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Phase separation and critical phenomena in the charge ordered system
Solid State Communications 145 (2008) 109–113 www.elsevier.com/locate/ssc
Phase separation and critical phenomena in the charge ordered system G. Pawłowski ∗ , T. Ka´zmierczak Institute of Physics, A. Mickiewicz University, ul. Umultowska 85, 61-614 Pozna´n, Poland Received 30 September 2007; accepted 15 October 2007 by A.H. MacDonald Available online 18 October 2007
Abstract This study is devoted to the phase separation problem in the charge ordered system, presented in a possibly simplest model form. The extended Hubbard model in the atomic limit (t → 0) on a square lattice is analysed using Monte Carlo simulation for the grand canonical ensemble. The model exhibits very rich structure of the phase diagram at finite temperatures as well as in the ground state. Besides the commensurate types of charge order one can observe a mixed phase in T > 0 and states with phase separations or stripes in low temperatures. The results show thus a possible classical nature of the phase separation and striped states for charge order. c 2007 Elsevier Ltd. All rights reserved.
PACS: 71.10.Fd; 71.45.Lr; 64.60.Ak Keywords: D. Charge-order; D. Phase separation
1. Introduction The phase separation phenomenon is a characteristic feature of strongly-correlated electron systems [1]. The presence of such states has been confirmed by the analysis of many model systems. Particularly suitable for this purpose has been the Hubbard model and its extensions. A number of studies for different limiting cases of the Hubbard model (especially in the limit of strong- and weak-coupling) have indicated the presence of states with phase separation [2–6] or striped states [7,8]. Analysis of the extended Hubbard model (EHM) with additional nearest-neighbour Coulomb repulsion leads to the appearance of additional separated states including chargeordered phases [9–11]. The paper is a contribution to the works on the phase separation in the Hubbard model and its extensions and concerns the atomic limit. The zero-overlap limit of the extended Hubbard model, when the Hamiltonian does not contain a hopping integral but only part of the Coulomb interactions, was introduced in 1971 by Bari [12] to describe the charge-orderings in a very narrow half-filled band insulators. Although it is one of the simplest cases of the microscopic ∗ Corresponding author. Tel.: +48 61 8295284; fax: +48 61 8257018.
E-mail address: [email protected] (G. Pawłowski). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.10.015
model, the results have shown a complex and highly nontrivial structure of the phase diagrams comprising the orderings with alternating charge density for the bipartite lattice [13–24]. The model, known also as the atomic limit of EHM (ALEHM), is described by the Hamiltonian X X X H=U n i↑ n i↓ + W ni n j − µ ni , (1) i
ij
i
where n i = n i↑ + n i↓ is the number of electrons on the ith site {0, ↑, ↓, ↑↓}, U represents the on-site and W the intersite Coulomb interaction (here restricted to the nearest-neighbour ones) and µ is the chemical potential. The model properties for general filling have been hitherto poorly recognised. Recently, a few interesting analyses of ALEHM, besides that concerning half-filled bands have been made. Mancini [25] has made a detailed analysis of the model for the 1D case. He has shown the presence of staggered charge order in the region 0 < µ < 4W , and the double peak structure of specific heat and compressibility. Misawa et al. [26] have analysed the critical behaviour in the vicinity of the tricritical points (MC simulation and Hartree–Fock approximation). Their results have been referred directly to the experimental ones obtained for the system (DI–DCNQI)2 Ag where the phase separation along the first-order transition line occurs. The authors gave the phase diagrams over a wide range
110
G. Pawłowski, T. Ka´zmierczak / Solid State Communications 145 (2008) 109–113
of chemical potentials (µ − µhalf ), both for the ground state and in finite temperatures. MC analysis for a square lattice (L = 10–40) for a full range of parameters of the AL-EHM has been reported by GP [27]. Application of the grand canonical ensemble (GCE) approach has permitted finding the n(µ) curves and determination of the phase borders of charge orderings as a function of electron concentration. An important finding reported in this work was detection of the presence of firstorder transitions beyond the half-filling, and consequently the presence of the tricritical point lines. The exclusive presence of orderings commensurable with the lattice (low and high charge order) for the general fillings in the form of spanning checkerboard clusters and stable mixed states (the intermediate charge order) in the range of intermediate concentrations has been shown. Also a preliminary analysis of the ground state has been made. On the basis of the stepwise-type isotherms n(µ) in low temperatures the possibility of phase-separation states has been concluded but not all types of such phases have been identified. This work is a continuation of the earlier analysis aimed at a comprehensive investigation of the vicinity of the ground state. The results presented have been obtained with exceptionally high accuracy. A Monte Carlo simulation for the grand canonical ensemble (GCMC) [29] was made for L = 20 square lattice with periodic boundary conditions. The algorithm includes three sequential update steps: insertion of the particle with randomly chosen spin direction, removal of the particle, and move of the particle (the canonical part). More details of the simulation and the analysis of the finite size effects has been presented in our previous work [27]. Analysis of the thermodynamic characteristics (the internal energy, specific heat, charge order susceptibility) has been performed in three different simulation paths. The first one describes the dependencies as a function of the chemical potential µ for a constant temperature; the second — the dependencies as a function of k B T for a settled average number of electrons in the system; and the third — the dependencies for a fixed value of the chemical potential as a function of k B T . The results obtained for all paths are coherent and permit determination of the accurate phase diagrams for the model. The model predicts many forms of different orderings. Before we define the specific phases, let us define the elementary ordering of the structure: LCO — Low Charge Order (. . . 1010. . . or . . . 1212. . . ), HCO — High Charge Order (. . . 2020. . . ), ICO — Intermediate Charge Order (. . . x0y0 . . . or . . . x2y2 . . ., where x, y = {1, 2}, excluding the case when for the whole ordered domain x = y). In the 2D case, the first — Low Charge Order is realized for a chessboard combination of electrons (holes) and empty (full) sites; the second — High Charge Order refers to electron pairs and empty sites. The last order is a simple combination of LCO and HCO structures, therefore this state can be called the Intermediate Charge Order.
