PHYSICAL REVIEW B VOLUME 50, NUMBER 17 1 NOVEMBER 1994-I Hopping theory of heat transport in disordered systems H. Bottger and Th. Damker Otto von -Gu-eriche Un-iversitat Magdeburg, Institut fur Theoretische Physih, D 890-16 Magdeburg, Germany (Received 7 June 1994; revised manuscript received 11 July 1994) Heat transport is studied in a simple model system of Anderson localized optical (carrier) phonons vrhich perform thermally activated hopping due to anharmonic interaction vrith delocalized acoustic phonons. The corresponding kinetic equations (rate equations) are derived by using the density- matrix formalism. The calculated hopping contribution to the heat conductivity exhibits a linear increase with temperature at lower temperatures and (depending on the choice of parameters) eventually reaches a "saturated" value at higher temperatures. Thus, unlike other authors, me do not need a special mechanism, such as lifetime broadening of the optical phonon states, to explain the transition to the saturation region. Furthermore, we show that particle (carrier) number nonconservation leads to a quenching of the hopping mechanism. I. INTRODUCTION In recent years, transport of vibrational energy in dis- ordered insulating solids has attracted great interest in both theoretical& — 2o and experimental2x — 28 studies. This interest has been much stimulated by the observation that the temperature dependence of the heat conductiv- ity, tc(T), of amorphous solids distinctively differs from that of crystals. In insulating crystals, at low T, e is a cubic function of T, and at high T it decreases with 1/T. This behavior can be well described by means of the Peierls-Boltzmann theory of a weakly interacting phonon gas (e. g. , Refs. 29, 30 and references therein). In amorphous solids, one can distinguish three characteris- tic temperature regimes: (i) low T (T & 1 K), where ~ is approximately a quadratic function of T, (ii) medium T (T 10 — 30K), where e is constant (" plateau" region), and (iii) high T (T ) 30K), where ~ rises smoothly and (mostly) reaches a limiting or "saturated" value. The behavior of ~ in the regime (i) can be well ex- plained by phonon scattering off two-level systems, ~ 2 whereas that in the regimes (ii) and (iii) is less well understood. Regime (ii) has evoked a wide variety of explanations, such as Rayleigh scattering, lo- calization, and inelastic scattering of phonons. Specific attention is at present paid to the regime (iii), where interesting physical phenomena, such as difFusive motion of nonpropagating modes& ' 2'~9'2o'26 or hopping of vibrational excitations, ' ' ' are emerging as possible transport mechanisms. The diffusive mechanism is based on the idea that disorder is sufficient to cause a phonon mean &ee path of the order of the interatomic spacing, which implies vibrational modes do not propa- gate, but disorder is insufficient to cause all the states to localize. This mechanism does not require anharmonic- ity. By contrast, the hopping mechanism assumes local- ized vibrational states and needs anharmonicity to allow energy transport between such states. Both the mechanisms open new conduction channels, in addition to the conventional channel of heat transport, by propagating long-wavelength acoustic modes. Both mechanisms cannot be described within the &amework of the Peierls-Boltzmann theory. Which mechanism is actually operative in a given real material is in general an open problem. The current experimental and theoretical situation is not yet quite satisfactory. More experimental data are needed and the transport mechanisms require further theoretical inspection. It is the purpose of this paper to examine the vibra- tional hopping transport on the basis of a rate equation approach. Before describing this approach in more de- tail, let us throw some light on previous studies related to the subject of this paper. In an early simulation evidence was seen that in a dis- ordered vibrational system anharmonicity can enhance the value of e at higher T. The vibrational hopping mechanism was first theoreti- cally studied ' in a &actal system by considering local- ized vibrational states (fractons) interacting with acous- tic modes. The corresponding transition probabilities be- tween localized states were calculated by means of the golden rule. The heat conductivity e was found to in- crease at high T linearly with T. It was argued that ~ saturates if T is high enough so that the lifetime of the localized modes becomes comparable with the period of their vibrations. The results were applied to regime (iii) mentioned above. A linear increase of x with T over a large temperature range, without saturation at high T, was recently ob- served in certain complex silica glasses and interpreted in terms of vibrational hopping. Results of simulations of r(T) in quasicrystalline systems seem also to support a hopping model. Thermal conductivity experiments on boron carbides suggest hopping of vibrational energy, too. ' In experiments with high-energy nonequilib- rium phonons, produced by intense pulsed optical exci- tations, evidence was seen for localized vibrational states in amorphous silicon, the necessary prerequisite for the hopping mechanism. It may be argued that vibrational hopping may also be 0163-1829/94/50(17)/12509(11)/$06. 00 50 12 509 1994 The American Physical Society