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Magnetic polarons in magnetic low carrier density systems
Journal of
ALLOY~ AND COMPOUND5 ELSEVIER
Journal of Alloys and Compounds 219 (1995) 290-295
Magnetic polarons in magnetic low carrier density systems T. Kasuya, Y. Haga, T. Suzuki Physics Department, Tohoku University, Sendai 980, Japan
Abstract
In magnetic low carrier density systems, in addition to the strong intra-atomic correlation, a strong correlation due to the long-range Coulomb interaction becomes important, causing a tendency to Wigner crystal or liquid system. In magnetic systems, the tendency to localization is further enhanced by magnetic polarization, and the magnetic polaron liquid and lattice are fairly easily formed. This is most clearly seen in CeP and CeAs; a detailed description is given in this paper. A weak Fs hole magnetic polaron is formed in a liquid state even at room temperature. The population of the magnetic polaron decreases with decreasing population of 4f/'8 but persists even at the lowest temperatures. An applied field and pressure change the weak magnetic polaron liquid to a strong magnetic polaron lattice with nearly saturated moment. Keywords: Magnetic polaron; Wigner crystal; Magnetic low carrier density systems
1. Introduction
Strongly correlated low carrier density systems with f and d electrons are attracting much attention. We are most interested in the 4f electron systems, in particular Ce and Yb monopnictides, because we have succeeded in preparing high quality single crystals, and various detailed experimental and theoretical investigations have been carried out, including study of the de Haas van Alphen effect (dHvA), revealing many novel characteristics of low carrier density systems. We have found that, in addition to the strong intra-atomic correlation due to the short-range Coulomb interaction, a strong correlation due to the long-range Coulomb interaction also becomes important, causing the tendency to localization to a Wigner crystal, or its melted form the Wigner liquid, or a kind of strongly correlated Fermi liquid. One remarkable result is an enhanced Kondo temperature TK in the low carrier density systems. This was most clearly shown for CeSb. In CeSb, a detailed Hartree-Fock like band calculation taking one 4f state withj~ = 5/2 corresponding to the ferromagnetic ordering below the Fermi energy EF to be consistent with the 4f resonance photoemission spectra (4f RPES), reproduced the dHvA effect very well, including the mass enhancement factor which was evaluated from the usual electron-magnon interaction (for a review of CeSb see Ref. [1]) in good agreement with experimental results. The value of p-f mixing was estimated very accurately using the above process. Using this information for p-f
0925-8388/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved
SSDI 0 9 2 5 - 8 3 8 8 ( 9 4 ) 0 5 0 4 3 - 0
mixing and the band density of states, the 4f RPES were calculated, giving perfect agreement with experimental results for all Ce-monopnictides with the twopeak structure separated about 2 eV. These two peaks were assigned to bonding and antibonding Kondo states which were created as the final states to screen the 4f hole created through the RPES [2]. This confirmed the common sense view that Hartree-Fock band calculation can reproduce the low frequency phenomena, in particular the Fermi surface topology, fairly well but cannot reproduce the high frequency phenomena, which can be calculated correctly using the one-particle Green function, for example. In the same sense as the above, the usual static Kondo temperature TK can be calculated accurately giving a very low value, many orders of magnitude smaller than expected from the typical Kondo behavior of resistivity, more than 10 K. This serious disagreement is thought to be due to the following. The bonding Kondo state observed in the 4f RPES corresponds to the extremely localized Kondo state because, owing to high frequency or a short lifetime, the f-symmetry conduction band hole created through p-f mixing to screen the 4f hole cannot move far from the central site where the 4f hole is created but is localized mainly at the nearest neighbor sites. This is the reason for the large Kondo temperature TKo= 1 eV, in which all 14 4f states are involved in screening of the hole at the central site. Note that as shown in another paper [2], the screening can be also done by the 5d electrons and then we have the d-symmetry bonding Kondo state
T. Kasuya et al. / Journal of Alloys and Compounds 219 (1995) 290-295
with much larger Kondo temperature TKd0of the order of several electronvolts. The Zhang-Rice state, described for high Tc materials, is nothing but a kind of d-symmetry bonding Kondo state [1]. For the low frequency phenomena, the above mentioned strongly local Kondo state can relax, extending far and causing a lower Kondo temperature since TK is determined mainly through the amplitude at the nearest neighbor sites because of the short-range character of c-f mixing, where c is a conduction band electron in general. The static Kondo temperature TK derived from static phenomena such as the coefficient y of the T-linear specific heat and the static magnetic susceptibility at the lowest temperature X0, as well as that derived from the temperature dependence of resistivity as mentioned before, corresponds to the low frequency limit, where the fsymmetry conduction electron extends very far, and thus is described as a scattered state, and the tail of the state, or the character very near Er, gives the above mentioned characteristic Kondo behavior. Therefore, the large difference in the values of TK from calculation and experiment indicates dearly that owing to the longrange Coulomb interaction relevant to the low carrier density system, the Kondo electron cannot relax very far but is made to remain local, keeping a higher value of TK. Indeed, the doubling in intensity at nearest neighbor sites, corresponding to doubling the effective density of states at EF, is enough to explain the disagreement between theory and experiment, and such a modification is reasonably obtained. This is the mechanism of the anomalously high TK value of low carrier density systems even if the band density of states at EF is very low [3]. A more dramatic phenomenon occurs in Kondo insulators, where the number of carriers is the same as the number of magnetic atoms, or of the Kondo state. This corresponds to the strong Kondo limit. This was first proposed for StuB6 as follows [4]. Owing to strong intra-atomic Coulomb interaction between the 4f and 5d electrons, determined to be about 10 eV from atomic spectra, the conduction electrons with 5d character, which are present in the same number a s Sm 3+, are localized at each Sm 3÷ forming the atomic-like state for Sm3++ 5d. In this treatment, however, the Kondo effect was not considered and thus no Kondo singlet. The strong Kondo limit was only studied based on a simple model, typically a localized electron with s = 1/2 and a conduction electron with s--1/2 coupled through the intra-atomic exchange interaction. Then we have a singlet state but it is no different from the atomic singlet state, pure magnetic singlet, and a simplified version of the first picture. The real strong Kondo limit in S m B r , as well as in SmS under pressure, is more complicated. The ground state of Sm 2÷ is the spin-orbit singlet for L = 3 and S = 3 with a small excitation energy, about 400 K, to the J = 1 excited
