Magnetization and energy gap of Kondo insulator YbB_{12} are calculated
theoretically based on the previously proposed tight-binding model composed of
Yb 5d$\epsilon$ and 4f $\Gamma_8$ orbitals. It is found that magnetization
curves are almost isotropic, naturally expected from the cubic symmetry, but
that the gap-closing field has an anisotropy: the gap closes faster for the
field in (100) direction than in (110) and (111) directions, in accord with the
experiments. This is qualitatively understood by considering the maximal
eigenvalues of the total angular momentum operators projected on each direction
of the magnetic field. But the numerical calculation based on the band model
yields better agreement with the experiment.Comment: 4 pages, 4 figures, to appear in J. Phys. Soc. Jp

Alekseev P. A.

Bouvet A.

Grandjean F.

Haen P.

Hundley M. F.

Iga F.

Jayaraman A.

Kasaya M.

Mignot J.-M.

Okamura H.

Saso T.

Slater J. C.

Sugiyama K.

Susaki T.

Takeda Y.

Takegahara K.

English

arXiv

Toshiyuki Izumi

Yoshiki Imai

Tetsuro Saso

Crossref

Theory for Magnetic Anisotropy of Field-Induced Insulator-to-Metal Transition in Cubic Kondo Insulator YbB12

Magnetization and energy gap of Kondo insulator YbB12 are calculated theoretically based on the previously proposed tight-binding model composed of Yb 5d epsilon and 4f Gamma(8) orbitals. It is found that magnetization curves are almost isotropic, naturally expected from the cubic symmetry, but that the gap-closing field has an anisotropy: the gap closes faster for the field in (100) direction than in (I 10) and (111) directions, in accord with the experiments. This is qualitatively understood by considering the maximal eigenvalues of the total angular momentum operators projected on each direction of the magnetic field. But the numerical calculation based on the band model yields better agreement with the experiment.Copyright(c)2007 The Physical Society of Japa

