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Conductivity and magnetic properties of one-dimensional Heisenberg-Kondo lattice
Synthetic Metals 103 (1999) 2149-2150
Conductivity
and magnetic
properties
of one-dimensional
T. Ogawa*, Institute
for
Molecular
Heisenberg-Kondo
K. Yonemitsu Okazaki 444-8585,
Science,
lattice
Japan
Abstract Using the one-dimensional quarter-filled Heisenberg-Kondo lattice model for a ?r-d electron system, CoPc(AsFe)c.s, we study the effects of long-range Coulomb interaction among x electrons on the conductivity. The density-matrix renormalization-group method is employed. As a first step, we regard the localized spins of d electrons as classical ones, and then, as quantum S=1/2 spins, which are coupled antiferromagnetically with each other. In the former case, d electrons produce a static staggered magnetic field for sr electrons. It is pointed out that the next-nearest-neighbor repulsion among A electrons and the magnetic coupling between x and d electrons are essential to open a charge gap. Keywords: Computer simulations, Many-body and quasiparticle theories, Metal-insulator phase transitions, Magnetic phase transitions, Organic conductors based on radical cation and/or anion salts
1. Introduction The phthalocyanine (PC) compounds CoPc(AsFs)s.a and NiPc(AsFs)o.s are r-d and 7r electron systems, respectively, with one-dimensional crystal structure [l]. The x orbitals of PC make a 3/4-filled band. NiPc(AsFs)a,s has nonmagnetic Nit2 ions and shows metallic behavior down to 40K. CoPc(AsFs)o.s has magnetic Cot2 (S = l/2) ions and is non-metallic. No clear evidence for lattice distortion is found. Therefore, magnetic coupling between A and d electrons would be responsible for the non-metallic ground state of the latter system. The magnetic property of the Heisenberg-Kondo lattice model is already reported [2]. The purpose of this study is to clarify how this difference in the conductivity occurs. Since previous studies show no insulating behavior in the partially-filled Heisenberg-Kondo lattice model [3,4], long-range Coulomb interaction is expected to be important. Then, we investigate the combined effects of long-range Coulomb interaction and the magnetic coupling between r and d electrons.
2. Model We add nearest- and next-nearest-neighbor teractions to the Heisenberg-Kondo lattice hole representation [5,6],
repulsive inmodel with the
* This work was supported by a Grant-in-Aid for JSPS Fellows from the Ministry of Education, Science, Sports and Culture, Japan.
+J&?,d.
$
+ JH.?f.
s’pl
.
(1)
1
where ct,. creates a x electron with spin u at site i, and s’rr = (c?tT, ciL)a’(ciT, t cil)* denotes the spin of the x electron. S> is the spin of the Co d electron at site i, which is coupled antiferromagnetically ( JH > 0) with the nearest-neighbor ones through the superexchange interaction. The d and x electrons are coupled ferromagnetically (Jr< < 0) since the corresponding orbitals are orthogonal. We obtain the ground states using the density-matrix renormalization-group (DMRG) method to study whether they have a finite charge gap or not in the thermodynamic limit, NL ---t oo, with NL being the number of sites. In this work, the charge gap is tentatively defined by dE(n,,
5”)
=
(E(n,
-=(ne,
+ 2, S’)
sz))/4
+ E(n,
,
- 2, Sz)
(2)
where E(n,, Sz) is the ground state energy for the total electron number, ner and the z-component of the total spin, 5”. Note that the total spin is not a conserved quantity when we take an anisotropic Heisenberg coupling below. The extrapolation to NL -f 00 is achieved by fitting of the data (NL < 120) to the fourth order with respect to I/NL. First, we regard the localized spins of d electrons as classical ones. Assuming that the antiferromagnetic interaction is strong enough, we fix the spins at a NCel ordered state. Then, the model becomes equivalent to the extended Hubbard model under a static staggered magnetic field, Ws = ]J1,-/41. The Brillouin zone is halved and the band becomes effectively half-filled. However, it is easily shown that the umklapp process~ between antiparallel spins is absent, gal = 0. The staggered magnetic field does not modify the charge excitation spectrum in the low-energy limit.
