PHYSICAL REVIEW B 69, 134423 ͑2004͒

Spin-polaron model: Transport properties of EuB6

Jayita Chatterjee, Unjong Yu, and B. I. Min Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea ͑Received 14 October 2003; revised manuscript received 2 February 2004; published 23 April 2004; corrected 12 July 2004͒

To understand anomalous transport properties of EuB6, we have studied the spin-polaron Hamiltonian incorporating the electron-magnon and electron-phonon interactions. Assuming a strong exchange interaction between carriers and the localized spins, the electrical conductivity is calculated. The temperature and

magnetic-field dependences of the resistivity of EuB6 are well explained. At low temperature, magnons domi- nate the conduction process, whereas the lattice contribution becomes significant at very high temperature due to the scattering with the phonons. Large negative magnetoresistance near the ferromagnetic transition is also

reproduced as observed in EuB6.

DOI: 10.1103/PhysRevB.69.134423 PACS number͑s͒: 75.47.Pq, 72.10.Di, 71.38.Ϫk

Hexaboride compounds have been studied extensively for mechanism is valid for the high-temperature phase of EuB6. their unusual transport and magnetic properties. Among However, no T5/2 behavior is observed and, moreover, the those, EuB6 has attracted special interest due to its exotic crossover temperature is too high to be applicable to EuB6. magnetic properties, such as two consecutive phase transi- The diverse properties of EuB6 are expected to come pri- tions at low temperature ͑15.5 K and 12.6 K͒, very low car- marily from the exchange interactions between the carriers 20 Ϫ3 rier density (ϳ10 cm ), metallic ferromagnetism below and Eu2ϩ 4f local moments. Indeed, the large energy and 3 Tc , and large negative colossal magnetoresistance at the field dependence of the spin-flip Raman-scattering peak re- 1,2 transition of 15 K. It was proposed that the higher transi- flects the substantial FM exchange interaction (Jcf tion temperature is a metallization temperature due to an in- ϳ 2ϩ 0.1 eV) between the carrier and the Eu spins in EuB6. crease in the number of itinerant electrons and the lower one The magnetic polaron formation is favored by the large fer- is a bulk Curie temperature of long-range ferromagnetic romagnetic J ,4 as a trapped charge lowers its energy by 2 cf ͑FM͒ order. Sample dependence is an important issue in polarizing the local moments of Eu2ϩ. EuB as it has a very small number of intrinsic carriers. The 6 The low-carrier density magnetic systems such as EuB6 higher transition temperature is seen to be much more share many properties with the colossal magnetoˆresistance 2 sample dependent than the lower one. manganites, such as the insulator-metal transition concomi- 3,4 Raman-scattering measurements indicated the sponta- tant with the FM transition, large negative magnetoresis- neous formation of magnetic polarons, involving FM clusters tance, and the formation of magnetic clusters near T . Dis- 2ϩ 3 c of Eu spins just above Tc . It was conjectured that the tinctly from manganites, however, Eu-based systems do not transition at higher temperature arises from the mutual inter- manifest the structural complexities originating from the action between the magnetic moments and the conduction strong electron-lattice coupling associated with Jahn-Teller electrons which leads to the formation of bound magnetic effect. Due to the structural simplicity, EuB6 is an ideal sys- polarons. The bound magnetic polaron corresponds to a com- tem for investigation of the interplay between magnetic and posite object of localized charge carrier and its induced transport properties. An essential aspect for a theoretical un- alignment in a background of local moments. On the other derstanding of the properties of EuB is the temperature de- 5 6 hand, Hirsch proposed a model that the magnetism of EuB6 pendences of the resistivity. So it would be of great interest is driven by the effective mass reduction or the band broad- to study the temperature and field dependence of resistivity ening upon spin polarization. The origin of this effect is the of EuB6. For this purpose, we consider the spin-polaron bond-charge Coulomb repulsion which, in a tight-binding Hamiltonian taking into account the magnetic excitations model, corresponds to the ‘‘off-diagonal’’ nearest-neighbor ͑magnons in the linear spin-wave approximation͒ together exchange and pair hopping matrix elements of the Coulomb with lattice excitations ͑phonons͒, interaction. But the parameters for the exchange and pair hopping considered in his model are unphysical and the Eu † † 4 f states were treated as delocalized electrons in contrast to ϭϪ † ϩ ␻ ជ ជ ϩ ⍀ ជ ជ H ͚ tci␴c j␴ ͚ qa ជ aq ͚ pb ជ b p 6 ␴ ជ q ជ p the band-structure results. The Ruderman-Kittel-Kasuya- ͗i, j͘, q p Yosida interaction among the localized 4 f electrons through ជ Jq ជ ជ † itinerant electrons is also considered to be an origin of fer- Ϫ iq Ri͑ † ϩ † ជ ͒ ͚ e • ci↑ci↓aϪ ជ ci↓ci↑aq 7 ជ 2 q romagnetism in EuB6. More recently, a fluctuation-induced i,q hopping model is proposed as a transport mechanism for the ជ ជ † spin polaron in a paramagnetic background of fluctuating ϩ ជ ip Ri͑ † ϩ † ͒͑ ជ ϩ ͒ ͑ ͒ ͚ g pe • ci↑ci↑ ci↓ci↓ b p bϪ ជ , 1 8 ជ p local moments. In this case, temperature dependence of the i,p 5/2 resistivity is obtained to be proportional to T for kBT տ ӷ JFM and to T for kBT JFM , where JFM is coupling be- where ci␴ is the annihilation operator for the conduction ␴ ជ ជ tween the local moments. They claimed that this transport electron with spin at site i, aq and b p are the annihilation

