Table Of ContentStrong-coupling theory of magnetic-exciton-mediated superconductivity in UPd Al
2 3
P. McHale
Max-Planck-Institut fu¨r Physik komplexer Systeme, N¨othnitzer Str. 38, 01187 Dresden, Germany
P. Thalmeier
Max-Planck-Institut fu¨r Chemische Physik fester Stoffe, N¨othnitzer Str. 40, 01187 Dresden, Germany
4
0 P. Fulde
0 Max-Planck-Institut fu¨r Physik komplexer Systeme, N¨othnitzer Str. 38, 01187 Dresden, Germany
2 (Dated: February 2, 2008)
n Thereiscompellingevidencefrominelastic-neutron-scatteringandtunnelingexperimentsthatthe
a
heavy-fermionsuperconductorUPd2Al3 canbeunderstoodasadualsystemconsistingofmagnetic
J excitons, arising from crystal-field-split U4+ levels, coupled to delocalised f-electrons. We have
6 computedthesuperconductingtransition temperatureandthemassrenormalisation arising from a
2 dualmodelwithmaximalspinanisotropyusingastrong-couplingapproach. Wefindaninstabilityto
twopossibleopposite-spin-pairingstateswitheven-orodd-paritygapfunctions. Eachhasalinenode
]
l perpendiculartothec-direction,inagreementwithNMRrelaxation-rate,specific-heatandthermal-
e conductivitymeasurements. Inaddition,bothhavetotalspincomponentSz=0,compatiblewiththe
-
r observationofapronouncedKnightshift andHc2 Paulilimiting. Forparametervaluesappropriate
st to UPd2Al3, we determine the dependence of the superconducting transition temperature Tc on a
. phenomenological coupling constant g andwe investigate theassociated mass enhancement and its
t
a anisotropy.
m
PACSnumbers: 74.20.Mn,74.20.Rp
-
d
n
o I. INTRODUCTION
c
[
Heavy-fermion superconductors are complicated materials which possess a fascinating and rich variety of physical
1 properties and which have stimulated the creation of an equally diverse range of theories in an effort to understand
v them. These materials are dominated by strong electronic correlations giving rise to magnetic and superconducting
0
instabilities. Accordingly, most theories ignore phononic degrees of freedom.
2
Oneofthe mostextensivelystudiedmodelsofheavy-fermionsuperconductivityis aphenomenologicalmodelbased
5
ontheexchangeofspinfluctuationsbetweenheavyquasiparticles. Thelatteroriginateinperiodicresonantscattering
1
0 of conduction electrons close to the Fermi level as described within the Kondo lattice model of Ce compounds1,2.
4 Pairing via spin fluctuations can be considered as a one-component model to the extent that the spin fluctuations
0 originate in the system of heavy quasiparticles and any interaction between conduction electrons and more localised
/ electrons, or conduction electrons and phonons, is neglected. In spite of its simplicity this model has proved to
t
a be very useful in our attempts to understand heavy-fermion compounds on the border of magnetic long-range order,
m
suchasCePd Si 3,CeIn 3 andCeRhIn 4. Itssuccessprobablyderivesfromthefactthatthespin-fluctuation-induced
2 2 3 5
- effective interaction tends to dominate all other channels of interaction when a material is tuned close to the border
d
of magnetism by doping or, as in the Ce compounds mentioned above, by applying pressure. The spin-fluctuation
n
o mechanisminthe Kondolattice may alsobe appropriateforthe Ce-basedheavy-fermionsuperconductorsat ambient
c pressure, CeCu2Si2 and CeCoIn54,5.
: It has become increasingly evident that the Kondo-lattice model is, however, not adequate in the case of heavy-
v
i quasiparticle formation in some Uranium heavy-fermion compounds6,7, where 5f electrons are partly localised and
X partly itinerant. The former occupy crystalline-electric-field (CEF) split 5f2 states. The latter, more itinerant, 5f
r electrons have a strongly enhanced effective mass due to a coupling to virtual excitations between CEF states. This
a
notion of heavy-quasiparticle formation is especially appropriate for UPd Al and it is the starting point of our
2 3
theoretical model for heavy-fermion superconductivity in this compound. Specifically, we consider a novel effective
pairingmechanismwhichisbasedonthevirtualexchangeofcollectiveCEFexcitationsknownas‘magneticexcitons’.
They are propagating bosonic modes in contrastto the overdampedmodes which give rise to pairing in the itinerant
spin-fluctuation model.
