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Spin excitons in heavy fermion semiconductors
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Journal of Magnetism and Magnetic Materials 226}230 (2001) 127}128
Spin excitons in heavy fermion semiconductors Peter S. Riseborough* Department of Physics, Polytechnic University, 6 Metro-Tech Center, Brooklyn, NY 11201, USA
Abstract The heavy fermion semiconductors such as Ce Bi Pt , YbB , and SmB may be modeled as indirect hybridization gap semiconductors. The magnitudes of the gaps are smaller than the gaps predicted by local density functional electronic structure calculations, suggesting strong renormalizations due to electronic correlations. The inelastic neutron scattering spectra Im[(q, )] are examined within the context of this model. Anomalously sharp and temperature dependent peaks are seen at energies within the indirect gap in the inelastic neutron scattering experiments on SmB and YbB . The in-gap features are identi"ed as a spin exciton excitations, which are induced by residual anti-ferromagnetic interactions between the renormalized quasi-particles. 2001 Elsevier Science B.V. All rights reserved. Keywords: Kondo insulators; Neutron scattering; Spin exciton
1. Introduction Anomalous features have been observed in inelastic neutron scattering experiments on the heavy fermion semiconductors SmB [1] and YbB [2,3]. The features have the form of a narrow peak with excitation energies less than that of the gap inferred from thermodynamic, transport and infrared conductivity measurements. The excitation energy disperses with the neutron momentum transfer q, and has a minimum at q values which correspond to the corner of the "rst Brillouin zone. The heavy fermion semiconductor materials in which these sharp features were observed share the characteristics of being strongly mixed valent [4]. On the other hand no similar in gap features have been reported in the inelastic neutron scattering experiments on almost integer valent heavy fermion semiconductor Ce Bi Pt [5]. In this manuscript we identify these excitations as the bound states of an electron}hole pair with non-zero total angular momentum. These narrow and dispersive magnetic excitations should soften as the strength of the magnetic interactions increase and a magnetic instability is produced. In the next section we shall calculate the dynamic
* Tel.: #1-718-260-3675; fax: #1-718-260-3139. E-mail address: [email protected] (P.S. Riseborough).
susceptibility of the Anderson lattice model in the limit of an in"nite coulomb repulsion between pairs of electrons residing in the f orbitals of the same ion. We shall treat the coulomb interaction using the slave boson method [6]. In the mean "eld approximation [7], the electronic dispersion relations correspond to those of a conduction band hybridized with a #at non-dispersive f band. These form two hybridized bands separated by a direct gap of the order of the renormalized hybridization matrix elements
2. The spin exciton The dynamic magnetic susceptibility (q;) is calculated from the R.P.A. like expression (q;) (q;)" , (1) 1!J(q) (q;) where (q;) is the irreducible susceptibility asso ciated with particle}hole excitations. The irreducible
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 0 8 8 - 9
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P.S. Riseborough / Journal of Magnetism and Magnetic Materials 226}230 (2001) 127}128
12t, hence this indicates that the exchange interaction should be large due to perfect nesting at half "lling. A more precise evaluation of J(q) at q"Q yields a value of J , which is large compared with J(0)"J (). 15 15 The spin-exciton manifests itself as a pole in Im[(q;#i)], when Im[ (q;#i)]"0. The energy dispersion of the pole is found from
Fig. 1. The dependence of (q;) for q"0 and q"Q. The imaginary parts are shown by solid lines, the q"Q response has peaks at the indirect gap energy, while at q"0 the response peaks at the direct gap energy. The horizontal dotted line corresponds to J(Q) calculated with E "#0.1 eV, the number of f holes 0.86 appropriate for YbB . The unhybridized conduc tion band density of states () was estimated from the experi mentally determined speci"c heat of LuB .
susceptibility is calculated from the slave boson mean "eld Greens functions dressed by the self energy and vertex function, corresponding to the emission and absorption of a slave boson [8]. Im[ (q;#i)] shows a q independent threshold at the indirect gap. At the corner of the zone, q"Q["(, , )], Im[ (Q;#i)] shows a square root singularity like variation close to threshold, while for q"(0, 0, 0), (0;) shows a slight peak at the direct gap, but has a long low energy tail due to boson assisted process which extends down to the indirect gap [9]. The exchange interaction J(q) is found from process involving the exchange of two slave bosons [10]. If the slave boson propagators are approximated as having simple poles at i"$E the interaction can be ap proximated as,
J(q)+
< (q), E
(2)
where (q) is the static susceptibility of the unhybridized d band. This represents an RKKY like interaction between the f moments in which the Schrie!er}Wolf exchange interaction, J "</E , polarizes the con15 duction band and then interacts with another f moment. The d band is modeled by a tight-binding band of width
1!J(q)Re[ (q;)]"0. (3) The sharp rise in Im[ (Q;#i)] at the threshold for the continuum of spin #ip particle}hole excitations, through the Kramers}Kronig relation, produces a large peak in the real part. It is noteworthy that the shape of the calculated spectral densities at q"0 and q"Q are similar to the results of optical absorption [11] and large momentum transfer scattering cross-section measurements [5] on polycrystalline Ce Bi Pt . The combined e!ect of the large Re[ (q;)] and J(Q) produces a bound state within the gap. This is most likely to happen in strongly mixed valent materials, where E is within 0.1 eV from . As shown in Fig. 1, when q is decreased towards the zone center, the value of J(Q) is diminished and the peak in Re[ (Q;#i)] becomes washed out. This re sults in a dispersion of the bound state energy which collapses into the continuum at a critical wave vector q . The existence of a bound state depends crucially on the f valence and on E . It could occur in Ce Bi Pt if the number of f electrons is such that 1'n '0.95. How ever, a bound state is more likely to occur in the Yb or Sm semiconductors, although less highly enhanced, where the magnitude of E is expected to be of the order of 10 eV, instead of 2}3 eV typical of Ce compounds.
