Kondo-lattice screening in a d-wave superconductor

0.26 MB·English

by  Daniel E. Sheehy

Kondo-lattice screening in a d-wave superconductor
Daniel E. Sheehy1,2 and J¨org Schmalian2
1 Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803
2 Department of Physics and Astronomy, Iowa State University and Ames Laboratory, Ames IA 50011
(Dated: February 1, 2008)
WeshowthatlocalmomentscreeninginaKondolatticewithd-wavesuperconductingconduction
electrons is qualitatively different from the corresponding single Kondo impurity case. Despite the
conduction-electron pseudogap, Kondo-lattice screening is stable if the gap amplitude obeys ∆ <
√TKD, in contrast to the single impurity condition ∆ < TK (where TK is the Kondo temperature
for∆=0andD isthebandwidth). Ourtheoryexplainstheheavyelectron behaviorinthed-wave
8
0 superconductorNd2−xCexCuO4.
0
2
n I. INTRODUCTION ∆/D
a 00..0022
J
The physical properties of heavy-fermion metals are

1 commonly attributed to the Kondo effect, which causes ∆c √TKD
∼
3
thehybridizationoflocal4-f and5-f electronswithitin-

erantconductionelectrons. TheKondoeffectforasingle local
] screened
l magnetic ion in a metallic host is well understood1. In moment
e
- contrast,thephysicsoftheKondolattice,withonemag- 00..0011 Kondo lattice
r neticionpercrystallographicunitcell,isamongthemost
t
s challenging problems in correlated electron systems. At
.
t the heart of this problem is the need for a deeper un-
a screened impurityH TK
derstanding of the stability of collective Kondo screen- Hj ∼
m
ing. Examples are the stability with respect to com- J
-
d peting ordered states (relevant in the context of quan- 00..0055 00..11 00..1155 00..22D
n tum criticality2) or low conduction electron concentra-
o tion (as discussed in the so-calledexhaustion problem3). FIG. 1: The solid line is the critical pairing strength
c In these cases, Kondo screening of the lattice is believed ∆c for T 0 [Eq. (33)] separating the Kondo screened
→
[ (shaded) and local moment regimes in the Kondo-lattice
to be more fragile in comparison to the single-impurity model Eq. (4). Following well-known results6,7 (see also Ap-
3 case. In this paper, we analyze the Kondo lattice in a
pendixA),thesingle-impurity Kondoeffectisonlystablefor
v host with a d-wave conduction electron pseudogap4. We
5 demonstrate that Kondo lattice screening is then sig- ∆.Dexp(−2D/J)∼TK (dashed).
1
nificantly more robust than single impurity screening.
8
The unexpected stabilization of the state with screened
1 afixedpointvalueJ∗ =r/ρ0 emergesforfinitebutsmall
moments is a consequence of the coherency of the hy-
. r. Here, ρ is the DOS for ω = D with bandwidth D.
4 0
bridized heavy Fermi liquid, i.e. it is a unique lattice ef-
0 Kondo screening only occurs for J∗ and the transition
fect. We believe that our results are of relevance for the
7 from the unscreened doublet state to a screened singlet
observed large low temperature heat capacity and sus-
0 ground state is characterized by critical fluctuations in
: ceptibilityof Nd2−xCexCuO4,anelectron-dopedcuprate time.
v superconductor5.
i Numerical renormalization group (NRG) calculations
X
The stability of single-impurity Kondo screening has demonstrated the existence of a such an impurity quan-
r beeninvestigatedbymodifyingthepropertiesofthecon- tum critical point even if r is not small but also
a
ductionelectrons. Mostnotably,beginningwiththework revealed that the perturbative renormalization group
of Withoff and Fradkin (WF)6, the suppression of the breaks down, failing to correctly describe this critical
single-impurity Kondo effect by the presence of d-wave point9. For r = 1, Vojta and Fritz demonstrated that
superconductingorderhasbeenstudied. Avarietyofan- the universal properties of the critical point can be un-
alyticandnumerictoolshavebeenusedtoinvestigatethe derstood using an infinite-U Anderson model where the
singleimpurityKondoscreeninginasystemwithconduc- level crossing of the doublet and singlet ground states
tion electron density of states (DOS) ρ(ω) ω r, with is modified by a marginally irrelevant hybridization be-
variable exponent r (see Refs. 6,7,8,9,10,11∝,12|).| Here, tweenthosestates10,11. NRGcalculationsfurtherdemon-
r = 1 corresponds to the case of a d-wave superconduc- strate that the non-universal value for the Kondo cou-
tor, i.e. is the impurity version of the problem discussed pling at the critical point is still given by J r/ρ ,
∗ 0
in this paper. For r 1 the perturbative renormaliza- even if r is not small8. This result applies to the≃case of
tiongroupoftheordi≪nary13 Kondoproblem(r =0),can brokenparticle-holesymmetry,relevantforourcompari-
begeneralized6. WhiletheKondocouplingJ ismarginal, sonwiththeKondolattice. Inthecaseofperfectparticle
2
hole symmetry it holds that8 J for r 1/2.
∗
→∞ ≥
NTmheearnesfiuelltdJt∗h≃eorr/yρ6,0wmhaiychalosothbeerwobisteaifnaeildsftroompraoplaerrglye T/TK
1
describe the criticalbehaviorof the transition,in partic-
ular if r is not small. The result for J as the transition
∗ 0.8
betweenthe screenedandunscreenedstates reliesonthe
assumptionthat the DOS behaves as ρ(ω) ω r allthe local
∝| | 0.6 moment
way to the bandwidth. However, in a superconductor
with nodes we expect that ρ(ω) ρ is essentially con-
≃ 0 0.4
stant for ω > ∆, with gap amplitude ∆, altering the screened
| |
predictedlocationofthe transitionbetweenthescreened
Kondo lattice
0.2
and unscreened states. To see this, we note that, for en-
ergiesabove∆,the approximatelyconstantDOSimplies
∆/D
the RG flow will be governed by the standard metallic 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Kondo result1,13 with r = 0, renormalizing the Kondo
coupling to J = J/(1 Jρ lnD/∆) with the effective FIG. 2: (Color online) The solid line is a plot of the Kondo
0
bandwidth ∆ (see Ref.−9). Then, we can use the above temperature TK(∆), above which V =0 (and Kondo screen-
ingisdestroyed),normalized toitsvalueat ∆=0[Eq.(14)],
result in the erenormalizedsystem, obtaining that Kondo
as a function of the d-wave pairing amplitude ∆, for the
screening occurs for Jρ &r which is easily shown to be
0 case of J = 0.3D and µ = 0.1D. With these parameters,
equivalent to the condition ∆.∆ with −
∗ TK(0) = 0.0014D, and ∆c, the point where TK(∆) reaches
e∆ =e1/rT , (1) zero,is0.14D [givenbyEq.(33)]Thedashedlineindicatesa
∗ K spinodal,alongwhichthetermproportionaltoV2 inthefree
where energy vanishes. At very small ∆ <2.7 10−4D, where the
×
1 transition is continuous, the dashed line coincides with the
TK =Dexp , (2) solid line.
