Table Of ContentAngular dependence of in-plane upper critical field H in d-wave superconductors
c2
Jing-Rong Wang and Guo-Zhu Liu
Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
The angular dependenceof in-planeuppercritical field Hc2 is widely used to identify theprecise
gapsymmetryofunconventionald-wavesuperconductors. Apartfromthewellstudiedorbitaleffect
of externalmagnetic field,Hc2 is also believed to bestrongly influencedbythePauli paramagnetic
effect in some heavy fermion compounds. After calculating Hc2 in the presence of both theorbital
4 andPaulieffects,wefindthatitsconcreteangulardependenceisdeterminedbythesubtleinterplay
1 ofthesetwoeffects. AninterestingandunexpectedresultisthatHc2mayexhibititsmaximalvalues
0 along the nodal or antinodal directions, depending on the specific values of a number of physical
2 parameters. We perform a systematical analysis on how the fourfold oscillation pattern of Hc2 is
affected by these parameters, and then apply our results to understand the recent experiments of
n
a Hc2 in two heavy fermion compounds CeCoIn5 and CeCu2Si2.
J
PACSnumbers:
3
1
I. INTRODUCTION other experiment observes the maxima along the [110]
n] direction30. This discrepancyisstillaopenpuzzlewhich
o Unconventional superconductors usually refer to the need to be resolved22. In the case of CeCu2Si2, many
c superconductorsthose cannot be understood within the earlierexperimentssuggestadx2−y2-wavegap31,32. Nev-
- ertheless, a recent measurement17 observes the maxima
r conventional Bardeen-Cooper-Schrieffer (BCS) theory.
p Notable examples are high-T cuprates1, heavy fermion of Hc2 along the [100]direction, which is arguedto infer
u superconductors2–4, organic csuperconductors5 and iron a dxy-wave gap according the corresponding theoretical
.s based superconductors6–9. The unconventional super- analysis17. Apparently, more research efforts are called
at conductivityisusuallydrivenbystrongelectron-electron for to solve these puzzles, which have motivated us to
m revisit this issue more systematically.
interaction, and the electron pairing mechanism usually
- has a magnetic origin. In the past decades, identifying Now suppose an external magnetic field is introduced
d the precise gap symmetry of unconventional supercon- toasuperconductor. Inprinciple,thisfieldcancoupleto
n
ductorshas attractedgreattheoreticalandexperimental the charge and spin degrees of freedom of the electrons
o
efforts since such efforts may lead to important progress viatheorbitalandZeemanmechanismsrespectively. The
c
[ in seeking the microscopic pairing mechanism. former mechanism is described by the minimal coupling
Many unconventional superconductors are believed to between the momentum of electrons and the vector po-
1
tential, andcan leadto the well-knownAbrikosovmixed
havea d-waveenergygap,whichis differentfromthatof
v
state in type-II superconductors. The latter mechanism,
6 isotropic s-wave superconductors. However, it is not an
usually called Pauli paramagnetic or Pauli limiting ef-
5 easytasktodeterminetheprecised-wavegapsymmetry.
9 A powerfulandfrequently used approachis to probe the fect, is known to be important in some heavy fermion
2 angulardependenceofvariousobservablequantities,such compounds17,33–35. Which one of these two effects plays
1. as upper critical field10–17, specific heat18–21, and ther- a dominant role is determined by a number of physical
0 mal conductivity19,23–25. In this paper, we are mainly factors. When both of them are important, novel and
4 interestedinthe behaviorsofin-planeuppercriticalfield interesting properties may emerge.
:1 Hc2 in heavy fermion superconductors. This issue has Since the middle of 1990s, the in-plane Hc2 has been
v recentlybeenaddressedwiththe aimtoidentify thepre- applied to identify the gap symmetry in layered uncon-
i cise gap symmetry of some heavy fermion compounds, ventional superconductors10–17. Early theoretical cal-
X
especially CeCoIn516 and CeCu2Si217. Despite the in- culations have showed that the in-plane Hc2 exhibits a
ar tensive theoretical and experimental efforts, it remains fourfold oscillation in d-wave superconductors10,11. The
unclear whether the gap symmetry of these compounds presence of such a fourfold oscillation has already been
is dx2−y2-waveordxy-wave. Thesetwo gapsaredifferent verifiedinmanyunconventionalsuperconductors,includ-
from each other primarily in the positions of gap nodes. ing high-Tc cuprate superconductors12,13, LuNi2B2C14,
Inprinciple,theirpositionscanbeclarifiedbymeasuring heavy fermion compounds CeCoIn516 and CeCu2Si217.
