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Gap-anisotropic model for the narrow-gap Kondo insulators
Physica B 259—261 (1999) 190—192
Gap-anisotropic model for the narrow-gap Kondo insulators Juana Moreno *, P. Coleman Abdus Salam International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy Serin Laboratory, Rutgers University, P.O. Box 849, Piscataway, NJ 08855-0849, USA
Abstract A theory is presented which accounts for the dynamical generation of a hybridization gap with nodes in the Kondo insulating materials CeNiSn and CeRhSb. By assuming that Coulomb interactions drive both charge and shape fluctuations of the Cerium f-state to high energies we are able to derive an effective three-channel Kondo lattice model. Interference amongst the different channels causes the hybridization to develop nodes and allows us to account for the formation of a nodal semi-metal. 1999 Elsevier Science B.V. All rights reserved. Keywords: Kondo insulators; Neutron scattering; Thermal conductivity
The smallest gap Kondo insulators, CeNiSn and CeRhSb, appear to develop gapless excitations in the ground state. Ikeda and Miyake [1] recently proposed that the insulating ground state of these materials develops in a crystal field state with an axially symmetric hybridization potential that vanishes along a single crystal axis. However, there is no a priori reason to suppose that such an anisotropic crystal field state would be selected by the crystal. Moreover, neutron scattering results show that the crystal electric fields are negligibly small compared to the Kondo energy; scales [2]. This led Kagan et al. to suggest that the unusual anisotropies in these systems must be generated by inter-site interactions [3,8]. We propose a novel mechanism for the formation of a nodal Kondo insulator. Our key assumption is that Coulomb interactions between the Cerium f-state and its environment act to suppress both f-charge fluctuations and fluctuations in the ionic shape. The residual shapepreserving dynamics at each cerium site resembles a three-channel Kondo model [4]. Each channel corresponds to one of the three degenerate orbital configura-
* Correspondence address: Department of Physics & Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA. Fax: #1-847-491-9982; e-mail: [email protected].
tions of a j" Cerium ion under monoclinic symmetry. Interference between the Kondo effect at different sites selects the particular linear combination of channels taking part in the Kondo effect. Remarkably, the Ikeda—Miyake (IM) state appears as the most stable mean-field solution, but we also find a new stable state with an octahedrally symmetric node distribution that appears to fit the properties of these Kondo insulators more precisely. If Coulomb interactions suppress shape fluctuations of the f-site, then only pure spin exchange survives, to give H"H !J cR ( j) f ( j) f R( j)c ( j), (1) ?? ? @ ?@ H ? where a"1, 2, 3 and a"$1 label the crystal field partial wave states according to “shape” and pseudo-spin, respectively. H " k ekcRk ck describes a featureless con N N N duction sea. By using a mean-field approach we arrive at the following Hamiltonian: » H*"H #» [ N? , ˆ (k)cR k f #h.c.]#N A N ?I J k
(2)
where cˆ "(c , c , c ) is a unit vector which selects the particular linear combination of channels partaking in the Kondo effect, and N? (k)" c 1kK p"aa2 describes the Aˆ ? ? overlap between this channel and a Bloch wave "kp2. The
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 8 8 3 - 7
J. Moreno, P. Coleman / Physica B 259—261 (1999) 190—192
mean field ground-state energy is then found to be
» *E "2N(0)» ln #F[cˆ ] , (3) ¹ ) where ¹ "(eD)e\ ,(, N(0) is the conduction elec) tron density of states. The term F[cˆ ] is given by
F[cˆ ]"
d)kK U ˆ (kK ) ln [U ˆ (kK )], A 4n A
(4)
where U ˆ (kK )"() " N? (k)" contains all details of the A ? N cˆ gap anisotropy. The weak logarithmic singularity inside the integrand minimizes the ground-state energy in gapless Kondo insulating configurations. The IM state has the lowest free-energy. But this theory also identifies a new locally stable state. This state is almost octahedral. Like the IM state, the hybridization drops exactly to zero along the zˆ -axis. But, in marked difference with the IM
Fig. 1. Normalized thermal conductivity versus temperature along the zˆ -axis (solid line) and in the basal plane (dashed line). Top: for the Ikeda—Miyake state. Bottom: for the quasi-octahedral scenario. Insets show density of states as a function of the energy.
