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Phonon softening in Peierls transition in an anisotropic triangular lattice
ARTICLE IN PRESS Physica E 42 (2010) 1239–1242
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Phonon softening in Peierls transition in an anisotropic triangular lattice Chiduru Watanabe a,b,1, Yoshiyuki Ono a,b, a b
Department of Physics, Faculty of Science, Toho University, Miyama 2-2-1, Funabashi, Chiba 274-8510, Japan Center for Materials with Integrated Properties, Toho University, Miyama 2-2-1, Funabashi, Chiba 274-8510, Japan
a r t i c l e in fo
abstract
Article history: Received 27 August 2009 Received in revised form 10 November 2009 Accepted 25 November 2009 Available online 5 December 2009
The 2D Peierls transition in an anisotropic triangular lattice is studied from the view point of phonon softening. The model is obtained by introducing a coupling in one of the diagonal directions to the isotropic square lattice, for which the multimode Peierls phase, involving not only the nesting vector component but also other Fourier components of distortion with wave vectors parallel to the nesting vector, is predicted. The results indicate that the multimode type Peierls transition can be expected as far as the diagonal coupling is weak compared to the nearest neighbor coupling in the square lattice. & 2009 Elsevier B.V. All rights reserved.
Keywords: 2D multimode Peierls transition Phonon softening Anisotropic triangular lattice
1. Introduction The electron–lattice (e–l) coupling in low dimensional systems plays an essential role in determining electronic properties. It is well-known that, in 1D systems, the e–l coupling induces the spontaneous Peierls distortion at low temperatures [1]. In the case of the 2D square lattice with a half-filled electronic band, the Fermi line takes a square shape having corners at ðp; 0Þ, ð0; pÞ, ðp; 0Þ and ð0; pÞ, which indicates the existence of a nesting vector Q ¼ ðp; pÞ. Here and in what follows, the lattice constant is set to be unity. The existence of the nesting vector suggests the realization of the Peierls transition with a similar mechanism as in 1D cases [2–6]. About a decade ago, it has been pointed out that the lowest energy state of a square-lattice system is not the single-mode Peierls state, where the Peierls distortion consists of only the Q component, but the multimode Peierls (MMP) state, where the distortion involves not only the Q component but also other Fourier components with fractional wave vectors of Q [7]. Furthermore it has been shown that there are non-equivalent energetically degenerate many MMP states with different combinations of the fractional wave vectors of Q [8]. In spite of clearness of the theoretical prediction, there is no report on the experimental evidence of the MMP state. In order to confirm the stability of the MMP state, the effects of the anisotropy and the electron–electron interactions on the MMP state have been
Corresponding author at: Department of Physics, Faculty of Science, Toho University, Miyama 2-2-1, Funabashi, Chiba 274-8510, Japan. E-mail address: [email protected] (Y. Ono). 1 Present address: Information Technology Education Center, Tokai University, Kitakaname 1117, Hiratsuka, Kanagawa 259-1292, Japan.
1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.11.145
investigated [9–14]. In all these studies, we have found the finite regions of parameters where the lowest energy state is the MMP state. Finite temperature effects have also been studied in a series of works [15–18], which indicate varieties of phase transitions at finite temperatures depending on values of parameters. More recently, Horita and one of the present authors (YO) have studied the Peierls phase in a model derived from the square lattice by introducing the electronic transfer in one of the diagonal direction [19]. This study was stimulated by the fact that some organic conductors have such a structure. It has been found that an MMP state remains to be the lowest energy state as long as the diagonal transfer is within a weak region compared to the horizontal and vertical transfers and that the degeneracy of many MMP states seen in the isotropic square lattice can be lifted by the introduction of the diagonal transfer. In the present work, the Peierls transition at a finite temperature is studied in terms of phonon softening for a modified square lattice or an anisotropic triangular lattice similar to the model treated in Ref. [19]. The model and formulation of calculations are briefly explained in the following section. The results are described in Section 3. The last section is devoted to conclusion and discussion.
