Physics in low carrier strongly correlated systems: Kondo insulator magnetic polaron and high Tc

N

ELSEVIER

PHYSICA Physica B 215 (1995) 88-98

Physics in low carrier strongly correlated systems: Kondo insulator magnetic polaron and high Tc Tadao Kasuya Physics Department, Tohoku University, Aoba-ku, Sendal 980, Japan Received 16 August 1994

Abstract

It is shown that in low carrier density systems a strong tendency to form the Wigner crystal, or electron localization, is clearly seen through long-range Coulomb interaction. The Kondo insulator is one typical example, in which the Kondo electron is localized and an exciton-like excitation is observed. The magnetic polaron liquid and lattice is another typical example and is seen most typically in Ce and Yb monopnictides. A detailed study on CeSb is shown. In this system, an enhanced Kondo state due to tendency of localization coexists. High Tc CuO 2 layered materials are also another typical example belonging to the present category. A description by the magnetic polaron picture is given. It is shown that, due to strong antiferromagnetic short-range order, a modified magnetic polaron with a singlet core is formed at high temperature and at lower temperature a pair magnetic polaron is formed. The superconductivity occurs due to their Bose condensation.

1. Introduction

The Kondo lattice systems have attracted much attention because of various novel characteristics as well as from the stand point of the most fundamental physical understanding. At higher temperature than the Kondo temperature, T > TK, the behavior is usually similar to that of dilute Kondo system. At lower temperature, T < TK, however, interaction among each Kondo state becomes important and various interesting novel properties in various types of coherent Kondo state are not yet well understood. In particular, the most controversial issue is the Fermi surface, i.e. whether and in what situation the 4f electrons contribute the Fermi surface as the 4f bands and how such change occurs from the localized character to the itinerant character, if it occurs continuously [1]. To understand the above problems better, we have studied the low carrier density systems because the typical Kondo state should be destroyed anyway in the low carrier density systems and it should give important information to investigate how it changes. Through such studies, it becomes clear that the most important character

in the low carrier density systems is the tendency to form the Wigner crystal, or its melting form, Wigner liquid, or a kind of strongly correlated Fermi liquid. This character had been masked by various defects, which causes easily magnetic impurity state or trapped magnetic polaron as seen typically in Eu-chalcogenides with defects or Th3P 4type rare earth chalcogenides with many rare earth vacancies [2], However, progress in the preparation of high-quality single crystals succeeded to reveal the intrinsic characteristics in the low carrier density systems. There are two main branches. One is the Kondo insulator systems, in which StuB6, YbB12 and probably TmSe had been the typical examples but many other examples such as CeNiSn have been discovered recently. In these systems, the conduction electron mixing with the 4f electron to form the singlet Kondo state is thought to localize. Then, exciton-like excitations are characteristics in these systems [3, 4] and indeed a detailed measurement on StuB 6 through inelastic neutron scattering by RossatMignod, Mignot and Alekseev shown this kind of excitation very clearly [5]. This topic is given in the paper by Mignot in this issue [6].

0921-4526/95/$09.50 O 1995 Elsevier Science B.V. All rights reserved

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T. Kasuya / Physica B 215 (1995) 88-98

Another typical example is formation of magnetic polaron liquid and lattice and typically observed in Ce- and Yb-monopnictides because high-quality single crystals of them were prepared successively in the T. Suzuki group in Tohoku University and interesting novel intrinsic properties have been investigate successfully [7]. Among them, CeSb has been studied most extensively after the pioneering work by Rossat-Mignod. In the following, present situation of our understanding on this material is shown in detail. The high T~ materials with the CuO 2 layer structure is another typical example of strongly correlated low carrier density systems. In this respect, Rossat-Mignod and the present author shared the same idea that the similar physical picture as in the f electron systems should be applicable to the present system. In this respect, a sharp peak structure in YB%Cu30 7_~, YBCO, found by Rossat-Mignod et al. by neutron scattering measurement [8], similar to that in SmB 6 mentioned above, as well as peculiar temperature dependence of spin correlation length in La2_~Sr~CuO4 [,9], LSCO, were the most crucial experimental evidences to support magnetic polaron formation on the CuO 2 layer. In the following, this picture [10, 11] will be extended further to explain many other experimental results.