291
state. However, the ground configuration of S m 3 + is J = 5 / 2 for L--5 and S--5/2 with a small magnetic moment due to near cancellation of the moments on L and S. Because TK for the Sm 3+ Kondo state is not very high, about 100 K, the valence fluctuation occurs mainly between the two ground configurations. Then, because the crystal field splitting in Sm 3÷ is very small, all six 4f electrons forming Sm 2÷ can mix with the conduction band electrons and thus form the Kondo singlet. In the strong Kondo limit, each conduction band electron of f-symmetry is localized on the nearest neighbor sites and in each instant of time one such electron is localized at each Sm 3÷ Kondo state. The insulating character with energy gap is guaranteed by the Luttinger theory [5]. When all the Sm are divalent, SmB6 should be an insulator with a small gap between the top of the valence band and the bottom of the conduction band, which is the bonding band among the 5d(Sm) and the s-p antibonding bands in the cage made by B6 molecules, similar to SrBr. When the valence fluctuation on some Sm increases and a dear Kondo state is formed, the insulating character persists as long as the process is continuous. This argument is only applicable, however, for the ground state, or the T ~ 0 limit. When the temperature T increases beyond TK, or the gap energy A, the system changes gradually to the usual metallic mixed valence system with a population of Sm2÷ of about 65%, which is connected to the metallic state of pure Sm 3÷ [6]. Such a cross-over is seen commonly in all Kondo insulator systems. A sharp exciton like spectrum is also a common feature related to the gap [7,8]. When the carrier number decreases further, the Wigner crystal like localization becomes the main feature, and helped by the magnetic polarization, as well as the lattice distortion in the usual phonon polaron, the magnetic polaron liquid state is stabilized. This state can be seen most dearly in CeP and CeAs and some detailed theoretical analyses have been performed. In this respect, in the following we show the recent progress of our study on these materials from the standpoint of magnetic polaron liquid and lattice formation, including some unpublished recent results.
2. Magnetic polarons in CeP and CeAs All the rare earth monopnictides, RX with X = N, P, As, Sb or Bi, crystallize in the NaCI type structure. The top of the valence band formed mainly from the p-state on X is at the F-point of the Brillouin zone and is split into Fs and F6 states through spin-orbit interaction in the p-state. The bottom of the conduction band is at each X-point of the Brillouin zone, formed mainly from the 5d states on Ce and the orthogonal plane wave states. They overlap slightly forming low
T. Kasuya et al. / Journal of Alloys and Compounds 219 (1995) 290-295
292
carrier semimetals. The p-f mixing interaction is determined fairly accurately from the band calculation, and becomes stronger as X changes from Bi to N [9]. The Fermi surface is very well reproduced by the band calculation when the occupied and unoccupied 4f levels are set to be consistent with PES and inverse PES, IPES or BIS and the band overlap is adjusted to be consistent with the carrier number [10]. In CeN, because the p--f as well as d-f mixings increase so much, the bonding Kondo state is pushed up above EF causing in CeN a valence fluctuation state with a monovalent character, where the 4f states are treated in the usual band scheme. In CeP and CeAs too, the bonding Kondo state is very near Ev, so that an applied pressure can induce the valence fluctuation state [11]. The temperature dependence of resistivity is shown in Fig. 1 for various CeX, YbX and LaSb [12]. Because of the well known lanthanide contraction, the p--f mixing in YbX is less than one-half that in CeX and thus the resistivity in YbX is fairly normal, at least at higher temperatures. However, the resistivity in CeSb shows typical Kondo behavior with a larger resistivity at room temperature. It is now well established that the magnetic 200[-
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polaron liquid is formed below 70 K, which is revealed most clearly by the anomalous lattice shrinkage due to an anomalous increase in the /'8 population, as well as an additional peak in the resistivity [1,13]. When pressure is applied, the anomalous peak in the resistivity increases, shifting to higher temperature, and at 7 GPa applied pressure the magnitude and the temperature dependence of resistivity become similar to those of CeP and CeAs. Furthermore, a sharp increase in resistivity occurs, beginning at 60 K, and is attributed to Wigner crystallization of magnetic polaron. A similar anomaly is observed in CeBi and CeAs under pressure but is much weaker than that in CeSb. These facts indicate clearly that a magnetic polaron liquid is formed in CeP and CeAs even at room temperature. The carrier number at room temperature is estimated from Hall constant measurements to be about 0.02 per Ce atom, and thus each polaron extends over about 50 Ce sites because at high temperature the magnetic polarization is weak and thus each hole making a polaron extends as much as possible until it overlaps with other polarons, preferentially with different symmetries, to save kinetic energy. Therefore, at room temperature the main force localizing holes is the long-range Coulomb repulsive force among the holes making the Wigner liquid. Because the strength of/'8 p--f mixing is evaluated to be about 1 eV from band calculation [14], the effective field on the F8o state of each Ce atom is evaluated to be about 100 K, more than half the crystal field splitting, /'8o is