Gaussian superconducting fluctuations, thermal transport, and the Nernst effect

4 Pages·0.16 MB·English

by  Iddo Ussishkin

Gaussian superconducting fluctuations, thermal transport, and the Nernst effect
Iddo Ussishkin, S. L. Sondhi, and David A. Huse
Department of Physics, Princeton University, Princeton, NJ 08544
(Dated: April22, 2002)
Wecalculatethecontributionofsuperconductingfluctuationstothermaltransportinthenormal
3
state, at low magnetic fields. We do so in the Gaussian approximation to their critical dynamics
0
which is also the Aslamazov-Larkin approximation in the microscopics. Our results for the ther-
0
mal conductivity tensor and the transverse thermoelectric response are new. The latter compare
2
favorably with thedata of Ong and collaborators on theNernst effect in thecuprates.

n
a
J The study offluctuations insuperconductors[1]hada expectwillbethecasenearatwodimensionalKosterlitz-

1 marked revival after the discovery of the cuprate super- Thouless transition. At higher temperatures, where the

conductors. It was realized early that their short coher- vorticesandanti-vorticesarestronglyoverlapping,a dif-
]
n encelengthsproducealargeregimeofstrongfluctuation ferent simplification involving Gaussian fluctuations be-
o phenomena[2]. Morerecentlythe“pseudogap”regionof comes available and we will take recourse to that in the
c thecupratephasediagramhascomeintosharpfocus,and present paper.
-
r onelineofthoughtattributesitsfeaturestostrongsuper- Gaussian computations.—Ourcalculationscanbe car-
p
conducting fluctuations [3, 4]. Among the most striking ried out in two equivalent ways. First, they involve fol-
u
experimentalfindingsthatplausiblysupportthislinkare lowing Aslamazov and Larkin [7] and keeping the Feyn-
s
. recentmeasurementsbyOng’sgroup[5]wheretheNernst man diagrams that now bear their names and arise in a
t
a effect becomes sizeable far above T in the underdoped Gaussian treatment of the quantum functional integral.
m c
regime of the high-temperature superconductors. As the Second, as was noted, e.g., in the case of paraconductiv-
- Nernst effect is smallin ordinarymetals andlargein the ity [1], one can keep Gaussian fluctuations in a stochas-
d
vortexstateofsuperconductorstheaprioricaseforcred- tic time-dependent Ginzburg-Landau equation (TDGL)
n
o iting superconducting fluctuations is strong. that is designed to recover the equilibrium Ginzburg-
c Landau free energy. At this level, with the coefficients
In this Letter we calculate the contribution of Gaus-
[
siansuperconductingfluctuations to the transversether- of the TDGL derived from BCS microscopics near the