2. CeSb, local Kondo and magnetic polaron The first extensive study on CeSb had been performed by Rossat-Mignod et al. by using neutron scattering method. The most striking result is its very complicated magnetic-phase diagram as shown in Fig. 1 including a devil's staircase-like series [,-12]. The first question is, of course, what kind of mechanism is responsible to such a fascinating-phase diagram. An extended A N N N I model can explain the overall feature [-13]. But explanation based on a more fundamental physical picture is required. It is established by the neutron scattering measurement that in the paramagnetic region without magnetic field the ground state is the F v doublet and the excitation energy to F a is 37 K, independent on temperature T [14]. Below 17 K, the A F P phases appear in the first-order transition and after a devil's staircase-like series, the A F phase appears, in which a ferromagnetic plane with nearly saturated moment, 2.1#B, directing perpendicular to the plane stacks in the sequence of up, up, down, down. In the A F P phases, the so-called para plane and the ferro plane described above form various types of stacking. When magnetic field is applied, the A F phase changes to the full ferro phase F through the A F F phases as shown in Fig. 1, while the A F P phases change to the F P phases. On the other hand, the Kasuya group chose CeSb as a typical example of low carrier density Kondo system

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and, because high-quality single crystals were prepared by Suzuki and Sera, detailed experimental studies have performed and revealed various unusual intrinsic characteristics. Among them, de Haas van Alphen, dHvA, effect gave the most important information. After serious dispute, we have now the final result as shown in Fig. 2 [15]. To understand this result, we should start from the reference system LaSb. Then it was shown that when the energy gap between the valence and the conduction band is adjusted so as to fit the experimental data for the" carrier number, and the vacant 4f level is put at the energy level observed by the inverse photoemission spectra, IPES or BIS, the usual band calculation gives a nearly perfect agreement [16]. Note that the nonadjusted band calculation gives usually too small band gap, or too large band overlapping, and too low 4f level position. An example is shown in Fig. 3 [17]. In this calculation the 4f level is 3 eV too low and the bottom of the conduction band is 0.3 eV too low from the Fermi energy. Then, the band calculation for the ferromagnetic CeSb in the sense of the nonrestricted H a r t r e ~ F o c k approximation, i.e. to put the 4f level of 2.1fib below the Fermi energy EF to be consistent with PES and all others above EF to be consistent with BIS, fits the dHvA data very well as shown also in Fig. 2 [18]. This dispersion curve is shown in Fig. 3 together with that for LaSb. The valence band of LaSb is formed mainly by the 5p(Sb) states with the top at the F-point of Brillouin zone split by the spin-orbit interaction on the Sb site into F8 and F6. The lower part of the conduction band is formed mainly by 5d(La) with the bottom at each X point of the Brillouin zone. Note that at the bottom X point substantial amount of orthogonal plane waves,

T. Kasuya / Physica B 215 (1995) 88-98

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O P W , mix to form the bonding state. When the wave vector q changes from Xz along the qz axis, the dispersion is weak and the effective mass is about the order of 2, a typical value for d band, but when q changes from X z perpendicularly to the qz axis, the dispersion is very sharp, changing nearly linearly with (q± - Xz) with one order of magnitude smaller effective mass at EF than that along qz, because the strong p - d mixing acts nearly proportionally to ( q ± - Xz) enhanced by a small energy separation [19] between p and d bands as seen in Fig. 3. Therefore at Ev the conduction electron has substantial a m o u n t of ~ f mixing matrix even though no c-f matrix remains at Xz. As shown later, this is one of the main mechanisms to causes a large K o n d o temperature T~ [20]. In CeSb of ferromagnetic ordering, a special band, we call it Fs0 hereafter, compatible with the occupied 4f state is pushed up through the strong p & mixing interaction causing a large hole Fermi surface, the

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(b) Fig. 3. (a) The result of the usual band calculation for LaSb. See the text for explanation. (b) The result of the adjusted band calculation for the ferromagnetic CeSb. The flat line at 0.10 Ry is due to the occupied 4f state (Ref. [18]). The unoccupied 4f levels are about 4.5 eV above the Fermi energy.

13, branch in Fig. 2. O n the other hand, the intraatomic 5 ~ 4 f exchange interaction is strongly anisotropic, which causes a strong exchange splitting on the conduction electrons near X z with the xy-type symmetry. Because the Fs0 holes extend up to Xz, the up-spin conduction electrons mix strongly with the Fs0 holes through strong p d mixing interaction and then both the Fermi surfaces for the hole and up-spin conduction electrons are canceled out and disappear around Xz and only a small Fermi surface for the down-spin conduction electrons shown in Fig. 2 as the a-branch remains. Effective mass is also obtained on the whole observed Fermi surface giving rather weak enhancement [15]. Before, the effective mass on the large hole Fermi surface, the 134 branch, was thought to be large because this Fermi

