Peierls Structural Transition in Q1D Organic Crystals of TTT 2 I 3 for Different Values of Carrier Concentration

The Peierls structural transition in the TTT 2 I 3 (tetrathiotetracene-iodide) crystal, for different values of carrier concentration is studied in 3D approximation. A crystal physical model is applied that considers two of the most important hole-phonon interactions. The first interaction describes the deformation potential and the second one is of polaron type. In the presented physical model, the interaction of carriers with the structural defects is taken into account. This is crucial for the explanation of the transition. The renormalized phonon spectrum is calculated in the random phase approximation for different temperatures applying the method of Green functions. The renormalized phonon frequencies for different temperatures are presented in two cases. In the first case the interaction between TTT chains is neglected. In the second one, this interaction is taken into account. Computer simulations for the 3D physical model of the TTT 2 I 3 crystal are performed for different values of dimensionless Fermi momentum k F , that is determined by variation of carrier concentration. It is shown that the transition is of Peierls type and strongly depends on iodine concentration. Finally, the Peierls critical temperature was determined.


Introduction
During last year, organic materials have attracted increasing attention due to more diverse and unusual proprieties [1] [2] [3]. Recently, it was shown that How to cite this paper: Andronic, S. and highly conducting Quasi-One-Dimensional (Q1D) organic crystals can have different promising thermoelectric applications [4] [5]. It was predicted in [6] that the values of dimensionless thermoelectric figure of merit ~4 could be realized after optimization of carrier concentration in TTT 2 I 3 organic crystals. On the other hand, it is well known that organic nanomaterials have large potential applications in electronic, sensing, energy-harnessing and quantum-scale systems [7]. We mention that, the most studied organic crystals are those of tetrathiotetracene-iodide (TTT 2 I 3 ) of p-type, tetrathiofulvalinium tetracyanoquinodimethane (TTF-TCNQ) of n-type and tetrathiotetracene tetracyanoquinodimethane (TTT(TCNQ) 2 ) of n-type. Q1D organic materials of TTT 2 I 3 , were synthesized independently [8] [9] [10] [11] with the aim to detect superconductivity in such a low-dimensional conductor. At the same time, these crystals show a metal-dielectric transition with decreasing temperature. Such transition has been observed in the Q1D charge transfer compound TTF-TCNQ in [12] [13]. This first experimental confirmation of the structural transition was predicted earlier by Rudolf Peierls [14] in 1D conductors. According to Peierls, for some lowered temperatures, the one-dimensional metallic crystal with a half filled conduction band has to pass in a dielectric state with a dimerized crystal lattice. This temperature T p is called the Peierls critical temperature. Later, different authors [15]- [22] studied this phenomenon in different Q1D organic crystals. In TTF-TCNQ crystals, the metal-insulator transition takes place at 54 K into TCNQ stacks and at 38 K into TTF stacks. We recently studied the Peierls transition of TCNQ stacks [23]. The renormalized phonon spectrum was analyzed for different temperatures. We observed that with lowering temperature, some modifications in the phonon spectrum take place. For certain temperature, the renormalized phonon frequency becomes equal to zero for a given value of the phonon wave vector.
We found that at 54 K the Peierls transition takes place.
The crystal of TTT 2 I 3 is a charge transfer compound. The orthorhombic crystal structure consists of segregated chains or stacks of plane TTT molecules and of iodine chains. This compound is of mixed valence. Two molecules of TTT give one electron to iodine chain formed of 3 I − ions that play the role of acceptors. Only TTT chains are conductive and the carriers are holes. The electrons on iodine ions are in a rather localized states and do not participate in the transport. TTT 2 I 3 crystal has the following lattice constants 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. We investigated the Peierls transition in a 2D physical model for a TTT 2 I 3.1 crystal [24]. In [25] the Peierls transition in the TTT 2 I 3 crystals with the intermediate value of carrier concentration in 2D approximation was studied. It was applied a complete physical model [26], that considers two hole-phonon interaction mechanisms.
The conductivity properties of TTT stacks are very sensitive to defects and impurities [27]. This is caused by the purity of initial materials and the conditions of crystal growth. In TTT 2 I 3 crystals, with the lowering of temperature the Advances in Materials Physics and Chemistry conductivity firstly grows, reaches a maximum and 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) (temperature dependence of electrical conductivity) after the maximum.
This paper reports on study of the Peierls transition in quasi-one-dimensional organic crystals TTT 2 I 3 . The main aim is to demonstrate that the sharp decrease of temperature dependence of electrical conductivity σ(T) is determined by the structural transition in the TTT chains. Thus, we analyze the behavior of Peierls transition in organic crystals of TTT 2 I 3 for different values of carrier concentration in 3D approximation. In the frame of the physical model we consider two hole-phonon interaction mechanisms. The first mechanism is of the deformation potential type and is determined by the variation of the transfer energy of a carrier from one molecule to the nearest one, caused by acoustic lattice vibrations. The second one is similar to that of polaron type, and it is determined by the variation of the polarization energy of molecules surrounding the conduction electron caused by the same acoustic vibrations. The dynamical interaction of carriers with the defects is also considered. We mention that, the Peierls structural transition in crystals of tetrathiotetracene-iodide, was not reported yet by other authors.
We start by analyzing the curves presented in Figure 1 [27]. We show that the The paper is structured as follows. Section 2 describes three-dimensional model of the crystal. Section 3 presents the results of computer simulations. The conclusions are formulated in Section 4.