Fig. 1. Isotherms for U/4W = 0.8 and k B T /W = 0.01 ÷ 0.66. The box shows the range of PS at low temperatures. Table 1 Stability ranges of the possible ordered states in the ground state for U ≤ 4W Chemical potential
Electron concentration
State
µ<0 µ=0 0<µ
n = 0.0 0 < n < 0.5 n = 0.5 0.5 < n < 1 n = 1.0
NO Domains of LCO LCO Domains of LCO–HCO HCO
Besides these orderings we observed the disordered states denoted as NO. For n = 1 there is some uniform disordered state, realized as a Mott state (. . . 1111. . . ). Because of the electron–hole symmetry of the model, the range of the hole orderings has not been discussed in this paper (1 < n ≤ 2). 2. The results In this paper we analyse in detail the model properties for U/4W = 0.8. For this value both types of commensurate orderings (LCO, HCO), phase separation states and intermediate orderings (ICO, for T > 0) are possible. Moreover, in finite temperatures and in the concentration range close to n ≈ 1 there are first-order phase transitions and tricritical point [27]. Fig. 1 presents the isotherms for U/4W = 0.8. At low temperatures the stepwise character of the curve µ(n) is observed. The stability ranges of the possible ordered states for the ground state is given in Table 1. At T = 0 only three electron concentrations are available n = 0.0; 0.5; 1.0, for which stable structures can be formed (NO, LCO, HCO). For µ = 0, µ = U there is a jumpwise change in concentration which determines the presence of phase separation states in the ground state. In the range 0 < n < 0.5 and 0.5 < n < 1.0 only the states with homogeneous domain separation occur. Let us note that in these states the total energy E (T = 0) of the system is fixed and depends only on the chemical potential (not on the particle concentration): µ = 0 : E(0) = 0, µ = U : E(0) = −N U/2,
G. Pawłowski, T. Ka´zmierczak / Solid State Communications 145 (2008) 109–113
(a) n = 0.899.
(b) n = 0.930.
(c) n = 0.922.
111
(d) n = 0.998.
Fig. 2. The characteristic projections of the PS states at low temperatures. Denotations: grey squares — one electron per site, black — two electrons per site and white — empty site.
where N is the number of sites. As follows from the MC simulations in very low temperatures for incommensurate filling, the states are highly unstable (respecting the number of electrons) and undergo continuous fluctuations. The phase separation is present also beyond the ground state. In the low temperatures there are distinct anomalies of n(µ). The evidence of phase separation by jumpwise change of electron concentration for µ/4W & U/4W is observed (marked on Fig. 1 by dotted lines). A closer look on the anomalies appearing in low temperatures reveals their complex structure. Fig. 2 presents the characteristic projections of the PS states at low temperatures for concentrations in the range 0.86 . n < 1.0. There are several distinct geometric types of phase separation. Fig. 2(a) presents the stripes of HCO states. The clearly visible bands contain the single occupied sites and separate the HCO regions. Moving along towards higher concentrations we find canted domains of HCO (Fig. 2(b)). In lower temperatures there are oblique domains separated by the Mott regions (Fig. 2(c)). The concentration range close to n ≈ 1 comprises also the HCO and Mott domains with straight edges perpendicular to the system axis (Fig. 2(d)). The anomalous PS period was studied in particular detail for the systems of different sizes L = 10–60 and for each size it revealed similar and repeatable properties. The MC simulations were also performed for different initial states (full, empty and randomly filled). The character of thermodynamical dependencies and their microscopic snapshot in the form of projections of selected states were similar for different initial states. In order to exclude the errors related to the simulation itself we applied independent algorithms based on different generators of random numbers. For the same model parameters and same temperature the results were qualitatively the same. In Fig. 3 we have the low-temperature dependencies of the charge susceptibility χC (µ). For µ & 0 there are peaks of susceptibility testifying to the second-order phase transition (NO ↔ LCO). In the range µ/4W ≈ U/4W only for k B T /W = 0.06 there is well-visible first-order transition (LCO ↔ HCO) leading to the PS state. Besides, there are only local maxima without distinct peaks. The extremes attest only to a considerable increase in concentration from n = 0.5 to n = 1.0. The lack of significant critical effects for µ & U means that
Fig. 3. Charge susceptibility for U/4W = 0.8.