2. Model and formulation The model Hamiltonian treated in this work is given by ( " # 3 X X X K ½ti ai vi ðrÞðcry þ ei ;s cr;s þ h:c:Þ þ ðvi ðrÞÞ2 H¼ 2 r s i¼1 i Mh ðu_ x ðrÞÞ2 þ ðu_ y ðrÞÞ2 ; þ 2
ð1Þ
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where the distortions vi ðrÞ (i ¼ 1; 2; 3) are expressed in terms of displacements {ux ðrÞ; uy ðrÞ} as follows: v1 ðrÞ ¼ ux ðr þ e1 Þux ðrÞ;
ð2Þ
v2 ðrÞ ¼ uy ðr þ e2 Þuy ðrÞ;
ð3Þ
1 v3 ðrÞ ¼ pffiffiffi ½ux ðr þe3 Þux ðrÞ þuy ðr þ e3 Þuy ðrÞ; 2
ð4Þ
and the vectors ei (i ¼ 1; 2; 3) are defined in the form, e1 ¼ ex ¼ ð1; 0Þ;
e2 ¼ ey ¼ ð0; 1Þ;
e3 ¼ ex þ ey ¼ ð1; 1Þ:
ð5Þ ð6Þ
y cr;s
and cr;s are usual creation and annihilation operators of an electron with spin s at site r, ti and ai representing the electronic transfer for the lattice without distortion and the electron–lattice coupling constant in the direction i, respectively. The parameters K and M indicate the lattice force constant and the ionic mass, respectively; the force constant is assumed to be independent of the coupling direction. For the sake of simplicity, we consider only the case with t1 ¼ t2 t0 and a1 ¼ a2 a. For the diagonal coupling, we restrict ourselves to the case where t3 and a3 are expressed as t3 ¼ gt0 and a3 ¼ ga, g indicating the strength of the diagonal coupling. According to the standard linear mode analysis technique [15], we express the dynamical displacements in the form, ud ðr; tÞ ¼ eiot ud ðr; oÞ;
½d ¼ x; y;
ð7Þ
and substitute them into the equation of motion, 2
M
d ud @/HS ¼ : @ud dt 2
ð8Þ
Here / S means the average over electronic states. Furthermore by introducing the Fourier decomposition about the space dependence, we end up with the following eigenvalue equation for the phonon modes:
o2 uðq; oÞ ¼ WðqÞuðq; oÞ;
ð9Þ
where q is the wave vector for a phonon, and the elements of the 2 2 matrix WðqÞ are given in the following form: W i;j ðqÞ ¼
4 X f ðek Þf ðek þ q Þ fai ½sinðk ei Þsinððk þqÞ ei Þ ek ek þ q MN2 k;s pffiffiffi þ a3 ½sinðk e3 Þsinððk þ qÞ e3 Þ= 2g faj ½sinðk ej Þ pffiffiffi sinððk þ qÞ ej Þ þ a3 ½sinðk e3 Þsinððk þqÞ e3 Þ= 2g pffiffiffi 2K ð10Þ þ f½1cosðq ei Þdi;j þ ½1cosðq e3 Þ= 2g; M
i and j standing for x or y. Here N2 is the total number of sites, and we have assumed periodic boundary conditions for the system size N N. The electronic dispersion is given by
ek ¼ 2t0 ðcoskx þ cosky Þ2t3 cosðkx þ ky Þ;
ð11Þ
and f ðek Þ is the Fermi distribution function, f ðek Þ ¼
1 ; exp½ðek mÞ=kB T þ 1
ð12Þ
with the temperature T, the Boltzmann constant kB and the chemical potential m; the chemical potential is determined numerically from the half-filling condition for each temperature. Because of lack of the electron–hole symmetry, m is not fixed at zero as in the square-lattice case. For each q, there are two eigenvalues given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð13Þ o27 ¼ 12fW xx þ W yy 7 ðW xx W yy Þ2 þ 4W 2xy g:
The data shown in the following section are obtained by substituting Eq. (10) into Eq. (13). In what follows, the frequency pffiffiffiffiffiffiffiffiffiffiffiffiffi is scaled by the bare optical phonon frequency oQ 4K=M , and the strength of the e–l coupling is expressed by the dimensionless coupling constant l a2 =Kt 0 .
