Field-induced valence fluctuation in YbB_{12}
R. Kurihara, A. Miyake, M. Tokunaga, A. Ikeda, Y. H. Matsuda, A. Miyata, D. I. Gorbunov, T. Nomura, S. Zherlitsyn, J. Wosnitza, F. Iga
FField-induced valence fluctuations in YbB R. Kurihara , A. Miyake , M. Tokunaga , A. Ikeda , Y. H. Matsuda , A. Miyata ,D. I. Gorbunov , T. Nomura , , S. Zherlitsyn , J. Wosnitza , and F. Iga Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan Hochfeld-Magnetlabor Dresden (HLD-EMFL) and W¨urzburg-Dresden Cluster of Excellence ct.qmat,Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany College of Science, Ibaraki University, Mito 310-8512, Japan
We performed high-magnetic-field ultrasonic experiments on YbB up to 59 T to investigate thevalence fluctuations in Yb ions. In zero field, the longitudinal elastic constant C , the transverseelastic constants C and ( C − C ) /
2, and the bulk modulus C B show a hardening with a changeof curvature at around 35 K indicating a small contribution of valence fluctuations to the elasticconstants. When high magnetic fields are applied at low temperatures, C B exhibits a softening abovea field-induced insulator-metal transition signaling field-induced valence fluctuations. Furthermore,at elevated temperatures, the field-induced softening of C B takes place at even lower fields and C B decreases continuously with field. Our analysis using the multipole susceptibility based on a two-band model reveals that the softening of C B originates from the enhancement of multipole-straininteraction in addition to the decrease of the insulator energy gap. This analysis indicates thatfield-induced valence fluctuations of Yb cause the instability of the bulk modulus C B . I. INTRODUCTION
Since the electronic and magnetic properties of mate-rials are mainly determined by valence electrons, a pre-cise knowledge about the valence state is important inmaterial science. Especially for 4 f -electron systems, thevalence determines the total angular momentum J , thelocalized (or delocalized) 4 f -electron character, and cor-responding wave functions. A non-integer valence stateappears in some rare-earth compounds with Ce, Sm, Eu,and Yb ions. In such materials, valence fluctuationsdue to hybridization between conduction electrons and4 f electrons play a key role in their physical properties.YbB is one of the valence fluctuating materials withsuch a c - f hybridization, a high-Kondo temperature, andinsulating character [1, 2].YbB has the UB -type crystal structure belongingto the F m m ( O h ) space group [1]. The Γ ground stateof the 4 f electrons based on Yb configuration in thecrystal electric field (CEF) has been proposed [3, 4]. Thealmost degenerated Γ and the Γ states at 270 K (23meV) were considered as excited states [3, 4]. TheseCEF states based on the J = 7 / ionsdetermined by NMR measurements [5]. In contrast, anonmagnetic ground state has been suggested from thetemperature-independent magnetic susceptibility at lowtemperatures [6, 7]. Indication for a strongly hybridizedelectronic state was found using bulk-sensitive x-ray pho-toelectron spectroscopy showing a slight deviation fromthe valence Yb [8]. The hybridization between 5 d con-duction electrons and 4 f localized electrons has been pro-posed as a candidate mechanism for an observed band-gap opening [9, 10]. The contribution of the B-2 p elec-trons to the c - f hybridization is also discussed as a resultof ddσ hopping through B clusters.In addition to the CEF scheme, several characteris-tic energies related to the insulating character have been studied in YbB . Both in a polycrystal and single crys-tal, resistivity measurements show evidence for two acti-vation energies of ∼
30 and 65 K [6, 7]. A density of stateswith two-double peaks was proposed as a mechanism oftwo activation energies [11]. NMR and specific-heat datahave been described by a simple two-band model, eachband having a bandwidth of 55 K, and with an energy gapof 140 K at the Fermi energy [6, 12]. High-resolution pho-toemission spectroscopy suggested a hybridization gap of170 K (15 meV) below 150 K and strongly hybridizedcharacter below 60 K [13].In YbB , various high-magnetic-field studies were per-formed to elucidate the mechanism of the formation ofthe energy gap. High-field magnetoresistance measure-ments indicated that the energy gap of 30 K closes around45 T while the other gap remains up to higher fields [11].Magnetization measurements revealed metamagnetic be-havior at insulator-metal (IM) transitions at B IM = 47 Tfor B (cid:107) [001] and 54 T for B (cid:107) [110] and B (cid:107) [111] [14]. An-other magnetization anomaly indicating the saturationof magnetization appears at 102 T [15]. The energy shiftof the 4 f band due to the Zeeman effect was proposed toexplain the closing of the band gap of 170 K. Synchrotronx-ray absorption spectra showed the field independenceof the L edge indicating no considerable change of theYb valence in the field-induced metal phase [16]. Specific-heat measurements revealed a discontinuous enhanced ofSommerfeld coefficient, γ ∼
60 mJ/mol · K , and a corre-sponding Kondo temperature of 220 - 250 K above theIM transition, suggesting that the high-field phase is avalence-fluctuating Kondo metal [17]. These high-fieldexperiments indicate a contribution of the c - f hybridiza-tion to the opening of the energy gap in YbB . Mag-netic quantum oscillations in the insulating phase havealso been focused to understand the insulating characterof YbB [18].To further investigate the valence fluctuations causedby the c - f hybridization in YbB , we focused on ul- a r X i v : . [ c ond - m a t . s t r- e l ] F e b volume strain ε B electric hexadecapole H (a) Yb B (b)
