PHONON TRANSPORT IN AMORPHOUS SILICON NANOWIRES. D.V. Crismari

PHONON TRANSPORT IN AMORPHOUS SILICON NANOWIRES D.V. Crismari E. P. Pokatilov Laboratory of Physics and Engineering of Nanomaterials and Synergetics Moldova State University, Chisinau, Republic of Moldova (Received 5 November 01) Abstract Among the perspective research directions in modern physics, an important role is played by the investigation of amorphous nanostructures [1-3]. The effect of the drop in lattice thermal conductivity in these compounds can be used in thermoelectric applications [4, 5]. It is difficult both theoretically and practically to make in fact a distinction between truly amorphous solids and crystalline solids if the crystal sizes are very small [6]. Even amorphous materials have a certain short-range order at the atomic length scale due to the nature of chemical bonding. Furthermore, in very small crystals, a large fraction of the atoms are located at the crystal surface or near it; relaxation of the surface and interfacial effects distort the atomic positions and decrease the structural order. 1. Introduction The amorphous state of a matter is characterized in general by the atomic structure with a short-range order, because it is characteristic of crystal structures that long-range order is absent [7-9]. One of the most available materials is amorphous silicon (a-si); it is a noncrystalline allotropic form of silicon [10]. It can be deposited in thin films at low temperatures onto a variety of substrates, as well as, for example, in the form of nanowires. It provides some unique capabilities for a variety of electronics and phononics. Silicon is a fourfold coordinated atom that is normally tetrahedrally bonded to four neighboring silicon atoms. In crystalline silicon (c-si) this tetrahedral structure continues over a large range, thus forming a well-ordered crystal lattice. In amorphous silicon this long range order is not present. The atoms rather form a continuous random network. Moreover, not all the atoms within amorphous silicon are fourfold coordinated. Due to the disordered nature of the material, some atoms have a dangling bond. These dangling bonds physically represent defects in the continuous random network and may cause anomalous electric and thermal behaviors.. VFF model and results on phonon properties In this study, theoretical researches of phonon transport in amorphous silicon nanowires (NWs) are presented. Phonon energy dispersions are calculated (see Fig. 1) at various degrees of amorphization of these nanometer structures with dimensions of only a few monolayers (MLs). Amorphization of crystalline quasi-one-dimensional compounds was achieved by varying the parameters of interatomic interactions within the Valence Force Field (VFF) model of crystal lattice vibrations [11, 1]. In the calculation of the energy spectrum of amorphous nanowires, the following mechanisms of interatomic interactions was taken into account: the two-particle

stretching ( V r D. V. Crismari j r 0 rij ), the three-particle bending ( V r ijk ), stretching- stretching ( Vrr rr rij rjk ), stretching-bending ( Vr r r0 rij ijk ), and the four- jk, particle bending-bending ( V r ). 0 j, k, l ijk In the work as a result, there was used VFF model in phonon energy spectrum calculations, where the atomic force field V describes the short-range covalent bond: r rr r jkl V V V V V V, (1) jk, jk, the interaction parameters mentioned above r, rr,, r, and are short-range force constants, r 0 is the equilibrium interatomic distance. In the given expression for the potential energy, subscripts j, k, and l denote the atoms, with which the considered atom i forms various types of interatomic interactions. The force constants ( i, j) are introduced by V definition ( ( i, j) ), and they are used to form the equations of motion u ( i) u ( j), () M u q i D i i u q i D q i j u q j j, (, ) (, ) (, ) (,, ) (, ) where q is the wave vector, and u are eigenvalues and eigenvectors of the confined disordered system, M is the atomic mass, D are elements of the dynamical matrix (, ) D ( i, i) ( i, j); D ( q, i, j) ( i, j) e iqn j i. j j (a) (b) 355

