Ferromagnetism and Kondo Insulator Behavior in the Disordered Periodic Anderson Model

0.15 MB·English

by  Unjong Yu

Ferromagnetism and Kondo Insulator Behavior in the
Disordered Periodic Anderson Model
Unjong Yu1, Krzysztof Byczuk1,2, and Dieter Vollhardt1
1Theoretical Physics III, Center for Electronic Correlations and Magnetism,
Institute for Physics, University of Augsburg, D-86135 Augsburg, Germany
2Institute of Theoretical Physics, University of Warsaw, ul. Hoz˙a 69, 00-681 Warszawa, Poland
(Dated: February 4, 2008)
The effect of binary alloy disorder on theferromagnetic phases of f-electron materials is studied
within the periodic Anderson model. We find that disorder in the conduction band can drastically
8 enhance the Curie temperature Tc due to an increase of the local f-moment. The effect may be
0 explainedqualitativelyandevenquantitativelybyasimpletheoreticalansatz. Theemergenceofan
0 alloy Kondo insulator at non-integerfilling is also pointed out.
2
PACSnumbers: 71.23.-k,75.20.Hr,75.30.Mb
n
a
J
Materialswithf-electronssuchastherareearths(e.g., motion of the Cu electrons [9]. Finally, alloying Ce with
5
2 cerium) or actinides (e.g., uranium) exhibit a wealth of La asin (Ce1−xLax)3Bi4Pt3 introduces disorderinto the
highly unusual thermodynamic, magnetic and transport f-electron system, which can trigger a transition from a

] properties [1]. The minimal microscopic model that can Kondoinsulatortoadirtymetal[10]. Clearlydisorderis
l
e account for this diverse physical behavior is the peri- animportantfeatureofmanyf-electronalloysandmust
-
r odic Anderson model (PAM) which describes a band therefore be included in any comprehensive theoretical
st of non-interacting electrons hybridizing with localized, study of such compounds.
. interacting f-electrons [2]. Depending on the position Previous investigations of the disordered PAM fo-
t
a of the f-electron energy εf relative to the conducting cussed,in particular,onthe effect ofnonmagnetic impu-
m
band, and on the strength of the hybridization V and rities on the heavy Fermi liquid or the Kondo insulating
- the local Coulomb interaction U, the PAM is able to re- state [11], and on the disorder-driven non-Fermi liquid
d
n produce heavy fermion, intermediate valence, and local behavior in Kondo alloys [12]. Grenzebach et al. [13] re-
o moment behavior. In the local moment regime and for cently presenteda detailed study of transportproperties
c large U the PAM reduces to the so called Kondo lattice of the disordered PAM within the dynamical mean-field
[
model, which may be employedas an effective model for theory(DMFT) [14]togetherwithathoroughdiscussion

1 manganites exhibiting colossal magnetoresistance [3], or of the development of the field. The effect of disorder in
v fordilutedmagneticsemiconductorswhichshowpromise the f-electrons on the ferromagnetic phase was investi-
4
for applications in spintronics [4]. At low enough tem- gatedby Meyer [15], who found that the Curie tempera-
3
peratures the PAM also describes magnetically ordered ture is always reduced.
9
3 phases. While antiferromagnetic order is well-known to In this Letter we report results of a detailed study
. occur close to half-filling [2], ferromagnetic solutions are of ferromagnetism in the PAM in the presence of alloy
1
found far away from half-filling [5]. Indeed, ferromag- disorder. In particular, we show that T can be sub-
0 c
8 netism has been observed in various f-compounds [6]. stantially enhanced by disorder in the conduction elec-
0 trons. We also predict Kondo insulator behavior away
Alloys of f-electron materials also display intriguing
v: properties. For example, by changing the stoichiomet- from half-filling at particular values of the alloy concen-
i tration. Quite generally,the Hamiltonian of the PAM in
X ric composition of Ce(Pt1−xNix)2Si2 the systems can be
the presence of local disorder has the form
tuned from the local moment regime at x = 0 to the
r
a intermediate valence regime at x = 1 [7]. Alloying in-
evitably introduces disorder into the system. In general, H = tijc†iσcjσ+ εfifi†σfiσ +εcic†iσciσ
disorder is expected to reduce the tendency towards fer- Xi,jσ Xiσ (cid:16) (cid:17)
romagnetic long-range order of the f-electrons and thus + Vc† f +V∗f† c +U nf nf ,(1)
iσ iσ iσ iσ i↑ i↓
lower the Curie temperature Tc. On the other hand, Xiσ (cid:16) (cid:17) Xi
in certain cases disorder is even known to improve the
stability of ferromagnetism. For example, disorder in where c† (c ) and f† (f ) are creation (annihilation)
iσ iσ iσ iσ
the conduction electron band caused by the substitu- operators of conduction (c) and localized (f) electrons
tion of Rh by Co in URh Co Ge leads to a maxi- with spin σ at a lattice site i. Here the on-site ener-
1−x x
mum in T at x 0.6 [8]. Similarly, a maximum in gies εf and εc are random variables and V is the local
c ≈ i i
T is observed in CeCu Si Ge at x 1.5 which may hybridization between f- and c-electrons. The hopping
c 2 2−x x
≈
be attributed to an enhanced exchange interaction be- amplitudeofthec-electronsisgivenbyt . TheCoulomb
ij
tween the f-electron moments induced by the diffusive interaction U acts only between f-electrons on the same
2
site. in general, independent parameters the values x = 0, 1
The alloy will be modeled by a bimodal probability correspond to a non-disordered system even if ∆y = 0.