FIG. 1. Schematic view of the volume strain and the hex-adecapole in YbB . (a) Crystal lattice around the Yb ion[4] and volume strain ε B with the irrep Γ of O h . Orangearrows indicate the isotropic deformation of the lattice. (b)Hexadecapole H with Γ symmetry obtained as a result ofan isotropic change of the Yb ionic radius. trasonic measurement. Since a valence change causesan isotropic change of the ionic radii, an isotropic vol-ume change of the crystal lattice is induced and the CEFHamiltonian, H CEF , of Eq. (B1) (see Appendix B) ischanged to H CEF + ( ∂H CEF /∂ε B ) ε B . Here, ε B is the vol-ume strain with the irreducible representation (irrep) Γ of the O h symmetry. This additional term to the CEF isdescribed as a coupling between ε B and a hexadecapole H with Γ in YbB . The schematic view of ε B and H in YbB are shown in Figs. 1(a) and 1(b), respectively.Based on simple Landau theory for elasticity, the totalfree energy consists of a lattice and an electronic part isgiven by [19] F = 12 C ε + 12 αH − g B H ε B . (1)Here, g B is the coupling constant between the strain and H , C is the bulk modulus without multipole contri-bution, and α is a coefficient. The 1st and 2nd termson the right-hand side of Eq. (1) correspond to the en-ergy loss due to the deformation of the lattice and theincrease of the hexadecapole moment, respectively. The3rd term corresponds to the energy gain of the electronicstate due to the hexadecapole-volume strain interaction.The response of the hexadecapole appears as a result ofthe decrease in the bulk modulus as C − g /α .As shown in previous reports [20–22], ultrasonic mea-surements are a powerful tool to detect valence fluctua-tions. In particular, in the Kondo insulator SmB , thedecrease in the bulk modulus C B with decreasing tem-peratures, namely the elastic softening of C B , has beenrevealed as a result of valence fluctuations between Sm and Sm [23]. The relation between the energy gap of c - f hybridized bands and the elastic softening is also dis-cussed in terms of the interaction between 4 f electronsand the bulk strain ε B with full symmetry Γ . Severaltheoretical studies have proposed such a contribution ofthe c - f hybridization to the elasticity [24–27]. Therefore, we measured relevant elastic constants in zero and highfields searching for an elastic softening related to the va-lence fluctuations in YbB .This paper is organized as follows. In Sec. II, experi-mental details of sample preparation and ultrasonic mea-surements in pulsed magnetic fields are explained. In Sec.III, we present the results of our ultrasonic experiments ofYbB . In zero field, an increase in the elastic constantswith decreasing temperatures, namely elastic hardening,accompanying curvature changes reveals some contribu-tion of valence fluctuations to the elasticity. In contrastto zero field, a field-induced softening of the bulk mod-ulus C B appears, which indicates field-induced valencefluctuations due to c - f hybridization. In Sec. IV, we an-alyzed the measured elastic constants using a multipole-susceptibility model. The field-induced valence fluctu-ations can be described in terms of the hexadecapole-volume strain coupling. Our analysis also confirms thedecrease of the energy gap in high fields. We summarizeour results in Sec. V. II. EXPERIMENT
Single crystals of YbB were grown using the floating-zone method [7]. Laue x-ray backscattering was usedto align, cut, and polish samples with (110), (1¯10),(¯110), (¯1¯10), (001), and (00¯1) faces and the size of 1 . × .
030 mm × .
763 mm. An ultrasound pulse-echomethod with a numerical vector-type phase-detectiontechnique was used to measure the ultrasound velocity v [28]. The elastic constant C = ρv was determinedfrom v and the calculated mass density ρ = 4 .
828 g/cm using the lattice constant a = 7 . plates with a 36 ◦ Y-cut and41 ◦ X-cut (YAMAJU CERAMICS CO.) were employedto generate longitudinal ultrasonic waves with the fun-damental frequency of approximately f = 30 MHz andtransverse waves with 18 MHz, respectively. As indi-cated in Fig. 2, higher-harmonic frequencies were usedto obtain high-resolution data. A room temperature vul-canizing rubber (Shin-Etsu Silicone KE-42T) was used toglue the LiNbO on the sample. The direction of ultra-sonic propagation, q , and the direction of polarization, ξ , for the elastic constant C ij are indicated in Fig. 2.Two nondestructive pulsed magnets were used: one witha pulse duration of 36 ms installed at the Institute forSolid State Physics, the University of Tokyo using a Hecryostat, and another magnet with a pulse duration of150 ms at the HLD-EMFL in Dresden using a He cryo-stat.
III. RESULTSA. Temperature dependence of elastic constants
To gain more information on the Yb valence in YbB ,we investigated the three elastic constants C , C , and C T = ( C − C ) /
2. Their relations to the symmetrystrain and electric multipole are summarized in Table I[19, 29]. Figure 2 shows the temperature dependence ofthe elastic constants in zero field. We observed the elastichardening of C , C , and C T with lowering tempera-tures. We also observed the elastic hardening of C from300 K (see Appendix A). All elastic constants exhibitan additional hardening and a characteristic curvaturechange in the vicinity of T (cid:63) = 35 K. As shown by thesolid curves in Fig. 2, the elastic constants would exhibita monotonic increase with decreasing temperature [31]if we do not consider multipole contributions, describedin the following Sec. IV [19]. Therefore, the additionalfeatures in the elastic constants of YbB indicate themultipole contribution to elasticity.To describe the origin of the anomaly in each elasticconstant of YbB , we focus on the multipole effect ofthe CEF wave functions of localized 4 f electrons takeninto account the presence of Γ , Γ , and Γ states [3, 4].Since the direct product of the Γ quartet is reducedas Γ ⊗ Γ = Γ ⊕ Γ ⊕ Γ ⊕ ⊕ [29, 30], wededuce that the Γ ground-state wave functions carrythe electric quadrupoles O u and O v with irrep Γ and O yz , O zx , and O xy with irrep Γ as summarized in TableI. In addition, the Γ quartet also provides the electrichexadecapole H = O + 5 O with irrep Γ . Becausethe magnetic multipole degrees of freedom do not couplewith the strain, we ignore magnetic dipoles with irrep Γ and magnetic octupoles with irreps Γ , Γ , and Γ . Thisgroup-theoretical consideration indicates that the elasticsoftening of ( C − C ) / and C withirrep Γ is due to a multipole-strain interaction describedas H MS = − g Γ γ O Γ γ ε Γ γ . (2)Here, g Γ γ is a coupling constant and Γ γ denotes the ir-rep. We show how to calculate the multipole suscepti-bility based on the CEF wave functions in Appendix B.Because the calculated multipole susceptibility for Γ -and Γ -type quadrupoles shows a divergent increase fordecreasing temperatures, a divergent elastic softening istheoretically expected in C and C T . However, our ex-perimental results show no softening in all measured elas-tic constants. Therefore, the CEF approach based on alocalized 4 f character does not apply to the elasticity ofYbB in zero field.The other possible scenario describing the additionalcontribution around T (cid:63) is a result of the charge freez-ing of Yb without long-range ordering as previously dis-cussed in the samarium compounds Sm Se and Sm Te [20, 21]. Since the charge freezing would be character-ized by a frequency-dependent ultrasound response, we ij ε ii y (z, x)z (x, y) ε yz (zx, xy) xy ε x -y x y ε B z FIG. 2. Temperature dependence of (a) the longitudinalelastic constant C , the transverse elastic constants (b) C and (c) C T = ( C − C ) /
