Temperature dependent band structure of the Kondo insulator

4 Pages·0.17 MB·English

by  C. Groeber

Temperature dependent band structure of the Kondo insulator
C. Gr¨ober and R. Eder
Institut fu¨r Theoretische Physik, Universit¨at Wu¨rzburg, Am Hubland, 97074 Wu¨rzburg, Germany
(February 1, 2008)
We present a Qantum Monte Carlo (QMC) study of the temperature dependent dynamics of
theKondoinsulator. Workingat theso-called symmetrical point allows to perform minus-sign free
8 QMCsimulationsandthusreachtemperaturesoflessthan1%oftheconductionelectronbandwidth.
9 Study of the temperature dependence of the single particle Green’s function and dynamical spin
9 correlationfunctionshowsasurprisinglyintricatelowtemperaturebandstructureandgivesevidence
1 for two characteristic temperatures, which we identify with the Kondoand coherence temperature,
respectively. In particular, the data show a temperature induced metal-insulator transition at the
n
a coherence temperature.
J
71.27.+a,71.30.+h,71.10.Fd
8

]
l The theoretical description of the Kondo lattice re- of the system overa wide temperature range. Previouslt
e
- mainsanoutstandingproblemofsolidstatephysics. This a QMC study for the 2 dimensional model at half-filling
r model,orvariationsofit,maybeviewedastheappropri- was performed by Vekic at al. [5], more recently a study
t
s ate one for understanding such intensively investigated of the temperature dependence of static susceptibilities
.
t classes of materials as the heavy electron metals [1,2] for the strong coupling versionof the modelhas been re-
a
m and the Kondo insulators [3]. Experimental results indi- ported by Shibata et al. [6].
cate that the electronic structures of Kondo lattice com- It is widely believed that the Kondo lattice has two dis-
-
d pounds undergo quite dramatic changes with tempera- tinctcharacteristictemperatures. AttheKondotemper-
n
ture [4]. It is the purpose of the present manuscript to ature, T , the f-electrons start to form loosely bound
K
o
reportaQMCstudyoftheelectronicstructureoftheso- singlets with the conduction electrons. This manifests
c
[ called Kondo insulator, which shows that this model in- itself in a deviation of the spin susceptibility from the

deed undergoes a quite profound changeof its unexpect- high-temperature Curie form due to the ‘quenching’ of
1
edly intricate band structure as temperature increases. the f-electron magnetic moment and an increase of the
v
2 We are using a one dimensional (1D) ‘tight-binding ver- dc-resistivity due to resonant scattering from the newly
6 sion’ of the model with L unit cells and 2 orbitals/unit formed low energy bound states [1]. The second (and
0 cell: lower) characteristic temperature is the coherence tem-
1
perature, T , where the local singlets establish long
0 H =−tX(c†i+1,σci,σ +H.c.)−V X(c†i,σfi,σ+H.c.) range cohereconhce amongstthemselves so as to participate
8
i,σ i,σ
9 inthequasiparticlebandsofaFermi-liquidlikeelectronic
t/ −ǫfXni,σ+UXfi†,↑fi,↑fi†,↓fi,↓. (1) state. Experimentally the coherence temperature is sig-
a i,σ i naledbytheonsetofadecreaseofthedcresistivitywith
m temperature [1], and the formation of the ‘heavy bands’
Herec† (f† )createsaconductionelectron(f-electron)
- i,σ i,σ which (judging by the volume of the Fermi surface) in-
nd in cell i, ni,σ=fi†,σfi,σ. Throughout we consider the case corporatethef-electrons[7]. Whilethereisasyetnoex-
of ‘half-filling’ i.e. two electrons/unit cell and, as an perimental proof, one might expect on the basis of these
o
c important technical point, we restrict ourselves to the considerations, that at temperatures above Tcoh the f-
v: symmetric case, ǫf=U/2. The latter choice, while prob- electronsdo notparticipateinthe Fermisurfacevolume,
i ably not leading to any qualitative change as compared whereas they do so below.