2 moelectric response above T in the low magnetic field mean-field Tc, the two computations are identical. The
c
v TDGL,however,hasasecondinterpretation—asamodel
limit—the simplest computation that can be used for
4
of the critical dynamics traditionally assumed for super-
a quantitative comparison with these experimental re-
8
conductors (model A) [8] which should have a wider va-
4 sults. This is already a non-trivialexercise as it requires
4 a proper subtraction of bulk magnetizationcurrents that lidity,withthecoefficientsnolongerconstrainedbyBCS
0 enter naive expressions [6]. We also give results for the microscopics. In the following we will mostly use the
2 TDGL description, while appealing to the microscopics
thermal conductivity tensor, thus presenting a full pic-
0 to justify the form of the current operators [9].
ture for Gaussian thermal transport in the system.
/
t The stochastic TDGL is defined by the Ginzburg-
a Our result for the transverse thermoelectric response
Landau free energy
m
dependsonlyonthesuperconductingcoherencelengthξ,
- making it particularly suitable for comparison with ex- b ¯h2 e∗ 2
nd periment. We present a comparison with data from dif- F = dx a|ψ|2+ 2|ψ|4+ 2m∗ ∇−i¯hcA ψ ,
o ferent samples of La2−xSrxCuO4 (LSCO). We find that Z " (cid:12)(cid:12)(cid:18) (cid:19) (cid:12)(cid:12) #(1)
c the Gaussian Nernst effect, using the actual T , is quan- (cid:12) (cid:12)
v: titatively comparableto the measuredsignalinc the opti- where a=a0(T −Tc), and the tim(cid:12)e evolution (cid:12)
i mallydopedandoverdopedsamples. Fortheunderdoped ∂ e∗ δF
X (τ +iτ′) +i φ ψ =− +ζ. (2)
sample the measuredsignalinthe “pseudogap”regionis ∂t ¯h δψ∗
r larger and requires postulating a suppression of the ac- (cid:18) (cid:19)
a
tual T from the mean-field T , consistent with the su- Equation (2) describes the decay of the order parame-
c c
perconducting fluctuations scenario, in order to achieve ter configuration towards its minimal free energy, inter-
an understanding of its magnitude, as we illustrate by a rupted by thermal fluctuations introduced through the
self-consistent Hartree computation. white noise ζ with correlator
Before proceeding, we note that the traditional de- ∗ ′ ′ ′ ′
hζ (r,t)ζ(r,t)i=2Tτδ(r−r)δ(t−t). (3)
scription of the Nernst effect is of thermally driven vor-
′
tices producing a transverse voltage via phase slips. In The relaxation time of the order parameter is τ, and τ
afluctuationregimethis descriptionis onlyconvenientif leadstotheprecessionofthe orderparameterduringthe
′
the vortices provide an effective parameterization, as we relaxation process (in this convention, τ/¯h and τ /¯h are
2
dimensionless). In the presence of particle-hole symme-
′ TABLE I: Contribution of Gaussian fluctuations to ther-
try τ =0.
moelectric and thermal transport coefficients. Here, ℓ =
To calculate the response of the system, Eq. (2) is (h¯c/eB)1/2 is the magnetic length, ξ = ¯h/(2m∗a)1/2 isBthe
to be solved in the presence of the appropriate driving Ginzburg-Landau coherence length, TΛ is a cutoff tempera-
field (electric field or temperature gradient). Note that ture, and e∗ = 2e is used. Exact coefficients are given for
−
a temperature gradient appears in the TDGL both in diverging terms only (for non-diverging contributions, singu-
the parameter a and in the noise correlator, Eq. (3). At lar behavior is indicated without specifying prefactors).
the level of Gaussian fluctuations the TDGL is a lin-
two dimensions three dimensions
ear equation, with a solution of the the form ψ(r,t) =
′
dr′dt′G(r,t;r′,t′)ζ(r′,t′), where G is the Green func- αSCa 1 e τ ln TΛ c c √T T
tRion of the TDGL. This expression may then be used to xx 2π ¯h τ (cid:16)T −Tc(cid:17) 0− 1 − c
calculate the electric and heat currents, 1 e ξ2 1 1 e ξ 1
αSC
je =−i2em∗¯h∗ ψ∗ ∇−i¯he∗cA ψ +c.c., (4) κSxCy c c6π(T¯h ℓ2BT∝)lnT −TTcΛ 12cπ ¯hcℓ2B(T∝ √TT)3−/2Tc
(cid:28) (cid:18) (cid:19) (cid:29) xx 0− 1 − c T Tc 0− 1 − c
′ (cid:16) − (cid:17)
jQ=−2¯hm2∗ (cid:28)(cid:18)∂∂t −ie¯h∗φ(cid:19)ψ∗(cid:18)∇− i¯hec∗A(cid:19)ψ(cid:29)+c.c. (5) κSxCy 41π meB∗c ¯hττ2 ln(cid:16)T T−ΛTc(cid:17) c0−c1√T −Tc
Ullah and Dorsey [10] used precisely the same model. aPreviouslyconsideredinRefs.[10,13,16].
Currents and fields.—As model A has no conservation
laws, the traditional route to identifying currents is not
leads to the same results obtained using the TDGL ap-
available. Whileitispossibletoconstructhydrodynamic
proach, Eqs. (1)–(5), where the response to a tempera-
arguments that justify the forms in (4) and (5) [11], for
ture gradient is obtained directly. Ignored here are the
ourpurposesitissufficienttoappealto themicroscopics
Maki-Thompson and density of state corrections to the
oftheAslamazov-Larkincontribution,andshowhowthe
normal state response [1, 15].
heat current arises from the appropriate vertex in the