2, and (d) the bulk modulus C B calculated from C and C T . The dotted lines indicate the fitof C , C T , and C B in the framework of the phenomenologicaltwo-band model discussed in Sec. IV. The solid lines indicatethe temperature dependence of the elastic constants withoutmultipole contribution. The strains ε ij for C , C , C T , and C B are schematically shown in the inset in each panel. Thevertical arrows in each panel indicate the characteristic tem-perature T (cid:63) discussed in Sec. IV. measured the elastic constants and ultrasonic attenuationcoefficients for several frequencies. However, we did notobserve any frequency dependence neither in the elasticconstants nor in the ultrasonic attenuation coefficientsbetween 30 and 160 MHz.Therefore, we focus on the contribution of valence fluc- TABLE I. Symmetry strains, electric multipoles, and elastic constants corresponding to the irreducible representations (irreps)of the space group O h .Irrep Symmetry strain Electric multipole Elastic constantΓ ε B = ε xx + ε yy + ε zz O + 5 O (= H ) C B = ( C + 2 C ) / ε u = (2 ε zz − ε xx − ε yy ) / √ O u (=3 z − r ) C T = ( C − C ) / ε v = ε xx − ε yy O v (= x − y ) Γ ε yz O yz C ε zx O zx ε xy O xy FIG. 3. Magnetic-field dependence of the relative variation of the elastic constants ∆ C ij /C ij = [ C ij ( B ) − C ij ( B = 0)] /C ij ( B =0) at several temperatures for B (cid:107) [001]. Field dependence of (a) the longitudinal elastic constant C , the transverse elasticconstants (b) C and (c) C T , and (d) the bulk modulus C B . The data sets are shifted consecutively along the ∆ C/C axesfor clarity. The vertical arrows indicate the insulator-metal transition field B IM . The horizontal arrows show the field-sweepdirections. tuations to the elasticity caused by the c - f hybridization.Figure 2(d) shows the temperature dependence of thebulk modulus C B = ( C + 2 C ) / C − C T / calculated from the experimental results of C and C T . C B exhibits as well a hardening with an ad-ditional contribution in the vicinity of 35 K. This resultfor YbB is in contrast to the significant softening of C B due to Sm valence fluctuations observed in SmB . In Sec.IV, we will discuss the origin of the additional contribu-tion in terms of the multipole susceptibility based on atwo-band model to confirm the contribution of valencefluctuations to the elastic constants in zero field. B. Magnetic-field dependence of elastic constants
To investigate the valence properties of YbB in mag-netic fields, we measured the elastic constants C , C ,and C T up to 59 T for B (cid:107) [001]. Figure 3 shows themagnetic-field dependence of the relative variation of theelastic constants ∆ C ij /C ij at several temperatures. Weobserved a field-induced IM transition and elastic soften-ing for each elastic constant in the Kondo-metal phase.Below 10 K, this softening appears rather abruptly abovethe insulator-metal transition field B IM . B IM are com-parable with results of a previous magnetocaloric-effect FIG. 4. Temperature-field phase diagram of YbB for B (cid:107) [001]. The filled red circles indicate the insulator-metaltransition field B IM . The filled red diamond indicates thecharacteristic temperature T (cid:63) (see text for details). The colorcode shows the value of the bulk modulus C B . study [17]. Since C B contains C and C T , our exper-imental results show as well the softening of C B in theKondo-metal phase.Above 10 K, no sharp anomaly corresponding to theKondo-metal phase transition is visible any more. How-ever, C still shows a significant softening in magneticfield contrary to the other elastic constants (Fig. 3).In particular, at 40 K, C exhibits a large softening of0 .
30% at 59 T while C T shows a softening of only 0 . C at 100 K is also in contrast to thehardening observed for C T .Between 10 and 30 K, a clear hysteresis appears in thepulsed-field data of C and C T (Fig. 3). This is approx-imately the temperature range where the additional con-tribution to the elastic constants is detected (Fig 2). Asshown in a previous magnetocaloric-effect study in adi-abatic condition below 7 K [17], the temperature of thesample is reduced by the application of a magnetic field.Because of the quasi-adiabatic experimental conditions,the final temperature after the field pulse fields mightbe higher than assumed which may cause the hysteresis.Therefore, the hysteresis of elastic constants C and C T can also be attributed to the magnetocaloric effect.We also looked for the quantum oscillation in YbB [18]. In principle, such quantum oscillations may appearas well in bulk sensitive ultrasound properties. However,we were not able to resolve any acoustic de Haas-vanAlphen effect at least 0.6 K. This result may imply aweak electron-phonon interaction for the studied acousticmodes.We calculated C B = C − C T / C and C T (Fig. 3). In- FIG. 5. Temperature dependence of the bulk modulus C B ofYbB at various magnetic fields for B (cid:107) [001] extracted fromthe up-sweep data shown in Fig. 3, except for the zero-fielddata. The dotted line indicates the fit describing C B in theframework of the phenomenological two-band model. Theinset shows C B below 10 K. deed, C B shows a very similar behavior as the individualelastic constants with a clear anomaly at B IM below 10K and hysteresis between 10 and 30 K. B IM determinedby our ultrasonic measurements are shown in Fig. 4. C B exhibits a small softening below 50 T at 20 K and45 T at 30 K. By contrast, above 40 K, C B shows mono-tonic softening with increasing fields. In particular, thelargest softening of 0.52% is observed in C B at 40 K. Thefield-induced elastic softening of C B is summarized in thecontour plot in Fig. 4.For further understanding of the field-induced elasticsoftening in YbB , we plotted the temperature depen-dence of C B for various magnetic fields (Fig. 5). Asshown in the inset of Fig. 5, C B exhibits a softening ofabout 0 .