T. Kasuya / Physica B 215 (7995) 88-98 surface was difficult to observe causing controversial debate on the physical picture of CeSb. Now, it is clear that even the largest observed effective mass on the large hole Fermi surface is 4.3mo, mo being the free electron mass. The value of 7, the coefficient for the T-linear specific heat, is estimated to be the order 1 mJ/mol K 2, much smaller than that obtained by the specific heat measurement by Kwon [13J before but in good agreement with the theoretically predicted value [21]. Then, a more careful measurement has been performed on a larger sample and it becomes clear now that the value ofy is really not large but small [22] to be consistent with the result of dHvA measurement, as well as the theoretical estimation. The theoretical evaluation of y-value was done by the usual method familiar to evaluate the effect of electron-phonon interaction. The effective electron-magnon interaction is obtained as shown below through the selfconsistent second-order perturbation for the c-f mixing interaction, that is a unitary transformation called the Coqblin-Schrieffer transformation [23]: 1.t'~ i ( k - k ' ) R n +'V'v'/t'~ H c m = A~ r,v~' l ~ ' { 1 .t~,~)e C~k~,

. ~,k',

(1)

in which c~k+,C~k are the usual creation and annihilation operators for the vth band state with wave vector, k, X~,u is the operator to change 4f state at site R , from v to # and I with many suffixes is the coupling parameter. Then, another unitary transformation gives self-energy of magnon, the magnon dispersion, and the mass enhancement of conduction electrons. There are five excitations from the Jz = -52ground state corresponding to 2.1#B of moment. Among them, the neutron scattering measurement gives only the dispersion for the transition from Jz = ~ to 3. The most part ofJz = 3 is included in one of the 4fF7 doublet, and the remaining is included in one of the 4fF8 quartet. So far, the latter transition is not yet clearly observed by the neutron scattering measurement. When the carrier number is very small, all the carriers concentrate near good symmetry points in the Brillouin zone and thus each carrier has its own specific crystal symmetry. In this case, only available matrix is of v' = v and i f = #. This means that the crystal symmetry is conserved and thus we call this type of interaction as the symmetry exchange interaction, very much different from the usual spin exchange interaction. Only in the doublet magnetic system, the above two interactions become the same. In the present system, CeSb, the formula for the symmetry exchange is approximately applicable even in the ferromagnetic F8o alignment, that is the dz = ~ alignment. Then it is seen that the effective mass enhancement is strongest in the large Fso hole Fermi surface. To evaluate it, we need the dispersion, or the energy levels because of the weak dispersion, of all five excitation spectra. In Ref. [21], it was assumed that all five excita-

91

tions have the same energy as the Fso-F7 excitation observed by the neutron scattering. However, it is a better approximation to assume that the excitation energy to other F8 is larger by 37 K, the crystal field splitting. In this model, the agreement with the experimentally obtained enhancement factor becomes better, nearly perfect agreement. Note that in Ref. [21] a more direct method was used to evaluate the enhancement, as well as the magnon dispersion, but is essentially the same as the present more conventional description. The magnon dispersion observed by neutron scattering [24] also caused puzzle because the dispersion along the plane gives the maximum at the zone center and the minimum at the boundary, completely opposite from the result in the usual ferromagnetic plane. This is also clear in the present model because what we observe is the dispersion of the motion of the excited F7 on the ferromagnetic F8o plane. Then, again in Eq.(1), the transition from the electron pocket at Xx to the Fso hole is the main contribution which gives the minimum of the magnon dispersion at Xx. A detailed numerical calculation was also done by Kaneta [19] giving essentially the same result as the above. However, because we should sum up the contribution from the whole Brillouin zone and, furthermore, because of the small Fermi surface due to low carrier, with small Fermi surface, the error of the numerical calculation is still fairly large. Anyway, the results described above means that CeSb is a rather normal magnetic system in the same sense as in Fe and Ni is which the nonrestricted Hartree-Fock, HF, band model gives the fundamental stand point and the ~ second-order perturbation for the electron-magnon interaction gives all the necessary physical quantities including the magnon dispersion. However, in CeSb, differently from Fe and Ni, the Kondo effect is seen clearly in the transport properties. In Fig. 4, resistivity p as a function of temperature T is shown for various Ce- and Yb-monopnictides [25]. In the Yb-compounds, in general, the p-f mixing interaction is less than half that in the equivalent Ce-compounds due to the well-known Lanthanides contraction and thus the Kondo temperature TK in Yb-monopnictides is expected to be many orders of magnitude smaller than that in Ce-monopnictides due to the exponential character of TK. Indeed, the resistivity in Yb-monopnictides is quite normal, at least at high temperature showing similar behavior as that of LaSb. On the other hand, p in CeSb (1 atm) shows a typical Kondo behavior with a much larger value of p. This caused a puzzle, however, because even in CeSb the calculated value of TK is many orders of magnitude smaller than that expected from the behavior in Fig. 4, the order of 10 K, due to a small density of state at EF because of the low carrier characteristic. This puzzle has been solved recently [20] as follows. First, as

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Kasuya / Physica B 215 (1995) 88-98