Fig. 4. Specific heat for a settled hni for U/4W = 0.8.
the CO stability region spreads over the concentration range 0.5 < n ≤ 1.0 (the LCO ↔ ICO ↔ HCO sequence). The plots in Figs. 4 and 5 illustrate the temperature dependence of the specific heat for a fixed concentration and chemical potential. The maxima C(T ) correspond to the phase transitions between charge order and disordered state. Near the half-filled band there is a jump in the specific heat typical for first-order phase transitions. In low temperatures there is a well-visible singularity in Cµ in the vicinity of µ = U (Fig. 5), related to the multicritical phase transition between the CO phase and the phase separation state.
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Fig. 5. Specific heat at a fixed µ for U/4W = 0.8.
Fig. 6. The compressibility K n 2 for U/4W = 0.8.
An objective pointer of a phase separated state existence is the evolution of the compressibility K of the system. For a system of a variable number of particles the adequate definition 1/K = n 2 (∂µ/∂n) says that at a fixed chemical potential the number of particles in an open system can fluctuate freely when K → ±∞, which is equivalent to a significant dynamics of the phase separation states. At the same constant total energy of the system the number of domains and their distribution change. Hence, the phase separation states are highly unstable and are subjected to continuous fluctuations of local density. Fig. 6 shows the temperature dependence of the lines with K n 2 = a constant found numerically on the basis of n(µ). As follows, at low temperatures we can expect phase separation states as the compressibility increases systematically with decreasing temperature. Fig. 7 gives the phase diagram for U/4W = 0.8. The critical points (denoted by stars and crosses) are obtained on the grounds of the thermal singularities of the specific heat (C V , Cµ ) and the charge order susceptibility χ (µ). The phase transition between LCO (or ICO) and NO state is of the secondorder. The transition HCO ↔ NO is of the first-order. Approximately for µ/4W = 1.2 and n = 0.93 there is a tricritical point (TCP). The region LCO–ICO–HCO corresponds to the whole CO phase. At finite temperatures there are no phase transitions between the above structural types of CO [27].
Fig. 7. Phase diagram for U/4W = 0.8. The critical points are denoted by crosses and stars. The solid lines were drawn for constant values of the chemical potential (−0.25 ≤ µ/4W ≤ 1.4).
At low temperatures there is the phase separation state, which contains domains of HCO and single occupied sites (Mott domains). It is difficult to draw very preciously the PS ↔ CO boundary. On the grounds of the known several critical points the most probably area of PS has been plotted. At low temperatures the electron concentration undergoes freely oscillations between 0.86 . n < 1 (for fixed chemical potential). The maximum temperature for such behaviour is of an order of k B T /W = 0.07. The solid lines on Fig. 7 were drawn for constant values of the chemical potential. The results permitted a conclusion that the thermodynamical characteristics of the system, even for very low temperatures was qualitatively different than that in the ground state. Then again, the very intriguing properties at T = 0, when the system is mainly phase separated, can be correctly understood only by knowing the results at finite temperatures. 3. Summary Properties of the extended Hubbard model in the atomic limit on the square lattice for the full range of electron concentration have been studied (taking into regard the electron–hole symmetry the analysis is restricted to n ≤ 1.0). Detailed analysis was made for U/4W = 0.8. The phase diagram as a function of concentration and temperature evidences a complex structure of the system’s orderings. Besides the commensurate phases (LCO, HCO) the diagram shows the mixed phase (ICO) and phase separate states. The mechanisms leading to the PS phenomenon are of classical nature because of the model assumed. This realisation leads to an important conclusion that the separated or striped states do not appear as a result of the long-range interactions as the latter do not occur in the model analysed. The main reason for the appearance of regular separated spatial forms is the fact that the fully homogeneous orderings are available only for particular concentrations (n = 0.0; 0.5; 1.0). On the other hand, a strong tendency of the system to self-order in low temperatures must lead to a domain structure for the intermediate fillings. A specific geometric configuration of
