Lilach Goren - Physics@Technion

zx
Enhancement of the Superconducting
Transition Temperature
in Cuprate Heterostructures
Lilach Goren
Ehud Altman
PRB 79, 174509 (2009)

temperature
Experiment
R (!)
R (!)
15
10
5
MI SC
I (nA)
1
0
-1
4.2 K
-50 0 50
Bias (mV)
x = 0.35
doping (x)
(a)
0
0 75 150
STO
225 300
O. Yuli et al., PRL 101, 057005 (2008)
(c) bare
(d)
A. 10 Gozar
x =
et
0.18
al., Nature 455, 782 (2008)
x = 0.12
bilayer
5
20 K
20 K
90 nm x
40
20
0
0 20 40 60
10
(b)
x = 0.10
21K
bare
bare
bilayer
28K
bilayer
can Tc be increased by coupling to a metallic layer?
10 nm x=0.35
90 nm x
21 K
T c (K)
30
20
10
0
(a)
bilayer
underdoped
0.05 0.10 0.15 0.20 0.25
Sr doping (x)
z
40
20
0
2
(b)
0.05
FIG. 2: (a) Tc vs. x of the bilayers (op
films (solid symbols) measured in this w
films grown on LaSrAlO4 (open symbol

Tc of underdoped cuprates
∆
Tc
Campuzano et al. (1999)
ergy scale on carrier density:
the spectra (T 15 K) at the
. The inset shows Tc vs doping.
, and the peak and hump binding
g state along with their ratio (c),
he empirical relation between Tc
1 2 82.6x 2 0.162 ergy scale on carrier density:
the spectra (T 15 K) [25] at with the
. squares The inset represent showslower Tc vs bounds. doping.
, and the peak and hump binding
g state along with their ratio (c),
•
are similar (Fig. 3c), unlike the dispersi
Intensity (arb.units) Intensity (arb.units)
(a)
O72K
(a)
Tc ∝ ρs(T U83K = 0)
U55K
• small stiffness O72K
ρs(T = 0) U83K ∼ x
0.16
0.08
0
U55K Binding Energy (eV)
Mode Energy (eV) Mode Energy (eV)
0.04
0.05 0.03
0.04 0.02
0.03 0.01
(b)
z
0.020
35 50 65
i l l
T ( K )
T (K)
0.01
(b)
for a YBCO film at optimal doping and four
FIG. 5. Doping dependence of the mode
stages of deoxygenation. Tc and A -2 (0) decrease
at p,0 showing as oxygen the is removed. decrease Irregularities in near theTc energ in s
the underdoped stages are the result 0 of thermal s
peak 0.16 and dip 0.08with
phase fluctuations underdoping. 0 and sample inhomogeneity. 35 Peak 50 and 65 p
t
obtainedBinding by independent Energy (eV) polynomial fits anU
m
ation and sample inhomogeneity effects become F
for the effects significant. of energy Our model resolution. reproduces these (b) when Do
A0(0 ) _~ 300K is fixed and Too ~- F 2/5 in the
FIG. the collective 5. Doping mode dependence energy inferred of thefrom mode d
experimental range, 0.4 < F < 1. The gap ra- Ac
tio, at that p,0 inferred showingAo(O)/kBTco, increases with underdoping,
from neutron the decrease data (for in the theenerg latted
but the gap ratio defined from the maximum
a
peak compiled and in dipRef. with gap on the [5], underdoping. contributing FS segments, Bi2212 results Peak Am~ = of Refs ands
AO (0) cos ( ~ (1 - F ) ) , increases only slightly.
r
obtained by independent The model provides polynomial a basis for interpretation fits of an
7(T) [5]. Figure 2 shows the normalized specific 3 R
for the effects of energy resolution. (b) Do
Tc is suppressed due to low carrier density
5 6 2 B.R. Boyce et al./Physica C 341
ρs
25
20
10
Boyce et al. (2000)
optimaliy doped
. . . . . . . . . I . . . . . . . . . i . . . . . . . . . r . . . . . . . . . i . . . . . . . . .
under doped
20 40 60 80
Figure 1. The inverse squared penetration depth
100
U
F
a
d