Moldavian Journal of the Physical Sciences, Vol. 11, N4, 01 (c) (d) Fig. 1. Phonon energy spectra of the crystalline silicon 8MLx8ML nanowire (a) and the same amorphous nanowires (bd) with different increasing variations of random force distribution in [100] crystallographic direction calculated within VFF model. It can first be noted from Fig. 1 that the maximum energy of optical phonons increases from about 64 mev in the case of the crystalline nanowire to more than 85 mev for the amorphous nanostructure with 90% disordering independently of the force parameters of interatomic interactions. Thus, in the investigated quasi-one-dimensional nanostructures, the number of the so-called heavy low-velocity quanta of vibrational motion of the atoms increases. In addition, on closer inspection, it is clear that, if in the nanowire (a) even phonons with energies around 53 mev have got visible nonzero group velocities, then for the amorphous nanoelement (d) the entire range of the spectrum from about 35 mev and above is almost nondispersive; it will be shown below that this could not but affect the lattice thermal conductivities, their falling in amorphous size-limited nanometer structures. Calculations showed that for the studied nanostructures the following formulaic (3) and numerical (4) representations of limit repetitions ( rep in indicated expressions) can be written for reliable determination of the phonon densities of states (subscript notation DOS, see Figs., 3) and the lattice thermal conductivities (index, see Figs. 4, 5) of amorphous nanowires at the highest studied scatter in force parameters of interatomic interactions ( amorph 90% ). For other degrees of amorphization with weaker disordering of the dynamic matrix, these repetitions are obviously more than sufficient in determining the investigated physical quantities: for all nanostructures lim rep lim rep 1 (3) for all nanostructures amorph 0% DOS amorph 0% NW 8x8MLs lim repdos 10 amorph 90% ; NW 8x8MLs lim rep 15 amorph90% (4) 356

D. V. Crismari Fig.. Comparison of the curves of phonon densities of states at random samples of 1, 60, 10, and 180 attempts for amorphization of the nanowire by +/- 90 % variation of interaction constants. In the calculations of phonon densities of states for a smoother and more accurate stroke of density dependences, it is necessary to increase the partition of the phonon energy s scale and repeated calculations in the case of amorphous nanowires. It is shown (see Fig. 3) that the curves of phonon densities considerably vary in amorphous nanowires compared to crystalline nanowires and get a smoother ride. The acoustic density range slightly varies with increasing degree of amorphization and forms the so-called boson peak at a high disorder of dynamic matrices [1-3], while the optical part of the dependence flattens and broadens to higher energies. Fig. 3. Comparison of the phonon density of states for the silicon amorphous nanowires with different degrees of disorder in force-constant distribution. 357

Moldavian Journal of the Physical Sciences, Vol. 11, N4, 01 3. Results on thermal conductivity of the disordered system The thermal conductivity of a system is atomically determined by revealing how the atoms composing the system interact. There is an approach for calculating the thermal conductivity of a system. The approach is based on the relaxation time approach. Due to the anharmonicity within the crystal and amorphous potentials obtained by the variation of dynamical matrixes, the phonons in the system are known to scatter. There are two main mechanisms for scattering: boundary scattering, where the phonon interacts with the boundary of the system, and phononphonon scattering, where the phonon breaks into two lower energy phonons or collides with another phonon and merge into one higher energy phonon. The heat flux along a nanowire can be calculated according to the expression [13-15], (5) W v( s, q) ( q) N[ q, ( q)] v( s, q) ( q) n[ q, ] s s s s s, q s, q where v is the energy carried by one phonon and N(, q) is the number of phonons in the flux. Taking into account the one-dimensional density of states and switching from summation to integration, we can obtain the phonon thermal conductivity in nanowires in the relaxation time approximation [16, 17]: 1 Es ( q) exp( ) kt max B ph ( ) ( ) ( ) ( ) Es V Nb tot s de kbt dxd, (6) y, s E ( ) 0 s q (exp( ) 1) kt B where V ( ) is the average group velocity of phonons, N ( ) is the number of phonon branches crossing frequency, and tot ( s ) is the total phonon relaxation time. In thermal conductivity calculation, the following phonon scattering mechanisms in nanowires are considered: phonon-phonon scattering and boundary scattering. The total phonon relaxation time is calculated by the formula: 1/ 1/ 1/, (7) tot U B where U is the relaxation time for Umklapp processes; B is the relaxation time for surface scattering generalized for the case of a quantum wire: 1 1 p 1 1 vs ( q) ( ), (8) 1 p d d B x y where d x and d y are the cross-section sizes of the nanowire; vs ( q) are the group velocities of phonon modes with wave vector q ; p is the surface scattering parameter characterizing the surface quality of the nanowire, i.e., its roughness, and calculated by comparing the heat b 358