G. Pawłowski, T. Ka´zmierczak / Solid State Communications 145 (2008) 109–113
domains e.g. stripes or rectangulars, becomes optimum with respect to a degree of ordering for a certain concentration range. Despite the lack of clear criteria (in particular those of topological nature), it seems obvious that the irregular (chaotic) geometric structures will be much weakly ordered (of lower order parameter values) and therefore are not energetically preferable at low temperatures. At high temperatures, with the dominant thermal excitations, there are mixed states and the domain structure disappears, consequently the degree of ordering significantly decreases. The CO amplitude undergoes irregular fluctuations in the whole system, and the order parameter tends gradually to zero [27]. The phase diagram for U/4W = 0.8 is representative for the range U < 4W . The cases of U/4W = 0.05; 0.2; 0.4; 06 have been also partially studied. The general conclusion following from our study is that the phase separation in finite temperatures occurs in the whole region 0 < U < 4W , with only the range of PS region changing. Preliminary analyses have also been made for U ≤ 0, however, this situation is qualitatively different. The attractive on-site interaction prefers electron pairing. However, the ALEHM analysis of this region seems less justified as even low values of the transfer integral ti j & 0 would lead to new results and the appearance of a whole gamut of quantum orderings (e.g. superconductivity), which radically decrease the CO stability [28]. Acknowledgments The important advice by S. Robaszkiewicz is gratefully acknowledged. This work was supported in part by the Polish State Committee for Scientific Research, Grant no: 1 P03B 084 26; 2004–2006.
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References [1] E. Dagotto, Nanoscale Phase Separation and Colossal Magnetoresistance, Springer-Verlag, Berlin, 2002. [2] P.B. Visscher, Phys. Rev. B 10 (1974) 943. [3] V.J. Emery, S.A. Kivelson, H.Q. Lin, Phys. Rev. Lett. 64 (1990) 475; Phys. Rev. B 42 (1990) 6523. [4] M. Marder, N. Papanicolaou, G.C. Psaltakis, Phys. Rev. B 41 (1990) 6920. [5] A. Tandon, Z. Wang, G. Kotliar, Phys. Rev. Lett. 83 (1999) 2046. [6] R. Zitzler, Th. Pruschke, R. Bulla, Eur. Phys. J. B 27 (2002) 473. [7] A. Moreo, D. Scalapino, E. Dagotto, Phys. Rev. B 43 (1991) 11442. [8] S.R. White, D. Scalapino, Phys. Rev. B 61 (2000) 6320. [9] P.G.J. van Dongen, Phys. Rev. Lett. 74 (1995) 182; Phys. Rev. B 54 (1996) 1584. [10] N.-H. Tong, S.-Q. Shen, R. Bulla, Phys. Rev. B 70 (2004) 085118. [11] H. Seo, J. Merino, H. Yoshioka, M. Ogata, J. Phys. Soc. Jpn. 75 (2006) 51009-1; H. Seo, Y. Motome, T. Kato, J. Phys. Soc. Jpn. 76 (2007) 013707. [12] R.A. Bari, Phys. Rev. B 3 (1971) 2662. [13] D. Ihle, B. Lorenz, Phys. Status Solidi (b) 60 (1973) 319. [14] G. Beni, P. Pincus, Phys. Rev. B 9 (1974) 2963. [15] R.S. Tu, T.A. Kaplan, Phys. Status Solidi (b) 63 (1974) 659. [16] B. Lorenz, Phys. Status Solidi (b) 106 (1981) K17. [17] T.M. Rice, L. Sneddon, Phys. Rev. Lett. 47 (1981) 689. [18] S. Robaszkiewicz, T. Kostyrko, Physica B & C 112 (1982) 389. [19] R. Micnas, S. Robaszkiewicz, K.A. Chao, Phys. Rev. B 29 (1984) 2784. [20] R.J. Bursill, C.J. Thompson, J. Phys. A 26 (1993) 4497. [21] J. Je¸drzejewski, Physica A 205 (1994) 702. [22] M. Bartkowiak, J.A. Henderson, J. Oitmaa, P.E. de Brito, Phys. Rev. B 51 (1995) 14077. [23] C. Borgs, J. Je¸drzejewski, R. Kotecky, J. Phys. A 29 (1996) 733. [24] E. Halvorsen, M. Bartkowiak, Phys. Rev. B 63 (2000) 014403. [25] F. Mancini, Eur. Phys. J. B 47 (2005) 527. [26] T. Misawa, Y. Yamaji, M. Imada, J. Phys. Soc. Jpn. 75 (2006) 064705. [27] G. Pawłowski, Eur. Phys. J. B 53 (2006) 471. [28] G. Pawłowski, S. Robaszkiewicz, Physica A 229 (2001) 475. [29] D.P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, Cambridge, 2000.
