Heat transport through interfaces with and without misfit dislocation arrays
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In spite of its large lattice mismatch, Bi grows epitaxially in (111) orientation and almost free of defects on Si substrates. On Si(111), the Bi film is under compressive strain of less than 2% and shows a 6–7 registry to the Si(111)-(7 7) substrate. On Si(001), the compressive lattice strain of 2.3% results in the formation of an array of misfit dislocations with a periodicity of 20 nm. We studied the cooling process of ultrathin bismuth films deposited on Si(111) and Si(001) substrates upon excitation with short laser pulses. With ultrafast electron diffraction, we determined the thermal boundary conductance rK from the exponential decay of the transient film temperature. Within the error bars of 7%, the experimentally determined thermal boundary conductances are the same for both substrates and thus independent of the presence of a periodic array of misfit dislocations and the different substrate orientation.
I. INTRODUCTION
The heat transport through a heterosystem consisting of a thin film deposited on a substrate is drastically reduced compared to the heat transport in bulk systems. The interface between the two materials acts as a barrier for thermal diffusion. This resistance of the interface is described by a thermal boundary conductance rK, which was discovered by Kapitza.1–3 This results in a discontinuity of the temperature at the interface: Q_ ¼ rK DT
:
ð1Þ
rK correlates the net heat flow Q_ across the interface to the temperature jump DT at the interface and is basically determined by the energy transmission probability C across the interface.1–3 Two well-accepted models for the calculation of the transmission probability are the acoustic mismatch model (AMM) and the diffuse mismatch model (DMM).1–3 Both models only consider phonons for the heat transport. If the dominant phonon wave length is larger than the interface roughness, the phonons are treated as elastic waves, which are reflected and refracted at the interface. The transmission probability is given by the ratio of energy of the transmitted wave compared to the incoming wave. Using the acoustic equivalents of the Fresnel equations in optics, the transmission probabilities are calculated.2–4 The DMM must be used if the interface roughness is larger than
the dominant phonon wave length. In this case, diffuse elastic scattering at the interface is assumed, and the transmission probability is given by the density of states of the two adjacent media—the film and the substrate.2,3 Verification of the above mentioned theories requires a suitable experimental model system. The heat transport across the interface should only be governed by phonons. Such model systems are thin bismuth films on silicon substrates. In these systems, the energy transport by electrons and holes is prevented by the formation of a Schottky barrier at the Bi/Si interface.5,6 Additionally, optical phonons in Bi with a frequency of 2.12 THz7 cannot couple to the optical phonons of Si with a much higher frequency of 10–15 THz.8 Due to the low group velocity of optical phonons, the
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