Theory of Spin Exciton in the Kondo Semiconductor YbB 12 Alireza Akbari1,4, Peter Thalmeier2, and Peter Fulde1,3 1Max Planck Institute for the Physcis of Complex Systems, D-01187 Dresden, Germany 2Max Planck Institute for the Chemical Physics of Solids, D-01187 Dresden, Germany 3Asia Pacific Center for Theoretical Physics, Pohang, Korea 4Institute for Advanced Studies in Basic Sciences, 45195-1159 Zanjan, Iran (Dated: 15 January 2009) The Kondo semiconductor YbB12 exhibits a spin and charge gap of approximately 15 meV. Closetothegapenergynarrowdispersivecollectiveexcitationswereidentifiedbypreviousinelastic 9 neutronscatteringexperiments. Wepresentatheoreticalanalysisoftheseexcitations. Startingfrom 0 aperiodicAndersonmodelforcrystallineelectricfield(CEF)split4fstateswederivethehybridized 0 quasiparticlebandsinslavebosonmean-fieldapproximationandcalculatethemomentumdependent 2 dynamicalsusceptibilityinrandomphaseapproximation(RPA).Weshowthatasmalldifferencein the hybridization of the two CEF (quasi-) quartets leads to the appearance of two dispersive spin n resonanceexcitationsatthecontinuumthreshold. Theirintensityislargestattheantiferromagnetic a (AF) zone boundary point and they have an upward dispersion which merges with the continuum J lessthanhalfwayintotheBrillouinzone. Ourtheoreticalanalysisexplainsthemostsalientfeatures 1 of previously unexplainedexperiments on themagnetic excitations of YbB12. 2 PACSnumbers: 71.27.+a,75.40.Gb,71.70.Ch ] l e - Theso-calledKondoinsulatorsorsemiconductorslike, tional heavy-fermion superconductors below the quasi- r t e.g., CeNiSn, SmB and YbB represent a special class particle continuum threshold at 2∆ where ∆ is the s 6 12 0 0 . of strongly correlated electrons[1]. In these compounds gapamplitude[12]. The upper peak is muchbroaderand t a the conduction electrons hybridize with nearly localized showslittle dispersion. Itis alsorapidly suppressedwith m 4f electrons. The Coulombrepulsion of the latter results increasingtemperature. Ithasbeenassociatedwithcon- - in a small energy gap[2] of order 10 meV at the Fermi tinuum excitations [13] also visible in a broadmaximum d n level[3, 4]. in the optical conductivity[8] around 38 meV. o Attemperatureshigherthanthegapenergythesema- Furthermore INS experiments on Yb1−xLuxB12 com- c terials behave like Kondo metals exhibiting their typical pounds for different Lu concentrations have indicated [ spin fluctuation spectrum. But a low temperatures a that the disruption of coherence on the Yb sublattice 1 spinandchargegapopensindicatingtheformationofan primarily affects the narrow peak structures occurring v insulating singlet ground state[1, 5]. This may be con- at 15 20 meV in pure YbB12 compound, whereas the 7 − cluded from the total suppression of the local moment spin gap and the broad magnetic signal around 38 meV 4 2 inthesusceptibilityandfromthesemiconductingbehav- remain almost unchanged[14]. 3 ior of the resistivity, respectively [6]. The gap formation These intriguing experimental results have commonly 1. may also be seendirectly in the dynamicalsusceptibility been interpreted in a qualitative way within the spin ex- 0 and finite frequency conductivity as probed in inelastic citonscenario[7,9,15]butanalternativemodelwasalso 9 neutronscattering(INS)andopticalconductivityexper- proposed[16]. Howevernoanalysisofthe formerwasat- 0 iments. In cubic YbB12 the spin [7] and charge [8] gap tempted sofar although it is of fundamental importance : v obtained in this way are approximatelyequal to 15 meV to understand the microscopic origin and fine structure i but in general they need not be identical. of the spin gap in Kondo