Abstract
We present a predictive Boltzmann model for the cross-plane thermal conductivity in superlattices. The developed model considers particle-like phonons exhibiting wave characteristics at the interfaces and makes the assumption that the phonon heat transport in a superlattice has a mixed character. Exact Boltzmann equation comprising spatial dependence of phonon distribution function is solved to yield a general expression for the lattice thermal conductivity. The intrinsic phonon scattering rates are calculated from Fermi’s golden rule, and the model vibrational parameters are derived as functions of temperature and crystallographic directions by using elasticity theory-based lattice dynamics approach. The developed theory is then adapted to calculate the cross-plane thermal conductivity of superlattices. It is assumed that the phonons of wavelengths comparable or smaller than the superlattice period or the root mean square irregularity at the superlattice interfaces may be subject to a resistive scattering mechanism at the interfaces, whereas the phonons of wavelengths much greater than the superlattice period undergo ballistic transmission through the interfaces and obey dispersion relations determined by the Brillouin zone folding effects of the superlattice. The accuracy of the concept of mixed phonon transport regime in superlattices is demonstrated clearly with reference to experimental measurements regarding the effects of period thickness and temperature on the cross-plane thermal conductivity of Si/Si0.7Ge0.3 and Si0.84Ge0.16/Si0.76Ge0.3 superlattices.
Highlights
It is assumed that the phonons of wavelengths comparable or smaller than the superlattice period or the root mean square irregularity at the superlattice interfaces may be subject to a resistive scattering mechanism at the interfaces, whereas the phonons of wavelengths much greater than the superlattice period undergo ballistic transmission through the interfaces and obey dispersion relations determined by the Brillouin zone folding effects of the superlattice
We have first calculated the dependence of temperature and period thickness on the cross-plane thermal conductivities of Si/Si0.7Ge0.3 and Si0.84Ge0.16/Si0.76Ge0.3 superlattices by assuming that the phonon heat transport is either purely incoherent or purely coherent
The derived formalism accounts for the interplay between the phonon intrinsic scattering and the phonon scattering by the sample boundary in the full temperature range
Summary
Understanding the laws that govern the phonon heat transport at atomic interfaces in crystalline superlattices has long been viewed as a key step toward efficient thermal-management strategy for high-performance superlattice-based thermoelectric, microelectronic, and optoelectronic devices.[1,2,3] several experimental studies on the phonon thermal conductivity in superlattice structures have been carried out,[4,5,6,7,8,9,10,11,12,13] and many numerical techniques describing the phonon heat transport in multi-layer systems have been developed.[14,15,16,17,18,19,20,21,22,23,24]. We tackle this issue and present an original approach for the determination of the phonon cross-plane thermal conductivity and heat transport regime in superlattices It consists of the interpolation between two Boltzmann models: an incoherent Boltzmann transport model assuming that the cross-plane thermal conductivity of the superlattice is a weighted average of the thermal conductivities of the two bulk materials that form the superlattice period with the additional contribution of the interface thermal resistance, and a coherent Boltzmann transport model based on the assumption that the superlattice is a bulk material, free of interfaces, characterized by phonon dispersion relations determined by the Brillouin zone folding effects of the superlattice. We show that the phonons in superlattices can carry thermal energy in a coherent and incoherent fashion, and the probability of each is determined by the phonon dispersion relations in unfolded Brillouin zone, the period thickness, and the interface asperities
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