3. Results of calculations In Fig. 1, o2 is plotted as a function of the temperature T for some special values of q in the case with N ¼ 8; l ¼ 0:5 and g ¼ 0:1 in order to discuss typical behaviors of the phonon frequencies at relatively small values of g. In the same figure, the data for o2þ ðQ Þ are also depicted by a dash-dotted curve for comparison. Here Q ¼ ðp; pÞ is perpendicular to Q . Although Q and Q are equivalent since their difference is equal to a reciprocal lattice vector, their fractions are not equivalent. From Fig. 1, it is found that the phonon modes, which are softened first with lowering the temperature from the high temperature region, are o -modes with wave vectors Q and integer multiples of 2Q =N (in the present case, Q =4, Q =2 and 3Q =4).2 Furthermore, they are softened simultaneously at a critical temperature Tc , at which no other mode is softened though the data for other modes are omitted here. Analyses of the eigenvectors indicate that those modes softened at Tc are transverse. On the other hand, the softening of the o þ mode with Q , which is longitudinal, occurs at a temperature lower than Tc . These facts are consistent with the phonon softening behavior in the case of the isotropic square lattice, where all the transverse modes with Q and integer times 2Q =N are softened simultaneously at a certain critical temperature along with the longitudinal mode with Q . Because of symmetry, this is the case for Q and integer times 2Q =N in the isotropic square lattice3. Fig. 1 indicates that an introduction of the diagonal coupling lifts the degeneracy between integer times 2Q =N and integer times 2Q =N and between the longitudinal and transverse modes with Q . The above results, however, is not in accordance with the behavior of the system at absolute zero of temperature [19], where the lowest energy state is the MMP state involving distortion components with Q and Q =2 (not Q =2) when the diagonal coupling is in the weak region. This discrepancy suggests that there might be a rich variety of phase transitions in the low temperature region below Tc . In fact, it is pointed out in the case of an anisotropic square-lattice system that there are complicated rich structures of phase transitions in the low temperature region [20]. Next we show in Fig. 2 the temperature dependence of the square frequencies for the case with N ¼ 8; l ¼ 0:5 and g ¼ 0:7 as a typical example for the strong diagonal coupling case. In this figure, o2 is plotted by solid lines for several wave vectors parallel to Q or Q . For reference, some curves of o2þ are also given by dash-dotted lines. When the temperature is lowered from the high temperature region, the mode first softened at Tc is the one with the wave vector Q =2. At Tc , no other mode is softened, which suggests that the low temperature phase would be a single-mode Peierls state involving the distortion with the wave vector Q =2. This can be understood from the electronic dispersion Eq. (11). If the last term of Eq. (11) is dominating, the Fermi line consists of two straight lines parallel to Q and separated by the wave vector 2 Though we have shown only the case with N ¼ 8 partly in order to avoid too many curves in one figure, the essential features are not changed for different system sizes. 3 It should be noted that a vanishing g does not lead to the isotropic square lattice because we did not introduce the factor g in the diagonal lattice coupling term for the sake of simplicity. This point will be discussed in the last section.
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tures lower than the crossing point. In the cases of other modes shown in Fig. 2, o modes are transverse and o þ modes are longitudinal, which is also the case in Fig. 1.
4. Conclusion and discussion
Fig. 1. The temperature dependence of square eigenfrequencies for various wave vectors in the case with N ¼ 8, l ¼ 0:5 and g ¼ 0:1.
Fig. 2. The temperature dependence of square eigenfrequencies for various wave vectors in the case with N ¼ 8, l ¼ 0:5 and g ¼ 0:7.
