The superconducting phase diagram in a model for tetragonal and cubic systems with strong antiferromagnetic correlations

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by  J. J. Deisz

The superconducting phase diagram in a model for tetragonal and cubic systems with
strong antiferromagnetic correlations
J.J. Deisz
Department of Physics
University of Northern Iowa
4
Cedar Falls, IA 50614
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0 (Dated: February 2, 2008)
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We calculate the superconducting phase diagram as a function of temperature and z-axis

n anisotropy in a model for tetragonal and cubic systems having strong antiferromagnetic fluctua-
a tions. The formal basis for our calculations is the fluctuation exchange approximation (FLEX)
J appliedtothesingle-bandHubbardmodelnearhalf-filling. Fornearlycubiclattices,twosupercon-

0 ductingphasetransitionsareobservedasafunctionoftemperaturewiththelow-temperaturestate
3 having the time-reversal symmetry-breaking form, dx2−y2 ±id3z2−r2. With increasing tetragonal
distortion the time-reversal-symmetry-breaking phase is suppressed giving way to only dx2−y2 or
n] d3z2−r2 single-component phases. Based on these results, we propose that CeIn3 is a candidate for
exhibitinga time-reversal symmetry-breakingsuperconducting state.
o
c
-
r The discovery of unusual superconductivity in the perature and tetragonal anisotropy for electrons paired
p
heavyfermionsystemsfocusedmuchearlyeffortonelec- viaspinfluctuationsinnearlyantiferromagneticsystems.
u
s tronic pairing mechanisms and the possibility that the The phase diagram is obtained numerically through the
. resulting superconducting states have lower symmetry propergeneralizationofthefluctuationexchangeapprox-
t
a
than the underlying crystalline lattice. The competi- imationforthesinglebandHubbardmodeltothesuper-
m
tion between superconductivity and magnetic ordered conducting state, a generalization which is necessary to
- states suggests that spin fluctuations are a prime can- resolve the relative stability of nearly degenerate uncon-
d
n didate to form the glue that binds electrons into Cooper ventional pairing states. We find that for cubic lattices
o pairs. Power law dependencies of thermodynamic prop- the stable superconducting state has the time-reversal
c erties and phase diagrams that include multiple super- symmetrybreakingformdx2−y2±id3z2−r2. Smalltetrag-
[
conducting states,forexample the H−T phase diagram onal distortions lift the degeneracy between dx2−y2 and
1 of UPt3 and the x−T phase diagram of U1−xThxBe13 d3z2−r2 pairing states leading to two superconducting
v [1], point to the lower symmetry of the superconduct- phase transitions as a function of decreasing tempera-
4
ing orderparameter. As a consequenceofhavingalower ture,butasthedegeneracyisliftedfurtherthroughlarger
4
symmetryincomparisontothatforthelattice,theorder tetragonaldistortionsthesecondtransitionissuppressed.
6
1 parameterofanunconventionalsuperconductorcanhave On the basis of these results, we suggest that the low-
0 moredegreesoffreedomthanexhibitedinaconventional temperature superconducting state in simple cubic sys-
4 superconductorleadingtophasediagramsinvolvingmul- tems with strong antiferromagnetic correlations, such as
0
tiple superconductingstates,butmicroscopicmodelsare CeIn3, are strong candidates for realizing multiple su-
/
t neededto shedlightonthe connectionbetweenthe pair- perconductingphasesincludingalow-temperaturephase
a
m ing interactionand the pairing states that are produced. with broken time-reversal symmetry.
Themicroscopicbasisforourcalculationsisthesingle-
- Recently, the interplay between crystal lattice sym-
d metry and spin-fluctuation induced pairing instabilities band Hubbard model,
n
o wKausroekxiplaonrdedAboykMi [o3n].thTouhxeyanfidndLotnhzaatriscphin[2]flaunctduAatriiotna-, H =−X(cid:16)tijc†i,σcj,σ+h.c.(cid:17)+UXc†i,↑ci,↑c†i,↓ci,↓,
c i,j,σ i
v: induced pairing into the unconventional dx2−y2 state is (1)
most effective for producing large transition tempera-
i where tij represents the electron hopping amplitude be-
X tures (T ) for quasi two-dimensional lattices, a result
c tweensitesiandj andU istheon-siteinteractionenergy
r that is consistent with the observation that the highest
between up and down spin electrons. The values of t
a ij
T values occur in the quasi two-dimensional cuprates
c reflect the underlying lattice structure. For tetragonal
and the more recent finding that T in the cubic heavy
c lattices, the hopping parameter for unit displacements
fermion system CeIn3 [4] is about one order of mag- in the x or y directions, t , is distinct from the same
xy
nitude less than is observed in a collection of related
for unit displacements along the z axis, t . For simplic-
z
quasi two-dimensional compounds, such as the series
ity we set the hopping integrals equal to zero for larger
CenTmIn3n+2m where T = Rh or Ir and n = 1 or 2 and displacements in which case the non-interacting electron
m=1 [5].