Phase separation and critical phenomena in the charge ordered system G. Pawłowski ∗ , T. Ka´zmierczak Institute of Physics, A. Mickiewicz University, ul. Umultowska 85, 61-614 Pozna´n, Poland Received 30 September 2007; accepted 15 October 2007 by A.H. MacDonald Available online 18 October 2007
Abstract This study is devoted to the phase separation problem in the charge ordered system, presented in a possibly simplest model form. The extended Hubbard model in the atomic limit (t → 0) on a square lattice is analysed using Monte Carlo simulation for the grand canonical ensemble. The model exhibits very rich structure of the phase diagram at finite temperatures as well as in the ground state. Besides the commensurate types of charge order one can observe a mixed phase in T > 0 and states with phase separations or stripes in low temperatures. The results show thus a possible classical nature of the phase separation and striped states for charge order. c 2007 Elsevier Ltd. All rights reserved.
PACS: 71.10.Fd; 71.45.Lr; 64.60.Ak Keywords: D. Charge-order; D. Phase separation
1. Introduction The phase separation phenomenon is a characteristic feature of strongly-correlated electron systems [1]. The presence of such states has been confirmed by the analysis of many model systems. Particularly suitable for this purpose has been the Hubbard model and its extensions. A number of studies for different limiting cases of the Hubbard model (especially in the limit of strong- and weak-coupling) have indicated the presence of states with phase separation [2–6] or striped states [7,8]. Analysis of the extended Hubbard model (EHM) with additional nearest-neighbour Coulomb repulsion leads to the appearance of additional separated states including chargeordered phases [9–11]. The paper is a contribution to the works on the phase separation in the Hubbard model and its extensions and concerns the atomic limit. The zero-overlap limit of the extended Hubbard model, when the Hamiltonian does not contain a hopping integral but only part of the Coulomb interactions, was introduced in 1971 by Bari [12] to describe the charge-orderings in a very narrow half-filled band insulators. Although it is one of the simplest cases of the microscopic ∗ Corresponding author. Tel.: +48 61 8295284; fax: +48 61 8257018.
E-mail address: [email protected] (G. Pawłowski). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.10.015
model, the results have shown a complex and highly nontrivial structure of the phase diagrams comprising the orderings with alternating charge density for the bipartite lattice [13–24]. The model, known also as the atomic limit of EHM (ALEHM), is described by the Hamiltonian X X X H=U n i↑ n i↓ + W ni n j − µ ni , (1) i
ij
i
where n i = n i↑ + n i↓ is the number of electrons on the ith site {0, ↑, ↓, ↑↓}, U represents the on-site and W the intersite Coulomb interaction (here restricted to the nearest-neighbour ones) and µ is the chemical potential. The model properties for general filling have been hitherto poorly recognised. Recently, a few interesting analyses of ALEHM, besides that concerning half-filled bands have been made. Mancini [25] has made a detailed analysis of the model for the 1D case. He has shown the presence of staggered charge order in the region 0 < µ < 4W , and the double peak structure of specific heat and compressibility. Misawa et al. [26] have analysed the critical behaviour in the vicinity of the tricritical points (MC simulation and Hartree–Fock approximation). Their results have been referred directly to the experimental ones obtained for the system (DI–DCNQI)2 Ag where the phase separation along the first-order transition line occurs. The authors gave the phase diagrams over a wide range
110
G. Pawłowski, T. Ka´zmierczak / Solid State Communications 145 (2008) 109–113
of chemical potentials (µ − µhalf ), both for the ground state and in finite temperatures. MC analysis for a square lattice (L = 10–40) for a full range of parameters of the AL-EHM has been reported by GP [27]. Application of the grand canonical ensemble (GCE) approach has permitted finding the n(µ) curves and determination of the phase borders of charge orderings as a function of electron concentration. An important finding reported in this work was detection of the presence of firstorder transitions beyond the half-filling, and consequently the presence of the tricritical point lines. The exclusive presence of orderings commensurable with the lattice (low and high charge order) for the general fillings in the form of spanning checkerboard clusters and stable mixed states (the intermediate charge order) in the range of intermediate concentrations has been shown. Also a preliminary analysis of the ground state has been made. On the basis of the stepwise-type isotherms n(µ) in low temperatures the possibility of phase-separation states has been concluded but not all types of such phases have been identified. This work is a continuation of the earlier analysis aimed at a comprehensive investigation of the vicinity of the ground state. The results presented have been obtained with exceptionally high accuracy. A Monte Carlo simulation for the grand canonical ensemble (GCMC) [29] was made for L = 20 square lattice with periodic boundary conditions. The algorithm includes three sequential update steps: insertion of the particle with randomly chosen spin direction, removal of the particle, and move of the particle (the canonical part). More details of the simulation and the analysis of the finite size effects has been presented in our previous work [27]. Analysis of the thermodynamic characteristics (the internal energy, specific heat, charge order susceptibility) has been performed in three different simulation paths. The first one describes the dependencies as a function of the chemical potential µ for a constant temperature; the second — the dependencies as a function of k B T for a settled average number of electrons in the system; and the third — the dependencies for a fixed value of the chemical potential as a function of k B T . The results obtained for all paths are coherent and permit determination of the accurate phase diagrams for the model. The model predicts many forms of different orderings. Before we define the specific phases, let us define the elementary ordering of the structure: LCO — Low Charge Order (. . . 1010. . . or . . . 1212. . . ), HCO — High Charge Order (. . . 2020. . . ), ICO — Intermediate Charge Order (. . . x0y0 . . . or . . . x2y2 . . ., where x, y = {1, 2}, excluding the case when for the whole ordered domain x = y). In the 2D case, the first — Low Charge Order is realized for a chessboard combination of electrons (holes) and empty (full) sites; the second — High Charge Order refers to electron pairs and empty sites. The last order is a simple combination of LCO and HCO structures, therefore this state can be called the Intermediate Charge Order.