6
distribution function, P(y ) = xδ(y y +∆y)+(1 Hence δy = x(1 x)∆y is a natural parameter for the
i i 0
− − −
x)δ(y y ), where y = εc, εf are independent, ran- disorder strength of alloy disorder.
i − 0 i i i
dom variables with reference values y = εc, εf. The The PAM with binary alloy disorder is solved within
0 0 0
alloy concentration is characterized by the parameter x DMFT [14], which becomes exact in the limit of infi-
and the splitting between the atomic levels of the alloy nite dimensions [16]. In DMFT the disordered PAM is
components by the energy ∆y = ∆c, ∆f, respectively. mapped onto independent impurities, i.e., for each ran-
While the concentration x and energy splitting ∆y are, dom variable y the action has the form
i
{ }
Sloc[fσ,cσ;{yi}]=Xσn (fσ∗n,c∗σn)(cid:18)iωn+Vµ−εfi iωn+µV−∗εci −ησn (cid:19)(cid:18)fcσσnn (cid:19)+UZ0β dτ nf↑(τ)nf↓(τ). (2)
The function η describes an effective dynamical hy- hybridized two-band system in several interesting ways.
σn
bridization of the c-electrons with the bath. It is the To this end we consider the disorder to act only on the
same for all random variables y and is determined by c-electrons or the f-electrons, respectively. In the case
i
{ }
the self-consistencyequations to be discussedbelow. We of c-electron disorder the diagonal elements of the local
start with the local matrix Green function Green function (5) are given by
T f (τ)f†(0) T f (τ)c†(0)
Glσoc(τ;{yi})=−(cid:18)hhTττcσσ(τ)fσσ†(0)ii hhTττcσσ(τ)c†σσ(0)ii(cid:19),(3) Gcσcn = (Gσccn)−1−x |V|2Gσfnf + (Gσccn)−1−1|−V|x2Gσfnf −∆c
where Tτ is the time-ordering operator. Since the Green Gff = x + 1−x .(7)
Hfuenrcetiwone p(e3r)fodrempeanrditshmonet{icyia}veirtagisinag,ria.en.d,otmhefauvnecrtaigoend. σn (Gσfnf)−1−|V|2Gσccn (Gσfnf)−1− (Gσccn|)V−|12−∆c
Green function is given by
Thecaseoff-electrondisorderisobtainedbyexchanging
f c in Eq. (7). Large energy splitting ∆c leads to
Gloc(τ)= dy P(y )Gloc(τ; y ). (4) ↔
σ Z i i σ { i} a band splitting of the conduction electrons as in the
{Yyi} single band model [19], i.e., each alloy subband contains
In the absence of interactions one then obtains the re- 2xNL and 2(1 x)NL states, respectively, where NL is
−
sults of the well-known coherent potential approxima- thenumberoflatticesites. Atthe sametime,thef-level
tion (CPA) [17]. Effects of Anderson localizationare ne- does not split. Altogether the alloy with hybridized c-
glectedinthiscasebutcanbeincorporatedbyemploying and f-electrons can therefore be a band insulator even
the geometric average[18]. The self-consistency requires for total densities different from integer values (2 or 4)
the averaged local Green function (4) to be the same as [19].