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Fig. 4. Temperature dependence for resistivity p is shown for LaSb, YbAs, YbP and CeSb (1 atom) for the left-hand scale and for CeA, CeP and CeSb under 7 GPa of pressure for the righthand side scale. A more complete picture for CeSb (7 GPa) is shown in Fig. 7. The data was taken from Ref. [25]. shown before, even conduction electrons at EF have a substantial amount of p(X) character causing a fairly strong c-f mixing interaction. Second, because of the low carrier, the Coulomb interaction is not sufficiently screened and the f-character conduction electron created through the c- f mixing has strong tendency of localization increasing the amplitude at the nearest-neighbor sites, through which the amplitude of the c-f mixing is determined. Through these two effecl~s, the effective density of state is doubled, which is enough to bring TK to the expected value. The above estimation is, however, for the F8 Kondo state, which governs the high-temperature phenomena. The most controversial issue has been whether the F7 Kondo state exists at low temperature. It is clear that TK7 for the pure ['7 doublet Kondo state should be very low but the combined bonding Kondo state, written in its simplest form

(~bZ Cvif +i ---->fib Z c8jfffj)l 0>,

(2)

.may have a large TK value, where cvi and csj are the annihilation operators for the conduction electrons with the FTi and Fsj symmetries, respectively, f~i and f~-j are the creation operators for the 4f electrons with F7i and F8/ symmetries, respectively, c% and fib are parameters and t0 > means a state with no 4f electron. Experimental evidences to suggest the F7 Kondo state at low-temperature Kondo state are (i) very small induced moment in the para layer in Fig. 1 with the field of 3 T, (ii) a fairly reduced entropy value at temperature just above the magnetic transition temperature Tm = 17 K, and furthermore (iii) a large 78 value, 20 mJ/mol, the coefficient of the T-linear specific heat in the ferromagnetic F8o ordered region, measured by Kwon [13]. For the last point, as shown before, a more careful measurement down to 0.1 K by N. Sato indicates 7 is rather normal denying the Kondo effect. As studied before, the mass enhancement is also normal denying Kondo effect. Of course, at the lowest temperature, all the 4f states are F8o with the full moment, which denies any Kondo effect to be consistent with the new experimental facts. Therefore, main objections are the first two facts. For (i), a careful measurement of magnetization was performed by Sera with the result shown in Fig. 5. The value of moment at 5 K and 8.5 T is saturation value. The value for 5 K and 3.5 T is exactly ½ of that at 8.5 T. It is also observed that the moment in the F-phase does not change with the magnetic field. Then it is clear that the moment on the plane antiparaUel to the magnetic field is also field independent. These facts indicate clearly that the effective field on the ferro layer is much stronger than the applied field. Based on these informations, it is possible to estimate the moment on the para plane. The estimated value is, indeed, small in agreement with the result of neutron scattering. However, the field dependence, from 1.6 to 3.5 T, is quite normal as the I"7 moment. These facts give the following picture. In the para phase under consideration, an exchange field of about 2.5 T acts to be antiparallel to the field direction. Origin of this exchange field is thought to be as follows. The above consideration was done mainly on the FP2 phase in which the double Fao and double para layers stack alternatively. On the double Fso layers, the holes with the nearly same F8o symmetry gather to gain the energy of the p-f mixing and the conduction electrons at around Xz are also gathered to screen the charge of the F8o holes. Because the up-spin electrons are more strongly trapped due to the d-f exchange interaction, population of the down-spin electrons is dominating on the double F7 layers causing negative exchange field. On the other hand, the moment of F7 in the FP' in the FP' phase estimated from the 8.5 T curve is large and positive, nearly 0.6#B per Ce, indicating a strong positive exchange field. In the FP' phase a single F7 layer is sandwiched by triple F80 layers and then substantial positive spin

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Fig. 5. Magnetic moment M of CeSb is shown as a function of temperature for various differentvalues of applied field as shown on each line. The solid lines are to guide the eye. The absolute value of M for 8.5 T of field at 4.5 K is 2.14#B. See Fig. 1 for the corresponding phase. The data was taken from Ref. 1-27]. polarization is expected for the electrons even on the Fv layer [26]. On the second point, formation of magnetic polaron at about 60 K is now thought to be in the main origin. Experimental result for the lattice extension is shown in Fig. 6 [27]. Because of the strong l~f mixing interaction, the lattice shrinks proportionally to the population of Fs. Indeed, the magnetic part obtained by subtracting that for LaSb shown by dashed lines are well explained by the above model giving the excitation energy level of Fs in good agreement with the result of neutron scattering for Cel -xLa~Sb ofx >0.26. On the other hand, the result for CeSb is clearly anomalous indicating anomalous increasing of F8 population begins to occur at 60-70 K. Note that the dot-dash line is the expected curve by using the crystal field splitting of 37 K obtained by neutron scattering. This change is naturally thought to be due to the magnetic polaron formation. Indeed the resistivity shown in Fig. 4 also shows an anomalous peak beginning at 60 K, similar to the peaks in CeP and CeAs in which the magnetic polaron formation is detected in various ways [26, 28, 29]. When static pressure is applied, the peak

