Exciton effects in a scaling theory of intermediate valence and Kondo systems

0.13 MB·English

by  V. Yu. Irkhin

Exciton effects in a scaling theory of intermediate valence and Kondo systems
V. Yu. Irkhin and M. I. Katsnelson
Institute of Metal Physics, Ekaterinburg 620219, Russia
Department of Physics, Uppsala University, Uppsala 751 21 Sweden
An interplay of the Kondo scattering and exciton effects (d−f Coulomb interaction) in the
intermediate valence systems and Kondo lattices is demonstrated to lead to an essential change of
4 the scaling behavior in comparison with the standard Anderson model. In particular, a marginal
0
regime can occur where characteristic fluctuation rate is proportional to flow cutoff parameter. In
0
this regime the “Kondo temperature” itself is strongly temperature dependent which may give a
2
keytotheinterpretation of controversial experimentaldata for heavyfermion and related systems.

n
a PACSnumbers: 71.27.+a,71.28.+d,75.30.Mb
J

6
2 There is an interesting class of rare-earth compounds states for a scattering center which is important for the
such as Ce Bi Pt , SmB , SmS under pressure (the formation of the Kondo resonance [11]. In the IV case
4 3 3 6
] “golden” phase), TmSe, YbB which were called ear- the divalent and trivalent states are degenerate by defi-
l 12
e lier intermediate valence (IV) compounds and now are nition, so that this analogy is not surprising. Therefore
- treated as “heavy-fermion (HF) semiconductors”, or it is natural to consider the formation of the Kondo res-
r
t “Kondo insulators” (for a review, see Refs.1, 2). Vari- onance for the IV compounds taking into account both
s
. ousnamesemphasizedifferentpeculiaritiesofthesecom- spin and charge fluctuations, or, equvalently, both the
t
a pounds. As for electron energy spectrum, most of them “Kondo” and exciton (“Falicov-Kimball” ) effects. This
m are narrow gap semiconductors with an anomalously is the aim of the present work. Since there is no clear
- small energy scale (the gap width), of the order of tens demarcation between the IV and Kondo systems, it will
d
or hundreds of Kelvins (see Ref.2 and a review of ear- be shown that the excitonic effects may be relevant also
n
lier experiments in Ref. 3). At the same time, they do for the latter case.
o
demonstrate intermediate valence of the rare earth ions To investigate effects of interaction of current carriers
c
[ (usually between 2+ and 3+) in a number of properties, withlocalmomentsweusetheHamiltonianoftheasym-

e.g., lattice constants (which are intermediate between metric infinite-U SU(N) Anderson model with inclusion
1
those for isostructural compounds with di- and trivalent oftheFalicov-Kimballinteraction(on-sited−f Coulomb
v
ions),corelevelspectra(whichareamixtureofthespec- repulsion G),
1
0 tra of di- and trivalent ions with comparable weights),
5 and many others [1, 4, 5]. H = [t c† c +V c† f +f† c
k km km km km km km
1 Xkm (cid:16) (cid:17)
40 eneArsgywseclallaesisfoarktehyepHoiFntmtoetuanlds,ertshteanodriagninomofatlohuisspsmroapl-l +Effk†mfkm]+G fi†mfimc†im′cim′ (1)
0 iXmm′
ertiesoftheIVcompounds. FortheHFmetalsitiscom-
/
t monly accepted now that they are the Kondo lattices, where the on-site f −f Coulomb interaction is put to
a
m which means that this energy scale (the Kondo temper- infinity, so that doubly occupied states are suppressed,
- ature TK) is a width of the Kondo resonance owing to fi†m = |imihi0| are the Hubbard operator (|imi and |i0i
d spin-dependent scattering of conduction electrons by f- aresingle-occupiedandemptystates),weneglectforsim-
n electron centers [6]. As a result of an interplay of the plicity the k-dependence of the hybridization V. Note
o Kondo effect and interatomic magnetic interactions, the that similar calculations can be performed for realistic
c
: TK valueforalatticecanstronglydifferfromthatforan rare-earth ions, including the case of two magnetic con-
v isolated impurity [7], spin fluctuations being of crucial figurations, see Ref. 12.
