Table Of ContentMagnetic-Field Dependences of Thermodynamic Quantities
in the Vortex State of Type-II Superconductors
Koichi Watanabe and Takafumi Kita
Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
5
0 Masao Arai
0 National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan
2 (Dated: February 2, 2008)
n We develop an alternative method tosolve theEilenberger equations numerically for the vortex-
a lattice states of type-II superconductors. Using it, we clarify the magnetic-field and impurity-
J
concentration dependences of the magnetization, the entropy, the Pauli paramagnetism, and the
7 mixingofhigher Landaulevels in thepairpotential for two-dimensional s-and dx2−y2-wavesuper-
1 conductorswith the cylindrical Fermi surface.
] PACSnumbers: 74.25.Op,71.18.+y
n
o
c
- I. INTRODUCTION the method may be more advantageous for the cal-
r culations of thermodynamic quantities in finite mag-
p
netic fields when realistic Fermi surfaces are used as in-
u Recent experiments1,2,3,4,5 have shown that magnetic-
s puts. It will also be convenient for microscopically cal-
. field dependences of thermodynamic quantities in the culating responses of the vortex-lattice state to external
t
a vortex state of type-II superconductors provide unique perturbations,32 such as vortex-lattice oscillations.
m informationonthepairingsymmetryandgapanisotropy.
Thismethodisappliedheretocalculatemagnetic-field
On the theoretical side, however, calculations of those
- dependencesofthemagnetization,theentropy,thePauli
d quantities still remain a fairly difficult task to perform.
paramagnetism,andthepairpotentialatvarioustemper-
n ThequasiclassicalequationsderivedbyEilenberger6pro-
o vide a convenientstartingpoint forthis purpose. Pesch7 aturesforthetwo-dimensionals-anddx2−y2-wavesuper-
c conductorsinthecleananddirtylimits. Thesequantities
obtained a compact analytic solution to them based on
[ have been obtained near H for the s-wave pairing.33,34
the lowest Landau-level approximation for the pair po- c2
Ourpurposehereistoclarifytheoverallfielddependence
1
tential. It has been used recently to discuss the field
v of those quantities.
dependences of the thermal conductivity,8 the density
5 Thispaperisorganizedasfollows. SectionII givesthe
of states,9,10,11 and thermodynamic quantities.12 On the
0 formulation. Section III presents numerical results. Sec-
4 other hand, Klein13,14 obtained a full numerical solution
tionIVsummarizesthepaper. Weusek =1throughout.
1 for the vortex-lattice state using a standard procedure B
0 to solve ordinary differential equations. This numeri-
5 cal approach has been used extensively by Ichioka et
0 al.15,16,17,18,19,20,21,22,23 to clarify the field dependences II. FORMULATION
/
t of the pair potential and the density of states for type-II
a
superconductors with various energy-gap structures. It A. Eilenberger equations
m
should be noted, however, that those numerical studies
-
d all adopted simplified model Fermi surfaces instead of WetaketheexternalmagneticfieldHalongthez axis
n complicated Fermi surfaces for real materials. Indeed, and express the vector potential as6,27,35,36,37,38
o recenttheoreticalstudies10,24,25,26 haveclarifiedthat de-
c tailed Fermi-surface structures are indispensable for the A(r)=Bxyˆ+A˜(r). (1)
:
v quantitative description of the vortex state.
Xi With these backgrounds, we here develop an alterna- HereBistheaveragefluxdensityproducedjointlybythe
tive method to solve the Eilenberger equations for the external current and the internal supercurrent, and A˜ is
r
a vortex-lattice states. A key point lies in expanding the the spatiallyvaryingpartofthemagneticfieldsatisfying
pairpotentialandthequasiclassicalf functionintheba- ∇ A˜ dr=0. We choosethe gaugesuch that∇ A˜ =0.