05% below 7 K down to ∼ C B shows significant softeningfrom 100 K down to 40 K in high fields, which is in con-trast to the hardening of C B in zero field. The softeningof C B is similar to that found for SmB caused by c - f hybridization-driven valence fluctuations correspondingto hexadecapole-strain interaction [23].The experimental results of the temperature depen-dence and the magnetic field dependence of C B of YbB cannot be described by the localized 4 f -electron model(see Appendix C). In the following Sec. IV, therefore, wediscuss our observations in terms of multipole-strain in-teraction and the multipole susceptibility for a two-bandmodel. D ( E ) E ∆ = 70 K ∆ W = 55 K WE F D : independent of ED D ( E ) E ∆ = 37 K ∆ W = 55 K W DDB = 0 T B = 58 T(a) (b)upperbandlowerband E u E l FIG. 6. Schematic view of the two-band model assuminga constant DOS over each band (red rectangular). (a) DOSat zero field with an energy gap 2∆ = 140 K and bandwidth W = 55 K. (b) DOS at 58 T. The energy gap 2∆ = 74 Kand the bandwidth W = 55 K is determined using Eq. (9).The blue dotted curves indicate the schematic view of a morerealistic DOS of YbB . IV. DISCUSSION
We discuss the origin of the elastic anomalies of YbB in terms of a two-band model assuming a constant den-sity of states (DOS) with respect to energy. This modelhas successfully reproduced the elastic softening observedin the Kondo compounds SmB and CeNiSn [23, 34].In YbB , this phenomenological model also gives qual-itative explanation for the temperature dependence of C , C T , and C B in zero field and for C B in high fields.Field-induced valence fluctuations are included by thehexadecapole-strain interaction. By that, the essentialparameters for the explanation of our experimental re-sults are identified (Table II).We introduce a two-band model, which is schematicallyshown in Fig. 6. In this model, we deal with the two c - f hybridized bands: an upper band above the Fermi energy E F with an energy E u0 , k and a lower band below E F with E l0 , k . The DOS with energy dispersion of each band issimplified to the rectangular form. The bandwidth W ,the DOS D , and the band gap 2∆ are set as shown inFig. 6. We assume that the multipole-strain interactionfor the electrons in the two bands can be written as [33] H MS = − (cid:88) k (cid:32) c † k , u c † k , l (cid:33) T (cid:18) d u k , Γ γ h k , Γ γ h ∗ k , Γ γ d l k , Γ γ (cid:19) (cid:18) c k , u c k , l (cid:19) ε Γ γ . (3)For the multipole-strain interaction H MS of Eq. (2), thediagonal term d l(u) k , Γ γ for the electrons in band u(l) indi-cates a renormalized multipole-strain coupling constantdescribed as g Γ γ (cid:104) u(l) | O Γ γ | u(l) (cid:105) . The off-diagonal term h k , Γ γ is written as g Γ γ (cid:104) u(l) | O Γ γ | l(u) (cid:105) . c k , u(l) and c † k , u(l) are annihilation and creation operators of an electron inthe band u(l) with wave vector k , respectively. Consider-ing the Anderson Hamiltonian describing c - f hybridiza-tion, we deduce that the multipole-strain interaction ofEq. (3) originates from electron-phonon interaction con-sisting of c - f and f - f terms [27]. The multipole-straininteraction of Eq. (3) for the two-band model provides asecond-order perturbation for the upper band, the lowerone, and the band gap. The perturbation energies of eachband and the perturbation energy gap are described as[23, 34] E u k (cid:0) ε Γ γ (cid:1) = E u0 , k − d u k , Γ γ ε Γ γ + (cid:12)(cid:12) h k , Γ γ (cid:12)(cid:12) k ε γ , (4) E l k (cid:0) ε Γ γ (cid:1) = E l0 , k − d l k , Γ γ ε Γ γ − (cid:12)(cid:12) h k , Γ γ (cid:12)(cid:12) k ε γ , (5)∆ k (cid:0) ε Γ γ (cid:1) = ∆ k − (cid:16) d u k , Γ γ − d l k , Γ γ (cid:17) ε Γ γ + (cid:12)(cid:12) h k , Γ γ (cid:12)(cid:12) k ε γ . (6)Here, 2∆ k = E u0 , k − E l0 , k is the energy gap between theupper band and the lower one. The total free energy F is written as [35] F = 12 C γ ε γ + nE F (cid:0) ε Γ γ (cid:1) − k B T (cid:88) s (=u , l) , k ln (cid:40) (cid:34) − E s k (cid:0) ε Γ γ (cid:1) − E F (cid:0) ε Γ γ (cid:1) k B T (cid:35)(cid:41) . (7)Here, C γ is the elastic constant due to the phonon partwith the irrep Γ γ , n is the total number of conductionelectrons, E F ( ε Γ γ ) is the Fermi energy in the deformedsystem, and k B is the Boltzmann constant. The firstterm on the right-hand side of Eq. (7) corresponds to thelattice part. The second and third terms correspond tothe free energy of the conduction electrons. The secondderivative of the total free energy with respect to thestrain ε Γ provides the elastic constant C Γ γ ( T ) describedas C Γ γ ( T ) = C γ + (cid:88) s, k ∂ E s k ∂ε γ f s k − k B T (cid:88) s, k (cid:18) ∂E s k ∂ε Γ γ (cid:19) f s k (1 − f s k )+ 1 k B T (cid:80) s, k (cid:104) ∂E s k ∂ε Γ γ f s k (1 − f s k ) (cid:105) (cid:80) s, k f s k (1 − f s k ) . (8)Here, f s k = { E s , k − E F ) /k B T ] } − is theFermi distribution function. ∂E s k (cid:0) ε Γ γ (cid:1) /∂ε Γ γ | ε Γ γ → and ∂ E s k (cid:0) ε Γ γ (cid:1) /∂ε γ | ε Γ γ → are written as ∂E s k /∂ε Γ γ and ∂ E s k /∂ε γ , respectively. The conservation law forthe total electron number with respect to the strain, ∂n/∂ε Γ γ = (cid:80) k ∂f k /∂ε Γ γ = 0, is employed to calculateEq. (8). The second term on the right-hand side of Eq.(8) corresponds to van Vleck term, which originates fromthe off-diagonal element h k , Γ γ in the multipole-strain in-teraction of Eq. (3). The third and fourth terms are theCurie terms ( ∼ /T ) related to the diagonal elements d l k , Γ γ and d u k , Γ γ . In this two-band model, the matrix ele-ments of a multipole and the band gap are independenton the wave vector k . The temperature dependence ofthe elastic constant is obtained by replacing the sum overthe wave vector (cid:80) k by the energy integral using the DOSof the two-band model shown in Fig. 6 as [36], C Γ γ ( T ) = C γ − D (cid:16) d uΓ γ − d lΓ γ (cid:17) (cid:20) tanh (cid:18) ∆ + W k B T (cid:19) − tanh (cid:18) ∆2 k B T (cid:19)(cid:21) + D (cid:12)(cid:12) h Γ γ (cid:12)(cid:12) k B T ∆ ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cosh (cid:16) ∆2 k B T (cid:17) cosh (cid:16) ∆+ W k B T (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (9)Here, we adopt the background elastic constant C γ = A − B/ ( e C/T −
1) [31]. C γ , D ( d uΓ γ − d lΓ γ ) , and D | h Γ γ | in Eq. (9) are treated as fit parameters. The second andthird terms in Eq. (9) correspond to Curie and van Vleckterm, respectively.The analysis by the multipole susceptibility of Eq. (9)reveals the contribution of valence fluctuations to theelastic constant in zero field. Fits to the temperature de-pendence of the elastic constants C , C T , and C B in zerofield are shown in Fig. 2. The fit parameters are summa-rized in Table II. Here, we adopt ∆ = 70 K and W = 55K at 0 T as determined by the analysis of specific-heatdata of YbB based on the rectangular two-band model[7]. The temperature dependence of the elastic constants C , C T , and C B can be well described by our model.The energy gap ∆, the bandwidth W , and the coeffi-cient of the Curie term, D ( d uΓ γ − d lΓ γ ) , are necessary toreproduce the additional contribution in the vicinity of T (cid:63) = 35 K. In contrast, the van Vleck contribution is notneeded to explain the experimental results. Our resultsindicate the importance of the multipole-strain interac-tion [Eq. (3)] to the elastic constants. In particular, thebroad increase of C B below ∼
40 K seems to be the resultof the isotropic change of the ionic radii caused by valencefluctuations due to the c - f hybridization. We also triedto fit C B to adopt ∆ = 30 K as determined by the high-field magnetoresistance [11]. However, we are not able toreproduce the curvature change in C B around 35 K (seeAppendix D).The multipole susceptibility also provides the renor-malized multipole-strain coupling constant and the in-teraction anisotropy. For the volume strain ε B , thefirst-order coefficient of the energy gap is described as d ∆( ε B ) /dε B | ε B → = ( d uB − d lB ) / ε B to the hydro-static pressure P , because P = C B ε B . In addition, weassume that ∆ k in Eq. (6) corresponds to the activationenergy E determined by resistivity measurements. Thus,based on the hydrostatic pressure dependence of the re-sistivity of YbB [37], we can estimate the renormalizedhexadecapole-strain coupling constant | d uB − d lB | /k B tobe 326 K by dE/dP = − .
