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Description
3.23 Electrical, Optical, Magnetic Properties of Materials (previously Physics and Chemistry of Materials) • All EOM properties derived from electrons in materials – Electronic: response of ‘free’ electrons in material – Optical: response of bound electrons and effect of electron transitions in material – Magnetic: response of electron spin and electron motion in material • Determining energy of electrons from electric and magnetic fields will determine electron response and hence properties • Real materials: effect of imperfections on these materials
©1999 E.A. Fitzgerald
323a1
Origin of Conduction Range of Resistivity
Why?
©1999 E.A. Fitzgerald
323a2
Response of material to applied potential I
I
V
V=f(I) Rectification, Non-linear, Non-Ohmic
R
Linear, Ohmic
V eV=IR
Metals show Ohmic behavior Microscopic origin?
©1999 E.A. Fitzgerald
323a3
Remove geometry of material
V I W L V=IR=IR/L R=L/(σA)
J=σE
In general,
! ~! J = σE
Isotropic material Anisotropic material
All material info
In cubic material,
©1999 E.A. Fitzgerald
J x σ xx J y = σ xy J z σ xz
E
J
E
J
σ xy σ xz E x σ yy σ yz E y σ yz σ zz E z 323a4
Microscopic Origin: Can we predict Conductivity of Metals? • Drude model: Sea of electrons – all electrons are bound to ion atom cores except valence electrons – ignore cores – electron gas J=σE=-nev, by definition of flux through a cross-section n=number of electrons per volume v=velocity of the carriers due to electric field--> drift velocity Therefore, σ=-nev/E and we define v=-µE µ is mobility, since the electric field creates a force on the electron F=-eE
σ = neµ
©1999 E.A. Fitzgerald
323a5
Current Density and Electron Density
J n=# electrons/volume A dx
©1999 E.A. Fitzgerald
J
Volume=A*dx Charge=dQ=-enAdx dx=vdt J=(1/A)(dQ/dt)=-nev define µ: v=-µE therefore J=-nev=σE if σ=neµ 323a6
Does this microscopic picture of metals give us Ohm’s Law? E F=-eE F=ma m(dv/dt)=-eE v=-(eE/m)t J=σE=-nev=ne2Et/m σ=ne2t/m
v,J,σ,I t E
Constant E gives ever-increasing J
t
No, Ohm’s law can not be only from electric force on electron! ©1999 E.A. Fitzgerald
323a7
Hydrodynamic representation of e- motion
p=momentum=mv
dp( t ) p( t ) =− + F1 ( t ) + F2 ( t ) +... dt τ Response (ma)
Drag
Driving Force
Restoring Force...
dp(t ) p( t ) ≈− − eE dt τ Add a drag term, i.e. the electrons have many collisions during drift 1/τ represents a ‘viscosity’ in mechanical terms
©1999 E.A. Fitzgerald
323a8
In steady state,
dp(t ) =0 dt −t τ
p(t ) = p∞ (1 − e ) p∞ = − eEτ
p -eEτ
τ
t
If the environment has a lot of collisions, vavg=-eEτ/m mvavg=-eEτ
Now we have Ohm’s law ©1999 E.A. Fitzgerald
ne 2τ σ= m
µ=
eτ m 323a9
Predicting conductivity using Drude ntheory from the periodic table (# valence e- and the crystal structure) ntheory=AVZρm/A, where AV is 6.023x1023 atoms/mole ρm is the density Z is the number of electrons per atom A is the atomic weight For metals, ntheory~1022 cm-3 If we assume that this is correct, we can extract τ
©1999 E.A. Fitzgerald
323a10
Extracting Typical τ for Metals
•
τ~10-14 sec for metals in Drude model
©1999 E.A. Fitzgerald
323a11
Thermal Velocity • So far we have discussed drift velocity vD and scattering time τ related to the applied electric field x • Thermal velocity vth is much greater than vD
x
x
L=(vD+vth)t
1 2 3 mvth = kT 2 2
vth =
3kT m
Thermal velocity is much greater than drift velocity
©1999 E.A. Fitzgerald
323a12
Example: Conductivity Engineering • Objective: increase strength of Cu but keep conductivity high ne 2τ σ= m " = vτ
µ=
eτ m
Scattering length connects scattering time to microstructure
Dislocation (edge)
l decreases, τ decreases, σ decreases ©1999 E.A. Fitzgerald
e-
323a13
Example: Conductivity Engineering • Can increase strength with second phase particles • As long as distance between second phase< l, conductivity marginally effected
L
L
L+S
α+L
β+L
α
S
α+β
Cu
Sn
X
Cu β
microstructure
S
Material not strengthened, conductivity decreases ©1999 E.A. Fitzgerald
β
L α
dislocation L>l
Dislocation motion inhibited by second phase; material strengthened; conductivity about the same 323a14
Example: Conductivity Engineering • Scaling of Si CMOS includes conductivity engineering • One example: as devices shrink… – vertical field increases – τ decreases due to increased scattering at SiO2/Si interface and increased doping in channel need for electrostatic integrity (ionized impurity scattering) – Scaling continues ‘properly’ if device shrinks fast enough to negate decrease in scattering time G S
SiO2 D
Evert Ionized impurities (dopants)
©1999 E.A. Fitzgerald
323a15
Determining n and µ: The Hall Effect Vx, Ex +++++++++++
---------
! ! ! ! F = qE + qv × B
Bz
Fy = −evD Bz
Ey
I, Jx
In steady state,
EY = vD BZ = E H , the Hall Field
Fy = −eE y
Since vD=-Jx/en, 1 E H = − J x BZ = RH J X BZ ne RH = −
©1999 E.A. Fitzgerald
1 ne
σ = neµ 323a16
A&M treatment using full equations of motion ! ! ! ! dp p ! p ! p ! xB (CGS units) = − + F = − − e E + dt mc τ τ
Separate into x & y coupled equations: ! ! ˆ ˆ p = p xi + p y j , B = Bz kˆ dp x p = − x − eE x − ω c p y τ dt dp y py 0= =− − eE y + ω c p x τ dt eB ωc = (CGS) mc
(pxxB is in +y direction, pyxB is in -x direction)
0=
px ω c p y − eτ e py ωc py Ey = − + eτ e ne 2τ σo = m Ex = −
Since ©1999 E.A. Fitzgerald
σ o E x = ω cτJ y + J x σ o E y = −ω cτJ x + J y In steady state,
Ey = −
σ o Ex = J x Jy = 0
1 ω cτ Jx = − J x B z = RH J x B z nec σo
323a17
Experimental Hall Results on Metals
• Valence=1 metals look like free-electron Drude metals • Valence=2 and 3, magnitude and sign suggest problems
©1999 E.A. Fitzgerald
323a18
Response of free e- to AC Electric Fields • Microscopic picture
EZ = EO e − iωt
e-
B=0 in conductor, dp(t ) p (t ) =− − eE0 e −iωt dt τ
try
! ! ! ! and F ( E ) >> F ( B )
p (t ) = p0 e − iωt
p0 − iωp 0 = − − eE 0 τ p0 =
eE0 iω −
1 τ
ω>>1/τ, p out of phase with E eE0 p0 = iω
ω → ∞, p → 0
ω<<1/τ, p in phase with E
p0 = −eE0τ ©1999 E.A. Fitzgerald
323a19
Complex Representation of Waves sin(kx-ωt), cos(kx-ωt), and e-i(kx-ωt) are all waves e -i(kx-ωt) is the complex one and is the most general imaginary A θ Acosθ
iAsinθ real
e iθ=cosθ+isinθ
©1999 E.A. Fitzgerald
323a20
Response of e- to AC Electric Fields • Momentum represented in the complex plane imaginary p (ω>>1/τ)
p
p (ω<<1/τ)
real
Instead of a complex momentum, we can go back to macroscopic and create a complex J and σ
J (t ) = J 0 e
− iωτ
nep0 ne 2 E0 J 0 = nqv = = 1 m m ( − iω ) τ
ne 2τ σ0 ,σ 0 = σ= 1 − iωτ m ©1999 E.A. Fitzgerald
323a21
Response of e- to AC Electric Fields •
•
Low frequency (ω<<1/τ) – electron has many collisions before direction change – Ohm’s Law: J follows E, σ real High frequency (ω>>1/τ) – electron has few collisions when direction is changed – J imaginary and 90 degrees out of phase with E, σ is imaginary
Qualitatively: ωτ<<1, electrons in phase, re-irradiate, Ei=Er+Et, reflection ωτ>>1, electrons out of phase, electrons too slow, less interaction,transmission ε=εrε0 εr=1 τ ≈ 10
−14
3x1010 cm / sec sec,νλ = c,ν = ≈ 1014 Hz −8 5000 x10 cm
E-fields with frequencies greater than visible light frequency expected to be beyond influence of free electrons ©1999 E.A. Fitzgerald
323a22
Response of light to interaction with material • Need Maxwell’s equations – from experiments: Gauss, Faraday, Ampere’s laws – second term in Ampere’s is from the unification – electromagnetic waves! SI Units (MKS) ! ∇•D = ρ ! ∇•B = 0 ! ! ∂B ∇xE = − ∂t ! ! ! ∂D ∇xH = J + ∂t ! ! ! ! D = ε 0 E + P = εE ! ! ! ! B = µ 0 H + µ 0 M = µH
Gaussian Units (CGS) ! ∇ • D = 4πρ ! ∇•B = 0 ! ! 1 ∂B ∇xE = − c ∂t ! ! 4π ! 1 ∂D J+ ∇xH = c c ∂t ! ! ! D = E + 4πP ! ! ! B = H + 4πM
µ = µr µ0 ;ε = ε rε 0 ©1999 E.A. Fitzgerald
323a23
Waves in Materials • Non-magnetic material, µ=µ0 • Polarization non-existent or swamped by free electrons, P=0 ! ! ∂B ∇xE = − ∂t ! ! ! ∂E ∇xB = µ 0 J + µ 0 ε 0 ∂t
! ! ∂∇xB ∇x(∇xE ) = − ∂t ∂ ∂E [ µ 0 J + µ 0ε 0 ] ∂t ∂t 2 ∂E ∂ E ∇ 2 E = µ 0σ + µ 0ε 0 2 ∂t ∂t − ∇2E = −
©1999 E.A. Fitzgerald
For a typical wave,
E = E0 e i ( k •r −ϖt ) = E0 e ik •r e −iϖt = E ( r )e − iϖt ∇ 2 E ( r ) = −iϖµ 0σE (r ) − µ 0ε 0ω 2 E ( r ) ω2 Wave Equation ∇ E ( r ) = − 2 ε (ω ) E (r ) c iσ ε (ω ) = 1 + ε 0ω 2
E (r ) = E0 e ik •r
ω2 k = 2 ε (ω ) c ω c v= = k ε (ω ) 2
323a24
Waves in Materials • Waves slow down in materials (depends on ε(ω)) • Wavelength decreases (depends on ε(ω)) • Frequency dependence in ε(ω) ε (ω ) = 1 +
iσ iσ 0 = 1+ ε 0ω ε 0ω (1 − iωτ )
ε (ω ) = 1 +
iω p2τ
ω − iω 2τ
ne 2 ω = Plasma Frequency ε 0m 2 p
For ωτ>>>1, ε(ω) goes to 1 For an excellent conductor (σ0 large), ignore 1, look at case for ωτ<<1 ε (ω ) ≈
©1999 E.A. Fitzgerald
iω 2pτ
ω − iω 2τ
≈
iω 2pτ
ω
323a25
Waves in Materials
For a wave
k=
σ0 ω ω i ε (ω ) = c c ωε 0
k=
σ 0ω σ 0ω ω 1+ i σ 0 i = + c 2 ωε 0 2ε 0 c 2 2ε 0 c 2
E = E0 e i ( kz −ωt )
Let k=kreal+kimaginary=kr+iki
E = E 0 e i [kr z −ωt ]e
− ki z
The skin depth can be defined by
1 δ = = ki
2ε o c 2 2 = σω o σ o µ oω CGS
MKS
δ ©1999 E.A. Fitzgerald
323a26
Waves in Materials For a material with any σ0, look at case for ωτ>>1
ε (ω ) = 1 −
ω<ωp, ε is negative, k=ki, wave reflected
ω 2p ω2
ω>ωp, ε is positive, k=kr, wave propagates
R
ωp
©1999 E.A. Fitzgerald
ω
323a27
Success and Failure of Free e- Picture •
•
Success – Metal conductivity – Hall effect valence=1 – Skin Depth – Wiedmann-Franz law Examples of Failure – Insulators, Semiconductors – Hall effect valence>1 – Thermoelectric effect – Colors of metals
K/σ=thermal conduct./electrical conduct.~CT 1 2 τ Κ = c v vtherm 3 3k b T 3 ∂E 2 cv = = nk b ; vtherm = m ∂T v 2 13 3 nk b2Tτ 3k bT Κ = nk b τ = 3 2 2 m m
ne 2τ σ = m 2
Therefore :
Luck: cvreal=cvclass/100; vreal2=vclass2*100 ©1999 E.A. Fitzgerald
Κ 3 kb = T σ 2 e
~C!