0163-1829/2004/69͑13͒/134423͑5͒/$22.5069 134423-1 ©2004 The American Physical Society JAYITA CHATTERJEE, UNJONG YU, AND B. I. MIN PHYSICAL REVIEW B 69, 134423 ͑2004͒ operators for a localized spin ͑magnon͒ with momentum qជ ˜ ϭϪ ͓ ͑ Ϫ ͒ † † ϩ ͑ Ϫ ͒ ជ HLF ͚ t cosh xi x j Xi X jci,␴c j,␴ sinh xi x j and for the phonons with wave vector p, respectively, and ͗i, j͘,␴ ␻ ជ ⍀ ជ q and p are the magnon and phonon frequencies, respec- ␻ ϳ † tively. q JFMS where JFM is the direct FM exchange in- ϫ † † ͔ϩ ␻ ជ ជ ϩ ͑ ͒ Xi X jci,␴c j,Ϫ␴ ͚ qa ជ aq O nl↑nl↓ ជ q teraction between Eu 4 f spins (S). The last two terms are q the interaction between the itinerant electrons and localized 2 g ជ spins and that between the local density of the carrier (ni) † p ϩ ͑ † † ͒ϩ ⍀ ជ ជ Ϫ ͑ ͒ O cl↑cm↓cm↑cl↓ ͚ pb ជ b p ͚ ni and localized charge vibrations phonons , respectively. The ជ p ជ ⍀ ជ z Ϫ p i,p p non-spin-flip-interaction term ͓S (ni↑ ni↓)͔ which just iÞ j 2 shifts the on-site energy is not considered, as we are inter- g ជ p ជ ជ Ϫ ជ Ϫ ip (Ri R j) ͑ ͒ ested only in the transport properties. Without loss of gener- ͚ nin je • , 4 ជ ⍀ ជ ality, the Coulomb interaction can be neglected in the very i, j,p p low carrier density limit. 2ϩ In EuB6, it was observed that clusters of Eu spins are where formed via the strong ferromagnetic c-f exchange interaction.3 Hence one can assume a strong exchange cou- ជ Jq ជ ជ † pling which induces the locally ferromagnetically ordered ϭ iq Ri ជ Ϫ ͒ ͑ ͒ xi ͚ e • ͑aq aϪ ជ , 5 ជ ␻ ជ q spin clusters in the ground state. Formation of magnetic po- q q larons is possible in the low-carrier-density magnetic system if the degrees of spin disorder is sufficient to localize the 4 g ជ ជ ជ carrier, but not too high to prevent the local FM alignment. p ip R † X ϭexp͚ͫ e • i͑b ជ Ϫb ជ ͒ͬ. ͑6͒ ͒ i p Ϫ As for the electron-phonon (e-ph interaction, the study of ជ ⍀ ជ p boron isotope effect suggests that the polarons formed in p p EuB6 are predominantly magnetic, with a negligible lattice contribution.4 It is, however, not so certain that the magnetic The magnon part remains unchanged after the transforma- tions. In the transformed Hamiltonian, only the hopping term polaron in EuB6 do not couple to the lattice degrees of free- dom of Eu ions. Further, thermal-conductivity measurement