THEORY OF VALENCE TRANSITIONS IN YTTERBIUM-BASED COMPOUNDS

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1 Chapter 1 THEORY OF VALECE TRASITIOS I YTTERBIM-BASED COMPODS V. Zlatić, and J. K. Freericks Institute o Physics, 1 Zagreb, Croatia Department o Physics, Georgetown niversity, Washington, DC 257, SA Isaac ewton Institute, Cambridge CB3 EH, K Abstract The anomalous behavior o YbInCu and similar compounds is modeled by the exact solution o the spin one-hal Falicov-Kimball model in ininite dimensions. The valence-luctuating transition is related to a metal-insulator transition caused by the Falicov-Kimball interaction, and triggered by the change in the -occupancy. 1. ITRODCTIO The intermetallic compounds o the YbInCu amily exhibit an isostructural coniguration to the low-temperature mixed-valent state with Yb ions luctuating between and conigurations [1]. The transition is particularly abrupt in high-quality stoichiometric YbInCu samples [2] with a transition temperature equal to K at ambient pressure; the susceptibility and the resistivity transition rom high-temperature state with trivalent Yb ions in the drop at by more than one order o magnitude in cooling, while the volume expansion is small,. The valence change inerred rom! by using the usual ionic radii o Yb #" and Yb #" is about %$'&()+*, which is consistent with the valence measurements by the,-#-#- -edge absorption [1, 4]. The critical temperature depends strongly on external pressure, magnetic ield, and alloying [5, 6]. A recent review o the experimental data is given in Re. [7] and here we just recall the main points which motivate our choice o model. The integer-valent phase (/. ) is characterized by a Curie-Weiss susceptibility [1, 6] with very small Curie-Weiss temperature The Curie constant corresponds to the ree moment o one magnetic -hole in a 567 spin-orbit state with 8:9;&<& =>?8'@. The electrical resistance is large and has 1

2 L L 2 a small positive slope; it remains almost unchanged in magnetic ields up to 3 T [5]. In some systems, like Yb ACB Y B InCu, the magnetoresistance is slightly negative, while in YbInCu (or YbIn ACB Ag B Cu or DEF+*< ) it is slightly positive [5]. The Hall constant is large and negative, indicating a small number o carriers [4, 8]. The thermoelectric power has a rather small slope which one inds in a semiconductor with a nearly symmetric density o states [9]. Recent data on the optical conductivity o YbInCu [1] shows the absence o a Drude peak at high temperatures and a pronounced maximum o the #" optical spectral weight at about 1 ev. The high-temperature ESR data or Gd embedded in YbInCu resemble those ound in integer-valence semi-metallic or insulator hosts [11]. Thus, the high-temperature phase indicates the presence o a well deined local moment but gives no signature o the Kondo eect. The overall behavior o the high-temperature phase is closer to that o a semi-metal or paramagnetic small-gap semiconductor than to a Kondo metal. The mixed-valent phase (HGI ) behaves like a Pauli paramagnet with moderately enhanced susceptibility and speciic heat coeicient [6]. The electrical resistance and the Hall constant are one order o magnitude smaller than in the high-temperature phase [4, 8]. The thermoelectric power [9] has a very large slope typical o a valence luctuator with large asymmetry in the density o states. The susceptibility, the resistivity and the Hall constant do not show any temperature dependence below, which is also typical o valence luctuators. The optical conductivity shows a major change with respect to the high-temperature shape. The peak around 1 ev is reduced, the Drude peak becomes ully developed, and an additional structure in the mid-inrared range appears quite suddenly below [1]. A large density o states at the chemical potential 8 is indicated by the ESR data as well [12]. Thus, the transition at seems to be rom a paramagnetic semimetal to a valence luctuator. In contrast to usual valence-luctuators, which are quite insensitive to the magnetic ield, the YbInCu amily o compounds also exhibit metamagnetic transitions when FGJ. The Yb moment is ully restored at a critical ield KMLO KL QP, with a Zeeman energy 8'@ comparable to the thermal energy R@S. The metamagnetic transition deined by the magnetoresistance or the magnetization data [7] gives an H-T phase boundary K L KV QPT LXW K *ZY [\]P. The zero-temperature ield K is related to _*]=` [7]. To account or these eatures we need a model in which the non-magnetic, valence-luctuating, metallic ground state can be destabilized by increasing temperature or magnetic ield. Above the transition, we need a paramagnetic semiconductor with an average -occupancy that is not changed much with respect to the ground state. The correct model or this system is a periodic Anderson model supplemented with a large Falicov-Kimball (FK) interaction term. The temperature or ield induced transition suggests that one should place the narrow -level just above the chemical potential 8. The hybridization keeps