the symmetry of the hole under consideration and indicates hereafter the special symmetry extending over the x-y plane; only the special 4f /"8 state with the same symmetry as the hole is pulled down. Because of the high temperature, the population of 4f Fso is not large. In the F8 hole polaron, the main current carrier is electrons on the conduction band. Each electron is scattered by the Coulomb and magnetic interaction, mostly the intra-atomic d-f multipole-anisotropic-exchange interaction, in the magnetic polaron nearly equally. The excited F8 Kondo state is also expected to exist because it is not much disturbed by the energy shift of 4f F8o. The scattering of electrons through the/'8 Kondo state is evaluated to be at least a factor 2 smaller than each of the above mentioned scatterings. With decreasing temperature, the population of 4f /'8, except Fso, decreases and thus the population of 4f £80 increases or remains nearly constant down to 100 K, as seen clearly in the temperature dependence of the Hall constant shown in Fig. 2 [12]. The resistivity shows a gradual increase correspondingly. Below 100 K, the thermal population of 4f Fso also begins to decrease, which causes a decrease in the number of magnetic polarons, accelerating further the decreasing population of 4f/"80. This is clearly seen in the Hall constant as shown in Fig. 2. Below 20 K, the carrier
T. Kasuya et al. / Journal of Alloys and Compounds 219 (1995) 290-295 '
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number is 0.007 and 0.002 in CeP and CeAs respectively, corresponding to values without the magnetic polaron. These numbers are obtained from dHvA measurement [15] as well as the Hall constant at the lowest temperature. The resistivity also shows a fairly sharp decrease below 100 K. This is understood as follows. The conductivity due to electrons may be written in the usual way as o'~ = n~e21~/m~ v ~ ,., ( e l / ~ ' ) ( v . / W ) 2
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The Fermi surface of electrons at each X-point of the Brillouin zone is strongly anisotropic. For example at Xz, the dispersion perpendicular to the z-axis is very sharp because of the strong p--d mixing mentioned before and the dispersion is given rather well by the k-linear relation, instead of the usual k 2 relation [16]. Therefore, the current is mainly carried by this part, v . , and then the second part of the formula is obtained. Because v ~ is nearly independent of the carrier number, the resistivity depends on the carrier number very weakly. However, the magnetic scattering due both to magnetic polaron and the Fs Kondo state, as well as phonon scattering, decrease sharply below 100 K causing a sharp decrease in the resistivity. Below 20 K, because the carrier number becomes very small, the impurity localization effect is seen clearly in both the resistivity and Hall constant, showing a strong dependence on sample quality. Below the N6el temperature TN, 7 K and 10 K for CeAs and CeP respectively, corresponding to an antiferromagnetic ordering of 4f F7, the intensity of random potential
293
decrease and the localization effect become negligible in a high quality sample as shown in Figs. 1 and 2. Then, even the dHvA effect is observed giving a fairly precise number of carriers as mentioned before. Then, as shown in Fig. 3 [17], a fairly large transversal magnetoresistance is observed indicating good stoichiometry. In contrast, the longitudinal magnetoresistance shows a strong decrease around 10 K, wiping out the peak structure due to impurity localization, because of the weakened random potential. The important question is whether the weak magnetic polaron liquid exists at low temperature where the population of 4f Fs0 is small. The best answer to this question is obtained from specific heat measurement under various applied fields as shown in Fig. 4 [18]. The magnetic specific heat can be obtained by subtracting that for LaP. The large magnetic specific heat at low temperature cannot be fitted by the simple crystal field model with A78= 170 K because it gives a value one order of magnitude smaller. However, the magnetic polaron model, in which the 4f Fso state is pulled down to the level of about 80 K above the 4f F7 level, can explain the specific heat in the paramagnetic region fairly well [19]. It is important to note that to calculate the 4f Fso level, the total energy difference due to the change in a single site from F7 to Fso should be evaluated to take into account the interaction between 4f Fso in different sites• This keeps the 4f /'so level fairly low even in a low 4f/'so population• The same model can explain anomalous temperature dependences in the magnetic susceptibility and the Knight shift [20,21]. When a magnetic field or pressure is applied at low temperature, the weak magnetic polaron liquid changes to a strong magnetic polaron lattice, in which a nearly full moment of 2gB is achieved to take full advantage of the p--f mixing interaction. Then, the effective mass of the polaron becomes very large causing localization and formation of a kind of Wigner lattice• The situation i ....
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T. Kasuya et al. I Journal of Alloys and Compounds 219 (1995) 290-295
294
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m~e as mentioned before compared with m l , are accumulated on the double F8~ layers but the down spin electrons are extended more on the /"7 layers because of the negative exchange interaction on the double Fs~ layers. Exactly the same mechanism is expected to be applicable in CeP and CeAs and neutron scattering measurements recently performed by Kohgi et al. [22] confirmed this expectation. An interesting new aspect is the persistence of an 11 layer period of the double layer ordering and this ordering is naturally attributed to the RKKY like effect due to the Fermi surfaces of the electrons at Xx and Xy of the Brillouin zone. When the field strength is increased, the magnetic polaron is more stabilized, which causes a larger carrier number and greater 4f F~ population. This may have two effects, decreasing the period from the 11 layer period and causing the formation of a larger multilayer than the double layer. A very interesting finding is that this process seems to occur discontinuously at the field strength which satisfies the dHvA condition for electrons at the Xz point [23]. More studies are needed, however, to confirm the above picture. There are many other interesting aspects of the neutron scattering results and they will be discussed in a separate paper.