191
state, it almost vanishes along the (1, 1, 0) and (1, !1, 0) directions in the basal plane. The inset in Fig. 1 shows the density of states predicted by the two candidates. Although both are gapless, the v-shaped pseudo-gap of the quasi-octahedral state is far more pronounced than that of the IM state, and it is closer in character to the observed tunneling density of states [5,9]. A more direct probe of the anisotropy is provided by the thermal conductivity [6]. In these calculations it is essential to include the momentum dependence of the hybridization potential in the evaluation of the quasiparticle current. Previous calculations [1] underestimated the anisotropy by neglecting these important contributions. The single node in the IM state leads to a pronounced enhancement of the low-temperature thermal conductivity along the nodal zˆ -axis. By contrast, in the quasioctahedral state the distribution of minima in the gap gives rise to a modest enhancement of the thermal conductivity in the basal plane. Experimental measurements [6] have shown than near 5 K the enhancement is much more pronounced in i than in i or i . This fact tends to V X W favor the quasi-octahedral scenario. Another respect in which the two states differ is in the spectrum of low-energy spin excitations. Fig. 2 displays Im sXX(q, u) at fixed momentum transfer q"(0.5, 0, 0). The IM state shows a broad maximum. However, the quasi-octahedral state displays a sharp peak with much higher intensity. This peak is associated to transitions between quasiparticle states along the (1, 1, 0) and (1, !1, 0) directions.
Fig. 2. Imaginary part of the magnetic susceptibility for IM state (solid lime) and quasi-octahedral state (dot-dashed line). In the inset Re/sXX(q, u"0) as a function of momentum transfer.
192
J. Moreno, P. Coleman / Physica B 259—261 (1999) 190—192
The inset of Fig. 2 shows that the static susceptibility of both states is weakly dependent on q. In this respect both states manifest the correct physical behavior [7]. However, the quasi-octahedral state displays a sharp leading edge in the spectrum, a feature which is more closely in accord with experiments than the broad hump predicted for the axial state. To conclude, we have proposed a mechanism for the dynamical generation of a nodal hybridization gap in the narrow-gap Kondo insulators. By supposing that Coulomb interactions drive the shape fluctuations of the Cerium f-state to high energies we derive an effective three-channel Kondo lattice model. Interference between the Kondo effect in the different channels selects a state with a nodal anisotropy. Our theory predicts two stable states, one axial, the other quasi-octahedral in symmetry. The quasi-octahedral solution appears to be the most promising explanation of the various properties of these materials.
We are grateful to Gabriel Aeppli, Frithjof Anders, Yoshio Kitaoka, Toshiro Takabatake and Adolfo Trumper of enlightened discussions. This research was partially supported by NSF grant DMR 96-14999.
References [1] H. Ikeda, K. Miyake, J. Phys. Soc. Japan 65 (1996) 1769. [2] P.A. Aleksee et al., JETP 79 (1994) 665. [3] Y. Kagan, K.A. Kikoin, N.V. Prokof’ev, JETP Lett. 57 (1993) 600. [4] T.S. Kim, D.L. Cox, Phys. Rev. Lett. 75 (1995) 1622. [5] T. Ekino et al., Phys. Rev. Lett. 75 (1995) 4262. [6] M. Sera et et al., Phys. Rev. B 55 (1997) 6421. [7] A. Schro¨der et al., electronic archive xxx.lanl.gov/condmat/9611132. [8] Yu. Kagan, K.A. Kikoin, A.S. Mishchenko, Phys. Rev. B 55 (1997) 12348. [9] D.N. Davydov et al., Phys. Rev. B 55 (1997) R7299.