809 K/GPa = − . × − K / (J / m ) for C B = 20 . × J/m (Table II). Thisassumption also provides the DOS in zero field to be D = 2 . × K − m − from D ( d uB − d lB ) = 3 . × J/m in Table II. Accordingly, the coupling constant foreach elastic mode was calculated (Table II). The couplingconstant for C is approximately 5 and 3 times smallerthan the coupling constant for C T and C B , respectively.Therefore, the dominant interaction is caused by the bulkstrain with Γ and the symmetry-breaking strain withΓ . This result is useful to elucidate the quantum states,which carry the multipole degrees of freedom.While valence fluctuations are caused by hexadecapole-strain interactions in YbB , the contribution of the fluc-tuations to the elasticity is unexpectedly small in zerofield. As shown in Fig. 2, C B does not exhibit a soft-ening in YbB . This result is quite different from the3.8% softening in C B observed for SmB . Furthermore,the coupling constant | d uB − d lB | /k B = 326 K of YbB isapproximately 4 times smaller than 1280 K reported for C B of SmB [23].In contrast to zero field, strong valence fluctuationsare revealed in applied magnetic fields. A fit to thetemperature-dependent data of C B at 58 T is shown inFig. 5 (dashed line). The fit parameters at 58 T arealso summarized in Table II. In this analysis, we did notchange C from that in zero field. We fixed the band-width W = 55 K as the previously proposed rigid-bandmodel [15], . The softening with the minimum at 40 Kis reproduced qualitatively. Notably, the coefficient ofthe Curie term D ( d uB − d lB ) is enhanced from 3 . × J/m at 0 T to 10 . × J/m at 58 T. Thus, the quan-tum state contributing to the Curie term of YbB mightapproach that of SmB in magnetic fields. We stressthat the hexadecapole-strain interaction originates fromthe coupling between the isotropic volume change of thecrystal lattice and the change of ionic radii due to va-lence fluctuations. Therefore, the larger D ( d uB − d lB ) in magnetic fields indicates the enhancement of valencefluctuations of Yb.A reduced energy gap is a plausible result of the IMtransition. Our analysis reveals that the energy gap2∆ = 140 K at 0 T is reduced to 74 K at 58 T. Thismay be attributed to the Zeeman effect that changes theenergy of the 4 f states (see Appendix C). However, thistwo-band model cannot describe the gap closing in highfields. Since the DOS is approximated as constant, wecannot describe the IM transition due to the overlap ofthe edge of DOS at the Fermi energy as schematicallyillustrated in Fig. 6. An analysis using more realisticDOS as proposed in a previous study [10, 15] is needed TABLE II. Fit parameters determined by the analysis of the elastic constants C , C T , and C B in zero field using the multipolesusceptibility given in Eq. (9). Parameters describing C B at 58 T are also listed. D = 2 . × K − m − in zero field iscalculated from | d uB − d lB | /k B . | d uB − d lB | /k B of C B at 58 T is derived from the rigid-band approximation. The parameters forSmB are reproduced from Ref. [23]. C Γ γ ∆ (K) W (K) D ( d uΓ γ − d lΓ γ ) (cid:0) J / m (cid:1) A (cid:0) J / m (cid:1) B (cid:0) J / m (cid:1) C (K) | d uΓ γ − d lΓ γ | /k B (K) C
70 55 0.352 17 .
744 1 . ×
32 106 C T
70 55 8.58 11 .
865 1 . ×
85 526 C B (0 T) 70 55 3.30 20 .
146 9 . × C B (58 T) 37 55 10.3 20 .
146 9 . × C B (SmB ) 160 150 25.6 1280 to describe the field-induced metal state in high fields.Since the DOS in zero field is estimated by using thepressure dependence of the activation energy of YbB ,we cannot apply D to estimate the coupling constant | d uB − d lB | /k B in high fields. Nevertheless, if we estimatethe coupling constant assuming a field-independent rigid-band model with energy gap, the field-enhanced valueof 576 K can be obtained. The increase in the elasticsoftening due to the increase in the coupling constant isalso consistent with a previous theoretical study of theelectron-phonon coupling mediated by conduction elec-trons and f -electrons [27]. Although the model needs tobe improved, this interpretation seems to be plausible.For further discussion of the field-enhanced valencefluctuations in YbB , we estimated the valence changeof Yb in high fields. Previous studies on SmB have re-vealed a valence change from 2 . ± .
01 at 300 K to2 . ± .
01 at 60 K [38] and a softening of C B by 3 . C B in SmB and YbB are described by the hexadecapole susceptibility based onthe two-band model, we assume that the valence changeper elastic softening applies to YbB as well. Since thecontribution of the hexadecapole-strain interaction to theelastic softening in YbB , namely the coefficient of theCurie term D ( d uB − d lB ) , is 2 . (Table II), the contribution of valence fluctuationsto the elastic softening in YbB is reduced by a factorof 2 .
5. At 1.4 K, in the high-field Kondo-metal phase,the 0.09% softening from B IM to 58 T (Fig. 3) indicatesa small valence change of only approximately − . − . .
