Thermal conductivity in a mixed state of a superconductor at low magnetic fields

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by  A. A. Golubov

Thermal conductivity in a mixed state of a superconductor at low magnetic fields
A. A. Golubov1 and A. E. Koshelev2
1 Faculty of Science and Technology and MESA+ Institute of Nanotechnology,
University of Twente, 7500 AE, Enschede, The Netherlands and
2 Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
(Dated: January 25, 2011)
Weevaluateaccuratelow-field/low-temperatureasymptoticsofthethermalconductivityperpen-
dicular to magnetic field for one-band and two-band s-wave superconductors using Keldysh-Usadel
1
formalism. We show that heat transport in this regime is limited by tunneling of quasiparticles
1
betweenadjacent vorticesacross anumberof local pointsand thereforewidely-used approximation
0
ofaveragingoverthecircularunitcellisnotvalid. Inthesingle-bandcase,weobtainparameter-free
2
analytical solution which provides theoretical lower limit for heat transport in the mixed state. In

n the two-band case, we show that the heat transport is controlled by the ratio of gaps and diffu-
a sion constants in different bands. Presence of a weaker second band strongly enhances thethermal
J
conductivityat low fields.

4
2
I. INTRODUCTION superconductors, such as borocarbides (LuNi2B2C)11,
] Sr RuO 13, C Yb14, heavy-fermion compounds CeIrIn ,
n 2 4 6 5
o The thermal conductivity of a metal in the super- CeCoIn5 and UPt313, MgB215, pnictides16,17 and iron-
c silicides18. Being very sensitive to the gap structure,
conducting state is markedly different from its value
- thermal transport at low magnetic fields varies consid-
r in the normal state. The physical reason is that the
p erably for different compounds.
Cooper pairs give no contribution to the transport of
u
heat. Therefore heat transport occurs solely due to the
s
. quasiparticle excitations controlled by the energy gap.
t
a Due to this reason, measurements of the thermal con- In this paper we reconsider more accurately the prob-
m ductivity were extensively used during last decades as a lemofthethermalconductivityacrossthemagneticfields
- tool to probe the energy gap symmetry in various new fors-wavesuperconductors. Thecaseofs-wavesupercon-
d
superconducting materials. ductor is a standard reference, all other situations are
n
Application of a magnetic field H provides a way to comparedwiththiscase. Surprisingly,anaccurateresult
o
c generatequasiparticlesina type-II superconductor. In a for the low-field asymptotics of the thermal conductiv-
[ s-wave superconductor at T T , the quasiparticles are ity was never derived. The widely-used circular unit cell
≪ c
localized near the vortex cores and thermal conduction approximation does not give correct result in low mag-
2
perpendicular to the magnetic field is due to tunneling netic fields because it misses essential physics. Namely,
v
7 between adjacent vortices. Thermal conductivity in the as the local thermal conductivity is strongly inhomoge-
1 mixed state of superconductors was extensively studied neous,theheattransportislimitedbytunnelingbetween
5 in the past experimentally1–3 and theoretically4,5. Early adjacentvorticesacrosscertainlocalpointsinthevortex
3 theoretical work4,5 addressed superconductors in diffu- lattice unit cell (bottlenecks). This leads to general low-
.
9 sive regime (dirty limit) in the mixed state with a tem- field asymptotics of the electronic thermal conductivity,
0 peraturegradientappliedtransversetothemagneticflux. κ exp( β B /B), where B is the upper critical
c2 c2
0 Theelectronicthermalconductivitywascalculatedusing fiel∝d.2 Sur−prisingly, the theoretical value of the numeri-
1 p
thequasiclassicaltime-dependentsuperconductivitythe- cal constant β is not available neither for clean nor for
:
v ory, assuming homogeneous temperature gradient, using dirtys-wavesuperconductors. Forcleancase,weprovide
i the circular-cell approximation, and averaging over the estimateforthisnumericalconstantusingasymptoticsof
X
unit cell. This approach was later extended to study the Bogolyubov wave functions of the localized states at
r
a heat transport in various unconventional superconduc- zero energy and microscopic value of the upper critical
tors. Theory for d-wave superconductors was developed field. In the dirty case we were able to perform more
in Ref. 6 for zero magnetic field and in Refs. 7 and quantitative analysis using the Keldysh-Usadel formal-
8 for the mixed state. More recently, with the dis- ism. We calculate the thermal conductance at low tem-
covery of magnesium diboride (MgB ), theory was ex- peratureandlowmagnetic fieldforsingle-andtwo-band
2
tendedtothecaseofmultibandsuperconductivity9. Dis- superconductorsinthe dirty limit. In this regime we ob-
covery of pnictides, with possibly unconventional pair- tain parameter-free analytical solution in a single-band
ing mechanism due to spin fluctuations leading to the casewhichprovidestheoreticallowerlimitforheattrans-
so-called s pairing state, motivated extension of heat port in the mixed state. We find that in dirty case the
±
transport theory10 taking into account multiband su- low-field thermal conductivity is drastically suppressed
perconductivity and resonant interband impurity scat- in comparison with clean case. Further, we generalize
tering. Many recent experimental studies reconsider old the developed formalism to a two-band superconductor,
superconductors, such as NbSe 12, and addressed novel taking MgB as an example.