X to other ratios of ǫf/U, has the crucial advantage that TurningtotheKondoinsulatorwenotethattheelectron
r at half-filling the model acquires particle-hole symme- count for these systems is such that the ‘Fermi surface’
a
try, i.e. the Hamiltonian becomes invariant under the comprisingboth, conductionandf-electrons,wouldpre-
transformation αi,σ → exp(iQ· Ri)α†i,σ, where α=c,f ciselyfilltheBrillouinzonesothatthesystemisa‘nomi-
and Q=(π,π,...) (this holds for bipartite lattices with nal’bandinsulator. Ifincreasingtemperaturecausesthe
only nearest neighbor hopping). Particle-hole symme- f-electronsto‘dropout’oftheFermisurfacevolume,this
try in turn implies that the QMC-procedure does not should manifest itself as an insulator-to-metaltransition
suffer from the notorious ‘minus-sign problem’ anymore, because the volume of the collapsed Fermi surface is no
so that reliable simulations for temperatures as low as longersufficient to coverthe entire Brillouinzone. Tran-
βt=30, corresponding to ≈ 0.8% of the conduction elec- ferring the above scenario for heavy Fermion metals to
tron bandwidth, can be performed without problems. the Kondoinsulator one wouldtherefore expect that the
This allows to scan the dynamical correlation functions
1
systemremainsametalaboveT ,withaFermisurface a transfer of spectral weight from the low energy peaks
coh
thatexcludesthef-electrons,andbecomesinsulatingbe- atthe gap-edgesinto the sidebands is takingplace. This
low T , when the f-electrons participate in the Fermi is accompaniedby a narrowingof the gap,and atβt=10
coh
surface volume to turn the system into a ‘nominal’ band the gap practically closes; in the spectrum for this tem-
insulator. In fact insulator-to-metal transitions which peraturethetwolow-energypeaksseemtohavecollapsed
are induced by temperature [4], or hydrostatic pressure into a single one right at µ - we believe that these are in
[8] have been observed experimentally. As will be seen factstilltwopeaks,whichhoweveraretoocloselyspaced
below, our data for the Kondo insulator are remarkably to be resolved. Increasing T even further (βt=5), the
consistent with such an interpretation. low energy peaks disappear completely. The c-electron
To begin with, we consider the T-dependence of the sin- spectraldensity no longershows anyindicationof a gap,
gleparticlespectralfunction. Thisis definedas(α=c,f) whereasthe f-likesidebandsstayatarelativelyhighen-
ergy away from µ. The extreme low energy states thus
1
Aα(k,ω)= Xeβωνµ|hν|αk,σ|µi|2δ(ω−ωνµ), (2) havepure c-character(within the resolutionofthe QMC
Z procedure)and the systemis a metal with a c-likeFermi
ν,µ
surface. We therefore interpret the temperature where
wherethesumisoveralleigenstates|νiofH−µN inthe the gap closes as the analogue of the coherence temper-
grand canonical ensemble, Z denotes the partition func- ature. With increasing T the energy of the sidebands is
tion and ωνµ the difference of the energies of the states loweredandatβt=3theycollapseintoasinglepeakright
ν and µ. Figure 1 shows the angle-integratedspectral at µ. This probably indicates a second transition and
at the relatively large value of βt=1 another reconstruc-
0.4 tion of the band structure has taken place, namely the
(cid:12)t=30
0.3 cf--eelleeccttrroonn disappearance of the f-like sidebands. The upper and
0.2
0.1 lower Hubbard band for the f-electrons are now quite
0
(cid:12)t=10 broad, and actually the possibility that there are very
0.3
0.2 low-intensityf-likefeaturesnearµcannotbecompletely
0.1
ruled out. However, the overall trend is quite obviously
0
(cid:12)t=5
D(!) 00..32 a strong decrease of the sidebands. Interpreting the lat-
terasthe lattice-analogueoftheKondo-resonanceinthe
0.1
0 impurity case [11],the temperature of the secondtransi-
(cid:12)t=3
0.3
tionshouldcorrespondtotheKondotemperatureT . In
0.2 K
0.1 thepresentcaseT isveryhighbecauseoftherelatively
K
0
(cid:12)t=1 large value of the c-f hybridization, V=t. We also note
0.3
0.2 thatthe closingofa gapin thef-likedensity ofstates in
0.1 2 dimensions was previously found by Vekic et al. [5].
