PROMISING LOW-DIMENSIONAL ORGANIC MATERIAL FOR IR DETECTORS

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1 OPTICS PROMISING LOW-DIMENSIONAL ORGANIC MATERIAL FOR IR DETECTORS A. CASIAN, V. DUSCIAC, V. NICIC Technical University of Moldova, Chisinau MD-2004, Rep. of Moldova Received September 25, 2008 The thermoelectric power factor is calculated and analyzed in quasi-onedimensional organic crystals of tetrathiotetracene-iodide, TTT 2 I 3, in the frame of model described in Phys. Rev. 66, (2002). When the crystal parameters are optimized, values of the power factor of the order of W/mK 2 are predicted which are almost 10 times higher than those of Bi 2 Te 3, a widely used thermoelectric material. Key words: IR detector, thermoelectric power factor, quasi-one-dimensional crystal, organic crystal, tetrathiotetracene-iodide. 1. INTRODUCTION Detectors of infrared (IR) radiation have large application in industry, in systems of control and monitoring, in scientific investigations and many other domains. However, the photoelectric detectors are sensitive only up to wave lengths of 1.3 µm. But the Earth atmosphere has only several windows of transparency in the larger wave length region of spectrum. In order to detect the IR radiation in the diapason of longer waves the thermoelectric detectors are widely used. In the last years, the requirements of infra red detectors for long wave length spectrum have grown considerable. Therefore the search of new, more sensitive thermoelectric non selective detectors is in attention of investigators. But in order to improve the parameters of such detectors new more efficient thermoelectric materials are necessary. An important parameter of material to be used as sensitive element in a thermoelectric IR detector is the thermoelectric power factor, P = σs 2, where σ is the electrical conductivity, and S is the thermopower (Seebeck coefficient). Values of P are necessary as high as possible. However, the requirements to increase simultaneously σ and S in the same material are contradictory. In ordinary materials the increase of electrical conductivity leads to the decrease of Paper presented at the 9 th International Balkan Workshop on Applied Physics, July 7 9, 2008, Constanta, Romania. Rom. Journ. Phys., Vol. 55, Nos. 1 2, P , Bucharest, 2010

2 206 A. Casian, V. Dusciac, V. Nicic 2 thermopower and vice-versa. In order to overlap these difficulties, it is necessary to search and investigate new materials and structures with more complicated electronic and phonon spectra. All the more, that from the theoretical point of view there is no an upper limit for the growth of the thermoelectric power factor. Recently, great values of the power factor have been obtained in low dimensional superlattice structures with quantum wells [1]. The increase of S in such structures is determined by the increase of electronic density of states. As a result the Fermi level is moved down and the electronic gas which in the bulk material would be degenerate at given concentration, in quantum wells becomes non degenerate or slightly degenerate. This leads to the growth of thermopower S. In the same time a possibility appears to slightly increase the carriers concentration in such a way that to obtain a simultaneous increase of σ and S and, respectively, an increase of the power factor P. Values of power factor P of the order of ( ) 10 3 W/mK 2 were measured in n-type PbTe/PbEuTe QWs [2], which are more than 1.7 times higher than the value of W/mK 2 measured in the best bulk widely used thermoelectric material Bi 2 Te 3. Even higher values of P for these structures have been predicted in [3 4]. Higher values of P = W/mK 2 have been measured in p-type PbTe/PbEuTe QWs [5], whereas the highest calculated values of P for such QWs with optimal parameters are ~ W/mK 2 [6]. But the technology to obtain such structures is rather complicate and expensive. A rigorous theory of the thermoelectric power factor in composite structures has been presented in [7]. In the last years the organic materials attract more and more attention as materials with more diverse and often unusual properties. In [8, 9] we have demonstrated that in highly conducting quasi-one-dimensional organic crystals one can expect to obtain, under certain conditions, increased values of the thermoelectric power factor. The aim of this paper is to investigate the opportunities of concrete crystals of tetrathiotetracene-iodide, TTT 2 I 3, type. 2. CRYSTAL MODEL The tetrathiotetracene-iodide crystals, TTT 2 I 3, have well pronounced quasione-dimensional (Q1D) properties. They are formed of segregate chains or stacks of tetrathiotetracene (TTT) and iodine [10] packed into a 3D crystalline structure. The interactions of molecules along stacks are much greater than between molecules of different stacks. Therefore, the crystals have needle-like form with the length of 6 12 mm, and the thickness of µm. In principle, the crystals have complicated structure. They are formed from molecules and the latter from atoms. But due to the fact that the intramolecular interactions are much stronger than the intermolecular ones, on can neglect in the weak fields the intramolecular processes and consider one electronic state per molecule. TTT 2 I 3 is a compound of