semiconductors. In this com- X In addition unpolarized [9] and polarized [7] INS has municationweshowindetailhowthespinexcitonbands r a foundaninterestingdispersivefine structurearoundthis in YbB12 arise on the background of a single-particle threshold energy. Three excitation branches have been continuumatthespingapedge. Wediscusstheoriginof identified with energies 15, 20 and 38 meV, respectively the splitting into two modes, its connection to CEF ef- by analyzing the spectral function of the dynamical sus- fects as well as their spectral shape and dispersion. Our ceptibility. Since the lower two INS peaks are narrow investigations clarify the underlying microscopic physics and mostly centered at the zone boundary L-point with of these intriguing and for a long time unexplained ob- Q = (π,π,π) they may be associated with the forma- servations. tion of a collective heavy quasiparticle spin resonance Our starting point is the hybridization-gap picture exciton appearing around the spin gap threshold [7, 9] basedonthe periodic Andersonmodelwhichis the most and driven by heavy quasiparticle interactions. The col- widely accepted for the description of Kondo semicon- lective modes remain visible in the 20 meV region up ductors. Usingthemean-fieldslavebosonapproximation to T = 159 K [10, 11]. Similar spin resonance phe- for CEFsplit 4f states of Yb we calculate the hybridized nomena appear as result of feedback effect in unconven- bands. With an empirical model for the quasiparticle 2 20 5 f1 4 15 OS3 D2 10 1 t0 5 0 -0.1 0 0/.1t 0.2 0.3 ωt/ 0 -5 IRIRmmee 0000;;;;qqqq ==== 0Q0Q 0 0.2 0.4 0.6 /t q/Q FIG. 1: Dynamical susceptibility in direction (1,1,1) for q= 0 (direct gap) and q = Q (indirect gap) versus energy, for FIG.2: Contourplotofrealpartofnoninteractingdynamical d(V¯ege=ne0r.a3t0et,qJuQasi=par0t)i.cleInbseatndsshowwisththVe1d=ens1ittyaonfdstδaVtes=fo0r (suV¯s1c=ep0ti.b3i0ltit,yJQfor=d0e)geinnetrhateedbiraencdtisonwi(t1h,1V,11)=. 1t and δV =0 twoCEF-splitquasiparticlebands(ǫ¯ =0.08tandb=0.41). f1 The green curvecorresponds to the band V1 =t and the red one to theband V2 =V1+0.13t. hole Hilbert space is then H = (ǫ +∆ )f† f + ǫ d† d interactions we evaluate the momentum dependent dy- f γ iγ iγ k kγ kγ Xiγ Xkγ namical magnetic susceptibility in RPA. Its imaginary part is proportional to the INS spectrum. We obtain + Ns−1/2 (Vkγeik·Rifi†γdkγbi+c.c), (1) sharp resonance features around the continuum thresh- Xikγ oldandwavevectorsnottoofarfromthe zoneboundary L-point. Away from this point the resonance peaks dis- Here γ = (Γ,m) where Γ=1,2 denotes the quartet and perse upwards in energy and broaden. They merge into m =1 4 is the orbital degeneracy index. Furthermore the single particle continuum less than halfway into the the loc−al constraint Q˜ = b†b + f† f = 1 has to i i i γ iγ iγ Brillouin zone (BZ), which describes the basic experi- be respected for all i. Therefore Pthe total Hamiltonian mental facts. In addition our calculation suggests that including the constraint is H λ (Q˜ 1), where λ − b i i− b CEFsplitting andassociatedCEForbitaldependence of istheLagrangemultiplier. HerethPef† createf-holesat iγ hybridization are responsible for the observed splitting latticesiteiinCEFstateγ,andthe d† createthe holes into two dispersive resonance modes. kγ intheconductionbandwithwavevectorkandCEFstate The Yb electronic configuration is 4f13 correspond- indexγ. Thef-orbitalenergyisǫf,while∆γ =∆Γ isthe CEF excitation energy, and N is the number of lattice ing to a single hole in the 4f-shell[17]. Therefore we s sites. Finally V is the hybridization energy between 4f consider the Anderson lattice model with