On themomagnetic effects due to the supeconducting fluctuations: Reply to arXiv:1012.4361 by Serbyn, Skvortsov, and Varlamov

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by  Andrei Sergeev

On themomagnetic effects due to the supeconducting fluctuations:
Reply to arXiv:1012.4361 by Serbyn, Skvortsov, and Varlamov
∗
A. Sergeev
Research Foundation, University at Buffalo, Buffalo, New York 14260
M. Yu. Reizer
5614 Naiche Rd. Columbus, Ohio 43213
1
1 V. Mitin
0 Electrical Engineering Department, University at Buffalo, Buffalo, New York 14260
2
In a recent Reply [1] to our Comment [2], Serbyn, Skvortsov, and Varlamov raised a question of
n
microscopic description, which we did not touch in [2], and criticized our work [3]. They hopefully
a
agreed with one of our key result [3] that the effective heat current vertex for fluctuating Cooper
J
pairs (the Aslamzov-Larkin block in the diagram technique) is modified in the magnetic field, so
1
”theheatcurrentisproportionaltothegauge-invariantmomentum”[1]. However,theystatedthat
2
in [3] we have overlooked thesame correction to theelectric current vertex and in thisway we lost

theirhugethermomagneticeffectthatdoesnotrequireanyparticle-holeasymmetryand,therefore,
]
n prevails over the ordinary thermomagnetic effects by at least five orders in magnitude in ordinary
o superconductorsnearTc and strongly dominates in thetemperature rangeup to∼100Tc. Herewe
c address theircriticism with all details and show that ourcalculations in [3] are correct.
-
r
p PACSnumbers:
u
s
. In our Comment [2] on ”Giant Nernst Effect due to by the magnetic field,
t
a Fluctuating Cooper Pairs in Superconductors” [4], we
m highlightedthatthemagnetizationcurrentsdonottrans- Bh = ω B q+ 2eAH , (1)
- fer the heat. Therefore, the amendment of the heat 2e (cid:18) c (cid:19)
d current by ”circular magnetization heat current, jQ =
n M where (ω,q) are the energy and momentum of Cooper
c(M×E),” (M is the magnetization) that was used in
o pairs,near the transitionB is some constant,and AH is
c [4] is wrong. In fact, the term (M×E) is the magne-
the vector potential of the magnetic field.
[ tization part of the Poynting vector [2]. Without such
Our opponent agree that we ”correctly obtain that the
correction the results of [4] contradict the third law of
1 heat current is proportional to the gauge-invariant mo-
thermodynamics. Besides this, we also stressed [2] that
v mentum” [1] (see Eq. 2 in [1]).
6 theGaussianmodelisfully applicabletoordinarysuper-
Atthesametime,ouropponentsstatedthatwe”failto
8 conductors, for which [4] predicts the fluctuation correc-
1 tion to β to be at least ǫ /T ∼ 105 times bigger than β include AH in the electric vertex and draw the diagrams,
F extracting AH from the propagators and from the heat
4 for noninteractingelectrons. Moreover,far above T this
. −2 c vertex only” [1]. Obviously, they assumed the electric
1 huge effect was predicted to decrease as T and, there-
current vertex in the magnetic field has a form
0 fore, it would dominate in the wide temperature range
1 up to ∼ 100Tc, i.e. up to the room temperatures [5]. 2e
1 Certainly, such huge effects are not known for ordinary Be =B q+ AH . (2)
: (cid:18) c (cid:19)
v superconductors(Nb,Al,Sn,andetc),whichjustslightly
Xi above Tc show the same values and temperature depen- They claimed that we did not take into account the sec-
denciesasnonsuperconductingmetals. Itisalsowellun- ond term in this equation.
r
a derstood that large thermomagnetic effects are observed Here we clarify our calculations. Below we will show
in materials with small Fermi energy, i.e. with the large that in the gauge we used in [3] this term does not con-
particle-hole asymmetry [6,7]. tribute to the thermomagnetic coefficient. In a general
case,thistermsgivesagaugeinvariantexpressionforthe
correlator of electric and heat currents.
In their Reply [1], Serbyn, Skvortsov, and Varlamov
Let us present electric and magnetic fields as
did not address our generalobjections [2] to their Letter
[4] and instead criticized our microscopic calculations in Ω
E=i AE H=i[k×AH], (3)
the previous paper [3].
c
then the thermalcurrentin the thermomagneticeffect is
The key result of our work [3] is that the heat current proportional to
vertexforfluctuatingCooperpairs(theAslamzov-Larkin
block in the diagram technique, see Fig. 1) is modified E×H=ΩAH(k·AE)−Ωk(AE ·AH). (4)
2
thepropagatorsofCooperpairs,heatcurrentvertex,and
FIG.1: FluctuationALdiagramsdescribingtheheatcurrent-
electric current vertex. The corresponding diagrams are
electric current correlator in crossed electric and magnetic
presentedinFig. 1. Inthe diagram(A), AH isextracted
fields. Wavy lines stand for the fluctuation propagators and
from the propagators,in the diagram (B) it is extracted
straight lines stand for the electron Green functions, which
fromthe heatcurrentvertex(Eg. 6),andinthe diagram
form the AL blocks.
(C) it is extracted from the electric current vertex (Eq.
7).
Our opponents accused us in overlooking of the dia-
2eB(A(cid:1)H(cid:215) q(cid:1)) gram (C) [1]:
c ”The relevant diagrams for thermoelectric response
contain three sources of the vector potential/magnetic-
=2ceB(AxHqx+AyHqy) field dependence: (A) Green functions/propagators, (B)
heat current vertex, and (C) electric current vertex. The
w resulting expression is gauge invariant only if all these
2eBqy (A) Bqx three sources are taken into account in a consistent fash-
ion and within a specific gauge. In Ref. [3] it is claimed
thatthecontribution(B)cancelsthecontribution(A)cal-
culatedbyUssishkininRef. [8]. Howevertheconsistency
betweencalculationsandgaugechoices inthetwopartsof
the same physical quantity is not discussed. Most impor-
tantly, contribution(C) is notevenmentionedbySergeev
et al., which makes their conclusion erroneous.”
(B) We have a simple answer to this criticism. In [3] we
w
c BAyH Bqx used the gauge
AH =(0,−AH,0), k=(−k,0,0). (8)
Obviously,in this gauge the diagram(C) gives zero con-
tribution, because AH = 0. This gauge is widely used
x
for calculations of the Nernst and Hall coefficients, as in
this case the second term in Eq. 5 is zero and [E×H]
y
is equal to −ΩAH(k ·AE) (see Eq. 5).
y x x
We can choose another gauge with
w (C) AH =(AH,0,0), k=(0,−k,0). (9)
2eBqy 2ceBAxH
In this case the diagram (B) gives zero contribution, be-
causeAH =0. However,itiseasytoseethatnowthedi-
y
agram(C)givesexactlythesamecontributionsasthedi-
agram(B)inthepreviousgauge. Nowtheterm[E×H]
y
i s g i v e n b y Ωk (AE ·AH) (see Eq. 5).
y x x
As usually,letusputE alongx-axisandHalongz-axis, Obviously,in a generalcase the diagrams (B) and (C)
then E×H will be in the negative direction of y-axis. providethermoelectriceffect,whichisproportionaltothe
Then Eq. 4 can be presented as gauge invariant expression for [E×H] given by Eq. 5.
Inconclusion,weexplainwithalldetailsthatin[3]we
[ E × H ] = − Ω A H ( k · A E ) + Ω k ( AE ·AH). (5)
y y x x y x x correctly ignore the diagram (C), because in the gauge

