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Possible effect of exciton condensation on properties of ferromagnetic metals
P O S S I B L E E F F E C T O F EXCITON C O N D E N S A T I O N ON P R O P E R T I E S O F F E R R O M A G N E T I C METALS M. I. K A T S N E L S O N and S. V. VONSOVSKII Institute of Metal Physics, Ural Science Research Centre, USSR Academy of Sciences, GSP-170, Sverdlovsk, USSR
In the half-filled Hubbard band in the metallic phase we consider the possibility of electron separation into Fermi liquid and exciton liquid. In the orbital degeneracy case excitons form that possess a magnetic moment. These can contribute considerably to paramagnetic susceptibility, lead to additional resistivity, and can produce ferromagnetic drops.
Much progress has recently been made in the spin fluctuation theory of itinerant magnets [1-5] which combines the best features of the localized and band models (see [6, 7]). Along with developing semiphenomenological techniques, it is of interest to try to understand the microscopic causes due to which electrons separate into itinerant and localized ones and to give a clear physical image of spin fluctuations. We have recently proposed an interpolation scheme between the cases of strong and weak coupling in the half-filled Hubbard model describing the possibility of partial exciton condensation in the metallic phase [8]. Consider briefly the results in [8] relating to the simple cubic lattice case. Use was made of the Hubbard Hamiltonian (which is, as shown in [8], a particular case of the Shubin-Vonsovskii polar model [9])
H = t3 E
CioCi+Ao + "[-
iAo
U E niTnit, i
(l)
where ci~+ and cio are the creation and annihilation operators of the electron on site i with spin projection a, A the numbers of next nearest neighbours in site i, and nio = cio+cio. The state of complete ferromagnetic saturation I~0> in the half-filled Hubbard model is not the ground state. To consider the state decay channels we solve the Schrrdinger equation for states with one inverted spin [~> = ~ a(i,j)ci~cj~ld~o>.
(2)
tJ
There exist two types of such states: (1) current-carrying states corresponding to the independent propagation of a double (twice occupied site) and a hole (their energy counted from the energy of state ]qb0> lies in the range ( U - 1 2 I t3[, U + 12[ t3 [)) and (2) currentless states (excitons) representing a bound pair of a double and a hole with
energy E < 0 (this is what proves the instability of state I~0>) and quasimomentum ~:. The energy of the exciton becomes minimal when r0 = (±Tr, ±~r, ±~r). In the case of 61t31 << U w e obtain a correct description of the antiferromagnetic state if it is regarded as a set of excitons with quasimomentum K0 that have formed independently. It is reasonable to describe this system as an exciton crystal where the amplitude of zero temperature oscillations is small compared to the lattice period. With 61/31 << U the exciton is just a single spin and the "zero temperature oscillations" are due to multiplicity indeterminacy in the Nrel anticollinear antiferromagnet [10, 11]. Free carriers arise in the system when [ fl I > fl~ = 0.1264 U. At [ t3 [ --~/3¢ + 0 their concentration is of the order of (1 fll/fl~ 1)3/2. However, excitons occur in the metallic phase too, their quasimomentum is determined by the condition I/9 Isinx0i/2 = - t3~
(3)
(i = x, y, z). Since the problem contains no small parameters when I t I - t o , the amplitude of zero temperature oscillations of excitons becomes of the order of the lattice period, the exciton superstructure collapses, and the excitons form a liquid similar to that described by Keldysh [12]. It should be emphasized that not only the quasimomentum but also the velocity of excitons in the condensate is other than zero. When It31 >> tc the number of excitons and their quasimomenta tend to zero and one may speak of an exciton gas. In the Hubbard degenerate model case the production of triplet excitons in the case of small It1 (see [13, 14] and Cyrot's paper in [7]) is energetically more advantageous (because of the Hund exchange). In the metallic phase they should not bring about orbital antiferromagnetism, in contrast to [13, 14], owing to the melting of the exciton
Journal of Magnetism and Magnetic Materials 15-18 (1980) 275-276 ©North Holland
215
M. I. Katsnelson and S. V. Vonsovskii/ Excitons in ferromagnetic metals
276
superlattice. Excitons possessing the magnetic moment/x do not obey the Fermi statistic and therefore their contribution to the magnetic susceptibility per exciton at temperature T is
above show that the formation of a superstructure is not indispensable (just like ferromagnetism, excitons also can exist in the crystal structure prevailing in the paramagnetic phase).