Gap-anisotropic model for the narrow-gap Kondo insulators Juana Moreno *, P. Coleman Abdus Salam International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy Serin Laboratory, Rutgers University, P.O. Box 849, Piscataway, NJ 08855-0849, USA
Abstract A theory is presented which accounts for the dynamical generation of a hybridization gap with nodes in the Kondo insulating materials CeNiSn and CeRhSb. By assuming that Coulomb interactions drive both charge and shape fluctuations of the Cerium f-state to high energies we are able to derive an effective three-channel Kondo lattice model. Interference amongst the different channels causes the hybridization to develop nodes and allows us to account for the formation of a nodal semi-metal. 1999 Elsevier Science B.V. All rights reserved. Keywords: Kondo insulators; Neutron scattering; Thermal conductivity
The smallest gap Kondo insulators, CeNiSn and CeRhSb, appear to develop gapless excitations in the ground state. Ikeda and Miyake [1] recently proposed that the insulating ground state of these materials develops in a crystal field state with an axially symmetric hybridization potential that vanishes along a single crystal axis. However, there is no a priori reason to suppose that such an anisotropic crystal field state would be selected by the crystal. Moreover, neutron scattering results show that the crystal electric fields are negligibly small compared to the Kondo energy; scales [2]. This led Kagan et al. to suggest that the unusual anisotropies in these systems must be generated by inter-site interactions [3,8]. We propose a novel mechanism for the formation of a nodal Kondo insulator. Our key assumption is that Coulomb interactions between the Cerium f-state and its environment act to suppress both f-charge fluctuations and fluctuations in the ionic shape. The residual shapepreserving dynamics at each cerium site resembles a three-channel Kondo model [4]. Each channel corresponds to one of the three degenerate orbital configura-
* Correspondence address: Department of Physics & Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA. Fax: #1-847-491-9982; e-mail: [email protected].
tions of a j" Cerium ion under monoclinic symmetry. Interference between the Kondo effect at different sites selects the particular linear combination of channels taking part in the Kondo effect. Remarkably, the Ikeda—Miyake (IM) state appears as the most stable mean-field solution, but we also find a new stable state with an octahedrally symmetric node distribution that appears to fit the properties of these Kondo insulators more precisely. If Coulomb interactions suppress shape fluctuations of the f-site, then only pure spin exchange survives, to give H"H !J cR ( j) f ( j) f R( j)c ( j), (1) ?? ? @ ?@ H ? where a"1, 2, 3 and a"$1 label the crystal field partial wave states according to “shape” and pseudo-spin, respectively. H " k ekcRk ck describes a featureless con N N N duction sea. By using a mean-field approach we arrive at the following Hamiltonian: » H*"H #» [ N? , ˆ (k)cR k f #h.c.]#N A N ?I J k
(2)
where cˆ "(c , c , c ) is a unit vector which selects the particular linear combination of channels partaking in the Kondo effect, and N? (k)" c 1kK p"aa2 describes the Aˆ ? ? overlap between this channel and a Bloch wave "kp2. The
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 8 8 3 - 7
J. Moreno, P. Coleman / Physica B 259—261 (1999) 190—192
mean field ground-state energy is then found to be
» *E "2N(0)» ln #F[cˆ ] , (3) ¹ ) where ¹ "(eD)e\ ,(, N(0) is the conduction elec) tron density of states. The term F[cˆ ] is given by
F[cˆ ]"
d)kK U ˆ (kK ) ln [U ˆ (kK )], A 4n A
(4)
where U ˆ (kK )"() " N? (k)" contains all details of the A ? N cˆ gap anisotropy. The weak logarithmic singularity inside the integrand minimizes the ground-state energy in gapless Kondo insulating configurations. The IM state has the lowest free-energy. But this theory also identifies a new locally stable state. This state is almost octahedral. Like the IM state, the hybridization drops exactly to zero along the zˆ -axis. But, in marked difference with the IM
Fig. 1. Normalized thermal conductivity versus temperature along the zˆ -axis (solid line) and in the basal plane (dashed line). Top: for the Ikeda—Miyake state. Bottom: for the quasi-octahedral scenario. Insets show density of states as a function of the energy.