2 2
2
II. TUNNELING OF QUASIPARTICLES nonequilibrium properties of dirty superconductors. Be-
BETWEEN VORTEX CORES IN CLEAN low we will reproduce the main relations of this formal-
ISOTROPIC SUPERCONDUCTOR ism needed for our derivations. Within this formalism a
superconductor is described by the Green’s function
The electronic transverse thermal conductivity in
mixedstateatlowtemperaturesandfieldsisdetermined GˆR GˆK
G= , (5)
by the probability quasiparticle tunneling between the 0 GˆA
(cid:18) (cid:19)
cores of neighboring vortices. In this section we eval-
uate this quantity with exponential accuracy for clean where in dirty limit the retarded (advanced) Green’s
isotropic superconductors. Even though this estimate is functions GR(A) satisfy the Usadel equation24
verystraightforwardandcouldbedonelongtimeago,to
our great surprise, we did not succeed to find it in the ~D (GˆR(A) GˆR(A))= iEτˆ +∆ˆ,GˆR(A) . (6)
3
literature. − ∇ ∇
h i
The Bogolyubov wave function of the localized state
Here E is quasiparticle energy, D is the electronic diffu-
in the vortex core at E =0 decays as19
sion coefficient, ∆ is the pair potential
Ψ(r) exp( r/ξ ), ξ =v /∆ (1)
∆ ∆ F
∝ − 0 ∆
where ∆ is the superconducting gapandvF is the Fermi ∆ˆ = ∆∗ 0 , (7)
velocity. Therefore,the probabilityoftunneling between (cid:18) (cid:19)
the vortex cores separated by distance a = 2Φ0/√3B GˆK = GˆRfˆ fˆGˆA, fˆ is the distribution function,
can be estimated as q GˆR =τˆ3G+τˆ1−F, GˆA = τˆ3GˆR†τˆ3, where G and F are
−
the normal and anomalous Green’s functions and τˆ are
8Φ i
P Ψ(a)2 exp( 2a/ξ )=exp 0 (2) Pauli matrices.
∆
∝| | ∝ − −s√3Bξ∆2 ! Thermal current is given by
The upper critical field B for a clean isotropic super-
c2 N D
conductor at T =0 is given by20 J = 0 ETr τˆ GˆR GˆK +GˆK GˆA dE, (8)
th 3
4 ∇ ∇
e2 Φ v Z h (cid:16) (cid:17)i
0 F
B = , ξ = (3)
c2 4γ2πξ02 0 2πTc wWhreitriengN0fˆi=s tfheˆ1n+orfmaτˆld,enwshiteyreoffstaatnedsftheaFreermodidleavnedl.
with γ = exp(0.5772)= 1.781 being the Euler constant. L T 3 L T
even in energy components of the distribution function,
Using also the BCS relation πT =γ∆, we evaluate
c one can rewrite the thermal current in the form25
2Φ 16π
0
2 = 1.486.
