system by enhancing the effective spin polarization of magnetite. The slightly low x-value at which inverse TMR was observed may be due to the difference in grain diameter between commercial Fe3O4 and reduced Fe3O4-II. Therefore, more detailed study is needed for mixed systems of Fe3O4-II and CrO2.

5.4. Summary Magneto-resistance ratios were measured for (Fe3O4)1-x(CrO2)x granular systems between 77 K and 300K, where x is the volume fraction of CrO2 in the measured sample. In these systems, an averaging effect on the coercive force Hc caused by randomly magnetic

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

31

anisotropy was observed. The expected inverse TMR effects were observed near or just below the percolation threshold of xc ~0.3, where the number of CrO2 grains completely enclosed by Fe3O4 grains became maximum. In the range 0.4 ≤ x ≤ 1, MR behaviors were characterized by low-resistive CrO2/I/CrO2 junctions. For x ≤ 0.2, high-resistive Fe3O4/I/Fe3O4 junctions were dominant in the magnetic conduction behaviors. The oxygen reduction of commercial Fe3O4 was useful for obtaining samples with visible and large inverse TMR effects. The percolation threshold xc of CrO2/I/CrO2 junctions seemed to decrease to near 0.2. A detailed study of this system near xc is now being carried out to realize low-cost, stable, and high-performance spintronics devices using half-metallic oxides.

6. Magneto-resistance of Nano-Fe3O4/Ag Granular System In this study, RT-MR effects in Fe3O4/Ag nano-granular systems were investigated, expecting the large RT-MRR with well reproducibility by physical and chemical stabilization of the conduction paths by the metallic Ag for the polarized conduction electrons among Fe3O4 nano-grains.

6.1. Sample Preparation Four kinds of samples with Fe3O4 nano-grains were prepared by the following procedures. First, the precursor of Fe3O4 nano-particles were precipitated by mixing aqueous solution of FeCl2.4 H2O, 2FeCl3.6H2O and aqueous ammonia. Then the obtained Fe3O4 nano-particles were mixed with commercial Ag (samples I) or Ag2O (samples II) powders. They were sintered at 473 K in the Ar atmosphere with 1.5% H2 gas. The prepared samples had the Ag volume fraction of x*=VAg /(VFe3O4+VAg) between 0 and 0.35. On the other hand, the samples III were directly precipitated by mixing the aqueous solution of FeCl2.4H2O, 2FeCl3.6H2O and the aqueous ammonium solution of Ag2O, but solvable Ag volume fraction x was limited to 0.13. At x* larger than 0.13, the Ag2O powder particles partially deposited in the mixed solution. Therefore, for samples III, only the precipitates in 0<x*<0.13 were sintered under the same condition as former samples I and II. The samples C were obtained from the sintered powder mixtures of commercial Fe3O4 and Ag at 773 K in the atmosphere of H2:Ar=0.02:0.98. The x-ray diffraction (XRD) patterns of samples II were given in Fig.6.1 for the (311) main peak of Fe3O4 and (111) one of Ag. From the half widths of diffraction peaks, the mean particle diameters d were estimated from Scherrer's formula for Fe3O4(dFe3O4) and Ag (dAg). The x* dependences of dFe3O4 and dAg were shown in Fig.6.2. In samples C, dFe3O4 and dAg were nearly constant at about 30 nm, which was the same order to the precursor of commercial powders, though the agglomeration of the grains can not be excluded in them. The dFe3O4-values of samples I were 15~20 nm in x*<0.1 and about 30 nm in x*>0.1. The dAg-values were 20~25 nm in x<0.1 and 30~35 nm in x*>0.1.

Yoshihide Kimishima

32

Intensity (arb. units)

Samples II 2000

Ag (111)

1600 1200 800 400

Fe3O4 (311)

x* = 0.35 0.26 0.19 0.13 0.09 0.05 0.03 0

0

35

36

37 2θ (deg)

38

39

Figure 6.1. XRD main peaks of (311) for Fe3O4 and (111) for Ag in samples II for values of x* between 0 and 0.35.

30 20

Fe3O4

d (nm)

10 I II III C

0 30 20 10

Ag

0 0.0 0.1 0.2 0.3 0.4 x* Figure 6.2. x*-dependences of the particle diameters of dFe3O4 and dAg, determined from XRD results.

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

33

The dFe3O4-values of samples II were 10~15 nm, and the dAg-values were 15~25 nm in x*<0.4. These results mean that the samples II formed a good nano-particle system of Fe3O4 and Ag in whole x-region between 0 and 0.4, though the samples III included smaller Fe3O4 particles in 0< x*<0.13. Therefore we mainly paid attention to the MR-behaviors of samples II as one of the typical nano-Fe3O4 and Ag granular systems. The transmission electron microscopy (TEM) images of sample II at x*=0.05 and 0.19 were shown in Figs.6.3(a) and (b), where we used a crushing method to prepare the electron microscopy specimens. Some samples were ground under CCl4, then, dispersed on carbon coated micro-grids. Specimens were examined with Hitachi HF-3000S field-emission analytical TEM operating at 300 kV. Since the very dense Debye rings were obtained, it was very difficult to choose a ring of Ag- or Fe3O4-phase by an objective aperture in dark-filed imaging. TEM contrast might be affected also by crystal orientation and thickness. Therefore we judged that it was difficult to obtain a good image-formation by dark field, and qualitative analysis of the grain was performed by energy dispersive x-ray spectroscopy (EDX). The EDX spectrum showed that the most darkened parts corresponded to Ag-particles, and the other to Fe3O4- particles. As shown in Fig.3(a), the Ag-particles of d~20 nm are isolated in the connected Fe3O4-particles of d=10~20nm at x*=0.05. At x*=0.19, larger or agglomerated Ag-particles began to appear, as shown in Fig.6.3(b). These results were not contradictory to the above XRD results.

Ag

100 nm

(a)

Ag

100 nm

(b) Figure 6.3. TEM images of samples II for x*= 0.05 (a) and 0.19 (b).

Yoshihide Kimishima

34

6.2. Results and Discussion MR measurements were done by usual 4-terminals method in DC-field between -1T and 1T. To avoid the nonlinear effect between voltage V and current I, resistance R was measured at constant current I of 1mA. Room temperature MRR-curves of samples II are partly shown in Fig.6.4. They show slight hysteresis due to the small coercive force of Hc. The MRR-curves sharply drop in low field region and show no saturation at ± 1T, which correspond to the magnetization curves up to 1 T by SQUID magnetometer.

Sam ples II 0

MRR (%)

-1

x* 0.05

-2

-4

0.13 0.35 0

-5

0.19

-3

-6 -1.0

-0.5

0.0 0.5 μ 0 H (T)

1.0

Figure 6.4. Field dependences of the room-temperature MR ratios (MRR) of samples II.

0

MRR1T(%)

-1 -2

I II III C

-3 -4 -5 0.0

0.1

0.2 x*

0.3

0.4

Figure 6.5. x*-dependences of the MR ratios (MRR) at 300 K and 1 T for all of the samples.

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

35

In Fig.6.5, MRR-values at 300K and 1T were plotted for all of the samples. It should be noted that the |MRR| of samples I-III were commonly larger than the highest |MRR|-value of 1.4 % in samples C. The |MRR| of samples II reaches maximum value of 5.0 % at x*= 0.19. For the samples I, largest |MRR| of 3.9% was also observed at x=0.19. Both of maximum |MRR|-values are larger than the |MRR| of 3.5% in the nano-Fe3O4 sample of x=0. These results may show the physical and chemical stabilization of intermediate conduction paths between Fe3O4 nano-particles. 2

ρ0 (Ωcm)

10

1

I II III C

10

0

10

-1

10

-2

10

0.0

0.1

0.2

0.3

0.4

x* Figure 6.6. x*-dependences of the 0-field resistivity ρ 0 at 300 K for all of the samples.

The ρ 0 of room temperature resistivity at 0 field were shown in Fig.6.6 for all of the

samples. ρ 0 is nearly equal to the maximum resistivity for each sample because of the very

small hysteresis in the ρ -H curves. For samples II, |MRR| and ρ 0 decreases and increases with increasing x* in x*<0.05, and becomes minimum and maximum at x=0.05, respectively. Above x=0.05, |MRR| increases and ρ 0 decreases with increasing of x, indicating visible peak at x*=0.19. Above x*=0.19, |MRR| and ρ 0 decrease monotonously according to Ag-content.

Samples I showed nearly the same behaviors with samples II. As for the samples III in 0<x*<0.13, minimum |MRR| and maximum ρ 0 appeared at x*= 0.08 and x*= 0.02, respectively. In the samples C, ρ 0 was nearly constant in x<0.2, and rapidly decreases in x>0.2. The |MRR| decreases with increasing x* below 0.2. At x*=0.2, |MRR| had an weak peak value of 1.3%, when ρ 0 also showed a broad peak. In x*>0.2, both of |MRR| and ρ 0 began to decrease with increasing x*. The weak peaks of |MRR| near x*=0.2 are also found for samples I and II. Since the linear relation between voltage V and current I was confirmed in the resistivity measurement for x*>0.2, the above experimental results around x*=0.2 can be interpreted as the percolation behavior of Ag conducting paths with the percolation threshold xc of about 0.2. Similar results of peak in |MRR| just below x*c were reported for the γ -Fe2O3/Ag

Yoshihide Kimishima

36

granular nano- composites [69]. Ordinary, the percolation problems were considered for the system including two kinds of conductive spheres with same diameter. Therefore it is not easy to apply the numerical results of percolation theory [67,68] to the present system with different diameters of dFe3O4 and dAg. However the x*c-value of 0.2 might reflect the theoretical percolation threshold of 0.198 obtained from the site model calculation for the closed packed fcc lattice [70]. Here we discuss the |MRR|- and ρ 0-behaviors of samples I , II and III at x around 0.05, where the |MRR| and ρ 0 become minimum and maximum, respectively. These results are probably explained by the spin injection and accumulation [71] in the Ag nano-particles. In a low field region, the magnetic moments in Fe3O4 particles are randomly oriented and spin injection to Ag particles dose not occur. However, in a high field, magnetic moments are aligned along the external field and the spin polarized electrons are injected to Ag nanoparticles. If dAg is less than the spin diffusion length ds, injected electron spins are accumulated in the Ag conducting paths. As a result, reduction in the density of state D(EF) by the increase in the Fermi energy EF of polarized electrons comes to be possible. Then the resistivity increases by magnetic field, and it shows the positive magneto-resistance effect. Actually, such a effect has been observed in Fe3O4-Ag composite films [72] when dAg is nearly equal to ds. If dAg>ds, the orientation of injected spins are relaxed in the middle of the Ag path. Even if dAg10 nm are satisfied, the absolute value of "negative" MRR at x*~0.05 may be reduced by the positive MRR. The increasing of ρ 0 in 0-field may be due to the spin injection from the remanent magnetic state in Fe3O4. No anomalous behaviors of ρ 0 and |MRR| of samples C near x*=0.05 can be explained

by larger dFe3O4 than dAg which maintains directly conducting paths of Fe3O4/Fe3O4. Even if the Ag content increases up to about x*=0.2, small Ag particles may be incapable of connecting large Fe3O4 particles due to the large interstices. On the other hand, in samples IIII, the relation of dFe3O4
6.3. Summary Room temperature magneto-resistances were measured and discussed for the four kinds of Fe3O4/Ag granular systems. Most of the systems commonly showed weak peaks of magnetoresistance ratio |MRR| and 0-field resistivity ρ 0 near the percolation threshold xc of about 0.2. The maximum |MRR| of about 5%, which is fairly high value for the granular MR of Fe3O4, was observed for nano-Fe3O4/Ag system prepared by sintering the mixture of Fe3O4 and Ag2O (samples II). Meanwhile the possibilities of spin injection from Fe3O4 nano-particles and spin accumulation in Ag particles were suggested for the samples including Fe3O4 nanoparticles (samples I , II and III) at x around 0.05.

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

37

It was also revealed that the site percolation problem for the mixed system, including two kinds of conductive particles with different diameters, is important to explain the MRR and ρ 0 behaviors in the present granular systems. In the Fe3O4/Ag nano-particle system, the MR behavior near Verwey transition temperature Tv~120 K is another interesting problem. The |MRR| experiments near Tv are now in progress for the present system and shall be reported elsewhere.

7. Study on Anomalous Magneto-resistance in Nano-Fe3O4/Ag Granular System In Section 6, room temperature magneto-resistance (MR) was measured and discussed for nano-Fe3O4/Ag granular system with nominal Ag volume fraction x* of VAg /(VFe3O4+VAg) between 0 and 0.35. In the system, prepared by sintering the mixture of nano-Fe3O4 and Ag2O, weak peak of magneto-resistance ratio |MRR| and 0-field resistivity ρ0 were observed near the percolation threshold x*c of about 0.2. The maximum |MRR| of about 5 % at 1 T was fairly high value for the granular MR of Fe3O4. Meanwhile the possibilities of spin injection from Fe3O4 nano-particles and spin accumulation in the Ag particles were suggested for the samples of x*=0.05 in Section 6. We often observed anomalously large MR behavior in the above system, but the reason has not been known. Our method to prepare the precursor of Fe3O4 nano-particles is as following. At first, aqueous solutions of FeCl2 · 4H2O and 2FeCl3 · 6H2O are mixed by magnetic stirrer. Next, aqueous ammonium solution (NH4OH) is dripped into the FeCl2· 4H2O+2FeCl3 · 6H2O aqueous solution. Then the Fe3O4 nano-particles are precipitated with NH4Cl which should be washed out several times by pure water. The nano-Fe3O4/Ag system was prepared by sintering the mixture of nano-Fe3O4 and Ag2O at 473 K for two hours in the Ar atmosphere with 2 % H2 gas as described in Section 6. We calculated the x* of nominal volume fraction of Ag from the mole ratio of the precursors of Fe3O4 and Ag2O, based on the hypothesis that the produced sample contained only Fe3O4 and Ag. However, if the above washing process was insufficient, NH4Cl is left among the Fe3O4 nano-particles and partly changes to AgCl in the sintering process. Therefore we studied the washing times dependence of magneto-resistance for nano-Fe3O4/Ag system in the purpose of revealing the influence of NH4Cl impurity in the precursor of Fe3O4 nano-particles. We selected the samples of x*=0, 0.05 and 0.20 samples in the present study, because characteristic MR behaviors were observed at x* = 0.05 and 0.20 in the previous studies in Section 6.

7.1. Sample Preparation The method of sample preparation was already described in the earlier section. We qualitatively controlled the amount of NH4Cl by the washing times of dirty Fe3O4 precipitate which was produced from mixing the FeCl2 · 4H2O+2FeCl3 · 6H2O aqueous solution and NH4OH. One washing process was composed of immersing the precipitate in 500 ml pure water, stirring it by magnetic stirrer for 5 minutes and separating the washed precipitate from

Yoshihide Kimishima

38

dirty water by the centrifugal separator. After drying the washed precipitate, Fe3O4 powder was pressed into the pellet with 10 mm diameter and sintered at 473 K for two hours in the atmosphere of 2%H2/Ar gas. We observed the CuK α x-ray powder diffraction (XRD) patterns of the sintered samples, and performed the energy dispersive x-ray spectroscopy (EDX) experiments to know the mole ratio of Fe, Ag and Cl, from which we could obtain the volume fraction of Fe3O4, Ag, and NH4Cl or AgCl, semi-quantitatively. Hereafter we use x as the volume ratio of Ag, and y as that of NH4Cl for x*=0 sample and of AgCl for x*=0.05 and 0.20 samples.

x* = 0 100 4th (y=0) 50 0

Intensity

3rd (y=0.03) 50 0 Fe3O4

(311)

2nd (y=0.07)

50 0 50

1st washing (y=0.5)

NH4Cl (110)

* 0 30

40

50

60

2θ (deg) Figure 7.1. XRD patterns of sintered Fe3O4 samples washed for 1 - 4 times by pure water, where y is the volume fraction of NH4Cl.

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

39

As shown in Fig.7.1, a visible XRD peak of NH4Cl is seen for the sample after 1st washing (y=0.5). The 2 times washed sample (y=0.07) and 3 times washed sample (y=0.03) show no apparent peak of NH4Cl, but we found from EDX that very little amount of NH4Cl is included in them. The 4 times washed sample (y=0) is almost pure sintered Fe3O4 nanoparticles. Here y is the volume fraction of NH4Cl. In Fig.7.2, XRD peaks are shown for the sintered samples of x* = 0.05. These samples were prepared by sintering the mixture of 1 ~ 4 times washed nano-Fe3O4 and commercial Ag2O at 473 K for two hours in the atmosphere of 2%H2/Ar gas. As shown in this figure, large amount of AgCl exists in the 1 time washed sample (x=0.04 and y=0.19). This AgCl impurity does not disappear even for 3 times washed sample (x=0.02 and y=0.02).

x* = 0.05 100 4th (x=0.04, y=0) 50 0 Fe3O4 (311)

Ag (111)

Intensity

50

3rd (x=0.06, y=0.02)

0 AgCl (200)

2nd (x=0.02, y=0.06)

50 0

1st washing (x=0.04, y=0.19)

50 0 30

40

50

60

2θ (deg) Figure 7.2. XRD patterns of sintered samples of x* = 0.05 washed for 1 - 4 times by pure water, where x is the volume fraction of Ag, and y is that of AgCl.

Yoshihide Kimishima

40

In Fig.7.3, XRD patterns are shown for the sintered samples of x*=0.20 which were washed for 1 ~ 4 times by pure water. Larger amount of AgCl than that in x*=0.05 system exists in 1 time washed sample (x=0.07 and y=0.30). The amount of AgCl impurity rapidly decreases at 2 times washed sample (x=0.15 and y=0.07), and no XRD peak of AgCl impurity is found for 3 times washed sample (x=0.16 and y=0.02).

x* = 0.2 100 4th (x=0.17, y=0.00) 50 0 Fe3O4 (311)

Intensity

50

Ag (111)

3rd (x=0.16, y=0.02)

0 2nd (x=0.15, y=0.07)

50 0

AgCl (200)

1st washing x=0.07,y=0.30

50 0 30

40

50

60

2θ (deg) Figure 7.3. XRD patterns of sintered x* = 0.20 samples washed for 1 - 4 times by pure water, where x is the volume fraction of Ag, and y is that of AgCl.

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

41

In Fig.7.4, the image of transmission electron microscopy (TEM) for 1 time washed x*=0.20 sample (x=0.07 and y=0.30). The gray particles with the diameter of 10-20 nm are the nano-Fe3O4 ones, and large darkened particles with the diameter of 50-200 nm are AgCl particles enclosing Ag core particles. We think that very thin films or very fine particles of AgCl, which are not clear in this figure, can be assumed at the grain boundaries between the Fe3O4 nano-particles, because the relatively large TMR in this sample, as will be mentioned later, should be only due to an insulating impurity of AgCl, being observed in XRD and EDX measurements.

Fe3O4

x*=0.2, x=0.07, and y=0.30 Figure 7.4. TEM image of sintered x*=0.20 samples washed for one time by pure water, where x is the volume fraction of Ag, and y is that of AgCl by EDX.

7.2. Results and Discussion MR measurements were done by usual four-terminals method in DC magnetic field between -1 T and 1 T. To avoid the nonlinear effect between voltage V and current I, resistance R was measured at constant current I of 1mA. Room temperature MR ratios (MRR) are shown in Fig.7.5, 7.6 and 7.7 for x*=0, 0.05 and 0.20 samples, respectively. As for x*=0 samples, one time washed sample (y=0.50) was insulator and showed very high resistivity above 10 M Ω probably by NH4Cl impurity between Fe3O4 nanoparticles. Therefore we could not measure MR for this sample. The two, three and four times washed samples with the NH4Cl volume fraction y of 0.07, 0.03 and 0, respectively, showed |MRR| of 2.5- 4% at 1 T as shown in Fig.7.5. Here the magnetoresistance ratio MRR is defined by [ ρ (H)- ρ (Hp)] / ρ (Hp) of Equation (2), where ρ was the resistivity and ρ (Hp) was the maximum one near 0 field.

Yoshihide Kimishima

42

x* = 0,

300 K

0 -2 4th (y=0) -4

MRR (%)

0 -2 3rd (y=0.03) -4 0 -2 -4 -10000

2nd washing (y=0.07) -5000

0

5000

10000

Applied field (Oe) Figure 7.5. Room temperature |MRR|-curves of x*=0 samples except for one time washed sample.

MR measurements were done by usual four-terminals method in DC magnetic field between -1 T and 1 T. To avoid the nonlinear effect between voltage V and current I, resistance R was measured at constant current I of 1mA. Room temperature MR ratios (MRR) are shown in Fig.7.5, 7.6 and 7.7 for x*=0, 0.05 and 0.20 samples, respectively. As for x*=0 samples, one time washed sample (y=0.50) was insulator and showed very high resistivity above 10 M Ω probably by NH4Cl impurity between Fe3O4 nanoparticles. Therefore we could not measure MR for this sample. The two, three and four times washed samples with the NH4Cl volume fraction y of 0.07, 0.03 and 0, respectively, showed |MRR| of 2.5- 4% at 1 T as shown in Fig.7.5. Here the magnetoresistance ratio MRR is defined by [ ρ (H)- ρ (Hp)] / ρ (Hp) of Equation (2), where ρ was the resistivity and ρ (Hp) was the maximum one near 0 field. For x*=0.05, one time washed sample (x=0.04 and y=0.19) and two times washed sample (x=0.02 and y=0.06) had relatively high |MRR| of about 4 % at 1 T as shown in Fig.7.6, where x is the volume fraction of Ag, and y is that of AgCl. However three times washed sample (x=0.06 and y=0.02) and four times washed one (x=0.04 and y=0) show low |MRR| of about 2% at 1T, which agrees with the previous result in Section 6. For the lowering of |MRR| in x*=0.05 sample, compared to that of x*=0 sample, we supposed the possibility of spin accumulation. The present results clearly mean that AgCl impurities contribute to the enhancement of TMR effects in Fe3O4 granular system. Thus lowering of |MRR| is reconfirmed as the intrinsic property of pure Fe3O4 /Ag system.

Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

x * = 0 .0 5 ,

43

300 K

0 -2 4 th (x = 0 .0 4 , y = 0 ) -4 0

MRR (%)

-2 3 rd (x = 0 .0 6 , y = 0 .0 2 ) -4 0 -2 -4 0

2 n d (x = 0 .0 2 , y = 0 .0 6 )

-2 -4

1 st w a sh in g (x = 0 .0 4 , y = 0 .1 9 )

-1 0 0 0 0

-5 0 0 0

0

5000

10000

A p p lie d fie ld (O e ) Figure 7.6. Room temperature |MRR|-curves of x* = 0.05 for one, two, three and four times washed samples.

x* = 0.2, 300 K 0 -2 -4

4th (x=0.17, y=0)

-6 0

MRR (%)

-2 -4

3rd (x=0.16, y=0.02)

-6 0 -2 -4 -6 0

2nd (x=0.15, y=0.07)

-2 -4 -6 -10000

1st washing (x=0.07, y=0.30) -5000

0

5000

10000

Applied field (Oe) Figure 7.7. Room temperature |MRR|-curves of x*=0.20 for one, two, three and four times washed samples.

Yoshihide Kimishima

44

Similar |MRR| behaviors were also observed for the x*=0.20 samples as shown in Fig.7.7. The one time washed sample (x=0.07 and y=0.30) and two times washed one (x=0.15 and y=0.07) including AgCl impurity showed 5-6 % of high |MRR|, and the three times washed sample (x=0.16 and y=0.02) and four times washed one (x=0.17 and y=0) showed relatively low |MRR| of about 3.5 %. Therefore it can be assumed that the previous |MRR| result of 5% at 300 K and 1T in x*=0.20 sample was due to the remainder of AgCl impurity by the insufficient washing. The |MRR| results of x*=0.05 and 0.20 samples in the present study suggest that Fe3O4/AgCl/Fe3O4 conducting paths of TMR are effective for the enhancement of MR in Fe3O4 nano-granular system. y-dependences of |MRR| at 300 K and 1 T are depicted in Fig.7.8 for the x* = 0, 0.05 and 0.20 samples. From these figures, we know that the y=0.06-0.07 samples result in high |MRR|values for x*=0.05 and 0.20 samples. It means that there is an adequate amount of AgCl for high MRR in this system. In the one and two times washed samples of x*=0.05 and 0.20 with visible amount of AgCl, the AgCl impurities may form the optimum insulating barriers for the tunneling of polarized electrons from nano-Fe3O4. Therefore very thin films with (sub-) nm thickness or very fine particles with (sub-)nm diameter of AgCl should also exist and contribute to the large TMR in the one time washed x*=0.20 and y=0.30 sample, though they were not clearly observed in the TEM image.

|MRR| (%)

6

4 x* = 0 0.05 0.20

2

0 0.0

0.1

y

0.2

0.3

Figure 7.8. y-dependences of |MRR| at 300 K and 1 T for the x*=0, 0.05 and 0.20 samples.

The above conjecture was also confirmed from the I-V measurements. The I-V characteristics of x*=0.2 and y=0.30 sample showed the nonlinear behavior of I ∝ V1.11 in the ranges 100 nA
Room Temperature Magneto-Resistance Effects of Half-Metallic Oxides

45

7.3. Summary The remainder of AgCl was confirmed in the production process of nano-Fe3O4/Ag granular system. The AgCl impurities contributed to large negative MR-ratio of about -5.5 % at 300 K and 1 T for x*=0.20 sample. It was suggested that Fe3O4/AgCl/Fe3O4 paths were effective for the large MR ratio in nano-Fe3O4 system.

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In: Spintronics: Materials, Applications and Devices Editors: G.C. Lombardi and G.E. Bianchi

ISBN: 978-1-60456-734-2 © 2009 Nova Science Publishers, Inc.

Chapter 2

WEAK HIGH-TEMPERATURE BULK FERROMAGNETISM IN MN-DOPED CDGEAS2 SEMICONDUCTORS Vyacheslav G. Storchak1*, Jess H. Brewer2,3, Dmitri G. Eshchenko4,5, Scott L. Stubbs3, Stephen P. Cottrell6, Andrey A. Nikonov1, Oleg E. Parfenov1 and Sergey F. Marenkin7 1

Russian Research Centre “Kurchatov Institute", Kurchatov Sq. 1, Moscow 123182, Russia 2 Canadian Institute for Advanced Research 3 Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C., Canada V6T 1Z1 4 Physik-Institut der Universität Zürich, CH-8057 Zürich, Switzerland 5 SµS, Paul Scherrer Institute, CH-5232, Villigen, Switzerland 6 ISIS Facility, Rutherford Appleton Laboratory, Oxfordshire OX11 OQX, UK 7 Institute for General and Inorganic Chemistry, Moscow 119991, Russia

Abstract Modern information technology utilizes electrons’ charge degrees of freedom to process information in semiconductors and their spin degrees of freedom to store information in magnetic materials. Materials in which the two degrees of freedom interact strongly offer applications known as “spintronics”. The chalcopyrite semiconductor CdGeAs2 is a promising candidate because it exhibits ferromagnetism up to ~350 K when doped with manganese. We have studied CdGeAs2:Mn with 18 and 36 mole% Mn using high transverse magnetic field muon spin relaxation, X-ray diffraction, magnetization, AC susceptibility, Auger spectroscopy and electronic transport measurements. We observe a muon spin precession frequency shift characteristic of a ferromagnetic medium, indicating that the ferromagnetism in CdGeAs:Mn is a bulk phenomenon. However, the net contribution to the magnetic field from dipole and

*

E-mail address: [email protected] or [email protected] (Corresponding author)

50

Vyacheslav G. Storchak, Jess H. Brewer, Dmitri G. Eshchenko et al. contact hyperfine interactions is found to be at least 3-4 orders of magnitude weaker than in ferromagnetic d-metals like Fe, Co and Ni.

Modern electronics and information technology is based on fundamental dichotomy: it utilizes electrons' charge degrees of freedom to process information in semiconductors and their spin degrees of freedom to store information in magnetic materials. An important step toward understanding of new phenomena and creating new technologies would be exploring the combination of the two degrees of freedom in a single material. When electrons’ spin and charge degrees of freedom strongly influence each other in the same material, it is possible to manipulate magnetically stored information with electric fields and/or modify fast logic gates by changing the magnetization of their components. In metal multilayers, such effects are manifest in giant magnetoresistance, where the orientations of the macroscopic magnetization in adjacent layers determine the electrical resistance of the structure [1, 2]. Unfortunately the spin of the carriers has so far played a minor role in semiconductor devices, mainly because the semiconductors used in information technology, such as Si and GaAs, are nonmagnetic, while magnetic semiconductors, such as Eu and Cr chalcogenides, which exhibit pronounced spin-related phenomena [3], are incompatible with modern charge-based electronics. To date, the mainstream approach for integrating magnetic and semiconducting properties has been to incorporate magnetic ions (typically Mn) into nonmagnetic semiconductors to realize diluted magnetic semiconductors (DMS) [4, 5]. In the early work [4], II-VI semiconductors such as CdTe, CdSe, ZnTe etc. were widely studied because of the appreciable solubility of divalent Mn+2 (d5), which occupies the isovalent cation site. However, this isovalent doping typically leads to antiferromagnetic (AFM) interactions in IIVI hosts. More recently, III-V-based materials have attracted a great deal of attention following the discovery [6] of ferromagnetism (FM) in InMnAs film grown using molecular beam epitaxy (MBE) which allows a non-equilibrium enhancement of the otherwise low solubility of transition metals in III-V hosts. Subsequent detection [7] of FM in GaMnAs films — the paradigmatic DMS, with the advantage of also being compatible with GaAs technology — has led to intense study of the FM in these materials. The exciting spintronics potential of GaMnAs (and several other materials based on group III-V semiconductors) arises not just from its simultaneous semiconducting and ferromagnetic properties, but also from the important feature that FM in these materials is the result of long-range coupling between the Mn atoms mediated by the charge carriers [holes, generated by Mn+3 (d4 → d5 + h) substitution at the trivalent cation site] of the host: when Mn is substituted for the Ga in GaAs lattice, it acts as an acceptor, providing holes that mediate a ferromagnetic interaction between the local moments of the open d-shells in the Mn atoms. This basic mechanism of ferromagnetism in diluted magnetic semiconductors has been established experimentally by the strong correlation between the ferromagnetic transition temperature Tc and the carrier concentration. The experimental finding is that the establishment of the long-range collective ferromagnetic order requires that at least about 2% of Ga ions are substituted on their regular lattice sites by magnetic Mn ions, to provide a sufficiently high density of carriers and magnetic ions. In order to incorporate so high concentration of Mn in the GaAs lattice without forming inclusions of the thermodynamically more stable magnetic MnAs phase, GaMnAs must be grown by molecular beam epitaxy at relatively low temperatures (of about 200-300° C). Subsequent annealing at relatively low

Weak High-Temperature Bulk Ferromagnetism…

51

temperatures removes interstitial Mn defects which act as dobble donors and thus deminish the hole concentration. The annealing procedure is found to be important as it systematically increases the ferromagnetic transition temperature and transforms the temperature dependence of magnetization into the form expected for homogeneous collinearly ordered ferromagmets. This demonstrates the critical imporatnce of defects that can dramatically modify the carrier concentration in the semiconducting host which is necessary to be high enough to effectively mediate interaction between magnetic moments of Mn ions. In any case such an unusual synergy between magnetism and semiconductivity may make it possible to modulate magnetic behavior by controlling the carrier transport properties (for example, using external electric field or light) and vice versa. The interplay between electrical and magnetic properties of III-V materials (InMnAs and GaMnAs) has already been reported in literature as electric field control of FM [8] and magnetic field driven giant Hall resistance jumps [9]. However, the still-low solubility of Mn ions in thin films of III-V group semiconductors grown by molecular beam epitaxy severely limits the FM transition temperature (-150 K); above the critical concentration (about 10% in GaAs) Mn atoms tend to cluster and phase separate [10]. Although there is an emerging consensus that the FM originates from the interaction between itinerant carriers and localized electrons, both the microscopic theory of FM and the nature of the ground state in these materials are the subjects of considerable debate. Both strong- and weak-coupling models (double-exchange [11] and kinetic approach [12], respectively) encounter certain difficulties [13] most probably caused by the absence of experimental data on the distribution of the magnetic field in DMS on the microscopic scale. So far the FM properties of DMS have been studied mostly using traditional macroscopic average techniques such as magnetometry, resistivity and anomalous Hall effect, optical measurements etc. Conclusions about the strength of the FM in a given material have typically been made based on the dimensions of the hysteresis loop in the magnetization “M vs. H” curve. These techniques, however, being integral, cannot provide information on the distribution of magnetic fields in the DMS on the microscopic scale. Modern microscopic techniques, such as ferromagnetic resonance, Mössbauer spectroscopy or perturbed angular correlations are kind of selective, sampling the response of one particular species (limited to several elements) of atoms in the close vicinity of these atoms. Meanwhile, techniques which are able to provide such information on the magnetic field distribution, such as muon spin relaxation (µSR) or polarized neutron scattering, are usually limited to studies of bulk materials. Fortunately, recent progress in the synthesis of novel DMS [14, 15] has led to growth of bulk chalcopyrite II-IV-V2 group semiconductors doped with Mn, namely CdGeP2:Mn [15], ZnSnAs2:Mn [17] and CdGeAs2:Mn [18], which exhibit FM with Curie temperatures well above room temperature. These ternary semiconductors are genealogically related to zincblende III-V group semiconductor compounds: their structure can be derived from that of the III-V crystal by alternately replacing the group-III atoms with group-II and group-IV elements. The remarkable advantage of II-IV-V2-Mn systems with respect to III-V-based dilute magnetic semiconductors is that Mn can easily substitute for group-II cations due to natural tendency of Mn to adopt a divalent Mn state [4]. By itself, this substitution leads to AFM interactions as in II-VI group semiconductors [4, 19, 20]. However, when it substitutes for a group-IV cation, Mn contributes two holes --- carriers which cause strong FM coupling between Mn atoms no matter which site is adopted by Mn ions [19]. In CdGeAs2:Mn, X-ray

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Vyacheslav G. Storchak, Jess H. Brewer, Dmitri G. Eshchenko et al.

diffraction measurements of lattice constants [18] have led to the conclusion that Mn ion substitutes for Cd atoms at molar concentrations below about 15%; at higher concentrations, Mn substitutes for Ge atoms until at around 35 mole% Mn the system starts to segregate and phase separate chemically. The idea of using II-cation sites to increase significantly Mn solubility and IV-cation sites to generate carriers resulted in extensive experimental [14, 15, 16, 17, 18] and theoretical [19, 20, 21] studies of Mn-doped II-IV-V2 semiconductors. However, the great body of knowledge obtained to date still lacks information about the magnetic field distribution on a microscopic scale. Neutron scattering cannot provide this information for II-IV-V2-based dilute magnetic semiconductors containing Cd, because Cd absorbs neutrons, but it may be obtained using muon spin relaxation technique, which is extremely sensitive to local magnetic fields [22]. In a µSR experiment one follows the spin polarization of 4 MeV positive muons stopped in the sample one at a time by accumulating histograms of the times of decay by positron detection in different directions. In an external magnetic field applied transverse to the muon polarization, each muon experiences the net effect of external and local magnetic fields, precessing at the characteristic frequency

ν = γµB, where γµ

= 135.54 MHz/T is the muon gyromagnetic

ratio and B is the magnitude of the magnetic field at the muon site. Muons stop randomly in the sample and so experience different fields, manifest in a Fourier transform of the time spectra as a distribution of precession frequencies (a “lineshape”). Measuring the µSR lineshape thus enables us to map out the distribution of magnetic fields in the sample. In this chapter we present our muon spin relaxation studies of II-IV-V2 chalcoperite semiconducting CdGeAs2 polycrystals doped with Mn (18 mole% Mn and 36 mole% Mn). In both samples, X-ray diffraction spectra (wavelength l=1.54 Å) taken using a Siemens D-500 X-Ray Diffractometer show the majority signal coming from the ordered phase of CdGeAs2 with some unidentified contributions at the level of a few mass %. However, X-ray diffraction techniques may be insensitive to small (characteristic dimensions of 100 Å) inclusions of foreign magnetic phases (like metallic MnAs) [23]. Although studies of our samples using Auger spectroscopy showed a homogeneous distribution of Mn on a freshly cleaved surface, this method also cannot provide information on a length scale of less than 50 Å.

Figure 1. µSR spectrum (left) and the Fourier transform of the muon spin precession signal (right) in CdGeAs2:36%Mn at T=2.9 K in an external magnetic field of H=1.0023 T (broad line). The narrow line corresponds to muons stopped in the reference sample (CaCO3).

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53

Both CdGeAs2:Mn samples stick firmly to a permanent magnet at room temperature — an inexpensive test for ferromagnetism but one which cannot determine if the magnetism is truly a bulk phenomenon. The magnetic field dependence of our samples’ room temperature magnetization levels off above about 1 T [23], characteristic of classical ferromagnets. Temperature dependent magnetization (measured by SQUID and PPMS) and magnetic susceptibility measurements display a rather smooth transition from FM to a paramagnetic (PM) state above about 320 K. Time-differential muon spin relaxation experiments were performed in high magnetic fields (up to 7 T) on the M15 and M20 surface muon channels at TRIUMF (Canada) using the HiTime and Helios spectrometers and at Paul Scherrer Institute (PSI, Switzerland) on the GPS spectrometer. Figure 1 shows the muon spin relaxation spectrum and the lineshape of a typical muon spin precession signal in CdGeAs2:36%Mn. The narrow line originates from muons stopped in the nonmagnetic reference sample (CaCO3); the broad line (from muons stopped in CdGeAs2:Mn) displays a significant negative field shift relative to that of the reference sample. To understand the nature of this shift one has to appreciate the origin of the net magnetic field seen by the muon in a FM medium according to the concept of a Lorentz sphere [22]:

Bμ = Bext - Bdem + BL + Bc + Bdip where

(1)

Bext is the external magnetic field (Bshift = Bμ - Bext ); Bdem = 4πNm (N is the

demagnetization factor of the sample and

m

is its magnetization per unit of volume) and

BL = 4π/3 m are the demagnetization fields from the finite sample and the empty Lorentz

Figure 2. Magnetization vs. muon spin relaxation technique: magnetic field dependence of

- Bdem

+ BL(circles) [the sum of the second and the third terms in equation (1)] and the magnetic field shift experienced by the muon (stars) in CdGeAs2:18%Mn at T=300 K.

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Vyacheslav G. Storchak, Jess H. Brewer, Dmitri G. Eshchenko et al.

sphere, respectively; Bc is the contact hyperfine field at the muon site and Bdip is the net magnetic field from magnetic dipoles inside the Lorentz sphere. Separation of the entire sample into two regions — inside the Lorentz sphere, where the sources of magnetic field are treated individually on a microscopic scale, and outside this sphere, where the continuum approach is applied — is a familiar approximation justified by the short-range nature of the magnetic interaction. Figure 2 shows the magnetic field dependence of (circles) and

Bshift

Bmag= 4π(1/3 – N)m

(squares) determined by measurements of the magnetization [23];

N=0.805 is tabulated in [24] for a square plate (8x8 mm, thickness 0.82 mm). The estimated uncertainty of this N value is ≈ 10%. Here Bshift is mainly determined by the macroscopic magnetization of our sample, Bmag. The difference between Bshift and Bmag gives the sum of the last two terms in equation (1), which reflects the microscopic origin of the magnetic fields in a dilute magnetic semiconductor. This difference varies between about 5 G and 20 G. Such a low value of

Bshift - Bmag does not exceed 10% of the absolute value of the magnetic field shift and thus may originate from the uncertainty in N. In order to determine this difference more precisely, we measured the magnetic field shift in a spherical sample (N=1/3) of CdGeAs2:18%Mn. Such a shape allows one to set Bmag=0 and get rid of the macroscopic magnetization effect on the magnetic field experienced by the muon. Measurements of this shift in a spherical sample of 3.35(5) mm diameter in a magnetic field of H = 0.64 T give a value of Bc + Bdip = 1.6±1.5 G. This value is at least 3-4 orders of magnitude smaller than the contact and dipole magnetic fields in classical ferromagnetic metals like Fe, Co, Ni, Gd etc. [22] and should be compared with theoretical estimates using recent calculations [13, 19, 20]; unfortunately these estimates are not yet available. In any case, such a low value of Bc+Bdip is more consistent with magnetic field-induced magnetism than with the spontaneous magnetization characteristic of classical ferromagnets. Figure 3 shows the temperature dependence of the magnetic field shift in two CdGeAs2 samples. In the entire measured temperature range, muon spin relaxation spectra in both samples exhibit a single line shifted with respect to the external magnetic field, implying a fairly uniform spatial distribution of the magnetic moments in CdGeAs2:Mn — an essential point which can not possibly be questioned using macroscopic magnetometry. Both samples show significant reduction of the magnetic field shift upon transition to the PM state at about 320 K. The magnetic field shift is reduced by about an order of magnitude when the Mn content is increased from 18% to 36%. Both magnetization and susceptibility measurements in these two samples show a corresponding reduction of magnetization. Although one would expect an increase of the magnetization at higher Mn content, the peculiar behavior observed may be connected to qualitative modification of the transport properties: Hall effect and thermopower measurements reveal the majority carriers in the 18% sample to be holes with a

Weak High-Temperature Bulk Ferromagnetism…

55

concentration of ~2.1020 cm-3 at T>Tc, while in the 36% sample the majority carriers are electrons with a corresponding concentration of ~4.1020 cm-3.

Figure 3. Temperature dependence of the magnetic field shift in two CdGeAs2 samples doped with 18% Mn (stars) and 36% Mn (circles). The applied magnetic field is H=1 T.

Extremely weak contact and dipole interactions do not allow one to exclude a direct exchange between Mn ions as a possible mechanism for FM in our II-IV-V2 DMS. On the other hand, as in III-V DMS, this ferromagnetism may be mediated by holes, the key feature that allows magnetic properties to be altered electrically [8, 9]. This occurrence of carrier-mediated ferromagnetism at high temperatures in a magnetically and electrically dilute system is very much unusual. In many II-VI group semiconductors, Mn moments interact very weakly each other (unless they are nearest neighbors) and fluctuate randomly in orientation down to very low temperature. Similarly, in III-V group semiconductors the Mn d-electrons are comparatively weakly incorporated into the bonding orbitals of the corresponding semiconductor. However, the occurrence of robust ferromagnetism in III-V systems doped with Mn clearly evidences that the coupling is much stronger than in II-VI-based semiconducting systems. Manganese atoms substitution introduces both local magnetic moments and valence-band hole sthat hybridize with Mn dorbitals of the same spin in III-V matrix. The key issue of the microscopic theories of the magnetism in III-V dilute magnetic semiconductors is whether the carrier end up tightly bound to corresponding Mn acceptor or as a more spatially extended wavefunction. As the magnetic moments in a dilute system are far apart from each other, their coupling responsible for long-range carrier-mediated ferromagnetism necessarily requires acceptor level states extended over at least a few lattice constant. This mechanism adopted for III-V group Mndoped semiconductors probably applies to genealogically related II-IV-V2 dilute magnetic semiconductors doped with magnetic ions. The two possible mechanisms described above have to be set against another possibility: small inclusions of FM phases of MnAs in CdGeAs2:Mn that might be too dilute to show up

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Vyacheslav G. Storchak, Jess H. Brewer, Dmitri G. Eshchenko et al.

in X-ray diffraction spectra or Auger spectra but could account for the entire magnetic moment of the diluted magnetic semiconductor. Figure 4 shows the temperature dependence of the full linewidth at half maximum height (FWHM) of the field distribution experienced by the muon ensemble. One might argue that considerably faster relaxation in CdGeAs2:Mn sample with lower Mn doping is the result of fast inhomogeneous diffusion of the muon in the sample with 18% Mn with its consequent trapping close to Mn ion. However, strong energy level shifts produced by Mn dopants obviously preclude any possible muon diffusion in heavily doped CdGeAs2 [26]. Instead, the linewidth in the 18%Mn sample is found to be in good agreement with a model of randomly distributed polarized Mn moments [27]. From these muon spin relaxation data we can not say if the Mn moments are uniformly diluted or segregated in small clusters, as both cases give zero average magnetic field (excluding geometrical contributions) and Lorentzian line shapes with the FWHH dependent only on the total number of magnetic impurities per unit volume [27].

Figure 4. Temperature dependence of the full width at half-height of the magnetic field distribution seen by muons in two CdGeAs2 samples doped with 18% Mn (stars) and 36% Mn (circles). The applied magnetic field is H=1 T.

In conclusion, µSR studies reveal a bulk character of FM in CdGeAs2:Mn at room temperature. The FM observed is very weak: at least 3-4 orders of magnitude weaker than in classical FM metals. This fact requires further theoretical study. This work was supported by the Kurchatov Institute, the Canadian Institute for Advanced Research, the Natural Sciences and Engineering Research Council and the Royal Society of London.

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References [1] G.A. Prinz, “Magnetoelectronics”, Science 282, 1660-1663 (1998). [2] S.A. Wolf, D.D. Awshalom, R.A. Buhrman, J.M. Daughton, S. von Molnar, M.L. Roukes, A.Y. Chtchelkanova and D.M. Treger, ”Spintronics: A Spin-Based Electronics Vision for the Future”, Science 294, 1488-1495 (2001). [3] E.L. Nagaev, “Colossal Magnetoresistance and Phase Separation in Magnetic Semiconductors” (London: Imperial College Press, 2002). [4] J.K. Furdyna, “Diluted Magnetic Semiconductors”, J. Appl. Phys. 264, R29-R64 (1988). [5] H. Ohno, “Making Nonmagnetic Semiconductors Ferromagnetic”, Science 281, 951-956 (1998). [6] H. Ohno, H. Munekata, S. von Molnar and L.L. Chang, ”Magnetotransport properties of p-type (In,Mn)As diluted magnetic III-V semiconductors”, Phys. Rev. Lett. 68, 26642667 (1992). [7] H. Ohno, A. Chen, F. Matsukura, A. Oiwa, A. Endo, S. Katsumoto and Y. Iye, “(Ga,Mn)As: A new diluted magnetic semiconductor based on GaAs”, Appl. Phys. Lett. 69, 363-365 (1996). [8] H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno and K. Ohtani, “Electric-field control of ferromagnetism”, Nature 408, 944-946 (2000). [9] H.X. Tang, R.K. Kawakami, D.D. Awschalom and M.L. Roukes, ”Giant Planar Hall Effect in Epitaxial (Ga,Mn)As Devices”, Phys. Rev. Lett. 90, 107201 (2003). [10] F. Matsukura, H. Ohno and T. Dietl, “III-V Ferromagnetic Semiconductors”, in Handbook of Magnetic Materials, (Ed. K.H.J. Buschow), 2002. [11] C. Zener, “Interaction between the d-Shells in the Transition Metals. II. Ferromagnetic Compounds of Manganese with Perovskite Structure”, Phys. Rev. 82, 403-405 (1951); T. Dietl, H. Ohno, F. Matsukura, J. Gibert and D. Ferrand, ”Zener model description of ferromagnetism in zinc-blende magnetic semiconductors”, Science 287, 1019-1022 (2000). [12] C. Zener, “Interaction between the d-Shells in the Transition Metals”, Phys. Rev. 81, 440-444 (1951). [13] S.C. Erwin and I.Zutic, “Tailoring ferromagnetic chalcopyrites”, Nature Materials 3, 410-414 (2004). [14] G.A. Medvedkin, T. Ishibashi, T. Nishi, K. Hayata, Y. Hasegawa and K. Sato, ”Room temperature ferromagnetism in novel diluted magnetic semiconductor Cd1-xMnxGeP2”, Jpn. J. Appl. Phys. 39, L949-L951 (2000); [15] G.A. Medvedkin, K.Hirose, T. Ishibashi, T. Nishi, V.G. Voevodin and K. Sato, ”New magnetic materials in ZnGeP2-Mn chalcopyrite system”, J. Cryst. Growth 236, 609-612 (2002). [16] S. Cho, S. Choi, G.-B. Cha, S.C. Hong, Y. Kim, Y.-J. Zhao, A.J. Freeman, J.B. Ketterson, B.J. Kim, Y.C. Kim and B.-C. Choi, ”Room-Temperature Ferromagnetism in (ZnMn)GeP Semiconductors”, Phys. Rev. Lett. 88, 257203 (2002). [17] S. Choi, G.-B. Cha, S.C. Hong, S. Cho, Y. Kim, J.B. Ketterson, S.-Y. Jeong and G.-C. Yi, ”Room-temperature ferromagnetism in chalcopyrite Mn-doped ZnSnAs single crystals”, Sol. St. Commun. 122, 165-167 (2002).

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[18] V.M. Novotortsev, V.T. Kalinnikov, L.I. Koroleva, R.V. Demin, S.F. Marenkin, T.G. Aminov, G.G. Shabunina, S.V. Boichuk and V.A. Ivanov, ”High- ferromagnetic semiconductor CdGeAs2{Mn}”, Russ. J. Inorg. Chem. 50, 492-497 (2005). [19] Y.-J. Zhao, W.T. Geng, A.J. Freeman and T. Oguchi, ”Magnetism of chalcopyrite semiconductors: CdMnGeP2”, Phys. Rev. B63, 201202 (2001). [20] P. Mahadevan and A. Zanger, ”Room-Temperature Ferromagnetism in Mn-Doped Semiconducting CdGeP2”, Phys. Rev. Lett. 88, 047205 (2002). [21] S. Picozzi, “Engineering ferromagnetism”, Nature Materials 3, 349-350 (2004). [22] A. Schenck, Muon Spin Rotation: Principles and Applications in Solid State Physics (Adam Hilger, Bristol, 1986); J.H. Brewer, “Muon Spin Rotation/Relaxation/ Resonance”, in Encyclopedia of Applied Physics 11, 23 (VCH Publishers, New York, 1994). [23] C.S. Barrett and T.B. Massalski, Structure of Metals (Pergamon Press, Third revised edition, 1980). [24] V.G. Storchak, D.G. Eshchenko, H. Luetkens, E. Morenzoni, R.L. Lichti, S.F. Marenkin, O.N. Pashkova and J.H. Brewer, ”Room temperature ferromagnetism in III-V and II-IVV dilute magnetic semiconductors”, Physica B374-375, 430-432 (2006). [25] P.G. Akishin and I.A. Gaganov, ”The macroscopic demagnetizing effects in cylindrical and rectangular box samples”, J. Mag. Mag. Mater. 110, 175-180 (1992). [26] V.G. Storchak and N.V. Prokof'ev, “Quantum diffusion of muons and muonium atoms in solids”, Rev. Mod. Phys. 70, 929-978 (1998). [27] V.Yu. Pomjakushin, A.M. Balagurov, T.V. Elzhov, D.V. Sheptyakov, P. Fischer, D.I. Khomskii, V.Yu. Yushankhai, A.M. Abakumov, M.G. Rozova, E.V. Antipov, M.V. Lobanov, S.J.L. Billinge, ”Atomic and magnetic structures, disorder effects, and unconventional superexchange interactions in A2MnGaO5+δ(A = Sr, Ca) oxides of layered brownmillerite-type structure”, Phys. Rev. B66, 184412 (2002).

In: Spintronics: Materials, Applications and Devices ISBN 978-1-60456-734-2 c 2009 Nova Science Publishers, Inc. Editors: G.C. Lombardi and G. E. Bianchi

Chapter 3

F ERROMAGNETISM IN S EMICONDUCTORS D OPED WITH N ON - MAGNETIC E LEMENTS Y.P. Feng∗, H. Pan, R.Q. Wu, L. Shen Department of Physics, National University of Singapore J. Ding, J.B. Yi Department of Materials Science and Engineering, National University of Singapore Y.H. Wu Department of Electrical and Computer Engineering, National University of Singapore

Abstract Materials with specific magnetic properties are required for spin generation, manipulation, and detection for spintronics applications. Transition metal (TM) doped semiconductors and oxides, which were expected to meet this requirement, are facing problems due to formation of secondary phases and clusters. To overcome this problem, researchers have gone beyond traditional magnetic TM doped semiconductors and oxides and have started looking for alternative dopants. Some breakthroughs have been made recently. In this article, we review progress made along this direction and discuss possible origins of the unexpected ferromagnetism observed in materials without magnetic elements. We focus on dilute magnetic semiconductors and oxides obtained by doping with elements with a full d-shell or 2p light elements. It has been demonstrated that room temperature ferromagnetism can be achieved in such systems.

1.

Introduction

Electron has two fundamental properties: charge and spin. The ability to generate, control and detect the motion of charges in either free space or in solid state forms the basis of modern electronics. Today’s communication, information processing and data storage technologies are dominated by semiconducting materials. Compared to charge, it is rather ∗

E-mail address: [email protected]

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Y.P. Feng, H. Pan, R.Q. Wu et al.

difficult to generate, control and detect electron spin. Applications of spin-based devices are so far limited to information storage, and most of the spin-related materials and devices still rely primarily on the spontaneous ordering of spins in the form of different types of magnetic materials. This situation is expected to change with successful development of spin-based electronics, or spintronics, the new kind of electronics that seeks to exploit, in addition to the charge degree of freedom, the spin of the carriers.[1] To make spintronic devices, the primary requirement is to have a system that can generate a current of spin polarized electrons, and a system that is sensitive to the spin polarization of the electrons for detection of electron spin. Most devices also have a unit in between that changes the current of electrons depending on the spin states. The simplest method of generating a spin polarized current is to inject the current through a ferromagnetic material. Compared to ferromagnetic metals, if a magnetic semiconductor can be used as a spin injector into a nonmagnetic semiconductor, it would facilitate the integration of spintronics and semiconductor-based electronics. Comparable resistivities of magnetic and nonmagnetic semiconductors could provide efficient spin injection. Therefore, to enable a host of new microelectronics device applications, it is necessary to develop materials which satisfy the following requirements. (i) Since the ambition is to use materials and devices at room temperature, spin-polarized charge carriers should be available at room temperature which requires the ferromagnetic transition temperature to be above room temperature. (ii) The mobile charge carriers should respond strongly to changes in the ordered magnetic state so that the ferromagnetism can be electrically tuned. (iii) The material should retain fundamental semiconductor characteristics when doped. In other words, one needs magnetic semiconductors exhibiting room temperature ferromagnetism. Ideally, a semiconductor can be made magnetic by including ions that have a net spin into a semiconductor host.[2] In such an alloy, a stoichiometric fraction of the constituent atoms is replaced by magnetic ions, typically magnetic transition metal (TM) atoms. The doped semiconductors are referred as dilute magnetic semiconductors (DMSs) because only a small amount of magnetic ions is required to make the semiconductor magnetic. In recent years, considerable work has been devoted to the study of DMS materials. However, despite of the large effort and the fact that many materials have been found to display roomtemperature ferromagnetism,[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] the origin of ferromagnetism in these materials continues to be debated. Some studies reported strong evidence of phase separation and formation of ferromagnetic clusters suggesting a nonintrinsic behavior which is not suitable for technological applications. Many problems remain to be solved before the materials can be used to build spintronics devices. Besides magnetic TM doped semiconductors, researchers are now looking for alternative dopants in order to overcome problems encountered by the former and in hope to produce practically useful DMSs. Some breakthroughs have been made recently. In this article, we review progress made along this direction and discuss possible origins of the unconventional ferromagnetism observed in materials without magnetic element. The current status in TM doped semiconductors and oxides is briefly discussed in the next section. We focus on DMSs obtained by nonmagnetic impurity doping in semiconductors and oxides and review the current progress in Section III (DMSs obtained by doping with elements with a full d-shell) and Section IV (DMSs obtained by doping with 2p light elements). Finally in Section V, some concluding remarks are given.

Ferromagnetism in Semiconductors Doped with Non-magnetic Elements

2.

