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Physica B 403 (2008) 977–981

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How do holes get heavy and superconduct?

N. Harrisona,�, S.E. Sebastianb, C.D. Batistaa, S.A. Trugmana

aLos Alamos National Laboratory, Los Alamos, NM 87545, USAbCavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK

Abstract

We discuss the implications of recent de Haas–van Alphen effect (magnetic quantum oscillation) measurements on cubic CeIn3, which

reveal pockets of f-electron holes centered at Q=2 in the Brillouin zone (where Q ¼ ½p; p;p� is the antiferromagnetic modulation vector).

This system had previously been identified as a local-moment magnet in which the f-electrons are completely localized. We discuss the

Lifshitz Fermi surface topological changes in these pockets that takes place in strong magnetic fields and that is likely to take place as a

function of pressure. We further discuss the implications for pressure-tuned superconductivity and the relevance of the experimental

findings to CeRhIn5 and cuprate superconductors.

r 2007 Elsevier B.V. All rights reserved.

PACS: 71.27.+a; 74.25.Dw; 74.70.Tx; 74.72.�h; 75.20.Hr; 75.50.Ee

Keywords: Heavy fermions; Superconductivity; Antiferromagnetism; Quantum criticality; F-electrons; Holes; Kondo; Lifshitz transitions

The question of the origin of unconventional super-conductivity has remained unanswered for almost 30 years[1]. Key findings common to many f-electron systems are (i)that the superconductivity involves Cooper pairs of heavyquasiparticles [2–5] with a strong f-electron character and(ii) that the order parameter is likely to possess nodes[6–10]. More recent discoveries of unconventional super-conductivity at a putative antiferromagnetic quantumcritical point (QCP) [11–13] raise the possibility of spinfluctuation-mediated pairing [14]. Observations of uncon-ventional superconductivity coexisting with local momentantiferromagnetism [15–17] make our attempt to under-stand pairing and delocalization of f-electrons all the morechallenging. This is particularly true for those compoundscontaining Ce for which there is only one 4f-electron to beshared between local moment antiferromagnetic anditinerant superconducting order parameters.

Given the nexus of antiferromagnetism and unconven-tional superconductivity, the delocalization of f-electronsmust be considered in relation to antiferromagnetism.Currently, there are two contending theoretical ideas for

e front matter r 2007 Elsevier B.V. All rights reserved.

ysb.2007.10.274

ng author. Tel.: +1505 665 3200; fax: +1 505 667 4311.

ss: nharrison@lanl.gov (N. Harrison).

how this can occur in 4f-electron systems [18]. The firstinvolves the formation of a spin density-wave (SDW), asoriginally proposed by Hertz and Millis [19,20], in whichbroken translational symmetry (with wavevector Q) givesrise to a reconstructed Fermi surface in a qualitativelysimilar manner to Cr (simplified minimal model depicted inFig. 1a). The heavy Fermi liquid incorporating itinerantf-electrons is considered to survive within the SDWphase, providing a natural explanation for Cooper pairformation involving itinerant f-electrons. However, thisidea cannot explain the existence of large ordered momentswithin the antiferromagnetic phase [21–23], with theconcomitant observation of a ‘‘small Fermi surface’’(not incorporating the f-electrons) in magnetic quantumoscillations experiments [24–26]. The second idea involvesthe local coupling of charge degrees of freedom to theantiferromagnetic QCP, such that the f-electrons becomelocalized inside the antiferromagnetic state [27] (depicted inFig. 1b). Observations of a transformation in the Fermisurface from a ‘‘large Fermi surface’’ (itinerant f-electron)model within the heavy Fermi liquid regime under pressureto a comparatively small Fermi surface within theantiferromagnetic phase are reported to be consistentwith this scenario [26]. This model appears to apply best

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Fermi liquid

QQ

Fermi liquid

ordering

AFM/SDW

ordering

local moment AFM

Fermi liquid

ordering

strong coupling AFM

Q

f-holes

Fig. 1. The two scenarios (a) and (b) for Fermi surface transformation at

a QCP, after Coleman et al. [18], together with a third scenario compatible

with CeIn3 (c). (a) corresponds to the weak coupling SDW scenario in

which a large Fermi surface comprising itinerant f-electrons prevails

throughout, modified only by Fermi surface reconstruction due to the

modulation vector Q. (b) corresponds to the local QCP scenario in which

the f-electrons become localized within the antiferromagnetic (AFM)

phase leading to a small Fermi surface not comprising the f-electrons. (c)