323a28
Wiedmann-Franz ‘Success’
Thermoelectric Effect Exposed Failure when cv and v2 are not both in property
E = Q∇T 3 − nk b cv nk = 2 =− b Thermopower Q is Q = − 3ne 3ne 2e
Thermopower is about 100 times too large! ©1999 E.A. Fitzgerald
323a29
Improvements? What are ion cores doing... • Scattering idea seems to work • any effect of crystal (periodic) lattice? • Diffraction – proves periodicity of lattice – proves electrons are waves – proves strong interaction between crystal and electrons (leads to band structures=semiconductors and insulators) – useful characterization technique • Course: bias toward crystalline materials: many applications: materials related to either end of spectrum (atomic/molecular or crystalline) extended
localized Point defects, atoms, molecules
Polymers, α Si
Bands; properties of many solids with or without extended defects
Diffraction is a useful characterization in all these materials ©1999 E.A. Fitzgerald
323a30
Diffraction • Incoming λ must be on the order of the lattice constant a or so (λ<~ few tenths of a nanometers) • x-rays will work (later, show electrons are waves also and they can be used for diffraction also) • x-rays generated by core e- transitions in atoms – distinct energies: E=hc/λ; E~ 10keV or so (core e- binding energies) Collimator crystal (decreases spread in θ and λ) Thermionic emission
e-
detector
λCu θ
Cooled Cu target ‘single-crystal’ diffraction ©1999 E.A. Fitzgerald
θ
sample
sample
‘double crystal’, ‘double axis’ diffraction
nλ=2dsinθ detector
Add a channel crystal (also called analyzer crystal) after the sample and it is called triple axis diffraction 323a31
Four Views of Diffraction • Bragg, Von Laue, Reciprocal lattice, Atomic origin • Start with review of Bragg diffraction Constructive interference: nλ=2dsinθ In cubic system, d=
a h2 + k 2 + l 2
In any system,
(2π )2 2 d hkl
!2 ! 2 2! 2 !2 2 2 = g = h b1 + k b2 + l b3
g is a reciprocal lattice vector, and bi are the primitive basis vectors describing the reciprocal lattice ©1999 E.A. Fitzgerald
323a32
Example of Diffraction from Thin Film of Different Lattice Constant • InGaAs on GaAs deposited by molecular beam epitaxy (MBE) • Can determine lattice constant (In concentration) and film thickness from interference fringes InxGa1-xAs GaAs
X-ray intensity
©1999 E.A. Fitzgerald
Interference fringes from optical effect
GaAs
In0.05Ga0.95As
323a33
Example: Heavily B-diffused Si • B diffusion from borosilicate glass • creates p++ Si used in micromachining • gradients created in B concentration and misfit dislocations
Si
Si:B Graded region
Si:B Graded Si:B
Si
©1999 E.A. Fitzgerald
323a34
Double axis vs. Triple Axis • •
More common double axis: greater signal but lower resolution Example: GaAsP Graded structure on GaAs substrate (for high-volume LEDs) with rotational inhomogeneity in composition/thickness due to production equipment
Double Axis Diffraction ©1999 E.A. Fitzgerald
Triple Axis Diffraction 323a35
Von Laue Representation • Equivalent to Bragg, more general • More convenient in picturing 3-D and often more convenient mathematically • leads to Ewalt sphere construction ! ! k is the wave-vector vector of initial x-rays,
! k
nˆ ! R ⋅ nˆ
k
magnitude =2π/λ
! k'
! k ' is the wave-vector of the diffracted x-rays, magnitude = 2π/λ
For constructive interference, ! ! R ⋅ nˆ + R ⋅ nˆ ' = mλ ! k' ! 2πnˆ 2πnˆ ' R ⋅ − = 2πm λ λ ! ! ! ˆ n' ! R ⋅ k − k ' = 2πm ! ! R ⋅ nˆ ' ! ! i∆k ⋅ R R ⋅ ∆k = 2πm or e =1 Red points are part of a periodic crystal lattice; If the difference in k and k’ is a multiple of R represents a translation vector of the lattice 323a36 a lattice spacing, then diffraction occurs ©1999 E.A. Fitzgerald
! R
( ) (
(
)
)
Equivalence of Von Laue and Bragg ! ! R ⋅ ∆k = 2πm ! 2πn 2πn 2πm ∆k = ! = ! = d R cosθ R'
! k'
! k ! k
! ∆k
! k'
! ∆k
! k θ B
! k'
! ∆k
Bragg plane
©1999 E.A. Fitzgerald
Where R’ is defined as the translation vector along ∆k
If Von Laue condition holds, then reciprocal lattice points exist at ends of ∆k 1 ! 1 2πm ∆k mλ 2 sin θ B = ! = 2 d = 2π 2d k λ
323a37
Reciprocal Lattice • Von Laue condition defines a set of diffraction points in 3D for 3-D crystal ! ! ! ! ! ! ! (k − k ' ) ⋅ R = ∆k ⋅ R = G ⋅ R = 2πm
! k θ B
! ! ∆k = G
Where G is termed a reciprocal lattice vector A vector representation of this relation is:
! k'
a2 × a3 V a ×a b2 = 2π 3 1 V a ×a b3 = 2π 1 2 V V = a1 ⋅ (a2 × a3 ) b1 = 2π
Where ai are primitive vectors in the real lattice, and the bi are the corresponding primitive vectors of the reciprocal lattice
bi ⋅ a j = 2πδ ij
or ©1999 E.A. Fitzgerald
δ ij = 0 when i ≠ j δ ij = 1 when i = j
323a38
Ewalt Sphere Construction •
A way to picture the generation of a diffraction image from the reciprocal lattice
•Draw reciprocal lattice •Draw incident k with vector head on reciprocal point •All k’ are represented by a circle defined by k as it is rotated about its tail (magnitude of k and k’ are the same since they are defined by λ) •All reciprocal lattice points that fall on the circle or sphere (3-D) satisfy diffraction condition •Diffraction pattern: reciprocal space
©1999 E.A. Fitzgerald
323a39
Atomic Origin of Diffraction • Important for understanding structure factors, nanostructures • Why is reciprocal space the Fourier transform of the real lattice? Start off with ‘two-beam analysis’ Recall δ=∆k•R 2
! ! k
incident wave, k, is a plane wave:
R
Ψ = Ψo e
!! ik ⋅r
1
Reflected/scattered waves are spherical waves
f Ψ1 = Ψo e i ( kD +δ ) D
Ψ1 = D
! k'
f Ψo e ikD D Therefore, for D>>|R|,
ΨTOT
ΨTOT
ΨTOT
f is the atomic scattering factor ©1999 E.A. Fitzgerald
[ [
] ]
fΨ0 ikD e 1 + e i ∆k ⋅ R = D = Ψ ( D ) f 1 + e i ∆k ⋅ R From atom 1
From atom 2 323b1
Atomic Origin of Diffraction • Generalize for " atoms
ΨTOT = Ψ ( D) f ∑ e i∆k ⋅r" "
Where rl points to each atom from the reference atom
Thus, ΨTOT is the Fourier Transform of Ψ(D) for small rl:
ΨTOT
Break rl into rg and di, where units of rg point to the unit cell (Bravais lattice), and di point to the basis Can generate all crystal structures this way
di rg
rl
i∆k ⋅r" ( ) dr" ≈ f ∫Ψ D e
ΨTOT is the amplitude of the diffracted wave I, the intensity, is proportional to the square of the amplitude
I ∝ ΨTOT ∝ f 2 ∑ e i∆k ⋅rl
2
2
rg ©1999 E.A. Fitzgerald
"
323b2
Atomic Origin of Diffraction • Origin of the lattice and structure factors Summing over rg in three co-ordinate directions with crystal length in each direction being M1,M2,M3: M1 M 2 M 3
ΨTOT = ∑∑∑ f j e m
n
((
i k rg + d j
))
M2
= G⋅S
M3 M1
o
Where ∆k is written now as k for brevity and convention
M1 M 2 M 3
G ≡ lattice factor = ∑∑∑ e m
2
I∝G S
n
o
ik ⋅rg
1 1 1 sin sin sin M k r M k r M k r ⋅ ⋅ ⋅ 1 2 3 g g g 2 2 2 = sin 1 k ⋅ r sin 1 k ⋅ r sin 1 k ⋅ r g g g 2 2 2 M very large
2
M small 2π/rg
©1999 E.A. Fitzgerald
323b3
Atomic Origin of Diffraction • The structure factor: connected to the crystal structure
S ≡ structure factor =
n = #basis atoms
∑fe
ik ⋅d j
j
j
Example: Body-centered cubic crystal structure Basis:
Sk = e
ik ⋅0
+e
d1 = 0 d2 =
1 ik ⋅ a ( xˆ + yˆ + zˆ ) 2
2π (n1 xˆ + n2 yˆ + n3 zˆ ) a n +n +n S k = 1 + e iπ (n1 + n2 + n3 ) = 1 + (− 1) 1 2 3
k=
d2
a (xˆ + yˆ + zˆ ) 2
d1
Sk=2, if sum even Sk=0 if sum odd (e.g. <111> disappear) Extra scatterer in basis interferes with simple cubic diffracted waves
Note: in structures with different atoms, e.g. GaAs,
Sk = ∑ f j e j
©1999 E.A. Fitzgerald
ik ⋅d j 323b4
Wave-particle Duality: Electrons are not just particles • Compton, Planck, Einstein – light (xrays) can be ‘particle-like’ • DeBroglie – matter can act like it has a ‘wave-nature’ • Schrodinger, Born – Unification of wave-particle duality, Schrodinger Equation
©1999 E.A. Fitzgerald
323b5
Light has momentum: Compton • No way for xray to change λ after interacting classically • Experimentally: Compton Shift in λ • Photons are ‘particle-like’: transfer momentum tor c te de
xray
I
θ ∆λ graphite
e∆λ =
©1999 E.A. Fitzgerald
λ
h (1 − cosθ ) = λc (1 − cosθ ) mo c
323b6
Light is Quantized: Planck • Blackbody radiation: energy density at a given ν (or λ) should be predictable • Missing higher frequencies! (ultra-violet catastrophe) L
L ρ(ν)dν=energy per volume being emitted in ν+dν
hollow cube, metal walls
ρ (ν )dν =
N (ν )dν ⋅ E wave volume
N(ν) is the number density, i.e. number of waves in ν+dν (#/frequency) Heat to T
Finding N(ν): Inside box, metal walls are perfect reflectors for the E-M waves
Ei = E oi e i (ωt − kz ) ; E r = E or e i (ωt + kz )
[
Perfect reflection, Eoi=-Eor
]
Etot = E oi e iωt e − ikz − e ikz = −2iE oi e iωt sin kz
©1999 E.A. Fitzgerald
323b7
Light is Quantized: Planck Real{Etot } = 2 E oi sin ωt sin kz
Standing Waves E-field inside metal wall is zero (due to high conductivity)
0
L
Therefore, sinkz must equal zero at z=0 and z=L sin kL = 0; kL = 2πn; k =
λ=
In 3-D,
ν=
©1999 E.A. Fitzgerald
2πn L
Also, since k=2π/λ,
n x2 + n y2 + n z2
2L
=
2L 2L or λ = n λ
Note that the wavelength for E-M waves is ‘quantized’ classically just by applying a confining boundary condition
2L
c n x2 + n y2 + n z2
n=
! cn 2L
! n = n x iˆ + n y ˆj + n z kˆ
323b8
Light is Quantized: Planck nz
ν(nx,ny,nz) ny
1 state (i.e. 1 wavelength or frequency) in (c/2L)3 volume in ‘n-space’ 2 possible wave polarizations for each state
c/2L
nx
(Note also that postive octant is only active one since n is positive: shows as 1/8 factor below) Using the assumption that ν>>c/2L, 1 4πν 3 8 L3ν 3π 8 3 = N= 3 3c 3 1 c 2 2L
dN 8 L3ν 2π N (ν ) = = dν c3
©1999 E.A. Fitzgerald
323b9
Light is Quantized: Planck Now that N(ν), the number of E-M waves expected in ν+dν, has been determined simply by boundary conditions, the energy of a wave must be determined for deriving ρ(ν) 8πν 2 L3 kT 3 N (ν )E wave 8πν 2 kT c = = ρ (ν ) = volume L3 c3
The classical assumption was used, i.e. Ewave=kbT This results in a ρ(ν) that goes as ν2
At higher frequencies, blackbody radiation deviates substantially from this dependence
©1999 E.A. Fitzgerald
323b10
Missing higher frequencies
ν2
•Low ν OK: E=kbT •At high ν, E goes to zero (i.e. no waves up there!)
~3µm
©1999 E.A. Fitzgerald
~1µm
~0.75µm
323b11
Light is Quantized: Planck •Classical E=kbT comes from assumption that Boltzmann distribution determines number of waves at a particular E for a given T •Since N(ν) can not the problem with ρ(ν), it must be E •E must be a function of ν in order to have the experimental data work out Origin of E=kbT Boltzmann distribution is P ' (E ) = Ae
−
E k bT
∞
Normalized distribution is
P(E ) = ∫ P ' (E )dE = 1; A = 0
1 k bT
Average energy of particles/waves with this distribution: ∞
E=
∫ EP' (E )dE 0 ∞
∫ P' ( E )dE
∞
= if P(E) is normalized = ∫ EP(E )dE = kbT 0
0
©1999 E.A. Fitzgerald
323b12
Light is Quantized: Planck •If P(E) were to decrease at higher E, than ρ(ν) would not have ν2 dependence at higher ν •P(E) will decrease at higher E if E is a function of ν •Experimental fit to data suggests that E is a linear function in ν, therefore E=nhν where h is some constant nhν
nhν − kbT ∑0 k T e b ∞
E=
∞
1 ∑0 k T e b 8πν 2 ρ (ν ) = 3 c
−
=
nhν k bT
e
hν e
hν k bT
hν
−1
hν k bT
−1
Note: the integrals need to be removed in the average and replaced with sums since the spacing of energies becomes greater as E increases
h determined by an experimental fit and equals
At small hν/kbT, ehν/kT~1+hν/kbT and ρ(ν)~kbT At large hν/kbT, ~hνe-hν/kT which goes to 0 at high E
©1999 E.A. Fitzgerald
323b13
Light is Quantized: Planck •
Lessons from Planck Blackbody – waves which are confined with boundary conditions have only certain λ available: quantized – E=nhν, and therefore E-M waves must come in chunks of energy: photon E=hν. Energy is therefore quantized as well – Quantized energy can affect properties in non-classical situations; classical effects still hold in other situations
©1999 E.A. Fitzgerald
323b14
Light is always quantized: Photoelectric effect (Einstein) • •
Planck (and others) really doubted fit, and didn’t initially believe h was a universal constant Photoelectric effect shows that E=hν even outside the box
I,E,λ metal block
Maximum electron energy, Emax
Emax=h(ν-νc)
!
eνc
ν
For light with ν<νc, no matter what the intensity, no e©1999 E.A. Fitzgerald
323b15
Light is always quantized: Photoelectric effect (Einstein) Ein=hν
vacuum
E
∆E EF
Evac=Ein-∆E Emax=Ein-∆E=hν-hνc Ein=hν!