−Jρ
(cid:18) 0(cid:19)
is the Kondo temperature of the system in the absence
ofpseudogap(whichweareusingheretoclarifythe typ- with D the conduction electron bandwidth. Below, we
ical energy scale for ∆∗). Setting r = 1 to establish the shallderivea moredetailed expressionfor∆c; inEq.(3)
implication of Eq. (1) for a d-wave superconductor, we we are simply emphasizing that ∆ is large compared to
c
see that, due to the d-wave pseudogap in the density of T [and, hence, Eq. (1)].
K
states, the conduction electrons can only screen the im-
Inaddition,wefindthatfor∆<∆ ,therenormalized
c
purity moment if their gap amplitude is smaller than a
mass only weakly depends on ∆, except for the region
critical value of order the corresponding Kondo temper-
close to ∆ . We give a detailed explanation for this en-
c
ature T for constant density of states. In particular,
K hanced stability of Kondo lattice screening, demonstrat-
for ∆ large compared to the (often rather small) energy
ingthatitisadirectresultoftheopeningofahybridiza-
scaleT ,thelocalmomentisunscreened,demonstrating
K tion gap in the heavy Fermi liquid state. Since the re-
the sensitivity of the single impurity Kondo effect with
sult was obtained using a large-N mean field theory we
respect to the low energy behavior of the host.
stress that such an approachis not expected to properly
Given the complexity of the behavior for a single im-
describe the detailed nature close to the transition. It
purity in a conduction electron host with pseudogap, it
should,however,giveacorrectorderofmagnituderesult
seemshopelessto study the Kondolattice. We willshow
for the location of the transition.
belowthatthis mustnotbethe caseandthat,moreover,
To understand the resilience of Kondo-lattice screen-
Kondo screeningis stable far beyondthe single-impurity
ing, recall that, in the absence of d-wave pairing, it is
result Eq. (1), as illustrated in Fig. 1 (the dashed line
well known that the lattice Kondo effect (and concomi-
in this plot is Eq. (1) with ρ = 1/2D). To do this,
0
tant heavy-fermion behavior) is due a hybridization of
we utilize a the large-N mean field theory of the Kondo
the conduction band with an f-fermion band that rep-
lattice to demonstrate that the transition between the
resents excitations of the lattice of spins. A hybridized
screened and unscreened case is discontinuous. Thus, at
Fermi liquid emerges from this interaction. We shall see
least within this approach, no critical fluctuations oc-
that, due to the coherency of the Fermi liquid state, the
cur (in contrast to the single-impurity case discussed
resulting hybridized heavy fermions are only marginally
above). Moreimportantly,ourlarge-N analysisalsofinds
affectedbytheonsetofconduction-electronpairing. This
that the stability regime of the Kondo screened lattice
weak proximity effect, with a small d-wave gap ampli-
is much larger than that of the single impurity. Thus,
tude ∆ ∆T /D for the heavy fermions, allows the
the screenedheavy-electronstate is more robustandthe f ≃ K
Kondoeffectinalatticesystemtoproceedviaf-electron-
local-moment phase only emerges if the conduction elec-
dominatedheavy-fermionstatesthatscreenthelocalmo-
tron d-wave gap amplitude obeys
ments, with such screening persisting up to much larger
∆>∆ T D T , (3) values of the d-wave pairing amplitude than implied by
c K K
≃ ≫
p
3
the singleimpurityresult6,7,asdepictedinFig.1(which are subject to a pairing interaction:
applies at low T). A typical finite-T phase diagram is
J
shown in Fig. 2. H= ξkc†kαckα+ 2 Si·c†iασαβciβ +Upair. (4)
Our theory directly applies to the electron-doped k,α i,α,β
X X
cuprate Nd Ce CuO , possessing both d-wave
2−x x 4
superconductivity14,15 with T 20K and heavy Here, J is the exchange interaction between conduction
c
fermion behavior below5 T 2≃ 3K. The latter electrons and local spins and ξk = ǫk µ with ǫk the
is exhibited in a large lineKar ∼heat −capacity coefficient conduction-electronenergyandµthech−emicalpotential.
γ 4J/(mol K2) together with a large low-frequency The pairing term
≃ ×
susceptibility χ with Wilson ratio R 1.6. The lowest
crystal field state of Nd3+ is a Kram≃ers doublet, well Upair =− Ukk′c†k↑c†−k↓c−k′↓ck′↑, (5)
separated from higher crystal field levels16, supporting kX,k′
Kondo lattice behavior of the Nd-spins. The supercon-
is characterized by the attractive interaction between
ducting Cu-O-states play the role of the conduction
conduction electrons Ukk′. We shall assume the latter
electrons. Previous theoretical work on Nd Ce CuO
discussed the role of conduction electron co2−rrxelatxions174. stabilizes d-wave pairing with a gap ∆k = ∆cos2θ with
θtheanglearoundtheconduction-electronFermisurface.