corresponds to an intermediate strong-coupling scenario in which the f-

electrons almost become localized within the antiferromagnetic phase but

remain partially present in the form of small hole pockets situated near

Q=2, with large effective masses determined by antiferromagnetic

correlations. The suppression of Kondo screening within the AFM phase

causes the hybridization between f-electrons and the small Fermi surface

of conduction electrons to become perturbatively weak. Here we need to

consider a highly simplified two band model: the larger section of Fermi

surface on the left is predominantly f-like in character and becomes mostly

gapped within the antiferromagnetic phase on the right leading to small

pockets (this is loosely equivalent to the a-sheet of CeIn3 breaking up

within the AFM phase, leaving behind small f-hole pockets). The other

section of Fermi surface is predominantly composed of conduction

electrons, enabling it to survive mostly intact within the antiferromagnetic

phase (this is loosely equivalent to what happens with the d-sheet in CeIn3[25]). Subtler changes in its topology may result from changes in the

strength of hybridization and/or band-filling effects.

[110] [100] [111] [110]

2.0

1.5

1.0

0.5

F (

kT

)

-30 0 30 60 90

B ( )

140

120

100

80

600 10 20 30 40

B B

B (

mJm

ol-1

K-2

)

Fig. 2. The f-electron hole pockets of CeIn3. (a) Shows two branches of

the r-orbit originally observed by Endo et al. [29] (squares) with the purple

lines showing the magnetic field orientation dependence expected for the

ellipsoids of revolution shown in (b). (c) Shows the contribution of these

holes to the Sommerfeld coefficient, according to a simple model that is

able to explain the experimentally observed magnetic quantum oscillation

data [30].

N. Harrison et al. / Physica B 403 (2008) 977–981978

to non-superconducting materials such as CeCu6 [27]and YbRh2Si2 [28], but does not formally explain largeeffective mass enhancements occur within a local moment

antiferromagnetic phase in which the Kondo effect shouldno longer be operative [26]. Such an explanation is anessential prerequisite for understanding the coexistence ofsuperconductivity and antiferromagnetism [16,17,29].Recent quantum oscillation measurements on CeIn3

have the potential to resolve the above dilemma [30]. CeIn3is an ideal system to study for several reasons. Its cubicsymmetry and simple antiferromagnetic structure [21]make it immune to local quantum criticality [31] andmetamagnetism [32], causing its magnetic behavior to bemuch simpler than anisotropic systems. It also has a largeresidual Sommerfeld coefficient (g � 120mJmol�1 K�2)within the antiferromagnetic phase at ambient pressure[33], making it a good candidate for studying the origin ofenhanced effective masses within a local moment anti-ferromagnetic phase [29]. We have found that approxi-mately 80% of g can be explained by the existence of heavypockets situated at ½p=2;p=2; p=2� in the cubic Brillouinzone, giving rise to several quantum oscillation frequencies.Fig. 2a shows two frequency branches observed by Endoet al. [29], which can be explained by Fig. 2b. Theirtopology corresponds to oblate ellipsoids of revolutionwith minor axes projected along h1 1 1i. The highestfrequency branch (dotted line in Fig. 2a) has only recentlybeen observed [30]. The topology of these pockets corres-ponds precisely to that expected for a three-dimensional

ARTICLE IN PRESSN. Harrison et al. / Physica B 403 (2008) 977–981 979

antiferromagnet doped with holes in which Q ¼ ½p;p;p�[30]. The competition between field-dependent secondnearest-neighbor antiferromagnetic and nearest-neighborferromagnetic correlated hopping processes in a magneticfield causes the effective mass to diverge in a magnetic field,giving rise to observed effective masses as large as 100 timesthe free electron mass at � 40T [30]. Fig. 2c shows thecontribution to g according to the model that reproducesthe experimental data [30]. The field-induced divergence at� 40T, caused by Lifshitz topological transitions in whichthe ellipsoids coalesce, is weakened by a depopulation ofthese pockets.