TheinterplayofconductionelectronsandCEFexcitationsisawellstudiedsubject. Initially,researchwasfocusedin
two areas8: transportanomalies and superconducting pair breaking orenhancement due to conduction-electronscat-
teringfromdiluteCEF-splitimpurities;andperiodiclatticesofCEFionsinteractingwithoneanotherviaconduction-
electron polarisation. In the latter case attention was focused on collective effects within the CEF system due to the
RKKY interactions mediated by itinerant electrons. Later, White and Fulde9 showed that the inverse effect, viz.
2
conduction-electron mass enhancement via virtual emission and absorption of magnetic excitons, is also important.
They demonstrated that this mechanism explains the enhanced effective electron masses in praseodymium metal.
It is then natural to ask whether the exchange of magnetic excitons between quasiparticles can also mediate
superconductivity. Until recently, no theoretical work had been carried out to answer this question, mainly for two
reasons. Firstly, when the effects on superconductivity of paramagnetic impurities with CEF-split energy levels were
studiedindetail10,itwasfoundthatforrare-earthionsthepair-breakingtransitionmatrixelementsusuallydominate
thepair-enhancingmatrixelements. Secondly,nogoodexampleofasuperconductingcompoundwiththisdualnature,
i.e., magnetic excitons arising from localised f electrons coupled to delocalised f electrons, was known. Recently,
however, evidence has been accumulating that the heavy-fermion compound UPd Al (T = 14.3K, T = 1.8K, see
2 3 N c
Ref. 11) is the first example where such a mechanism is responsible for superconductivity. Since this mechanism
is pair breaking in the s-wave channel (as was already known from the impurity models mentioned above), the
superconducting gapfunction must changesign asa function of k (i.e., the gapmust havea node) as is the case with
spin-fluctuation-mediated superconductivity.
An early indication that UPd Al is a localised-delocalised f system came from magnetic-susceptibility measure-
2 3
ments. Graueletal12 measuredthedcmagneticsusceptibilityinUPd Al andarguedthattheanisotropywhichthey
2 3
observed arose from a tetravalent configuration of the Uranium ions, i.e. U4+ (5f2). Since then Knight-shift13 and
optical14 measurements have supported the notion that UPd Al contains both localised and delocalised f electrons.
2 3
However,the most direct evidence in favour of the dual-system hypothesis comes from inelastic neutron scattering
(INS) and tunnelling experiments. A dispersive crystal-field excitation (magnetic exciton) was observed by Mason
andAeppli15 inINSexperimentswithin the AF phase. Laterhigher-resolutionexperiments16–18 revealeda resonance
feature which appears in the INS spectrum upon entering the superconducting state. This result demonstrates that
a strong interaction exists between the localised and delocalised components of the f-electron system. Pioneering
tunnellingexperiments19 performedonUPd Al -AlO -Pbtunnelling junctionsallowedexperimenterstoview,forthe
2 3 x
firsttime,thetunnelling densityofstates(DOS)ofaheavy-fermionsuperconductor. Strong-couplingfeaturesappear
intheDOSaround1meVclosetothesuperconductinggapenergyreinforcingtheviewthatthe exchangedbosonsare
the magnetic excitons. Takentogether,INS and tunnelling experiments lead Sato and co-workers20 to the conclusion
that superconductivity arises in this material from an effective interaction between itinerant electrons mediated by
magnetic excitons. Using a model two-component Hamiltonian they were able to explain qualitative features of the
INS scattering spectrum and the superconducting tunnelling spectrum.
Subsequently,theoriginofthemagneticexcitonsandtheirglobaldispersion,asmeasuredinRef.15,wasinvestigated
inmoredetail21. Usinganxy-typeinteraction,whereonlytheσ componentsoftheconduction-electronspincouples
to the magneticexcitons,the effective non-retardedpairpotenti±alwasderived. The gapequationsweresolvedwithin
a weak-coupling approach demonstrating that the highest T is obtained by an odd-parity state. In this model,
c
however, the structure of the pairing amplitudes in spin space is complicated, making it unsuitable for going beyond
the non-retarded approximation.
In this paper, therefore, we investigate an alternative model with a simplified interaction between localised and
itinerant5felectrons. ThisinteractionisoftheIsingtype,i.e.,onlytheσ componentoftheconduction-electronspin
z
canscattermagnetic excitons. In this case,the gapequationsnaturally divide into those forequal- andopposite-spin
pairing, in contrast to the usual ‘singlet’ and ‘triplet’ classification that arises in spin-rotation-symmetric models.
In the present work we treat this simplified dual model for UPd Al in a more sophisticated strong-coupling ap-
2 3
proach using a mapping to an electron-boson Hamiltonian and solving the E´liashberg equations for the frequency-
andmomentum-dependentself-energyandgapfunctions. Thelevelschemewhichweusehere,andwhichgivesriseto
theIsing-typeinteraction,isquiterealistic: recentband-structurecalculationsbasedonthislevelschemeaccountwell
for experimental de Haas-van Alphen frequencies in UPd Al 7. We demonstrate that our model can yield a super-
2 3
conducting transition temperature and a mass renormalisation which are consistent with experiment for reasonable
values of the coupling constant.