× ·
sis functions of the vortex-lattice states, thereby trans- The Eilenberger equation for the even-parity pairing
R
forming the differential equations into algebraic equa- without Pauli paramagnetism is given by6,25
tions. This method has been powerful for (i) solving
the Ginzburg-Landau equations27 and the Bogoliubov- ε + ~ g +1~v ∂ f = ∆φ+ ~ f g. (2a)
de Gennes equations28,29,30,31 and (ii) obtaining quan- n 2τh i 2 F· 2τh i
(cid:18) (cid:19) (cid:18) (cid:19)
titative agreements on the upper critical field H of
c2
Nb, NbSe , and MgB with Fermi surfaces from first- Hereε =(2n+1)πT (n=0, 1, 2, )istheMatsubara
2 2 n
principles electronic-structure calculations.25,26 Thus, energy with T the temperat±ure,±τ i·s·t·he relaxation time
2
by nonmagnetic impurity scattering, and denotes Here a and a† are the boson operators:
h···i
the Fermi-surface average:
hgi≡ dSF(2gπ()ε3nN,k(0F),rv) , (cid:20) aa† (cid:21)= √lc2(cid:20) −cc1∗1 iicc∗22 (cid:21)(cid:20)∇y −2∇πixBx/Φ0 (cid:21) , (6a)
Z | F| with l Φ /2πB and c c∗+c∗c =2, and A˜ and v¯
withdSF aninfinitesimalareaontheFermisurface,N(0) are decfi≡ned by0 1 2 1 2 F+
the density of states per spin and per unit volume at p
the Fermi energy in the normal state, and vF the Fermi √2πl
velocity. The operator ∂ in Eq. (2a) is defined by A˜ i c(c1A˜x+ic2A˜y), (6b)
≡− Φ
0
2π
∂ ∇ i A,
≡ − Φ
0
v¯ c v +ic v , (6c)
F+ 2 Fx 1 Fy
withΦ hc/2ethe fluxquantum,39 ∆(r) is the pairpo- ≡
0
≡
tential, and φ(k ) specifies the gapanisotropysatisfying respectively. The constants (c ,c ) can be fixed appro-
F 1 2
φ(k ) =1. Finally, the quasiclassical Green’s functions priately to make the subsequent calculations efficient. A
F
fh and gi for ε >0 are connected by g=(1 ff†)1/2 with convenient choice25 is to impose the condition that the
n
f†(ε ,k ,r)=f∗(ε , k ,r). − gradient term in the Ginzburg-Landau equation be ex-
n F n F
Equation(2a)hast−obesolvedsimultaneouslywiththe pressed in terms of a†a without using aa and a†a†, i.e.,
self-consistency equation for the pair potential and the the pair potential near T be described in terms of the
c
Maxwell equationfor A˜, which are givenrespectively by lowest Landau level only. Alternatively, one may follow
Graseret al.10 tochangethem ateverytemperatureand
magnetic field so as to make the free-energy within the
∞
T ∆(r)
∆(r)ln c0 =2πT φ(k )f(ε ,k ,r) , lowest-Landau-levelapproximation smallest.