s√3Bc2ξ∆2 s√3γe2 ≈ Jth =N0 E[DL(E)∇fL(E)+ImJEfT]dE, (9)
Z
This allows us to represent the tunneling probability (2)
where
as
16π Bc2 Bc2 DT =D (ReG)2+(ReF)2 ,
P exp exp 1.486 . (4)
∝ −s√3γe2 B !≈ − r B ! D =Dh(ReG)2 (ImF)2i
L
−
This result also determines the low-field asymptotics of
h i
the electronic thermal conductivity of cleanisotropic su- are the energy-dependent spectral diffusion coefficients
perconductor. Theconstantinthe exponentisobviously and ImJ is the spectral supercurrent given by ImJ =
E E
sensitive to anisotropy of Fermi surface. A more quanti- 1Tr[τˆ (GˆR GˆR GˆA GˆA)]=ImFRReFR χ,whereχ
4 3 ∇ − ∇ ∇
tativeanalysiswhichwouldincludealsoevaluationofthe is the superconducting phase. In the mixed state all
preexponential factor requires much more complicated quantities are spatially inhomogeneous (coordinate de-
microscopic kinetic theory for clean limit. In the follow- pendences are dropped for brevity). Functions f and
L
ingsectionswemakequantitativecalculationsofthermal f satisfy the following kinetic equations22,23
T
transport at low fields for dirty superconductor.
(D f )+ImJ f =2Rf , (10)
T T E L T
∇ ∇ ∇
(D f )+ImJ f =0. (11)
III. THE FORMALISM: THERMAL ∇ L∇ L E∇ T
TRANSPORT IN DIRTY SUPERCONDUCTORS
WITHIN QUASICLASSICAL KELDYSH-USADEL where R = 1Tr ∆ˆ(GˆR+GˆA) . In thermal equilibrium
4
MODEL
fL = tanh(E/2khBT) and fT =i0. The thermal conduc-
tivity components are defined as
Our study is based on the quasiclassical Keldysh-
Usadel formalism21–23 which was developed to describe κ = J / T , (12)
α th,α α
h i h∇ i
3
the boundary regions between the cells, see Fig. 1. This
means that thermal transport occurs via “bottlenecks“,
saddle points of D (E,r). Our purpose is to evaluate
L
average thermal conductivity limited by these bottle-
necks. Since in the vicinity of bottlenecks, the spectral
supercurrent ImJ vanishes by symmetry, the expres-
E
sion for localenergy currentsimplifies and has the form:
J (r) N dEED f .
th 0 L L
≈ ∇
We start with evaluation of the diffusion constant
R
D (E,r) which is determined by the real part of the
L
Green’s function θ(E,r). For an isolated vortex, at dis-
tances r ξ from its core we can present ∆ and θ as
θ(r) = θ≫+ θ˜(r), ∆(r) = ∆ + ∆˜(r) where ∆ and
0 0 0
θ are the equilibrium values at zero magnetic field,
0
tanθ = i∆ /E and θ˜(r) = θ˜ (r)+iθ˜(r) is small cor-
0 0 r i
FIG. 1. Left: Graylevel map of the local thermal resistance rection. For E ∆ , θ π/2+iE/∆ , therefore the
∝ 1/DL(0,r) in the vortex lattice. Light regions correspond energy-diffusion≪cons0tant0in≈this region D0 (E,r) Dθ˜2.
to low thermal resistance. Arrows illustrate heat flow in the L ≈ r
The real and imaginary parts of θ˜ obey the following
bottleneckregions. Right: Three-dimensionalplotofthelocal
thermal resistance in the bottleneck region marked by rect- equations
angle in theleft picture. E2+∆2
D 2θ˜ D 0p2θ˜ 2 ∆2 E2θ˜ =0, (15)
∇ r− E2 ∆2 r− 0− r
− 0 q
where T istheaveragetemperaturegradientandα= E2+∆2
h∇ i D 2θ˜ D 0p2θ˜ 2 ∆2 E2θ˜
(x,y,z). ∇ i− E2 ∆2 i− 0− i
In the following, we shall use the standard θ- − 0 q
parametrization, GˆR(r) = τˆ3cos[θ(r)] + τˆ1sin[θ(r)] in = 2E∆˜ D E∆0 p2, (16)
which the Usadel equation has the following form26,27 ∆20−E2 − ∆20−E2
D 2θ p2cosθsinθ +2∆cosθ+2iEsinθ =0 (13) wherepp=1/r isthe gauge-invariantphasegradient. For
∇ − an isolated vortex θ˜ and θ˜ have qualitatively different
r i
whe(cid:0)re p(r) = φ (2π(cid:1)/Φ0)A is superconducting mo- behavior at large distances for E < ∆0: θ˜i decays as
mentum (withi∇n ci−rcular cell approximation p = 1/r p2 1/r2 and θ˜r decays exponentially
− ∝
r/r2, r = Φ /πH). The selfconsistency equation has
thesforms 0 θ˜v,r(r)=Cvexp(−kξr), (17)
p kξr
T T ∆
∆ln = sinθ (14) k (E)=√2 ∆p2 E2 1/4/√D.
