quantsim09 - Warwick
A non
non--equilibrium network model for magneto
transport in the quantum Hall effect regime
J.Oswald and C. Uiberacker
Institute of Physics, University of Leoben, Austria
Outline:
Motivation/Introduction
Basics of the network model
Examples of simulation Results
Conclusions
Motivation
Magnetic field sweep
Rxx
RBH
I
RH
Rxx
classical
Introduction
Landau levels and the edge channel picture
B
D.B. Chlovskii, B.I. Shklovskii, L.I.
2
Glatzman, Phys.Rev. B46, 4026 (1992)
e
I    U H
h
q
c 
B
mc
1 h
RH   2
 e
n ... Filling factor = # edge channels
screening
edge
edge
The edge channel picture suggests
dissipation-less current flows directly at
the edges
edge
bulk
edge
Experiment: E. Ahlswede, J. Weis, et al, Physica B 298(1-4), 562(2001)
The network model - Basics
Saddles of the random potential correspond to the
nodes of the network
Saddle point
RH
Network grid
LL broadening in the bulk due
to
random potential fluctuations
I
RL
J. Oswald and M. Oswald,
J. Phys.: Condens. Matter 18, R101R138 (2006)
The network model – the network setup
In reality we get channel pairs from adjacent loops at
the saddles, which become the nodes
Handling of the nodes
Net current flow in directed 1D channel pairs
m1*=
= ?m1
Scattering region
Reservoir
Reservoir
m2
IL
IL= e2/h*(m1-m2) 2
ILIL(m
-m1-m
)2*)
= e /h*(m
1
2
IR
m1
m2* = ?m2
Dissipation: PV = IR* (m1-m2*)
Disspation from longitudinal values → PV = IL* (m1-m1*)
UL
Reservoir
m1*
IL
Reservoir
IR
IL= e2/h*(m1-m2*)
2
m2*
2
IL= e2/h*(m1-m2*)
IR= e2/h*(m1-m1*)
P = IR/IL = R/T
m1* = (m1+P.m2)/(1+P)
= (m1 - m1*)/(m1 – m2*)
m2* = (m2+P.m1)/(1+P)
Handling of the nodes
Transfer of chemical potentials
m2 = (m1+P.m3)/(1+P)
m4 = (m3+P.m1)/(1+P)
P = IR/IL = R/T
 2   T R  1 
   
 
  4   R T  3 
The network model - Layout
Tunneling between magnetic bound states
EF > ELL
EF < ELL
hill
valley
  L2  E F ( x, y ) eB 
P ( x, y )  exp 
 
eV
h 

L is the period and V is the amplitude of the
representative periodic potential modulation, EF
is the Fermi energy relative to the saddle energy.
EF < ELL
EF > ELL
2
1
2
P>1
P'
P
4
3
P<1
1
3
4
P' = P -1 < 1
The network model - Basics
The role of a random potential
  L2  E F ( x, y ) eB 
P ( x, y )  exp 
 