Q =2, i.e. Q =2 becomes the nesting vector in this limit. The first softened Q =2mode is found to be transverse. Below Tc , we have to consider a static distortion with the wave vector Q =2 and with a transverse nature. The structure of the low temperature region will be treated elsewhere. Here we simply mention that the fact that the mode with Q =2 is dominating in the case of relatively large values of g is consistent with the previous study at zero temperature [19]. In Fig. 2, the modes with 3Q =4 show peculiar behavior. There is a crossing between o and o þ . In order to explain this behavior, we should notice that, when q is parallel to Q or Q , W xx and W yy are equal to each other because of symmetry. Therefore the square eigenfrequencies for those special wave vectors are given by o27 ¼ W xx 7 jW xy j, which indicates also that the corresponding modes are longitudinal or transverse depending on the sign of W xy . In fact, the o mode for 3Q =4 in Fig. 2 is transverse at higher temperatures and longitudinal at tempera-
The 2D Peierls transition in an anisotropic triangular lattice is discussed from the view point of phonon softening. The model system is constructed from the square lattice by introducing couplings in one of the diagonal directions. When the anisotropic parameter g, describing the strength of the diagonal coupling relative to the normal square-lattice type coupling, is small, phonon modes with fractional wave vectors of Q ½ ¼ ðp; pÞ are softened simultaneously at a certain critical temperature Tc along with the Q ½ ¼ ðp; pÞ (or equivalently Q ) mode as the temperature is lowered from the higher region. This fact suggests the realization of the MMP state below Tc . On the other hand, for larger values of g, only one mode with the wave vector Q =2 is softened with lowering temperature at a certain temperature while other modes are not yet softened, which suggests a singlemode Peierls state below the softening temperature. Although we have not yet systematically studied the gdependence of the phonon dispersion, a preliminary rough estimation of the border value of g separating the weak and strong coupling regions shows that it lies around 0.6. Although we do not show explicit data because of the limitation of the allotted space, we have analyzed the gdependence of the phonon softening temperatures. Generally speaking, the phonon softening temperatures are increasing functions of g, while those for some modes show a maximum as functions of g. Even if we can expect an MMP state for small values of g, the present phonon softening study seems not consistent with the previous zero temperature study [19]. A similar inconsistency between the phonon softening and zero temperature studies was found in the anisotropic square lattice model [9–11]. Thereby, the existence of rich variety of phase transitions is suggested in the low temperature region [20]. It will be plausible to expect a complicated structure of various phase transitions between the phonon softening temperature and the absolute zero temperature. In this paper we have shown only the results for N ¼ 8. Similar calculations were carried out for larger system sizes up to N ¼ 32. Essential features are found not to be changed at least up to this size. In the present model we did not introduce the factor g to the diagonal lattice coupling (i.e. the K-term) for simplicity. Therefore, even if we put g ¼ 0, the system is not reduced to the isotropic square lattice. Nevertheless, in the case of wave vectors parallel to Q , the effect of the diagonal lattice coupling vanishes as will be seen in the last term of Eq. (10). In this sense the results for the modes with fractions of Q coincide with those of the isotropic square lattice when g is set to be zero. In organic materials there are several systems which might correspond to the model treated in the present work. For example, ytype BEDT-TTF organic conductors [21] or some of PdðdmitÞ2 type complex salts [22] might be essentially regarded as anisotropic triangular lattices. There would remain a possibility to find the MMP phase in layered organic materials. In the present work, we did not consider the effect of electron– electron interactions. It is inevitable to take account of them when we compare the theoretical results with experiments. An investigation in this direction will be done in the future. Study of the low temperature phase should also be done along with the detailed analyses of dependences of the lowest energy state on the parameters g and l. This is left for future works.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
R.E. Peierls, Quantum Theory of Solids, Clarendon Press, Oxford, 1955. S. Tang, J.E. Hirsh, Phys. Rev. B 37 (1988) 9546. K. Machida, M. Kato, Phys. Rev. B 36 (1987) 854. R.T. Scalettar, N.E. Bickers, D.J. Scalapino, Phys. Rev. B 40 (1989) 197. S. Mazumdar, Phys. Rev. B 39 (1989) 12324. S. Tang, J.E. Hirsch, Phys. Rev. B 39 (1989) 12327. Y. Ono, T. Hamano, J. Phys. Soc. Japan 69 (2000) 1796. T. Hamano, Y. Ono, J. Phys. Soc. Japan 70 (2001) 1849. Y. Ono, S. Chiba, T. Hamano, Synth. Met. 152 (2005) 405. C. Watanabe, S. Chiba, Y. Ono, AIP Conf. Proc. 850 (2006) 1321.