Tc of underdoped cuprates
∆
Tc
Campuzano et al. (1999)
ergy scale on carrier density:
the spectra (T 15 K) at the
. The inset shows Tc vs doping.
, and the peak and hump binding
g state along with their ratio (c),
he empirical relation between Tc
1 2 82.6x 2 0.162 ergy scale on carrier density:
the spectra ky (T 15 K) [25] at with the
. squares The inset represent showslower Tc vs bounds. doping. kx
, and the peak and hump binding
g state along with their ratio (c),
are similar (Fig. 3c), unlike the dispersi
Intensity (arb.units) Intensity (arb.units)
(a)
O72K
(a)
Tc ∝ ρs(T U83K = 0)
U55K
• small stiffness O72K
ρs(T = 0) U83K ∼ x
• Low energy quasiparticles of d-wave SC: Ek
•
0.16
0.08
0
U55K Binding Energy (eV)
Mode Energy (eV) Mode Energy (eV)
0.04
0.05 0.03
0.04 0.02
0.03 0.01
(b)
z
0.020
35 50 65
i l l
T ( K )
T (K)
0.01
(b)
FIG. 5. Doping dependence of the mode
at p,0 showing the decrease in the energ
0
peak 0.16 and v dip 0.08with
underdoping. 0
35 Peak 50 and 65
obtainedBinding by independent Energy (eV) polynomial fits anU
for the effects of energy resolution. (b) Do
FIG. the collective 5. Doping mode dependence energy inferred of thefrom modeA
at that p,0 inferred showing from neutron the decrease data (for in the theenerg latte
peak compiled and in dipRef. with[5], underdoping. Bi2212 results Peak of Refs and
obtained by independent polynomial fits an
3
for the effects of energy resolution. (b) Do
2 f (k − kn) 2 || + v2 ∆ (k − kn) 2 for a YBCO film at optimal doping and four
stages of deoxygenation. Tc and A -2 (0) decrease
as oxygen is removed. Irregularities near Tc in s
the underdoped stages are the result of thermal s
phase fluctuations and sample inhomogeneity. p
⊥
t
m
ation and sample inhomogeneity effects become F
significant. Our model reproduces these when
A0(0 ) _~ 300K is fixed 2 lnand 2 Too
vf
~- F 2/5 in the
d
ρs(T experimental )=ρ0 range, − 0.4 < F α2 < 1. The T
gap rac
tio, Ao(O)/kBTco, increases π with v∆underdoping,
d
but the gap ratio defined from the maximum
a
gap on the contributing FS segments, Am~ =
s
AO (0) cos ( ~ (1 - F ) ) , increases only slightly.
The model provides Lee a basis & Wen for interpretation (1997)
r
of
7(T) [5]. Figure 2 shows the normalized specific
R
Tc is suppressed due to low carrier density
5 6 2 B.R. Boyce et al./Physica C 341
ρs
25
20
10
Boyce et al. (2000)
optimaliy doped
. . . . . . . . . I . . . . . . . . . i . . . . . . . . . r . . . . . . . . . i . . . . . . . . .
under doped
20 40 60 80
Figure 1. The inverse squared penetration depth
100
U
F
a
d

How can we increase Tc?
T=0:
underdoped cuprates
large pairing gap
small stiffness
ρs(T = 0) ∝ x
S. Kivelson, Physica B 11, 61 (2002)
E. Berg D. Orgad and S. Kivelson, PRB 78, 094509 (2008)
T>0:
underdoped cuprates
ρs(T )=ρ0 − B T
ky
kx
metal
proximity to the metal enhances the SF stiffness
v∆
no gap
large carrier density
z
4

How can we increase Tc?
T=0:
underdoped cuprates
large pairing gap
small stiffness
ρs(T = 0) ∝ x
S. Kivelson, Physica B 11, 61 (2002)
E. Berg D. Orgad and S. Kivelson, PRB 78, 094509 (2008)
metal
no gap
large carrier density
T>0: underdoped cuprates
bilayer
ρs(T )=ρ0 − B T
proximity to the metal enhances the SF stiffness
v∆
ky
the proximity gap is smaller than the gap is Tc higher?
kx
ρs(T ) = ˜ρ0 − ˜ B T
z
˜v∆
4

temperature
Low energy effective model
AF
hk =
SC
doping (x)
Nambu-Gorkov Hamiltonian for the quasiparticles
in the bilayer:
⎛
⎞
ξ (1)
k ∆k ˜t⊥(x) 0
⎜ ∆k −ξ
⎜
⎝
(1)
−k 0 −˜t⊥(x)
˜t⊥(x) 0 ξ (2)
k 0
0 −˜t⊥(x) 0 −ξ (2)
⎟
⎠
−k
L. Goren and E. Altman, PRB 79, 174509 (2009)
ky
kx
z
5

temperature
Low energy effective model
AF
hk =
SC
doping (x)
Nambu-Gorkov Hamiltonian for the quasiparticles
in the bilayer:
⎛
⎞
h eff
2 =
ξ (1)
k ∆k ˜t⊥(x) 0
⎜ ∆k −ξ
⎜
⎝
(1)
−k 0 −˜t⊥(x)
˜t⊥(x) 0 ξ (2)
k 0
0 −˜t⊥(x) 0 −ξ (2)
⎟
⎠
−k
Effective low energy Hamiltonian of the metallic layer:
⎛
k −
2 ˜t⊥ ξ Ek
(1)
k
2 ˜t⊥
∆k
Ek
⎜
⎝ ξ(2)
−ξ (2)
k +
L. Goren and E. Altman, PRB 79, 174509 (2009)
2 ˜t⊥
∆k
Ek
2 ˜t⊥ ξ Ek
(1)
k
⎞
⎟
⎠
ky
˜v∆ v∆
2xt 2 ⊥
δE 2
kx
fermi surface
mismatch
z
5

temperature
Low energy effective model
AF
SC
doping (x)
ρ1 = ρ1(0) −
ρ2 = ρ2(0) −
2 ln 2 vf
α2
π v∆
L. Goren and E. Altman, PRB 79, 174509 (2009)
+
2 ln 2
π
˜α2 ˜vf
˜v∆
T
T − c vf
v∆
T
! s
ky
500
400
300
200
100
0
0 10 20
T
30 40
˜v∆ v∆
2xt 2 ⊥
δE 2
kx
fermi surface
mismatch
z
5