II. MODEL
Band structure calculations based on the supposition that two of the three Uranium 5f-electrons are localised
reproduce the observed de Haas-van Alphen frequencies in UPd Al very well7. These calculations suggest a level
2 3
schemeforthelocalisedU5fstateswhichweadopthere. Accordingtothejj-couplingschemetheU4+(5f2)ionshave
total angular momentum J = 4. The twofold degeneracy of the ionic ground state is lifted by a crystalline electric
field. We consider only the excitation between the non-degenerate ground state Γ = 1(J =3 J = 3 ) and
| 4i 2 | z i−| z − i
the first excited state Γ = 1(J =3 + J = 3 ). The CEF energy splitting is of the order ∆ 6 meV, as
| 3i 2 | z i | z − i ≃
obtained from fitting the magnetic-exciton dispersion obtained from INS15 measurements to theoretical results21.
Although such a model can give rise to induced magnetism via mixing of the states Γ and Γ , we take the
3 4
| i | i
3
view here that the underlying antiferromagnetic (AF) order is not an important consideration in the description of
heavy-fermionsuperconductivity in UPd Al . AF order mainly leads to a folding down of the conduction bands into
2 3
the AF Brillouin zone. Also, the magnetic-exciton dispersion in the ordered phase is not appreciably different from
that in the paramagnetic phase, due to the fact that T ∆21. Furthermore, in mean-field theory, the AF order
N
≪
parameter will only slightly modify the superconducting T due to a reconstructionof the conduction-electron states
c
close to the AF Bragg planes.
Accordingly we consider a three-dimensional lattice of localised 5f2 CEF states and itinerant f-electrons. In the
subspace Γ , Γ we may write the CEF part of the Hamiltonian as
3 4
{| i | i}
H =∆ S , (1)
CEF iz
i
X
where S denotes a pseudospin (S=1). In this representation we interpret the CEF ground state as having S = -1
2 z 2
andenergy ∆/2andthe excitedCEFstate ashavingS = 1 andenergy∆/2. Inthe pseudospinrepresentation,the
− z 2
only non-zero component of the physical total angular momentum J is the J -component:
z
0 1
J = γ 2 =γS , (2)
z 1 0 x
(cid:18) 2 (cid:19)
where γ =6. Then the full two-component Hamiltonian may be written as
H = ǫkc†k,σck,σ+∆ Siz −J SxiSxi+δ−I σizSix. (3)
k,σ i iδ i
X X X X
The third term in H is a nearest-neighbour superexchange interaction between localised 5f states. The last term is
the exchangeinteractionbetweenlocalised5fCEFstatesandthe delocalised5fconductionelectrons;the conduction-
electron spin operator is
σ =ψ (r )σ ψ (r ). (4)
i α† i αβ β i
After carryingouta Holstein-Primakofftransformation(validatlow temperaturesT ∆)andanadditionalBogoli-
≪
ubov transformation involving the resulting Holstein-Primakoff bosons (see the Appendix), H takes the form
H = ǫkc†k,σck,σ+ ωq(α†qαq+1/2)−I drψα†(r)σαzβψβ(r)φ(r) (5)
k,σ q Z
X X
where
1 1 ∆
φ(r)= √V 2λq αq+α†−q eiqr; λ2q = ωq. (6)
Xq (cid:16) (cid:17)
ω is the dispersion of those bosons which have creation and destruction operators α and α respectively (see the
q †q q
Appendix).