F n F
T n=0(cid:20) εn −h i(cid:21) UsingEq.(6a),wecanmakeupasetofbasisfunctions
X (2b) to describe arbitrary vortex-lattice structures as27
−∇2A˜(r)=−i16π2ecN(0)T n∞=0hvFg(εn,kF,r)i, (2c) ψNq(r)=sc1a22π√lcπV n=−NXNf/f/22+1exp(cid:20)iqy(cid:18)y+ lc22qx(cid:19)(cid:21)
X
na na
with T the transition temperature for τ= . exp i 1x y+l2q 1y
c0 ∞ × l2 c x− 2
Finally, the free-energy functional corresponding to (cid:20) c (cid:16) (cid:17)(cid:21)
Eq. (2) is given by6,33 c c x l2q na 2
exp 1 2 − c y − 1x
(∇ A)2 T × "− 2 (cid:18) c1lc (cid:19)#
F =F + dr × +N(0)∆(r)2ln
s n 8π | | T 1 x l2q na
Z (cid:26) c0 H − c y− 1x . (7)
∞ ∆(r)2 ×√2NN! N(cid:18) c1lc (cid:19)
+2πTN(0) | | I(ε ,k ,r) , (3)
n F
n=0(cid:20) εn −h i(cid:21)(cid:27) Here N =0,1,2, denotes the Landau level, q is an
X ···
arbitrary chosen magnetic Bloch vector characterizing
where F is the free energy in the normal state and I is
n the broken translational symmetry of the vortex lattice
defined by
and specifying the core locations, and V is the volume
f f† + f f† of the system. The quantities a1x, a1y and a2 are the
I ≡∆∗f+∆f†+2εn(g−1)+~ h i4τh i components of the basic vectors a1 and a2 in the xy
plane, respectively, with a yˆ and a a =2πl2, 2 de-
+~ghgi−1 ~f†vF·∂f −fvF·∂∗f† . (4) notesthenumberoftheflu2xkquantum1xin2thesyscteNmf,and
2τ − 2(g+1) H (x) ex2 d Ne−x2 istheHermitepolynomial. The
N ≡ −dx
Indeed,functionaldifferentiationsofEq.(3)withrespect basis functions are both orthonormal and complete, sat-
to f, ∆, and A˜ lead to Eqs. (2a), (2b), and (2c), respec- isfying aψNq(cid:0)=√N(cid:1)ψN−1q and a†ψNq=√N+1ψN+1q.
tively.
C. Algebraic Eilenberger equations
B. Operators and basis functions
We now expand ∆, f, and A˜ in the basis functions of
We first express the gradient operator in Eq. (2a) as the vortex lattice as
∞
v ∂ = v¯F∗+(a+A˜)−v¯F+(a†+A˜∗). (5) ∆(r)=√V ∆NψNq(r), (8a)
F
· √2lc NX=0
3
and g=(1 ff†)1/2 on those points by Eqs. (8) with re-
−
stricting the summations to those satisfying N N and
∞ ≤ c
f(εn,kF,r)=√V fN(εn,kF)ψNq(r), (8b) |K|≤Kc, where Nc and Kc are some cutoffs. Now, the
integrationscanbeperformedbythetrapezoidalrule;its
N=0
X convergenceis excellent for periodic functions. Also, the
summation over n in Eq. (9b) is restricted in the actual
A˜(r)= A˜ eiK·r, (8c) calculations to those satisfying εn εc. The conver-
K | | ≤
gence can be checked by increasing N , N , K , and
K6=0 int c c
X ε . Finally, the vortex-lattice structure can be fixed by
c
where K is the reciprocal-lattice vector.27 Substituting requiring that the free energy (3) be minimum.
Eq. (8) into Eq. (2) and using the orthogonality of the
basis functions, Eq. (2) is transformed into a set of alge-
braic equations for f , ∆ , and A˜ as E. Thermodynamic quantities
N N K
{ } { } { }
εnfN +β¯∗√N+1fN+1 β¯√NfN−1 Once ∆, f, and A˜ are determined as above, we can
−
1 f g g f calculate thermodynamic quantities of the vortex-lattice
= √V ψN∗q ∆φg+~h i 2−τh i −β¯∗A˜+β¯A˜∗ dr,(9a) state. Specifically, the magnetization M due to super-
Z (cid:18) (cid:19) current and the entropy S are obtained by34
s
∆NlnTc0 =2πT ∞ ∆N φ(kF)fN(εn,kF) , (9b) −4πM = B1V dr (∇×A˜)2
T nX=0(cid:20) εn −h i(cid:21) Z ∞(cid:20) f†v ∂f fv ∂∗f†
+2π2N(0)T F· − F· , (11a)
g+1
16π2N(0)T ∞ nX=0(cid:28) (cid:29)
A˜ = βg(ε ,k ,r) e−iK·rdr,
K − (Kl B)2V h n F i
c nX=0Z N(0)
(9c) S =S dr ∆(r)2
s n
with − T | |
Z (cid:26)
∞
β¯ ~(c2vFx+ic1vFy), β ~(c1vFx+ic2vFy). (10) −2πT [hI(εn,kF,r)i+2εnhg−1i] , (11b)
≡ 2√2lc ≡ 2√2lc nX=0 (cid:27)
Together with the equation to determine Hc2 derived respectively, where Sn 2π2VN(0)T/3 is the entropy in
recently,25 the above coupled equations form a basis thenormalstateandIi≡sdefinedbyEq.(4). Also,whenit
for efficient numerical calculations of the Eilenberger is much smaller than the diamagnetismby supercurrent,
equations for vortex-lattice states with arbitrary Fermi- the magnetization M due to Pauli paramagnetism can
sP
surface structures. be calculated by34
2πT ∞ ∂ g A˜ 2
D. Numerical procedures M =M 1 dr h i 1+ ∇× ,
sP nP
" − V n=0Z ∂εn(cid:18) B (cid:19) #
X
(11c)
For a given vortex-lattice structure specified by the
where M 2VN(0)µ2B with µ the Bohr magneton.