~2
X
T+O
(4)
where O is of the order of the fluctuating disordering field acting on the exciton (in this case, of the order of the exchange interaction of the exciton with the Fermi liquid subsystem formed by unbound electrons). If 8 is small compared to the Fermi energy E F we obtain an enhancement of the susceptibility at T << O and a Curie-Weiss type behaviour at T ~> 0 << E F, i.e. features peculiar to spin fluctuation theory. Furthermore, excitons are capable of producing ferromagnetic drops to be detected using neutron scattering and probably spin wave resonance techniques. Finally, the presence in the ground state of an exciton gas (or liquid) upon which current carriers may be scattered would lead to the occurrence of residual resistivity independent of the impurity concentration and other defects in the specimen. Also, excitons should lead to the occurrence of new optical absorption bands whose interpretation, however, is hardly possible since the picture of optical spectra of metals is intricate. Let it be noted in conclusion that the theory of exciton ferromagnetism was considered earlier [15] for those cases when some peculiarities of the electron energy spectrum (flat portions of the Fermi surface, etc.) lead to exciton instability with the interaction being as weak as desired and use can be made of the H a r t r e e - F o c k approximation. In this model, exciton ferromagnetism should be accompanied by the formation of spin and charge density waves and by the occurrence of structural instabilities. The qualitative considerations about the melting of the exciton superlattice which we have given
References I. K. K. Murata and S. Doniach, Phys. Rev. Lett. 29 (1972) 285; K. K. Murata, Phys. Rev. B12 (1975) 282. 2. T. Moriya and A. Kawabuta, J. Phys. Soc. Jap. 34 (1973) 639. 3. I. E. Dzialoshinskii and P. S. Kondratenko, ZhETF 70 (1976) 1987. 4. T. Moriya, Physiea 86-88B (1976) 356; Physica, 91B (1977) 235. 5. T. Moriya and Y. Takahashi, J. Phys. Soc. Jap. 45 (1978) 397; T. Moriya, Solid State Commun. 26 (1978) 483. 6. C. Herring, Magnetism vol. 4 (Academic Press New York, 1966). 7. Itinerant-Electron Magnetism, Proc. Int. Conf., Oxford (1976), Physica 91B (1977). 8. S. V. Vonsovskii and M. 1. Katsnelson, J. Phys. C12 (1979) 2055; Collected Papers devoted to 100th anniversary of A. F. Ioffe, Leningrad, to be printed. 9. S. P. Shubin and S. V. Vonsovskii, Proc. Roy. Soc. A145 (1934) 159. 10. S. V. Vonsovskii, Magnetism, vol. 2 (J. Wiley, New York, 1974). il. P. W. Anderson, Phys. Rev. 83 (1951) 1260; S. V. Vonsovskii and M. S. Svirskii, ZhETF 57 (1969) 251. 12. L. V. Keldysh, in: Collected Papers: Eksitony v poluprovodnikakh, (Nauka, 1971) p. 5. 13. K. 1. Kugel and D. I. Khomskii, ZhETF 64 (1973) 1429. 14. C. L. Caen-Lyon and M. Cyrot, J. Magn. Magn. Mat. 5 (1977) 142. 15. B. A. Volkov and Yu. V. Kopaev, Pis'ma ZhETF 19 (1974) 168; B. A. Volkov, Yu. V. Kopaev and A. I. Rusinov, ZhETF 68 (1975) 1899; B. A. Volkov, A. I. Rusinov and R. Kh. Timerov, ZhETF 70 (1976) 1130; T. G. Tratas, FTT 20 (1978) 1766; B. A. Volkov Trans. FIAN 104 (1978) 3.