Izumi Toshiyuki

Imai Yoshiki

佐宗 哲郎

Saitama University Cyber Repository of Academic Resources

Typeset with jpsj2.cls <ver.1.1.1>Theory for Magnetic Anisotropy of Field-Induced Insulator-to-Metal Transition inCubic Kondo Insulator YbB12Toshiyuki Izumi∗, Yoshiki Imai and Tetsuro SasoDivision of Materials Sciences, Graduate Course of Science and Engineering, Saitama University, Shimo-Ohkubo 255, Sakura-ku,Saitama City, Saitama, 338-8570(Received )Magnetization and energy gap of Kondo insulator YbB12 are calculated theoretically basedon the previously proposed tight-binding model composed of Yb 5d and 4f Γ8 orbitals. Itis found that magnetization curves are almost isotropic, naturally expected from the cubicsymmetry, but that the gap-closing ﬁeld has an anisotropy: the gap closes faster for the ﬁeldin (100) direction than in (110) and (111) directions, in accord with the experiments. This isqualitatively understood by considering the maximal eigenvalues of the total angular momen-tum operators projected on each direction of the magnetic ﬁeld. But the numerical calculationbased on the band model yields better agreement with the experiment.KEYWORDS: Kondo insulator, YbB12, magnetization, field-induced insulator-metal transition, magneticanisotropy1. IntroductionYbB121) is the most typical Kondo insulator, since ithas a cubic crystal structure (Yb and B12 forms NaClstructure), and many experiments: transport,2) calori-metric,3) photoemission,4,5) optical,6,7) neutron scatter-ing,8–10) etc. are already reported. We believe that mostof them can be understood in terms of the band insula-tor model with strong Coulomb repulsion between f elec-trons. In fact, based on the LDA+U band calculation, itwas found that the conduction bands near the Fermi levelare composed of the 5d orbitals on Yb atoms, and thesimple tight-binding band reproduces the LDA+U cal-culation rather well.11) Some of the conduction bandsare doubly degenerate (four-fold if spin degeneracy is in-cluded). A gap opens by the hybridization of the four-fold degenerate Γ8 states with the above-mentioned con-duction bands with degeneracy. It was pointed out thatthe gap can not open if the crystal-ﬁeld groud state of4f were Γ7 doublet. Therefore, proper consideration ofthe degeneracy is a key to understanding the opening ofa gap in the Kondo insulators.11)Based on this band model, one of the authors12) calcu-lated frequency- and temperature-dependence of the op-tical conductivity and obtained semi-quantitative agree-ment with the experiment.7) He succeeded in reproduc-ing the mid-infrared peak and the prominent tempera-ture dependence of the low frequency part, which is dueto the many-body eﬀect.In contrast to YbB12, it is not clear whether otherwell-known or classical “Kondo insulators”, SmS13) andTmSe,14) belong to the same category as YbB12, sinceSm and Tm has more complicated 4f states than Yb. Inaddition, TmSe has odd number of electrons per unit cell,so that it can not be a band insulator if 4f electrons aremixed valent. Other Kondo insulators, which may be-long to the same category as YbB12, will be Ce3Bi4Pt315)and Ce-skutterudites, e.g. CeFe4As12.16) Nevertheless,YbB12 can be a prototype of the Kondo insulator from∗ Present address: Japan Radio Co. Ltd.the point of view of the simpleness of the band structure,suﬃcient number of experiments and the existence of thecompact theoretical model.Despite the above-mentioned success, YbB12 still hasmany features to be understood correctly and quan-titatively. One of them is the magnetic properties.Sugiyama, et al.2) ﬁrst found that the activation en-ergy of the resistivity of this insulator vanishes at aroud50 Tesla when magnetic ﬁeld is applied. This is called asﬁeld-induced insulator-to-metal transition and the criti-cal ﬁeld is denoted as Bc hereafter. Iga, et al.17) foundthat the magnetization curve for each direction doesnot show an anisotropy, but the critical ﬁeld Bc does.Namely, Bc(110) and Bc(111) is larger than Bc(100) by10 percent (Bc(100) =47T, Bc(110)  Bc(111) =53T).The transitions are ﬁrst order since hysterisis is observed.Using the simple periodic Anderson model, one of thepresent authors showed that the overall feature of themagnetization curve of YbB12 can be explained and thetransition can be ﬁrst order if the Coulomb repulsionis taken into account.18,19) In addition, he found thatthe renormalization factor (which renormalizes the gap)due to strong correlation does not largely change by themagnetic ﬁeld in the insulating phase, so that the renor-malized up- and down-spin bands are shifted by the ﬁeldas if they are rigid bands and the renormalized bandgap is closed also as if in the case of the rigid bands.18)But the anisotropic feature of the f-electrons under cubiccrystalline ﬁeld was not taken into account.In the present study, we investigate theoretically theorigin and mechanism of the anisotropy of the ﬁeld-induced insulator-to-metal transition. In §2, we brieﬂyexplain the previously introduced tight-binding bandmodel. Calculations of magnetization and energy gapunder magnetic ﬁeld will be presented in §3. The lastsection is devoted to the discussions and conclusions.12 Author Name2. Tight-Binding Energy BandThe energy dispersions of the conduction bands ofYbB12 are well approximated by the tight-binding modelof 5d orbitals of Yb on the fcc lattice and expressed bythe following simple expressions:11)Eαβk = Edε + 3(ddσ) cos(kα/2) cos(kβ/2), (1)where (ddσ) is a Slater-Koster integral,20) (α, β) denotes(x, y), (y, z) or (z, x). The lowest band along Γ-K-X(110)and Γ-X(100) are doubly degenerate (four-fold if the spindegeneracy is included). Introducing the hybridizationbetween these d bands and the 4f j = 7/2 Γ8 states, weobtained the energy bands with a gap.In eq.