x
Strange consequence of Compton plus E=hν: light has momentum but no mass λ=
©1999 E.A. Fitzgerald
hc h = since E = cp for a photon E p
323b16
DeBroglie: Matter is Wave • • • • •
His PhD thesis! λ=h/p also for matter To verify, need very light matter (p small) so λ is large enough Need small periodic structure on scale of λ to see if wave is there (diffraction) Solution:electron diffraction from a crystal
Nλ=2dsinθ For small θ, θ~λ/d, so λ must be on order of d in order to measure easily
©1999 E.A. Fitzgerald
323b17
DeBroglie: Matter is Wave Proof electron was wave by transmission and beackscattered experiments, almost simultaneously Diffraction Spots
Transmission
Backscattered
film
©1999 E.A. Fitzgerald
323b18
DeBroglie: Matter is Wave Modern TEM
©1999 E.A. Fitzgerald
Modern SEM
323b19
Imaging Defects in TEM utilizing Diffraction • •
The change in θ of the planes around a defect changes the Bragg condition Aperture after sample can be used to filter out beams deflected by defect planes: defect contrast
©1999 E.A. Fitzgerald
323b20
Imaging Defects and Man-made Epitaxial Structures in TEM utilizing Diffraction
Si0. 5Ge0. 5/Ge superlattice (each layer ~100A) Si0.25Ge0.75
Si1-xGex Layers (each layer about 3000A) 39 ©1999 E.A. Fitzgerald
323c1
Unification: Wave-particle Duality • Need to reconcile old classical world with new results from duality • Introduce concept that classical results are averages over large population of quanta • Automatically introduces the concept of uncertainty E-M from Maxwell: I~|E|2~Nhν macro micro I is an average over quantized behavior Quantum micro
Statistical Mechanics averaging
Thermodynamics, Properties macro
•Schrodinger and Born introduce Ψ, the wave-function, to unify particles and waves •|Ψ|2 is the physical quantity: probability density
©1999 E.A. Fitzgerald
323d1
Unification: Wave-particle Duality Ψ must be able to represent everything from a particle to a wave (the two extremes)
particle
wave
Ψ = Ae
Ψ = ∑ an e i (k n x −ω nt )
i (kx −ωt )
n
n = ∞ to create a delta function in ψ2
k and p known exactly
generalized i (k n x −ω n t )
Ψ = ∑ an e n
©1999 E.A. Fitzgerald
323d2
Normalization of Wavefunctions • Probability of finding particle over a confining volume is 1 * Ψ ∫ Ψdr = 1
V
Example:
©1999 E.A. Fitzgerald
i
1 ik!⋅r! e Ψ= V 1 −ik!⋅r! * e Ψ = V 1 * ∫V Ψ Ψdr = V V∫ dr = 1
Ψ |Ψ|2 real
Ψ∗
323d3
Consequence of Wave-Particle Duality and Quantization
©1999 E.A. Fitzgerald
323d4
Consequence of Wave-Particle Duality and Quantization
©1999 E.A. Fitzgerald
323d5
Consequence of Electrons as Waves on Free Electron Model • Boundary conditions will produce quantized energies for all free electrons in the material • Two electrons with same spin can not occupy same electron energy (Pauli exclusion principle) Imagine 1-D crystal for now Traveling wave picture
Standing wave picture
ei(kx-ωt) e-i(kx+ωt)
ei(kx-ωt)+ e-i(kx+ωt) =e-iωt(eikx+ e-ikx) = e-iωt(2coskx)
Since material is usually big and electron small, traveling wave picture used ©1999 E.A. Fitzgerald
323d6
Consequence of Electrons as Waves on Free Electron Model Traveling wave picture
0 L
Standing wave picture
Ψ ( x) = Ψ ( x + L)
0
L
e ikx = e ik ( x + L ) e ikx = 1 2πn k= L
Just having a boundary condition means that k and E are quasi-continuous, i.e. for large L, they appear continuous but are discrete ©1999 E.A. Fitzgerald
323d7
Representation of E,k for 1-D Material " 2k 2 p 2 E= = 2m 2m
states electrons
E All e- in box accounted for
EF m=+1/2,-1/2
En+1 En En-1
Quasi-continuous
dE " 2 k = dk m " 2k ∆E = ∆k m
∆k=2π/L
kF
kF Total number of electrons=N=2*2kF*L/2π ©1999 E.A. Fitzgerald
k 323d8
Representation of E,k for 1-D Material 2k F L π 2m − 12 dN dk 1 2 m g (E) = E = = dk dE L π " 2 k π" N=
2mE "2k 2 ;k = E= 2m " dE " 2 k = dk m
g(E)=density of states=number of electrons per energy per length
n=
N 2k F 2 2mEF nπ or k F = = = 2 L "π π
•n=is the number of electrons per unit length, and is determined by the crystal structure and valence •The electron density, n, determines the energy and velocity of the highest occupied electron state at T=0
©1999 E.A. Fitzgerald
323d9
Representation of E,k for 2-D Material
E=
" 2 (k x2 + k y2 )
E(kx,ky)
2m
ky
kx
©1999 E.A. Fitzgerald
323d10
Representation of E,k for 3-D Material kz
ky
Ε(kx,ky,kz)
kF
E=
m π2
2mE "3
Fermi Surface or Fermi Sphere
k F = (3π 2 n )
1 3
"k vF = F m ©1999 E.A. Fitzgerald
2m
kx
2π/L g (E) =
" 2 (k x2 + k y2 + k z2 )
" 2 k F2 EF = 2m
TF =
EF kB 323d11
So how have material properties changed? •
•
The Fermi velocity is much higher than kT even at T=0! Pauli Exclusion raises the energy of the electrons since only 2 e- allowed in each level Only electrons near Fermi surface can interact, i.e. absorb energy and contribute to properties
TF~104K (Troom~102K), EF~100Eclass, vF2~100vclass2
©1999 E.A. Fitzgerald
323d12
Effect of Temperature (T>0): Coupled electronic-thermal properties in conductors (i.e. cv) • • •
Electrons at the Fermi surface are able to increase energy: responsible for properties Fermi-Dirac distribution NOT Bolltzmann distribution, in which any number of particles can occupy each energy state/level
Originates from:
EF ...N possible configurations T=0
T>0
1
f = e ©1999 E.A. Fitzgerald
If E-EF/kbT is large (i.e. far from EF) than
( E − EF ) k bT
+1
f =e
− ( E − EF ) k bT
323d13
Fermi-Dirac Distribution: the Fermi Surface when T>0 f(E) fBoltz
1
kbT kbT
T=0 T>0
0.5
All these e- not perturbed by T
Boltzmann-like tail, for the larger E-EF values
µ~EF
E
Heat capacity of metal (which is ~ heat capacity of free e- in a metal):
∂U cv = T ∂ v
U ~ ∆E ⋅ ∆N ~ kbT ⋅ [g (E F )⋅ kbT ] ~ g (E F )⋅ (kbT )
2
∂U 2 cv = = 2 ⋅ g ( E F ) ⋅ kb T ∂T v ©1999 E.A. Fitzgerald
U=total energy of electrons in system
Right dependence, very close to exact derivation 323d14
Heat Capacity (cv) of electrons in Metal • Rough derivation shows cv~const. x T , thereby giving correct dependence • New heat capacity is about 100 times less than the classical expectation
Exact derivation: cvclass cvquant
©1999 E.A. Fitzgerald
π2 2 cv = ⋅ kb T ⋅ g (E F ) 3
3 nkb 3 E 2 = 2 F ~ 100 @ RT = 2 π kbT π kbT nkb 2 EF
323d15
Remaining Issues • •
• • •
Electron wave picture has fixed some thermal/electrical properties and electron velocity issues Still can not explain: – Hall coefficients – Colors of metals – Insulators, Semiconductors Can not ignore the ions (i.e. everything else but the valence electrons that we have been dealing with so far) any longer! Whatever we modify, can not change the electron wave picture that is now working well for many materials properties HOW DO THE VALENCE ELECTRON WAVES INTERACT WITH THE IONS AND THEIR POTENTIALS?
©1999 E.A. Fitzgerald
323d16
Electrons in a Periodic Potential • Rigorous path: HΨ=EΨ • We already know effect: DeBroglie and electron diffraction • Unit cells in crystal lattice are 10-8 cm in size • Electron waves in solid are λ=h/p~10-8 cm in size • Certain wavelengths of valence electrons will diffract!
©1999 E.A. Fitzgerald
323d17
Diffraction Picture of the Origin of Band Gaps
• Start with 1-D crystal again λ~a 1-D
a
nλ = 2d sin θ nλ = 2a 2π k= λ πn k= a ©1999 E.A. Fitzgerald
d=a, sinθ=1
Take lowest order, n=1, and consider an incident valence electron moving to the right π i x π ki = ;ψ i = e a a
π −i x π Reflected wave to left: k o = − ;ψ o = e a a 2π ∆k = ki − k o = a
Total wave for electrons with diffracted wavelengths: ψ = ψ i ±ψ o π x a π ψ a = ψ i −ψ o = i 2 sin x a ψ s = ψ i + ψ o = 2 cos
323d18
Diffraction Picture of the Origin of Band Gaps Probability Density=probability/volume of finding electron=|ψ|2 π x a π = 4 cos 2 x a
ψ a = 4 sin 2 2
ψs
2
a
a
•Only two solutions for a diffracted wave •Electron density on atoms •Electron density off atoms •No other solutions possible at this wavelength: no free traveling wave ©1999 E.A. Fitzgerald
323d19
Nearly-Free Electron Model • • •
Assume electrons with wave vectors (k’s) far from diffraction condition are still free and look like traveling waves and see ion potential, U, as a weak background potential Electrons near diffraction condition have only two possible solutions – electron densities between ions, E=Efree-U – electron densities on ions, E= Efree+U Exact solution using HΨ=EΨ shows that E near diffraction conditions is also parabolic in k, E~k2
©1999 E.A. Fitzgerald
323d20
Nearly-Free Electron Model (still 1-D crystal) states
" 2k 2 p 2 E= = 2m 2 m
E
Away from k=nπ/a, free electron curve
Near k=nπ/a, band gaps form, strong interaction of e- with U on ions
Eg=2U
dE " 2 k = dk m " 2k ∆E = ∆k m
Diffraction, k=nπ/a
Quasi-continuous
∆k=2π/L
-π/a
0
π/a
k
∆k=2π/a=G=reciprocal lattice vector ©1999 E.A. Fitzgerald
323d21
Consequences of Diffraction on E vs. k curves • At k=π/a, there must be also a k=-π/a wave, since there is absolute diffraction at this k Standing wave at • True for every k=nπ/a diffraction condition • Creates a parabola at every nG
E
k=-π/a in reference parabola
−2π/a
-π/a G
©1999 E.A. Fitzgerald
0 −2π/a
k=π/a in reference parabola
π/a -π/a
2π/a k 0 323d22
Extended-Zone Scheme • Bands form, separated by band gaps • Note redundancy: no need for defining k outside +-π/a region
E Band 2 Band gap Eg
Band 1 −2π/a
©1999 E.A. Fitzgerald
-π/a
0
π/a
2π/a
k
323d23
Reduced-Zone Scheme • Only show k=+-π/a since all solutions represented there
−π/a ©1999 E.A. Fitzgerald
π/a 323d24
Real Band Structures • GaAs: Very close to what we have derived in the nearly free electron model • Conduction band minimum at k=0: Direct Band Gap
©1999 E.A. Fitzgerald
323d25
Real Band Structures • Ge: Very close to GaAs, except conduction band minimum is in <111> direction, not at k=0 • Indirect Band Gap
©1999 E.A. Fitzgerald
323d26
Trends in III-V and II-VI Compounds Larger atoms, weaker bonds, smaller U, smaller Eg, higher µ, more costly!
Band Gap (eV) SiGe Alloys
©1999 E.A. Fitzgerald
Lattice Constant (A)
323d27
Schrodinger Equation • For quantitative understanding, need to develop equation to calculate energy of wavefunctions (electrons) in different environments • ‘Newton’s Laws’ for wave-particles h = λ ; E = hν • Equation was postulated p • New equation needs: p2 E= +U – Duality 2m – Energy conservation Ψ = ∑ ciψ i – Analogous to operations with E-M waves i (superposition) – Solutions must be waves
Ψ = Ae
i (kx −ωt )
323e1 ©1999 E.A. Fitzgerald
Schrodinger Equation h λ = ; E = hν p p2 E= +U 2m
Ψ = ∑ ciψ i i
Ψ = Ae i (kx −ωt )
h2 + U (x, t ) = hν 2 2mλ 2π k= ; ω = 2πν λ ! 2k 2 + U (x, t ) = !ω 2m Equation must be linear in Ψ: no powers of Ψ, Ψ in every term
∂Ψ ∝ −ω ∂t ∂ 2Ψ 2 ∝ − k ∂x 2
∂Ψ α∂ 2 Ψ + U (x, t )Ψ = β 2 ∂x ∂t
try
Ψ = Ae i (kx −ωt ) − !2 α= ; β = i! 2m
− ! 2 ∂ 2 Ψ (x, t ) ∂Ψ (x, t ) ( ) ( ) ! U x t x t i + Ψ = , , ∂x 2 ∂t 2m − !2 2 ∂Ψ (r , t ) ∇ Ψ (r , t ) + U (r , t )Ψ (r , t ) = i! ∂t 2m kinetic
potential
total
323e2 ©1999 E.A. Fitzgerald
Time-Independent S.E. Ψ = Ae i (kx −ωt ) = Ae ikx e −iωt = ψ (x )Φ (t ) insert into S.E. - !2 2 ∇ ψ (x ) + U (x )ψ (x ) = Eψ (x ) 2m general S.E. or T.I.S.E. can be abreviated Hψ = Eψ where H is an implied operator − !2 2 ∇ +U H= 2m H is the Hamiltonian, the energy operator 323e3 ©1999 E.A. Fitzgerald
General Representation of Electron Wave Functions in Periodic Lattice • Often called ‘Bloch Electrons’ or ‘Bloch Wavefunctions’ Away from Bragg condition, ~free electron
E
! 2k 2 − !2 2 − !2 2 ikx ∇ +Uo ≈ ∇ ; ψ ≈e ;E = H= m m 2 2 2m Near Bragg condition, ~standing wave electron − !2 2 H= ∇ + U o ≈ U o (x ); ψ ≈ cos Gx or sin Gx = u (x ); E = U o (x ) 2m
Since both are solutions to the S.E., general wave is
Ψ = ψ freeψ lattice = e ikxu (x ) 0
π/a
k
termed Bloch functions 323e4
©1999 E.A. Fitzgerald
Block Theorem • If the potential on the lattice is U(r) (and therefore U(r+R)=U(r)), then the wave solutions to the S.E. are a plane wave with a periodic part u(r) that has the periodicity of the lattice
Ψ (r ) = e ik ⋅r u (r ) u (r ) = u (r + R )
Note the probability density spatial info is in u(r): 2 Ψ *Ψ = Ψo u * (r )⋅ u (r ) An equivalent way of writing the Bloch theorem in terms of Ψ: Ψ (r + R ) = e ik (r + R )u (r + R ) = e ik (r + R ) Ψ (r + R ) = e ik ⋅R Ψ (r )
Ψ (r ) e ik ⋅r
323e5 ©1999 E.A. Fitzgerald
Implications of Bloch Electron Waves Bloch electrons are not free electrons p in general ! pˆ Ψ = −i!∇Ψ = −i!∇ e ik ⋅r u (r ) = !kΨ − [i!∇u (r )]e ik ⋅r
k≠
(
)
generally not zero
!k
=“crystal momentum”
Velocity: no degradation, no time dependence, =group velocity of wave packet v(k ) =
dω 1 1 ∇ k E (k ) (v = ∇ kω = = ∇ k E (k )) ! dk !