We define the normal electron and exciton Green’s functions as follows:
G (r r,τ τ ) = Tψ (r,τ)ψ (r,τ ) (7)
αβ − ′ − ′ −h α β ′ ′ i
D(r r,τ τ ) = Tφ(r,τ)φ(r ,τ ) . (8)
′ ′ ′ ′
− − −h i
Note that φ(r) S (r), as can be seen immediately by comparing the terms I in Eqs. (3) and (5), and so D is
x
∼ ∼
essentially a pseudospin susceptibility. Furthermore, as in the theory of phonon-mediated superconductivity, φ(r) is
real,φ(r)=φ (r), andcommutes with itself, [φ(r),φ(r )]=0,conditions whichallowus to use Wick’s Theorem. The
† ′
E´liashberg equations for the conduction-electron self-energy Σ(p,iω ) and gap function Φ(p,iω ) which follow from
n n
the Hamiltonian Eq. (5) are
T
Σ(p,iω ) = K(p p,iω iω )G(p,iω ) (9)
n N − ′ n− n′ ′ n′
pX′ωn′
T
Φ(p,iω ) = p K(p p,iω iω )G(p,iω )2Φ(p,iω ). (10)
n N − ′ n− n′ | ′ n′ | ′ n′
pX′ωn′
4
N is the total number of lattice sites. The kernel K is given by
I2∆ 1
K(q,iν )= I2D0(q,iν )= . (11)
n − n 2 ν2+ω2
(cid:18) (cid:19) n q
WehavedecidednottorenormaliseD0 bytheinteractionI betweenmagneticexcitonsandelectrons. Thisinteraction
is, however, already included in D0 to the extent that we model the exciton dispersion ω by the true experimental
q
dispersion. A particularly noteworthy feature of this formulation is that the effective interaction is dominated by
its static part and is strongly peaked in q-space at the antiferromagnetic wavevector Q = (0,0,π). This strong
c
dependence on wavevector contrasts with the practically wavevector-independent interaction in the usual phonon-
mediated superconductivity. The difference arises because, in the phonon problem, the quantity corresponding to λ
q
has an additional factor, ω , leading to a wavevector-independent static phonon propagator. A strong interaction
q
betweenthecollectivemodesofthelocalisedmomentsandtheheavyconductionelectronsatQ=(0,0,π)hasactually
c
been observed17,18 in UPd Al . The electron Green’s function is related to the electron self-energy via the Dyson
2 3
equation
G−1(p,iωn)=iωn (ǫp µ) Σ(p,iωn). (12)
− − −
The prefactor p in Eq. (10) is the expectation value of the Ising spin-spin interaction σˆzσˆz in the spin part of the
pair wave function χ = S, S (S here should not be confused with the pseudospin introduced earlier):
z
| i | i
p= χσˆzσˆz χ . (13)
h | | i
In the opposite-spin pair (OSP) states,
1 ( )
χ = √2 |↑↓i−|↓↑i , (14)
| i ( √12(|↑↓i+|↓↑i)
p reduces to
p= σˆzσˆz = 1. (15)
h↑↓| |↑↓i −
Ontheotherhand,foreachoftheequal-spinpair(ESP)states, χ = and ,pis+1. Notethatspinrotational
| i |↑↑i |↓↓i
symmetry is brokenin a maximal(Ising-type)wayin Eq.(10). Consequentlythe usualclassificationofpairing states
into an (even-parity) singlet with S =0 and an (odd-parity) triplet with S =1 is no longer valid. In our model, the
1,0 pair state no longerhas the same energyas that of the 1, 1 pair states. The result is a different classification
| i | ± i
into an ESP doublet and two OSP singlet states belonging to irreducible representations of the D group (Table I).
6h
Finally, the conduction-bandfilling,defined asthe ratioofthe number ofelectronsto the number oflattice sites,is
1
n= n , (16)
k
N
k
X
where n is the quasiparticle occupation factor:
k
n = n +n =2T G(k,iω ). (17)
k k k n
↑ ↓
Xωn
Experimentally, the magnetic exciton has its strongest dispersion ω along the c-direction. Neglecting the weaker
q
dispersion in the a-b plane allows us to reduce the three-dimensional problem to one dimension:
T dp
Σ(p ,iω ) = K(p p ,iω iω ) ′ G(p ,p ,iω ) (18)
z n Nz pX′zωn′ z − ′z n− n′ Z (2π⊥)2 ′⊥ ′z n′
T dp
Φ(p ,iω ) = p K(p p ,iω iω )Φ(p ,iω ) ′ G(p ,p ,iω )2 (19)
z n Nz pX′zωn′ z− ′z n− n′ ′z n′ Z (2π⊥)2| ′⊥ ′z n′ |
1 dk
n = Nz A (2π⊥)2nk, (20)
Xkz Z
5
provided we can carry out the integrals along the perpendicular direction analytically. Here, N is the number of
z
lattice sites in the z-direction and A is the area of the (hexagonal) unit cell in the plane. We model the exciton
dispersion by
ω(q )=ω [1+βcos(q )]; 0<β 1, (21)
z ex z
≃
This form of the dispersion along the c-direction describes qualitatively the observed excitation branch15. Here
2βω 8 meV is the overall dispersion width and (1-β)ω 1 meV is the exciton gap at the AF wave vector
ex ex
Q=(0,0≃,π). The quantity ω is a characteristic exciton energy.≃We choose β = 0.8 and ω 5 meV 60 K. The
c ex ex ≃ ≃
detailed RPA theory of the magnetic-exciton dispersion and a fit to the INS data of Ref. 15 was given in Ref. 21.