(1), we locate the energy level of 5dε orbitalsat Edε=1.0 Ry, and set (ddσ)= 0.06 Ry. This value of(ddσ) is the eﬀective one due to hopping between near-neighbor Yb sites through B12 clusters.The hybridization between these 5d bands and the 4fstates is described by the eﬀective (dfσ) integrals21) be-tween the nearest-neighbor Yb sites. We retain only thenearest neighbor (dfσ) bonds as the simplest model. Welocate the 4f Γ8 states at EΓ8 = 0.88 Ry and choose(dfσ)=0.015 Ry. We also include (dfπ) = −0.0075 Ryand (ﬀσ) = −0.003 Ry to adjust to the LDA+U cal-culation. Furthermore, the ﬁlled bands below the gapare shifted down by ∆E = −0.011 Ry relative to thebands above the gap. This treatment is in accord withthe spirit of the LDA+U treatment. These parametersare the same as those in our previous paper12) to ﬁt theoptical conductivity.7)Using these parameters, we obtain the dispersioncurves which reproduce the LDA+U bands around thegap rather well.11) They have an indirect gap of about0.0069 Ry (= 1089 K) between X and L points and thedirect gap of 0.018 Ry (= 0.245 eV). The former is aboutsix times larger than the best estimated value of the in-direct gap in the optical experiment, Eg =174 K.7) How-ever, in the present realistic band model, we can not usethis value. If we use a band model with the correct valueof the indirect gap, it is diﬃcult to reproduce the gap inthe density of states, since we must use a very ﬁne meshto calculate the three-dimensional k-summation over theBrillouin zone.Note that the band calculation can treat the f-stateonly by the j-j coupling scheme. One hole in j = 7/2Γ8 state corresponds to the Γ8 state with the total angu-lar momentum J = 7/2 of Yb3+ (the f-electron numbernf = 13). In the present case, the number of ﬁlled 4fΓ8 electrons is about 3.1 and 0.9 hole exists in the bandsabove the gap, whereas j = 5/2, j = 7/2 Γ6 and Γ7 statesare far below Γ8 and are implicitly treated as completelyﬁlled. Thus the present model approximately reproducesthe experimental valence of Yb+2.9.3. Calculation of Magnetization and EnergyGap under Magnetic FieldWe apply the magnetic ﬁeld B = µ0H along (100),(110) and (111) directions. The Hamiltonian reads0.80.911.11.2Energy(Ry)Γ K X W L Γ XFig. 1. The tight-binding band for YbB12 under B=500T in (100)direction. The X points in the left and right denote the equiva-lent (110) and (100) points (in unit of 2π/a), respectively.H =∑α,βi,jtαβij c†iαcjβ +∑igcµBsi ·H +∑igJµBji ·H,(2)where tαβij denotes the hopping matrix elements betweenthe ten orbitals {α, β = xy ↑, yz ↑, zx ↑, xy ↓, yz ↓, zx ↓,Γ(1)±8 ,Γ(2)±8 } on i and j sites and is expressed by theSlater-Koster parameter given in the last section.12) µBis the Bohr magneton, gc = 2 and gf = 8/7 are the g-factors of conduction and f electrons. si and ji are thespin and the total angular momentum operators for theconduction and f electron at i site, respectively. We haveomitted the eﬀect of magnetic ﬁeld on the orbital motionexcept that through j. The eﬀect of orbital motion canbe included by attaching the phase factor to the hoppingmatrix elements, but it will be smaller than those con-sidered here through j, since the latter will be enhancedby the strong correlation.In the present model with the larger gap than the ex-periment, the gap closes at the large ﬁeld of about 500T. We show the band structure at this ﬁeld in (100) di-rection in Fig.1. Four bands lie below the gap and sixbands above the gap and all the degeneracies under zeroﬁeld are lifted.Diagonalizing the Hamiltonian, the Bloch function isobtained asψkn(r) =1N∑i,αukn,αeik·Riφα(r −Ri), (3)where φα(r − Ri) is the local orbital α at the site Ri,n denotes the diagonalized band index, ukn,α the coef-ﬁcient and N the total number of unit cell. Using this,the magnetization M is calculated byM =1N∑k,nf(Ekn) [gcµB〈ψkn|S|ψkn〉+ gfµB〈ψkn|J |ψkn〉] , (4)Title of the Article 3where S =∑i si, J =∑i ji, f(E) is the Fermi functionand Ekn the n-th band energy. Note that the applica-tion of the magnetic ﬁeld lowers the symmetry, so thatthe sum over 1/48 of the ﬁrst Brillouine zone is not al-lowed. In the actual calculation, we summed over wholeof the ﬁrst and the second Brillouine zones because theprogramming becomes much easier: we can simply makesummation over a cube with twice the volume of the ﬁrstBrillouine zone of fcc lattice. We divide this cube intoat most 4003 mesh points. Temperature is set equal tozero throughout the paper.Also for simplicity, we have summed only over the low-est four ﬁlled bands in the calculation of the magnetiza-tion, so that the calculation makes sense only for B < Bc(critical ﬁeld) in each ﬁeld direction. For B > Bc, onehas to determine the chemical potential for each valueand direction of the ﬁeld and sum up over more thanfour bands. We did not do it since we are interested onlyin the anisotropy problem in the present paper.M/µB = |M |/µB is plotted in Fig.2 for the three di-rections of the ﬁeld. In accord with the general theoremof the group theory that the linear response tensor mustbe proportional to the unit tensor (isotropic), the presentresult shows that the magnetization is isotropic up to 200T. (Indirect) energy gap vs. magnetic ﬁeld is plotted inFig.3. Since we take large value of the initial gap, thecritical value of the ﬁeld in each direction is also large:Bc(100) = 464T, Bc(110) = 501T, Bc(111) = 506T.These values will become smaller if we use the correctvalue of the band gap. For B < 300T, Eg(111) becomesslightly smaller than Eg(110), but the diﬀerence is verysmall and it seems out of range of the precision of thepresent calculation. It seems reasonable to consider thatEg(100) < Eg(110) < Eg(111) for all the ﬁeld, as will bediscussed in the next section.0 100 200 300 400 50000.050.1B [T]M/μB (111) (110) (100)Fig. 2. Magnetization curves under the magnetic ﬁeld in (100),(110) and (111) directions.4. Discussions and ConclusionsIn order to understand an origin of the anisotropy ofthe critical ﬁeld, we deﬁne the angular momentum oper-ators projected onto each direction: j100 = j ·u100 = jx,j110 = j · u110 = (jx + jy)/√2, j111 = j · u111 =(jx+jy +jz)/√3, where u100, u110 and u111 are the unit0 100 200 300 400 500 60000.0020.0040.0060.008B [T]E g(B) [Ry](100)(111)(110)Fig. 3. Closing of the energy