the angular dependence of H . Unfortunately, experi- IntheearlycalculationsofWonet. al.10 andTakanaka
c2
mental studies have not yet reached a consensus on this et. al.11 who solely considered the orbital effect, H is
c2
issue. In the case of CeCoIn , currently most of experi- found to exhibit its maxima along the antinodal direc-
5
ments suggest that the gap symmetry should be dx2−y2- tions where the d-wavesuperconducting gapis maximal.
wave20,26,27, however there is currently still an experi- The subsequent analysis of Weickert et. al.16 includes
mental discrepancy in the concrete angular dependence boththe orbitalandPauliparamagneticeffects, but still
ofH inCeCoIn : someexperimentsfindthatthe max- finds the maxima of H along the antinodal directions.
c2 5 c2
ima of H are along the [100] direction16,28,29, whereas A similar conclusion is drawn in a recent work19, where
c2
2
|∆dx2−y2| psylamymedetariseisgnoiffiCcaenCtorInoleainndthCeeCdeuteSrim.ination of the gap
5 2 2
90 1 In this paper, motivated by the recent progress and
120 60
0.8 the existing controversy, we analyze the angular depen-
0.6 denceofin-planeH anditsconnectionwiththed-wave
150 30 c2
0.4 gapsymmetrybyconsideringtheinterplayoforbitaland
0.2 Paulieffectsinthecontextsofheavyfermioncompounds.
180 0 Aftercarryingoutsystematicalcalculations,wewillshow
that the maxima of angle-dependent H (θ) are not al-
c2
waysalongtheantinodaldirectionswhenboththeorbital
210 330 and Pauli effects are important. The concrete fourfold
oscillation pattern of H (θ) is determined by a number
c2
240 300 of physical parameters, including temperature T, criti-
270
caltemperature T , gyromagneticratio g, fermionveloc-
c
|∆d | ity v0, and two parameters that characterize the shape
xy
of the underlying Fermi surface. Each of these parame-
90 1
120 60 ters can strongly affect the angular dependence of Hc2.
0.8
Among the above six relevant parameters, the temper-
0.6
150 30 ature T is particularly interesting, due to that in any
0.4
given compound t is the only free parameter and all the
0.2
other parameters are fixed at certain values. If we vary
180 0 temperature T but fix all the rest parameters, H (θ) is
c2
found to exhibit its maxima along the nodal directions
atlowertemperaturesandalongtheantinodaldirections
210 330
athighertemperatures. Thismeanstheangle-dependent
H (θ) is shifted by π/4 as temperature increases across
c2
240 300
certain critical value.
270
Our results can be used to clarify the aforementioned
experimentalpuzzle abouttheangulardependenceofin-
FIG. 1: Shapesof dx2−y2-wave and dxy-wave gaps. plane Hc2. Since Hc2(θ) shifts by π/4 as some of the
relevant parameters are changed, the seemingly contra-
the authors also show that increasing the Pauli effect dictoryexperimentalresultsreportedinRefs.16,28,29 may
reduces the difference in H between nodal and antin- bewellconsistent. Ontheotherhand,sincetheconcrete
c2
odal directions. There seems to be a priori hypothesis behavior of Hc2(θ) is very sensitive to the specific values
in the literature that a larger gap necessarily leads to a of several parameters, one should be extremely careful
larger magnitude of Hc2, which means Hc2 and d-wave when judging the gap symmetry by measuring Hc2.
gapshouldalwayshavetheir maxima andminima atex- In Sec.II, we derive the equation for Hc2 after includ-
actly the same azimuthal angles. If such a hypothesis is ing both the orbital and Pauli paramagnetic effects. In
correct, it would be straightforward to identify the pre- Sec.III, we present numerical results for Hc2 in three
cise gap symmetry. For instance, if the experimentally cases, i.e., pure orbital effect, pure Pauli paramagnetic
observed H displays its maxima along the [100] direc- effect, and interplay of both orbital and Pauli effects.
c2
tion,the gappossessesadx2−y2 symmetry. Onthe other WeshowthatHc2displayscomplicatedangledependence
hand, if the maxima are observed along the direction due to interplay of orbital and Pauli effects. In Sec.IV,
[110], the gap symmetry should be d -wave. To make a we discuss the physical implications of our results and
xy
comparison, we show the angular dependence of dx2−y2- make a comparison with some relevant experiments.
and d -wave gaps in Fig. 1.