Xi importanceforthe HFbehavior. Itisverynaturaltoex- Following to Ref. 10, we treat the coherent and inco-
pectthattheseeffectsareimportantalsofortheIVcom- herentcases. In the firstcase dispersion in the spectrum
r
a pounds. Atthesametime,valenceorcharge fluctuations of f-electrons occurs. For simplicity this is supposed
should be also considered. They are determined in part to be proportional to the conduction electron spectrum,
bytheCoulomb(“Falicov-Kimball”)interactionbetween E =E +λε ,ε ∝t , λ=−1 (the f-bandhas a hole
fk f k k k
conduction and localized electrons [8]. Taking into ac- character). In the incoherent regime λ = 0, so that the
count these interactions together with the hybridization f-electronsremainlocalized. Notethatinthepresenceof
processes it is possible to describe the IV state as a kind the energy gapwe alwaysdeal with the coherentregime.
ofexcitoncondensation[3,9]. NotethatintheIVregime Therenormalizationofthe CoulombparameterGand
the spinless Falicov-Kimball model with hybridization is the hybridization V is obtained, similar to Ref. 10, from
formally equivalent to the anisotropic Kondo problem, the two-particle Green’s function
differentvalentstatesplayingtheroleofpseudospin“up”
and “down” states [10]. It is the degeneracy of quantum Fmσm′(E)=hhfi†mcim′|c†im′fimiiE (2)
2
whichdeterminesthevertex. Weobtainthesingularcor- D = ρ−1 = 1). Then we have the closed equation for
rection with the structure w(C). In particular, for the incoherent regime
δFmσm′(E)=GFmσm′(E) E−ntqm+′ E (3)
Xq q f ∂w(C) ρV2(0) D 2ρG(0)
=1+(N −1) (11)
where nkm =hc†kmckmi is the Fermi function. In the co- ∂C C (cid:12)(cid:12)(cid:12)w(C)(cid:12)(cid:12)(cid:12)
herent regime, a similar correction occurs from the dis- (cid:12) (cid:12)
When E lies sufficiently below the Fermi level (the
persion of f-states. The correction (3) contains a loga- f
Kondo regime), the quantity |w(C)| can become small
rithmic Kondo-like divergence owing to charge fluctua-
with decreasing |C|. We can use this condition to define
tions, which is cut at E (the latter quantity plays the
f the boundary between IV and Kondo cases. Formal def-
role of the external field in the equivalent anisotropic
inition of IV systems is the absence of solutions to the
Kondo model). Unlike the renormalization of E , the
f equationw(C)=0whichjustdeterminestheKondores-
renormalizations of V and G do not contain the degen-
onance (cf. Refs. 6, 17). Physically, Kondo lattice has a
eracy factor of N.
three-peakdensityofstates(twoHubbardbandsandthe
TherenormalizationofE owingtospin-flipprocesses
f Kondo resonance), which is similar to the “doped Mott
is obtained in the second order in hybridization (cf. [12,
insulator”(note that inthe dynamicalmean-fieldtheory
13, 18])
(DMFT) the Hubbard model is reduced to the Ander-
δE =V2 nqm′ +nδG (4) son impurity model [15]). On the other hand, IV state
fm
E −t is similar to the phase of strongly correlated metal: the
m′X6=m,q F q
Kondo peak as a separate solution is absent.
We have taken into account in Eq.(4) the Hartree renor- For G(0) = 0, the boundary condition for the Kondo
malizationoff-levelenergy,whichoccursinthecoherent state is |Ef(0)| > Γ = (N −1)ρV2(0). In the opposite
case, E → E +Gn, n being the concentration of con- IV case |w(C)| remains finite. For G(0) 6= 0, V(C) in-
f f
ductionelectrons;weputinnumericalcalculationsn=1, creases during the renormalization process, and the ef-
which corresponds to the Kondo regime. fective level width Γ(C) becomes larger, so that the IV
To derive the scaling equations for the effective model region becomes more wide. A temperature dependence
parameters we use the poor-man scaling approach [14]. of the energy gap (an increase with decreasing temper-
Pickingoutintheintegralswiththe Fermifunctions(3), ature) in IV compounds was observed experimentally in
(4) the contributions from the energy layer C < E < SmB6 [16], YbB12 and Ce4BiPt3 [2]. According to our
C+δC nearthe FermilevelE =0andreplacingE → treatment, the dependence of the effective hybridization
F f
E (C),V →V(C),G→G(C) we obtain (cf. Ref. 10) V(C)isnon-monotonous: itpassesthroughamaximum.