gap by the magnetic ﬁeld in (100),(110) and (111) directions. Diﬀerence between (110) and (111)curves is almost invisible.vectors in each direction. Making the matrix represen-tation of these operators in the Γ8 space and diagonalizethe 4 × 4 matrices, we obtain the maximal eigenvalue foreach direction as follows: 1.833 in (100), 1.635 in (110)and 1.500 in (111). From this result, one can understandthat the f-states above and below the gap is shifted by theﬁeld faster in this order. (See Fig.4.) There is, however,a diﬀerence between the above explanation and the ex-periment. Namely, the critical ﬁelds in (110) and (111)directions are almost the same, whereas Bc(100) is 10per cent smaller in the experiment. Our numerical cal-culation based on the realistic band model presented inFig.3. successfully reproduces this experimental feature.It may be because the percentage of each Γ8 componentis diﬀerent at each k point, so that the above-mentionedexplanation using the maximal eigenvalues of projectedangular momentum operators holds only qualitatively.EEg(100) (110) (111)BFig. 4. Schematic ﬁgure to explain how the gap is closed when themagnetic ﬁeld is applied in each direction.The other problem is that the anisotropy of the gap-4 Author Nameclosing is not clearly visible in Fig.3 in the low ﬁeld regionwhere the magnetization is isotropic (B < 200 T). Thismay be partly because of insuﬃciency of the number ofthe k-mesh points. We anticipate, however, that theclear anisotropy observed in the high ﬁeld range in Fig.3is a real one and may survive also in the low ﬁeld regionin accord with the experiment.One more problem in our theory is that the calcu-lated magnetization is too small. Value of the magne-tization right before the metal-to-insulator transition isabout 0.3 ∼ 0.4 µB/Yb in the experiment,17) but the cal-culation shows only about 0.1 µB/Yb even at 500 T, andM/µB ∼ 0.01 at H = 50 T. Namely, about 30∼40 timesenhancement is necessary. It is clear that the slope of themagnetization curve will be enhanced by a many-bodyeﬀect,18) but a quantitative calculation using the presentmodel with full anisotropy is not easy. Nevertheless, itwas already clariﬁed that the renormarization factor doesnot strongly depend on the magnetic ﬁeld until the gapis closed by the ﬁeld at least for the simple periodic An-derson model18) as mentioned in the Introduction. Sincethe magnetization must be isotropic in the linear region,the enhancement factor must also be isotropic. Thus,the many-body eﬀect will give only a quantitative eﬀecton the slopes of the magnetization curves and may notaﬀect the anisotropy of the critical ﬁeld.In conclusion, using the tight-binding model composedof Yb 5d and 4f Γ8 orbitals, which reproduces theLDA+U band structure near the energy gap, we have cal-culated the magnetization curves and the collapse of theenergy gap of the Kondo insulator YbB12 when the mag-netic ﬁeld is applied in (100), (110) and (111) directions.Since the initial gap was taken several times larger thanthe experimental one due to a technical reason in thenumerical calculation, the critical values of the ﬁelds atwhich the gap vanishes are too large. We, however, havesuccessfully explained the order of the direction of thecritical ﬁeld: Bc(100) < Bc(100) ≈ Bc(111). Althoughmany-body eﬀect is not included, we have disscussed thatthe results may not change even if the strong correlationis taken into account. Of course, more perfect calcu-lation should be performed including correlation and asmall initial gap, but we believe that the present studycan be an important step to understand the propertiesof the Kondo insulators from the microscopic point ofview, especially, from the model based on the LDA(+U)band calculation plus correlation.AcknowledgmentOne of the authors (T. S.) acknowledges F. Iga forcorrespondense on experiments.1) M.Kasaya, F. Iga, M.Takigawa and T.Kasuya: J.Mag.Magn.Mater. 47 & 48 (1985) 429.2) K. Sugiyama, F. Iga, M. Kasaya, T. Kasuya and M. Date: J.Phys. Soc. Jpn. 57 (1988) 3946; F. Iga, et al.: JJAP Series 11(1999) 88; J. Mag. Magn. Mater. 226-230 (2001) 85; J. Mag.Magn. Mater. 226-230 (2001) 137.3) F. Iga, M. Kasaya and T. Kasuya: J. Mag. Magn. Mater. 76& 77 (1988) 156.4) T. Susaki, A. Sekiyama, K. Kobayashi, T. Mizokawa, A. Fu-jimori, M. Tsunekawa, T. Muro, T. Matsushita, S. Suga, H.Ishii, T. Hanyu, A. Kimura, H. Namatame, M. Taniguchi, T.Miyahara, F. Iga, M.Kasaya, and H.Harima: Phys.Rev. Lett.77 (1996) 4269.5) Y. Takeda, M. Arita, M. Higashiguchi, K. Shimada, H. Na-matame, M. Taniguchi, F. Iga, and T.Takabatake: Phys. Rev.B 73 (2006) 033202.6) H. Okamura, S. Kimura, H. Shinozaki, T. Nanba, F. Iga, N.Shimizu and T. Takabatake : Phys. Rev. B 58 (1998) R7496.7) H. Okamura, T. Michizawa, T. Namba, S. Kimura, F. Iga andT. Takabatake: J. Phys. Soc. Jpn. 74 (2005) 1954.8) P. A. Alekseev: Appl. Phys. A 74 Suppl. (2002) 562.9) A. Bouvet, T. Kasuya, M. Bonnet, L. P. Regnault, J. Rossat-Mignod, F. Iga, B. Fak and A. Severing: J. Phys.: Condes.Matter 10 (1998) 5667.10) J.-M.Mignot, P.A.Alekseev, K.S.Nemkovski, L.-P.Regnault,F. Iga and T. Takabatake: Phys. Rev. Lett. 94 (2005) 247204.11) T. Saso and H. Harima: J. Phys. Soc. Jpn. 72 (2003) 1131.12) T. Saso, J. Phys. Soc. Jpn. 73 (2004) 2894.13) A.Jayaraman, V.Narayanamurti, E.Bucher and R.G.Maines:Phys. Rev. Lett. 25 (1970) 1430.14) P. Haen, F. Lapierre, J. M. Mignot, R. Tournier and F.Holtzberg: Phys. Rev. Lett. 43 (1979) 304.15) M. F. Hundley, P. C. Canﬁeld, J. D. Thompson, Z. Fisk andJ. M. Laurence: Phys. Rev. B 42 (1990) 4862.16) F. Grandjean, A. Gerard, D. J. Braung, W. Jeitschko: J. Phys.Chem. Solids 45 (1984) 877.17) F. Iga, T. L. Bihan, D. Braithwate, J. Flouquet, M. Idiri, K.Kindo, N. Takamoto, I. Oguro, N. Shimizu, S. Hiura and T.Takabatake: JJAP Series 11 (1999) 88.18) T. Saso and M. Itoh: Phys. Rev. B 53 (1996) 6877.19) T. Saso: J. Phys. Soc. Jpn. 66 (1997) 1175.20) J. C. Slater and G. F. Koster, Phys. Rev. 94 (1954) 1498.21) K. Takegahara, Y. Aoki and A. Yanase, J. Phys. C 13 (1980)853.