Tc Tc ω (cid:18) − ω(cid:19) ξ 0−
X Evaluationofnumericalc(cid:0)onstantre(cid:1)quiressolutionoffull
The diffusion coefficients in the θ-parametrization nonlinear problem. Numerical solution by the method
are given by the expressions D (E,r) = described27 providesC 4.2. Inthevortexlatticeaway
T v
Dcosh2(Im[θ(E,r)]), D (E,r)=Dcos2(Re[θ(E,r)]). from the core regions θ˜≈can be represented as a sum of
L r
contributions from individual vortices
exp( k r R)
IV. THERMAL TRANSPORT IN THE VORTEX θ˜ (r) θ˜ (r R)= C − ξ| − | ,
lat,r v,r v
STATE AT LOW FIELDS ≈XR − XR kξ|r−R|
(18)
p
where R are the vortex coordinates. In particular, for
We consider a superconductor in low magnetic field,
triangular lattice R= (a/2+ma+na/2,n√3/2) where
B B . We assume an ideal triangular vortex lat-
c2
≪
tice of vortex lines and study thermal transport in the a= 2Φ /√3B is the lattice constant and m and n are
0
direction perpendicular to the field. The local ther-
integqers.
mal conductivity is mostly determined by the diffusion
At small field thermal transport is determined by en-
constant DL(E, r) at low energies. This quantity is ergy flow via bottlenecks, saddlepoints of D (E,r) at
L
very inhomogeneous in the vortex state. It has max-
boundaries of the lattice unit cell. Near the bottleneck
ima atthe vortexcores andrapidly drops awayfromthe
point,(x,y)=(0,0),wecankeeponlycontributionfrom
cores reflecting localization of quasiparticles in the core
twoneighboringvorticeslocatedat( a/2,0)whichgives
regions. This means that the local thermal resistance ±
2C
1/DL(0,r) is maximal at the boundaries of the lattice θ˜ (r) v exp( k a/2)cosh(k x)exp( k y2/a)
∝ lat,r ξ ξ ξ
unit cell. In such situation the temperature is mostly ≈ k a/2 − −
ξ
homogeneous within the unit cells and only changes in (19)
p
4
and Using also quasiequilibrium approximation
for the gradient of f (E,r), f (E,r)
L x L
DL(E,r)≈D8kCav2 exp(−kξa)cosh2(kξx)exp(−2kξy2/a). −hecaotsflho−w2(Ene/a2rktBhTe)b(Eot/t2leknBeTck2)J∇xT((rr)),∇weκ o(rb)tainT(trh≈)e,
ξ th,x x x
≈ − ∇
(20) where the local thermal conductivity κ (r) is given by
x
∞ exp( k a) 2k y2 E/2k T2
κ (r) 8C2DN dEE − ξ cosh2(k x)exp ξ B .
x ≈ v 0Z−∞ kξa ξ (cid:18)− a (cid:19)cosh2(E/2kBT)
At low temperatures T < ∆ ξ/a the main contribution vortex-lattice state
0
totheenergyintegralcomesfromtheregionE .T. This
10√2π5/2C2 exp( k a)
allows us to neglectthe energy dependence of kξ(E) and κ/T ≈ 3√3 vkB2N0D −k ξa0
replace k (E) k = 2∆ /D = 1/ξ . In this case, ξ0
ξ ξ0 0 ∆
using −∞∞x2c→osh−2x dxp=π2/6, we obtain 10√2π5/2C2 expp− (8π/√3)Bc2/B
R κ (r)= 16π2C2k2TN Dexp(−kξ0a) = 3√3 vkB2N0D (cid:18)(8πq/√3)Bc2/B 1/4 (cid:19)
x 3 v B 0 k a (26)
ξ0 (cid:0) (cid:1)
2k y2
cosh2(kξ0x)exp ξ0 . (21) where we used relations Bc2 = Φ0kξ20/4π and kξ0a =
× − a
(cid:18) (cid:19) (8π/√3)B /B.