In order to explain the presence of f-hole pockets inCeIn3, it is necessary to invoke a scenario in which therenormalized lower ‘‘Hubbard’’ band of a strongly coupledantiferromagnet barely intersects the chemical potential at½p=2;p=2;p=2�, causing it to lose a few electrons to thesurrounding conduction band. This situation is depicted asa ‘‘third’’ intermediate scenario between the weak couplingSDW and local moment antiferromagnetic regimes inFig. 1c (albeit using a highly trivialized schematic). Theeffective hole doping is very small in CeIn3, amounting toonly 3% of volume of the cubic Brillouin zone [30]. Such anelectronic structure is the three-dimensional analog ofthat observed in the antiferromagnetic regime of the quasitwo-dimensional antiferromagnetic cuprate Ca2CuO2Cl2[34], indicating a close similarity in the electronic structuresof d- and f-electron antiferromagnets at a microscopic level(i.e. holes are observed to aggregate at ½p=2;p=2; q� inphotoemission experiments on cuprate antiferromagnets).Both the cuprates [37] and CeIn3 [11] become super-conducting as antiferromagnetism is suppressed by pres-sure or by chemical substitution of many holes. Similaritiesbetween these two classes of systems have previouslyonly been made at a relatively phenomenological level[14,17,35,36].

Given that the hole pockets of CeIn3 account for thelions share of the density of electronic states, the emergenceof superconductivity within the antiferromagnetic phaseunder pressure [11,16] is likely to involve the formation ofCooper pairs primarily at these locations in k-space. Thesituation is less clear within the underdoped region of thecuprates due to uncertainty as to the nature of the groundstate at higher levels of hole doping [37]. However, at highpressures or much higher levels of doping, one can expecttheir electronic structures once again to become similar(in making comparisons between these systems, allowancesmust be made for the differences in dimensionality betweenCeIn3 and the cuprates and for the additional presence oflight conduction electron bands in CeIn3). A qualitativelysimilar Lifshitz Fermi surface topological transition to thatobserved under high magnetic fields in CeIn3 should alsooccur as the antiferromagnetic order parameter vanisheswith the Neel temperature TN under pressure. Closureof the antiferromagnetic gap will cause the holes in Fig. 1cto coalesce with holes in neighboring Brillouin zonesperpendicular to Q, eventually causing the large itinerant

f-electron Fermi surface to reappear just inside theantiferromagnetic phase as TN! 0. Hybridization effectsbetween the f-electrons and conduction electrons are alsoexpected to be fully restored. The singular points on theFermi surface caused by the Lifshitz transition can thencontribute more effectively to likely superconductivitywithin the antiferromagnetic phase. The occurrence of amaximum transition temperature Tc of CeIn3 at a pressuremarginally lower than that pc at which TN! 0 alsosuggests the possible role of a Lifshitz transition [16,31].Divergences caused by Lifshitz transitions become more

significant for two-dimensional Fermi surfaces, owing tothe distribution of the singularity over larger (linear)regions of the Fermi surface in k-space (orthogonal to thelayers). Their occurrence as a function of band filling in thecuprates (manifested as a van Hove singularity) has alreadybeen considered as a possible factor in the enhancementof superconductivity [38,39], and may also be relevantwithin the antiferromagnetic phase of CeRhIn5 where thesuperconducting transition temperature is observed to beconsiderably higher than that of CeIn3 [13,17].At ambient pressure, the Sommerfeld Coefficient of

CeRhIn5 (g � 60mJmol�1 K�2) is lower than that of CeIn3and consistent with a scenario in which the f-electrons arealmost completely localized [40]. This is a likely reason whyheavy pockets of holes are not seen in quantum oscillationexperiments on CeRhIn5 at ambient pressure [41]. Oncepressure is applied, however, g increases, becomingsignificantly larger (� 200mJmol�1 K�2) as superconduc-tivity is established within the antiferromagnetic phase atp � 15 kbar. Observations of a large staggered momentpersisting to pressures far beyond 15 kbar implies that asimple Kondo mechanism cannot be responsible for theincrease in g. There is an obvious need to explore smallerpotentially heavier sections of the Fermi surface topologyof CeRhIn5 as a function of p. Quantum oscillationexperiments on CeIn3 [25], CeRh2Si2 [24] and CeRhIn5[26] have thus far only concentrated on the large sections ofFermi surface and the extent to which changes can bereconciled with the scenarios depicted in Figs. 1a and b. Onthe basis of comparisons with band structure calculations,these changes are usually reported to be consistent with alocalized-to-itinerant transition: i.e. Fig. 1b.The available experimental evidence (for example in

CeIn3), however, indicates the need to consider pockets off-holes, which are not captured in the simple pictureof a localized-to-itinerant phase transition at pc at whichTN! 0. In particular, the d-sheet, which is predominantlycomposed of regular conduction electrons, is observed toevolve in a continuous fashion through the critical pressurepc, maintaining the same shape throughout [25] (i.e. thequantum oscillation frequencies retain approximately thesame relative ratios). The absence of a mass divergence atthe QCP at pc does not support a continuous evolution ofthe Fermi surface from an electron sheet to a hole sheet ashas been suggested as evidence for the localized-to-itineranttransition picture [25]. A much simpler interpretation is