In the half-filled Hubbard band in the metallic phase we consider the possibility of electron separation into Fermi liquid and exciton liquid. In the orbital degeneracy case excitons form that possess a magnetic moment. These can contribute considerably to paramagnetic susceptibility, lead to additional resistivity, and can produce ferromagnetic drops.
Much progress has recently been made in the spin fluctuation theory of itinerant magnets [1-5] which combines the best features of the localized and band models (see [6, 7]). Along with developing semiphenomenological techniques, it is of interest to try to understand the microscopic causes due to which electrons separate into itinerant and localized ones and to give a clear physical image of spin fluctuations. We have recently proposed an interpolation scheme between the cases of strong and weak coupling in the half-filled Hubbard model describing the possibility of partial exciton condensation in the metallic phase [8]. Consider briefly the results in [8] relating to the simple cubic lattice case. Use was made of the Hubbard Hamiltonian (which is, as shown in [8], a particular case of the Shubin-Vonsovskii polar model [9])
H = t3 E
CioCi+Ao + "[-
iAo
U E niTnit, i
(l)
where ci~+ and cio are the creation and annihilation operators of the electron on site i with spin projection a, A the numbers of next nearest neighbours in site i, and nio = cio+cio. The state of complete ferromagnetic saturation I~0> in the half-filled Hubbard model is not the ground state. To consider the state decay channels we solve the Schrrdinger equation for states with one inverted spin [~> = ~ a(i,j)ci~cj~ld~o>.
(2)
tJ
There exist two types of such states: (1) current-carrying states corresponding to the independent propagation of a double (twice occupied site) and a hole (their energy counted from the energy of state ]qb0> lies in the range ( U - 1 2 I t3[, U + 12[ t3 [)) and (2) currentless states (excitons) representing a bound pair of a double and a hole with
energy E < 0 (this is what proves the instability of state I~0>) and quasimomentum ~:. The energy of the exciton becomes minimal when r0 = (±Tr, ±~r, ±~r). In the case of 61t31 << U w e obtain a correct description of the antiferromagnetic state if it is regarded as a set of excitons with quasimomentum K0 that have formed independently. It is reasonable to describe this system as an exciton crystal where the amplitude of zero temperature oscillations is small compared to the lattice period. With 61/31 << U the exciton is just a single spin and the "zero temperature oscillations" are due to multiplicity indeterminacy in the Nrel anticollinear antiferromagnet [10, 11]. Free carriers arise in the system when [ fl I > fl~ = 0.1264 U. At [ t3 [ --~/3¢ + 0 their concentration is of the order of (1 fll/fl~ 1)3/2. However, excitons occur in the metallic phase too, their quasimomentum is determined by the condition I/9 Isinx0i/2 = - t3~
(3)
(i = x, y, z). Since the problem contains no small parameters when I t I - t o , the amplitude of zero temperature oscillations of excitons becomes of the order of the lattice period, the exciton superstructure collapses, and the excitons form a liquid similar to that described by Keldysh [12]. It should be emphasized that not only the quasimomentum but also the velocity of excitons in the condensate is other than zero. When It31 >> tc the number of excitons and their quasimomenta tend to zero and one may speak of an exciton gas. In the Hubbard degenerate model case the production of triplet excitons in the case of small It1 (see [13, 14] and Cyrot's paper in [7]) is energetically more advantageous (because of the Hund exchange). In the metallic phase they should not bring about orbital antiferromagnetism, in contrast to [13, 14], owing to the melting of the exciton
Journal of Magnetism and Magnetic Materials 15-18 (1980) 275-276 ©North Holland
215
M. I. Katsnelson and S. V. Vonsovskii/ Excitons in ferromagnetic metals
276
superlattice. Excitons possessing the magnetic moment/x do not obey the Fermi statistic and therefore their contribution to the magnetic susceptibility per exciton at temperature T is
above show that the formation of a superstructure is not indispensable (just like ferromagnetism, excitons also can exist in the crystal structure prevailing in the paramagnetic phase).