30
T(K) Fig. 4. S p e c i f i c h e a t o f C e P s h o w n as a f u n c t i o n o f t e m p e r a t u r e f o r v a r i o u s fixed v a l u e s o f a p p l i e d m a g n e t i c field. T h e v a l u e o f L a P is a l s o s h o w n as r e f e r e n c e ; f r o m R e f . [18].
is essentially the same as for the antiferro-para (AFP) and the ferro-para (FP) phases in CeSb [1] and the ferromagnetic layer of full moment stacks in various ways together with/'7 layers. In CeSb, the ordering at low temperature changes with applied field from the up, up, down, down structure to the up, up, up, down and finally to the perfect ferro ordering. This sequence was reproduced very well by Hartree-Fock like band calculation and the result can be interpreted as follows. For the z-axis ordering, called/'8, for both the stacking sequence and the direction of moment, for example, the corresponding holes have a large effective mass mz and the usual effective mass m . causing a stable layer structure. The band calculation indicates that the double ferromagnetic layer is the most appropriate and to save kinetic energy the double layers should stack antiferromagnetically. Indeed, when the ferromagnetic ordering is forced, the c-axis lattice distance expands to weaken the increase in kinetic energy. When a double / ' 7 layer enters between two double F~ layers with parallel spins, a typical example in the FP phase, an antiferromagnetic exchange field of about 2.7 T is induced on the F7 layers [3]. This is interpreted as follows. To screen the plus charge of holes, the electrons near Xz of the Brillouin zone, which also have a large
3. Discussion and conclusion
In the introduction, the recent progress in the study of strongly correlated low carrier density systems was surveyed. It was shown that the tendency to Wigner crystal localization due to the long-range Coulomb interaction is the most important feature. In magnetic systems this is further accelerated by magnetic polarization, as well as by lattice distortion in the sense of the usual phonon polaron, and a weak magnetic polaron liquid or crystal is formed. The Kondo state can coexist with the weak magnetic polaron liquid with an enhanced form owing to the tendency to localization. A more drastic phenomenon occurs in Kondo insulator systems, in which the number of carriers is equal to the number of magnetic atoms and the localized Kondo state connecting to the strong limit of the Kondo lattice is realized. At low temperature, a transition to the strong magnetic polaron lattice occurs, where a nearly saturated localized moment appears within each polaron to gain the full energy of the c-f mixing or exchange interaction. In CeP and CeAs, a weak magnetic polaron liquid is formed even at room temperature and detailed experimental as well as theoretical studies have been done. These were reviewed in some detail in the second section. It was shown that indeed various anomalous behaviors can be explained consistently by the above model.
T. Kasuya et aL / Journal of Alloys and Compounds 219 (1995) 290-295
The magnetic polaron model was also applied to Yb monopnictides and some 3d systems, in particular to the high Tc CuO: layer systems, with satisfactory results explaining crucial experimental results [1,24]. To study low carrier density systems, in particular their intrinsic characteristics obtained by subtracting impurity effects, the long-range Coulomb interaction should be considered seriously as the universal fundamental interaction.
Acknowledgments In the study of CeP and CeAs, the collaboration with N. M6ri, Y. Okayama, M. Kohgi and T. Osakabe was essential. Valuable comments and discussions with them are highly appreciated.
References [1] For the review of CeSb, T. Kasuya, Memorial symposium for J. Rossat-Mignod, in Physica B, in press. [2] M. Takeshige, O. Sakai and T. Kasuya, Z Phys. Soc. Jpn., 60 (1991) 666. [3] T. Kasuya, M. Sera and T. Suzuki, J. Phys. Soc. Jpn., 62 (1993) 3364.
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[4] J.C. Nickerson, R.M. White, K.N. Lee, R. Bochmann, T.H. Geballe and G.H. Hull, Jr., Phys. Rev. B, 3 (1971) 2030. [5] J.M. Luttinger, Phys. Rev., 119 (1960) 1153. [6] T. Kasuya, J. Phys. Soc. Jpn., 63 (1994) 397. [7] T. Kasuya, Europhys. Lett., 26 (1994) 277. [8] T. Kasuya, Europhys. Lett., 26 (1994) 283. [9] D.M. Wieliczka, C.G. Olson and D.W. Lynch, Phys. Rev. B, 29 (1984) 3028. [10] O. Sakai, M. Takeshige, H. Harima, K. Otaki and T. Kasuya, J. Magn. Magn. Mater., 52 (1985) 18. [11] N. M6ri, Y. Okayama, H. Takahashi, Y. Haga and T. Suzuki, Jpn. J. Appl. Phys. Ser., 8 (1993) 182. [12] T. Kasuya, Y. Haga, Y.S. Kwon and T. Suzuki, Physica B, 186--188 (1993) 9. [13] M. Sera, Thesis, Dr. of Science, Tohoku University, 1982. [14] T. Kasuya, Y. Haga, T. Suzuki, Y. Kaneta and O. Sakai, Z Phys. Soc. Jpn., 61 (1992) 3447. [15] T. Kasuya, T. Suzuki and Y. Haga, Z Phys. Soc. Jpn, 62 (1993) 2549. [16] T. Kasuya, M. Sera and T. Suzuki, J. Phys. Soc. Jpn., 62 (1993) 2561. [17] Y.S. Kwon, Thesis, Dr. of Science, Tohoku University, 1991. [18] Y. Haga, T. Suzuki and T. Kasuya, Z Phys. Soc. Jpn., in press. [19] T. Kasuya, Y. Haga and T. Suzuki, Z Phys. Soc. Jpn., 62 (1993) 3376. [20] T. Kasuya and Y. Haga, Solid State Commun., 93 (1995) 307. [21] T. Kasuya, Z Phys. Soc. Jpn., 63 (1994) 4318. [22] M. Kohgi, T. Osakabe, K. Kakurai, T. Suzuki, Y. Haga and T. Kasuya, Phys. Rev. B, 49 (1994) 7068. [23] M. Date, A. Yamagishi, H. Hori and K. Sugiyama, Jpn. Z Appl. Phys. Set., 8 (1993) 195. [24] T. Kasuya, Physica C, 223 (1994) 233; 224 (1994) 191.