basic vectors a1 and a2 in Eq. (7), the coupled equa- The quanntPit≡y B B
tion (9) may be solved iteratively in order of Eqs. (9a),
(9b), and(9c)by adopting a standardtechnique to solve
∂g 1 ∂f† ∂f
nonlinear equations.40 A convenient starting point is to = f + f† (12)
put ∆(N0) =δN0∆(T) 1−B/Hc2, fN =0, and A˜K =0 ∂εn −2(1−ff†)1/2 (cid:18) ∂εn ∂εn (cid:19)
on the right-hand of Eq. (9a), where ∆(T) is the angle-
p in Eq. (11c) may be obtained either by numerical differ-
averaged energy gap in zero field. In this connection, it
entiationsordirectlyfromthe equationofdifferentiating
may be worth noting that the tridiagonal matrix con-
Eq. (2a) with respect to ε :
structed fromthe coefficients off onthe left-hand side n
N
of Eq. (9a) can be inverted analytically.25 The Fermi- ~ 1 ∂f ~ ∂ g
surface integrals can be performed as described in ε + g + ~v ∂ + 1+ h i f
n F
h···i 2τh i 2 · ∂ε 2τ ∂ε
Sec. IV of Ref. 25. In contrast, integrations over r in (cid:18) (cid:19) n (cid:18) n(cid:19)
Eqs. (9a) and (9c) may be carried out as follows: At the ~ ∂g ~ ∂ f
= ∆φ+ f + h ig. (13)
beieKg·irnnaitngeqoufalelyachspacacelcdulNation,Nwe pdriespcarerteeψpNoqin(trs) iannda (cid:18) 2τh i(cid:19)∂εn 2τ ∂εn
int int
unit cell. We then construc×t ∆(r), f(ε ,k ,r), A˜(r), This equation can be solved similarly as Eq. (2a).