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Fig. 6. Temperature dependence of the relative change Ag/~'of the lattice distance is shown for various values of x in Cel _~La~Sb by the solid line. The dashed curve is obtained by subtracting the change in LaSb. The dot-dash line for CeSb is the expected curve calculated by using the crystal field splitting of 37 K. The data was taken from Ref. [27]. grows rapidly beginning at higher temperature because the p-f mixing is enhanced and at 3 G P a of pressure another sharp peak appears beginning at 40 K as shown in Fig. 7 [30]. The peak of resistivity becomes the highest, 4.5mf~cm, at 7 G P a of pressure. This anomalous increase is attributed most probably to the formation of magnetic polaron Wigner crystal [25]. Sharp drop of resistivity beyond the peak is attributed to the formation of the layered Fso ordering studied before, which is now recognized as the transition from the weak magnetic polaron to the strong magnetic polaron with near saturation moment. The latter is stabler at low. temperature and, because the effective mass of the polaron becomes heavy, they are easily localized and form a lattice, another type of Wigner crystal but is conductive. In a weak magnetic polaron in CeSb, the electrons with the same spin direction as the 4fF8 moment gather and create exchange field of more than 10 K for 4fFT, causing a splitting of more than 10 K in the I " 7 doublet. This is the origin of the small entropy in the paramagnetic region. This effect is seen in CeP and CeAs in

94

T. Kasuya / Physica B 215 (1995) 88-98

3. Magnetic polarons in high Tc materials

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various ways, such as anomalous specific heat as well as neutron scattering [31]. More experimental study is needed for CeSb. Another interesting phenomenon to support strong tendency of Wigner crystal is seen in an anomalously large longitudinal magnetoresistivity Pll as shown in Fig. 8 [27, 32]. On the other hand, Pll in LaBi and CeBi are quite normal, no change at all. The carrier numbers in LaBi and CeBi are larger than those in LaSb and CeSb and thus a critical concentration of carrier seems to be in between them. One-dimensionaltype Fermi surface under magnetic field is thought to induce a multi-Q charge density wave. More study is needed.

Both the heavy fermions in the f electron systems and the CuO2 layered high T c material belong to the strongly correlated electron systems. Furthermore, CeSb studied before and the high Tc materials belong to the low carrier density systems in which another type of strong correlation due to the long-range Coulomb interaction becomes important and then magnetic polarons in a liquid or a lattice style are easily formed. On this stand point, it is worthwhile to investigate the high Tc materials on the stand point of magnetic polaron. Such a study began when the present author stayed in Saclay in the last early summer to collaborate with Rossat-Mignod and two letter papers have been published 1-10,11]. Here, a brief review including some new developments is given. High T¢ material, in particular LSCO, has very similar electronic structure to that of CeSb studied above, nearly filled p-valence band with small number of holes and strongly correlated 3d magnetic state. However, high T¢ materials are different in the point that the exchange interaction between 3d spins is very strong and, because of the Very good two dimensionality in the CuO2 plane, strong short-range order exists up to 1000 K. Due to this strong short-range Order, various types of magnetic polaron different from those studied above are possible to exists and the most crucial experiment to determine the type is the insulator-like behavior in LSCO for x near 0.125, the so-called ~ anomaly. This is similar to the sharp peak of resistivity in CeSb under pressure and thus naturally interpreted as the formation of a magnetic polaron lattice. On this respect, this phenomenon is studied at first. Furthermore, this was not studied in detail in the previous letters. The essential physics in high T¢ materials is played commonly on the CuO2 layer and thus we consider mainly the CuO2 layer, monolayer in LSCO and double layer with a Y layer between them in YBCO studied in detail by neutron scattering method by Rossat-Mignod et al. [8]. In the CuO2 layer, Cu form the simple square lattice and one O atom sits in between two Cu atoms. Therefore, x 2 - yZ-type 3d state and the 2p~ state on O form the bonding and antibonding states and the Fermi energy is just in the middle of the antibonding band in LSCO for x = O, LCO, and in other materials in the insulating state. Because of the strong antiferromagnetic short-range order, it may be better to choose the nearest-neighbor long-range antiferromagnetic state as the first starting state. Then, the initial square Brillouin zone is reduced to a square of a half size. In the large Brillouin zone scheme, the energy gap appears along the boundary of the reduced Brillouin zone as shown in Fig. 9. When the strong correlation effect among the 3d electrons is explicitly included, the upper part of the