ALLOY~ AND COMPOUND5 ELSEVIER
Journal of Alloys and Compounds 219 (1995) 290-295
Magnetic polarons in magnetic low carrier density systems T. Kasuya, Y. Haga, T. Suzuki Physics Department, Tohoku University, Sendai 980, Japan
Abstract
In magnetic low carrier density systems, in addition to the strong intra-atomic correlation, a strong correlation due to the long-range Coulomb interaction becomes important, causing a tendency to Wigner crystal or liquid system. In magnetic systems, the tendency to localization is further enhanced by magnetic polarization, and the magnetic polaron liquid and lattice are fairly easily formed. This is most clearly seen in CeP and CeAs; a detailed description is given in this paper. A weak Fs hole magnetic polaron is formed in a liquid state even at room temperature. The population of the magnetic polaron decreases with decreasing population of 4f/'8 but persists even at the lowest temperatures. An applied field and pressure change the weak magnetic polaron liquid to a strong magnetic polaron lattice with nearly saturated moment. Keywords: Magnetic polaron; Wigner crystal; Magnetic low carrier density systems
1. Introduction
Strongly correlated low carrier density systems with f and d electrons are attracting much attention. We are most interested in the 4f electron systems, in particular Ce and Yb monopnictides, because we have succeeded in preparing high quality single crystals, and various detailed experimental and theoretical investigations have been carried out, including study of the de Haas van Alphen effect (dHvA), revealing many novel characteristics of low carrier density systems. We have found that, in addition to the strong intra-atomic correlation due to the short-range Coulomb interaction, a strong correlation due to the long-range Coulomb interaction also becomes important, causing the tendency to localization to a Wigner crystal, or its melted form the Wigner liquid, or a kind of strongly correlated Fermi liquid. One remarkable result is an enhanced Kondo temperature TK in the low carrier density systems. This was most clearly shown for CeSb. In CeSb, a detailed Hartree-Fock like band calculation taking one 4f state withj~ = 5/2 corresponding to the ferromagnetic ordering below the Fermi energy EF to be consistent with the 4f resonance photoemission spectra (4f RPES), reproduced the dHvA effect very well, including the mass enhancement factor which was evaluated from the usual electron-magnon interaction (for a review of CeSb see Ref. [1]) in good agreement with experimental results. The value of p-f mixing was estimated very accurately using the above process. Using this information for p-f
0925-8388/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved
SSDI 0 9 2 5 - 8 3 8 8 ( 9 4 ) 0 5 0 4 3 - 0
mixing and the band density of states, the 4f RPES were calculated, giving perfect agreement with experimental results for all Ce-monopnictides with the twopeak structure separated about 2 eV. These two peaks were assigned to bonding and antibonding Kondo states which were created as the final states to screen the 4f hole created through the RPES [2]. This confirmed the common sense view that Hartree-Fock band calculation can reproduce the low frequency phenomena, in particular the Fermi surface topology, fairly well but cannot reproduce the high frequency phenomena, which can be calculated correctly using the one-particle Green function, for example. In the same sense as the above, the usual static Kondo temperature TK can be calculated accurately giving a very low value, many orders of magnitude smaller than expected from the typical Kondo behavior of resistivity, more than 10 K. This serious disagreement is thought to be due to the following. The bonding Kondo state observed in the 4f RPES corresponds to the extremely localized Kondo state because, owing to high frequency or a short lifetime, the f-symmetry conduction band hole created through p-f mixing to screen the 4f hole cannot move far from the central site where the 4f hole is created but is localized mainly at the nearest neighbor sites. This is the reason for the large Kondo temperature TKo= 1 eV, in which all 14 4f states are involved in screening of the hole at the central site. Note that as shown in another paper [2], the screening can be also done by the 5d electrons and then we have the d-symmetry bonding Kondo state
T. Kasuya et al. / Journal of Alloys and Compounds 219 (1995) 290-295
with much larger Kondo temperature TKd0of the order of several electronvolts. The Zhang-Rice state, described for high Tc materials, is nothing but a kind of d-symmetry bonding Kondo state [1]. For the low frequency phenomena, the above mentioned strongly local Kondo state can relax, extending far and causing a lower Kondo temperature since TK is determined mainly through the amplitude at the nearest neighbor sites because of the short-range character of c-f mixing, where c is a conduction band electron in general. The static Kondo temperature TK derived from static phenomena such as the coefficient y of the T-linear specific heat and the static magnetic susceptibility at the lowest temperature X0, as well as that derived from the temperature dependence of resistivity as mentioned before, corresponds to the low frequency limit, where the fsymmetry conduction electron extends very far, and thus is described as a scattered state, and the tail of the state, or the character very near Er, gives the above mentioned characteristic Kondo behavior. Therefore, the large difference in the values of TK from calculation and experiment indicates dearly that owing to the longrange Coulomb interaction relevant to the low carrier density system, the Kondo electron cannot relax very far but is made to remain local, keeping a higher value of TK. Indeed, the doubling in intensity at nearest neighbor sites, corresponding to doubling the effective density of states at EF, is enough to explain the disagreement between theory and experiment, and such a modification is reasonably obtained. This is the mechanism of the anomalously high TK value of low carrier density systems even if the band density of states at EF is very low [3]. A more dramatic phenomenon occurs in Kondo insulators, where the number of carriers is the same as the number of magnetic atoms, or of the Kondo state. This corresponds to the strong Kondo limit. This was first proposed for StuB6 as follows [4]. Owing to strong intra-atomic Coulomb interaction between the 4f and 5d electrons, determined to be about 10 eV from atomic spectra, the conduction electrons with 5d character, which are present in the same number a s Sm 3+, are localized at each Sm 3÷ forming the atomic-like state for Sm3++ 5d. In this treatment, however, the Kondo effect was not considered and thus no Kondo singlet. The strong Kondo limit was only studied based on a simple model, typically a localized electron with s = 1/2 and a conduction electron with s--1/2 coupled through the intra-atomic exchange interaction. Then we have a singlet state but it is no different from the atomic singlet state, pure magnetic singlet, and a simplified version of the first picture. The real strong Kondo limit in S m B r , as well as in SmS under pressure, is more complicated. The ground state of Sm 2÷ is the spin-orbit singlet for L = 3 and S = 3 with a small excitation energy, about 400 K, to the J = 1 excited
