Расчет энергетических зон, спиновой поляризации и транспорта в наноструктурах со спин-орбитальным взаимодействием: Методические указания к лабораторной работе




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I.

#!+ #

 1.
$ )#) !#3$  '#-# $#-,#- %$ # 0   

1.1.      -   

- !"#$ (%) &'!()$*+ &$, &',!-.$$  !"/ + !+ &$* $))+   $0!"#* 1$2 V (r ) , ,!, +"$)+ &$, 1"$, /$+,2 "3 − ∇V (r) , +4&!$ +3 &,"!)( Hˆ & SO 2!("# ! /!+ 04 &)! [1]

Hˆ SO =

5ˆ 1 ˆ [∇U , p]σ , 2 2 4m0 c

(1)

2)$ m - (!++! +&)* /!+ 04, pˆ - $!  (6"#+!, ! $!  σ7ˆ 0 (!$ +3 ,!, &$,  σ8ˆ = (σˆ , σˆ , σˆ ) + ,($ !(, 3&"3-.(+3 x y z (! 0!( 9!6" [1]. : ++ $($ $)0, 3 * & !+! ;(6"4 (1)  )!"$$  $,+ 6, (4 (!$( '!/$$ + 3* 9"!,! < = 1. +" $) )+ #  $0!"#* 1$2 6+"&"$! +,!/,!( = !!($ & !'0! &"' $2 2!0, !,(, ,!, 2$ $ $$>) & "6&),&* + 6, 6$ [2],  !,!3 !+(($ 3 !'4&!$ +3 + 6, 6*  '!/!$ +3 SIA (Structure Inversion Asymmetry). 9 1 ( & % &'!()$*+ & )(6$ &,"!), !'4&!$(4* 2!("# !(+ ?!@4 (A.B. ?+ !@!, 1960) [3], ,  $  !!&" $ ∇V (r ) &)"#   Oz ($$ "$)6-.* &): Hˆ R = α (σˆ x pˆ y − σˆ y pˆ x ).

(2)

C+ ! ! α & ;(6"$ (2), !'4&!$(!3 !!($ ( ?!@4, $)$"3$ "/6 &,"!)! )!2 ! % &'!()$*+ &3  & /4> &$ "6&),&4> + 6, 6!> ($$ '!/$$ & $)$"!> % + + + (1 D5) ⋅ 10 −9 eV ⋅ cm . 9 ,"#,6 &(  >E)$$( &,"!) ?!@4 & )$*+ &$ 3'! &$@$* !+(($  + 6, 64, &$"/! !!($ ! &'!(  + '($$ &$@$2 1"$, /$+,2 "3 & $)$"!> α (E$ ($3 # 3  50%  $ &!/!"#2 '!/ $3 [4]. -3-

62* &,"!) & % &'!()$*+ &$ (E$ )! # &6 $33 !+(($ 3 !+"E$3 ! (& & 
($ "6&),&* + 6, 64, ,2)! ,+ !"" $ ($$ 0$ ! &$+. !"/$ !,* !+(($ , '!/!$(* ,!, BIA + &$ + &6$ +"!2!$((6 $++$"#>!6'! (Bulk Inversion Asymmetry), + * (G. Dresselhaus, 1955) & 2!("# !$, ($-.( " $)6-. &) [5]: Hˆ D = β (σˆ x pˆ x − σˆ y pˆ y ).

(3)

 β , $)$"3-.* &$"/6 1 2 &,"!)!, !'4&!- !!($ ( 9!!( $ $++$"#>!6'!. : "#@+ &$ ++"$)6$(4> "6&),&4> !+ 6, 6 "/!  @$3 )&6> &,"!)& α β ~ 1.5 2.2 .  + $"#!3 &$ *  $&!" !!($ & % &'!()$*+ &3 4", & /!+ + , "6/$ 9)4 & $)!&> 1,+$($ !>  '($$- ;  ,! & ,&! &4> 3(!>

InAs/AlGaSb [6].

1.2.     