0
-8 -6 -4 -2 0 2 4 6 8
!-(cid:22)=t Togetamoredetailedpicture,weconsiderthek-resolved
FIG. 1. Angle integrated spectral density D(ω) for an single particle spectral function, shown in Figure 2 for
L=16 Kondo lattice for different temperatures. Parameter some temperatures in between the two transitions. For
values are U/t=8, V/t=1. βt=30 the c-electrons show a standard cos(k) band, al-
beit with a clear gap at k0=π/2, the Fermi momentum
F
density, D (ω)=(1/L) A (k,ω). At the lowest tem- of nonhybridized conduction electrons. At this momen-
α Pk α
perature, βt=30, D (ω) shows the behaviour expected tum the band changes its spectral character and ‘bends
α
for a Kondo insulator: the c-electron density is roughly over’ into a practically dispersionless f-like band, which
consistentwiththe standard1D tight-binding density of can be followed up to k=π. This kind of band structure
states,with indicationsofthe twovan-Hovesingularities is familiar from various studies [9,10,12]. The weak and
at±2t. Aroundµ,however,the spectraldensity showsa practically dispersionless f-like sidebands are at some-
small but unambiguous gap, which demonstrates the in- whathigher energyand, atveryhighenergies,thef-like
sulating nature of the ground state. The f-like spectral Hubbardbands. The width ofthe Hubbardbands seems
densityshowsverysharplowenergypeaksattheedgesof todependstronglyonmomentum-thisisadeficiencyof
thisgap,aswellashigh-intensity‘Hubbardbands’atap- the QMC and maximum entropy method, which is most
proximately±U/2. Therearealsotwoweak‘side bands’ accurate near µ. As seen in the k-integrated spectra,
at approximately±0.8t. With the exception ofthese, all raising the temperature leads to a transfer of spectral
featuresareconsistentwithexactdiagonalization[9]and weight from the ‘flat band’ forming the single-particle
d→∞ results [10] at T=0. This suggests that the side- gap into the Kondo resonance-like sidebands. At βt=5,
bands arealreadyaneffect ofthe finite temperature and where the gap has closed, the c-electron spectrum shows
the further development with T confirms this. Inspec- averyconventionalcos(k)-bandwithno moreindication
tionof the series βt=30,10showsthat withincreasingT of any gap. At k0 there is now one symmetric and un-
F
2
split peak right at µ - as it is required by particle-hole component of the spin-operator for α-electrons. This is
symmetry for a metallic system. The system thus has a shown in Figure 3. At βt=30, the f-like SCF shows an
intense branch of low energy excitations with a tiny but
(cid:12)t=30 f(cid:0)electron clearly resolved spin wave-like dispersion. The spectral
c(cid:0)electron
weight of this branch is sharply peaked at k=π, indi-
k=((cid:25)) cating relatively long ranged and strong antiferromag-
netic spin correlations. Fitting the equal (imaginary)
time f-like spin correlation function in real space to the
k=((cid:25)2) expression S(r)=A(e−r/ζ+e−(L−r)/ζ) we obtain the val-
ues ζ=4.61,4.67 and 2.18 for βt=30,20 and 10. The
dominant feature in the c-like SCF on the other hand,
k=(0)
(cid:12)t=5 is a free electron-like particle-hole continuum. Interest-
k=((cid:25)) inglyenough,thereisalsoareplicaofthef-electronspin
wave branch in the c-like SCF. This shows that at low
excitation energies c and f electrons behave as a single
k=((cid:25)2) ‘all-electron fluid’, whose excitations have composite f-c
character. Thefreeelectroncontinuumitselfdoeshavea
gapof≈0.6tatk=π-thiscorrespondstoapproximately
k=(0)
(cid:12)t=1
k=((cid:25))
(cid:12)t=30,f-elec. rescaledby1=10
f(cid:0)electron
c(cid:0)electron
k=((cid:25)2) k=((cid:25))
k=((cid:25)2)
k=(0)
-8 -6 -4 -2 0 2 4 6 8
!-(cid:22)=t
FIG. 2. Momentum resolved single-particle spectral func- k=(0)
(cid:12)t=5
tion A(k,ω)at differenttemperatures. Parameters as in Fig-
ure1, themomentumk increases in stepsof2π/L=π/8from k=((cid:25))
thebottom of each panel.
