Received 16 March 2016; accepted 19 April 2016; published 22 April 2016
Quasi-one-dimensional (Q1D) organic materials have received much interest during the last years due to more diverse and often unusual properties manifested by them  -  . Besides, their properties can be easily manipulated and controlled by molecular chemistry methods. It has been also mentioned that the highly conducting Q1D organic crystals may have very prospective thermoelectric applications   . The charge transfer compound TTF-TCNQ (tetrathiafulvalene-tetracyanoquinodimethane) is the most investigated Q1D organic crystal with the high electrical conductivity  . Its structure consists of segregated chains or stacks of TTF and TCNQ molecules. The metallic properties are due to partial transfer of electrons from TTF to TCNQ molecules.
With the decrease of temperature, two processes undergo. From one side, the lattice distortion is favorable because it diminishes the electronic energy of the crystal, lowering the Fermi energy. From other side, the lattice distortion increases the elastic lattice energy. At some temperature, when the first process prevails over the second one, a Peierls structural transition undergoes.
It is established that in TTF-TCNQ at 54 K the Peierls transition takes place into interacting TCNQ stacks with the opening of a band gap in the electronic spectrum above the Fermi energy and a strong reducing of electrical conductivity. Although the development of the pre-transitional 1D structural fluctuations is observed on a large temperature range below 150 K   , these fluctuations lead to modification of the phonon spectrum with the lowering temperature   . If at some temperature, the renormalized phonon frequency becomes equal to zero, a periodic lattice distortion with the respective wave vector occurs and the Peierls structural transition is observed.
In the last time, the physical model of Q1D organic crystals was completed by an additional electron-phonon interaction mechanism. It takes into account the fluctuations due to acoustic longitudinal phonons of the polarization energy of molecules surrounding the conduction electron  -  . The molecules of TCNQ are relatively big and their polarizability is large too.
In  , the Peierls structural transition in Q1D crystals of TTF-TCNQ type was investigated in this more complete physical model of the crystal, but in strictly 1D approximation. It was considered the case when the conduction band is half filled and Fermi dimensionless quasi momentum, kF = π/2. The Peierls critical temperature was determined.
In  , the Peierls transition has been studied also in the 1D physical model. But now the renormalized acoustic phonon frequencies W(q) as functions of wave number q were calculated: 1) when the conduction band is half filled and the dimensionless Fermi momentum kF = π/2 and 2) when the concentration of conduction electrons is reduced and the band is filled up to a quarter of the Brillouin zone, kF = π/4. It was shown that a more detailed calculation considerably modifies the dependences W(q). In both cases, the critical temperature of transition was determined.
The 2D physical model for the real crystals of TTF-TCNQ was investigated in  . The polarization operator as a function of temperature was calculated numerically for different values of the parameter d, which describes the ratio of the transfer energy in the direction transversal to conductive chains to the transfer energy along the chains. The Peierls critical temperature was determined in two cases: 1) when the dimensionless Fermi momentum is kF = 0.59 π/2, and 2) when the carrier’ concentration is slightly modified and the Fermi momentum kF = 0.59 π/2 ± δ, where δ represents the variation of the Fermi momentum. In this paper, the numerical modellings were performed for the values of the sound velocity at low temperatures taken from  : vs1 = 3.4 × 105 cm/s along chains and vs2 = 5.25 × 105 cm/s in transversal direction. However, more exact calculations of the Peierls critical temperature in TCNQ chains have shown that in order to achieve the experimental result, it is necessary to choose vs2 = 0.5vs1. The last value has been used later on.
The aim of this paper is to present a detailed modeling of the Peierls transition in TTF-TCNQ crystals in the frame of above mentioned more complete physical model. The effects of interchain interaction on the dispersion of renormalized phonons and on Peierls critical temperature are analyzed. For the simplicity, we consider the 2D approximation.