Fig. 1. Isotherms for U/4W = 0.8 and k B T /W = 0.01 ÷ 0.66. The box shows the range of PS at low temperatures. Table 1 Stability ranges of the possible ordered states in the ground state for U ≤ 4W Chemical potential
Electron concentration
State
µ<0 µ=0 0<µ
n = 0.0 0 < n < 0.5 n = 0.5 0.5 < n < 1 n = 1.0
NO Domains of LCO LCO Domains of LCO–HCO HCO
Besides these orderings we observed the disordered states denoted as NO. For n = 1 there is some uniform disordered state, realized as a Mott state (. . . 1111. . . ). Because of the electron–hole symmetry of the model, the range of the hole orderings has not been discussed in this paper (1 < n ≤ 2). 2. The results In this paper we analyse in detail the model properties for U/4W = 0.8. For this value both types of commensurate orderings (LCO, HCO), phase separation states and intermediate orderings (ICO, for T > 0) are possible. Moreover, in finite temperatures and in the concentration range close to n ≈ 1 there are first-order phase transitions and tricritical point [27]. Fig. 1 presents the isotherms for U/4W = 0.8. At low temperatures the stepwise character of the curve µ(n) is observed. The stability ranges of the possible ordered states for the ground state is given in Table 1. At T = 0 only three electron concentrations are available n = 0.0; 0.5; 1.0, for which stable structures can be formed (NO, LCO, HCO). For µ = 0, µ = U there is a jumpwise change in concentration which determines the presence of phase separation states in the ground state. In the range 0 < n < 0.5 and 0.5 < n < 1.0 only the states with homogeneous domain separation occur. Let us note that in these states the total energy E (T = 0) of the system is fixed and depends only on the chemical potential (not on the particle concentration): µ = 0 : E(0) = 0, µ = U : E(0) = −N U/2,
G. Pawłowski, T. Ka´zmierczak / Solid State Communications 145 (2008) 109–113
(a) n = 0.899.
(b) n = 0.930.
(c) n = 0.922.
111
(d) n = 0.998.
Fig. 2. The characteristic projections of the PS states at low temperatures. Denotations: grey squares — one electron per site, black — two electrons per site and white — empty site.
where N is the number of sites. As follows from the MC simulations in very low temperatures for incommensurate filling, the states are highly unstable (respecting the number of electrons) and undergo continuous fluctuations. The phase separation is present also beyond the ground state. In the low temperatures there are distinct anomalies of n(µ). The evidence of phase separation by jumpwise change of electron concentration for µ/4W & U/4W is observed (marked on Fig. 1 by dotted lines). A closer look on the anomalies appearing in low temperatures reveals their complex structure. Fig. 2 presents the characteristic projections of the PS states at low temperatures for concentrations in the range 0.86 . n < 1.0. There are several distinct geometric types of phase separation. Fig. 2(a) presents the stripes of HCO states. The clearly visible bands contain the single occupied sites and separate the HCO regions. Moving along towards higher concentrations we find canted domains of HCO (Fig. 2(b)). In lower temperatures there are oblique domains separated by the Mott regions (Fig. 2(c)). The concentration range close to n ≈ 1 comprises also the HCO and Mott domains with straight edges perpendicular to the system axis (Fig. 2(d)). The anomalous PS period was studied in particular detail for the systems of different sizes L = 10–60 and for each size it revealed similar and repeatable properties. The MC simulations were also performed for different initial states (full, empty and randomly filled). The character of thermodynamical dependencies and their microscopic snapshot in the form of projections of selected states were similar for different initial states. In order to exclude the errors related to the simulation itself we applied independent algorithms based on different generators of random numbers. For the same model parameters and same temperature the results were qualitatively the same. In Fig. 3 we have the low-temperature dependencies of the charge susceptibility χC (µ). For µ & 0 there are peaks of susceptibility testifying to the second-order phase transition (NO ↔ LCO). In the range µ/4W ≈ U/4W only for k B T /W = 0.06 there is well-visible first-order transition (LCO ↔ HCO) leading to the PS state. Besides, there are only local maxima without distinct peaks. The extremes attest only to a considerable increase in concentration from n = 0.5 to n = 1.0. The lack of significant critical effects for µ & U means that
Fig. 3. Charge susceptibility for U/4W = 0.8.