Fermi surface as expected for unhybridized c-electrons, k=((cid:25)2)
i.e. the f-electrons indeed have ‘dropped out’ of the
Fermi surface volume. The dispersionless f-like side-
bands are at a relatively high energy. The tiny low en- k=(0)
(cid:12)t=1
ergy‘foot’seeninsomeofthec-likespectramayindicate k=((cid:25))
a very weak mixing of the c-electrons into the Kondo-
resonance but apparently this no longer leads to a gap.
Rather, the Kondo resonance is now essentially decou- k=((cid:25)2)
pled from the Fermi surface physics. Finally, for βt=1,
even these Kondo resonance-like sidebands have disap-
k=(0)
peared and the only f-like peaks in the spectral func- 0 1 2 !-3(cid:22)=t 4 5 6
tion are the upper and lower Hubbard bands. At this
FIG.3. DynamicalspincorrelationfunctionS(k,ω)fordif-
high temperature the f-electrons do not participate in
ferent temperatures. Parameters are as in Figure 1.
the low energy physics at all. The expectation value of
−V (c† f + H.c.) decreases by ≈ 30% between
Pi,σ i,σ i,σ
twice the single particle-gap in A(k,ω) [13]. We pro-
βt=30andβt=1-whilethereisappreciablemixingeven
ceed to βt=5, where the single-particle gap has closed.
at high temperature, this does obviously not lead to co-
The f-like SCF still shows a low-energy peak with fi-
herent band formation any more.
nite excitation energy, which however is practically dis-
We proceed to the dynamical spin correlation function
persionless, both with respect to its energy and with re-
(SCF), defined as
specttoitsspectralweight. Inotherwords,themagnetic
S(k,ω)= Z1 Xe−βEµ|hν|Sαz(k)|µi|2δ(ω−ωνµ), (3) fin-seuxlcaittoartitornanbseictioomnetshupsraicstricealalltyedimtomaobdirlaes.tiTchcheamngeetaol-f
ν,µ
the spin correlation function [5] - the question whether
where Sz(k) is (the Fourier transform of) the z- the longer ranged spin correlations below the transition
α
3
are the driving force behind the gap formation [13] or spincorrelationfunctionshowsnomoreindicationofthe
whether the change of the spin correlations is merely low energy resonance - the overall picture is completely
a ‘byproduct’ of the collapse of the single-particle gap, the same as for V/t=1 and βt=1, with the sole excep-
remains to be clarified. In any way, the almost com- tion that all features are much sharper due to the lower
pletely localized spin dynamics of the f-electrons natu- temperature. Here we do not pursue the issue of the de-
rallyshouldleadtoaCurie-lawforthestaticspinsuscep- tailed parameter dependence of the characteristsic tem-
tibility at temperatures aboveT . A crossoverfrom an peratures-itisquiteobvious,however,thatlowervalues
coh
activation-gap dominated susceptibility at low tempera- of V shift the characteristic temperatures of the system
tures and a Curie law at high temperatures has indeed towardslowervalues,butotherwiseleavethephysicsun-
been observed by Shibata et al. [6]. In the c-like SCF changed.