61

Magnetic Transition Metal Doped Semiconductors

Magnetic transition metal (TM) doped III-V semiconductors such as (Ga,Mn)As,[18, 19] (In,Mn)As[20, 21] and (Ga,Mn)N[19, 22] belong to the most widely studied magnetic semiconductors, and the mechanism of ferromagnetism in such systems is relatively well understood.[19, 22] In particular, (Ga,Mn)As has become one of the best-understood ferromagnets, and it is now regarded as a textbook example of a rare class of robust ferromagnets with dilute magnetic moments coupled by delocalized charge carriers. An extensive review on ferromagnetism in (III,Mn)V semiconductor was given by Jungwirth et al. recently.[19] It is known that magnetically doped III-V semiconductors are ferromagnetic for a wide range of carrier concentrations, from the insulating to highly conducting regimes, owing to different mechanisms.[23] Here, Mn acts both as an acceptor and as a source of local moments. It has been proposed that if the local exchange between the carriers and the magnetic ions is large enough, an impurity band is formed in the energy gap of the host semiconductor.[24, 25] In a highly insulating system, the Fermi level is well below the mobility edge of the impurity band. In this regime ferromagnetism can be explained as the result of percolation of bound magnetic polarons.[26] In the more conducting samples, itinerant carriers would mediate ferromagnetism via a Ruderman-Kittel-KasuyaYosida (RKKY) mechanism.[27] The highest Curie temperature achieved in the (Ga,Mn)As system is 173 K which was obtained in Mn-doped GaAs prepared using low temperature annealing techniques.[28, 29, 30] This, however, is still too low for actual applications. In order to make DMS a real technology, materials with a Curie temperature higher than room temperature are needed. Higher Curie temperature was obtained in heterostructures consisting of Mn δ-doped GaAs and p-type AlGaAs layers by varying the growth sequence of the structures followed by low-temperature annealing.[31] However, it is not certain whether the Curie temperature in such systems can be pushed up to above room temperature.[32] Mn doping in other III-V semiconductors were also investigated. (III,Mn)Sb based DMSs belong to the same category as (Ga,Mn)As and (In,Mn)As. However, their Tc ’s are expected to be lower compared to the corresponding arsenides due to weaker p-d exchange and smaller magnetic susceptibility of itinerant holes in the antimonides.[33, 34] Mn-doped phosphide or nitride DMSs were predicted to be high Tc ferromagnetic semiconductors based on the kinetic-change model.[33] Solubility of Mn in these materials is also larger than that in arsenides. However, the nature of magnetic interaction in (III,Mn)P and (III,Mn)N is more complex and not completely understood.[35] Studies based on the mean-field Zener model predicts that DMSs with a Tc above room temperature can be obtained with a right combination of host material, carrier concentration, and magnetic impurity (type and density).[33] In particular, with 5% of Mn and 3.5 × 1020 cm−3 of holes in wide-gap semiconductors, such as GaN, ZnO and C, these materials should be ferromagnetic at room temperature. First-principles calculations also predict a rather stable ferromagnetism for these materials.[36, 37] Stimulated by these theoretical predictions, intensive research has been carried out to explore high Tc DMSs.[38, 39, 6, 40] Among the different types of materials that have been investigated, oxide-based DMSs have attracted special attention, in particular TiO 2,[17] ZnO[41] and SnO 2[42] based materials. However, despite considerable theoretical and experimental efforts, the nature of their

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electronic structure and the origin of ferromagnetism observed in some DMS is still under debate. Particularly in TM doped oxides, there has been no consensus on the origin of ferromagnetism. Whether this is an extrinsic effect due to direct interaction between the local moments in magnetic impurity clusters (or nanoclusters) or is indeed an intrinsic property caused by exchange coupling between the spin of carriers and the local magnetic moments is still not clear. This is a very important issue because spintronics requires the carriers to be polarized and this can only be guaranteed if ferromagnetism is intrinsic. Experimental evidence for carrier-mediated ferromagnetism in oxide based DMSs is not yet conclusive. Let’s take ZnO as an example. As a direct, wide band gap semiconductor, ZnO has potential applications in UV photonics and transparent electronics. It offers significant potential in providing charge, photonic and spin-based functionality. Theoretical predictions suggest that room temperature carrier-mediated ferromagnetism should be possible in ZnO, albeit for the p-type material. Unfortunately, the realization of the p-type ZnO has proved difficult. ZnO-based DMS was first reported for Co-doped thin films by Uedo et al. [41] The average magnetic moment per Co atom they obtained in Zn 0.85Co0.15O films was 2 µB . Since this report, there have been hundreds of reports on ZnO based DMSs.[38, 39, 6, 40] The intensive experimental investigations so far have only produced widely diverging results, ranging from non-ferromagnetic or ferromagnetic with extrinsic origins to intrinsic ferromagnetism with various Curie temperatures.[38, 39, 40] Although observation of ferromagnetism in ZnO has been reported frequently in literature by magnetometry measurement, so far there has been no report on correlated ferromagnetism in magnetic, optical and electrical measurements. Many reports have raised the serious doubts on the magnetism of magnetically-doped ZnO.[43] Even though some studies suggest that with sufficient carrier density, room temperature ferromagnetism can be achieved in (Zn,Co)O,[43, 44, 45, 46, 47] crucial experiment such as optical magnetic circular dichroism that is designed to give signature of dilute ferromagnetism failed to clarify the issue.[49, 48, 16] The large disparity in experimental results is mainly caused by the fact that the properties of TM doped ZnO tend to be very sensitive to the preparation methods and conditions, which in turn makes it difficult to conduct systematic studies. Results of various studies on other oxides have been equally controversial. While some support carrier mediation of ferromagnetism, others show evidence of ferromagnetism contributed by magnetic clusters. It appears that there exist several competing ferromagnetic mechanisms and their dominances vary from material to material and under different conditions. Recently, Calder´on and Das Sarma analyzed different models for carrier-mediated ferromagnetism in dilute magnetic oxides and proposed that a combination of percolation of magnetic polarons at lower temperature and RKKY ferromagnetism at higher temperature as the reason for the high critical temperature measured in these materials.[23] In dilute magnetic oxides, carriers, usually electrons, are provided by the oxygen vacancies that are believed to act as shallow donors.[50, 51] This is different from III-V semiconductors, like (Ga,Mn)As, where the carriers (holes) are provided by the magnetic impurities themselves which act also as donors (or acceptors). Since the binding energy of the electrons on the oxygen vacancies is not large enough to keep the electrons bound up to the high temperature reported for the Tc (∼ 700 K), they suggested that thermally excited carriers also mediate ferromagnetism via the RKKY mechanism at sufficiently high temperatures, complementing the bound polaron picture.

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The observed ferromagnetism in magnetic TM doped DMSs could be due to a number of different origin.[52] Carrier-mediated ferromagnetism in spatially uniform ferromagnetic DMSs is certainly possible and the mechanisms are relatively well understood. However, in a composite material, precipitation of a known ferromagnetic, ferrimagnetic or antiferromagnetic compound can account for magnetic characteristics at high temperature. Even in the absence of precipitates of foreign compound, alloys can phase separate into nanoscale regions with small and large concentrations of the magnetic constituent, and high-temperature magnetic properties are dominated by the regions with high magnetic ion concentrations. In many carrier-doped DMSs, a competition between long range ferromagnetic and short-range antiferromagnetic interactions and/or the proximity of the localization boundary lead to an electronic nanoscale phase separation. Questions then can be asked: Is the FM observed in DMSs really intrinsic? Are there really strong magnetic interactions between well-separated magnetic dopants? Available experimental techniques are unable to provide exclusive evidence to differentiate the various mechanisms and have difficulties in unambiguous determination of the origin of magnetic signals in these materials. Theoretical studies also predicted that clusters of magnetic elements are energetically favored in some DMSs. For example, through first-principles calculation, Cui et al. [53, 54] predicted that the magnetic Cr dopants have a clustering tendency in (Ga,Cr)N. Magnetic secondary clusters have been shown to be ferromagnetic. Ferromagnetism of the DMSs can thus arise from magnetic secondary clusters.[55, 56, 48, 57, 9] These extrinsic magnetic behaviors are undesirable for practical applications in spintronics. In a recent study, Prater et al. found that upon high-temperature annealing in oxygen, (Zn,Co)O samples which showed magnetic ordering above room temperature as-deposited became insulating and the magnetization drops, suggesting that the observed magnetic behavior of the oxide is directly related to the presence of intrinsic defects, notably oxygen vacancies and Zn interstitials.[58] The origin of ferromagnetism in TM doped DMSs is still controversial due to the possibility of magnetic secondary phases and other source of magnetism, uncertainty of magnetic interactions.

3.

DMS without Magnetic Elements

It is important to note that the vast majority of the DMSs that have been investigated are semiconductors or oxides doped with magnetic d atoms having open dn shells (1 ≤ n ≤ 9) or various combinations of several magnetic atoms of different kinds. A possible way to avoid problem related to magnetic precipitate is to dope a nonmagnetic matrix with nonmagnetic impurity atoms. Because the dopants, and hopefully their oxides, are not magnetic, DMSs obtained this way can be free of ferromagnetic precipitates and hence unambiguous DMSs. Copper-doped ZnO was among the first such systems that have been investigated. In a systematic theoretical study of ZnO doped with 3d TM elements using the Korringa-KohnRostoker (KKR) Green’s function method based on the local density approximation, Sato and Katayama-Yoshida initially predicted that ZnO doped with 25% Cu is nonmagnetic.[36] However, it was realized later that this was due to the effect of a small supercell (high Cu concentration) in the calculation, such that it was necessary to place Cu atoms both ˚ and on above/below each other in adjacent basal planes (with a separation of 5.20 A)

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˚ Furadjacent cation positions within a single basal plane (with a separation of 3.25 A). ther theoretical studies at lower doping concentrations predicted that ZnO doped with 6.25%[59] and 3.125%[60] Cu are ferromagnetic. In these first principles studies, based on local spin density approximation (LSDA)/LSDA+U and generalized gradient approximatin ˚ Feng[61] (GGA)/GGA+U, respectively, the Cu atoms were separated by at least 6.1 A. later examined the effect of Cu separation and the stability of the FM state in (Zn,Cu)O and ˚ along the c-axis the FM state is clarified that when the Cu atoms are separated by 5.20 A ˚ within the basal plane the AFM favored, but when the Cu atoms are separated by 3.25 A state is favored, but the AFM state has higher total energy compared to the FM state at larger Cu separation. Therefore, it was concluded that ferromagnetic semiconductors can be obtained by doping Cu into ZnO. Incidentally, the initial experiment on Cu-doped ZnO thin films prepared by combinatorial laser molecular-beam epitaxy method failed to detect ferromagnetism down to 3 K.[62] However, ZnO thin films doped with 0.3% Cu prepared with pulsed-laser deposition were shown to be ferromagnetic by magnetic circular dichroism spectra.[48] Further studies devoted to Cu-doped ZnO, with samples of different Cu concentrations, by Lee et al. [63] provided additional experimental confirmation of ferromagnetism in Cu-doped ZnO. More recently, Buchholz et al. reported room temperature FM in p-type ZnO thin films but nonferromagnetic behavior in n-type Cu-doped ZnO at room temperature.[64] Ferromagnetism in Cu-doped ZnO was reexamined recently by Ye et al.[65] using accurate full-potential linearized augmented plane-wave and DMol calculations based on density functional theory. Each Cu dopant was found to carry a magnetic moment of 1 µB . For ZnO doped with 12.5% Cu, the FM state was found energetically favored by 43 meV compared to the AFM state which lead to an estimated Tc of about 380 K. More importantly, ferromagnetism was predicted for both n-type and p-type samples. Subsequently, Hou et al. observed ferromagnetism in carefully prepared n-type Cu-doped ZnO thin films.[66] In this study, a series of Cu-doped ZnO thin films was prepared on glass substrate by dc reactive magnetron sputtering, to reduce the influence of ferromagnetic impurities. All the films, ranging from 2 to 12% Cu doping, were found ferromagnetic at room temperature and the moment per Cu ion was found to decrease with increasing Cu concentration and nitrogen doping. The results confirm that ferromagnetism can be mediated by itinerant electrons in Cu-doped ZnO. Besides ZnO, GaN is another wide gap semiconductor that was predicted to hold the possibility of a DMS with a Curie temperature above room temperature.[33] Motivated by the success of ferromagnetism in Cu-doped ZnO, Wu et al.[67] carried out first-principles calculations based on spin density functional theory to study the magnetic properties of GaN doped with 6.25% of Cu. Using a supercell which consists of 2 × 2 × 2 wurtzite GaN units, with one or two Ga atoms substituted by Cu, they found that each Cu dopant induces a magnetic moment of 2.0 µB which is larger than that of Cu-doped ZnO.[65] Furthermore, the calculated band structure (Fig. 1) shows that Cu-doped GaN is half metallic with the majority spin being semiconducting and the minority spin being metallic with sufficient unfilled states above the Fermi level. Calculations based on two Cu atoms at the largest ˚ in the supercell indicated that the FM state is the ground state possible separation of 6.2 A and its energy is 50 meV lower than that of the AFM state. Based on this significant energy difference, Wu et al. predicted that Cu-doped GaN should be a room temperature DMS,

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Figure 1. Band structure of the majority spin (a) and the minority spin (b) of GaN doped with 6.25% Cu, calculated using DFT/GGA. The Fermi level is set to zero. Reprinted with permission from Ref. [67], R. Q. Wu et al. Appl. Phys. Lett. 89, 062505 (2006). Copyright @ American Physical Society. The above prediction was confirmed recently by Lee et al. using an implantation and subsequent annealing process.[68] In their experiment, 1 MeV Cu 2+ ions were implanted into GaN with a dose of 1 × 1017 cm−2 at room temperature which was followed by rapid thermal annealing of the samples at 700, 800, and 900 ◦ C, respectively, for 5 min. Both samples annealed at 700 and 800 ◦ C were found ferromagnetic at room temperature. Figure 2 shows the magnetization-field (M −H) curves at room temperature (300 K) of as-implanted and annealed samples. The sample annealed at 900 ◦ C did not show ferromagnetism, possibly due to clustering of Cu during the annealing process. Even though the estimated saturation magnetization of 0.057 µB and 0.27 µB per Cu atom for the samples annealed at 700 and 800 ◦ C, respectively, are much smaller than the value (2 µB /Cu atom) predicted by the first-principles calculations,[67], the experimental evidence of room temperature ferromagnetism of Cu implanted GaN is encouraging. This, however, was cautioned by Rosa and Ahuja[69] who investigated structural and electronic properties of Cu-doped GaN using density functional theory (DFT) within GGA. They considered two configurations where the Cu atoms are separated along the [0001] ˚ (far configuration) and 3.22 A ˚ (close configuration). Their results of direction by 5.23 A total energy calculations indicated that the close configuration has a lower energy (by 0.4 eV/cell) and therefore more stable than the far configuration. Due to atomic relaxation, the spin polarization on the Cu atom in the GaN lattice is rather small, leading to rather weak ferromagnetic behavior. Ferromagnetism in Cu-doped ZnO and GaN can be explained based on the p-d hy-

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Figure 2. Magnetization-field (M − H) curves at room temperature (300 K) for the asimplanted GaN:Cu sample, and samples annealed at 700 and 800 ◦ C after the Cu implantation. Reprinted with permission from Ref. [68], J.-H. Lee, et al. Appl. Phys. Lett. 90, 032504 (2007). Copyright @ American Institute of Physics. bridization mechanism.[70, 71] Here the d orbitals of Cu hybridizes strongly with the p orbitals of its neighboring anions (oxygen or nitrogen) of the host semiconductor, resulting in spin polarization of the neighboring anions with large magnetization. They couple ferromagnetically or anitiferromagnetically with the dopant. Other dopants in turn couple to the spin polarized anions in the same way for an energy gain, resulting in an indirect FM coupling among dopants. Very recently, room temperature ferromagnetism was also found in Cu-doped GaN nanowires.[72] The saturation magnetic moments in the hot-wall chemical vapor deposited single-crystalline GaN nanowires doped with 1% and 2.4% Cu were measured to be higher than 0.86 µB /Cu at 300 K. The author attributed the ferromagnetism to p-d hybridization between Cu and N ions, which induces delocalized magnetic moments and long-range coupling. Besides Cu, other IIA non-magnetic elements such as Pd[73] have been considered as possible dopants for DMSs.

4.

DMS by Anion Doping with 2p Light Elements

Carbon can exist in a number of polymorph such as graphite, diamond, graphene, nanotubes, and fullerene. None of these show magnetism. However, various recent studies indicate that defects in carbon system can lead to magnetization.[74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87] Motivated by this idea, Pan et al. considered carbon as a possible dopant for ZnO based DMS.[88] Using first-principles method based on DFT within LSDA,

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Density of States

they considered various possible point defect forms of carbon in ZnO and concluded that substitution of carbon for oxygen results in spin polarization. A magnetic moment of 2.0 µB was found for each carbon, contributed mainly by the carbon p orbitals. The neighboring Zn atoms and the second nearest neighboring oxygen atoms also contribute a small part to the overall magnetic moment. Strong coupling between the carbon p orbitals, oxygen p orbitals, and the zinc d orbitals were found, as shown by the projected density of states (PDOS) shown in Fig. 3. The interaction results in the splitting of the carbon 2p orbitals near 2.3 eV. The spin-up bands are fully occupied while the spin-down bands are partially filled (Fig. 4). Further calculations showed that ferromagnetic coupling between magnetic moments of different impurity sites is energetically favored compared to the AFM state. This energy difference of 63 meV per pair of C dopants is significant enough to make Zn(O,C) a room temperature ferromagnet.

4 3 2 1 0 −1 −2 −3 −4 1 0.5 0 −0.5 −1 −1.5 2 1 0 −1 −2 −3 100 0 −100 −200 −10

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Figure 3. Total (top panel) and local density of states for the carbon dopant, nearest neighbor Zn atom and the next nearest neighbor O atom, calculated using first-principles method based on DFT within LSDA. The Fermi level is indicated by the dashed vertical line. To verify the theoretical prediction, Pan et al. prepared C-doped ZnO films using pulsed-laser deposition. Three samples with estimated carbon concentrations of 0, 1 and 2.5%, respectively, were prepared. Careful characterization of the samples showed that the pure ZnO film without carbon doping is nonmagnetic, whereas both C-doped ZnO films show ferromagnetism at room temperature (inset in Fig. 5). Based on the measured temperature dependence of magnetization (Fig. 5), the Curie temperatures of both films should be higher than 400 K. The measured saturation magnetization (Fig. 6) yields a magnetic moment per carbon in the range of 1.5 − 3.0 µB which is in good agreement with the theoretical prediction. XPS measurement also revealed existence of carbon in carbide form

Y.P. Feng, H. Pan, R.Q. Wu et al. 4

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3

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2 1 0

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Figure 4. Band structure of C-doped ZnO, calculated using first-principles method based on DFT within LSDA. One oxygen atom is replaced by carbon in a 72-atom supercell ( 3×3×2 primitive cells). The Fermi level is indicated by the dashed horizontal line. which is an indication that carbon substitutes for oxygen in ZnO. Ferromagnetism in C-doped ZnO is significant not only because it is a room temperature DMS, but also because it represents a whole class of new materials whose magnetic properties are determined solely by the interaction between the carriers in p rather than d or f atomic shells. In addition to C-doped ZnO, room temperature ferromagnetism was observed recently in nitrogen doped ZnO.[89] Using pulsed laser deposition, Yu et al. prepared nitrogen embedded ZnO films. The presence of nitrogen ions in the films was confirmed by the secondary ion microscopic spectrum and by Raman experiments. The films were found ferromagnetic at room temperature. The authors attributed the unexpected ferromagnetism to electron transfer from the completely filled d-orbits of Zn to the defect state. The concept of anion doping was first proposed by Kenmochi et al. in 2004 for creating DMS based on CaO[90, 91]. Using the KKR method within the LSDA, Kenmochi et al. studied the electronic structure and magnetic properties of B-, C- or N-doped CaO, as well as Ca vacancies.[90] In particular, they compared the total energies of FM state and spinglass state in each case. It was found that the FM state is more stable than the spin-glass state in Ca(O,C) and Ca(O,N). However, the spin-glass state is more stable than the FM state in Ca(O,B), while the Ca vacancies do not induce any magnetic moment. Figure 7 shows their calculated total DOS and the partial density of 2p-states at B, C and N sites in the ferromagnetic state. A deep impurity band is pushed up into the band gap of CaO. The large exchange-splitting energy between majority spin states and minority states leads to a high-spin ground state. Based on the partially occupied narrow and highly-correlated deep-impurity bands, Kenmochi et al. proposed Zener’s double-exchange as mechanism for the origin of the ferromagnetism in Ca(O,C) and Ca(O,N). Dinh et al.[92] extended the above work to alkaline-earth-metal-oxide, MgO, CaO, BaO and SrO. They discussed the origin of the ferromagnetism through the calculation of the electronic structure and exchange coupling constant by using the pseudo-potentiallike self-interaction-corrected local spin density. The Monte Carlo method was also used to predict the Curie temperature. It was shown that the stability of half-metallic ferromagnetism induced by C in the alkaline-earth-metal-oxide host materials is improved by taking the electron self-interaction into account, compared with the standard local density approx-

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Figure 5. Ms (T )/Ms(5K) versus temperature for samples doped with 1% (target concentration 1%) and 2.5% (target concentration 5%) of carbon, respectively. The solid lines are a guide for the eye. The inset shows the hysteresis loop of the sample with 2.5% carbon taken at 300 K. Reprinted with permission from Ref. [88], H. Pan, et al. Phys. Rev. Lett. 99, 127201 (2007). Copyright @ American Physical Society.

imation (LDA) case, and the C’s 2p electron states in the bandgap become more localized resulting in the predominance of the short-ranged exchange interaction. All alkaline-earthmetal-oxides were found half-metallic ferromagnetic. It was proposed that the ferromagnetic double exchange mechanism is predominant for all materials considered, except for Mg(O,C) at high C concentration. Tc of Mg(O,C) is predicted to be the highest at 10% C doping, while that of other alkaline-earth-metal-oxides increases monotonously. Very recently, the method was further explored theoretically in BeO by Shein et al.[93] and in SrO by Elfimov et al.[94] Results of first-principles calculation based on DFT within GGA, carried out by Shein et al.[93] using a 72-atom supercell Be 36 O35X (X = B, C, N), suggest that in the case of a partial substitution of boron, carbon, or nitrogen atoms for oxygen atoms in the BeO system, a spontaneous spin polarization of the 2p states of impurity atoms takes place, and the Be(O,X) systems become either a semiconducting magnet [Be(O,B)] or half-metallic magnets [Be(O,C) and Be(O,N)], in which conduction is due to the 2p spin states of the anion alone. In a very recent publication,[94] Elfimov et al. presented a theoretical argument, based on first-principles calculations within LSDA and LSDA+U, combined with some experimental support that substitution of nitrogen for oxygen in simple band insulators (e.g. SrO) is sufficient to produce DMS. The substitution of nitrogen for oxygen in simple nonmagnetic oxides leads to holes in nitrogen 2p states which form local magnetic moments. Because of the very large Hund’s rule coupling of nitrogen and oxygen 2p electrons and the rather extended spatial extent of the wave functions these

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Figure 6. The room-temperature saturation magnetization Ms and the magnetic moment per carbon in the carbide state (carbide carbon) as a function of the measured carbon concentration of the C-doped specimens. The carbon concentration in the sample was estimated by SIMS measurements, while the percentage of carbon in the carbide state was estimated by XPS measurements. Reprinted with permission from Ref. [88], H. Pan, et al. Phys. Rev. Lett. 99, 127201 (2007). Copyright @ American Physical Society. materials are predicted to be ferromagnetic metals or small band gap insulators. The theoretical calculations with regard to the basic electronic structure and the formation of local magnetic moments were supported by experimental studies conducted on Sr(O,N) grown by solid-state chemistry methods. Meanwhile, Feng and coworkers extended their theoretical studies on 2p light element doping to other semiconductors and oxides and predicted ferromagnetism in a number of systems, including C-doped CdS,[95] C-doped AlN,[96], N-doped ZnO,[97], etc. Similar behaviors as that of C-doped ZnO were found in these materials. The mechanism for ferromagnetism in these anion doped magnetic materials is still not understood for their particular electronic structure and magnetic properties. DMSs produced by anion doping typically have a low doping concentration but enough mobile carriers. Only 2p electrons of dopants and hosts contribute to the magnetism. The origin of ferromagnetism in these materials challenges our current understanding of ferromagnetism of DMS. The existing theories of DMS cannot be applied because they are based on d and f orbitals but there are no such orbitals in materials doped with light elements. There are several important differences between the 2p and the 3d orbitals which determine the different magnetic properties of DMS doped with 2p light element (anion) and 3d TM (cation). First, the anion 2p bands of the light element (LE) are usually full in ionic states, leaving no room for unpaired spins compared to 3d bands of TM. Secondly, the spin-orbit interaction of p

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Figure 7. Total DOS per unit cell in CaO with Ca vacancies, and DOS (solid lines) and partial density of p-states (dashed lines) at B [1(b)], C [1(c)], and N [1(d)] site per atom in Ca(O,X) (X = B,C and N) in the ferromagnetic state, respectively. The impurities are doped up to 5 at%. The Fermi energy is set to zero. The upper and lower sides of the figures stand for the DOS of up and down spin states respectively. Reprinted with permission from Ref. [90], K. Kenmochi, et al. Jpn. J. Appl. Phys. 43, L934 (2004). Copyright @ The Japan Society of Applied Physics. states is considerably reduced compared to that of d states since it scales with the fourth power of the atomic number. Consequently, spin relaxation of DMS doped with 2 p light elements is expected to be suppressed by up to two orders of magnitude in comparison with 3d cation doped DMS [98]. Thirdly, valence electrons in p states are more delocalized than those in d or f states and have much larger spatial extensions which could promote longrange exchange interactions. Therefore, despite suffering from low solubility[88, 99, 100], DMS doped with 2p light elements can be weak ferromagnets in a highly ordered and low doping concentration. It is reasonable to assume that alignment of magnetic moments in 2 p LE doped DMS is achieved through the p-p coupling interaction between the impurity p states and the host p states at the top of the valence band, similar to p-d hybridization in some of the TM doped DMSs. This interaction follows essentially from quantum mechanical level repulsion, which “pushes” the minority states upward, crossing the Fermi level. Consequently, the p states split into more stable threefold t2 states which are either fully occupied or completely empty. The symmetry and wave function of the impurity 2 p state are similar to those of the top valence band of the III-nitride and II-oxide which consists mostly of anion p orbitals. Therefore, a strong p-p coupling interaction between the impurity state and valence band state is allowed near the Fermi level. Substitution of C for N in AlN or C/N for O in ZnO introduces impurity moments as well as holes. Different from Mn ions in (Ga,Mn)As which polarizes spin of holes in opposite direction, the spin density near each anion impurity in 2p LE doped DMS tends to align parallel to the moment of the impurity ion under the p-p interaction. The strong p-p interaction leads to stronger coupling between impurity and carrier spin orientations. Sufficiently dense spin-polarized carriers are able to effectively mediate an indirect, long-range ferromagnetic coupling between the 2 p LE dopants. The spatially extended p states of the host and the impurity are able to extend the p-p interaction and spin alignment to a large range and thus to facilitate long-range magnetic

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coupling between the impurities. This model based on p-p coupling interaction gives a reasonable explanation to the experimentally observed ferromagnetism in ZnO doped with a small amount of carbons [88]. In this model, free carriers play an essential role in mediating the spin alignment in such DMS.

5.

Concluding Remarks

The discovery of ferromagnetism in semiconductors and oxides doped with non-magnetic elements opens a new pathway to new magnetic materials without conventional magnetic elements. Following this approach, new materials may be developed to meet the materials requirements of spintronics which TM doped semiconductors or oxides promised to but so far unable satisfactorily to deliver. As pointed out by Shein et al.,[93] magnetization of a nonmagnetic host as a result of its doping by nonmagnetic 2p impurities can be expected for a wide class of related systems. As the hosts, one can take such well-known ionic insulators as III-V, II-VI semiconductors/oxides. The dopants should be chosen so that the orbital energies of their p states be higher than the p band of oxygen (anion) of the matrix, so that these states be localized in the energy gap of the initial crystal to ensure the conditions for their spontaneous spin polarization. Understanding the origin of ferromagnetism in systems consisting of only elements with s and p electrons is essential for further development. Research in this direction is still in the infant stage and no complete theory has been put forward to explain the unexpected magnetism in such systems. First-principles methods, particularly that based on the density functional theory, have played a very important role in the study of DMSs and is expected to continue to play such a role in providing theoretical understanding to the phenomena and in predicting new DMS materials. Different from other computational approaches, the firstprinciples method solves the quantum mechanical problem self-consistently and it does not require any experimental input or empirical parameters. It is, therefore, ideal for studying physical properties of new materials and for predicting new materials. With advances in computational algorithms and availability of high performance computing resources, firstprinciples method can now be used to model and study more realistic and complicated systems. First-principles methods are thus indispensable in the study of DMSs. They will continue to be used to provide theoretical understanding and the physical insights for the magnetic materials and to provide guidance to experimental studies. Magnetic ordering is a collective phenomenon. Monte Carlo simulation is useful in studying the statistical behavior, and particularly in predicting Curie temperature. With input from first-principles calculations, Monte Carlo simulation will continue to play an important role in further study of DMSs. Theoretical modeling is essential in establishing a theory of ferromagnetism in DMSs. Existing theories such as RKKY, polaron percolation, etc. each works well for some materials but cannot explain the magnetic ordering in other materials. There may be different mechanisms of ferromagnetic coupling in different materials and it would be a challenging task to establish a complete ferromagnetic theory that works for all DMSs. To reach this goal, it is also necessary to improve experimental techniques in materials growth and characterization, to provide unambiguous evidence for origin of magnetism in DMSs. In order to draw meaningful conclusions, it is crucial

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to conduct systematic studies using different characterization methods. A concerted effort between theorists, computational scientists and experimentalists shall lead to a complete understanding of magnetic behavior of DMSs and practically useful DMSs.

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In: Spintronics: Materials, Applications and Devices Editors: G. C. Lombardi and G. E. Bianchi

ISBN: 978-1-60456-734-2 © 2009 Nova Science Publishers, Inc.

Chapter 4

MAGNETIC RESONANCE AND SPIN-WAVE (MAGNON) EXCITATIONS IN FERROMAGNETIC SEMICONDUCTOR (GA,MN)AS Xinyu Liu, Y.-Y. Zhou and J.K. Furdyna Department of Physics, University of Notre Dame, Notre Dame, IN 46556

Eric C.T. Harley and L.E. McNeil Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3255

Abstract We review recent studies of spin (or magnetization) dynamics in thin layers of the ferromagnetic semiconductor (Ga,Mn)As, including Brillouin light scattering, ferromagnetic resonance, and time-resolved magneto-optical measurements carried out to investigate the various spin dynamic phenomena in this material. In most experiments the analysis of spin wave (magnon) modes in terms of the Landau-Lifshitz-Gilbert equation and specific boundary conditions allows one to determine the value of the exchange stiffness constant, along with the bulk and surface anisotropy in this material. The relationships between the results obtained by nonlinear optics and magnetic resonance, and between frequency and time-domain experiments are also discussed.

I. Introduction Ferromagnetic III-Mn-V semiconductors such as (Ga,Mn)As continue to hold the interest of the scientific community, both for their fundamental scientific interest and for their compatibility with semiconductor technology that may lead to novel nonvolatile spintronic applications.[1,2,3] Spintronics is an emerging field of technology which – by using the spin degree of freedom of electrons in addition to their charge – holds many advantages that may be exploited computation, memory storage, and read-out. Here the speed of manipulating spin alignment is expected to be crucial in all these operations. It is therefore essential to

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investigate the time dependence of spin processes (spin dynamics) and their relationship with itinerant charge carriers in solids on which spintronic devices will be based. It is now well established that the ferromagnetic ordering of the magnetic moments of Mn ions in (Ga,Mn)As is induced by the holes arising from the presence of those same Mn ions.[4,5] However, our understanding of the fundamental magnetic excitations in this material – which are intimately related to the exchange interaction between the Mn ions and the holes – is still far from complete.[6,7] Recently extensive studies of magnetization dynamics in (Ga,Mn)As films have been carried out by two complementary approaches – by experiments in the frequency domain, such as Brillouin light scattering (BLS) [8] and ferromagnetic resonance (FMR) [6,9,10,11]; and by optical real-time techniques, such as the ultra-fast magneto-optical Kerr effect (MOKE).[12,13,14,15,16,17] Among these works, comprehensive models of spin dynamics and of the coupling of magnetic excitations to light have been proposed to explain the experimental results and to allow one to gain accurate values of the exchange, bulk, and surface anisotropy constants which ultimately determine the spin dynamics in a ferromagnet.[15,10] These results and their analysis may lead to the identification of universal principles that characterize all carrier-mediated ferromagnetic semiconductors. In particular, the understanding of these principles may hold the key to extraordinarily fast manipulation of spins based on the combination of semiconductor physics and magnetism that is unique to these materials. In this chapter we review recent BLS, FMR, and time-resolved magnetooptic Kerr rotation experiments used to study the fundamental dynamic magnetic excitations – referred to as spin waves or magnons – in ferromagnetic (Ga,Mn)As films. In addition to providing considerable insight into the properties of bulk and surface magnetic excitations in (Ga,Mn)As, these methods have allowed us to establishing hitherto unrecognized relationships between nonlinear optics and magnetic resonance, as well as between frequency and time-domain experimental techniques. The spin dynamics in a ferromagnet can be described using the Landau-Lifshitz-Gilbert (LLG) equation as follows:[18,19]

∂M α ∂M = −γM × H eff + M × , ∂t M ∂t

(1)

where γ is the gyromagnetic constant, M is the magnetization, Heff is the effective magnetic field within the specimen, and α is a phenomenological damping parameter. In this equation, the first and second terms represent the precessional motion and energy dissipation, respectively. Note that the effective field Heff consists of a superposition of the external magnetic field and of contributions from the anisotropy and the exchange fields. In particular, Heff in a system with magnetic anisotropy can be described by [20,15]

H eff = −∇Φ +

D ∇ 2 M − ∇ M FA + H , gμ B M

(2)

where the first term is the dipolar force (Φ being defined by the relation ∇ Φ = 4π∇ ⋅ M ), D is the exchange stiffness constant, FA is the magnetic anisotropy contribution to the free2

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energy density, and H is the external magnetic field. It can be shown from Eqs. (1) and (2) that the local magnetic anisotropy, short-range exchange interactions and long-range dipolar forces compete in determining the static and dynamic properties of magnetic materials. Two approaches have been considered to solve Eqs. (1) and (2) for the spin wave (magnon) modes, so as to enable comparison of the theoretical models with experimental data observed in the ferromagnet. The first approach was developed by Damon and Eschbach (DE), in which exchange processes are ignored.[21] This theory, purely magnetostatic in nature, was developed for flat, unbounded films or slabs with the magnetic field parallel to the film plane, and has yielded various propagating modes with wave vectors k|| in the film plane. The most striking result of this approach is the existence of two categories of spin wave mode: the volume wave, and the surface wave. These modes are identified by the nature of the wave vector component k⊥ perpendicular to the film surface. Specifically, the volume modes have a real k⊥, and therefore exhibit oscillatory behavior throughout the film thickness. The surface modes, on the other hand, have an imaginary k⊥, thus exhibiting exponential decay along the thickness of the film, and are therefore localized at either the top or at the bottom surface of the magnetic layer. On the other hand, when the dipolar force is ignored (or in the case of k|| = 0), the spin wave (magnon) modes can be attributed entirely to exchange interactions. Here exchange also brings into play the explicit effect of surface boundary conditions (e.g., the Rado-Weertman condition [22]) on the dynamic components of the magnetization vector, through pinning effects.[23,24,25] In this approach the actual pattern of the dynamic magnetization eigenmodes in a ferromagnetic film is selected by the conditions at the boundaries, which depend on the surface energy density FS. In this case the calculated results show a progression from almost-pinned to unpinned standing waves as one proceeds from large positive values of FS to zero as well as the surface character of magnon modes with an imaginary k⊥ for negative values of FS.[15] The outline of this article is as follows. In Sec. II we discuss the results for spin waves (magnons) obtained by both BLS and FMR measurements. The experimentally observed field, angular, and wave vector dependences of magnon modes will be presented, along with their analysis in terms of both the dipolar and the exchange-dominated surface mode models. Section III will deal with time-domain magnetic precession results measured by pump-probe techniques using subpicosecond laser pulses. Special attention will be given to the thermal origin of light-induced magnetic precession in (Ga,Mn)As, as well as to magnetic excitations that arise from a mixture of bulk and surface modes. In Sections IV and V we will discuss the relationships among these experiments, and the conclusions that follow from the results presented in this paper.

II. Spin Waves (Magnons) in (Ga,Mn)As A. Magnetic Brillouin Light Scattering (BLS) Magnetic BLS measurements of (Ga,Mn)As were carried out at Argonne National Laboratory using the experimental setup shown in Fig. 5.4 of Ref. [8]. As discussed in that Reference, a Fabry-Perot interferometer in 5-pass mode and a PMT detector were used to disperse and measure the light scattered from the samples. The samples were mounted in a liquid He flow-

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through cryostat. The cryostat was evacuated, and the sample cooled down to 20~30 K. For the BLS experiments an electromagnet mount was used, which was designed so that a magnetic field could be applied either parallel or perpendicular to the in-plane magnon wave vector k||. The optical excitation was carried out by a polarized Ar+ ion laser beam incident at an angle of 9° to the normal direction of the sample. This geometry resulted in a horizontal in-plane magnon wave vector in the laboratory frame. The scattered light was collected in the normal direction, thus keeping as much of the elastically scattered beam as possible out of the spectrometer. The scattered light was analyzed using the crossed polarization orientation in order to further cut down on elastically scattered light, while still letting the magnon signal through.

Figure 1. Dependence of magnon frequency on Mn concentration at a temperature of 30 K. The magnetic field Happ is applied along the

[ 1 10]

direction, and in-plane magnon wave vector k|| is along

[1 1 0] . (After Harley, Ref. [8]). Based on a single domain mode and using Eqs. (1) and (2), the BLS data can be used for determining the magnetic anisotropy constants of thin (Ga,Mn)As films. In particular, BLS measurements carried out for a wide range of Mn concentrations showed that the maximum magnon frequency v occurs at concentrations around 5%, which is coincident with Mn concentrations giving the maximum Curie temperature, as seen in Fig. 1. Figure 1 shows the magnon frequency v as function of Mn concentration x for a series of Ga1-xMnxAs samples with x ranging from 0.014 to 0.09 and with thicknesses of about 100 nm. Note that all BLS measurements were carried out with the external magnetic field Happ applied along the [ 1 10] crystallographic direction of the (Ga,Mn)As specimen, and with the in-plane magnon wavevector k|| along [1 1 0] . Clearly the relationship between v and x is similar to that of TC

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and x: both v and TC increase as Mn concentration increases up to around 5%, and drop as x continues to increase further. The relation between TC and x has been well explained by the change of the hole concentration p [26], which is strongly affected by Mn interstitials.[27] Furthermore, it should be noted that the magnon frequency v strongly depends on the magnitude of the effective magnetic anisotropy, thus suggesting that the effective anisotropy of (Ga,Mn)As films increases with their Curie temperature. Finally, based on the empirical relationship TC ~ p1/3 (as discussed, e.g., in Ref. [1]), we can immediately conclude that there exists a tight relationship between the magnetic anisotropy and the hole concentration p. In this regard the observed BLS data may provide useful insights into the origin of magnetic anisotropy in Ga1-xMnxAs films.

Figure 2. Angle and wavevector dependence for magnon modes in 100nm Ga0.964Mn0.036As film at T = 20 K. Curves through the data points are best fits based on the Damon-Eschbach model. The data points shown are an average of Stokes and anti-Stokes peak positions. (After Harley, Ref. [8]).

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Figure 2 shows the field, angle, and wave vector dependences of magnon modes for a Ga0.964Mn0.036As film. We note immediately that field dependences of the magnon frequencies are different for the easy axis [100] and for the hard axis [110], thus suggesting the existence of a strong cubic anisotropy in the sample plane. Specifically, when the field is applied along the easy axis (i.e., [100] or [010]), the magnetization always remains oriented along the same direction. In that case, the magnon frequency increases monotonically with increasing field, as is clearly seen in Fig. 2(a). On the other hand, if the field is applied along the hard axis [110], the magnetization will gradually rotate from the easy <100> axis to the field direction as the field is increased. Such magnetization rotation results in a small variation of magnon frequency between 10-60 kA/m, as is seen in Fig. 2(b). Additionally, the fitting of magnon frequencies in the figure also suggests the existence of an in-plane uniaxial anisotropy in the (Ga,Mn)As film. Secondly – and quite importantly – there is a strong dependence of the magnon frequency on k|| for both the easy axis and the hard axis. Such k|| dependence of magnon frequencies provides a strong indication that DE modes play a crucial role in BLS. Finally, a careful examination of asymmetric Stokes and anti-Stokes peaks in BLS spectra not only reveals that surface modes contribute to the observed BLS peaks, but also points out that the magnetostatic environment slightly differs between the top and bottom sample surfaces.[8] All these observations provide valuable input for forming a comprehensive picture of spin dynamics in (Ga,Mn)As.

B. Spin Wave Resonances (SWRs) in (Ga,Mn)As Spin wave resonances were investigated at 9.46 GHz using a Bruker electron paramagnetic resonance (EPR) spectrometer at the University of Notre Dame. In these measurements the applied dc magnetic field H was in the horizontal plane, and the microwave magnetic field was vertical. The sample was placed in a suprasil tube inserted in a liquid helium continuous flow cryostat, which could achieve temperatures down to 4.0 K. A detailed description of the apparatus can be found in Ref. [10]. Although the experiment included both a study of FMR and spin waves, here we will restrict our attention primarily to the multi-mode spin wave resonance (SWR) spectra and their dependence on the orientation of the dc magnetic field H with respect to the crystallographic axes of the sample and its macroscopic morphology. The SWR spectra observed on thin (Ga,Mn)As films with thicknesses in the range from 100 to 200 nm exhibited certain universal features.[28,29,30] These are illustrated by the data shown in Fig. 3, taken at T = 4 K on a 120 nm thick Ga0.92Mn0.08As film for several magnetic field orientations. One sees in the figure that the SWR spectrum evolves as H is rotated from the out-of-plane orientation (H||[001], θH = 0o) to the in-plane orientation (H|| [1 1 0] , θH = 90o). In particular, for H||[001] the spectrum consists of at least 4 well resolved Portis-type [31] SWR lines separated by equal magnetic field increments. One should add that the number of SWR modes observed in this condition increases with increasing thickness of the film. As one rotates H away from the perpendicular orientation, the SWR modes successively disappear, and eventually – at some critical angle θc (19º in Fig. 3) – the multi-SW spectrum vanishes except for a single narrow resonance line. This line corresponds to the so-called uniform FMR mode. For angles θH > θc the multi-mode nature of the SW spectrum reemerges, generally containing two or three broad resonances. Further analysis identifies one of these (on the high-field side) as an exchange-dominated non-propagating surface mode

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[23], also known as a surface spin excitation mode. It has been shown that this mode occurs in the case of unpinned surface spins.[32] The angular dependences of the SWR mode positions are shown in Fig. 4. Following the convention used by Kittel [33] and assuming symmetric boundary conditions for the (Ga,Mn)As film, we label the successive resonances by odd integers, n = 1, 3, 5…, starting from the high-field mode. Since the intensity of SWRs decreases rapidly with increasing wave vector k⊥, the main (i.e., the strongest) mode is believed to have a small k⊥, and can thus be assumed to lie very close to the theoretical field position of the uniform-mode (which by definition has k⊥ = 0). Its angular dependence has therefore been treated simply as a uniform mode in the process of determining bulk magnetic anisotropy parameters of thin (Ga,Mn)As films.[10] The SWR spectra and their angular dependence shown in Figs. 3 and 4 can be qualitatively explained using the Rado-Weertman boundary condition. In this case the spin pinning condition at each film surface can be described by the surface energy density FS. One can show that when FS > 0, the spins are pinned at the surface, and a series of bulk SWR modes with real wave vectors k⊥ is observed. For H||[001] the dispersion of these waves is modified to obey a linear law, corresponding to the so-called Portis model. As H tilts away from the [001] direction, FS decreases, causing the surface spin pinning to fade away. At the critical-angle orientation corresponding to FS = 0 (e.g., for θH = θc = 19° in Fig. 3) only one resonance peak remains, corresponding to the uniform mode with k⊥ = 0.

Figure 3. Spin wave resonance spectra observed for the 120 nm Ga0.92Mn0.08As specimen at T = 4 K, at various orientations θH for H between the

[ 1 10]

and [001] directions in the out-of-plane

configuration. The arrows indicate the surface spin wave mode. (After Liu et al., Ref. [11]).

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Figure 4. Resonance fields Hn of spin wave modes n = 1 to 9) as a function of the dc magnetic field orientation for the out-of-plane configuration. Inset: Dependence of Hn measured for H||[001] on the mode number, n - 1. The line in the insets is a linear fit. (After Liu et al., Ref. [11]).

Finally, when H continues to approach the [110] or [1 1 0] directions, a weak higher-field mode appears several hundred Oersteds above the strongest mode (see Figs. 3 and 4). This mode is identified as a non-propagating surface mode (i.e., k⊥ is imaginary for that mode), consistent with the case FS < 0, and representing the condition when the surface spins are unpinned. Note that the identification of a surface mode in the SWR spectrum is based on the following experimental facts: (a) the resonance field of the mode is above the theoretical uniform-mode position; (b) a critical-angle orientation is observed at a specific θc; and (c) the intensity ratio of the first two modes (n = 1; n = 3) I1/I3, the field separation between these two modes ΔH1,3, and the relation between I1/I3 and ΔH1,3 are consistent with the predictions of the model.[11] The angular dependence of pinning conditions can be qualitatively represented by the angular dependence of ∂H R ∂H 2 ⊥ , where HR is the field position of the uniform mode and H2⊥ is the perpendicular uniaxial anisotropy field. Note that the critical angle coincides with the angle where ∂H R ∂H 2 ⊥ equals zero. In other words, the critical angle occurs very near the orientation at which the resonance field becomes insensitive to small changes in magnetic anisotropy. As a result, the SWR data suggest that the magnetic anisotropy fields H2⊥ in the bulk must differ by some finite amount from those at the surface.[34] The fact that there exists a distinct surface anisotropy field different from the bulk is essential in determining surface spin pinning, which in turn determines the character of SWR spectra in (Ga,Mn)As film.

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III. Light-Induced Magnetic Precession in (Ga,Mn)As A. Time-Resolved Magnetooptic Kerr Rotation Recently measurements of light-induced magnetization precession in (Ga,Mn)As films – including its dependence on temperature and on photo-excitation intensity – were carried out by J. Qi et al. at Vanderbilt University in the absence of an external magnetic field.[16] In these experiments time-resolved magnetooptic Kerr rotation (TRKR) was measured on a 300 nm thick ferromagnetic Ga0.965Mn0.035As with p ≈ 1020 cm-3 and TC ≈ 70 K via the pump-probe technique, employing a Coherent MIRA 900 Ti:Sapphire laser to produce ~150-fs-wide pulses in the 720 nm (1.719 eV) to 890 nm (1.393 eV) wavelength range with a repetition rate of 76 MHz. The pump beam (with typical light pulse energy of 0.065 nJ and a fluence of 0.85 μJ/cm2) was incident normal to the film surface, while the probe was at an angle of about 30o from the surface normal. The pump beam could be adjusted to have either linear, rightcircular (σ+), or left-circular (σ-) polarization, while the probe beam was linearly polarized. This configuration produced a combination of polar and longitudinal MOKE responses, with the former dominating.[35] The time-resolved magnetooptic Kerr rotation signal was detected using a balanced photodiode bridge in combination with a lock-in amplifier.

0.2

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Figure 5. (a) Magnetooptic Kerr rotation measurements for Ga0.965Mn0.035As excited by linearly polarized and circularly polarized light at a temperature of 20 K. (b) Oscillation (open circles) arising from excitation by linearly-polarized light. The solid line is the fitted result. (c) Fourier transform of the oscillation in (b). (After Qi et al., Ref. [16])

Figure 5 shows typical light-induced TRKR results obtained on a (Ga,Mn)As film for two temperatures. The amplitude of the time-resolved magnetooptic Kerr rotation signal was

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found to be symmetric with respect to both right- or left-circularly polarized excitation. In particular, at temperatures T < 40 K (i.e., below ~TC/2), Qi et al. observed an oscillatory component superimposed on the probe signal, indicating a precessing magnetization (see Fig. 5(a)). It is important to note that the oscillations were observed not only with σ+ and σ- pump polarizations, but also with linearly polarized pump light. The phase difference among the oscillations for σ+ and σ- pump excitation is less than ~5o. This negligible phase difference implies that the oscillatory behavior is not due to non-thermal circular-polarization-dependent carrier spin dynamics.[36] After the initial few picoseconds the oscillations can be fitted well by the following equation (see Fig. 5(b)): θK(t) = A0exp(-t/τ)cos(ωt+ϕ), where A0, ω, τ, and ϕ are the amplitude, precession frequency, decay time, and initial phase of the oscillation, respectively. The fitted precession frequencies for these data are shown in Fig. 5(c); and the fitted parameters obtained for a series of pump intensities and temperatures are summarized in Fig. 6.

A0 (μdeg)

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200 I=I0

150

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100

20

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50 30

28

20

21

10

14 10 20 30 40 Temperature (K)

2 4 6 8 10 Pump Intensity(I0)

Figure 6. Amplitude A0 and angular frequency ω as a function of temperature T0 at constant pump intensity I = I0 (left-hand panels); and as a function of pump intensity in units of I0 at T0=10 K, where I0=0.065 nJ/pulse (right-hand panels). (After Qi et al., Ref. [16])

The observed optically-excited magnetization precession seen in Fig. 5 can be understood as follows. Excitation of the (Ga,Mn)As system by an optical pulse can induce a transient change Δp in the local hole concentration and transient change ΔT in the local temperature, the latter affecting both the carrier temperature ΔTe/h and the lattice temperature ΔTl, and thus also causing a transient change in magnetic anisotropy parameters. Below the Curie temperature, the direction of the in-plane magnetic easy axis depends on the interplay between the uniaxial anisotropy constant Ku and the cubic anisotropy constant Kc. After optical excitation the in-plane easy axis will thus acquire an instantaneous new orientation

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determined by the local transient temperature T+ΔT and the local transient hole concentration p+Δp, provided that ΔT or Δp is sufficiently strong. This transient change in the magnetic easy axis – due to the change in the minimum of the magnetic free energy of the Mn spin system caused by the photoexcitation – will trigger a precessional motion of the magnetization around the new effective magnetic field (i.e., transient magnetic anisotropy field in this case). When the fluctuations of the magnetization orientation around the easy axis are small, the magnetization dynamics is usually described by the LLG equation, which can thus be applied to the experimental data of Qi et al. at low-pump intensities (e.g., 0.065 nJ/pulse). The LLG equation predicts an oscillatory behavior of the magnetization due to the precession of local Mn moments around the transient magnetic anisotropy field, the precession frequency being proportional to this transient anisotropy field. The magnitude of the transient anisotropy field will decrease as the combined magnitude of the equilibrium transient temperature, T + ΔT, increases, primarily due to the decrease in Kc. This leads to a decrease of the precession frequency ω as the temperature (either T or ΔT) increases, or as the pump intensity increases, as shown by the results in Fig. 6. As can also be seen in Fig. 6, the amplitude of the oscillations A0 increases as the ambient temperature decreases or as the pump intensity increases. This result is in accord with the fact that the relative changes in the local temperature ΔT/T and in the local carrier density Δp/p, which determine the magnitude of the light-induced tilt in the easy axis [4,37] increase as T decreases or as the pump intensity increases. It is important to note that in the experiment of Qi et al. the amplitude of the oscillations saturates as the pump intensity exceeds about 4I0 (I0 = 0.065 nJ/pulse) at T = 10 K. Thus the observed saturation may indicate that the magnetic easy axis is stabilized at pump intensity larger than 4I0. Using the value of specific heat of 1 mJ/gK for GaAs [38] at pump intensity ~4I0, Qi et al. estimated that the increase of the local hole concentration Δp/p is about 0.4%, and the local temperature increase ΔT/T is about 160%. This leads to the transient local temperature T+ΔT close to TC/2. The data of Qi et al. thus agree with earlier results showing that the magnetic easy axis is already along the [ 1 10] direction when the temperature of (Ga,Mn)As is close to or higher than TC/2.[37] To sum up, the results of Qi et al. point to a relatively simple thermal origin of light-induced magnetic precession in (Ga,Mn)As arising directly from the temperature dependence of the magnetic anisotropy.

B. Excitation of Mixed Bulk and Surface Magnon Modes Prior to the investigation described in the preceding section Wang et al. at the University of Michigan carried out a systematic investigation of light-induced spin dynamics in (Ga,Mn)As in the presence of an in-plane magnetic field, also using laser pulses to generate and to probe coherent magnetization precession.[14,15] these authors have developed a comprehensive model of magnetic eigenmodes and their coupling to light, which allows one to gain accurate values of the exchange, bulk and surface anisotropy constants from time-domain data, establishing a hitherto unrecognized relationship between nonlinear optics and FMR. Using a Ti-sapphire laser that provided ~ 70 fs pulses at the central wavelength of 800 nm with the repetition rate of 82 MHz, pump-probe magnetooptic Kerr effect experiments were performed between 4.0 and 6.0 K in the Voigt geometry. Laser beams of energy density per pulse ~ 0.2

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μJ/cm2 (pump) and 0.03 μJ/cm2 (probe) were incident on the films along the [001] (i.e., the sample normal) direction. In this configuration the pump pulses induce a coherent magnetic precession, thus modifying the reflection of the probe pulses that follow at a chosen time delay. Such time-domain magnetooptic Kerr effect measurements then give the pumpinduced shift in the angle of the reflected probe field polarization, δθ, as a function of the time delay between the pump and the probe pulses.[14]

Figure 7. Results observed at T = 4 K for an as-grown 120 nm (Ga,Mn)As film. The top panel shows Voigt-geometry Kerr effect data (open circles) at H0 = 0.17 T. The applied field is parallel to the inplane [100] direction. The black curve is a linear prediction fit, which gives two modes with periods of 99.7 and 83.9 ps. Their contributions to the fitted signal are shown, respectively, by the blue and red curves. The bottom panel shows the Fourier transform of the fit and, in the inset, the calculated mz component for the three lowest eigenmodes. (After Wang et al., Ref. [15])

Selected results of Wang et al. are summarized in Figs. 7 and 8. The top graph in Fig. 7 shows Kerr effect data at T = 4K for an as-grown 120-nm-thick (Ga,Mn)As film after subtraction of an exponentially decaying background, indicating the spin relaxation of the photoexcited electrons. The observed oscillations are ascribed to the precession of the magnetization around its equilibrium direction. Wang et al. used linear prediction methods to fit their time-domain data, whose Fourier transform (see bottom panel of Fig. 7) reveals two

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modes, labeled S0 and S2. With the exception of the thinnest film (with thickness L = 25 nm), these two modes were observed in the Kerr effect spectra on all the samples. The dependence of their frequencies on the applied magnetic field is shown in Fig. 8, together with FMR results (see Section II.B of this paper) for the L = 71 nm slab.

Figure 8. Measured magnetic-field dependence of precession mode frequencies (top panel) and of the ratio A2/A0 (bottom panel). Solid curves are fits using the theoretical expressions discussed in the text. The FMR spectrum for a 71-nm-thick (Ga,Mn)As sample observed at 9.46 GHz with H||[100] is shown in the inset. (After Wang et al., Ref. [15])

The bottom graph gives the dependence of the amplitude ratio A2/A0 of the two oscillations. Further analysis shows that the value of this ratio is critical for determining the magnetic parameters, and especially the surface energy FS. Here it is important to note that the observation of S2 is in itself a strong indication that FS < 0, because the amplitude ratio depends strongly on FS, but is much less sensitive to other magnetic constants, as indicated by Fig. 4 in Ref. [15], which shows that the amplitude ratio A2/A0 is very small for FS > 0 but dramatically increases for FS < 0. One can show that this negative contribution to the energy due to surface anisotropy results in magnetic excitations that are a mixture of bulk waves and surface modes.

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IV. Discussion We have shown that both FMR and time-resolved MOKE data can be successfully interpreted in terms of magnetic exchange and surface anisotropy. In terms of this approach the angular and field dependences of multi-mode spin wave (magnon) frequencies and their intensity ratios can then be used to obtain accurate values of fundamental magnetic parameters, such as the bulk magnetic anisotropy, the g-factor, the exchange stiffness constant, and the surfacespin pinning conditions. The success of this approach appears, however, to contradict the features observed in BLS measurements. This apparent contradiction can be understood as follows. FMR reported so far were carried out with the applied magnetic field in the out-ofplane geometry, and time-resolved MOKE measurements were done in the Voigt geometry with the in-plane wave vector k||=0. Under these conditions the dipole-dipole interaction in Eq. (2) can be neglected in the analysis, the properties of spin wave (SW) modes observed in (Ga,Mn)As films can then be formulated entirely in terms of exchange interactions and the surface anisotropy. Additionally, it should be emphasized that the magneto-exchange waves observed in these experiments are confined to the interior of the films, and are nonpropagating. In BLS measurements, on the other hand, the in-plane wave vector k|| excited by the incident light must be taken into account. The observed wave vector dependence of magnon frequencies then suggests that dipole-dipole interactions in Eq. (2) cannot be neglected even if the magnetization of the ferromagnetic (Ga,Mn)As material is very small. For those reasons, in order to fully understand the magnetic excitations in (Ga,Mn)As, it appears that both dipolar and exchange interactions should be considered as one attempts to develop a comprehensive theory of spin dynamics in (Ga,Mn)As and related materials. It is therefore important that further systematic experiments be carried out in this area, with special attention given to identifying the actual boundary conditions. In this connection we should note that in the present review we have only considered symmetric boundary conditions when dealing with exchange-dominated spin wave modes. This is justified, since only symmetric spin wave modes are observed in both FMR and in time-resolved MOKE experiments. The symmetry of the boundary conditions suggests that the major source of surface anisotropy comes from the abrupt step in the Mn ion or in carrier (hole) concentration at the two boundaries of a (Ga,Mn)As film, rather than other forms of discontinuities, such as the oxidation of the top surface of the sample. On the other hand, when considering the BLS results (asymmetric Stokes and anti-Stokes peaks) we should mention that – although the surface anisotropy might also be symmetric – in the geometry of the BLS measurement the magnetostatic environment at the two surfaces is itself asymmetric. Returning to the SWR problem in (GaMn)As, it is interesting to note that although the dynamic surface anisotropy provides a qualitative explanation for the angular dependence of the SWR spectrum, an alternate model – the Portis volume inhomogeneity model [31] – is required to obtain a realistic description of the SWR spectrum observed for H||[001], as seen in Figs. 3 and 4. One should note that the general exchange model (i.e., the term

D ∇ 2 M in Eq. (2)) is formulated in terms of the Heisenberg localized-spin model that gμ B M assumes nearest-neighbor exchange interactions and a Zeeman Hamiltonian in its standard form. This theory thus only includes exchange interactions between nearest neighbors,

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assuming these interactions to be short-range. In the case of (Ga,Mn)As, the assumption of a “short range” interaction is not fully justified, since neither the pd exchange between the holes and localized Mn moments nor the super-exchange interaction between the Mn spins can be treated as a short-range process. Thus the effects of surface anisotropy are not only felt (directly) by the spins on the surface, but the bulk spins can also be affected by the surface anisotropy through nonlocal exchange interactions.