f
Now we consider in more detail the incoherent case
which should be realized for diluted systems (the An-
∂E (C) ∂G(C)
f = −ρ(N −1)V2(C)+n (5) derson’s localization prevents coherence at low tempera-
∂lnC ∂ln|C|
tures). Topresentnumericalresults(Figs.1-4)weusethe
∂V(C) 1−λ variable ξ =ln|D/C|. As follows from (9), in the Kondo
= − ρV(C)G(C) (6)
∂ln|C−E | 1+|λ| case the hybridization parameter V(C) decreases practi-
f
callybyajumpwhenweapproachthepointC =E (C),
∂G(C) 2λ f
= ρG2(C) (7) seeFig.2. Withfurtherdecreasing|C|,aconsiderablere-
∂ln|C−E | 1+|λ|
f gion arises where we have to high accuracy C ∼= E (C)
f
where ρ is the bare conduction-electron density of states (Fig.1). More exactly, we have ∂w(C)/∂C ≃ 0, so that
at E . Earlier [3] we have considered the exciton effects we obtain from (11) near the maximum of w(C)
F
with neglecting spin fluctuations. We will see that the
renormalization(4) results in new essential effects. w(C)≃−D (N −1)ρV2(0)/C 1/2ρG(0). (12)
We have from Eqs.(6) and (7) (cid:12) (cid:12)
(cid:12) (cid:12)
α In this regime V(C) = |C/[(N −1)ρ]|1/2, and the
G(C) V(C) 2λ
= ,α=− (8) effective s − f exchange parameter is ρI(C) =
G(0) (cid:20)V(0)(cid:21) 1+|λ|
ρV2(C)/E (C)=−1/(N −1)=const.
f
so that G(C) = G(0) in the incoherent regime. We de- In a standardconsideration,the condition C =Ef(C)
rive from (6) in the incoherent and coherent cases, re- determines an energy scale for a crossover to the regime
spectively of a heavy-fermion (Kondo) local Fermi liquid [13]. The
“marginal” situation with E (C) ∼= C in a whole in-
f
V(C) = V(0)|D/w(C)|ρG(0) (9) terval of the cutoff parameter C means an essentially
non-Fermi-liquid (NFL) picture. A similar mechanism
V(C) = V(0)/[1+G(0)ρln|w(C)/D|] (10)
of NFL behavior in magnetic Kondo lattices was pro-
where w(C) = C −E (C), D is a cutoff parameter of posedinRefs.7,19whereasoftbosonwasobtainedwith
f
theorderofbandwidth(weputinnumericalcalculations the characteristic energy ω(C) ∼= |C|. Note also that a
3
(cid:19)(cid:17)(cid:21) (cid:16)( (cid:19)(cid:17)(cid:19)(cid:28) 9
(cid:3)(cid:3)I
(cid:19)(cid:17)(cid:20)(cid:24)
(cid:19)(cid:17)(cid:19)(cid:27)
(cid:19)(cid:17)(cid:20)
(cid:19)(cid:17)(cid:19)(cid:26)
x
(cid:19)(cid:17)(cid:19)(cid:24)
(cid:19)(cid:17)(cid:19)(cid:25)
x
(cid:20)(cid:17)(cid:24) (cid:21) (cid:21)(cid:17)(cid:24) (cid:22) (cid:22)(cid:17)(cid:24) (cid:23) (cid:23)(cid:17)(cid:24)
(cid:20) (cid:21) (cid:22) (cid:23) (cid:24)
FIG. 1: Scaling trajectories −Ef(ξ) for V(0) = 0.1,G(0) = FIG. 4: Dependences V(ξ) for the same parameter values as
0.1,N = 2 in the incoherent case as compared to the in Fig.3 (for thecurvesfrom below toabove)
curve |C|/D = exp(−ξ) (dashed line). The parameter
values (for the curves from below to above) are E (0) =
f
−0.08,−0.1,−0.14.
a natural replacement |C|→T).
Physically, the regime where a typical energy scale
is just the temperature means a classical (Maxwell-
9
Boltzmann) electron gas (interacting with local mo-
(cid:19)(cid:17)(cid:22) menta): heat capacity is approximately constant etc. Of
course, in a pure form this behavior is never observed
(cid:19)(cid:17)(cid:21)(cid:24) since,accordingtoourresults,ittakesplaceonlyinare-
strictedtemperatureinterval. However,wehaveastrong
deviation from a simple scaling picture where we just
(cid:19)(cid:17)(cid:21)
enter strong-coupling regime, a characteristic “Kondo”
temperature being T-independent. In particular, the
(cid:19)(cid:17)(cid:20)(cid:24)
x Wilson number is nearly constant, but differs consider-
ably from that in the singlet state (I =−∞).
ef
A similar “marginal” region in the dependence E (C)
f
(cid:20) (cid:21) (cid:22) (cid:23) (cid:24) occursnotonly forthe Kondo,but alsoforIV statenear
FIG. 2: Dependences V(ξ) for the same parameter values as thecriticalline(insuchasituation,thedependencew(C)
in Fig.1 (for thecurves from below to above) hasashallowminimum,andV(C)asharpmaximum,see
Figs.1-2).