bandwidth is equal to W0 =8txy+4tz.
In this Letter we report on model calculations of the An approximation scheme must be employed for cal-
superconducting phase diagram as a function of tem- culations based on this model. We use the fluctuation
2
exchange approximation (FLEX) of Bickers, White, and focus here is on results for the superconducting tran-
Scalapino [6], a numerically-based scheme that is con- sition temperature and the associated order parameter
serving in the sense described by Baym [7]. FLEX pro- symmetry. We obtain these results by determining the
vides aself-consistentdescriptionofboththe quasiparti- spatial dependence of the pair wave function, ψ(r), for
cles and the magnetic-fluctuation induced pairing inter- thestablesuperconductingstateandtheassociatedpair-
action for a given on-site interaction strength, tempera- ing amplitude, m . These quantities are obtained from
p
ture,andbandfilling. Itisexpectedthatresultsobtained theself-consistentresultfortheanomalousGreen’sfunc-
withFLEXarequantitativelyaccuratefor,atbest,weak- tion, F, via
teov-einntwerimtheindiathteiscroaunpglientgh,eir.ee.aUre/Wno0ta<∼bl1e[q8u,a9l]i.taHtiovweefvaeirl-, F↓↑(τ →0−,r)≡hcr=0,↑cr,↓i=mpψ(r). (2)
ures such as, as will be discussed shortly, violations of Other than requiring that ψ(r) is even as a function of
the Mermin-Wagner-Hohenberg theorem for d-wave su- r (i.e. we restrict ourselves to singlet-pairing), no as-
perconducting states. sumption is made on the symmetry of the pairing state.
In contrast to other works, our formulation of FLEX To permitany possiblesinglet-pairingstate to emergein
includes the entire set of fluctuation diagrams for the these calculations, we initialize the self-consistent FLEX
electron self-energy in both the normal and supercon- equations with a small, spatially random pairing field
ducting states, i.e. we include all possible combinations to induce a pairing amplitude in all possible symmetry
of particle-like, hole-like and anomalous Green’s func- channels. Thesmallfieldisremovedastheself-consistent
tions. At the formal level, this ensures that response procedure projects the wavefunction of the most stable
functions andGreen’sfunctions areobtainedina consis- pairing state, ψ(r).
tent manner such that conservation laws derivable from We emphasize the mean-field nature of the FLEX
symmetries of the Hamiltonian are obeyed. While Tc is phasediagramford-wavesuperconductivityinthesingle-
determined by self-energy diagrams which contain only band Hubbard model. The mean-field T values essen-
c
one anomalous Green’s function, a more complex set of tially represent the temperature at which superconduct-
diagrams with three anomalous Green’s functions con- ing order emerges locally. The true thermodynamic T
c
tribute to fourth order terms in the Ginzburg-Landau is determined by phase fluctuations that are not present
expansion for the free energy in terms of the supercon- in FLEX. The neglect of these fluctuations is especially
ducting order parameter and these terms are known to problematic in the two-dimensionallimit where they are
determine the relative stability of multicomponent pair- known to eliminate the possibility of a finite tempera-
ing states [1]. We note that the results we obtain for Tc ture phase transition. Nonetheless, it is well known that
intwodimensionsareinagreementatthe10%levelwith a relatively weak interplanar coupling can stabilize su-
those obtainedearlier[10] suggestingthat self-energydi- perconductivity in a quasi-two-dimensional systems and
agramsthatareomittedinmosttreatmentsofFLEXdo the mean-field phase diagram is revealing with respect
not play a large role in determining Tc, at least in the to determining the conditions under which the tendency
two-dimensional limit. toward superconducting order is greatest.