Theory for magnetic anisotropy of field-induced insulator-to-metal transition in cubic kondo insulator YbB_<12>

http://jpsj.ipap.jp/link?JPSJ/76/084715/ | http://jpsj.ipap.jp/link?JPSJ/76/084715/Magnetization and energy gap of Kondo insulator YbB12 are calculated theoretically based on the previously proposed tight-binding model composed of Yb 5d epsilon and 4f Gamma(8) orbitals. It is found that magnetization curves are almost isotropic, naturally expected from the cubic symmetry, but that the gap-closing field has an anisotropy: the gap closes faster for the field in (100) direction than in (I 10) and (111) directions, in accord with the experiments. This is qualitatively understood by considering the maximal eigenvalues of the total angular momentum operators projected on each direction of the magnetic field. But the numerical calculation based on the band model yields better agreement with the experiment.Copyright(c)2007 The Physical Society of Japantextapplication/pdfjournal articl

Izumi, Toshiyuki

Imai, Yoshiki

佐宗, 哲郎

サソウ, テツロウ

Typeset with jpsj2.cls <ver.1.1.1>Theory for Magnetic Anisotropy of Field-Induced Insulator-to-Metal Transition inCubic Kondo Insulator YbB12Toshiyuki Izumi∗, Yoshiki Imai and Tetsuro SasoDivision of Materials Sciences, Graduate Course of Science and Engineering, Saitama University, Shimo-Ohkubo 255, Sakura-ku,Saitama City, Saitama, 338-8570(Received )Magnetization and energy gap of Kondo insulator YbB12 are calculated theoretically basedon the previously proposed tight-binding model composed of Yb 5dε and 4f Γ8 orbitals. Itis found that magnetization curves are almost isotropic, naturally expected from the cubicsymmetry, but that the gap-closing field has an anisotropy: the gap closes faster for the fieldin (100) direction than in (110) and (111) directions, in accord with the experiments. This isqualitatively understood by considering the maximal eigenvalues of the total angular momen-tum operators projected on each direction of the magnetic field. But the numerical calculationbased on the band model yields better agreement with the experiment.KEYWORDS: Kondo insulator, YbB12, magnetization, field-induced insulator-metal transition, magneticanisotropy1. IntroductionYbB121) is the most typical Kondo insulator, since ithas a cubic crystal structure (Yb and B12 forms NaClstructure), and many experiments: transport,2) calori-metric,3) photoemission,4,5) optical,6,7) neutron scatter-ing,8–10) etc. are already reported. We believe that mostof them can be understood in terms of the band insula-tor model with strong Coulomb repulsion between f elec-trons. In fact, based on the LDA+U band calculation, itwas found that the conduction bands near the Fermi levelare composed of the 5dε orbitals on Yb atoms, and thesimple tight-binding band reproduces the LDA+U cal-culation rather well.11) Some of the conduction bandsare doubly degenerate (four-fold if spin degeneracy is in-cluded). A gap opens by the hybridization of the four-fold degenerate Γ8 states with the above-mentioned con-duction bands with degeneracy. It was pointed out thatthe gap can not open if the crystal-field groud state of4f were Γ7 doublet. Therefore, proper consideration ofthe degeneracy is a key to understanding the opening ofa gap in the Kondo insulators.11)Based on this band model, one of the authors12) calcu-lated frequency- and temperature-dependence of the op-tical conductivity and obtained semi-quantitative agree-ment with the experiment.7) He succeeded in reproduc-ing the mid-infrared peak and the prominent tempera-ture dependence of the low frequency part, which is dueto the many-body effect.In contrast to YbB12, it is not clear whether otherwell-known or classical “Kondo insulators”, SmS13) andTmSe,14) belong to the same category as YbB12, sinceSm and Tm has more complicated 4f states than Yb. Inaddition, TmSe has odd number of electrons per unit cell,so that it can not be a band insulator if 4f electrons aremixed valent. Other Kondo insulators, which may be-long to the same category as YbB12, will be Ce3Bi4Pt315)and Ce-skutterudites, e.g. CeFe4As12.16) Nevertheless,YbB12 can be a prototype of the Kondo insulator from∗ Present address: Japan Radio Co. Ltd.the point of view of the simpleness of the band structure,sufficient number of experiments and the existence of thecompact theoretical model.Despite the above-mentioned success, YbB12 still hasmany features to be understood correctly and quan-titatively. One of them is the magnetic properties.Sugiyama, et al.2) first found that the activation en-ergy of the resistivity of this insulator vanishes at aroud50 Tesla when magnetic field is applied. This is called asfield-induced insulator-to-metal transition and the criti-cal field is denoted as Bc hereafter. Iga, et al.17) foundthat the magnetization curve for each direction doesnot show an anisotropy, but the