3 3 Low-dimensional organic material for IR detectors 207 mixed valence. Two molecules of TTT give one electron to the iodine chain which is formed from I3 ions. The overlapping of molecular wave functions along TTT stacks is important, but between stacks it is small. Thus, the carriers are holes and are moving mainly along the stack where they were formed and rarely hope from one stack to another. The conductivity along stacks is of band type and between stacks is of hopping type. The latter is small and can be neglected. The conductivity of iodine chains is also very small and is neglected too. In such a way, we obtain a crystal model formed from 1D conducting stacks or chains of molecules. In real Q1D crystals even weak interaction between 1D chains is of major importance, so as it determines different phase transitions at different critical temperatures. However, for temperatures higher than the maximum critical temperature it is possible to describe the transport phenomena neglecting the interaction between chains [11]. We will take into account two electron-phonon interactions [12]. The first is similar to that of deformation potential and is determined by the fluctuations caused by intermolecular vibrations of the energy w of the electron transfer from one molecule to another along the stack. The coupling constant is proportional to the derivative w of w with respect to the intermolecular distance. The second interaction is similar to that of polaron and is determined by the fluctuations of polarization energy of molecules surrounding the carrier. The coupling constant is proportional to the average polarizability of molecule α 0. The consideration of both these interactions together is very important, because under certain conditions the interference between them can take place. Using the band scheme in the tightbinding and nearest approximations, the Hamiltonian of 1D crystal chain can be presented in the form [12] ε k k ωq q q k k q q q k q k, q H = ( k) a a + b b + A( k, q) a a ( b + b ), (1) In order to obtain the Hamiltonian of whole crystal it needs to sum up (1) on + all chains into the basic region of crystal. Here ak and a k are the annihilation and creation operator of a carrier in the state with the wave vector projection k and + energy ε ( k) = 2w(1 coska) (for holes w < 0), b q and b q are the respective operators for longitudinal acoustic phonons with the wave vector projection q and 1 frequency ω q = 2v s a sin qa / 2, where a and vs are the lattice constant and the sound velocity along the chains. The lattice dynamics of the 1D crystal is similar to the needle-like one. There are three branches of vibrations. However, it can be shown that the carriers interact only with the longitudinal acoustic phonons. The first term in (1) is the energy operator of holes, the second term is the energy operator of free phonons and the third term is the hole-phonon interaction. The matrix element of interaction A( k, q) is given by the expression

4 208 A. Casian, V. Dusciac, V. Nicic 4 1/2 2i w Akq (, ) = [sinka sin(( k aa ) ) + γ sin qa], (2) 1/2 (2 NMω ) q γ = 2 e α / a w, (3) where N is the number of molecules in the basic region of the chain, M is the mass of molecule, γ is a dimensionless parameter which represents the ratio of amplitudes of above mentioned interactions. The first two terms in (2) describe the first interaction and the third term describes the second interaction. To the Hamiltonian (1) it needs to add the Hamiltonian of carriers interaction with impurities which has a usual form. For simplicity we will consider the impurities neutral and pointlike. 3. THERMOELECTRIC POWER FACTOR Under previous conditions, for the investigation of carriers transport in 1D systems it is possible to apply the kinetic equation [11]. Let consider that a weak external electric field and temperature gradient is applied along the chains. The linearized kinetic equation takes the form of Boltzmann equation. At room temperature the scattering processes can be considered elastic. In this case the kinetic equation is solved exactly and the electrical conductivity σ, the thermopower S and the power factor P can be expressed through the transport integrals R n σ = R 0, S = R 1 / etr0, P = σs 2, (4) where 2 2 n Rn = e ( E EF) ν ( E) τ( E) ρ( E) f 0 ( E) de. (5) 0 Here e is the electron charge, T is the temperature, v ( E) = 2 a E( E) 2 is the square of the carriers velocity as a function of energy E, = 4w, E F is the ρ is the density of electronic states per unit volume and energy Fermi energy, ( E) ( E) ( 2 z / πabc) E ( E) ρ =, (6) a, b and c are the lattice constants, z is the number of chains through the transversal τ = τ E L E is the relaxation time, where section of the unit cell, ( ) ( ) ( ) τ E 0 ( ) 1/2 2 s 0 ( E) = 2 2 8akTw 0 Mv E E is the relaxation time for the case, when γ = 0 and D = 0, i.e. when only the first electron-phonon interaction exists, and 1/2 (7)

