1. FSU Physics Department
Pairing instability of composite fermions in double layer quantum Hall system
Huan D. Tran
Quantum Hall effect
Composite fermion
FQHE in double layer electron system
Proposal work
4/30/2019
FSU Physics Department

2. FSU Physics Department
Quantum Hall Effect
4/30/2019
FSU Physics Department

3. Quantum Hall Effect (cont.)
Strong B
Low T
IQHE: n = 1, 2, … (‘80)
FQHE: n =1/3, 2/5, … (‘82)
(more than 50)
4/30/2019
FSU Physics Department

4. Quantum Hall Effect (cont.)
Störmer, RMP 71, 875 (‘99)
4/30/2019
FSU Physics Department

5. Quantum Hall Effect (cont.)
IQHE, physics of non-interacting electron system
Quantized cyclotron
orbits (Landau levels)
n: number of filled Landau levels
4/30/2019
FSU Physics Department

6. Quantum Hall Effect (cont.)B1 B2 RH B
n = 2
n: number of electrons per flux quantum
4/30/2019
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7. Quantum Hall Effect (cont.)
Disorder  localized & extended states
Localized & extended states  plateaus & steps
Laughlin, PRB 23, 5632 (‘81) 
4/30/2019
FSU Physics Department

8. Quantum Hall Effect (cont.)
FQHE observed at non-integer filling
Tsui et.al., PRL 48, 1559 (‘82)
Non-interacting electrons  No energy gap!
Electron interaction must be considered
4/30/2019
FSU Physics Department

9. Quantum Hall Effect (cont.)
Laughlin wavefunction, theory of filling factors 1/(2p+1)
Laughlin, PRL 50, 1395 (‘83)
The filling factors 1/3, 1/5, 1/7, … are explained
Remaining filling factors 2/5, 3/7, 4/7, … need further treatment
4/30/2019
FSU Physics Department

10. FSU Physics Department
Composite Fermions
Jain, PRL 63, 199 (‘89)
Y  -Y
Roughly speaking, each electron captures 2m flux quanta to become a composite fermion denoted by 2mCF
More rigorously, a gauge transform is needed to form composite fermions from electrons
4/30/2019
FSU Physics Department

11. Composite Fermions (cont.)B B*
n = 2/5
n* = 2
CFs see an effective (smaller) magnetic field
FQHE of electrons  IQHE of 2mCFs
4/30/2019
FSU Physics Department

12. Composite Fermions (cont.)
FQHE of electrons  IQHE of 2mCFs
FQHE of electrons at n=p/(2mp±1)  IQHE of 2mCFs at n*=p
Jain’s series p/(2mp±1) (1/3, 2/5, 3/5, 3/7, …)
4/30/2019
FSU Physics Department

13. Composite Fermions (cont.)
n = ½ state
B* = 0 B
Halperin, Lee, and Read, PRB 47, 7312 (‘93)
CFs behave as though there is NO field at all
No QHE at this filling factor
Experiments: Fermi sea, SdH oscillation,…
4/30/2019
FSU Physics Department

14. Composite Fermions (cont.)
n = ½ state
B* = 0
dr1
B0
CFs see strongly fluctuating effective magnetic field “B”
Gauge field: a “B” =  × a
4/30/2019
FSU Physics Department

15. Double-Layer Electron System (DLES)
Layer index is introduced as an additional degree of freedom
Inter-layer correlation leads to new physics, i.e. new FQHE states…
States of system depend critically on d and tunneling
Layer spacing d and tunneling t12 (DSAS) are important parameters d
t12
4/30/2019
FSU Physics Department

16. FSU Physics Department
DLES (cont.)
Double-layer electron system at ntot=1
B = (hc/e) n d
Each layer now has n = ½
4/30/2019
FSU Physics Department

17. FSU Physics Department
DLES (cont.)
Double-layer composite fermion metal at ntot=1 d
B* = 0 RL
QHE exists at this system
4/30/2019
FSU Physics Department

18. FSU Physics Department
DLES (cont.)
a(1) & a(2)  a(+) & a(-)
B* = 0
dr1
dr2
a(1)
a(2)
In-phase gauge-field fluctuations of DLCFM at ntot=1
4/30/2019
FSU Physics Department

19. FSU Physics Department
DLES (cont.)
B* = 0
dr1
dr2
Out-of-phase fluctuation, a new feature of DLES
4/30/2019
FSU Physics Department

20. FSU Physics Department
DLES (cont.)
BCS-like pairing of CFs
B* = 0
dr1
dr2
4/30/2019
FSU Physics Department

21. FSU Physics Department
DLES (cont.)
Bonesteel et. al., PRL 77, 3009 (‘97)
Murphy et.al., PRL 72, 728 (‘94)
Keeping only a(-) at n = 1
No tunneling, large d, small q
TC and D  1/d2
Paired QH state at zero tunneling!
No QHE observed at n = 1, zero tunneling, d/l0 ≥ 2
4/30/2019
FSU Physics Department

22. FSU Physics Department
My Project
The formation of composite fermion pairing in DLES, stable or unstable?
Consider the CF pairing state using the Eliashberg equations with as few approx. as possible.
How does a(+) affect critical temperature TC and energy gap D of the pairing state?
Coupling parameter l will be carried out with contributions from full spectrum of q.
4/30/2019
FSU Physics Department

23. FSU Physics Department
My Project (cont.)
Experimental regions
4/30/2019
FSU Physics Department

24. FSU Physics Department
My Project (cont.)
The presence of both a(+) and a(-) lead to composite pairing, which may result quantum Hall effect
The pairing is suppressed very strongly (more than 3 orders) when the a(+) is taken. This may give an explanation for the absence of experimental observation
4/30/2019
FSU Physics Department

25. Thank you for your patience!
4/30/2019
FSU Physics Department