To estimate the Kondo temperature we can use Hal-
regime with the rate of the order parameter fluctuations dane’s arguments for the Anderson model with N = 2
1/τ (T) ∝ T is typical near a quantum phase transi- [13]. ThegeneralizationtoarbitraryN canbeperformed
φ
tion [21]. Our situation is reminiscent of this regime in as (cf. Ref. 20)
the sense that the characteristic valence fluctuation fre-
quencymax(ρV2(T),|Ef(T)|)isproportionaltoT (after TK ≃D(ρ|I|N)1/Nexp(1/ρNI),I =V2/Ef (13)
Thisexpressionisformallybasedonperturbationtheory
(two-loopscaling). However,Haldane noted that replac-
(cid:19)(cid:17)(cid:21) (cid:16)( ing in this formula both D and Ef by the characteristic
(cid:3)(cid:3)I energy scale T∗ = −C, which is determined from the
equation C = E (C), yields the correct estimation for
(cid:19)(cid:17)(cid:20)(cid:24) f
the Kondo temperature. As demonstrated above, in the
presence of exciton effects (Falicov-Kimball interaction
(cid:19)(cid:17)(cid:20) G), a situation is possible where this equation holds ap-
proximatelyinawholeenergyinterval. The energyscale
x T∗ where the “marginal” regime starts is considerably
(cid:19)(cid:17)(cid:19)(cid:24)
changed by the exciton effects.
InthecoherentcasethelastterminEq.(5)resultsina
smearing of the singularity, especially for small N. How-
(cid:21) (cid:22) (cid:23) (cid:24) ever, with increasing N the dependence E (C) (Fig.3)
f
FIG. 3: Scaling trajectories −E (ξ) for V(0) = 0.05,G(0) = becomes qualitatively similar to that in the incoherent
f
0.05,N = 6,Ef(0) = −0.09,−0.11,−0.15 in the coherent case. On the other hand, the dependence V(C) (Fig.4)
case. is essentially modified even for N → ∞. At C ∼= E (C)
f
4
we have frombothineleasticneutronscatteringandnuclearmag-
netic resonance (NMR) have the form Γ(T) ∝ T1/2. In
1=−(N−1)ρV2(C)/C−n[G(0)/V(0)]∂V(C)/∂C (14) the Kondo resonance model such a behavior takes place
above the Kondo temperature T , but in some cases
K
Solution to this Riccati equation is obtained in terms (CeB , CePd B , YbBe ) the dependence T1/2 takes
6 3 0.6 13
of the imaginary-argument Bessel and Macdonald func- place at very low temperatures of a few K [22]. Further,
tions Ip(x) and Kp(|(N −1)ρC|1/2) with x = 2|(N − the characteristic energy scale from linear heat capacity
1)ρV2(0)C|1/2/[nG(0)], p=0,1. For large N, the quan- term γ is of order of tens of Kelvins, whereas the tem-
tity ∂V(C)/∂C and consequently V2(ξ)/Ef(ξ) turn out peraturewhereγ startstodeviatefromconstantisjusta
to be practically ξ-linear near the maximum of w(C). fewofKelvins. Thusthereexistsnouniqueenergyscale.
Thus we have a “classical” electron liquid with singular We have demonstrated that, indeed, the infrared behav-
interactionswhichhavelogarithmicenergydependences. ior canbe essentially different fromthe simple Anderson
With further decreasing |C| we have from Eq.(6) modelowingtospindynamics(seeRefs.7,19)andcharge
fluctuations (exciton effects) considered in the present
∂V(C)/∂C ∝|C|−1/2 =|Ef(C)|−1/2 (15) work.
so that the correction to V2(C)/E (C) is proportional
f
to |C|−1/2.
The d−f Coulombinteractioncanstronglyrenormal-
ize the hybridization, which leads to the increase of the Of course, the estimations performed are qualitative
characteristicenergyscale. Inthe incoherentregimeit is since they are based on the continuation of perturbative
a width of the resonance, in the coherent one the width Gellmann-Lowscalingfunctiontothestrongcouplingre-
of (indirect) gap or pseudogap [3]. The renormalization gion. Atthesametime,thestatementthatexcitoneffects
ofthefluctuationrateρV2(C)canbeverystrong(about cannot be described by universal temperature indepen-
order of magnitude for realistic parameters). The corre- dentT seemstobe reliableitself. Recently,directways
K
spondingtemperaturedependencescanbefoundinboth of observing the Kondo resonance (STM) were proposed
the regimes by the RG approach with the replacement [23]. As we know, they have been not yet applied to
|C|→T . the IV systems. It would be interesting to compare the
In standard treatments of HF systems, one picks results for T of these new experimental methods with
K
usually anomalous magnetic contributions to thermo- those from investigating thermodynamic properties.
dynamic properties and compares them with exact re- TheresearchdescribedwassupportedinpartbyGrant
sults in the one-impurity Kondo problem. In partic- No. 747.2003.2 from the Russian Basic Research Foun-
ular, the dependences of the crystal-field level width dation (Support of Scientific Schools).
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