critical field Bc does.Namely, Bc(110) and Bc(111) is larger than Bc(100) by10 percent (Bc(100) =47T, Bc(110)  Bc(111) =53T).The transitions are first order since hysterisis is observed.Using the simple periodic Anderson model, one of thepresent authors showed that the overall feature of themagnetization curve of YbB12 can be explained and thetransition can be first order if the Coulomb repulsionis taken into account.18,19) In addition, he found thatthe renormalization factor (which renormalizes the gap)due to strong correlation does not largely change by themagnetic field in the insulating phase, so that the renor-malized up- and down-spin bands are shifted by the fieldas if they are rigid bands and the renormalized bandgap is closed also as if in the case of the rigid bands.18)But the anisotropic feature of the f-electrons under cubiccrystalline field was not taken into account.In the present study, we investigate theoretically theorigin and mechanism of the anisotropy of the field-induced insulator-to-metal transition. In §2, we brieflyexplain the previously introduced tight-binding bandmodel. Calculations of magnetization and energy gapunder magnetic field will be presented in §3. The lastsection is devoted to the discussions and conclusions.12 Author Name2. Tight-Binding Energy BandThe energy dispersions of the conduction bands ofYbB12 are well approximated by the tight-binding modelof 5dε orbitals of Yb on the fcc lattice and expressed bythe following simple expressions:11)Eαβk = Edε + 3(ddσ) cos(kα/2) cos(kβ/2), (1)where (ddσ) is a Slater-Koster integral,20) (α, β) denotes(x, y), (y, z) or (z, x). The lowest band along Γ-K-X(110)and Γ-X(100) are doubly degenerate (four-fold if the spindegeneracy is included). Introducing the hybridizationbetween these d bands and the 4f j = 7/2 Γ8 states, weobtained the energy bands with a gap.In eq.(1), we locate the energy level of 5dε orbitalsat Edε=1.0 Ry, and set (ddσ)= 0.06 Ry. This value of(ddσ) is the effective one due to hopping between near-neighbor Yb sites through B12 clusters.The hybridization between these 5d bands and the 4fstates is described by the effective (dfσ) integrals21) be-tween the nearest-neighbor Yb sites. We retain only thenearest neighbor (dfσ) bonds as the simplest model. Welocate the 4f Γ8 states at EΓ8 = 0.88 Ry and choose(dfσ)=0.015 Ry. We also include (dfπ) = −0.0075 Ryand (ffσ) = −0.003 Ry to adjust to the LDA+U cal-culation. Furthermore, the filled bands below the gapare shifted down by ∆E = −0.011 Ry relative to thebands above the gap. This treatment is in accord withthe spirit of the LDA+U treatment. These parametersare the same as those in our previous paper12) to fit theoptical conductivity.7)Using these parameters, we obtain the dispersioncurves which reproduce the LDA+U bands around thegap rather well.11) They have an indirect gap of about0.0069 Ry (= 1089 K) between X and L points and thedirect gap of 0.018 Ry (= 0.245 eV). The former is aboutsix times larger than the best estimated value of the in-direct gap in the optical experiment, Eg =174 K.7) How-ever, in the present realistic band model, we can not usethis value. If we use a band model with the correct valueof the indirect gap, it is difficult to reproduce the gap inthe density of states, since we must use a very fine meshto calculate the three-dimensional k-summation over theBrillouin zone.Note that the band calculation can treat the f-stateonly by the j-j coupling scheme. One hole in j = 7/2Γ8 state corresponds to the Γ8 state with the total angu-lar momentum J = 7/2 of Yb3+ (the f-electron numbernf = 13). In the present case, the number of filled 4fΓ8 electrons is about 3.1 and 0.9 hole exists in the bandsabove the gap, whereas j = 5/2, j = 7/2 Γ6 and Γ7 statesare far below Γ8 and are implicitly treated as completelyfilled. Thus the present model approximately reproducesthe experimental valence of Yb+2.9.3. Calculation of Magnetization and EnergyGap under Magnetic FieldWe apply the magnetic field B = µ0H along (100),(110) and (111) directions. The Hamiltonian reads0.80.911.11.2Energy(Ry)Γ K X W L Γ XFig. 1. The tight-binding band for YbB12 under B=500T in (100)direction. The X points in the left and right denote the equiva-lent (110) and (100) points (in unit of 2π/a), respectively.H =∑α,βi,jtαβij c†iαcjβ +∑igcµBsi · H +∑igJµBji · H,(2)where tαβij denotes the hopping matrix elements betweenthe ten orbitals {α, β = xy ↑, yz ↑, zx ↑, xy ↓, yz ↓, zx ↓,Γ(1)±8 ,Γ(2)±8 } on i and j sites and is expressed by theSlater-Koster parameter given in the last section.12) µBis the Bohr magneton, gc = 2 and gf = 8/7 are the g-factors of conduction and f electrons. si and ji are thespin and the total angular momentum operators for theconduction and f electron at i site, respectively. We haveomitted the effect of magnetic field on the orbital motionexcept that through j. The effect of orbital