2. The Physical Model in 2D Approximation
The Hamiltonian of the 2D crystal in the tight binding and nearest neighbor approximations has the form:
In Equation (1) the first term represents the energy operator of free electrons in the periodic field of the lattice, are the creation and annihilation operators of such electron, where k is two-dimensional wave vector with projections (kx, ky). The carrier energy is measured from the bottom of conduction band.
where and are the transfer energies of a carrier from one molecule to another along the chain (x direction) and in perpendicular direction (y direction). In Equation (1) are creation and annihilation operators of an acoustic phonon with two-dimensional wave vector q and frequency. The second term in the Equation (1) is the energy of longitudinal acoustic phonons
where and are limit frequencies for oscillations in x and y directions. The third term in Equation (1) represents the electron-phonon interactions. Two interaction mechanisms are considered. The first interaction is determined by the fluctuations of transfer energies and, due to the intermolecular vibrations (acoustic phonons). This interaction is similar to that of deformation potential, and the coupling constants are proportional to the derivatives and of and with respect to the intermolecular distances, ,. The second interaction is of polaron type. This interaction is conditioned by the fluctuations of the polarization energy of the molecules surrounding the conduction electron. The coupling constant of this interaction is proportional to the average polarizability of the molecule. This interaction is important for crystals composed of large molecules such as TCNQ, so as is roughly proportional to the volume of molecule.
The square module of matrix element from Equation (1) has the following form:
where N is the number of molecules in the basic region of the crystal, M is the mass of the molecule;, parameters γ1 and γ2 have the sense of the amplitudes ratio of second electron-phonon interaction to the first one along chains and in transversal direction
From exact series of perturbation theory for the phonon Green function we sum up the diagrams containing closed loops of two electron Green functions that make the most important contribution. This is the random phase approximation. We obtain for the Fourier component of the phonon Green function:
is the free phonon Green function, δ → 0+ and Ω(q) is the renormalized phonon frequency. is the phonon polarization operator, q is the wave vector of longitudinal acoustic phonons. The real part of the dimensionless polarization operator is presented in the form:
Here, is the number of elementary cells in the basic region of the crystal, the number of molecules is, where r = 2 is the number of molecules in the elementary cell, is the Planck constant. The temperature enters in our expressions only through the Fermi distribution functions and. Ω(q) is determined by the pole of function and is obtained from the transcendent dispersion equation
This equation can be solved only numerically.
3. Results and Discussions
The calculations were performed for the following parameters: w1 = 0.125 eV, eV∙Å−1, a = 12.30 Å, b = 3.82 Å, kF = 0.59π/2   , vs1 = 3.4 × 105 cm/s, vs2 = 0.5vs1 cm/s, d = 0.015, M = 3.7 × 105me (me is the mass of the free electron), γ1 = 1.37, r = 2. The parameter γ2 is determined from the relation: γ2 = γ125b5/(a5d).
Figure 1 and Figure 2 show the dependences of renormalized phonon frequencies Ω(qx) as functions of qx for different temperatures and different values of qy. The same dependences for initial phonon frequency ω(qx) are presented too. It is seen that the values of Ω(qx) are diminished in comparison with those of frequency ω(qx) in the absence of electron-phonon interaction. This means that the electron-phonon interaction diminishes the values of lattice elastic constants. In addition, it is observed that with a decrease of temperature T the curves change their form, and in dependencies Ω(qx) a minimum appears. This minimum becomes more pronounced at lower temperatures. After all, at certain temperature Ω(qx) attains zero for qx = 0.58π. At this temperature, the structural Peierls transition takes place. The deviation of Ω(qx) = 0 from qx = 2kF is caused by T ≠ 0 and kF ≠ π/2.
Figure 1(a) shows the case, when qy = 0. At T = 59.7 K the transition occurs in TCNQ chains alone. The crystal lattice along TCNQ chains changes from the initial state with the lattice constant b to a new crystalline state with constant 4b, that is four times larger. At this temperature, the electrical conductivity is strongly diminished, so as a gap in the carrier spectrum is fully opened just above the Fermi energy. In addition, it is seen that the slope of Ω(qx) at small qx is diminished in comparison with that of ω(qx). This means that the electron-phonon interaction has reduced also the sound velocity in a large temperature interval.