323e6 ©1999 E.A. Fitzgerald
Implications of Bloch Electron Waves Energy Bands!, and E invariant to shifts in k by G k = k '+G
Ψ (r + R ) = ei (k '+ G )⋅R Ψ (r ) = e ik 'R Ψ (r )
Ψ invariant to shift in G
HΨ (r ) = E (r )Ψ (r ) HΨ (r + R ) = E (r + R )Ψ (r + R )
Ψ (r + R ) = e ikr Ψ (r )
(
)
∴ HΨ (r + R ) = H e ikr Ψ (r ) = E (r )e ikr Ψ (r ) = E (r )Ψ (r + R ) Ψ(r) can have E(r), E(r+R),…--> Energy Bands E
Implies a maximum in E even though k continues to increase
323e7 ©1999 E.A. Fitzgerald
k
k+G
k
Implications of Bloch Electron Waves Since E is invariant to k shifted by G (same as diffraction argument): ! 2 (k x ± mx Gx ) E= 2m
2
1-D E E(k)+E(Gy and/or Gz) E=
3-D 2 2 2 ! 2 [(k x + mx Gx ) + (k y + m y G y ) + (k z + mz Gz ) ] 2m
E(k) E(k+G)
E(k-G)
−2π/a -π/a
0 G
π/a
kx 2π/a
Note: key difference in 1-D to 3-D in graph on left is that we are looking along kx, but in 3-D the value can change due to Gy and/or Gz
323e8 ©1999 E.A. Fitzgerald
General Derivation of NFE • We know that the wavefunctions will look like plane waves • We know that lattice potential from the lattice ions have the periodicity of real space and G in k-space
Ψ (r ) = ∑ cq e iqr q
U (r ) = ∑ U G e iGr G
HΨ = E Ψ ! 2 2 iqr − !2 2 ∇ Ψ = ∑ q cq e 2m q 2m
iGr iqr UΨ = ∑ U G e ∑ cq e q G
turn the crank!
©1999 E.A. Fitzgerald
! 2q 2 − E cq + ∑ U G 'cq −G ' = 0 G' 2m
323e9
Proof of Bloch Theorem from General Derivation • For a given k, central equation shows that the coefficients cq that matter are those that are a multiple of G from k
Ψk = ∑ cq eiqr ≈ ∑ ck −G ei (k −G )r q
G
In our diffraction example, k=π/a, and cq and cq-G were 1 at the diffraction condition,
Ψ
k=
π a
=e
π i r a
+e
π −i r a
Continuing,
−iGr Ψk = e ∑ ck −G e = e ikr u (r ) G u (r ) = ∑ ck −G e −iGr Bloch waves are indeed the general solution ikr
G
323e10 ©1999 E.A. Fitzgerald
Central Equation near the Bragg Plane
E
•Evaluate at q=+ or -1/2 G •Assume since U(r) goes as 1/r, that only nearest two potentials are relevant
1/2 G
For q=1/2 G,
For q=-1/2 G,
E(q) G’=-G
G’=G ! 2 1 2 c 1 + U G c 1 = 0 G E − 2m 2 G − G 2 2
E(q-G)
! 2 1 2 − G − E c 1 + U − G c 1 = 0 2m 2 − G G 2 2
EG/2
0
π/a G
reciprocal lattice points
2π/a kx
E-G/2
(EG / 2 − E )c1/ 2G + U G c−1/ 2G = 0 U −G c1/ 2G + (E−G / 2 − E ) = 0 Also, U G = U −G 323e11
©1999 E.A. Fitzgerald
Central Equation near the Bragg Plane (EG / 2 − E ) UG
E UG
(E−G / 2 − E )
=0
(EG / 2 − E )(E−G / 2 − E ) − U G2 = 0 + EG / 2 − EG / 2 ) E (E 2 ± −G / 2 + E = −G / 2 U G 2 2 2
At the Bragg plane, EG/2=E-G/2
E = EG / 2 ± U = EG / 2 ± U G 2 G
Plugging back into one of the equations,
cG / 2 = ± c−G / 2
UG
EG/2
0
UG
π/a
ψ q = cq eiqr ; ψ G / 2 = cG / 2ei (G / 2) r ψ q −G = ±cq ei (q −G )r ; ψ −G / 2 = cG / 2 e −i (G / 2) r Ψ = ψ G / 2 ± ψ −G / 2 Ψa & Ψb
©1999 E.A. Fitzgerald
323e12
Central Equation just off the Bragg Plane E
∆q
1/2 G
E=
E(q)
E=
E(q-G) ∆E
0
π/a
2π/a 2
! 2G 2 ( ) ∆ q !2 ! 2G 2 1 2m 2 (∆q ) + ±U ± E= U 2m 16m 8 ! 2 (1 + ζ ) (∆q )2 + U o E= 2m ! 2G 2 ! 2G 2 ζ =± ; Uo = U + 16mU 16m ©1999 E.A. Fitzgerald
E=
Eq + Eq − G 2 Eq + Eq −G 2 Eq + Eq −G 2
±
(E
− E q −G )
2
q
+U 2
4
1 ∆E ± 1+ 4 U
2
∆E << 1 U
1 (∆E ) ±U ± 8 U
2
2
G G ! ∆q + ! 2 ∆q − 2 2 ∆E (∆q ) = Eq (∆q ) − Eq −G (∆q ) = − 2m 2m ! 2G ∆E (∆q ) = ∆q 2m Eq (∆q ) + Eq −G (∆q ) ! 2 G2 2 (∆q ) + ∴ = 2 2m 8 2
Let m*=m/(1+ζ)
! 2 (∆q ) E ±Uo = 2m *
2
Can make band edges near band gaps free-electron-like 323e13
2
Central Equation Far from Bragg Condition • •
Far from the Bragg condition, there is little effect of the periodic ion potential In the language of the central equation, all the cq-G are small except for one of them
!2 2 !2 2 q cq − cq E + U o ceff = 0 ≈ E − q 2m 2m !2 2 ∴E ≈ q 2m
The electrons are free electrons if the wavevector is far from a wavevector value that diffracts strongly
323e14 ©1999 E.A. Fitzgerald
Chemistry Approach • • • •
Let’s build the solid atom by atom! For many atoms, arrive at same band picture that we have achieved Important since systems with small numbers of atoms need to be treated differently understand matter from a few atoms to the infinite solid
•Recall that we have always been dealing with 1 e- in a periodic potential --> easier problem for S.E. •We will eventually run into the many-bodied problem --> many electrons, many atoms •We will make approximations to continue, and arrive at similar results to the NFE
Hψ atomic = Eatomicψ atomic HΨmolecule = Emolecule Ψmolecule HΨchain = Echain Ψchain
HΨsolid = (Evalence e −'s + Eions )Ψsolid ©1999 E.A. Fitzgerald
323e15
Review of H atom • •
S.E. in spherical co-ordinates Separation of variables; separation constants are basically quantum numbers; 3 for 3 dimensions Creates atom structure for periodic table
•
z r Atomic number (atomic charge)=Z
U=
e-
θ
+ φ
µ=
y
− Ze 2
− Ze 2 = 2 2 2 r x +y +z 1 1 1 + m M
Problem easier if we choose U(r)
Reduced mass keeps problem a single body problem
−! 2 ∇ ψ + U (r )ψ = Eψ 2m
x
Where the del is in spherical co-ordinates:
∂2 ∂ ∂ 1 ∂ 2 ∂ 1 1 ∇ = 2 + θ sin r + 2 2 r ∂r ∂r r sin θ ∂φ 2 r 2 sin θ ∂θ θ 2
323e16 ©1999 E.A. Fitzgerald
Review of H atom ψ = R(r )Θ(θ )Φ (φ ) Hψ = Eψ Do separation of variables; each variable gives a separation constant φ separation yields ml θ gives " r gives n
After solving, the energy E is a function of n − µZ 2 e 4 − 13.6eV E= = n2 (4πε o )2 2! 2 n 2
ml and " in Φ and Θ give Ψ the shape (i.e. orbital shape) The relationship between the separation constants (and therefore the quantum numbers are:)
n=1,2,3,… " =0,1,2,…,n-1 ml=- ", - "+1,…,0,…, "-1, (ms=+ or - 1/2)
0
U(r)
-13.6eV
323e17 ©1999 E.A. Fitzgerald
Relationship between Quantum Numbers
s
s
p
s
p
d
Origin of the periodic table 323e18 ©1999 E.A. Fitzgerald
Hydrogen Wavefunctions
323e19 ©1999 E.A. Fitzgerald
Physical Nature of Orbitals (Ψ2)
ψ 2 = ψ *ψ = P (r , l , ml ) Probability density
1s
Radial Probability Density = R*R4πr2dr Look at only probability of finding electron in a shell of thickness dr at r from the nucleus
Bohr radius
r
Ψ can be negative; cross over is 0 in Ψ2 2s
Compare to Bohr ∞
∞
r = ∫ rP(r , l , ml )dr = ∫ rR * R 4πr 2 dr 0
0
n 2 ao r= Z
rBohr =
2p
1 l (l + 1) 1 + 2 1 − n 2
2
3s 3p
2
n ao ! ; ao = 2 = 0.52 A µe Z
3d
323e20 ©1999 E.A. Fitzgerald
Physical Nature of Orbitals (Ψ2)
•
Θ and Φ functions give shape
l=3 (f shell)
323e21 ©1999 E.A. Fitzgerald
Multielectron Atoms • Recall previous wavefunctions were for single electron in different possible orbits • What about a real atom with mulitple electrons? – Screening, so E=-13.6Z2/n2 doesn’t work well always – empirically: Z=Zeff • Can solve exactly by putting other electrons in U through an electron density, ρ Pick a starting Ψ and ρ H Ψ=Ε Ψ
New cq, define new Ψ, ρ: check for minimum E ©1999 E.A. Fitzgerald
Determine cq, E 323e22
What about building up our solid: multiatoms? • Simplest case: H2+, hydrogen molecular ion (1e-, 2 protons) R
U molecule
1 1 1 = e − − + r1 r2 R 2
+
+ r1
r2
•ions repel •electrons attract ions together when in between ions (bond)
e-
•Born-Oppenheimer Approximation !protons fixed (R const.): get energy of electron !determine proton-proton energy separately and superimpose 1 1 U el = e − − r1 r2 2
Solve HΨmol=EΨmol
Emol
e2 = Eel (R ) + R
323e23 ©1999 E.A. Fitzgerald
Multiatoms:H2+ • Assume that the molecular wavefunction for the electron is a linear combination of the atomic wavefunctions (LCAO)
Ψ = c1φ1 + c2φ 2 1s : φ1 =
− r1 ao
1
πa
3 o
1
e ; φ2 =
πa
3 o
e
− r2 ao
HΨ = EΨ Ψ * HΨ = EΨ *Ψ E=
* Ψ ∫ HΨdV * Ψ ∫ ΨdV
=
Ψ* H Ψ Ψ* Ψ
ˆ (c φ + c φ )dV ( ) c c H φ + φ ∫ = ∫ (c φ + c φ )(c φ + c φ )dV * 1 1
* 1 1
* 2 2
* 2 2
1 1
1 1
2 2
2 2
323e24 ©1999 E.A. Fitzgerald
Multiatoms:H2+ We will get three kinds of integrals that we will abbreviate:
Coulomb Integrals
H11 = ∫ φ1 Hˆ φ1dV ; H 22 = ∫ φ 2 Hˆ φ 2 dV
Bond Integrals
H12 = ∫ φ1 Hˆ φ 2 dV
Overlap Integrals
S12 = ∫ φ1φ 2 dV
* * Assume φ1 = φ1 ; φ 2 = φ 2 (i.e. real functions)
and
* φ ∫ i φi dV = 1 (they are normalized)
c12 H11 + c22 H 22 + 2c1c2 H12 E= c12 + c22 + 2c1c2 S12
Apply variational method: pick Ψ’s to get the lowest E; minimize E with respect to c1 and c2 323e25
©1999 E.A. Fitzgerald
Multiatoms:H2+ ∂E = 0 = c1 (H11 − E ) + c2 (H12 − ES12 ) ∂c1 ∂E = 0 = c1 (H12 − ES12 ) + c2 (H 22 − E ) ∂c2 For identical atoms, H = H11 = H 22 ≡ atomic integrals c1c2 (H − E ) − c1c2 (H12 − ES12 ) = 0 2
EA = H +
2
EA
H12 − HS12 1 + S12
H − HS12 E B = H − 12 1 − S12
anitbonding
H
H EB
bonding
323e26 ©1999 E.A. Fitzgerald
Multiatoms:H2+ Use E’s in original equations:
c1 ± c2
ΨB = c1 (φ1 + φ 2 ) ΨA = c1 (φ1 − φ 2 )
Note: 2 atoms, 2 levels Bonding
Anti-Bonding
Ψ2
323e27 ©1999 E.A. Fitzgerald
Multiatoms:H2+ E
Bonding
E
Nuclear repulsion
Anti-Bonding Nuclear repulsion
2
e R
e2 R
Ro R K (R − Ro )2 2 F = − K (R − Ro )
Electron energy
E=
R
Electron energy
Atoms connected by ‘springs’ 323e28 ©1999 E.A. Fitzgerald
LCAO: Adding more atoms • Assume H2 solution similar to H2+ • Let’s use Li as the atom to build our solid Molecular orbital notation
Ψ1σ * = c(φ1 − φ 2 ) Ψ1σ = c(φ1 + φ 2 )
Representation of Ψ2 1σ*
anitbonding
+
Sign of wavefunction
1s
1s 1σ
bonding
+
323f1 ©1999 E.A. Fitzgerald
LCAO: Adding more atoms • Li: 1s22s1 • Look at Li2
+
2σ*
-
E 2s
2s
+
2σ
Li2
1σ* 1s
1σ
1s 1s are ‘buried’ inside above pictures and more atomic-like (experience less delocalization since they are not valence electrons)
323f2 ©1999 E.A. Fitzgerald
LCAO: Adding more atoms • Conservation of states • more nodes for higher energy molecular wavefunctions nodes
Ψ4 = c41φ1 − c42φ 2 + c43φ3 − c44φ 4
Li4
Ψ3 = c31φ1 − c32φ 2 − c33φ3 + c34φ 4 Ψ2 = c21φ1 + c22φ 2 − c23φ3 − c24φ 4 Ψ1 = c11φ1 + c12φ 2 + c13φ3 + c14φ 4
3 2 1 0
+ - + +
-
+ -
+ +
j=4 j=3
2s
j=2
2s
4 states, 4 valence e- in extended orbital
j=1
323f3 ©1999 E.A. Fitzgerald
LCAO: Adding more atoms • Generalize to N Li atoms N
Ψ j = ∑ c jiφi ; c ji = i =1
2 πji jπ sin ; E j = α + 2 β cos N +1 N +1 N +1 bond integral
atom index
orbital index
known as the Debye-Huckel model examples: N=2 α-β
N=large
N=4 j=4
α+β j=1
α-1.6β α-0.6β α+0.6β α+1.6β
α-2β EF α+2β 323f4
©1999 E.A. Fitzgerald
Generaliztion of Debye-Huckel for large N • Same as “tight-binding” model N a
Index i=Z/a
N+1~N=L/a
Z
L=Na
cj ≈
E
2 πjz sin N L
j ⇒ N, k ⇒
π L
πj 2 sin (kz ) ⇒c~ L N E = α + 2 β cos(ka )
k=
α
2β 2β
-π/a
π/a
k
vg =
1 ∂E 2 aβ =− sin ka ! ∂k ! 323f5
©1999 E.A. Fitzgerald
Bonding of other elements across periodic table • LiN : all spherical shells • across periodic table (1s2 2s2 2px), encounter p-orbitals • leads to π-bonds as well as σ-bonds
+
-
-
+
2pz
+
+
+
2py
+ More wiggles, higher energy
3σ
-
=
+
+
+
+-+-
=
+ -
=
+-
-
-
3σ*
πy
-
+ -
+
+ ©1999 E.A. Fitzgerald
-
+
2pz
-
2py
=
+-
+
πy* 323f6
Bonding of other elements across periodic table • Example: Oxygen (O2) •From Hund’s rules: put 1e- in each π orbital first (maximize spin) •O2 is paramagnetic
3σ∗ πx∗, πy∗ 2p
2p
πx, πy 3σ
•For O2 and F2, the 3σ is lower than the π-orbitals •For Li2 to N2, the πorbitals are actually lower than the 3σ
2σ* 2s
2s 2σ 1σ* 1s
1σ
1s
•O has eight electrons, thus O2 distributes 16 electrons over 10 molecular levels 323f7
©1999 E.A. Fitzgerald
Visualization of π orbitals Example: 4 Carbon chain, unsaturated (C4H6)
H C
H C H E
Nodes 3 2 1 0
E=α+2βcoska ©1999 E.A. Fitzgerald
+-+-+-+ + - + - + + - + + -
β may change from σ to π, but principle the same
σ bonds π bonds
H C H C H
π bonds extend over entire chain
Ψ4=c1φ1-c2φ2+c3φ3-c4φ4
α-1.6β
Ψ3=c1φ1-c2φ2-c3φ3+c4φ4
α-0.6β
Ψ2=c1φ1+c2φ2-c3φ3-c4φ4
α+0.6β
Ψ1=c1φ1+c2φ2+c3φ3+c4φ4
α+1.6β
π electrons: each carbon contributes 1 e- to the extended π
323f8
Bonding of atoms with different levels/orbitals • Example: LiF
5.4 eV
F px
σ*
18.6 eV
Li 2s
Li 2s
0
+
+
+
Zero overlap
F 2p
σ
+ +
+
-
=
-
Ψ = c1φ Li + c2φ F
σ bond
Ψσ * ⇒ c1 >> c2
Li
+F ionic
Ψσ ⇒ c2 >> c1 323f9
©1999 E.A. Fitzgerald
Hybridization • Two ways to look at it – solutions to S.E. that minimize E are Ψ’s that look like combinations of s and p orbitals – orbital ‘distorts’ to reach out and bond Examples
Hybrid
H Be H
Linear, sp
H
+
+
+
Be
H
- -
+ +
H
H sp hybridization
+
Tetrahedral, sp3
B
-
H H H C H H
Planar,
H
-
H B
sp2
+ 1s2s
+
+
-
+ 323f10 ©1999 E.A. Fitzgerald
Bonding and Hybridization • • •
Energy level spacing decreases as atoms are added Energy is lowered as bonding distance decreases All levels have E vs. R curves: as bonding distance decreases, ion core repulsion eventually increases E
E
Debye-Huckel
p s
R NFE picture, semiconductors
hybridization 323f11
©1999 E.A. Fitzgerald
Properties of non-free e• Electrons near the diffraction condition are not approximated as free • Their properties can still be viewed as free e- if an ‘effective mass’ m* is used !2k 2 E= 2mec*
! 2k 2 E= 2m
!2 m = 2 ∂ E ∂k 2 * ec
! 2k 2 E= 2mev* !2 m = 2 ∂ E ∂k 2 * ev
−π/a
Note: These electrons have negative mass!