Three-Dimensional Physical Model of the Crystal
In this Section we will present the physical model of the TTT 2 I 3 crystal that was described in more detail in [24]. The Hamiltonian of the 3D crystal model in the tight binding and nearest neighbor approximations is presented in the following Let's analyze each term of Equation (1)   2 1 cos 2 1 cos 2 1 cos , (2) where 1 w , 2 w and 3 w are the transfer energies of a hole from one molecule to another along the chain (x direction) and perpendicular to it (y and z directions), respectively.
It is well known, that the Peierls transition occurs at low temperatures. In this case, the interaction of electrons with optical phonons can be neglected. Thus, the spectrum of acoustic phonons of a simple one-dimensional chain can be described by [28] ( ) where 1 ω , 2 ω and 3 ω are the limit frequencies in the x, y and z directions, respectively. As was mentioned above, in the frame of this model we consider We notice that in Equation (4) several parameters are used for the detailed analysis of the crystal, considering the two hole-phonon interaction mechanisms mentioned above. Namely, M is the mass of the TTT molecule. N is the total number of molecules in the main region of the crystal. 1 The parameters 1 γ , 2 γ and 3 γ describe the ratio of amplitudes of the polaron-type interaction to the deformation potential one in the x, y and z directions, respectively. These parameters can be obtained from the following expressions To explain the behavior of the electrical conductivity from Figure In Equation (6)  , .
x k k q q A B q n n ω ε ε k n is the Fermi distribution function. Finally, we mention that Equation (9) can be solved only numerically.