we used in [3] the diagram (C) is zero. In any other
TofindthethermomagneticcoefficientusingtheKubo

gaugethis diagramgivesnonzerocontribution. The sum
method, one should calculate the correlator of the heat
ofthecontributionsofthediagrams(B)and(C)givesthe
currentdirectedalongy andthe electriccurrentdirected
gauge-invariant term, which cancels the contribution of
along x,
the diagram (A) in zero order in the particle-hole asym-
ω 2e ω ω metry (PHA). The nonzero thermomagnetic coefficient
Bh = B q+ AH = Bq + BAH (6)
y 2e (cid:18) c (cid:19) 2e y c y arises only in the second order in PHA [3].
y
Thus, we confirm that in the Fermi liquid with the
2e 2e
Be = B q+ AH =Bq + BAH. (7) particle-holeexcitations,thethermomagneticcoefficients
x (cid:18) c (cid:19) x c x arealwaysproportionaltothesquareoftheparticle-hole
x
asymmetry. Therefore, huge thermomagnetic effects ob-
Weagreewith[1],thatcalculatingthermomagneticco- served in high-T cuprates can be associated with the
c
efficient of fluctuating pairs one should extract AH from larger particle-hole asymmetry due to the Fermi surface
3
reconstruction or due to a non-Fermi liquid state, such We are grateful M.N. Serbyn, M.A. Skvortsov, and
as the vortex liquid. A.A. Varlamov for detailed discussion of our work [3].
∗ Electronic address: [email protected]ffalo.edu in Superconductors, Master Theses (2009),
1 M.N. Serbyn, M.A. Skvortsov, A.A. Varlamov, http://chair.itp.ac.ru/biblio/masters/2009/serbin diplom 2009.pdf
arXiv:1012.4361. 6 K.Behnia, M.A.Measson, Y.Kopelevich,Phys.Rev.Lett.
2 A. Sergeev, M.Yu. Reizer, and V. Mitin, arXiv:0906.2389, 98, 076603 (2007).
accepted toPhys. Rev.Lett. 7 J. Chang, R. Daou, Cyril Proust et al., Phys. Rev. Lett.
3 A. Sergeev, M.Yu. Reizer, and V. Mitin, Phys. Rev. B 77, 104, 057005 (2010).
064501 (2008). 8 I. Ussishkin et al., Phys. Rev. Lett. 89, 287001 (2002); I.
4 M.N.Serbyn,M.A.Skvortsov,A.A.Varlamov,andV.Gal- Ussishkin,Phys. Rev.B 68, 024517 (2003).
itski, Phys. Rev.Lett. 102, 067001 (2009).
5 M.N. Serbyn, Fluctuation Nernst Effect