π/a 323f12
©1999 E.A. Fitzgerald
1-D Crystal Metals and Insulators • • • • •
How do band gaps affect properties of materials? Only electrons near EF participate in properties If EF is in the middle of the band, free e- and metallic behavior If EF is near the band gap, changes in materials properties may occur Need to find out where EF is! Where kF=π/a if we 2k L 2 L want to see how many N= F = a π electrons are in first band Note: L/a is the number of unit cells in the 1-D crystal; therefore, the number of electrons per unit cell, which depends on valence and the crystal structure, determine where EF is with respect to the band gap
EF with 2e- per unit cell EF with 1e- per unit cell
−π/a ©1999 E.A. Fitzgerald
π/a
323f13
1-D Crystal Metals and Insulators • 2e- per unit cell: EF at band edge: 2 possibilities – Band gap >> kT: electrons at band max can not accept energy from electric fields; no conduction, insulating behavior – Band gap near kT: some thermal fluctuations large enough to allow population of second band; carriers are there, but less than for free e-, semimetal • 1e- per unit cell: EF in middle of band: free e-, metallic Note: crystal structure (number of atoms per primitive cell) and valence (number of conduction electrons per atom), combined with band gap size, determine the electronic properties 323f14 ©1999 E.A. Fitzgerald
Higher Dimensions (2 and 3-D) • 1-D: E(kx); 2-D: E(kx,ky); 3-D: E(kx,ky,kz)
‘Fermi Surface’ E Zone center
ky First zone
π/a First zone
0
kx −π/a
−π/a
0
π/a
π/a
3-D
−π/a
π/a 2-D
1-D Zone center ©1999 E.A. Fitzgerald
−π/a 0
323f15
Higher Dimensions: Visualizing the Fermi Sphere and Bragg Plane Intersections • Use Ge band diagram to demonstrate how to show EF intersecting two important directions
EF 323f16 ©1999 E.A. Fitzgerald
Bands and Zones • Examine 2-D simple cubic crystal, or a slice through a 3-D cubic crystal • Bragg planes exist at bisection of every pair of reciprocal lattice points • As kf increases, Ef increases and crosses Bragg planes; when crossing, going to next band
323f17 ©1999 E.A. Fitzgerald
Bands and Zones 1st band Electron surface
Hole surface
G
2nd band
Reduced zone
Extended zone
323f18 ©1999 E.A. Fitzgerald
Carrier Dynamics at Fermi Surfaces ∂E
• Recall velocity is proportional to vg ∝ ∂k • Carriers have mass inversely proportional to curvature • Bulk electronic properties average over all surfaces
∂2E m* ∝ 2 ∂k
[010] •In magnetic field, carriers move along the Fermi Surface •Can measure the Fermi Surface this way
vg m l*
m t* v total since F~vxB [100] vg from E-field
Example: Si 323f19 ©1999 E.A. Fitzgerald
Metals and Insulators • Covalent bonds, weak U seen by e-, with EF being in mid-band area: free e-, metallic • Covalent or slightly ionic bonds, weak U to medium U, with EF near band edge – EF in or near kT of band edge: semimetal – EF in gap: semiconductor
• More ionic bonds, large U, EF in very large gap, insulator 323f20 ©1999 E.A. Fitzgerald
Insulators • Very large band gaps=no conduction electrons at reasonable temperatures • All electrons are bound • Optical properties of insulators are derived from the electric field being able to temporarily move electrons: polarization • We will return to the interaction of E-field with bound electrons in Dielectrics Section 323f21 ©1999 E.A. Fitzgerald
Semiconductors • Band gaps not large enough to prevent all carriers from getting into next band: three mechanisms – photon absorption – thermal – impurity (i.e. doping)
• Carriers that make it to the next band are like free carriers with new mass, m* 323f22 ©1999 E.A. Fitzgerald
Semiconductors: Photon Absorption • When Elight=hν>Eg, an electron can be promoted from the valence band to the conduction band E Ec near band gap E=hν k Creates a ‘hole’ in the valence band
Ev near band gap 323f23 Note: Most absorption near the band gap since the density of states is highest there ©1999 E.A. Fitzgerald
Holes and Electrons • • •
Instead of tracking electrons in valence band, more convenient to track vacancies of electrons, or ‘holes’ Also removes problem with negative electron mass: since hole energy increases as holes ‘sink’, the mass of the hole is positive as long as it has a positive charge Both carriers at the band edge can be thought of as classical free carriers like the Drude model had, as we shall see
Decreasing electron energy
Decreasing hole energy Decreasing electron energy 323f24 ©1999 E.A. Fitzgerald
Conductivity of Semiconductors • Need to include both electrons and holes in the conductivity expression
ne 2τ e pe 2τ h + σ = neµ e + peµ h = * me mh*
p is analogous to n for holes, and so are τ and m*
Note that in both photon stimulated promotion as well as thermal promotion, an equal number of holes and electrons are produced, i.e. n=p
323f25 ©1999 E.A. Fitzgerald
Thermal Promotion of Carriers •
EF
E
•
We have already developed how electrons are promoted in energy with T: Fermi-Dirac distribution Just need to fold this into picture with a band-gap
gc(E)~E1/2 in 3-D
Eg
f(E)
1
gv(E)
©1999 E.A. Fitzgerald
Despite gap, at non-zero temperatures, there is some possibility of carriers getting into the conduction band (and creating holes in the valence band)
g(E)
323f26
Density of Thermally Promoted of Carriers Number of electrons per volume in conduction band
n=
∞
∫ f ( E ) g ( E )dE
Ec
Density of electron states per volume per dE Fraction of states occupied at a particular temperature 1
f (E) = e 1 g c (E) = 2π 2
2m 2 ! ∞
* e
≈e
(E − E F )
3 2
−
(E − EF ) k bT
when ( E − EF ) >> kbT
+1
k bT
1 2m n = 2 2 2π !
(E − E g )
* e
1 2
π Since ∫ x e dx = , then 2 0 1 2 −x
©1999 E.A. Fitzgerald
e
1 2
EF ∞ k bT
∫ (E − E ) e g
−E k bT
dE
Eg
− Eg
m k T kbFT kbT e e n = 2 2 ! 2π * e b
EF − Eg
n = NCe
3 2
3 2
E
NC
k bT 323f27
Density of Thermally Promoted of Carriers • A similar derivation can be done for holes, except the density of states for holes is used • Even though we know that n=p, we will derive a separate expression anyway since it will be useful in deriving other expressions 1 g v (E) = 2π 2
2m 2 !
* h
3 2
1 (− E )2
0
p=
∫f
h
( E ) g v ( E )dE , where f h = 1 − f ( E )
−∞ 3 2
−E
m k T kbTF e p = 2 2 π ! * h b 2
p = Nve
− EF k bT
323f28 ©1999 E.A. Fitzgerald
Thermal Promotion • Because electron-hole pairs are generated, the Fermi level is approximately in the middle of the band gap • The law of mass action describes the electron and hole populations, since the total number of electron states is fixed in the system mh* 3 + kbT ln * n = p gives E F = 2 4 me Eg
Since me* and mh* are close and in the ln term, the Fermi level sits about in the center of the band gap 3 2
(
kT p or n = ni = 2 b 2 me*m 2π!
)e
3 * 4 v
− Eg 2 k bT
323f29 ©1999 E.A. Fitzgerald
Law of Mass Action for Carrier Promotion 3
(
kT ni2 = np = 4 b 2 me*m 2π!