the lattice Green function, i.e., We now include the interaction U between the f-
electrons and investigate its influence on the alloy sub-
Gσn =Xk (cid:18)iωn+µV−Σfσn iωn+µ−V∗ǫk−Σcσn (cid:19)−1.(5) bthaendpsr.eseTnhcee oefffedcitsiovredetrw,oE-oqrsb.it(a2l-6i)m, pisursiotylvepdrobbylefimniitne
temperature determinant quantum Monte-Carlo (QMC)
The local self-energies appear in the k-integrated Dyson simulations. Ferromagnetic instabilities are detected by
equationΣ = −1 G ,where isthelocalGreen
σn Gσn− σn Gσn the divergence of the static spin susceptibility and by a
function of the non-interacting bath electrons, with non-vanishing value of the magnetization [20]. In the
iω +µ εf V∗ numerical examples presented below the DOS for the
Gσ−n1 =(cid:18) n V − 0 iωn+µ−εc0−ησn (cid:19). (6) n√o4n-inεt2e/r2aπct,inwghce-reeletchteroennserhgaysuthneitmisotde=l f1o.rmInNt0h(eε)fo=l-
−
Eqs. (2-6) form a general, closed set of equations, which lowing we fix the interaction and the hybridization at
determine all local, dynamical correlation functions of U = 1.5 and V = 0.6, respectively, and include disorder
the disordered PAM. either in the f-electron or c-electron system.
To understand the effect of alloy disorder on the As shown in Fig. 1 the computed Curie temperature
physics described by the PAM it is instructive to inves- for the transition to the ferromagnetic state is a non-
tigate the case U = 0 first, since alloy disorder affects a monotonic function of the alloy concentration x. In par-
3
ticular, the behavioris quite differentfor disorderacting 0.0 1 2 ∆f = 1.0 ∆c = 1.0
on the f- or the c-electrons.
f-electron disorder: In agreement with Meyer [15] the Tc 0.0 0 8
0.004
presence of f-electron disorder always reduces the Curie FM F M FM
0
temperaturerelativetoitsnon-disorderedvaluesatx=0 0.0 1 2 ∆ f = 1.7 ∆ c = 1.7
or 1. For strong enough disorder T eventually vanishes,
e.g., at x=0.28and x=0.75,respcectively, for ∆f =1.7 Tc 0.0 0 8
0.004
(left panel of Fig. 1). This is due to the splitting of the FM FM
0
f-electron band at large ∆f which increases the double 0.0 1 2 ∆ f = 2.0 ∆ c = 2.0
occupation of the lower alloy subband; this reduces the
local moment of the f-electrons and thereby Tc. Tc 00..00 00 84
c-electron disorder: By contrast, c-electron disorder FM FM FM
0
leads to a much more subtle dependence of Tc on con- 0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1
centration x. Namely, for increasing energy splitting ∆c x x
there are, in general, three different features observed,
the physical origin of which will be discussed in more FIG. 1: Curie temperature Tc as a function of alloy concen-
detail later: (i) at x = 1, i.e., in the non-disordered trationxandenergysplitting∆f (leftcolumn)and∆c (right
case, Tc is reduced; (ii) a minimum develops in Tc at column) for ntot =1.3 and ε0c−ε0f =3.25. Strong c-electron
x = n 1 > 0; (iii) T is enhanced over its non- disorder enhances Tc compared to its values at x=0 or 1.
tot c
−
disordered values at x = 0 or 1. Altogether this leads
to a global maximum in T vs. x. While the decrease
c 0.3 ∆c = 0.0
of Tc at x = 1 is a simple consequence of the reduction 0.2
of the energy difference between the f-level and the c- 0.1
electronband, εc εf =εc εf ∆c, for increasing∆c, 0
the latter effects a−re more0s−ubt0le−. 0.2 ∆c = 2.0

To understand the minimum in Tc vs. x we com- 0.1
)
puted the evolution of the spectral functions Nc(ω) at ω 0
(
x = 0.3 for ∆c increasing from 0 to 6 (Fig. 2). There cN 0.2 ∆c = 4.0
is an opening of a gap at the chemical potential sig- 0.1
nalling a metal-insulator transition in this system. This
0
is caused by the splitting of the c-electron band due to ∆c = 6.0
0.2
binaryalloydisorderandthecorrelationsbetweenthef-
0.1
electrons. Namely, for energy splittings ∆c much larger
than the width of the c-electron band the total number 0
−4 −2 0 2 4 6 8 10
of available low-energy states is reduced from 4N to ω − µ
L
[4 2(1 x)]N = 2(1+x)N , whereby the filling ef-
L L
− −
fectively increases by a factor of 4/[2(1+x)], such that FIG.2: Spectralfunctionofc-electronsfordifferent∆catx=
neff = 2n /(1+x), if n < 2(1+x). For the filling 0.3(otherparametersasinFig.1)obtainedwithinQMCand
tot tot tot
n = 1.3 studied in the Figs. 1 and 2 the concentra- maximalentropyatT =1/60. Byincreasing∆c apseudogap
titoont x = 0.3 is a special case since then neff = 2. The opens which becomes a real gap for T →0.
tot
system is then effectively at half-filling and behaves as a
Kondo insulator at large U, ∆c, and low temperatures.