T. Kasuya / Physica B 215 (1995) 88-98

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.X-

63

Fig. 9. The large and the small Brillouin zones are shown. The large and small circles are the peak positions for the neutron scattering corresponding to the antiferromagnetic ordering and the satellite, respectively, antibonding band above the gap may be called unoccupied 3dx2_y2 band, while the lower part of the antibonding band may be called simply the valence p-band, or the p-d mixing valence band, similar to the p-valence band in CeSb. Indeed, when the 3d PES is performed, the 3d bonding and antibonding Kondo peaks appear, similar to the 4f bonding and antibonding peaks in CeSb [33, 34]. Therefore, the top part of the valence band may be called the d-p bonding Kondo band. In the high Tc society, this state is called Zhang-Rice state. Note that there are no difference for the final states whether the hole is created at first on p or d state. Only the intensity distribution changes. Note also that because of the strong short-range order and strong inter-site exchange interaction, the bonding Kondo stVate on each site is not the perfect singlet but is polarized depending on the site. When a hole is created at O-site, this p-hole polarizes the two nearest neighbor d spins in the same spin direction as that of the p-hole through the p-d mixing, exactly the same as the F8 p-f mixing effect in CeSb. Because the p-d mixing is much stronger than the p-f mixing, the formation of magnetic polaron is evaluated to be possible. The most probable magnetic polaron Wigner crystal for x equal ½ is thought to be that as shown in Fig. 10. Each up-spin polaron is surrounded by four down-spin polarons and in each up-spin polaron the moment of each down spin site, there are two such sites, is reduced strongly to near zero because of the strong competition

x

O-x.

O

x

x

x\o/x x

x

\/

0

x

×

Fig. 10. A model for the 8-site magnetic polaron lattice is shown on the upper side. A model for the 5-site magnetic polaron lattice is shown on the lower side. between the intersite antiferromagnetic exchange interaction and the ferromagnetic l>d mixing and furthermore the bonding Kondo effect. This will be treated later again. The effective bonding energy is roughly evaluated to be the order of 1000 K and thus each polaron survives even at high temperature. To fit the satellite structure observed by neutron scattering as shown in Fig. 9, some modification is necessary. In the present situation, the up-spin p-hole is difficult to penetrate into the down-spin polaron causing a decrease in the kinetic energy. To release this situation, spin rotation of 2r~a and 2rcl3 is introduced to the nearest-neighbor polaron in the right and the upper directions, respectively. Or it may be possible to multiply the modulation factor cosgx(~nx + tiny), where nx and % are the number of lattice point along x and y direction, respectively. Anyway, it is possible to fit the overall feature of the satellite structure

96

T. Kasuya / Physica B 215 (1995) 88-98

assuming the equal distribution of four different domains as obtained easily from Fig. 10. However, in the present model, in general, each satellite spot can have a further structure. Experimentally a sharp peak is observed at around x =18 and this is favorable for the magnetic polaron Wigner crystal. No other clear model exists to explain the present phenomena, F o r x smaller than 0.125, the above magnetic polaron can move around and the magnetic polaron liquid state similar to that in CeP and CeAs is realized. When x is not very much different from 0.125, the short-range order between the magnetic polarons is preserved and thus the satellite structure survives even though the intensity decreases and the width increases. When the magnetic polaron moves, the spin direction of polaron rotates following the surrounding situation. Therefore, the spin correlation between the magnetic polaron is destroyed easily with decreasing value of x and another peak of neutron scattering with the center at (½, 1) with the width corresponding to the spin correlation length for the magnetic polaron radius, which is approximately x -1/2, is developed. This is indeed consistent with the result of the neutron scattering measurement [9]. When x decreases beyond 0.04, impurity localization effect becomes significant and thus the above simple rule begins to deviate but the essential feature persists to the samples of lower x values as shown in Fig. 11 [10]. On the other hand, when x increases beyond 0.125, the size of the magnetic polaron should decrease, probably to five site polaron. This size of polaron requires a large kinetic energy for localization and thus becomes unstable causing the system to the normal Fermi liquid state. This is also consistent with the experimental evidence. 0.06

"""

• ;.

,

,

,

,

,

0.04

/

o
~.~=0.0

.-

/..~.//,fy //~=o

---....._-:-.L~-/ O-

"

o

'

"

I

.

,

200

.l'

, 400

.

,

, 600

T ( K )

Fig. 11. Inverse of the spin correlation length, ~- 1, for various La2_xSr~CuO4 are plotted as functions of temperature. The data are from Ref. [10]. The original data are from Ref. [9].