)e
3 * 2 h
− Eg kbT
− Eg
;
ni2 = N C NV e kbT
•Note that re-arranging the right equation leads to an expression similar to a chemical reaction, where Eg is the barrier •NCNV is the density of the reactants, and n and p are the products E [N C NV ] → [n]+ [p ] g
[n][p ]
[N C NV ]
− Eg
=e
kbT
=
[ni ]2
[N C NV ]
•Thus, a method of changing the electron or hole population without increasing the population of the other carrier will lead to a dominant carrier type in the material •Photon absorption and thermal excitation produce only pairs of carriers: intrinsic semiconductor •Increasing one carrier concentration without the other can only be achieved with impurities, also called doping: extrinsic semiconductors 323f30 ©1999 E.A. Fitzgerald
Intrinsic Semiconductors • •
Conductivity at any temperature is determined mostly by the size of the band gap All intrinsic semiconductors are insulating at very low temperatures
Recall:
ne 2τ e pe 2τ h σ = neµ e + peµ h = + * me mh*
σ int = ni e(µ e + µ h ) ∝ e
− Eg 2 k bT
This can be a measurement for Eg
For Si, Eg=1.1eV, and let µe and µh be approximately equal at 1000cm2/V-sec (very good Si!) σ~1010cm-3*1.602x10-19*1000cm2/V-sec=1.6x10-6 S/m, or a resistivity ρ of about 106 ohm-m max
•One important note: No matter how pure Si is, the material will always be a poor insulator at room T •As more analog wireless applications are brought on Si, this is a major issue for system-on-chip applications –E-M waves lose strength since e- are responding to wave: loss and low Q resonant circuits 323f31 ©1999 E.A. Fitzgerald
Extrinsic Semiconductors •
•
Adding ‘correct’ impurities can lead to controlled domination of one carrier type – n-type is dominated by electrons – p-type if dominated by holes Adding other impurities can degrade electrical properties Impurities with close electronic structure to host isoelectronic
hydrogenic
x x
Ge
Impurities with very different electronic structure to host
x x
x
x
P
+
deep level
-
x x
x
x
x
Si
Si
Ec Ev ©1999 E.A. Fitzgerald
Au x
Ec ED
Ec
Ev
Ev
EDEEP 323f32
Hydrogenic Model • •
For hydrogenic donors or acceptors, we can think of the electron or hole, respectively, as an orbiting electron around a net fixed charge We can estimate the energy to free the carrier into the conduction band or valence band by using a modified expression for the energy of an electron in the H atom me 4 13.6 En = 2 2 2 = − 2 n 8ε o h n 4
(in eV)
e2 2 =e εr
me m *e 4 1 13.6 m* 1 En = 2 2 2 → 2 2 2 2 = − 2 n m ε2 8ε o h n 8ε o h n ε r
•Thus, for the ground state n=1, we can see already that since ε is on the order of 10, the binding energy of the carrier to the center is <0.1eV •Expect that many carriers are then thermalize at room T •Experiment: •B acceptor in Si: .046 eV •P donor in Si: 0.044 eV •As donor in Si: 0.049 323f33 ©1999 E.A. Fitzgerald
The Power of Doping •
•
Can make the material n-type or p-type: Hydrogenic impurities are nearly fully ionized at room temperature – ni2 for Si: ~1020cm-3 – Add 1018cm-3 donors to Si: n~Nd – n~1018cm-3, p~102 (ni2/Nd) Can change conductivity drastically – 1 part in 107 impurity in a crystal (~1022cm-3 atom density) – 1022*1/107=1015 dopant atoms per cm-3 – n~1015, p~1020/1015~105 σ/σi~(p+n)/2ni~n/2ni~105! Impurities at the ppm level drastically change the conductivity (5-6 orders of magnitude)
323f34 ©1999 E.A. Fitzgerald
Expected Temperature Behavior of Doped Material (Example:n-type) • 3 regimes ln(n)
Eg/2kb
Eb/kb
Intrinsic
Extrinsic Freeze-out 1/T
323f35 ©1999 E.A. Fitzgerald
Doping Statistics and Law of Mass Action • •
If electron population rises, hole concentration decreases (and reverse) Dopant dominates conductivity, assuming dopant concentration much greater than intrinsic
n − p = 0 for intrinsic ∆ = n − p ≠ 0 for extrinsic np = ni2 p = n−∆
n(n − ∆ ) = ni2 n 2 − n∆ − ni2 = 0 ∆ ∆ n n= ± 1 + 4 i 2 2 ∆
2
ni << 1, n = ∆ ∆ 323g1 ©1999 E.A. Fitzgerald
Contrasting Semiconductor and Metal Conductivity ne 2τ σ= m
• Semiconductors – changes in n(T) can dominate over τ – as T increases, conductivity increases • Metals – n fixed – as T increases, τ decreases, and conductivity decreases
323g2 ©1999 E.A. Fitzgerald
General Interpretation of τ • Metals and majority carriers in semiconductors – τ is the scattering length – Phonons (lattice vibrations), impurities, dislocations, and grain boundaries can decrease τ 1 1 1 1 1 = + + + + ... τ τ phonon τ impur τ disl τ gb τi =
li 1 = vth vthσ i N i
liσ i N i = 1
where σ is the cross-section of the scatterer, N is the number of scatterers per volume, and l is the average distance before collisions
The mechanism that will tend to dominate the scattering will be the mechanism with the shortest l (most numerous), unless there is a large difference in the cross-sections
Example: Si transistor, τphonon dominates even though τimpur gets worse with scaling 323g3 ©1999 E.A. Fitzgerald
Limiting Scattering Mechanism • • •
As temperature is decreased, phonon scattering ceases to be main scattering mechanism Low T can be used to determine ultimate material quality Expected dependence in metals: ρ~T
σ
σ extrinsic intrinsic
defect-limited Phonon-limited
T
T
323g4 ©1999 E.A. Fitzgerald
Estimate of T dependence of conductivity • τ~l for metals • τ~l/vth for semiconductors • First need to estimate l=1/Nσ
1 N ionσ ion
l ph =
σ ion ∝ π x 2 x=0
x2 =
∫
+∞
Ψ * x 2 Ψdx
−∞ +∞
∫
−∞
Use Ψ for harmonic oscillator, get:
Ψ *Ψdx
!ω
k x2 = E = e
!ω kT
−1
Average energy of harmonic oscillator
323g5 ©1999 E.A. Fitzgerald
Estimate of T dependence of conductivity !ω
k x2 = E = e !ω = kθ kθ E = θ eT −1
!ω kT
−1
For a metal:
θ T
Therefore, <x2> is proportional to T if T large compared to θ:
θ T x2 ∝ T
e ≈ 1+
l∝
1 1 1 ∝ 2 ∝ T σ x
σ cond ∝ µ ∝ τ ≈
l 1 1 = = vF vF N ionπ x 2 vF N ionπT
For a semiconductor, remember that the carriers at the band edges are classical-like:
l τ= = vth ©1999 E.A. Fitzgerald
1 3 − l ∝ T1 ∝ T 2 3kT T2 * m
3 − eτ µ = * ∝T 2 m 323g6
Example: Electron Mobility in Ge µ~T-3/2 if phonon dominated (T-1/2 from vth, T-1 from x-section σ)
At higher doping, the ionized donors are the dominate scattering mechanism
323g7 ©1999 E.A. Fitzgerald
Other Interpretation of τ •
Minority carriers in semiconductors – can think of τ as the time to recombination: recombination time – does not affect Drude model in any way
τ, l
Ec Recombination event: electron disappears
E
Ev x
Imagine p-type material, so there are many more holes than electrons holes=majority carrier electrons=minority carrier
τ is referred to in this context as the minority carrier lifetime
323g8 ©1999 E.A. Fitzgerald
Recombination and Generation (Ec to Ev) •
•
Generation – intrinsic: photon-induced or thermally induced, G=#carriers/vol.-sec – extrinsic:deep levels due to traps – Go is the equilibrium generation rate Equilibrium: Recombination R=Gtherm – intrinsic: across band gap, R=# carriers/vol.-sec – extrinsic: deep levels due to traps – Ro is the equilibrium recombination rate, which is balanced by Go Non-equilibrium intrinsic recombination
n-type material
R=
∆p p ; τh = o τh Ro
Where po is the equilibrium minority carrier concentration
p-type material
R=
n ∆n ; τh = o τe Ro
Where no is the equilibrium minority carrier concentration ©1999 E.A. Fitzgerald
Non-equilibrium extrinsic recombination
R=
∆p τh
τh =
1 σ h vth N t
Where σh is the capture cross section for holes and Nt is the concentration of recombination centers
∆n R= τe
τe =
1 σ e vth N t 323g9
Proof that Minority Carrier Dominates Recombination (non-equilibrium) Equilibrium recombination: Ro = number carriers recombining/vol - sec = Bno po B=
Ro no po
Non-Equilibrium recombination: n = no + ∆n p = po + ∆ p R = Bnp =
Ro (no po + po ∆n + no ∆p + ∆n∆p ) = Ro 1 + ∆n + ∆p no po no po
Assume low-level injection, ∆n, ∆p << no,po
n-type, ∆n/no <<1, R=∆p/τh, τh=po/Ro p-type, ∆p/po <<1, R=∆n/τe, τe=no/Ro 323g10 ©1999 E.A. Fitzgerald
Other Recombination Pathways • Deep levels in semiconductors act as carrier traps and/or enhanced recombination sites Ec Edeep
Recombination through deep level
E
Ev x
•Barrier to capture carrier is Eg/2 •Since the probability of the carrier transition is ~e-∆E/kT, trapping a carrier with a deep state is very probable •A trapped carrier can then help attract another carrier, increasing recombination through the deep state
1 1 1 = + τ τ int rinsic τ deep 323g11 ©1999 E.A. Fitzgerald
Trap-dominated Recombination • e.g. n-type material
τh =
R=
1 σ h vth N t
∆p = σ h vth N t ∆p τh
323g12 ©1999 E.A. Fitzgerald
Effect of Traps (Defects) on Bands • Trapping (Fermi level in defect) creates depleted regions around defect N E F = E g + kbT ln d NC
•EF position in semiconductor away from traps in n-type material •EF pulled to mid-gap in defect/trap area
Edonor Etrap
Ec EF
Ev EF pulled to trap level in defect
©1999 E.A. Fitzgerald
Depleted regions; internal electric field
323g13
Response of Semiconductors to Stimulus • • • • •
Can apply E-field, light light in, G increase T increase, G increase Defects increase, R increase and G increase in E-field Stimulus removed: return to equilibrium
Example: GaAs surface states
Surface: Fermi level ‘pinned’ to defect level
Without applied voltage or concentration gradient, EF is flat
Ec EF
Ev
∂n ∂V and ∂x ∂x determine properties
323h1 ©1999 E.A. Fitzgerald
Key Processes: Drift and Diffusion Electric Field: Drift
I h = epvd A; J h = epµ h E I e = envd A; J e = enµ e E Concentration Gradient: Diffusion
J h = −eDh ∇p J e = eDe∇n
J hTOT = epµ h E − eDh∇p J eTOT = enµ e E + eDe∇n 323h2 ©1999 E.A. Fitzgerald
Continuity Equations •
For a given volume, change in carrier concentration in time is related to J
∂n ∂n ∂n ∂n ∂n + = + − ∂t ∂t drift ∂t diff ∂t R ∂t G ∂n 1 ∂n ∂n = ∇ ⋅ J TOT − + ∂t e ∂t R ∂t G ∂p ∂p ∂p 1 = − ∇ ⋅ J TOT − + ∂t ∂t R ∂t G e 1-D, ∂ 2n ∂E ∂n + De 2 − Bnp + G = nµ e ∂x ∂x ∂t ∂2 p ∂E ∂p + Dh 2 − Bnp + G = − pµ h ∂x ∂x ∂t
323h3 ©1999 E.A. Fitzgerald
Minority Carrier Diffusion Equations • •
In many devices, carrier action outside E-field controls properties--> minority carrier devices Only diffusion in these regions
∂n ∂ n = De 2 − Bnp + G ∂t ∂x ∂p ∂2 p = Dh 2 − Bnp + G ∂t ∂x ∆p in n - type, − Bnp = − τh 2
in p - type, − Bnp = −
∆n τe
Assuming low-level injection, ∂n ∂no ∂∆n ∂∆n = + ≈ ∂t ∂t ∂t ∂t therefore
∂∆n ∂ 2 ∆n ∆n = De − + G in p - type material 2 ∂x ∂t τe ∂∆p ∂ 2 ∆p ∆p = Dh − + G in n - type material 2 ∂t ∂x τh
323h4 ©1999 E.A. Fitzgerald
Use of Minority Carrier Diffusion Equations •
Example: Light shining on a surface of a semiconductor x
n-type ∆p(x)?
hν
0
∂ 2 ∆p ∆p ∂∆p = Dh − +G 2 =0 in bulk τh ∂t ∂x
G at x=0 (assume infinite absorption coefficient to simplify example)
∂ 2 ∆p ∆p = ∂x 2 Dhτ h
Steady state solution
Now use boundary conditions of the problem:
@ x = ∞, ∆p = 0 ∴A= 0
try ∆p = Ae ax + Be − ax a Ae + a Be 2
a=
©1999 E.A. Fitzgerald
ax
1 Dhτ h
2
− ax
Ae ax + Be − ax = Dhτ h
∆p = Be
−
x Dhτ h
@ x = 0, ∆p = Gτ h ∴ B = Gτ h ∆p = Gτ h e
−
x Lh
Units of length: minority carrier diffusion length, Lh ∆p
Gτh J h = eDh
∂∆p ∂x
323h5
x
The p-n Junction (The Diode) • •
Derivation of ideal diode equation covered in the SMA Device Course Development here introduces the fundamental materials concepts n-type material in equilibrium
p-type material in equilibrium p~Na
n~Nd
n~ni2/Na
p~ni2/Nd
N E F = −kbT ln a NV
N E F = E g + kbT ln d NC
Ec
Ec EF
EF Ev
Ev
What happens when you join these together? 323h6 ©1999 E.A. Fitzgerald
Joining p and n p
n Ec EF
Ev Carriers flow under driving force of diffusion until EF is flat
-
-
+ + + +
Holes diffuse Electrons diffuse
323h7 ©1999 E.A. Fitzgerald
-
-
-
-
-
+ + + + + + + +
Holes diffuse Electrons diffuse
-
- - - - -
+ + + + E
+ + + +
An electric field forms due to the fixed nuclei in the lattice from the dopants Therefore, a steady-state balance is achieved where diffusive flux of the carriers is balanced by the drift flux
323h8 ©1999 E.A. Fitzgerald
-
- - - - -
+ + + +
xp
+ + + + Metallurgical junction
xn W: depletion or space charge width
ρ N d xn = N a x p
E
ρ ( x) E=∫ dx ε
V = ∫ E ( x)dx
Vbi
V
xp =
2ε r ε oVbi Nd e Na (Nd + Na )
xn =
2ε r ε oVbi Na e Nd (Nd + Na )
W=
2ε r ε oVbi N a + N d e Nd Na
323h9
©1999 E.A. Fitzgerald
What is the built-in voltage Vbi? p
n
np eVbi
nn pp
Ec EF
eVbi Ev
pn eVbi=EFn-EFp E Fp
p = − k b T ln NV
N = − k b T ln a NV
∴Vbi =
E Fn
p = − kbT ln n NV
k bT N a N d ln 2 e ni
ni2 = − kbT ln NV N d
We can also re-write these to show that eVbi is the barrier to minority carrier injection:
pn = p p e
− eVbi k bT
n p = nn e
− eVbi k bT
323h10 ©1999 E.A. Fitzgerald
Qualitative Effect of Bias • • •
Applying a potential to the ends of a diode does NOT increase current through drift The applied voltage upsets the steady-state balance between drift and diffusion, which can unleash the flow of diffusion current “Minority carrier device” np eVbi
nn pp
eVbi pn
n p = nn e
− e (Vbi −Va ) k bT
Ec EF
pn = p p e
+eVa -eVa
Ev − e (Vbi −Va ) k bT
323h11 ©1999 E.A. Fitzgerald
Qualitative Effect of Bias • •
Forward bias (+ to p, - to n) decreases depletion region, increases diffusion current exponentially Reverse bias (- to p, + to n) increases depletion region, and no current flows ideally
Forward Bias
Reverse Bias
np
np
eVbi-e|Va|
nn
nn pp
Ev
pn
EF
eVbi+e|Va| Ev pn
+
Va
-
qV a qV De ni2 Dh ni2 kbTa k T b e − 1 = J o e − 1 J = q + Le N a Lh N d
Di k bT = µi q ©1999 E.A. Fitzgerald
eVbi+e|Va| Ec
eVbi-e|Va|
pp
Solve minority carrier diffusion equations on each side and determine J at depletion edge
Ec EF
I V=f(I) Rectification, Non-linear, Non-Ohmic
Linear, Ohmic V
Li = Diτ i V=IR
323h12
Devices Solar Cell/Detector
Reverse Bias/Zero Bias
Jedrift Ec EF
Jhdrift
Ev
LED/Laser Jediff Ec EF Ev
Jhdiff
Laser •population inversion •reflectors for cavity
323h13 ©1999 E.A. Fitzgerald
Transistors Bipolar (npn) emitter
Jdiff
base
Jdrift
collector Ec EF
Barrier, controlled by VEB
VBC
VEB
Ev
emitter
base
collector 323h14 ©1999 E.A. Fitzgerald
Transistors FET
source
n
gate
x p
drain
n
x=metal is a MESFET x=metal/poly Si/oxide is a MOSFET
CMOS
323h15 ©1999 E.A. Fitzgerald
Other means to create internal potentials: Heterojunctions • •
Different semiconductor materials have different band gaps and electron affinity/work functions Internal fields from doping p-n must be superimposed on these effects: Poisson Solver (dE/dx=V=ρ/ε) Vacuum level
ϕ1
ϕ2
EF
Thin films ©1999 E.A. Fitzgerald
Substrate
Eg1
Eg2
Potential barriers for holes and electrons can be created inside the material 323h16
Quantum Wells EC n=3
hν
n=2 n=1
EV
L If we approximate well as having infinite potential boundaries: k=
nπ L
for standing waves in the potential well
! 2k 2 h 2n 2 = E= 2m* 8m* L2
We can modify electronic transitions through quantum wells 323h17
©1999 E.A. Fitzgerald
Strain and the Band Alignment in GeSi/Si GeSi strained to Si substrates
Si strained to Relaxed GeSi/Si
Si
GeSi
GeSi
Si
GeSi Si
Advantages of Relaxed Structure: •both electron and hole confinement for n & p channels •no alloy scattering for electrons •transverse electron mass (lighter in-plane) •modulation doping: no direct ionized impurity scattering ©1999 E.A. Fitzgerald
323h18
Graded, Relaxed SiGe Si0.70Ge0.30
Si
_______1µ µm XTEM of graded structure
_____17µ µm Plan-view EBIC of graded structure ρt=7x105cm-2
323h19 ©1999 E.A. Fitzgerald
Impact of Relaxed SiGe on Future Systems
ε-Si Relaxed SiGe
ε-enhanced CMOS
Heterostructure CMOS (HCMOS) (buried channel)
end-of-roadmap, largest volume
GaAs-based SiGe HEMTs, MODFETs detectors, light emitters on Si wireless to the masses
All-FET Wireless or High Performance System on a Chip
Wireless Circuit Boards, Optical Interconnects 323h20
©1999 E.A. Fitzgerald
NMOS Results •
Strained Si greatly enhances mobility in real SiO2/Si surface channels Effective Electron Mobility vs. Effective Vertical Field no well implant
2 Effective electron mobility (cm /Vs)
1200 1100 1000 900
Bulk Si 10%SiGe 15%SiGe 20%SiGe 25%SiGe 30%SiGe Universal
800 700 600 500 400 300 200 100 0 0E+0
©1999 E.A. Fitzgerald
1E+5
2E+5
3E+5
4E+5
5E+5
6E+5
7E+5
Effective vertical field (V/cm)
8E+5
9E+5
1E+6
323h21
Dielectric and Optical Properties • As with conductivity, we will start with macroscopic property and connect to the microscopic • All aspects of free electrons have been covered: only bound electrons left • Capacitance, Optical properties --> ε,n --> molecules and atoms Review of capacitance and connection to dielectric constant First, no material in capacitor to
Vo
Q = CV dq dV =i=C dt dt
V C i
Vo
V ©1999 E.A. Fitzgerald
t
323h22
The Capacitor + -+ - I=0 always ++ - in capacitor + -+ t d/2
d/2
+V
ρ
E
∇⋅E =
E=
ρ εo
−d 2
∫
−d −t 2
V=
Q ρ ρt dx = = εo ε o Aε o
d 2
∫ Edx =
−d 2
V
Qd Aε o
Q Qd =V = C Aε o ©1999 E.A. Fitzgerald
C=
εo A d
323h23
The Capacitor • • •
The air-gap can store energy! If we can move charge temporarily without current flow, can store even more Bound charge around ion cores in a material can lead to dielectric properties
•Two kinds of charge can create plate charge: •surface charge •dipole polarization in the volume •Gauss’ law can not tell the difference (only depends on charge per unit area)
323h24 ©1999 E.A. Fitzgerald
Material Polarization + + -+++ E’
E
E' = P D = ε o E + P = εE
ε = ε rε o P εr = 1+ = 1+ χ εoE
+ ++ ++ + +
- - - C=
ε rε o A d
P is the Polarization D is the Electric flux density or the Dielectric displacement χ is the dielectric or electric susceptibility
All detail of material response is in εr and therefore P ©1999 E.A. Fitzgerald
323h25
Origin of Polarization • • •
We are interested in the true dipoles creating polarization in materials (not surface effect) As with the free electrons, what is the response of these various dipole mechanisms to various E-field frequencies? When do we have to worry about controlling – molecular polarization (molecule may have non-uniform electron density) – ionic polarization (E-field may distort ion positions and temporarily create dipoles) – electronic polarization (bound electrons around ion cores could distort and lead to polarization)
• •
Except for the electronic polarization, we might expect the other mechanisms to operate at lower frequencies, since the units are much more massive What are the applications that use waves in materials for frequencies below the visible?