52% softening of C B .Such a valence change at 40 K may be detectable byhigh-field synchrotron x-ray measurements [39].Our results seem to be in conflict with the localizedtendency of 4 f states in the magnetic fields [40–42]. Fora comprehensive understanding of the results, we discussthe Zeeman mixing and hybridization between Yb andB electrons in addition to the c - f hybridization due tothe 5 d and 4 f electrons of the Yb atoms. In YbB ,the contribution of the Γ and Γ states to the ground state is enhanced by the Zeeman effect (see appendixC, Fig. 9). Thus, we expect that magnetic fields re-duce the anisotropy of the electronic states due to thecontributions of the Γ , Γ , and Γ wave functions inYbB . In addition, as shown in Fig. 1(a), the Yb ionof YbB is surrounded by a highly isotropic cage madeup of 24 borons. This indicates an isotropic hybridizationbetween the Yb 4 f electrons and the B 2 p electrons in ad-dition to the 5 d -4 f hybridization. Thus, we suggest thatthe valence fluctuations are induced by the interatomic p - f hybridization due to the isotropic wave function in highfields. Furthermore, a field-induced p - f hybridization isconsistent with the enhancement of the hexadecapole-strain interaction in high fields. In general, the matrixelement of the hexadecapole H for the wave function ψ is given by (cid:82) d r ψ ∗ H ψ . Therefore, a spatially expandedwave function, which is expected due to the interatomictype p - f hybridization, might enhance the renormalizedmultipole-strain coupling d u(l)B in Eq. (3). Our assump-tion is consistent with the isotropic resistivity in the low-temperature Kondo-metal phase [14]. Although the crys-tal structure and magnetic character are different fromthose of YbB , the similar mechanisms of field-induced p - f hybridization and delocalization of 4 f electrons havebeen proposed in the heavy-fermion compound CeRhIn to describe the emergence of an anisotropic electronicstate in high fields [43–46]. V. CONCLUSION
In the present work, we investigated valence fluctua-tions of YbB in zero and high fields by use of ultra-sonic measurements. In zero field, the additional elas-tic hardening of C , C , C T = ( C − C ) /
2, andthe bulk modulus C B = ( C + 2 C ) / c - f hybridization are suggested to beenhanced by the field-induced elastic softening of C B . Wefound signatures of strong field-induced valence fluctua-tions in the vicinity of 40 K. Our phenomenological anal-ysis of the temperature dependence of C B based on thetwo-band model reveals that both, the additional contri-bution in zero field and the field-induced elastic softening,are reasonably described by the hexadecapole suscepti-bility. In particular, the field-induced elastic softening isattributed to the enhancement of the hexadecapole-straincoupling. This result indicates that the magnetic field en-hances an isotropic volume change of the crystal latticeand the change of ionic radii due to valence fluctuations.Therefore, we propose field-induced valence fluctuationsdue to c - f hybridization in YbB . In particular, we pro-pose that the p - f hybridization between Yb-4 f and B-2 p electrons plays a key role in high fields. The observeddecrease of the energy gap in magnetic fields is explainedby the energy shift of the 4 f electrons due to the Zeemaneffect.Our study shows that ultrasonic measurements are use-ful to detect valence fluctuations. As suggested by a theo-retical work [48], such measurements may play a key rolein the study of valence quantum criticality. We expectthat field-induced valence fluctuations appear in othervalence-fluctuating compounds. ACKNOWLEDGMENT
The authors thank Yuichi Nemoto and MitsuhiroAkatsu for supplying the LiNbO piezoelectric transduc-ers. We also thank Keisuke Mitsumoto and ShintaroNakamura for valuable discussions. This work was partlysupported by JSPS Bilateral Joint Research Projects(JPJSBP120193507) and Grants-in-Aid for young scien-tists (KAKENHI JP20K14404). We acknowledge thesupport of the HLD at HZDR, member of the Euro-pean Magnetic Field Laboratory (EMFL), the DeutscheForschungsgemeinschaft (DFG) through the W¨urzburg-Dresden Cluster of Excellence on Complexity and Topol-ogy in Quantum Matter − ct.qmat (EXC 2147, projectNo. 390858490), and the BMBF via DAAD (project No57457940). Appendix A: Temperature dependence of elasticconstant C Figure 7 shows the temperature dependence of the elas-tic constant C in a wide temperature range of up to 300K. C exhibits an increase with decreasing temperaturesfrom 300 K. This result indicates the elastic hardening ofthe bulk modulus C B from 300 K down to low tempera-tures. Appendix B: Multipole susceptibility for CEF wavefunctions in zero field
Here, we present the CEF wave functions, the multi-pole matrices, and multipole susceptibility of YbB as-suming the localized 4 f electrons. We show that themultipole susceptibility cannot describe our experimen-tal results in Fig. 2. !" ( )) * + ) ) * , - . / )’ FIG. 7. Temperature dependence of the longitudinal elasticconstant C . To calculate the multipole susceptibility of YbB , weuse CEF wave functions of the 4 f electrons for Yb with the total angular momentum J = 7 /
2. The CEFHamiltonian H CEF under O h symmetry is written as H CEF = B (cid:0) O + 5 O (cid:1) + B (cid:0) O − O (cid:1) . (B1)Here, B and B are the CEF parameters. The matrixelements of O , O , O , and O for | J z (cid:105) are listed in Ref.[49]. The wave functions diagonalizing H CEF are givenby [4] (cid:12)(cid:12) Γ ± (cid:11) = − (cid:114) (cid:12)(cid:12)(cid:12)(cid:12) ± (cid:29) + (cid:114) (cid:12)(cid:12)(cid:12)(cid:12) ∓ (cid:29) , (B2) (cid:12)(cid:12) Γ ± (cid:11) = 12 (cid:12)(cid:12)(cid:12)(cid:12) ± (cid:29) + √ (cid:12)(cid:12)(cid:12)(cid:12) ∓ (cid:29) , (B3) (cid:12)(cid:12) Γ ± (cid:11) = (cid:114) (cid:12)(cid:12)(cid:12)(cid:12) ± (cid:29) + (cid:114) (cid:12)(cid:12)(cid:12)(cid:12) ∓ (cid:29) , (B4) (cid:12)(cid:12) Γ ± (cid:11) = − √ (cid:12)(cid:12)(cid:12)(cid:12) ± (cid:29) + 12 (cid:12)(cid:12)(cid:12)(cid:12) ∓ (cid:29) , (B5)where (cid:12)(cid:12) Γ ± (cid:11) and (cid:12)(cid:12) Γ ± (cid:11) are the ground-state wavefunctions and (cid:12)(cid:12) Γ ± (cid:11) and (cid:12)(cid:12) Γ ± (cid:11) are the degenerate ex-cited states. The matrix elements of H CEF for thewave functions given in Eqs. (B2) - (B5) provide theeigenenergy of each CEF state described as E Γ =120 ( B + 168 B ), E Γ = 120 (7 B − B ), and E Γ = −
120 (9 B + 126 B ). The energy gap ∆ CEF = 23 meV= 270 K between the ground state and the excited statesΓ and Γ [4] provides the CEF parameters B = − . B = − .