In the wide temperature range, the above type spin correlation persists and then the dominant motion of the magnetic polaron is described approximately such as keeping the p-hole spin direction and rotating the 3d spins. In this case, the speed of the polaron, that is the mobility, is roughly determined by the corresponding spin wave velocity vs(qc), where qo corresponds to that of the spin correlation length [35]. As vs(q~) decreases with increasing temperature, the mobility # should decrease but no guarantee for the T-linear relation of the resistivity. In general, however, T-linear resistivity is obtained in the strong coupling regime. Because the Lorentz force has the same effect as the electric field, the normal Hall effect inversely proportional to the p-hole population is expected at low temperature in agreement with the experimental result [36]. At high-temperature, population of usual fermion increases causing gradual change to the normal state. On the other hand, the Fermi surface determined from the sharp change of the distribution in the k-space seen by PES should be that corresponding to the large Fermi surface in the large Brillouin zone, in agreement with the experimental fact [37]. There is no mystery on this problem. In YBCO, a transition to a spin gap state was observed by Rossat-Mignod et al. clearly at low temperature [8] and we assigned it as the transition to a pair magnetic polaron state with the spin singlet state. In LSCO, the spin gap transition has not yet been detected by neutron measurement but various other evidences are accumulating indicating that such a transition also exists in LSCO at rather higher temperature [38]. In the previous paper, we assigned the sharp increase in qc = ~-1 in Fig. 11 as the thermal population of magnetic polaron from the impurity localized state. From the recent evidences, however, the temperature for the sharp change corresponds well with the transition to the spin gap state, that is the pair magnetic polaron State in our assignment. If our assignment is correct, the value of ~ should increase by root 2, which agrees fairly well with the experimental result even for the x = 0.02 sample, even though the impurity effect is substantial and thus the spin correlation length at the higher plateau is longer than x-o.5 due to a smaller number of free magnetic polaron. Because the pair magnetic polaron is a Boson, the Bose condensation, that is the transition to the superconductivity, should occur at a lower temperature if the impurity effect is not too serious. The spin gap character is naturally derived from the spin singlet character. Because the impurity effect is more significant in LSCO than in YBCO the gap character is easily masked. Essential feature is same in YBCO, but there are some minor differences. The most favourable superconducting state in YBCO seems to occur at 06,92 and beyond this concentration of oxygen, the system seems to be in the

T. Kasuya / Physiea B 215 (1995) 88-98

overdoped region. Because the hole concentration on each CuOz layer is evaluated to be about 0.2 per CuO2, the 5-site magnetic polaron is expected to be the dominant one, as shown in Fig. 10. Unlike LSCO, however, such a crystallization never occurs in YBCO. The reason is thought to be as follows. The first question is why the 5-site polaron is stable in YBCO. This is because in YBCO the dispersion along (½, ½) is weaker than that in LSCO and thus the localization energy is not very high. In YBCO, too, the competition between the three types of interaction mentioned before is strong but in the schematic picture it is possible to describe the situation such that the strong bonding Kondo singlet state is formed between the p-hole and the central d-spin and the ferromagnetically aligned outer layer covers it. In this case, because the p-hole is well-localized within the polaron, there should be some vacant space to absorb the zero point vibration of the Wigner crystal. Therefore, the tight crystallization shown in Fig. 10 is unstable. If there is some vacant space, the crystallization becomes difficult and the Wigner liquid state is more stable. On the other hand, two central sites in the 8-site polaron is similar to the problem of two Kondo sites problem studied extensively by mar, y groups including our group [39]. At high temperature, a strong effective antiferromagnetic coupling is induced between the spins in these two sites, and they form the usual magnetic singlet. Then outside this singlet the usual p - d magnetic polaron exists. In this situation, the p-hole can go further beyond the polai'on for the zero-point vibration. The spin rotation mentioned before is, in this sense, necessary to absorb the zero-point vibration, making the Wigner crystal shown in Fig. 10 stable. This is also the reason that no satellite appears in YBCO near the optimum doping. The singlet pair magnetic polaron in LSCO is thought to be stabilized at low temperature with increasing strength of antiferromagnetic correlation. In each polaron, the state in which one spin among the two sites directs down and another spin forms the singlet bonding Kondo state with the p-hole and two types of them make bonding state becomes stable. This state is stabilized further by forming the pair magnetic polaron because then two bonding Kondo states move around the up- and down-spin sublattices. Therefore, this is described well by the ~ J model. The main force for the pair formation is, in this sense, the long-range Coulomb correlation. It is interesting to note that, as was suggested by Maekawa, the short-range Coulomb repulsion may have the similar effect in the 2-D systems. In YBCO, the same thing occurs between two magnetic polarons situated on the upper and lower layers, respectively. Then, the spin correlation within the layer does not change and the strong correlation among the upper and lower layer spins persists in good agreement with the experimental result [8]. One of the most

97

crucial experimental evidence to support the singlet pair magnetic polaron is a sharp peak at 40 meV observed by neutron scattering [8]. This peak appears only in the purest sample of 06.92 below 70 K. Rossat-Mignod called it an exciton with nearly zero width. In the present mode, this corresponds to the excitation of a phole to a nonbound state. Because of the Coulomb interaction with the left charge, the excited p-hole forms an exciton. As the number of free hole increases, the Coulomb interaction is easily screened and the exciton peak disappears changing to a broad peak with lower energy because the screening becomes more effective reducing the excitation energy.

