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Charge dynamics of the coupled anisotropic t–J ladders: a model for α′-NaV2O5
Physica B 281&282 (2000) 646}647
Charge dynamics of the coupled anisotropic t}J ladders: a model for a@-NaV O 2 5 S. Nishimoto, Y. Ohta* Department of Physics, Faculty of Science, Chiba University, Inage-ku, Chiba 263-8522, Japan
Abstract We study the charge dynamics of a@-NaV O , a coupled anisotropic ladder system at quarter "lling, by using an 2 5 exact-diagonalization technique on small clusters of the trellis-lattice t}J}< model. We derive values of model parameters, and calculate the optical conductivity and dynamical density correlation function, to show that the charge ordering is caused by the observed anomalous #uctuations of the valence state of V ions. E!ects of Na-de"ciency are also discussed. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: a@-NaV O ; Charge ordering; t}J ladder; Valence #uctuation 2 5
Vanadium bronze a@-NaV O is a compound which 2 5 shows a charge-ordering phase transition at ¹ "34 K # associated with opening of the spin gap [1], where the charge dynamics due to anomalous #uctuations of the valence state of V ions are an intriguing issue [2}9]. In this paper, we use a Lanczos diagonalization of small clusters to study our proposed model of coupled anisotropic t}J ladders at quarter "lling [4], and discuss the charge dynamics of the system [6}8].
"lling (0.5 electrons per site) or less to simulate a@Na V O where there are (1!x)/2 electrons per 1~x 2 5 V ion. We use values of the ladder parameters obtained in Ref. [4]: t "0.298, t "0.140, J "0.0487, and M @@ M J "0.0293 in units of eV. We assume t "0.05 and @@ xy J "0.0025 eV for simplicity, and the values of < , < , xy xy @@ and < are varied for simulating various situations. M 2. Optical conductivity
1. Generic model From the mapping of the d}p model, we propose that a generic model for low-energy electronic states of the materials be the trellis-lattice t}J}< model, i.e., the coupled anisotropic t}J ladders [4]. The Hamiltonian contains the hopping (t ), exchange (J ), and Coulombic ij ij (< ) parameters de"ned as t , J , and < for the rungs of ij M M M the ladders, as t , J , and < for the legs of the ladders, @@ @@ @@ and as t , J , and < for the zigzag-chain bonds conxy xy xy necting the ladders. We consider the case at quarter
* Corresponding author. Tel.: #81-43-290-2755; fax: #8143-290-2874. E-mail address: [email protected] (Y. Ohta)
We have pointed out [6] that the experimental features in the optical conductivity spectra at a frequency range of uK0.6}2.5 eV [10,11], including the positions of the main peak and its shoulders, as well as the anisotropy in the spectral weight, observed over a wide temperature range above and below ¹ , can be reproduced by our # model with reasonable parameter values, if we assume that the system is in the charge disproportionated ground state or in the situation where the valence state is slowly #uctuating [6,12] (see Fig. 1). The same conclusions are obtained recently by a "nite-temperature calculation of the model [13]. The observed valence state of V ions at ¹'¹ is # known to depend on experiments: the structural studies at room temperature consistently indicate the uniform valence of V4.5`, while the optical measurements indicate the disproportionated valence of V4` and V5`. The
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 0 3 4 - 0
S. Nishimoto, Y. Ohta / Physica B 281&282 (2000) 646}647
Fig. 1. Doping dependence of the optical conductivity p(u) for the trellis-lattice t}J model without imbalance of site energies but with nonzero < terms. Upper and lower panels are for the quarter "lling and 25% doping levels, respectively. Solid and dotted curves correspond to the polarizations EDDa and EDDb, respectively.
