Transport properties of junctions and lattices via solvable model

Nikolai Bagraev (impurity.dipole@pop.ioffe.rssi.ru), A.F. Ioffe Physico-Technical Institute,St. Petersburg, Russia

Lev Goncharov (Lev.goncharov@mail.ru), Department of Physics of St. Petersburg State University, Russia

Gaven Martin (G. J. Martin@massey.ac.nz), New Zealand Institute of Advanced Study, New Zealand

Boris Pavlov (pavlovenator@gmail.com) Department of Physics of St. Petersburg State University, Russia

Adil Yafyasov (yafyasov@bk.ru), Department of Physics of St. Petersburg State University, Russia

Quantum networks

• Quantum network constructed on the surface of semiconductor as a union of quantum wells Ωk and quantum wires ωi

• Transport of electrons through the network described by the Schrödinger-type equation

δ

Quasi-1D quantum wires ωi width δ

Quantum wells Ωk

2 2

2 2: , :

( )

mE mV

V

2

Quasi-one-dimensional quantum wiresSplitting of variables allows to modify equation on wires

2 2 2

2 2

( )( , ) ( , ) ( )

2 sin , ( , ) ;

nn

n n n s s sn

x nx y x y xx

ne y x y e P e e

2

2

Spectra on the semi-infinite wires

2

2

4

2

2

9

2

2

16

3

Δ1

Δ2

Δ3

ΔTλF

•Consider the Fermi level inside the first spectral band Δ1

•Assume the temperature to be low, so that essential spectral interval ΔT is inside Δ1

2 2

2.

2 2

2.

. .

: ,

: ,

ss open ch

ss closed ch

s s s ss open ch s closed ch

nn channel n isopen P P

nn channel n is closed P P

K P K P

Λ1

Λ2

Λ3

Λ4

4

Dirichlet-to-Neumann map

Γ1

Γ2

Γ3

i

Ω

\

( ) :

0 :

V DNn

u Vu uuu DN un

u u

DN DNP DN P P DN PDN

DN DNP DN P P DN P

5

Scattering matrix and intermediate DN-map

• Intermediate DN-map DNΛ is a finite-dimensional DN-map of Schrödinger problem with partial-zero boundary condition:

3

1

1( ) ( )

k

iK x iK x iK xk l l

l

e e e Se e se

S iK DN iK DNIwith DN DN DN DN

DN K

Exact finite-dimensional equation on scattering matrix

Γ1

Γ2

Γ3

\0

0

u Vu uu

P u

P u matching condition

6

Singularities of intermediate DN-map

• Inherited singularities from DNΓ

• Zeros of DN--+K-

7

IDN DN DN DNDN K

DNΛ may have singularities of two types:

But singularities of first type compensate each other

1

1 1

1

, , ( )

s

s s

s s

s

ss s ss

DN K DNn n

IL T Q T Tn K K

IJ P K PK K

( )I IDN K K K J T J T

K K I L Q

Silicon-Boron two-dimensional structure

• Experiment shows high mobility of charge carriers on double-layer quasi-two-dimensional silicon-boron structure

• Boron atoms at high concentration form sublattice in silicon matrix

9

- no boron

- B++B-

Model description

Consider the boron sublattice as a periodic quantum network

•Elements of the network are connected by aid of rather long and narrow links

10

Model description• Separation of variables and cross-section

quantization on the links generate infinite number of spectral channels

• Only finite number of spectral channels is open (oscillating solutions of Schrödinger equation on the links)

• Closed channels (exponentially decreasing solutions of Schrödinger equation on the links) could be omitted

• Matching on the open channels only allow to reduce the infinite-dimension matching problem to finite-dimensional one

11

Statement of periodic spectral problem( ) , 0

, 0

l l l l l

l ll l

V x P P

P P P Pn n

Intermediate problem for single element of the lattice

( )

0

V x

P

P matching condition

: uDN u P u En

12

Spectral problem with partial quasi-periodic boundary condition on the pairs of opposite slots

• Now we can exclude links and make respective changes in boundary conditions

• Boundary data and boundary currents then are connected by intermediate partial DN-map DNΛ (but not traditional partial DN-map)

13

Spectral problem with partial quasi-periodic boundary condition on the pairs of opposite slots

2

2

, ,

, ( ) ,

scaled energyof cross-section confinment

a - length of the links

ls

s s

ls

s s

ip al l l l

l open l openchannels channels

ip al l l l

l open l openchannels channels

ls l l

e

en n

with

p and

Γ2+

Γ2-

Γ1+Γ1

-

Excluding links and correct boundary conditions

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Assumption• To simplify following calculations consider a case, when only one

spectral channel in cylindrical links is open, so• That simplify above boundary conditions as:

1 1s s s s sP e e e e

1

1

2

2

ss s

s

s s

ip a

ip a

P e P

P e Pn n

1 1

2 2

1

2 21 1 1 1

1 1 1 11 1 2 2 1 1 2 2

2 2 2 22 2

2 22 2

2

1

,

100

ip a ip a

ip a ip a

ip a

e e

P P P Pe e

e

1

1 1

2

2 1 22 2

0 00 01

, , ,0

1 0 1

ip a

ip a ip a

e

e e

15

Dispersion relation• Connect boundary data and boundary currents with DNΛ

1 1 2 2 1 1 2 2

1 1 1 1 2 2

2 1 1 2 2 2

1 2 1 211 12

2 1 221 22

, 0

0

0

, ,det

, ,

s t

DN

in consideration of

DN DN

DN DN

and note theconditionof existenceof non trivial Bloch function

DN DN

DN DN

20

16

Double periodic quasi-2D lattice• Assume, that two boron sublattices interact by

means of tunneling through the slot Γ0

Ωu is a period of first sublattice and Ωd is a period of the second one

17

Double-lattices quasi-periodic boundary conditions

• These conditions impose a system of homogeneous linear equations on

• And we can note the condition of existence of non-trivial Bloch functions

0 0

0

0

2, ,

, ,2

00

00

00

ss s

s

s s

u u

d

d

ip au d u d

u d u dip a

u

uu

d d

d

P e P

P e Pn n

and tunneling boundary condition

P Pn

PPn

1 2 0 0 1 2, , , , ,u u d du d

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Dispersion relation, ,

, 11 ,

, , 1 ,11 12 , 10

, , , 2 ,21 22 , 20

, 1 , 2 ,01 , 02 , 00

1 , 1 1 , 2, 11 , , 12 ,,

2 , 1 2, 21 , , 22

:

,

( ) ,

, ,

( )

u d s u d ts t u d u d

u d u d u du d

u d u d u d u du d

u d u d u du d u d

u d u du d u d u d u du d

T u d uu d u d u d

d DN

d d DN

DN p d d DN

DN DN DN

DN DNDN p

DN DN

, 2

,d

u d

And dispersion relation is

2 det det det det 0u d u dT TDN DN DN DN

19

Dispersion relation2 det det det det 0u d u d

T TDN DN DN DN

det 0 det 0u dT TDN and DN

20

• If β→∞, linear system splits in two independent blocks

• If β is finite, then intersection of terms transforms to quasi-intersection

21

N.T. Bagraev, A.D. Bouravleuv, L.E. Klyachkin, A.M. Malyarenko, V.V.Romanov, S.A. Rykov: Semiconductors, v.34, N6, p.p.700-711, 2000.

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