3 } K t } } Theory o valence transitionsin ytterbium-based compounds 3 the -count inite below the transition, while large - correlations allow only the luctuations between zero- and one-hole (magnetic) conigurations. The low-temperature phase is close to the valence luctuating ixed point and shows no Kondo eect. However, because o the Falicov-Kimball term, there is a critical -occupation at which there is a transition into the high-temperature state with a large gap in the d- and -excitation spectrum. The $'& is driven to criticality either by temperature or magnetic ield. In the high-temperature phase the hybridization can be neglected because the -level width is already large due to thermal luctuations, and quantum luctuations are irrelevant. nortunately, the above model would be diicult to solve in a controlled way, and here we consider a simpliied model in which the hybridization is neglected at all temperatures. This leads to a spin-degenerate Falicov-Kimball model which explains the collapse o the non-magnetic metallic phase at or K L, and gives a good qualitative description o the high-temperature paramagnetic phase. However, the deiciency o the simpliied model is that it yields a negligible -count in the metallic phase and predicts a large change in the Yb valence at or K L. It is clear that we can not obtain the valence luctuating ground state and maintain the average -occupancy below the transition without hybridization-induced quantum luctuations. In what ollows, we describe the model, explain the method o solution, and present results or static and dynamic correlation unctions. 2. CALCLATIOS The Hamiltonian o the Falicov-Kimball model [13] consists o two types o electrons: conduction electrons (created or destroyed at site a by bdc eg or b eg ) and localized electrons (created or destroyed at site a by eh c or eg ). The conduction electrons can hop between nearest-neighbor sites on the D-dimensional lattice, with a hopping matrix YZi ekj n o lyzi;mo ; we choose a scaling o the hopping matrix that yields a nontrivial limit in ininite-dimensions [14]. The -electrons have a site energy pq&, and a chemical potential 8 is employed to conserve the total number o electrons $:r;stu$:r v tu$'&ws tu$'&wv[x$zy+{ y. The Coulomb repulsion &<& between two -electrons is ininite and there is a Coulomb interaction } between the b - and -electrons that occupy the same lattice site. An external magnetic ield ~ couples to localized electrons with a Landé g-actor. resulting Hamiltonian is [15, 16] J ej YZi ej Y68' ekj P b eg c b j t e \\ b eh c b eg eh\ c eg t ~q e ˆ e pƒ&(y 8 P &<& e e cs e s b eg c b eg t Š eg c eg e cv e v The eh c eg P (1.1)

4 ² ª š ¾ Å P ¾ ª * Œ š P ª Y } ¾ * Å } 4 The model can be solved in the ininite-dimensional limit by using the methods o Brandt-Mielsch [15]. We consider the hypercubic lattice with Gaussian density o states Œ P)Žw hy ^i m n i m, and take i;m as the unit o energy (i m _* ). Our calculations are restricted to the homogeneous phase. The local conduction-electron Green s unction satisies Dyson s equation PT3 t 8 Ỹ where is a complex variable and š P Y b O (1.2) is the local sel energy which does not depend on momentum [14]. In ininite dimensions, š is deined by a sum o skeleton diagrams, which depend on the local d-propagator but not on i ej. The exact sel-energy unctional or the FK model is obtained by calculating the thermodynamic Green s unction [17] o an atomic system coupled to an ž external time-dependent ield Ÿ ž PT_Y * ' rg&[] ª where the S-matrix or the -ield is AC«' g b ž± P b ž c AZµ ³PTx ª r µ r ¹ ]º g» r ¼ ½ º» r ½ º h» and K Ÿ keeping just a single lattice site. The exact solution or ¾ ¹? CÀÁ requency aãâ ¾ xa $Ätx*\P is given by, ¾ Az ¾ t ¾ Å Az P ² ³P (1.3) (1.4) is obtained rom the Hamiltonian (1.1) by removing the hopping and at Matsubara where Å and Å are the -occupation numbers ( Å Æ*qY and [15] with Ä 8 PÇ 8 P «?È?É; Ê 8 P't The bare Green s unction satisies ¾ aãâ ¾ P s ¾ AC«ºÌË ACÈ» * aâ ¾ tî8ïỹ «?È?É; Ê ¾ aâ ¾ P 8 Y, Å with ¾ the Fourier transorm o the external time-dependent ield. The sel-energy unctional š ¾ v ¾ (1.5) ) (1.6) P (1.7) (1.8) can now be obtained [15] by using the Dyson equation or the atomic propagator, ¾ ¾ Az YÐ ¾ Az (1.9)