291
state. However, the ground configuration of S m 3 + is J = 5 / 2 for L--5 and S--5/2 with a small magnetic moment due to near cancellation of the moments on L and S. Because TK for the Sm 3+ Kondo state is not very high, about 100 K, the valence fluctuation occurs mainly between the two ground configurations. Then, because the crystal field splitting in Sm 3÷ is very small, all six 4f electrons forming Sm 2÷ can mix with the conduction band electrons and thus form the Kondo singlet. In the strong Kondo limit, each conduction band electron of f-symmetry is localized on the nearest neighbor sites and in each instant of time one such electron is localized at each Sm 3÷ Kondo state. The insulating character with energy gap is guaranteed by the Luttinger theory [5]. When all the Sm are divalent, SmB6 should be an insulator with a small gap between the top of the valence band and the bottom of the conduction band, which is the bonding band among the 5d(Sm) and the s-p antibonding bands in the cage made by B6 molecules, similar to SrBr. When the valence fluctuation on some Sm increases and a dear Kondo state is formed, the insulating character persists as long as the process is continuous. This argument is only applicable, however, for the ground state, or the T ~ 0 limit. When the temperature T increases beyond TK, or the gap energy A, the system changes gradually to the usual metallic mixed valence system with a population of Sm2÷ of about 65%, which is connected to the metallic state of pure Sm 3÷ [6]. Such a cross-over is seen commonly in all Kondo insulator systems. A sharp exciton like spectrum is also a common feature related to the gap [7,8]. When the carrier number decreases further, the Wigner crystal like localization becomes the main feature, and helped by the magnetic polarization, as well as the lattice distortion in the usual phonon polaron, the magnetic polaron liquid state is stabilized. This state can be seen most dearly in CeP and CeAs and some detailed theoretical analyses have been performed. In this respect, in the following we show the recent progress of our study on these materials from the standpoint of magnetic polaron liquid and lattice formation, including some unpublished recent results.
2. Magnetic polarons in CeP and CeAs All the rare earth monopnictides, RX with X = N, P, As, Sb or Bi, crystallize in the NaCI type structure. The top of the valence band formed mainly from the p-state on X is at the F-point of the Brillouin zone and is split into Fs and F6 states through spin-orbit interaction in the p-state. The bottom of the conduction band is at each X-point of the Brillouin zone, formed mainly from the 5d states on Ce and the orthogonal plane wave states. They overlap slightly forming low
T. Kasuya et al. / Journal of Alloys and Compounds 219 (1995) 290-295
292
carrier semimetals. The p-f mixing interaction is determined fairly accurately from the band calculation, and becomes stronger as X changes from Bi to N [9]. The Fermi surface is very well reproduced by the band calculation when the occupied and unoccupied 4f levels are set to be consistent with PES and inverse PES, IPES or BIS and the band overlap is adjusted to be consistent with the carrier number [10]. In CeN, because the p--f as well as d-f mixings increase so much, the bonding Kondo state is pushed up above EF causing in CeN a valence fluctuation state with a monovalent character, where the 4f states are treated in the usual band scheme. In CeP and CeAs too, the bonding Kondo state is very near Ev, so that an applied pressure can induce the valence fluctuation state [11]. The temperature dependence of resistivity is shown in Fig. 1 for various CeX, YbX and LaSb [12]. Because of the well known lanthanide contraction, the p--f mixing in YbX is less than one-half that in CeX and thus the resistivity in YbX is fairly normal, at least at higher temperatures. However, the resistivity in CeSb shows typical Kondo behavior with a larger resistivity at room temperature. It is now well established that the magnetic 200[-
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polaron liquid is formed below 70 K, which is revealed most clearly by the anomalous lattice shrinkage due to an anomalous increase in the /'8 population, as well as an additional peak in the resistivity [1,13]. When pressure is applied, the anomalous peak in the resistivity increases, shifting to higher temperature, and at 7 GPa applied pressure the magnitude and the temperature dependence of resistivity become similar to those of CeP and CeAs. Furthermore, a sharp increase in resistivity occurs, beginning at 60 K, and is attributed to Wigner crystallization of magnetic polaron. A similar anomaly is observed in CeBi and CeAs under pressure but is much weaker than that in CeSb. These facts indicate clearly that a magnetic polaron liquid is formed in CeP and CeAs even at room temperature. The carrier number at room temperature is estimated from Hall constant measurements to be about 0.02 per Ce atom, and thus each polaron extends over about 50 Ce sites because at high temperature the magnetic polarization is weak and thus each hole making a polaron extends as much as possible until it overlaps with other polarons, preferentially with different symmetries, to save kinetic energy. Therefore, at room temperature the main force localizing holes is the long-range Coulomb repulsive force among the holes making the Wigner liquid. Because the strength of/'8 p--f mixing is evaluated to be about 1 eV from band calculation [14], the effective field on the F8o state of each Ce atom is evaluated to be about 100 K, more than half the crystal field splitting, /'8o is the