323h26 ©1999 E.A. Fitzgerald
Application for different E-M Frequencies
Methods of detecting these frequencies
Other satellite, 10-50GHz λ=3cm-6mm (‘mm wave’)
Cell phones λ=14-33cm
Fiber optics λ=1.3-1.55µm
DBS (TV) λ=2.5cm ‘MMIC’, pronounced ‘mimic’ mm wave ICs
In communications, many E-M waves travel in insulating materials: What is the response of the material (εr) to these waves? 323h27 ©1999 E.A. Fitzgerald
Wave Eqn. With Insulating Material and Polarization " " ∂B ∇xE = − ∂t " " " " " " ∂D nonmag " " ∂ ε o E + P insulating " ∂E → ∇xB = ε → ∇xB = J + ∇xH = J + ∂t ∂t ∂t " " " " ( D = ε 0 E + P = εE )
(
)
∂2E εr ∂2E ∇ E = µ 0ε 0ε r 2 = 2 2 c ∂t ∂t 2
E = E0 ei ( k •r −ϖt ) = E0 eik •r e −iϖt = E (r )e − iϖt
ω 2ε r E ( r ) ∇ E (r ) = − c2 c2 2 2 ω = k εr c c ω= → k k optical n εr 2
So polarization slows down the velocity of the wave in the material 323h28
©1999 E.A. Fitzgerald
Compare Optical (index of refraction) and Electrical Measurements of ε Material
Optical, n2
Electrical, ε
diamond
5.66
5.68
Only electrical polariztion
NaCl
2.25
5.9
Electrical and ionic polariztion
H2O
1.77
80.4
Electrical, ionic, and molecular polariztion
Polarization that is active depends on material and frequency 323h29 ©1999 E.A. Fitzgerald
Microscopic Frequency Response of Materials • Bound charge can create dipole through charge displacement • Hydrodynamic equation (Newtonian representation) will now have a restoring force • Review of dipole physics: -q
-
! d
! ! Dipole moment: p = qd
+
+q
! p
Applied E-field rotates dipole to align with field:
! ! ! Torque τ = pxE
! !
! !
Potential Energy U = − p ⋅ E = p E cosθ 323i1 ©1999 E.A. Fitzgerald
Microscopic Frequency Response of Materials •
For a material with many dipoles:
! ! ! ! ! P = Np = NαE ( p = αE )
(polarization=(#/vol)*dipole polarization) α=polarizability χ=
! ! p = αE
! P
! , so χ = Nα εo E
Actually works well only for low density of dipoles, i.e. gases: little screening
For solids where there can be a high density: local field Eext Eloc
For a spherical volume inside (theory of local field), ! ! ! P Eloc = E ext + 3ε o 323i2
©1999 E.A. Fitzgerald
Microscopic Frequency Response of Materials •
We now need to derive a new relationship between the dielectric constant and the polarizability D = ε r ε o Eext = ε o Eext + P P = ε r ε o Eext − ε o Eext 2 + εr Eloc = Eext 3 Plugging into P=NαEloc:
ε r ε o E ext − ε o E ext = Nα
(ε r − 1)ε o = Clausius-Mosotti Relation: ©1999 E.A. Fitzgerald
3
ext
Nα (ε r + 2 ) 3
ε r − 1 Nα α = = ε r + 2 3ε o 3νε o Macro
(ε r + 2 ) E
Micro
Where v is the volume per dipole (1/N) 323i3
Different Types of Polarizability Highest natural frequency
• Atomic or electronic,αe • Displacement or ionic, αi • Orientational or dipolar, αo
Lowest natural frequency
Lightest mass
Heaviest mass
α = αe + αi + αo As with free e-, we want to look at the time dependence of the E-field:
E = Eo e − iωt
∂ 2 x m ∂x − eE − Kx m 2 = τ ∂t ∂t Restoring Force Response
Driving Force
Drag
m"x" = −eE − Kx x = xo e −iωt m(−ω 2 ) xo = −eEo − Kxo xo =
ωo = ©1999 E.A. Fitzgerald
eEo K m ω 2 − m K m
=
eEo m ω 2 − ω o2
(
)
So lighter mass will have a higher critical frequency
323i4
Classical Model for Electronic Polarizability •
Electron shell around atom is attached to nucleus via springs
+
+
! E K
K Zi electrons, mass Zim
! E ! p
r
Z i m"r" = − Kr − Z i eEloc , assume r = ro e − iωt 323i5 ©1999 E.A. Fitzgerald
Electronic Polarizability ro =
ro =
eEo K ; = ω oe m ω 2 − ω oe2 mZ i
(
)
p = qd = − Z i er; p = po e po =
2
)
Zie2 αe = m ω 2 − ω oe2
)
(
ω >> ω oe , α e = 0 Zie2 ω << ω oe , α e = mω oe2
K m ω 2 − mZ i
If no Clausius-Mosotti, Nα e NZ i e 2 εr = 1+ = 1+ = n2 2 2 εo ε o m ω − ω oe
− i ωt
Zie Eo = α e Eo m ω 2 − ω o2
(
eEo
(
)
εr NZ i e 2 1+ ε o m(ω oe2 )
1
ωoe
ω 323i6
©1999 E.A. Fitzgerald
QM Electronic Polarizability • • •
At the atomic electron level, QM expected: electron waves QM gives same answer qualitatively QM exact answer very difficult: many-bodied problem E1 E0
e2 f10 E1 − E0 α e (ω ) = ω = ; 10 # m ω102 − ω 2
f10 is the oscillator strength of the transition (ψ1 couples to ψo by E-field) For an atom with multiple electrons in multiple levels:
f E − E0 e2 α e (ω ) = ∑ j ≠ 0 2 j 0 2 ; ω j 0 = j # m j10 − ω
323i7 ©1999 E.A. Fitzgerald
Ionic Polarizability • • •
Problem reduces to one similar to the electronic polarizability Critical frequency will be less than electronic since ions are more massive The restoring force between ion positions is the interatomic potential
E(R) Parabolic at bottom near Ro
Nucleii repulsion
k ( R − Ro ) 2 E= 2 ∂E = k ( R − Ro ) F= ∂R
Ro
-
+
R Electron bonding in between ions
F = kx ⇒ σ ij = Cijkl ε kl 323i8
©1999 E.A. Fitzgerald
Ionic Polarizability Eloc •2 coupled differential eqn’s •1 for + ions •1 for - ions
+
-
+
-
"" = u""+ − u""− w = u+ − u− , w M=
p u-
u+
Ionic materials always have ionic and electronic polarization, so:
α tot
e2 = αi + αe = α+ + α− + M ω oi2 − ω 2
(
1 1 1 + M+ M−
"" = −2 Kw + eEloc Mw Eloc = Eo e −iωt , w = wo e −iωt wo =
)
eEo 2K ω = , oi M ω oi2 − ω 2 M
(
)
po = ewo = α i Eo e2 αi = M ω oi2 − ω 2
(
) 323i9
©1999 E.A. Fitzgerald
Ionic Polarizability •
Usually Clausius-Mosotti necessary due to high density of dipoles e2 ε r − 1 Nα tot 1 α α = = + + − + εr + 2 M (ω oi2 − ω 2 ) 3ε o 3ε o v
By convention, things are abbreviated by using εs and ε∞: ε s −1 e2 1 ω << ω oi , = α + + α − + ε s + 2 3ε o v M (ω oi2 ) ε ∞ −1 n2 −1 1 [α + + α − ] ω >> ω oi , = 2 = ε ∞ + 2 n + 2 3ε o v
εr εs n2=ε∞
∴ εr = ε∞ +
ε∞ − ε s 2 2 ε∞ + 2 = ω ω , T oi ω2 + ε 2 s −1 2 ωT
ωT
ω 323i10
©1999 E.A. Fitzgerald
Orientational Polarizability • No restoring force: analogous to conductivity O - p
H +
θ
O
O p=0
Analogous to conductivity, the molecules collide after a certain time t, giving:
α αo = DC 1−iωτ
+q
C
H
-q For a group of many molecules at some temperature: f =e
−U k bT
=e
pE cosθ k bT
After averaging over the polarization of the ensemble molecules (valid for low E-fields):
α DC
p2 ~ 3kbT
323i11 ©1999 E.A. Fitzgerald
Dielectric Loss • • •
For convenience, imagine a low density of molecules in the gas phase C-M can be ignored for simplicity There will be only electronic and orientational polarizability ε r = 1 + χe + χ o = n2 +
Nα DC 3ε o (1 − iωτ )
ωτ << 1, , ε r = ε so = n 2 +
Nα DC 3ε o
ε so − n 2 ∴ εr = n + 1 − iωτ 2
ε’,ε’’ αe+αi
εso
We can write this in terms of a real and imaginary dielectric constant if we choose:
n2
ε r = ε '+iε ' ' ε so − n 2 ε so − n 2 ε'= n + ωτ ; ε ''= 2 2 2 2 1+ ω τ 1+ ω τ 2
©1999 E.A. Fitzgerald
αe -2
0 logωτ
Water molecule: τ=9.5x10-11 sec, ω~1010 microwave oven, transmission of E-M waves
+2 323i12
Dielectric Constant vs. Frequency • Completely general ε due to the localized charge in materials
ε
molecules
αo
ions
αi n2 1
electrons
αe 1/τ
ωT
ωoe
ω
Dispersion-free regions, vg=vp 323i13 ©1999 E.A. Fitzgerald
Dispersion •
Dispersion can be defined a couple of ways (same, just different way) – when the group velocity ceases to be equal to the phase velocity – when the dielectric constant has a frequency dependence (i.e. when dε/dω not 0)
ω
Dispersion-free Dispersion
ω= vp =
vp =
c k εr
ω ∂ω c = = = vg k ε r ∂k
ω ∂ω c = ≠ = vg k ε r (ω ) ∂k
k
323i14 ©1999 E.A. Fitzgerald
Characteristics of Optical Fiber • Snell’s Law θ1 n1
Boundary conditions for E-M wave gives Snell’s Law:
n2 Refraction
n1 sin θ1 = n2 sin θ 2
θ2 θ1 θ2
n1 n2
Internal Reflection: θ1=90°
θ 2 = θ c = sin −1
n1 n2
Glass/air, θc=42°
323i15 ©1999 E.A. Fitzgerald
Characteristics of Optical Fiber • Attenuation – Absorption • OH- dominant, SiO2 tetrahedral mode – Scattering • Raleigh scattering (density fluctuations) αR~const./λ4 (<0.8 µm not very useful!) • Dispersion – material dispersion (see slide i13) x – modal dispersion •Light source always has ∆λ •parts of pulse with different l propagate at different speeds
Black wave arrives later than red wave y n1 Solution: grade index ©1999 E.A. Fitzgerald
n2
n
323i16
Characteristics of Optical Fiber
323i17 ©1999 E.A. Fitzgerald
Characteristics of Optical Fiber
323i18 ©1999 E.A. Fitzgerald
Characteristics of Optical Fiber
LED ∆λ=30nm Laser ∆λ=3nm Laser ∆λ=0.1nm
323i19 ©1999 E.A. Fitzgerald
Ferroelectrics • ‘Confused’ atom structure creates metastable relative positions of positive and negative ions
323i20 ©1999 E.A. Fitzgerald
Ferroelectrics • Each unit cell a dipole! • Very large Ps (saturated polarization, P(E=0) • No iron involved; ‘Ferro’ since hysterisis loop analogous to magnetic materials E Two equal-energy atom positions
Can flip cell polarization by applying large enough reverse E-field to get over barrier
P Ps ∆E ‘normal’ dielectric Ro
Ec
E
R
323i21 ©1999 E.A. Fitzgerald
Magnetic Materials • The inductor
1 ∂B (CGS) c ∂t 1 ∂ 1 ∂Φ B EdS BdS ∇ × = − = − ∫∫ c ∂t ∫ ∫ c ∂t Φ B ≡ magnetic flux density
∇× E = −
(
)
∫ ∫ ∇ × EdS =∫ E ⋅ d! (Green's Theorem) V = ∫ E ⋅ d! = −
1 ∂Φ B (explicit Faraday's Law) c ∂t
Φ B = LI (Q = CV) ∂I ∂Φ B =L ∂t ∂t ∂NΦ B ∂I VEMF = − = −L ∂t ∂t ∂I ∂V V =L (recall I = C for the capacitor) ∂t ∂t ∂I Power = VI = LI ∂t 1 1 Energy = ∫ Power ⋅ dt = ∫ LIdI = LI 2 = NΦ B I 2 2 1 2 capacitor CV 2 323j1
©1999 E.A. Fitzgerald
The Inductor ∇× B =
4π 1 ∂E J+ c c ∂t
∫ ∫ ∇ × BdS = ∫ B ⋅ d! =
4π 4π J dS I ⋅ = ∫ ∫ c c
4π In c N = n ⋅ length = nl Nφ B N ( BA) 4π 2 L= n lA = = I I c
B=
Insert magnetic material Magnetic dipoles in material can line-up in magnetic field
B = H + 4πχH = H + 4πM
∂M = χ µ = 1 + 4πχ ∂H B = 4πM + 1 B = µH
M = χH ©1999 E.A. Fitzgerald
B magnetic induction χ magnetic susceptibility H magnetic field strength (applied field) M magnetization 323j2
H and B • H has the possibility of switching directions when leaving the material; B is always continuous
H
B p
M
M
At p: ©1999 E.A. Fitzgerald
q
B H M=0
H
At q: B M
323j3
Maxwell and Magnetic Materials • Ampere’s law ∫ H ⋅ d! = I = 0 • For a permanent magnet, there is no real current flow; if we use B, there is a need for a fictitious current (magnetization current) • Magnetic material inserted inside inductor increases inductance 4π In A Φ B = BA ~ 4πMA = 4πχHA = 4πχ c NΦ B (4π ) 2 L= n lAχ = I c 2
L increased by ~χ due to magnetic material
Material Type
χ
Paramagnetic
+10-5-10-4
Diamagnetic
-10-8-10-5
Ferromagnetic +105 323j4 ©1999 E.A. Fitzgerald
Microscopic Source of Magnetization • No monopoles • magnetic dipole comes from moving or spinning electrons Orbital Angular Momentum
µ
µ is the magnetic dipole moment " " Energy = E = − µ ⋅ H = − µ H cos θ
I
A e-
L What is µ? For θ=0,
E = − µH ≈ −Φ B I since energy ~ LI 2 and for 1 loop L =
ΦB I
Φ B = ∫ ∫ H ⋅ dS ~ HA
∴ µH = Φ B I = HAI and ∴ µ = IA e ω A = πr 2 c 2π e µ = − ωr 2 2c I =−
©1999 E.A. Fitzgerald
323j5
Microscopic Source of Magnetization • Classical mechanics gives orbital angular momentum as: " " " L = r × p = mr 2ω e e# LQM = − LZ = − µ B LZ µL = − Example for l=1: 2mc 2mc e# = µ B E(H=0) mc 2 LZ = m! = −!,...,0,...!