xy
It is necessary to emphasize that the above hypothe-
sized connection between in-plane H and d-wave gap, II. EQUATION FOR IN-PLANE UPPER
c2
thoughintuitively reasonable,is actuallynot alwayscor- CRITICAL FIELD Hc2
rect. Whenthereisonlyorbitaleffect,themaximaofH
c2
andd-wavegaparealongthesamedirectionsinallcases. Heavyfermioncompoundsareknowntohavealayered
In the presence of Pauli paramagnetic effect, however, structure, which is analogous to cuprates. However, the
there is indeed no guarantee that such a connection is inter-layer coupling is not as weak as that in cuprates.
valid. Inordertoclarifythedetailedconnectionbetween It is convenient to consider a rippled cylinder Fermi sur-
theprecisegapsymmetryandtheangulardependenceof face,schematicallyshowninFig.2. Thefermionvelocity
H , we will considerthe influence of the interplay of or- has three components k , where k denote the two
c2 x,y,z x,y
bital and Pauli effects on H more systematically. This components in the superconducting plane. Here, we use
c2
problem is important because in-plane H has recently t torepresenttheinter-layerhopingparameterandcthe
c2 c
3
k One can write the generalized derivative operator as
z
Π(R) = i R+2eA(R)
− ∇
= i∂ e i∂ e
x x y y
− −
π/c +( i∂z+2eH( xsinθ+ycosθ))ez.
− −
Followingthe generalmethods presentedinRefs.37–44,
k we obtain the following linearized gap equation:
y
T +∞ πT π dχ 2π dϕ
ln( )∆(R) = dη
kx − Tc Z0 sinh(πTη)Z−π 2π Z0 2π
1
γ2(kˆ) 1 cos η h′+ v (kˆ)
× α (cid:26) − (cid:20) (cid:18) 2 F
Π(R))] ∆(R), (4)
· }
where χ=k c. The function ∆(R) is
z
1
−π/c ∆(R)= 2eH 4 e−eH(xsinθ−ycosθ)2. (5)
(cid:18) π (cid:19)
Here we do not include Landau level mixing16,17,39 for
FIG. 2: Schematic diagram for a rippled Fermi surface. simplicity, which will not affect our conclusion. For the
chosen Fermi surface, the Fermi velocity vector is36
unit sizealongz-direction,andthen write the dispersion v (kˆ)=v cosϕe +v sinϕe +v sinχe . (6)
F a x a y c z
as18,19,23,36
The Fermi velocity component along the c-axis is v =
c
k2+k2
ε(k)= x y 2t cos(k c). (1) 2tcc. The two-component in-plane velocity vector has a
c z
2m − constantmagnitudev ,definedasv =v 1+λcos(χ),
a a 0
wherev = kF0 withtheFermimomentumpk beingre-
Introducing a constant magnetic field H to the system 0 m F0
leads to fruitful behaviors. For type-II superconductors, lated to the Fermi energy ǫF by kF0 = √2mǫF. The
the field H weaker than lower critical field H cannot shape of rippled cylinder Fermi surface is characterized
c1
penetrate the sample due to the Meissner effect. As H by a velocity ratio vc/v0 = λγ, where λ = 2tc/ǫF and
exceeds Hc1 and further increases, the superconducting γ = ckF0/2. As will shown below, both λ and γ can
pairingis graduallydestructedby the orbitaleffect. The strongly affect the behavior of Hc2. Moreover,we define
superconductivity is entirely suppressed once H reaches h′ = −gµ2BH, where µB is Bohr magneton and g is the
the upper critical field H , which can be obtained by gyromagnetic ratio. The orbital effect of magnetic field
c2
solving the corresponding linearized gap equation. In is reflectedin the factorvF(k) Π(R), whereasthe Pauli
somesuperconductors,thePauliparamagneticeffectcan paramagneticeffectisreflected·inthefactorh′. Thecon-
alsoclosethegapbybreakingspinsingletpairs,andmay crete behavior of Hc2 is determined by the interplay of
even be more important than the orbital effect16,17. In these two effects.
order to make a general analysis, we consider both of InEq.(4),theinfluenceofgapsymmetryisreflectedin
these two effects in the following. thefunctionγα(k). Forisotropics-wavepairing,γs(kˆ)=
Toproceed,itisusefultorewritethein-planemagnetic 1; for dx2−y2-wave pairing, γd(kˆ) = √2cos(2ϕ); for dxy-
field H in terms of a vector potential A. Let us choose wave pairing, γ (kˆ)=√2sin(2ϕ).