Conclusion In this chapter we have reviewed recent studies on spin (or magnetization) dynamics in thin layers of the ferromagnetic semiconductor (Ga,Mn)As. Recently, frequency- and time-domain techniques, including BLS, FMR, and time-resolved MOKE measurements had been carried out to investigate the various spin dynamic phenomena in this material, including magnetic field, angular, and wave vector dependences of spin wave (magnon) modes, as well as the behavior of magnetization precession in real time. In most experiments the analysis of spin wave (magnon) modes in terms of the LLG equation and specific boundary conditions allows one to determine the value of the exchange stiffness constant, along with the bulk and surface anisotropy in the magnetic material of interest. In particular, surface magnon modes, which are characterized by an imaginary k⊥, are observed in both frequency- and time-domain experiments. However, in terms of magnetostatic and magneto-exchange interactions, understanding of the origin of these surface modes is still far from complete. In this regard, there is a pressing need for further systematic investigation of magnetic excitations in (Ga,Mn)As, with special attention to both dipolar and exchange processes. As a different approach to the problem of spin dynamics in (Ga,Mn)As, ultrafast lightinduced excitation of collective magnetization precession has also been studied in this material by time-resolved magnetooptic Kerr rotation measurements. This approach is important because – in addition to its fundamental interest – it offers extremely important prospects for ultra-fast manipulation of magnetic properties of (Ga,Mn)As – a feature that is of great interest to the development of spintronic applications based on this material. Here the observed temperature-dependent coherent oscillations provide a measure of ultrafast changes in the in-plane orientation of magnetization in a given ferromagnetic domain due to laserinduced transient changes of magnetic anisotropy. Moreover, the occurrence of magnetization precession in the absence of an applied magnetic field up to some distinct critical temperature that is significantly lower than TC is attributed to the existence of a second-order magnetic phase transition. Finally, results obtained from the field-dependence of magnon frequencies and of oscillation-amplitude ratios reveal the presence of spin-wave excitations that are a mixture of bulk and surface modes, which form when the magnetic surface anisotropy is negative. The analysis of this phenomenon therefore provides real-time confirmation of the properties of spin excitations that had been observed by frequency-domain measurements. Since the ultrafast control and manipulation of magnetization in (Ga,Mn)As-based ferromagnetic semiconductor structures is critical for fabrication of ultrahigh-speed spintronic devices – particularly those of interest in quantum information technology – the results reported in this review should be of considerable importance for designing such devices and for assessing their viability.

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Acknowledgment We would like to thank D. M. Wang, Y. H. Ren, R. Merlin, M. Grimsditch, J. Qi, Y. Xu, I. E. Perakis, N. Tolk, J. Wang and D. S. Chemla for their stimulating and fruitful discussions, and their invaluable contributions to the understanding of various aspects of magnetic excitations in (Ga,Mn)As presented in this paper. We would also like to thank Dr. M. Grimsditch for the use of his BLS apparatus. This work was partly supported by NSF Grant DMR06-03752 and by ARO Grant DAAG55-98-D-0001.

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In: Spintronics: Materials, Applications and Devices Editors: G. C. Lombardi and G. E. Bianchi

ISBN: 978-1-60456-734-2 © 2009 Nova Science Publishers, Inc.

Chapter 5

SPIN TRANSFER TORQUE EFFECT AND ITS APPLICATIONS Haiwen Xi, Xiaobin Wang, Yuankai Zheng and Xiaohua Lou Seagate Technology, 7801 Computer Avenue South, Bloomington, Minnesota 55435, USA

Abstract In this chapter, we give a review of spin transfer torque effect in magnetic devices. Spin transfer torques arise from the interaction between the spins of conduction electrons and the local magnetic moments. Phenomena of spin transfer torques include current induced magnetization reversal and microwave excitation in magnetic nanostructures and current driven domain wall motion in magnetic wires, which will be discussed in detail. We will also discuss the implications of the effect in microelectronic applications and associated problems.

Keywords: Spin transfer torques; spin momentum transfer; magnetization dynamics; magnetic domain wall motion; magnetic multilayers; magnetic tunnel junctions

I. Introduction The research of understanding and manipulating the spin degrees of freedom of electrons in solid-state systems, including homogenous materials and hybrid structures, over the past decades has given birth to a field called “spintronics”. The tremendous interest of the field is based on the successful application of the giant magnetoresistive (GMR) effect, which was discovered in 1988 (Baibich et al., 1988 and Binasch et al., 1989) and has fostered a revolution in data storage and magnetic field sensing technology in the past decade (Prinz, 1998 and Chappert, Fert, and van Dau, 2007). In the magnetic thin films with ferromagnetic (FM) layers separated by nonmagnetic (NM) layers, the electronic current is unbalanced in terms of spin polarity and the resistance varies with the magnetization configuration of the structures.

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One may think that, as a reverse process of the GMR effect, an unbalanced current with spin-up electrons outnumbering spin-down electrons, or vice versa, can change the magnetization configuration. In late 1990s, Berger (1996) and Slonczewski (1996) proposed, independently, that when a spin-polarized current flows through an FM conductor, the magnetic moments of the FM layer experience a torque exerted by the spin current. The consequence of the spin current induced torque was later termed “spin transfer torque” (STT) effect. It has also been called “spin momentum transfer” (SMT) effect in many occasions. The basic principle of spin transfer torques is illustrated in Fig. 1. Let us first consider a ferromagnetic trilayer structure with a nonmagnetic layer separating a thin FM layer with the magnetization free to rotate from a thick FM layer with the magnetization fixed in space. The trilayer is assumed to be in the non-collinear configuration, meaning that the magnetization of the free layer make a non-zero angle with respect to the magnetization of the fixed layer. When a non-spin-polarized electron current is injected into the trilayer from the fixed FM layer, the electrons are spin polarized by the fixed FM layer because the transmissions of spin-up electrons and spin-down electrons are different. The spin-up electrons, which have the spin orientation in the direction of the fixed layer magnetization, are transmitted through the layer while many of the spin-down electrons are reflected back, i.e., the spin specular effect. After the spin filtering, the spin-polarized current travels through the NM layer and into the free FM layer, where the spin current interacts with the magnetic moments of the free layer due to the s-d exchange coupling. Since the spin of the incident electrons are not aligned with the free layer magnetization, the electron spin will precess about the direction of the magnetization. Likewise, the free layer magnetization precesses about the electron spin. Eventually when the electrons exit the free layer, the electron spin is aligned by the free layer magnetization. Meanwhile, in order to conserve the angular momentum, the free layer magnetization is pulled towards the direction of the incident electron spin that acts as applying a torque on the free layer magnetization (Slonczewski, 1996, Bazaliy, Jones, and Zhang, 1998, and Stiles and Zangwill, 2002). The more rigorous theoretical consideration of the spin transfer torques in the trilayers shall take into account reflection of the electrons at the FM/NM interfaces and the angular momentum carried away in both the reflection and transmission processes (Waintal et al., 2000 and Xia et al., 2002). The technique to calculate the spin transfer torque is first compute the transmission and reflection amplitudes of the current density for the spin-up and spindown electrons (Waintal et al., 2000). The spin transfer torque on the FM layer is equal to the net spin current transferred from the electrons to the FM, given as the difference between the transmitted spin current and the sum of the incident and reflected spin currents. In fact, both the fixed FM layer and the free FM layer will experience spin transfer torques. Usually in the study of the spin transfer torque effect, the fixed FM layer is assumed to be so rigid that the spin torque will not affect its magnetization. Figure 1(a) shows the situation of electron current flowing from the fixed layer to the free layer. The spin torque on the free layer can be understood by the above discussion. In the case of electron current flowing from the fixed layer to the free layer as shown in Fig. 1(b), the current is first spin-polarized by the free layer and then flows into the fixed layer. A fraction of the electrons are reflected back to the free layer. Since the spin-up electrons are more likely to be transmitted, the reflected electrons will have an average spin polarization opposite to the direction of the fixed layer magnetization. As the reflected electrons interact with the free layer, we will see that the spin torque in this case is opposite to that in the case of electron

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current flowing from the fixed layer to the free layer. As will be shown, it makes it possible to utilize the spin transfer torque in magnetic memory application. (a)

θ

Electron injection Free FM Fixed FM

(b)

θ Electron injection Free FM Fixed FM

Figure 1. Schematic of an FM/NM/FM trilayer in the non-collinear magnetization configuration with an electron current flowing (a) from the fixed FM layer to the free FM layer and (b) from the free FM layer to the fixed FM layer.

Spin transfer torque can be described by a Stoner-model approach as well. Spin-up electrons and spin-down electrons incident at the FM/NM interface see different potential heights, which is determined by the exchange splitting in the FM layer. Therefore, the scattering at the interface and, consequently, the reflection and transmission for the spin-up and spin-down electrons are different. Detailed discussion of spin transfer torques in this model can be found in the paper of Stiles and Zangwill (2002). The essence of spin transfer torques is simply the interaction between electron spin and local magnetic moment. However, observations of the spin transfer torque effect were made less than ten years ago, thanks to the advance of the nano-fabrication technology. Sun’s (2006) simple calculation indicates that spin torque strength becomes greater than the oersted field strength of the current only when the size of the magnetic structure is 100 nm or less. Also note that the oersted field generated by the current is circular with highest field strength on the edge of the magnetic device. Spin torque is fairly uniform across the device. Now we know that the magnetization dynamics induced by spin-polarized currents can be either perturbative behavior in the forms of spin wave excitations or global coherent behavior such as irreversible magnetization reversals and rotations. Both kinds of the magnetization behavior have been experimentally demonstrated in patterned magnetic nanostructures (Tsoi et al., 1998, Tsoi et al., 2000, and Myers et al., 1999). Since the spin transfer torque is still under intensive study and new results keep coming out, a comprehensive review in the subject dose not seem possible at the present time.

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Nevertheless, there have been several publications by Bass et al. (2004), Sun (2006), and Ralph and Buhrman (2006) that interested readers will find helpful for understanding the spin transfer torque and related phenomena. In this chapter, we will try our best to give a broad review in the subject. The two unique phenomena arising from the spin transfer torque, irreversible magnetization reversal/switching and sustained magnetization oscillation, will be discussed in detail. Following these is the review and discussion of the current-induced magnetic domain wall motion and of the magnetization instability, which are an effect of the interaction between electric currents and non-uniform magnetization structures. In the final section of the chapter, we will explore the possibility to utilize the spin transfer torque effect in magnetic random access memory (MRAM) and domain-wall memory, radio-frequency oscillation generation, and dc-ac conversion in the near future and spinwave generation and amplification further down the road.

II. Current-Induced Magnertization Reversal A. Expression for Spin-Torque Induced Magnetic Dynamics In the ferromagnetic metallic systems, the response times of conduction electrons and magnetic moments are different (Sun, 2006). Response time of the spin electrons determined by the spin relaxation is usually in the order of picoseconds. Response time for local magnetic moments described by the moment precession is a little less than a nanosecond. Therefore, in the magnetic systems with the interaction between spin electrons and local moment, the dynamics of the magnetic moments can be decoupled from that of spin electrons, which are considered to be always in quasi-equilibrium during the transport process with instantaneous response to the local magnetic moments. To describe the magnetization dynamics of the free layer, the Landau-Lifshitz-Gilbert (LLG) equation can be used with modified terms to reflect the spin-moment interaction, dM M dM . = −γM × (H eff + bn s ) − γg (n m , n s ) BM × (M × n s ) + α × dt M s dt

(1)

M is the magnetization vector of the free layer with a saturation magnetization Ms. nm is its unit vector. ns is the spin polarization direction of the conduction electrons that is determined by the magnetization of the fixed layer. γ is the gyromagnetic ratio and α is the Gilbert damping constant. Heff is the effective field, which includes external magnetic field, anisotropy field, and demagnetizing field, etc. Heide (2001) proposed a non-equilibrium exchange interaction (NEXI) mode to study the angular momentum exchange between the spin electrons and local moments. The interaction is a volume effect and is reflected by a force exerted throughout the magnetic layer. The force range is controlled by the spin diffusion length of the conduction electrons. When the longitudinal spin diffusion length is very long, e.g., about 60 nm for Co, the NEXI effect is small in thin film devices. Zhang, Levy, and Fert (2002) proposed another mechanism based on the exchange interaction between the two ferromagnetic layers. The

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main interaction is from the transverse spin accumulation. The transverse spin accumulation has a much shorter length of about 2 nm. For both mechanisms, the interactions on the free layer magnetization can be expressed by the field term, −γM × bn s , in Eq. (1). A third origin for this term is that the transverse spin angular momentum incident onto the interface of the free layer is not completely absorbed by the free layer. As a consequence, the in-plane component of the spin transfer torque, the second term in Eq. (2), is reduced from the ideal case and a field term, also termed perpendicular component of the spin torque, appears. The effects have been experimentally observed in both metallic spin-valves (Zimmler et al., 2004) and Al2O3-barrier magnetic tunnel junctions (MTJ) (Petit et al., 2007). Both results show that the effective field term is much smaller than the torque term. Calculations also predict the perpendicular component is much smaller than the inplane component (Stiles and Zangwill, 2002, Xia et al., 2002, and Zwierzycki et al., 2005). Therefore, we will ignore the field term (b = 0) and focus on the in-plane spin transfer torque term. In the second term in Eq. (1), B is the in-plane spin transfer torque strength expressed by B=

h ηI , 2e M sV

(2)

where η is the spin-polarization ratio and I is the current. Therefore, ηI is the spin current amplitude. V is the volume of the free layer and then MsV is the total magnetic moment of the free layer. Eq. (2) implies that the angular moment of the spin current is transferred to the local magnetic moments, which might not be the case, according to the discussion in the previous paragraph. g (n m , n s ) is an angular factor for spin transfer torque. There are a number of angular dependencies derived from different theoretical approaches (Slonczewski, 1999 and 2002, Waintal et al., 2000, Stiles and Zangwill, 2002, Bauer et al., 2003, and Shpiro, Levy, and Zhang, 2003). They all look similar but have some minor difference. Several experimental observations also reveal that spin torque does change with angle made by the free layer magnetization and the pin polarization direction (Albert et al., 2001 and Mancoff et al., 2003). As will be shown later, the angular dependence may be reflected in magnetization reversal induced by spin torque.

B. Comparison between Field Reversal and Spin Torque Reversal From the modified LLG equation above, one can see that the torque of the effective field causes the free layer magnetization to precess along the equilibrium direction defined by the effective field. The damping term pull the magnetization towards the equilibrium. In the simple picture described by Fig. 2, the current-induced spin torque acts either with or against the damping, depending on the spin current direction. When the spin torque is in the same direction as the damping term, it will make the magnetization to converge to the equilibrium faster. Otherwise, the magnetization convergence is slower. When the spin torque is greater than the damping term but with opposite direction, magnetization reversal can occur.

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Heff Damping

Field torque

Spin torque

M

Figure 2. Schematic of the field torque, damping, and spin transfer torque on the magnetization in precession along the effect field direction. Current is applied in the direction so that the spin torque works against the damping.

It is known that uniaxial anisotropy energy Ku for the free layer determines its stability. When a magnetic field H is applied to the free layer, the magnetic energy of the free layer can either increase or decrease, depending on the field direction. If the field is applied along the easy axis, the energy barrier for the two stable states of a single-domain free layer can be expressed to be ΔE = K uV (1 ± h) 2 ,

(3)

where h = H / H K and H K = 2 K u / M s . When the field is greater than the uniaxial anisotropy field H K , the energy barrier disappears and magnetization reversal can happen. However, when a spin current is applied to the free layer, the spin torque only forces the free layer to oscillate in the energy valley corresponding to one of the stable states. There is no energy change. However, when the oscillation is large enough, the energy barrier is overcome and, as a result, it can cause the magnetization to reverse. The oscillation is similar to the activation caused by thermal effect. An effective temperature for the activation induced by spin transfer torque can be written as (Li and Zhang, 2004) T ' = T /(1 − I / I c 0 ) .

(4)

The intrinsic chital current I c 0 will be explained soon in this section. Fig. 3 shows the calculated magnetization dynamics of the free layer induced by field torque and spin torque for comparison. In the calculation, the free layer magnetization is initially slightly disturbed from the equilibrium by an angle of 0.1°.

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(a)

1.5

mx

my

mz

1.0

M

0.5

t (nsec)

0.0 0.0

0.5

1.0

1.5

2.0

-0.5 -1.0 -1.5 1.5

(b)

1.0

M

0.5

t (nsec)

0.0 0.0

0.5

1.0

1.5

2.0

-0.5 -1.0 mx

my

mz

-1.5

Figure 3. Calculated magnetization reversal dynamics (a) driven by external magnetic field greater than HK and (b) driven by electric current greater than a critical value that will be discussed in the next subsection. Magnetization is described in the Cartesian coordinates and easy axis is along the z-axis. Note that the magnetization precession and reversal induced by spin current take a longer time to develop.

C. Characteristics of Current-Induced Magnetization Reversal Figure 4 shows a typical R−I curve measured in a spin-valve nanodevices with the sharp transitions in resistance marked as the spin-current induced magnetization reversal. Sun (2000) examined the stability condition of the magnetization dynamics governed by Eq. (1) and found that magnetization reversal occurs when the current is greater than the critical value, I c±0 = ±

1 ⎛ 2e ⎞⎛ α ⎞ ⎜ ⎟⎜ ⎟ M sV (2πM s + H K ± H ) . g (±) ⎝ h ⎠⎜⎝ η ⎟⎠

(5)

I c±0 is the intrinsic critical currents for the free layer magnetization reversing from the

parallel (anti-parallel) alignment to the anti-parallel (parallel) alignment with respect to the fixed layer. g (±) is the angular factor of spin torque for the parallel (anti-parallel) alignment. Condition (5) for magnetization reversal is verified by numerical calculations (Li and Zhang, 2003 and Xi, Lin, and Wilomowski, 2005).

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Figure 4. Reprehensive hysteresis loop of differential resistance vs. applied current in a ferro-magnetic trilayer with a pillar structure. Copy from Albert et al. (2000).

Critical current density J c±0 , i.e., the critical current per unit area through the trilayers, is commonly used to characterize the spin-torque effect. From Eq. (5), we can estimate the critical current density is around 2 ×107 A/cm2, choosing a typical Co free layer with a thickness of 3 nm, a saturation magnetization of 1400 emu/cm2, a damping constant of 0.02, and a spin polarization efficiency of 0.2. The unambiguous experimental observation of spin torque induced magnetization reversal (switching) in MTJs is believed to be reported in the study of Myers et al. (1999) in magnetic metallic layers with point contact. Prior to this, current induced magnetization reversal was claimed to be found in manganite trilayer junctions at low temperatures (Sun, 1999). However, the critical current densities for magnetization reversal appeared to be too low, implying that other magnetization reversal mechanisms may have involved. Since Myers et al., spin-torque induced magnetization reversal has been broadly observed in many lithographically patterned pillar structures at various temperatures including room temperature (e.g., Katine et al., 2000, Grollier et al., 2001 and 2003, Fabian et al., 2003, Mancoff et al., 2003, Ozyilmaz et al., 2003, Sun et al., 2003, and Urazhdin et al., 2003 and 2004). In the next, we will review and analyze the published results with regard to the relationship, Eq.(5). (a) Asymmetry in critical currents According to Eq. (5), the critical current depends on the angular prefactor g (±) of the spin torque term. It is understood by the fact that the spin accumulation for the parallel and anti-parallel configuration of the magnetizations in trilayers is different. Therefore, the spin transfer torque is different as well. The asymmetry in critical current with respective to the current polarity (or magnetization alignment) has been investigated (Deac et al., 2005) in ferromagnetic multilayers with Cu and Ag as nonmagnetic spacers. It was found that g (−) is about twice as g (+) , in agreement with theory. It suggests that the critical current for the free layer magnetization switching from the parallel alignment to the anti-parallel alignment is greater than the one otherwise. However, real experiments are confounded by many factors and, in fact, the trend does not hold for the systems in many other reports. (b) Role of spin current efficiency Spin transfer torque effect may sometimes be considered as the reverse effect of the GMR or TMR effect, implying a strong correlation between the two phenomena. Fig. 5 shows

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the dependence of the critical current density on the resistance change of spin-valves (Jiang et al., 2004). The critical current density decreases with increasing resistance change, indicating the consistent role of the spin-polarization ratio in both GMR effect and spin torque effect. Similar relationship between the critical current density and the resistance change or the MR ratio was found in other spin-valves (Urazhdin et al., 2004 and 2005) and MTJs (Diao et al., 2005). However, there is discrepancy in the spin polarization ratio between the GMR/TMR and spin-induced magnetization reversal measurements. η in Eq. (5) is better termed spincurrent efficiency. It was also found that a Ru layer (Jiang et al., 2004) or FeMn layer (Urazhdin et al., 2004) in contact with the free layer are a strong majority spin scatterer and then enhances spin transfer effect. This situation has been simulated by a circuit model in attempt to reduce switching current (Manschot, Brataas, and Bauer, 2004). However, a recent experimental study (Kurt et al., 2006) did not confirm the theory and possible reasons for the discrepancy could be finite temperature effect and spread-out current. (c) Problems about damping constant The damping constant is both material-related and structure-related (Heinrich, 2005). The damping constant can be caused by magneto-elastic scattering (Rossi, Heinonen, and McDonald, 2005), two-magnon scattering from inhomogeneity (McMichael, Twisselmann, and Kunz, 2003), and electron-hole pair (Kambersk, 1976, Sinova et al., 2004, and Gilmore, Izerda, and Stiles, 2007). It has been known that different measurements give different damping constant (e.g., Fuchs et al., 2007). For multilayer structure, the damping constant is also related to the adjacent conduct layer through the spin pumping or spin accumulation effect (Tserkovnyak, Brataas, and Bauer, 2002, Mizukami, Ando, and Miyazaki, 2002, and Heinrich et al., 2003). When the longitudinal spin diffusion length is much longer than the

Figure 5. Dependence of critical current density on the resistance change of spin-valves. Data is taken from Jiang et al. (2004) for solid triangles and Urazhdin et al. (2204) for open triangles. Copy from Jiang et al. (2004).

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thin film thickness, the precession of the magnetization of the ferromagnetic layer will transfer the spin into the normal metal adjacent to the ferromagnetic layer. This is the pumping effect. The spin electron can flow in both directions due to the spin diffusion effect after spin accumulation in normal metal layer. The spin electrons flowing back can enhance the damping. For metals with large spin diffusion length such as Cu, the damping constant enhancement is low. For metal with small diffusion length such as Pt and Pd, the damping constant increases. Since the bulk material has much lower damping constant than the thin films, we can engineer the thin film to reduce the damping constant to reduce the critical current. (d) Dependence on saturation magnetization From Eq. (7), one can see that the intrinsic critical current is proportional to M s2 when the uniaxial anisotropy field and applied field are much less. Experimental results show the critical current can be significantly reduced by lowering the saturation magnetization of the free layer in spin-valves (Yagami et al., 2004). In the spin-valves, the other parameters can be easily controlled. However, for the magnetic tunneling junctions (MTJ) that will be discussed later in this section, experiments are more complicated, especially for textured MgO-barrier junctions. In most cases, the spin current efficiency and tunnel magnetoresistance (TMR) change significantly with the magnetizations of the ferromagnetic electrodes. MgO texture is strongly dependent on adjacent ferromagnetic structures (Yuasa and Djayaprawira, 2007). In addition, TMR and spin current efficiency are dependent on the MgO structure. In order to fully use both the low saturation magnetization and high spin current efficiency to reduce critical current, engineering on materials and structures is still a challenge. (e) Dependence on free layer thickness As mentioned above, the FM layer thickness will influence damping constant and spin current efficiency. For instance, the damping constant decreases as the FM thickness

ΔJC (107 A/cm2)

40

30

20

10

0 0

1

2

3

4

5

6

tFM (nm)

Figure 6. Critical current density, ΔJC = JC+ − JC−, as a function of the free layer thickness for current sweep rate of 300 mA/sec (open circles) and 0.1 mA/sec (solid circles). The difference in critical current density between the current sweep rates indicates the thermal effect that will be discussed in the next subsection. Figure is redrawn from Albert et al. (2002).

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increases if the adjacent normal metal has short spin diffusion length. Meanwhile, MR ratio increases with increasing FM thickness due to the bulk spin-dependent scattering in spinvalves. All these effects are in fact not so significant. The relationship between the critical current and free layer thickness is simpler compared to the relationship to other parameters. Fig. 6 shows the critical current density dependence on the free layer thickness (Albert et al., 2002). The linear relationship between the FM thickness and the critical current density agrees well with Eq. (5). (f) Dependence on applied magnetic field Dependence of the critical switching current on the applied magnetic field has been investigated in many magnetic nanostructures (Albert et al., 2000, Lacour et al., 2004, Urazhdin, Pratt Jr., and Bass, 2004, Huai et al., 2004, Urazhdin et al., 2005, and Ochiai et al., 2005). Fig. 7 shows an example of the field dependence, which is fairly a linear function in a certain field range and seems to agree with Eq. (5). However, the sensitivity of the critical current to the applied magnetic field is too large. The reason for this is the thermal effect that will be shown in the next subsection. The applied field can change the energy barrier significantly for the free layer and make it easier or harder to switch. One can also note that beyond a certain field, one of the critical currents approaches to the other rapidly with increasing magnetic field. After the critical currents meet, there is no longer magnetization reversal observed. In this situation, the area closed by the lines of the critical currents can be considered as the region for bistable states of the free layer magnetization. Outside the area, the magnetization is in either the parallel or anti-parallel alignment with the fixed layer, or a steady precession state. A complete phase diagram for the free layer magnetization will be illustrated in the next section. 8 6

IC (mA)

4 2 0 -2 -4 -6 0

200

400

600

800

1000

H (Oe)

Figure 7. Critical currents (open circles for IC+ and solid circles for IC−) as functions of applied magnetic field. The lines are guides to eyes. Figure is redrawn from Albert et al. (2000).

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D. Thermal Effect on Current-Induced Magnetization Reversal For nano-magnetic device at a finite temperature T, the thermal activation can be described by an effective field Hth added to the Landau-Lifshitz-Gilbert equation (Heinonen and Cho, 2004) 1 dM = −M × (H eff + H th ) − αM × M × H eff . γ dt

(6)

The effective thermal field Hth can be expressed as H th (t ) =

2αk B T I ran (t ) , γM s dVΔt

(7)

where I ran (t ) is a Gaussian random function with < I ran,i (t ) >= 0 and < I ran,i (t ) I ran, j (t ) >= δ ij , where i, j = x, y, z . The modified LLG equation describes the dynamics of the macro-spin M in a potential well ΔE by a thermally activated motion and a finite life time. The lifetime of the macro-spin under thermal activation τ approximately follows the Boltzmann statistics, ⎛ ΔE ⎝ k BT

τ = τ 0 exp⎜⎜

⎞ ⎟⎟ . ⎠

(8)

τ0 is the inverse of the reciprocal attempt frequency, τ 0 ≈ 1 /(γH K ) . When an external field is applied on the free layer, the external field will change the thermal barrier height as mentioned above. The thermally activated lifetime τ is now written as ⎞ ⎛K V τ = τ 0 exp⎜⎜ u (1 ± h) 2 ⎟⎟ , (9) k T ⎠ ⎝ B following Eq. (3). After introducing a spin current in the magnetic system, the thermal activation lifetime of the micro-spin system can be written to be ⎛ ΔE ⎞ ⎟⎟ , ⎝ k BT ' ⎠

τ = τ 0 exp⎜⎜

(10)

with T ' being the effective temperature of the macro-spin. Recalling Eq. (4), the averaged critical switching current with pulse width of τp is derived to be

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⎡ k T ⎛ τ p ⎞⎤ I c± = I c±0 ⎢1 − B ln⎜⎜ ⎟⎟⎥ . ⎢⎣ ΔE ± ⎝ τ 0 ⎠⎥⎦

(11)

The thermal effect was first studied by Koch, Katine, and Sun (2004) and later derived and formalized by Li and Zhang (2004) and Apalkov and Visscher (2005) from the FokkerPlank equation. When both magnetic field and spin current are applied on the micro-spin system, the thermally activated lifetime can be expressed as ⎛ I ⎞⎤ (1 ± h) 2 ⎜⎜1 − ± ⎟⎟⎥ . I c 0 ⎠⎥⎦ ⎢⎣ k B T ⎝ ⎡ K uV

τ ± = τ 0 exp ⎢

(12)

When the electric current approaches or exceeds the intrinsic (zero-temperature) critical current I c±0 , the spin torque can result in a fairly fast magnetization switching even without the help of thermal excitation. The switching process is a precession mode, which is strongly dependent on the initial condition of the macro-spin. The switching time can be derived to be

τ ±−1 =

αγ ( I − I c±0 ) , M sV ln(π / 2θ 0 )

(13)

where θ0 is the initial deviation of the macro-spin from its equilibrium.

Figure 8. Critical switching current density as a function of the current pulse width. The current driven magnetization dynamics is classified as precession switching, thermally-activated reversal, and mixed dynamic reversal by current pulse width. Copy from Diao et al. (2007).

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The time-dependent critical switching current is shown in Fig. 8. For long current pulses, magnetization reversal is dominated by the thermal activation and the switching current relates to the current pulse width by Eq. (11). It has been proved by numerous studies (e.g., Yagami et al., 2004 and 2005, Higo et al., 2005, and Pakala et al., 2005). When the current pulse width is close to or shorter than τ 0 , the precession process is dominant. Then the critical switching current increases with decreasing pulse width. Devolder et al. (2005) have intensively investigated the current-induced magnetization switching in the sub-nanosecond region. When the current pulse width is in between, the magnetization reversal may involve both thermal activation and intrinsic precession dynamics. This regime has not been well studied yet.

E. Magnetic Tunnel Junction and Other Nanostructures Liu et al. (2003) were the first to report current induced magnetization reversal in AlOx-based MTJs. In the MTJs with lateral size of microns, the oersted field generated by the current may play an important role in magnetization reversal. Shortly, unambiguous experimental observation of spin torque induced magnetization reversal (switching) in MTJs was reported by Huai et al. (2004). In the last three years, spin-torque induced magnetization reversal has been broadly observed in MTJs by many research groups (Ding et al., 2005, Fuchs et al., 2005 and 2006, Higo et al., 2005, Huai et al., 2005, Pakala et al., 2005, Diao et al., 2005, 2006 and 2007, Finocchio et al., 2006, Inokuchi et al., 2006, Kubota et al., 2006, Zhang et al., 2006, and Yoshikawa et al., 2007). The importance of the current-induced magnetization reversal lies in the potential applications in nonvolatile magnetic memory technology. In general, the spin torque induced magnetization reversal in MTJs is very similar to that in metallic spin-valves. However, it can be observed that, in the reports to date, the critical current (density) in MTJs is generally lower than that in ferromagnetic metallic structures. It could be explained by that the current flowing through an oxide barrier with thickness fluctuation is high non-uniform (Sousa et al., 2004). The space-varying current will generate local heating and also local spin torque. It has been demonstrated that current through a nano-aperture can reversal the free layer magnetization in a non-uniform mode in metallic trilayers and then the critical current can be significantly reduced (Ozatay et al., 2006). Another explanation lies in possible existence of spin torque component perpendicular to the plane made by the free layer magnetization and spin polarization direction, on top of the in-plane component (Xia et al., 2002 and Theodonis et al., 2006). The perpendicular component may have a non-linear dependence on the bias voltage in MTJs and have a torque strength that may not be ignored. However, recent study (Sankey et al., 2008) shows that the perpendicular component can be as large as 30% of the in-plane component in CoFeB/MgO/CoFeB MTJs at low bias in resonance measurements. Similar results were observed in Al2O3-barrier MTJs (Petit et al., 2007). Nevertheless, its effect on magnetization reversal is yet to be understood. Current-induced magnetization reversal was also found in ferromagnetic semiconductor (Ga,Mn)As/GaAs/(Ga,Mn) tunnel junctions at low temperatures (Chiba et al., 2004). The critical switching current density was about 105 A/cm2. The low saturation magnetization and high spin polarization of the ferromagnetic semiconductors are not enough to account for the low critical current density. Spin-orbit coupling is much more significant in ferromagnetic semiconductors than in transition metal ferromagnets and may play a more important role in spin-transfer torque. To date, this remains as a problem and deserves further study.

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F. Applications and Associated Issues In many situations, MTJs are better than metallic GMR stacks for finding applications in the information era. Higher resistance of MTJs matches to silicon-based electronics better and higher MR ratio will provide higher output signal. The downside is that MTJs usually exhibits large noise, compared with GMR stacks. Magnetic memory using MTJs as memory cells has been commercialized but technical difficulties remain. The present MRAM (Engel et al., 2005) is using ampere field of electric current to switch the free layer magnetization. It requires large current and consumes a lot of power and the scalability of the technology is questionable. In addition, it may have the halfselect issue. Spin-torque driven magnetic switching overcomes these issues. It eliminates field generating current lines and uses low current passing memory cell to switch the device and then reduces the memory system complexity. High and low resistance states are controlled by the direction of write current flow. When a sufficient electric current flows from the fixed layer to free layer (or electrons flow from the free layer to the fixed layer), the free layer magnetization is aligned antiparallel to fixed layer, resulting in a high resistance state. Vice versa, when an electric current flows from the free layer to the fixed layer, the free layer aligns parallel to the fixed layer, resulting in a low resistance state. MRAM using spin torque switching has been demonstrated in the last 2 years (Hosomi et al., 2005 and Kawahara et al., 2007). Spin-torque driven MRAM is particularly advantageous over the conventional field-driven MRAM in scalability since the critical current for magnetization switching decreases with the lateral size of the MTJ stacks. However, the current provided by CMOS transistors is about 300 μA on the present technology and will be less, following technology scaling. Critical current for magnetization switching of the memory cells should match the value or be lower. Typical switching current density reported in MTJs is on the order of ~ 107 A/cm2, which implies a current of at least a few hundreds of μA for a MTJ memory cell at 90nm technology node. Besides, switching current needs to be low in order to avoid MTJ breakdown for device reliability concern. Therefore, reduction of switching current for MTJs is extremely important for realizing spin torque MRAM with a memory capacity of practical use. There are many ways to reduce switching current (density), mainly from material selection and structure engineering. Eq. (6) serves as a guideline for this purpose. Spin polarization ratio was found (Diao et al., 2005) to be higher in MgO-based MTJs than in AlOx-based MTJs, which accounts for the lower switching current in the MgO-based MTJs. There is a report (Jiang et al., 2004) that shows the critical current density can be reduced by a factor of 5 by adding non-magnetic cap layer, such as Pt and Ru, next to the free layer. These materials have a short spin diffusion length, which effectively increases spin transfer efficiency. Similarly, switching current reduction was also found in MgO-based MTJs with Ru-SAF free layer (Ochiai et al., 2005). MTJs with a composite free layer consisting of a nano-current-channel layer inside can be switched with lower current density (Meng and Wang 2006). In the recent study of Diao et al. (2007), MgO-based MTJs with dual barrier and fixed layers were fabricated. The free layer in the middle will experience spin torques from both side, which added together constructively. A record-low switching current density, J c 0 = 1.0 ×106 A/cm2, was reported. There is still tremendous effort in reducing switching current density for MTJs.

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One of the directions to reduce critical switching current is to introduce perpendicular magnetic anisotropy in the MTJs stacks so that the perpendicular anisotropy can cancel out the demagnetizing field shown in Eq. (5). Magnetization dynamics driven by spin current should be similar to that of magnetic devices with in-plane magnetization configuration. While the thermal stability is retained, the critical switching current can be significantly reduced (Klostermann et al., 2007). Spin torque switching was observed in metallic multilayer (Mangin et al., 2006). Recently, it has also been reported in MgO-barrier MTJs with perpendicular anisotropy and magnetization.

III. Current-Induced Magnetic Oscillation The effect of spin transfer torques is not only the magnetization reversal. In fact, spin wave excitation and magnetization precession induced by spin current was predicted first in the theoretical studies of Berger (1996) and Slonczewski (1996 and 1999). In this section, we will focus on this effect and discuss the current status of the research on magnetic dynamics and spin wave generation induced by spin transfer torque effect and its application for microwave generator and wireless communication.

A. Magnetic Oscillations in I-V Measurements Similar to the observation of magnetization reversal driven by spin transfer torque, experiments of spin transfer torque excitations were usually performed in similar systems: a magnetic layer with larger saturation magnetization as a spin polarizer and a magnetic free layer with a smaller saturation magnetization. Such systems include metallic nanocontacts, magnetic multilayer with a mechanical point contact, and magnetic nanopillars as shown in Fig. 9. An external magnetic field is typically applied to tilt the free layer magnetization out of the film plane. As shown in Fig. 2, a dc current flowing through the device generates a torque, which can suppress or enhance the conventional damping torque, depending on the current direction. When the damping torque is balanced by the spin transfer torque, the free layer magnetization will be driven into a steady precession state. The free layer magnetization dynamics will be reflected in the resistance of and therefore, the voltage across the device, due to the giant magnetoresistance effect. Certainly the large demagnetizing field in thin film devices will complicate the magnetization dynamics. Current-induced magnetic excitation was first demonstrated by Tsoi et al. (1998, 2000, and 2002) on [Co/Cu]n magnetic multilayers. Current was injected through a sharpened Ag tip in the film so that the structure is shown by Fig. 9(a). During dV/dI−I measurements, a strong magnetic field was applied to the multilayer film. Fig. 10 shows the derivative contact resistance as a function of the injected current for different magnetic fields. Because the multilayer film is so large, the sudden changes, such as spikes or dips, in the dV/dI−I curves ca not be explained by the magnetization reversal. Also, uniform magnetization precession does not seem possible in the point contact structure where the current is highly non-uniform. Therefore, those features in the curves may be indicators of spin wave excitation by current. The results also showed that the sudden changes in the derivative resistance only appear in one direction of the current flow, implying current polarity dependence. Furthermore, the

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current for spin wave excitation appears to be a linear function of the applied magnetic field. There is no theory to link the features in the dV/dI−I curves directly to spin wave excitation though. In the study of Tsoi et al. (1998), microwave absorption observed at the currents for the sudden resistance change might be evidence for spin wave excitation. Brillouin light scattering technique has also been used to detect spin wave excitation induced by spin current in Fe/Cr/Fe trilayers at room temperature (Rezende et al., 2000). (a)

(b)

FM2 NM FM1

(c) FM1

TE

NM FM2

FM2 NM FM1 BE

Figure 9. Spin transfer torque excitation measurement geometry. (a) Magnetic trilayer or multilayer in contact with a mechanical point contact, where FM1 and FM2 are ferromagnetic metal layers and NM is non-magnetic spacer layer; (b) Magnetic trilayer or multilayer with a nanocontact. The black region is isolating dielectric; (c) Lithographically patterned nanopillar device with a bottom electrode (BE) and a top electrode (TE).

Figure 10. Derivative contact resistance as a function of the dc current injected through a point contact into a magnetic multilayer film at different magnetic fields. Copy from Tsoi et al. (2002).

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Similar results have been observed by other groups such as Myers et al. (1999) in Co/Cu/Co trilayer structure with a Cu nano-contact isolated by silicon nitride shown in Fig. 9(b) and Rippard, Pufall, and Silva (2003) in Co/Cu multilayers with a mechanical point contact. Rippard, Pufall, and Silva have studied systematically the current and field dependent dV/dI peak. They confirmed the linear dependence of critical dV/dI peak current and magnetic field and compared with the theoretical expression (Slonczewski, 1999) ⎤ ⎛ e ⎞⎛ M t ⎞ ⎡ 23.4 D + 6.3αr 2 (− M s + H ex + H )⎥ , I c = ⎜ ⎟⎜⎜ s FM ⎟⎟ ⎢ ⎝ h ⎠⎝ η ⎠ ⎣ 2hγ ⎦

(14)

where D is the magnetic exchange stiffness, tFM is the magnetic free layer thickness, r is the effective contact size and Hex is the exchange field. Simplifying Eq. (14), one can find that the onset of spin wave excitation is linearly dependent on the exchange energy density MsD/ħγ, which varies greatly for different ferromagnetic materials. This dependence points out that the spin wave excitation critical current can be minimized by material engineering, such as doping Cu to dilute the exchange coupling of NiFe alloy such as Ni40Fe10Cu50. They have also found that magnetic excitation can be driven by spin transfer torque at zero magnetic field (Pufall, Rippard, and Silva, 2003). The effective field for magnetic dynamics is provided by the exchange field of the ferromagnetically-coupled Co/Cu multilayer. The possible simplification and improvement for spin wave excitation indicated by these works are useful for future spin transfer driven application discussed later. Lithographically patterned nanopillars shown in Fig. 9(c) with lateral dimension of ~100 nm have been found to be better systems to study spin transfer torque effect because of the limited effect of the exchange coupling in the patterned magnetic films (Katine et al., 2000 and Urazhdin et al., 2003). Furthermore, the smaller magnetic moment of the nanopillar gives a smaller critical current to observe the spin transfer torque effect. For point contact systems, critical current as high as ~109 A/cm2 is usually 10 times larger than that (~108 A/cm2) in a nanopillar. In the nanopillar trilayer structure, the change in resistance is only expressed as a spike in the dV/dI−I curve. Accompany with the spike are a gradual increase in the dc resistance. Katine et al. (2000) therefore argued that the state in the resistance transition does not correspond to a full reversal of the free layer, but an intermediate steady state. Similar result was observed in magnetic trilayers with current-confined spacers (Peng et al., 2005). Unlike in the point contact structure, the current flowing in the nanopillar structure is fairly uniform. Therefore, the magnetization excitation by spin current can be either spin waves or uniform magnetization precession with k = 0. Detailed discussion will be given in the next subsection. All the experiments discussed above were done using asymmetric structures, which have a thin magnetic free layer and a thick layer with an almost fixed magnetization. The magnetic excitation feature, i.e., dV/dI spikes, is only observed for one current polarity. According to Newton’s third law, the spin transfer torque experienced by the magnetic free layer should also have an effect on the spin current and the magnetic fixed layer. However, this interaction was not clearly observed in the early study of spin transfer excitations. Tsoi, Sun, and Parkin (2004) have used symmetric magnetic nanopillars with comparable free and “fixed” layer thicknesses to study the spin transfer driven excitation and found that the critical magnetic excitation current is linearly dependent on the applied magnetic field and the thickness of

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excited layer, in consistent with previous measurements and theory. Interestingly, Tsoi, Sun, and Parkin have also observed excitations at both current directions and confirmed that although both magnetic layers and feel spin transfer torque in either current direction, only one layer where electrons flow out can be excited by the spin current. This behavior is explained using the Berger model, in which spin accumulation is the driving force for magnetic excitations. Therefore, the work by Tsoi, Sun, and Parkin indicates that energy level, as well as the pure spin torque, determines the threshold current for magnetic excitation. Similar observations have been reported in point contact structures (Tsoi, Tsoi, and Wyder, 2004).

B. Magnetic Oscillations in Frequency Measurements Although the resistance measurement techniques by many groups on confined magnetic multilayer system have provided convincing support of spin transfer torque induced magnetic dynamics and excitation, direct evidence should be observed in the frequency-domain measurements. Therefore, the experimental work on spin transfer driven magnetic dynamics has been focused on frequency-domain measurements to understand the effect of bias current and magnetic field.

(a)

(b)

Figure 11. (a) Microwave spectrum after Johnson noise subtraction for different dc current at a fixed magnetic field. The inset indicates resonance peaks at both f and 2f. (b) Magnetic dependence of the microwave oscillations of a small amplitude signal (top line) and a large amplitude signal (bottom line). Copy from Kiselev et al. (2003).

The first direct frequency-domain spin wave excitation measurement was conducted by Kiselev et al. (2003) on a 130 nm × 70 nm elliptical nanopillar with a trilayer structure of Co (40 nm)/Cu (10 nm)/Co (3 nm). In the presence of a certain in-plan magnetic field, one begins to see peaks in dV/dI−I curve, the indication of magnetic excitation as discussed earlier. In the frequency-domain measurement by using a heterodyne mixer circuit,

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resonance peaks appear in the power spectrum as shown in Fig. 11(a), clearly indicating the microwave excitation driven by the spin transfer torque effect. At moderate current and field, the frequency of the spin wave excitation has a red shift at higher current and increases linearly at high field. The magnetic field dependence of the microwave oscillation with small amplitude (see Fig. 11(b)) can be well fit to Kittel’s formula for ferromagnetic resonance (FMR) with some adjustment in the effective magnetization on the free layer. It is attributed to small-angle elliptical precession of the free layer confined in the film plane around the equilibrium direction. The precession angle estimated from measurements is about 10°. With increasing current, a microwave oscillation with 2 orders of magnitude higher amplitude appears. Resonance frequency abruptly shifts from that for the small amplitude oscillation (see Fig. 11(b). Kiselev et al. attributed it to a large-angle in-plane precession mode, according to single-domain micromagnetic simulation. The trajectory of this precession mode is non-elliptical and clamshell-like. The free layer magnetization has significant antiparallel component with respect to the applied field and, therefore, the fixed layer magnetization, resulting in the increase of microwave emission power. At an even higher current, a third precession mode, out-of-plane precession, is found in the macrospin simulation. In this regime, magnetization precesses out of the film plane with the precession frequency increasing with current. For a current above the out-of-plane precession regime, Kiselev et al. noticed a sharp drop of microwave emission power associated with the disappearance of free layer dynamics. However, some small amplitude and large frequency peaks appear in the spectrum at this current range. These peaks are not the harmonics of lowcurrent precession frequency and found to be related to the dynamics of fixed layer. This type of feature has not been widely observed in experiments by other groups and not well understood. Almost at the same time, Rippard et al. (2004) looked the magnetic excitation in CoFe/Cu/NiFe trilayer with a point contact structure using similar technique. The outcome was similar to that of Kiselev et al. but difference was also evident. Rippard et al. did not see the microwave oscillation, which was attributed by Kiselev et al. to the out-of-plane precession mode. It might be related to the precession trajectory instability and relatively low current densities in their devices due to the point contact geometry. Harmonics of the in-plane precession frequency were unambiguously observed by Rippard et al. in a broader frequency range. However, the bias dependence of the ratio, as well as the bias dependence of the spin wave emission power and linewidth has been found to be complicated and difficult to explain using macrospin LLG simulation. Rippard et al. speculated that the microwave oscillation might be spin wave (k ≠ 0) excitation induced by the injected current via point contact into the patterned thin film device. Rezende, de Aguiar, and Azevedo (2005 and 2006) used the four-magnon interaction theory to quantitatively explain the current and field dependencies of the microwave oscillation frequency and amplitude. However, the observed high-order harmonics of the oscillations at 2f, 3f, and … do not agree with the spin wave modes defined by the thin film geometry. Instead, they may arise from the uniform large-angle in-plane precession or out-of-plane precession of the free layer magnetization (Kiselev et al., 2003 and Xi and Zhi, 2004). There has been since a lot of experimental and theoretical effort to understand the current-induced magnetization dynamics in the nanoscale magnetic devices (Kiselev et al., 2004, Rippard et al., 2004, Pufall et al., 2005, Bertotti et al., 2005, Berkov and

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Gorn, 2005, Russek et al., 2005, Ozyilmaz et al., 2005, Montigny and Miltat, 2005, Slavin and Tiberkevich, 2005, and Mancoff et al., 2006). The complexity of the current-induced magnetization dynamics can be seen in the phase diagram. Kiselev et al. (2003) constructed the first phase diagram in their study of microwave oscillation in nanopillars and many experimental and theoretical effort followed. Fig. 12 shows calculated magnetization modes for a spin-valve nanopillar in the presence of current and in-plane magnetic field. In some regime, multiple modes co-exist. It implies the difficulty in experiments to identify the dynamic modes of the magnetization and explains the discrepancies between the data and the macrospin model. Other sources for the discrepancies can be attributed to the possible domains in the free layer, thermal fluctuation (Xiao, Zangwill, and Stiles, 2005 and Russek et al., 2005), and the interaction with the fixed layer that is not considered in the model. Phase diagram is different for different structure (e.g., Bazaliy, 2007)

Figure 12. Phase diagram of the current-induced magnetization dynamics in a spin-valve nanopillar structure. Calculation is conducted for 3 °K. The boundary of the P/AP region indicates current and field induced magnetization reversal. Copy from Xiao, Zangwill, and Stiles (2005).

The dynamics of magnetic excitation in a perpendicular field or a field with large outof-plane angle is more complicated than in an in-plane field. It has been intensively investigated (Rippard et al., 2004 and 2006, and Kiselev et al., 2004). However a lot of experimental observations cannot be explained by macrospin LLG simulation. One reason is that the applied field can dramatically change the orientation of thin magnetic film magnetization, which is aligned in the film plane at zero or small fields and becomes out of plane at high fields. It has been found that the spin wave excitation power usually increases by two orders when the applied field is changed from the in-plane geometry to the out-ofplane geometry. For a field smaller than the out-of-plane saturation field of the free layer,

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the observed microwave frequency increases linearly with the applied field, in consistent with Kittel’s formula for small-angle magnetization precession. A clear frequency jump has been experimentally observed for a higher current but cannot be reproduced in micromagnetic simulation. Nevertheless, the frequency jump is believed to be the change of magnetization dynamics mode from small-angle precession to large-angle precession about the applied field (Rippard et al., 2004 and Kiselev et al., 2004). This argument is further supported by the fact that a dV/dI peak occurs at the same current for the frequency jump. In a perpendicular field, microwave frequency shows a blue shift with repesct to bias current, i.e., increasing with increasing current, in contrast with the red shift for the case in an in-plane field. The physics of this blue shift can be understood by the decrease of the demagnetization field when the magnetization precession is tilted toward the film plane by the larger spin transfer torque. A second frequency jump occurs when the current is sufficient to pull the free layer magnetization to be antiparallel to the applied field and turns the dynamics into the third mode. Because of the reorientation of the demagnetization field, microwave frequency changes accordingly. In a magnetic field larger than the saturation fields of both magnetic layers, the magnetizations of the free and fixed layers are aligned in parallel. When the bias current is turned on, the spin transfer devices show a direct transition from parallel to antiparallel state (Rippard et al., 2004). No steady-precession dynamics is expected in the structure. Since the macrospin model is not inadequate to describe some of the experimental observations, Acremann et al. (2006) have used the x-ray time-resolved spin-sensitive imaging technique to study the magnetization dynamics and found out that the switching of magnetization driven by spin transfer torque exhibits a complex “C”-like vortex motion. These observations have been reproduced in a micromagnetic simulation by Ito et al. (2007). Further investigation of these results will gradually complete the model of spin transfer torque and enhance our understanding of spin torque driven dynamics.

C. Magnetic Oscillations in Real-Time Measurements Recently, Krivorotov et al. (2005) have shown that it is possible to directly detect the time evolution of magnetic precession using time-domain measurements. The measurement setup is similar to the previous frequency-domain measurements, but with the device voltage coupled to a fast sampling oscilloscope. Because of the signal-to-noise requirement for sub-nanosecond time-resolved measurement, the spin transfer device needs to have stable and repeatable initial magnetization configuration for signal averaging. In order to achieve such required phase coherence, Krivorotov et al. used a multilayer structure with the fixed layer exchanged biased 45o from the free layer easy axis. Due to shape anisotropy and demagnetization field, the equilibrium angle between two layer magnetizations is estimated to be ~30o. The structure was first tested using dV/dI and frequency-domain measurement with an in-plane field along the exchange bias direction to confirm the microwave excitations. Clear sinusoidal-like voltage oscillation was observed in the timedomain measurement, providing the strongest evidence of microwave excitation.

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(a)

(b)

(c)

(d)

Figure 13. (a) Frequency spectra of magnetic oscillation for different current; (b) and (c) Signals for the magnetization relaxes toward equilibrium at different currents; and (d) Dependence of the effective damping constant on dc bias current. Copy from Krivorotov et al. (2005).

Using this technique, Krivorotov et al. (2005) were also able to study the time dependent evolution of magnetization switching at zero field and bias dependent switching time. They initialized the device in the low-resistance state and applied a short pulse to intentionally switch the free layer into high-resistance state and watch the dynamic evolution. Coherent oscillation of voltage and then coherent magnetization precession are observed. The result shown in Fig. 13 confirms that the magnetization reversal is driven by coherent magnetization precession rather than magnon excitation or spin wave heating. From the decay time τd, damping constant can be calculated. First, the experiment verified the spin torque theory (Berger 1996) that says the damping constant is not intrinsic in magnetic multilayer. It can be changed by input dc current. Krivorotov et al. have observed about 5-fold decrease of damping current when the current approaches the critical switching current. Second, the result shows discrepancy in the value of damping constant between different measurement technique such as spin-torque induced magnetization switching and ferromagnetic resonance. Further study is needed to understand the problem.

D. Frequency Modulation and Phase Locking As discussed above, the microwave frequency driven by spin transfer can be tuned by the dc bias current. It would be interesting to see the effect of an ac current in addition to the dc bias

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on the microwave excitation. Pufall et al. (2005) first found that an ac current resulted in a frequency modulated spectral output with sidebands generated near the microwave resonance frequency when the frequency of the ac current is much lower than the microwave frequency. Another study conducted by the same group (Rippard et al., 2005) shows that when the frequency of the ac current is close to the natural oscillation frequency of the device, a CoFe/Cu/NiFe trilayer with a point contact structure in the study, the output frequency is locked by the ac current. During the frequency locking, the output amplitude is significantly enhanced and the resonance linewidth is improved. When the dc bias current varies over the frequency-locking regime, the phase of the output signal frequency changes about ± 90° with respect to the input ac current until the oscillator breaks free. Li, Li, and Zhang (2006) predicted the ac current can also drive the magnetization into a chaotic mode. In an independent study of Tsoi, Tsoi, and Wyder (2004), microwave excitation spectrum in dV/dI curve was found to be amplified by a microwave magnetic field. Most interesting results recently have come out to demonstrate the mode locking of two nano-oscillators in proximity (Kaka et al., 2005 and Mancoff et al., 2005). Mode locking of microwave frequencies between two proximate nano-oscillators is an effect of nonlinear interaction (Slavin and Kabos, 2005, Slavin and Tiberkevich, 2006, Rezende, de Aguiar, and Azevedo, 2005, and Rezende et al., 2007), originating from the spin wave propagation in the magnetic free layer. Pufall et al. (2006) have shown that when the continuous free layer is cut off using focused ion beam, the mode-locking disappeared. The total microwave emission power becomes incoherent sum of two individual nano-oscillators, instead of the coherent sum for the uncut case. Therefore, spin wave propagation can be understood as the origin of mode locking, other than the ampere field from the applied current. Fig. 14 shows the structure of a two nano-contact device in the study of Kaka et al. (2005). The two nanooscillators are lithographically patterned metallic nanocontacts on the same magnetic free layer of a CoFe/Cu/NiFe trilayer thin film and have a diameter of ~40 nm and center-tocenter separation of ~500 nm. The two contacts are separated current biased, making each contact an independently controlled oscillator. Device geometry and circuit enable the characterization of nano-oscillators individually as well as simultaneously.

Figure 14. Schematic of the two-nano-contact device structure in the study of Kaka et al. (2005). Copy from the publication.

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Figure 15. Spectrum from both contact oscillators as the dc bias current through contact B varies from 7 mA to 12 mA while the dc bias current through contact A is fixed at 8 mA. Copy from Kaka et al. (2005).

Figure15 shows the frequency spectrum of the two contact oscillators with oscillator A is constantly biased at 8 mA and the dc current through oscillator B is ramped from 7 mA to 12 mA. The oscillation frequency of B increases gradually at a moderate field that is tilted 75o from the sample plane, while the oscillation frequency of A decreases slightly. When the oscillation frequencies of A and B become comparable, steady precession modes of A and B are frequency-locked. Only one peak appears in the spectrum with a linewidth reduced by at least an order of magnitude from the unlocked case. The narrow linewidth indicates the insensitivity of locked mode to thermal noise and fluctuation. It is a result of mutual feedback between the two contact oscillators and implies coupling of the spin waves in the free layers. At a higher current for oscillator B, the locking disappears because of the large difference in natural oscillation frequencies of A and B. Above the critical current of unlocking, experimental results show that two nano-oscillators behavior entirely uncorrelated. Due to the ampere field from the current in oscillator B, the oscillation frequency of A shows small biasdependent decrease. The power emitted from two individual nano-oscillators is also measured and show significant dissimilarity. More interesting observation is that the total microwave power by oscillators A and B varies when an extra phase shift is added to one signal, indicating that the phase of two oscillation signals is also locked. When the device microwave emission power is maximized with a constant phase shift, the total power PC becomes the coherent sum of the individual powers, i.e., PC = PA + PB + 2 PA PB .