0 -µBH
µs
Spin Moment
e e# S = − g0 S z = − g0µ B SZ mc 2mc 1 S Z = mS = ± g 0 = 2 for electron spin 2
µs = −
+µBH
+(1/2)µBg0H
Q. M .
E(H=0) -(1/2)µBg0H
323j6 ©1999 E.A. Fitzgerald
Total Energy Change for Bound Electron in Magnetic Field • Simple addition of energies if spin-orbital coupling did not exist " " E = − µ ⋅ H = µ B (LZ + g 0 S Z )H = µT H But spin-orbit coupling changes things such that:
µT ≠ µ B (LZ + g 0 S Z ) = µ B J Z QM definitions:
L = #LZ = #m!
S = #S Z = #ms J = L+S J = #J Z
µ T = gµ B J Z
3 1 S (S + 1) − L(L + 1) + J (J + 1) 2 2 " " " " E = − µT ⋅ H = gµ B J ⋅ H = gµ B J Z H g=
323j7 ©1999 E.A. Fitzgerald
Total Energy Change for Bound Electron in Magnetic Field • Kinetic energy from Lorentz force has not been included pH = −
e " " r×H 2c
Lorentz for circular orbit
(
)(
p2 e2 " " " " r×H ⋅ r ×H Energy change = = 2 2m 8mc
)
For the plane perpendicular to H and assuming circular orbit: e2 2 2 e2 r H = x 2 + y 2 )H 2 ( Energy change = 2 2 8mc 8mc e2 2 2 2 H x y ∴ ∆ETOT = gµ B HJ Z + + ( ) 2 8mc Numbers:
µBH for H=10-4 Gauss =10-4 eV
for H=10-4 Gauss =10-9 eV
The Lorentz effect is minimal with respect to magnetic moment interaction, if it exists 323j8 ©1999 E.A. Fitzgerald
Atoms with Filled Shells • J=0 (L=0, S=0) • Only Lorentz contribution • Leads to diamagnetism
e2 2 2 2 H x y ( ) ∆E = + 2 8mc
Need to sum over all e- in atom:
∆Eatom
e2 2 2 2 H r = ∑ i (for a spherical shells) 12mc 2 3 i
N ∂2E χ =V ∂H 2 1 ∂E M = − V ∂H e2 N χ =− 6mc 2 V
∂M χ= ∂H e2 N 2 ∑i ri = − 6mc 2 V r Z i 2
e2 N 2 r Zi χ =− 6mc 2 V
~ -10-5 323j9
©1999 E.A. Fitzgerald
Atoms with Partially Filled Shells • J not zero • Need Hund’s rules from QM – Fill levels with same ms to maximize spin – maximize L (first e- goes in largest l) – J=|L-S| for n<=(2l+1), J=|L+S| for n>(2l+1) • Conventional notation: (2 S +1) X J • J=0 when L and S are not zero is a special case – 2nd order effect--> perturbation theory • Partially filled shells give atoms paramagnetic behavior L
0
1
2
3
4
5
6
X
S
P
D
F
G
H
I
∆E = gµ B HJ Z (+10-2-10-3>10-9 eV for diamagnetic component) 323j10 ©1999 E.A. Fitzgerald
Sc Ti V Cr Mn Fe Co Ni Cu Zn
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb 323j11 ©1999 E.A. Fitzgerald
Temperature Dependence of Paramagnetism •
Temperature dependence determined by thermal energy vs. magnetic alignment energy (same derivation as for molecular polarizability in the case of electric dipoles)
f =e
−U k bT
=e
µZ
pE cosθ k bT
µ fdΩ ∫ = ∫ fdΩ Z
for electric dipoles;
f =e
−U k bT
=e
µH cosθ k bT
for magnetic dipoles;
For low H fields and/or low T, µ 2 H µ B2 J 2 H = µZ = k T 3 b 3kbT N µ B2 J 2 H M= V 3kbT N µ B2 J 2 χ= V 3kbT
χ QM
1 N µ B2 g 2 J ( J + 1) = kbT 3V
χ∝
1 T
Curie’s Law 323j12
©1999 E.A. Fitzgerald
Effect of De-localized electrons on Magnetic Properties • Pauli Paramagnetism – dues to the reaction of free e- to magnetic field ∆E = g 0 µ B H
E(H=0)
M = − µ B (n+ − n− ) g + (E ) =
1 g (E ) 2 E(H=0) 1 g − (E ) = g (E ) 2
µ = − g0 µ B S
(n+ is the density of free electrons parallel to the H field) g + (E ) =
1 g (E + µ B H ) 2 ∆E = g 0 µ B H 1 g − (E ) = g (E − µ B H ) 2
For exact solution, need to expand about Ef for n+ and n-
Only e- near Fermi surface matter: ∆E = g 0 µ B H
(n+ − n− ) = ∆n ≈ g (EF ) ∆E ∆n ≈ g (EF )µ B H ©1999 E.A. Fitzgerald
2
M = µ B2 Hg (EF ), χ = µ B2 g (EF )
Note: Pauli paramagnetism has no T dependence, whereas Curie paramagnetism has 1/T dependence 323j13
Effect of De-localized electrons on Magnetic Properties • Landau paramagnetism – – – –
Effect of bands/Fermi surface on Pauli paramagnetism F=qvxB for orbits orbit not completed under normal circumstances however, average effect is not zero
χ Land
1 = − χ Pauli 3
323j14 ©1999 E.A. Fitzgerald
Ferromagnetism • Most important but not common among elements • Net magnetization exists without an applied magnetic field • To get χ~104-105 as we see in ferromagnetism, most moments in material must be aligned! • There must be a missing driving force NOT dipole-dipole interaction: too small Edipole =
" " 1 " " −4 ˆ ˆ µ ⋅ µ − µ ⋅ µ ⋅ ( )( ) [ ] r r eV 3 ~ 10 1 2 1 2 r3
Spin Hamiltonian and Exchange
H spin = −∑ J ij S i S j
J ij ≡ exchange constant
Assuming spin is dominating magnetization,
() ( )(
)
()
" " " " " " " 1 H =− ∑ S R ⋅ S R' J R − R' − gµ B H ∑ SR " " " 2 R,R' R
323j15 ©1999 E.A. Fitzgerald
Exchange E~-JS1S2 J negative, E~+S1S2--> Energy
if
J positive, E~-S1S2--> Energy
if
Fe, Ni, Co ---> J positive! Other elements J is negative Rule of Thumb: r interatomic distance ≡ > 1 .5 2ra 2(atomic radius)
J is a function of distance! 323j16 ©1999 E.A. Fitzgerald
Ferromagnetism M
M (T ) ∝ (TC − T )
β ≈ 0.33 - 0.37
χ (T ) ∝ (TC − T )
γ ≈ 1.3 - 1.4
β
−γ
B=H+4πM Br, Ms
Domain rotation
Irreversible boundary displacement reversible boundary displacement
‘normal’ paramagnet
TC
Hc
H
T Easy induction, “softer”
Magentic anisotropy hardness of loop dependent on crystal direction comes from spin interacting with bonding 323j17 ©1999 E.A. Fitzgerald
Domains in Ferromagnetic Materials B N
S N S N
S
N S N S
M
Magnetic domain
Magnetic energy 1 = ∫ B 2 dV 8
Domain wall or boundary
Flux closure No external field
323j18 ©1999 E.A. Fitzgerald
3.23 Electrical, Optical, Magnetic Properties of Materials Prof. Eugene A. Fitzgerald Rm 13-4053, x8-7461,
[email protected] Prof. Yoel Fink, x3-4407,
[email protected] Grader: ? Class: Tuesday/Thursday 8:30-10, 4-237 Recitation: to be announced (3 recitations) Problem Sets: one a week, 1-3 problems, Due outside Prof. Fink’s office Exams: 3 exams, NO final Grading: problem sets 10%, each exam 30% Outline {first numbers refer to chapters to be read in A&M, second reference after comma refers to text and chapters for background reading in the topic area} Dates are approximate Conductivity and Bands Sept. 7 Origin of Ohms Law {1, O 4} Hydrodynamic representation of electrons in solids {1, O 4} Sept. 12 Hall effect {1, O 4} Sept. 14 AC response of electron particles; The plasma frequency; E-M waves in materials {1 O 4} Sept. 19,21 Electron waves and diffraction {4 5 6 , C 2 3 4} Origin of structure factor, Diffraction {4 5 6, C 2 3 4} Sept. 26 Wave-particle duality: DeBroglie, Compton, Planck, Einstein {-, E&R 1 2 3} Sept. 28, Oct. 3 Electron waves in solids {2, O 5} Quantized electron energy {2, O 5} Density of electron states {2, O 5} Boltzmann and Fermi-Dirac distributions {2, O 5} Heat capacity of electron waves {2, O 5} Oct. 5 Nearly free electrons in solids
Origin of band gaps and band structure {9, O 5} Schroedinger equation and application to electron waves in solids {8, E&R 5 O 5} Oct. 12 Bloch waves and general solution from S.E. Oct. 17 First Exam Oct. 19 H atom {-, E&R 7} Chemistry approach: H2+, tight-binding model {-, O 5} Bonding: connection to mechanical properties Oct. 24,26,31 Buliding materials atom by atom: Debye-Huckel model Electronic structure of polymer chains Hybridization Metals and insulators Bands and zones Carriers in bands Effective mass {-,K 8} Nov. 2 Intrinsic/Extrinsic Semiconductors {28, P&N vol. 1} Electrical activity of defects Hydrogenic model of extrinsic semiconductors {28, P&N vol. 1} Nov. 7 Carrier Scattering, Recombination, and Generation in semiconductors Defects: Traps/R-G centers Nov. 9,14 Drift, diffusion, and the continuity equations {-, P&N vol. 1} The ideal diode: depletion region, built-in voltage, and operation {-, P&N vol. 2} Band offsets Electrons in V(x): step, well, infinite well {-, S&W 3} Quantum wells Resonant tunnel diode operation 2DEG in Si/SiGe and transistor applications Nov. 16 Second Exam Dielectric and Optical Properties of Materials: The Capacitor {27, O 8} Nov. 21 Application of Maxwell’s equations to capacitance Dielectric constant and polarizability Dielectric response at optical frequencies Nov. 28,30 Hydrodynamic representation of polarizability Local field and Clausius-Mossetti Orientational, electronic, and ionic polarizability
Pyroelectrics and ferroelectrics Defects and dielectric loss Dispersion, attenuation in optical fibers Dec. 5 ‘Photonic Band Gaps’ Magnetic Properties of Materials: The Inductor {31 32 33, O 9} Dec. 7 Application of Maxwell’s equations to inductance Magnetization: paramagnetism, diamagnetism, ferromagnetism {31} Microscopic origin of magnetization {31} Central QM equation for magnetization and Hund’s rules {31} Pauli paramagnetism {31} Exchange and ferromagnetism {32 33} Mean-field theory {32 33} Applications: bubbles, magneto-optic storage Defects and domain pinning Dec. 12 Third Exam Texts Main text: Ashcroft and Mermin, Solid State Physics {A&M} Highly recommended: Omar, Elementary Solid State Physics {O} Cullity, Elements of X-ray Diffraction {C} Eisberg and Resnick, Quantum Physics {E&R} Kittel, Introduction to Solid State Physics {K} Pierret and Neudeck, Modular Series on Solid State Devices {P&N} Solymar and Walsh, Lectures on the Electrical Properties of Materials {S&W} Pankove, Optical Processes in Semiconductors {P} Livingston, Electronic Properties of Engineering Materials