c2
Nearthebottleneckregionthelocalthermalconductivity qIntroducing the thermal conductivity in the normal
has a form κ(x,y) = κ F (x)F (y), where the function state κ = π2k2N DT, we rewrite the final result in
0 x y N 3 B 0
F (x)=cosh2(k x)hasminimumatx=0andF (y)= the form
x ξ0 y
exp 2k y2/a has maximum at y =0. The total flow
ξ0
pgievre(cid:0)nu−nbiyt length(cid:1)along the field through the bottleneck is κ 10 2πC2exp(cid:18)−q(8π/√3)Bc2/B(cid:19)
κN ≈ r 3 v (8π/√3)Bc2/B 1/4
∞
1/4
Ith ≈κ0Fx(x) Fy(y)dy ∇xT. (22) 130 B (cid:0)exp 3.81 B(cid:1)c2 . (27)
(cid:20)Z−∞ (cid:21) ≈ (cid:18)Bc2(cid:19) − r B !
Energy conservation requires that I has to be x- Thisparameter-freeanalyticalresultprovidestheoretical
th
independent. Therefore,thetemperaturedrop∆T across lowerlimitfortheheattransportinthemixedstateinan
the bottleneck can be evaluated as isotropic dirty s-wave superconductor at low field. The
constant3.81inthe exponentissignificantlyhigherthan
∞ F−1(x)dx the constant 1.486which we evaluatedfor the clean case
∆T −∞∞x Ith (23) (4)meaningthatthescatteringdrasticallysuppressesthe
≈ κ F (y)dy
R0 −∞ y quasiparticle thermal conductivity at low fields.
R
meaningthatthetotalthermalconductancethroughthis
region K =Ith/∆T can be evaluated as V. DISCUSSION OF EXPERIMENT
∞
K =κ −∞Fy(y)dy = 8π5/2Cv2k2TN Dexp(−kξ0a). Ratherlimited setofexperimentaldatais availableon
0 R−∞∞Fx−1(x)dx 3√2 B 0 akξ0 etelemctpreornaitcucroenstarnibdultoiownmtoagthneettihcefirmeldasl,cosinndcuecpthivointyonatcloonw-
(24)
R p tributionshouldbe accuratelysubtractedin this regime.
Evaluating the total average energy flow density
To our knowledge, the only experimental data on elec-
tronic thermal conductivity which clearly demonstrate
I +I /4 5 ∆T
J = th th = K , (25) low-field/low-temperature behavior expected for an s-
th
a√3/2 2√3 a wave superconductor are reported for Nb in Refs. 1 and
2 and for V Si in Ref. 12. In these experiments the
3
we obtain the final result for the low-field/low- samples were in the clean limit. In Ref. 2 the elec-
temperature limit for the thermal conductivity in the tronic thermal conductivity was fitted by the expression
5
κ exp( β B /B)forvaluesofB uptoaboutB /3, In the presence of two electronic bands with different
c2 c2
∝ −
with β = 1.66. This number is slightly higher than our energygaps,low-temperaturebehaviorofthermaltrans-
p
estimate of β =1.486 in Eq. (4). This difference can be portisdeterminedbythebandwithlowergap(π-band).
explained by the influence of impurity scattering. The Still, the generalization from single-band to two-band
shape of field dependence of thermal conductivity for case involves not simply renormalization of the energy
V SireportedinRef. 12isinqualitativeagreementwith gap, but also correction to the asymptotic behavior of
3
thepredictedexponentialdependence,howeverthequan- Green’sfunctioninthe π -bandandtothe upper critical
titativeanalysiswasnotmadeandthevalueofβ wasnot field.