[11] C. Watanabe, S. Chiba, Y. Ono, J. Phys. Soc. Japan 76 114704. [12] S. Chiba, Y. Ono, J. Phys. Soc. Japan 73 (2004) 2777. [13] S. Chiba, Y. Baba, Y. Ono, J. Phys. Soc. Japan 75 (2006) 034705. [14] S. Chiba, Y. Ono, AIP Conf. Proc. 850 (2006) 1323. [15] S. Chiba, Y. Ono, J. Phys. Soc. Japan 72 (2003) 1995. [16] S. Chiba, Y. Ono, Physica E 22 (2004) 152. [17] T. Hamano, Y. Ono, Physica E 22 (2004) 156. [18] S. Chiba, Y. Ono, J. Phys. Soc. Japan 73 (2004) 2473. [19] Y. Horita, Y. Ono, J. Phys. Soc. Japan 78 (2009) 024711. [20] C. Watanabe, Doctor’s Thesis, Toho University, 2008 (in Japanese). [21] H. Mori, S. Tanaka, T. Mori, Phys. Rev. B 57 (1998) 12023. [22] M. Tamura, R. Kato, Sci. Technol. Adv. Mater. 10 (2009) 024304.
(2007)
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Phonon softening in Peierls transition in an anisotropic triangular lattice Chiduru Watanabe a,b,1, Yoshiyuki Ono a,b, a b
Department of Physics, Faculty of Science, Toho University, Miyama 2-2-1, Funabashi, Chiba 274-8510, Japan Center for Materials with Integrated Properties, Toho University, Miyama 2-2-1, Funabashi, Chiba 274-8510, Japan
a r t i c l e in fo
abstract
Article history: Received 27 August 2009 Received in revised form 10 November 2009 Accepted 25 November 2009 Available online 5 December 2009
The 2D Peierls transition in an anisotropic triangular lattice is studied from the view point of phonon softening. The model is obtained by introducing a coupling in one of the diagonal directions to the isotropic square lattice, for which the multimode Peierls phase, involving not only the nesting vector component but also other Fourier components of distortion with wave vectors parallel to the nesting vector, is predicted. The results indicate that the multimode type Peierls transition can be expected as far as the diagonal coupling is weak compared to the nearest neighbor coupling in the square lattice. & 2009 Elsevier B.V. All rights reserved.
Keywords: 2D multimode Peierls transition Phonon softening Anisotropic triangular lattice
1. Introduction The electron–lattice (e–l) coupling in low dimensional systems plays an essential role in determining electronic properties. It is well-known that, in 1D systems, the e–l coupling induces the spontaneous Peierls distortion at low temperatures [1]. In the case of the 2D square lattice with a half-filled electronic band, the Fermi line takes a square shape having corners at ðp; 0Þ, ð0; pÞ, ðp; 0Þ and ð0; pÞ, which indicates the existence of a nesting vector Q ¼ ðp; pÞ. Here and in what follows, the lattice constant is set to be unity. The existence of the nesting vector suggests the realization of the Peierls transition with a similar mechanism as in 1D cases [2–6]. About a decade ago, it has been pointed out that the lowest energy state of a square-lattice system is not the single-mode Peierls state, where the Peierls distortion consists of only the Q component, but the multimode Peierls (MMP) state, where the distortion involves not only the Q component but also other Fourier components with fractional wave vectors of Q [7]. Furthermore it has been shown that there are non-equivalent energetically degenerate many MMP states with different combinations of the fractional wave vectors of Q [8]. In spite of clearness of the theoretical prediction, there is no report on the experimental evidence of the MMP state. In order to confirm the stability of the MMP state, the effects of the anisotropy and the electron–electron interactions on the MMP state have been
Corresponding author at: Department of Physics, Faculty of Science, Toho University, Miyama 2-2-1, Funabashi, Chiba 274-8510, Japan. E-mail address: [email protected] (Y. Ono). 1 Present address: Information Technology Education Center, Tokai University, Kitakaname 1117, Hiratsuka, Kanagawa 259-1292, Japan.