    

"%() *%+) (2) , -.!!"/0%#1% (3) & 23  4 !"#$% &' &1%, 4( !5&, (6&. 664 7.,!#5!5U & 9% ,"/56,%6 (": $%!5,;) & (%66 7"#7.&(6,'& !5.#'5#.. >'5,&64 %!!4 m , % 7.56;,%"/6%: =6.9,:4 7!5:66%, 5 '&%65&) !!5:6,: $%!5,;) ! !7,6 ,5%"/6) &1%, ( !5&, 7,!)&%85!: 9% ,"/5 6,%6 2 p ˆ Hˆ = + α (σˆ x pˆ y − σˆ y pˆ x ) + β (σˆ x pˆ x − σˆ y pˆ y ). 2m

(4)

?#( .+%5/ !5%;,6%.6 #.%&66, @.A(,69.% Hˆ Ψ = EΨ ! # & 9% ,"/56,%6 &0(:5 %5.,;) B%#",, 9% ,"/56,%6 (4). B!'"/ ' :&":8,!: %5.,;% , &5. 9 7.:('%, &"6&%: >#6';,: ("C6% )5/ (�' 76656) !7,6. , 7,!)&%8, $%!5,;# ! !7,6 1 2 [1]. DC6 &,(5/, $5 9% ,"/5 6,%6 (4) !(.C,5 ",646) , '&%(.%5,$6 7 , 7#"/!# !"%9% ). 2!5&66) , D>#6';,: , 5%',0 7.%5..+6 &, '%' ,1&!56, :&":85!: 7" !', &"6). ) #( 7=5 # ,!'%5/4 , !5%;,6%.649 #.%&66,: @.A(4,69.%.& &,( 7. ,1&(6,: 7"!' &"6) 6% 6,1&!56) (�' 76656) !7,6 : J G Hc E ikr H 1 E Ψ = e I F. (5) c2

-4-

B(!5%&":: >#6';,8 (5) & !5%;,6%.6 #.%&66,4 @.A(,69..% ! 9% ,"/56,%6 (4), ) 7"#$% !,!5 # (� ",6 6)0 (6 (6)0 #.%&66,4 6% ' 7665) !7,6.% c : 1, 2    k2  − E  c1 + zc2 = 0   2m  2   k * − E  c2 = 0,
z c1 +  2m

(6)

9( z = α (k y + ik x ) + β (k x + ik y ). !"&, .%&6!5&% 6#"8 7.(",5": !,!5 ) (6) 71&":5 6%45, =6.95,$!',4 !7'5.: k2 2 2 Eλ (k ) = + λ (αk y + β k x ) + (αk x + βk y ) , 2m

λ = ±1.

(7)

B!" =59 C6 6%45, ' 7665) !7,6.% c1, 2 , 5. . &"6 >#6';,8, '5.%: , 5 &,(: Ψλk =

e

ikr

  

1

2 λe

iθ (k )

,

θ (k ) = Arg[αk y + βk x − i (αk x + βk y )],

9( λ = ±1. (8) % ', .%1 , =6.95,$!',4 !7'5. (7) , &"6&) >#6';,, (8) 9% ,"/546,%6% (4) 0%.%'5.,1#85!: 6%. '&%65&)0 $,!" (λ , k ). %' , & "6& &'5., ,.", , 7#"/!, 1(!/ 7,!4 )&%5 7"!'#8 &"6# 5%' C, '.56 5!#5!5&, !7,6- ,5%"/69 &1%, ( !5&,:. &) :&":5!: (,!'  '&%65 & $,!" λ = ±.%1, !5&5!5 ( &5&: 4 ,!0(69 7%.%",$!'9 !7'5.%, !7"A66)0 23 &1%, 4( !5&, . - !5&,5"/6, ,1 (7) "9' &,(5/, $5 & 5!#5!5&, 23 &1%, ( !5&,:, 5. . 7., α = β = 0 , ) 7"#$, 7.!54 7%.%",$!',4 1%'6 (,!7.!,,. 3(6, ,1 6%," &%C6)0 &7.!&, &16,'%8,0 7., .%!! 5.6,, : 1%(%$%  .%!7.("6,, !7,6& & 7. !5.%6!5& !,!5 ! #$A5 !7,6%, :&":5 ! '&%65 &)0 $,!". %', 7.!5.%6!5& & 6%+ !"#$% :&":5!: 7"!'!5/ (k x , k y ) , 7#"/!& (&# .69 ="'5.669 9%1%, 6% '5.4 C6 7!5.,5/ .%!7.("6,: !7,6& (": '%C(4 &5&, !7'5.% (7) ! λ = ±1. -5-

%C(4 5$' (k , k ) 5&$%5 6%. !.(6,0 16%$6,4 7.';,4 !7,6% x y (%66  '&%65& !!5:6,,: i = x, y , z & σ i (λ , k ) = Ψλk | σˆ i | Ψλk

.