The transport coefficients are defined by the standard
microscopic theory. [12]
linear response relations,
The microscopic theory is perhaps best recast in a
functional integralapproach. By means ofthe Hubbard- je σ α E
tr = . (7)
Stratonovich decoupling of the interaction, the expecta- jQ α˜ κ −∇T
(cid:18) tr (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
tion value of a current operator is expressedas an imag-
Theoff-diagonalthermoelectrictensorsobeytheOnsager
inary time functional integral over the pairing field,
relations α˜ = Tα. Below, we calculate the contribution
hji= R DψDDψψ¯Dhjψ¯iψeψ−¯φSeef−f(ψSe,ψf¯f(,φψ),ψ¯,φ), (6) oeκlfSeCGc)t.ariuWcsseaiandndostnuhopetecrtcohonenrsmidduaecrlthicnoegrnedflutuhccettiuvealiettyciotrntiescnatsloocrtoshne(dαtuhScCetrivmaintody-
where S (ψ,ψ¯,φ) isRthe effective action for the pairing tensor σSC, which has been studied extensively [1].
eff
field ψ, in presence of a potential φ, and hjiψψ¯φ is the Before proceeding, we note that particle-hole symme-
current of a non-interacting Fermi gas driven by elec- try (τ′ = 0) implies that σSC = αSC = κSC = 0. In
xy xx xy
tric and pairing fields. Ignoring the φ dependence of hji thefollowingwebreakparticle-holesymmetryonlywhen
is the Aslamazov-Larkinapproximation;the expectation considering these coefficients. In addition, we calculate
value is the vertex in their diagram and its long wave- the longitudinal coefficients in the absence of magnetic
length, low frequency form is the TDGL current. For field B, and the transverse coefficients to linear order in
the electric current this was done in the original calcu- B (validfor Bξ2 ≪hc/e∗). The finalresultsin bothtwo
lation [7]. Using the microscopic heat current operator, and three dimensions are presented in Table I. The gen-
one obtains a relation between heat current and electric eralization of these results to a layered superconductor
currentvertices,JQ =−ωJe/2e,independentofdisorder. using a Lawrence-Doniachmodel is straightforward.
Thiscorrectsafactorof2inthecalculationofReizerand The thermoelectric response αSC may be calculated in
xx
Sergeev [13]. The heat current in the TDGL, Eq. (5), is two ways, either as the heat current response to an elec-
now immediate by this result. tricfieldorastheelectriccurrentresponsetoatempera-
To obtain a response to a temperature gradient in ture gradient. In this way Onsagerrelations are verified.
the microscopic theory, one calculates the response to a The result has a logarithmic divergence at T in two di-
c
“gravitationalfield”,coupledtotheenergydensityinthe mensions [10, 13, 16].
Hamiltonian, and then uses an Einstein relation to ob- The calculation of the transverse thermoelectric coef-
taintheresponsetoatemperaturegradient[14]. Themi- ficient αSC raises the issue of equilibrium magnetization
xy
croscopic calculation of the Aslamazov-Larkin diagrams currents. To illustrate this point, consider the result of
3
calculating both the heat current response to an electric
field and the electric current response to a temperature
gradient (in two dimensions), (cid:3) 0:4 x = 0:12
m
jQ 1 e∗2BT 1 V T(cid:10) x = 0:17
y =− , (8) K
E 16π m∗ca T −T (cid:2) 0:3 x = 0:20
x 0 c )
n
(cid:23)
je 1 e∗2B 2T 3 (cid:0) 0:2
y
= − . (9) (cid:23)
(−∇T) 48π m∗ca (T −T )2 T −T (
x 0 (cid:18) c c(cid:19) xx
(cid:27)
Equation (8) agrees with the calculations of Ullah and 0:1
Dorsey [10]. However, the two results, Eqs. (8)–(9), give
a different answer for the transverse thermoelectric re-
sponse, and are clearly incompatible with Onsager rela- 0:2 0:4 0:6 0:8 1:0
tions. The reason for this apparent discrepancy is that (T (cid:0)T
)=T
Eqs. (4)–(5) give the total currents in the system, con-
FIG. 1: Points are σ (ν νn) for different samples of
taining contributions from both transport and magneti- xx −
La2−xSrxCuO4 [22], with x=0.12 (underdoped,Tc =29K),
zation currents. In the context of superconductivity this
x=0.17(nearoptimaldoping,T =36K),andx=0.2(over-
c
issue was raised in Refs. [17, 18], but was later ignored doped, T = 27 K). Solid line is theoretical value of αSC/B,
c xy
in works regarding fluctuation contributions [10, 19, 20]. using ξ(0) = 30 ˚A and an anisotropy of γ = 20. Dashed line
ab
HerewefollowCooper,Halperin,andRuzin[6]whocon- is obtained using a Hartree approximation (see text).
sideredthe generalproblemofmagnetizationcurrentsin
magneto-thermoelectrictransportindetail(theyencoun- diverge in any dimension, although it is singular. (In
tered it in the context of the quantum Hall effect). Ref.[21]adivergingresultwasobtainedduetothewrong
In the presence of a magnetic field, the system has formofthe heatcurrentvertex.) Thetransversethermal
magnetization currents in equilibrium. These currents conductivity κSC has a logarithmic divergence at T in
xy c
are divergence-free, and as a consequence do not make two dimensions (similar to αSC). In contrast with an