0379-6779/%X$ - see front matter 0 1999 Elsevier Science S.A. All rights reserved. PII: SO379-6779(98)00618-3
T. Ogawa,
2150
K. Yonemitsu
I Synthetic
Therefore, the on-site repulsion, U, is not effective on opening a charge gap. The nearest-neighbor repulsion, VI, can open a charge gap, but it must be strong enough.- This is explained by writing the interaction in momentum space: it has a small factor, cos p, where p is about 2kp for low-energy excitation processes. Then, the next-nearest-neighbor repulsion, I/;, plays a very important role. The umklapp scattering strength is given, in the standard notation, by
Metals
103 (1999)
2149-2150
insulator with extrinsic orbit couplings, etc.
effects such as impurities
or spin-
r
V*=lO
In the original Heisenberg-Kondo lattice model with localized S = l/2 spins, quantum fluctuations make the antiferromagnetic order of localized spins short-ranged. Thus, we search the possibility by numerical calculations.
.
J,z=50 J,=-4 0 0
0.04
0.08
0.12
i/N,
3. Results First, we show the charge gap of the extended Hubbard model under the staggered magnetic field W, = 1 in the thermodynamic limit, It is confirmed that a charge gap does not open by only on-site repulsion, U, in any strength, or by small nearest-neighbor repulsion, VI < 2 (not shown). When the nearest-neighbor repulsion is strong, VI > 2, a 4kF charge density wave is formed, leading to an insulator [7]. The next-nearest-neighbor repulsion, Vz, opens a charge gap, no matter how small V, is, as shown in Fig. 1.
0
2
4
6
8
10
Fig. 1. Charge gap of the extended Hubbard model for U = Vi = 0 under the staggered magnetic field W, = 1. Next, we calculate the charge gap of the HeisenbergKondo lattice model, where the antiferromagnetic coupling between localized S = l/2 spins is generally taken to be anisotropic ( Jgy 5 J:). Since numerical performance is not yet tuned enough, the precision is limited. Fig. 2 shows the results for NL = 80 and the number of density matrix eigenstates retained per block, m 2 120. The results in the Ising limit, Jiy = 0, are almost the same as in the corresponding extended Hubbard model under the staggered magnetic field. With increasing Jiy, the charge gap decreases. It is not yet certain whether the charge gap survives in the isotropic limit, Jzy = Jg = 50: and in the thermodynamic limit. It is highly plausible, however, the next-nearest-neighbor repulsion easily makes the system an
Fig. 2. Charge gap of the anisotropic Heisenberg-Kondo lattice model for U = VI = 0, Vz = 10, Jr< = -4, and J; = 50.
4. Summary We study the ground state of the one-dimensional extended Hubbard model under a staggered magnetic field and the (anisotropic) Heisenberg-Kondo lattice model, which are extended to include on-site, nearest-neighbor and nextnearest-neighbor Coulomb repulsions. For the former system, it is demonstrated that the next-nearest-neighbor repulsion brings about umklapp processes at quarter filling with the help of the staggered field, so that a charge gap opens and the system becomes an insulator. For the latter system, the Ising-like coupling between ?r and d electrons helps opening a charge gap. However, it is beyond the present numerical precision whether the charge gap survives or not in the isotropic Heisenberg limit. The authors are grateful to J. Kishine for helpful discussions. References K. Yakushi, H. Yamakado, T. Ida, A,Ugawa, Solid State Commun 54 (1991) 919; K. Yakushi, T. Hiejima, H. Yamakado, Materials and Measurements in Molecular Electronics (ed. by K. Kajimura, S. Kuroda, Springer Proceedings in Physics) 81 (1996) 2b3. K.Yonemitsu, J. Kishine, T. Ogawa, Rev. High Pressure Sci. Technol. 7 (1998) 490. A.E. Sikkema, I. Afleck, S.R. White, Phys. Rev. Lett. 79 (1997) 929. D. Poilblanc, S. Yunoki, S. Maekawa, E. Dagotto, Phys. Rev. 3 56 (1997) 1645. B. Guay, L.G. Caron, C. Bourbon&s, Synth. Met. 29 (1989) E557. A. Mishima, Synth. Met. 55-57 (1993) 1815. S. Yunoki, J. Hu, A.L. Malvezzi, A. Morio, N. Furukawa, E. Dagotto, Phys. Rev. L&t. 80 (1998) 845.