(9)

B(!5%&":: & &).%C6, (9) &"6 >#6';,8 (8), ) 7"#$% , $5 σ z = 0 5C(!5&66 (": "8 )0 16%$6,4 (λ , k ), 5. . !7,6) 7":.,1&%6) & 7"!'!5, (&# .69 ="'5. 669 9%1%.  7665) C 7.';,4 !7,6& & 7"!'!5, (k x , k y ) .%1#85 &'5.6 7" ! !"(#84 !5.#'5#.4:
σ λk = (σ x ,σ y ) = λ (cosθ , sin θ ),

(10)

9( >%1% θ (k ) 7.(":5!: & (8).   

                   !" #$ % &' &(% ), *+, -. #/ ",') #"' 0!','&1' 1     &2* & + '3 453, . '. β = 0 , α = 3 ⋅ 10 −9 eV ⋅ cm " 677'* "#$ 1     59'+ 3! :' "1 ,&1 '!+"" 3 0&(% ', %  # $ (7) 6$ m = 0.067m0 . 8/ 1.3.

k2 E R (λ , k ) = + λk , (11) λ = ±1. 2m          !( (!  ," 0'! "" " " "  &  + ' ! - * /*$ #*      (11) # / #  # $ # ) 0!" ',' " ,&1 5'"; ' ' 0' ! & " k = (kx , k y ) # $ $  .1  # #  *  λ = ±1. < # $ 0'* !  0!" 6  (, &'$ ,&1 $ +&1,$ " # 0!"1 "1   !'  $"1 ,"  0'! "$$3; *!"#3;  . =$'4$11 ($":$11) #' #2  0' * ! $  " .1  #' #(' λ = −1, #$( !'$$11 (#'!;$11) #' #2 0" 3# ' 1 #3! :'$"'  1.       (11) >λ = ) %' "1 0"  3; 0!' " ,&1 + "&2  " 453 0!','&1@ $  $      1  $ $ $ # ) *? #3! :'$"' (10) 7 / θ (k ) "/ (8). <&(%' $$1 ;' "%' * 1 * ! "$    A 0!"#','$ $ " .2. =",$, %  ,&1 5'"; #'* !$+ 0&1 ( ) σ = σ , σ λ k x y    ) ' ' 0' ! ! 0!','&' "1 # # *  1 $ 0"$# "'@ #", ,$!,$3; #";!') λ =± 0! "#0&:$3 $ 0! #&'$"' #! 9'$"1.

-6-



 



.1.    (11)      


 
             k = (k x , k y )       
 


      λ
= ±1.  

    
     
 
 
.     
    α = 3 ⋅ 10 −9 eV ⋅ cm , m = 0.067m0 .


    

   
 

 .2.      σ λk = (σ x ,σ y )           
    
  
   .1 ( ) λ = −1 (b)        

!

   


         

    λ = ±1 λ = 1.

   
   


   
 "
.

-7-

      ) '!' ,  59'( &(% @ +, ,  !''  0!" ( (@ & ,3     # < *    *  $ #  $$  # #* , ) ",' "' !' '&2; (/ .  & + '+ 453 "  %&' -. #/   #  * *   $
, *    (,' !  !" 2 " ( ?"@, *+, α = 3 ⋅ 10 −9 eV ⋅ cm , 0 ! ' !    # !' '&2; (/ β = 2 ⋅ 10 −9 eV ⋅ cm , " 0-0!':$'(
    m=0.067m0 .  '!+' "%' ") 0' ! ) 5'"; #' #' λ = ± 1 0* / $ $ " .3. :$ $ * * (7) ,&1  !"1 0' ! #",' 2, %  # 59') " ( ?"" αβ ≠ 0  (  #(' "'    * k x → − k x , k y → −k y , $ # "&( '!'3 ! '! [1] ;! $1' 1 ?'$ !      "$#'! "" / *$ ,&1 ," 0'! "" E (k ) = E (− k ) .

   (7)    λ = ±1      .3.        #             !  "    # #   . $  #             %     &

 k x → − k x , k y → − k y ,                     α = 3 ⋅ 10 −9 eV ⋅ cm , E (k ) = E (− k ) . '

β = 2 ⋅ 10 −9 eV ⋅ cm , m = 0.067 m0 .