To access sufficiently large system sizes with modest Our primary result, shown in Figure 2, is the calcu-
computationalresources,we combine this formulationof lated superconducting phase diagram as a function of
FLEXwith the dynamicalcluster approximation(DCA) the scaled temperature, kBT/W0, and the ratio of in-
[11]. Essentially, in the DCA large lattice calculations terplanar to intraplanar hopping, t /t for fixed values
z xy
are made feasible by approximating correlation effects of density (n = 0.85 electrons per site) and the ratio
via a smaller embedded cluster. When combined with of the on-site interaction energy to bare electron band-
theDCA,therearethreenumericalparametersinFLEX: width (U/W0 =0.5). We focus on three features in Fig-
thenumberofMatsubarafrequencypointsused(m),the ure 2: the low-temperature stability of the time-reversal
latticesize(NL)andtheDCAclustersize(Nc). Weshow symmetry breaking state dx2−y2 ± id3z2−r2 for lattices
the dependence of results for the pairing amplitude, mp with cubic and nearly cubic symmetry, two distinct su-
(defined below), versus temperature on these numerical perconducting transitions as a function of temperature
parametersinFigure1foratwo-dimensionallattice. For fornearlycubiclattices,and,inagreementwithprevious
the range of these parameters that we can access, the results, maximal T values occurring in the quasi-two-
c
most significant variation in these curves is through the dimensional limit.
DCA cluster size (bottom graph in Figure 1). Nonethe- We firstfocus onthe cubic limit, t /t =1, forwhich
z xy
less,wefindthattheerrormadeintheresultforTcisonly Tc is at a local minimum. We find that, in agreement
onthe order 10%whenusing a 42 DCA cluster. Inwhat with Arita, Kuroki and Aoki [3], that the stable super-
follows, we use m = 16384 frequency points, NL = 323 conducting state belongs to the two-fold degenerate rep-
lattices and Nc =43 DCA clusters. resentation Γ+3 of the cubic group which is described by
Although a variety of electronic and thermodynamic basis functions of the form dx2−y2 and d3z2−r2. Sym-
properties can be calculated with the FLEX, our main metry considerations allow for either single-component
3
or multicomponent states of the form dx2−y2, d3z2−r2, experimental data for superconductors with cubic sym-
dx2−y2 ±d3z2−r2, or dx2−y2 ±id3z2−r2. Weak-coupling- metry. Data for the compound PrO4Sb12 is consistent
based arguments suggest that the most stable state is with the superconducting phases observed in this cal-
the one for which the superconducting gap is most com- culation [13]. For example, specific heat measurements
plete on the Fermi surface as this will tend to maxi- showevidenceofadoublesuperconductingtransition[14]
mize the condensation energy. This argument favors the and muon-spin relaxation data is consistent with a low-
dx2−y2±id3z2−r2 pairingstateasithaspointnodeswhile temperature time-reversalsymmetry breakingstate [15].
the others have line nodes [12]. However, the strong collective mode that forms the glue
FLEXincorporatesfeedbackeffects betweenquasipar- between paired electrons is due, most likely, to electric
ticles, the order parameter and the pairing interaction quadrupolar rather than magnetic degrees of freedom
and,thus,doesnotnecessarilygeneratethepairingstate [16]. Nonetheless, it is interesting to note the possibil-
expected from weak-coupling theory. Nonetheless, we ity that two different pairing mechanisms may tend to
indeed find that FLEX produces the dx2−y2 ±id3z2−r2 generatethesametime-reversalsymmetrybreakingstate
pairing state in this case. This state breaks time re- in cubic symmetry and, thus, the phenomenology devel-
versal symmetry leading to unusual phenomena such as opedforthesuperconductingstateinPrO4Sb12 [13]may
as bulk magnetic effects associated with the supercon- apply also for magnetically-pairedsuperconductors.
ducting pairs [1]. Assuming that quasiparticles are well- The alloy series U1−xThxBe13 displays a complex su-
defined in the superconducting state, such pairing is ex- perconducting phase diagram as a function of thorium
pectedtogeneratethermodynamicpropertiesthatreflect concentration, x. For 0.2 <∼ x <∼ 0.4 values, a double
thepointnodestructureofthegapfunction,suchashav- superconducting transition is observed in the electron
ing a T3 low-temperature specific heat. specific heat [17] and muon spin resonance data points
For slight deviations from cubic symmetry, i.e. to the appearance of an internal magnetic field below
t /t = 1±ǫ, ǫ ≪ 1, T increases due to the improved the second transition. Sigrist and Rice interpreted the
z xy c
stability of either the dx2−y2 or d3z2−r2 pairing state. magnetic anomaly in terms of a multicomponent time-
Thetwopairingstatesarenolongerdegenerateforǫ6=0 reversal symmetry breaking superconducting state [18].