(15)

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It also confirms that the phase locking is time-independent. Thus, we can conclude that two proximate nano-oscillators can be locked in frequency and phase, resulting in a significant enhancement of microwave emission. Slavin and Tiberkevich (2006) has theoretically defined the conditions of the oscillation frequency and separation between the contact oscillators for phase locking. Grollier, Cros, and Fert (2006) have conducted micromagnetic simulations on a string of electrically connected oscillators. The oscillators can be locked in both frequency and phase when the natural oscillation frequencies are close to one another. The coupling arises from the microwave components of the current induced by an oscillator to the others. As a result, synchronization of the nano-oscillators can be achieved and the output power can be enhanced significantly.

E. Potential Applications and Others Magnetization oscillation induced by spin current discussed above is in the magnetic thin film devices with magnetizations of the magnetic layers in the film plane. In fact, it can be obtained in other magnetization configurations (Kent, Ozyilmaz, and del Barco, 2004, Xi, Gao, and Shi, 2004 and 2005, Xi and Shi, 2004, and Xi, Shi, and Gao, 2005). In particular, the magnetization dynamics is very simple in the configuration with the fixed layer magnetization perpendicular to the film plane while the free layer magnetization in the film plane. Under a small dc bias current, the free layer magnetization stays in the initial equilibrium direction defined by the in-plane anisotropy. When the dc current increases to a certain critical value, the free layer magnetization transitions to a steady precession mode, tilting up from the film plane and then rotating in the plane parallel to the film. This kind of behavior has recently been found in the magnetic nanopillar with a (Co/Pt)n multilayer as the perpendicular fixed layer (Houssameddine et al., 2007). When the dc current is too high, the spin torque will break the free layer magnetization into domains. Therefore, the magnetization dynamics is no longer periodic but chaotic (Xi, Shi, and Gao, 2005). Hoefer et al. (2005) and Consolo et al. (2007) have conducted calculations for devices of this magnetization configuration with current injected in the thin film via point contact. Spin waves will be generated by the spin current and propagate in the film plane. Interestingly, spin waves were found to be excited by non-polarized current through point contact into a single ferromagnetic film (Ji, Chien, and Stiles, 2003). The experimental results were qualitatively explained by a model in which the spins of the conduction electrons interact with a non-uniform magnetization (Xi et al., 2007). The interaction was taken from the study of Li and Zhang (Li and Zhang, 2004, Zhang and Li, 2004, and Li, He, and Zhang, 2005) and the calculation was conducted using perturbation theory. Since the current density at the point contact is so high, the current can possibly reverse the magnetization of a nano-domain near the point contact. This situation was found in the structure where a non-polarized current was injected into a single ferromagnetic layer exchange-biased by an antiferromagnetic film (Chen et al., 2004). However, theoretical study is yet needed to verify the results. The microwave generated by spin torque in magnetic devices have several unique characteristics. First, the microwave can be excited by a dc bias current and the frequency can be tuned by bias current and applied magnetic field. Second, the typical dimension of such devices is on the nanometer scale. It has been suggested to use the spin transfer devices as a source of microwave generation and a dc-ac nano-converter. However, the microwave

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emission power of the nano-oscillators is generally less than 1 nW, much smaller than the power for any possible microwave applications. One of the methods to boost output power is mode locking of nano-oscillators as suggested in the study of Kaka et al. (2005). A second issue is that the impedance of typical metallic nano-oscillators is too low to match the peripheral circuit of the systems. Nano-oscillators using MTJs will definitely help in this regard and microwave oscillation has been found in MgO-based MTJs (Nazarov et al., 2006). The frequency modulation demonstrated by Pufall et al. (2005), ac current control by Rippard et al. (2005), and mode locking by Kaka et al. (2005) and Mancoff et al. (2005) open a field for the nano-oscillators to be used as microwave sources, modulators, couplers, frequency receivers and mixers. It is noteworthy that the Q-factor of the microwave oscillation can be as good as 104 when in mode-locking. Therefore, they will find application in nanometer microwave circuits for synchronization and phase control, as well as short distance chip-to-chip or intra-chip wired and wireless communications. Nevertheless, the device engineering is in an early stage. Besides the issues of the nano-oscillators, integration with CMOS posts a great challenge at the present time.

IV. Current-Induced Domain Wall Motion A. Brief History Current-induced domain wall (DW) was first predicted and observed in seminal works of Berger (Berger, 1978 and 1984, Hsu and Berger, 1982, Freitas and Berger, 1885, and Huan and Berger, 1988). His earliest predictions on current-induced domain wall motion grasp the fundamental physics mechanism of current spin and domain wall magnetization interactions. Domain wall magnetization rotates electron spin direction through exchange interaction and then inversely electron spin creates a reaction torque on domain wall. Based upon this physics mechanism, a dc current would apply a force on domain wall and induce domain wall motion. Subsequently, Frietas and Berger (1985), and Hung and Berger (1988) observed wall displacement caused by spin polarized current torque in NiFe film. In these experiments, large current pulses about 200 nsec were sent through 30 ~ 40 nm thick NiFe films containing Neel walls. Domain wall displacements in the same direction as that of the drift electron-like carriers were observed, consistent with the prediction of electron spin and domain magnetization interaction. The current density required to move domain wall in these experiments were about 1~5 ×107 A/cm2. After the pioneer work of Berger, the current-driven domain wall motions have been demonstrated and confirmed on magnetic thin films and magnetic nanostructures. In the following, we will review recent experimental observations of spin torque induced domain wall motion and then theoretical description of the phenomena. First, it is critical to design the magnetic structures for forming domain walls and methods. Methods of domain-wall detection and quantitative measurements of current induced domain wall motion are with equal importance. Current induced domain wall motion on a patterned rectangular NiFe film was observed by Gan et al. (2000) using magnetic force microscopy (MFM) image technique to detect the displacement of the Bloch domains occurred when current density reached order of 107 A/cm2. Sahli and Berger (1993 and 1994) reported similar current induced Bloch domain wall motion on NiFe thin film with varying thickness from 120 − 740 nm. Rectangular or

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exponential current pulses of duration ~ 0.1 μsec were used. Kerr contrast microscopic was used to detect domain wall motion. The phase diagrams for domain wall motion with respect to current and magnetic field were mapped out. Quantitative information on the critical current to move domain walls as a function of film thickness was obtained. Koo, Kraff, and Gomez (2002) detected current-switched bistable remanent domain walls on ferromagnetic islands using MFM technique. These islands are rectangular elements with thickness of 50 to 100 nm and lateral dimensions on the order of several microns. The closure configuration could be set into either the 4 or 7 domain configuration by applying 10 ns current pulses at density on the order of 107 A/cm2.

B. Domain Wall Motion in Straight Wires The subject of current induced magnetic domain wall motion revives in the last five years due to its potential applications that will be discussed later and the tremendous progress in microand nano-fabrication techniques. Many of the observations and measurements on the current induced domain wall motion were done in nanowires. Compared to thin film, nanowires had a much larger ratio of thickness to width, resulting in thinner walls and, consequently, a stronger interaction of wall and current. Figure 16 shows a typical magnetic wire for the study. It was a 100 nm wide CoFe stripe connected to a large diamond-shaped pad at the left made by electron-beam lithography (Tsoi, Fontana, and Parkin, 2003). Two constrictions of 50 nm wide in the stripe were used to trap the injected domain wall from diamond pad. MFM image was taken to identify the domain wall and the resistance measurement was used to determine domain wall propagation. It was observed that domain wall moved by the electron current flow.

Figure 15. Scanning electron microscopy (SEM) (a) and MFM (b) images of a magnetic wire used for the study of domain wall motion driven by current. Domain walls are seen to be are trapped in the constrictions. Copy from Tsoi, Fontana, and Parkin (2003).

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Florez, Krafft, and Gomez (2004) have measured the current induced domain wall motion in a strip with notches. Experiments were carried out on the NiFe patterns fabricated over thermally grown SiO2 substrates through the e-beam lithography followed by a standard liftoff process. The structures consisted of 300 nm wide, 40 nm-thick NiFe wires, containing two constrictions each roughly 220 nm wide. The elements were asymmetrical with a nucleation pad on one side and a sharp end on the other, ensuring domain wall nucleation from only one direction and the reproducibility. Experiments were performed by applying concurrent magnetic fields and electrical currents to the sample and using MFM to detect resulting domain wall motion. Direct observation of current-induced magnetization reversal was achieved and dependence of critical current on the applied magnetic field amplitude was obtained. Kimura et al. (2003) reported domain wall depinning in straight nanowires and critical current versus bias field relationship was obtained. In this measurement, the NiFe wire was 250 nm wide and 30nm thick and connected to a 1 μm × 5 μm pad. Resistivity measurements to detect domain wall motion were performed at 3 °K using a low-noise four terminal dc measurement system with external magnetic fields ranging from −1000 Oe to 1000 Oe applied along the wire. The domain wall was pinned at the junction of the pad and the wire prior to the current injection. When a polarized spin current with amplitude beyond a critical value was injected along the direction of the domain-wall propagation, the pinned domain wall was freed and pushed into the wire. The critical current decreased monotonically with increasing magnetic field. In the study of Lepadatu and Xu (2004), a set of necked wires with a fixed length of 400 um and width of 1 um was defined on a silicon substrate using electron beam lithography and lift-off technique. A constriction was defined halfway along the wire, forming point contacts of nominal widths ranging from 50 to 350 nm, with an increment of 50 nm. Resistance change that indicates domain wall formation and depinning was studied to correlate with the constriction width and applied current. A 180°-degree domain wall was formed at the constriction for the contact widths smaller than 300 nm. The critical current density was around 107 A/cm2. Hayashi et al. (2006) injected and pinned four well-defined magnetic states, vortex and transverse domain walls with two chiralities for each, at a notch on NiFe nanowires that were 100 ~ 300 nm wide and 10 nm thick. MFM image and resistance measurement were taken to detect domain wall structure and movement. Depinning of these different domain walls by current was observed. It was found that, while the magnetic fields required to depin these four domain wall states were substantially different, the critical current densities to depin these states were nearly the same in small fields. Beach et al. (2006) have conducted detailed measurements of domain wall propagation after depinning. Using high-bandwidth scanning Kerr polarimetry, they were able to obtain the dependence of domain wall velocity on electric current and magnetic field after depinning shown in Fig. 17. First the domain wall velocity increases linearly in a small magnetic filed until it reaches a maximum value at H = 6 Oe. Then the domain wall becomes unstable and oscillates back and forth, causing the average velocity to drop (Yang et al., 2007). At high fields beyond the Walker breakdown regime, the velocity of the domain wall recovers to be a linear function of applied field. Fig. 17 also shows that the field dependence of the wall velocity is different for polarity of the injected current. The asymmetry may indicate the subtle role of the spin torque on a vortex domain

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wall. In addition, the study tried to understand the roles of adiabatic and nonadiabatic torques of the current, which will be discussed later in this section.

Figure 17. Domain wall velocity as a function of applied magnetic field for different current. Inset shows the low field region. Copy from Beach et al. (2006).

Current induced domain wall motions were also generated in spin valve stripes. Grollier et al. (2003) presented experiments in which a CoO/Co/Cu/NiFe exchange-biased spin valve was partly switched by current-induced domain wall motion. Resistance measurement was used to identify domain wall state. At zero or very low field, the domain wall displacement was in opposite directions for opposite dc currents, and back-and-forth motions between two pinning centers were obtained. The critical current density to move domain wall was of the order of 106 A/cm2. Lim et al. (2004) showed that a single current pulse as short as 0.4 ns could trigger domain wall displacement in spin-valve stripes of 0.3 mm width inserted into a coplanar waveguide. Domain wall position was also determined by resistance measurement. In zero magnetic field, domain wall displacement occurred in the same direction as the conduction electron current above a critical current density on the order of 106 A/cm2. The distance traveled by the domain wall along the stripe increased with the current pulse amplitude and applied field strength. However, it did not depend on the pulse duration in the range between 0.4 and 2 ns.

B. Domain Wall Motion in Nonlinear Structures Besides straight wires with constriction, also as known as notch and pinning site, to pin the injected domain wall, nanowires with well-designed shapes were also used. The nonlinear shape of the wires allows using magnetic field to generate domain walls and storing domain walls after the field is removed. L-shaped nanowires have been used by a few groups (Yamaguchi et al., 2004 and Yang and Erskine, 2007) to study the domain wall motion. The

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NiFe magnetic wires in the study by Yamaguchi et al. were 10 nm thick and 240 nm wide. A single domain wall was introduced in the wire by a diamond pad at one end of the wire. MFM was used to visualize domain wall position that changed back and forth by positive and negative current pulse. They concluded that only a small fraction of the spin angular momentum was used to move the vortex domain wall. Because the current density was so high, Joule heating effect on the domain wall and its motion needed to be addressed (Yamaguchi et al., 2005). Tsai et al. (2005) have observed current induced domain wall motion in patterned Ushaped nanowires. Semi-circular arc of U shape pattern was used to pin the domain wall. The magnetic domain structure was observed by MFM and the magnetoresistance was measured at 77 °K by four-channel dc measurement system. The influence of the positive and negative dc current injection on domain wall motion with density on the order of 107 A/cm2 were investigated. It was found that critical current varied dramatically when the bias field was close to the switching field but only gradually when the field was far away from. Klaui et al. (2005) investigated current induced domain wall motion in NiFe wires with a zigzag geometry. The shape anisotropy from zigzag geometry forced the magnetization to form domains of alternating directions in adjacent segments after application of a magnetic field in transverse direction to the wire. Current induced propagation of head-to-head as well as tail-to-tail domain walls in the NiFe wires were observed with a mean velocity of 0.3 m/s in the direction of the electron flow. High resolution images of the wall structure after consecutive pulse injections showed a transformation from a vortex wall to a distorted transverse wall due to the current (Klaui et al., 2005 and 2006). The change in wall velocity was correlated with a change in the domain wall spin structure. Domain wall velocity was also found to depend on the width of the wires (Jubert et al., 2006). S-shape patterned wires that exhibit some features similar to both the zigzag wires and the ring structures have been used to study domain wall motion as well (Chen et al., 2006).

(a)

Figure 18. (a) SEM image of a ring structure with eight nonmagnetic contacts; (b) Schematic of the ring in the onion state with a head-to-head domain wall located at the notch; and (c) image of the complete ring structure. Copy from Kloui et al. (2003).

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Current induced domain wall motions were extensively studied on ring structures. The symmetric shape of the ring had particular benefits of forming various domain wall structures when applying external magnetic field and current. Klaui et al. (2003 and 2005) reported current controlled domain wall motions on NiFe ring structures 5 to 34 nm thick with 2 nm Au capping and sub micrometer lateral dimensions. The outer and inner radii of the ring structure were around 950 nm and 550 nm. Notch features were included in the ring to achieve reliable and reproducible spin structures of the domain walls. Photoemission electron microscopy (PEEM) is used to image domain wall and resistance measurement between two of the eight contacts was used to determine domain wall position. Fig. 18 shows the ring structure used in the study of Kloui et al. (2003 and 2005). Current pulses of up to 20 us width and current densities up to 5 ×108 A/cm2 were used to obtain information on critical current density versus film thickness, switching field amplitude versus current density. There exist primarily transverse head-to-head domain walls in thin ring structures while vortex walls appear in thick rings. The experiments demonstrated that using current pulses without any externally applied magnetic fields could control both displacements of vortex and transverse domain walls. When both field and current were applied to move domain wall, the switching field decreases as current increases. Laufenberg et al. (2006) presented experimental studies of domain wall motion induced by current pulses as well as by magnetic fields at temperatures between 2 °K and 300 °K in a 110 nm wide and 34 nm thick NiFe ring. Experimental data showed that critical current density increased with increasing temperature, in contrast to that of the field induced domain wall motion. This observation was explained by a reduction of the spin torque efficiency with temperature. Recently, Bedua et al. (2007) studied spin torque current induced resonance for vortex and transverse wall spin structures on magnetic ring. For constrained geometry, magnetic domain walls could behave like quasi-particle. Resonance could be excited if an ac current was applied to the domain walls. Sharp reduction in the depinning field were observed at the resonant frequencies corresponding to the transverse wall moving in the attractive potential well and the vortex core of the vortex wall moving within the wall spin structure. Domain wall motion by fast current pulses has been also investigated in a racetrack-like ring structure (Meier et al., 2007).

C. Domain Wall Motion in Other Magnetic Media There were several measurements on domain wall motions on ferromagnetic semiconductor (FMS) stripes (Chiba et al., 2006, Tang et al., 2006, and Yamanouchi et al., 2006 and 2007). These measurements demonstrated that in ferromagnetic semiconductor structure, magnetization reversal through domain-wall switching could be induced in the absence of a magnetic field using current pulses with densities as low as 105 A/cm2. Yamanouchi et al. (2006) reported domain-wall motion with velocity spanning over 5 orders of magnitude up to 22m/s in (Ga,Mn)As wires grown by molecular beam epitaxy (MBE). Magneto-optical-Kerr microscopic (MOKE) was used to determine domain position and speed. The ultra-low critical current density for domain wall motion was attributed to spin-current assisted creep. Tang et al. (2006) investigated the transport dynamics of individual magnetic domain walls in ferromagnetic semiconductor (Ga,Mn)As stripes. The domain wall position was determined by planar Hall resistance measurements. They explained the obtained experimental data of

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domain wall motion was explained by thermally assisted flow for low fields and viscous flow for high fields and disapproved spin torque effects in (Ga,Mn)As devices. Similar measurements conducted by Yamanouchi et al. (2006) indicated domain wall distortion triggered by the current pulse edges were not likely to be the mechanism of the observed current induced domain wall motion, after considering the Joule heating and ampere field effect. They suggested that the spin angular momentum transfer through p–d exchange interaction between localized Mn spin and itinerant hole appeared to be a possible mechanism behind the observed results. It has yet to be understood from further study.

E. Physical Origin of Spin Torques In the next subsection, we will discuss the theoretical effort in the field of domain wall motion. Theoretical understanding of electric current and domain wall interactions is still in progress. In macroscopic spin transfer torque theory, the usual Landau-Lifshitz-Gilbert (LLG) equation of magnetization dynamics is augmented with spin transfer torque terms. Phenomenological approach to construct these spin torque terms can be based upon the fact that current breaks the inversion symmetry. This implies that, in the long-wavelength limit, terms proportional to gradient of magnetization in current direction are allowed in the LLG equation. Based on the assumption of locality implied by the gradually varying magnetization of a thick domain wall, the spin transfer torque can be separated fairly generally to adiabatic term proportional to (n e ⋅ ∇)M and a nonadiabatic term proportional to M × (n e ⋅ ∇)M , where M is the magnetization and n e is a unit vector in current flow direction (Bazaliy, Jones, and Zhang, 1998, Zhang and Li, 2004, Fernandez-Rossier et al., 2004, and Thiavill et al., 2005). Spin torque induced domain wall motion can be analyzed based upon macroscopic models with adiabatic and non-adiabatic spin torque terms. Although the explicit solution of the LLG equation with spin torque terms is difficult and in general requires micromagnetic simulations (Thiavill et al., 2004 and 2005, and Martinez et al., 2007), analytical approximations can be obtained based upon either “nonlinear mode analysis” or “collective coordinate approach”. The Landau-Lifshitz equation without damping and spin torque term is a completely integrable system and its eigenmodes can be linear modes such as spin waves and nonlinear excitation modes such as domain walls and magnetic solitons. Inverse scattering is an effective technique to formulate these complete sets of nonlinear eigenmodes in magnetic systems (Takhtajan, 1977). Spin torque terms and damping term bring strong interactions of these nonlinear magnetic modes that can lead to nontrivial dynamics. Analytical treatment of these rich dynamic behaviors can be based upon analysis of the interactions of these basic nonlinear modes (e.g., Li, Liang, and Li, 2004, He and Liu, 2005, and Liu, Liu, and Ge, 2005). When these numerical or analytical solutions were compared to experimental measurements of current induced domain wall motion, they indicated that adiabatic term itself could not explain the measured critical current density and domain velocity. Non-adiabatic term and thermal fluctuations at finite temperatures were important in explaining the measured critical current density and domain wall velocity.

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F. Theoretical Models and Analyses Microscopic treatment of the current induced spin torque is of challenge and still in controversial. A summary of current microscopic theories against historical experimental measurement data can be found in the study of Berger (2006). Zhang and Li (2004) proposed a phenomenological microscopic model, of which the essence is to describe the current and magnetization interactions through two types of electrons: electrons near Fermi surface providing spin-dependent transport and electrons below Fermi sea involving the magnetization dynamics. Although it is impossible to unambiguously separate electrons of transport from electrons of magnetization, this simplified “s−d coupling” picture provides a starting point to model complex interactions between the current spin and the local magnetization. Based upon the relaxation time approximation and linear response function of the conduction electron spin in the presence of time- and spatial- varying local moments, the magnetization dynamics for spin torque induced domain wall motion can be described by: ⎛ M ∂M ⎞ ∂M M ∂M M ⎛ M ∂M ⎞ ⎟ − c J ⎜⎜ ⎟⎟ . = −γM × H eff + α × − bJ ×⎜ × × ∂t Ms ∂t M s ⎜⎝ M s ∂x ⎟⎠ ⎝ M s ∂x ⎠

(16)

Estimation showed that the influence of s−d interaction the gyromagnetic ratio and damping parameter is negligible (Zhang and Li, 2004). The third term is the adiabatic spin torque term with a magnitude bJ = ηje u B / eM s (1 + ξ 2 ) , where η is the polarization efficiency, u B the Bohr magneton, j e the current density and ξ = τ ex / τ sf the ratio between the s−d

exchange time and the spin-flip relaxation time. The last term is the nonadiabatic term. Zhang and Li also illustrated that the physics origin of this nonadiabatic term is related to the spatial mistracking of spins between conduction electrons and local magnetization. The magnitude of c J in nonadiabatic term is ζ times smaller than magnitude of adiabatic term bJ . For ferromagnetic material, estimation of ζ is around 0.01. This means that the

magnitude of nonadiabatic term is usually much smaller than that of the adiabatic term. However, it was also pointed out that the nonadiabatic term is essentially important in understanding measured current induced domain wall motion. The terminal velocity of a domain wall was argued to be independent of the adiabatic spin torque strength but rather controlled by the small nonadiabatic term. Xiao, Zangwill, and Stiles (2006) argued against the existence of a nonadiabatic torque from spin-flip scattering mechanism used by Zhang and Li (2004). Their analysis indicated that nonadiabatic corrections to the spin-transfer torque occur only for domain walls with small widths and therefore the nonadiabatic torque decreases exponentially with increasing domain wall width. Benakli, Hohlfeld, and Rebei (2007) generalized the Zhang-Li formulism to thin domain walls, considering tensor form of the spin diffusion. They also tried to determine the contribution of the conduction electrons to the damping in ferromagnetic metals in a self-consistent manner. The tensor formality of the spin diffusion was previously obtained in the study of Rebei and Hitchon (2005), where the s−d interaction was treated nonperturbatively from first principles in real space. A set of coupled equations of motion involving the non-uniform magnetization, the spin current, and the two-point correlation functions of the magnetization are established. In these studies, the diffusion tensor terms

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provide a larger contribution to the drive torque than to the damping process, leading to an overall increase of the domain wall velocity. Interesting results were shown regarding the prediction of current induced domain wall velocity. First, the off-diagonal terms of the diffusion tensor enhances the domain wall velocities. Second, the dependence of the domain wall velocity on the domain wall width was found to be non-linear and strongly influenced by the non-adiabatic behavior of the conduction electrons through the non-diagonal corrections of the diffusion tensor. Tserkovnyak et al. (2006) considered spin-conserving and spin-dephasing impurity scattering using a self-consistent adiabatic local-density approximation to study the magnetization dynamics. Gilbert damping and spin-transfer torques was obtained based on a quantum kinetic equation. The macroscopic damping in weak itinerant ferromagnets is related to the microscopic spin dephasing and thus Gilbert damping and a current-driven dissipative torque scaled identically and competed. Duine et al. (2007) treated the current-induced torques and thermal fluctuations in itinerant ferromagnets based on a functional formulation of the Keldysh formalism. Berger (2007) has considered the general relationships between electron theories of damping, of current-induced drive torques on domain walls, and of the domain wall resistance. He showed that any mechanism of the Gilbert damping of wall motion was based on the same electron processes as for the current-induced drive torque on the domain wall. This argument leads to a classification of most existing electron theories of damping and drive torques, and of wall resistance, into only three different kinds: electron spin relaxation, energy of mistracking, and anisotropic s−d exchange. In all cases, the drive torque was derived from the damping torque by replacing the wall speed with an electron drift speed. Using Lagrangian formalism, Tatara and Kohno (2004) developed a theory for a continuum planar and axial anisotropies ferromagnet. The current exhibits spin transfer effect and momentum transfer effect. Neglecting spatial spin wave modes, the dynamic equations for domain wall center position and the magnetization angle were obtained for thick adiabatic walls and thin abrupt walls. For thick adiabatic walls, the spin-transfer effect of the current is dominant and there is a threshold spin current, below which the wall could not be driven. This threshold current is non-zero even in the absence of a pinning potential and is controlled by magnetic anisotropy. Beyond the threshold, domain wall velocity varies with current by v DW ∝ I 2 − I 02 . It is noteworthy there is discrepancy in the dependence of domain wall

velocity on current between Tatara and Kohno (2004) and Li and Zhang (2004). Present experimental observations are not sufficient to verify the exact current dependence. On the other hand, for a thin domain wall, its motion is driven by the momentum-transfer effect arising from electron reflection. Then, the depinning current is given in terms of domain wall resistivity. Jung and Lee (2007) extended the study by including domain wall width variations. They showed that wall width variations modify critical switching current as well as domain wall velocity. The model also shows oscillating domain wall velocity solutions for high spin current, a kind of behavior similar to Walker breakdown. Spin wave modes in domain wall driven by current in a mesoscopic system were studied by Ohe and Kramer (2006). The results suggest that nonadiabatic effects due to spin-wave excitation can dynamically distort the local spin configuration and lead to a finite velocity of the wall even for currents below the critical value. Calculated domain wall velocity as a function of the current (density) is

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shown in Fig. 19. It is also believed that this provided a microscopic mechanism to account for the contribution to the shape of the domain wall.

Figure 19. Domain wall velocity V as a function of the s-d exchange constant Hsd, which is proportional to current density within the adiabatic approximation. Two cases with the Fermi energy of the conduction electrons greater and lower than the exchange coupling are shown. The solid line is the formula from Tatara and Kohno (2004). Copy from Ohe and Kramer (2006).

Developing a theory to explain current induced domain wall in ferromagnetic semiconductors is of particular interesting because the critical current density required to move domain walls in ferromagnetic semiconductor is 2 to 3 orders of magnitude lower than that in ferromagnetic metals. Nguyen, Skadsem, and Brataas (2007) proposed hole currentdriven domain wall dynamics in (Ga,Mn)As. They showed that the spin orbit coupling caused significant reflection of holes at the domain wall, even in the adiabatic limit when the domain wall is much thicker than the Fermi wavelength. The reflection of holes is a result of spin accumulation and mistracking between current-carrying spins and the domain wall magnetization. It increases the out-of-plane nonadiabatic spin-transfer torque and consequently the current-driven domain wall mobility by 3 to 4 orders of magnitude. This can explain the low critical current and high mobility of current induced domain wall in semiconductor ferromagnetic structure. Duine, Nunez, and MacDonald (2007) developed a theory of thermal fluctuation effects on current induced domain wall motion. The averaged drift velocity of the domain wall as a function of the applied current was obtained, based on the stochastic LLG equation at finite temperatures. Results of the model agree qualitatively well to experimental measurements. Estimation of the bending energy of the domain wall also shows that these degrees of freedom were thermally accessible. At any non-zero temperature, average drift of domain wall velocity initially varies linearly with current. Thermal activation of rigid domain-wall motion can play an important role in these magnetic semiconductor experiments.

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G. Potential Applications Information can be carried by magnetic domain walls. Current induced domain wall motion provides an approach to transfer and manipulate information. It is better than magnetic field driving since current is localized. Parkin (2004) proposed a magnetic shift register, also known as racetrack device, to use the phenomenon for data storage. A series of domain walls are stored in a U-shape magnetic stripe. Current is injected into the stripe to move domain walls. Domain walls are generated by a device that is similar to magnetic recording write head and detected by a magnetic sensor in a position close to the stripe. Mass data can be stored in the racetrack device. However, read and write of the racetrack memory are essentially sequential. With regard to current status of the study in domain wall motion, the critical current (density) is still too high for application. Also, motion and storage of more than one domain wall are yet to be demonstrated. Current-induced domain wall motion can be used to make random access memory (Numata et al., 2007). Each memory cell consists of a planar U-shape magnetic stripe with two bended corners. A domain wall is moved by an injected current from one corner to the other. A MTJ is integrated with the magnetic stripe to detect the magnetization of the center part of the stripe, which is associated with the domain wall position. Fast write and read have been demonstrated. However, write current is still as high as that for present commercialized MRAM. This domain wall magnetic memory is believed to be more scalable than the magnetic field-driven MRAM. Nevertheless, further study and engineering in materials and devices are needed before it becomes a real product. With this, we conclude the review on spin transfer torque effect and its applications in microelectronics.

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In: Spintronics: Materials, Applications and Devices ISBN 978-1-60456-734-2 c 2009 Nova Science Publishers, Inc. Editors: G. C. Lombardi and G. E. Bianchi

Chapter 6

B LOCH S PHERE , S PIN S URFACES AND S PIN P RECESSION IN Q UANTUM W ELLS A. Dargys∗ Semiconductor Physics Institute, A.Goˇstauto 11, LT-01108 Vilnius, Lithuania

Abstract The Bloch sphere represents all allowed spin states. It is often used to visualize dynamical spin properties since the points on the Bloch sphere, in contrast to Hilbert space, are directly related with the experiment. In semiconductors, where spin-orbit interaction may rearrange energy bands at critical points of the Brillouin zone, the spin is not a good quantum number and a more general form of the spin surface is required in this case to describe free electron spin properties in bulk semiconductors and quantum wells (QWs). In the first part of this chapter the main properties of the Bloch sphere are summarized, with a particular emphasis on the free charge carrier spins in bulk semiconductors. Starting from the Kramers theorem for spin split bands it is shown that the spin surface, an analogue of the Bloch sphere, remains a useful concept in the analysis of dynamical properties of free carrier spins in crystalline solids as well. In the second part, to illustrate typical shapes of the spin surfaces and how they may be calculated, the properties of electron and hole spin surfaces in HgTe/HgCdTe quantum wells are considered in detail. The most interesting case, when the energy bands are inverted and the energy gap is negative, is analyzed using the eight-band k · p Hamiltonian. Characteristic shapes of the spin surfaces are presented in a form of graphs. It is shown that, depending on the free electron (hole) wave vector magnitude and symmetry of the considered 2D energy subband, the spin surface may be an ellipsoid, disk, line, or reduced to the Bloch sphere. In the third part, the spin properties of a wider class of nanostructures ( n- and ptype GaAs/AlAs QWs, hollow cylindrical QWs) are briefly reviewed. The role of the spin surface as well as of natural quantization axis in the design of spintronics devices is discussed.

Pacs:85.75.-d; 71.70.Ej; 75.10.Hk; 85.75.Hk Keywords: Spintronics; Spin-orbit coupling; Spin models; Spin FET ∗

E-mail address: [email protected]

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A. Dargys

Introduction

Spintronics requires active and passive components, such as spin FETs, guides, filters, bends, couplers etc, to transmit and control the spin polarization in a well-defined way [1, 2, 3]. The quantum mechanical spin normally is controlled by magnetic field, however, in semiconductors the spin can also be manipulated via spin-orbit (SO) interaction, the strength of which in semiconductor nanostructures can be controlled by an external voltage. In spintronics the preference is given to electric rather than magnetic fields, since microand nanomagnets have relatively large stray fields (such magnet acts like a single, giant dipole [4]) and, in general, the magnetic field is more inertial and, thus, less attractive in the ultrafast spintronics. Recently, it was experimentally demonstrated [5, 6] that the spin precession of hot electrons can modulate the conductivity of relatively long silicon channels, when the charge carrier movement is in a nonballistic regime. In the quantum mechanics the main attention usually is focused on spin eigenstates and how to construct a correct eigenfunction in a many electron problem when spin is included [7]. This is a very important step, for example, in quantum chemistry to get the meaningful results. However, in semiconductor spintronics usually one deals with the energy bands that represent single electron states. Furthermore, from the applied solid state physics point of view, similarly as in EPR or NMR dynamical spectroscopies, the superposition spin states appear to be more important than pure eigenstates, since the control of the superposition by external fields brings into play new specific quantum effects, for example Rabi oscillations or possibility to control wave function dephasing [8, 9, 10, 11]. In the standard EPR and NMR spectroscopies the evolution of the spin superposition can be visualized on the Bloch sphere that represents all allowed 12 -spin states. This one-to-one mapping can be achieved because 12 -spin in the two dimensional Hilbert space possesses SU (2) group symmetry which is a covering group of the rotational SO(3) group in the Euclidean space. This one-to-one mapping property allows the Schr¨odinger-Pauli equation to be replaced by a classical precession equation of a spinning top, as a result the complex spin dynamics that evolves in the abstract Hilbert space now can be represented on the Bloch sphere embedded in an easy-to-grasp three-dimensional spin, or magnetization space. As known, in crystalline solids, in the absence of SO interaction the energy bands are doubly degenerate. This property follows from the Kramers theorem [12]. If SO interaction is included, the Kramers pair degeneracy is lifted and spin splitting in energy bands appears [13]. Similarly as in the EPR and NMR spectroscopies, one can make a superposition of spin-split band states, which from the applied spintronics point of view are very important, since the control of their superposition allows to change the direction of magnetization in the spintronics devices. However, in the presence of SO interaction the spin is not a good quantum number and, as a result, the superpositions cannot be represented on the Bloch sphere anymore. On the other hand, the Kramers theorem guarantees that such superpositions can be represented on a closed surface (spin surface) in a three-dimensional (3D) spin space. A concrete shape of the spin surface will depend on symmetry properties of the selected Kramers pair, strength of the SO interaction and electron or hole wave vector. The properties of spin surfaces in bulk semiconductors were considered earlier [14, 15, 16, 17, 18, 19]. Here we shall mainly concentrate on two-dimensional (2D) semiconductors, because they are most suited for the realization of a ballistic spin FET

Bloch Sphere, Spin Surfaces and Spin Precession

143

originally proposed by S. Datta and B. Das [20]. A critical analysis of the main difficulties encountered in the construction of 2D spin FETs was discussed recently in Ref. [19], where it was pointed out that the knowledge of the spin surface of a ballistic carrier moving in the 2D channel of the spin FET allows the experimenter to foresee not only possible trajectories of the magnetization vector but also allows to make some conclusions about matching conditions of the free electron or hole spins at the interface between the 2D channel and spin injector. In Sec. 2. the salient features of the Bloch sphere will be discussed, with the accent to those related to bulk Dresselhaus and Rashba Hamiltonians. In Sec. 3. we shall be interested in spin properties of the inverted band 2D semiconductors represented by HgTe/Hg 1−xCdx Te compounds [21], where the giant SO splitting in electronic 2D structures has recently aroused much interest due to possible application in spintronics [22, 23, 24, 25, 26, 27]. The spin splitting of up to 30 meV has been measured, which is almost an order of magnitude larger than in A 3B5 compounds. It will be shown that due to strong SO interaction the spin surfaces in both electronic and hole 2D channels of HgTe/Hg1−xCdx Te may assume various forms. In Sec. 4. the spin surfaces of n- and p-type GaAs/AlAs QWs will be considered. Apart from planar geometry, QWs of tubular geometry, for example represented by carbon nanotubes, have recently aroused much interest. The spin properties of semiconducting cylinders, or tubes, where the electron wave function is confined to the shell of the cylinder, will be analyzed.

2.

Bloch Sphere

2.1.

A Simple 12 -Spin System

A straightforward way to introduce the Bloch sphere is to resort to the Pauli-Schr¨odinger equation, a nonrelativistic equation that, apart from electron coordinate r and time t, also takes into account the electron spin, i~

∂|Ψi = HP S |Ψi. ∂t

where |Ψi ≡ |Ψ(r, t)i =

|Ψ↑ (r, t)i |Ψ↓ (r, t)i

(1)

.

(2)

The equation (1) describes the evolution of the two component spinor, the components of which have opposite spins indicated by up and down arrows. The full form of the Hamiltonian in (1) is 1 2 (p − eA) I + µB σ·B + V (r)I , (3) HP S = 2m where V (r) is the coordinate-dependent potential, I is (2 × 2) unit matrix, µB is the Bohr magneton, µB = e~/(2m), and σ = (σx , σy , σz ) is the vector of Pauli matrices wherein 0 1 0 −i 1 0 , σy = , σz = . (4) σx = 1 0 i 0 0 −1

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A. Dargys

The spin operator is proportional to the vector Pauli matrix: S=

~ σ. 2

(5)

The term µB σ·B = µB (σx By + σy By + σx Bz ) ≡ HZ

(6)

is called the Zeeman Hamiltonian. In the presence of the magnetic field B, it describes the spin splitting of degenerate energy levels E↑ and E↓, the spins of which are pointing in opposite directions with respect to magnetic field direction, Fig. 1a. The vector potential A, which is related to magnetic induction by B = curlA, describes orbital electron motion in the magnetic field. The energy difference ∆E = E↑ − E↓ is usually proportional to the magnetic field strength and is known as the Zeeman spin splitting energy. È\ bÈ\+aÈ¯\

!!!

HÈ\+äÈ¯\L 2

E Xsz \

E¯ a)

Xsx \

È¯\

Xsy\

b) Figure 1. a) Two-level system represented by up | ↑i and down | ↓i spin states. b) Bloch sphere in the spin space. The north and south poles on the sphere represent | ↑i and | ↓i states. The general spin state is represented by a superposition |ψi = b| ↑i + a| ↓i, where a 2 2 and b are complex coefficients that satisfy √ |a| + |b| = 1. On the equator there is shown the superposition state ψ = (| ↑i + i| ↓i)/ 2 , which is important in the spin FET operation. As known, the spin can be flipped over by time-dependent magnetic field. In case of the electron spin this process is called the electron paramagnetic resonance (EPR). In case of nuclear spin it is called nuclear magnetic resonance (NMR). Historically, the origin of the two-state spin model shown in Fig. 1 comes from dynamic experiments with localized spins that gave birth to a number of new quantum effects such as Rabi oscillations, quantum echoes, level saturation etc [8, 9, 10, 11]. After invention of the laser the Bloch-sphere model was carried over to the so-called two-level atoms to describe the wave function evolution under ultrashort laser pulse excitation [29, 30]. If the magnetic field is homogeneous and does not depend on time, the orbital electron motion and spin flipping described by Eqs. (1)-(3) are not linked together, and the spinor (2) can be factorized into the product of orbital and spin motions: |Ψi = ψ(r, t)

| ↑, ti | ↓, ti

,

(7)

Bloch Sphere, Spin Surfaces and Spin Precession

145

where ψ(r, t) satisfies the standard Schr¨odinger equation, while the two-component spinor obeys the following Pauli evolution equation ∂ | ↑, ti | ↑, ti = HZ . (8) i~ | ↓, ti ∂t | ↓, ti In the absence of SO interaction the latter describes the spin flipping processes when the magnetic field changes with time. When Bkz and, in addition, B is constant, the above two equations are decoupled: ∂ | ↑, ti = µBz | ↑, ti, ∂t ∂ i~ | ↓, ti = −µBz | ↓, ti. ∂t

(9a)

i~

(9b)

Their solutions are | ↑, ti = e−iµBz /~ | ↑i, | ↓, ti = eiµBz /~ | ↓i. In the σz representation (4), the spin eigenstates can be written as column vectors 1 0 | ↑i = , | ↓i = . 0 1

(10)

(11)

From these equations one finds that the average Cartesian spin components are: h↑, t|S|↑, ti = h↑ |S|↑i = (~/2)h↑ |σ|↑i =

~ (0, 0, 1), 2

(12)

and similarly ~ (0, 0, −1). (13) 2 This is a well-known result. In the presence of the magnetic field Bkz the natural quantization axis is parallel to z axis and the measurement outcome will be +~/2 if the spin is in | ↑i state. The positive sign means that the spin points in the positive z direction, along the magnetic field. If the spin is in | ↓i state, the outcome will be −~/2. These two spin states can be visualized in the spin space hSi by two points as shown in Fig. 1b. In general case, the quantum mechanics allows the electron to be in any superposition of spin eigenstates: (14) |ψi = C1 (t)|r, t, ↑i + C2 (t)|r, t, ↓i, h↓, t|S|↓, ti =

where C1 (t) and C2 (t) are the complex functions of time. We shall assume that the electron energies E↑ and E↓ in spin eigenstates are known and the spin variables can be factored out, |r, t, ↑↓i = e−iE↑↓ t/~ | ↑↓i.

(15)

Since the Hilbert space is linear, there exists a linear and unitary transformation which relates different spinors of the Hilbert space. By this reason the coefficients at the moment t can be expressed from the coefficients at the moment t = 0, C1 (t) = α(t)C1 (0) + β(t)C2 (0),

(16a)

C2 (t) = γ(t)C1(0) + δ(t)C2 (0).

(16b)

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A. Dargys

If the vector C(t) with components C1 (t) and C2 (t) is introduced, the system can be rewritten in the matrix form C(t) = A(t)C(0), (17) α(t) β(t) A(t) = , (18) γ(t) δ(t) From the normalization condition hψ|ψi = 1 follows that the coefficients Ci at all moments should satisfy the relation C1∗ (t)C1 (t) + C2∗ (t)C2 (t) = 1. From the latter follows that, in a general case, the vector C(t) can be represented in the form: sin(ϑ(t)/2)eiφ1(t) C1 (t) = . (19) C(t) = C2 (t) cos(ϑ(t)/2)eiφ2(t) where the functions ϑ(t) and φi (t) depend on the concrete form of the Hamiltonian. Then, with the help of (16)-(19) one finds that the evolution matrix is cos(∆ϑ/2)ei∆φ11 sin(∆ϑ/2)ei∆φ12 , (20) A(t) = − sin(∆ϑ/2)ei∆φ21 cos(∆ϑ/2)ei∆φ11 where ∆ϑ = ϑ(t) − ϑ(0) and ∆φij = ∆φi (t) − ∆φj (0). The matrix A describes the evolution in the Hilbert space. It can be checked that A satisfies the unitarity condition, A−1 (t) = A† (t), and Eq. (17). Since the physical information is carried by a phase difference rather than by an absolute phase, the differences will be replaced by ϑ and φ. Then, we can rewrite the spinor and evolution matrix in the forms sin(ϑ/2)e−iφ/2 , (21) C= cos(ϑ/2)eiφ/2 sin(ϑ/2)ei(φ+φ0)/2 cos(ϑ/2)ei(φ−φ0 )/2 , (22) A(t) = − sin(ϑ/2)e−i(φ+φ0)/2 cos(ϑ/2)e−i(φ−φ0)/2 The obtained vector and matrix fully describe the evolution of the spin. The matrix belongs to SU (2) symmetry group. From all this follows that 12 -spin evolution is characterized by time dependence of two parameters, θ(t) and φ(t), |ψ(t)i = cos

ϑ(t) iφ(t)/2 −iE↓ t/~ ϑ(t) −iφ(t)/2 −iE↑ t/~ e e e | ↑i + sin e | ↓i. 2 2

(23)

A concrete dependence of ϑ and φ on time is determined by the Hamiltonian of the problem. Any 2 × 2 matrix can be expressed as a sum of the unit and Pauli matrices multiplied by an appropriate coefficients. For example, the matrix (18) can be rewritten in the form A=

α−δ γ +β γ−β α+δ I+ σz + σx − i σy . 2 2 2 2

(24)

If averages of all matrices that appear in (24) are calculated using (23) one finds hIi = C1∗ C1 + C2∗ C2 = 1, hσx (t)i = hσy (t)i = hσz (t)i =

C1∗ C2 + C2∗ C1 = sin ϑ(t) cos φ(t), i(C2∗ C1 − C1∗ C2 ) = sin ϑ(t) sin φ(t), C1∗ C1 − C2∗ C2 = cos ϑ(t),

(25a) (25b) (25c) (25d)

Bloch Sphere, Spin Surfaces and Spin Precession

147

from which follows that hσ(t)i2 = hσx (t)i2 + hσy (t)i2 + hσz (t)i2 = 1 at all moments, in other words the right-hand sides of (25) describe a unit sphere in the spin space. Thus, every spinor can be represented by a point on the sphere and the evolution of the spinor is the movement of the point on sphere surface. Equation (23) shows that the values ϑ = 0 and ϑ = π correspond to north and south poles in the Fig. 1b, which describe pure | ↑i and | ↓i states. On the poles the parameter φ is undetermined. All the remaining points on the sphere correspond to coherent superposition states. The sphere is called F. Bloch’s sphere. By the way, all possible light polarizations can be represented on the Bloch sphere too. However, in the latter case the sphere bears H. Poincar´e’s name [28]. Since the average spin is expressed through Pauli matrices the end of the average spin vector hSi, which describes electron spin contribution to sample magnetization, is also represented on the sphere. The Bloch sphere is also a useful object in the quantum computing. Since the qubit (or quantum bit) can be represented by 12 -spin, the Bloch sphere at the same time represents all possible qubit states. Thus, it can be used to visualize the evolution of the qubits under the control of quantum gates [31]. As known, the classical bit has only two states. It is known that both the half-spin and the two-level atom belong to the same SU (2) group. Therefore, the Bloch sphere also can be used to represent two-level system dynamics. The only difference between the systems lies in the interpretation of the poles: in the spin case they are interpreted as spin eigenstates, while in the two-level system they are interpreted as the population difference, also known as the inversion [29, 30]. From what has been said follows that there exists one-to-one correspondence between the quantum states in the Hilbert space and the points on the Bloch sphere. The latter representation is more transparent, especially if one has to deal with an ensemble of spins, where dephasing between ensemble elements comes into play. This is illustrated in Fig. 2, where the classical E. L. Hahn experiment on the spin echo is depicted by an ensemble of spin vectors on the Bloch sphere. The situation is more complicated if one deals with higher order spins, for example 32 - or 52 -spins [32, 33], which lie, respectively, in four and six dimensional Hilbert spaces. These states cannot be unambiguously mapped onto a rotational three dimensional space SO(3). However, in case of crystalline solids, as we shall see, the Kramers theorem comes for help.

2.2.

Kramers Pair

Till now the electron was considered as a localized object, similarly as in the EPR and NMR spectroscopies, where the electron movement is restricted by molecular or nuclear potentials. In crystalline solids the electrons are delocalized and, because of the periodicity of the lattice potential, they satisfy the Bloch theorem: ϕk (r) = eik·r uk (r), where uk (r) is lattice periodic function and k is the wave vector. In the presence of the periodic potential the spin can be introduced into wave function with a help of the Kramers theorem, which states that the spin pair (spinor) with the opposite spins, ϕk↑ and ϕ−k↓ , constructed from ϕk (r) should satisfy the relation [12] Kϕk↑ = ϕ−k↓ ,

(26)

where K is the time reversal transformation. The two wave functions in (26) make up the Kramers pair. It can be shown [12] that the energy bands that are connected with the

148

A. Dargys z’

Π 2

M

y’

x’

aL

z’ Τ

B1

x’

z’

z’

bL

M y’

y’

x’

z’

cL z’

2Τ

Π

M

x’

B1

y’ dL

x’

y’ eL

x’

y’ fL

Figure 2. The principle of E. L. Hahn spin echo experiment presented as an ensemble of spin vectors evolving on the Bloch sphere (only the equator of the Bloch sphere is shown). The spins are subjected to a constant magnetic field B0 kz. The primes on the coordinate axes (x0, y 0, z 0) indicate that the coordinate system is rotating around z = z 0 axis with the angular frequency ω equal to spin splitting energy ∆E divided by Planck’s constant. a) Initially all spins are in the same eigenstate | ↑i (along z axis), as a result the magnetization M in sample is maximal. b) Resonant control-field pulse B1 rotates the magnetization M by angle π/2 around x0 . This is equivalent to unitary √ transformation that brings the eigenstate | ↑i to the superposition state (| ↑i + i| ↓i)/ 2 . c) The spins are freely precessing around z axis until moment t = τ . Since in a real experiment the precession frequencies ω of the individual spins are slightly different, for example, due to small inhomogeneity in the magnetic field B0√, the spin states at t = τ will be in different superpositions (e∆ϕ/2 | ↑i + ie−∆ϕ/2 | ↓i)/ 2, where ∆ϕ is the phase differences between individual spin wave functions. This is represented by a fan of spins on the equator of the Bloch sphere. d) The control field rotates all spins by angle π around x0 axis once more, as a result the dephasing angles change to new values. e) Further evolution during interval τ brings all √ spins to identical quantum superposition states (−| ↑i + i| ↓i)/ 2 at the moment t = 2τ . Since now all spins in the ensemble appear in the same quantum state the magnitude of the magnetization will be maximal and equal to the initial magnetization at t = 0 but pointing in a different direction, in this case it is perpendicular to the initial direction. In the experiment, the revival of the magnetization induces the signal (echo) in the receiver coil. f) The dephasing of the individual spins begins once more. Kramers pair are degenerate and satisfy E↑(k) = E↓(−k), E↓ (k) = E↑(−k) .

(27)

Bloch Sphere, Spin Surfaces and Spin Precession

149

Thus, when the spin is taken into account the bands become doubly degenerate in a sense that each energy value occurs twice, in diametrically opposed ( k and −k) points of the Brillouin zone. The double degeneracy related with spin is allowed at the same k only if lattice possesses spatial inversion symmetry. For example, this happens in centrosymmetric silicon, all energy bands of which are doubly degenerate at every point of the Brillouin zone. In semiconductors which do not possess spatial inversion symmetry, for example in A3 B5 or A2 B6 compounds, the presence of a finite spin-orbit interaction will give rise to spin splitting in Kramers pairs as shown in Fig. 3a in case of the conduction band, even in the absence of the magnetic field. In n-type bulk semiconductors the spin splitting is included via Dresselhaus Hamiltonian HD [34], ~2k2 I + γσ·χ ≡ H0 + HD , (28) 2m∗ where m∗ is the effective mass and χ is the vector consisting of the components: χx = kx (ky2 − kz2), χy = ky (kz2 − kx2 ), χz = kz (kx2 − ky2 ). The Hamiltonian (28) yields the following spectrum shown in the Fig.3a, H(k) =

~2k2 ± γχ 2m∗ q ~2 k 2 ± γ k2 (kx2 ky2 + ky2kz2 + kz2 kx2 ) − 9kx2 ky2 kz2 , = 2m∗ where k = |k| and χ = |χ| .

k

-k¯

Energy

Energy

D = E↑↓

k¯

-k

-k¯

a)

k

k¯

-k Wave vector

(29)

Wave vector

b)

Figure 3. Character of the SO splitting related to a) Dresselhaus and b) Rashba Hamiltonians. The solid and dashed lines make up the Kramers pair and correspond to components, ϕk↑ and ϕ−k↓ , of the same spinor. The Hamiltonian (28) can be used to calculate the spin surface of conduction band electrons. For this purpose it is enough to parametrize the Kramers pair in the energy representation, for example in the form (23), find an appropriate unitary transformation matrix U that connects the energy and σz representations (we shall remind that the Hamiltonian (28) was constructed in σz representation) and, finally, calculate the quantum mechanical averages of spin components. If the following parametrized form of the spinor in the energy representation is used (30) |ϕi = (sin ϑ, cos ϑeiφ ) ,

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A. Dargys

then after transformation back to σz representation, |ψi = U |ϕi, the following average spin components hσi i = hψ|σi|ψi are found [35] 1 sin 2ϑ χx cos 2ϑ + (χx χz cos φ + χχy sin φ) , (31) hσxi = 2χ χ⊥ 1 sin 2ϑ χy cos 2ϑ + (χy χz cos φ − χχz sin φ) , (32) hσy i = 2χ χ⊥ 1 (33) (χz cos 2ϑ − χ⊥ sin 2ϑ cos φ) , hσz i = 2χ q where χ⊥ = χ2x + χ2y . By fixing the values of ϑ and φ one selects a concrete direction of the spin in a real physical space. If Eqs.(31)-(33) are plotted parametrically in the spin space one can show that they describe a unit sphere. This means that independent of values of parameters ϑ and φ the spin components satisfy hσxi2 + hσy i2 + hσz i2 = 1.

(34)

From this follows that spin dynamics of bulk conduction band electrons in zinc-blende semiconductors is restricted to Bloch sphere, similarly as in EPR or NMR spectroscopy. However, the analogy is not exact. The spin quantization axis in EPR, as known, is parallel to magnetic field. As we shall see, for spins in the conduction band the natural quantization axis should be referenced with respect to electron wave vector, Fig. 4. If the spin surface is known, the natural quantization axis (QA) is represented by a line that connects the two opposite poles, Fig.1. The coordinates of the vectors the ends of which lie on the poles can be found from Eqs.(31)-(33) assuming that ϑ = 0 and ϑ = π/2, hσiQA = ±

1 (χx , χy , χz ), 2χ

(35)

The obtained pole vectors (35) are independent of φ. From (35) follows that the direction of the natural quantization axis is described by the unit vector 1QA = χ/|χ|. Since k · 1QA = 0, it means that the natural quantization axis is perpendicular to electron propagation direction. This situation is illustrated in Fig. 4, where mutual orientation of the wave vector, natural quantization axis and spin precession trajectory are shown. However, for high symmetry directions, for example when kk[001] or kk[111], the quantization axis remains undefined, since for these crystallographic directions one has 1QA = (0, 0, 0). As can be seen from the dispersion (29), in these high symmetry directions the bands appear to be doubly degenerate. The situation here reminds the electron spin in a free space without magnetic field, where the energies due to spin are doubly degenerate for all directions of the spin and, therefore, the direction of the natural quantization axis is undefined. The evolution of the average spin hSi = ~hσi/2 on the spin surface can be described by the precession equation, which in case of spherical spin surface coincides with the classical equation for a toy top, dhSi/dt = −Ω × hSi, (36) where in the considered case the precession vector is given by ~Ω = 2γχ = 2γ kx (ky2 − kz2 ), ky (kz2 − kx2 ), kz (kx2 − ky2 ) . The module of ~Ω is equal to the conduction band spin splitting energy ~|Ω| = ∆E = E↑D − E↓D and is given by the doubled

Bloch Sphere, Spin Surfaces and Spin Precession

151

XS\

Wt+Φ J k

Quantization axis Figure 4. Mutual orientation of electron wave vector k, natural quantization axis and spin precession trajectory on the Bloch sphere for electron in zinc-blende semiconductors. square root term in the dispersion (29). The quantity Ω = |Ω| is the angular precession frequency, Fig. 4. If the bands are degenerate then ∆E = 0, and the precession, as follows from (36), is absent. Because of the simplicity of spin dynamics on the spherical surface, the most of papers that have appeared till now are devoted to analysis of electron rather than hole spin FETs [1]. In case of 2D holes, as we shall see in Sec. 3. and 4., the spin surface is not spherical. ky

ky

kx

kx

R. eigenspins

a)

D.eigenspins

b)

Figure 5. The electron eigenspin field in the wave vector plane (kx − ky ) for a) Rashba and b) Dresselhaus Hamiltonians in zinc-blende semiconductors. In heterojunctions or asymmetric QWs the structural asymmetry may induce an additional SO-related spin splitting. The corresponding Hamiltonian is called the Rashba Hamiltonian [36]. The structural asymmetry in the QW, for example, may be due to different arrangement of lattice atoms on opposite QW walls, or due to space charge in the

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A. Dargys

QW [2]. The splitting may be also induced by external electric field perpendicular to QW plane or appear even in an individual heterostructure. If n is the normal to the QW plane the Rashba Hamiltonian is HR = α(σ × k)·n ,

(37)

˚ where the constant α for various semiconductors is of the order of α = (0.02 − 0.2) eVA [2]. If the QW lies in x − y plane and n is parallel to z axis, n = (0, 0, 1), then the sum of kinetic H0 = ~2(kx2 + ky2 )/2m∗ and Rashba Hamiltonians reduces to

H(kx, ky ) = H0 + HR =

~2 2 2m∗ (kx

+ ky2) −iα(kx + iky )

iα(kx − iky ) ~2 2 2 2m∗ (kx + ky )

!

.