References [1] P.M. Alekseev, J.-M. Mignot, J.M. Rossat-Mignod, V.N. Lazukov, I.P. Sadikov, Physica B 186}188 (1993) 384. [2] A. Bouvet, T. Kasuya, M. Bonnet, L.P. Regnault, J. Rossat-Mignod, F. Iga, B. Fak, A. Severing, J. Phys. CM. 10 (1998) 5667. [3] E.V. Nefeodova, P.M. Alekseev, J.M. Mignot, V.N. Lazukov, I.P. Sadikov, Y.B. Paderno, N.Y. Shitsevelova, R.S. Eccleston, Phys. Rev. B 60 (1999) 13507. [4] T. Kasuya, J. Phys. Soc. Japan 65 (1998) 2548. [5] A. Severing, J.D. Thompson, P.C. Can"eld, Z. Fisk, P.S. Riseborough, Phys. Rev. B 44 (1991) 6832. [6] P. Coleman, Phys. Rev. B 35 (1987) 5072. [7] P.S. Riseborough, Phys. Rev. B 45 (1992) 13984. [8] P.S. Riseborough, Phys. Rev. B 58 (1998) 15534. [9] P.S. Riseborough, Physica B 199}200 (1994) 466. [10] S. Doniach, Phys. Rev. B 35 (1987) 1814. [11] B. Bucher, Z. Schlesinger, P.C. Can"eld, Z. Fisk, Phys. Rev. Lett. 71 (1993) 1748.
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112
Journal of Magnetism and Magnetic Materials 226}230 (2001) 127}128
Spin excitons in heavy fermion semiconductors Peter S. Riseborough* Department of Physics, Polytechnic University, 6 Metro-Tech Center, Brooklyn, NY 11201, USA
Abstract The heavy fermion semiconductors such as Ce Bi Pt , YbB , and SmB may be modeled as indirect hybridization gap semiconductors. The magnitudes of the gaps are smaller than the gaps predicted by local density functional electronic structure calculations, suggesting strong renormalizations due to electronic correlations. The inelastic neutron scattering spectra Im[(q, )] are examined within the context of this model. Anomalously sharp and temperature dependent peaks are seen at energies within the indirect gap in the inelastic neutron scattering experiments on SmB and YbB . The in-gap features are identi"ed as a spin exciton excitations, which are induced by residual anti-ferromagnetic interactions between the renormalized quasi-particles. 2001 Elsevier Science B.V. All rights reserved. Keywords: Kondo insulators; Neutron scattering; Spin exciton
1. Introduction Anomalous features have been observed in inelastic neutron scattering experiments on the heavy fermion semiconductors SmB [1] and YbB [2,3]. The features have the form of a narrow peak with excitation energies less than that of the gap inferred from thermodynamic, transport and infrared conductivity measurements. The excitation energy disperses with the neutron momentum transfer q, and has a minimum at q values which correspond to the corner of the "rst Brillouin zone. The heavy fermion semiconductor materials in which these sharp features were observed share the characteristics of being strongly mixed valent [4]. On the other hand no similar in gap features have been reported in the inelastic neutron scattering experiments on almost integer valent heavy fermion semiconductor Ce Bi Pt [5]. In this manuscript we identify these excitations as the bound states of an electron}hole pair with non-zero total angular momentum. These narrow and dispersive magnetic excitations should soften as the strength of the magnetic interactions increase and a magnetic instability is produced. In the next section we shall calculate the dynamic
* Tel.: #1-718-260-3675; fax: #1-718-260-3139. E-mail address: [email protected] (P.S. Riseborough).