di!erence in the time scale of the measurements is responsible for this discrepancy, and thus the #uctuations of the valence state of V ions and their quantitative description are an important issue [6}8,13]. 3. Dynamical density correlation We calculate the dynamical density correlation functions of the dimerized t}J chain and coupled anisotropic t}J ladders (trellis lattice) at quarter-"lling, i.e., the systems regarded as a network of pairs (dimers or rungs) of sites coupled weakly via the hopping and exchange interactions. We thereby demonstrate that the intersite Coulomb repulsions between the pairs induce a lowenergy collective mode in the charge excitations of the systems where the internal charge degrees of freedom of the pairs play an essential role. We discuss implications to the electronic states of a@-NaV O , i.e., #uctuations of 2 5 the valence state of V ions and phase transition as a charge ordering [7]. 4. E4ects of Na-de5ciency A Na-de"cient phase of a@-Na V O exists in the 1~x 2 5 composition range 0)x)0.2 [14]. To clarify the e!ects of Na-de"ciency, we study our model with the "lling of electrons less than 1 [8]. Our calculated results suggest 4 that the long-range charge order seems to be destroyed at least at 12.5% doping levels (single-hole results) as the equal-time charge correlation functions indicate, but we note that the short-range correlations still remain strong. The spectral-weight transfer due to doping observed in the calculated single-particle spectra operates just as in
Fig. 2. Schematic phase diagram of a@-Na
647
V O . 1~x 2 5
the case of doping of Mott insulators, and large energyscale reconstruction of the electronic states due to doping can also be observed in the calculated optical conductivity spectra. Small Drude weight obtained is also relevant. All of these results seem to be due to strong quantum #uctuations of charges in relation to the presence of the rung (or dimer) structure of the system. In summary, we have studied the coupled anisotropic t}J ladders for a@-NaV O , putting particular emphasis 2 5 on its charge degrees of freedom. Schematic phase diagram is shown in Fig. 2. As for the charge-ordering pattern at ¹(¹ , we point out that a delicate balance of # long-range Coulombic forces is important as our calculated Madelung energies indicate [8]. Acknowledgements We acknowledge the use of computer centers of the Institute for Solid State Physics, University of Tokyo, the Institute for Molecular Science, Okazaki, and Yukawa Institute for Theoretical Physics, Kyoto University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
T. Ohama et al., Phys. Rev. B 59 (1999) 3299. P. Thalmeier, P. Flude, Europhys. Lett. 44 (1998) 242. H. Seo, H. Fukuyama, J. Phys. Soc. Japan 67 (1998) 2602. S. Nishimoto, Y. Ohta, J. Phys. Soc. Japan 67 (1998) 2996. M.V. Mostovoy, D.I. Khomskii, unpublished. S. Nishimoto, Y. Ohta, J. Phys. Soc. Japan 67 (1998) 3679. S. Nishimoto, Y. Ohta, J. Phys. Soc. Japan 67 (1998) 4010. S. Nishimoto, Y. Ohta, in preparation. J. Riera, D. Poilblanc, Phys. Rev. B 59 (1999) 2667. S.A. Golubchik et al., J. Phys. Soc. Japan 66 (1997) 4042. A. Damascelli et al., Phys. Rev. Lett. 81 (1998) 918. S. Nishimoto, Ph.D. Thesis, Chiba University, 1999. M. Cuoco et al., cond-mat/9906169. M. Isobe, Y. Ueda, J. Alloys Compounds 262}263 (1997) 180.
Charge dynamics of the coupled anisotropic t}J ladders: a model for a@-NaV O 2 5 S. Nishimoto, Y. Ohta* Department of Physics, Faculty of Science, Chiba University, Inage-ku, Chiba 263-8522, Japan
Abstract We study the charge dynamics of a@-NaV O , a coupled anisotropic ladder system at quarter "lling, by using an 2 5 exact-diagonalization technique on small clusters of the trellis-lattice t}J}< model. We derive values of model parameters, and calculate the optical conductivity and dynamical density correlation function, to show that the charge ordering is caused by the observed anomalous #uctuations of the valence state of V ions. E!ects of Na-de"ciency are also discussed. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: a@-NaV O ; Charge ordering; t}J ladder; Valence #uctuation 2 5
Vanadium bronze a@-NaV O is a compound which 2 5 shows a charge-ordering phase transition at ¹ "34 K # associated with opening of the spin gap [1], where the charge dynamics due to anomalous #uctuations of the valence state of V ions are an intriguing issue [2}9]. In this paper, we use a Lanczos diagonalization of small clusters to study our proposed model of coupled anisotropic t}J ladders at quarter "lling [4], and discuss the charge dynamics of the system [6}8].