d
thea-axisasx-coordinateandb-axisasy-coordinate,and
AlthoughthelinearizedgapequationEq.(4)isformally
then write down a vector potential
general and valid in many superconductors, its solution
is determined by a number of physical effects and as-
A=(0,0,H( xsinθ+ycosθ)), (2)
− sociated parameters. For instance, the behavior of Hc2
where θ denotes the angle between a-axis and field H. maybe stronglyinfluencedby the concreteshapesofthe
For conventional s-wave superconductors, the gap is Fermi surface. The Fermi surface has different spatial
isotropic and the upper critical field H is certainly dependence in various superconductors, which naturally
c2
θ-independent. In the case of d-wave superconductors, leads to different forms of fermion dispersion and Fermi
however, the gap is strongly anisotropic, thus Hc2 be- velocity vF. Such a difference certainly affects the equa-
comes θ-dependent. Now the field H takes the form tion of Hc2. For spherical Fermi surface, Fermi velocity
v depends on the azimuthal angle ϕ within the basal
F
H = A=(Hcosθ,Hsinθ,0). (3) plane and the angle between z-axis and v . Therefore,
F
∇×
4
40
38 t=0.1 θ=0(deg)
35 θ=45(deg)
36
30
34
25
Tesla) 32 Tesla) 20
H(c230 H(c2
15
28
10
26
5
24 (a)
0
0
(b)
t=0.9
4 −2
3.95 −4
H(Tesla)c2 3.9 ∆ H(Tesla)c2 −6
3.85 −8
3.8 −10
−12
0 20 40 60 80 100 120 140 160 180 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
θ t
FIG.3: Fourfold oscillation ofθ-dependentHc2 attworepre- FIG. 4: t-dependence of Hc2 with Tc = 1K, v0 = 3000m/s,
sentative temperatures t=0.1 and t=0.9. λ=0.5, and γ =1.
the equation of Hc2 contains the integrations over these where
two variables41,43,44. For cylindrical Fermi surface, the
1
direction of vector vF solely depends on the azimuthal a± = 2√eH [−isinθ∂x+icosθ∂y ∓∂z
angle ϕ, so there is only the integration over angle ϕ in
the equationof H 42. For rippled Fermi surface, the di- 2ieH(xsinθ ycosθ)], (10)
c2 ± −
rection of v depends on the azimuthal angle ϕ and the 1
F a = [ i∂ cosθ i∂ sinθ], (11)
coordinate χ along z-axis, then the integrations over ϕ 0 √2eH − x − y
andχenterintotheequationofH ,asshowninEq.(4).
c2 which satisfy
Inaddition,therearetwoindependentparametersλand
γ which can characterize the rippled Fermi surface in [a−,a+]=1,[a±,a0]=0. (12)
Eq. (4). Notice that once λ = 0, the rippled cylindrical
Fermi surface reduces to the cylindrical Fermi surface. In the following analysis, we take dx2−y2-wave pairing
The influence of Fermi surface on H is rarely studied asanexampleandassumethatγ (kˆ)=√2cos(2ϕ). The
c2 d
in the literature. In this paper, we adopt rippled Fermi results in the case of d -wave pairing can be obtained
xy
surfaceandshowthatH canexhibitdifferentbehaviors analogously. It is easy to examine that the qualitative
c2
under different parameters. conclusionwill be not changed. After averagingoverthe
Tofacilitateanalyticalcomputation,wecanchoosethe groundstate∆ (R)onbothsidesofEq.(4)andinserting
0
direction of field H as a new z′-axis and define the dx2−y2-wave gap γd(kˆ) = √2cos(2ϕ), we obtain the
eee′x′y′ ===e−execszionsθθ−+eeycsoinsθθ . (7) follolnwtin=g inte+gr∞al eqduuation1for Hcocs2(,hu) π dχ 2π dϕ
z x y − Z0 sinh(u)(cid:26) − Z−π 2π Z0 2π
In the coordinate frame spanned by (e′x,e′y,e′z), we have [1+cos(4θ)cos(4ϕ)]
a new velocity vector ×
exp ρu2 λ2γ2sin2(χ)
vF(kˆ) = vasin(θ−ϕ)e′x−vcsin(χ)e′y ×+(1+(cid:2)−λcos((cid:0)χ))sin2(ϕ) , (13)
+v cos(θ ϕ)e′, (8) (cid:1)(cid:3)(cid:9)
a − z where t = T , h = gµBHc2 and ρ = v02eHc2 . One can
and a new generalized derivative operator Tc 2πkBT 8π2kB2T2
analyze the detailed behavior of H , especially its de-
c2
Π(R) = √eH (a++a−)e′x−i(a+−a−)e′y pendence on various physical parameters,systematically
+√2a(cid:2)0e′z , (9) bthyesnoelvxitnsgectthiiosni.ntegral equation. This will be done in
i
5
3.5 3.2 t=0.1
3.18
3
3.16
2.5 H(Tesla)c233..1124
sla) 2 3.1
e
T
H(c21.5 3.08
3.06
1 1.526 t=0.9
1.525
0.5 1.524
0 Tesla) 11..552223
0 0.2 0.4 t 0.6 0.8 1 H(c21.521
1.52
1.519
FIG. 5: t-dependence of Hc2 with Tc = 1K, λ = 0.5, γ = 1,
1.518
and g=1.