Results and Discussions
In the subsequent analysis, we perform the computer simulations for the 3D physical model of the crystal. The following set of parameters are used [29] Figure 2 for k F = 0.512•π/2 and D = 1.026 eV•Å −1 . Other parameters: q y = π, q z = π. Figure 6. The same as in Figure 2 for k F = 0.508•π/2 and D = 1.045 eV•Å −1 . Other parameters: q y = 0, q z = 0. Advances in Materials Physics and Chemistry Figure 7. The same as in Figure 2 for k F = 0.508•π/2 and D = 1.026 eV•Å −1 . Other parameters: q y = π, q z = π. Figure 8. The same as in Figure 2 for k F = 0.502•π/2 and D = 1.057 eV•Å −1 . Other parameters: q y = 0, q z = 0. Figure 9. The same as in Figure 2 for k F = 0.502•π/2 and D = 1.055 eV•Å −1 . Other parameters: q y = π, q z = π. renormalized phonon spectrum Ω(q x ) as a function of q x , when q y = 0 and q z = 0 (the interaction between TTT chains is neglected). The dimensionless Fermi momentum k F = 0.517•π/2. Parameter D (D = 1.036 eV•Å −1 ) determines the intensity of hole interaction with a defect. The Peierls transition begins at T = 35 K. At this temperature, the electrical conductivity is strongly diminished (for comparison see Figure 1(a)), so as a gap in the carrier spectrum is fully opened just above the Fermi level.
If the interaction between transversal chains is taken into account (q y ≠ 0 and q z ≠ 0), the temperature at Ω(q x ) = 0 is diminished. Figure 3 shows the case Advances in Materials Physics and Chemistry when q y = π and q z = π, k F = 0.517•π/2 and D = 1.033 eV•Å −1 . It was observed that parameter D decreases, or the hole interaction with a defect is smaller in this case. Analyzing the figure, one can see that the transition is completed at T = 9.8 K.
According to [27], the electrical conductivity significantly decreases and achieves zero at T ~ 10 K. Thus, our calculations show that the transition is of Peierls type and takes place at this temperature.
In Figures 4-7 we show the behavior of Peierls structural transition in organic crystals of TTT 2 I 3 for intermediate values of carrier concentration. Figure 4 shows the case where the Fermi momentum decreases and has a value of k F = 0.512•π/2. The interaction between TTT chains is neglected. Parameter D has a value of D = 1.042 eV•Å −1 . From the figure it is evident that the Peierls transition begins at T = 50 K. When the interaction between TTT chains is taken into account (q y = π and q z = π), the temperature decreases considerably and the transition is completed at T = 11 K (see Figure 5). Now, the hole interaction with a defect is smaller D = 1.026 eV•Å −1 . Figure 6 shows the case when the carrier concentration additionally decreases and k F = 0.508•π/2. The Peierls transition begins at T = 62 K and is finished at T = 12 K (see Figure 7). Parameter D has a value D = 1.045 eV•Å −1 , when q y = 0 and q z = 0, and decreases to D = 1.026 eV•Å −1 , when q y = π and q z = π. Figure 8 and Figure 9 describe the Peierls transition in TTT 2 I 3 crystals for the lowest value of carrier concentration. Figure 8 describes the case when q y = 0 and q z = 0 and the dimensionless Fermi momentum k F = 0.502•π/2. Parameter D has a value of D = 1.057 eV•Å −1 . The interaction between TTT chains is neglected. In this case the Peierls transition begins at T = 90 K. At this temperature, the electrical conductivity achieves a maximum. With the lowering temperature, the electrical conductivity decreases. Figure 9 shows the case where the interaction between TTT chains is taken into account (q y = π and q z = π), D = 1.055 eV•Å −1 and k F = 0.502•π/2. As is observed from the figure, the transition is completed at T = 19.7 K. It is evident from Figure 1(d) that the electrical conductivity significantly decreases and achieves zero at T ~ 20 K. Also, it is observed that the parameter D decreases, or the hole interaction with a defect is smaller in this case.
The Peierls structural transition in Q1D organic crystals of TTT 2 I 3 , strongly depends on iodine concentration. From figures presented above and from Table   1 it was observed that with a decrease of the carrier concentration, the Peierls critical temperature increases.

Conclusion
The Peierls structural transition in Q1D organic crystal of TTT 2 I 3 for different values of carrier concentration was studied in 3D approximation. In the frame of the crystal model, two of the most important hole-phonon interaction mechanisms are considered: of the deformation potential type and of the polaron type. Advances in Materials Physics and Chemistry  The interaction of holes with the structural defects in direction of TTT chains was taken into account too. The renormalized phonon spectrum has been calculated in the random phase approximation. Computer simulations for renormalized phonon spectrum, Ω(q x ), for different temperatures are presented in two cases: 1) when the interaction between TTT chains is neglected (q y = 0, q z = 0) and 2) when the interaction between TTT chains is taken into account (q y = π, q z = π).
It was observed that the interchain interaction, reduces the transition temperature. The hole-phonon interaction and the interactions with structural defects diminish Ω(q x ) and reduce the sound velocity in a wide temperature range.
When the interaction between TTT chains is taken into account, the parameter D decreases, or the hole interaction with a defect is smaller in this case. It was shown that the Peierls transition temperature strongly depends on iodine concentration. In this paper we analysed the dependences of the renormalized phonon spectrum for the following values of the carrier concentration.
1) When k F = 0.517•π/2 and the hole concentration achieves the higher value in TTT 2 I 3 crystal, the Peierls transition begins at T ~ 35 K in TTT chains and considerably reduces the electrical conductivity. Due to interchain interaction, the transition is completed at T ~ 10 K.
2) When the carrier concentration achieves an intermediate value, for k F = 0.512•π/2, the Peierls transition begins at T ~ 50 K in TTT chains and is finished at T ~ 11 K.
3) When k F = 0.508•π/2, the transition begins at T ~ 62 K in TTT chains and is completed at T ~ 12 K. 4) For the lowest values of hole concentration, when k F = 0.502•π/2, the Peierls transition begins at T ~ 90 K in TTT chains. The electrical conductivity is considerably reduced. Due to interchain interaction, the transition is completed at T ~ 20 K.
Analyzing the behavior of the Peierls transition in the cases presented above, it can be observed that, with a decrease of the carrier concentration, the Peierls S. Andronic, A. Casian