contains nonlinear functions of spin-wave operators. As we are interested in the transport properties in the very low car- on EuB6 reveals that the e-ph scattering is strongly favored 2 rier density limit, we will not consider the renormalized in- below Tc to describe the T variation of the thermal conductivity.9 Therefore, to explore the role of lattice in the teractions between carriers. To calculate the electrical con- Ϫ † Ϫ † magnetic polaron formation, it is worthwhile to consider the ductivity, dynamics of cosh(xi xj)Xi Xj and sinh(xi xj)Xi Xj e-ph interaction in the Hamiltonian. in the hopping term of Eq. ͑4͒ should be properly treated. At To decouple the electron-magnon interaction term, we low temperature, transport is described by an effective band- employ the following canonical transformation: width with a background of magnons and phonons. The ef- fective mass of the carrier increases as a result of the inter- action with the magnons. The effect of the background is Ϫ ˜ ϭ R1 R1 H e He , included by calculating the thermal average of cosh(xi Ϫ † Ϫ † xj)Xi Xj and sinh(xi xj)Xi Xj . In conduction, there are two independent processes: elas- ជ Jq ជ ជ † tic and inelastic. For the elastic conduction process, thermal ϭ iq Ri͑ † ϩ † ͒͑ ជ Ϫ ͒ ͑ ͒ R1 ͚ e • ci↑ci↓ ci↓ci↑ aq aϪ ជ . 2 ជ ␻ ជ q average of the terms mentioned above is calculated as in the i,q q conventional polaron problem,12 assuming all magnons and phonons to be independent: In the presence of e-ph coupling, spread and depth of lattice deformation can be studied by using the Lang-Firsov ͑LF͒ 10 † 2 mag 2 ph transformation, ͗ ͑ Ϫ ͒ ͘ϭ ͓Ϫ͉ ជ͉ ជ ͔ ͓Ϫ͉ ជ ͉ ជ ͔ cosh xi x j Xi X j exp Vq Nq exp U p N p ϫexp͓Ϫ 1 ͉͑V ជ͉2ϩ͉U ជ ͉2͔͒, ͑7͒ Ϫ 2 q p ˜ ϭ R2 ˜ R2 HLF e He , † mag ph ͗ Ϫ ͘ϭ ជ ជ and sinh(xi xj)Xi Xj 0. Here Nq and N p are the magnon ជ and phonon numbers, respectively, g p ជ ជ † ϭϪ ip R j ជ Ϫ ͒ ͑ ͒ R2 ͚ e • n j͑b p bϪ ជ . 3 ជ ⍀ ជ p j,p p mag Ϫ1 ជ ϭ͕ ͓␻ ជ ͑ ͔͒Ϫ ͖ Nq exp q / kBT 1 , Note that, for weak to intermediate e-ph coupling, the varia- 11 tional LF transformation would give more satisfactory re- ph Ϫ1 N ជ ϭ͕exp͓⍀ ជ /͑k T͔͒Ϫ1͖ , ͑8͒ sults than LF transformation. In the present work, we con- p p B sider the LF transformation for simplicity. As a result, the transformed Hamiltonian becomes, and

