Transcript Slide 1

КВАНТОВЫЙ ТРАНСПОРТ В
ПОЛУПРОВОДНИКОВЫХ МИКРОСТРУКТУРАХ
1.ГЕТЕРОСТРУКТУРЫ. Home made quantum mechanics
2.ОТКУДА БЕРЕТСЯ СОПРОТИВЛЕНИЕ ПРИ Т=0. Формула
Ландауэра-Буттикера
3. Как считать. ТРАНСПОРТ ЧЕРЕЗ КВАТОВЫЕ ДОТЫ
Полупроводниковые гетероструктуры
Полупроводниковые гетероструктуры
gates
U
2DEG
z
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SupriyoDatta
Special Issue: Physics of electronic transport in
single atoms,
molecules,and related nanostructures,
Nanotechnology 15 (2004) S433
Проводимость Ландауэра
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Rolf Landauer (1957)
Проводимость Ландауэра  T=0
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S и T матрицы
S-mattix
j  imag ( * )  k[| A |2  | B |2 ]  k[| C |2  | D |2 ].
Ток сохраняется
| A |2  | D |2 | B |2  | C |2 .
C 
*
*
 A
C B   B    A D  S S  D . S  S  1
 
 
*
S S  1
*
Унитарность S-матрицы
Т-матрица
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Амплитуда трансмиссии
T-matrix
T-matrix
Resonant tunneling, LED
LED
LED
LED
Multichannel conductance
eikn x
n ( y)
2 kn
отражается
e  ikm x
m rnmm ( y) 2 k
m
eikm x
m tnmm ( y ) 2 k
m
T  I inc / I out , I inc   dyjx ( x  xL , y ), I out   dyjx ( x  xR , y ),
2
2e
Tn  | tnm |2 . T  | tnm |2  Sp(TT  ). G=
Sp(TT + )
h
m
n,m
Quantum point contacts (QPC)
QPC
From A. Cserti, J. Appl. Phys. (2006)
QPC
Подход эффективного гамильтониана
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1. М. С. Лифшиц, ЖЭТФ (1957).
2. U.Fano, Phys. Rev. 124, 1866 (1961).
3. H. Feshbach,, Ann. Phys. (New York) 5 (1958) 357; 19 (1962) 287.
4. C. Mahaux, H.A. Weidenmuller, (Shell-Model Approach to Nuclear
Reactions), (1969).
5. I.Rotter, Rep. Prog. Phys., 54, 635 (1991).
6. S.Datta, (Electronic transport in mesoscopic systems) (1995).
7. Sadreev and I. Rotter, JPA (2003).
8. Sadreev, JPA (2012).
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Coupled mode theory (оптика)
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H.A.Haus, (Waves and Fields in Optoelectronics) (1984).
C. Manolatou, et al, IEEE J. Quantum Electron. (1999).
S. Fan, et al, J. Opt. Soc. Am. A20, 569 (2003).
S. Fan, et al, Phys. Rev. B59, 15882 (1999).
W. Suh, et al, IEEE J. of Quantum Electronics, 40, 1511 (2004).
Bulgakov and Sadreev, Phys. Rev. B78, 075105 (2008).
Coupled mode theory
Одно модовый резонатор
CMT
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Х. Хаус, Волны и поля в оптоэлектронике
Одно-модовый резонатор
da
 (i0   )a  kSin
dt
Sout  CSin   a, W=|a|2 .
dW d | a |2

 2 | a |2   | Sout |2   2 | a |2
dt
dt
a(t )  a exp(it ), (i  i0   )a  kSin ,
  0  i , 2 a  kSin
Инверсия по времени
d | a |2
k2
2
 2 | a | 
| Sin |2
dt
2
d | a |2
| Sin |  | Sout | 
 2 | a |2  2 ( a* Sin  aSin* ).
da dt
 (i0   )a  2 Sin
2 a  Sout  CS
,
dtin
2
2
*
2 | a |2 | Sout |2 C 2 | Sin |2 C ( Sin Sout
 Sin* S out ),
Sout   Sin  2 a
*
2 (a Sin  aSin* )  Sin Sout
 Sin* S out  2C | Sin |2 ,
*
*
 | Sout |2 C 2 | Sin |2  (C  1)( Sin Sout
 Sin* Sout )  2C | Sin |2 | Sin |2  | Sout |2
 = 2
k 
C  1
CMT
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Много-модовый резонатор
IEEE J. Quantum Electronics, 40, 1511 (2004)
Зарядовые эффекты
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1. Кулоновские взаимодействия в 1d
проволоке.
2. Кулоновская блокада в квантовых дотах
The reason for the spin precession is that the spin operators do not commutate with the
SOI operator, which leads to spin evolution for the electron transport. In particular the
SOI has a polarization effect on particle scattering processes, and this effect was
considered for different geometries of confinement of the 2DEG:
S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).
E.N.Bulgakov, K.N.Pichugin, A.F.Sadreev, P.Streda, and P.Seba,
Phys. Rev. Lett. 83, 376 (1999).
A.Voskoboynikov, S.S.Liu, and C.P.Lee, Phys. Rev. B 58, 15397 (1998), Phys.
Rev. B 59, 12514 (1999).
A.V.Moroz and C.H.W.Barnes, Phys. Rev. B 60, 14272 (1999).
F.Mireles and G. Kirczenow, Phys. Rev. B 64, 024426 (2001).
L.W.Molenkamp and G.Schmidt, cond-mat/0104109.
Let it be 1d or quasi one-dimensional wire.
E  En 
2
2m
k y2   k y
kx  0
Particular solutions of the Shrödinger equation are
1 1
1 1 
k y  k y1 ,|1 
;
k

k
,|
2

y2
  y
 ;
1
2 
2  1
E
The total solution
ky1
1 1 ik1 y y
1 1  ik2 y y
 ( x, y)  n ( x)
  e  n ( x)
 e
2 1
2  1
ky
2
The angle of spin presession
  2m  L