-8-

      0!','&'$"1 0"$# ,&1 , $$) / , %" #' ' ,$" "/ *$ (!#        0 1$$) 6$'!+"" E (λ , k ) = const 0* / $3 $ " .4, 0!" 6  " .4( )          #'% ' $":$') #' #" 0'* ! λ = −1,  " .4(b) 0" 3# ' ! 0!','&' "' $    ,&1 #'!;$') #' #" 0'* !  λ = 1 . .5! 9 ' $  '51 #$"  $"' !  /&"%"' # ) 7!' " 0&9  ," *)$ (! 0 1$$ 6$'!+"" $ " .4( ) " " .4(b). /&"%"' 0&9 ," '%'$ " E = const ,&1 λ = −1 " λ = 1 "''  '  " ,&1 #    !  !'$$) ! $'' / , %" β = 0 , * * 6  :$ #",' 2 $ " .1, 0!#,1      "  '!+' "%' + " '%' "1 0 &' "1 !3  (! / 6$ *  * $  $ E = const   #$ 7  . 8/ $ 0! #&'$"1  0"$# :$ ,'&  !"1   ! 0!','&'$"1  2 #3#,, %  "' 0"$#  ! : ' " "' !"@ / *$ ," 0'! "" # k -0! ! $ #'.

      

   .4. α = 3 ⋅ 10 −9 eV ⋅ cm , β = 2 ⋅ 10 −9 eV ⋅ cm         #       

    

   

  E (λ , k ) = const                   λ = 1 .  () 
  λ = −1 (b)                             k   
     

.

-9-


         2.     

:           2.1.  

!"#

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«'%,+&246 '2)74 7,;4» [9]. C27 &:. %&'()*+)A :4/&7. :4''3&7:)(, 84*495 & :4'')6+,, 2&(+ +4 &%:4+,>) 3)<*5 &/(4'70@ /)8 D? 284,3&*)A'72,6 , &/(4'70@ ' D? 284,3&*)A'72,)3 E41/., 7. ). %:, α ≠ 0 , Aβ = 0 .A $ *4++&A -(42) 3. &/&/F,3 :)85A (0747 :4/&7. [9] +4 /&()) &/F, '(594 :4'')6+,6 +4 -:4+,>) *25= &/(4'7) ' :48(,9+.3, %4:43)7:43, E41/. α ≠ α , β = 0 , %:, G7&3 1 2 :)*)() :) :4 *)7 ( 9)+ (0747 7. % H α 1 → 0 /5 %& 5 85 /& [9]. 74+&2;4 84*49, '=)347,9)';, %&;484+4 +4 E,'.5. E4''347:,24)7'6 &' :4'')6+,, *25=;&3%&+)+7+.= '%,+&:&2 2,*4 (8), 62(6@F,='6 84*494 & 72)++.3, '&'7&6+,63, *253):+&-& G();7: &++&-& -484. E4'')6+,) '&/' +4 -:4+,>) x = 0 *25= &/(4'7)A ' %4:43)7:43, '%,+-&: /,74(0+&-& %:4,3 &,'=*) &*,7 A'72,6 α 2 ()2&3 %&(5%:&'7:4+'72) , α 2 %:42&3 %&(5%: &'7:4+'72). 28 & H:, 8+49)+,, β =10 +4/)-4@F46 2&(+4 ,3))7 2,* 2 N K ikr L 1 I e (12) x < 0 : Ψin = M J , θ (k ) = Arg[k y − ik x ], λ = ±1, 2 λe iθ (k ) , =4:4;7):,85)7'6 ;&3%&+)+743, ,3%5(0'4 (k , k ) , 9,'(&3 λ = ±1, x y A :,+4*()<+ +494@F,3 )O 70 ; ):=+) (, +,<+)A 94'7, '%);7:4 &/&8 % &' 2 /& -43,(07&+,4+4 E41/., &%,'.24)3&-& B&:3A5(&A (11). H&';&(0;5 '7:5;75:4, ,8&/:4
- 10 -

(13)

4 ϕ )'70 5-&( %4*)+,6. &3%&+)+74 ,3%5(0'4 k %:, G7&3 '&&72)7'72)++& x :42+4 k 2 − k 2 %:, *4++&3 +4/&:) ;24+7&2.= 9,')( (λ , E , k ). y

y


.5.                -    x = 0       α1  α 2 .            λ = −1  λ = 1            ! !             λ . 2.2. "#$%&'&() *$+ $,( #$--')&()

$ ',(5 7&-&, 97& :)1)+,) 5:42+)+,6 .:O*,+-):4 (8) *(6 84*49, '& '%,+*2) 2)72, ' λ = ±1, %:&1)*146 , &:7:4,@ *25= 2)72)A & +), ) 7+.3, 2& ; /5 5 % ' 82 ' &GBB,>,)+743, r1,2 . /3)++&, &7:4
Ψr = r1

e

− ik1 x x +ik y y

2

5 2 − ik x +ik y 3 1 0 e 2x y 4 iθ1 1 + r2 −e 2

5 2 3 1 0 4 iθ 2 1 , e

-*) k1, 2 = 2mE + (mα1 )2 ± mα1 , k x1, 2 = (k1, 2 ) 2 − k y2 , , θ = Arg[k − ik ]. 1, 2 y x1, 2

