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Transport Calculation of Excitonic Superfluid in Bilayer System

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Excitonic superfluid in QH-bilayer system (CalTech, Princeton, Stuttgart) ... Supercurrent is a constant over spac: ?~ 1/t ~ 1.4 a.u. ~ 17 sites. No tunnling : ... – PowerPoint PPT presentation

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Title: Transport Calculation of Excitonic Superfluid in Bilayer System


1
Transport Calculation of Excitonic Superfluid in
Bilayer System
  • Jung-Jung Su
  • Allan MacDonald

2
Motivation Approach
3
NEGF Basics
4
Description of Our Model
  • Two interacting wires
  • Mean-field Theory

Hamiltonian in lattice representation
HTB tight-binding Hamiltonian S
self-energy of the leads Sin self-energy of
interaction
Self energy of leads for open boundary condition
tc, tv hopping of c and v bands
Exchange interaction btw the layers, obtain
self-consistently
Vx delta-function interaction strength Rho
density matrix
5
4-lead Result I (Equlibrium)
N 300, a0.08, overlap20 Ryd, Vx8.0 Ryd V0
order parameter
order parameter
Excess population
Excess population
Normal State Inside
Normal State Outside
6
4-lead Result II (Tunneling Case )
.
1
2
N
V/2
V/2
N 300, a0.08, overlap20 Ryd, Vx8.0 Ryd V0.5
Ryd
.
N
1
2
-V/2
-V/2
  • No Self-Consistent Solution !!

order parameter
  • Although in the MF level electron-electron
    interaction appears like interlayer tunneling,
  • the self-consistent solution shows number
    conservation in each layer as in the original
    2-body Hamiltonian

Excess population
Normal State Inside
7
4-lead Result III ( Parallel Flow )
.
1
2
N
V/2
-V/2
N 300, a0.08, overlap20 Ryd, Vx8.0 Ryd V0.5
Ryd
.
N
1
2
V/2
-V/2
  • No Current out of the leads

order parameter
  • Zero phase gradient of order parameter suggested
    that no supercurrent

Excess population
Normal State Inside
8
Result IV ( Naive Counter-Flow, no
tunneling)
.
1
2
N
V/2
-V/2
N 300, a0.08, overlap20 Ryd, Vx8.0 Ryd V0.5
Ryd
.
N
1
2
-V/2
V/2
Estimated Supercurrent
  • Normal current exist b.c. of finite size effect

9
4-lead Result VI (Counter-Flow, with tunneling)
.
1
2
N
V/2
-V/2
N 300, a0.08, overlap20 Ryd, Vx8.0 Ryd V0.5
Ryd, t0.5
.
N
1
2
-V/2
V/2
Phase of off-diagonal block density matrix
  • What does tunneling do?
  • Bare tunneling is trying to keep the phase a
    constant.
  • The length required by tunneling to turn finite
    phase gradient into zero

? 1/t1/2 1.4 a.u. 17 sites
  • Zero Super current at the center

10
4-lead Result VII (Counter-Flow, with tunneling)
.
1
2
N
V/2
-V/2
N 300, a0.08, overlap20 Ryd, Vx8.0 Ryd V0.5
Ryd, t0.05
.
N
1
2
-V/2
V/2
  • Both Supercurrent and normal current go to zero
    at the center. However, no significant difference
    can be measured from the leads

11
2-lead Result I ( no shorting)
.
1
2
N
V/2
N 300, a0.08, overlap20 Ryd, Vx8.0 Ryd V0.5
Ryd
.
2 lead version of the previous capacitance
calculation
N
1
2
-V/2
With tunnling t 0.5 Ryd
No tunnling no self-consistent solution
gt Consistent with previous result
top
bottom
Supercurrent is a constant over spac ? 1/t
1.4 a.u. 17 sites
12
2-lead Result I ( shorting)
.
1
2
N
V/2
N 300, a0.08, overlap20 Ryd, Vx8.0 Ryd V0.5
Ryd
.
N
1
2
-V/2
With no tunnling
With tunnling t 0.5 Ryd
top
bottom
top
bottom
  • With tunneling large enough, the shorting case
    gives similar result as no shorting

13
Conclusion and Future
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