xx
any contribution to the net current flows that are mea- electric field, a temperature gradient does not give rise
suredin atransportexperiment. The totalcurrents[cal- to a magnetization heat current in the present case.
culated by Eqs. (4)–(5)] are thus a sum of transport and Cuprate data.—We now compare our results for αSC
xy
magnetization parts, with experiment. Consider the measurement of the
Nernst effect: The sample is placed in a temperature
je =je +je , jQ =jQ +jQ . (10)
tr mag tr mag gradient (−∇T) k xˆ in the presence of a magnetic field
In the present case , following the arguments of Ref. [6], Bkˆz. ThetransverseelectricfieldEy ismeasuredinthe
the magnetization electric and heat currents which con- absence of any transport electric current. Imposing the
tribute to the total currents in Eqs. (8)–(9) are condition je =0 in Eq. (7) gives Nernst signal,
tr
∂M E 1 α σ −α σ
je =c ×(−∇T), jQ =cM×E, (11) ν = y = xy xx xx xy . (12)
mag ∂T mag (−∇T)B B σ2 +σ2
xx xy
where M is the equilibrium magnetization. The fluctua-
Taking the values for σ and α from the Gaussian contri-
tioncontributionto the magnetizationis foundeither by
butions, we find that in this approximation the Nernst
thermodynamics[1]orbycalculatingthecurrentflowing
effecttendstoaconstantatT ,ν(T )=αSC/σSCB. How-
in the system in equilibrium in the presence of a varying c c xy xx
ever,forcomparisonwithexperimentthefluctuationcon-
system parameter (such as the magnetic field or T ).
c tributions, σSC and αSC, should be added to the normal
By subtracting the magnetization currents, Eq. (11),
state contribution, σn and αn. Moreover, we will con-
weobtainthe resultforαSC (seeTableI),whichdiverges
xy sider temperatures which are not too close to T , such
as the conductivity, αSC ∝σSC ∝(T −T )(d−4)/2. As ex- c
xy xx c that σSC ≪σn ≈σ. Equation (12) may then be written
pected, Onsager relations are recovered by this subtrac-
in the low magnetic field limit as
tion. We note that the correction due to magnetization
currentsis notsmall: In responseto anelectric field, the 1 αSC
magnetizationcurrentaccountsfortwothirdsofthetotal ν ≈νn+ xy, (13)
B σ
xx
heatcurrentintheweakmagneticfieldlimit. Inresponse
to a temperature gradient, the magnetization current is The normal state Nernst effect νn is generally small due
more singular than the transport current at T . to a cancellation between the two terms in Eq. (12).
c
Finally, we consider the thermal conductivity tensor Eq.(13)isnowusedforacomparisonwithexperiment
κ. The longitudinal thermal conductivity κSC does not in the low magnetic field limit, presented in Fig. 1 for
xx
4
data measured on three samples of LSCO [22]. The ex- by the self-consistent solution of a˜ = a+bh|ψ|2i. That
perimentaldatashowsσ (ν−νn),whereν andσ was thisisplausibleisdemonstratedinFig.1,wherewefitthe
xx xx
measuredateachtemperature,while νn is deducedfrom datafortheunderdopedsample(usingatwo-dimensional
an extrapolation from the high temperature regime. As Hartree approximation, with TMF = 49 K, T = 200 K,
c Λ
νn is very small in this material, this extrapolation is b/a =0.63¯h2/m∗, and ξ(0) =30 ˚A).
0 ab
expected to introduce a relatively small error. We thank TomLubensky andSteve Kivelsonfor valu-
On the theoretical side, the result for αSxCy (see Ta- ableinput,andPhuanOngandYayuWangfornumerous
ble I) is generalized for a layered superconductor using enlighteningdiscussionsontheirwork. Wealsothankthe
the Lawrence-Doniachmodel, Rueff Wormser Fund, the PackardFoundation, and NSF
DMR-9978074,9802468 and 0213706.
1 e ξ2 1
αSC = ab . (14)
xy 6π ¯hl2s 1+(2ξ /s)2
B c
Here, s is the interlayer spacping, and ξab (ξc) is the co- [1] For reviews see A. I. Larkin and A. A. Varlamov,in The
herence length in the direction parallel (perpendicular) Physics of Superconductors, Vol.1,editedbyK.-H.Ben-
to the planes. Note that the only fitting parameters are nemann and J. B. Ketterson (Springer, Berlin, 2003), p.
the coherence length ξ(0) (ξ =ξ(0) T /(T −T )) and 95; W. J. Skocpol and M. Tinkham, Rep. Prog. Phys.
ab ab ab c c 38, 1049 (1975).
the anisotropy γ = ξ /ξ . In Fig. 1 αSC/B is plotted
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using ξ(0) =30 ˚A and γ =20 (cf., e.g., Ref. [23]). B 43, 130 (1991).
ab
As this comparison suggests, Gaussian superconduct- [3] V. J. Emery and S. A. Kivelson, Nature (London) 374,
ing fluctuations are sufficient to explain the magnitude 434 (1995).
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c
tional and High-Temperature Superconductors, edited by
limit in the optimally doped and overdoped samples.
G. Iadonisi, J. R. Schrieffer, and M. L. Chiofalo (IOS
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level our results are applicable in this case as well.
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c