explicitly extracted. In a field B = B /20 the value of The Usadel equation for the π-band reads
c2
κ 2.5 10−3κ provided in Ref. 12 exceeds our dirty-
N
lim≈it est·imate, Eq. (27), by about two orders of mag- Dπ 2θπ p2cosθπsinθπ +2∆πcosθπ+2iEsinθπ=0,
∇ −
nitude. This discrepancy can be naturally attributed to (28)
(cid:0) (cid:1)
the fact that measured V3Si samples were in the clean where Dπ and ∆π(r) are the diffusion constant and gap
limit. The magnitude of thermal conductivity at not for the π-band. Similar to a single-band case, asymp-
very small magnetic fields is in good qualitative agree- totics of the Green’s function in the π-band at large dis-
ment with calculations made for the clean limit using tance from the vortex core is given by
the Landau-level expansion and assuming homogeneous
temperature gradient28. To our knowledge, there are no θ˜ (r)=C exp(−kπr). (29)
π,r π
dataavailableonelectronicthermalconductivityofdirty √kπr
s-wavesuperconductorsinthelow-field/low-temperature
with
limit. In available measurements of alloys1 and dirty Nb
samples3 the thermalconductivityatlowfieldsandtem- k (E)=√2 ∆2 E2 1/4/ D .
peratures is dominated by phonons and separating elec- π π0− π
tronic contribution is a challenging task. The main exponential(cid:0)dependenc(cid:1)e ofpthe zero-energy
Green’sfunctionatthebottleneckpointis exp( k a),
π0
∝ −
where k =k (0)= 2∆ /D , which gives
π0 π π0 π
VI. EXTENSION TO A TWO-BAND k pa= B /B, (30)
π0 π
SUPERCONDUCTOR
2∆ 2Φ
p π0 0
B = . (31)
π
Dπ √3
Here we extend the above formalism to a two-band
superconductor. At present, the most established exam- Therefore,themagneticfielddependenceofthermalcon-
ple of such system is MgB2, which is characterized by ductivity is determined by the field scale Bπ and can be
two electronic bands: π-band and σ-band, see, e. g. re- presented in the form similar to Eq. (27),
cent review Ref. 29. The quasi-two-dimensional σ-band
is characterized by stronger superconductivity than the exp B /B
κ 2π π
three-dimensionalπ-band. Heattransportinatwo-band 10 C2 − , (32)
superconductorwasstudiedtheoreticallyinRef.9assum- κπN ≈ r 3 π ((cid:16)Bπ/pB)1/4 (cid:17)
ing clean limit conditions and using the averaging over
unit cell method4,5. In the dirty limit, theory of density where κ = π2k2N D T is the partial π-band contri-
πN 3 B π π
of states in the mixed state and the upper critical field bution to the normal-state thermal conductivity.
for MgB was developed in Refs. 30 and 31. Below we To proceed further, we have to find relation between
2
extend the calculations of the heat transport presented the π-bandfieldscaleB andthe upper criticalfieldB
π c2
above, to the case of a diffusive two-band superconduc- for a two-band superconductor. The upper critical field
tor, taking MgB as an example. at low temperatures, T T , is given by31
2 c
≪
W +W lnr (W +W lnr )2
B (0)=a Bs (0), a =exp 1 2− x + 1 2− x +W lnr , (33)
c2 c2 c2 c2 − 2 s 4 1 x
 
where indices 1 and 2 correspond to the σ and π bands, Λ is the coupling-constant matrix, r = / ,
αβ x σ π σ,π
D D D
are the diffusion constants in σ and π bands,
(Λ Λ )/2+ (Λ Λ )2/4+Λ Λ ∆ Φ
W = ∓ 11− 22 11− 22 12 21, Bs (0)= BCS 0 (34)
1,2 Λ Λ Λ Λ c2 2π
11 p22− 12 21 Dσ
6
isthe single-banduppercriticalfieldforthe σ-band,and and important fingerprint of the two-band superconduc-
∆ = πe−γET 1.764T (γ 0.5772 is the Euler tivity in MgB . This small scale was experimentally ob-
BCS c c E 2
≈ ≈
constant). served not only in the thermal conductivity15, but also
To make estimates for MgB , we use the follow- in the specific heat32 and flux-flow resistivity33. For the
2
ing coupling matrix elements30,31: Λ 0.81, Λ magneticfield appliedalongc-axisitis 3-5times smaller
11 22
≈ ≈
0.278, Λ 0.115, Λ 0.091, which givesW 0.088 than B . In this case, the low-field regime described by
12 21 1 c2
≈ ≈ ≈
and W 2.56. With such coupling matrix the two- Eq.(32)isexpectedatfields<B /30 100G.Unfortu-
2 c2
band BC≈S model gives30 ∆ 0.3∆ 0.177πT . nately, most experimental data of Ref.≈15 are presented
π0 σ0 c
≃ ≃
Since the parameter W is small, typically the inequal- for higher magnetic fields.