That sheet of the Fermi surface of UPd Al thought to be most important in bringing about heavy-fermion
2 3
behaviour21 has the shape of a corrugated cylinder with its axis aligned along the c-direction of the hexagonal
lattice. We thereforechoosetomodelthe electrondispersionasthe sumofastronglydispersivepartinthe planeǫ
k⊥
and a weakly dispersive part in the c-direction ǫ . We approximate the hexagonal unit cell in the plane (k ,k ) by
kz x y
a circle with radius k , chosen so that the hexagon and circle have the same area. (The areas must be the same in
0
order that the maximum value of the band filling is two in both cases.) Then, assuming a parabolic dispersionin the
plane ǫ = ǫ w2 (0 < w = k /k < 1), we may carry out the integrals in Eqs. (18), (19) and (20) analytically. In
k⊥ 0
the followingw⊥e measureallen⊥ergies,temperatures andfrequencies in units of ǫ . The reducedE´liashbergequations
are ⊥
T
Σ(pz,iωn) = N K(pz−p′z,iωn−iωn′)Gz(p′z,iωn′) (22)
z pX′zωn′
T
λ(T)Φ(pz,iωn) = pN K(pz−p′z,iωn−iωn′)Mz(p′z,iωn′)Φ(p′z,iωn′) (23)
z pX′zωn′
2T (z 1)2+(z )2
′ ′′
n = 1 ln − , (24)
− N (z )2+(z )2
z kzX,ωn>0 (cid:20) ′ ′′ (cid:21)
where
g
K(q ,iν ) = (25)
z n 2 2
(ω /ω ) +(ν /ω )
qz ex n ex
1 (z 1)2+(z )2
′ ′′
G′z(p′z,iωn) = −2ln (z−)2+(z )2 (26)
(cid:20) ′ ′′ (cid:21)
1 z z
G′z′(p′z,iωn) = − tan−1 −z ′ −tan−1 −z′ (27)
(cid:26) (cid:18) ′′ (cid:19) (cid:18) ′′(cid:19)(cid:27)
1 1 z z
Mz(p′z,iωn) = z tan−1 −z ′ −tan−1 −z′ . (28)
′′ (cid:26) (cid:18) ′′ (cid:19) (cid:18) ′′(cid:19)(cid:27)
and
λ(T)=1 for T =T . (29)
c
We have lumped together numerical prefactors and the original coupling constant I into a new coupling constant g
having dimensions of energy:
I2∆ 1 k2 1
g = 0 , (30)
2 2c2π ω2
(cid:18) (cid:19) ex
where c is the lattice constant in the z-direction. The value of g is not calculated; rather it is considered a model
parameter motivated by experiment. The complex number z has a real part
z′ =− ǫp′z −µ+Σ′(p′z,iωn) (31)
and an imaginary part (cid:0) (cid:1)
z =ω Σ (p ,iω ). (32)
′′ n− ′′ ′z n
6
TABLEI:Spinandorbital structureof thepossible gap functionswhich aresolutions ofthe E´liashberg equationsfor thedual
model of UPd2Al3 used in this paper.
p |χi=|S, Szi D6h repres. spin pairing Φ(pz)
-1 |0,0i = √12(|↑↓i−|↓↑i) Γ+1 (A1g) OSP cos(cpz)
-1 |1,0i = √12(|↑↓i+|↓↑i) Γ−1 (A1u) OSP sin(cpz)
+1 |1,±1i = |↑↑i,|↓↓i Γ−1 (A1u) ESP sin(2cpz)
We choose a simple tight-binding form for ǫp′z
ǫp′z = α2 cos(p′z); α≪1 (33)
where α determines the degree of corrugationof the FS cylinder along c.
ThemomentumconvolutionsinEqs.(22)and(23)wereevaluatedwiththeaidofafast-Fourier-transformalgorithm
ona32 32lattice. The correspondingfrequencysumswerecarriedoutusingthe renormalisationgrouptechniqueof
Paoand×Bickers22 whichallowsaconsiderablereductionofthecomputationaleffort. Between240and480Matsubara
frequencieswerekeptateachstageoftherenormalisationgroupprocedure. Therenormalisationprocedurewasstarted
at a temperature T = 0.01ǫ and the frequency sum cut-off used was Ω 15ǫ . The renormalisation procedure
0 c
restricts us to discrete tempe⊥ratures so that the point at which the condition≈in E⊥q. (29) is met must be determined
by interpolation. The discrete temperatures were sufficiently close that a linear interpolation was adequate.