motion canbe included by attaching the phase factor to the hoppingmatrix elements, but it will be smaller than those con-sidered here through j, since the latter will be enhancedby the strong correlation.In the present model with the larger gap than the ex-periment, the gap closes at the large field of about 500T. We show the band structure at this field in (100) di-rection in Fig.1. Four bands lie below the gap and sixbands above the gap and all the degeneracies under zerofield are lifted.Diagonalizing the Hamiltonian, the Bloch function isobtained asψkn(r) =1N∑i,αukn,αeik·Riφα(r − Ri), (3)where φα(r − Ri) is the local orbital α at the site Ri,n denotes the diagonalized band index, ukn,α the coef-ficient and N the total number of unit cell. Using this,the magnetization M is calculated byM =1N∑k,nf(Ekn) [gcµB〈ψkn|S|ψkn〉+ gfµB〈ψkn|J |ψkn〉] , (4)Title of the Article 3where S =∑i si, J =∑i ji, f(E) is the Fermi functionand Ekn the n-th band energy. Note that the applica-tion of the magnetic field lowers the symmetry, so thatthe sum over 1/48 of the first Brillouine zone is not al-lowed. In the actual calculation, we summed over wholeof the first and the second Brillouine zones because theprogramming becomes much easier: we can simply makesummation over a cube with twice the volume of the firstBrillouine zone of fcc lattice. We divide this cube intoat most 4003 mesh points. Temperature is set equal tozero throughout the paper.Also for simplicity, we have summed only over the low-est four filled bands in the calculation of the magnetiza-tion, so that the calculation makes sense only for B < Bc(critical field) in each field direction. For B > Bc, onehas to determine the chemical potential for each valueand direction of the field and sum up over more thanfour bands. We did not do it since we are interested onlyin the anisotropy problem in the present paper.M/µB = |M |/µB is plotted in Fig.2 for the three di-rections of the field. In accord with the general theoremof the group theory that the linear response tensor mustbe proportional to the unit tensor (isotropic), the presentresult shows that the magnetization is isotropic up to 200T. (Indirect) energy gap vs. magnetic field is plotted inFig.3. Since we take large value of the initial gap, thecritical value of the field in each direction is also large:Bc(100) = 464T, Bc(110) = 501T, Bc(111) = 506T.These values will become smaller if we use the correctvalue of the band gap. For B < 300T, Eg(111) becomesslightly smaller than Eg(110), but the difference is verysmall and it seems out of range of the precision of thepresent calculation. It seems reasonable to consider thatEg(100) < Eg(110) < Eg(111) for all the field, as will bediscussed in the next section.0 100 200 300 400 50000.050.1B [T]M/μB (111) (110) (100)Fig. 2. Magnetization curves under the magnetic field in (100),(110) and (111) directions.4. Discussions and ConclusionsIn order to understand an origin of the anisotropy ofthe critical field, we define the angular momentum oper-ators projected onto each direction: j100 = j ·u100 = jx,j110 = j · u110 = (jx + jy)/√2, j111 = j · u111 =(jx +jy +jz)/√3, where u100, u110 and u111 are the unit0 100 200 300 400 500 60000.0020.0040.0060.008B [T]Eg(B) [Ry](100)(111)(110)Fig. 3. Closing of the energy gap by the magnetic field in (100),(110) and (111) directions. Difference between (110) and (111)curves is almost invisible.vectors in each direction. Making the matrix represen-tation of these operators in the Γ8 space and diagonalizethe 4 × 4 matrices, we obtain the maximal eigenvalue foreach direction as follows: 1.833 in (100), 1.635 in (110)and 1.500 in (111). From this result, one can understandthat the f-states above and below the gap is shifted by thefield faster in this order. (See Fig.4.) There is, however,a difference between the above explanation and the ex-periment. Namely, the critical fields in (110) and (111)directions are almost the same, whereas Bc(100) is 10per cent smaller in the experiment. Our numerical cal-culation based on the realistic band model presented inFig.3. successfully reproduces this experimental feature.It may be because the percentage of each Γ8 componentis different at each k point, so that the above-mentionedexplanation using the maximal eigenvalues of projectedangular momentum operators holds only qualitatively.EEg(100) (110) (111)BFig. 4. Schematic figure to explain how the gap is closed when themagnetic field is applied in each direction.The other problem is that the anisotropy of the gap-4 Author Nameclosing is not clearly visible in Fig.3 in the low field regionwhere the magnetization is isotropic (B < 200 T). Thismay be partly because of insufficiency of the number ofthe k-mesh points. We anticipate, however, that theclear anisotropy observed in the high field range in Fig.3is a real one and may survive also in the low field regionin