Careful investigations show that the single ion Kondo
We are particularly interested in the low-temperature
temperature slightly increases in systems with elec-
strong-couplingphaseofthismodel,whichcanbestudied
tronic correlations18,19, an effect essentially caused by
by extending the conduction-electron and local-moment
the increase in the electronic density of states of the
spin symmetry to SU(N) and focusing on the large-N
conduction electrons. However, the fact that these con-
limit24. In case of the single Kondo impurity, the large-
duction electrons are gapped has not been considered,
N approachisnotabletoreproducethecriticalbehavior
even though the Kondo temperature is significantly
atthetransitionfromascreenedtoanunscreeenedstate.
smaller than the d-wave gap amplitude ∆ 3.7meV
≃ However, it does correctly determine the location of the
(See Ref. 20). We argue that Kondo screening in
transition,i.e. thenon-universalvalueforthestrengthof
Nd Ce CuO with T ∆ can only be understood
2−x x 4 K ≪ the Kondo coupling where the transition from screened
in terms of the mechanism discussed here.
to unscreened impurity takes place8. Since the location
We add for completeness that an alternative sce-
ofthe transitionandnot the detailednature ofthe tran-
nario for the large low temperature heat capacity of
sition is the primary focus of this paper, a mean field
Nd Ce CuO is based on very low lying spin wave
2−x x 4 theory is still useful.
excitations21. While such a scenario cannot account for
Although the physical case corresponds to N =2, the
a finite value of C(T)/T as T 0, it is consistent
large-N limityieldsavaliddescriptionoftheheavyFermi
→
with the shift in the overall position of the Nd-crystal liquid Kondo-screened phase25. We thus write the spins
field states upon doping. However, an analysis of the in terms of auxiliary f fermions as S σ f† f
spin wave contribution of the Nd-spins shows that for i · αβ → iα iβ −
δ /2, subject to the constraint
αβ
realistic parameters C(T)/T vanishes rapidly below the
Schottky anomaly22, in contrast to experiments. Thus f† f =N/2. (6)
webelievethatthe largeheatcapacityandsusceptibility iα iα
α
of Nd Ce CuO at low temperatures originates from X
2−x x 4
Kondo screening of the Nd-spins. To implement the large-N limit, we rescale the ex-
Despite its relevance for the d-wave superconductor change coupling via J/2 J/N and the conduction-
Nd2−xCexCuO4, we stress that our theory does not ap- electron interaction as Uk→,k′ → s−1Uk,k′ [where N ≡
ply to heavy electron d-wave superconductors, such as (2s + 1)]. The utility of the large-N limit is that the
CeCoIn (see Ref. 23), in which the d-wave gap is not (mean-field) stationary-phase approximation to is be-
5 H
a property of the conduction electron host, but a more lievedtobe exactatlargeN. Performingthismeanfield
subtleaspectoftheheavyelectronstateitself. Thelatter decoupling of yields
H
gives rise to a heatcapacity jump at the superconducing
s
otruarntshiteioorny∆∆CC((TTc))thaγtTis hcoomldps.arable to γTc, while in H= ξkc†kmckm+V fk†mckm+h.c. +λfk†mfkm
c ≪ c k,mX=−sh (cid:16) (cid:17) i
s
∆†kc−k−mckm+h.c. +E0, (7)
−
II. MODEL k,mX=1/2(cid:16) (cid:17)
with E a constant in the energy that is defined below.
0
The principal aim of this paper is to study the screen- Thepairinggap,∆k,andthehybridizationbetweencon-
ing of local moments in a d-wave superconductor. Thus, ductionandf-electrons,V,resultfromthemeanfieldde-
weconsidertheKondolatticeHamiltonian,possessinglo- coupling of the pairing and Kondo interactions, respec-
cal spins (Si) coupled to conductionelectrons (ckα) that tively. The hybridization V (that we took to be real)
4
measures the degree of Kondo screening (and can be di-
rectlymeasuredexperimentally26)andλisthe Lagrange E
∆
multiplier that implements the aboveconstraint,playing 1.0
the roleofthe f-electronlevel. Thefree energyF ofthis
single-particle problem can now be calculated, and has
the form: 0.5
NV2 Nλ
F(V,λ,∆k)= J − 2 +s ∆k∆k′Uk−k1′ (8) 0.6 0.8 1.0 1.2 1.4 Dξ
kk′
X
1 1
+N 4(ξk+λ)− 2Ekα−T ln 1+e−βEkα , -0.5
k,α=±(cid:18) (cid:19)
X (cid:0) (cid:1)
whereT =β−1 isthetemperature. Thefirstthreeterms -1.0
arethe explicit expressionsfor E0 inEq.(7), andEk± is
FIG. 3: The dashed line is the lower heavy-fermion band
Ek± = √12 ∆2k+λ2+2V2+ξk2± Sk, (9) (ucnrpoassirinedg(z∆er=o a0t)ctahseehaneadvtyh-feersmoliidonlinFeesrmariesuErfak−ce)forfo∆r kth=e
Sk = (∆2kq+ξk2 λ2)2+4V2 (ξk+pλ)2+∆2k , 0.1D, showing a small f-electron gap ∆fk ≃.±014D.
−
describingthe bands ofour d-wave(cid:2)pairedheavy-fer(cid:3)mion
E
system.
D
The phase behavior of this Kondo lattice system for
givenvaluesofT,J andµisdeterminedbyfindingpoints 0.4
at which F is stationary with respect to the variational
parameters V, λ, and ∆k. For simplicity, henceforth we 0.2 6
take∆k asgiven(andhavingd-wavesymmetryasnoted
above) with the goal of studying the effect of nonzero √TKD ξ
∼
pairingontheformationoftheheavy-fermionmetalchar- -1 -0.5 0.5 1 D
?
acterized by V and λ that satisfy the stationarity condi-
tions -0.2
∂F
=0, (10a) -0.4
∂V
∂F
=0, (10b)
∂λ FIG. 4: Plot of the energy bands E+(ξ) (top curve) and
E−(ξ) (bottom curve), defined in Eq. (13), in the heavy
with the second equation enforcing the constraint,
Fermi liquid state (for ∆ = 0), for the case V = 0.2D and
Eq.(6). We shallfurthermorerestrictattentionto µ<0 λ = 0.04D, that has a heavy-fermion Fermi surface near
(i.e., a less than half-filled conduction band). ξ=Dandanexperimentally-measurablehybridizationgap26
Before we proceedwe point out that the magnitude of (the minimum value of E+ E−, i.e., the direct gap) equal
−
the pairing gap near the unpaired heavy-fermion Fermi to 2V √TKD. Note, however, the indirect gap is λ TK.
