PRZEGLĄD ELEKTROTECHNICZNY, ISSN 0033-2097, R. 93 NR 8/2017 61 Bohdan ANDRIYEVSKY 1 , Włodzimierz JANKE 1 , Aleksy PATRYN 1 , Mirosław MALIŃSKI 1 , Vasyl' STADNYK 2 , Mykola ROMANYUK 2 Politechnika Koszalińska, Wydział Elektroniki i Informatyki (1), The Ivan Franko NU of Lviv, Faculty of Physics (2) doi:10.15199/48.2017.08.16 Ab initio molecular dynamics calculations of heat conductivity for silicon related materials Abstract. Results of the ab initio molecular dynamics for pure silicon and phosphorus doped silicon crystals have been presented. The relation between the phonon lifetime  and the root mean square deviation  of atoms based on the condition of the interferometric minimum has been proposed. The relation approximates adequately the temperature dependence of the heat conductivity of pure silicon. However, that relation has not reproduced properly the reference experimental magnitude of the phonon conductivity coefficient of silicon for the phosphorus content n P = 510 20 cm -3 . This result indicates that the additional kind of the phonon scattering on the local phosphorus stimulated defects should be taken into consideration. Streszczenie. Przedstawiono wyniki obliczeń z pierwszych zasad dynamiki molekularnej niedomieszkowanych i domieszkowanych fosforem kryształów krzemu i zaproponowano wzór relacji między czasem życia fononów  i odchyleniem standardowym atomów , bazujący na interferencyjnym warunku minimum fal fononowych. Zaproponowany wzór adekwatnie opisuje temperaturową zależność współczynnika przewodnictwa cieplnego niedomieszkowanego krzemu. Jednak zaproponowany wzór nie odtwarza zadowalająco referencyjnej doświadczalnej wartości przewodności cieplnej krzemu dla koncentracji fosforu n P = 510 20 cm -3 . Ten wynik wskazuje na to, że dla adekwatnego odtwarzania wartości doświadczalnych należy uwzględnić dodatkowy kanał rozpraszania fononów, związany z lokalnymi około fosforowymi defektami. (Obliczenia z pierwszych zasad przewodności cieplnej materiałów na bazie krzemu metodą dynamiki molekularnej). Słowa kluczowe: krzem, dynamika molekularna, współczynnik przewodności cieplnej, czas życia fononów Keywords: silicon, molecular dynamics, thermal conductivity coefficient, phonon lifetime Introduction Thermal effects occurring in the active electronic elements and devices lead to their temperature increase and may change their functional characteristics. To keep these characteristics within the determined frameworks, one has to construct the corresponding devices taking into account the heat conductivity characteristics of the corresponding active and neighboring materials. Because of the large variety of these materials in modern electronics the problem of their heat characteristics determination is extremely important [1]. There are several theoretical techniques reported in the literature [2], which are used for determination of the phonon thermal conductivity. One of the often applied techniques is the molecular dynamics (MD), which can be separated into two main groups: (1) equilibrium and (2) non- equilibrium methods. In an equilibrium MD simulation, the system under investigation has a constant average temperature and an average heat flux of zero [2]. However, at each instant of time a finite heat flux exists due to instantaneous fluctuations of temperature. The popular Green-Kubo method [3, 4], based on the general fluctuation-dissipation theorem [5], relates the lattice thermal conductivity of the system to the time required for such fluctuations to dissipate. The corresponding algorithms however are not implemented in the more popular ab initio packages, for example in VASP code [6]. In the present study, we propose a new approach for the theoretical first-principles calculations of the phonon lifetime  and thermal diffusivity D of a material on the basis of the standard ab initio molecular dynamics (AMD) study. This approach has been applied to the study of heat conductivity in pure silicon and phosphorus doped silicon crystals. Some subsidiary calculations of this type were performed with diamond, sapphire (Al 2 O 3 ) and silver. On the basis of the calculated thermal diffusivity D one can obtain the corresponding data for the coefficient of thermal conductivity , if the information on the heat capacity C p of the material studied is available. This approach uses the relation for the thermal conductivity  derived from the kinetic theory of phonons propagation [7], (1)  = (1/3)C p V 2  , where  is a density, V is the velocity of acoustic waves, and τ is the phonon lifetime. Method and calculation details The equilibrium-type ab initio molecular dynamics (AMD) calculations of the crystal have been performed in the framework of the density functional theory (DFT) using the VASP package [6]. The projector augmented-wave (PAW) method with a cutoff energy of 400 eV for the plane waves was employed [6, 8] together with the corresponding pseudopotentials. For the exchange and correlation terms, the gradient corrected Perdew-Burke-Ernzerhof (PBE) functional was used. The AMD calculations of silicon related crystals were performed at the macro-canonical NVT ensemble for different temperatures at the optimized crystal structure of the super cell 333. Most results of AMD calculations have been obtained for the simulation time up to 15 ps with the time steps of 1.5 fs. For the post MD analysis the nMoldyn 3.0 program was used [9]. From the relation (1), the value of heat diffusivity D is determined as, (2) D = (1/3)V 2  = (1/3)LV , where L is the phonon mean free path. The values V,  and L are interrelated according to the known relation L = V. Analysis of the reference data of the temperature dependences of the values from the relations (1) and (2) in the range near T = 293 K has revealed that the dependence (T) determines mainly the dependence of the thermal conductivity (T). Relative temperature changes of the values , C, and V are about one order of magnitude smaller. It is known that the phonon lifetime  and the corresponding phonon mean free path L are determined by the phonon scattering in solids, caused by the thermal atomic vibrations and crystal structure defects. Here, we propose the method of the phonon lifetime  estimation based on the results of the molecular dynamics calculations. We demonstrate that the root-mean-square deviation of atoms obtained from the molecular dynamics calculations determines the phonon lifetime  and its