КВАНТОВЫЙ ТРАНСПОРТ В ПОЛУПРОВОДНИКОВЫХ МИКРОСТРУКТУРАХ

1.ГЕТЕРОСТРУКТУРЫ. Home made quantum mechanics2.ОТКУДА БЕРЕТСЯ СОПРОТИВЛЕНИЕ ПРИ Т=0. Формула

Ландауэра-Буттикера 3. Как считать. ТРАНСПОРТ ЧЕРЕЗ КВАТОВЫЕ ДОТЫ

Полупроводниковые гетероструктуры

U

z

gates

2DEG

Полупроводниковые гетероструктуры

SupriyoDatta Special Issue: Physics of electronic transport in

single atoms, molecules,and related nanostructures, Nanotechnology 15 (2004) S433

Проводимость Ландауэра

Rolf Landauer (1957)

Проводимость Ландауэра T=0

S и T матрицы

* 2 2 2 2

2 2 2 2

* * * *

( ) [| | | | ] [| | | | ].

| | | | | | | | .

D . 1

j imag k A B k C D

A D B C

C AC B A S S S S

B D

Ток сохраняется

S-mattix

Унитарность S-матрицы1S S

Т-матрица

Амплитуда трансмиссии

T-matrix

T-matrix

Resonant tunneling, LED

LED

LED

LED

Multichannel conductance

( )2

nik x

n

n

ey

k

( )2

mik x

nm mm m

et y

k

( )

2

mik x

nm mm m

er y

k

отражается

/ , ( , ), ( , ), inc out inc x L out x RT I I I dyj x x y I dyj x x y 2

2 2 +

,

2e| | . | | ( ). G= Sp(TT )

hn nm nmm n m

T t T t Sp TT

Quantum point contacts (QPC)

QPCFrom A. Cserti, J. Appl. Phys. (2006)

QPC

Подход эффективного гамильтониана

Coupled mode theory (оптика)

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).

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 Х. Хаус, Волны и поля в оптоэлектронике

0

2

( )

, W=|a| .

in

out in

dai a kS

dt

S CS a

Одно-модовый резонатор

22 2 2 2| |

2 | | | | | |out

dW d aa S a

dt dt

= 2 0

0

( ) exp( ), ( ) ,

, 2in

in

a t a i t i i a kS

i a kS

Инверсия по времени2 2

2 2| |2 | | | |

2 in

d a ka S

dt

k

22 2 2 * *

2 2 2 2 * *

* * * * 2

2 2 2 * *

| || | | | 2 | | 2 ( ).

2 ,

2 | | | | | | ( ),

2 ( ) 2 | | ,

| | | | ( 1)( ) 2 |

in out in in

out in

out in in out in out

in in in out in out in

out in in out in out in

d aS S a a S aS

dt

a S CS

a S C S C S S S S

a S aS S S S S C S

S C S C S S S S C S

2 2 2| | | | |in outS S

1C

0( ) 2

2

in

out in

dai a S

dt

S S a

CMT Много-модовый резонатор

IEEE J. Quantum Electronics, 40, 1511 (2004)40, 1511 (2004)

Зарядовые эффекты

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.

22

2n y yE E k km

Let it be 1d or quasi one-dimensional wire. 0xk

1 2

1 11 1,|1 ; ,| 2 ;

1 12 2y y y yk k k k

1 21 11 1

( , ) ( ) ( )1 12 2

y yik y ik y

n nx y x e x e

Particular solutions of the Shrödinger equation are

The total solution

2m L

The angle of spin presession

E

ky1 ky2

Spin evolution for movement along curvilinear wire

2

2

2

2

2

cos(2 1) cos(2 1)

1

sin(2 1) cos(2 1)

1

2 sin 2

1

x

y

z

2

22 1 , , 2

1 1m R

For the straight wire R L (β→∞) we again obtain a simple spin precession

cos 2 , 0, sin 2x y z

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 atransmission 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 moreelegant way by use of spin dependentS-matrix theory by Kisilev and Kim (cond-mat/411070) and Zhai and Hu (to be published)

Numerical results, ,

[ ]x y z

P

Different way to definespin polarization via Transmission probabilities

'T

''

( )T TP

T

Bulgakov et al, PRL, 83, 376 (1999)Mireles and Kirczenow, PRB66, 214415 (2002)Hu and Zhai (to be published)

Spin transistorE.N.Bulgakov and A.F.Sadreev, Phys. Rev. B 66, 075331 (2002)

T-shaped ballistic spin filterKiselev 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

*

0

0 *

* 2 * 20

( 1)

; ;

0 /( ); 2 ;

/ 0

2 , / 2 ;

[ , ] 0; [ , ] 0,

/ 2; ;

( )( 1/ 2) ;

( )

SL

SLz z

z

z z z y

im

z m m m i m

z x iy H H V

zH V r V

z

m R E m R

J H K H

J L K i C

u r eJ m

v r e

Resonant transmission through the QD,weak coupling

Radiation field with circular polarization

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 half-integer 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.

( ) (sin ,cos ,0)A t A t t ��������������

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(itJz) to give rise to the following effective Hamiltonian:

*

2;z

iedAH H J

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

3/ 2 1/ 2 47.12 29.33

for 0.75

E E

Chaotic billiards with account of spin-orbit interaction (SOI)

Bulgakov and Sadreev, JETPLett.78, 911 (2003); PRE 70, 56211 (2004)

For E ; imag( );

( )2

b

b b

b

jx iy

Distributions of u iv

t iw

0.25

| |( ) exp

4 2

jP j

j j

02

( ) ( / ).j

P j K j

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).

2

2

- ,

- ,

L

L

2; ;L i L Lx y

4 2 2 2

1 2

2 2 2 21 1

2 2 2 22 2

1. = , =

[ ( 2 ) ] 0;

;

( / 2 / 2 1 4 / ) ;

( / 2 / 2 1 4 / )

L L

L L L L

a b

Exact relations for arbitrary QD with SOI

Strong SOI 2

There are two characteristic scales in solution :

1

and21/ + 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