n F
4
III. RESULTS
-4
x10
3.0
We now present numerical results for two-dimensional
systemswiththecylindricalFermisurfacewhichisplaced 2.5 T/Tc=0.9
in the xy plane perpendicular to H. We haveconsidered Hc22.0 T/Tc=0.7
a couple of energy gaps: 4pM/1.5 TT//TTcc==00..53
-
1.0
1 :s-wave
φ(k )= . (14)
F (√2(kˆF2x−kˆF2y) :dx2−y2-wave 0.5
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Then it is convenient to set c1=c2=1 in Eq. (6). There B/Hc2
is another parameter in the system corresponding to the
Ginzburg-Landau parameter κ. We have fixed it by (a)
(~c/2e)∆(0) 10 : s-wave
κ0 ≡ 4πN(0)~2vF2 =(cid:26) 7 : dx2−y2-wave , (15) 3.0x10-4
with ∆(p0) the angle-averaged energy gap at T =0. It 2.5 T/Tc=0.9
followsfromEq.(46)ofRef.33thatEq.(15)corresponds Hc22.0 T/Tc=0.7
M/ T/Tc=0.5
to κ = 49 and 40 for the s- and dx2−y2-wave pairings 4p1.5 T/Tc=0.3
in the clean limit, respectively, and κ=1300 for the s- -
wave pairing with τ = 0.01~/∆(0). The vortex-lattice 1.0
structure has been fixed as hexagonal (square) for the 0.5
s-wave(dx2−y2-wave)pairingso thatfinite contributions 0.0
in the expansion (8a) come only from N = 0,6,12, 0.0 0.2 0.4 0.6 0.8 1.0
(N=0,4,8, ) Landau levels. ··· B/Hc2
···
Equation(9)fortheabovemodelhasbeensolvedwith
the procedure of Sec. IID over 0.05H B H . The (b)
c2 c2
≤ ≤
value Ncut is chosen as 30 (16) for the s-wave (dx2−y2- FIG. 1: The magnetization by supercurrent as a function of
wave) pairing. On the other hand, we have set εc = B/Hc2 in the clean limit. (a) s-wave; (b) dx2−y2-wave. The
20T (50T ) at T = 0.9T (0.3T ). These values have temperatures are T/Tc = 0.9, 0.7, 0.5, and 0.3 from top to
c c c c
bottom in the high-field region.
been enough to get the convergence. Using ∆, f, and
A˜ thus obtained, we have calculated the magnetization
by supercurrent, the entropy, and the magnetization by
Pauli Paramagnetism by Eqs. (11a), (11b), and (11c),
contribution is still 0.1∆ even around B = 0.1H .
respectively. ∼ 0 c2
Thisfactimplies thatthe Peschapproximation7 isexcel-
Figure 1 presents magnetic-field dependence of the
lent down to H 0.1H for the two-dimensional cases
magnetizationbysupercurrentforthes-anddx2−y2-wave with the isotropi∼c Fermci2surface. This may not be the
pairings in the clean limit at several temperatures. In
case for systems with complicated Fermi surfaces, how-
both cases, the initial slope at B = H gradually de-
c2 ever, as suggested by the fact that there is already an
creases as the temperature is lowered, implying a mono-
amount of higher-Landau-level contributions at H in
tonic increase of the Maki parameter41 κ (T) as T 0. c2
2 → those cases.24,25,26 The Peschapproximationmay be im-
This feature ofκ (T)has alsobeen predicted inthe case
2 provedtosomedegreebytheprocedureofGraseretal.10
of the three-dimensional spherical Fermi surface with
tochangec andc inEq.(6a)ateverytemperatureand
thes-wavepairing.33,41,42,43 Unlikethethreedimensional 1 2
magnetic field so that the free-energy is smallest.
case,33,42,43 however, the slope in these two-dimensional
Figures 3 and 4 plot the entropy S and the magne-
cases remains finite and does not approach0 evenin the s
tization M due to Pauli paramagnetism, respectively,
clean limit of T 0, in agreement with a previous cal- sP
culationofκ .33→Thecurvesatlowtemperaturesbecome as a function of B/Hc2 for the s- and dx2−y2-wave pair-
2 ings in the clean limit at several temperatures. To see
more and more concave upward, thereby compensating
the initialreductionofthe magnetization. The tempera-
ture variation is slightly larger for the dx2−y2-wave pair-
ing than the s-wave pairing. TABLEI:TheexponentαSofEq.(16a)forthes-anddx2−y2-
Figure 2 shows the expansion coefficients ∆ in Eq. wave pairings in the clean limit calculated by the best fit to
N
(8a) as a function of B/Hc2 for the s- and dx2−y2- thenumerical data of Fig. 3.