Fig. 4. Specific heat for a settled hni for U/4W = 0.8.
the CO stability region spreads over the concentration range 0.5 < n ≤ 1.0 (the LCO ↔ ICO ↔ HCO sequence). The plots in Figs. 4 and 5 illustrate the temperature dependence of the specific heat for a fixed concentration and chemical potential. The maxima C(T ) correspond to the phase transitions between charge order and disordered state. Near the half-filled band there is a jump in the specific heat typical for first-order phase transitions. In low temperatures there is a well-visible singularity in Cµ in the vicinity of µ = U (Fig. 5), related to the multicritical phase transition between the CO phase and the phase separation state.
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Fig. 5. Specific heat at a fixed µ for U/4W = 0.8.
Fig. 6. The compressibility K n 2 for U/4W = 0.8.
An objective pointer of a phase separated state existence is the evolution of the compressibility K of the system. For a system of a variable number of particles the adequate definition 1/K = n 2 (∂µ/∂n) says that at a fixed chemical potential the number of particles in an open system can fluctuate freely when K → ±∞, which is equivalent to a significant dynamics of the phase separation states. At the same constant total energy of the system the number of domains and their distribution change. Hence, the phase separation states are highly unstable and are subjected to continuous fluctuations of local density. Fig. 6 shows the temperature dependence of the lines with K n 2 = a constant found numerically on the basis of n(µ). As follows, at low temperatures we can expect phase separation states as the compressibility increases systematically with decreasing temperature. Fig. 7 gives the phase diagram for U/4W = 0.8. The critical points (denoted by stars and crosses) are obtained on the grounds of the thermal singularities of the specific heat (C V , Cµ ) and the charge order susceptibility χ (µ). The phase transition between LCO (or ICO) and NO state is of the secondorder. The transition HCO ↔ NO is of the first-order. Approximately for µ/4W = 1.2 and n = 0.93 there is a tricritical point (TCP). The region LCO–ICO–HCO corresponds to the whole CO phase. At finite temperatures there are no phase transitions between the above structural types of CO [27].
Fig. 7. Phase diagram for U/4W = 0.8. The critical points are denoted by crosses and stars. The solid lines were drawn for constant values of the chemical potential (−0.25 ≤ µ/4W ≤ 1.4).
At low temperatures there is the phase separation state, which contains domains of HCO and single occupied sites (Mott domains). It is difficult to draw very preciously the PS ↔ CO boundary. On the grounds of the known several critical points the most probably area of PS has been plotted. At low temperatures the electron concentration undergoes freely oscillations between 0.86 . n < 1 (for fixed chemical potential). The maximum temperature for such behaviour is of an order of k B T /W = 0.07. The solid lines on Fig. 7 were drawn for constant values of the chemical potential. The results permitted a conclusion that the thermodynamical characteristics of the system, even for very low temperatures was qualitatively different than that in the ground state. Then again, the very intriguing properties at T = 0, when the system is mainly phase separated, can be correctly understood only by knowing the results at finite temperatures. 3. Summary Properties of the extended Hubbard model in the atomic limit on the square lattice for the full range of electron concentration have been studied (taking into regard the electron–hole symmetry the analysis is restricted to n ≤ 1.0). Detailed analysis was made for U/4W = 0.8. The phase diagram as a function of concentration and temperature evidences a complex structure of the system’s orderings. Besides the commensurate phases (LCO, HCO) the diagram shows the mixed phase (ICO) and phase separate states. The mechanisms leading to the PS phenomenon are of classical nature because of the model assumed. This realisation leads to an important conclusion that the separated or striped states do not appear as a result of the long-range interactions as the latter do not occur in the model analysed. The main reason for the appearance of regular separated spatial forms is the fact that the fully homogeneous orderings are available only for particular concentrations (n = 0.0; 0.5; 1.0). On the other hand, a strong tendency of the system to self-order in low temperatures must lead to a domain structure for the intermediate fillings. A specific geometric configuration of