0 20 40 60 80 100 120 140 160 180
θ
III. NUMERICAL RESULTS OF Hc2 AND
PHYSICAL IMPLICATIONS FIG. 6: Angular dependence of Hc2 at t = 0.1 and t = 0.9
with Tc = 1K, v0 = 3000m/s, λ = 0.5, γ = 1, and g = 1.
The fourfold oscillation patterns are apparently different at
Inthissection,wefirstpresentthenumericalsolutions low and high temperatures.
of Eq. (13), then discuss the physical implications of the
results,andfinally compareour results withsome recent
experiments. From Eq. (13), we know the behavior of
A. Pure orbital effect
H (θ) is determined by six physical parameters:
c2
T :Zero-field critical temperature, (14)
c
First, we consider only the orbital effect by setting
t=T/T , (15)
c the gyromagnetic factor g = 0. In this case, the factor
v = 2ǫ /m, (16) cos(hu)appearinginEq.(13)isequaltounity,cos(hu)=
0 F
g :gypromagnetic ratio, (17) 1. WeassumethatTc =1K,v0 =3000m/s,λ=0.5,and
γ = 1, which are suitable parameters in heavy fermion
λ=2t /ǫ , (18)
c F compounds.
γ =k c/2. (19)
F0 After carrying out numerical calculations, we plot the
Amongthis setofparameters,λandγ arerelatedto the angular dependence of Hc2(θ) in Fig. 3 at two represen-
shape of rippled cylinder Fermi surface. We notice that tative temperatures t = 0.1 and t = 0.9. It is easy to
the influence of these two parameters are rarely investi- seefromFig.3thatHc2(θ)exhibits afourfoldoscillation
gatedinpreviousworksonHc2. Thecriticaltemperature pattern. Moreover, the maxima of Hc2 are always along
T andthe gyromagneticratiog will be takenas varying theantinodaldirectionsforanyvaluesoftherelevantpa-
c
parameters. rameters,whichmeanstheangulardependenceoforbital
The detailed behavior of Hc2 can be clearly seen from effect-induced Hc2 is exactly the same as that of d-wave
its angular dependence. In addition, it is also interest- gap. This is consistent with the original theoretical pre-
ing to analyze the difference of H between its values dictions of Won et. al.10 and Takanaka et. al.11. An
c2
obtained at θ =45◦ and θ =0◦: important feature that needs to be emphasized is that
the positions of peaks are temperature independent, as
◦ ◦
∆H =H (θ =45 ) H (θ =0 ), (20)
c2 c2 c2 clearly manifested in both Fig. 3 and Fig. 4.
−
since the maxima and minima of H always appear at The above properties can also be elaborated by the
c2
◦
these two angles. H exhibits its maxima at θ = 45 if detailed t-dependence of H and ∆H are presented in
c2 c2 c2
◦
∆H >0 and at θ =0 if ∆H <0. Fig. 4. H is an monotonously decreasing function of
c2 c2 c2
In order to demonstrate the influence of the orbital parameter t, valid for all values of θ. This is easy to
effect and that of the Pauli paramagnetic effect on the understand since the magnitude of the superconducting
angulardependence ofin-plane H , wefind it helpful to gap always decreases monotonously with growing tem-
c2
consider three cases separately: pure orbital effect; pure perature. Moreover, the difference ∆H is negative for
c2
Pauli effect; interplay of orbital and Pauli effects. all values of t.
6
3.5 10
θ=0(deg)
3 θ=45(deg) 9
8
2.5 7
Tesla) 2 Tesla) 56
H(c21.5 H(c2 4
1 3
2
0.5
(a) 1
0 0
0.1 0.03
(b)
0.025
0.08
0.02
0.015
0.06
∆ H(Tesla)c2 0.04 ∆H(Tesla)c2 0.00.00510
0.02
−0.005
−0.01
0
−0.015
−0.02 −0.02
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3
t T(K)
c
FIG. 7: t-dependence of Hc2 with Tc = 1K, v0 = 3000m/s, FIG. 8: Tc-dependence of Hc2 with t = 0.5, v0 = 3000m/s,
λ=0.5, γ =1, and g=1. λ=0.5, γ =1, and g=1.