5 5 Low-dimensional organic material for IR detectors 209 4w L( E) = (8) γ ( E E0 ) + 4w D describes the interference of both interactions and has the form of Lorentzian as a function of energy E. The quantity E 0 = 2w( γ 1) / γ is the resonance energy, which corresponds to the maximum of τ ( E). The dimensionless parameter D in (8) describes the carriers scattering on impurities 2 ( 0 ) / im s D= n I d Mv a k Tw, (9) where n im is the linear concentration of impurity, I and d are the effective height and width of impurity potential. The expression for thermoelectric power factor P has been modelled in the TTT 2 I 3 crystals as a function of Fermi energy and different values of parameters γ and D. 4. RESULTS AND DISCUSSION The following values of crystal parameters were taken: the mass of molecule M = m e (m e is the mass of free electron), w = 0.17 ev, w = 0.22 evå 1 the lattice constants a = Å, b = Å, c = 4.96 Å, c is the direction of chains, the sound velocity along chains v s = cm/s. Different degrees of crystal purity have been considered and, accordingly, for the parameter D the values 0.2, 0.06 and 0.03 have been taken as more accessible. In TTT 2 I 3 the parameter γ is unknown, because the molecule polarizability in the crystal is unknown. Therefore we have calculated the power factor P as a function of Fermi energy for different values of γ and above mentioned values of D. The results of thermoelectric power factor modelling are presented in Figs Calculations are made for room temperature T 0 = 300K as a function of dimensionless Fermi energy ε F, expressed in unities of 2w which is one half of conduction band width. In Fig.1 the case when γ = 1.8 is presented. This value of γ Fig. 1 Thermoelectric power factor P as a function of dimensionless Fermi energy for γ = 1.8.

6 210 A. Casian, V. Dusciac, V. Nicic 6 corresponds to molecule polarizability α 0 of the order of 41 Å 3. Note that in antracene α 0 = 25.3 Å 3. But the TTT molecule is bigger and α 0 must be greater too. From Fig. 1 it is seen that the power factor as a function of the Fermi energy has a maximum and P max achieves W/mK 2 at ε F = 0.22 (E F = ev) in the purest crystals with the smaller D = In the crystal with D = 0.06 maximum achieves W/mK 2. Even in the crystal with the lowest degree of purity (D = 0.2) P max = W/mK 2, a value higher than in Bi 2 Te 3. The increase of power factor P is determined by the increase of electrical conductivity σ, due to the contribution of carriers with increased relaxation time from the states near the maximum of Lorentzian in (8), and to simultaneous increase of the Seebeck coefficient S, due to sharp dependence of relaxation time on carrier energy E. In the case when D = 0.03, we have σ = Ω 1 cm 1 and S = 157 µv/k. When D = 0.06, σ = 10 4 Ω 1 cm 1 and S = 139 µv/k, and for D = 0.2, σ = Ω 1 cm 1 and S = 111 µv/k. Note that for existing crystals grown from the solution [10] σ varies from 800 up to 1800 Ω 1 cm 1, whereas for more perfect crystals grown from gas phase σ varies from 10 3 up to 10 4 Ω 1 cm 1 [13]. But in these crystals S varies only slightly, from 40 to 45 µv/k. It is explained by the fact that in these crystals σ depends mainly on carrier mobility which grows when the impurity concentration decreases. But S depends mainly on band filling and less on impurity concentration. Thus, in order to increase P it is necessary not only to increase the degree of crystal perfection, but also to optimize the carriers concentration. Concretely, to obtain P max ~ W/mK 2 the carriers concentration n must be diminished from cm 3 ( ε F = 0.4) in stoichiometric crystals up to cm 3, i.e. n must be diminished by approximately 1.5 times. This concentration can be obtained by synthesis of non stoichiometric crystals with the deficit of iodine. It would be interesting to verify this prediction. In the Fig. 2 the case when γ = 2 is presented (α 0 ~ 45 Å 3 ). It is seen that the Fig. 2 Thermoelectric power factor P as a function of dimensionless Fermi energy for γ = 2.

