Fluctuation conductivity in disordered superconducting films:
Transverse Transport
/
the Hall and Nernst Effects
Usadel equation for fluctuation corrections
Alexander Finkel‘stein
Fluctuation Conductivity in Disordered Superconducting Films:
Konstantin Tikhonov (KT) TA&MU
Karen Michaeli (KM) Pappalardo Fellow at MIT
and Georg Schwiete (GS) FU Berlin
“Fluctuation Hall conductivity in Superconducting Films”
N. P. Breznay, KM, KT, AF, and Aharon Kapitulnik
submitted
” The Hall Effect in Superconducting Films”
KM, KT, and AF
PRB accepted, arXiv 12036121
“Fluctuation Conductivity in Disordered Superconducting Films”
KT,GS, and AF PRB 85, 174527 2012
Outlook for two parts of the talk ( I,
II
):
I:
Effect of fluctuations is more pronounced for the
transverse components of the transport (e.g., the Hall and Nernst effects)
as compared to the longitudinal components:
II:
 j      E 
    ~


 jE      T 
We developed an approach to the calculation of fluctuation conductivity in the
framework of the Usadel equation. The approach has clear technical advantages
compared to the standard diagrammatic techniques.
We generalized results for fluctuation corrections to arbitrary (B,T) and
compared various asymptotic regions with previous studies.
The approach has also been applied to the calculation of Hall conductivity (and also
checked by comparison with the diagrammatic calculation).
We hope that the formalism proves useful for studies of fluctuations out-of3
equilibrium and in superconductor-normal metal hybrid systems.
The Nernst Coefficient
The Nernst signal

Y. Wang, et al 2005
Ey
xT  B
 j      E 
    ~


 jE      T 
 xy xx   xx xy
eN 

 xT
 xy2   xx2
Ey
twice off-diagonal effect / usually “twice“ small /
this appeared not true for the superconducting fluctuations
Under the approximation of the constant density of states:
2
v
  xT
jey  2e 2 C  F
d T
d k
f 0  k 

 2 d 0  k  k  0
For a non-constant density of states  c
T
F
example of “twice” smallness
This fact makes the Nernst effect very favorable for studying
fluctuations
a-la para-conductivity (e.g., Aslamazov-Larkin).
There is no Drude terms to compete with !
are superconducting fluctuations
Nernst Effect – Conventional
Superconductors
The strong Nernst signal above Tc cannot be
explained by the vortex-like fluctuations.
The Nernst signal
Nb0.15Si0.85
A. Pourret, et al 2007
the fluctuations of the order parameter cause the effect.

Ey
xT  B
Why the Nernst Signal Created by the Superconducting
Fluctuations is so strong, even stronger than in the Hall effect?
 j      E 
 j        T 


 h 
c
 c

F
T
T
c  c 
4eDH
c
twice off-diagonal effect /
usually “twice“ small /
not true for the discussed problem
no need for “particle-hole” asymmetry
in the fluctuation propagator to get the
transverse thermo-electric coefficient xy
(unlike xx or xy,
which are only “once” transverse )
“Particle-Hole” asymmetry:
LR ()  LA ()
Agreement with the experiment
(no fitting parameters; TC and diffusion coefficient
were taken from independent measurements)
Karen Michaeli & AF
αxy
Experimental data from A. Pourret, et al 2007
Nb0.15Si0.85 film of thickness
35nm
TC  380mK
2
cm
D  0.187
sec
“Fluctuations of the superconducting order parameter as an origin of the Nernst Effect”
EPL, 86 (2009);
Phys Rev B 80 (2009)
“Quantum kinetic approach for studying thermal transport in the presence of electronelectron interactions and disorder” Phys Rev B 80 (2009)
Serbin et al. Phys. Rev. Lett. 2009
8
the Hall Signal Created by the Superconducting Fluctuations
N. P. Breznay et al. submitted
9
Fluctuation corrections to conductivity due to SC
fluctuations: phenomenology
Shortcomings
Advantage: physical transperancy
10
The Hall effect very close to Tc;
result that can be obtained by the phenomenological approach
A. Aronov, S. Hikami, and A.Larkin (1995)
LR ()  LA ()
11
KM, KT, and AF submitted , arXiv 12036121
12
the Hall Signal Created by the Superconducting Fluctuations
KM, KT, and AF PRB accepted, arXiv 12036121
Two types of the contributions
depending on the mechanism of deflection in the transverse direction:
quasiparticles or superconducting modes
The standard set of the diagrams
(but in the case of Hall, lot of cancellations!)
plus the overlooked one,
which is a reminiscent of the DOS correction to the Hall conductivity.
flux technique (M. Khodas and A.F. 2003)
13
B-T Phase Diagram
14
B-T Phase Diagram
T
B-induced QCP
ordered
QCP
r
15
Transverse transport in the vicinity of the critical points;
there are regions where Hall correction does not depend on
C
Hall effect

e 
1
C 

ln
T
/
T
C
 
2
C
signH   T
e2
1

ln  H / H C T  
e2
1
ln
ln  H / H C 
4eHD
C 
c
16
2
The Nernst Coefficient
 xy xx   xx xy
eN 