∼ ∼
surface (located at ξ =V2/λ) is remarkably small. Tay-
lor expanding Ek− near this point, we find
III. KONDO LATTICE SCREENING
Ek− ≃ Vλ22 ξ−V2/λ−λ∆2k/V2 2+∆2k 1/2, (11) A. Normal conduction electrons
h(cid:0) (cid:1) i
2
giving a heavy-fermion gap ∆fk =(λ/V) ∆k [with am-
A useful starting point for our analysis is to recall
2 2
plitude ∆ = ∆(λ/V) ]. We show below that (λ/V)
f the well-known27 unpaired (∆ = 0) limit of our model.
≪ 1 such that ∆fk ≪ ∆k. In Fig. 3, we plot the Byminimizing the correpsondingfreeenergy[simply the
lower heavy-fermion band for the unpaired case ∆k = 0 ∆=0limitofEq.(8)],oneobtains,atlowtemperatures,
(dashed line) along with ±Ek− for the case of finite ∆k that the Kondo screening of the local moments is repre-
(solidlines)inthevicinityoftheunpairedheavy-fermion
sented by the nontrivialstationarypoint of F at V =V
0
Fermisurface,showingthesmallheavy-fermiongap∆fk. and λ=λ =V2/D, with
Thus,wefindaweakproximityeffectinwhichtheheavy- 0 0
fermion quasiparticles, which are predominantly of f-
character, are only weakly affected by the presence of D+µ 1
V exp , (12)
0
d-wave pairing in the conduction electron band. ≃s 2ρ0 (cid:18)−2Jρ0(cid:19)
5
Here we have taken the conduction electron density of
states to be a constant, ρ = (2D)−1, with 2D the V
0 D
bandwidth. The resulting phase is a metal accommo-
dating both the conduction and f-electrons with a large
density of states λ −1 near the Fermi surface at 0.03
0
ǫk ≃ µ+V02/λ0, r∝evealing its heavy-fermion character.
In Fig. 4, we plot the energy bands
0.02
1
E±(ξk)= ξk+λ (ξk λ)2+4V2 , (13)
2 ± −
(cid:18) q (cid:19) 0.01
of this heavy Fermi liquid in the low-T limit.
With increasing T, the stationary V and λ decrease
∆
monotonically, vanishing at the Kondo temperature
0.02 0.04 0.06 0.08 D
2eγ 1
T = D2 µ2exp , (14)
K π − − ρ J FIG.5: (Coloronline)Main: Mean-fieldKondoparameterV
0
p (cid:2) (cid:3) asafunctionofthed-wavepairingamplitude∆,forexchange
2eγ D µ coupling J = 0.30D and chemical potential µ = 0.1D, ac-
= − λ . (15) −
π sD+µ 0 cvoiardaindgirteoctthmeinapimprizoaxtiimonatoeffoErqm.u(l8a)Eatq.T(3=1)1(0s−o4liDd l(inpeo)inatns)d,
Here, the second line is meant to emphasize that T is thelatterexhibitingafirst-ordertransitionnear∆=0.086D.
K
of the same order as the T = 0 value of the f-fermion
chemical potential λ , and therefore T V , i.e., T
0 K 0 K
≪
is smallcomparedto the zero-temperaturehybridization that a d-wave superconductor can only screen a local
energy V . spin if the pairing strength is much smaller than T ,
0 K
It is well established that the phase transition-like be- can also be derived within the mean-field approach to
haviorofV atT isinfactacrossoveronceN isfinite1,24. the Kondo problem, as shown in Appendix A (see also
K
Nevertheless,the large-N approachyieldsthe corrector- Ref.7). Withinthisapproach,acontinuoustransitionto
derofmagnitudeestimateforT andprovidesaveryuse- theunscreenedphase(whereV2 0continuously)takes
K
→
ful description of the strong coupling heavy-Fermiliquid place at ∆ ∆ .
∗
≃
regime, including the emergence of a hybridization gap Thus, calculations for the single impurity case indi-
in the energy spectrum. catethat Kondoscreeningis rathersensitive toa d-wave
pairing gap. The question we wish to address is, how
does d-wavepairing affectKondo screeninginthe lattice
B. d-wave paired conduction electrons case? Infact,wewillseethattheresultsarequitediffer-
entinthe Kondolattice case,suchthatKondoscreening
Next, we analyze the theory in the presence of d-wave persists beyond the point ∆ . To show this, we have nu-
∗
pairingwithgapamplitude ∆. Thus,weimaginecontin- merically studied the ∆-dependence of the saddle point
uously turning on the d-wave pairing amplitude ∆, and of the free energy Eq. (8), showing that, at low temper-
study the stability of the Kondo-screened heavy-Fermi atures, V only vanishes, in a discontinuous manner, at
liquid state characterizedby the low-T hybridizationV , much larger values of ∆, as shown in Fig. 5 (solid dots)
0
Eq.(12). AswediscussedinSec.I,inthecaseofasingle for the case of J = 0.30D, µ = 0.1D and T = 10−4D
−
Kondoimpurity,itiswellknownthatKondoscreeningis (i.e.,T/T .069). InFig.2,weplotthephasediagram
K
≃
qualitatively different in the case of d-wave pairing, and as a function of T and ∆, for the same values of J and
the single impurity is only screened by the conduction µ, with the solid line denoting the line of discontinuous
electrons if the Kondo coupling exceeds a critical value transitions.