(38)

The spectrum of this Hamiltonian is R = E↑↓

~2kk2 2m∗

± αkk ,

(39)

q kx2 + ky2 , and up and down spin indices where kk is the in-plane wave vector, kk = correspond to plus and minus signs. The character of the spectrum (39) is shown in Fig. 3b. Now the spin splitting of the Kramers pair takes place along the wave vector axis. As a result two energy minima appear. Typically, the splitting energy ∆E = E↑R − E↓R = 2αkk is of the order ∆E . 0.01 eV. The eigenspinors of the Rashba Hamiltonian, which are related to the poles on the spin surface, now are 1 | ↑i = √ 2

−ikk /(kx + iky ) 1

1 , | ↓i = √ 2

ikk/(kx + iky ) 1

.

(40)

R | ↑↓i. With these spin eigenThey satisfy the eigenvalue equation H(kx, ky )| ↑↓i = E↑↓ function one finds the following average spins:

h↑ |σ| ↑i = (−kx , ky , 0)/kk = (− cos ϕ, sin ϕ, 0),

(41a)

h↓ |σ| ↓i = (kx, −ky , 0)/kk = (cos ϕ, − sin ϕ, 0),

(41b)

where ϕ is the azimuthal angle. The last two equations are visualized on the kx − ky plane in Fig. 5a, from which it is clear that the natural quantization axis is perpendicular to the wave vector. In Fig. 5b we have also plotted the eigenspins of the Dresselhaus Hamiltonian. The latter field also demonstrates why the quantization axis is undefined along [100]-type directions. It can be shown that the parametrized superposition of the Rashba Hamiltonian basis states (40) can be represented on the Bloch sphere too. Of course, in the superposition states that are characterized by different wave vectors the spins of individual electrons will be precessing around their own natural quantization axes which will be parallel to respective eigenspins in the Fig. 5.

Bloch Sphere, Spin Surfaces and Spin Precession

153

Figure 6. The trajectory of the total angular momentum hJi of the valence band hole in the spin space under excitation of the heavy-light-heavy interband transitions by a resonant 2π electric field pulse. The representation of spin dynamics in the spin space may be helpful in case of interband transitions as well, when the charge carrier under external perturbation undergoes the interband excitation, i.e. the transition between different Kramers pairs [37, 38, 39]. Figure 6 shows the evolution of the total angular momentum in the spin space under harmonic excitation F1 cos ωhl t, where the frequency ωhl lies in the Terahertz frequency range and close to the heavy-light resonance transition energy, ~ωhl ≈ Eh − El = 6.99 meV, and F1 ≈1 kV/cm. The hole is coherently excited form the heavy-mass band to light-mass band and then back to heavy-mass band by 2π pulse of length 24.2 ps. As can be seen from Fig. 6, the endpoint of the hJi vector at first spirals out and then returns back to its initial position in a slightly different trajectory due to small deviation of ~ωhl from the exact resonance in the simulation. The time needed for the spin vector to make a full cycle equals one period of the intervalence Rabi oscillations. At intermediate times the hole is in the superposition of heavy- and light-mass states, with difference ∆mJ between the projections mJ of the total angular momentum J equal ∆mJ = 1. Till now it was tacitly assumed that the precession of the average spin hSi occurs in the time-domain. The spin surface may be also addressed to visualize spatial precession of hSi. The latter is important for proper operation of the spin FET [20]. In Fig. 7 the solid lines show the Kramers pair that was split by SO interaction. The superposition state can be constructed from any two states represented by two points on the respective lines. The resulting spinor can be written in a form |ψ(y, t)i ≈ ei(ω0 +∆ω/2)t+i(k0 +∆k/2)y cos ϑ| ↑i + e

(42)

i(ω0 −∆ω/2)t+i(k0 −∆k/2)y iφ

e sin ϑ| ↓i ,

where the coordinate y runs along the channel of the FET. In the superposition (42), it was assumed that the splittings ∆k and ∆ω = ∆E/~ are symmetric with respect to the degeneracy point (k0 , ω0), where ω0 = E0/~. The spinor (42) is normalized. The parameters ϑ and φ allow to vary the contribution of spin-up | ↑i and spin-down | ↓i eigenvectors in the mixed state. We shall assume that the natural quantization axis is parallel to x axis, i.e.

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A. Dargys

E0

DE

Energy

Dk

È\ È¯\ k0 Wave vector

Figure 7. The dashed line shows the dispersion of a doubly spin-degenerate band. The SO interaction splits this band into up and down spin Kramers pair (solid lines) with respect to natural quantization axis. In the superposition that is represented by two points on the spin-split bands the important parameters are ∆E and ∆k, which determine the magnitude of the spin splittings along the energy and wave vector axes at the Fermi energy E0. The splitting ∆E describes time resolved spin precession frequency, similarly as in the EPR or NMR spectroscopies. The splitting ∆k determines the spatial spin precession, or the pitch of a helix trajectory drawn by a tip of the spin along the FET channel. To have a nanometer or shorter channel length in the FET, ∆k should be relatively large. it lies in the x − y plane of the channel (cf. Fig.5). Then, using the spinor (42) and the Pauli matrices Σ transformed so that the spin quantization axis coincides with the x axis, 1 1 −1 † √ Σ = U σU , where U = 2 1 1 , one finds the following average spin as a function spin direction: hS(y, t)i = hψ(y, t)|Σ|ψ(y, t)i = 1 2

(43)

(cos 2ϑ, − sin(∆ωt + ∆ky − φ) sin 2ϑ, − cos(∆ωt + ∆ky − φ) sin 2ϑ) .

At ϑ = π/4 and arbitrary φ the spin lies in the y − z plane. If in the superposition state, in addition, one satisfies ∆ω = ∆E/~ = 0 (see Fig. 7), then from the expression (43) follows that the average spin is time-independent and at the distance y = L makes up a fixed angle with z axis, 1 (44) hS(L, t)i = (0, sin ∆kL, cos ∆kL) . 2 The Eq. (44) describes time-independent spatial helix as a function of the spin injectorcollector spacing L. This is a typical regime of the spin FET [20]. The pitch of the spin helix (precession length) can be controlled by external electric field via ∆k, i.e. via the magnitude the Kramers pair splitting along horizontal, wave vector axis. In the spin FET the initial and final spins should be either parallel (in the conducting) or antiparallel (in nonconducting) states, respectively. If in the superposition one has ∆ω 6= 0, there, in addition, will be rotation of the spin around the quantization axis with the precession frequency ∆ω.

Bloch Sphere, Spin Surfaces and Spin Precession

155

This is undesirable for a normal spin FET operation, since the direction of the magnetization at the drain electrode will not be spatially fixed. At the finite lattice temperature T , when ∆ω < kT , spin thermalization may destroy spin polarization, therefore, the electron collisions with spin scatterers should be as rare as possible to minimize the thermalization effect. In A3 B5 semiconductors ∆E = 1 − 10 meV [2]. When the energy bands are degenerate, as already has been mentioned, the quantization axis will have no privileged direction and, as a result, the location of the eigenstate poles on the spin surface will be undefined. The external magnetic field B also induces the splitting in the Kramers pair and fixes the direction of the quantization axis, in this case parallel to B. However, if both the SO interaction (Rashba or Dresselhaus) and the magnetic field are present the quantization axis will not necessarily be aligned along the magnetic field. To find the direction of the natural spin quantization axis in this case the full quantum mechanical problem should be solved. However, the effect of the magnetic field on the shape of spin surfaces will not be discussed in this chapter.

3.

Spin Surfaces: Inverted Band HgTe/CdTe QW

CdTe and HgTe are semiconductors with zinc-blende lattice structure. In bulk Hg 1−xCdx Te alloys the energy gap is inverted, if x < 0.18. At these compositions the conduction band of Γ6 symmetry is separated from the valence band by a negative gap energy Eg as shown in Fig. 8a and the alloy, in fact, becomes a semiconductor with zero energy gap. The in±1/2 ±1/2 and Γ8 bands in Fig. 8a, in normal semiconductors such as GaAs represent, verted Γ6 respectively, the conduction and valence bands. In HgTe/CdTe quantum wells, Fig. 8b, two distinct heterostructure regimes are possible [22, 23, 24, 25, 26, 27]. If HgTe well is thin enough the first electronic 2D subband E1 is pushed up high enough, above the first heavy-hole subband. The QW in this case is said to switch to normal regime. However, at thicker HgTe wells (& 6 nm) the first 2D heavy-hole subband H1 appears to be energetically above E1. In this case one has the inverted band regime in QWs as well. Here we shall investigate the properties of spin surfaces in relatively wide wells, when the inverted energy subbands appear in the QW and, as we shall see, the spin surfaces become strongly deformed both in the conduction and valence 2D subbands. The spin splitting of the 2D subbands may arise either from bulk asymmetry (absence of the inversion symmetry in constituent layer materials) or from structural asymmetry (asymmetry with respect to the center of the QW). The latter may be induced by the stress fields, space charge in the QW or external electric fields in the direction perpendicular to the QW (nonflat band case), or may be inherent to the QW itself (the Rashba spin splitting), for example, due to different arrangement of lattice atoms in the interfaces. In the latter case the QW structure is not symmetric although the energy bands, as mentioned, may remain flat[2]. Other sources of asymmetry may include asymmetric doping profiles, different alloy compositions of the confining material on either side of the well. Large electric fieldinduced spin splitting effect in narrow-gap HgCdTe was observed for the first time in the oscillatory magnetoconductivity experiments by R. Walrab et al [40]. Later, this was confirmed in various experiments with HgTe/CdTe QW’s [23, 27, 25, 41, 42, 43], where the spin splitting energy as high as 30 meV was observed. Since in narrow-gap compounds the

156

A. Dargys

bulk asymmetry induced splitting was found to be small [26, 42], below we shall limit ourselves to the structural QW asymmetry, either due to Rashba Hamiltonian or due to external electric field. 0.25 0 Energy HeVL

-0.25

G±12 8 G±32 8 G±12 6

E HeVL

Eg

1 Ec , G6

0.75 -0.5

0.5

-1

0.25

HgTe kÈÈ@110D

-0.75 G±12 7

-0.25 -0.5

-1.25

CdTe Eg

5

CdTe HgTe Eg 10

15

20

25

30

z HnmL

E v, G8

b)

-0.6-0.4-0.2 0 0.2 0.4 0.6 Wave vector Hnm-1 L

a) Figure 8. a) The inverted band structure of bulk HgTe, where band symmetry nomenclature is indicated too. Due to double degeneracy of the bands, they are described by (8 × 8) Hamiltonian. b) Conduction (dashed line) and valence (solid line) band edge profiles in CdTe/HgTe/CdTe quantum well. In CdTe the gap Eg is positive, while in HgTe it becomes negative, as shown by up and down arrows, respectively. The inverted 2D subbands appear when the width of the QW is larger than approximately 6 nm.

3.1.

Basis Functions and Hamiltonians

At k = 0 the fundamental energy bands have symmetries Γ6 , Γ7 , Γ8 in the double point group notation. This notation will be retained at nonzero wave vectors as well. The 2D subband structure calculation method used below is based on the k · p method and the envelope function approach introduced by M. G. Burt [44] and applied by B. A. Foreman [45] to take boundary conditions in zinc-blende-type semiconductors correctly. The corresponding basis functions in the total angular momentum representation |J, mJ i are (see Fig. 8b for band symmetry nomenclature) √ |Γ6 , +1/2i = (1/ 3)|S ↑i, √ |Γ6 , −1/2i = (1/ 3)|S ↓i, √ |Γ8 , +3/2i = (1/ 2)|(X + iY ) ↑i, √ |Γ8 , +1/2i = (1/ 6) (|(X + iY ) ↓i − 2|Z ↑i) , (45) √ |Γ8 , −1/2i = −(1/ 6)|(X − iY ) ↑i + 2|Z ↓i, √ |Γ8 , −3/2i = −(1/ 2)|(X − iY ) ↓i, √ |Γ7 , +1/2i = (1/ 3) (|(X + iY ) ↓i + |Z ↑i) , √ |Γ7 , −1/2i = (1/ 3) (|(X − iY ) ↑i − |Z ↓i) ,

Bloch Sphere, Spin Surfaces and Spin Precession

157

where Γi , mJ in the kets indicate, respectively, the representation and projection of the angular total momentum J. The orbital functions |Xi, |Y i and |Zi transform as Cartesian coordinates x, y and z, and |Si is the totally symmetric function. The concrete expressions for the orbital functions are not required. The up and down arrows indicate two spin eigenstates with respect to z axis, which is assumed to be parallel to [001]-type crystallographic axis and perpendicular to the QW plane. The basis (45) is orthonormalized. The Hamiltonian of the well and barrier materials with [001] growth direction parallel to z axis takes the following form in the above basis [45, 41], H0 =  T 0   T  0  −1  √ P k− 0 q2  −1  2 P kz √ P k−  3 q6  2  √1 P k+  6 3 P kz  1 √ P k+  0 2   −1 −1  √ P kz √ P k− 3  3 −1 √ P k+ √13 P kz 3

q

0

−1 √ P k+ 6

√1 P k− q6 2 3 P kz

U +V

−S −

R

0

† −S −

U −V

C

R

R†

C†

U −V

S+

0

R† √ 2V q − 3 S˜†

S+ q † − 32 S˜+ √ − 2V

U +V √ 2R

−1 √ P k+ 2

† √1 S 2 −

√ − 2R†

2 3 P kz

2 −

0 √1 P k− 2

†

† √1 S 2 +

−1 √ P kz 3 −1 √ P k+ 3 1 √ S− 2

√ 2V q − 32 S˜+ √ † 2R U −∆ C†

−1 √ P k− 3 √1 P kz 3√



    − 2R   q  3˜  − 2 S−  , √ − 2V    √1 S +  2    C  U −∆ (46)

where three diagonal blocks correspond to conduction, valence and SO split-off bands, respectively. The coordinate z, as mentioned, is perpendicular to the QW plane. The in-plane wave vector is characterized by k± = kx ± iky and kk2 = kx2 + ky2 , while the perpendicular to the QW wave vector kz should be replaced by kz = −(i/~)∂/∂z. The other symbols are ~2 [(2F + 1)kk2 + kz (2F + 1)kz ], 2m0 ~2 (γ1kk2 + kz γ1 kz ), U = Ev (z) − 2m0 ~2 (γ2kk2 − 2kz γ2 kz ), V =− 2m0 ~2 √ 2 2 3 µk+ − γk− , R=− 2m0 ~2 √ S± = − 3k± ({γ3, kz } + [κ, kz ]), 2m0 ~2 √ 1 3k± ({γ3, kz } − [κ, kz ]), S˜± = − 2m0 3 ~2 k− [κ, kz ], C= m0 µ = (γ3 − γ2)/2 , T = Ec (z) +

γ = (γ3 + γ2)/2 .

(47) (48) (49) (50) (51) (52) (53) (54) (55)

158

A. Dargys

Here [A, B] = AB − BA is the commutator, {A, B} = AB + BA is the anticommutator. P is the Kane momentum matrix element. The conduction Ec and valence Ev band edges (see Fig. 8b), the spin-orbit splitting energy ∆, and the valence band parameters γ1, γ2 , γ3, κ and F are functions of the coordinate z. It is assumed that the band structure parameters change abruptly at the interface. The matrix elements of the Hamiltonian (46), which describe the Γ7 and Γ8 states and their coupling are similar to those described by Foreman [45]. The parameter F is related to the conduction-band-edge mass, mc /m0 = 1/(2F + 1). The parameter κ can be expressed through the valence band parameters, κ = (−1 − γ1 + 2γ2 + 3γ3)/3. Since in the bulk the commutator [κ, kz ] is equal zero, √ ~2 3γ3k± kz . Thus, C 6= 0 at the it follows that in the bulk C = 0 and S ± = S˜± = − m 0 QW interfaces only. The parameter µ = (γ3 − γ2)/2 describes the valence band warping. The Hamiltonian (46) gives doubly spin-degenerate bands (Kramers pairs). For bulk CdTe parameters the Hamiltonian gives heavy- and light-mass bands at the symmetry point Γ8 . For HgTe parameters the same Hamiltonian gives the heavy-mass and inverted light-mass bands (the latter in fact becomes an electron-like band, Fig. 8a) in the following denoted as ±3/2 ±1/2 ±1/2 and Γ8 , respectively. The band Γ6 , which in A3 B5 compounds plays the role Γ8 of the conduction band, is inverted in HgTe and shifted down in energy by 0.303 eV. The spin split-off valence band is below by 1.08 eV with respect to Γ8 point.

Table 1. Band structure parameters of HgTe/Hg1−x Cdx Te quantum wells. Eg is the energy gap, ∆ is the spin-orbit splitting energy, Λ is the valence band offset, Ep = 2m0 P 2 /~2 is the energy related to the Kane momentum matrix element P , F is related to the conduction band effective mass, γi’s are valence band parameters, ri ’s are the Rashba coefficients. Band parameter Eg (eV) ∆ (eV) Λ (eV) Ep (eV) mc /m0 F γ1 γ2 γ3 rc (eV nm) rv (eV nm) rs (eV nm) rvs (eV nm)

HgTe −0.303 1.08 0 18.8 0.02 24.5 4.1 0.5 1.3

Hg0.3Cd0.7Te 1.006 0.961 0.404 18.8 0.045 10.6 2.26 −0.046 0.411 0.1 0.1 0.1 0.1

CdTe 1.606 0.91 0.577 18.8 0.096 4.7 1.47 −0.28 0.03

The band structure parameters that were used in the present calculations are listed in Table 1. The parameters of Hg0.3Cd0.7Te alloy were obtained by linear interpolation, except for the energy bandgap. In Hg 1−x Cdx Te the dependence of the gap Eg on the lattice

Bloch Sphere, Spin Surfaces and Spin Precession

159

temperature T and composition x was approximated by the empirical formula [46] Eg (eV) = − 0.303(1 − x) + 1.606x − 0.132x(1 − x)+ [6.3(1 − x) − 3.25x − 5.92x(1 − x)]10−4T 2 . 11(1 − x) + 78.7x + T

(56)

In the following, the energy will be referenced with respect to zero-gap in the bulk, Fig. 8b. From this figure one reads ( Eg (CdTe) − Λ > 0, in the barrier;, (57) Ec (z) = in the well, Eg (HgTe) < 0,

Ev (z) =

(

−Λ < 0, in the barrier, 0, in the well,

(58)

where Λ is the valence band offset at the barrier-well interface. In the numerical calculations the well width, d = 12 nm, was used. The needed Rashba Hamiltonian for the eight-band zinc-blende semiconductor was considered by the invariant method in R. Winkler’s book [47], where it was shown that the most general form of the Rashba Hamiltonian for the zinc-blende lattice may have as many as ten free parameters. Estimation of the magnitude of these parameters for various semiconductors shows that only five-six of them are important. Below, in writing the Rashba 6c6c , r 8v8v , r 7v7v and r 8v7v in the noHamiltonian only the leading ones were included ( r41 41 41 51 tation of Ref. [47]). Here the reduced Hamiltonian, when the Rashba field is parallel to z 11 11 33 31 31 33 11 11 axis, will be used. In the basis |JmJ i = | 2 2 i, | 2 2 i, | 2 2 i, | 2 2 i, | 2 2 i, | 2 2 i, | 2 2 i, | 2 2 i the Rashba Hamiltonian assumes the form HR =   0 0 0 0 0 0 0 irck−  −irc k+ 0 0 0 0 0 0 0   √   0 0 −i 23 rv k− 0 0 − √i6 rvs k− 0   0   √  0 i 3 i √ rvs k−  r k 0 −ir k 0 0 − 0 v + v −   2 3 2 √ ,  3 i   0 0 0 irv k+ 0 −i 2 rv k− − 3√2 rvs k+ 0   √   i 3 i √  0 0 0 0 0 − 6 rvs k+  0 2 rv k+    i  0 √ 0 √i6 rvs k+ 0 r k 0 0 ir k vs − s −   3 2 i i √ √ 0 0 0 r k 0 r k −irs k+ 0 3 2 vs + 6 vs − (59) where rc , rv and rs are the Rashba coefficients for conduction, valence and SO split-off bands, and rvs includes the coupling between the bands. In the Hamiltonian (59) the leading terms are linear in the wave vector k± = kx ± iky . The calculations were also repeated when the spin splitting of the bands is induced by an externally applied voltage Vb , the electric field of which is perpendicular to the QW

160

A. Dargys

plane. The respective potential was assumed to change linearly with the coordinate z in the diagonal Hamiltonian of the following form HR = eVb (z/L)I,

(60)

where L is the total transverse length of the structure and I is (8 × 8) unit matrix. The full Hamiltonian of the problem is equal to the sum of both Hamiltonians: H = H0 + HR .

(61)

Cadmium and mercury chalcogenides are compounds that do not possess the inversion symmetry. The inversion asymmetry in well and barrier materials may bring an additional contribution to the spin splitting due to cubic in the wave vector terms (the Dresselhaus contribution [34]; see also Eq. (28)). The available experimental data indicate that the spin splitting in the bulk mercury telluride should be very small [26, 42]. Thus, the spin precession due to Dresselhaus contribution, if any, should be negligible too. On the other hand, the spin splitting in n-type HgTe single quantum wells due to Rashba mechanism in combination with the inverted band structure was found to be very large [40, 22, 43, 42]. By this reason we have neglected the Dresselhaus contribution altogether and included only the Rashba Hamiltonian either in form (59) or in form (60) in the spin splitting of the Kramers pairs.

3.2.

Boundary Conditions and Method of Solution

At the well-barrier interface the wave functions should remain continuous. Also the derivatives with respect to coordinate z should be continuous. As shown by Burt [44] and Foreman [45] the correct rather than ad hoc symmetrization of the operators in the Hamiltonian provides an unambiguous determination of the boundary conditions at the interface. The related derivative matrix was presented in [41]. It should be noted that the required boundary conditions are automatically satisfied through the (correct) operator ordering in the Hamiltonian (46), since this Hamiltonian apart from the symmetrized terms contains additional off-diagonal elements [κ(z), kz ], which are equal zero in the bulk. Therefore, in the numerical calculations the approximation of derivatives by finite differences and subsequent expansion of the Hamiltonian on a large mesh automatically takes into account the correct boundary conditions. A more thorough discussion of this point can be found in [48]. The method of finite differences was used to discretize the wave function and its derivatives. After the discretization the resulting large Hamiltonian had a block-diagonal structure. The dimensions of individual blocks were same as those of the original (8 × 8) Hamiltonian. In the expanded Hamiltonian the diagonal blocks represented zn mesh points and the adjacent (8 × 8) upper and (8 × 8) lower blocks represented zn+1 and zn−1 mesh points, respectively. The wave functions of the first and last points on the mesh have been equated to zero what is equivalent to infinite walls at the extreme ends of the sample. At a given wave vector the spectrum of such a Hamiltonian is discrete. Only those energy levels and wave functions that lie in the quantum well are physically meaningful and consistent with a finite length of QW barriers, where wave functions of the energy levels are decaying exponentially. The discretized and expanded in this way Hamiltonian gives correct eigenvalues and eigenfunctions if the discretization step is small enough and the magnitudes of

Bloch Sphere, Spin Surfaces and Spin Precession

161

the exponentially decaying wave functions near the extreme ends of the sample are small enough.

a)

b)

Figure 9. a) Dispersion of the lowest electronic, E2 and H1, and hole, H2 and H3, subbands and b) dependence of the spin splitting energy on the wave vector in HgTe/Hg 0.3Cd0.7Te QW, when the Rashba Hamiltonian (59) is used. kk[10]. Singular value decomposition (SVD) method was used to find the eigenvalues and corresponding eigenfunctions of the total Hamiltonian (61). For the solutions to be meaningful, the SVD algorithm requires all eigenvalues to be positive [49]. Therefore, a constant energy was added to diagonal elements of the matrix (61) to shift all spectrum to positive values. This has no influence on the eigenfunctions and spin properties.

3.3.

Spectrum and Probability Distribution

Figures 9 and 10 show the spectrum and spin splitting of the two lowest conduction subbands, E2 and H1, and two highest valence subbands, H2, H3 and H4. In these and in subsequent figures the QW width was assumed to be d = 12 nm and the total length of the structure L = 36 nm. The required band parameters are given in the Table 1. In figures the energy is referenced with respect to the degeneracy point in the bulk, where the heavy-mass and inverted light-mass bands meet at k = 0 (see Fig. 8a). In Fig. 9 the spin splitting comes from the Rashba Hamiltonian (59), while in Fig. 10 from an external bias Vb [Eq. (60)]. It should be stressed that in the former case the bands remain flat and the spin splitting is induced by structural asymmetry of the QW that is taken into account in the Rashba Hamiltonian (59). In both figures the energy of the first heavy-hole subband H1 is inverted (electron like) and lies above the first electronic subband E1 (not shown in the figure). The E1 subband lies below H3 and H4 subbands and is intermixed with lower lying valence subbands. Although H1 subband is electron-like and at a first glance is similar to

162

A. Dargys

a)

b)

Figure 10. a) Dispersion of the lowest electronic, E2 and H1, and hole, H2-H4, subbands and b) dependence of spin splitting energy on the wave vector in HgTe/Hg 0.3Cd0.7Te QW, when the bias voltage [see the Hamiltonian (60)] of the amplitude Vb = 0.25 V was applied over the structure of length 36 nm. kk[10]. other conduction subbands, however, as we shall see from the spin surfaces its valence band origin is fully reflected in its spin properties. Figures 9b and 10b show that both Hamiltonians, Eqs. (59) and (60), give relatively large splitting in Kramers pairs. The dependence of the splitting on wave vector is different, although the overall shapes of the spectra are similar. According to experiments [24] the Rashba coefficient in mercury tellurides is r = (0.05 − 0.2) eV nm. The values rc = rv = rs = rvs ≡ r = 0.1 eV nm were used in the present calculations. In Fig. 10 the average electric field in the structure is about 70 kV/cm. No space charge effects were included and the external electric field was assumed to be the same everywhere. However, in both cases the magnitude of the spin splitting is relatively large as compared to that in other semiconductors. This conclusion remains true even if space charge effects and electric field redistribution in the nanostructure are taken into account by Poisson equation [43, 42]. Figure 11 shows the probability density distribution of the electron and hole charge in two main conduction (E2, H1) and valence (H2, H3) subbands of the QW, when the splitting is induced by Rashba SO interaction, Eq. (59). In the panels, the pairs of probability distributions are related to spin-split subbands. As should be, it was observed that the number of maxima and minima grows with the subband index, however, the minima do not reach zero as is the case with simple quantum systems. The spin splitting magnitude has only negligible effect on the probability distributions shown in the Fig. 11 for SO coupling values used here. In the absence of the SO splitting all bands become doubly degenerate and the direction of quantization axis for a ballistic carrier spin becomes undefined. Then, the individual probability distributions in the up and down spin subbands lose their meaning.

Bloch Sphere, Spin Surfaces and Spin Precession

163

Figure 11. Probability distribution of charge density in the subbands E2, H1, H2, and H3 at the wave vector kk[11] and |k|=0.15 nm−1 . The solid and dashed lines correspond to up and down spin-split energy subbands. The vertical lines indicate the interfaces between the well and barriers. r = 0.1 eVnm.

3.4.

Spin and Orbital Matrices

The spherical symmetry of the spin surface is retained in case of an isolated spin. In the elementary, A3B5 , A2B6 and a large number of other compounds the SO interaction brings about rearrangement in the band structure, particularly in the valence band. As a result, the spin surfaces of the valence bands may be strongly deformed. If in addition the band inversion takes place, as we shall see, the spin surfaces of the conduction band may be deformed too. To calculate spin surfaces of Kramers pairs one must know spin operators (matrices) written in the same representation as the considered Hamiltonian (61). The method of construction of the required spin matrices was described in Ref. [50]. In the basis 11 11 33 31 31 33 11 11 |JmJ i = | 2 2 i, | 2 2 i, | 2 2 i, | 2 2 i, | 2 2 i, | 2 2 i, | 2 2 i, | 2 2 i one finds the following Carte-

164

A. Dargys

sian components of the vector of spin matrices, S = (Sx, Sy , Sz ), 

      Sx =       



       Sy =       

1 2

0 1 2

0 0 0 0 0 0 0

0 0 0 0 0 0

0 − 2i i 0 2 0 0 0 0 0 0 0 0 0 0 0 0



      Sz =       

1 2

0 0 0 0 0 0 0

0 0 0

0 0 1 √ 2 3

1 √ 2 3

0 0 √1 6

1 3

1 3

0

0 0

1 √ 2 3 1 − 3√ 2

1 √ 3 2

0 0 0

0 0

2 3

0 0

− 2√i 3 0 i 3

0 0

√i 6

i √ 3 2

0

0 − 12 0 0 0 0 0 0

0

0

i √

0 0 1 2

0 0 0

0 0 0

1 0 6 0 0 0 0√ 0 − 32 0 0

0

0 0 0 − 3i 0 i √ 2 3 i √ 3 2

0 0 0 0

0 0 √1 6

1 √ 3 2

0

1 √ 2 3

0 0 − √16

0 0 0 0 − 2√i 3 0 0

0

√i

0 0 0 0 − 16 0 0√ − 32

0 0 0 0 0 − 12 0 0

6

1 − 3√ 2 0 0 − 16

0 − √16 − 16 0

0 0 − √i6 0 − 3√i 2 0 0 − 6i

0 0 0√ − 32 0 0 − 16 0



0 0 0

      ,      

0 0 0 − 3√i 2 0 − √i6 i 6

0

0 0 0 0√ − 32 0 0 1 6

(62)



       ,      

(63)



      .      

(64)

These matrices satisfy standard commutation relations Sx Sy −Sy Sx = iSz , etc. The square of the vector S gives the diagonal matrix S2 = (3/4)I, where I is (8 × 8) unit matrix. Similarly, one can introduce the orbital surfaces that are related with the electron orbital motion and which are related to momentum operator L. As in the spin matrix case the introduction of such an object is justified by the fact that in semiconductors the energy bands are doubly degenerate, or nearly degenerate, and the distance to other energy bands is relatively large as compared to the spin splitting energy. The respective components of the vector orbital matrix, L = (Lx , Ly , Lz ), can be calculated in a similar manner as for the

Bloch Sphere, Spin Surfaces and Spin Precession

165

spin matrix. In the above mentioned basis and ordering they are 

      Lx =        

       Ly =       

0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 √ 0 3 0 0 0 0 0 − √16 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 √i3 0 0 0 0 0 − √i6 0 0 

      Lz =       

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 √1 3

0

2 3

2 3

0

0 0

√1 3 1 √ 3 2

1 − 3√ 2

0 0 − √i

3

0 2i 3

0 0 − 3√i 2 0 0 1 0 0 0 0 0

0 0 0

0 0 0 1 3

0 0 √ 2 3

0

0 0 0 0 − 2i 3 0 √i 3 i √

−3 0 0 0 0 0 − 13 0 0 √ 2 3

2



0 0 0 0

0 0 − √16 0

√1 3

1 √ 3 2

0 0

0 0

√1 6 2 3

√1 6

2 3

0

0 0 0 0 − √i3 0 0 − √i6

0 0 0 0 0 −1 0 0

0 0 0 √ 2 3

0 0 2 3

0

0 0 0 1 − 3√ 2 0

0 0 √i 6

0 i √ 3 2

0 0 0 i √ 3 2

0 √i

0 0

6 −2i 3

2i 3

0

0 0 0 0 √

      ,      

(65)



       ,      

(66)



       2 . 3  0    0  − 23

(67)

As should be, these matrices also satisfy the standard commutation relations: Lx Ly − Ly Lx = iLz , etc. The appearance of zeros in the upper (4 × 4) diagonal blocks is in accord with the fact that the conduction band is made up of s-type orbitals. The total angular momentum, J = L + S, is equal to the sum of the above Li and Si matrices. Since the Hamiltonian (61) is in the Jz representation of the total angular momentum J, the matrix Jz = Lz + Sz , as can be checked by adding the matrices (67) and (64), is diagonal with the elements on the diagonals that represent, respectively, the conduction J = 1/2, heavy-light J = 3/2, and split-off J = 1/2 multiplets.

3.5.

Parametrization of 2D Spinors

As shown in Sec. 2. it is convenient to go over to the energy representation where the ith band diagonal elements Ei↑(k) and Ei↓(k) are Kramers pairs. To construct the spin surface from the ith Kramers pair we shall write the relevant superposition in the energy

166

A. Dargys

representation in the following way J J )i + sin ϑeiφ |ϕ (Γm |ϕ(Γi)i = cos ϑ|ϕ(Γ−m i i )i,

(68)

where mJ = 1/2 or 3/2, and the parameters ϑ and ϕ define the mixing strength and phase difference in the superposition. The diagonalization of the total Hamiltonian can be performed with an appropriate (8 × 8) unitary matrix U : h i (69) U †HU ≡ HE = diag 2EΓ±1/2 , 2EΓ±3/2 , 2EΓ±1/2 , 2EΓ±1/2 , 8

8

6

7

where H is the total Hamiltonian (61) and “ diag” indicates the diagonal matrix with Kramers pairs originating from the same point group. In the energy representation (69), in the order of decreasing eigenenergies, the parametrized spinors have the following form ±1/2

|ϕ(Γ8

) = (cos ϑ, sin ϑeiφ , 0, 0, 0, 0, 0, 0),

±3/2 |ϕ(Γ8 ) ±1/2 |ϕ(Γ6 ) ±1/2 |ϕ(Γ7 )

(70a)

= (0, 0, cos ϑ, sin ϑeiφ , 0, 0, 0, 0),

(70b)

(0, 0, 0, 0, cos ϑ, sin ϑeiφ , 0, 0),

(70c)

= (0, 0, 0, 0, 0, 0, cosϑ, sin ϑeiφ ).

(70d)

=

The values of parameters ϑ and φ have no influence on the eigenenergies. The selection of the concrete values of ϑ and φ fixes the mixing ratio and phase, and at the same time fixes the magnitude and direction of the average spin of the carrier that has the wave vector k and is in a selected Kramers pair. The required unitary matrix U can be constructed from the eigenspinors of the considered Hamiltonian as discussed in the Appendix in book [51]. If U is known, the spinor in the initial Jz representation then will be J J ; ϑ, φ)i = U |ϕ(Γ±m ; ϑ, φ)i. |ψ(Γ±m i i

(71)

In the following we shall be interested in the quantum mechanical average spin J J J ; ϑ, φ)i = hψΓ±m ; (ϑ, φ)|S|ψΓ±m ; (ϑ, φ)i, hS(Γ±m i i i

(72)

which represents the spin surface of the Kramers doublet, where S is given by matrices (62)–(64). As mentioned, the parametrized spinor |ψi represents all possible superpositions of up and down spin eigenstates in the selected Kramers pair. The same parametrized J ; ϑ, φ)i. spinors (70) was used to calculate the orbital surfaces hL(Γ±m i

3.6.

Spin and Orbital Surfaces

Figures 12 and 13 show the spin surfaces, when k||[10] and k||[11], for three energy subbands and their evolvement when the magnitude of the in-plane wave vector k is increased. In the figures, the parallels and meridians that visualize the spin surfaces were drawn, respectively, either at ϑ = const or at φ = const. An imaginary line (in the figures it is not shown) that pierces the surfaces and connects the two singular points (poles) represents the true, or natural quantization axes of the problem. It is seen that in all cases the quantization axis lies in the QW plane and is perpendicular to k. The eigenenergies that correspond to

Bloch Sphere, Spin Surfaces and Spin Precession -0.5

XSy \ 0.5 0

-0.5

0.5

XSy \ 0.5 H1 0

-0.5

0.5 H1 XSz \ 0

-0.5

-0.5

0 XSx \

-0.5 -0.5

0.5

XSy \ 0.5 0

-0.5

0.5

0 XSx \

-0.5 0.5

XSy \ 0.5 0

-0.5

0.5

-0.5

0 XSx \

0.5

-0.5

0.5

0 XSx \

-0.5 0.5

XSy \ 0.5 0

-0.5

0.5

0 XSx \

H2

-0.5 -0.5

0.5

XSy \ 0.5 0

XSz \ 0

-0.5 -0.5

0.5

H1 XSz \ 0

-0.5

0 XSx \

0.5

E2 XSz \ 0

H2

-0.5 -0.5

XSy \ 0.5 0

XSy \ 0.5 0

XSz \ 0

-0.5 -0.5

0.5

H1 XSz \ 0

-0.5

0 XSx \

0.5

E2 XSz \ 0

H2 XSz \ 0

-0.5 -0.5

XSy \ 0.5 H2 0

0.5

E2 XSz \ 0

167

0 XSx \

-0.5 0.5

0 XSx \

0.5

Figure 12. Spin surfaces and their horizontal projections for E2, H1 and H2 subbands (columns). The QW lies in x − y plane. The direction of the wave vector kk[10] is shown by lines. |k| = 0.06, 0.15 and 0.3 nm−1 , respectively, for upper, middle and bottom rows. the two poles were plotted in Figs. 9a and 10b as a function of the wave vector magnitude. If the spin splitting is neglected ( HR = 0) the subbands become degenerate and, as a result, the direction of the quantization axis becomes undefined. In the numerical calculations its direction, nonetheless, will be determined by the used algorithm. In the present calculations the quantization axes (not the orientation of spin surfaces in the spin space) has automatically aligned with hSz i when the Rashba Hamiltonian HR was switched off. It should be stressed once more that the spin surface is a universal characteristic: neither its shape nor its orientation in the spin space depend on a concrete value of the spin splitting energy ∆E or wave vector difference ∆k as long as ∆E E0 and ∆k k0. Thus, any two points that lie on the Kramers pair in the vicinity of the degeneracy point E0(k0) in Fig. 7, if parametrized, will represent the same spin surface and, thus, all allowed directions and magnitudes of the injected spins into the spin FET channel. In the experiment, a concrete direction on the spin surface and the magnitude of hSi will be automatically determined by used spin injector. The first columns in Figs. 12 and 13 show that in electronic E2 subband the shape of the spin surface, independent of electron energy, is very close to the Bloch sphere. Therefore, in the E2 subband the electron spin dynamics for all k’s to a high accuracy can be described by the precession equation dhSi/dt = −Ω × hSi, (73) where Ω is the precession vector. The module of Ω is equal to spin splitting energy ∆E

168

A. Dargys

-0.5

XSy \ 0.5 0

-0.5

0.5

XSy \ 0.5 H1 0

-0.5

0.5

0.5

E2 XSz \ 0

H1 XSz \ 0

-0.5

-0.5

0 XSx \

-0.5 -0.5

0.5

XSy \ 0.5 0

-0.5

0.5

0 XSx \

-0.5 0.5

XSy \ 0.5 0

-0.5

0.5

-0.5

0 XSx \

0.5

-0.5

0.5

0 XSx \

-0.5 0.5

XSy \ 0.5 0

-0.5

0.5

0 XSx \

H2

-0.5 -0.5

0.5

XSy \ 0.5 0

XSz \ 0

-0.5 -0.5

0.5

H1 XSz \ 0

-0.5

0 XSx \

0.5

E2 XSz \ 0

H2

-0.5 -0.5

XSy \ 0.5 0

XSy \ 0.5 0

XSz \ 0

-0.5 -0.5

0.5

H1 XSz \ 0

-0.5

0 XSx \

0.5

E2 XSz \ 0

H2 XSz \ 0

-0.5 -0.5

XSy \ 0.5 H2 0

0 XSx \

-0.5 0.5

0 XSx \

0.5

Figure 13. The same as in Fig. 12 but for kk[11].

(see Figs. 9b and 10b) divided by Planck’s constant. The vector Ω lies in the QW plane and is perpendicular to k. The spherical symmetry of E2 spin surface comes from s-like ±1/2 character of the orbitals, in Fig. 8b represented by Γ6 . In the next section we will show that in normal gap A 3 B5 QWs, where the contribution of p orbitals is small, the sphericity of electronic spin surfaces is satisfied to high accuracy. From what has been said it may be concluded that the free electron spin behavior in E2 subband (for example, the precession represented by circles around the quantization axis, or spin echoes of the ensemble of spins) will be exactly the same as in EPR or NMR spectroscopies, where as discussed in the previous section the dynamics of an ensemble of spins can be completely described by average spin trajectories on the Bloch sphere. The middle columns in Figs. 12 and 13 show that in the electronic H1 subband there is a large admixture of p-type atomic orbitals, because the shape of the spin surface strongly depends on |k|. This property was found earlier for holes in bulk A 3 B5 semiconductors, wherein the valence band is made up of p-atomic orbitals [52, 53]. At small wave vectors the spin surface is needle-like which shrinks to line at k = 0. However, it blows up to the Bloch sphere if the free carrier energy becomes large enough. This indicates that H1 indeed originates from the bulk valence-band spinors characterized by strong SO interaction. At small energies the electrons injected into H1 subband will have almost linear polarization, i.e. the spin precession will occur on a line perpendicular to the QW plane. From this follows that to have an efficient spin injection into this subband the injector should have ver-

Bloch Sphere, Spin Surfaces and Spin Precession XLy \ 0.5 -0.50

XLy \ 0.5 -0.50 E2

0.5

XLy \ 0.5 -0.50

H1

H1

0.5 XLz \ 0

XLz \ 0

-0.5

-0.5

-0.5

-0.5 0 XLx \ 0.5

-0.5 0 XLx \ 0.5

XLy \ 0.5 -0.50

XLy \ 0.5 -0.50 H1

0.5 XLz \ 0

XLz \ 0

-0.5

-0.5

-0.5

-0.5 0 XLx \ 0.5

XLy \ 0.5 -0.50

-0.5 0 XLx \ 0.5

XLy \ 0.5 -0.50 E2

0.5

XLy \ 0.5 -0.50 H1

0.5 XLz \ 0

XLz \ 0

-0.5

-0.5

-0.5

-0.5 0 XLx \ 0.5

H2

0.5

XLz \ 0

-0.5 0 XLx \ 0.5

H2

0.5

XLz \ 0

-0.5 0 XLx \ 0.5

H2

-0.5 0 XLx \ 0.5

XLy \ 0.5 -0.50 E2

0.5

H2

0.5

XLz \ 0

169

-0.5 0 XLx \ 0.5

Figure 14. Orbital surfaces and their horizontal projections for E2, H1 and H2 subbands (columns). The QW lies in x − y plane. The direction of the wave vector kk[11] is shown by lines. The magnitude of k is, respectively, for the upper, middle and bottom rows: |k| = 0.06, 0.15 and 0.3 nm−1 .

tical rather than horizontal polarization with respect to the QW plane. Furthermore, in this case the average spin precession trajectories of the injected electrons cannot be described by Eq. (73). As follows from spin surface shape for H1 subband, now the trajectories will be ellipsoids. At high energies, when the spin surface is spherical (the lowest panels in middle columns), the precession will resume its normal character. The precession equations (analogues of the Eq. (73)) for ellipsoidal spin surfaces for particular directions of k in bulk A 3 B5 compounds were considered in Refs. [52, 53]. The strong deviation of spin surface from spherical shape explains why in the experiments it is difficult to observe free hole EPR. In the H2 subband (the fundamental hole subband), as the third column shows, the spin surface is needle-shaped at low hole energies and disc-shaped at high energies. In the latter case all allowed spin polarizations will lie on a flat surface. However, the surface does not possess rotational symmetry with respect to |k|. From Figs. 12 and 13 it can be concluded that the orientation of the spin surface in the spin space and the orientation of wave vector in the Brillouin zone are linked together. The change of the spin surface form is mainly determined by the subband type and length of k.

170

A. Dargys

Figure 14 shows the orbital surfaces calculated under similar circumstances. For an ideal s-type atomic orbital one should have L = 0 and, consequently, the orbital surface should shrink to a point. However, as it is seen from the first column in Fig. 14, due to very small admixture of p-orbitals, the surface has a finite volume, although of a small magnitude. The second and third columns show that the shapes of orbital surfaces for other Kramers pairs strongly depend on |k|. We cannot draw, however, any analogy with the electron orbitals in atomic physics, since in the latter case at L = 1 the energy levels are triply rather than doubly degenerate in the absence of the spin orbit interaction. Since the total angular momentum is J = L + S, the Figs. 13 and 14 can be used to construct the corresponding surfaces for the total angular momentum hJi = hLi + hSi. The latter visualizes all allowed values and directions of the average J. Spin properties of free charge carriers in lead chalcogenides, PbTe and PbSe, where the energy bands are made up of s, p and d orbitals were analyzed in paper [54]. The ellipsoidal spin surfaces were observed in both the conduction and valence bands of these materials. The spin polarization properties of ballistic holes in wurtzites represented by GaP, where apart from the SO interaction the crystal field splitting dominates in the band edge splitting, were analyzed in Ref. [15].

4.

Spin Surfaces: AlAs/GaAs QW and QT

As we have seen, the SO interaction may play a dual role: it may rearrange the energy spectrum, for example this happens in valence bands of elementary and A 3 B5 semiconductors, and at the same time the SO interaction may split the Kramers pairs to spin-split subbands with opposite spins. The effect of band rearrangement on the shape of spin surface usually is strong. On the other hand the effect of spin splitting due to Rashba and/or Dresselhaus Hamiltonians usually is very small and as a rule can be neglected if only a general spin surface shape is required. Thus, to have only a general knowledge about spin surface, in principle, one may neglect the spin splitting and solve a simpler problem. The inclusion of Rashba and Dresselhaus Hamiltonians in the problem fixes the natural quantization axis around which the spin precession takes place. In the last section we have included the interaction between conduction and three valence bands. If the interaction between particular Kramers pairs is weak, the general shape of the spin surface can be calculated using a simpler set of energy bands. In this section the properties of free 2D carrier spins, when the interaction between s-type conduction and p-type valence bands is negligible will be considered. Usually such an approach of decoupled conduction and valence bands is acceptable for semiconductors having large and positive energy gaps. For example, in case of GaAs, InP or CdTe in the analysis of general spin properties it is enough to restrict oneself to either conduction or valence band only. In this section three examples where decoupled conduction-valence band approach is valid are briefly discussed: electron and hole QWs in planar GaAs/AlAs heterostructure, and electron QW in the hollow cylinder, or quantum tube (QT), where the electron wave function is concentrated around the tube circumference. The tubular geometry is typical to carbon nanotubes [55, 56, 57]. The self-rolled bilayer system, as proposed by V. Ya. Prinz et al [58, 59, 60], is another example. These nanotubes support the electron motion on a curved surface.

Bloch Sphere, Spin Surfaces and Spin Precession

4.1. 4.1.1.

171

Spin Surfaces in Planar AlAs/GaAs QWs n-type QW

To simplify the problem, in the analysis of spintronics devices one frequently starts from a simpler Hamiltonian which has the following form in the σz representation H = H0 + α(σx ky − σy kz ) + β(σxkx − σy ky ).

(74)

The kinetic term H0 is diagonal and describes the doubly degenerate and parabolic conduction band. The second and third terms describe Rashba and Dresselhaus contributions, respectively. In the bulk semiconductors the Dresselhaus Hamiltonian is proportional to the third power of the wave vector (see the Hamiltonian (28)). In (74), the Dresselhaus contribution (the last term) is proportional to the first power. It has been obtained after kz components have been replaced by average values of the operators kz = −i∂/∂z and kz2 = −∂ 2/∂z 2 , and after some simplifying assumptions (smallness of the in-plane wave vector k, insignificance of the spin splitting anisotropy in the QW plane) have been applied [61, 62]. To evaluate the deviation of the spin surface of 2D electrons from the Bloch sphere, in Ref. [17] the full Dresselhaus Hamiltonian with the cubic in ki terms included, H = H0 + HD = H0 + γσ · χ,

(75)

where H0 = k2/(2m∗)I and χx = kx (ky2 − kz2 ) etc, was addressed. The spin surface of the n-th QW energy subband consists of well (w) and barrier (b) contributions 1 hSin = hψn |σ|ψni 2 Z ∞ Z d (76) hψwn |σ|ψwn i dz + hψbn |σ|ψbn i dz, = 0

d

m∗

and γ enter in the well ψwn and barrier where 2d is the well width. Different values of ψbn spinors. The required spinors were found numerically from the eigenvalue equation. Figure 15 shows the spin surface of the lowest energy subband at kk[11], |k| = 8 nm−1 and 2|k|d = 0.567. The parameters of the symmetric Al0.3Ga0.7As/GaAs/Al 0.3Ga0.7 As heterostructure were used in the calculation. The surface shown in the Fig. 15, in fact, is a spheroid, the rotation axis of which connects the opposite poles and can be described by the formula (77) |hSin|2 = a + b cos(4ϑ), where a ≈ 1/4 and b a. In Al0.3Ga0.7As/GaAs QW, the spheroid can be approximated by |hSi0|2 = 0.24999+0.00001 cos(4ϑ) when the electron is in the ground state. The poles at ϑ = 0 and ϑ = π/2 correspond to spin eigenvalues ± 12 . Equation (77) shows that the largest deviation from the sphericity is observed in the mixed state at equal contributions of the up and down spin eigenstates ( ϑ = π/4) in the superposition. The deviation from the sphere of the first excited subband was found to be larger and equal |hSi1|2 = 0.24993 + 0.00007 cos(4ϑ). If k is rotated in the QW plane, then the spheroid will rotate in the opposite direction around the axis that is perpendicular to the QW plane. From this we conclude that 2D electron spin surfaces, when the SO interaction is included, in principle, do not possess spherical symmetry. However, the nonsphericity is small and to a high precision the spin surfaces can be approximated by the Bloch sphere.

172

A. Dargys XSy\

0.5 0

-0.5 0.5

XSz\ 0

-0.5 -0.5 0 XSx \

0.5

Figure 15. Spin surface of 2D electrons in A 3 B5 quantum wells. The thick line shows the in-plane wave vector k. The quantization axis (not shown) is perpendicular to k and connects the two poles on the sphere.

4.1.2.

p-type QW

In A3B5 and A2B6 compounds in the presence of the SO interaction the valence bands suffer a considerable rearrangement what is reflected in a complex structure of dispersion ±1/2 band in the Fig. 8 a) is far away from and spin properties. If the SO split-off band ( Γ7 the other bands and the energy gap is positive and large enough, one can reduce the problem to heavy- and light-mass bands described by 4 × 4 Luttinger-Kohn Hamiltonian. Below in considering the hole spin properties we shall limit ourselves to sum of the Luttinger-Kohn HLK and Rashba HR Hamiltonians:

Hv = HLK + HR.

(78)

The four-band Luttinger-Kohn Hamiltonian in the total angular momentum basis mJ = 3 1 1 3 2 , 2 , 2 , 2 has the following form [63],



HLK

 Hhh b c 0  b∗ Hlh 0 c  , =  c∗ 0 Hlh −b  0 c∗ −b∗ Hhh

(79)

Bloch Sphere, Spin Surfaces and Spin Precession

173

where Hhh = Hlh = b = c =

~2 ∂ ∂ 2 2 (γ1 − 2γ2) + V (z), (γ1 + γ2 )(kx + ky ) − 2m0 ∂z ∂z ~2 ∂ ∂ 2 2 (γ1 + 2γ2) + V (z), (γ1 − γ2 )(kx + ky ) − 2m0 ∂z ∂z √ 2 ∂ 3~ ∂ + γ3 , (−ky − ikx) γ3 −i 2m0 ∂z ∂z √ 2 3~ γ2 (kx2 − ky2) − 2iγ3kx ky . 2m0

(80) (81) (82) (83)

V (z) = V0, if z is in the barrier, and V (z) = 0, if z is in the well. The contribution to the spin splitting of hole states due to structural inversion asymmetry was approximated by the matrix [47, 64] √   3 0 0 0 2 k− √   3 k 0 k− 0   √ (84) HR = r1  2 + , 3   0 0 k+ 2 k− √

0

0

3 2 k+

0

where k± = kx ± ky . In calculations the Rashba interaction strength was assumed to be equal r1 = 0.005 eV nm. The depths of the well was V0 = 0.0956 eV, the value which corresponds to Ga 0.21Al0.79As/GaAs heterostructure. The valence band parameters for respective materials were: for GaAs, γ1 = 6.85, γ2 = 2.1, γ3 = 2.9; for Ga0.21Al0.79As, γ1 = 6.14, γ2 = 1.8, γ3 = 2.56. The QW width was 10 nm and the total width of the structure was large enough (30 nm) so that the probability density practically was zero at the ends of the structure. Figure 16 shows the dispersions of the first three 2D valence bands. At k = 0, the Hamiltonian (78) is diagonal and, as it is seen, the energies are doubly degenerate. At k 6= 0, the spin splitting is induced by Hamiltonian (84). In all cases the valence band edge in the well and barrier regions remains flat. The magnitude of the splitting ∆E = E↑(k) − E↓(k) is shown in Fig. 17. It is seen that the splitting is cubic in k for HH1 and linear in k for a HL1 subbands at small splitting energies only. At high energies the deviation from both cubic and linear dependencies is strong. In the four-band approximation the spin and total angular momentum matrices are proportional, S = J/3, therefore it is enough to consider one of them. To find the spin surface, the heavy- and light-mass band spinors |ϕi that are related to corresponding Kramers pair were parametrized in a standard form |ϕi = cos ϑ|ϕ↑i + sin ϑeiφ |ϕ↓i.

(85)

Upper and lower panels in Fig. 18 show the spin surfaces of the two lowest 2D energy subbands at three wave vectors parallel to [01] direction. It is clear that the deviation of surfaces from spherical shape is large. When the wave vector is rotated in the QW plane the spin surfaces synchronously rotates about hJz i axis, however, their overall shape remains nearly the same. Secondly, as already discussed, the two poles on the spin surfaces correspond to two orthogonal eigenstates in (85) and are related to the spin-split energy subbands

174

A. Dargys

1 0.8 0.6 0.4 0.2 0

EnergyHmeVL

EnergyHmeVL

Figure 16. Dependence of 2D hole energy on the wave vector parallel to [01] direction. Only three main subbands degenerate at k = 0 are shown. The letters on the curves correspond to respective panels in Fig. 18.

Al.21 Ga.79 AsGaAs 0

0.01

0.02 k H1ÞL

0.03

a)

3 2.5 Al.21 Ga.79 AsGaAs 2 1.5 1 0.5 0 0 0.01 0.02 k H1ÞL

0.03

b)

Figure 17. Dependence of the spin splitting energy ∆E on hole wave vector for a) HH1 and b) HL1 subbands. The well depth is 0.0956 eV.

with opposite spins. In the Fig. 18 the dependence of the spin surface on the wave vector can be explained by different contributions of heavy- and light-mass bands. Since for a proper operation of the spin FET the contributions of the up and down spin states in the superposition should be equal, the corresponding points must lie on the equators of surfaces in Fig 18. The important message in 2D hole case is that the magnitude of the average spin strongly depends on the hole wave vector length |k| = (kx2 + ky2 )1/2. The well depth has relatively small effect. Singular shapes appear in those cases when spin

Bloch Sphere, Spin Surfaces and Spin Precession XJy\ 1 -1 0

XJy\ 1 -1 0

XJy\ 1 -1 0

1

1

1

XJz\ 0

XJz\ 0

XJz\ 0

-1

-1

-1

-1

0 XJx \

-1 1

a)

XJy\ 1 -1 0

0 XJx \

-1 1

b)

XJy\ 1 -1 0 1

1

XJz\ 0

XJz\ 0

XJz\ 0

-1 -1

0 XJx \

d)

1

c)

1

f)

-1 -1

1

0 XJx \

XJy\ 1 -1 0

1

-1

175

0 XJx \

-1 1

e)

0 XJx \

Figure 18. The spin surfaces and their projection on hJx i − hJy i plane of HH1 (a, b, c) and LH1 (d, e, f) subbands in Al 0.21Ga0.79As/GaAs QW. The direction of the wave vector ˚ −1 (a, d), 0.012 A ˚ −1 (b, e), 0.024 A ˚ −1 (c, f). is shown by lines: ky = 0 and kx = 0.003 A The respective wave vectors are designated by letters on the dispersion curves in Fig. 16, too. The QW lies in the x − y plane. surface shrink to line or plane. For example, this happens in HH1 subband at small hole energies, k ≈ 0 . In this case the injection from the ferromagnetic contacts with spins in the QW plane will be difficult, since the spin magnitudes in 2D channel and in ferromagnetic source will not match each other. However, for perpendicular polarization the spin matching, in principle, is possible. In Ref. [65] a fourteen-fold anisotropy of the electrical spin injection efficiency between directions perpendicular and parallel to hole current flow in the ferromagnetic-semiconductor heterostructures was observed experimentally. The detected anisotropy may be easily explained by needle-like shape of the spin surface. In case of LH1 subband an opposite situation is observed, where a needle-like spin ˚ −1 in Fig.16f. In this case an efficient surface appears at large wave vectors, |k| = 0.024 A injection will be possible if the source and drain magnetizations are aligned with the QW plane. If more than four bands are included in the initial Hamiltonian (78), the mentioned singular lines or planes will transform into needle- or disk-shaped spin surfaces.

4.2.