a f-hole in kγ andconductionholes. Inthefollowing,thekdependence a j = 7/2 state, including the CEF effect, i.e., H = t of the hybridization energy is neglected, i.e., V = V . H +H +H +H . Here H describes the lattice of kγ γ f d f−d C f This is justified for a fully gapped Kondo insulator like thelocalized,CEF-split4f-holes,H theconductionelec- d YbB whereV doesnotvanishalonglinesinkspace. trons and H is the hybridization between both. Fi- 12 kγ f−d Furthermoretouseonlyaminimumsetofmodelparam- nallyH isthe Coulombinteractionwithanon-sitehole repulsioCn Uff. Our model assumes the limit Uff eters,wereplaceVγ =VΓ,m byVΓ =(1/2)( m|VΓ,m|2)21 where doubly occupied (hole) states (4f12) are excl→ud∞ed which is the average over each set of quartPet states. We and the two possible Yb configurations are either 4f14 useanearest-neighbortightbindingmodelwithhopping or 4f13. The one without a 4f hole, i.e., 4f14 can be t for the conduction electron bands ǫk. The spectral accounted for by an auxiliary boson b† [18]. In cubic function of the experimental dynamical susceptibility of i YbB exhibitstwosharppeaks[7]. Thereforeitisessen- symmetry the j =7/2multiplet is split by the CEF into 12 tialthatthetwoCEFquartetshavetwodifferentaverage aquartetΓ groundstateandtwoexciteddoubletstates. 8 ThelattermaybetreatedasaquasiquartetΓ′ according hybridization energies VΓ (Γ=1,2). 8 toINSresultsathighertemperatures[19]. Thetwoquar- The mean-field approximation to Eq. (1) is obtained tets (index Γ = 1,2) have energies ∆ = 0 and ∆ > 0. by taking b = b . Minimizing the ground state energy 1 2 i h i The model Hamiltonian in the restricted zero- and one- withrespecttobandtheLagrangemultiplier λ leadsto b 3 the equations 30 25 JJQQ==00.065t λ b= V W ; nf +b2 =1; n= (nd +nf),(2) b γ γ γ γ γ Xγ Xγ Xγ 20 15 where the following expectation values are introduced Wγ = N1s khfk†γdkγi, ndγ = N1s khd†kγdkγi and nfγ = 10 N1s khfk†γPfkγi. In Eq. (2), n iPs the density of holes 5 perPsite which defines the chemical potential µ. The 0 mean-field Hamiltonian can be diagonalized. One ob- 0.05 /0t.1 0.15 tains, H = E (k)a† a , where the hy- MF kγ,± γ,± kγ,± kγ,± FIG.3: Theimaginarypartofthesusceptibilityforthenonin- P bridized bands have energies Eγ,±(k) = 21(cid:20)ǫk + ǫ¯fγ ± tfoerraVc1tin=gt(,JqQ==Q0)aannddδinVte=ra0c.t1in3tg.cTashee(JJQQγ1 =areJQsl2ig=ht0ly.0s6u5bt)- (ǫ ǫ¯ )2+4V¯2 whicharestillfourfold(m=1-4)de- critical leading to a finite intrinsic resonance line width. q k− fγ γ(cid:21) generate. Thecorresponding4f-weightfunctionsofthese quasiparticle bands are given by Af (k) = Ad (k) = interaction Jˆq and the non-interacting quasiparticle sus- γ,± γ,∓ 1[1 ǫ¯fγ−ǫk ]. From the mean-field solution we ceptibilityχˆ0(q,ω)are2×2matricesintheCEFquartet 2 ± √(ǫ¯fγ−ǫk)2+4V¯γ2 index Γ=1,2. also obtain The exchange interaction Jˆ between quasiparticles q V¯ f(E (k)) f(E (k)) is assumed to be peaked at the AF wave vector Q = γ γ,+ γ,− Wγ = N E (k)−E (k) ; (π,π,π), i.e., the L-point because there the most pro- s Xk γ,+ − γ,− nouncedmagneticresponseisfound. InprincipleJˆ may q nf/d = 1 Af/d(k)f(E (k)), (3) becalculatedtoorder(1/N2)[20,21]butthisisstrongly γ Ns Xk,± γ,± γ,± modeldependent. WechoosetoparameterizeJˆqinasim- pleway: TheinteractionispeakedatQorQ=√3π and where V¯γ = Vγb, and ǫ¯fγ = ǫf +∆γ −λb. In the zero it has the Lorentzian form JqΓΓ′ = (q−QΓ)2Q2+Γ2 JQΓΓ′, temperature limit, T = 0, the upper bands are empty. h Qi where Γ has the meaning of an inverse AF correlation Thenthe Fermifunctions reduce to f(E (k))=0, and Q γ,+ length. Eachelementoftheirreduciblesusceptibilityma- f(E (k)) = ϕ ( ϕ = 4n). Under the condition, γ,− γ γ γ trix is calculated