susceptibility of the Anderson lattice model in the limit of an in"nite coulomb repulsion between pairs of electrons residing in the f orbitals of the same ion. We shall treat the coulomb interaction using the slave boson method [6]. In the mean "eld approximation [7], the electronic dispersion relations correspond to those of a conduction band hybridized with a #at non-dispersive f band. These form two hybridized bands separated by a direct gap of the order of the renormalized hybridization matrix elements
2. The spin exciton The dynamic magnetic susceptibility (q;) is calculated from the R.P.A. like expression (q;) (q;)" , (1) 1!J(q) (q;) where (q;) is the irreducible susceptibility asso ciated with particle}hole excitations. The irreducible
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 0 8 8 - 9
59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87
128
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
P.S. Riseborough / Journal of Magnetism and Magnetic Materials 226}230 (2001) 127}128
12t, hence this indicates that the exchange interaction should be large due to perfect nesting at half "lling. A more precise evaluation of J(q) at q"Q yields a value of J , which is large compared with J(0)"J (). 15 15 The spin-exciton manifests itself as a pole in Im[(q;#i)], when Im[ (q;#i)]"0. The energy dispersion of the pole is found from
Fig. 1. The dependence of (q;) for q"0 and q"Q. The imaginary parts are shown by solid lines, the q"Q response has peaks at the indirect gap energy, while at q"0 the response peaks at the direct gap energy. The horizontal dotted line corresponds to J(Q) calculated with E "#0.1 eV, the number of f holes 0.86 appropriate for YbB . The unhybridized conduc tion band density of states () was estimated from the experi mentally determined speci"c heat of LuB .
susceptibility is calculated from the slave boson mean "eld Greens functions dressed by the self energy and vertex function, corresponding to the emission and absorption of a slave boson [8]. Im[ (q;#i)] shows a q independent threshold at the indirect gap. At the corner of the zone, q"Q["(, , )], Im[ (Q;#i)] shows a square root singularity like variation close to threshold, while for q"(0, 0, 0), (0;) shows a slight peak at the direct gap, but has a long low energy tail due to boson assisted process which extends down to the indirect gap [9]. The exchange interaction J(q) is found from process involving the exchange of two slave bosons [10]. If the slave boson propagators are approximated as having simple poles at i"$E the interaction can be ap proximated as,
J(q)+
< (q), E
(2)
where (q) is the static susceptibility of the unhybridized d band. This represents an RKKY like interaction between the f moments in which the Schrie!er}Wolf exchange interaction, J "</E , polarizes the con15 duction band and then interacts with another f moment. The d band is modeled by a tight-binding band of width
1!J(q)Re[ (q;)]"0. (3) The sharp rise in Im[ (Q;#i)] at the threshold for the continuum of spin #ip particle}hole excitations, through the Kramers}Kronig relation, produces a large peak in the real part. It is noteworthy that the shape of the calculated spectral densities at q"0 and q"Q are similar to the results of optical absorption [11] and large momentum transfer scattering cross-section measurements [5] on polycrystalline Ce Bi Pt . The combined e!ect of the large Re[ (q;)] and J(Q) produces a bound state within the gap. This is most likely to happen in strongly mixed valent materials, where E is within 0.1 eV from . As shown in Fig. 1, when q is decreased towards the zone center, the value of J(Q) is diminished and the peak in Re[ (Q;#i)] becomes washed out. This re sults in a dispersion of the bound state energy which collapses into the continuum at a critical wave vector q . The existence of a bound state depends crucially on the f valence and on E . It could occur in Ce Bi Pt if the number of f electrons is such that 1'n '0.95. How ever, a bound state is more likely to occur in the Yb or Sm semiconductors, although less highly enhanced, where the magnitude of E is expected to be of the order of 10 eV, instead of 2}3 eV typical of Ce compounds.
References [1] P.M. Alekseev, J.-M. Mignot, J.M. Rossat-Mignod, V.N. Lazukov, I.P. Sadikov, Physica B 186}188 (1993) 384. [2] A. Bouvet, T. Kasuya, M. Bonnet, L.P. Regnault, J. Rossat-Mignod, F. Iga, B. Fak, A. Severing, J. Phys. CM. 10 (1998) 5667. [3] E.V. Nefeodova, P.M. Alekseev, J.M. Mignot, V.N. Lazukov, I.P. Sadikov, Y.B. Paderno, N.Y. Shitsevelova, R.S. Eccleston, Phys. Rev. B 60 (1999) 13507. [4] T. Kasuya, J. Phys. Soc. Japan 65 (1998) 2548. [5] A. Severing, J.D. Thompson, P.C. Can"eld, Z. Fisk, P.S. Riseborough, Phys. Rev. B 44 (1991) 6832. [6] P. Coleman, Phys. Rev. B 35 (1987) 5072. [7] P.S. Riseborough, Phys. Rev. B 45 (1992) 13984. [8] P.S. Riseborough, Phys. Rev. B 58 (1998) 15534. [9] P.S. Riseborough, Physica B 199}200 (1994) 466. [10] S. Doniach, Phys. Rev. B 35 (1987) 1814. [11] B. Bucher, Z. Schlesinger, P.C. Can"eld, Z. Fisk, Phys. Rev. Lett. 71 (1993) 1748.
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