Spin evolution for movement along curvilinear wire
cos(2  1)   2 cos(2  1)
  x 
1  2
sin(2  1)   2 cos(2  1)
  y 
1  2
2 sin 2
  z 
1  2
2 
1  , 
2

1 1  2
,   2 m R
For the straight wire R
simple spin precession
L (β→∞) we again obtain a
  x  cos 2,   y  0,   z  sin 2
Two-dimensional curved waveguide
Spin evolution in the 2d curved waveguide R=d, β = 1
ε=25, the first-channel transmission
ε=39.25, near an edge of the
second-channel transmission
We prove that for a
transmission through
arbitrary billiard with two attached
leads there is no spin polarization, if
electrons incident in the single
energy subband and were spin
unpolarized
The same result was obtained in more
elegant way by use of spin dependent
S-matrix theory by Kisilev and Kim
(cond-mat/411070) and Zhai and Hu
(to be published)
Numerical results
P


 x, y , z
[ 

   ]
Different way to define
spin polarization via
Transmission probabilities
T '
Bulgakov et al, PRL, 83, 376 (1999)
Mireles and Kirczenow, PRB66, 214415 (2002)
Hu and Zhai (to be published)
(T T

P 
T



'
'

)
Spin transistor
E.N.Bulgakov and A.F.Sadreev, Phys. Rev. B 66,
075331 (2002)
T-shaped ballistic spin filter
Kiselev and Kim, Appl. Phys. Lett.
(2001)
QD with Rashba SOI - exact solution
Bulgakov and Sadreev, JETP Lett. 73, 505 (2001)
Tsitsishvili, Lozano, and Gogolin, PRB, 70, 115316
(2004) + mag. field
z  x  iy; H  H 0  VSL ;
 0
H0  
 V (r ); VSL  2  
*

/

z

  2m* R, E0  2 / 2m* R 2 ;
 
z z*
 / z 
;
0 
[ J z , H ]  0; [ K , H ]  0,
J z  Lz   z / 2; K  i y C ;
 u (r )eim 
J z  m  (m  1/ 2) m ;  m  

i ( m 1) 
 v ( r )e

Resonant transmission through the QD,
weak coupling
Radiation field with circular polarization
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It is well known in atomic spectroscopy that atomic
spectroscopy that circularly polarized radiation field can
transmit an electron from a multiplet state with a halfinteger total angular momentum to a continuum with a
definite spin polarization (Delone and Krainov, Sov.
Phys. Usp. 127, 651 (1979).
We consider similar phenomenon for the electron
ballistic transport in quantum dots and in microelectronic
devices with bound states.
A(t )  A(sin t,cos t,0)
Similar to the two-level system, an effect of this radiation field can
be considered exactly by transformation to the rotating coordinate
system by the unitary operator exp(iwtJz) to give rise to the following
effective Hamiltonian:
2iedA  
 
H  H Jz 
  * ;
c  z z 
Therefore the radiation field with circular polarization effects the QD
like an external magnetic field, i.e., lifts the Kramers degeneracy.
This phenomenon firstly was considered by Ritus for an atom
(Sov. Phys. JETP 24, 1041 (1967)).
Second, it obviously follows that the radiation field mixes only states
M and M‘ differing by M = ±1.
Effect of radiation field with circular polarization
The transmission probability through QD
  E3/ 2  E1/ 2  47.12  29.33
for   0.75
Chaotic billiards with account of spinorbit interaction (SOI)
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Bulgakov and Sadreev,
JETPLett.78, 911 (2003);
PRE 70, 56211 (2004)
For

E
 b

 ; j  imag(  );

b
b
 ( x  iy) b 
2

Distributions of
    u  iv 
 


t

iw
  

0.25
  | j | 
P( j ) 
exp  

4 j 
2

j




P( j ) 
j

2
Saichev et al, J. Phys. A35, L87 (2002);
Barth and Stockmann, Phys. Rev. E 65, 066208 (2002).
Kim et al, Progr. Theort. Phys. Suppl. 150, 105 (2003).
Sadreev and Berggren, Phys. Rev. E70, 26201 (2004).
K 0 ( j /  ).
Exact relations for arbitrary QD with SOI
-    L    ,
2
-2    L   ,
1.  =
 L
,  =
L
L


 i ;  2   L L;
x y

 LL
 LL
[ 4  (  2  2 ) 2   2 ]  0;
  a1  b2 ;


 21  (   2 / 2   2 / 2 1  4 /  2 )1;
 22  (   2 / 2   2 / 2 1  4 /  2 )2
Strong SOI


2
There are two characteristic scales in solution :



1 and
 1/  + 2
1/
1
Statistics of the eigenfunctions
  ( x, y)   u( x, y)  iv( x, y) 


.
  ( x, y)   t ( x, y)  iw( x, y) 
Comparison of numerical statistics with
analytical distributions for strong 