the particle-hole continuum persists and the gap near In summary, we have studied the temperature evolution
k=π is now very small or zero (the absence of the gap of various dynamical correlation functions of the Kondo
in the single particle spectrum suggests it to be zero). insulatorandfoundindicationsfortwodistinctelectronic
There is no more distinguishable peak in the c-like SCF crossovers. At the low temperature crossover the single-
which would correspond to the dispersionless f-like spin particle gap closes so that the system becomes a metal,
resonance - this suggests that the f-electrons now are the magnetic correlations on the f-sites become local-
largely decoupled from the c-like band, as indicated by ized, and the ‘spin gap’ closes due to c-like spin excita-
their non-participationinthe Fermisurface. At the very tions. While c and f electrons seem to form a coherent
high βt=1, there is still some (very weak) indication of ‘all-electronfluid’belowthecrossovertemperature,thec
the low energy f-electron spin resonance, but the inten- andf-likefeaturesinthecorrelationfunctionsabovethis
sity is low and the resonance is now relatively broad. It temperature are decoupled. We therefore interpret this
should be noted that the relative change of the f-like temperature as the analogue of the coherence tempera-
magneticmomentislessthan2%overtheentiretemper- ture in heavy-Fermion metals. At the high-temperature
ature range we studied - temperature thus affects only crossoverboth,thedispersionlessf-likeKondo-resonance
the coupling of these moments to the conduction elec- in the single-particlespectrum and the f-likelow-energy
trons. spinexcitationdisappear. Theonlyremainingf-likefea-
FortherelativelylargevalueofV=1thehigherofthetwo ture in the single particle spectrum are the high-energy
crossover temperatures (i.e. the Kondo temperature) is ‘Hubbard bands’, correspondingto the ‘undressed’ tran-
alreadyratherhigh,βt≥1. Basedontheimpurityresults sitions f1 → f0 and f1 → f2. We therefore interpret
[14]onemightexpectthatforsmallerV the Kondotem- this second temperature as the Kondo temperature of
perature is lower, and to check this, we have performed the system.
0.6
0.5 Vf-=elte=ctr1o:n0
0.4 c-electron
0.3
0.2
0.1
0
0.5 V=t=0:50 [1] G. R.Stewart, Rev.Mod. Phys. 56, 755 (1984).
!) 0.4
D( 0.3 [2] P. Fulde, J. Keller, and G. Zwicknagl, Solid State Phys.
0.2 41, 1 (1988).
0.1
0 [3] G. Aeppli and Z. Fisk, Comments Condens. Matter
0.5 V=t=0:25 Phys., 16 155 (1992).
0.4
0.3 [4] Z. Schlesinger et al.,Phys. Rev.Lett. 71, 1748 (1993).
00..21 [5] M. Vekic et al.,Phys. Rev.Lett. 74, 2367 (1994).
0 [6] N. Shibata et al.,cond-mat/9712315.
-8 -6 -4 -2 0 2 4 6 8
!-(cid:22)=t [7] L. Tailefer et al., J. Magn. Magn. Mater. 63& 64 372
FIG. 4. Single particle spectral density D(ω) at βt=5 for (1987).
different V/t. All other parameters are as in Figure 1. [8] J. Cooley et al.,Phys. Rev.Lett. 74, 1629 (1995).
[9] K. Tsutsui et al., Phys.Rev.Lett. 76, 279 (1996).
[10] A. M. Tahvildar-Zadeh, M. Jarrel, and J. K. Freericks,
simulations at fixed βt=5, but with variable hybridiza-
e-print cond-mat/9710136.
tion. Figure 4 shows the results for the single particle
[11] O. Gunnarson and K. Sch¨onhammer, Phys. Rev. B. 28,
spectral density. The weight of the Kondo resonance-
4315 (1983).
like sidebands for fixed temperature decreases with V,
[12] R. Eder, O. Stoica, and G. A. Sawatzky, Phys. Rev. B.
and for V/t=0.25 they have disappeared completely. In 55, 6109 (1997); see also cond-mat/9711248.
the k-resolved c-electron spectrum (not shown) there is [13] H. Tsunetsugu, M. Sigrist, and K. Ueda, Rev. Mod.
now only a very sharp nearest neighbor-hopping band Phys. 69, 809 (1997).
with a clear Fermi level crossing at k0 and the f-like [14] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).
F
4