7 7 Low-dimensional organic material for IR detectors 211 maximums are a little increased, and are slightly displaced to higher values of ε F. The highest maximum is ~ W/mK 2 and corresponds to ε F = 0.28 or n = cm 3. In this case the carriers concentration must be diminished only by 1.3 times. As compared with the previous case, σ is increased up to Ω 1 cm 1, but S is diminished up to 141 µv/k. The further increase of γ up to 2.2 does not change considerably the behavior of P. 5. CONCLUSIONS The thermoelectric power factor P has been modeled in quasi-one-dimensional organic crystals TTT 2 I 3 with different degree of purity. Two electron-phonon interaction mechanisms and the scattering of carriers on impurities are taken into account. The kinetic equation was solved and P was expressed through the transport integrals. The latter were calculated numerically. It is shown that P as a function of Fermi energy has a maximum. In the crystals with the lowest degree of purity (D = 0.2) P max = W/mK 2, a little higher value than in Bi 2 Te 3, a widely used thermoelectric material. In the same time, for more pure crystals grown from gas phase (D = 0.03) values of P max ~ W/mK 2 are predicted which are almost 10 times higher than in Bi 2 Te 3, but these values are expected at carriers concentrations which are by times lower than in stoichiometric crystals. It would be interesting to verify these theoretical predictions by experimental work. REFERENCES 1. M.S. Dresselhause, J.P. Heremance. In: Thermoelectric Handbook; Macro to Nano, Ed. by Rowe, CRC press, Boca Raton, FL, (2006), Chap T.C. Harman, P.J. Taylor, M.P. Walsh, and B.E. LaForge. Quantum dot superlattice thermoelectric materials and devices, Science, Vol. 297, 2229 (2002). 3. A. Casian, I. Sur, H. Scherrer, Z. Dashevsky, Thermoelectric properties of n-type PbTe/PbEuTe quantum wells, Phys. Rev. B, 61, (2000). 4. A. Casian, Z. Dashevsky, V. Kantser, H. Scherrer, I. Sur, A. Sandu, Theoretical modelling of the thermoelectric properties in AIVBVI quantum well structures, Phys. Low-Dim. Struct., 5/6, 49 (2000). 5. T.C. Harman, D.L. Spears, D.R. Calawa, S.H. Groves, and M.P. Walsh, In: Proc. of 16 th Int. Conf. on Thermoel., Dresden, Germany, 1997, p I. Sur, A. Casian, and A. Balandin, Electronic thermal conductivity and thermoelectric figure of merit of n-type PbTe/PbEuTe quantum wells, Phys. Rev. B, 69, (2004). 7. D.A. Broido and N. Mingo. Theory of the thermoelectric power factor in nanowire-composite matrix structures, Phys. Rev. B 74, , A. Casian, J. Stockholm, V. Dusciac, R. Dusciac, Iu. Coropceanu, In Proc. of 25 th Inter. Conf. on Thermoel., Wien, Austria, 2006, p A. Casian, V. Dusciac, High values of the thermoelectric power factor expected in quasi-onedimensional organic crystals, J. of Thermoel., 1, 29, 2007.

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