wave pairings in the clean limit at T = 0.3T . Com- T/Tc 0.9 0.7 0.5 0.3
c
paredwith the case near T ,27 the mixing of higher Lan- αS (s-wave) 0.73 0.69 0.66 0.63
c
dau levels develops from higher fields. However, the αS (d-wave) 0.72 0.66 0.59 0.52
5
1.0 1.0
D0
=0)0.8 D6 S)S00.8
D/D(T=0,HN000...246 D12 (S-S)/(S-SS0n00..46 TTT///TTTccc===000...357
0.2
T/Tc=0.9
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
B/Hc2 B/Hc2
(a)
(a)
1.0 1.0
H=0) 0.8 DD04 S)s0 0.8
D(T=0, 00..46 D8 S)/(S-s0n 0.6 T/Tc=0.3
D/N 0.2 (S-s 0.4 TT//TTcc==00..57
0.0 0.2 T/Tc=0.9
-0.2 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
B/Hc2 B/Hc2
(b)
(b)
tpFiaIoGinri.no2gf:BwTh/hHeercee2xtpihnaentschiuoernvceclseoaeanffireclii∆men0itt,sa∆∆t6N,Tain=ndE0q.∆3.T1(2c8:af)r(oaam)sast-ofwupanvtcoe- (FbI)Gd.x32:−yT2h-weaevnet.roTphyeStsemaspaerfautnucrteisonaroefTB//THc =c2.0.(9a,)0s.-7w,a0v.5e,;
and 0.3 from bottom to top. They are normalized to vary
∆bo4t,toamnd; ∆(b8)frdoxm2−yto2p-wtaoveboptatoirmingnewarheHrec2t.he curves are ∆0, from 1 at B=Hc2 to 0 at B=0.
the field dependence clearly, the entropy is normalized
M M B αχ
by using S S (B = 0) and S S (B = H ) as sP sP0
s0 s n s c2 − = . (16b)
≡ ≡
(Ss−Ss0)/(Sn−Ss0); itvariesfrom1to 0for Hc2≥B≥0. Mn−MsP0 (cid:18)Hc2(cid:19)
The same normalization is adopted for M . All curves
sP
To confirm the numerical results, we first estimated α
deviate upwards from the linear behavior B/H and S
c2
∝ and α for the s-wave pairing by using only the data of
become more and more convex upward as T 0. This χ
→ 0.85H B 0.95H . Althoughnotpresentedhere,the
tendencyismoreconspicuousforthedx2−y2-wavepairing c2≤ ≤ c2
procedureexcellentlyreproducedthevaluesofaprevious
due to the residual low-energy density of states. To see
calculation near H ,34 as they should.
thebehaviorquantitatively,wehavefittedournumerical c2
data by the formulas: Table I shows the exponent αS obtained by the best
fit to the data of 0.05H B 0.95H in Fig. 3. The
c2 c2
≤ ≤
S S B αS value of the dx2−y2 pairing at T/Tc=0.9 is almost the
s s0 sameasthe correspondings-waveresult. Asthetemper-
− = , (16a)
Sn−Ss0 (cid:18)Hc2(cid:19) ature is decreased, however, the dx2−y2-wave exponent
decreasesmore rapidly so that the curve in Fig. 3(b) be-
comes more convex upward; this is due to the residual
low-energy density of states of the dx2−y2-wave pairing.
TABLE II: The exponent αχ of Eq. (16b) for the s- and Table II presents another exponent αχ obtained by the
dx2−y2-wavepairingsinthecleanlimitcalculatedbythebest best fit to the data of 0.05Hc2 B 0.95Hc2 in Fig. 4.
fit to thenumerical data of Fig. 4. Each value is fairly close to th≤e co≤rresponding one for
T/Tc 0.9 0.7 0.5 0.3 α , as may be expected from the fact that both quan-
S
αχ (s-wave) 0.71 0.67 0.63 0.63 tities probe the zero-energy density of states. In this
αχ (d-wave) 0.70 0.63 0.56 0.49 context, Ichioka et al.17 calculated the field dependence
6
-7
x10
1.0 3.5
M)sP00.8 3.0 T/Tc=0.9
M-nP Hc22.5 T/Tc=0.7
M)/(sP000..46 TT//TTcc==00..35 -4pM/ 12..50 TT//TTcc==00..53
M-sP T/Tc=0.7 1.0
( 0.2 T/Tc=0.9
0.5
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
B/Hc2 B/Hc2
FIG.5: MagnetizationbysupercurrentasafunctionofB/Hc2
(a) forthes-wavepairingwithτ=~/∆(0). Thetemperaturesare
T/Tc=0.9, 0.7, 0.5, and 0.3 from top to bottom.