Conductivity
and magnetic
properties
of one-dimensional
T. Ogawa*, Institute
for
Molecular
Heisenberg-Kondo
K. Yonemitsu Okazaki 444-8585,
Science,
lattice
Japan
Abstract Using the one-dimensional quarter-filled Heisenberg-Kondo lattice model for a ?r-d electron system, CoPc(AsFe)c.s, we study the effects of long-range Coulomb interaction among x electrons on the conductivity. The density-matrix renormalization-group method is employed. As a first step, we regard the localized spins of d electrons as classical ones, and then, as quantum S=1/2 spins, which are coupled antiferromagnetically with each other. In the former case, d electrons produce a static staggered magnetic field for sr electrons. It is pointed out that the next-nearest-neighbor repulsion among A electrons and the magnetic coupling between x and d electrons are essential to open a charge gap. Keywords: Computer simulations, Many-body and quasiparticle theories, Metal-insulator phase transitions, Magnetic phase transitions, Organic conductors based on radical cation and/or anion salts
1. Introduction The phthalocyanine (PC) compounds CoPc(AsFs)s.a and NiPc(AsFs)o.s are r-d and 7r electron systems, respectively, with one-dimensional crystal structure [l]. The x orbitals of PC make a 3/4-filled band. NiPc(AsFs)a,s has nonmagnetic Nit2 ions and shows metallic behavior down to 40K. CoPc(AsFs)o.s has magnetic Cot2 (S = l/2) ions and is non-metallic. No clear evidence for lattice distortion is found. Therefore, magnetic coupling between A and d electrons would be responsible for the non-metallic ground state of the latter system. The magnetic property of the Heisenberg-Kondo lattice model is already reported [2]. The purpose of this study is to clarify how this difference in the conductivity occurs. Since previous studies show no insulating behavior in the partially-filled Heisenberg-Kondo lattice model [3,4], long-range Coulomb interaction is expected to be important. Then, we investigate the combined effects of long-range Coulomb interaction and the magnetic coupling between r and d electrons.
2. Model We add nearest- and next-nearest-neighbor teractions to the Heisenberg-Kondo lattice hole representation [5,6],
repulsive inmodel with the
* This work was supported by a Grant-in-Aid for JSPS Fellows from the Ministry of Education, Science, Sports and Culture, Japan.
+J&?,d.
$
+ JH.?f.
s’pl
.
(1)
1
where ct,. creates a x electron with spin u at site i, and s’rr = (c?tT, ciL)a’(ciT, t cil)* denotes the spin of the x electron. S> is the spin of the Co d electron at site i, which is coupled antiferromagnetically ( JH > 0) with the nearest-neighbor ones through the superexchange interaction. The d and x electrons are coupled ferromagnetically (Jr< < 0) since the corresponding orbitals are orthogonal. We obtain the ground states using the density-matrix renormalization-group (DMRG) method to study whether they have a finite charge gap or not in the thermodynamic limit, NL ---t oo, with NL being the number of sites. In this work, the charge gap is tentatively defined by dE(n,,
5”)
=
(E(n,
-=(ne,
+ 2, S’)
sz))/4
+ E(n,
,
- 2, Sz)
(2)
where E(n,, Sz) is the ground state energy for the total electron number, ner and the z-component of the total spin, 5”. Note that the total spin is not a conserved quantity when we take an anisotropic Heisenberg coupling below. The extrapolation to NL -f 00 is achieved by fitting of the data (NL < 120) to the fourth order with respect to I/NL. First, we regard the localized spins of d electrons as classical ones. Assuming that the antiferromagnetic interaction is strong enough, we fix the spins at a NCel ordered state. Then, the model becomes equivalent to the extended Hubbard model under a static staggered magnetic field, Ws = ]J1,-/41. The Brillouin zone is halved and the band becomes effectively half-filled. However, it is easily shown that the umklapp process~ between antiparallel spins is absent, gal = 0. The staggered magnetic field does not modify the charge excitation spectrum in the low-energy limit.