symmetry of the hole under consideration and indicates hereafter the special symmetry extending over the x-y plane; only the special 4f /"8 state with the same symmetry as the hole is pulled down. Because of the high temperature, the population of 4f Fso is not large. In the F8 hole polaron, the main current carrier is electrons on the conduction band. Each electron is scattered by the Coulomb and magnetic interaction, mostly the intra-atomic d-f multipole-anisotropic-exchange interaction, in the magnetic polaron nearly equally. The excited F8 Kondo state is also expected to exist because it is not much disturbed by the energy shift of 4f F8o. The scattering of electrons through the/'8 Kondo state is evaluated to be at least a factor 2 smaller than each of the above mentioned scatterings. With decreasing temperature, the population of 4f /'8, except Fso, decreases and thus the population of 4f £80 increases or remains nearly constant down to 100 K, as seen clearly in the temperature dependence of the Hall constant shown in Fig. 2 [12]. The resistivity shows a gradual increase correspondingly. Below 100 K, the thermal population of 4f Fso also begins to decrease, which causes a decrease in the number of magnetic polarons, accelerating further the decreasing population of 4f/"80. This is clearly seen in the Hall constant as shown in Fig. 2. Below 20 K, the carrier
T. Kasuya et al. / Journal of Alloys and Compounds 219 (1995) 290-295 '
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number is 0.007 and 0.002 in CeP and CeAs respectively, corresponding to values without the magnetic polaron. These numbers are obtained from dHvA measurement [15] as well as the Hall constant at the lowest temperature. The resistivity also shows a fairly sharp decrease below 100 K. This is understood as follows. The conductivity due to electrons may be written in the usual way as o'~ = n~e21~/m~ v ~ ,., ( e l / ~ ' ) ( v . / W ) 2
(1)
The Fermi surface of electrons at each X-point of the Brillouin zone is strongly anisotropic. For example at Xz, the dispersion perpendicular to the z-axis is very sharp because of the strong p--d mixing mentioned before and the dispersion is given rather well by the k-linear relation, instead of the usual k 2 relation [16]. Therefore, the current is mainly carried by this part, v . , and then the second part of the formula is obtained. Because v ~ is nearly independent of the carrier number, the resistivity depends on the carrier number very weakly. However, the magnetic scattering due both to magnetic polaron and the Fs Kondo state, as well as phonon scattering, decrease sharply below 100 K causing a sharp decrease in the resistivity. Below 20 K, because the carrier number becomes very small, the impurity localization effect is seen clearly in both the resistivity and Hall constant, showing a strong dependence on sample quality. Below the N6el temperature TN, 7 K and 10 K for CeAs and CeP respectively, corresponding to an antiferromagnetic ordering of 4f F7, the intensity of random potential
293
decrease and the localization effect become negligible in a high quality sample as shown in Figs. 1 and 2. Then, even the dHvA effect is observed giving a fairly precise number of carriers as mentioned before. Then, as shown in Fig. 3 [17], a fairly large transversal magnetoresistance is observed indicating good stoichiometry. In contrast, the longitudinal magnetoresistance shows a strong decrease around 10 K, wiping out the peak structure due to impurity localization, because of the weakened random potential. The important question is whether the weak magnetic polaron liquid exists at low temperature where the population of 4f Fs0 is small. The best answer to this question is obtained from specific heat measurement under various applied fields as shown in Fig. 4 [18]. The magnetic specific heat can be obtained by subtracting that for LaP. The large magnetic specific heat at low temperature cannot be fitted by the simple crystal field model with A78= 170 K because it gives a value one order of magnitude smaller. However, the magnetic polaron model, in which the 4f Fso state is pulled down to the level of about 80 K above the 4f F7 level, can explain the specific heat in the paramagnetic region fairly well [19]. It is important to note that to calculate the 4f Fso level, the total energy difference due to the change in a single site from F7 to Fso should be evaluated to take into account the interaction between 4f Fso in different sites• This keeps the 4f /'so level fairly low even in a low 4f/'so population• The same model can explain anomalous temperature dependences in the magnetic susceptibility and the Knight shift [20,21]. When a magnetic field or pressure is applied at low temperature, the weak magnetic polaron liquid changes to a strong magnetic polaron lattice, in which a nearly full moment of 2gB is achieved to take full advantage of the p--f mixing interaction. Then, the effective mass of the polaron becomes very large causing localization and formation of a kind of Wigner lattice• The situation i ....
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T. Kasuya et al. I Journal of Alloys and Compounds 219 (1995) 290-295
294
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m~e as mentioned before compared with m l , are accumulated on the double F8~ layers but the down spin electrons are extended more on the /"7 layers because of the negative exchange interaction on the double Fs~ layers. Exactly the same mechanism is expected to be applicable in CeP and CeAs and neutron scattering measurements recently performed by Kohgi et al. [22] confirmed this expectation. An interesting new aspect is the persistence of an 11 layer period of the double layer ordering and this ordering is naturally attributed to the RKKY like effect due to the Fermi surfaces of the electrons at Xx and Xy of the Brillouin zone. When the field strength is increased, the magnetic polaron is more stabilized, which causes a larger carrier number and greater 4f F~ population. This may have two effects, decreasing the period from the 11 layer period and causing the formation of a larger multilayer than the double layer. A very interesting finding is that this process seems to occur discontinuously at the field strength which satisfies the dHvA condition for electrons at the Xz point [23]. More studies are needed, however, to confirm the above picture. There are many other interesting aspects of the neutron scattering results and they will be discussed in a separate paper.