andthe initialtransitionfromthe superconductingtran- However, Kromer et al. [19] use specific heat and lat-
sitionisintoasinglecomponentstate. However,because tice expansion data to argue for the appearance of spin
of the near degeneracy of these states, a second super- density wave below T . The interpretation of the phase
c
conducting transition occurs into the dx2−y2 +id3z2−r2 diagram for these compounds is still being debated [20].
statewithdecreasingtemperature. Thelow-temperature WesimplynotethatthescenariodescribedbySigristand
stateisnodelessbecaused3z2−r2 basisfunctionsaremem- Riceisconsistentwithpairingstatesthatareobservedin
bers of the symmetric group for tetragonal lattices and, the model that we consider here and should their inter-
consequently, are intrinsically mixed with contributions pretationbeincorrectitwouldpointtotheimportanceof
frombasis functions with s-wavesymmetry. However,s- processes that are neglected in spin-fluctuation models.
waveterms in the pair wavefunctionarerelatively small. The cubic superconductor CeIn3 is, perhaps, the most
For example, the s-like terms haveamplitudes which are likely candidate system for connecting with these model
about 4% of those for d3z2−r2 for tz/txy = 1.1. Con- calculations. OnaccountofthevicinityoftheNe´elstate,
sequently, the gap is correspondingly small for a set of thereisastronglikelihoodthatelectronpairingisrelated
points onthe Fermisurface that correspondto the point to the strong antiferromagnetic correlations in this sys-
nodes that are found for cubic lattices. tem. Tothebestofourknowledge,noevidencehasbeen
As is expected, we find that for lattices favoring in- presented either for a double superconducting transition
plane conduction, i.e. 0 ≤ tz/txy < 1, the dx2−y2 pair- as a function of temperature nor for a low-temperature
ing state is most stable, while the d3z2−r2 (+ s-wave) timereversalsymmetrybreakingstateinthiscompound.
pairing state is most stable when interplanar motion is In light of weak coupling arguments [12], the FLEX re-
enhanced, i.e. t /t > 1. It is apparent from Figure 2 sults presentedhere andthe anomaliesobservedin other
z xy
that the largest T values are obtained in the quasi-two- unconventional cubic superconductors, the presence or
c
dimensionallimit,i.e. t /t →0,inagreementwiththe absence of time-reversalsymmetry breaking in this com-
z xy
resultsreferredto earlier[2, 3]andinlinewith thetrend pound will be revealing with respect to the applicability
observedinthecompoundsCeIn3[4]andCenTmIn3n+2m of simple models, such as the one considered here, for
where T = Rh or Ir and n = 1 or 2 and m = 1 [5]. providinga minimalmicroscopicbasisfor understanding
There also is a region of enhanced Tc for tz/txy > 1 up superconductivity in CeIn3 and related compounds.
to approximately tz/txy ∼ 2 at which point Tc becomes In summary, we have performed fluctuation exchange
vanishingly small. We note that tz/txy = 2 corresponds approximation calculations for the Hubbard model near
to the point at which the interplanar bandwidth equals half-fillingto modelthe phasediagramoftetragonaland
the bandwidth corresponding to in-plane motion. cubicsuperconductingsystemswhosepairingismediated
It is interesting to consider these results in light of by antiferromagnetic spin fluctuations. Near cubic sym-
4
these calculations, we propose that CeIn3 is a candidate
0.03 m = 1024 materialforexhibitingbrokentime-reversalsymmetryin
m = 2048
0.02 the superconducting state.
m = 4096
0.01 We are especially gratefulto D.W. Hess for many use-
0 fulconversations. Theauthorextendshisappreciationto
0.03 N = 162 theDepartmentofPhysicsattheUniversityofCincinnati
L
mp0.02 NNL == 362422 for their hospitality during the summer of 2002 and to
L M. Jarrell and T. Meier for useful conversations during
0.01 N = 1282
L that time. The author gratefully acknowledges partial
0
support from the Graduate College of the University of
0.04
N = 42 Northern Iowa.
0.03 Nc = 82
0.02 Nc = 162
0.01 c
0
0.001 0.002 0.003 0.004
k T / W
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