from the quasiparticle states as [21]: n = 2 or ϕγ = 1, Pwhich holds as long as the chemical potential is within the hybridization gap, we obtain the χΓΓ′(q,ω) = Af (k+q)Af (k) following mean-field equations from Eqs. (2): 0 Γ,± Γ′,∓ × Xk,± ǫ¯f1 −ǫf =ΓX=1,22VDΓ2 lnqq((DD−+¯ǫǫ¯ffΓΓ))22++44VV¯¯ΓΓ22+−DD−−ǫǫ¯¯ffΓΓ; Thenon-diagonale(cid:20)lefmE(EΓen,Γ∓t,±s(k(ok)f−+thEeqΓ)in)′,t±−e(rfka(c+EtioΓq′n),∓m−(kaω)t)ri(cid:21)x,c(o4r)- 2b2 = ( (ǫ¯ +D)2+4V¯2 (ǫ¯ D)2+4V¯2). responding to interactions of quasiparticles with differ- ΓX=1,2 q fΓ Γ −q fΓ − Γ ent CEF symmetry are neglected, implying JqΓΓ′ = JΓ(q)δΓΓ′. Then the RPA susceptibility is simply a sum Here ǫ¯ = ǫ¯ +∆ , V¯ = V¯ +δV¯ = b(V +δV), and of two contributions χΓΓ(q,ω) from the two sets of hy- f2 f1 2 2 1 1 D = 6t is half the conduction band width. The den- bridized bands: sity of states of the conduction band is assumed to be χ(q,ω)= [1 J (q)χΓΓ(q,ω)]−1χΓΓ(q,ω). (5) rectangular (g(ǫ) = 1/2D; D < ǫ < D and zero − Γ 0 0 − XΓ otherwise). By solving the set of equations numerically one can find the ǫ¯ and b values. In order to be in We now discuss the results of numerical calculations f1 the Kondo limit and have an insulating state with small basedon the previous analysis. In Fig. (1) we have plot- hybridization gap the parameters should fulfill the con- ted the real and imaginary part of χΓΓ(q,ω) without 0 dition ∆ < δV < V , ǫ < D. In the absence of CEF CEF splitting (∆ = δV = 0) versus energy for wave 2 1 f 2 | | effects, by choosingǫ = 0.75t,V =t, δV =0, ∆ =0 vectorsq=0andq=Q. OnenoticesthatImχΓΓ(Q,ω) f − 1 2 0 we found ǫ¯ = 0.05t and b = 0.30 from the mean-field has a stronglow-energypeak due to a smallindirect gap f1 solutions which will be used in Fig. 1. while ImχΓγ(0,ω) has a small peak at much higher en- 0 The dynamic magnetic susceptibility is calculated ergy due to a large direct gap. The broad structure of within RPA approximation. Since we have two CEF the former is due to noninteracting single-particle exci- quartetsthespinresponsehasthematrixformχˆ(q,ω)= tations and the q and ω dependence is depicted in the [I Jˆ χˆ (q,ω)]−1χˆ (q,ω), where the unit matrix I, the inset of Fig. 4. q 0 0 − 4 realistic band model might give a larger extension than the one seen in Fig. 2. Due to the CEF effect the f-levels split into two (pseudo-) quartets (∆ > 0) which hybridize differently. 2 ForδV >0theresonanceωΓ=2associatedwiththeγ =2 r hybridized bands moves to higher energy and a second ωt/ peak in addition to the one at ωrΓ=1 appearsin the spec- tral function. This is clearly seen in Fig. 3 where the ωt/ CEF split resonance peaks at q = Q appear around the thresholdenergyofthenon-interactingcontinuumstates. Inthisfigureweusesubcriticalvaluesfortheinteraction constants. Thereforetheresonancepeaksarerightabove q/Q the continuum threshold and have a finite intrinsic line q/Q width. If the interaction constants are slightly increased the FIG. 4: Contour plot of imaginary part of RPA dynamical resonancesmovebelowthecontinuumandturnintotrue susceptibilitywithLorentzianinteractionJΓ(q)forCEF-split spin exciton poles without intrinsic line widths (within quasiparticle bands, ǫ¯f1 = 0.08t and b = 0.41. Here V1 = RPA). Their dispersion is shown in the main panel of t and V2 = V1 +0.13t (JQ1 = 0.125t, JQ2 = 0.143t and Fig. 4. Away from the L-point (Q = √3π) they dis- ΓJΓQ(Q=)2=wh1i/cRhes[aχtΓ0isΓfy(Qth,ωerr)e]s)o,nianndceirceocntidointio(1n,1in,1R).PAThfoerpmeualkas: perse upwards and merge into the continuum. We iden- tify these spin resonance modes with the observed ex- at the zone boundary (q = Q) appear at ω1 = 0.047t and ω2 = 0.063t. By choosing t = 0.32eV (D = 1.92 eV) then perimental peaks at 15 and 20 meV [7, 9] and we have ω1 = 15meV and ω2 = 20meV which are comparable with chosen parameters such that their energy splitting and experimental results. The inset shows the contour plot of dispersion are reproduced. Our numerical calculations imaginary part of dynamical susceptibility of noninteracting show that the best fit to experiments is obtained for degenerate bands for comparison (V1 =1t, δV =0.13t, V¯1 = δV = 0.13t where ¯ǫ = 0.08t and b = 0.41. These spin 0.41t, andJQ =0)inthedirection(1,1,1). Thecolorscaleof excitonpeaks separaf1tewith increasingCEFsplitting ∆ theinset is the same as Fig.2. 2 and hybridization energy difference δV. Therefore the influence of the latter is strong since it directly affects the hybridization gap and hence the noninteracting sus- The density of states for noninteracting quasiparti- ceptibility and resonance condition. We note that an in- cles ργ(ω)= N1s k,±δ(ω−Eγ,±(k)) including the CEF crease in JQ (or a decrease of the hybridsation gap) will splitting for thePtwo sets of bands with ∆2 = 0.01t and lead to a decrease of the spin exciton mode frequencies δV = 0.13t is plotted in the inset of Fig. (1). The two at Q. For J = 0.179t the lowest mode would become Q1 hybridizationgapsare differentdue to a finite δV. How- soft. This softening signifies the instability of the para- ever the latter is kept small enough to ensure that the magnetic state and the onset of AF order in a Kondo chemical potential is within the gap. semiconductor. This is not observed in YbB at am- 12 When the AF interaction J (q) is turned on, the bient pressure. We suggest that an investigation of the Γ imaginary part becomes ImχΓΓ(q,ω)=F(α ,η )/J (q) pressure dependence of spin exciton mode frequencies at Γ Γ Γ where αΓ = JΓ(q)ImχΓ0Γ(q,ω), ηΓ = 1 − Q would give important clues how close YbB12 is to J (q)ReχΓΓ(q,ω), and F(α ,η ) = α /(η2 +α2). In AF order. Finally we mention that our present model Γ 0 Γ Γ Γ Γ Γ that case the spectrum for q = Q moves to lower en- does not include the broad excitations at 38 meV. As ergies and a narrow double-peak structure, i.e., the col- has been suggested in Ref. 13 it might be due to a con- lective spin resonanceexcitations appear. Their energies tinuumofadditionalbandstateswhichdonottakeplace ωΓ are determined by the solution of η = 0. If they in the resonance formation. Their inclusion would re- r Γ are lying within the indirect hybridization gap one has quire a multi-orbital conduction band model. α 0. Then the spectral function is a delta function We thank P. A. Alekseev and I. Eremin for helpful dis- Γ → πδ(η )/J (q) at the resonance energy ω . The disper- cussions. Γ Γ r sion of the resonance, is determined by the real part of χΓΓ(q,ω) presented in Fig. 2. The plot shows that for 0 q < Q the maximum of the spectral function follows a ridge which decreases in height and bends to higher en- [1] G.Aeppli,andZ.Fisk,CommentsCond.Mat.Phys.16, ergy. This turns into an upward dispersion of the res- 155 (1992). onance pole. Its endpoint in the BZ is limited by the [2] M. F. Hundleyet al., Phys.Rev.B 42, 6842 (1990). extension of the ridge in Reχ0. The latter is fixed for [3] M. Kasaya et al., J. Magn. Magn. Mater. 31-34, 437 the simple hybridization band model used here. A more (1983). 5 [4] F. Iga, N. Shimizu and T. Takabatake, J. Magn. Magn. [13] J.-M. Mignot et al., Physica B 383 16 (2006) Mater. 177-181, 337 (1998). [14] Alekseev et al., J. Phys.: Condens. Matter 16, 2631 [5] P.S. Riseborough, Adv.Phys. 49, 257 (2000). (2004). [6] T. Susaki et al., Phys. 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