Cylindrical QW

In the cylindrical QW or quantum tube (QT), the confining potential topology is different from that of the planar QW. In the QT the electron is squeezed between cylindrical walls and must satisfy a periodic boundary condition. The probability to find the electron in the center of the QT is close zero, even in the ground state. Figure 19 shows the potential profile of QT which is equal zero everywhere, V (r) = 0, except inside a cylindrical shell, where for simplicity it was assumed to be constant: V (r) = const < 0. In the limiting case of an infinite confinement the electron wave functions will be bounded by cylinder walls with inner and outer radius r1 < R < r2, where R is some average radius. Due to spatial quantization, the wave function in the radial direction will be quantized. If the potential

176

A. Dargys j

aL R

z

r

r

bL

Figure 19. a) Cross section of the quantum tube with the coordinate system used. b) The profile of the potential in the direction perpendicular to z axis (tube axis) and the ground state wave function. is constant, all radial modes will have the same velocity proportional to longitudinal wave vector kz . For arbitrary potential there may be some dispersion between different modes. In the following we will restrict ourselves to the case of a single, fundamental radial mode and neglect all higher order modes. The electron spin properties on curved surfaces were also considered in Refs [66, 67, 68]. The projection of the total angular momentum on the QT axis z is equal to sum of orbital and spin components, Jz = Lz + Sz = Lz ± 12 ≡ j. The spinor should satisfy the periodic boundary condition on the circumference of the cylinder: ψ(ϕ) = ψ(ϕ + 2π). From what has been said follows that the two-component basis spinors can be expressed in the form (86) Ψ(kz , ϕ, n) = eikz z ψn (ϕ), where ψn (ϕ) =

1

ei(j− 2 )ϕ φ1 0

or

0 1

ei(j+ 2 )ϕφ2

(87)

where φ1 and φ2 are the eigenfunctions independent of the azimuthal angle ϕ. The index n represents a particular radial mode and will be suppressed in the following. To satisfy the periodic boundary conditions, j should assume the values j = ± 12 , ± 32 , ± 52 . . . . The kinetic part of the Hamiltonian takes into account the electron motion along and around the cylinder of the average radius R, ~2 H0 = 2m∗

0 kz2 + kϕ2 2 0 kz + kϕ2

,

(88)

where kϕ is the angular wave vector kϕ = −

i ∂ , R ∂ϕ

(89)

Bloch Sphere, Spin Surfaces and Spin Precession

177

which should include the periodic boundary conditions. In the following a dimensionless energy and wave vector will be used: ε = 2m∗ R2E/~2 and k = Rkz . Therefore, in the basis (87) the normalized Hamiltonian (88) assumes the form: 2 k + (j − 1/2)2 0 . (90) H0 = 0 k2 + (j + 1/2)2 The SO interaction will be included via Rashba Hamiltonian which is proportional to the normalized interaction constant α ˜ = 2αm∗ R/~2. The required Pauli matrices in the cylindrical coordinate system (r, ϕ, z) are 0 e−iϕ 0 −ie−iϕ 1 0 , σ , σ = = . (91) σr = ϕ z eiϕ 0 ieiϕ 0 0 −1 Then, with a unit vector n normal to the cylinder surface, the Rashba Hamiltonian (37) reduces to −kϕ −ie−iϕ kz . (92) HR = α (σϕ kz − σz kϕ ) = α ieiϕ kz kϕ The exponents e±iϕ take into account the rotation of the spinor around z axis. The eigenenergies are invariant to such rotation. The sum of kinetic and Rashba Hamiltonians gives the following QT Hamiltonian in the dimensionless units 2 ˜ − 12 ) iαke ˜ −iϕ k + (j − 12 )2 + α(j , (93) H= −iαke ˜ iϕ k2 + (j + 12 )2 − α(j ˜ + 12 ) the spectrum of which is ε(k) = j 2 + k2 +

1 − 2α ˜ p 2 ± j (1 − 2α) ˜ + (j 2 + k2)α ˜2 . 4

(94)

When α ˜ = 1 the spectrum simplifies to ε(k) = (k2 ± k) + j 2 + 1/4 ,

(95)

which, apart from the constant (j 2 + 1/4) that can be eliminated by a shift of the reference energy, coincides with the Rashba spectrum for a planar QW (cf. Eq. (39)). From this follows that at α ˜ = 1 the precession length of the spin in the cylindrical spin FET will be independent of electron energy, similarly as it is in the Datta-Das transistor [20]. After parametrization of the eigenfunctions in the energy representation and transformation back to the initial representation (87), as described in the previous sections, one obtains the following components of the average spins (91) in the cylindrical coordinate system hσr i = − sin φ sin 2ϑ, ˜ ) cos φ sin 2ϑ + kα ˜ cos 2ϑ , hσϕ i = −d−1 R j(1 − α j(1 − α ˜ ) cos 2ϑ − k α ˜ cos φ sin 2ϑ , hσz i = d−1 R

(96) (97) (98)

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A. Dargys

where dR = [j 2(1 − 2α) ˜ + (j 2 + k2 )α ˜2]1/2. The square of the average spin hσi reduces to hσi2 = hσr i2 + hσϕi2 + hσz i2 = 1,

(99)

which means that the spin surface of electron that propagates between the walls of the QT is spherical and independent of the electron energy, or band parameters. It can be shown that this property is preserved even after inclusion of the Dresselhaus Hamiltonian in the problem [68]. By choosing the parameter ϑ and φ values one selects a particular point on the sphere where the electron spin is pointing. However, from the obtained result, Eqs. (96)(98), follows that for a selected ϑ or φ the spin direction depends on the magnitude of electron wave vector kz , Rashba constant α, angular momentum j, and average radius of the cylinder R. The natural quantization axis can be deduced from pole coordinates, which can be found from the conditions ϑ = 0 and ϑ = π/2, kα ˜ j(1 − α ˜) ϑ=0 . (100) hσ iQA = 0, ∓ , ± ϑ = π/2 dR dR This expression shows that in general case the natural quantization axis lies on the surface of a cone, the apex angle of which depends on band parameters and cylinder radius. It should be noted that the inclusion of SO interaction does not fix the quantization axis on the cone surface. In the absence of SO interaction, when α ˜ = 0, the quantization axis becomes parallel to z (cylinder axis). The most interesting situation realizes when α ˜ = 1, then the quantization axis becomes perpendicular to the cylinder axis. In this case, as noted earlier (see Eq. (95)), the spectrum coincides with that of the Rashba Hamiltonian for a planar QW. In real semiconductor one usually has α ˜ < 1. From the commutators of the Pauli matrices (91) with the Hamiltonian (93) one can find the time-dependence of the vector spin matrix, i dσ = [σ, H] , dt ~

(101)

and then, after averaging, the classical precession equation. Using the parametrized spinors (87) one can obtain the following system of differential equations that depends on spin components (hσr i, hσϕi, hσz i) and semiconductor band parameters, ˜ )hσϕ i − 2kαhσ ˜ z i cos ϕ , hσ˙ r i = 2j(1 − α

(102)

˜ r i + 2kαhσ ˜ z i sin ϕ , hσ˙ ϕ i = −2j(1 − α)hσ

(103)

˜ r i cos ϕ − hσϕ i sin ϕ] . hσ˙ z i = 2kα[hσ

(104)

The obtained system of precession equations can be rewritten in a more transparent form if the precession vector is introduced:   kα ˜ sin ϕ Ω = 2  kα ˜ cos ϕ  . (105) j(1 − α) ˜ The angle ϕ reflects the indeterminacy of the natural quantization axis on the cone. Since the spin splitting energy does not depend on ϕ, without losing of generality it can be assumed that ϕ = 0. In this case the quantization axis will be fixed in (ϕ, z) plane. Using

Bloch Sphere, Spin Surfaces and Spin Precession

179

Eq. (105) and hSi = ~hσi/2 the system (102)-(104) can be cast into standard form for a spinning top: dhSi/dt = −Ω × hSi . (106) The obtained precession equation can be solved analytically. The module of the precession vector Ω gives the precession frequency, which coincides with the spin splitting energy of the spectrum (94), p ˜ + (j 2 + k2 )α ˜2 . (107) |Ω| = 2 j 2(1 − 2α) Α=0.044

1 0.5 0 XS \ z -0.5 -1 -0.5 0 XSr \ 0.5

1 -1

-0.5

0

0.5

1

-1

XSj\

Figure 20. The spin surface and the family of spin trajectories represented by thick curve on the spin surface for wave vectors in the range k = 0.7 − 1.5 and α ˜ = 0.044, j = 1/2. The spin surface was drawn for k = 1. The following properties of the precession equation (106)-(107) are to be noted. Firstly, the precession frequency is not symmetric with respect to sign of the Rashba constant. Secondly, as follows from (105) the natural quantization axis is independent of band parameters in two cases: when α ˜ = 0 and α ˜ = 1. The first one is trivial and corresponds to absence of SO interaction. More interesting is the second case. Since 2kα ˜ = 4kz m∗ αR2/~2, this regime can be realized by trimming the radius of the quantum tube. When α ˜ = 1, the precession frequency, as seen from Eq. (107), is proportional to k. This property, as already mentioned, is important for a proper operation of the spin FET. In Fig. 20, thick line on the equator shows the family of trajectories calculated with equations (102)-(104) at j = 1/2, α ˜ = 0.044 and wave vectors varied in the range k = 0.7 − 1.5 (these normalized values correspond, for example, to the following QT parameters: α = 5 × 10−9 eVcm, m∗ = 0.066m0, R = 20 nm, kz = (0.035 − 0.075) nm−1 , ~2/(2m∗R2 ) = 0.057 eV). At t = 0 the electron was assumed to be in a balanced superposition of states with opposite eigenspins. The reference Bloch sphere was drawn for k = 1, α ˜ = 0.044, j = 1/2. The individual spin trajectories are circles that enclose different quantization axes related to particular k values. The scatter of the trajectories on the spin surface at different wave vectors is represented by thickness of the resulting line.

5.

Conclusion

The present study of spin properties in semiconductors with zinc-blende lattice shows that the spin surfaces of free carriers in QWs, in general, have an ellipsoidal form. The lengths

180

A. Dargys

of the fundamental axes and their orientation in the spin space depend on 2D carrier inplane wave vector magnitude. In the semiconductors with a positive bandgap, for example in GaAs, the electron spin surfaces were found to be very close to spherical in both bulk and 2D structures and, thus, in modelling spintronics devices, for example, in modelling the electron spin evolution in spin FET channels [69], spin lifetimes [70], or in optimizing the spin control by external electric fields [38, 39], may be replaced by the Bloch sphere. In all these cases, to a high degree of accuracy one can assume that the precession trajectories of the spin vector around the quantization axis are circles. However, the properties of the natural precession axis will depend on the topology of the QW. Until now the most of investigations were devoted to planar QWs, where the natural precession axis lies in the QW plane. Here we have also considered the cylindrical quantum well, where the precession axis was found to lie on the conical surface. The influence of SO interaction on the shape of 3D and 2D hole spin surfaces, in general, was found to be strong. The deformation of the spin surfaces mainly comes from the rearrangement of the valence band edge. The inclusion of the Dresselhaus and Rashba SO interactions has only negligible influence on a general shape of the spin surface. Depending on the subband symmetry and wave vector magnitude, the surface may assume the form of ellipsoid, spheroid of rotation, or even shrink to a line. The ratio of the fundamental axes strongly depends on the band type (heavy, light or split-off) and wave vector magnitude. However, the dependence on the wave vector direction was found to be weak and to the first approximation one may assume that the spin surface rotates synchronously with the wave vector. The spin dynamics on such deformed surfaces is more complicated. In the bulk zinc-blende semiconductors, the hole spin dynamics and precession equation was investigated in Ref. [52]. In case of the inverted energy bands, for example in HgTe, where the bandgap is negative due to strong mixing of orbitals of different symmetry, the spin surfaces may strongly deviate from the Bloch sphere in both the conduction and valence bands. The employment of the Bloch sphere in calculating spin lifetime or spin dynamics may be unacceptable in this case. Since the 2D carrier spin properties in the inverted band case also depend on the well width, every specific case should be considered individually. The knowledge of spin surfaces may be very useful in visualizing the spin dynamics in spintronics devices and qubit evolution in quantum gates, or in predicting the spin matching conditions at the interface of different materials, for example, between the ferromagnetic spin injector and the semiconductor channel. Of a particular interest may be needle-like spin surfaces, which can be matched the spins of a particular polarization only. In addition, the knowledge of the spin surface allows one to exclude the forbidden spin trajectories in the spin FET channels or spin guides. In classical Monte Carlo simulation methods the knowledge of the spin surface allows one to select the magnetization vectors for the individual charge carriers in an ensemble of classical electrons and to fix the natural quantization axis around which the precession takes place.

Acknowledgement The work was partly supported by Lithuanian State Science and Studies Foundation under contract C-07004.

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References ˇ c, I.; Fabian, J.; Das Sarma, S. Rev. Mod. Phys. 2004, vol. 76, 323-410. [1] Zuti´ [2] Silsbee, R. H. J. Phys.: Condens. Matter 2004, vol. 16, R179-R207. [3] Jungwirth, T.; Sinova, J.; Maˇsek, J.; MacDonald, A. H. Rev. Mod. Phys 2006, vol. 78, 809-864. [4] Dennis, C. L.; Borges, R. P.; Buda, L. D.; Ebels, U.; Gregg, J. F.; Hehn, M.; Jouguelet, E.; Ounadjela, K.; Petej, I.; Prejbeanu, I. L.; Thornton, M. J. J. Phys.: Condens. Matter 2002, vol. 14, R1175-R1262. [5] Appelbaum, I.; Monsma, D. J. Appl. Phys. Lett. 2007, vol. 90, 262501-1-3. [6] Huang, B.; Monsma, D. J.; Appelbaum, I. Appl. Phys. Lett. 2007, vol. 91, 072501-1-3. [7] Pauncz, R. Spin Eigenfunctions (Construction and Use); Plenum Press: New York and London, 1979. [8] Farrar, T. C.; Becker, E. D. Pulse and Fourier Transform NMR (Introduction to Theory and Methods); Academic Press: New York, 1971. [9] Waugh, J. S. New NMR Methods in Solid State Physics; Massachusetts Institute of Technology: Cambridge, Massachusetts, 1978. [10] Slichter, C. P. Principles of Magnetic Resonance; Springer-Verlag: Berlin, 1980. [11] Ernst, R. R.; Bodenhausen, G.; Wokaun, A. Principles of Nuclear Magnetic Resonance in One and Two Dimensions; Oxford University: Oxford, 1987. [12] Kittel, C. Quantum Theory of Solids; John Wiley & Sons: New York, 1963. [13] Zawadzki, W.; Pfeffer, P. Semicond. Sci. Technol. 2004, vol. 19, R1-R17. [14] Dargys, A. Phys. Status Solidi B 2004, vol. 241, 2954-2961. [15] Dargys, A. Phys. Rev. B 2005, vol. 72, 045220-1-10. [16] Dargys, A. Acta Physica Polonica A 2005, vol. 107, 46-55. [17] Dargys, A. Phys. Status Solidi B 2006, vol. 243, R54-R56. [18] Dargys, A. Semicond. Sci. Technol. 2007, vol. 22, 497-501. [19] Dargys, A. Lithuanian J. Phys. 2007, vol. 47, 185-194. [20] Datta, S.; Das, B. Appl. Phys. Lett. 1990, vol. 56, 665-667. [21] Weiler, M. H. In Semiconductors and Semimetals; Willardson, R. K., Beer, A. C., Eds., Vol. 16; Academic Press: New York, 1981; pp. 119-191.

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[41] Novik, E. G.; Pfeuffer-Jeschke, A.; Jungwirth, T.; Latussek, V.; Becker, C. R.; Landwehr, G.; Buhmann, H.; Molenkamp, L. W. Phys. Rev. B 2005, vol. 72, 0353211-12. [42] Zhang, X. C.; Ortner, K.; Pfeuffer-Jeschke, A.; Becker, C. R.; Landwehr, G. Phys. Rev. B 2004, vol. 69, 115340-1-7. [43] Zhang, X. C.; Pfeuffer-Jeschke, A.; Ortner, K.; Hock, V.; Buhmann, H.; Becker, C. R.; Landwehr, G. Phys. Rev. B 2001, vol. 63, 245305-1-8. [44] Burt, M. G. J. Phys.: Condens Matter 1992, vol. 4, 6651-6690. [45] Foreman, B. A. Phys. Rev. B 1993, vol. 48, 4964-4967. [46] Laurenti, J. P.; Camassel, J.; Bouhemadou, A. J. Appl. Phys. 1990, vol. 67, 6454-6460. [47] Winkler, R. Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems; Springer-Verlag: Berlin-Heidelberg, 2003. [48] Boxberg, F.; Tulkki, J. Rep. Prog. Phys. 2007, vol. 70, 1425-1471. [49] Golub, G. H.; Van Loan, C. F. Matrix Computations; The John Hopkins University Press: Baltimore and London, 1989. [50] Abolfath, M.; Jungwirth, T.; Brum, J.; MacDonald, A. H. vol. Phys. Rev. B 2001, 63, 054418-1-14. [51] Tinkham, M. Group Theory and Quantum Mechanics; McGraw-Hill, Inc: New York, 1964. [52] Dargys, A. Solid-State Electron. 2007, vol. 51, 93-100. [53] Dargys, A. Proc. SPIE “Advanced Optical Materials, Technologies, and Devices”, 2006, vol. 6596, 6596O1-6. [54] Dargys, A. Phys. Scr. 2006, vol. 75, 519-524. [55] Saito, R.; Dresselhaus, G.; Dresselhaus M. S. Physical Properties of Carbon Nanotubes; Imperial College Press: London, 1999. [56] Sch¨onenberger, C. Semicond. Sci. Technol. 2006, vol. 21, S1-S9. [57] Cottet, A.; Kontos, T.; Sahoo, S.; Mann, H. T.; Choi, M.-S.; Belzig, W.; Bruder, C.; Morpurgo, A. F.; Sch¨onenberger, C. Semicond. Sci. Technol. 2006, vol. 21, S78-S95. [58] Prinz, V. Ya.; Seleznev, V. A.; Gutakovsky, A. K.; Chehovskiy, A. V.; Preobrazhenskii, V. V.; Putyato, M. A.; Gavrilova, T. A. Physica E 1984, vol. 6, 828-831. [59] Schmidt, O. G.; Eberl, K. Nature 2001, vol. 410 (March), 168. [60] Friedland, K. J.; Hey, R.; Kostial, H.; Riedel, A.; Ploog, K. H. Phys. Rev. B 2007, vol. 75, 045347-1-4.

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[61] Eppenga, R.; Schuurmans, M. F. H. Phys. Rev. B 1988, vol. 37, 10923-10926. [62] de Andrada e Silva, E. A. Phys. Rev. B. 1992, vol. 46, 1921-1924. [63] Luttinger, J. M.; Kohn, W. Phys. Rev. 1955, vol. 97, 869-883. [64] Winkler, R. Phys. Rev. 2000, vol. 62, 4245-4248. [65] Young, D. K.; Johnston-Halperin, E.; Awschalom, D. D.; Ohno, Y.; Ohno, H. Appl. Phys. Lett. 2002, vol. 80, 1598-1500. [66] Magarill, L. I.; Romanov, D. A.; Chaplik, A. V. JETP 1998, vol. 86, 771-778 [Zh. Eksp.Teor. Fiz. 1998 vol. 113, 1411-1428]. [67] Chaplik, A. V.; Romanov, D. A.; Magarill, L. I. Superlattices and Microstructures 1998, vol. 23, 1231-1235. [68] Dargys, A. Lithuanian J. Phys. 2008, vol. 48, No 1. [69] Bournel, A.; Dollfus, P.; Cassan, E.; Hesto, P. Appl. Phys. Lett. 2000, vol. 77, 23462348. [70] Song, P. H.; Kim, K. W. Rev. Phys. B 2002, vol. 66, 035207-1-8. [71] Dargys, A. Phys. Rev. B 2002, vol. 66, 165216-1-8.

In: Spintronics: Materials, Applications and Devices ISBN 978-1-60456-734-2 c 2009 Nova Science Publishers, Inc. Editors: G.C. Lombardi and G. E. Bianchi °

Chapter 7

E LECTRICAL AND M AGNETIC P ROPERTIES OF N ANO - SCALE π- JUNCTIONS Samanta Piano Physics Department, CNR-Supermat Laboratory, University of Salerno, Via S. Allende, 84081 Baronissi (SA), Italy

Abstract The physics of the π phase shift in ferromagnetic Josephson junctions enables a range of applications for spin-electronic devices and quantum computing. In this respect our research is devoted to the evaluation of the best materials for the development and the realization of the quantum devices based on superconductors and at the same point towards the reduction of the size of the employed heterostructures towards and below nano-scale. In this chapter we report our investigation of transitions from 0 to π states in Nb Josephson junctions with strongly ferromagnetic barriers of Co, Ni, Ni80 Fe20 (Py) and Fe. We show that it is possible to fabricate nanostructured Nb/ Ni(Co, Py, Fe)/Nb π-junctions with a nano-scale magnetic dead layer and with a high level of control over the ferromagnetic barrier thickness variation. In agreement with the theoretical model we estimate, from the oscillations of the critical current as function of the ferromagnetic barrier thickness, the exchange energy of the ferromagnetic material and we obtain that it is close to bulk ferromagnetic materials implying that the ferromagnet is clean and S/F roughness is minimal. We conclude that S/F/S Josephson junctions are viable structures in the development of superconductor-based quantum electronic devices; in particular Nb/Co/Nb and Nb/Fe/Nb multilayers with their low value of the magnetic dead layer and high value of the exchange energy can readily be used in controllable two-level quantum information systems. In this respect, we discuss applications of our nano-junctions to engineering magnetoresistive devices such as programmable pseudo-spin-valve Josephson structures.

1.

Introduction

Conventional electronics is based on the transport of electrical charge carriers, however the necessity to have more versatile and efficient devices on micro- and nano-scale has diverted the attention towards the spin of the electron rather than its charge: this is the essence of spintronics. Although spintronic devices were traditionally built with semiconductors

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Samanta Piano

and/or ferromagnetic materials, nowadays hybrid structures constituted by superconductors (S) and ferromagnetic (F) materials are emerging as promising alternatives. They serve as candidates to realize and investigate systems useful both as testgrounds to probe fundamental physics questions, and as realistic elements for application in the spintronics industry thanks to the improved degree of control on the spin polarized current. In hybrid S/F structures it has been verified that, due to the simultaneous presence of these two competitive orders, peculiar effects appear: the superconductivity is reduced by the spin polarization of the F layer, while due to the proximity effect the Cooper pairs can enter in the superconductor resulting in a oscillatory behavior of the density of states. The last effect opens the doors towards new and exciting developments for the realization of quantum electronic devices. In fact in S/F/S systems, due to the oscillations of the order parameter, some oscillations manifest in the Josephson critical current as a function of the ferromagnetic barrier thickness evidencing the presence of two different states, 0 and π, corresponding to the sign change of the Josephson critical current. Our research, presented in this chapter, fits in this rapidly developing and exciting area of condensed matter physics [1]. Within the context of the spin polarized devices based on the interplay between superconductivity and ferromagnetism, we have fabricated S/F/S nano-structured Josephson junctions constituted by a low temperature superconductor, Niobium (Nb), and strong ferromagnetic metal, Nickel(Ni), Ni80 Fe20 (Py), Cobalt (Co) and Iron (Fe), and we have investigated their magnetic and electrical properties. These structures have evidenced a small magnetic dead layer and oscillations of the critical current as a function of the ferromagnetic barrier. These oscillations show excellent fits to existing theoretical models. We also determine the Curie temperature for Ni, Py and Co. In the case of Co and Fe we estimate the mean free path to confirm that the oscillations are in the clean limit and from the temperature dependence of the Ic RN product we show that its decay rate exhibits a nonmonotonic oscillatory behavior with the ferromagnetic barrier thickness. We investigate the presence of the Shapiro steps on the I vs V curve applying a microwave and the effect of the magnetic field on the maximum supercurrent. In this last case we find oscillations of the maximum of the supercurrent corrisponding to a Fraunhofer pattern. Finally, in the case of Co barrier, focusing in detail on a single 0-π phase transition we show evidence for the appearance of a second harmonic in the current-phase relation at the minimum of the critical current. For the high value of the exchange energies and small magnetic dead layer the S/F/S structures with Co or Fe barrier can be considered as good candidates for the realization of quantum devices.

2.

F/S Junctions: Basic Aspects

The aim of this section is to give an outline of the physics of the FerromagneticSuperconducting (F/S) interfaces (for a review see [2]). Andreev reflection plays an important role to understand the transport process in F/S junctions. The Andreev reflection near the Fermi level preserves energy and momentum but does not preserve spin, in other words the incoming electron and the reflected hole have opposite spin. This is irrelevant for the transport in N/S junctions (N stands for normal metal) due to the spin-rotation symmetry. On the other hand at F/S interfaces, since the spin-up and spin-down bands in F are different, the spin flipping changes the conductance profile. In particular in fully spin-polarized

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Figure 1. Andreev reflection process for a metal with spin polarization P = 0 (a) and P = 100% (c); experimental measurements of a N/S interface (b) and F/S junction (d). Figure adapted from Ref.[3]. metals all carriers have the same spin and the Andreev reflection is totaly suppressed; therefore, at zero bias voltage the normalized conductance becomes zero (see Fig. 1). In general, for arbitrary polarization P , it can easily be shown that GF S (0) = 2(1 − P ) GF N

(1)

where, in terms of the spin-up N↑ and spin-down N↓ electrons, the spin polarization is defined as (N↑ − N↓ ) P = , (2) (N↑ + N↓ ) From this analysis it has been shown that the point contact measurements based on the Andreev reflection process give a quantitative estimation of the polarization of the ferromagnetic material [3, 4]. In fact from the reduction of the zero bias conductance peak it is possible with a modified Blonder-Tinkham-Klapwijk model to estimate the polarization P . In a ferromagnet in proximity to a superconductor, the coherence length ξF , due to the presence of the exchange field Eex , is given by: s D~ , (3) ξF = 2(πKB T + iEex ) where D is the diffusive coefficient. If the exchange energy is large compared to the temperature, Eex > kB T , the coherence length is much shorter than in the case of the N/S proximity effect. In addition to the reduced coherence length compared to typical N/S structures, a second characteristic property arises from the complex nature of the coherence length: the induced pair amplitude oscillates spatially in the ferromagnetic metal as

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a consequence of the exchange field acting upon the spins of the two electrons forming a Cooper-pair [5, 6, 7, 8] (see Fig. 2). This oscillation includes a change of sign and by using appropriate values for the exchange energy and F-layer thickness, negative coupling can be realized. We refer to the state corresponding to a positive sign of the real part of the order parameter as “0-state” and that corresponding to a negative sign of the order parameter as “π-state”. ψ

0

π

0

Figure 2. Oscillatory behavior of the exponential decay of the superconducting order parameter at the F/S interface.

Demler et al. [9] gave a qualitative picture of this oscillatory character. They considered a Cooper pair transported adiabatically across an F/S interface, with its electron momenta aligned with the interface normal direction. The pair entering in the F region decays exponentially on the length scale of the normal metal coherence length. Then the up-spin electron, oriented along the exchange field, decreases its energy by h = Eex /~, where Eex is the exchange energy of the F layer. On the other hand, the down-spin electron increases its energy by Eex . To compensate this energy variation, the up-spin electron increases its kinetic energy, while the down-spin electron decreases its kinetic energy. As a result the Cooper pair acquires a center-of-mass momentum Q = 2Eex /vF , which implies the modulation of the order parameter with period πvF /Eex , where vF is the Fermi velocity. As a consequence of the oscillations of the order parameter, a similar oscillatory behavior is observed for the density of states. This behavior can be explained by considering the spin effect on the mechanism of Andreev reflections [2]. The process is illustrated in Fig. 3 using the energy-momentum dispersion law of the normal metal: in the case of a N/S interface an incoming electron in a normal metal N with energy lower than the superconducting energy gap ∆ from the Fermi level can be reflected into a hole at the N/S interface (Andreev Reflection) [10]. If the normal layer is very thin the density of states in N is close to that of the Cooper pair reservoir. The situation is strongly modified if the normal metal is ferromagnetic. As Andreev reflections invert spin-up into spin-down quasiparticles and vice versa, the total momentum difference includes the spin splitting of the conduction band: 4PF ' Q. The density of states is modified in a thin layer on the order of ξF . In particular, the interference between the electron and hole wave functions produces an oscillating term in the superconducting density of states with period Eex /~vF . This effect has been observed experimentally by Kontos et al. [11]

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E

k-

∆PF ~ Q

k+

k

Figure 3. Schematic of the Andreev reflection process in a F/S junction, the momentum shift, ∆pF , is dominated by the spin splitting of the up and down bands.

2.1.

Theory of the Josephson π-junctions

Due to the spatially oscillating induced pair amplitude in F/S proximity structures it is possible to realize negative coupling of two superconductors across a ferromagnetic weak link (S/F/S Josephson junctions). In this case of negative coupling, the critical current across the junction is reversed when compared to the normal case giving rise to an inverted current-phase relation. Because they are characterized by an intrinsic phase shift of π, these junctions are called Josephson π-junctions. One of the manifestations of the π phase is a non-monotonic variation of the critical temperature and the critical current, with the variation of the ferromagnetic layer thickness (tF ) [2]. Referring to the current-phase relation Ic = I0 sin φ, for a S/I/S Josephson junction the constant I0 > 0 and the minimum energy is obtained for φ = 0. In the case of a S/F/S Josephson junctions the constant I0 can change its sign from positive to negative indicating the transition from the 0-state to π-state. Physically the changing in sign of I0 is a consequence of a phase change in the electron pair wave function induced in the F layer by the proximity effect. Experimentally, measurements of Ic are insensitive to the sign of I0 hence the absolute value of I0 is measured, so we can reveal a non-monotonic behavior of the critical current as a function of the F layer thickness. The vanishing of critical current marks the transition from 0 to π state. The dependence of the critical current on the thickness of the ferromagnetic layer in S/I/F/S junctions has been experimentally investigated by Kontos et al. [12]. The quantitative analysis of the S/F/S junctions is rather complicated, because the ferromagnetic layer can modify the superconductivity at the F/S interface. Then other parameters, such as the boundary transparency, the electron mean free path, the magnetic scattering, etc can affect the critical current. It is outside the purpose of this chapter to derive explicitly the expression of the critical current as a function of the F layer, for the interested reader we refer to this review [2]. The majority of experimental studies have concentrated on weak ferromagnets where Eex ∼ KB Tc , where Tc is the superconducting critical temperature, resulting in multiple oscillations in Ic with temperature and tF . In the case of strong ferromagnets, where TCurie À Tc , only oscillations of Ic with tF , and not with temperature, are observed. In this chapter we will present the study of the oscillations of the critical current as a function of tF for S/F/S Josephson junctions with strong ferromagnetic barriers.

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Samanta Piano The generic expression of the critical current as a function of F layer is given by: ¯ ¯ ½ ¾ ¯ sin tF −d1 ¯ t0 − tF ¯ ξ2 ¯ Ic RN (tF ) = Ic RN (t0 )¯ , ¯ exp ¯ sin tFξ−t0 ¯ ξ1

(4)

1

where t1 is the thickness of the ferromagnet corresponding to the first minimum and Ic RN (t0 ) is the first experimental value of Ic RN (RN is the normal state resistance), and ξ1 and ξ2 are the two fitting parameters. Eq. 4 ranges in the clean and in the dirty limit. In particular, in clean limit, tF < L where L is the mean free path of the F layer, ξ2 = vF ~/2Eex . In this way, known ξ2 and estimating the Fermi velocity from reported values in literature, one can calculate the exchange energy of the ferromagnetic barrier. Then, in the case of clean limit the oscillations of IC RN vs tF can be modeled by a simpler theoretical model [13] given by: Ic RN (tF ) = Ic RN (t0 )

| sin(2Eex tF /~vf ) | , 2Eex tF /~vf

(5)

where in this case the two fitting parameters are vF and Eex . On the other hand, in dirty limit tF > L we can model the oscillations [14] by the following formula: Z 1 X µ ∆2 dµ |, (6) Ic RN (tF ) = Ic RN (t0 ) | Re 2 2 ∆ + ωm −1 sinh(kω tF /µL) ωm >0

where ∆ is the superconducting order parameter, ωm is the Matsubara frequency and is given by ωm = πT kB (2m + 1) where T is the transmission coefficient and m is an integer number. kω = (1 + 2 | ωm | τ /~) − 2iEex τ /~ and µ = cos θ where θ is the angle the momentum vector makes relative to the distance normal to the F/S interface. L is given by vF τ and τ is the momentum relaxation time. In this case the fitting parameters are vF , Eex and the mean free path L of the ferromagnetic layer.

3.

Josephson Junction Fabrication

To investigate the physics of the ferromagnetic π-junctions and their possible applications for spin-electronic devices we have realized and characterized Nb (250 nm thick) / Ferromagnetic layer / Nb (250 nm thick) Josephson junctions. As ferromagnetic layer we have used: Ni, Py, Co and Fe. In the case of Co, Fe and Py their respective thicknesses tF were varied from ' 0.5-5.5 nm while, in the case of Ni, tNi was varied from ' 1.0 to 10 nm. To assist processing in a focused ion beam (FIB) microscope, a 20 nm normal metal interlayer of Copper (Cu) was deposited inside the outer Nb electrodes, but 50 nm away from the Fe barrier. We remark that the 20 nm of Cu is a thickness smaller than its coherence length, so it is completely proximitized into the Nb, and it does not affect the transport properties of the Josephson junction. Refer to Fig. 4 for an illustration of our heterostructures [1]. The heterostructures employed in this chapter, have been deposited by d.c. magnetron sputtering. In a single deposition run, multiple silicon substrates were placed on a rotating holder which passed in turn under three magnetrons: Nb, Cu and the ferromagnetic layer

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Nb 200 nm Cu 20 nm Nb 50 nm FM Nb 50 nm Cu 20 nm Nb 200 nm Nb: 250 nm Nb

Nb: 250 nm

Ni:

Py: 0.5 – 5 nm

0.5 – 12 nm

Nb: 250 nm Nb

Nb: 250 nm

Nb: 250 nm Nb

Nb Nb: 250 nm

Co: 0.5 – 5 nm

Fe: 0.5 – 5.5 nm

Nb: 250 nm Nb

Nb Nb: 250 nm

Figure 4. A schematic picture of the S/F/S heterostructure used in this chapter. FM

Nb

0°

θ

Figure 5. An illustration of the sputtering process: the ferromagnetic interlayer thickness is varied as a function of the angle θ from the pre-sputter position.

(Fe, Co, Py or Ni). The speed of rotation was controlled by a computer operated stepper motor with a precision angle of better than 3.6◦ and each sample was separated by an angle of at least 10◦ . Prior to the deposition each target material was calibrated and the deposition rates were estimated with an Atomic Force Microscopy (see table 1 for a summary of the deposition parameters for all target materials presented in this chapter). In the case of the ferromagnetic materials (Co, Py, Fe and Ni) the rates of deposition, and hence tF , were obtained by varying the speed of each single chip which moves under the ferromagnetic target while maintaining constant power to the magnetron targets and Ar pressure. This was achieved by knowing the chip position relative to the target material (θ) and by programming the rotating flange such that a linear variation of ferromagnetic thickness with θ, d(tF )/dθ, was achieved. tF is inversely proportional to the speed of deposition, tF ∝ 1/Vt , and hence it can be shown that in order to achieve a linear variation of tF (θ)

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Table 1. A summary of the deposition parameters for all materials sputtered. tF refers to the expected film thickness. Target material Nb Cu Ni Co Py Fe

Rate (nm/W at 1 rpm) 6.89 4.39 2.12 1.8 1.64 2.0

Power (W) 90 30 40 40 40 40

Speed range (rpm) .028 0.22 0.20-4.2 0.36-3.6 0.33-3.3 0.22-2.2

tF (±0.2 nm) 250 20 0.5 - 10.5 0.5 - 5.0 0.5 - 5.0 0.5 -5.0

one programmes the instantaneous speed, at position θ and time t seconds (i.e. V (θ)t ), of deposition according to Ã !−1 Vi Vf Vi Vf θ − , (7) V (θ)t = Vf − Vi Vf − Vi 2π where Vi is the initial speed and Vf is the final speed in units of rpm. This method of varying tF guaranteed, in all cases, that the interfaces between each layer were prepared under the same conditions while providing precise control of the F layers [15]. Fig. 5 shows an illustration of the sputtering process. For each run, simultaneously, our heterostructures have been deposited on 5 × 5 mm2 and 5 × 10 mm2 SiO2 substrates. The first ones have been used for the magnetic measurements, the second ones for the realization of Josephson junctions.

3.1.

X-ray Measurements

To confirm our precise control over the ferromagnetic thickness variation we performed low angle X-ray reflectivity of a set of calibration Nb/Co/Nb thin films where the Nb layers had a thickness of 5 nm and the Co barrier thickness was varied from 0.5 nm to 5.0 nm [15]. A series of low angle X-ray scans were made and the thickness of the Co layer (tCo(observed) ) was extracted by fitting the period of the Kiessig fringes using a simulation package. It was found that our expected Co barrier thicknesses, tCo(expected) , was well correlated with tCo(observed) with a mean deviation of ±0.2 nm. In the inset of Fig. 6 we show an example of the low angle x-ray data plotted with the equivalent simulation data where tCo(observed) was extracted, while in Fig. 6 we report a comparison of tCo(observed) with tCo(expected) and its deviation.

4.

Nanoscale Device Process

We can summarize the realization of the Josephson junctions in three different steps: (i) patterning the films using optical lithography to define the micron scale tracks and the contact pads. Our mask permits etching of at least 14 devices, which allows us to measure

Electrical and Magnetic Properties of Nano-scale π-junctions

Intensity (cps)

10

6

10 10

X-ray data

4

Simulation 3

Expected

2

Observed (X-ray) 10

1

0.4 10

5

0

0.0

0.5

1.0

1.5

2.0

0.3

4

3

t

Co

0.2

2 0.1 1

0

|Deviation (E-O)| (nm)

Omega-2

(nm)

193

0.0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Sample number

Figure 6. tCo(observed) plotted with tCo(expected) extracted from simulation data, in the inset: low angle x-ray data plotted with the equivalent simulation for sample 1 where tCo(expected) =5 nm and tCo(observed) =5±0.2 nm. The mean deviation is estimated to be ±0.2 nm.

numerous devices and to derive good estimates of important parameters, like, for example, characteristic voltage (see Fig. 7 to see an illustration of the mask); (ii) broad beam Ar ion milling (3 mAcm−2 , 500 V beam) to remove unwanted material from around the mask pattern, thus leaving 4µm tracks for subsequent FIB work; (iii) FIB etching of micron scale tracks to achieve vertical transport [16, 17, 15]. In particular to realize our devices we have used a three-dimensional technique [18]. The wedge holder used in this chapter was designed by D.-J. Kang and it is schematically shown in Fig. 8; it is constituted by three sample lodgings, one in horizontal and two at 45◦ . Once loaded the sample on the 45◦ lodging, we can rotate the stage of the FIB at 45◦ , in this way the beam is perpendicular to the surface of the sample, and the first cut is done; then the sample holder is rotated of 180◦ around an axis normal to the sample stage, to permit the vertical cut to be done (see Fig. 8). This setup allows to load two samples for each run. The fabrication procedure is shown in Fig. 9 [19]. A first box with area 4 × 2µm2 is milled with 150 pA to realize tracks of about 700 nm (b). The time of milling is about 1 − 3 minutes, and this milling can be calibrated using the stage-current/end-point detection measurement. Fig. 10 shows how the milling of different layers can be distinguished, the first peak corresponds to the Nb, the two small peaks correspond to the two Cu layers, and finally the intensity decreases approaching to the SiO2 substrate. The sidewalls of the narrowed track are then cleaned with a beam current of 11 pA. This removes excessive gallium implantation from the larger beam size of the higher beam currents. The cleaning takes ∼ 25-30 seconds per device. The track width is then ≤ 500 nm. The sample is then tilted to θ = 90◦ , and the two cuts are made with a beam current of 11 pA to give the final device (Fig. 9 (d)) with a device area in the range of 0.2 − 1 µm2 . This technique permits to achieve vertical transport of the current: in Fig. 9 (e) we show a schematic picture of the

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Figure 7. (a)“Cam 39” mask design for a 10 × 5 mm2 chip used in this chapter, showing contact pads for wirebonding. (b) Central detail: on each chip we realize 14 possible junctions with thinnest track widths of 4µm (c). (a)

(b)

(c)

sample

45° rotation 180° rotation

Figure 8. Schematic of the sample holder used in FIB (a) showing two axes of rotation, 45◦ , to horizontal milling (b) and 180◦ to vertical milling. current path. In Fig. 11 the final FIB image of a Nb/Cu/Co/Cu/Nb device is presented.

5.

Magnetic Measurements

In this section we report magnetic measurements of Nb Josephson junctions with strongly ferromagnetic (F) barriers: Ni, Ni80 Fe20 (Py), Co and Fe. From measurements of the magnetization saturation (MS ) as a function of the F thickness, our heterostructures have shown a magnetic dead layer ranging between 0.5 nm and 1.7 nm. Then we give an estimation of

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195

(a)

(b)

(c)

(d)

(e)

Figure 9. FIB procedure for device fabrication: the initial trilayer after the photolithography and ion-milling (a) is cut with 150 pA beam (b) and then with 11 pA beam (c-d). (e) At the end two side cuts are realized to create the final device structure with a device area in the range of 0.2 − 1 µm2 achieving vertical transport of the current. (Figure adapted from ref. [20]).

Nb

360

Cu markers

stage current(pA)

340 320 300 milling starts SiO

2

substrate

Ferromagnetic layer

280 260 0

10

20

30

40

50

60

70

time(s)

Figure 10. End Point Detector. The graph shows the stage current as a function of the milling time, we can distingue the Nb layer, the two Cu layers, the Ferromagnetic barrier and when the current decreases we have reached the substrate.

the Curie temperature of the ferromagnetic layer.

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Figure 11. FIB image of a Nb/Cu/Co/Cu/Nb device, the two light gray lines are the Cu markers.

5.1.

Measurement of the Magnetic Dead Layer

In this section we explain the magnetic properties of Nb-Ni-Nb, Nb-Py-Nb, Nb-Co-Nb and Nb-Fe-Nb heterostructures as a function of F thickness [16, 15, 21, 17].

S

2

M (emu/m )

4.0

t

Ni

=4.0 nm

1.0

3.5

0.5

3.0

0.0

-0.5

2.5

-1.0

2.0

-200 -150 -100 -50

1.5 1.0

0

50

100 150 200

H(Oe)

Dead Zone = 1.72 nm Experimental data

0.5

Least-squares fit

0.0 0

1

2

3

4

5 t

6

7

8

9

10

11

(nm)

Ni

Figure 12. Saturation magnetization versus thickness of the Ni barrier, inset: hysteresis loop for tNi = 4 nm. To this aim we have studied, using a VSM at room temperature, the hysteresis loop of our heterostructures in order to follow the evolution of the magnetization as a function of the applied magnetic field. In Fig. 12 (inset), Fig. 13 (a), 14 (a) and (b), we show a collection of hysteresis loops for different thicknesses of the F barrier. We notice that both the MS and the width of the hysteresis loop, as expected, decrease with decreasing F layer thickness, and the ferromagnetic order disappears when the F barrier goes to zero. Furthermore from the hysteresis loops we have measured the saturation magnetization as a function of the F

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barrier to extrapolate the magnetic dead layer. -5

4.0x10

-5

t

3.0x10

t

1.4

=5.0 nm

2.0x10

t

=4.1 nm

Least-squares fit

1.2

Py

=3.5 nm

Py

1.0

-5

1.0x10

2

(emu/m )

M(emu)

Experimenta data

Py

-5

0.0

S

-5

M

-1.0x10

0.8 0.6

-5

0.4

-5

0.2

-2.0x10 -3.0x10

(b)

(a)

-5

0.0

-4.0x10

-500 -400 -300 -200 -100

0

100

H(Oe)

200

300

400

Dead zone=0.5 nm

500

0.0

0.5

1.0

1.5

2.0

2.5 t

3.0

3.5

4.0

4.5

5.0

5.5

6.0

(nm)

Py

Figure 13. (a) Hysteresis loops for different Py thicknesses. (b) Saturation magnetization versus thickness of the Py barrier.

Figure 14. Hysteresis loops for different Co (a) and Fe (b) thicknesses. Saturation magnetization versus thickness of the Co (c) and Fe (d) barrier. From the linear fit (gray line) the magnetic dead layer is extrapolated. The presence of a magnetic dead layer has been reported in other studies of S/F/S heterostructures [22, 23, 15] and it can be explained as a loss of magnetic moment of the heterostructures. The magnetic dead layer can be due to numerous factors, as, for example, lattice mismatch at the Nb-F interface resulting in a reduction in the ferromagnetic atomic volume [24] and crystal structure which leads to a reduction in the exchange interaction between neighboring atoms. This loss in exchange interaction manifests itself as a loss in magnetic moment due to a collapse in the regular arrangement of electron spin and magnetic moment and can lead to the suppression in TCurie and Eex [25, 26, 27, 28]. Another factor can be the inter-diffusion of the ferromagnetic atoms into the Nb. Like in the case of lattice

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mismatch this would result in a breakdown of the crystal structure at the interface leading to a reduction in the exchange interaction. The knowledge of the magnetic dead layer is a crucial point within the implementation of π-technology, to guarantee the reproducibility and the control of the devices. Extrapolating with a linear fit the saturation magnetization as a function of F thickness, we have obtained the estimation of the magnetic dead layer: ∼1.7 nm for Ni (see Fig. 12), ∼ 0.5 nm for Py (see Fig. 13 (b)) and ∼ 0.75 nm for Co (see Fig. 14 (c)). 6 Co

5

Py

Ms

2

(emu/ m )

Ni

4

Fe

3 2 1 0

0

1

2

3

4

5 t

F

6

7

8

9

10

11

(nm)

Figure 15. Saturation magnetization vs Ni, Py, Co and Fe thickness at T= 300K. In the case of Fe the saturation magnetization was measured for three different deposition runs (see Fig. 14 (d)). For each deposition we have obtained similar saturation magnetization and by extrapolating the least-squares fit of these data we have estimated a Fe magnetic dead layer of ∼ 1.1 nm [17]. In Fig. 15 we summarize the magnetic moment vs thickness for Ni, Py, Co and Fe. From these data we can extrapolate the slope of MS vs tF . From the theoretical model [29, 30] the predicted slopes are: ' 0.60 emu/cm3 for Ni, ' 0.52 emu/cm3 for Py, ' 1.42 emu/cm3 for Co and ' 2.6 emu/cm3 for Fe. On the other hand from our experimental data the slopes of MS vs tF for these ferromagnetic materials are suppressed from these expected bulk values: 0.27 emu/cm3 for Ni, 0.27 emu/cm3 for Py, 1.0 emu/cm3 for Co and 0.83 emu/cm3 for Fe. We can argue that, in our case, we are not considering bulk materials, as reported from Slater and Pauling [29, 30], instead our systems are constituted by F/S sandwiches, so the presence of a superconducting layer, Nb, can induce a weakening of the ferromagnetic properties of the F layer and a possible diffusion of the Nb into the F barrier.

5.2.

Calculation of the Curie temperature

We have measured the thermal variation of the saturation magnetization, M (T ), of Co, Ni, and Py when sandwiched between thick Nb layers [15]. To be certain, M (T ) with temperature was not weakened by a thermally activated diffusion of ferromagnetic atoms into Nb, or vice versa; at the interface we measured both the magnetization when warming and cooling. The warming and cooling data agreed for all three barrier systems up to a

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Figure 16. Thermal variation of the magnetization for Co, Py and Ni (black dots) with the best fitting curves to extrapolate the Curie temperature of Ni (571K), Py (800 K) and Co (1200K). Red squares are cooling data for Ni. temperature of 620 K. Above this temperature, MS was found to drop by virtue of thermally activated diffusion. We have modeled the warming and cooling data of M s (10K < T < 620K) with the following formula M (T )/M (0) = (1 − T /TCurie )β , where M (0) is the saturation magnetization at absolute 0 K, T is the measuring temperature, and β and TCurie are fitting parameters. This gives TCurie values of 1200 K for Co, 571 K for Ni, and 800 K for Py (see Fig.16) in agreement with the bulk values. Data for Ni are the most reliable because we have a full data set. However, in any case these measurements provide that, in our metallic systems, interdiffusion at the ferromagnetic surface cannot be ruled out because the ferromagnetic layer is known to form a variety of magnetic and nonmagnetic alloys with Nb.

6.

Transport Measurements

In this section we present the I(V ) vs V curves for the Nb/F/Nb Josephson junctions varying the thickness of the F barrier. All these materials show multiple oscillations of the Josephson critical current with barrier thickness implying repeated 0-π phase-transitions in the superconducting order parameter. The critical current oscillations have been modeled

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with the clean and dirty limit theoretical models, from this analysis we have extrapolated the exchange energy and the Fermi velocity of the ferromagnetic barrier.

6.1.

Experimental Data: Critical Current Oscillations

400

I( A)

200

0

3.5 nm

-200

3.3 nm 2.9 nm 2.6 nm

-400

-80

2.2 nm

-60

-40

-20

0

20

40

60

80

V(mV)

Figure 17. I − V characteristics for Nb/Co/Nb Josephson junctions with different Co thickness. For several junctions on each chip we have measured the current versus voltage from which the critical current (Ic ) and the normal resistance (RN ) of the Josephson junction have been extrapolated. In Fig. 17 we show an example of I vs V curves for different Co thicknesses, we notice that for each curve we have roughly the same resistance, that means the same area of the Josephson junctions. This is an indication of the good control on the fabrication of the devices with the FIB. The Ic RN products, as a function of F barrier thickness, exhibit a decaying, oscillatory behavior, in agreement with the theoretical predictions. The oscillations of Ic RN as a function of Ni, Py, Co and Fe thicknesses at 4.2 K are shown in Figs. 18 (a)-(b) and Figs. 19 (a)-(b) [16, 17, 15]. In the case of Py, the clean limit model, Eq. 5, closely matches the experimental data up to a thickness of ' 2 nm, while in the case of Ni the oscillations are explained by the clean limit theory up to ' 7 nm. Above these values a better fit is obtained using a formula for a diffusive and high Eex ferromagnet, Eq. 6. For Eq. 5 the fitting parameters are the Eex and the Fermi velocity vF . In the case of Eq. 6 the fitting parameters are: vF , the mean free path L, the superconducting energy gap ∆ and Eex . But L and ∆ are not free parameters, because they are fixed by the theoretical predictions. In this way the only free parameters are, as for the clean limit equation, the Fermi velocity and the exchange energy. Using Eq. 6 we obtain the best fitting values for Ni: vF (Ni) = 2.8 × 105 m/s and LNi ' 7 nm; while for Py vF (Py) = 2.2 × 105 m/s, LPy ' 2.3 nm and ∆ = 1.3 meV. These values are consistent with the ones used in Eq. 5 and elsewhere [31, 22]; while for the exchange energy we estimate Eex (Ni) ' 80 and Eex (Py) ' 201 meV. Eex (Ni) is close

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Figure 18. Oscillations of the IC RN product as a function of the thickness of the Ni (a) and Py (b) barrier. The solid green line is the theoretical fit in agreement with Eq. 5, dash-dot blue line is referred to Eq. 6. to other reported values by photoemission experiments [32], but smaller than that reported by other authors [31]. The smaller than expected Eex (Ni) is a consequence of impurities and possibly interdiffusion of Ni into Nb. Anyway, from the magnetic measurements the extrapolated value of the TCurie provides evidence that our Ni is of acceptable quality. The Eex (Py) is consistent with the expected value and is approximately twice that measured in Nb/Py/Nb junctions deposited with epitaxial barriers where Eex ' 95 meV [22]. On the other hand for Co and Fe the thicknesses range all in the clean limit, so the experimental data have been modeled with the general formula, Eq. 4, and ξ1 and ξ2 are the two fitting parameters. We can see that the experimental data are in good agreement with the theoretical model. In particular for the Co data, from the theoretical fit shown in Fig. 19(a) we find that the period of oscillations is T = 1.91 nm, hence ξ2 ∼ 0.30 nm and ξ1 ∼ 3.0 nm. In the clean limit, ξ2 = vF ~/2Eex . In this way, known ξ2 and estimating the Fermi velocity of being vF = 2.8 × 105 m/s, as reported in literature, we can calculate the exchange energy of the Co: Eex = ~vF /2T ≈ 309 meV. For Fe, from the theoretical fit we obtain ξ1 = 3.8 nm and ξ2 = 0.25 nm. So the

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Figure 19. Ic RN vs tCo (a) and tFe (b) with the best fitting theoretical model in agreement with Eq. 4, dash-dot violet line, and Eq. 5 solid green line.

period of the oscillations is T = 1.6 nm. Known ξ2 and vF = 1.98 × 105 m/s [33], we can calculate the exchange energy of the Iron: Eex = ~vF /2T ≈ 256 meV. To confirm that our oscillations are all in the clean limit (meaning that the considered Co and Fe thickness are always smaller than the Co and Fe mean free path), we have modeled our data with the simplified formula which holds only in this limit, Eq. 5, where in this case Eex and vF are the two fitting parameters. From the theoretical fit we obtain Eex (Co) = 309 meV, vF (Co) = 2.8×105 m/s and Eex (Fe) = 256 meV, vF (Fe) = 1.98×105 m/s. We remark that the best fits are obtained with exactly the same values as previously reported from Eq. 4. Both models provide excellent fits to our experimental data showing multiple oscillations of the critical current in a tiny (nano-scale) range of thicknesses of the Co and Fe barrier.

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Estimation of the Mean Free Path

With a simplified model that is obtained solving the linear Eilenberger equations [34] we can estimate the mean free path for Co and Fe. The general formula is: tanh

L ξef f

=

−1 ξef f −1 ξ0−1 + L−1 + iξH

(8)

−1 −1 −1 where ξef f is the effective decay length given by ξef f = ξ1 + iξ2 , ξo is the GinzburgLandau coherence length and ξH is a complex coherence length. In the clean limit 1 + −1 Lξ0−1 À 21 max{ln(1 + Lξ0−1 ), ln(LξH )}. The solution of Eq. 8 gives

ξ1−1 = ξ0−1 + L−1 , ξ0 =

vF ~ vF ~ , ξ2 = ξH = , 2πTc kB 2Eex

(9)

and the numerical solution is shown in Fig. 20 for Co and in Fig. 21 for Fe.

Figure 20. Estimation of Co mean free path. The dependence of ξ2 /ξ1 with inverse magnetic length, L/ξH , calculated for different ratios of L/ξ0 . Inset: inverse decay length, L/ξ2 = f (L/ξ0 ) for when L/ξH ' 0.1 In the case of Co [16], following this method we find from Fig. 20 that the experimental ratio ξ2 /ξ1 ' 0.1 corresponds to two inverse magnetic lengths of L/ξH ' 16.5 and L/ξH ' 18.7. By assuming L/ξ0 ' 0.1 and for the estimated parameters ξ1 ' 3 nm and ξ2 ' 0.3 nm we obtained, from the inset in Fig. 20 a mean free path L ' 5 nm. Furthermore to validate the determined mean free path of our Co thin film we have estimated LCo in a 50 nm thick Co film by measuring its resistivity as a function of temperature using the Van der Pauw technique. The transport in Co thin films is dominated by free-electron-like behavior [35] and hence the maximum mean free path is estimated from L = ~kF /ne e2 ρb , where ne is the electron density for Co and is estimated from the ordinary Hall effect to be 5.8 × 1028 cm−3 [36] and ρb is the residual resistivity. The residual resistance ratio ((ρT + ρb )/ρb ) was measured to be ' 1.41 and hence we calculate LCo at 4.2 K to be 10 nm. This verifies that our assumption that for Co the oscillations are all in the clean limit is well justified.

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Figure 21. Estimation of Fe mean free path. ξ2 /ξ1 vs L/ξH , calculated for different ratios of L/ξ0 = 0.2.Inset: L/ξH vs L/ξH to estimate the mean free path, L. The same method can be done to estimate the mean free path for Fe [17]. In this case the ratio ξ2 /ξ1 ' 0.06, so supposing L/ξ0 =0.2, we can extrapolate from the graphical solution (Fig. 21) a value of L/ξH ' 29. Considering the curve L/ξ2 vs L/ξH (see inset in Fig. 21) we obtain a value L/ξ2 of about 25 and we estimate the value of the mean free path of about 6nm. With this analysis we can remark that for Co and Fe the condition that all the thicknesses are in the clean limit is unambiguously fulfilled.

6.3.

Shapiro Steps and Fraunhofer Pattern

In Fig. 22 we show an example of typical I vs V characteristics for a sample with a barrier of 4.1 nm of Py (a) [21] and 0.8 nm of Fe (b) [17].

Figure 22. A typical I vs V curve of a Josephson junction with 4.1 nm of Py (a) and 0.8 nm of Fe (b) (black dash line) and the integer Shapiro steps in the voltage-current curve with an excitation at 13.86 GHz (a) and 14.01 GHz (red line). φ0 = h/2e For these I(V ) curves we also show the effect of an applied microwave field with an excitation at f = 13.86 GHz and f = 14.01 GHz, respectively. Constant voltage Shapiro steps appear due to the synchronization of the Josephson oscillations on the applied excitation [37]. As expected the steps manifest at voltages equal to integer multiples of hf /2e.