~2
X
T+O
(4)
where O is of the order of the fluctuating disordering field acting on the exciton (in this case, of the order of the exchange interaction of the exciton with the Fermi liquid subsystem formed by unbound electrons). If 8 is small compared to the Fermi energy E F we obtain an enhancement of the susceptibility at T << O and a Curie-Weiss type behaviour at T ~> 0 << E F, i.e. features peculiar to spin fluctuation theory. Furthermore, excitons are capable of producing ferromagnetic drops to be detected using neutron scattering and probably spin wave resonance techniques. Finally, the presence in the ground state of an exciton gas (or liquid) upon which current carriers may be scattered would lead to the occurrence of residual resistivity independent of the impurity concentration and other defects in the specimen. Also, excitons should lead to the occurrence of new optical absorption bands whose interpretation, however, is hardly possible since the picture of optical spectra of metals is intricate. Let it be noted in conclusion that the theory of exciton ferromagnetism was considered earlier [15] for those cases when some peculiarities of the electron energy spectrum (flat portions of the Fermi surface, etc.) lead to exciton instability with the interaction being as weak as desired and use can be made of the H a r t r e e - F o c k approximation. In this model, exciton ferromagnetism should be accompanied by the formation of spin and charge density waves and by the occurrence of structural instabilities. The qualitative considerations about the melting of the exciton superlattice which we have given
References I. K. K. Murata and S. Doniach, Phys. Rev. Lett. 29 (1972) 285; K. K. Murata, Phys. Rev. B12 (1975) 282. 2. T. Moriya and A. Kawabuta, J. Phys. Soc. Jap. 34 (1973) 639. 3. I. E. Dzialoshinskii and P. S. Kondratenko, ZhETF 70 (1976) 1987. 4. T. Moriya, Physiea 86-88B (1976) 356; Physica, 91B (1977) 235. 5. T. Moriya and Y. Takahashi, J. Phys. Soc. Jap. 45 (1978) 397; T. Moriya, Solid State Commun. 26 (1978) 483. 6. C. Herring, Magnetism vol. 4 (Academic Press New York, 1966). 7. Itinerant-Electron Magnetism, Proc. Int. Conf., Oxford (1976), Physica 91B (1977). 8. S. V. Vonsovskii and M. 1. Katsnelson, J. Phys. C12 (1979) 2055; Collected Papers devoted to 100th anniversary of A. F. Ioffe, Leningrad, to be printed. 9. S. P. Shubin and S. V. Vonsovskii, Proc. Roy. Soc. A145 (1934) 159. 10. S. V. Vonsovskii, Magnetism, vol. 2 (J. Wiley, New York, 1974). il. P. W. Anderson, Phys. Rev. 83 (1951) 1260; S. V. Vonsovskii and M. S. Svirskii, ZhETF 57 (1969) 251. 12. L. V. Keldysh, in: Collected Papers: Eksitony v poluprovodnikakh, (Nauka, 1971) p. 5. 13. K. 1. Kugel and D. I. Khomskii, ZhETF 64 (1973) 1429. 14. C. L. Caen-Lyon and M. Cyrot, J. Magn. Magn. Mat. 5 (1977) 142. 15. B. A. Volkov and Yu. V. Kopaev, Pis'ma ZhETF 19 (1974) 168; B. A. Volkov, Yu. V. Kopaev and A. I. Rusinov, ZhETF 68 (1975) 1899; B. A. Volkov, A. I. Rusinov and R. Kh. Timerov, ZhETF 70 (1976) 1130; T. G. Tratas, FTT 20 (1978) 1766; B. A. Volkov Trans. FIAN 104 (1978) 3.