B. Pure Pauli paramagnetic effect
behavior is very similar to that in the case of pure or-
bital effect. However, at a relatively lower temperature
We next consider the effects of pure Pauli paramag- t=0.1,themaximaofH arealongthenodaldirections
c2
netic effect by setting v0 =0, which leads to where the dx2−y2-wave gap vanishes. Two conclusions
+∞ 1 cos(hu) can be immediately drawn: Hc2 does not always exhibit
ln(t)= du − , (21) its maxima at the angleswhere the superconducting gap
− Z0 sinh(u) reachesits maximal value; the fourfoldoscillationcurves
which is completely independent of θ. The t-dependence ofHc2isshiftedbyπ/4astemperaturegrowsintherange
of Hc2 is shown in Fig. 5. Different from pure orbital of 0<T <Tc.
effect, Hc2 isnotamonotonousfunction: itrisesinitially FromFig. 7(a),we seethat Hc2 firstariseswithgrow-
with growing t, but decreases as t is larger than certain ing t and then decreases rapidly once t exceeds a critical
critical value t , which is roughly 0.5t under the chosen value. Apparently,suchanon-monotonoust-dependence
c
set of parameters. of Hc2 is a consequence of the interplay of both orbital
and Pauli paramagnetic effects. On the other hand, the
difference ∆H shown in Fig. 7(b) is positive for small
c2
C. Interplay of orbital and Pauli effects values of t but becomes negative for larger values of t.
Additiontotemperaturet,theconcreteangulardepen-
We now turn to the general and interesting case in dence of Hc2 is also strongly influenced by a number of
whichboththeorbitalandPauliparamagneticeffectsare otherphysicalparameters,includingcriticaltemperature
important. This case is broadly believed to be realized Tc, fermion velocity v0, gyromagnetic factor g, and two
in several heavy fermion compounds, such as CeCoIn Fermisurfacefactorsλandγ. Indeed,differentvaluesof
5
and CeCu Si . As aforementioned, the concrete behav- these parameters can lead to very different behaviors of
2 2
iors of θ-dependent Hc2 are influenced by a number of Hc2. In the following, we show how Hc2 and ∆Hc2 are
parameters. In order to illustrate the numerical results changedastheseparametersarevarying. Tosimplifythe
and their physical implications, we vary one particular analysis,wevaryoneparticularparameterandfixallthe
parameter while fixing all the rest parameters. In most other parameters in each figure.
of the following calculations, the gyromagnetic factor is T : First, we consider the influence of critical tem-
c
taken to be g = 1. The influence of various values of g perature T on H and ∆H , and show the results
c c2 c2
will be analyzed separately. in Fig. 8. It is well-known that T of heavy fermion
c
AsshowninFig. 6,underthecurrentlychosenparam- compounds is actually quite low, especially when com-
eters, the maxima of H locates along the antinodal di- paredwithcupratesandiron-basedsuperconductors. To
c2
rections at a relatively higher temperature t=0.9. This cover all possible heavy fermion compounds, we assume
7
3.3 40
3.2 35
3.1 30
3
25
H(Tesla)c222..89 H(Tesla)c220
15
2.7
2.6 10
2.5 5
2.4 0
0.2 2
0.15 0
0.1 −2
∆H(Tesla)c2 0.05 ∆H(Tesla)c2 −4
0 −6
−0.05 −8
−0.1
0 4000 8000 −10
0 0.5 1 1.5 2
v(m/s)
0 g
FIG.9: v0-dependenceofHc2witht=0.1,Tc =1K,λ=0.5, FIG. 10: g-dependence of Hc2 with t = 0.1, v0 = 3000m/s,
γ =1, and g=1.
Tc =1K, λ=0.5, and γ =1.