1
ity W lnr (W lnr )2/4 is valid. In this case We shall note that in available MgB single crystals
1 x 2 x 2
| | ≪ −
one can expand Eq. (33) with respect to W and obtain estimates suggest that the σ band is in the clean limit.
1
simple result However, our results should be qualitatively applicable
even in this case. The reasonis that dominant contribu-
W1lnrx tion comes from the π-band which is in the dirty limit,
a 1+ , (35)
c2
≈ W2 lnrx as argued in Ref. 30 where the low-energy DoS in the
−
vortex state of MgB was calculated.
meaning that the upper critical field is close to Bs (0) 2
c2 In summary, we have readdressed the problem of the
and is mostly determined by the coherence length of the
heat transport of a superconductor in the mixed state
σ-band. Using Eqs. (31), (33), and (34), we obtain the
at low temperatures and low magnetic fields, going be-
relation between B and B (0)
π c2 yond the circular unit cell approximation. In the clean
limit we estimated the numerical constant β in the low-
8π ∆
B = π0 DσB (0), (36) field asymptotics of the electronic thermal conductiv-
π c2
√3ac2∆BCS Dπ ity, κ exp( β Bc2/B), using the Bogolyubov wave
∝ −
functions of the localized states at zero energy. In the
whichpresentsthe mainresultofthis section. Thisscale p
dirty limit we have performed quantitative analysis of
hastobecomparedwiththescale(8π/√3)B (0)forthe
c2
heattransportusingKeldysh-Usadelformalismandhave
single-band case.
shown that heat transport is limited by tunneling be-
In order to determine the pre-exponential factor C
π
tween adjacent vortices across certain local points (bot-
in Eqs. (29) and (32), we have to calculate the Green’s
tlenecks). Intheisotropics-wavesuperconductorwehave
function θ˜ (r), which requires solution of the full two-
π,r obtained parameter-free analytical solution which pro-
band Usadel problem, as described in Ref. 30. An im-
vides theoretical lower limit for heat transport in the
portant parameter is the ratio of diffusion coefficients
mixed state. Based on this solution, one can conclude
in two bands r . For illustration, we consider two
x that low-field/low-temperature thermal conductivity in
cases here: r = 1 and 0.2, for which the ratios of
x the mixed state is drastically suppressed by impurity
the coherence lengths in the two bands are ξ /ξ =
π σ scattering. Wehaveextendedourresultsto the caseofa
(D /D )(∆ /∆ ) = 1.83 and 4.1. For these two
π σ σ0 π0 two-band superconductor, taking MgB as an example.
2
pcases we compute Bπ ≈ 0.32(8π/√3)Bc2(0), Cπ ≈ 3.6 Inthiscase,wepredictanenhancementofheattransport
for rx = 1 and Bπ 0.065(8π/√3)Bc2(0), Cπ 2.9 for withstrongdependenceontheratioofgapsanddiffusion
≈ ≈
rx =0.2. This gives constants in different bands.
1/4
κ B B (0)
c2
130 exp 2.15 , r =1
κπN ≈ (cid:18)Bc2(0)(cid:19) − r B ! x ACKNOWLEDGMENTS
1/4
B Bc2(0) The authors would like to acknowledge useful dis-
130 exp 0.98 , r =0.2
x
≈ (cid:18)Bc2(0)(cid:19) − r B ! cussions with N.B.Kopnin, V.M.Vinokur, and I.I.Mazin.
A.E.K. is supported by UChicago Argonne,LLC, opera-
We can see that the field scale in the two-band case is tor of Argonne National Laboratory, a U.S. Department
stronglyreducedincomparisonwiththesingle-bandcase of Energy Office of Science laboratory, operated under
leading to large enhancement of the thermal conductiv- contract No. DE-AC02-06CH11357. This work also was
ity at low fields. This reduction is mostly caused by the supported by the “Center for Emergent Superconduc-
smaller energy gap in the π band. Another factor which tivity”, an Energy Frontier Research Center funded by
may contribute is possible large value of the diffusion the U.S. Department of Energy, Office of Science, Of-
constantD . Thesmallerfieldscalecausedbythelarger fice of Basic Energy Sciences under Award Number DE-
π
coherence length in the π-band is an established feature AC0298CH1088.
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