Results - phenomenology
We take parameters from experiment with bulk pure material
•
•
•
•
ρ1(T = 0)
dρ1
dT
∆
vF
• δk (fermi surface mismatch)
L. Goren and E. Altman, PRB 79, 174509 (2009)
with good FS matching, Tc can be enhanced for t⊥>t/5
R (!)
R (!)
15
10
5
I (nA)
1
0
-1
4.2 K
-50 0 50
Bias (mV)
(a)
x = 0.35 STO
0
0 75 150 225 300
10
5
90 nm x
(c) bare
x = 0.18
bilayer
20 K
20 K
10 nm x=0.35
90 nm x
STO
0
0 20 40 60
40
20
0
0 20 40 60
10
[K]
100
(b)
x = 0.10
21K
(d)
x = 0.12
21 K
80
60
40
20
bare
bare
bilayer
28K
bilayer
32 K
0
0 20 40 60
Δ/2
T c (K)
30
20
10
0
t ⊥ =t/4
0
0 0.05 0.1 0.15 0.2
doping x (x)
(a)
t ⊥ =t/5
0.05 0.10 0.15 0.20 0.25
Sr doping (x)
z
0
T
c
Student Version of MATLAB
40
20
0
(b)
0.05 0.10
Sr d
FIG. 2: (a) Tc vs. x of the bilayers (open sym
films (solid symbols) measured in this work.
films grown on LaSrAlO4 (open symbols), an
symbols), as compiled from Refs. 9 and 15.
depicts the 6
Tc of bulk LSCO.

Results - phenomenology
We take parameters from experiment with bulk pure material
•
•
•
•
ρ1(T = 0)
dρ1
dT
∆
vF
• δk (fermi surface mismatch)
We also carried out a
microscopic slave boson
mean field theory calculation:
L. Goren and E. Altman, PRB 79, 174509 (2009)
with good FS matching, Tc can be enhanced for t⊥>t/5
max
R (!)
T c /T c,0
R (!)
1.5
15
10
5
-1
1
0
4.2 K
-50 0 50
t ⊥ =0.5t
x = 0.35
0
0.5 t =0.3t
⊥
10
5
I (nA)
1
0
Bias (mV)
0 75 150 225 300
t ⊥ =0.1t
(a)
90 nm x
STO
(c) bare
x = 0.18
bilayer
20 K
40
20
0
0 20 40 60
10
(b)
x = 0.10
21K
(d)
x = 0.12
bare
0 0.1 0.2
10 nm x=0.35
x
20 K
90 nm x
STO
0
0 20 40 60
[K]
100
21 K
80
60
40
20
bare
bilayer
28K
bilayer
32 K
0
0 20 40 60
Δ/2
T c (K)
30
20
10
0
t ⊥ =t/4
0
0 0.05 0.1 0.15 0.2
doping x (x)
(a)
t ⊥ =t/5
0.05 0.10 0.15 0.20 0.25
Sr doping (x)
z
0
T
c
Student Version of MATLAB
40
20
0
(b)
0.05 0.10
Sr d
FIG. 2: (a) Tc vs. x of the bilayers (open sym
films (solid symbols) measured in this work.
films grown on LaSrAlO4 (open symbols), an
symbols), as compiled from Refs. 9 and 15.
depicts the 6
Tc of bulk LSCO.

Zero temperature paramagnetism
expect:
for two underdoped layers
with doping x ~ y:
δρpara(0) ∝− xy(α1 − α2) 2
δk 2 (x, y)
δρpara =
ρs(T = 0) = n1
m ∗ 1
1
J1σ0
2
1
J1σ0
1
t 2 ⊥ kF
vF
+ n2
m ∗ 2
J2σ0
! / *01!,
)
"'(
"
&'(
. " 1!'&.
. " 1!').
. " 1!'(.
! " # $ %
*+ !+ ,
& " " -. "
2
measuring the T=0 stiffness reduction can reveal the doping dependence of the QP charge
1
+ 2
2 + 2×
2
J2σ0
1
J1σ0
2
J2σ0
1 2
2
z
7

Summary
• d-wave symmetry of the gap can be crucial in determining Tc enhancement
• Low energy effective theory of the bilayer explains enhancement of Tc but requires
rather too large interlayer tunneling
• Can effectively large interlayer coupling be achieved due to inhomogeneous interface?
(Work in progress)
MI SC
z
8