1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.11.145
investigated [9–14]. In all these studies, we have found the finite regions of parameters where the lowest energy state is the MMP state. Finite temperature effects have also been studied in a series of works [15–18], which indicate varieties of phase transitions at finite temperatures depending on values of parameters. More recently, Horita and one of the present authors (YO) have studied the Peierls phase in a model derived from the square lattice by introducing the electronic transfer in one of the diagonal direction [19]. This study was stimulated by the fact that some organic conductors have such a structure. It has been found that an MMP state remains to be the lowest energy state as long as the diagonal transfer is within a weak region compared to the horizontal and vertical transfers and that the degeneracy of many MMP states seen in the isotropic square lattice can be lifted by the introduction of the diagonal transfer. In the present work, the Peierls transition at a finite temperature is studied in terms of phonon softening for a modified square lattice or an anisotropic triangular lattice similar to the model treated in Ref. [19]. The model and formulation of calculations are briefly explained in the following section. The results are described in Section 3. The last section is devoted to conclusion and discussion.
2. Model and formulation The model Hamiltonian treated in this work is given by ( " # 3 X X X K ½ti ai vi ðrÞðcry þ ei ;s cr;s þ h:c:Þ þ ðvi ðrÞÞ2 H¼ 2 r s i¼1 i Mh ðu_ x ðrÞÞ2 þ ðu_ y ðrÞÞ2 ; þ 2
ð1Þ
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where the distortions vi ðrÞ (i ¼ 1; 2; 3) are expressed in terms of displacements {ux ðrÞ; uy ðrÞ} as follows: v1 ðrÞ ¼ ux ðr þ e1 Þux ðrÞ;
ð2Þ
v2 ðrÞ ¼ uy ðr þ e2 Þuy ðrÞ;
ð3Þ
1 v3 ðrÞ ¼ pffiffiffi ½ux ðr þe3 Þux ðrÞ þuy ðr þ e3 Þuy ðrÞ; 2
ð4Þ
and the vectors ei (i ¼ 1; 2; 3) are defined in the form, e1 ¼ ex ¼ ð1; 0Þ;
e2 ¼ ey ¼ ð0; 1Þ;
e3 ¼ ex þ ey ¼ ð1; 1Þ:
ð5Þ ð6Þ
y cr;s
and cr;s are usual creation and annihilation operators of an electron with spin s at site r, ti and ai representing the electronic transfer for the lattice without distortion and the electron–lattice coupling constant in the direction i, respectively. The parameters K and M indicate the lattice force constant and the ionic mass, respectively; the force constant is assumed to be independent of the coupling direction. For the sake of simplicity, we consider only the case with t1 ¼ t2 t0 and a1 ¼ a2 a. For the diagonal coupling, we restrict ourselves to the case where t3 and a3 are expressed as t3 ¼ gt0 and a3 ¼ ga, g indicating the strength of the diagonal coupling. According to the standard linear mode analysis technique [15], we express the dynamical displacements in the form, ud ðr; tÞ ¼ eiot ud ðr; oÞ;
½d ¼ x; y;
ð7Þ
and substitute them into the equation of motion, 2
M
d ud @/HS ¼ : @ud dt 2
ð8Þ
Here / S means the average over electronic states. Furthermore by introducing the Fourier decomposition about the space dependence, we end up with the following eigenvalue equation for the phonon modes:
o2 uðq; oÞ ¼ WðqÞuðq; oÞ;
ð9Þ
where q is the wave vector for a phonon, and the elements of the 2 2 matrix WðqÞ are given in the following form: W i;j ðqÞ ¼
4 X f ðek Þf ðek þ q Þ fai ½sinðk ei Þsinððk þqÞ ei Þ ek ek þ q MN2 k;s pffiffiffi þ a3 ½sinðk e3 Þsinððk þ qÞ e3 Þ= 2g faj ½sinðk ej Þ pffiffiffi sinððk þ qÞ ej Þ þ a3 ½sinðk e3 Þsinððk þqÞ e3 Þ= 2g pffiffiffi 2K ð10Þ þ f½1cosðq ei Þdi;j þ ½1cosðq e3 Þ= 2g; M
i and j standing for x or y. Here N2 is the total number of sites, and we have assumed periodic boundary conditions for the system size N N. The electronic dispersion is given by
ek ¼ 2t0 ðcoskx þ cosky Þ2t3 cosðkx þ ky Þ;
ð11Þ
and f ðek Þ is the Fermi distribution function, f ðek Þ ¼
1 ; exp½ðek mÞ=kB T þ 1
ð12Þ
with the temperature T, the Boltzmann constant kB and the chemical potential m; the chemical potential is determined numerically from the half-filling condition for each temperature. Because of lack of the electron–hole symmetry, m is not fixed at zero as in the square-lattice case. For each q, there are two eigenvalues given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð13Þ o27 ¼ 12fW xx þ W yy 7 ðW xx W yy Þ2 þ 4W 2xy g:
The data shown in the following section are obtained by substituting Eq. (10) into Eq. (13). In what follows, the frequency pffiffiffiffiffiffiffiffiffiffiffiffiffi is scaled by the bare optical phonon frequency oQ 4K=M , and the strength of the e–l coupling is expressed by the dimensionless coupling constant l a2 =Kt 0 .