III. RESULTS AND DISCUSSION
Scattering by isolated impurities with CEF excitations is usually pair breaking in the s-wave channel8 because the
dipolarexchangeinteractionswhichbreakthesingletstatearestrongerthanspin-conservingquadrupolarinteractions
between conduction electrons and CEF states. The s-wave state (where Φ(p ) is nodeless) is also not favourable in
z
our case of a periodic lattice of partly localised 5f-electrons which has dispersive CEF excitations. One can most
clearly see this in the following way. In the singlet channel p = 1. Because the kernel K(q,iν ) is strongly peaked
n
at q = π and ν =0, the gap equation may then be crudely ap−proximated in the following way:
z c n
π π
Φ(p ,iπT) C(p , )Φ(p ,iπT), (34)
z z z
≃− c − c
where C(p ,π) is a positive number. Therefore a finite gap Φ(p ,iπT) must change sign on translation through
z c z
Q = π, i.e., the gap must possess the symmetry Φ(p π)= Φ(p ), which excludes an s-wave gap function.
z c z − c − z
Howevera non-s-wavesuperconducting state with nodes may take advantageof the excitondispersionand become
stable. From numerical solutions of the linearised gap equations for our model we find that the instability with the
highestT is accidentallydoublydegenerate. The correspondinggapfunctions transformas the even-andodd-parity
c
OSP states, cos(cp ) and sin(cp ). The former is the usual singlet state; the latter is the S =0 part of the triplet
z z z
which would appear in a theory with full spin rotational symmetry. We also find an instability in the ESP channel
but at a much lower temperature. The corresponding pairing symmetry is Φ(p ) sin(2cp ). We summarise our
z z
∼
results in Table I.
The symmetries of the orbital pair wavefunctions can be understood in the following way. By comparing our self-
energy equations with those obtained from a four-fermion interaction we see that the effective interaction between
quasiparticles in our theory is
γ γ Vˆ γ γ =I2D0σz σz , (35)
h 2 4| | 1 3i γ1γ2 γ3γ4
or equivalently
Vˆ =I2D0σˆzσˆz. (36)
Hence the pairing interaction in some χ channel is
| i
V (q,iν )= χσˆzσˆz χ I2D0(q,iν ). (37)
χ n n
h | | i
We note that our equations go beyond previous calculations6,21 in that our formulation includes the full momentum
and frequency dependence of the effective interaction V (q,iν ).
χ n
7
In the OSP channel the static interaction is
I2∆/2
V(q )= I2D0(q )= , (38)
z − z ω(q )2
z
whichisstronglypeakedat π. Thecorrespondinginteractioninrealspaceisthereforeattractivewhenthequasiparti-
c
cles are seperated by a lattice spacing in the c-direction. Now cos(cp ) and sin(cp ) are peaked with equal amplitude
z z
in real space at z = c. Hence they are equally well suited to take advantage of the attractive part of the interaction
and haveequalsuperconducting transitiontemperatures. The ESP interactionis the negativeof the OSP interaction
and so it first becomes attractive at z = 2c in real space. The only odd-parity wavefunction which is peaked in real
space at this position is sin(2cp ).
z
Ultimately, the degeneracy of the OSP states is a peculiarity of the particular CEF level scheme we adopt for the
localised5f statesandofthe approximateformoftheir excitondispersionwhichwe assume(viz. stronglywavevector
dependent only in the z-direction). Both OSP states are, in fact, compatible with present experimental evidence.
OSP states, regardless of the value of total spin angular momentum S, give rise to a reduction in the paramagnetic
susceptibility on entering the superconducting state. This reduction comes about because oppositely paired spins
have no magnetisation. Furthermore, such a reduction in the Pauli susceptibility leads to an upper critical field H
c2
which, in some cases, can be smaller than that due to the Meissner effect. This effect is called Pauli limiting23.
A pronounced Knight-shift reduction at Pd sites below T (Refs. 13,24) and H Pauli-limiting25 have both been
c c2
observedinUPd Al . NotealsothatbothOSPgapfunctions havenodelinesperpendiculartothe c-direction. Inthe
2 3
even-parity case they are located at k = 1Q , i.e., at the Bragg planes and zone boundaries of the AF Brillouin
z ±2 z
zone,whereasintheodd-paritycasetheyarelocatedatk =0. Experimentally,theexistenceofnodelineswasinferred
z
fromNMRrelaxation-rate26,27,specific-heat28 andthermal-conductivitymeasurements29. Theseexperimentsdidnot,
however,locatethenodepositionalongthec-directionandsothecorrectgapsymmetryinUPd Al presentlyremains
2 3
anopenproblem. To distinguishbetweenthe possiblenodalgapfunctions,itis veryimportantto performfield-angle
dependent measurements of the specific heat or of the thermal conductivity at low temperature. As proposed in
Ref. 30, such measurements may be able to locate the node line in q-space.