1.0
M)sP00.8 gaps on different Fermi surfaces.3 Hence the agreement
M-nP between the experiment by Sonier et al.47 and the the-
M)/(sP00.6 T/Tc=0.3 orybyIchiokaet al.17 mightbeanartifactandshouldbe
M-sP0.4 TT//TTcc==00..57 crioensfiinrmcoerdpobryatminogrebodtehtaFileerdmei-xspuerrfaimceenatnsdagsawpesltlrauscttuhreeos-.
( 0.2 T/Tc=0.9 Inthiscontext,itisworthnotingthatnodetailedexper-
iments have been performed on the field dependence of
0.0
0.0 0.2 0.4 0.6 0.8 1.0 γ (B) evenfor the classictype-II superconductorsVand
B/Hc2 Nsb, although early experiments50,51 suggest similar up-
ward deviations from the linear behavior γ B/H .
(b) n c2
We nextfocusonthe s-wavepairinginthe dirtylimit.
FIG. 4: The magnetization MsP by Pauli paramagnetism as Figure 5 shows the magnetization for τ=0.01~/∆(0) as
a function of B/Hc2. (a) s-wave; (b) dx2−y2-wave. The tem- a function of B/H . Compared with the clean-limit re-
peratures are T/Tc = 0.9, 0.7, 0.5 and 0.3 from bottom to c2
sults ofFig. 1(a),we observeanextendedlinearity down
top. They are normalized to vary from 1 at B=Hc2 to 0 at
B=0. to B/Hc2 0.2 irrespective of the temperature. The de-
∼
crease of the initial slope for T 0 is as expected from
thetemperaturedependenceoft→heMakiparameterκ .41
2
By scaling this change of the initial slope, all the curves
almost fall onto a single curve. This is a marked feature
of the zero-energy density of states N(0) at T/T =0.5
c
to find N(0) (B/H )0.67 and N(0) (B/H )0.41 for inthe dirtylimitwhichisabsentintheclean-limitresult
c2 c2
∝ ∝ of Fig. 1(a).
the s- and dx2−y2-wave pairings, respectively. Our esti-
Figure 6 shows the field dependences of S and M
matesforα (T 0)andα (T 0)aresomewhatsmaller s sP
S → χ → for τ =0.01~/∆(0) at various temperatures. Table III
(larger) for the s-wave (dx2−y2-wave)pairing.
presentsthecorrespondingexponentsα andα obtained
Experiments on the T-linear specific-heat coeffi- s χ
from the data of 0.5H B 0.95H ; unlike the clean-
cient γ (B) have been performed for clean V Si,44 c2≤ ≤ c2
s 3 limit case, it has been impossible to fit the whole region
NbSe ,45,46,47,48 and CeRu .49 The quantity γ (B)/γ
2 2 s n by a single exponent, especially at intermediate temper-
coincides for T 0 with (S S )/(S S ) of Fig.
→ s− s0 n− s0 atures, as may be realized from Fig. 6. Compared with
3. Those data all show marked upward deviations from
Figs. 3(a) and 4(a), the curves are more monotonic with
thelinearbehaviorγ B/H ,indicatingthatitisacom-
n c2 thealmostlinearbehavior B/H . Lookingatthetem-
monfeatureamongcleansuperconductorsirrespectiveof ∝ c2
perature dependence more closely, however, we observe
the energy-gap symmetry. Sonier et al.47 thereby ex-
a change from a convex-upward behavior at high tem-
tracted the exponent 0.66 for the field dependence of
γ (B) as T 0, in good agreement with the result 0.67
s
by Ichioka e→t al.17 for the clean two-dimensional s-wave
model. However, a more recent experiment by Hanaguri TABLEIII:TheexponentsαS andαχ forthes-wavepairing
et al.48 reported a different exponent 0.5. It should also withτ=0.01~/∆(0)calculatedbythebestfittothenumerical
be noted that NbSe has three kinds of Fermi surfaces data of 0.5Hc2≤B≤0.95Hc2.