3. Results of calculations In Fig. 1, o2 is plotted as a function of the temperature T for some special values of q in the case with N ¼ 8; l ¼ 0:5 and g ¼ 0:1 in order to discuss typical behaviors of the phonon frequencies at relatively small values of g. In the same figure, the data for o2þ ðQ Þ are also depicted by a dash-dotted curve for comparison. Here Q ¼ ðp; pÞ is perpendicular to Q . Although Q and Q are equivalent since their difference is equal to a reciprocal lattice vector, their fractions are not equivalent. From Fig. 1, it is found that the phonon modes, which are softened first with lowering the temperature from the high temperature region, are o -modes with wave vectors Q and integer multiples of 2Q =N (in the present case, Q =4, Q =2 and 3Q =4).2 Furthermore, they are softened simultaneously at a critical temperature Tc , at which no other mode is softened though the data for other modes are omitted here. Analyses of the eigenvectors indicate that those modes softened at Tc are transverse. On the other hand, the softening of the o þ mode with Q , which is longitudinal, occurs at a temperature lower than Tc . These facts are consistent with the phonon softening behavior in the case of the isotropic square lattice, where all the transverse modes with Q and integer times 2Q =N are softened simultaneously at a certain critical temperature along with the longitudinal mode with Q . Because of symmetry, this is the case for Q and integer times 2Q =N in the isotropic square lattice3. Fig. 1 indicates that an introduction of the diagonal coupling lifts the degeneracy between integer times 2Q =N and integer times 2Q =N and between the longitudinal and transverse modes with Q . The above results, however, is not in accordance with the behavior of the system at absolute zero of temperature [19], where the lowest energy state is the MMP state involving distortion components with Q and Q =2 (not Q =2) when the diagonal coupling is in the weak region. This discrepancy suggests that there might be a rich variety of phase transitions in the low temperature region below Tc . In fact, it is pointed out in the case of an anisotropic square-lattice system that there are complicated rich structures of phase transitions in the low temperature region [20]. Next we show in Fig. 2 the temperature dependence of the square frequencies for the case with N ¼ 8; l ¼ 0:5 and g ¼ 0:7 as a typical example for the strong diagonal coupling case. In this figure, o2 is plotted by solid lines for several wave vectors parallel to Q or Q . For reference, some curves of o2þ are also given by dash-dotted lines. When the temperature is lowered from the high temperature region, the mode first softened at Tc is the one with the wave vector Q =2. At Tc , no other mode is softened, which suggests that the low temperature phase would be a single-mode Peierls state involving the distortion with the wave vector Q =2. This can be understood from the electronic dispersion Eq. (11). If the last term of Eq. (11) is dominating, the Fermi line consists of two straight lines parallel to Q and separated by the wave vector 2 Though we have shown only the case with N ¼ 8 partly in order to avoid too many curves in one figure, the essential features are not changed for different system sizes. 3 It should be noted that a vanishing g does not lead to the isotropic square lattice because we did not introduce the factor g in the diagonal lattice coupling term for the sake of simplicity. This point will be discussed in the last section.
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tures lower than the crossing point. In the cases of other modes shown in Fig. 2, o modes are transverse and o þ modes are longitudinal, which is also the case in Fig. 1.
