with Inherent Gap

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15 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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Pairing with Unconventional

Pair
-
symmetry in 2D Systems


with Inherent Gap

Renyuan Liao

Khandker Quader


Department of Physics

Kent State University, Ohio, USA

Condensed Matter Theory Workshop
-
29, Kyoto, Japan, 2005

Motivation


Evidence for superconductivity close to “
inherent gap
” in the
normal state: e.g.
“pseudogap” in high
-
Tc cuprates



Long
-
standing interest in
superconductor
-
insulator transition
(SIT)
and
semiconductor
-
superconductor transition



* Recent experiments on disordered
indium oxide film



show field
-
tuned
superconductor
-
insulator transition


(
Ref:
M. Steiner and A. Kapitulnik,cond
-
mat/0406227
)



* Recent experiments on very
thin film of YBCO


show
Kosterlitz
-
Thouless fluctuatations.


(
Ref: Y. Zuev, M. S. Kim and T. R. Lemberger, cond
-
mat/0410135
)


k

k+w

w

Exciton (
E
B
)

Excitonic Insulator

Δ
o

Ref: Jerome,Rice,Kohn ‘67

For
| E
B
| >
Δ
o

: Excitonic Insulator









I
k
w
k
*
k
*
k
exc
|
a
b
v
u
(
hole

electron

BCS
-
like, but no ODLRO !

Pairing in Semiconductors

ε
(k)

k

-
k

k

k

-
k

Δ
o

-

Δ
o

Refs: Kohmoto, Takada ’90:
inter
-
band


s
-
wave


Pistolesi, Nozieres ’99:
intra
-
band


s
-
wave

v
o
v
k
m
k
2
2





c
o
c
k
m
k
2
2






Intra
-
band pair :

















0
|
)
a
a
v
u
(
)
b
b
v
u
(
k
k
k
k
k
k
k
k
k
ra
int


Inter
-
band pair :

















0
|
)
b
a
v
u
(
)
b
a
v
u
(
k
k
k
k
k
k
k
k
k
er
int
Pairing

|
Δ
pair
| >
Δ
e
-
h

-

| V |

-

| V |

This Work


Explore
unconventional

(d
-

, p
-

, s*) pairing



in 2D systems with
“inherent” gap




Prototype: semiconductor gap: k,
ω

-
independent



Study:
--

Mean
-
field (BCS) pairing &


Phase fluctuations (Kosterlitz
-
Thouless)


--

undoped & doped cases


--

2D continuum & 2D lattice


--

zero and finite temperature


--

compare w. s
-
wave results


(Pistolesi & Nozieres; Kohmoto & Takada)




Undoped Semiconductor

Δ
o

-

Δ
o

μ

w
m

-

w
m

k

-
k

Const
)
(



2D DOS :

)
cos(
)
cos(
U
V
k
k






2
2
Intra
-
band Pairing Interaction :

)
cos(
k




2
d
-
wave Gap Function:

c
,
v
o
c
,
v
k
m
k
2
2





Particle Energies:

Δ
o
= 0 :
Usual BCS sc gap eqn for metals w. some U

U


Δ
m

Superconducting gap for the metal

Undoped Semiconductor



Non
-
zero
Δ
o

; T = 0 mean
-
field gap equation:




























m
m
)
(
Cos
)
(
Cos
d
d
U
)
(
Cos
)
(
Cos
d
d
U
m
P
2
2
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
0


Non
-
zero
Δ
o

; Finite
-
T mean
-
field gap equation:

































m
m
)
(
Cos
)
(
Cos
d
d
U
)
)
(
cos
tanh(
)
(
Cos
)
(
Cos
d
d
U
m
P
2
2
2
2
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
0
2
2





o
qp
--

quasiparticle gap in presence of
Δ
o


: superconducting gap in presence of semiconductor gap
Δ
o

)
,
(
f
m




0
-

w
m

w
m

μ

Δ
o

-

Δ
o

]
,
;
,
[
P
m
m











0
0
0
0
--

integration path

Undoped Semiconductor : Results

-

w
m

w
m

μ

Δ
o

-

Δ
o



Superconductivity for :