134423-2 ͑ ͒ SPIN-POLARON MODEL: TRANSPORT PROPERTIES OF EuB6 PHYSICAL REVIEW B 69, 134423 2004

ជ Jq ជ ជ ជ ␦ជ ជ ϭ iq Ri Ϫ iq ͒ Vq e • ͑1 e • , ␻ ជ q

ជ g p ជ ជ ជ ␦ជ ជ ϭ ip Ri Ϫ ip ͒ ͑ ͒ U p e • ͑1 e • , 9 ⍀ ជ p

␦ជ ϭ ជ Ϫ ជ ͗ ͘ where Ri R j and ••• denotes the thermal average. The temperature dependence in the above thermal average mag ph ជ ជ comes only through Nq and N p . If we assume at finite temperature that the metallic conduction of the elastic pro- ␴ 13 cess e is inversely proportional to the effective mass and the effective mass is inversely proportional to the effective bandwidth, then FIG. 1. The normalized resistivity ͑arbitrary units͒ for EuB in ␴ ͑ ͒ϰ͗ ͑ Ϫ ͒ † ͘ ͑ ͒ 6 e T tcosh xi x j Xi X j . 10 the low-temperature region with ␻ϭ15 K, ⍀ϭ168 K, ␣ϭ8.0, ␤ ϭ0.5, ⌬ϭ0.004 eV, bϭ0.002 eV. ‘‘ϩ’’ symbols represent the ex- For the inelastic process, the conduction is provided by perimental data in ␮⍀ cm ͑Ref. 17͒. the incoherent hopping with emitting and absorbing the mag- nons and phonons. The conductivity is given by the Kubo 12 formula, found very satisfactory agreement with the experimental re- sults. In the present model, the resistivity peak near 15 K ␴ ϭ 2 ␦2 ͒ ͑ ͒ in ne w /͑3kBT , 11 signifies the crossover from a high-temperature insulating state with localized and isolated carriers ͑magnetic polarons͒ where n is density of mobile carriers, and w is the transition to a low-temperature conducting state resulting from the probability rate given by the Fermi golden rule, overlap of the magnetic polarons. Polaron overlap is marked by the rapid drop in resistivity at the transition. Consider- ϭ 2 ប⌬͒ Ϫ⌬ ͒ ͑ ͒ w ͑t / exp͑ /kBT , 12 ation of only the electron-magnon interaction in the present model can reproduce the resistivity peak around 15 K. How- ⌬ϭ 1 ͚ ជ ␻ ជ͉ ជ͉2ϩ͚ ជ ⍀ ជ ͉ ជ ͉2 with the activation energy 4 ( q q Vq p p U p ). ever, the inclusion of phonons within this model makes the For simplicity, if we consider single magnon (␻) and pho- agreement better. At very low temperature, the contribution non (⍀) frequency, the total conductivity is given by the of magnons to the conductivity is more important than that sum of two independent processes, of phonons. With increasing temperature, the band conduc- tion dominates with enhanced effective mass of the carrier so ␴ ͒ ␴ ͒ϭ Ϫ␣ ␻͒Ϫ␤ ⍀͒ ϩ 2 ⌬ ͒ ͑T / ͑0 exp͓ N͑ N͑ ͔ ͓b /͑3 kBT ͔ that the resistivity increases very rapidly. In contrast, at very high temperature, the role of the lattice effect becomes sig- ϫexp͑Ϫ⌬/k T͒, ͑13͒ B nificant due to the scattering with the phonons,9 where the incoherent hopping dominates. where ␣ϭ͚ ជ͉V ជ͉2, ␤ϭ͚ ជ ͉U ជ ͉2, bϭet␦ͱn/ប␴(0), and q q p p Now let us investigate the effect of the external magnetic ␴(0) is the zero-temperature conductivity. ␻ϳ ⍀ field on the transport. One can include the effect of the ex- For numerical calculation, we consider Tc . As for , since no inelastic neutron scattering measurements have been ternal magnetic field H for FM magnons by replacing the ␻ ជ ␻ ជ ϩ ␮ magnon frequency q by ( q geff BH), where geff is the reported on EuB6, we tentatively use the renormalized fre- ϩ 14 effective g factor for the localized spins of Eu2 . g (ϭg* quency based on the Einstein phonon frequency for LaB6. eff 2ϩ ϩ ␹ ␮2 If we assume ⍀ϭ168 K for the localized mode of Eu ion, Jcf /g Bxc) takes into account the enhanced effective the model calculation reproduces the experimental features. field arising form the exchange interaction between the car- The activation energy is a function of ␻ and ⍀, and also a riers and Eu-4 f local moments in the presence of the mag- function of Jcf and e-ph coupling. We observed that nature of netic field. Here g* is the intrinsic g factor for the Eu-4 f the resistivity is sensitive to the energy parameters ⌬ and b. spin, ␹ and g are the susceptibility and the g factor for the The average distortion around Eu site is more than an order conduction electrons, respectively, and xc is the concentra- of magnitude smaller than that in the perovskites,15 and so tion of conduction electron. Taking the temperature- we choose ␤ to be much less than ␣. independent Pauli paramagnetic susceptibility for ␹ which is ͑ Figure 1 provides the calculated resistivity normalized to given by the density of states N(EF) of conduction ϭ 20 ϭ 2,21 T 0) as a function of temperature. The calculated resistivity electrons and xc 0.01 per unit cell, one can estimate describes well the qualitative features of the experimental the value of geff to be nearly equal to 6. Then, with the above 16–19 ␻ ជ resistivity. For comparison with the experimental data, field dependent q , the experimental results of large magne- we have multiplied our results by an arbitrary factor and toresistance can be achieved.