ThedashedlineinFig.2denotesthespinodalT ofthe
1 1 s
J∗ . (16) freeenergyF atwhichthequadraticcoefficientofEq.(8)
≃ ρ 1+lnD/∆
0 crosseszero. The significanceofT is that,if the Kondo-
s
For J < J , the impurity is unscreened. This result for to-local moment transition were continuous (as it is for
∗
J can equivalently be expressed in terms of a critical ∆ = 0), this would denote phase boundary; the T 0
∗ →
pairing strength ∆ , beyond which Kondo screening is limit of this quantity coincides with the single-impurity
∗
destroyed for a given J: critical pairing Eq. (17). An explicit formula for Ts can
be easily obtained by finding the quadratic coefficient of
1
Eq. (8):
∆ =Dexp 1 , (17)
∗
− ρ J
0
[equivalent to Eq. (1) for r =(cid:2) 1], whi(cid:3)ch is proportional 1 = tanhEk/2Ts(∆), (18)
to the Kondo temperature T . This result, implying J 2Ek
K k
X
6
with Ek ξk2 +∆2k, and where we set λ = 0 [which conduction electrons. Thus, for T = 0 the particle-
≡
must occur at a continuous transition where V 0, as particleresponsewillbesingular. Thisisthewellknown
p →
can be seen by analyzing Eq. (10b)]. As seen in Fig. 2, Cooper instability. For V =0 we obtain for example
the spinodaltemperature is generallymuch smallerthan
the true transition temperature; however, for very small N D2 µ2
χ (V =0)= log − , (23)
∆ 0,T (∆)coincideswiththeactualtransition(which ∆ 8 ∆2
s
→
becomes continuous), as noted in the figure caption.
whereweused∆asalowercutofftocontroltheCooper
Our next task is to understand these results within
logarithm. Below we will see that, except for extremely
an approximate analytic analysis of Eq. (8); before do-
small values of ∆, the corresponding Cooper logarithm
ing so, we stress again that the discontinuous transition
is overshadowed by another logarithmic term that does
from a screened to an unscreened state as function of T
not have its origin in states close to the Fermi surface,
becomes arapidcrossoverforfinite N. The largeN the-
but rather results from states with typical energy V
ory is, however, expected to correctly determine where ≃
√T D.
this crossovertakes place. K
Inordertoevaluateχ intheheavyFermiliquidstate,
∆
we start from:
1. Low-T limit v2 u2
G (k,ω)= k + k , (24)
cc
ω E+(ξk) ω E−(ξk)
According to the numerical data (points) plotted in − −
Fig. 5, the hybridization V is smoothly suppressed with where E± is given in Eq. (13) and the coherence factors
increasingpairingstrength∆beforeundergoingadiscon- of the hybridized Fermi liquid are:
tinuous jump to V =0. To understand, analytically, the
l∆im-ditepoefnFde,nic.ee.,ofthVeagtrolouwn-dT-s,twateeshenaellrgaynaEly.zeTthhee eTss=en0- u2k = 1 1 ξk−λ ,
tialquestionconcernsthestabilityoftheKondo-screened 2 − (ξk λ)2+4V2
−
state with respectto a d-wavepairinggap, characterized  q 
by the following ∆-dependent hybridization vk2 = 1 1+ ξk−λ . (25)
∆2 2 (ξk λ)2+4V2
V(∆)=V 1 , (19) −
0 − ∆2  q 
(cid:16) typ(cid:17) Inserting Gcc(k,ω) into the above expression for χ∆
with ∆ an energy scale, to be derived, that gives the yields
typ
typical value of ∆ for which the heavy-fermion state is
affected by d wave pairing. N D−µ v4 u4 4v2u2θ(E−)
To show th−at Eq. (19) correctly describes the smooth χ∆ = 8 dξ E + E + E +E .
Z−D−µ (cid:18) + | −| + − (cid:19)
suppressionof the hybrizationwith increasing ∆, and to (26)
obtainthescale∆typ,wenowconsiderthedimensionless We usedthatE+ >0is alwaysfulfilled, aswe considera
quantity less than half filled conduction band.
Considering first the limit λ = 0, it holds E (ξ) < 0
1 ∂2E −
χ , (20) and the last term in the above integral disappears. The
∆ ≡−2ρ ∂∆2
0 remaining terms simplify to
that characterizes the change of the ground state en-
N D−µ 1
ergy with respect to the pairing gap. Separating the χ (λ=0) = dξ ,
∆
amplitude of the gap from its momentum dependence, 8 Z−D−µ ξ2+4V2
i.e. writing ∆k = ∆φk, we obtain from the Hellmann- N D2 µ2
p
Feynman theorem that: = log − . (27)
8 4V2
1 ∂
χ = H , Even for λ nonzero,this is the dominant contribution to
∆
−2ρ0∆(cid:28)∂∆(cid:29) χ∆ in the relevant limit λ ≪ V ≪ D. To demonstrate
N this we analyze Eq. (26) for nonzero λ, but assuming
= −2ρ ∆ φk c†kmc†−k−m . (21) λ V asisindeedthe caseforsmall∆. Thecalculation
0 k ≪
X D E is lengthy but straightforward. It follows:
For ∆ 0 this yields
→ N λ D2 µ2 N λ D µ
χ∆ = N dω φ2kGcc(k,iω)Gcc( k, iω). (22) χ∆ = 8 (cid:18)1+ D(cid:19)log 4V−2 + 8 D log ∆|2|. (28)
2ρ 2π − −
0 Z k
X The last term is the Cooper logarithm, but now in the
Here, G (k,iω) is the conduction electron propagator. heavy fermion state. The prefactor λ/D T /D is a
cc K
≃
As expected, χ is the particle-particle correlator of the result of the small weight of the conduction electrons on
∆
7
the Fermi surface (i.e. where ξ V2/λ) as well as the
≃
reduced velocity close to the heavy electron Fermi sur-
face. Specificallyitholdsu2 ξ V2/λ λ2/V2 aswell
≃ ≃
as E− ξ ≃V2/λ ≃ Vλ22 ξ−(cid:0) Vλ2 . (cid:1) C/γ0T
1.2
Thus, except for extr(cid:16)emely s(cid:17)mall gap values where
(cid:0) (cid:1)
∆2 < D2 D −D/TK, χ is dominated by the λ = 0 1.0
4TK ∆
0.8
result, Eq.(cid:16)(27),(cid:17)and the Cooper logarithm plays no role
in our analysis. The logarithm in Eq. (27) is not origi- 0.6
nating from the heavy electron Fermi surface (i.e. it is
0.4
not from ξ r2 ). Instead, it has its origin in the inte-
gration over≃stλates where E < 0. The important term 0.2
−
v4 u4 in Eq. (26) is peaked for ξ 0 i.e. where T/D
2E+ − 2E− ≃ 0.0001 0.0002 0.0003 0.0004 0.0005
E (ξ 0) = V and is large as long as ξ . V. For
±
ξ 0 h≃olds v4± u4 1 . This peak|a|tξ 0 has FIG. 6: Plot of the low-temperature specific heat coefficient
≃ 2E+ ≃−2E− ≃ 32V ≃ C = ∂2F, for the case of λ = 10−2D, V = 10−1D, and
its origin in the competition between two effects. Usu- T −∂T2
µ = 0.1D, for the metallic case (∆ = 0, dashed line) and
ally, u or v are large when E± ≃ ξ. The only regime the c−ase of nonzero d-wave pairing (∆ = 0.1D, solid line).
