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