24 meV.The matrices of the hexadecapole H = O + 5 O withirrep Γ , the quadrupoles O u and O v with Γ , and O yz , O zx , and O xy with Γ for the wave functions (B2) - (B5)are calculated as0 H = (cid:12)(cid:12) Γ (cid:11) (cid:12)(cid:12) Γ − (cid:11) (cid:12)(cid:12) Γ (cid:11) (cid:12)(cid:12) Γ − (cid:11) (cid:12)(cid:12) Γ +6 (cid:11) (cid:12)(cid:12) Γ − (cid:11) (cid:12)(cid:12) Γ +7 (cid:11) (cid:12)(cid:12) Γ − (cid:11)
120 0 0 0 0 0 0 00 120 0 0 0 0 0 00 0 120 0 0 0 0 00 0 0 120 0 0 0 00 0 0 0 840 0 0 00 0 0 0 0 840 0 00 0 0 0 0 0 1080 00 0 0 0 0 0 0 1080 , (B6) O u = − √
35 0 0 00 6 0 0 0 − √
35 0 00 0 − − √ − − √ − √
35 0 0 0 0 0 0 00 − √
35 0 0 0 0 0 00 0 − √ − √ , (B7) O v = √
105 0 0 0 00 0 2 √ − √ √
105 0 0 √
105 0 0 0 2 √ √ −
30 0 √
105 0 0 0 0 00 − − , (B8) O yz = √ i
00 0 0 3 √ i − i − √ i − √ i − √ i − √ i
00 0 3 √ i − i
00 0 0 0 0 0 0 √ i −√ i √ i i i √ i −√ i , (B9) O zx = −√
35 00 0 0 3 √ −
40 0 0 0 − √ √
30 3 √ − √ − √ √ −√
35 0 0 − √ − √ √
35 0 0 , (B10)1 O xy = −√ i √ i i − √ i √ i − √ i − i √ i − i −√ i √ i i . (B11)Here, Stevens equivalent operators O u = 3 J z − J ( J + 1), O v = J x − J y , O yz = J y J z + J z J y , O zx = J z J x + J x J x ,and O xy = J x J y + J y J x , given by the components ofthe total angular momentum J x , J y , and J z , are used tocalculate the matrix elements. Considering the second-order perturbation processes for the i -th CEF state withenergy E i due to the multipole-strain interaction of Eq.(2), which is described as E i (cid:0) ε Γ γ (cid:1) = E i − g Γ γ (cid:104) i (cid:12)(cid:12) O Γ γ (cid:12)(cid:12) i (cid:105) ε Γ γ − g γ (cid:88) j (cid:54) = i (cid:12)(cid:12) (cid:104) i (cid:12)(cid:12) O Γ γ (cid:12)(cid:12) j (cid:105) (cid:12)(cid:12) E j − E i ε γ , (B12)the total free energy F , that consists of the CEF stateand the strain, is written as F (cid:0) T, ε Γ γ (cid:1) = 12 C γ ε γ − N k B T ln Z (cid:0) ε Γ γ (cid:1) . (B13)Here, N is the number of Yb ions per unit volume and Z ( ε Γ γ ) is the partition function written as Z ( ε Γ γ ) = (cid:80) i exp (cid:2) − E i ( ε Γ γ ) /k B T (cid:3) . Thus, the elastic constant andthe multipole susceptibility are calculated as C Γ γ = ∂ F∂ε γ = C γ − N g γ χ Γ γ , (B14) − g γ χ Γ γ = (cid:42) ∂ E∂ε γ (cid:43) − k B T (cid:40)(cid:42)(cid:18) ∂E∂ε Γ γ (cid:19) (cid:43) − (cid:28) ∂E∂ε Γ γ (cid:29) (cid:41) . (B15)Here, C γ is a background elastic constant, (cid:104) A (cid:105) is thethermal average using Boltzmann statistics written as (cid:104) A (cid:105) = (cid:80) i A i exp[ − E i /k B T ] /Z , and ∂E ( ε Γ γ ) /∂ε Γ γ | ε Γ γ → and ∂ E ( ε Γ γ ) /∂ε γ | ε Γ γ → are written as ∂E/∂ε Γ γ and ∂ E/∂ε γ , respectively. The first term on the right-hand side of Eq. (B15) corresponds to van Vleck termbeing constant at low temperatures and the second oneto Curie term showing the reciprocal temperature de-pendence. The calculated multipole susceptibility χ Γ γ isshown in Fig. 8.The hexadecapole susceptibility χ B would indicate amonotonic hardening of C B below 100 K down to lowtemperatures because the temperature dependence of the elastic constant C Γ γ is given by − χ B , i.e., the secondterm in Eq. (B14). The divergent behavior of χ Γ and χ Γ would predict an elastic softening of ( C − C ) / C at low temperatures, respectively. However, ourexperimental results of YbB in zero field cannot be de-scribed by the susceptibility based on CEF wave func-tions using this picture, i.e., localized 4 f electrons. FIG. 8. Temperature dependence of the electric multipolesusceptibility of YbB . (a) Hexadecapole susceptibility χ B of H = O + 5 O with Γ . This susceptibility providesthe temperature dependence of the bulk modulus C B . (b)Quadrupole susceptibility χ Γ of O u and O v with Γ related to( C − C ) / χ Γ of O yz , O zx ,and O xy with Γ related to C . As indicated in Eq. (B14), − χ Γ γ contributes to these elastic constants. (The curves for O u and O v as well as for O yz , O zx , and O xy virtually lie ontop of each other). Appendix C: Hexadecapole susceptibility for CEFwave functions in magnetic fields
In this section, the CEF wave functions, the hexade-capole matrix, and the hexadecapole susceptibility inmagnetic fields of YbB are presented assuming local-ized 4 f electrons. We show that the elastic softening of C B in high fields cannot be described by the hexadecapolesusceptibility χ B .To calculate the hexadecapole susceptibility in mag- netic fields, we consider the Zeeman Hamiltonian for B (cid:107) [001] given by H Zeeman = − g J µ B B J z . (C1)Here, g J is the Land´e g -factor, µ B is the Bohr magneton, B is the magnetic field, and J z is the magnetic dipole.Using the CEF wave functions of Eqs. (B2)-(B5), thematrix of H Zeeman of Eq. (C1) is written as H Zeeman = − B √ B B − √ B B √ B