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Figure 23. A typical Fraunhofer pattern for a device with 2.5 nm thick Fe barrier. The central peak is offset by 4.9 mT, which corresponds to a coercitive field of the Fe barrier, as confirmed by the hysteresis loop (red line). We have looked at the effect of a magnetic field on the maximum supercurrent Imax in our devices. We find that Imax oscillates with applied magnetic field, giving rise to a Fraunhofer pattern; however, we also find that Imax , which normally corresponds to the central peak of a Fraunhofer pattern, is offset from zero applied field to ±Hof f set , which is equal to ±4.9 mT. Fig. 23 shows a typical Fraunhofer pattern for a Fe barrier device with a barrier thickness of ' 2.5 nm. We compared the variation of Imax with applied field to the magnetic hysteresis loop of the same film (measured prior to patterning and device fabrication) at 20 K. The offset field is found to correspond approximately to the coercive field of the unpatterned film, which is ±Hcoercive ' 4.3 mT. The central peak is shifted by the coercive field in each direction, which is due to the changing magnetization of the ferromagnetic barrier. The side peaks are not hysteretic and displaced by the saturation moment of the barrier because the hysteresis loop is saturated and the barrier moment is constant for both field sweep directions. The coercive field and offset field in the Fraunhofer pattern do not exactly agree: firstly, the coercive field is approximated at 20 K; secondly, processing a device in a FIB microscope is likely to harden the Fe magnetic domains by virtue of Ga ion implantation. Similar results were measured for Co, Ni, and Py.

6.4.

Half Integer Shapiro Steps Close to the 0-π Transition

In this subsection, we present a study of a single transition from the 0 to π state in Co devices where tCo thickness varied from 1.8 to 2.5 nm [15]. The Co magnetic dead layer was estimated to be 1.2 nm; although larger than the previously estimated dead layer of 0.8 nm for the Co barriers, we found that the bulk magnetizations for the two sample sets are similar. Importantly, the magnetic data convincingly showed incremental increases in magnetic moment with increasing Co thickness. From current-voltage measurements, Ic and RN were extracted so that Ic RN could be determined and tracked as a function of tCo . The characteristic voltage decreases to a small voltage around a mean Co barrier thickness of 2.05 nm and then increases, implying a change in phase of π. Each datum point in Fig. 24(a) was measured, and the vertical error bars were derived from a combination of esti-

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Figure 24. (a) Ic RN versus tCo compared to the fit of Eq. 5 used as an eye guide only; (b) I vs V for tCo = 2.1 nm without (dashed blue line) and with (solid red line) microwaves at f = 16.69 GHz; this device exhibits a sudden drop in differential resistance at V /nφ0 f =0.5. The inset shows the details of the half integer Shapiro step. mating Ic and RN from the current-voltage curves and from a small noise contribution from the current source. Particularly for those devices with the smallest characteristic voltage, there was a considerable scatter in the obtained Ic RN values. For an eye guide only, we have modeled the transition with Eq. 5, assuming Eex (Co) = 309 meV. The curve is offset to fit the experimental data. The model ignores any influence of a second harmonic term in the current-phase relation. In general, the current-phase relation is periodic in φ, the phase difference; however, recent theoretical and experimental works [38] have looked at the possibility of observing higher order harmonics, Is = Ic1 sin φ+Ic2 sin(2φ), where Ic2 À Ic1 at the 0 to π crossover and the second harmonic dominates. When Is = Ic2 sin(2φ), one expects both integer and half-integer Shapiro steps in the current-voltage curves at particular microwave frequencies. In our experiment [15], by applying microwaves in the 13–17 GHz range to those devices near the transition, we have found that the device with the smallest characteristic voltage and with a Co barrier thickness of 2.1 nm exhibited current steps at both half-integer n = 1/2 and integer n = 1 values of V /nφ0 f , as plotted in Fig. 24(b). These results imply that this device is close to a minimum characteristic voltage and provides evidence for a second harmonic in the current-phase relation; anyway further investigations must be done before any conclusive remarks can be made.

6.5.

Temperature Dependence of the Ic RN Product

Further evidence of the oscillatory dependence of the characteristic voltage is given by measurements of its thermal variation. We notice that, for each Co thickness, the Josephson junction resistance remains approximately constant, slightly increasing when approaching the critical temperature of Nb (TC = 9.2K), as expected; this is shown in Fig.25 for tCo =2.2 nm. On the other hand the critical current, and hence the product IC RN , quickly decreases with increasing T and goes to zero at the critical temperature of the Nb. This is shown explicitly in Fig. 26 (a) for different Co thicknesses.

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Figure 25. Temperature dependence of the Nb/Co/Nb Josephson junction with 2.2 nm of Co.

Figure 26. (a) IC RN vs T for different tCo . (b) Oscillations of the decay rate αT (T0 = 4.2K) of IC RN , Eq. 10, with the temperature [stars; the blue line is a guide to the eye], as compared with the oscillations of the IC RN product itself in Nb/Co/Nb π junctions. As for the Co data, in Fig. 27 (a) we show the temperature dependence of the IC RN product for different thicknesses of the Fe barrier. We remark that also in this case the IC RN product, quickly decreases with increasing the temperature [17]. It is interesting to notice how the rate at which the critical current drops to zero, exhibits a non-monotonic dependence on the Co and Fe layer thickness. We can define the relative decay rate αT of the IC RN product with the temperature as in Ref. [31], namely ¯ ¯ ¯ d ln[IC RN (T )] ¯ ¯ ¯ αT (T0 ) = ¯ (10) ¯ dT T =T0 ¯ ¯ ¯ dIC RN (T ) ¯ 1 ¯ ¯ , = ¯ ¯ dT T =T0 IC RN (T0 ) calculated at T0 = 4.2 K. In Fig. 26 (b) and in Fig. 27 (b) we plot αT as a function of tCo and tFe , respectively, for consecutive thicknesses; we observe oscillations of αT in phase with the oscillations of the IC RN , in other words the smaller IC RN decays to zero

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Figure 27. (a) Temperature dependence of the IC RN for different thicknesses of the Fe layer. (b) Oscillations of the decay rate αT (T0 = 4.2K) of IC RN , Eq. 10, with the temperature [stars; the blue line is a guide to the eye], as compared with the oscillations of the IC RN product itself [spheres; the red line is the fit given by Eq. 4] in a window of Fe thicknesses. with increasing T more slowly than the greater IC RN . A satisfactory explanation for this intriguing phenomenon awaits further investigation.

7.

Conclusion

In summary, we have fabricated and investigated Nb/F/Nb Josephson junctions, with F standing for a strong ferromagnetic layer of Ni, Py, Co or Fe, varying the F barrier thickness. A magnetic dead layer lower than 1.7 nm is extrapolated by a linear regression of the thickness dependence of the saturation magnetization. From the I(V ) vs V curves we have measured the Josephson critical current IC and the normal resistance RN in order to follow the oscillations of the IC RN product as a function of the F thickness. In agreement with the theoretical models for the clean and dirty limit, we have fitted the experimental data estimating the exchange energy and the Fermi velocity of the F barrier (see table 2 for a summary of the Eex and vF ). Shapiro steps appear at integer multiples of the applied voltage. Then, for different thickness of the Co and Fe barrier, we have shown that the IC RN product decreases with increasing temperature, and in particular the decay rate presents the same oscillatory behavior as the critical current. With the results of this chapter, we have shown that we are able to produce nanostructured Nb/F/Nb π-junctions with a high control of the F thickness (within an accuracy of 0.2 nm). The low magnetic dead layer gives us the possibility to reproduce the transport properties in our heterostructures with a small thickness deviation. The estimated Eex is close to that of the bulk F, which implies that our films are deposited cleanly with only a small reduction in exchange energy. Interfacial roughness and possibly interdiffusion of F into Nb is assumed to account for the slightly smaller exchange energy. Moreover, in particular for Co and Fe, which exhibit oscillations all in the clean limit, the high exchange energy means small period of IC RN oscillations, enabling us to obtain the switch from 0 to π state in a very small range of thicknesses (. 3 nm). We can therefore conclude that Co and Fe barrier based Josephson junctions are viable structures to the development of superconductor-based quantum electronic devices. The electrical and magnetic proper-

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ties of Co and Fe are well understood and are routinely used in the magnetics industry: we have demonstrated that Nb/Co/Nb and Nb/Fe/Nb hybrids can readily be used in controllable two-level quantum information systems. Table 2. Summary of the exchange energy Eex and the Fermi velocity vF extrapolated from theoretical fits of the critical current oscillations and the magnetic dead layer tD . Barrier Ni Py Co Fe

Eex (meV ) 80 201 309 256

vF (×105 m/s) 2.8 2.8 2.8 1.98

tD (nm) 1.7 0.5 0.7 1.1

In this context, a feasible future development will be to realize pseudo spin valve devices using two F layers made of different ferromagnetic materials with the different coercive fields separated by a superconducting layer [39]. In this case the spacer layer is relatively thick, and is used to decouple the ferromagnetic layers to prevent them from switching at the same field. In such an instance the fundamental physics will play with the interlink between these two competing orders, and on the practical point of view it could be the starting brick for suitable devices. In this way the magnetization orientation of the F/S/F structure can be controlled by a weak magnetic field which by itself is insufficient to destroy superconductivity, but the small magnetic field enables the softer F layer to be aligned with it, while the harder material will not switch. This artificial structure allows active control of the magnetic state of the barrier. The magnetoresistance of the pseudo spin valve gives direct access to the information about relative orientation of the ferromagnetic layers, and the magnetic state of the barrier. The novelty, and strength of such devices, lies in the fact that these heterostructures can be realized either with classical ferromagnetic materials and low-TC superconductors, or with colossal magneto-resistance materials and high-TC cuprates. So what will be the most promising future technology??? Will it rely on liquid helium, or nitrogen???? We hope to contribute towards providing a proper answer to such dilemmas. On a far-reaching scope, the route to room-temperature applications would constitute a more formidable challenge for our research.

Acknowledgements I would like to thank M. G. Blamire, J.W.A. Robinson and G. Burnell for sharing with me the joy of working together on these important topics presented in this chapter and I thank M. G. Blamire and the University of Cambridge for the kind hospitality. I acknowledge the support of the European Science Foundation π-shift network. Last but not least, I thank Gerardo Adesso for his support and always active collaboration and Giovanni Battista Adesso for his brightness while I wrote this chapter. At the end thanks to Blinky Bill!!

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References [1] Piano, S. Ph.D. thesis, University of Salerno, 2007. [2] Buzdin, A. I. Rev. Mod. Phys. 2005, 77, 935. [3] Soulen, R.; Byers, J.; Osofsky, M.; Nadgorny, B.; T. Ambrose, S. C.; Broussard, P.; Tanaka, C.; Nowak, J.; Moodera, J.; Barry, A.; Coey, J. Science 1998, 282, 85. [4] Strijkers, G. J.; Ji, Y.; Yang, F. Y.; Chien, C. L.; Byers, J. M. Phys. Rev. B 2001, 63, 104510. [5] Buzdin, A. I.; Kuprianov, M. V. JETP Lett. 1990, 52, 488. [6] Buzdin, A. I.; Kuprianov, M. V. JETP Lett. 1991, 53, 321. [7] Radovic, Z.; Dobrosavljevi´ c-Gruji´ c, L.; Buzdin, A. I.; Clem, J. R. Phys. Rev. B 1988, 38, 2388. [8] Radovic, Z.; Ledvij, M.; Dobrosavljevi´ c-Gruji´ c, L.; Buzdin, A. I.; Clem, J. R. Phys. Rev. B 1991, 44, 759. [9] Demler, E. A.; G. B. Arnold, M. R. B. Phys. Rev. B 1997, 55, 15 174. [10] Andreev, A. F. Zh. Eksp. Teor. Fiz. 1964, 46, 1128. [11] Kontos, T.; Aprili, M.; Lesueur, J.; Genˆet, F.; Stephanidis, B.; Boursier, R. Phys. Rev. Lett. 2001, 86, 304. [12] Kontos, T.; Aprili, M.; Lesueur, J.; Genˆet, F.; Stephanidis, B.; Boursier, R. Phys. Rev. Lett. 2002, 89, 137007. [13] Buzdin, A. I.; Bulaevskii, L.; Panyukov, S. JETP Lett. 1982, 35, 179. [14] Bergeret, F. S.; Volkov, A. F.; Efetov, K. B. Phys. Rev. B 2001, 64, 134506. [15] Robinson, J. W. A.; Piano, S.; Burnell, G.; Bell, C.; Blamire, M. G. Phys. Rev. B 2007, 76, 094522. [16] Robinson, J. W. A.; Piano, S.; Burnell, G.; Bell, C.; Blamire, M. G. Phys. Rev. Lett. 2006, 97, 177003. [17] Piano, S.; Robinson, J. W. A.; Burnell, G.; Bell, C.; Blamire, M. G. Eur. Phys. J. B 2007, 58, 123. [18] Kim, S.-J.; Latyshev, Y. I.; Yamashita, T. Appl. Phys. Lett. 1999, 74, 1156. [19] Bell, C.; Burnell, G.; Kang, D.-J.; Hadfield, R. H.; Kappers, M. J.; Blamire, M. G. Nanotechnology 2003, 14, 630. [20] Blamire, M. G. Supercond. Sci. Technol. 2006, 19, S132.

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[21] Robinson, J. W. A.; Piano, S.; Burnell, G.; Bell, C.; Blamire, M. G. IEEE Trans. Appl. Supercond. 2007, 17, 641. [22] Bell, C.; Loloee, R.; Burnell, G.; Blamire, M. G. Phys. Rev. B 2005, 71, 180501(R). [23] Born, F.; Siegel, M.; Hollmann, E. K.; Braak, H.; Golubov, A. A.; Gusakova, D. Y.; Kupriyanov, M. Y. Phys. Rev. B 2006, 74, 140501(R). [24] Kitada, M.; Shimizu, N. J. Mat. Sci. Lett. 1991, 10, 437. [25] Aarts, J.; Geers, J. M. E.; Brck, E.; Golubov, A. A.; Coehoorn, R. Phys. Rev. B 1997, 56, 2779. [26] Zhang, R.; Willis, R. F. Phys. Rev. Lett. 2001, 86, 2665. [27] Leng, Q.; Han, H.; Hiner, C. J. Appl. Phys. 2000, 87, 6621. [28] Pick, S.; Turek, I.; Dreyss´e, H. Solid State Commun. 2002, 124, 21. [29] Slater, J. C. Phys. Rev. 1936, 49, 537. [30] Pauling, L. J. Appl. Phys. 1937, 8, 385. [31] Blum, Y.; Tsukernik, A.; Karpovski, M.; Palevski, A. Phys. Rev. Lett. 2002, 89, 187004. [32] Heinmann, P.; Himpsel, F.; Eastman, D. Solid State Commun. 1981, 39, 219. [33] Covo, M. K.; Molvik, A. W.; Friedman, A.; Westenskow, G.; Barnard, J. J.; Cohen, R.; Seidl, P. A.; Kwan, J. W.; Logan, G.; Baca, D.; Bieniosek, F.; Celata, C. M.; Vay, J.-L.; Vujic, J. L. Phys. Rev. ST AB 2006, 9, 063201. [34] Gusakova, D. Y.; Kupriyanov, M. Y.; Golubov, A. A. JETP Lett. 2006, 83, 487. [35] Gurney, B. A.; Speriosu, V. S.; Nozieres, J. P.; Lefakis, H.; Wilhoit, D. R.; Need, O. U. Phys. Rev. Lett. 1993, 71, 4023. [36] K¨otzler, J.; Gil, W. Phys. Rev. B 2005, 72, 060412(R). [37] Barone, A.; Patern`o, G. Physics and Applications of the Josephson effect; John Wiley & Sons: New York, US, 1982. [38] Sellier, H.; Baraduc, C.; Lefloch, F.; Calemczuk, R. Phys. Rev. Lett. 2004, 92, 257005. [39] Bell, C.; Burnell, G.; Leung, C. W.; Tarte, E. J.; Kang, D.-J.; Blamire, M. G. Appl. Phys. Lett. 2003, 84, 1153.

In: Spintronics: Materials, Applications and Devices ISBN 978-1-60456-734-2 c 2009 Nova Science Publishers, Inc. Editors: G. C. Lombardi and G. E. Bianchi

Chapter 8

F UNDAMENTALS OF H ALF -M ETALLIC F ULL -H EUSLER A LLOYS ¨ K. Ozdo˘ gan∗ Department of Physics, Gebze Institute of Technology, Gebze, Kocaeli, Turkey E. S¸as¸ıo˘glu† Institut f¨ur Festk¨orperforschung, Forschungszentrum J¨ulich, J¨ulich, Germany and Fatih University, Physics Department, B¨uy¨ukc¸ekmece, ˙Istanbul, Turkey I. Galanakis‡ Department of Materials Science, School of Natural Sciences, University of Patras, Patra, Greece

Abstract Intermetallic Heusler alloys are amongst the most attractive half-metallic systems due to the high Curie temperatures and the structural similarity to the binary semiconductors. In this review we present an overview of the basic electronic and magnetic properties of the half-metallic full-Heusler alloys like Co 2MnGe. Ab-initio results suggest that the electronic and magnetic properties in these compounds are intrinsically related to the appearance of the minority-spin gap. The total spin magnetic moment in the unit cell, Mt , scales linearly with the number of the valence electrons, Zt , such that Mt = Zt − 24 for the full-Heusler alloys opening the way to engineer new half-metallic alloys with the desired magnetic properties. Moreover we present analytical results on the disorder in Co 2Cr(Mn)Al(Si) alloys, which is susceptible to destroy the perfect half-metallicity of the bulk compounds and thus degrade the performance of devices. Finally we discuss the appearance of the half-metallic ferrimagnetism due to the creation of Cr(Mn) antisites in these compounds and the Co-doping in Mn 2VAl(Si) alloys which leads to the fully-compensated half-metallic ferrimagnetism. ∗

E-mail address: [email protected] E-mail address: [email protected] ‡ E-mail address: [email protected] †

¨ K. Ozdo˘ gan, E. S¸as¸ıo˘glu and I. Galanakis

214

1.

Introduction

Half-metallic ferromagnets, initially predicted by de Groot and his collaborators in 1983[1] using first-principles calculations are at the center of scientific research due to their potential applications. The minority-spin electrons in these compounds show a semiconducting electronic band-structure while the majority-spin electrons present the usual metallic behavior of ferromagnets. Thus such alloys would ideally exhibit a 100% spin polarization at the Fermi level and therefore they should have a fully spin-polarized current and should be ideal spin injectors into a semiconductor maximizing the efficiency of spintronic devices.[2] Heusler alloys are well-know for several decades and the first Heusler compounds studied had the chemical formula X 2 YZ, where X is a high valent transition or noble metal atom, Y a low-valent transition metal atom and Z a sp element, and crystallized in the L21 structure which consists of four fcc sublattices.[3, 4] Heusler compounds present a series of diverse magnetic phenomena like itinerant and localized magnetism, antiferromagnetism, helimagnetism, Pauli paramagnetism or heavy-fermionic behavior and thus they offer the possibility to study various phenomena in the same family of alloys.[3, 4] A second class of Heusler alloys have the chemical formula XYZ and they crystallize in the C1b structure which consists of three fcc sublattices; they are often called half- or semi-Heusler alloys in literature, while the L21 compounds are referred to as full Heusler alloys. The interest in these types of intermetallic alloys was revived after the prediction,[1] using first-principles calculations, of half-metallicity in NiMnSb, a semi-Heusler compound. The high Curie temperatures[3, 4] exhibited by the half-metallic Heusler alloys are their main advantage with respect to other half-metallic systems (e.g. some oxides like CrO 2 and Fe3 O4 and some manganites like La 0.7Sr0.3MnO3 ).[5] While for the other compounds the Curie temperature is near the room temperature, for the half-metal NiMnSb it is 730 K and for the half-metallic Co 2 MnSi it reaches the 985 K.[3] A second advantage of Heusler alloys for realistic applications is their structural similarity to the zinc-blende structure, adopted by binary semiconductors widely used in industry (such as GaAs on ZnS). Semi-Heusler alloys have been already incorporated in several devices as spin-filters,[6] tunnel junctions[7] and GMR devices.[8] Recently, also the half-metallic full-Heusler alloys have found applications. The group of Westerholt has incorporated Co 2 MnGe in the case of spin-valves and multilayer structures[9] and the group of Reiss managed to create magnetic tunnel junctions based on Co 2MnSi. [10, 11] A similar study by Sakuraba and collaborators resulted in the fabrication of magnetic tunnel junctions using Co 2 MnSi as magnetic electrodes and Al-O as the barrier and their results are consistent with the presence of half-metallicity for Co2 MnSi.[12] Moreover Dong and collaborators recently managed to inject spin-polarized current from Co2 MnGe into a semiconducting structure.[13] In the present contribution we review our most recent results on the electronic properties of the half-metallic full-Heusler alloys obtained from first-principles electronic structure calculations. In section 2. we summarize our older results on bulk compounds obtained using the full-potential version of the screened Korringa-Kohn-Rostoker (KKR) method[14] (see references [15, 16, 17] for an extended review). In the next sections we overview some of our results on the defects in the half-metallic full Heusler alloys obtained using the full– potential nonorthogonal local–orbital minimum–basis band structure scheme (FPLO)[18, 19] within the local density approximation (LDA) [20, 21, 22] and employing the coherent

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potential approximation (CPA) to simulate the disorder in a random way.[19] In section 3. we present the physics of defects in the ferromagnetic Heusler alloys containing Co and Mn like Co2 MnSi.[23, 24] The study of defects, doping and disorder is of importance to accurately control the properties of half-metallic full-Heusler alloys.[25] In section 4. we demonstrate the creation of half-metallic ferrimagnets based on the creation of Cr and Mn antisites in Co 2 (Cr or Mn)(Al or Si) alloys[26, 27] and we expand this study to cover the case of Co defects in ferrimagnetic Mn 2 VAl and Mn2VSi alloys leading to full-compensated half-metallic ferrimagnets (also-known as half-metallic antiferromagnets).[28] Finally in section 5. we summarize and conclude.

2.

Electronic and Gap Properties – Slater Pauling Behavior

The electronic, magnetic and gap properties of half-metallic full-Heusler compounds have been studied using first-principles calculations in reference [29]. These results have been extensively also reviewed in reference [15], in the introductory chapter of [16] and in a chapter in reference [17]. The reader is directed to them for an extended discussion. In this section we will only briefly overview these properties.

Total spin magnetic moment (Mt)

7

Co2MnSi Co2MnGe Co2MnSn

6 5

Co2CrAl Fe2MnSi

4 3

Co2VAl Fe2MnAl

2 1

Co2MnAl Co2MnGa

Fe2CrAl

0

Fe2VAl

-1 -2 -3 20

Slater-Pauling rule:

Mn2VGe

Mt=Zt-24

Mn2VAl 21

22

23

24

25

26

27

28

29

30

31

32

Total number of valence electrons (Zt) Figure 1. Calculated total spin moment per unit cell in µB as a function of the total number Zt of valence electrons per unit cell for the full Heusler alloys. The dashed line represents the Slater-Pauling behavior. The electronic and magnetic properties of the full-Heusler compounds are similar to the half-Heusler compounds[30] with the additional complication of the presence of 2 Co atoms per unit cell as in Co 2 MnGe. In the case of the semi-Heusler alloys there are exactly

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nine occupied minority-spin electronic states; one of s character and three of p-character provided by the sp atom and five bonding d-states originating from the hybridization between the transition metal atoms. In addition to these bands, in the case of full-Heusler alloys there are five states exclusively located at the Co sites and near the Fermi level and the Fermi level is located among these states so that three out of five are occupied and two of them unoccupied leading to small energy gaps.[29] Now there are 12 minority-spin occupied states and the compounds with 24 valence electrons like Fe 2 VSb are semiconductors. If vanadium is substituted by a higher-valent atom, spontaneous spin polarization occurs, and the exchange splitting shifts the majority states to lower energies. The extra electrons fill in only majority states and the total spin moment per unit cell in µB , Mt , which is the number of uncompensated spins is the total number of valence electrons, Zt , minus two times the 12,: Mt =Zt-24. This behavior is the so-called Slater-Pauling behavior and e.g. half-metallic Co2 CrAl (27 valence electrons) has a total spin moment of 3 µB and halfmetallic Co2 MnSi (29 valence electrons) a spin moment of 5 µB . This rule provides a direct connection between the half-metallicity and the total spin moment which can be easily determined experimentally. In figure 1 we have plotted the calculated total spin moments for several full-Heusler compounds as a function of the total number of valence electrons. The dashed line represents the Slater-Pauling rule of half-metallic full Heusler alloys. Since 7 majority bands (2×Co eu , 5×Mn d) are unoccupied, the largest possible moment is 7 µB and would occur if all majority d-states were occupied. Of course it would be impossible to get a compound with a total spin moment of 7 µB but even Mt = 6 µB is difficult to obtain. As it was shown by Wurmehl et al. [31], the on-site correlations in Co 2 FeSi play a critical role for this compound and calculations within the LDA+U scheme, rather than the LDA, give a spin moment of 6 µB . We should also note here that ferromagnetism is stabilized by the inter-sublattice interactions between the Mn(Cr) and Co atoms and between Co atoms belonging to different sublattices as shown by first-principles calculations.[32, 33] Before closing this section we should discuss also the role of the sp-elements in halfmetallic Heusler alloys. While the sp-elements are not responsible for the appearance of the minority gap, they are very important for the physical properties of the Heusler alloys and their structural stability. sp atoms provide one s and three p bands per spin which lay very low in energy and accommodate d-electrons of the transition metal atoms. Thus the effective d-charge, which is accommodated by transition-metal atomic d-hybrids, is reduced stabilizing the half-metallicity. These s- and p-states of the sp atom strongly hybridize with the transition metal d-states and the charge in these bands is delocalized and locally the sp atoms even lose charge in favor of the transition metal atoms.

3.

Defects in Full-Heuslers Containing Co and Mn

In this section we discuss results on the defects in the case of Co 2 MnZ alloys where Z is Al and Si.[23, 24] To simulate the doping by electrons we substitute Fe for Mn while to simulate the doping of the alloys with holes we substitute Cr for Mn and we have considered the cases of moderate doping substituting 5% and 10% of the Mn atoms and we present our results in figure 2 for the Co 2 MnSi and Co2 MnAl compounds. As discussed in the previous section the gap is created between states located exclusively at the Co sites. The states low in energy (around -6 eV below the Fermi level) originate from the low-lying p-states of

Fundamentals of Half-Metallic Full-Heusler Alloys

217

the sp atoms and the ones at around -9 eV below the Fermi level are the s-states of the spatom. The majority-spin occupied states form a common Mn-Co band while the occupied minority states are mainly located at the Co sites and the minority unoccupied states at the Mn sites. The extra electron in the the Co 2 MnSi alloy occupies majority states leading to an increase of the exchange splitting between the occupied majority and the unoccupied minority states and thus to larger gap-width for the Si-based compound. In the case of the Al-based alloy the bonding and antibonding minority d-hybrids almost overlap and the gap is substituted by a region of very small minority density of states (DOS); we will call it a pseudogap. In both cases the Fermi level falls within the gap (Co 2 MnSi) or the pseudogap (Co2 MnAl) and an almost perfect spin-polarization at the Fermi level is preserved. 2

Co2MnSi Co2[Mn1-xCrx]Si Co2[Mn1-xFex]Si

x=0.05

DOS(states/eV)

0

-2 2

x=0.1

0

-2 -0.4

-0.2

0

0.2

0.4

Energy-EF (eV) 2

Co2MnAl Co2[Mn1-xCrx]Al Co2[Mn1-xFex]Al

x=0.05

DOS(states/eV)

0

-2 2

x=0.1

0

-2 -0.4

-0.2

0

0.2

0.4

Energy-EF (eV)

Figure 2. Spin-resolved density of states (DOS) for the case of Co 2 [Mn1−x Crx ]Si and Co2 [Mn1−x Fex ]Si in the upper panel, and Co 2 [Mn1−x Crx ]Al and Co2 [Mn1−x Fex ]Al in the lower panel for two values of the doping concentration x. DOS’s are compared to the one of the undoped Co 2 MnSi and Co2MnAl alloys. The zero of the energy axis corresponds to the Fermi energy. Positive values of the DOS correspond to the majority-spin (spin-up) electrons and negative values to the minority-spin (spin-down) electrons. In the insets we present the DOS for a wider energy range.

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Doping the perfect ordered alloys with either Fe or Cr smoothens the valleys and peaks along the energy axis. This is a clear sign of the chemical disorder; Fe and Cr induce peaks at slightly different places than the Mn atoms resulting to this smoothening and as the doping increases this phenomenon becomes more intense. The important detail is what happens around the Fermi level and in what extent is the gap in the minority band affected by the doping. So now we will concentrate only at the enlarged regions around the Fermi level. The dashed lines represent the Cr-doping while the dotted lines are the Fe-doped alloys. Cr-doping in Co 2MnSi has only a marginal effect to the gap. Its width is narrower with respect to the perfect compounds but overall the compounds retain their half-metallicity. For Co2 MnAl the situation is reversed with respect to the Co 2MnSi compound and Cr-doping has significant effects on the pseudogap. Its width is larger with respect to the perfect compound and becomes slightly narrower as the degree of doping increases. In the case of Fe-doping the situation is more complex. Adding electrons to the system means that, in order to retain the perfect half-metallicity, these electrons should occupy high-energy lying antibonding majority states. This is energetically not very favorable in the case of Co2 MnSi and for these moderate degrees of doping a new shoulder appears in the unoccupied states which is close to the right-edge of the gap; a sign of a large change in the competition between the exchange splitting of the Mn majority and minority states and of the Coulomb repulsion. In the case of the 20% Fe doping in Co 2MnSi (not shown here) this new peak crosses the Fermi level and the Fermi level is no more exactly in the gap but slightly above it. Further substitution should lead to the complete destruction of the half-metallicity.[34] Recent ab-initio calculations including the on-site Coulomb repulsion (the so-called Hubbard U ) have predicted that Co 2 FeSi is in reality half-metallic reaching a total spin magnetic moment of 6 µB which is the largest known spin moment for a halfmetal.[31, 35] Fe-doping on the other hand in Co 2 MnAl almost does not change the DOS around the Fermi level. The extra-electrons occupy high-energy lying antibonding majority states but, since Co 2 MnAl has one valence electron less than Co 2 MnSi, half-metallicity remains energetically favorable and no important changes occur upon Fe-doping and further substitution of Fe for Mn should retain the half-metallicity even for the Co 2 FeAl compound although LDA-based ab-initio calculations predict that the limiting case of Co 2 FeAl is almost half-metallic.[34]

4.

Defects Driven Half-Metallic Ferrimagnetism

In the previous section we have examined the case of defects in half-metallic ferromagnets. Half-metallic ferrimagnetism (HMFi) on the other hand is highly desirable since such compounds would yield lower total spin moments than the corresponding ferromagnets. Well-known HMFi are the perfect Heusler compounds FeMnSb and Mn 2 VAl.[36] We will present in the first part of this section another route to half-metallic ferrimagnetism based on antisites created by the migration of Cr(Mn) atoms at Co sites in the case of Co 2 CrAl, Co2 CrSi, Co2 MnAl and Co2MnSi alloys. The ideal case for applications would be a fullycompensated ferrimagnet, also known as half-metallic antiferromagnet (HMA),[37] since such a compound would not give rise to stray flux and thus would lead to smaller energy consumption in devices. In the secobd part of this section we present a way to achieve HMA based on Co defects in half-metallic ferrimagnets like Mn 2 VAl.

Fundamentals of Half-Metallic Full-Heusler Alloys

219

[Co1-xCrx]2CrSi

4

DOS (states/eV)

0

-4 x=0 x=0.1

[Co1-xCrx]2CrAl

4

0

-4 -2

0

2

Energy-EF (eV)

Figure 3. Total density of states (DOS) as a function of the concentration x for the [Co1−x Crx ]2CrAl (upper panel) and [Co 1−x Crx ]2 CrSi (lower panel) compounds. We will start our discussion from the Cr-based alloys and using Co 2CrAl and Co2 CrSi as parent compounds we create a surplus of Cr atoms which sit at the perfect Co sites. In figure 3 we present the total density of states (DOS) for the [Co 1−x Crx ]2 CrAl and [Co1−x Crx ]2CrSi alloys for concentrations x= 0 and 0.1, and in table 1 we have gathered the spin moments for the two compounds under study. We will start our discussion from the DOS. The perfect compounds show a gap in the minority-spin band and the Fermi level falls within this gap and thus the compounds are half-metals. When the sp atom is Si instead of Al the gap is larger due to the extra electron which occupies majority states of the transition metal atoms[29] and increases the exchange splitting between the majority occupied and the minority unoccupied states. This electron increases the Cr spin moment by ∼0.5 µB and the moment of each Co atom by ∼0.25 µB about. The Cr and Co majority states form a common band and the weight at the Fermi level is mainly of Cr character. The minority occupied states are mainly of Co character. When we substitute Cr for Co, the effect on the atomic DOS of the Co and Cr atoms at the perfect sites is marginal. The DOS of the impurity Cr atoms has a completely different form from the Cr atoms at the perfect sites due to the different symmetry of the site where they sit. But although Cr impurity atoms at the antisites induce minority states within the gap, there is still a tiny gap and the Fermi level falls within this gap keeping the half-metallic character of the parent compounds. The discussion above on the conservation of the half-metallicity is confirmed when we compare the calculated total moments in table 1 with the values predicted by the Slater Pauling rule for the ideal half-metals. Since Cr is lighter than Co, substitution of Cr for Co decreases the total number of valence electrons and the total spin moment should also decrease. This is achieved due to the antiferromagnetic coupling between the Cr impurity atoms and the Co and Cr ones at the ideal sites, which would have an important negative contribution to the total moment as confirmed by the results in table 1. Thus the Cr-doped alloys are half-metallic ferrimagnets and their total spin moment is considerable smaller than

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Table 1. Atom-resolved spin magnetic moments for the [Co 1−xCrx ]2 CrAl, [Co1−x Crx ]2 CrSi, [Co1−xMnx ]2 MnAl and [Co1−x Mnx ]2 MnSi compounds (moments have been scaled to one atom). The two last columns are the total spin moment (Total) in the unit cell calculated as Co 2 × [(1 − x) ∗ m + x ∗ mCr or M n(imp)] + mCr(M n) + mAl or Si and the ideal total spin moment predicted by the Slater-Pauling rule for half-metals (see section 2.). With IMP we denote the Cr(Mn) atoms sitting at perfect Co sites. Compound Co2 CrAl [Co0.95Cr0.05]2 CrAl [Co0.9Cr0.1 ]2 CrAl Co2 CrSi [Co0.95Cr0.05]2 CrSi [Co0.9Cr0.1 ]2 CrSi Co2 MnAl [Co0.95Mn0.05]2 MnAl [Co0.9Mn0.1 ]2 MnAl Co2 MnSi [Co0.95Mn0.05]2 MnSi [Co0.9Mn0.1 ]2 MnSi

Co 0.73 0.71 0.69 0.95 0.93 0.91 0.68 0.73 0.78 0.98 0.99 0.99

IMP – -1.82 -1.85 – -1.26 -1.26 – -2.59 -2.49 – -0.95 -0.84

Cr/Mn 1.63 1.62 1.61 2.17 2.12 2.07 2.82 2.82 2.83 3.13 3.09 3.06

Al/Si -0.09 -0.09 -0.08 -0.06 -0.06 -0.05 -0.14 -0.13 -0.12 -0.09 -0.08 -0.07

Total 3.00 2.70 2.40 4.00 3.70 3.40 4.04 3.81 3.61 5.00 4.80 4.60

Ideal 3.00 2.70 2.40 4.00 3.70 3.40 4.00 3.80 3.60 5.00 4.80 4.60

the perfect half-metallic ferromagnetic parent compounds; in the case of [Co 0.8 Cr0.2]2 CrAl it decreases down to 1.8 µB from the 3 µB of the perfect Co2 CrAl alloy. Here we have to mention that if also Co atoms migrate to Cr sites (case of atomic swaps) the half-metallity is lost, as it was shown by Miura et al. [38], due to the energy position of the Co states which have migrated at Cr sites. Similar phenomena to the discussion above occur in the [Co 1−x Mnx ]2MnZ compounds varying the sp atom, Z, which is one of Al or Si. We have taken into account five different values for the concentration x; x= 0, 0.025, 0.05, 0.1, 0.2. In figure 4 we have drawn the total density of states (DOS) for both families of compounds under study and for two different values of the concentration x: the perfect compounds (x=0) and for one case with defects, x= 0.1. As mentioned above and in agreement with previous previous electronic structure calculations on these compounds Co 2 MnSi compounds present a real gap and Co 2 MnAl a pseudogap.[29, 23, 39, 40] When we create a surplus of Mn atoms which migrate at sites occupied by Co atoms in the perfect alloys, the gap persists and both compounds retain their half-metallic character as occurs also for the Cr-based alloys presented above. Especially for Co2MnSi, the creation of Mn antisites does not alter the width of the gap and the half-metallicity is extremely robust in these alloys with respect to the creation of Mn antisites. The atomic spin moments show behavior similar to the Cr alloys as can be seen in table 1 and the spin moments of the Mn impurity atoms are antiferromagnetically coupled to the spin moments of the Co and Mn atoms at the perfect sites resulting to the desired half-metallic ferrimagnetism.

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221

[Co1-xMnx]2MnAl

4

DOS (states/eV)

0

-4 x=0 x=0.1

[Co1-xMnx]2MnSi

4

0

-4 -2

0

2

Energy-EF (eV)

Figure 4. Total density of states (DOS) around the gap region for the [Co 1−x Mnx ]2MnAl and [Co1−x Mnx ]2MnSi alloys as a function of the concentration x : we denote x = 0 with the solid line and x = 0.1 with a dashed thick line. The ideal case of fully-compensated half-metallic ferrimagnet or half-metallic antiferromagnetic (HMA) can be achieved by doping with Co the Mn 2VAl and Mn2 VSi alloys which are well known to be HMFi. The importance of this route stems from the existence of Mn2 VAl in the Heusler L21 phase as shown by several groups.[41, 42, 43] Each Mn atom has a spin moment of around -1.5 µB and V atom a moment of about 0.9 µB .[41, 42, 43] All theoretical studies on Mn 2 VAl agree on the half-metallic character with a gap at the spin-up band instead of the spin-down band as for the other half-metallic Heusler alloys.[29, 44, 45, 46] Prior to the presentation of our results we have to note that due to the Slater-Pauling rule, these compounds with less than 24 valence electrons have negative total spin moments and the gap is located at the spin-up band. Moreover the spin-up electrons correspond to the minority-spin electrons and the spin-down electrons to the majority electrons contrary to the other Heusler alloys.[29] We have substituted Co for Mn in Mn 2 V(Al or Si) in a random way and in figure 5 we present the total and atom-resolved density of states (DOS) in [Mn 1−x Cox ]2 VAl (solid line) and [Mn 1−x Cox ]2 VSi (dashed line) alloys for x=0.1. The perfect compounds show a region of low spin-up DOS (we will call it a “pseudogap”) instead of a real gap. Upon doping the pseudogap at the spin-up band persists and the quaternary alloys keep the half-metallic character of the perfect Mn 2 VAl and Mn2 VSi compounds. Co atoms are strongly polarized by the Mn atoms since they occupy the same sublattice and they form Co-Mn hybrids which afterwards interact with the V and Al or Si states.[29] The spin-up Co states form a common band with the Mn ones and the spin-up DOS for both atoms has similar shape. Mn atoms have less weight in the spin-down band since they accommodate less charge than the heavier Co atoms. In table 2 we have gathered the total and atom-resolved spin moments for all the Co-

¨ K. Ozdo˘ gan, E. S¸as¸ıo˘glu and I. Galanakis

222

Mn

DOS (states/eV)

4 Total 0

-4 [Mn0.9Co0.1]2VAl

4 Co

V

[Mn0.9Co0.1]2VSi

0 -4 -6

-3

0

3-6

-3

0

3

Energy-EF (eV) Figure 5. Total and atom-resolved DOS for the [Mn 0.9Co0.1 ]2 VAl and [Mn0.9Co0.1]2 VSi compounds. Note that the atomic DOS’s have been scaled to one atom.

doped compounds as a function of the concentration. We have gone up to a concentration which corresponds to 24 valence electrons in the unit cell, thus up to x=0.5 for the [Mn1−x Cox ]2 VAl and x=0.25 for the [Mn 1−x Cox ]2VSi alloys. In the last column we have included the total spin moment predicted by the Slater-Pauling rule for the perfect halfmetals. A comparison between the calculated and ideal total spin moments reveals that all the compounds under study are half-metals with very small deviations due to the existence of a pseudogap instead of a real gap. Exactly for 24 valence electrons the total spin moment vanishes as we will discuss in the next paragraph. Co atoms have a spin moment parallel to the V one and antiparallel to the Mn moment, and thus the compounds retain their ferrimagnetic character. As we increase the concentration of the Co atoms in the alloys, each Co has more Co atoms as neighbors, it hybridizes stronger with them and its spin moment increases while the spin moment of the Mn atom decreases (these changes are not too drastic). The sp atoms have a spin moment antiparallel to the Mn atoms as already discussed in reference [47]. The most interesting point in this substitution procedure is revealed when we increase the Co concentration to a value corresponding to 24 valence electrons in the unit cell, thus the [Mn0.5Co0.5]2 VAl and [Mn0.75Co0.25]2 VSi alloys. The Slater-Pauling rule predicts for these compounds a zero total spin moment in the unit cell and the electrons population is equally divided between the two spin-bands. Our first-principles calculations reveal that this is actually the case. The interest arises from the fact that although the total moment is zero, these two compounds are made up from strongly magnetic components. Mn atoms have a mean spin moment of ∼-1.4 µB in [Mn0.5Co0.5]2 VAl and ∼-0.9 µB in [Mn0.75Co0.25]2 VSi. Co and V have spin moments antiferromagnetically coupled to the Mn ones which for

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223

Table 2. Atom-resolved spin magnetic moments for the [Mn 1−x Cox]2 VAl and [Mn1−x Cox]2 VSi compounds (moments have been scaled to one atom). The two last columns are the total spin moment (Total) in the unit cell calculated as 2 × [(1 − x) ∗ mM n + x ∗ mCo ] + mV + mAl or Si and the ideal total spin moment predicted by the Slater-Pauling rule for half-metals. The lattice constants have been chosen 0.605 nm for Mn2 VAl and 0.6175 for Mn2 VSi for which both systems are half-metals (see reference [47]) and have been kept constant upon Co doping. Compound Mn2VAl [Mn0.9Co0.1]2 VAl [Mn0.7Co0.3]2 VAl [Mn0.5Co0.5]2 VAl Compound Mn2VSi [Mn0.9Co0.1]2 VSi [Mn0.75Co0.25]2 VSi

Mn -1.57 -1.56 -1.48 -1.39 Mn -0.96 -0.93 -0.90

Co – 0.340 0.46 0.59 Co – 0.82 0.94

V 1.08 1.07 0.95 0.78 V 0.86 0.85 0.84

Al 0.06 0.07 0.05 0.02 Si 0.06 0.05 0.04

Total -2.00 -1.60 -0.80 ∼0 Total -1.00 -0.60 ∼0

Ideal -2.0 -1.6 -0.8 0 Ideal -1.0 -0.6 0

[Mn0.5Co0.5 ]2VAl are ∼0.6 and ∼0.8 µB , respectively, and for [Mn0.75Co0.25]2 VSi ∼0.9 and ∼0.8 µB . Thus these two compounds are half-metallic fully-compensated ferrimagnets or as they are best known in literature half-metallic antiferromagnets.

5.

Summary and Conclusions

In this chapter we have reviewed our results on the defects in half-metallic full-Heusler alloys. Firstly we have presented a short overview of the electronic and magnetic properties of the half-metallic full-Heusler alloys and have discussed in detail the Slater-Pauling behavior of these alloys (the total spin moment scales linearly with the total number of valence electrons as a result of the half-metallicity). We have also studied the effect of doping on the magnetic properties of the Co 2MnAl(Si) full-Heusler alloys. Doping simulated by the substitution of Cr and Fe for Mn overall keeps the half-metallicity and has little effect on the half-metallic properties of Co 2 MnSi and Co2 MnAl compounds. Afterwards, we have studied the effect of defects-driven appearance of half-metallic ferrimagnetism in the case of the Co 2 Cr(Mn)Al(Si) Heusler alloys. More precisely, based on first-principles calculations we have shown that when we create Cr(Mn) antisites at the Co sites, these impurity Cr(Mn) atoms couple antiferromagnetically with the Co and the Cr(Mn) atoms at the perfect sites while keeping the half-metallic character of the parent compounds. The ideal case of half-metallic fully-compensated ferrimagnets (also known as half-metallic antiferromagnets) can be achieved by doping of the half-metallic ferrimagnets Mn2 VAl and Mn2VSi. Co substitution for Mn keeps the half-metallic character of the parent compounds and when the total number of valence electrons reaches the 24, the total spin moment vanishes as predicted by the Slater-Pauling rule. Defects are a promising alternative way to create robust half-metallic ferrimagnets, which are crucial for magnetoelectronic

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applications.

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[14] N. Papanikolaou, R. Zeller, and P. H. Dederichs, J. Phys.: Condens. Matter 14, 2799 (2002). [15] I. Galanakis, Ph. Mavropoulos, and P. H. Dederichs, J. Phys. D: Appl. Phys. 39, 765 (2006). [16] Half-metallic alloys: fundamentals and applications, Eds.: I. Galanakis and P. H. Dederichs, Lecture notes in Physics vol. 676 (Berlin Heidelberg: Springer 2005). [17] ”Electronic and Magnetic Properties of the Normal and Quaternary Full-Heusler Alloys: The Quest for New Half-Metallic Ferromagnets”, I. Galanakis, in ”New Developments in Ferromagnetism Research”, V.N. Murray (ed.), (Nova Publishers, New York 2005), pp 79-97. [18] K. Koepernik and H. Eschrig, Phys. Rev. B 59, 3174 (1999). [19] K. Koepernik, B. Velicky, R. Hayn, and H. Eschrig, Phys. Rev. B 58, 6944 (1998). [20] P, Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). [21] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). [22] J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992). ¨ [23] I. Galanakis, K. Ozdo˘ gan, B. Aktas¸, and E. S¸as¸ıo˘glu, Appl. Phys. Lett. 89, 042502 (2006). ¨ [24] K. Ozdo˘ gan, E. S¸as¸ıo˘glu, B. Aktas¸, and I. Galanakis, Phys. Rev. B 74, 172412 (2006). [25] S. Picozzi and A. J. Freeman, J. Phys.: Condens. Matter 19, 315215 (2007). ¨ [26] K. Ozdo˘ gan, I. Galanakis, E. S¸as¸ıo˘glu, and B. Aktas¸, Phys. Stat. Sol. (RRL) 1, 95 (2007). ¨ [27] K. Ozdo˘ gan, I. Galanakis, E. S¸as¸ıo˘glu, and B. Aktas¸, Sol. St. Commun. 142, 492 (2007). ¨ [28] I. Galanakis, K. Ozdo˘ gan, E. S¸as¸ıo˘glu, and B. Aktas¸, Phys. Rev. B 75, 092407 (2007). [29] I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys. Rev. B 66, 174429 (2002). [30] I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys. Rev. B 66, 134428 (2002). [31] H. M. Kandpal, G. H. Fecher, C. Felser, and G. Sch¨onhense, Phys. Rev. B 73, 094422 (2006). [32] E. S¸as¸ıo˜glu, L. M. Sandratskii, P. Bruno, and I. Galanakis, Phys. Rev. B 72, 184415 (2005). [33] Y. Kurtulus, R. Dronskowski, G. D. Samolyuk, and V. P. Antropov, Phys. Rev. B 71, 014425 (2005).

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[34] I. Galanakis, J. Phys.: Condens. Matter 16, 3089 (2004). [35] S. Wurmehl, G. H. Fecher, H. C. Kandpal, V. Ksenofontov, C. Felser, and H. -J. Lin, Appl. Phys. Lett. 88, 032503. [36] R. A. de Groot, A. M. van der Kraan, and K. H. J. Buschow, J. Magn. Magn. Mater. 61, 330 (1986). [37] H. van Leuken and R. A. de Groot, Phys. Rev. Lett. 74, 1171 (1995). [38] Y. Miura, K. Nagao, and M. Shirai, Phys. Rev B 69, 144413 (2004). [39] M. Sargolzaei, M. Richter, K. Koepernik, I. Opahle, H. Eschrig, and I. Chaplygin, Phys. Rev. B 74, 224410 (2006). ¨ [40] K. Ozdo˘ gan, B. Aktas¸, I. Galanakis, and E. S¸as¸ıo˘glu, J. Appl. Phys. 101, 073910 (2007). [41] Y. Yoshida, M. Kawakami, and T. Nakamichi, J. Phys. Soc. Jpn. 50, 2203 (1981). [42] H. Itoh, T. Nakamichi, Y. Yamaguchi, and N. Kazama, Trans. Jpn. Inst. Met. 24, 265 (1983). [43] C. Jiang, M. Venkatesan, and J. M. D. Coey, Sol. St. Commun. 118, 513 (2001). [44] E. S¸as¸ıo˘glu, L. M. Sandratskii, and P. Bruno, J. Phys.: Condens. Matter 17, 995 (2005). [45] S. Ishida, S. Asano, and J. Ishida, J. Phys. Soc. Jpn. 53, 2718 (1984). [46] R. Weht and W. E. Pickett, Phys. Rev. B 60, 13006 (1999). ¨ [47] K. Ozdo˘ gan, I. Galanakis, E. S¸as¸ıo˘glu, and B. Aktas¸, J. Phys.: Condens. Matter 18, 2905 (2006).

In: Spintronics: Materials, Applications and Devices Editors: G. C. Lombardi and G. E. Bianchi

ISBN: 978-1-60456-734-2 © 2009 Nova Science Publishers, Inc.

Chapter 9

SPIN DRIFT–DIFFUSION AND ITS CROSSOVER IN SEMICONDUCTORS M. Idrish Miah1,2 1

Nanoscale Science and Technology Centre and School of Biomolecular and Physical Sciences, Griffith University, Nathan, Brisbane, QLD 4111, Australia. 2 Department of Physics, University of Chittagong, Chittagong, Chittagong – 4331, Bangladesh.

Abstract A two-component drift-diffusion equation for spin density is derived and is used to model spin transport in nonmagnetic semiconductors in different transport regimes. We study spin current (js) and find that drift (j) and diffusion (jdi) currents contribute to js in the down-stream and up-stream directions and that jdi decreases with the electric field (E) while j increases and there is a spin drift-diffusion crossover field (Ec) in the down-stream direction. It is also found that js increases in the degenerate regime, suggesting a possible way for the enhancement of js in semiconductors. However, js in the up-stream direction is found to be vanished when E is very large. We derive the expressions for the intrinsic spin diffusion length (δs) of a semiconductor in different electron statistical regimes, namely nondegenerate, degenerate and highly degenerate and show that δs can be obtained directly from Ec. The results of the investigation are also shown to be useful in identifying whether the process for a given E jdi ), spin drift-diffusion crossover ( j ≈ jdi ) or spin would be in the spin drift ( j diffusive (

j

jdi )

regime and in obtaining transport properties of the electron spin in

semiconductors, the crucial requirements for practical spintronic devices.

1. Introduction In addition to charge and mass, electrons have a third property called spin. But so far, electronics has neglected the spin. Electron spin (self-rotation, rotation of the electron on its axis) is a quantum mechanical property associated with its intrinsic angular momentum, and as an intrinsic magnetic moment is associated with spin, spin is closely related to magnetic

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M. Idrish Miah

phenomena. The idea to use the spin property of electrons in electronic devices can result in a promising trend for future spin-based electronics (spintronics) [1-4]. Spintronics refers to the study of the role played by electron spin in solid state physics, and possible devices, such as spin field effect transistors (SFETs), spin storage/ memory devices or spin quantum computers, that specifically exploit spin properties instead of, or in addition to, charge degree of freedom. It is generally expected that addition of spin degree of freedom in information processing will extend the functionality of conventional devices and allow development of novel electronic devices (spintronic devices), which can hold promise of e.g., reduced power consumption, faster operation and smaller size. Recent interest [5,6] has been motivated by successful examples of metallic spintronic devices, such as ferromagnetic metal-based reading heads for hard disc drives and magnetic random access memory [7]. These first metallic spintronic devices were sandwiched structures consisting of alternating ferromagnetic and nonmagnetic metal layers whose electric resistance depends strongly on the external magnetic field. Depending on the relative orientation of the magnetizations in the magnetic layers, the device resistance changes from small (parallel magnetizations) to large (antiparallel magnetizations). This change in resistance is called the giant magnetoresistance, a quantum mechanical effect discovered in 1986 in layered magnetic thin-film structures. In comparison with metal-based spintronics, utilization of semiconductors promises more versatile design due to the ability to adjust potential variation and spin polarization in the device by, for example, external voltage and device structure. Although different types of semiconductor devices have recently been proposed [4-12], the actual advantages of these devices as compared to the conventional electronic devices have not yet been clearly established. Even, for example, the most promising proposal [8] for an electronic analogue of an electro-optical modulator, that was later termed “SFET”, has been criticized [13]. One of the major hurdles in the development of spintronic devices has been the problem of efficiently injecting spin-polarized carriers into a semiconductor, transporting them over reasonable distances without spin-flipping or spin relaxation (because if spin relaxes too fast, the distance travelled by electron without losing its spin will be too short to perform any practical operation) and then detecting them. Moreover, in comparison with conventional microelectronic devices, detailed spin-dependent or spin transport parameters are still not well studied. In order to understand properly the spin transport and to successfully implement new device concepts, realistic models are very much needed. Much effort has thus been spent in understanding the generation/ injection, transport and detection of spin currents in semiconductors [4-6,8-20]. The generation or injection of spin current into a semiconductor has been achieved either by optical methods [5,6] or mostly by magnetic semiconductors or ferromagnetic contacts [5,16-18]. Although the detection of spin current has previously been achieved mainly through optical methods [5,19-21], electrical detection of spin current is very desirable for fully exploring the possibility of utilizing spin degree of freedom and for practical spintronic devices. Very recently, an electrical means of detecting spin current in semiconductors has also been reported [22-24]. Here we study spin transport and spin currents in nonmagnetic semiconductors using a two-component (spin-up and spin-down) drift-diffusion model. We first derive a driftdiffusion equation for spin density to examine the drift and diffusion contributions to the spin current (js). We find that there is a spin drift-diffusion crossover field (Ec), where the drift and diffusion contribute equally to js in the down-stream direction, suggesting a possible way to identify whether the process for a given electric field (E) would be in the spin drift, spin drift-

Spin Drift–Diffusion and Its Crossover in Semiconductors

229

diffusion crossover or spin diffusive regime, and that js in the up-stream direction vanishes when E is very large. We then derive the expressions for the intrinsic spin diffusion length (δs) of a semiconductor in different electron statistical regimes, namely nondegenerate, degenerate and highly degenerate and show that δs can be obtained directly from Ec. It is also found that js increases in the degenerate regime, suggesting a possible way for the enhancement of js in semiconductors.

2. Spin Drift–Diffusion Model We start from the following continuity equation that takes into account spin relaxation

∂cα ( −α ) (r , t ) ∂t

=

∇. jα ( −α ) (r ) e

−

cα ( −α ) (r , t ) − c−α (α ) (r , t )

τ s ,α ( −α )

,

(1)

where α is the spin index (α ≡ spin-up or ↑, - α ≡ spin-down or ↓) and cα ( −α ) (r , t ) ,

jα ( −α ) (r ) and τ α ( −α ),s denote, respectively, carrier density, current density and spin relaxation time with spin α (-α). Assuming that there is no space charge and that the material is homogeneous, the expression for the current density of a spin species, including the drift and diffusion contributions, can be written as

jα ( −α ) (r ) = σ α ( −α ) E + eCd ,α ( −α )∇cα ( −α ) (r , t ) ,

(2)

σ α ( −α ) = ecα ( −α ) μα ( −α ) ,

(3)

with

where

μα ( −α ) , σ α ( −α ) and Cd ,α ( −α ) are the mobility, Drude conductivity and diffusion

coefficient for carriers with spin α (-α). The first term on the right (Eq. (2)) is the drift current density and the second is the diffusion current density. We assume that the transport is unipolar (n-doped semiconductor) so that the electron-hole recombination can be neglected. Applying the local charge neutrality constraint ( cα + c−α = 0 ) carefully, we obtain

∂[cα (r , t ) − c−α (r , t )] c (r , t ) − c−α (r , t ) = μ E.∇[cα (r , t ) − c−α (r , t )] + Cd ∇ 2 [cα ( r , t ) − c−α (r , t )] − α ∂t τs ,

(4)

where we have assumed that the mobility, diffusion coefficient and spin relaxation time are equal for spin-up and spin-down carriers ( μα = μ −α = μ , Cd ,α = Cd , −α = Cd and

τ s ,α = τ s ,−α = τ s ) and (τ s ) −1 = (τ s ,α ) −1 + (τ s ,−α ) −1 . Defining the spin density as

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M. Idrish Miah

s (r ) = cα (r ) − c−α (r ) ,

(5)

we now obtain, from Eq. (4) in steady-state, i.e. when ∂ (cα − c−α ) / ∂t = 0 ,

∇ 2 s (r ) +

where

μ E.∇s(r ) Cd

=

s(r )

δ s2

,

(6)

δ s = Cdτ s is the intrinsic spin diffusion length. Eq. (6) is a drift-diffusion

homogeneous equation for the spin density, which includes the electric field effect in the second term on the left. This drift-diffusion-type model, also known as spin drift-diffusion model, has recently been used widely to model spin-polarized electron transport or spin transport in general in semiconductors [9-11,14,25,26]. In spin transport studies of semiconductors in the diffusive regime, the spin polarization is usually assumed to obey the same usual diffusion equation for the electrochemical potential as in metals (e.g. [27-31]), where there are no electric field effects and the spin polarization decays exponentially on a length scale of δs, from the point of injection as

∇ 2 Δϕ (r ) =

ϕ (r ) , δ s2

(7)

where Δϕ (r ) = ϕα (r ) − ϕ−α (r ) is the splitting of the electrochemical potentials

ϕα (ϕ−α ) of up-spin (down-spin) carriers owing to the imbalance of carrier densities in the ±α spin bands. In semiconductors, in contrast to metals, the field remains partially unscreened. Therefore, it is necessary to study the spin transport in the drift-diffusion regime. The spindependent electrochemical potential for electrons in a semiconductor is related to its density by

cα (r ) = cα0 {e β (ϕα −ϕ0 ) − 1} , where

(8)

β is the inverse of the thermal energy Ek = k BT , cα0 is the equilibrium value of cα

and φ0 is the value of the electrochemical potential that the system would have without spin polarization. φ0 is determined by ∇ϕ0 ( r ) = (e / σ ) j = eE , which, after integration by assuming that the material is homogeneous and combination with Eq. (8), reduces to an

ϕα (r ) = β −1 ln(1 + cα / cα0 ) + eE.r + Ω (where Ω is an integration constant), from which Eq. (6) can be shown in terms of Δϕ , which would

expression for φ for the individual spins:

be nonlinear in general, but linear and equivalent in the limit Δϕ

β −1 , i.e. in the

nondegenerate regime, indicating the advantage of using the spin density instead of the electrochemical potential splitting to describe spin transport in the degenerate regime by the drift-diffusion model.