In particular, itinerant ferromagnetism is unfavorable in ansatz can be microscopically justified within a static
thiscase,i.e.,T =0inthevicinityofx=0.3at∆c =2, mean-field theory [22]. The two functions Fc, Ff are
c
cf. Fig. 1. The metal to Kondo insulator transition at determined by Tc calculated within DMFT for the non-
non-integerfillinginthePAMpredictedhereisacounter- disorder case at fixed µ−ǫc0 or µ−ǫf0, respectively; they
part of the Mott-Hubbard metal to insulator transition areshowninFig.3(a)and3(b)foronesetofparameters.
at non-integral fillings in the one-band Hubbard model The prefactor Tc0 is determined by the requirement that
found in [19, 21]. thedimensionlessfunctionsFf andFc beequaltooneat
We now turn to the maximum in Tc vs. x. It their maxima. We note thatFf(µ−εf0) has a maximum
can be understood within the following model based when the f-level is half-filled (µ = εf +U/2), i.e., when
0
on an ansatz for the Curie temperature, T (U,V,µ) = the local moment is maximal.
c
Tc0(U,V,µ)Fc(µ−ε0c)Ff(µ−ε0f), which implies that the The Curie temperature in the presence of c-electron
formationoflocalf-electronmoments(Ff)isassumedto disorder can now be estimated by averaging over the c-
be independent fromthe c-electronmediated orderingof electron part, Fc, giving rise to the disorder-dependent
those moments (Fc). In fact, for the RKKY model this function c(x,µ ǫc)=[xFc(µ εc+∆c)+(1 x)Fc(µ
F − 0 − 0 − −
4
(a) (c)
0.9
0.9

cF00 .. 63 µ − εf0 = 0.75 c f , F 00 .. 63 F fc [1] CG(2..00RM1.).S;tLeV.waDarrmetg,aio,RregRvi,.eRMv.eovd.M.MoPdoh.dy.sP.P5hhy6ys,s..7754518,(,618927814(9)1;97(93199,)7.769)7;
0
−1.8 −1.5 −1.2 −0.9 −0.6 [2] P. A. Lee, T. M. Rice, J. W. Serene, L. J. Sham, and
µ − εc0 0 a nsa tz (d) J.W.Wilkins,CommentsCondens.MatterPhys.12,98
(b) 0.012 DMFT (1986).
0 . 9 0.009 [3] E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344, 1
fF0 . 6 µ − εc = −1.17 Tc0.006 [4] S(2e0e0,1e)..g., T. Jungwirth, J. Sinova, J. Maˇsek, J. Kuˇera,
0 . 3 0 0.003 A. H.MacDonald, Rev. Mod. Phys. 78, 809 (2006).

0 0.000 [5] A. N. Tahvildar-Zadeh, M. Jarrell, and J. K. Freericks,
0 0.4 0.8 1.2 0 0.2 0.4 0.6 0.8 1 Phys. Rev. B 55, R3332 (1997); R. Doradzin´ski and
µ − εf0 x J. Spal ek, Phys. Rev. B 56, R14239 (1997); D. Meyer
and W. Nolting, Eur. Phys. J. B 18, 385 (2000);
FIG. 3: (a) Fc(µ− εc0), (b) Ff(µ − εf0) appearing in the C. D. Batista, J. Bonˇca, and J. E. Gubernatis, Phys.
ansatz forTc (seetext)calculated for∆c =0. (c)Fc(µ−εc0) Rev. B 68, 214430 (2003).
and Ff(µ−εf0) for ∆c = 2.0; other parameters as in Fig. 1. [6] See, e.g., T. Durakiewicz, C. D. Batista, J. D. Thomp-
(d) Comparison of Tc obtained from the ansatz and within son, C. G. Olson, J. J. Joyce, G. H. Lander, J. E. Gu-
DMFT. bernatis, E. Guziewicz, M. T. Butterfield, A. J. Arko,
J. Bonˇca, K. Mattenberger, and O. Vogt, Phys. Rev.
Lett. 93, 267205 (2004).
εc)]. The linear dependence on the alloy concentration [7] F. C. Ragel and P. de V. du Plessis, J. Phys.: Condens.