412
0
4
2
1
.
)
ln
(
exp
m
o






---

d
-
wave

2
1



m
o
---

s
-
wave

Superconductor
-
insulator transition (SIT) at
Δ
*
o



As
Δ
o


Δ
*
o ;
T
C

→ 0

“sharp”
Δ
*
o

| Gain in interaction energy | < cost in KE in producing free carriers across gap


superconductivity is lost

0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.2
0.4
0.6
0.8
1.0





C



and T
c



0
/

M
T
KT
C
/

M


(T=0)/

M
T
C
/

M
d wave

|
No doping

Results
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.2
0.4
0.6
0.8
1.0

(T)/

M
T/

M


=1


=0.9


=0.1


=0.01


=0.001
d wave |
No doping
BCS limit

=(









close to SIT
At SIT: Tc=0, and





SIT: Superconducting - Insulating Transition
elongated plateau

?
)
(



0
Doped Semiconductor

μ

= 0

μ

<
Δ
o

μ

=
Δ
o

+
ε

δ

= 2
ρε

μ


d
-
wave Pairing Interaction






2
2
cos
cos
U
V




cos
k
Δ
o

Interplay of
ε
,
Δ
o

and


)
,
,
(
f
o





;
)
,
,
(
f
M
o





2
2







)
(
o
qp
o






qp
o



































m
m
)
(
Cos
)
(
Cos
d
d
U
)
(
E
tanh
)
(
E
)
(
Cos
d
d
U
m
P
2
2
2
2
2
2
2
2
2
2
2
0
2
2
0

Self consistent equation for order parameter
Δ
:


Conservation of particle number:

































m
m
d
)
(
E
)]
(
E
[
f
)
(
d
d
F
P
2
2
2
2
2
0
)
(
cos
)
(
)
(
E








2
2
2
2
)
e
/(
)
E
(
f
E
F
1
1



Doped System

μ


doping



Tra
n
sition at
Δ
o=
Δ
o*

: Not “sharp” as in undoped case



“Crossover”




Superconductor ↔ semi
-
insulator crossover at
Δ
o=
Δ
o*


w. large shift in
μ




Superconductivity for any value of
Δ
o ; low
-
TC sc for
Δ
o >
Δ
o*




On superconducting side (
Δ
o
<
Δ
o* ) : doping a small perturbation




Non
-
BCS

features in pairing for
Δ
o >
Δ
o* :



Doped System











)
T
(
);
T
(
T
)
T
(
C
qp
0
0
0
μ




BE
-
type pairs around crossover

0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0




















d wave |
doping
|T=0




corresponds to different doping levels
zero doping
crossover
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.1
0.2
0.3
0.4
0.5
Critical temperature T
C
as a function of

0


for different value of



C









=1



=0.1



=0.01



=0.001



=0
d wave |
doping
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0



0


















d wave |
Doping
|T=0
Chemical potential(

)
for different value of



as a function of




0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.5
1.0
1.5
2.0

c




















d
-
wave

C
T
)
(


0
Non
-
BCS
-
like:
Δ

o
>
Δ
o
*


)
(
T
C
0


Non
-
BCS
-
like:
Δ

o
>
Δ
o
*


?



0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.2
0.4
0.6
0.8
d wave | T=0








/


Phase diagram: d-wave pairing case
Critical point

*
0
Insulating phase
Semicondutor
SC
SM
Crossover region

μ
=
Δ
o
curve

ε

drives system away from sc
-
insulator critical point

d
-
wave

s
-
wave

Critical point

insulating phase


Kosterlitz
-
Thouless phase fluctuations



Compare mean
-
field with




x
-
y model:




KE of pairs with COM Q/2










2
2
2
)
x
(
x
d
J
H


J
T
KT
c
2




2
2
2
/
)
(
)
Q
(
/
Q
k
/
Q
k
k
k



























v
k
c
k
k
c
k
v
m
Q
u
m
Q
)
(
K
)
Q
(
K
H
2
2
2
2
4
4
0
BCS
C
T
KT
C
T



2
|
|

Bogoliubov Coefficients

k
k
,
u

0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.2
0.4
0.6
0.8
1.0





C



and T
c



0
/

M
T
KT
C
/

M


(T=0)/

M
T
C
/

M
d wave

|
No doping

Results
SUMMARY/COMMENTS



Superconductor
-
insulator transition in undoped systems at critical inherent gap




Doping drives away from critical point
→ sc
-
semi insulator crossover; sc persists




Non
-
BCS like features beyond crossover; quasiparticles less efficient in destroying


superconductivity compared to BCS case




Phase fluctuations responsible for pair
-
breaking;




Above qualitative features not as sensitive to superconducting order parameter


symmetry, or presence of a lattice; some important differences in details:



--

superconductivity vanishes for smaller critical inherent gap for OP w. nodes;


harder to pair w. non
-
s
-
wave symmetry


--

smaller Tc’s in d
-
wave case


--

for d
-
wave, need considerably larger doping to reach




o



BCS
C
KT
C
T
T



k,
ω


dependent “inherent” gap consideration




Understanding physical origin of “inherent” gaps is specific systems




Feedback of superconductivity on the “inherent” gap; coupled problem




Effect of including inter
-
band pairing interactions as well




Other “beyond mean
-
field” considerations

COMMENTS