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Transport properties of junctions and lattices via solvable model
Nikolai Bagraev ([email protected]), A.F. Ioffe Physico-Technical Institute,St. Petersburg, Russia
Lev Goncharov ([email protected]), Department of Physics of St. Petersburg State University, Russia
Gaven Martin (G. J. [email protected]), New Zealand Institute of Advanced Study, New Zealand
Boris Pavlov ([email protected]) Department of Physics of St. Petersburg State University, Russia
Adil Yafyasov ([email protected]), 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
( )( , ) ( , ) ( )
2sin , ( , ) ;
nn
n n n s s sn
x nx y x y x
x
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 uu
u 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 DN
Iwith DN DN DN DN
DN K
Exact finite-dimensional equation on scattering matrix
Γ1
Γ2
Γ3
\0
0
u Vu u
u
P u
P u matching condition
6
Singularities of intermediate DN-map
• Inherited singularities from DNΓ
• Zeros of DN--+K-
7
IDN DN DN DN
DN 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 T
n K K
IJ P K P
K 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
:u
DN 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 energy of 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
14
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
,
1
0
0
ip a ip a
ip a ip a
ip a
e e
P P P Pe e
e
1
1 1
2
2 1 2
2 2
0 0
0 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
0
0
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
PPn
PP
n
1 2 0 0 1 2, , , , ,u u d du d
��������������������������������������������������������
18
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.
23