G. Pawłowski, T. Ka´zmierczak / Solid State Communications 145 (2008) 109–113
domains e.g. stripes or rectangulars, becomes optimum with respect to a degree of ordering for a certain concentration range. Despite the lack of clear criteria (in particular those of topological nature), it seems obvious that the irregular (chaotic) geometric structures will be much weakly ordered (of lower order parameter values) and therefore are not energetically preferable at low temperatures. At high temperatures, with the dominant thermal excitations, there are mixed states and the domain structure disappears, consequently the degree of ordering significantly decreases. The CO amplitude undergoes irregular fluctuations in the whole system, and the order parameter tends gradually to zero [27]. The phase diagram for U/4W = 0.8 is representative for the range U < 4W . The cases of U/4W = 0.05; 0.2; 0.4; 06 have been also partially studied. The general conclusion following from our study is that the phase separation in finite temperatures occurs in the whole region 0 < U < 4W , with only the range of PS region changing. Preliminary analyses have also been made for U ≤ 0, however, this situation is qualitatively different. The attractive on-site interaction prefers electron pairing. However, the ALEHM analysis of this region seems less justified as even low values of the transfer integral ti j & 0 would lead to new results and the appearance of a whole gamut of quantum orderings (e.g. superconductivity), which radically decrease the CO stability [28]. Acknowledgments The important advice by S. Robaszkiewicz is gratefully acknowledged. This work was supported in part by the Polish State Committee for Scientific Research, Grant no: 1 P03B 084 26; 2004–2006.
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References [1] E. Dagotto, Nanoscale Phase Separation and Colossal Magnetoresistance, Springer-Verlag, Berlin, 2002. [2] P.B. Visscher, Phys. Rev. B 10 (1974) 943. [3] V.J. Emery, S.A. Kivelson, H.Q. Lin, Phys. Rev. Lett. 64 (1990) 475; Phys. Rev. B 42 (1990) 6523. [4] M. Marder, N. Papanicolaou, G.C. Psaltakis, Phys. Rev. B 41 (1990) 6920. [5] A. Tandon, Z. Wang, G. Kotliar, Phys. Rev. Lett. 83 (1999) 2046. [6] R. Zitzler, Th. Pruschke, R. Bulla, Eur. Phys. J. B 27 (2002) 473. [7] A. Moreo, D. Scalapino, E. Dagotto, Phys. Rev. B 43 (1991) 11442. [8] S.R. White, D. Scalapino, Phys. Rev. B 61 (2000) 6320. [9] P.G.J. van Dongen, Phys. Rev. Lett. 74 (1995) 182; Phys. Rev. B 54 (1996) 1584. [10] N.-H. Tong, S.-Q. Shen, R. Bulla, Phys. Rev. B 70 (2004) 085118. [11] H. Seo, J. Merino, H. Yoshioka, M. Ogata, J. Phys. Soc. Jpn. 75 (2006) 51009-1; H. Seo, Y. Motome, T. Kato, J. Phys. Soc. Jpn. 76 (2007) 013707. [12] R.A. Bari, Phys. Rev. B 3 (1971) 2662. [13] D. Ihle, B. Lorenz, Phys. Status Solidi (b) 60 (1973) 319. [14] G. Beni, P. Pincus, Phys. Rev. B 9 (1974) 2963. [15] R.S. Tu, T.A. Kaplan, Phys. Status Solidi (b) 63 (1974) 659. [16] B. Lorenz, Phys. Status Solidi (b) 106 (1981) K17. [17] T.M. Rice, L. Sneddon, Phys. Rev. Lett. 47 (1981) 689. [18] S. Robaszkiewicz, T. Kostyrko, Physica B & C 112 (1982) 389. [19] R. Micnas, S. Robaszkiewicz, K.A. Chao, Phys. Rev. B 29 (1984) 2784. [20] R.J. Bursill, C.J. Thompson, J. Phys. A 26 (1993) 4497. [21] J. Je¸drzejewski, Physica A 205 (1994) 702. [22] M. Bartkowiak, J.A. Henderson, J. Oitmaa, P.E. de Brito, Phys. Rev. B 51 (1995) 14077. [23] C. Borgs, J. Je¸drzejewski, R. Kotecky, J. Phys. A 29 (1996) 733. [24] E. Halvorsen, M. Bartkowiak, Phys. Rev. B 63 (2000) 014403. [25] F. Mancini, Eur. Phys. J. B 47 (2005) 527. [26] T. Misawa, Y. Yamaji, M. Imada, J. Phys. Soc. Jpn. 75 (2006) 064705. [27] G. Pawłowski, Eur. Phys. J. B 53 (2006) 471. [28] G. Pawłowski, S. Robaszkiewicz, Physica A 229 (2001) 475. [29] D.P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, Cambridge, 2000.