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We investigate the metal-insulator transition in quasi-one-dimensional organic crystals of tetrathiotetracene-iodide, TTT <sub>2</sub>I<sub>3</sub>, in the 2D model. A crystal physical model is applied which takes into account two the most important hole-phonon interaction mechanisms. One is similar to that of deformation potential and the other is of polaron type. The scattering on defects is also considered and it is crucial for the explanation of the transition. The phonon polarization operator and the renormalized phonon spectrum are calculated in the random phase approximation for different temperatures applying the method of Green functions. We show that the transition is of Peierls type. The effect of lattice distortion on the dispersion of renormalized acoustic phonons is analyzed.

Quasi-one-dimensional (Q1D) organic crystals of tetrathiotetracene-iodide, TTT_{2}I_{3}, were synthesized independently and nearly simultaneously [

TTT_{2}I_{3} is also a charge transfer compound. The orthorhombic crystal structure consists of segregated chains or stacks of plane TTT molecules and of iodine chains. The lattice constants are a = 18.40 Å, b = 4.96 Å and c =18.32 Å, which demonstrates a very pronounced crystal quasi-one-dimensionality. The highly conducting direction is along b. The compound is of mixed valence. Two molecules of TTT give one electron to iodine chain formed of ^{3} - 10^{4}) Ω^{−1}∙cm^{−1}, but in those grown from solution [^{3}) Ω^{−1}∙cm^{−1}. Such variation in σ of crystals, grown in different laboratories, shows that the conductivity properties of TTT stacks are highly sensitive to defects and impurities. It is caused by the purity of initial materials and the conditions of crystal growth. In all crystals, with the lowering of temperature the conductivity firstly grows, reaches a maximum after that falls. The temperature of the maximum, T_{max}, and the value of the ratio σ_{max}/σ_{300} depends on the iodine content. Crystals with a surplus of iodine, TTT_{2}I_{3.1}, have T_{max} ~ (34 - 35) K and very sharp fall of σ(T) after the maximum.

The aim of present paper is to demonstrate that this sharp decrease of σ(T) is determined by the Peierls structural transition in the TTT chains. At our knowledge, the Peierls transition in TTT_{2}I_{3} was not studied neither theoretically, nor experimentally. It is known that the Peierls structural transition is connected by the competition of two processes that take place, when the temperature is decreased. From one side, it is favorable that the lattice distorts because this diminishes the electronic energy of the crystal, lowering the Fermi energy. However, this distortion increases the lattice elastic energy. At some temperature, named the Peierls critical temperature T_{p}, when the first process prevails over the second one, a Peierls structural transition takes place.

The Peierls structural transition has been studied in many Q1D crystals [

Note that earlier the crystals of TTT_{2}I_{3} have been investigated as good candidates for thermoelectric applications [_{2}I_{3}.

We will apply a more complete crystal model [_{2}I_{3} at low temperature. The dispersion of renormalized phonons and the Peierls critical temperature are determined. For the simplicity, we consider the 2D physical model.

We apply the two-dimensional physical model described in [

where the first term is the energy operator of free holes in the periodic field of the lattice, _{x}, k_{y}). The energy of the hole

Here

where

The square module of matrix element

where M is the mass of the TTT molecule, N is the number of molecules in the basic region of the crystal, _{1} and γ_{2} have the sense of the amplitudes ratio of the second hole-phonon interaction to the first one along chains and in the transversal direction:

The analysis shows that the Hamiltonian from the Equation (1) can not explain the sharp decrease of electrical conductivity for temperatures lower than T_{max} = 35 K, even, when we vary the crystal parameters and consider only the first interaction mechanism. It is necessary to take into account also the dynamical interaction of carriers with the defects. The static interaction will give contribution to the renormalization of hole spectrum. The defects in TTT_{2}I_{3} crystals are created due to different coefficients of dilatation of TTT and iodine chains. The Hamiltonian of this interaction

Here

the derivative with respect to intermolecular distance from the energy of interaction of a carrier with a defect,

where the constant D = 1.05 and determines the intensity of hole interaction with a defect.

In order to investigate the Peierls transition, the method of temperature dependent retarded Green functions is applied [

for lattice displacements

where

The equation of motion for the operator u_{q} is deduced as follows:

On the base of Equation (8), one can obtain the first equation for the Green function

Further, one can obtain the equation of motion for the new Green function

tions. In order to cut up the chain, let’s consider that the hole-phonon interaction is weak and express the three-particle Green function through the one-par- ticle Green function as follows

Thus, a closed equation for the function

Now it is conveniently to pass to Fourier transformation of the function

As a result, it follows:

where

From Equations (12) and (13) it results the expression for the Fourier transformation of the lattice displacement Green function

In order to distinguish the retarded Green function it is needed to put,

aginary part of the renormalized lattice frequency.