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Fig. 3. vF plots. (a) Shows the orbitally averaged vF for the smallest cross-

section orbit of the f-hole pocket in CeIn3 versus the magnetic flux density

B according to the model that is able to describe the quantum oscillation

data very well [30]. BL is the field at which the Lifhsitz transition occurs

whereas Bc is the upper critical field of the antiferromagnet. (b) Shows vFfor the a3 (green circles) and b2 (blue circles) orbits in CeInRh5 versus

pressure p [26]. pc is the pressure at which TN ! 0.

N. Harrison et al. / Physica B 403 (2008) 977–981980

that the d-sheet of CeIn3 continues to be made up primarilyof conduction electrons at p4pc without a significantcontribution from the itinerant f-electrons once the largeFermi surface is recovered. The f-electron contributionmay not be captured by comparisons with bandstructurecalculations, as also implied by the a-sheet behavior. Thelarge size of the a-sheet observed under pressure comparedto that predicted by the itinerant f-electron bandstructurecalculations [25], in conjunction with the observation of amuch smaller a-sheet (� 30 %) in strong magnetic fieldsbetween 60 and 90T [42] (where the f-electrons are essentiallylocalized), suggests the itinerant f-electrons contribute moreto the a-sheet than they do to the d-sheet at p4pc.

Should f-hole pockets appear in CeRhIn5 under pres-sure, there are two ways in which they could contribute tothe Fermi surface topology. The first is that they formisolated sections of Fermi surface that are weakly coupledto the conduction bands due to a suppressed hybridizationwithin the antiferromagnetic phase in a qualitativelysimilar manner to CeIn3 at low magnetic fields [30]. Thesecond is that they exist within close proximity to theconduction electron bands in k-space, becoming hybridizedwith them and giving rise to ‘‘hot spots’’ as in CeIn3 instrong magnetic fields [30]. In this context, ‘‘hot spots’’ isthe term used to describe small regions of the Fermi surfacewhere the effective mass is strongly enhanced compared tothe remainder of the same sheet [43]. In either case, aLifshitz transition would be expected within the antiferro-magnetic phase as a function of p for at least one Fermisurface sheet.

Tracing the orbitally averaged Fermi velocity vF(obtained for orbits) provides a convenient method forinferring the existence of possible Lifshitz transition massdivergences indicated by a collapsing Fermi velocity. As amodel example, we plot in Fig. 3a vF ¼ e_

ffiffiffiffiffiffiffiffiffiffiffiAk=p

p=mn for

the f-hole pockets of CeIn3 (where Ak is the smallest cross-sectional area of the orbit in k-space), obtained using amodel that accurately describes the quantum oscillationsobserved at high magnetic fields [30]. vF can be seen to fall

to zero in an approximately linear fashion as the Lifshitztransition field BL is approached. In Fig. 3b we plot theFermi velocity for the a3 and b2 quantum oscillationfrequencies in CeRhIn5 versus p [26]. The near-lineardependence of vF for the b2 orbit is consistent with aLifshitz transition within the antiferromagnetic phaseinvolving this sheet. The b2 frequency can also be seen toincrease slightly with p as pc is approached [26], as might beexpected were the topology to change. Since this changeinvolves a large sheet of Fermi surface composed primarilyof conduction electrons, the pockets near ½p=2; p=2; q� in af-hole picture would have to become hybridized with theconduction bands near pc in order to explain theexperimental data. The a3 frequency, by contrast, showsno divergence, suggesting that it does not undergo aLifshitz transition in a qualitatively similar manner to thed-sheet of CeIn3. The absence of a divergence for p4pc

also suggests that an alternate picture to the local QCPscenario in CeRhIn5 needs to be considered [25].In summary, by enabling the correlations to be mapped

in k-space, quantum oscillation experiments have anenormous potential to resolve the origin of Cooper pairingin f-electron systems. Much more detailed experimentalstudies of the Fermi surface topologies of CeIn3 andCeRhIn5 under pressure are required to improve thecurrent understanding of the relation between the Fermisurface topology and unconventional superconductivity.

This work is supported by the National Science Founda-tion, the Department of Energy, the State of Florida,Trinity College (Cambridge) and the Institute for ComplexAdaptive Matter.

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