Tc varies in the range of [0,3K]. All the other parame- and ∆Hc2 on g is given in Fig. 10. First of all, taking
ters are fixed. Hc2 rises monotonously with growing Tc, g = 0 simply leads to the known results obtained in the
which is obviously owing to the monotonous increase of case of pure orbitaleffect presented in Sec.IIIA. Second,
the superconductinggap. Moreover,ifTc is smallerthan Hc2 decreases monotonously with growing g. An imme-
some critical value, ∆Hc2 is negative, which means the diate indication of this behavior is that increasing the
maxima of Hc2 are along the antinodal directions. For Pauli paramagnetic effect always tends to suppress Hc2
largerTc,∆Hc2 becomespositiveandthemaximaofHc2 in the presence of orbitaleffect. Finally, it is easy to ob-
are shifted to the nodal directions. Apparently, Tc has serve that ∆Hc2 is negative if g takes very small values
important impacts on the concrete angular dependence and positive when g becomes larger than certain critical
ofHc2. Inpassing,we pointout that the maxima ofHc2 value. Therefore, the gyromagnetic factor g also plays
will be shifted back to the antinodal directions for even a crucial role in the determination of the concrete angle
higher Tc (not shown in the figure). dependence Hc2.
v : We then consider the influence of fermion velocity λ: λ represents the ratio of inter layer coupling 2t
0 c
v on H and ∆H , and show the results in Fig. 9. In and the Fermi energy E . If t = 0, the corresponding
0 c2 c2 F c
thelimitv =0,theorbitaleffectisactuallyignoredand λ = 0, then the rippled cylindrical Fermi surface reduce
0
the Pauli effect entirely determines H . In such a limit, to the cylindrical Fermi surface. The dependence of H
c2 c2
H is angle independent, so ∆H = 0. For finite v , and ∆H on λ is as depicted in Fig. 11. H deceases
c2 c2 0 c2 c2
H becomesangledependentandexhibitsfourfoldoscil- monotonously with the growing λ. For given values of
c2
lation, as a consequence of the interplay between orbital other parameters shown in Fig. 11, the maxima of H
c2
and Pauli effects. As v is growing, H first increases are along the nodal directions for small λ, but along the
0 c2
and then decreases, which indicates that the enhance- antinodal directions for large values of λ.
ment of orbital effect does not necessarily suppress Hc2 γ: γ represents the ratio of two momentum kF0 and
once the Pauli paramagnetic effect is present. However, 1/2c,c isthe unitcellsizealongthirddirection. Thede-
as already discussed earlier, Hc2 deceases monotonously pendenceofHc2 and∆Hc2 onγ isshowninFig.12. Hc2
with growing v0 when the Pauli effect is completely ne- deceases monotonously with the growing γ. For given
glected. Furthermore, ∆Hc2 is negative for both small valuesofotherparametersshowninFig.12,the maxima
and large values of v0, but is positive for intermediate of Hc2 are along the nodal directions for small γ, but
valuesofv0. Therefore,theconcreteangulardependence along the antinodal directions for large values of γ.
of Hc2 is very sensitive to the values of fermion velocity. Fromalltheseresults,weseethatboththemagnitudes
g: We next consider the influence of the gyromag- and the detailed angular dependence of in-plane H are
c2
netic factor g, which characterizes the effective strength significantly influenced by a number of physical param-
of Pauli paramagnetic effect. The dependence of H eters. A particularly interesting feature is the fourfold
c2
8
3 3.5
2.9 3
2.8 2.5
Tesla) 2.7 Tesla) 2
H(c2 H(c2
2.6 1.5
2.5 1
2.4 0.5
0.03 0.03
0.02 0.025
0.01 0.02
Tesla) 0 Tesla) 0.015
H(c2−0.01 H(c2 0.01
∆ ∆
−0.02 0.005
−0.03 0
−0.04 −0.005
10−2 10−1 100 10−1 100 101
λ γ
FIG. 11: Variation with λ fixing t = 0.5, v0 = 5000m/s, FIG. 12: γ-dependence of Hc2 with t = 0.5, v0 = 3000m/s,
Tc =1K, γ =1 and g=1. Tc =1K, λ=0.5, and g=1.
oscillationpatternofangledependentHc2 canbe shifted maxima of Hc2 may be along either the nodal or the
by π/4 if one varies any one of these parameters. Hc2 antinodaldirection,depending onthe specificvaluesofa
may exhibit its maxima along either nodal or antinodal number of physical parameters, as a consequence of the
directions, depending on the specific values of relevant delicate interplay between orbital and Pauli effects. In-
parameters, which is apparently in sharp contrast in the accurate and even incorrect conclusions might be drawn
naivenotionthatHc2 alwaysdisplaysthesameanglede- if some of these parameters are not properly chosen. In
pendence of the d-wave superconducting gap. order to identify the precise gap symmetry of CeCoIn
5
or CeCu Si , one should first choose suitable values for
2 2
all the relevant parameters before probing the angular
D. Comparison with recent experiments dependenceofH anddeducingthegapsymmetryfrom
c2
experimental data.