191
state, it almost vanishes along the (1, 1, 0) and (1, !1, 0) directions in the basal plane. The inset in Fig. 1 shows the density of states predicted by the two candidates. Although both are gapless, the v-shaped pseudo-gap of the quasi-octahedral state is far more pronounced than that of the IM state, and it is closer in character to the observed tunneling density of states [5,9]. A more direct probe of the anisotropy is provided by the thermal conductivity [6]. In these calculations it is essential to include the momentum dependence of the hybridization potential in the evaluation of the quasiparticle current. Previous calculations [1] underestimated the anisotropy by neglecting these important contributions. The single node in the IM state leads to a pronounced enhancement of the low-temperature thermal conductivity along the nodal zˆ -axis. By contrast, in the quasioctahedral state the distribution of minima in the gap gives rise to a modest enhancement of the thermal conductivity in the basal plane. Experimental measurements [6] have shown than near 5 K the enhancement is much more pronounced in i than in i or i . This fact tends to V X W favor the quasi-octahedral scenario. Another respect in which the two states differ is in the spectrum of low-energy spin excitations. Fig. 2 displays Im sXX(q, u) at fixed momentum transfer q"(0.5, 0, 0). The IM state shows a broad maximum. However, the quasi-octahedral state displays a sharp peak with much higher intensity. This peak is associated to transitions between quasiparticle states along the (1, 1, 0) and (1, !1, 0) directions.
Fig. 2. Imaginary part of the magnetic susceptibility for IM state (solid lime) and quasi-octahedral state (dot-dashed line). In the inset Re/sXX(q, u"0) as a function of momentum transfer.
192
J. Moreno, P. Coleman / Physica B 259—261 (1999) 190—192
The inset of Fig. 2 shows that the static susceptibility of both states is weakly dependent on q. In this respect both states manifest the correct physical behavior [7]. However, the quasi-octahedral state displays a sharp leading edge in the spectrum, a feature which is more closely in accord with experiments than the broad hump predicted for the axial state. To conclude, we have proposed a mechanism for the dynamical generation of a nodal hybridization gap in the narrow-gap Kondo insulators. By supposing that Coulomb interactions drive the shape fluctuations of the Cerium f-state to high energies we derive an effective three-channel Kondo lattice model. Interference between the Kondo effect in the different channels selects a state with a nodal anisotropy. Our theory predicts two stable states, one axial, the other quasi-octahedral in symmetry. The quasi-octahedral solution appears to be the most promising explanation of the various properties of these materials.
We are grateful to Gabriel Aeppli, Frithjof Anders, Yoshio Kitaoka, Toshiro Takabatake and Adolfo Trumper of enlightened discussions. This research was partially supported by NSF grant DMR 96-14999.
References [1] H. Ikeda, K. Miyake, J. Phys. Soc. Japan 65 (1996) 1769. [2] P.A. Aleksee et al., JETP 79 (1994) 665. [3] Y. Kagan, K.A. Kikoin, N.V. Prokof’ev, JETP Lett. 57 (1993) 600. [4] T.S. Kim, D.L. Cox, Phys. Rev. Lett. 75 (1995) 1622. [5] T. Ekino et al., Phys. Rev. Lett. 75 (1995) 4262. [6] M. Sera et et al., Phys. Rev. B 55 (1997) 6421. [7] A. Schro¨der et al., electronic archive xxx.lanl.gov/condmat/9611132. [8] Yu. Kagan, K.A. Kikoin, A.S. Mishchenko, Phys. Rev. B 55 (1997) 12348. [9] D.N. Davydov et al., Phys. Rev. B 55 (1997) R7299.