0
can again be justified microscopically within a static Matter 16, 2647 (2004)
mean-field theory for the RKKY model, where T de- [8] S. Sakarya, N. T. Huy, N. H. van Dijk, A. de Visser,
c
M. Wagemaker, A. C. Moleman, T. J. Gortenmulder,
pendslinearlyontheDOSatthechemicalpotential[22].
J. C. P. Klaasse, M. Uhlarz, and H. v. L¨ohneysen,
T is now determined for each concentration x. We cal-
c arXiv:cond-mat/0609557.
culate µ, which is an implicit function of x, in the non-
[9] M. B. Silva Neto, A. H. Castro Neto, D. Mixson,
hybridized limit (V = 0) within a rigid band approxi- J.S.Kim,andG.R.Stewart,Phys.Rev.Lett.91,257206
mation [23]. The dependence of the resulting functions (2003).
c(x,µ εc) andFf(µ εf)onxareshowninFig.3(c) [10] M.F.Hundley,P.C.Canfield,J.D.Thompson,Z.Fisk,
fFor∆c =−2.00. Ingeneral−Ff0(µ ǫf)hasaglobalmaximum and J. M. Lawrence, Physica B 171, 254 (1991).
− 0 [11] A. Yoshimori and H. Kasai, Solid State Commun. 58,
atthosevaluesofxforwhichthef-levelishalf-filled[see 259 (1986); Zh.-Zh. Li and Y. Qiu, Phys. Rev. B 43,
Fig. 3(c)]. By contrast, Fc(x,µ−ǫc0) is characterizedby 12906(1991);T.Mutou,Phys.Rev.B64,245102(2001);
awideminimum,relatedtothe formationofthepseudo- S. Wermbter, K. Sabel, and G. Czycholl, Phys. Rev. B
gapintheinteractingDOSseeninFig.2. Thisminimum 53, 2528 (1996); P. Schlottmann, Phys. Rev. B 46, 998
reaches zero, i.e., c(x,µ εc) = 0, for a finite range (1992).
F − 0 [12] E. Miranda, V. Dobrosavljevi´c, and G. Kotliar, Phys.
of x values as shown in Fig. 3(c). The resulting T (x)
c Rev. Lett.78, 290 (1997).
obtained by the product of these two functions agrees
[13] C. Grenzebach, F. B. Anders, G. Czycholl, and Th. Pr-
remarkably well with the numerical result obtained by
uschke, arXiv:cond-mat/0711.1768.
DMFT as shown in Fig. 3(d). [14] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozen-
In conclusion, the interplay between the disorder in- berg, Rev. Mod. Phys. 68, 13 (1996); G. Kotliar and
duced splitting of the conduction band and many-body D. Vollhardt, Phys.Today 57, 53 (2004).
correlationsamongthe f-electronscanleadto aremark- [15] D. Meyer, Solid State Commun. 121, 565 (2002).
[16] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324
able enhancement of the Curie temperature in the peri-
(1989);
odic Anderson model. There are two competing effects [17] R. Vlaming and D. Vollhardt, Phys. Rev. B 45, 4637
determining Tc as the alloy concentration x is decreased (1992); V. Janiˇs and D. Vollhardt, Phys. Rev. B 46, 15
from x = 1: (i) a rise due to an increase of the local 712 (1992).
moment, and (ii) a decrease due to the opening of a gap [18] K. Byczuk,W. Hofstetter, and D. Vollhardt,Phys. Rev.
in the alloy Kondo insulator at non-integer filling. Al- Lett. 94, 56404 (2005).
together this leads to a global maximum in T vs. x. [19] K. Byczuk,W. Hofstetter, and D. Vollhardt,Phys. Rev.
c B 69, 45112 (2004).
Therefore experimental investigations of f-electron ma-
[20] M.Ulmke,Eur.Phys.J.B1,301(1998);K.Byczukand
terials with alloy disorder in the conducting band are D. Vollhardt, Phys.Rev. B 65, 134433 (2002).
expected to be particularly rewarding.
[21] K.Byczuk,M.Ulmke,andD.Vollhardt,Phys.Rev.Lett.
This work was supported in part by the Sonder- 90, 196403 (2003).
forschungsbereich 484 of the Deutsche Forschungsge- [22] See, e.g., T. Dietl, J. Cibert, D. Ferrand, and Y. Merle
meinschaft (DFG). d’Aubign´e, Mater. Sci. Eng. B 63, 103 (1999).
5
[23] Approximating the chemical potential by µ = xµx=1+ as ∆c is smaller than theband width.
(1−x)µx=0,leadstoqualitativelysimilar resultsaslong