(14)

(15) (16)

H:&1)*146 2&(+4 ' +),82)'7+.3, ;&GBB,>,)+743, t /5*)7 &7(,9470'6 &7 1, 2 A (,10 .:4<)+, +49)+,)3 4:43)7:4 41 . , +4%:42()+,)3 2* ,<)+,6 ()(14)-(16) 8 % E / 2 ' 24 +4%:42& 2*&(0 &', Ox : - 11 -

x > 0: 

Ψt = t1

e

ik3 x x +ik y y

2

  



1 −e

+ t2 iθ 3


k 3, 4 = 2mE + (mα 2 ) ± mα 2 , 2



e

ik3 x x +ik y y

2

  

1 e

iθ 4


,

(17)

k x 3, 4 = ( k3, 4 ) 2 − k y2

θ 3, 4 = Arg[k y − ik x 3, 4 ].

(18)

(19)



   (12)-(19)               .         r t !  "     1, 2 1, 2    #    x = 0 ,    " #   #   $ %     '       [1]. &        #  ' # ,        !  ' #  '   %      :

Ψ | xx><00 νˆx Ψ | xx><00

=0 = 0,

   #   ()  "'    " " * + !   "   [9]

νˆx =

(20)

α

  

pˆ x − ασˆ y . m

(21)

,      ""   " (12), #  (20),  "%   #     (14) (17),    !'  "#   %     '   #  '   %    r t ,  %"   1, 2

1, 2

!#    -'  ' (12). .    " "   ' + ,    " !   '      !  . = = 2.3. /012345 673898:29;8 8<;5<983<29;5 3 >?2@5A@5 321<5

B     t1, 2    ++# # (17)-(19). C  "    "     %   '       "    λ = ±1    -'  (12)    #      ' I D [9]    '  "   " k , 3, 4 

+   - "  '  '     "            * ! (11):

- 12 -

I (ϕ ) =

1

t1, 2 (ϕ )

dθ dϕ

2

,

(22)

 #   θ
   '  λ = 1   ++'       #   #    ϕ   tan θ  = k k . y

3, 4

    "    " #  ! [9]    

I  (ϕ )

      

       .     1 2 

                ,

               . .6. !     1 "      



 #$            $  $    #             , . . 1 2 , 

      "" #  "     . %&     .6.               

         '   ,   .             $  "           ,  ,       $    #   $ 

(#         .6. )            

      ϕ c ,     $ #     



 $                                  , . .  .    , 4,

α =0

α →0

α ≠0

λ = −1

λ =1

t >> t

α << α

k

    $    "     (18),     1 2 , . .     « #    »    « #   »,   ,   #     

       $             +  

   #           . * (#   #    #      .6. #    , # 

     x4            .   &((#  c   +     #         $         [9],  

       #      . 

θ

k

(ϕ )

I (ϕ )

ϕ =ϕ

- 13 -





 
  .6.         I           










     α =0 α ≠ 0 [9].       
  





λ = −1 (   I − ) λ = 1 (   I + ).          
  



   


  

ϕ    


   ϕ c ,        

   .  !   


     k x 4      (      )      . "  
 
     

  
 
    




, ,    
  

       (  ).

- 14 -

II.



 

 


     1

1.1.

1.2.

1.3.

     " $ #  "               

  

        -      (4) .     "         $ %    &  (7) (8)   " (#  #$ (    "      , (10) #  

   .            .1.1. ,   &  #  #                  -       (

 #$    #     " 

 #   

  !  .1), (       #   #

     .3.).  

 "    " 

   " 

     ! . ! "        #$ #    

           &  .      "      $ '       (10)    )    #  

            .        

" 

     ! ( .2)    

 "



   #$     ,          (  #    .4.).

  #





    2 2.1.

2.2.

2.3.

2.4.

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[4] J.B. Miller et al, Phys. Rev. Lett. 90, 076807 (2003). [5] G. Dresselhaus, Phys. Rev. 100, 580 (1955). [6] S.D. Ganichev et al, Phys. Rev. Lett. 92, 256601 (2004). [7] Semiconductor Spintronics and Quantum Computation, edited by D.D. Awschalom, D. Loss, and N. Samarth (Springer-Verlag, Berlin, 2002). [8] I. Žuti , J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [9] M. Khodas, A. Shekhter, and A.M. Finkel’stein, Phys. Rev. Lett. 92, 086602 (2004).



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