where u or v are still sizable while E remain small is
± Thisshowsthat,evenwithnonzero∆,thespecificheatcoef-
close to the bare conduction electron Fermi surface at
ficientwill appeartosaturateat alarge valueat low T (thus
ξ V (the position of the level repulsion between the
exhibiting signatures of a heavy fermion metal), before van-
| | ≃
two hybridizing bands). Thus, the logarithm is caused ishingat asymptotically lowT ∆ (=∆(λ/V)2=10−4D)
f
by states that are close to the bare conduction electron Each curve is normalized to the≪T =0 value for the metallic
Fermi surface. Although these states have the strongest case, γ0≃ 32π2ρ0V2/λ2.
response to a pairing gap, they don’t have much to do
with the heavy fermion characterof the system. It is in-
teresting that this heavy fermion pairing response is the stays finite, the simple relation Eq. (31) gives an ex-
same even in case of a Kondo insulator where λ=0 and cellent description of the heavy electron state. Above
the Fermi levelis inthe middle ofthe hybridizationgap. the small f-electron gap ∆ , these values of V and λ
f
The purpose of the preceding analysis was to derive yield a large heat capacity coefficient (taking N = 2)
an accurate expression for the ground-state energy E at γ 2π2ρ V2/λ2 and susceptibility χ 2ρ V2/λ2, re-
≃ 3 0 ≃ 0
small ∆. Using Eq. (20) gives: flectingtheheavy-fermioncharacterofthisKondo-lattice
systemeveninthepresenceofad-wavepairinggap. Ac-
E =E(∆=0) χ∆ρ0∆2, (29) cording to our theory, this standard heavy-fermion be-
−
havior (as observed experimentally5 in Nd Ce CuO )
2−x x 4
which, using Eq. (27) and considering the leading order
will be observed for temperatures that are large com-
in λ V and ∆ V, safely neglecting the last term
paredto the f-electrongap ∆ . However,for very small
≪ ≪ f
of Eq. (28) according to the argument of the previous
T ∆ ,the temperaturedependence ofthe heatcapac-
f
paragraph,and dropping overall constants, yields ≪
itychanges(duetothed-wavecharacterofthef-fermion
gap), behaving as C = AT2/∆ with a large prefactor
E V2 λ λ ρ ∆2 D2 µ2
N ≃ J −2+V2ρ0lnD+µ− 08 ln V−2 . (30) A ≃ (D/TK)2. This leads to a sudden drop in the heat
capacity coefficient at low T, as depicted in Fig. 6.
Using Eq. (10), the stationaryvalue of the hybridization ThesurprisingrobustnessoftheKondoscreeningwith
(toleadingorderin∆2)isthenobtainedviaminimization respecttod-wavepairingisrootedintheweakproximity
with respect to V and λ. This yields effect of the f-levels and the coherency as caused by the
formation of the hybridization gap. Generally, a pairing
∆2 gap affects states with energy ∆k from the Fermi en-
V(∆) V , (31)
≃ 0− 16V0 ergy. However, low energy states that are within TK of
theFermienergyarepredominantlyoff-electroncharac-
with the stationary value of λ = 2ρ V2, which estab- ter (a fact that follows from our large-N theory but also
0
lishes Eq. (19). A smooth suppression of the Kondo from the much more general Fermi liquid description of
hybridization from the ∆ = 0 value V [Eq. (12)] oc- theKondolattice28)andareprotectedbytheweakprox-
0
curs with increasing d-wave pairing amplitude ∆ at low imity. These states only sense a gap ∆fk ∆k and can
≪
T. This result thus implies that the conduction electron readily participate in local-moment screening.
gap only causes a significant reduction of V and λ for Furthermore,the opening of the hybridizationgap co-
∆ ∆ √T D. herently pushes conduction electrons to energies V
typ K
≃ ∝ ≃
In Fig. 5 we compare V(∆) of Eq. (31) (solid line) from the Fermi energy. Only for ∆ V √T D will
K
≃ ≃
with the numerical result (solid dots). As long as V the conduction electrons ability to screen the local mo-
8
ments be affected by d-wave pairing. This situation is the latter we set V = λ=0 in Eq. (8), obtaining (recall
very different from the single impurity Kondo problem Ek = ξk2+∆2k)
where conduction electron states come arbitrarily close
to the Fermi energy. Fp 1 1 4 D2 µ2
LM ρ (D+µ)2 ρ ∆2ln −
0 0
N ≃−2 − 4 ∆
p
T ln2 T ln 1+e−βEk , (36)
2. First-order transition − −
k
X (cid:2) (cid:3)
TheresultEq.(31)oftheprecedingsubsectionstrictly where we dropped an overall constant depending on the
applies for ∆ 0,althoughas seeninFig.5, inpractice conduction-band interaction.
