Title: Transport Calculation of Excitonic Superfluid in Bilayer System
1Transport Calculation of Excitonic Superfluid in
Bilayer System
- Jung-Jung Su
- Allan MacDonald
2 Motivation Approach
3 NEGF Basics
4Description 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
13Conclusion and Future