4. Conclusion and discussion
Fig. 1. The temperature dependence of square eigenfrequencies for various wave vectors in the case with N ¼ 8, l ¼ 0:5 and g ¼ 0:1.
Fig. 2. The temperature dependence of square eigenfrequencies for various wave vectors in the case with N ¼ 8, l ¼ 0:5 and g ¼ 0:7.
Q =2, i.e. Q =2 becomes the nesting vector in this limit. The first softened Q =2mode is found to be transverse. Below Tc , we have to consider a static distortion with the wave vector Q =2 and with a transverse nature. The structure of the low temperature region will be treated elsewhere. Here we simply mention that the fact that the mode with Q =2 is dominating in the case of relatively large values of g is consistent with the previous study at zero temperature [19]. In Fig. 2, the modes with 3Q =4 show peculiar behavior. There is a crossing between o and o þ . In order to explain this behavior, we should notice that, when q is parallel to Q or Q , W xx and W yy are equal to each other because of symmetry. Therefore the square eigenfrequencies for those special wave vectors are given by o27 ¼ W xx 7 jW xy j, which indicates also that the corresponding modes are longitudinal or transverse depending on the sign of W xy . In fact, the o mode for 3Q =4 in Fig. 2 is transverse at higher temperatures and longitudinal at tempera-
The 2D Peierls transition in an anisotropic triangular lattice is discussed from the view point of phonon softening. The model system is constructed from the square lattice by introducing couplings in one of the diagonal directions. When the anisotropic parameter g, describing the strength of the diagonal coupling relative to the normal square-lattice type coupling, is small, phonon modes with fractional wave vectors of Q ½ ¼ ðp; pÞ are softened simultaneously at a certain critical temperature Tc along with the Q ½ ¼ ðp; pÞ (or equivalently Q ) mode as the temperature is lowered from the higher region. This fact suggests the realization of the MMP state below Tc . On the other hand, for larger values of g, only one mode with the wave vector Q =2 is softened with lowering temperature at a certain temperature while other modes are not yet softened, which suggests a singlemode Peierls state below the softening temperature. Although we have not yet systematically studied the gdependence of the phonon dispersion, a preliminary rough estimation of the border value of g separating the weak and strong coupling regions shows that it lies around 0.6. Although we do not show explicit data because of the limitation of the allotted space, we have analyzed the gdependence of the phonon softening temperatures. Generally speaking, the phonon softening temperatures are increasing functions of g, while those for some modes show a maximum as functions of g. Even if we can expect an MMP state for small values of g, the present phonon softening study seems not consistent with the previous zero temperature study [19]. A similar inconsistency between the phonon softening and zero temperature studies was found in the anisotropic square lattice model [9–11]. Thereby, the existence of rich variety of phase transitions is suggested in the low temperature region [20]. It will be plausible to expect a complicated structure of various phase transitions between the phonon softening temperature and the absolute zero temperature. In this paper we have shown only the results for N ¼ 8. Similar calculations were carried out for larger system sizes up to N ¼ 32. Essential features are found not to be changed at least up to this size. In the present model we did not introduce the factor g to the diagonal lattice coupling (i.e. the K-term) for simplicity. Therefore, even if we put g ¼ 0, the system is not reduced to the isotropic square lattice. Nevertheless, in the case of wave vectors parallel to Q , the effect of the diagonal lattice coupling vanishes as will be seen in the last term of Eq. (10). In this sense the results for the modes with fractions of Q coincide with those of the isotropic square lattice when g is set to be zero. In organic materials there are several systems which might correspond to the model treated in the present work. For example, ytype BEDT-TTF organic conductors [21] or some of PdðdmitÞ2 type complex salts [22] might be essentially regarded as anisotropic triangular lattices. There would remain a possibility to find the MMP phase in layered organic materials. In the present work, we did not consider the effect of electron– electron interactions. It is inevitable to take account of them when we compare the theoretical results with experiments. An investigation in this direction will be done in the future. Study of the low temperature phase should also be done along with the detailed analyses of dependences of the lowest energy state on the parameters g and l. This is left for future works.
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C. Watanabe, Y. Ono / Physica E 42 (2010) 1239–1242
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