2
and one of them is quite different in structure from the T/Tc 0.9 0.7 0.5 0.3
cylinder. There also exists a recent experiment which αS 0.84 0.88 0.92 1.10
indicates existence of different superconducting energy αχ 0.84 0.89 0.99 1.15
7
mainanalyticformulasaregiveninSec.IICtogetherwith
the numerical procedure to solve them in Sec. IID. This
1.0
method, which directly extends the H equation25,26
c2
S)S00.8 to lower fields, has a potential applicability to systems
S-n withcomplicatedFermisurfacesand/orgapstructuresto
(S-S)/(SS000..46 TT//TTcc==00..97 cocaofrntrdhyuecormtuotorsdd.yetnaaimlediccqaulcaunltaittiioesnsfoornvtahreiofiuesldtydpeep-IeIndsuenpceers-
0.2 T/Tc=0.5
T/Tc=0.3 Using it, we have calculated the field dependences of
0.0 the magnetization by supercurrent, the mixing of higher
0.0 0.2 0.4 0.6 0.8 1.0
Landau levels in the pair potential, the entropy, and
B/Hc2
the Pauli paramagnetismfor the two-dimensionals- and
(a) dx2−y2-wavepairingsinthe cleananddirtylimits atvar-
ious temperatures. Previous results near H for the s-
c2
wave pairing33,34 have been reproduced adequately and
1.0 extendedtolowerfields toclarifythe overallfielddepen-
M)sP00.8 dences. The differences between the s- and dx2−y2-wave
M-nP pairings are quite small at high temperatures but de-
M)/(sP00.6 T/Tc=0.9 vtheelompagrrkaedduadlilffyearesntcheeintemthpeelroawtu-erneeirsglyowdeenresdit,yroeflfesctatitnegs
M-sP0.4 T/Tc=0.7 between the two cases. The field dependences of the en-
( 0.2 T/Tc=0.5 tropy and Pauliparamagnetismin the clean limit at low
T/Tc=0.3 temperatures present convex-upward behaviors for both
0.0 pairings. In contrast, the curves of the s-wavepairing in
0.0 0.2 0.4 0.6 0.8 1.0
thedirtylimitaremoremonotonicandfairlyclosetothe
B/Hc2
linear behavior, but also acquire downward curvature at
(b) low temperatures. As for the magnetization by super-
current, there is a wide region of linear field dependence
FIG. 6: (a) The entropy SS and (b) the magnetization MsP
by Pauli paramagnetism as a function of B/Hc2 for the s- from Hc2 both at high temperatures and in the dirty
wave pairing with τ = 0.01~/∆(0). The temperatures are limit. The region shrinks in the clean limit as the tem-
T/Tc=0.9, 0.7, 0.5 and 0.3 from top to bottom. perature is lowered, and the curve acquires pronounced
upward curvature. It is also found that the mixing of
higher Landau levels in the pair potential is small for
B &0.1H but develops rapidly as the field is further
c2
peratures to a convex-downward behavior at low tem- decreased.
peratures,in agreementwith a previous calculationnear
H .34 This feature also appears in the field dependence
c2
ofthezero-energydensityofstatesascalculatedrecently
by Miranovi´cet al.22 The convex-downwardbehavior at
T =0.3T may become more pronounced at lower tem-
c Acknowledgments
peratures to be observable experimentally.
This work is supported by a Grant-in-Aid for Scien-
IV. SUMMARY tific Research from the Ministry of Education, Culture,
Sports, Science, and Technologyof Japan,and also from
We have developed analternative method to solve the the21stcenturyCOEprogram“TopologicalScienceand
Eilenberger equations for the vortex-lattice state. The Technology,” Hokkaido University.
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