Spin Drift–Diffusion and Its Crossover in Semiconductors

231

3. Spin Transport and Spin Currents To understand the electric field effects on spin transport and to deduce the spin transport properties, we consider a system depicted in Fig. 1. We take the electric field along the negative y-direction and consider x to be the transverse direction. Suppose that a continuous spin unbalance is generated or injected at the centre of the system at x=0 and y=0 with spin polarization axis taken to be either z or –z. From the experimental standpoint, such a spin polarization can be created locally in the bulk of a semiconductor by, e.g. using ferromagnetic-metal scanning tunnelling microscopy (STM) tips, or optical excitations. For optical excitation of bulk zinc-blende semiconductors, such as GaAs, with photon energy just above the band gap, because of the selection rules [32] governing optical transitions from heavy-hole, or light-hole, states to conduction band states, left circular polarization (σ-) generates a density of spin-up electrons ( cα ) which is three times the density of spin-down electrons ( c−α ), and vice versa for right circularly polarized light (σ+). Hence the initial electron spin polarization (p0) generated by a circularly polarized beam in a zinc-blende semiconductor, defined as

p(r ) =

cα (r ) − c−α (r ) s(r ) = , cα (r ) + c−α (r ) cα (r ) + c−α (r )

(9)

is ±0.5 (50%). Optical excitation with σ+ (σ-) generates spins along the direction parallel (antiparallel) to the direction of the light propagation, i.e. along -z (+z) (Fig. 1).

Figure 1. A schematic representation of the system under investigation, where electron spins are generated or injected at the centre (x=0 and y=0), with spin polarization along either z or –z, under the action of applied electric field in the –y direction.

Once an electron spin imbalance is injected into a semiconductor, electrons experience spin-dependent interactions with the environment, i.e. with impurities and excitations or phonons, which cause spin relaxation. While the total current is conserved when the carrier electrons propagate through the system under investigation, the spin density or polarization decreases as the distance from the point of injection increases and the length scale associated with this decrease is governed by the down-stream and up-stream spin diffusion lengths,

232

M. Idrish Miah

which can be obtained from the solution of Eq. (6) for y>0 and y<0 respectively. The general solution restricting one-dimensional variation to the y-direction is found as

s ( y ) = R1 e

(

)

γ E2 +1/ δ s2 + γ E y

+ R2 e

−

(

)

γ E2 +1/ δ s2 −γ E y

,

γ E = μ E /(2Cd ) ,

with

(10) (11)

where R1 and R2 are the constants of integration to be determined by the initial conditions. If the spin density at the injection point, x=0 and y = 0 , is s0 = s (0) = cα (0) − c−α (0) , then Eq. (10) becomes

s ( y ) = s0 e

−

s ( y ) = s0 e where the quantity

(

γ E2 + 1/ δ s2 − γ E

(

(

) ( −1

⎡ ⎢⎣

)

γ E2 +1/ δ s2 −γ E y

)

γ E2 +1/ δ s2 +γ E y

, y>0

(12)

, y<0,

(13)

γ E2 + 1/ δ s2 + γ E

)

−1

⎤ ⎥⎦ is the down-stream [up-

stream] spin diffusion length, i.e. the distances over which the carriers move within the spin lifetime τs in the down-stream [up-stream] direction. In the absence of the electric field ( γ E =0 = 0 ) both spin diffusion lengths are equal to δs. With increasing the field the downstream diffusion length increases whereas the up-stream diffusion length decreases. Eqs. (12) and (13), which give the spin density distribution along the positive and negative y-directions, respectively, show that the stronger the field is, the further away (in the down-stream case) the spin density polarization can be dragged (or transported over distances significantly exceeding δs). We will, however, now show that extension of the (down-stream) spin diffusion length in this way does not translate to a significant increase in spin current owing to the competing effect of field on diffusion. The relation between the position dependence of the spin current density and the spin density or polarization can be obtained from Eq (2):

js (r ) = jα (r ) − j−α ( r ) = e( μ E + Cd ∇) s (r ) = ec ( μ E + Cd ∇ ) p (r ) ,

(14)

where the total charge current density is

j (r ) = jα (r ) + j−α (r ) = e(cα + c−α ) μ E = ecμ E .

(15)

The expressions for the down-stream and up-stream spin current densities in terms of the spin polarization can be obtained from Eqs. (12), (13) and (14)

Spin Drift–Diffusion and Its Crossover in Semiconductors −( js ( y ) = p0 ⎡ −ecμ Eyˆ − ecCd ( γ E2 + 1/ δ s2 − γ E ) yˆ ⎤ e ⎣ ⎦

( js ( y ) = p0 ⎡ −ecμ Eyˆ + ecCd ( γ E2 + 1/ δ s2 + γ E ) yˆ ⎤ e ⎣ ⎦ where E = − Eyˆ and p ( y ) = s ( y ) / c = p0 exp[−(

233

)

γ E2 +1/ δ s2 −γ E y

)

γ E2 +1/ δ s2 +γ E y

, y>0

(16)

, y<0,

(17)

γ E2 + 1/ δ s2 − γ E ) y ] . As seen from the

Eqs. (16) and (17), both down-stream and up-stream spin currents have two current contributions. The first term in the square bracket corresponds to the total drift current density and the second term, −ecCd

(

γ E2 + 1/ δ s2 − γ E

) ⎡⎢⎣ecC ( d

)⎦

γ E2 + 1/ δ s2 + γ E ⎤⎥ , to the total

down-stream [up-stream] diffusion current density, jdi ,d [ jdi ,u ] , both terms contributing additively to js in the down-stream case since they are in the same direction along -y. The spin drift current density ( js ,dr ), spin diffusion current density ( js ,di ) and the total spin current density are therefore

js ,dr ( y ) = −ecμ Ep ( y ) yˆ

js ,di ( y ) = ± ecCd

(18)

)

(

γ E2 + 1/ δ s2 ± γ E p ( y ) yˆ

js ( y ) = js ,dr ( y ) + js ,di ( y ) = ⎡ −ecμ Eyˆ ± ecCd ( γ E2 + 1/ δ s2 ± γ E ) yˆ ⎤ p( y ) , ⎣ ⎦

(19)

(20)

where +(–) for up-stream (down-stream) direction.

4. Spin Drift–Diffusion Crossover and Transport Regimes The jdi ,d decreases with E while j increases and if we define a spin drift-diffusion crossover field Ec as

ecμ Ec = ecCd

(

)

γ E2 + 1/ δ s2 − γ E , c

c

(21)

where the spin drift and spin diffusion mechanisms contribute equally to the spin current, we obtain an important relation

Ec =

1 Cd , 2 μδ s

(22)

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M. Idrish Miah

which in the nondegenerate regime ( Cd = μ / eβ ), becomes

1 1 . 2 eβδ s

Ec =

(23)

Eqs. (22) and (23) relate the spin drift-diffusion crossover field with intrinsic spin diffusion length of a semiconductor in the degenerate and nondegenerate regimes respectively. We now consider Eq. (16) and (17) in the special case of strong field. 1/ δ s ), In the down-stream direction (Eq. (16)), when E is very large ( γ E

γ E2 + 1/ δ s2 − γ E

0,

(24)

and the diffusion current is zero ( jdi ,d ( y ) = 0 ), so that only the drift current contributes to the spin current

js ( y ) ∝ j ( y )

js ,dr ( y ) = −ecμ Ep( y ) yˆ .

(25)

For the up-stream case (Eq. (17)), the drift and diffusion currents are in the opposite directions, so are in competition with each other. When E is very large,

γ E2 + 1/ δ s2 + γ E

2γ E = μ E / Cd ,

(26)

and the up-stream diffusion current is jdi ,u ( y ) = neμ Eyˆ and the spin current exactly vanishes (in the up-stream direction). These facts are clearly illustrated in Figs. 2 and 3, where j , jdi ,d and jdi ,u , calculated for GaAs with δ s = 2 μm [20], are plotted as a function of E and temperature (T) in the nondegenerate regime. As also seen from the Fig. 2, the diffusion current in the down-stream direction decreases with E (and increases with T) while the drift current increases (and decreases with T) and there is a Ec for a T, which importantly suggests a possible way to identify whether the process for a given E would be in the spin drift, spin drift-diffusion crossover or spin diffusive regime, defined as Spin drift:

j

jdi

(27)

Spin drift-diffusion crossover:

j ≈ jdi

(28)

Spin diffusive:

j

jdi .

(29)

Spin Drift–Diffusion and Its Crossover in Semiconductors

235

1.6 1.4

j (1/μm)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.5

jdi,d (1/μm)

0.4

0.3

0.2

0.1 0

5

10

15

20

E (mV/μm)

Figure 2. (Top) j as a function of E at different temperatures (solid line: 300 K; long dashed line: 200 K; short dashed line: 150 K). (Bottom) The same plot for jdi,d. In the plots, j and jdi,d are reduced by the common quantity ecCd and therefore are independent of unknown parameter Cd and carrier density. 1.8 1.6

jdi,u (1/μm)

1.4 1.2 1.0 0.8 0.6 0.4 0

5

10

15

20

E (mV/μm)

Figure 3. jdi,u, reduced by the quantity ecCd, as a function of E at different temperatures (solid line: 150 K; long dashed line: 200 K; short dashed line: 300 K).

236

M. Idrish Miah

Since Ec is quite small, in order for the diffusion contribution become significant, E must therefore be weak. However, the up-stream diffusion current increases with increasing E and decreases with T (Fig. 3).

5. Diffusion Analysis In order to quantify conditions for the enhancement of the diffusion contribution, we consider the relationship between the mobility and the diffusion coefficient. Since the diffusion and mobility of both spin and charge in doped semiconductors are approximately determined by the properties of a single (spin) carrier species, specifically, the lower-conductivity spin species, usually the minority spins (here -α) [25], one can relate the diffusion coefficient with the mobility through the expression derived for the - α species. Thus, for the electron diffusion coefficient [33], we obtain

2μ Cd = − eβ

∫

∫

∞

0

∞

0

S d f FD d ε −α

(30)

S d ∂ ε −α f FD d ε −α

where S d (ε −α ) is the density of states for the minority spin and with energy ε −α , measured

from the bottom edge of the conduction band for the spin −α , which for a bulk parabolic band is S d (ε −α ) = ( 3 / )

3

2ε −α / π 2 , f FD (ε −α ) is the Fermi-Dirac (FD) distribution

function and ∂ ε −α f FD is its partial derivative with respect to

ε −α . For degenerate systems,

Eq. (30) can be approximated [34,35] as

Cd =

Here

FD 2μ I p ( ρ F ) . eβ I FD p −1 ( ρ F )

(31)

ρ F = β (ε F − ε 0,−α ) is the reduced Fermi level, ε 0,−α is the bottom edge of the

conduction band for the spin −α ,

ε F is the Fermi level, I FD p ( ρ F ) are the FD integrals

(FDIs) with p = ±1/2 in general

I FD p (ρF ) =

1 Γ ( p +1)

∫

∞

0

ψ p dψ 1 + exp(ψ − ρ F )

(32)

with

I FD p '( ρ F ) =

d d ρF

FD I FD p ( ρ F ) = I p −1 ( ρ F ) .

(33)

Spin Drift–Diffusion and Its Crossover in Semiconductors

237

The FDIs of order ±1/2 are called the Fermi integrals (FIs), I p ( ρ F ) . Eq. (31) is also F

known as the generalized Einstein relation. The FIs are widely used in problems concerning semiconductors and metals, where FD statistics are involved (e.g. [25,33-35]). They can be evaluated analytically in two limiting cases: 1. When

ρF

−1 , I Fp ( ρ F ) ≈ e ρF ,

(34)

which applies to the Maxwell-Boltzmann (MB) statistics ( f MB ∼ e

− βε − α

) and suffices for the

description of electronic and transport properties of the nondegenerate systems, or in the lowdensity limit, and Eq. (31) gives simply the Einstein relation

Cd = 2. When

ρF

μ ; eβ

(35)

1 (the case for the highly degenerate systems), I Fp ( ρ F ) ≈

1 ( p +1) Γ ( p +1)

ρ Fp +1

(36)

and Eq. (31) reduces to

4 μρ F 4 μ (ε F − ε 0,−α ) = Cd = e 3 eβ 3

2μ 3

2

(3π 2 c) 2 / 3 , m* e

where m* is the effective electron mass, which shows that eβ Cd / μ

(37)

1 , meaning that Cd

increases in the degenerate regime by a large factor from the nondegenerate value limited by the Einstein relation (Eq. (35)), i.e.

CdD

CdND ,

(38)

where D and ND stand, respectively, for degenerate and nondegenerate regimes. Given that diffusion is considered to be driven by a gradient in the chemical potential, ϕ , caused by the increase in carrier density [25,33], this result can be seen as a consequence of the faster increase of ϕ with density owing to the Fermi pressure, because the Fermi energy strongly depends on carrier density. Eq. (38) suggests a possible way to enhance the spin current in semiconductors. An increase in E will increase the drift current but will in principle decrease the diffusion current, making it difficult to increase the spin current by increasing the external D

field. Since for degenerate systems Cd

CdND , a way to increase the spin current for a

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M. Idrish Miah

given E would be to use the degenerate regime by lowering the temperature or increasing carrier density. Increasing Cd in this way would increase the diffusive contribution without any change of the drift contribution. Although it is easy to adjust the value of eCd β / μ in a semiconductor by doping, in a metal it is difficult to control, suggesting that a semiconductor degenerate regime should also be considered for room-temperature device applications with higher doping density. The semiconductor degenerate regime extends to higher densities when the applied electric field is increased. In semiconductors, in contrast to metals, the electric field remains partially unscreened. However, by increasing the carrier density, the system will eventually enter the metallic regime, in which the drift term, i.e. the electric field effects in the transport processes can be neglected [25].

6. Relations between Ec and δs The expressions for δs in different electron statistical regimes of a semiconductor can be obtained from Eqs. (22), (31), (35) and (37). These, in the nondegenerate, degenerate and highly degenerate regimes, respectively, are

1 2eβ Ec

δs =

F 2 I p (ρF ) δs = eβ Ec I Fp −1 ( ρ F )

δs =

2 2 (3π 2 c) 2 / 3 , 3eη m Ec

(39)

(40)

(41)

where η is the ratio of m* to the free electron mass m (η=0.067 for GaAs [36], for example),

I Fp ( ρ F ) =

1 Γ ( p +1)

∫

∞

0

ψ p dψ , 1 + exp(ψ − ρ F )

(42)

and p = 1/2. For the highly degenerate systems, δs is independent of temperature. Eqs. (39) (41) show that δs can be obtained from Ec if the temperature, both the electron density and temperature and electron density, respectively, are known for the nondegenerate, degenerate and highly degenerate systems. Since our results are based on common semiconductor parameters and on the different energy scales or electron statistical regimes, they can be readily extended to a wide range of semiconductor materials and to various systems with different doping densities and at different temperatures.

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239

7. Conclusion We derived a drift-diffusion equation for spin density and modelled spin transport in nonmagnetic semiconductors in different transport regimes. Based on the spin drift-diffusion model, we studied js and found that j and jdi contribute to down-stream and up-stream js and that jdi decreases with E while j increases and there is a Ec in the down-stream direction. It was also found that js increases in the degenerate regime, suggesting a possible way for the enhancement of js in semiconductors. In the up-stream direction, js was found to be vanished when E is very large. We derived the expressions for δs of a semiconductor in different electron statistical regimes and showed that δs can be obtained directly from Ec. The results were also shown to be useful in identifying whether the process for a given E would be in the spin drift, spin drift-diffusion crossover or spin diffusive regime and in obtaining transport properties of the electron spin in semiconductors. The findings resulting from this investigation might have potential applications in semiconductor spintronics.

References [1] G. A. Prinz, Phys. Today 48, 58 (1995). [2] S. D. Sarma, Am. Sci. 89, 516 (2001). [3] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001). [4] S. D. Sarma, J. Fabian, X. Hu, I. Žutić, Solid State Communic. 119, 207 (2001). [5] I. Žutić, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004). [6] D.D. Awschalom, D. Loss, and N. Samarth, Eds., Semiconductor Spintronics and Quantum Computation (Springer, Berlin, 2002). [7] P. P. Freitas, S. Cardos, R. Sousa, W. Ku, R. Ferreira, V. Chu, J. P. Conde, IEEE Transac. Magn. 36, 2796 (2000). [8] S. Datta, B. Das, Appl. Phys. Lett. 56, 665 (1990). [9] I. Žutić, J. Fabian, S. D. Sarma, Phys. Rev. Lett. 88, 066603 (2002). [10] Z. G. Yu, M. E. Flatté, Phys. Rev. B 66, 201202-1 (2002). [11] Y. V. Pershin, V. Privman, Phys. Rev. Lett. 90, 256602-1 (2003). [12] J. C. Egues, G. Burkard, and D. Loss, Appl. Phys. Lett. 82, 2658 (2003). [13] S. Bandyopadhyay, and M. Cahay, Appl. Phys. Lett. 85, 1433 (2004). [14] L. Villegas-Lelovsky, Braz. J. Phys. 36, 851 (2006). [15] J. M. Kikkawa, D.D. Awschalom, Nature 397, 139 (1999). [16] M. Oestreich, J. Hübner, D. Hägele, P. J. Klar, W. Heimbrodt, W. W. Rühle, D. E. Ashenford, and B. Lunn, Appl. Phys. Lett. 74, 1251 (1999). [17] M. Ramsteiner, H. Y. Hao, A. Kawaharazuka, H. J. Zhu, M. Kästner, R. Hey, L. Däweritz, H. T. Grahn, and K. H. Ploog, Phys. Rev. B 66, 081304-1 (2002). [18] J. Wang, D. Y. Xing, and H.B. Sun, J. Phys.: Condens. Matter 15, 4841 (2003). [19] R. Fiederling, M. Keim, G. Reuscher, W. Ossau, G. Schmidt, A. Waag, and L. W. Molenkamp, Nature 402, 787 (1999). [20] J. Kikkawa, D.D. Awschalom, Nature 397, 139 (1999). [21] B. T. Jonker, Y. D. Park, B. R. Bennet, H. D. Cheong, G. Kioseoglou, and A. Petrou, Phys. Rev B 62, 8180 (2000).

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[22] M. I. Miah, Mater. Lett. 60, 2863 (2006). [23] M. I. Miah, J. Phys. D: Appl. Phys. 40, 1659 (2007). [24] X. Lou, C. Adelmann, C. A. Crooker, E. S. Garlid, J. Zhang, K. S. M. Reddy, S. D. Flexner, C. J. Palmstrøm, and P. A. Crowell, Nat. Phys. 3, 197 (2007). [25] R. A. Smith, ed., Semiconductors, Cambridge Univer. Press, New York, 1978. [26] M. I. Miah, and E. MacA. Gray, Solid State Sci. (press). [27] P.C. van Son, H. van Kempen, and P. Wyder, Phys. Rev. Lett. 58, 2271 (1987). [28] T. Valet, and A. Fert, Phys. Rev. B. 48, 7099 (1993). [29] G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip, and B. J. van Wees, Phys. Rev. B. 62, R4790 (2000). [30] S. Zhang, J. Appl. Phys. 89, 7564 (2001). [31] A. Fert, and H. Jaffres, Phys. Rev. B. 64, 184420 (2001). [32] D. T. Pierce, and F. Meier, Phys. Rev. B 13, 5484 (1976). [33] M. Shur, Physics of Semiconductor Devices, Prentice-Hall, New Jersey, 1990. [34] T. Y. Chang, A. Izabelle, J. Appl. Phys. 65, 2162 (1989). [35] X. Aymerich-Humet, F. Serra-Mestres, J. Millán, J. Appl. Phys. 54, 2850 (1983). [36] Semiconductors – basic data (O. Madelung Ed.), Second Ed. (Springer, New York, 1996).

Reviewed by: Professor V. I. Kityk, J.Dlugosz University, Poland.

In: Spintronics: Materials, Applications and Devices Editors: G. C. Lombardi and G. E. Bianchi

ISBN: 978-1-60456-734-2 © 2009 Nova Science Publishers, Inc.

Chapter 10

ELECTRONICS, SPINTRONICS AND ORBITRONICS CHARGE-BASED ELECTRONICS IS UNDER CONTROL – TECHNOLOGY WELCOMES ORBITRONICS FOR THIS MILLENIUM THOUGH SPINTRONICS IS NOT YET ESTABLISHED.

M. Idrish Miah1,2,* 1

Nanoscale Science and Technology Centre, Griffith University, Nathan, Brisbane, QLD 4111, Australia. 2 Department of Physics, University of Chittagong, Chittagong, Chittagong – 4331, Bangladesh.

Spin

e-

Charge

The control and manipulation of electronic charge by electric fields is well established for electronics and the progress in conventional electronics has proven that the spin of the electron was ignored in past in the mainstream of electronics, because the electron has spin as well as charge. Electron spin (self-rotation - rotation of the electron on its axis) is a quantum

*

E-mail address: [email protected]

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M. Idrish Miah

mechanical property associated with its intrinsic angular momentum, and as an intrinsic magnetic moment is associated with spin, spin is closely related to magnetic phenomena. Researchers [1-7] recently realized that spin of electrons can create a current, called spin current, as the movement of electrons creates a charge current, and adding the spin degree of freedom, i.e. the spin property of electrons to electronics could revolutionize today’s electronics, where the spin degree of freedom of the electron could also be utilized to sense, store, process and transfer information (which could extend the functionality of conventional devices), emerging a field called “spintronics” (“Spintronics” was coined by S. A. Wolf in 1996 and appeared in scientific journals from 1999) based on the control and manipulation of electron spin instead of, or in addition to, its charge, leading to novel electronic devices (spintronic devices) such as spin field effect transistors (SFETs), spin storage/ memory devices or spin quantum computers, which can hold promise of, e.g. reduced power consumption, faster operation and smaller size. Most essential requirements for such devices are the efficient injection of spin-polarized carriers into a semiconductor (such that they can be transported reliably over reasonable distances without spin-flipping or spin relaxation: if the spin relaxes too fast, the distance traveled by an electron without losing its spin will be too short to perform any practical operation) and then detecting them (since information is carried by the electron spin). Much effort on the requirements for spintronic devices reveals that the better understanding of spin currents in semiconductors is still a sufficiently challenging problem, though the first step in making the metal-based spin-technology amenable to industrial application came in 1988 with the invention of the very high performance hard disk drives and magnetic RAM utilizing giant magnetoresistive [8] magnetic multilayer (sandwiched structures consisting of alternating ferromagnetic and nonmagnetic metal layers) – composed of transition metals. These first metallic spintronic components were passive devices whose electric resistance depends strongly on the external magnetic field. Depending on the relative orientation of the magnetizations in the magnetic layers, the device resistance changes from small (parallel magnetizations) to large (antiparallel magnetizations). The proposal of utilizing electron spin for quantum information processing using spin as the binary system (quantum bits, in short qbits), say, ‘1’ for spin-up and ‘0’ for spin-down in quantum computing is a natural extension of spintronics since the electron provides an ideal condition for a qubit. In comparison with metal-based spintronics, utilization of semiconductors promises more versatile design due to the ability to adjust potential variation and spin polarization in the device by, e.g. external voltage and device structure. However, the first active semiconductor spintronic device was suggested by Datta and Das in 1990 [9], where they proposed an electronic analogue of an electro-optical modulator, that was later termed SFET, in a twodimensional electron gas (2-DEG) contacted with two ferromagnetic electrodes: one as a source for the injection of spin-polarized electrons and the other as an analyzer for electronspin polarization. The Datta-Das SFET device relies on the basic concept of modulating the transistor’s source-to-drain current by varying the Rashba interaction in the channel with a gate voltage. The injection and detection of spin-polarized electrons or spin currents in semiconductors have been achieved either optically or by electrical methods with varying degrees of success [5-7]. More sophisticated experiments toward all-electrical spintronic devices (similar to the SPINFET) using controllable parameters, such as device structure or gate voltage, able to

Electronics, Spintronics and Orbitronics

243

combine spin injection and detection are required. Such device and control strategies for realizing new high-performance devices taking advantage of the electron spin as well as of its charge might open the way to establish the semiconductor systems for spintronics. Given the intense work on injection and detection of spin currents in semiconductors and the recent breakthroughs in this field, there is no doubt that implementing semiconductor-based spintronic devices is only a matter of time. In reality, an electron in strongly correlated electron systems such as the transition-metal oxides, in which electrons interact between neighboring and often next-neighboring, has three attributes: charge (-e), spin (±1/2) and orbital which determine its behavior. Transition-metal oxides are often called correlated-electron systems because of many interacting electrons compared to doped-semiconductors. The orbital represents the electron’s probability-density distribution, as the shape of the electron cloud in a solid. A transition-metal ion in a crystal having perovskite structure, for example, is surrounded by six oxygen ions, which give rise to Coulomb potential of electrons from each lattice site [10]. While hybridization of d-electron of transition-metal ion with the oxygen p-electron states tends to delocalize the electrons, Coulomb repulsion tends to localize individual electrons at atomic lattice sites. The subtle balance results in a crystal-field potential, which partly lifts the degeneracy of the d-electron levels, where wave functions pointing toward oxygen ions have higher energy than those pointing in between them. For example, in the Mott-insulating state (conduction electrons are tied to the atomic sites) of a crystal, the d-electrons are almost entirely localized on the atomic sites, which make both the spin and orbital degeneracy. The degeneracy orbital states can be considered as a new degree of freedom that behaves like the spin degree of freedom, and the technology based on the orbital degree of freedom in the correlated-electron systems may be termed as “orbitronics” (The term “orbitronics” was first given by the Takura group). When an excitation (or a field) excites an orbital in a crystal at a particular lattice site, the orbital wave (and its quantized equivalent, “orbiton”) represents in principle the dynamical response of the orbital under excitation, and it propagates in the crystal through the interactions between orbitals at different transition-metal ions – this means modulations in the relative shape of the electron clouds in an orbitally ordered state could give rise to orbital waves. Typically, the orbital degree of freedom in the lattice and electronic response can be found in manganese oxide compounds with perovskite structure, where colossal magnetoresistance, i.e. a very large decrease of resistance, is observed upon the application of an external magnetic field, where conduction is known to be occurred through a process known as the double-exchange interaction [10,11]. In this type of compound, d-electrons on each transition-metal ion act like strongly coupled ferromagnets, where the angle between local spin moments on adjacent sites can determine the electron hopping from site to site, and by controlling the magnetic field intensity one might be able to control the orbital correlation which results in a control of electrical conduction due to varied electron hopping. Much intensive work is to be dedicated for searching the related possibility of controlling electrical conduction (or electric current) by changing the d-electron orbital shapes or states on a lattice site. Very recently, Tokura and co-workers [12,13] at the University of Tokyo in Japan have proposed that the orbital states of the electronic and magnetic phases of correlated-electron materials could be controlled by external fields such as electric, magnetic, stress, light or x-

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rays fields. Such control strategies for exploring the possibility of utilizing orbital degree of freedom might open the way to introduce the correlated-electron systems for a new type of electronics – orbitronics. We don’t know for sure where this will go, but we do know that the potential is very exciting. However, future electronics devices with increased performance, capability and functionality would rely on a complete understanding of the three control parameters, namely carriers’ charge, spin and orbital.

References [1] S. D. Sarma, Am. Sci. 89 (2001), 516. [2] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelkanova and D. M. Treger, Science 294 (2001), 1488. [3] H. – A. Engel, P. Recher and D. Loss, Solid State Commun. 119 (2001), 229. [4] D. D. Awschalom, D. Loss and N. Samarth, Eds., Semiconductor Spintronics and Quantum Computation (Springer, Berlin, 2002). [5] Žutić, J. Fabian and S. D. Sarma, Rev. Mod. Phys. 76 (2004), 323. [6] Y. Kato, R. C. Myers, A. C. Gossard and D. D. Awschalom, Science 306 (2004), 1910. [7] M. I. Miah, Mater. Lett. 60 (2006), 2863; J. Phys. D: Appl. Phys. 40 (2007), 1659. [8] G. Prinz, Phys. Today 48 (1995), 58. [9] S. Datta and B. Das, Appl. Phys. Lett. 56 (1990) 665. [10] C. Zener. Phys. Rev. 81 (1951), 440. [11] J. L. Cohn. J. Superconduc. Incorp. Novel Magnet. 13 (2000), 291. [12] Y. Tokura and N. Nagaosa. Science 288 (2000), 463. [13] Y. S. Lee, Y. Tokunaga, T. Arima and Y. Tokura, Phys. Rev. B 75 (2007), 174406.

INDEX A

Australia, 227, 241

B Aβ, 146 absorption, 112 acceptor, 50, 55, 61, 62 adiabatic, 125, 129, 130, 131, 132 Ag, vii, viii, 1, 2, 3, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 44, 104, 112 age, 208 AIP, 94 air, 17 alkaline, 69, 70 all-electric, 242 ammonia, 31 ammonium, 31, 37 amorphous, 11, 13, 15, 16 amplitude, 87, 90, 92, 94, 101, 115, 116, 120, 125, 126, 128, 162, 187, 189 Amsterdam, 182 angular momentum, x, 98, 100, 126, 128, 153, 156, 165, 170, 172, 173, 176, 178, 227, 242 Anion, 68 anomalous, vii, 1, 16, 36, 51 antibonding, 217, 218 antiferromagnetic, 2, 50, 63, 122, 219 application, 25, 97, 99, 112, 114, 123, 127, 133, 143, 186, 243 aqueous solution, 11, 17, 31, 37 aqueous solutions, 37 atmosphere, 30, 31, 37, 38 Atomic Force Microscopy (AFM), 50, 51, 64, 65, 68, 191 atomic physics, 170 atoms, 50, 51, 55, 58, 60, 63, 64, 65, 66, 68, 70, 144, 151, 155, 197, 198, 215, 216, 217, 218, 219, 220, 221, 222, 223

back, 98, 106, 125, 126, 150, 153, 177 bandwidth, 125 barrier, ix, 3, 5, 10, 101, 102, 106, 107, 108, 110, 111, 112, 157, 159, 160, 171, 173, 185, 186, 190, 192, 195, 196, 197, 198, 199, 200, 201, 202, 204, 205, 206, 207, 208, 209, 214 barriers, vii, ix, 1, 2, 3, 5, 6, 9, 10, 24, 25, 30, 44, 160, 163, 185, 189, 194, 201, 205 behavior, 6, 15, 24, 28, 30, 35, 36, 37, 44, 51, 54, 60, 63, 64, 73, 81, 88, 89, 93, 99, 114, 121, 122, 130, 131, 168, 186, 188, 189, 203, 208, 214, 216, 220 bending, 132 benefits, 127 Bohr, 130, 143 boundary conditions, viii, 79, 81, 85, 92, 93, 156, 160, 176, 177 British Columbia, 49 bulk materials, 51, 198

C candidates, 186 carbide, 68, 71 carbon, 33, 68, 69, 70, 71, 77, 143, 170, 183 carbon nanotubes, 170 Cartesian coordinates, 103 cation, 50, 51, 52, 64, 71 chalcogenides, 50, 160, 170 channels, 53, 142, 143, 180 circular dichroism, 62, 64 classical, 53, 54, 56, 142, 147, 150, 178, 180, 209 classification, 131 CLO, 24 clouds, 243

246

Index

clusters, viii, 29, 56, 59, 60, 62, 63 CMOS, 111, 123 Co, viii, ix, x, 24, 49, 54, 62, 63, 100, 104, 112, 113, 114, 115, 122, 126, 185, 186, 190, 191, 192, 193, 194, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223 coil, 148 Columbia, 49 communication, 46, 59, 112 community, 79 compensation, 30 condensed matter, 186 conductance, 186, 187 conduction, vii, ix, 1, 2, 5, 9, 10, 15, 17, 24, 27, 29, 31, 35, 70, 97, 100, 122, 126, 129, 130, 132, 149, 150, 155, 157, 158, 159, 161, 162, 163, 165, 170, 180, 188, 231, 236, 243 conductive, viii, 2, 24, 35, 36 conductivity, vii, 1, 10, 11, 17, 24, 142, 229, 236 confinement, 175 consensus, 51, 62 consumption, vii, 218, 228, 242 conversion, 100 Cooper pair, 186, 188 Coulomb, 5, 218, 243 coupling, 50, 51, 55, 62, 67, 68, 69, 70, 72, 73, 80, 90, 98, 110, 114, 121, 122, 129, 132, 141, 158, 159, 162, 188, 189, 219 CPA, 215 critical current, ix, 103, 104, 105, 106, 107, 109, 110, 111, 114, 121, 123, 125, 126, 127, 128, 129, 132, 133, 185, 186, 189, 190, 199, 200, 202, 206, 207, 208, 209 critical current density, 104, 105, 106, 107, 110, 111, 125, 126, 128, 129, 132 critical points, ix, 141 critical temperature, 62, 94, 189, 206 crystal structure, 10, 17, 23, 197, 198

D decomposition, 161 definition, 7, 85 deformation, 180 degrees of freedom, viii, 49, 50, 97, 132 demagnetization, 53, 118 deposition, 64, 68, 69, 190, 191, 192, 198 derivatives, 160 detection, viii, 50, 52, 59, 60, 123, 193, 228, 242 diamond, 68, 124, 126 dichotomy, 50

diffusion, x, 36, 56, 58, 100, 105, 106, 107, 111, 130, 197, 198, 199, 227, 228, 229, 230, 231, 232, 233, 234, 236, 237, 239 dipole, viii, 49, 54, 55, 92, 142 dispersion, 85, 150, 151, 154, 172, 175, 176, 188 displacement, 123, 126 distribution, 51, 52, 54, 56, 162, 163, 232, 236, 243 domain walls, 123, 125, 126, 127, 128, 129, 130, 131, 132 donors, 50, 62 dopant, 64, 65, 67, 68 doped, viii, 18, 49, 51, 52, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 218, 219, 222, 229, 236, 243 doping, viii, x, 10, 16, 17, 18, 21, 23, 50, 56, 59, 60, 61, 64, 68, 69, 70, 71, 73, 114, 155, 213, 215, 216, 217, 218, 221, 223, 238 drying, 37 duration, 123, 126

E electrical properties, 186 electron beam, 125 electron beam lithography, 125 electron density, 203, 238 electron gas, 242 electron microscopy, 25, 33, 40, 124, 128 electrons, viii, ix, x, 2, 5, 9, 10, 15, 17, 22, 24, 29, 31, 36, 44, 49, 50, 51, 54, 55, 60, 62, 65, 70, 71, 79, 91, 97, 98, 99, 100, 106, 111, 114, 122, 129, 130, 132, 142, 147, 149, 150, 152, 168, 169, 171, 172, 180, 187, 188, 213, 214, 215, 216, 217, 218, 219, 221, 222, 223, 227, 230, 231, 242, 243 emission, 33, 116, 120, 121, 122 employment, 180 energy density, 81, 85, 90, 114 epitaxy, 50, 51, 64, 128 EPR, 84, 142, 144, 147, 150, 154, 168, 169 eye, 69, 206, 207, 208

F fabricate, ix, 185 fabrication, 94, 99, 124, 193, 195, 200, 205, 214 feedback, 121 Fermi, 2, 5, 36, 61, 65, 66, 68, 69, 71, 72, 129, 132, 154, 186, 188, 190, 200, 201, 208, 209, 214, 216, 217, 218, 219, 236, 237 Fermi energy, 5, 36, 71, 132, 154, 217, 237 Fermi-Dirac (FD), 236, 237 ferromagnetic, viii, ix, 2, 6, 10, 12, 16, 17, 19, 23, 24, 25, 27, 49, 50, 51, 54, 57, 58, 60, 61, 62, 63, 64, 66, 68, 69, 70, 71, 72, 73, 79, 80, 81, 87, 92,

247

Index 93, 94, 97, 98, 100, 104, 106, 110, 113, 114, 115, 119, 122, 124, 128, 130, 132, 175, 180, 185, 186, 187, 188, 189, 190, 191, 192, 194, 195, 196, 197, 198, 199, 200, 205, 208, 209, 215, 220, 228, 231, 242 Ferromagnetic, 57, 79, 186, 195 field-dependent, 7 field-emission, 33 film, 50, 81, 83, 84, 85, 86, 87, 91, 92, 100, 106, 112, 113, 115, 116, 117, 118, 120, 122, 123, 124, 128, 192, 203, 205, 228 films, 2, 6, 8, 10, 36, 40, 44, 50, 51, 80, 81, 82, 83, 84, 85, 87, 90, 92, 97, 106, 114, 123, 192, 203, 208 filters, 214 finite differences, 160 finite volume, 170 first principles, 130 flow, 17, 84, 106, 111, 112, 114, 124, 127, 128, 129, 175 focused ion beam (FIB), 120, 190, 193, 194, 195, 196, 200, 205 focusing, 186 Fourier, 52, 87, 91, 181 Fox, 74 freedom, viii, 49, 50, 60, 79, 97, 132, 228, 242, 243, 244 FWHM, 56

G GaAs, ix, 50, 51, 57, 61, 90, 110, 141, 143, 155, 170, 171, 173, 174, 175, 180, 214, 231, 234, 238 GaP, 170 gas, 3, 17, 31, 37, 38, 242 Gaussian, 108 glass, 64, 69 government, iv grain, 2, 5, 15, 16, 18, 30, 33, 40 grain boundaries, 2, 15, 40 grains, vii, 1, 2, 3, 5, 6, 8, 9, 10, 29, 30, 31 graphite, 68 groups, 5, 110, 113, 115, 116, 126, 221 growth, 51, 61, 73, 157

H

heterojunctions, 151 heterostructures, ix, 61, 175, 185, 190, 192, 194, 196, 197, 208, 209 high temperature, 55, 63 Hilbert, ix, 141, 142, 145, 146, 147 Holland, 182 homogenous, 97 hybrid, 97, 186 hybridization, 67, 71, 216, 243 hybrids, 209, 216, 217, 221 hypothesis, 37 hysteresis, 34, 35, 51, 69, 104, 196, 205

I IBM, 139 identification, 80, 86 images, 33, 124, 127 imaging, 33, 118 implementation, 198 impurities, viii, 2, 25, 42, 44, 56, 62, 64, 71, 72, 73, 201, 231 indeterminacy, 178 indices, 152 induction, 144 industrial, 242 industrial application, 242 industry, 186, 209, 214 information systems, ix, 185, 209 injection, 36, 37, 60, 125, 127, 168, 175, 228, 230, 231, 232, 242 InP, 170 insulators, 70, 71, 73 integration, 60, 123, 230, 232 interactions, viii, 49, 50, 51, 55, 58, 63, 71, 81, 92, 93, 101, 123, 129, 180, 216, 231, 243 interstitial, 50 interstitials, 63, 83 intrinsic, x, 8, 42, 62, 63, 102, 103, 106, 109, 110, 119, 189, 227, 229, 230, 234, 242 ion beam, 120, 190 ionic, 23, 71, 73 ions, vii, 1, 18, 23, 24, 50, 51, 55, 60, 61, 67, 69, 72, 80, 243 IOP, 75

J H1, 155, 161, 162, 163, 167, 168, 169 H2, 30, 31, 37, 38, 86, 161, 162, 163, 167, 168, 169 Hall resistance, 51, 128 harmonics, 116, 206 heating, 110, 119, 127, 128 Heisenberg, 93 helix, 154

Josephson effect, 211 Joule heating, 127, 128 jumping, 24 Jung, 131, 136

248

Index

K kinetic energy, 188

L L1, 138 L2, 214, 221 Lagrangian, 131 laser, 64, 68, 69, 81, 82, 87, 90, 93, 144 lattice, 10, 17, 18, 24, 36, 50, 51, 55, 66, 89, 147, 149, 151, 155, 158, 159, 179, 197, 223, 243 law, 85, 114, 188 lift, 124, 125 linear, 15, 22, 25, 35, 44, 85, 86, 87, 91, 107, 110, 112, 113, 125, 129, 130, 145, 158, 159, 168, 173, 191, 197, 198, 203, 208, 230 linear regression, 208 liquid helium, 84, 209 localization, 63 location, 155 London, 56, 57, 95, 181, 183 low-density, 237 low-temperature, 61 L-shaped, 126

M magnetic field, viii, 2, 8, 11, 17, 24, 25, 27, 28, 30, 36, 41, 42, 49, 51, 52, 53, 54, 55, 56, 80, 81, 82, 84, 86, 87, 89, 90, 91, 92, 93, 94, 97, 100, 102, 103, 107, 109, 112, 113, 114, 115, 116, 118, 120, 122, 123, 124, 125, 126, 127, 128, 132, 133, 142, 144, 145, 148, 149, 150, 155, 186, 196, 205, 209, 228, 242, 243 magnetic moment, ix, x, 10, 24, 36, 51, 54, 55, 61, 62, 64, 65, 67, 68, 69, 70, 71, 80, 97, 98, 99, 100, 101, 114, 197, 198, 205, 218, 220, 223, 227, 242 magnetic properties, viii, x, 5, 18, 23, 51, 55, 59, 63, 65, 69, 71, 93, 196, 213, 215, 223 magnetite, 30 magnetometry, 51, 54, 62 magnetoresistance, 41, 42, 50, 106, 112, 127, 209, 228, 243 magnetron, 64, 190, 191 magnetron sputtering, 64 mainstream, x, 50, 241 manganese, viii, 49, 243 manganese, 55, 57 manipulation, vii, viii, x, 59, 80, 93, 94, 241, 242 mapping, 142 mask, 192, 193, 194 Massachusetts, 181

Matrices, 163 matrix, 55, 63, 73, 143, 144, 146, 149, 158, 160, 161, 164, 165, 166, 173, 178 MBE, 50, 128 measurement, 35, 62, 68, 93, 113, 115, 118, 119, 124, 125, 126, 127, 128, 129, 145, 193 media, 11, 17 memory, vii, 1, 3, 79, 99, 100, 110, 111, 133, 228, 242 mesoscopic, 131 metal ions, 243 metal oxides, 243 metals, viii, 49, 50, 54, 56, 60, 71, 106, 130, 132, 187, 219, 220, 222, 223, 230, 237, 238, 242 microscopy, 33, 123, 124, 128, 231 microwave, ix, 84, 97, 112, 115, 116, 117, 118, 119, 120, 121, 122, 123, 186, 204, 206 microwaves, 206 migration, 218 Minnesota, 97 minority, x, 2, 24, 65, 66, 69, 72, 213, 214, 216, 217, 218, 219, 221, 236 mobility, 61, 132, 229, 236 modeling, 73 models, 51, 62, 80, 81, 129, 141, 186, 200, 202, 208, 228 modulation, 123 molecular beam epitaxy, 50, 51, 128 molecular-beam, 64 momentum, x, 97, 98, 101, 126, 128, 131, 153, 156, 157, 158, 164, 165, 170, 172, 173, 176, 178, 186, 188, 189, 190, 227, 242 Monte Carlo, 69, 73, 180 morphology, 84 Moscow, 49 Mössbauer, 51 motion, ix, 59, 80, 89, 97, 100, 108, 118, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 144, 164, 170, 176 muon, viii, 49, 51, 52, 53, 54, 56

N nanoclusters, 62 nanodevices, 103 nanometer, 122, 123, 154 nanometer scale, 122 nanoparticles, 41, 42 nanostructures, ix, 97, 99, 107, 110, 123, 141, 142 nanotubes, 68, 170 nanowires, 67, 124, 125, 126, 127 National University of Singapore, 59 natural, ix, 51, 120, 121, 122, 141, 145, 150, 151, 152, 153, 154, 155, 166, 170, 178, 179, 180, 242

249

Index Nb, ix, 185, 186, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 206, 207, 208, 209 New Jersey, 240 New York, 58, 138, 181, 182, 183, 211, 240 Newton, 114 Ni, viii, ix, 49, 54, 57, 185, 186, 190, 191, 192, 194, 196, 198, 199, 200, 201, 205, 208, 209 nitride, 61, 72, 113 nitrogen, 64, 67, 69, 70, 209 NMR, 142, 144, 147, 150, 154, 168, 181 noise, 110, 115, 118, 121, 125, 206 non-magnetic, 67, 73, 111, 113 non-thermal, 88 non-uniform, 100, 110, 112, 122, 130 normal, 82, 87, 90, 106, 107, 152, 155, 168, 169, 177, 186, 188, 189, 190, 193, 200, 208 normalization, 146 North Carolina, 79, 94 Notre Dame, 79, 84 n-type, 64 nuclear, 144, 147 nucleation, 124

O optical, viii, 51, 62, 79, 80, 82, 89, 128, 192, 228, 231, 242 optics, viii, 79, 80, 90 oscillation, 87, 88, 94, 100, 102, 115, 116, 118, 119, 120, 121, 122, 123, 188 oscillations, ix, 88, 90, 91, 92, 93, 115, 116, 142, 144, 153, 185, 186, 188, 189, 190, 199, 200, 201, 202, 203, 204, 207, 208, 209 oscillator, 120, 121, 122 oxidation, 93 oxide, 2, 3, 4, 17, 23, 61, 62, 63, 69, 72, 110, 243 oxygen, viii, 2, 3, 10, 17, 23, 31, 62, 63, 67, 68, 69, 70, 73, 243

P parabolic, 171, 236 paramagnetic, vii, 1, 9, 10, 16, 53, 84, 144 parameter, 5, 27, 80, 130, 147, 158, 178, 186, 188, 190, 199, 235 particles, vii, viii, 1, 2, 5, 6, 11, 13, 14, 15, 16, 19, 25, 31, 33, 35, 36, 37, 38, 40, 44 patterning, 192, 205 periodic, 122, 147, 175, 176, 177, 206 permit, 193 perovskite, 243 perturbation, 122, 153 perturbation theory, 122 phase boundaries, 15

photoexcitation, 89 photon, 231 photonic, 62 photonics, 62 physical properties, 73, 216 physics, ix, 80, 117, 123, 130, 142, 170, 185, 186, 190, 209, 215, 228 pinning effect, 81 planar, 128, 131, 133, 143, 170, 175, 177, 178, 180 planetary, vii, 1, 11, 16, 17, 24 play, 23, 72, 73, 81, 84, 110, 132, 142, 147, 170, 209, 216 Poisson, 162 polarization, vii, 1, 2, 5, 9, 16, 21, 24, 27, 30, 52, 60, 66, 67, 68, 70, 73, 82, 87, 88, 90, 98, 100, 101, 104, 105, 110, 111, 130, 142, 155, 169, 170, 175, 180, 186, 187, 214, 216, 217, 228, 230, 231, 232, 242 polarized light, 87, 231 polycrystalline, 2, 8 population, 147, 222 powder, 2, 3, 5, 6, 8, 11, 12, 17, 25, 31, 37 power, vii, 71, 111, 115, 116, 117, 120, 121, 122, 171, 191, 228, 242 powers, 121 press, 240 pressure, 3, 191, 237 projected density of states, 68 property, vii, x, 1, 2, 9, 10, 22, 24, 42, 62, 142, 168, 178, 179, 187, 227, 242 pseudo, x, 69, 185, 209 PSI, 53 p-type, 11, 17, 23, 57, 61, 62, 64 pulsed laser, 69 pulsed laser deposition, 69 pulses, 81, 87, 90, 109, 123, 128 pumping, 105, 106

Q quantum, vii, ix, x, 10, 11, 17, 23, 72, 73, 94, 130, 141, 142, 144, 145, 147, 148, 149, 155, 156, 158, 160, 162, 166, 170, 172, 175, 176, 179, 180, 185, 186, 208, 209, 227, 241, 242 quantum computers, vii, 228, 242 quantum gates, 180 quantum well, ix, 141, 155, 156, 158, 160, 172, 180 qubit, 147, 180, 242

R radio, 100 radius, 18, 175, 176, 178, 179 Raman, 69

250

Index

random, vii, 1, 3, 24, 25, 100, 108, 133, 215, 221, 228 random access, vii, 1, 3, 24, 100, 133, 228 range, vii, ix, 1, 15, 16, 27, 30, 50, 54, 55, 61, 63, 67, 68, 71, 72, 81, 82, 84, 87, 93, 100, 107, 116, 126, 153, 179, 185, 192, 193, 195, 201, 202, 206, 208, 217, 238 reading, 228 recombination, 229 reflectivity, 192 regular, 50, 197 relationship, 80, 82, 83, 90, 104, 105, 107, 125, 236 relationships, viii, 79, 80, 81, 130 relaxation, viii, 49, 51, 52, 53, 54, 56, 66, 71, 91, 100, 129, 130, 131, 190, 228, 229, 231, 242 relaxation time, 129, 130, 190, 229 reliability, 111 reservoir, 188 resistance, vii, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 16, 17, 22, 23, 24, 27, 28, 30, 31, 33, 36, 37, 41, 42, 50, 51, 97, 103, 104, 105, 110, 111, 112, 113, 114, 115, 119, 124, 125, 126, 128, 131, 190, 200, 203, 206, 208, 209, 228, 242, 243 resistive, vii, 1, 5, 8, 30, 31 resistivity, vii, viii, 1, 2, 5, 6, 7, 11, 14, 15, 16, 17, 22, 24, 25, 27, 30, 35, 36, 37, 41, 42, 51, 131, 203 resolution, 127 resources, 73 rhombohedral, 3 room temperature, vii, viii, 1, 2, 8, 9, 24, 25, 27, 35, 37, 44, 51, 52, 56, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 104, 112, 196, 214 room-temperature, vii, 1, 4, 10, 34, 60, 71, 238 roughness, ix, 185, 208 Ruderman-Kittel-Kasuya-Yosida, 61 Russia, 49 Rutherford, 49

S sampling, 51, 118 scalability, 111 scalable, 133 Scanning electron, 124 scanning electron microscopy (SEM), 25, 29, 124, 127 scattered light, 82 scattering, viii, 2, 3, 10, 51, 52, 79, 80, 99, 105, 107, 112, 129, 130, 189 Schmid, 76 scientific community, 79 searching, 243 semiconductor, viii, x, 49, 50, 51, 54, 55, 57, 58, 60, 61, 62, 65, 67, 79, 80, 93, 94, 110, 128, 132, 142,

155, 159, 175, 178, 180, 214, 227, 228, 229, 230, 231, 234, 238, 239, 242, 243 semiconductors, viii, ix, x, 49, 50, 51, 52, 55, 57, 58, 59, 60, 61, 62, 63, 64, 71, 73, 79, 80, 110, 132, 141, 142, 149, 150, 151, 152, 155, 156, 162, 164, 168, 170, 171, 179, 180, 185, 214, 216, 227, 228, 230, 231, 236, 237, 239, 242, 243 sensing, 97 sensitivity, 107 series, 64, 82, 85, 88, 133, 192, 214 sharing, 209 short-range, 54, 63, 70, 81, 93 shoulder, 218 Siemens, 52 signals, 63, 121 silicon, 110, 113, 125, 142, 149, 190 similarity, x, 213, 214 simulation, 73, 116, 117, 118, 153, 180, 192, 193 simulations, 122, 129 Singapore, 59 single-crystalline, 67 singular, 166, 175 sintering, viii, 2, 3, 17, 30, 36, 37, 38 sites, 50, 52, 68, 69, 216, 217, 218, 219, 220, 223, 243 solid solutions, 17 solid-state, 71, 97 spacers, 104, 114 spatial, 54, 70, 71, 129, 130, 131, 149, 153, 154, 175 spectrum, 33, 52, 53, 69, 84, 86, 92, 93, 115, 116, 120, 121, 149, 152, 160, 161, 177, 178, 179 spheres, 35, 208 sputtering, 64, 190, 191, 192 SQUID, 6, 11, 17, 25, 34, 53 stability, 64, 69, 102, 103, 111, 216 statistics, 108, 237 steady state, 114 steel, 11, 17 stiffness, viii, 79, 81, 92, 93, 114 STM, 231 STO, 24 stochastic, 132 storage, vii, 1, 3, 24, 59, 60, 79, 97, 133, 228, 242 strategies, 243, 244 strength, 51, 99, 101, 110, 126, 130, 142, 144, 166, 173, 209 stress, 155, 243 substitutes, 51, 69 substitution, 50, 51, 55, 68, 70, 218, 219, 222, 223 substrates, 8, 124, 190, 192 suffering, 71 superconducting, 11, 188, 189, 190, 198, 199, 200, 209

251

Index superconductivity, 11, 17, 23, 186, 189, 209 superconductors, ix, 185, 186, 189, 209 superposition, 80, 142, 144, 145, 147, 148, 152, 153, 154, 165, 166, 171, 174 suppression, 3, 197 surface wave, 81 surplus, 219, 220 susceptibility, viii, 49, 53, 54, 61 switching, 30, 100, 104, 105, 107, 108, 109, 110, 111, 118, 119, 127, 128, 131, 209 Switzerland, 49, 53 synthesis, 51

T targets, 191 thermal stability, 111 thin film, 2, 8, 10, 40, 44, 51, 97, 100, 106, 112, 116, 120, 122, 123, 124, 192, 203 thin films, 2, 8, 10, 40, 44, 51, 97, 106, 123, 192, 203 TiO2, 8, 24, 62 Tokyo, 243 torque, ix, 97, 98, 99, 100, 101, 102, 103, 104, 109, 110, 111, 112, 113, 114, 115, 118, 119, 122, 123, 125, 128, 129, 130, 132, 133 transfer, ix, 69, 97, 98, 99, 100, 101, 102, 104, 106, 110, 111, 112, 113, 114, 115, 118, 119, 122, 128, 129, 130, 131, 132, 133, 138, 242 transformation, 127, 145, 147, 148, 149, 150 transistor, 177, 242 transistors, vii, 111, 228, 242 Transition Metal (TM), viii, 57, 59, 60, 61, 62, 63, 71, 73 transition temperature, 6, 36, 50, 51, 60 transmission, 5, 33, 40, 98, 99, 190 transmission electron microscopy (TEM), 33, 40, 41, 44 transparency, 189 transparent, 62, 147, 178 transport, viii, x, 24, 49, 51, 54, 100, 128, 129, 185, 186, 190, 193, 195, 203, 208, 227, 228, 229, 230, 231, 237, 238, 239 tubular, 143, 170

Turkey, 213 two-dimensional, 242

U uncertainty, 54, 63

V values, vii, 1, 4, 5, 6, 9, 11, 14, 15, 16, 17, 19, 25, 27, 30, 31, 32, 33, 34, 44, 80, 81, 90, 92, 147, 148, 150, 160, 161, 162, 166, 170, 171, 176, 178, 179, 188, 190, 198, 199, 200, 201, 202, 206, 217, 219, 220 vapor, 67 vector, ix, 81, 82, 84, 85, 92, 93, 100, 129, 141, 142, 143, 144, 146, 147, 149, 150, 151, 152, 153, 154, 156, 157, 159, 160, 161, 162, 163, 164, 166, 167, 168, 169, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 190 velocity, 125, 126, 127, 128, 129, 130, 131, 132, 176, 188, 190, 200, 201, 208, 209 visible, viii, 2, 31, 35, 38, 44 vortex, 118, 125, 126, 127, 128

W water, 37, 38, 39, 40, 41 waveguide, 126 wireless, 112, 123

X XPS, 68, 71 x-ray diffraction (XRD), 3, 11, 17, 18, 23, 25, 31, 32, 33, 37, 38, 39, 40

Z Zener, 57, 61, 69, 244 zinc (Zn), 51, 57, 62, 63, 64, 68, 69, 150, 151, 155, 156, 159, 179, 180, 214, 231

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