134423-3 JAYITA CHATTERJEE, UNJONG YU, AND B. I. MIN PHYSICAL REVIEW B 69, 134423 ͑2004͒

FIG. 2. Normalized resistivity vs temperature T ͑K͒ for different FIG. 3. Normalized magnetoresistance ͑MR͒ vs temperature magnetic fields H ͑in tesla͒. The inset shows the low-temperature ϭ ͑ (T/Tc with Tc 15 K) for different values of the magnetic field in variation of resistivity. tesla͒. The same parameters as in Fig. 1 are used. Inset: MR vs H ͑in tesla͒ at Tϭ15 K. ‘‘ϩ’’ symbols represent the experimental data ͑ ͒ The temperature dependence of the electrical resistivity Ref. 17 . with varying the external magnetic field is presented in Fig. 2. The field and temperature dependences of the electrical netic field. A large negative MR near the FM transition, as resistivity reveal that the charge transport is strongly corre- depicted in the inset, is indeed observed in EuB .17 The MR 2 6 lated with the higher transition temperature. In the presence approaches Ϫ1 in the high-field limit, reflecting that the of magnetic field, the transition is broadened and shifted to zero-field resistivity is completely suppressed, which is in higher temperature. In the inset shown is the resistivity in the excellent agreement with experimental features. low-temperature region with varying the external magnetic In conclusion, we have proposed the model which ac- field ranging from 0.05 to 0.5 T. At zero field, the resistance counts qualitatively well for several anomalous transport drops sharply just above the transition temperature. This re- properties observed in EuB . The strong exchange interac- sistivity peak is suppressed by the magnetic field in good 6 tion (J ) and the FM magnons in the present model can agreement with the observation.2 cf reproduce the temperature and magnetic-field dependences In Fig. 3, we plot the magnetoresistance MRϭ͓␳(H) „ of the resistivity in EuB . Large negative MR occurs even at Ϫ␳(0)͔/␳(0) predicted by the present model as a function 6 … temperatures higher than T , which is consistent with ex- of temperature. With increasing temperature, a large negative c periments. contribution to MR appears. We obtain the negative MR even at temperatures higher than Tc , which is consistent This work was supported by the KOSEF through the with the experiment.16 Inset of Fig. 3 provides the normal- eSSC at POSTECH and in part by the KRF ͑Grant No. KRF- ϭ ͑ ͒ ͒ ized MR at T Tc 15 K as a function of the external mag- 2002-070-C00038 .

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