2
2
 xT
 xy
  xx
Ey
 j  
 j   
 h 
  E 
   T 
 xy
eN 
 xx
αxx contributes negligible
in comparison to αxy
The Peltier coefficient is related
to the flow of entropy
c 
4eDH
c
According to the third
law of thermodynamics
 0
when
T 0
17
The Peltier Coefficient
near the quantum critical point
C  T
 H  T
ln 
 
H
T
 C    C
H
ln
 1
HC
e ln 3
 xy  
signH
2 ln  H / HC (T ) 
Since the transverse
signal is non-dissipative
the sign of the effect is
not fixed.
Transverse transport in the vicinity of the critical point is very peculiar
18
Fit of the data obtained by the Kapitulnik
group
N. P. Breznay, KM, KT,
AF, and Aharon Kapitulnik,
submitted
19
Usadel equation: the bridge between phenomenology
and diagrammatics
(Eilenberger 1968; Usadel 1970)
Start with action with electron-electron interaction in the Cooper channel decoupled via
D (Hubbard-Stratonovich transformation):
Single particle Hamiltonian:
where
There is a separation of scales:
Low energy physics in the diffusive limit is contained in the reduced function
20
Usadel equation: cont.
One can write closed (nonlinear) equation for the reduced g:
Current density can also be expressed in terms of g:
Averaging with respect to:
with
Closed scheme
Gaussian approximation
21
Usadel equation: solution
In the regime of Gaussian fluctuations, the solution of the Usadel equation can be
found by a perturbative expansion around the metallic solution:
with
Fermi distribution
scalar potential
GL action can be written as follows
22
For B=0 a similar formalism was developed by Volkov et al (1998)
and more recently by Kamenev and Levchenko (2007)
Three mechanisms of the corrections
d is the correction to the quasiparticle
density of states as would be measured by a
tunneling probe
dD is the renormalization of the diffusion
coefficient due to coherent Andreev scattering
js is the supercurrent density
f, f* etc. parametrize deviations of g from the metallic solution, f~CD
23
Fluctuation corrections to conductivity due to
superconducting fluctuations
Kubo formula
Disorder dressing
Both fermionic and
bosonic degrees present
B-T Phase Diagram for the longitudinal transport
III Asymptotic results for fluctuation conductivity
- contact with known limiting cases
I
II
IV
kOm
Resistance curves for different temperatures
II
“criticality”
zoomed image
Magnetotransport starting in the region of the QCP
and for large magnetic fields
The quantum critical regime
There are two distinct regimes:
Low temperature:
Sign change!
Classical regime:
We recover the result obtained by Galitski, Larkin (2001)
[In contrast to more recent study by Glatz, Varlamov, Vinokur (2011)]
Fluctuation conductivity in superconducting films
Effect of fluctuations is more pronounced for the
transverse components of the transport as compared to the longitudinal components:
Here we demonstrate a theoretical fit of the
recent data obtained by the A. Kapitulnik group
(Stanford) for the Hall conductivity in
superconducting Tantalum Nitride (TaNx)
films.* A large contribution to the Hall
conductivity near the superconducting transition
arising due to the fluctuations has been tracked
to temperatures well above Tc=2.75K and
magnetic fields well above the upper critical
field, Hc2. Quantitative agreement has been
found between the data and the calculations
based on the microscopic analysis of the
superconducting fluctuations in the disordered
films.
*Studying fluctuation effects in the Hall
conductivity is an experimental challenge in
systems with high carrier concentration and
large longitudinal resistance.
N. P. Breznay et. al
submitted Phys. Rev. B
Conclusion
We developed an approach to the calculation of fluctuation conductivity in the
framework of the Usadel equation. The approach has clear technical advantages
compared to diagrammatic techniques.
Calculation can be performed in the scalar gauge rather than with the tmedependent vector potential (no analytical cntinuation is needed).
We generalized results for fluctuation corrections to arbitrary (B,T) and compared
various asymptotic regions with previous studies
(where asymptotics are calculated separately).
The approach has also been applied to the calculation of the Hall conductivity.
The approach provides a more transparent physical structure.
We hope that the formalism proves useful for studies of fluctuations out-ofequilibrium and in superconductor-normal metal hybrid systems.
29
Magnetoresistance
Almost vertical 
Intersection point
0.35 K
0.76 K
Line of maxima in
magnetoresistance
TiN-film, Tc~0.6 K
Baturina et al. (2003)
Our fit of the data obtained by the Kapitulnik
group
N. P. Breznay et al. 2012
The quantum critical regime
There are two distinct regimes:
Low temperature:
Sign change!
Classical regime:
We recover the result obtained by Galitski, Larkin (2001)
[In contrast to more recent study by Glatz, Varlamov, Vinokur (2011)]