5 ~ Theory o valence transitionsin ytterbium-based compounds 5 and eliminating ¾ ¹? À Á lattice is achieved by adjusting ¾ in such a way that ¾ ¹? À Á rom Eqs. (1.5) and (1.9). The mapping onto the satisies the lattice Dyson equation (1.2). The numerical implementation o the above procedure is as ollows: We start with an initial guess or the sel energy š and calculate the local propagator in (1.2). sing (1.9) we calculate the bare atomic propagator ¾ and ind and. ext we obtain, and ind ¾ rom (1.5). sing ¾ and Å Å ¾, we compute the atomic sel energy and iterate to the ixed point. The iterations on the imaginary axis give static properties, like $'&, the - magnetization ÑÒ& i#p, and the static spin and charge susceptibilities. Having ound the -electron illing Å at each temperature, we iterate Eqs. (1.2) to (1.9) on the real axis and obtain the retarded dynamical properties, like the spectral unction, the resistivity, the magnetoresistance, and the optical conductivity. At the ixed point, the spectral properties o the atom perturbed by -ield coincide with the local spectral properties o the lattice. 3. RESLTS AD DISCSSIO We studied the model or a total electron illing o 1.5 and or several values o pq& and }. The main results can be summarized in the ollowing way..5.4 n (T) E =.7t* E =.6t* E =.5t* E =.2t* Ó Ô T/t* Õ Figure 1.1 rom top to bottom, and is given by -.7, -.6, -.5, and -.2, respectively. umber o the Ö -holes plotted versus 'Ø ÙÚ or ÛzØ ÙÚÝÜßÞ. The à Ø ÙÚ increases The occupancy o the -holes at high temperatures is large and there is a huge magnetic degeneracy. The -holes are energetically unavorable but are maintained because o their large magnetic entropy. In Fig.(1.1) we show $'& as a unction o temperature, plotted or } i m, and pƒ&^i m rom -.2 to -.7. Below a certain temperature, which depends on } and pq&, there is a rapid transition to the low-temperature phase. The transition becomes sharper and is pushed to lower temperatures as pƒ& decreases at constant }. However, we

6 Š æ ç ç ç ç } 6 restrict ourselves to continuous crossovers here, since the region with irst-order transitions leads to numerical instabilities. The uniorm -spin susceptibility is obtained by calculating the spin-spin correlation unction [15, 16] and is given by á âp_ãq$'& QP \, where ã ä 8 5MtE*\P is the Curie constant. The á QP ã is shown in Fig χ(t)/c 2. E =.7t* E =.6t* E =.5t* E =.2t* T/t* å Figure 1.2 niorm static magnetic susceptibility o the Ö -holes plotted versus 'Ø Ù Ú or Û:Ø Ù Ú Ü Þ. The values o à Ø#Ù Ú are the same as in Fig(1.1). The corresponding values o è Ø#Ù Ú are estimated rom the maximum o é³êh ìë, and are given by.3,.8,.15,.35, respectively. The è increases rom top to bottom. or } ^i m and or pq& as quoted in Fig.(1.1). The is obtained rom the QP ã and the values corresponding to various parameters maximum o the á used in this paper are quoted in the caption o Fig. (1.2). The high-temperature susceptibility ollows an approximate Curie-Weiss law, but the Curie-Weiss parameters depend on the itting interval. The interacting density o states Œír ÂP or the conduction electrons is shown in Fig.(1.3) or } ^i;mä and pƒ&^i mîïy[=, and or several temperatures. (The energy is measured with respect to 8.) The high-temperature DOS has a gap o the order o }, and the chemical potential is located within the gap. Below the transition $'& is small, the correlation eects are reduced, and Œdr ÂP assumes a nearly non-interacting shape, with large Œír 8 P and halwidth ð xi m. The transport properties o the high-t phase are dominated by the presence o the gap, which leads to a small dc conductivity with a weak temperature dependence. The transport properties o the paramagnetic phase are unrelated to the spin-disorder Kondo scattering (there is no spin-spin scattering in the FK model). Below the transition the conductivity increases and assumes large metallic values. The intraband optical conductivity ÂP is plotted in Fig.(1.4) as a unction o requency, or several temperatures. Above, we observe a reduced Drude peak around Â ) and a pronounced high-requency peak around Â. The