30
T(K) Fig. 4. S p e c i f i c h e a t o f C e P s h o w n as a f u n c t i o n o f t e m p e r a t u r e f o r v a r i o u s fixed v a l u e s o f a p p l i e d m a g n e t i c field. T h e v a l u e o f L a P is a l s o s h o w n as r e f e r e n c e ; f r o m R e f . [18].
is essentially the same as for the antiferro-para (AFP) and the ferro-para (FP) phases in CeSb [1] and the ferromagnetic layer of full moment stacks in various ways together with/'7 layers. In CeSb, the ordering at low temperature changes with applied field from the up, up, down, down structure to the up, up, up, down and finally to the perfect ferro ordering. This sequence was reproduced very well by Hartree-Fock like band calculation and the result can be interpreted as follows. For the z-axis ordering, called/'8, for both the stacking sequence and the direction of moment, for example, the corresponding holes have a large effective mass mz and the usual effective mass m . causing a stable layer structure. The band calculation indicates that the double ferromagnetic layer is the most appropriate and to save kinetic energy the double layers should stack antiferromagnetically. Indeed, when the ferromagnetic ordering is forced, the c-axis lattice distance expands to weaken the increase in kinetic energy. When a double / ' 7 layer enters between two double F~ layers with parallel spins, a typical example in the FP phase, an antiferromagnetic exchange field of about 2.7 T is induced on the F7 layers [3]. This is interpreted as follows. To screen the plus charge of holes, the electrons near Xz of the Brillouin zone, which also have a large
3. Discussion and conclusion
In the introduction, the recent progress in the study of strongly correlated low carrier density systems was surveyed. It was shown that the tendency to Wigner crystal localization due to the long-range Coulomb interaction is the most important feature. In magnetic systems this is further accelerated by magnetic polarization, as well as by lattice distortion in the sense of the usual phonon polaron, and a weak magnetic polaron liquid or crystal is formed. The Kondo state can coexist with the weak magnetic polaron liquid with an enhanced form owing to the tendency to localization. A more drastic phenomenon occurs in Kondo insulator systems, in which the number of carriers is equal to the number of magnetic atoms and the localized Kondo state connecting to the strong limit of the Kondo lattice is realized. At low temperature, a transition to the strong magnetic polaron lattice occurs, where a nearly saturated localized moment appears within each polaron to gain the full energy of the c-f mixing or exchange interaction. In CeP and CeAs, a weak magnetic polaron liquid is formed even at room temperature and detailed experimental as well as theoretical studies have been done. These were reviewed in some detail in the second section. It was shown that indeed various anomalous behaviors can be explained consistently by the above model.
T. Kasuya et aL / Journal of Alloys and Compounds 219 (1995) 290-295
The magnetic polaron model was also applied to Yb monopnictides and some 3d systems, in particular to the high Tc CuO: layer systems, with satisfactory results explaining crucial experimental results [1,24]. To study low carrier density systems, in particular their intrinsic characteristics obtained by subtracting impurity effects, the long-range Coulomb interaction should be considered seriously as the universal fundamental interaction.
Acknowledgments In the study of CeP and CeAs, the collaboration with N. M6ri, Y. Okayama, M. Kohgi and T. Osakabe was essential. Valuable comments and discussions with them are highly appreciated.
References [1] For the review of CeSb, T. Kasuya, Memorial symposium for J. Rossat-Mignod, in Physica B, in press. [2] M. Takeshige, O. Sakai and T. Kasuya, Z Phys. Soc. Jpn., 60 (1991) 666. [3] T. Kasuya, M. Sera and T. Suzuki, J. Phys. Soc. Jpn., 62 (1993) 3364.
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[4] J.C. Nickerson, R.M. White, K.N. Lee, R. Bochmann, T.H. Geballe and G.H. Hull, Jr., Phys. Rev. B, 3 (1971) 2030. [5] J.M. Luttinger, Phys. Rev., 119 (1960) 1153. [6] T. Kasuya, J. Phys. Soc. Jpn., 63 (1994) 397. [7] T. Kasuya, Europhys. Lett., 26 (1994) 277. [8] T. Kasuya, Europhys. Lett., 26 (1994) 283. [9] D.M. Wieliczka, C.G. Olson and D.W. Lynch, Phys. Rev. B, 29 (1984) 3028. [10] O. Sakai, M. Takeshige, H. Harima, K. Otaki and T. Kasuya, J. Magn. Magn. Mater., 52 (1985) 18. [11] N. M6ri, Y. Okayama, H. Takahashi, Y. Haga and T. Suzuki, Jpn. J. Appl. Phys. Ser., 8 (1993) 182. [12] T. Kasuya, Y. Haga, Y.S. Kwon and T. Suzuki, Physica B, 186--188 (1993) 9. [13] M. Sera, Thesis, Dr. of Science, Tohoku University, 1982. [14] T. Kasuya, Y. Haga, T. Suzuki, Y. Kaneta and O. Sakai, Z Phys. Soc. Jpn., 61 (1992) 3447. [15] T. Kasuya, T. Suzuki and Y. Haga, Z Phys. Soc. Jpn, 62 (1993) 2549. [16] T. Kasuya, M. Sera and T. Suzuki, J. Phys. Soc. Jpn., 62 (1993) 2561. [17] Y.S. Kwon, Thesis, Dr. of Science, Tohoku University, 1991. [18] Y. Haga, T. Suzuki and T. Kasuya, Z Phys. Soc. Jpn., in press. [19] T. Kasuya, Y. Haga and T. Suzuki, Z Phys. Soc. Jpn., 62 (1993) 3376. [20] T. Kasuya and Y. Haga, Solid State Commun., 93 (1995) 307. [21] T. Kasuya, Z Phys. Soc. Jpn., 63 (1994) 4318. [22] M. Kohgi, T. Osakabe, K. Kakurai, T. Suzuki, Y. Haga and T. Kasuya, Phys. Rev. B, 49 (1994) 7068. [23] M. Date, A. Yamagishi, H. Hori and K. Sugiyama, Jpn. Z Appl. Phys. Set., 8 (1993) 195. [24] T. Kasuya, Physica C, 223 (1994) 233; 224 (1994) 191.