The results of our numerical calculations of the superconducting transition temperature in the ESP and OSP
channels are shown in Fig. 1. The superconducting instability occurs first in the OSP channel. We have carried out
the calculations with ω =0.01ǫ . This excitonenergy correspondsto about 60Kfor anelectronic bandwidth of 0.5
ex
eV.One thereforeobtains atrans⊥itiontemperatureinthe OSPchannelcloseto the experimentalvalue11 (T =1.8K)
c
with a dimensionless coupling constant g/ǫ of about 0.5. At this point we remind the reader that this temperature
is in the range of validity of our theory T ⊥ ω ∆.
c ex
≪ ∼
That contribution to the mass renormalisation m /m (m is the band mass) arising from the momentum depen-
∗ b b
denceoftherealpartoftheselfenergyissmallandsom /m simplyreducestotheE´liashbergrenormalisationfactor,
∗ b
Z(p,iω ), which is practically momentum-independent. The weak dependence of Z(p,iω ) on p follows from the
n n
combined effect of a strong exciton dispersion along the c-axis and a weak electron dispersion in the same direction.
Our results for the mass renormalisation are shown in Fig. 2. Note that the mass renormalisation is approximately
linear in g I2 (see Ref. 31). Ab initio calculations of the band masses m /m (m is the free electronmass) for the
b 0 0
∼
γ ring and β ring of the cylindrical Fermi surface have been carried out by confining two of the three 5f electrons to
the Uranium ions7. These values, in conjunction with a mass enhancement of m /m of about 10, yielded values of
∗ b
m /m whichwereingoodagreementwithexperiment32. Wefindthatsuchalargevalueofthemassrenormalisation
∗ 0
correspondsto a value of the coupling constant(g/ǫ 2) which is roughly a factor of 4 largerthan that requiredto
reproduce the transitiontemperature (g/ǫ 0.5). ⊥On≈e shouldnote that the AF long-rangeordermay actto reduce
T , an effect which has not been considere⊥d≈here. Therefore the proper g may indeed be larger than 0.5.
c
We have also carried out our calculations in the case that the Fermi surface is purely cylindrical (α = 0). The
results are practically unchanged. This case, however, has the appealing feature that the bracketed factor in the
definition of the coupling parameter (see Eq. 30) is related in the following simple way to the (constant) density of
states per spin, N(ω),
I2∆ 1
g = N(µ)ǫ . (39)
2 ⊥ωe2x
The dimensionless coupling constant can now be written
1
2
g/ǫ = (λ λ ) λ , (40)
1 2 3
⊥ 2
8
ω ε ; α β
= 0.01 = -0.1; = 0.8
ex
0.08
0.06 | χ > = | 1,0 > or | 0,0 >
x | χ > = | 1,-1 > or | 1,+1 >
e
ω
/
c0.04
T
0.02
0
0 1 2 3 4 5 6 7 8 9 10
ε
g/
FIG. 1: The dependence of the superconducting transition temperature Tc on the coupling constant g [Eq. (30)]. The band
filling n is 0.6.
ω ε ; α β
= 0.01 = -0.1; = 0.8
ex
45
40
35
30
b
m 25
/
*m 20 m* /m
min b
15 *
m /m
max b
10
5
0
0 1 2 3 4 5 6 7 8 9 10
ε
g/
FIG. 2: The dependence of the mass renormalisation m∗/mb on the coupling constant g [Eq. (30)]. The band filling n is 0.6
as in Fig. 1. m∗min and m∗max are the extremal effective masses over all values of qz. m∗min occurs at qz = 0; m∗max occurs
at qz = πc. m∗ is practically constant over the Fermi surface for the range of values of g which we consider in this work
(0.5<g/ǫ <4).
⊥
9
where the dimensionless constants λ are defined by
i
λ = IN(µ) (41)
1
λ = ∆/ω (42)
2 ex
1
λ = . (43)
3
∆N(µ)
Of these three parameters,the first, λ , is not presently knownfor the Uranium 5f systems. The second,λ , is about
1 2
unity. The third, λ , is approximately167,if we makethe estimate N(µ) 1eV 1 per cell per spin. The value ofthe
3 −
≈
coupling parameter we require to get the experimentally observed transition temperature then leads to the estimate
λ 0.08. A value of λ which is a factor of 2 larger would yield a mass renormalisation in good agreement with
1 1
≈
experiment, but then T would be larger than the experimental value (1.8K) by roughly a factor of 1.6. As already
c
mentioned, this discrepancy may be alleviated by the effects of AF order. The values for λ obtained here for a dual
1
5f compound are of the same order of magnitude as for 4f systems9,31.