→
it agrees quite well with the numerical minimization of ThetermproportionaltoT inEq.(36)comesfromthe
the freeenergyuntilthe first-ordertransition. Tounder- fact that Ek− = 0 for V = λ = 0, and corresponds to
standthewayinwhichV isdestroyedwithincreasing∆, the entropy of the local moments. At low T, the gapped
we must consider the V 0 limit of the free energy. nature of the d-wave quasiparticles implies the last term
→
We start with the ground-state energy. Expanding E in Eq. (36) can be neglected (although the nodal quasi-
[the T 0 limit of Eq. (8)] to leading order in V and particlesgiveasubdominantpower-lawcontribution). In
→
zeroth order in λ (valid for V 0), we find (dropping derivingtheKondofreeenergyFK,Eq.(35),wedropped
overallconstants) → overall constant terms; re-establishing these to allow a
comparison to F , and setting F =F , we find
LM LM K
E ∆ 16ρ
N ≃−4ρ0V2ln ∆c + 3 ∆0V3, (32) 1ρ ∆2ln3 ∆c =T ln2, (37)
0
6 ∆
where we defined the quantity ∆
c that can be solvedfor temperature to find the transition
temperature T for the first-order Kondo screened-to-
1 K
∆c =4 D2 µ2exp , (33) local moment phase transition:
− −2ρ J
(cid:18) 0 (cid:19)
p ρ ∆2 ∆
at which the minimum value of V in Eq. (32) vanishes T (∆)= 0 ln3 c, (38)
K
6ln2 ∆
continuously, with the formula for V(∆) given by
that is valid for ∆ ∆ , providing an accurate ap-
c
V(∆) 1∆ln∆c, (34) proximation to the n→umerically-determined TK curve in
≃ 2 ∆ Fig. 2 (solid line) in the low temperature regime (i.e.,
near ∆ =0.14D in Fig. 2).
c
near the transition. According to Eq. (33), the equilib-
Equation(38) yields the temperature atwhich, within
rium hybridization V vanishes (along with the destruc-
mean-fieldtheory,thescreenedKondolatticeisdestroyed
tion of Kondo screening) for pairing amplitude ∆
c ∼ by the presence of nonzero d-wave pairing; thus, as long
√T D, of the same order of magnitude as the T = 0
K as ∆ < T (∆), heavy-fermion behavior is compatible
K
hybridization V , as expected [and advertised above in
0 with d-wave pairing in our model. The essential feature
Eq. (3)].
of this result is that T (∆) is only marginally reduced
K
Equation (33) strictly applies only at T = 0, appar-
fromthe ∆=0KondotemperatureEq.(2), establishing
ently yielding a continuous transition at which V 0
→ thestabilityofthisstate. Incomparison,accordingtoex-
for ∆ ∆ . What about T = 0? We find that, for
→ c 6 pectationsbasedonasingle-impurityanalysis,onewould
small but nonzero T, Eq. (33) approximately yields the
expect the Kondo temperature to follow the dashed line
correct location of the transition, but that the nature
in Fig. 2.
of the transition changes from continuous to first-order.
Away from this approximate result valid at large N,
Thus, for ∆ near ∆ , there is a discontinuous jump to
c the RKKY interaction between moments is expected to
the local-moment phase that is best obtained numeri-
lowerthelocal-momentfreeenergy,alteringthepredicted
cally, as shown above in Figs. 5 and 2. However, we can
location of the phase boundary. Then, even for T =
get an approximate analytic understanding of this first-
0, a level crossing between the screened and unscreened
order transition by examining the low-T limit. Since ex-
ground states occurs for a finite V. Still, as long as the
citations are gapped, at low T the free energy F of the
K ∆=0heavyfermionstateisrobust,itwillremainstable
Kondo-screened (V = 0) phase is well-approximated by
6 at low T for ∆ small compared to ∆c, as summarized in
inserting the stationary solution Eq. (34) into Eq. (32):
Figs. 1 and 2.
F 1 ∆
K ρ ∆2ln3 c, (35)
0
N ≃−6 ∆
IV. CONCLUSIONS
for F at ∆ ∆ . The discontinuous Kondo-to-local
K c
→
moment transition occurs when the Kondo free energy WehaveshownthatalatticeofKondospinscoupledto
Eq. (35) is equal to the local-moment free energy. For an itinerant conduction band experiences robust Kondo
9
screening even in the presence of d-wave pairing among supportedbytheAmesLaboratory,operatedfortheU.S.
theconductionelectrons. Theheavyelectronstateispro- Department of Energy by Iowa State University under
tected by the large hybridization energy V T . The Contract No. DE-AC02-07CH11358. DES was also sup-
K
≫
d-wave gap in the conduction band induces a relatively ported at the KITP under NSF grant PHY05-51164.
weak gap at the heavy-fermion Fermi surface, allowing APPENDIX A: SINGLE IMPURITY CASE
Kondo screening and heavy-fermion behavior to persist.
OurresultsdemonstratetheimportanceofKondo-lattice
For a single Kondo impurity a critical value J for
∗
coherency,manifestedby the hybridizationgap,which is
the coupling between conduction electron and impurity
absent in case of dilute Kondo impurities. As pointed
spin emerges, separating Kondo-screenedfrom local mo-
out in detail, the originfor the unexpected robustnessof
ment behavior for a single spin impurity in a d-wave su-
the screened heavy electron state is the coherency of the
perconductor, see Eq. (16). As discussed in the main
Fermi liquid state. With the opening of a hybridization
text, this is equivalent to a critical pairing Eq. (17)
gap, conduction electron states are pushed to energies
above which Kondo screening does not occur. The re-
of order √T D away from the Fermi energy. Whether
K sult was obtained in careful numerical renormalization
or not these conduction electrons open up a d-wave gap group calculations8,9. In the present section, we demon-
is therefore of minor importance for the stability of the
strate that the same result also follows from a simple
heavy electron state. large-N mean field approach. It is important to stress
Ourconclusionsarebasedonalarge-N meanfieldthe-
that this approach fails to describe the detailed critical
ory. In case of a single impurity, numerical renormaliza- behavior. However, here we are only concerned with the
tion group calculations demonstrated that such a mean
approximate value of the non-universal quantity J . In-
∗
field approach fails to reproduce the correct critical be-
deed, mean field theory is expected to give a reasonable
havior where the transition between screened and un-
value for the location of the transition.
