Pairing instability of composite
fermions in double layer
quantum Hall system
Huan D. Tran
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Quantum Hall effect
Composite fermion
FQHE in double layer electron system
Proposal work
6/29/2016
FSU Physics Department
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Quantum Hall Effect
 Ex    xx
 
 E  
 y   yx
 xy  jx 
 
 yy  j y 
VH
W
RH 
  xy
  xy
I
W
VL
L
RL 
  xx
I
W
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Quantum Hall Effect (cont.)
2
RH 
n e2
 Strong B
n  B 1
 Low T
B
RH 
en
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IQHE: n = 1, 2, … (‘80)
FQHE: n =1/3, 2/5, … (‘82)
(more than 50)
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Quantum Hall Effect (cont.)
Störmer, RMP 71, 875 (‘99)
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Quantum Hall Effect (cont.)
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IQHE, physics of non-interacting electron system


Ek  k  12 C
C 
eB
2
2
; l0 
me
eB
Quantized cyclotron
orbits (Landau levels)
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Be
N 
2
n
2n
Be
n: number of filled Landau levels
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Quantum Hall Effect (cont.)
B1
RH
 0 
2
e
n=1
B2
n=2
B
n: number of electrons per flux quantum
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Quantum Hall Effect (cont.)
• Disorder  localized &
extended states
• Localized & extended
states  plateaus & steps
Laughlin, PRB 23, 5632 (‘81)
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Quantum Hall Effect (cont.)
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FQHE observed at non-integer filling
Tsui et.al., PRL 48, 1559 (‘82)
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Non-interacting electrons  No energy gap!
Electron interaction must be considered
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Quantum Hall Effect (cont.)
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Laughlin wavefunction, theory of filling factors 1/(2p+1)
Laughlin, PRL 50, 1395 (‘83)
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The filling factors 1/3, 1/5, 1/7, … are explained
Remaining filling factors 2/5, 3/7, 4/7, … need further
treatment
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Composite Fermions
Jain, PRL 63, 199 (‘89)
Y  Y
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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
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Composite Fermions (cont.)
B
B*
n = 2/5
•
•
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n* = 2
CFs see an effective (smaller) magnetic field
FQHE of electrons  IQHE of 2mCFs
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Composite Fermions (cont.)
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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, …)
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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,…
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Composite Fermions (cont.)
n = ½ state
B* = 0
B0
d1
• CFs see strongly fluctuating effective magnetic field “B”
• Gauge field: a “B” =  × a
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Double-Layer Electron System (DLES)
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t12
d
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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
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DLES (cont.)
Double-layer electron system at ntot=1
B = (hc/e) n
d
Each layer now has n = ½
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DLES (cont.)
Double-layer composite fermion metal at ntot=1
B* = 0
d
RL
QHE exists at this system
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DLES (cont.)
a(1) & a(2)  a(+) & a(-)
B* = 0
d1
a(1)
d2
a() 


a ()


1 (1)
a  a ( 2)
2
1 (1)

a  a ( 2)
2
a(2)
In-phase gauge-field fluctuations of DLCFM at ntot=1
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DLES (cont.)
B* = 0
d1
d2
a() 


a ()


1 (1)
a  a ( 2)
2
1 (1)

a  a ( 2)
2
Out-of-phase fluctuation, a new feature of DLES
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DLES (cont.)
BCS-like pairing of CFs
B* = 0
d1
d2
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DLES (cont.)
Bonesteel et. al., PRL 77, 3009 (‘97)
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Keeping only a(-) at n = 1
No tunneling, large d, small q
TC and D  1/d2
Paired QH state at zero tunneling!
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Murphy et.al., PRL 72, 728 (‘94)
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No QHE observed at n = 1, zero
tunneling, d/l0 ≥ 2
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My Project
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6/29/2016
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.
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My Project (cont.)
Experimental regions
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My Project (cont.)
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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
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Thank you for your patience!
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