conduction theory with experimental data. D. V. Crismari Fig. 4. Comparison of the curves of thermal conductivities at random samples of 1, 5, 15, and 5 attempts for amorphization of the nanowire by +/- 90 % variation of interaction constants. Thermal conductivity is described by the Boltzmann equation with the relaxation time approximation in which phonon scattering is a limiting factor. The phonon mean free path has been directly associated with the effective relaxation length for processes without directional correlation. Thus, if V g is the group velocity of a phonon wave packet, then the relaxation length l is defined as follows: l V, where is the characteristic relaxation time. Thermal g conductivity of the silicon nanowire with the square cross-section of 8 x 8 monolayers, which is calculated at room temperature (see Fig. 5), decreases from about 4.5 W/m K for the crystalline silicon nanowire to.8 W/m K for the amorphous nanowire with a random scatter in values of parameters of interatomic interactions. Fig. 5. Lattice thermal conductivities of the crystal nanowire 8 x 8 MLs of silicon and nanowires with the following degrees of amorphization: +/- 5, 50, 90 % (samples of 15 attempts). 359

Moldavian Journal of the Physical Sciences, Vol. 11, N4, 01 4. Conclusions In conclusion it can be summarized that the drop in the lattice thermal conductivity of silicon crystalline nanowires and an even greater effect in amorphous quasi-one-dimensional nanostructures are explained by the quantization of the phonon spectrum and the strong scattering of phonons at the boundaries of nanowires. A stronger modification of the phonon properties can be achieved in silicon amorphous nanowires with disordered bond forces. The obtained results are in good agreement with the experimentally determined vibrational densities of states of disordered systems showing a low-frequency excess, the so-called boson peaks. Finally, it should be noted that, in this paper, the surface phonon scattering in amorphous nanostructures was considered to be approximately identical to crystalline nanowires, and further research will be dedicated to clarifying the effect of this scattering mechanism on the thermal conductivity of these nanostructures. Acknowledgments. This work was carried out with the partial financial support of the research project for young scientists 1.819.05.18F. References [1] R. Zorn, Phys. Rev. B 81, 05408 (010). [] W. Schirmacher, G. Diezemann, and C. Ganter, Phys. Rev. Lett. 81, 136 (1998). [3] F. Finkemeier and W. von Niessen, Phys. Rev. B 63, 3504 (001). [4] H. Wada and T. Kamijoh, Jpn. J. Appl. Phys. 35, L648 (1996). [5] Y.H. Lee, R. Biswas, C.M. Soukoulis, C.Z. Wang, C.T. Chan, and K.M. Ho, Phys. Rev. B 43, 6573 (1991). [6] A. Inoue and K. Hashimoto, Advances in Materials Research: Amorphous and Nanocrystalline Materials, Springer, Berlin, vol. 3, 001. [7] А. Feltz, Amorphous and Vitreous Inorganic Solids, Мir, Moscow, 1986. [8] М. Brodsky, D. Carlson, and G. Connell, Amorphous Semiconductors, Мir, Moscow, 198. [9] X. Liu, J.L. Feldman, D.G. Cahill, R.S. Crandall, N. Bernstein, D.M. Photiadis, M.J. Mehl, and D.A. Papaconstantopoulos, Phys. Rev. Lett. 10, 035901 (009). [10] R. Zallen, The Physics of Amorphous Solids, Wiley Interscience, New York, 1998. [11] D. Camacho and Y.M. Niquet, Physica E 4, 1361 (010). [1] R. Tubino, L. Piseri, and G. Zerbi, J. Chem. Phys. 56, 10 (197). [13] D.L. Nika, E.P. Pokatilov, A.S. Askerov, and A.A. Balandin, Phys. Rev. B 79, 155413 (009). [14] G.P. Srivastava, The Physics of Phonons, IOP, Philadelphia, 1990. [15] C.M. Bhandari and D.M. Rowe, Thermal Conduction in Semiconductors, Wiley, New York, 1998. [16] N. Mingo, Phys. Rev. B 68, 113308 (003). [17] N.D. Zincenco, D.L. Nika, E.P. Pokatilov, and A.A. Balandin, J. Phys.: Conf. Ser. 9, 01086 (007). 360