screened impurity takes place. However the mean field
Our starting point is the model Hamiltonian
theory yields the correct value for the strength of the
Kondo coupling at the transition. In our paper we are J
notconcernedwiththe detailednatureinthe nearvicin- H= ǫkc†kmckm++N fm†fmc†kmck′m′
ity of the transition. Our focus is solely the location of Xkm m,mX′,k,k′
tshcreeebnoeudndloacrayl bmeotwmeeennt tphheashee,aavnydFweremdiolieqxupidectantdhautna- −kk′ Ukk′c†k↑c†−k↓c−k′↓ck′↑. (A1)
X
meanfieldtheorygivesthecorrectresult. Onepossibility
to test the results of this paper is a combination of dy- with the corresponding mean-field action S =Sf +Sb+
namicalmeanfieldtheoryandnumericalrenormalization Sint with (introducingthe Lagrangemultiplier λandhy-
group for the pseudogap Kondo lattice problem. bridization V as usual, and making the BCS mean-field
In case where Kondo screening is inefficient and ∆ > approximationfor the conduction fermions):
√T D, i.e., the “local moment” phase of Figs. 1 and 2,
K
tchaellygroorudnedredst.aTtehoisfctahne hmaovmeeinnttesrwesitllinlgikeimlypbliecamtiaognnseftoir- Sf = dτ c†km(∂τ +ǫk)ckm+fm†(∂τ +λ)fm ,
Z m k
the superconductingstate. Examplesarereentranceinto X(cid:2)X (cid:3)
N
a normal phase (similar to ErRh4B4, see Ref. 29) or a Sb = dτ J V†V −λNq0 , (A2)
modified vortex lattice in the low temperature magnetic Z (cid:16) (cid:17)
phase. In our theory we ignoredthese effects. This is no Sint = dτ fm†ckmV +V†ckmfm + ∆k∆′kUk−k1′
problem as long as the superconducting gap amplitude
Z mk kk′
∆ is small compared to √T D and the Kondo lattice is X(cid:0) (cid:1) X
K J
wscerlelesncerdeesnteadte. wTihlluns,otthbeerseigginoinficoafnsttlaybailffiteyctoefdtbhyeiKncolnuddo- − ∆†kc−k−mckm+c†kmc†−k−m∆k , (A3)
ingthemagneticcouplingbetweenthef-electrons. Only mX=1/2(cid:0) (cid:1)
the nature of the transition and, of course, the physics where the λ integral implements the constraint Nq =
0
of the unscreened state will depend on it. Finally, our f†f , withq =1/2. Here,we havetakenthe large
m m m 0
theory offers an explanation for the heavy fermion state N limit, with N =2J +1.
in Nd Ce CuO , where ∆ T . P
2−x x 4 K The mean-fieldapproximationhavingbeen made, it is
≫
Acknowledgments — We are grateful for useful discus- now straightforward to trace over the fermionic degrees
sions with A. Rosch and M. Vojta. This research was of freedom to yield
N V 2 N
F = | | λNq T ln (iω λ Γ (iω))(iω+λ+Γ ( iω)) Γ (iω)Γ¯ (iω) , (A4)
0 1 1 2 2
J − − 2 − − − −
Xω h i
10
for the free energy contribution due to a single impurity which can be evaluated numerically to determine V and
inad-wavesuperconductor. Here,wedroppedanoverall λ as a function of T and ∆.
constant due to the conduction fermions only, as well as
The Kondo temperature T is defined by the temper-
thequadratictermin∆k (whichofcoursedeterminesthe K
atureatwhichV2 0continuously;atsuchapoint,the
equilibrium value of ∆k; here, as in the main text, we’re →
constraint Eq. (A11) requires λ 0. Here, we are in-
interested in the impact of a given ∆k on the degree of →
terested in finding the pairing ∆ at which T 0; thus,
Kondo screening), and defined the functions K →
this is obtained by setting T =V =λ=0 in Eq. (A10):
Γ (iω) = V 2 iω+ǫk , (A5)
1 | | (iω)2 E2
Xk − k 1 0 dz πρd(z)
Γ (iω) = V2 ∆k , (A6) J = π − z , (A12)
2 (iω)2 E2 Z−D−µ
k − k ∆
X = ρ log +ρ , (A13)
Γ¯ (iω) = (V†)2 ∆k . (A7) − 0 D+µ 0
2 (iω)2 E2
k − k
X
At this point we note that, for a d-wave superconduc- where, for simplicity, in the final line we approximated
tor, Γ2 = Γ¯2 = 0 due to the sign change of the d-wave ρd(z) to be given by
order parameter. The self-energy Γ (iω) is nonzero and
1
essentiallymeasuresthedensityofstates(DOS)ρ (ω)of
d
the d-wave superconductor. In fact, one can show that
the corresponding retarded function Γ (ω) satisfies ρ ω /∆, for ω <∆,
1R ρ (ω) 0| | | | (A14)
d
∞ ρ (z) ≃(ρ0, for |ω|>∆,
Γ (ω)= V 2 dz d , (A8)
1R
| | ω+iδ z
Z−∞ −
that captures the essential features (except for the nar-
with δ = 0+, so that the imaginary part Γ′′ (ω) =
1R row peak near ω = ∆) of the true DOS of a d-wave
π V 2ρ (ω) measures the DOS. Writing Γ (ω)
−V 2|G|(ω)d, we have for the free energy 1R ≡ superconductor, with ρ0 the (assumed constant) DOS of
| | the underlying conduction band.
N V 2 The solution to Eq. (A13) is:
F = | | λNq (A9)
0
J −
∞ dz V 2G′′(z)
+N n (z)tan−1 −| | ,
π F z λ V 2G′(z) 1
Z−∞ (cid:16) − −| | (cid:17) ∆∗ =(D+µ)exp 1− ρ J , (A15)
0
and for the stationarity conditions, Eq. (10), (cid:2) (cid:3)
1 = ∞ dz nF(z)G′′(z)(z−λ) ,(A10) showingadestructionoftheKondoeffectfor∆→∆∗,as
J π (z λ V 2G′(z))2+ V 4(G′′(z))2 V 0continuously,thusseparatingtheKondo-screened
Z−∞ − −| | | | →
∞ dz n (z)V 2G′′(z) (for ∆ < ∆∗) from the local moment (for ∆ > ∆∗)
F
q = | | (A,11) phases.
0 − π (z λ V 2G′(z))2+ V 4(G′′(z))2
Z−∞ − −| | | |
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5 T. Brugger, T. Schreiner, G. Roth, P. Adelmann, and G. (2000).
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