7 ó Theory o valence transitionsin ytterbium-based compounds T=.9 t* T=.18 t* T=.47 t* T=1.16 t* T=3.12 t* ρ d (ω).4 ñ ò ω/t* Figure 1.3 Interacting density o states plotted versus ôzø Ù Ú or ÛzØ#Ù Ú Ü Þ, à ( è Ø#Ù Ú Üú <ø=û Þ ), and or various temperatures, as indicated in the igure. Ø Ù Ú Üöõ' <ø ù shape o ÂP changes completely across. Below the Drude peak is ully developed and there is no high-energy (intraband) structure. However, i the renormalized -level is close to 8, the interband d- transition could lead to an additional mid-inrared peak. The ratio o the high-requency peak in Fig(1.4) and the corresponding value o ü+*<?i m, is } þ \ý. For the same value o and pƒ&ýÿy[¹7^i m ( I >?i m ) we obtain } \ ï*<>], while or pƒ&ýÿy[¹7]?i;m (ÒI i;m ) we ind } \ (not shown). I we 1.2 σ(ω).8 T=.9 t* T=.18 t* T=.47 t* T=1.56 t* ω/t* Figure 1.4 Optical conductivity plotted versus ôø Ù Ú or various temperatures. The Û, à è, are the same as in Fig.(1.3). Az estimate the -d correlation in YbInCu rom the 8 Ñ conductivity data [1], we obtain the experimental value }, and peak in the optical Æ* ev. Together and with K [7] this gives the ratio } \. I we take } ^i m adjust pƒ& ^i m so as to bring the theoretical value o in agreement with the the

8 L W 8 Az Az thermodynamic and transport data on YbInCu, we get a high-requency peak in ÂP at about 8 Ñ, 6 Ñ, and 15 Ñ, or pq&_y[¹7]?i m, pƒ&_y ¹7^i;m, and pƒ& _Y[=?i m, respectively. Az.4 m (h,t).2 T=.5t* T=.6t* T=.7t* T=.8t* T=.9t* T=.1t* T=.2t* Figure 1.5 The -electron magnetization is plotted as a unction o ]Ø#ÙÚ or various temperatures. The Û, à.1.2 h/t*, and è, are the same as in Fig.(1.3). The -electron magnetization ÑÒ& ~³P is plotted in Fig.(1.5) versus reduced magnetic ield ~³^i m, or several temperatures. Above the characteristic temperature m, the ÑÒ& ~³P curves exhibit typical local moment behavior. E? Below m we ind a metamagnetic transition at a critical ield KML ; the ÑÒ& ~³P is negligibly small below K L and the local moment is ully restored above K L. Taking the inlection point o the ÑÒ& ~P curves, calculated or several values 1 H c (T)/H c =3t* E =.5t* =4t* E =.3t* =4t* E =.2t* T/T v 1 Figure 1.6 several values o à ormalized critical ield is plotted as a unction o reduced temperature :Ø# Ú Ø#ÙÃÚ and ÛzØ ÙÚ W. The ull line represents ûzõîêh 'Ø# èú ë and ìú è ÜÏ è Ø. è or o and pƒ&, as an estimate o KLw âp we obtain the phase boundary which is shown in Fig.(1.6), together with the expression K L KV QP *[Y \ m P.

