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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 - PowerPoint PPT Presentation
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Transverse Transport Transverse Transport / /
the Hall and Nernst Effectsthe Hall and Nernst Effects
Usadel equation for fluctuation correctionsUsadel equation for fluctuation corrections
Alexander Finkel‘stein
Fluctuation conductivity in disordered superconducting films: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
Fluctuation Conductivity in Disordered Superconducting Films:Fluctuation Conductivity in Disordered Superconducting Films:
Outlook for two parts of the talk ( Outlook for two parts of the talk ( I, II ):):
3
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: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-of-equilibrium and in superconductor-normal metal hybrid systems.
T
E
j
j
E
~
2 2
y xy xx xx xyN
x xy xx
Ee
T
The Nernst CoefficientThe Nernst Coefficient
T
E
j
j
E
~
twice off-diagonal effect / usually “twice“ small /
this appeared not true for the superconducting fluctuations
Y. Wang, et al 2005
The Nernst signal
BT
E
x
y
Under the approximation of the constant density of Under the approximation of the constant density of states:states:
0
22 0
0
22
kk
kk
fd
T
T
d
vej d
xFC
ye
cF
T
For a non-constant density of states 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 Nernst Effect – Conventional
Superconductors Superconductors
A. Pourret, et al 2007
The strong Nernst signal above Tc cannot be explained by the vortex-like fluctuations.
The Nernst signal
BT
E
x
y
the fluctuations of the order parameter cause the effect.
85.015.0 SiNb
Why the Nernst Signal Created by the Superconducting Why the Nernst Signal Created by the Superconducting Fluctuations is so strong, even stronger than in the Hall Fluctuations is so strong, even stronger than in the Hall
effect?effect?
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:
cc
F
T
T
h
j E
j T
4c c
eDH
c
( ) ( )R AL L twice off-diagonal effect /
usually “twice“ small /
not true for the discussed problem
85.015.0 SiNb
sec187.02cmD
mKTC 380
Experimental data from A. Pourret, et al 2007
film of thickness nm35
α xy
Agreement with the experiment (no fitting parameters; TC and diffusion coefficient were taken from independent measurements)
“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 electron-electron interactions and disorder” Phys Rev B 80 (2009)
Serbin et al. Phys. Rev. Lett. 2009
Karen Michaeli & AF
8
N. P. Breznay et al. submitted
the Hall Signal Created by the Superconducting the Hall Signal Created by the Superconducting Fluctuations Fluctuations
9
10
Fluctuation corrections to conductivity due to SC fluctuations: phenomenology
Advantage: physical transperancy Shortcomings
The Hall effect very close to Tc; result that can be obtained by the phenomenological
approach( ) ( )R AL L A. Aronov, S. Hikami, A. Aronov, S. Hikami, and A.Larkin
(1995)
11
KM, KT, and AF submitted , arXiv 12036121
12
KM, KT, and AF PRB accepted, arXiv 12036121
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.
the Hall Signal Created by the Superconducting the Hall Signal Created by the Superconducting Fluctuations Fluctuations
Two types of the contributions depending on the mechanism of deflection in the transverse direction:quasiparticles or superconducting modes
flux technique (M. Khodas and A.F. 2003)
13
B-T Phase DiagramB-T Phase Diagram
14
T
r
ordered
QCP
B-induced QCP
B-T Phase DiagramB-T Phase Diagram
15
Hall effect
2 1
lnln /C
C
e
H H
22 1
ln /CC
e
T T
2 1
ln / C
esignH T
H H T
CTransverse transport in the vicinity of the critical points; there are regions where Hall correction does not depend on
4C
eHD
c
16
2 2
y xy xx xx xyN
x xy xx
Ee
T
The Nernst CoefficientThe Nernst Coefficient
αxx contributes negligible in comparison to αxy
xx
xyNe
h
j E
j T
The Peltier coefficient is related to the flow of entropy
According to the third law of thermodynamics
0 when 0T
c
eDHc
4
17
The Peltier Coefficient The Peltier Coefficient near the quantum critical pointnear the quantum critical point
C T ln 1C
H
H
ln
C C
H T
H T
ln 3
2 ln / ( )xyC
esignH
H H 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
20
Usadel equation: the bridge between phenomenology and diagrammatics
(Eilenberger 1968; Usadel 1970)
Single particle Hamiltonian:
Start with action with electron-electron interaction in the Cooper channel decoupled via (Hubbard-Stratonovich transformation):
where
There is a separation of scales:
Low energy physics in the diffusive limit is contained in the reduced function
21
Usadel equation: cont.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
22
Usadel equation: solutionUsadel 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
GL action can be written as follows
Fermi distribution scalar potentialscalar potential
23
Three mechanisms of the correctionsThree mechanisms of the corrections
is the correction to the quasiparticle density of states as would be measured by a tunneling probe
D 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~C
For B=0 a similar formalism was developed by Volkov et al (1998) and more recently by Kamenev and Levchenko (2007)
Fluctuation corrections to conductivity due to Fluctuation corrections to conductivity due to superconducting fluctuations superconducting fluctuations
Kubo formula
Disorder dressing
Both fermionic and
bosonic degrees present
B-T Phase Diagram for the longitudinal transportB-T Phase Diagram for the longitudinal transport
Asymptotic results for fluctuation conductivity- contact with known limiting cases
II
IIII
IIIIII
IVIV
““criticality” criticality” zoomed imagezoomed image
Magnetotransport starting in the region of the QCP Magnetotransport starting in the region of the QCP and for large magnetic fieldsand for large magnetic fields
Resistance curves for different temperatures Resistance curves for different temperatures
kOmkOm
IIII
The quantum critical regimeThe quantum critical regime
There are two distinct regimes:
Low temperature:
Classical regime: Sign change!
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
ConclusionConclusion
29
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 tme-dependent 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-of-equilibrium and in superconductor-normal metal hybrid systems.
Magnetoresistance
Baturina et al. (2003)
TiN-film, Tc~0.6 K
0.35 K
0.76 K
Line of maxima in magnetoresistance
Almost vertical Intersection point
N. P. Breznay et al. 2012
Our fit of the data obtained by the Kapitulnik group
The quantum critical regimeThe quantum critical regime
There are two distinct regimes:
Low temperature:
Classical regime: Sign change!
We recover the result obtained by Galitski, Larkin (2001) [In contrast to more recent study by Glatz, Varlamov, Vinokur (2011)]