00 0 0 − B −√ B √ B − B − √ B B √ B − B
00 0 0 −√ B B . (C2)Here, for the convenience, B in the matrix elements of Eq. (C2) is set as B = g J µ B B . The total Hamiltonian H total = H CEF + H Zeeman is diagonalized as H total = | (cid:105) | −(cid:105) | (cid:105) | −(cid:105) | (cid:105) | −(cid:105) | (cid:105) | −(cid:105) E +1 E − E +2 E − E +3 E − E +4
00 0 0 0 0 0 0 E − , (C3)Here, the eigen energis in the matrix of Eq. (C3) arewritten as E ± = ∆ CEF − B ± δE , (C4) E ± = ∆ CEF + 3 B ± δE , (C5) E ± = ∆ CEF − B ± δE , (C6) E ± = ∆ CEF + B ± δE . (C7)For convenience, δE i in Eqs. (C4) - (C7) is set as δE = (cid:114) ∆ E + 359 B , (C8) δE = (cid:114) ∆ E + 359 B , (C9) δE = (cid:113) ∆ E + 3 B , (C10) δE = (cid:113) ∆ E + 3 B . (C11)We also set ∆ E i in Eqs. (C8) - (C11) as follows:∆ E = ∆ CEF B, (C12)∆ E = ∆ CEF − B, (C13)3 FIG. 9. Magnetic-field dependence of the eigenenergy in H total of Eq. (C3) for B (cid:107) [001]. The color code shows theZeeman mixing ratio β i in the wave functions of Eqs. (C16)-(C23). ∆ E = ∆ CEF − B, (C14)∆ E = ∆ CEF B. (C15)The wave functions diagonalizing the matrix Eq. (C3)are written as | (cid:105) = α (cid:12)(cid:12) Γ +6 (cid:11) + β (cid:12)(cid:12) Γ (cid:11) , (C16) | −(cid:105) = β (cid:12)(cid:12) Γ +6 (cid:11) − α (cid:12)(cid:12) Γ (cid:11) , (C17) | (cid:105) = α (cid:12)(cid:12) Γ − (cid:11) − β (cid:12)(cid:12) Γ − (cid:11) , (C18) | −(cid:105) = − β (cid:12)(cid:12) Γ − (cid:11) − α (cid:12)(cid:12) Γ − (cid:11) , (C19) | (cid:105) = α (cid:12)(cid:12) Γ +7 (cid:11) + β (cid:12)(cid:12) Γ (cid:11) , (C20) | −(cid:105) = β (cid:12)(cid:12) Γ +7 (cid:11) − α (cid:12)(cid:12) Γ (cid:11) , (C21) | (cid:105) = α (cid:12)(cid:12) Γ − (cid:11) − β (cid:12)(cid:12) Γ − (cid:11) , (C22) FIG. 10. Electric hexadecapole susceptibility χ B of YbB for B (cid:107) [001]. (a) Magnetic-field dependence of χ B at severaltemperatures. (b) Temperature dependence of χ B below 100K in several magnetic fields. In the inset in (b), χ B is shownup to 300 K. | −(cid:105) = − β (cid:12)(cid:12) Γ − (cid:11) − α (cid:12)(cid:12) Γ − (cid:11) . (C23)Here, the coefficients α i and β i for i = 1 , , , α = ∆ E + δE (cid:113) (∆ E + δE ) + B , (C24) α = ∆ E + δE (cid:113) (∆ E + δE ) + B , (C25) α = ∆ E + δE (cid:113) (∆ E + δE ) + 3 B , (C26) α = ∆ E + δE (cid:113) (∆ E + δE ) + 3 B , (C27)4 β i = (cid:113) − α i . (C28)The magnetic-field dependence of the eigenenergies E ± i of Eqs. (C4)-(C7) are shown in Fig. 9. This result isconsistent with the previous calculation for YbB [15].The multipole susceptibility of Eq. (B15) in magneticfield is calculated using the wave functions of Eqs. (C16) - (C23), the energy of Eqs. (C4) - (C7), the multipolematrices of Eqs. (B6) - (B11), the second-order pertur-bation of Eq. (B12), and the free energy of Eq. (B13).In particular, we show the field-dependent hexade-capole susceptibility of H in Fig. 10. Here, the matrixof the hexadecapole H is written as H = (cid:32) | (cid:105) | −(cid:105) (cid:0) α + 1 (cid:1) α β α β − (cid:0) α − (cid:1) (cid:33) ⊕ (cid:32) | (cid:105) | −(cid:105) (cid:0) α + 1 (cid:1) − α β − α β − (cid:0) α − (cid:1) (cid:33) ⊕ (cid:32) | (cid:105) | −(cid:105)− (cid:0) α − (cid:1) − α β − α β (cid:0) α − (cid:1) (cid:33) ⊕ (cid:32) | (cid:105) | −(cid:105)− (cid:0) α − (cid:1) − α β − α β (cid:0) α − (cid:1) (cid:33) . (C29)The experimental results of the magnetic-field depen-dence of C B at 20, 40, and 50 K (Fig. 3) can be qualita-tively reproduced by the hexadecapole susceptibility χ B shown in Fig. 10(a). However, the experimental resultsof the elastic softening of C B in high fields (Fig. 5) cannotbe described by χ B shown in Fig. 10(b) since χ B indicatesa hardening of C B towards lower temperatures. There-fore, our experimental results of YbB in high magneticfields cannot be described by the susceptibility based onCEF wave functions of localized 4 f electrons. Appendix D: Hexadecapole susceptibility forsmaller energy gap !" ’ ( ) * ) + , - & . (! FIG. 11. Fit of the bulk modulus C B of YbB by thehexadecapole susceptibility with energy gaps ∆ = 70 and 30K. Figure 11 shows the fit of bulk modulus C B in YbB by the hexadecapole susceptibility with energy gaps ∆ =70 K and 30 K. We cannot describe the curvature changefor ∆ = 30 K, which corresponds to the activation en-ergy determined by the high-field magnetoresistance [11].This result indicates that the contribution of the largergap to the elasticity is dominant in zero field in YbB . [1] M. Kasaya, F. Iga, K. Negishi, S. Nakai, and T. Kasuya,J. Mag. Mag. Mat. - , 437 (1983). [2] T. Susaki, A. Sekiyama, K. Kobayashi, T. Mizokawa, A. Fujimori, M. Tsunekawa, T. Muro, T. Matsushita,S. Suga, H. Ishii, T. Hanyu, A. Kimura, H. Namatame,M. 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