eV
h 

G xx
R xx
e2
P
 2


R xx  R xy2
h 1  P2
Example 0
QHE sample with gate
across the Hall bar
R.M.Ryan, et al, Phys.Rev.B48, 8840 (1993)
Example 1
Single saddle on a 61x61 grid
2
2
1
1
Example 2
Array of 4 saddles in a 61x61 grid
Example 3
Voltage and current distribution in a standard quantum
Hall conductor
20000
Rxx ()
15000
10000
5000
0
4,0
n=2
n=3
4,5
5,0
5,5
B (T)
6,0
6,5
7,0
Example 4
Anti - Hall - bar embedded in a Hall-bar : Sample layout
E
IAB
F
5
A
1
3
C
6
4
I12
2
B
D
The layout for experimental investigations was developed by Mani et al.
(R.G.Mani, J. Phys. Soc. Jpn. 65, 1751 (1996)) and is also the basis for the
numerical simulations later on.
Example 4
Anti
Anti--Hall
Hall--bar within a HallHall-bar structure: Network layout
Example 4
Hall voltage
Experiment
Simulation
M. Oswald, J. Oswald and R.G. Mani, Phys. Rev. B 72, 035334 (2005)
J. Oswald, M.Oswald, Phys. Rev. B 74, 153315 (2006)
Example 4
Longitudinal voltage
Experiment
Simulation
M. Oswald, J. Oswald and R.G. Mani, Phys. Rev. B 72, 035334 (2005)
J. Oswald, M.Oswald, Phys. Rev. B 74, 153315 (2006)
Example 4
Bulk current distribution for the case of current
compensation (IA,B = -I1,2) in the plateau regime
M. Oswald, J. Oswald and R.G. Mani, Phys. Rev. B 72, 035334 (2005)
J. Oswald, M. Oswald, Phys. Rev. B 74, 153315 (2006)
Example 5 - Non-ideal contacts
Ideal contacts
Non - ideal contacts
The puzzle of current flow in the QHE:
Sample with narrow gate stripe in the middle
of the bulk
The electron cannel below the gate stripe (in red) is kept
always at that integer filling, which is closest to the actual
bulk filling of the non-gated region. The remaining bulk
makes the usual transitions from the conducting to the
insulating regimes while sweeping the magnetic field.
QHE sample with narrow gate stripe
Bulk region: near half filling > conducting
gate stripe
-contact
+ contact
Sample with gate stripe:
Gate region: integer filling > insulating
4
X
6
4
8
10
10
8
6
4
2
0
-2
-4
-6
-8
-10
2
8
6
10
Y
2
potential (arb. units)
+ contact
-contact
contact
Sample without gate stripe:
10
8
6
4
2
0
-2
-4
-6
-8
-10
2
potential (arb. units)
Simulation of the lateral potential
distribution with and without gate:
4
X
6
4
8
10
2
6
8
Y
10
QHE sample with narrow cut
Just a simple cut, hole or complete depletion at the position of
the gate does not have the right effect.
A cut in the bulk in longitudinal direction of the sample leads to the same
lateral potential distribution like in a non
non--gated sample
n = 0,5
CUT
n = 1,5
QHE sample with narrow gate stripe
Interpretation of current flow
I0
I0
gate stripe
The experimentally injected current gets concentrated to
the narrow gate region, where it flows dissipation-less
QHE sample with narrow gate stripe
Longitudinal Resistance and Hall resistance at different
length of the gate stripe
Simulation results: The gate stripe sharpens the plateau transitions
5000
26,0k
4500
4000
3000
2500
2000
1500
without gate
short gate
medium gate
long gate
22,0k
20,0k
Rxy
3500
Rxx
24,0k
without gate
short gate
medium gate
long gate
18,0k
16,0k
1000
14,0k
500
12,0k
0
10
11
12
13
B (T)
14
15
16
9
10
11
12
13
14
15
16
B (T)
The length of the long gate stripe is 85%, the medium gate stripe is 22% and
the short gate stripe is about 10% of the sample length.
QHE sample with narrow gate stripe
Simulation: sweep of the gate potential at fixed magnetic field
Magnetic field sweep without gate
Gate voltage sweep at fixed magnetic
field as indicated on the left
1500
1400
1300
4000
1200
1100
1000
3000
900
Rxx
Rxx
800
700
600
2000
500
400
300
1000
200
100
0
0
0
6
8
10
12
14
16
20
40
60
80
100
Gate potential (arb. units)
B (T)
From the experimental point of view it might be much easier to fix the
magnetic field somewhere in the tail of a resistance peak and performing a
gate voltage sweep. While approaching the right gate voltage the resistance
should drop to zero, indicating a narrowing of the resistance peak.
QHE sample with narrow gate stripe
Simulation: sweep of the gate potential at fixed magnetic field
Magnetic field sweep without gate
Gate voltage sweep at fixed magnetic
field as indicated on the left
R1
17000
28000
26000
16000
24000
22000
15000
18000
Rxy
Rxy
20000
16000
14000
14000
12000
13000
10000
8000
12000
6
8
10
B (T)
12
14
16
0
20
40
60
80
100
Gate potential (arb. units)
Fixing the magnetic field somewhere before the plateau value of the Hall
resistance is reached and making a gate voltage sweep. While approaching
the right gate voltage the Hall resistance should move on to the exact plateau
value, indicating a narrowing of the plateau transition.
Summary and Conclusion
Network model which describes exclusively the injected
excess currents and potentials close to the real experimental
conditions.
modeling of complex sample configurations possible
(geometry, gate electrodes, inhomogeneities)
describes edge and bulk properties at the same time
Dissipation-less current only in incompressible stripes, and no
current in the edge stripes
Outlook:
 Feed back of the excitation potental to the system
Use as a “non-equilibrium transport modul” which takes input
also from other equilibrium theory
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