accord with the experiment.One more problem in our theory is that the calcu-lated magnetization is too small. Value of the magne-tization right before the metal-to-insulator transition isabout 0.3 ∼ 0.4 µB/Yb in the experiment,17) but the cal-culation shows only about 0.1 µB/Yb even at 500 T, andM/µB ∼ 0.01 at H = 50 T. Namely, about 30∼40 timesenhancement is necessary. It is clear that the slope of themagnetization curve will be enhanced by a many-bodyeffect,18) but a quantitative calculation using the presentmodel with full anisotropy is not easy. Nevertheless, itwas already clarified that the renormarization factor doesnot strongly depend on the magnetic field until the gapis closed by the field at least for the simple periodic An-derson model18) as mentioned in the Introduction. Sincethe magnetization must be isotropic in the linear region,the enhancement factor must also be isotropic. Thus,the many-body effect will give only a quantitative effecton the slopes of the magnetization curves and may notaffect the anisotropy of the critical field.In conclusion, using the tight-binding model composedof Yb 5dε and 4f Γ8 orbitals, which reproduces theLDA+U band structure near the energy gap, we have cal-culated the magnetization curves and the collapse of theenergy gap of the Kondo insulator YbB12 when the mag-netic field is applied in (100), (110) and (111) directions.Since the initial gap was taken several times larger thanthe experimental one due to a technical reason in thenumerical calculation, the critical values of the fields atwhich the gap vanishes are too large. We, however, havesuccessfully explained the order of the direction of thecritical field: Bc(100) < Bc(100) ≈ Bc(111). Althoughmany-body effect is not included, we have disscussed thatthe results may not change even if the strong correlationis taken into account. Of course, more perfect calcu-lation should be performed including correlation and asmall initial gap, but we believe that the present studycan be an important step to understand the propertiesof the Kondo insulators from the microscopic point ofview, especially, from the model based on the LDA(+U)band calculation plus correlation.AcknowledgmentOne of the authors (T. S.) acknowledges F. Iga forcorrespondense on experiments.1) M.Kasaya, F. Iga, M.Takigawa and T.Kasuya: J.Mag.Magn.Mater. 47 & 48 (1985) 429.2) K. Sugiyama, F. Iga, M. Kasaya, T. Kasuya and M. Date: J.Phys. Soc. Jpn. 57 (1988) 3946; F. Iga, et al.: JJAP Series 11(1999) 88; J. Mag. Magn. Mater. 226-230 (2001) 85; J. Mag.Magn. Mater. 226-230 (2001) 137.3) F. Iga, M. Kasaya and T. Kasuya: J. Mag. Magn. Mater. 76& 77 (1988) 156.4) T. Susaki, A. Sekiyama, K. Kobayashi, T. Mizokawa, A. Fu-jimori, M. Tsunekawa, T. Muro, T. Matsushita, S. Suga, H.Ishii, T. Hanyu, A. Kimura, H. Namatame, M. Taniguchi, T.Miyahara, F. Iga, M. Kasaya, and H.Harima: Phys.Rev. Lett.77 (1996) 4269.5) Y. Takeda, M. Arita, M. Higashiguchi, K. Shimada, H. Na-matame, M. Taniguchi, F. Iga, and T.Takabatake: Phys. Rev.B 73 (2006) 033202.6) H. Okamura, S. Kimura, H. Shinozaki, T. Nanba, F. Iga, N.Shimizu and T. Takabatake : Phys. Rev. B 58 (1998) R7496.7) H. Okamura, T. Michizawa, T. Namba, S. Kimura, F. Iga andT. Takabatake: J. Phys. Soc. Jpn. 74 (2005) 1954.8) P. A. Alekseev: Appl. Phys. A 74 Suppl. (2002) 562.9) A. Bouvet, T. Kasuya, M. Bonnet, L. P. Regnault, J. Rossat-Mignod, F. Iga, B. Fak and A. Severing: J. Phys.: Condes.Matter 10 (1998) 5667.10) J.-M.Mignot, P.A.Alekseev, K.S.Nemkovski, L.-P.Regnault,F. Iga and T. Takabatake: Phys. Rev. Lett. 94 (2005) 247204.11) T. Saso and H. Harima: J. Phys. Soc. Jpn. 72 (2003) 1131.12) T. Saso, J. Phys. Soc. Jpn. 73 (2004) 2894.13) A.Jayaraman, V.Narayanamurti, E.Bucher and R.G.Maines:Phys. Rev. Lett. 25 (1970) 1430.14) P. Haen, F. Lapierre, J. M. Mignot, R. Tournier and F.Holtzberg: Phys. Rev. Lett. 43 (1979) 304.15) M. F. Hundley, P. C. Canfield, J. D. Thompson, Z. Fisk andJ. M. Laurence: Phys. Rev. B 42 (1990) 4862.16) F. Grandjean, A. Gerard, D. J. Braung, W. Jeitschko: J. Phys.Chem. Solids 45 (1984) 877.17) F. Iga, T. L. Bihan, D. Braithwate, J. Flouquet, M. Idiri, K.Kindo, N. Takamoto, I. Oguro, N. Shimizu, S. Hiura and T.Takabatake: JJAP Series 11 (1999) 88.18) T. Saso and M. Itoh: Phys. Rev. B 53 (1996) 6877.19) T. Saso: J. Phys. Soc. Jpn. 66 (1997) 1175.20) J. C. Slater and G. F. Koster, Phys. Rev. 94 (1954) 1498.21) K. Takegahara, Y. Aoki and A. Yanase, J. Phys. C 13 (1980)853.

Theory for Magnetic Anisotropy of Field-Induced Insulator-to-Metal Transition in Cubic Kondo Insulator YbB 12

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Theory for Magnetic Anisotropy of Field-Induced Insulator-to-Metal
  Transition in Cubic Kondo Insulator YbB_{12}

Theory for Magnetic Anisotropy of Field-Induced Insulator-to-Metal Transition in Cubic Kondo Insulator YbB_{12}

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