When the interaction between TCNQ chains is taken into account (qy ≠ 0), the Peierls critical temperature is diminished. Figure 1(b) and Figure 2 correspond to 2D physical model, qy ≠ 0. Figure 1(b) shows Ω(qx) for qy = π/4 and different temperatures. One can observe that Ω(qx) attains zero at T ~ 59 K, i.e. the transition takes place at this T.
When qy = 2kF (Figure 2(a)), the Peierls critical temperature some more decreases and has a value of T ~ 56 K. Figure 2(b) shows the dependences of Ω(qx) on qx for qy = π and different temperatures. It is observed that the transition temperature decreases still more and equals T ~ 54 K. Note that this value agrees with the experimental data.
Figure 1. Renormalized phonon spectrum Ω(qx) for γ1 = 1.37 and different temperatures. The dashed line is for the spectrum of free phonons. (a) qy = 0; (b) qy = π/4.
Figure 3 shows clearly that with the increase of parameter qy, the Peierls critical temperature decrease. So, for qy = 0, T ~ 59.7 K; for qy = π/4, T ~ 59 K; for qy = 2kF, T ~ 56 K and for qy = π, T ~ 54 K. This means that Peierls transition begins at T ~ 59 ÷ 60 K in TCNQ chains alone, but when the interaction between TCNQ chains is taken into account in 2D approximation, Peierls transition occurs completely at T ~ 54 K. It is expected that 3D approximation will not introduce important modifications.
Figure 4 shows the dependencies of the real part of dimensionless polarization operator as function of qx for different values of qy and different temperatures at Ω = 0. From the both graphs it is observed a very sharp peak near the value of unity.
Figure 2. Renormalized phonon spectrum Ω(qx) for γ1 = 1.37 and different temperatures. The dashed line is for the spectrum of free phonons. (a) qy = 2kF; (b) qy = π.
Figure 3. Renormalized phonon spectrum Ω(qx) for γ1 = 1.37, different values of qy and different temperatures.
Figure 4. Polarization operator as function of qx for: (a) qy = 0 and T = 59.7 K; (b) qy = π and T = 54 K.
Note that in above calculations we have used the value of vs1 = 3.4 × 105 cm/s at low temperatures from  , but for vs2 we have taken 0.5vs1 instead of vs2 = 5.25 × 105 cm/s measured in  . If the last value of vs2 is applied, it results that a complete Peierls transition should undergo at T = 15 K. Such big deviation of critical temperature from the observed one at T = 54 K suggests, that the sound velocity in the transversal direction cannot be larger than in longitudinal to chains direction and the most probably vs2 ~ 0.5vs1, as demonstrate our calculations of transition temperature. In addition, due to the quasi-one-dimensionality of the crystal, the chemical bonds in transversal direction must be weaken. This fact also suggests that vs2 should be less than vs1.
The behavior of phonons near Peierls transition has been studied in quasi-one-dimensional organic crystals of TTF-TCNQ in 2D approximation. A more complete crystal model is applied which takes into account the two most important electron-phonon interactions. One of them is of the deformation potential type. The coupling constants are proportional to the derivatives of the transfer energies and with respect to the intermolecular distances. The other interaction is similar to that of a polaron with the coupling constant proportional to the average polarizability of the molecule. The ratios of amplitudes of second electron-phonon interaction to the first one along chains and in transversal direction are noted by γ1 and γ2, respectively. Analytical expression for the polarization operator was obtained in random phase approximation. The numerical calculations for renormalized phonon spectrum, Ω(qx), for different temperatures are presented: 1) when qy = 0 and the interaction between TCNQ chains is neglected and 2) when qy ≠ 0 and interactions between the adjacent chains are considered. It has been established that Peierls transition begins at T ~ 59.7 K in TCNQ chains alone and reduces considerably the electrical conductivity. Due to interchain interaction, the transition is finished at T ~ 54 K. It is demonstrated that the electron-phonon interaction diminishes Ω(qx) with respect to initial frequency ω(qx) and reduces the sound velocity in a large temperature interval.