4. Conclusion

In the first part, it was shown that long-standing mysteries in CeSb initiated by Rossat-Mignod are now solved perfectly by the magnetic polaron liquid formation. With decreasing temperature, it can order in a magnetic polaron Wigner crystal and with further decreasing temperature to the strong magnetic polar0n with full polarization of the 4f moment. This polaron forms another type of Wigner crystal, a layered conductive ordering. The same picture is applicable also in CeP and CeAs, A detailed theoretical and experimental work was done by Takahashi [40] and by Sera [27] except the recent progress in the magnetic polaron model. In the second part, it was shown first that the situation in the high Tc CuO2 layer system is essentially the same as that in CeSb. Only difference is a strong short-range order of Cu spins in the CuO2 layer due to two dimentionality. In LSCO mysterious ~ anomaly was attributed to the Wigner crystallization of 8-site magnetic polaron. The spin correlation function offers the most crucial evidence to support the present model. The transition to the singlet pair magnetic polaron at lower temperature around 400 K was also,suggested by the data of spin correlation length. The satellite structure is attributed to a spin rotation effect to absorb the zero-point vibration of Wigner crystal. In YBCO, however, the zero-point vibration cannot be well-absorbed, which prevent the Wigner crystal formation as well as the satellite structure. This is due to the different characteristics of the 5-site magnetic polaron. A sharp exciton-like spectra is a crucial experiment to support the singlet pair magnetic polaron. Typical low carrier strongly correlated electron systems both in the 4f and 3d systems are now understood based on the same universal picture. The materials considered here will be the model systems for further deep study of the new interesting region.

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T. Kasuya / Physica B 215 (1995) 88-98

Acknowledgements F o r the first part of the paper, there are many colleagues who performed the experimental and theoretical investigation. The author particularly mentions J. Rossat-Mignod, H. Takahashi, O. Sakai and M. Sera for their interesting discussions and the collaborations. F o r the second part, the author appreciates valuable discussions and comments by J. Rossat-Mignod, S. Maekawa, Y. Endo and M. Sera. This paper is dedicated to late Dr. J. Rossat-Mignod.

References [1] See, for example, Y. Onuki, T. Goto and T. Kasuya, Material Science and Technology, Vol. 3A, ed. K.H.J. Buschow (VCH, Weinheim, 1992) p. 545. [2] See, for example, a review paper, T. Kasuya, J. Appl. Phys. 77 (1995) 3200. [3] T. Kasuya, Europhys. Lett. 26 (1994) 277. [4] T. Kasuya, Europhys. Lett. 26 (1994) 283. [5] P.A. Alekseev, J.-M. Mignot, J. Rossat-Mignod, V.N. Lazukov and I.P.'3adikov, Physica B 186-188 (1993) 384; also unpublished work. [6] J.-M. Mignot, Physica B 215 (1995) 99. [7] See for example, a review paper, T. Kasuya et al., Physica B 199&200 (1994) 585. [8] See the review paper, J. Rossat Mignod, L.P. Regnault, P. Bourqes, P. Burlet, C. Vettier and J.Y. Henry, Series Frontiers in Solid State Sciences in Magnetism and Superconductivity (World Scientific, Singapore, 1993). [9] B. Keimer, N. Belk, R.J. Birgeneau, A. Cassanhol C.Y. Chen, M. Greven, M.A. Kastner, A. Ahrony, Y. Endo, R.W. Erwin and G. Shirane, Phys. Rev. B 46 (1992) 14034. [10] T. Kasuya, Physica C 223 (1994) 233. [11] T. Kasuya, Physica C 224 (1994) 191. [12] J. Rossat-Mignod, J.M. Effantin, P. Burlet, T. Chattopadhyay, L.P. Regnault, H. Bartholin, C. Vettier, O. Vogt, D. Ravot and J.C. Achard, J. Magn. Magn. Mater. 52 (1985) 111. [13] T. Kasuya, Y.S. Kwon, T. Suzuki, K. Nakanishi, F. Ishiyama and K. Takegahara, J. Magn. Magn. Mater. 90&91 (1990) 389. [14] H. Heer, A. Furrer, W. Hfilg and O. Vogt, J. Phys. C 12 (1979) 5207. [15] R. Settai, T. Goto, S. Sakatsume, Y.S. Kwon, T. Suzuki and T. Kasuya, Physiea B 186-188 (1993) 176.

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