The real part of Equation (15) will determine the renormalized lattice frequency Ω(q), as the solution of the transcendent equation

where the principal value of the dimensionless polarization operator takes the form:

Here,

trix elements of the hole-phonon interaction from Equation (4), and of hole interaction with defects from Equation (6), the

Computer simulations are performed for the following parameters [^{5} m_{e} (m_{e} is the mass of the free electron), w1 = 0.16 eV, ^{−1}, a = 18.35 Å, b = 4.96 Å, c = 18.46 Å. The sound velocity at low temperatures is v_{s1} = 1.5・10^{5} cm/s along chains (in b direction), d = 0.015, γ_{1} = 1.7, and γ_{2} is determined from the relations: γ_{2} = γ_{1}b^{5}/(a^{5}d). For v_{s}_{2} in a transversal (in a direction) we have taken 1.35 × 10^{5} cm/s.

Figures 1-4 present the dependences of renormalized phonon frequencies Ω(q_{x}) as functions of q_{x} for different temperatures and different values of q_{y }. In the same graphs, the dependences for initial phonon frequency ω(q_{x}) are presented too. It is seen that the values of Ω(q_{x}) are diminished in comparison with those of frequency ω(q_{x}) in the absence of hole-phonon interaction. This means that the hole-phonon interaction and structural defects diminish the values of lattice elastic constants. Also, one can observe that with a decrease of temperature T the curves change their form, and in dependencies Ω(q_{x}) a minimum appears. This minimum becomes more pronounced at lower temperatures.

_{y} = 0. In this case the interaction between TTT chains is neglected. The Peierls transition begins at T = 35 K. 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(q_{x}) at small q_{x} is diminished in comparison with that of ω(q_{x}). This means that the hole-phonon interaction and structural defects have reduced also the sound velocity in a large temperature interval. When the interaction between transversal chains is taken into account (q_{y} ≠ 0), the temperature when Ω(q_{x}) = 0 is diminished.

_{x}) for q_{y} = π/4 and different temperatures. One can see that Ω(q_{x}) attains zero at T ~ 30 K. Figures 2-4 correspond to the 2D physical model, q_{y} ≠ 0.

When q_{y} = 2k_{F} (_{x}) = 0, decreases additionally and has the value of T = 21 K.

_{x}) on q_{x} for q_{y} = π and different temperatures. It is observed that the temperature, when Ω(q_{x}) = 0, decreases still more and equals T = 19 K, thus, our calculations show that at this temperature the Peierls transition is finished. A new superstructure must appear. Unfortunately, at our knowledge, such experiments were not realized. It would be interesting to verify experimentally our conclusions. As it is seen from [_{2}I_{3} temperature dependence of electrical conductivity.

_{x} for different values of q_{y} and different temperatures at Ω = 0. In _{y} = 0, and the interaction between TTT chains is not taken into account. It is observed a peak near the value of unity. This means that the Peierls transition begins at T = 35 K.

In _{x}, but for q_{y} = π. From this graph it is observed that, when the interaction between TTT chains is taken into account, the Peierls critical temperature decreases and transition is finished at T = 19 K.

We have investigated the behavior of phonons near Peierls structural transition in quasi-one-dimensional organic crystals of TTT_{2}I_{3} (tetrathiotetracene iodide)

type in 2D approximation. A more complete crystal model is applied which takes into account two the most important hole-phonon interactions. One interaction is of deformation potential type and the other is similar to that of polaron. The ratios of amplitudes of second hole-phonon interaction to the first one along chains and in transversal direction are noted by γ_{1} and γ_{2}, respectively. The interaction of holes with the structural defects in direction of TTT chains is taken into account too. Analytical expression for the polarization operator was obtained in random phase approximation. The method of retarded temperature dependent Green function is applied. The numerical calculations for renormalized phonon spectrum, Ω(q_{x}), for different temperatures are presented in two cases: 1) when q_{y} = 0 and the interaction between transversal chains is neglected and 2) when q_{y} ≠ 0 and interactions between the adjacent chains are considered. It has been established that Peierls transition begins at T ~ 35 K in TTT chains and reduces considerably the electrical conductivity. Due to interchain interaction the transition is finished at T ~ 19 K. It is demonstrated that the hole-phonon interaction and the interactions with the structural defects diminish Ω(q_{x}) and reduce the sound velocity in a large temperature interval.

Andronic, S. and Casian, A. (2017) Metal-Insulator Transition of Peierls Type in Quasi-One-Dimen- sional Crystals of TTT_{2}I_{3}. Advances in Ma- terials Physics and Chemistry, 7, 212-222. https://doi.org/10.4236/ampc.2017.75017