0379-6779/%X$ - see front matter 0 1999 Elsevier Science S.A. All rights reserved. PII: SO379-6779(98)00618-3
T. Ogawa,
2150
K. Yonemitsu
I Synthetic
Therefore, the on-site repulsion, U, is not effective on opening a charge gap. The nearest-neighbor repulsion, VI, can open a charge gap, but it must be strong enough.- This is explained by writing the interaction in momentum space: it has a small factor, cos p, where p is about 2kp for low-energy excitation processes. Then, the next-nearest-neighbor repulsion, I/;, plays a very important role. The umklapp scattering strength is given, in the standard notation, by
Metals
103 (1999)
2149-2150
insulator with extrinsic orbit couplings, etc.
effects such as impurities
or spin-
r
V*=lO
In the original Heisenberg-Kondo lattice model with localized S = l/2 spins, quantum fluctuations make the antiferromagnetic order of localized spins short-ranged. Thus, we search the possibility by numerical calculations.
.
J,z=50 J,=-4 0 0
0.04
0.08
0.12
i/N,
3. Results First, we show the charge gap of the extended Hubbard model under the staggered magnetic field W, = 1 in the thermodynamic limit, It is confirmed that a charge gap does not open by only on-site repulsion, U, in any strength, or by small nearest-neighbor repulsion, VI < 2 (not shown). When the nearest-neighbor repulsion is strong, VI > 2, a 4kF charge density wave is formed, leading to an insulator [7]. The next-nearest-neighbor repulsion, Vz, opens a charge gap, no matter how small V, is, as shown in Fig. 1.
0
2
4
6
8
10
Fig. 1. Charge gap of the extended Hubbard model for U = Vi = 0 under the staggered magnetic field W, = 1. Next, we calculate the charge gap of the HeisenbergKondo lattice model, where the antiferromagnetic coupling between localized S = l/2 spins is generally taken to be anisotropic ( Jgy 5 J:). Since numerical performance is not yet tuned enough, the precision is limited. Fig. 2 shows the results for NL = 80 and the number of density matrix eigenstates retained per block, m 2 120. The results in the Ising limit, Jiy = 0, are almost the same as in the corresponding extended Hubbard model under the staggered magnetic field. With increasing Jiy, the charge gap decreases. It is not yet certain whether the charge gap survives in the isotropic limit, Jzy = Jg = 50: and in the thermodynamic limit. It is highly plausible, however, the next-nearest-neighbor repulsion easily makes the system an
Fig. 2. Charge gap of the anisotropic Heisenberg-Kondo lattice model for U = VI = 0, Vz = 10, Jr< = -4, and J; = 50.
4. Summary We study the ground state of the one-dimensional extended Hubbard model under a staggered magnetic field and the (anisotropic) Heisenberg-Kondo lattice model, which are extended to include on-site, nearest-neighbor and nextnearest-neighbor Coulomb repulsions. For the former system, it is demonstrated that the next-nearest-neighbor repulsion brings about umklapp processes at quarter filling with the help of the staggered field, so that a charge gap opens and the system becomes an insulator. For the latter system, the Ising-like coupling between ?r and d electrons helps opening a charge gap. However, it is beyond the present numerical precision whether the charge gap survives or not in the isotropic Heisenberg limit. The authors are grateful to J. Kishine for helpful discussions. References K. Yakushi, H. Yamakado, T. Ida, A,Ugawa, Solid State Commun 54 (1991) 919; K. Yakushi, T. Hiejima, H. Yamakado, Materials and Measurements in Molecular Electronics (ed. by K. Kajimura, S. Kuroda, Springer Proceedings in Physics) 81 (1996) 2b3. K.Yonemitsu, J. Kishine, T. Ogawa, Rev. High Pressure Sci. Technol. 7 (1998) 490. A.E. Sikkema, I. Afleck, S.R. White, Phys. Rev. Lett. 79 (1997) 929. D. Poilblanc, S. Yunoki, S. Maekawa, E. Dagotto, Phys. Rev. 3 56 (1997) 1645. B. Guay, L.G. Caron, C. Bourbon&s, Synth. Met. 29 (1989) E557. A. Mishima, Synth. Met. 55-57 (1993) 1815. S. Yunoki, J. Hu, A.L. Malvezzi, A. Morio, N. Furukawa, E. Dagotto, Phys. Rev. L&t. 80 (1998) 845.