"lling (0.5 electrons per site) or less to simulate a@Na V O where there are (1!x)/2 electrons per 1~x 2 5 V ion. We use values of the ladder parameters obtained in Ref. [4]: t "0.298, t "0.140, J "0.0487, and M @@ M J "0.0293 in units of eV. We assume t "0.05 and @@ xy J "0.0025 eV for simplicity, and the values of < , < , xy xy @@ and < are varied for simulating various situations. M 2. Optical conductivity
1. Generic model From the mapping of the d}p model, we propose that a generic model for low-energy electronic states of the materials be the trellis-lattice t}J}< model, i.e., the coupled anisotropic t}J ladders [4]. The Hamiltonian contains the hopping (t ), exchange (J ), and Coulombic ij ij (< ) parameters de"ned as t , J , and < for the rungs of ij M M M the ladders, as t , J , and < for the legs of the ladders, @@ @@ @@ and as t , J , and < for the zigzag-chain bonds conxy xy xy necting the ladders. We consider the case at quarter
* Corresponding author. Tel.: #81-43-290-2755; fax: #8143-290-2874. E-mail address: [email protected] (Y. Ohta)
We have pointed out [6] that the experimental features in the optical conductivity spectra at a frequency range of uK0.6}2.5 eV [10,11], including the positions of the main peak and its shoulders, as well as the anisotropy in the spectral weight, observed over a wide temperature range above and below ¹ , can be reproduced by our # model with reasonable parameter values, if we assume that the system is in the charge disproportionated ground state or in the situation where the valence state is slowly #uctuating [6,12] (see Fig. 1). The same conclusions are obtained recently by a "nite-temperature calculation of the model [13]. The observed valence state of V ions at ¹'¹ is # known to depend on experiments: the structural studies at room temperature consistently indicate the uniform valence of V4.5`, while the optical measurements indicate the disproportionated valence of V4` and V5`. The
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 0 3 4 - 0
S. Nishimoto, Y. Ohta / Physica B 281&282 (2000) 646}647
Fig. 1. Doping dependence of the optical conductivity p(u) for the trellis-lattice t}J model without imbalance of site energies but with nonzero < terms. Upper and lower panels are for the quarter "lling and 25% doping levels, respectively. Solid and dotted curves correspond to the polarizations EDDa and EDDb, respectively.
di!erence in the time scale of the measurements is responsible for this discrepancy, and thus the #uctuations of the valence state of V ions and their quantitative description are an important issue [6}8,13]. 3. Dynamical density correlation We calculate the dynamical density correlation functions of the dimerized t}J chain and coupled anisotropic t}J ladders (trellis lattice) at quarter-"lling, i.e., the systems regarded as a network of pairs (dimers or rungs) of sites coupled weakly via the hopping and exchange interactions. We thereby demonstrate that the intersite Coulomb repulsions between the pairs induce a lowenergy collective mode in the charge excitations of the systems where the internal charge degrees of freedom of the pairs play an essential role. We discuss implications to the electronic states of a@-NaV O , i.e., #uctuations of 2 5 the valence state of V ions and phase transition as a charge ordering [7]. 4. E4ects of Na-de5ciency A Na-de"cient phase of a@-Na V O exists in the 1~x 2 5 composition range 0)x)0.2 [14]. To clarify the e!ects of Na-de"ciency, we study our model with the "lling of electrons less than 1 [8]. Our calculated results suggest 4 that the long-range charge order seems to be destroyed at least at 12.5% doping levels (single-hole results) as the equal-time charge correlation functions indicate, but we note that the short-range correlations still remain strong. The spectral-weight transfer due to doping observed in the calculated single-particle spectra operates just as in
Fig. 2. Schematic phase diagram of a@-Na
647
V O . 1~x 2 5
the case of doping of Mott insulators, and large energyscale reconstruction of the electronic states due to doping can also be observed in the calculated optical conductivity spectra. Small Drude weight obtained is also relevant. All of these results seem to be due to strong quantum #uctuations of charges in relation to the presence of the rung (or dimer) structure of the system. In summary, we have studied the coupled anisotropic t}J ladders for a@-NaV O , putting particular emphasis 2 5 on its charge degrees of freedom. Schematic phase diagram is shown in Fig. 2. As for the charge-ordering pattern at ¹(¹ , we point out that a delicate balance of # long-range Coulombic forces is important as our calculated Madelung energies indicate [8]. Acknowledgements We acknowledge the use of computer centers of the Institute for Solid State Physics, University of Tokyo, the Institute for Molecular Science, Okazaki, and Yukawa Institute for Theoretical Physics, Kyoto University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
T. Ohama et al., Phys. Rev. B 59 (1999) 3299. P. Thalmeier, P. Flude, Europhys. Lett. 44 (1998) 242. H. Seo, H. Fukuyama, J. Phys. Soc. Japan 67 (1998) 2602. S. Nishimoto, Y. Ohta, J. Phys. Soc. Japan 67 (1998) 2996. M.V. Mostovoy, D.I. Khomskii, unpublished. S. Nishimoto, Y. Ohta, J. Phys. Soc. Japan 67 (1998) 3679. S. Nishimoto, Y. Ohta, J. Phys. Soc. Japan 67 (1998) 4010. S. Nishimoto, Y. Ohta, in preparation. J. Riera, D. Poilblanc, Phys. Rev. B 59 (1999) 2667. S.A. Golubchik et al., J. Phys. Soc. Japan 66 (1997) 4042. A. Damascelli et al., Phys. Rev. Lett. 81 (1998) 918. S. Nishimoto, Ph.D. Thesis, Chiba University, 1999. M. Cuoco et al., cond-mat/9906169. M. Isobe, Y. Ueda, J. Alloys Compounds 262}263 (1997) 180.