IV. CONCLUSIONS AND OUTLOOK
MotivatedbyexperimentsonUPd Al ,wehaveexamined,forthefirsttime,amagnetic-excitonmodelofsupercon-
2 3
ductivity within a strong-couplingretarded framework. The model naturally explains the strong interaction between
the collective modes of the localised moments and the heavy conduction electrons which is observed at Q=(0,0,π)
c
in UPd Al . Solutions of the E´liashberg equations show that the model favours a superconducting instability for the
2 3
even- or odd-parity OSP states. Each state has line nodes perpendicular to the c-direction and total z-component of
spin S = 0, characteristics which are compatible with measurements of UPd Al . We find that a superconducting
z 2 3
transitiontemperatureandmassrenormalisation,eachingoodagreementwiththeirexperimentalvaluesinUPd Al ,
2 3
canbe obtainedusingreasonablevalues ofthe parametersin the theory. Takentogether,these results strengthenthe
argument in favour of a magnetic-exciton system in UPd Al .
2 3
The evidence for dual f-systems in other Uranium compounds does not weigh as heavily as it does in the case of
UPd Al . In UPt the number of itinerant f-electrons remains controversial. However band-structure calculations
2 3 3
based on the assumption that two of the three 5f-electrons are localised reproduce the observed de Haas-van Alphen
frequencies as well as the anisotropic heavy electron masses in this compound very well7.
Recently unconventional superconductivity with split-T superconductivity has been discovered in the Pr-based
c
skutterudite PrOs Sb 33,34. Thermodynamic and transport measurements suggest that the observed heavy-fermion
4 12
state arises from the interactionof electric quadrupole moments of the CEF-split 4f2 states of Pr3+ with the conduc-
tionelectrons33. Thegroundstateisprobablyasingletandthefirstexcitedstateisprobablyatriplet,approximately
0.5meVhigher,withlargeoff-diagonalquadrupolarmatrixelementsconnectingthetwostates. Thelargenessofthese
matrixelements,togetherwiththeverysmallsinglet-tripletsplitting,stronglysuggestsaneffectivemassrenormalisa-
tionviavirtualCEFexcitations. Finally,experimentsrevealthatthesuperconductingorderparameterisanisotropic,
making this compound another candidate for CEF-exciton-mediated superconductivity, but this time of quadrupolar
nature.
UPd Al providesastrongmotivationforthestudyofdualf-electronsystemssuchastheonediscussedinthiswork.
2 3
Wehaveconsideredadualmodelinwhichallparametersarefixedexceptthecouplingconstant. Itwillbeinteresting
to investigatethe dependence ofthe transitiontemperatureonconduction-bandfilling andonthe formofthe exciton
dispersion, and to look for a mechanism that can discriminate between the even- and odd-parity superconducting
OSP states.
Acknowledgments
P.M. would like to thank P. Monthoux for valuable discussions. P.T. would like to thank K. Maki, Y. Matsuda,
K. Izawa, T. Watanabe and G. Varelogiannis for helpful discussions. The authors acknowledge discussions with G.
Zwicknagl and A. Yaresko.
10
APPENDIX
In this appendix we sketch the derivation of Eq. (5). Our starting point is the set of Holstein-Primakoff transfor-
mations (valid for T ∆),
≪
1
S = +a a (A.1)
z †
−2
S+ = a
= S = 1(a+a ) (A.2)
S− = a† ) ⇒ x 2 †
Inserting these expressions into the second and third sums in Eq. (3) and Fourier transforming, we arrive at the
following form,
W (q) I
W1(q)a†qaq− 22 (a†qa†−q+aqa−q) − 2N c†kασαzβck+qβ(aq+a†−q), (A.3)
q (cid:20) (cid:21) kq
X X
where
W (q)=∆ W (q). (A.4)
1 2
−
The first sum can be diagonalised35, while at the same time introducing only a factor λ in the second sum. To do
q
this we use the Bogoliubov transformation
α u v a
k = k k k , (A.5)
α†k ! vk uk ! a†k !
− −
where u =cosh(θ ) and v =sinh(θ ). If θ is chosen such that
k k k k k
W (k)/W (k)=tanh(2θ ), (A.6)
2 1 k
−
then (ignoring constant additive terms) the first sum in Eq. (A.3) takes the form,
ω (α α +1/2) (A.7)
q †q q
q
X
where the dispersion ω satisfies
q
ω2 =W2(q) W2(q). (A.8)
q 1 − 2
Finally, it is easy to show that
aq+a†q =λq(αq+α†q), (A.9)
− −
where
λ =cosh(θ ) sinh(θ ). (A.10)
q q q
−
Using Eqs. (A.4) and (A.6), together with the identity
ω2
∆ 2W (q)= q, (A.11)
2
− ∆
we can then show that
∆
λ2 = . (A.12)
q ω
q
The interaction term in Eq. (5) follows immediately.
1 A. C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, 1993).