9 ~ L ~ Theory o valence transitionsin ytterbium-based compounds 9 ote, the qm values in Fig.(1.6) dier by more than an order o magnitude, while KV is only weakly parameter dependent. the ratio m R(h,T) 1.5 T=.3t* T=.4t* T=.7t* T=.12t* T=.22t* T=.42t* h/t* Figure 1.7 Field-dependent resistivity plotted versus Ø Ù Ú. The dierent symbols correspond to dierent temperatures, as indicated in the igure. The Û and à are the same as in Fig.(1.3). The metamagnetic transition is also seen in the ield-dependent electrical resistance QP which is plotted in Fig.(1.7) as a unction o ~^i m, or several temperatures. A substantial change in the QP across qm or K L is clearly seen. 4. SMMARY >From the preceding discussion it is clear that Falicov-Kimball model captures the main eatures o the experimental data or YbInCu and similar compounds. The temperature- and ield-induced anomalies are related to a metalinsulator transition, which is caused by large FK interaction and triggered by the temperature- or the ield-induced change in the -occupancy. At high temperatures, we ind a large gap in Œdr ÂP ; we expect a similar gap in the -electron spectrum as well. At low temperatures, both gaps are closed, and the renormalized -level renormalizes down to the chemical potential. Our calculations describe doped Yb systems with broad transitions but appear to be less successul or those compounds which show a irst-order transition. The numerical curves can be made sharper (by adjusting the parameters) but they only become discontinuous in a narrow parameter range. The main diiculty with the FK model is that it predicts a substantial change in the -occupancy across the transition and associates the loss o moment with the loss o -holes. But in the real materials the loss o moment seems to be due to the valence luctuations, rather than to the reduction o $'&. The description o the valence luctuating ground state would require the hybridization and is beyond the scope o this work. The actual situation pertaining to Yb ions in the mixed-valence state might be quite complicated, since one would have to consider an extremely

10 1 asymmetric limit o the Anderson model, in which the ground state is not Kondolike, there is no Kondo resonance, and there is no single universal energy scale which is relevant at all temperatures [18]. We speculate that the periodic Anderson model with a large FK term will exhibit the same behavior as the FK model at high temperatures. Indeed, i the conduction band and the -level are gapped, and the width o the -level is large, then the eect o the hybridization can be accounted or by renormalizing the parameters o the FK model. On the other hand, i the low-temperature state o the ull model is close to the valence-luctuating ixed point with the conduction band and hybridized -level close to the Fermi level, then the likely eect o the FK correlation is to renormalize the parameters o the Anderson model. Acknowledgments We acknowledge discussions with Z. Fisk, B. Lüthi, M. Miljak, M. Očko, and J. Sarrao. This research was supported by the ational Science Foundation under grant DMR Reerences [1] I. Felner and I. ovik., Phys. Rev. B 33, 617 (1986). [2] J.L. Sarrao et al., Physics B, 223&224, 366 (1996). [3] J. M. Lawrence et al., Phys. Rev. B,55, (1997). [4] A. L. Cornelius et al., Phys. Rev. B,56, 7993 (1997). [5] C. D. Immer et al., Phys. Rev. B,56, 71 (1997). [6] J.L. Sarrao et al., Phys. Rev. B,58, 49 (1998) [7] J.L. Sarrao, Physica B, 259&261, 129 (1999) [8] E. Figueroa et al., Solid State Commun. 16, 347 (1998) [9] M. Očko, J. Sarrao, Z. Fisk, unpublished. [1] S. R. Garner et al., preprint (2). [11] T. S. Altshuler et al., Z. Phys. B99, 57 (1995). [12] C. Rettori et al., Phys. Rev. B,55, 116 (1997). [13] L. M. Falicov and J. C. Kimball, Phys. Rev. Lett. 22, 997 (1969). [14] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989). [15]. Brandt and C. Mielsch, Z.Phys. B 75, 365 (1989);. Brandt and M. P. rbanek, ibid. 89, 297 (1992). [16] J. Freericks and V. Zlatić, Phys. Rev. B 58, 322 (1998). [17] L.P. Kadano and G. Baym, Quantum Statistical Physics (W. A. Benjamin, Menlo Park, CA), 1962 [18] H. B. Krishnamurti et al., Phys. Rev. B 21, 144 (198).

Jim Freericks (Georgetown University) Veljko Zlatic (Institute of Physics, Zagreb)

Theoretical description of the hightemperature phase of YbInCu 4 and EuNi 2 (Si 1-x Ge x ) 2 Jim Freericks (Georgetown University) Veljko Zlatic (Institute of Physics, Zagreb) Funding: National Science