As aforementioned, in the last several years the in- Amongtheabovesixrelevantparameters,thetemper-
plane Hc2 has been widely investigated with the aim ature t is particularly interesting, because in any given
to identify the precise superconducting gapsymmetry in compound t is the only free parameter and all the other
two heavy fermion compounds CeCoIn and CeCu Si . parameters are fixed at certain values. Our extensive
5 2 2
In this subsection, we make a comparison between our calculations show that there is always a π/4 difference
theoreticalanalysisandsome recentexperiments ofHc2. betweenHc2 andd-wavegapatsmalltandthatHc2 and
Our results are valuable to theoretical and experimental d-wave gap always exhibit exactly the same angular de-
researchof Hc2 in two main aspects. pendence once t exceeds certain critical value, provided
First, one should be very careful when fitting theoret- that the gyromagnetic factor g is sufficiently large. It
ical calculations with experimental data. In the current appears that the impact of Pauli effect on H is much
c2
literature, it is often taken for granted that the in-plane more important at low temperatures than at high tem-
H alwaysexhibitsexactlythesameangulardependence peratures. Ifoneattemptstodeducetheprecisegapsym-
c2
as that of the superconducting gap. In other words, the metry by fitting experiments of H , it would be better
c2
maxima of in-plane H are believed to be always along tomeasureH ataseriesofverydifferenttemperatures.
c2 c2
the antinodal directions where the d-wave gap is max- Otherwise, incorrect results might be obtained.
imal. According to this seemingly correct relationship, Second, our results may help to resolve some current
the superconducting gap symmetry is simply identified controversies with regard to the precise gap symmetry
as dx2−y2-wave (dxy-wave) if Hc2(θ) is found to exhibit of heavy fermion compounds. The gap symmetry of
◦ ◦
its maxima at θ = 0 (45 ). However, as showed in our CeCoIn has been investigated extensively by means of
5
extensive calculations, such a relationship is not always variousexperimentaltechniques. Settaiet. al.28reported
correct. In a Pauli-limited d-wave superconductor, the thatthemaximaofin-planeH arealong[100]direction
c2
9
through de Haas-van Alphen oscillation signal at 40mK. IV. DISCUSSION AND CONCLUSION
The cantilever magnetometer measurements at 20mK of
Murphy et. al.30 observed that the maxima of H are
c2
along [110] direction. Bianchi et. al.29 measured the In this paper, we have performed a detailed and sys-
specific heat and found the maxima of Hc2 along [100] tematical analysis of the unusual behaviors of in-plane
directionattemperatureshigherthan1K.After measur- upper critical field H in the contexts of Pauli-limited
c2
ingthemagneticfielddependenceofelectricresistivityat heavy fermion compounds. We show that the concrete
100mK, Weickert et. al.16 revealed that the maxima of angular dependence of H is determined by a delicate
c2
Hc2 are along [100] direction. Obviously, there seems to interplay of the orbital and Pauli paramagnetic effects.
be a discrepancy among the experimental results about The most interesting result is that H does not nec-
c2
the detailed angulardependence ofHc2, which is consid- essarily exhibit the same fourfold oscillation pattern as
ered as an open puzzle22 and complicates the search for thed-wavesuperconductinggap,whichisoftentakenfor
the precise gap symmetry. grantedinthe literature. Forcertainvaluesof aseriesof
Accordingtoourresults,however,probablysuchadis- physical parameters, H may display its maxima along
c2
crepancy does not exist at all, since the maxima of Hc2 thenodaldirectionswherethesuperconductinggapvan-
maybealongeither[100]or[110]directionwhensomeof ishes. We also have compared our theoretical analysis
therelevantphysicalparametersaremoderatelychanged. with some current measurements of in-plane H in two
c2
In particular,the maxima of Hc2 canshift by π/4 as the heavy fermion compounds CeCoIn5 and CeCu2Si2.
temperature increases beyond certain critical value. No-
ticethatthemeasurementsofRef.30 wereperformedata The theoretical results presented in this paper impose
temperature as low as 20mK.There is a goodpossibility an important restraint on the determination of the pre-
that the position of the maxima of Hc2 are shifted from cisegapsymmetryofPaulilimitedd-waveheavyfermion
[110] direction at low temperatures to [100] direction at superconductorsbymeansofmeasuringthein-planeH .
c2
higher temperatures. Although this possibility needs to One has to be extremely careful when trying to deduce
be further examined, it should be safe to say that the the gap symmetry from experiments of H .
c2
seemingly contradictory experimental results about in-
plane H of CeCoIn may be well consistent with each The authors are supported by the National Natural
c2 5
other. More careful and more systematical research are Science Foundation of China under grants No. 11074234
required to completely solve this problem. and No. 11274286.
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