Thermal interfacial transport in the presence of ballistic heat modes

Thermal interface (Kapitza) resistance expresses how hard it is for heat to flow across material junctions inside multilayer structures. This quantity plays a crucial role in the thermal performance of nanoscale devices but is still poorly understood. Here we show that conventional Fourier-based metrology overestimates metal/semiconductor resistances by up to threefold due to misinterpretation of ballistic heat flow modes. We achieve improved identification and a different physical insight with a truncated Lévy formalism. This approach properly distinguishes interfacial dynamics from nearby quasiballistic heat flow suppression in the semiconductor. Unlike conventionally extracted values, interface resistances obtained with our new approach are independent of laser modulation frequency, as physically appropriate, and much more closely approach theoretical predictions.

Figure 1

Time-domain thermoreflectance characterization of Al/InGaAs sample at room temperature. A theoretical model is fitted to lock-in measurements of the thermal transient of the metal transducer surface at laser modulation frequency $f_{\text{mod}}$. This provides the thermal resitivity $r_{\text{ms}}$ of the metal/semiconductor interface and thermal conductivity $k$ of the semiconductor. Conventional approaches use a regular Fourier diffusion model and perceive “effective” parameters that strongly vary with $f_{\text{mod}}$ due to quasiballistic effects. The truncated Lévy approach describes the quasiballistic transport as a non-Brownian superdiffusive process and extracts “intrinsic” parameters that are independent of frequency as physically appropriate.

Figure 2

Conventional metrology overestimates the thermal interface resistance when quasiballistic effects are present. Calculated interfacial heat flux versus time for a single energy impulse at the transducer surface. The truncated Lévy heat flux (curve B) is lower than the nominal Fourier prediction (curve A) due to incorporation of quasiballistic transport modes. A similar suppression can be obtained phenomenologically by a Fourier model with increased $r_{\text{ms}}$ (curve C), illustrating the overestimation tendency.

Figure 3

Normalized 1D energy density at several indicated times versus normalized position inside semi-infinite InGaAs for a $1\text{J}/{\mathrm{m}}^2$ impulse at the top surface, $x=0$. The Fourier response is Gaussian at all times. During the quasiballistic regime, truncated Lévy profiles show heat concentration near the source, reduced heating at intermediate distances, and “heavy tails” at large distances. As time progresses the transport gradually becomes more and more diffusive, and TL distributions converge to Fourier solutions at times of the order of $t_{\text{BD}}=u_{\text{BD}}^2/2D$ (a few microseconds in InGaAs).

Figure 4

Lévy barrier in conventional interface metrology. (a) Fourier identification of truncated Lévy simulations. The conventionally perceived interface resistivity is consistently larger than the actual value set in the simulation. Even when a thermally perfect interface is assumed (actual $r_{\text{ms}}\equiv 0)$, a significant resistivity $r_{\text{LB}}$ is perceived. We call this the “Lévy barrier” resistivity, as this quantity can be directly attributed to Fourier misinterpretation of quasiballistic transport effects. (b) Calculated Lévy barrier resistivity as a function of the ballistic parameters of a generic semiconductor alloy. The barrier becomes larger as the superdiffusive transport deviates more severely from Brownian dynamics (smaller $\alpha )$ and persists over longer distances (larger $u_{\text{BD}})$.

Figure 5

Theoretical and experimental interface resistivity versus temperature. (a) Conventional identification of Al/Si, which does not exhibit quasiballistic effects over the measurement frequency range, shows good agreement with theoretical predictions. (b) Truncated Lévy identification of Al/InGaAs brings the measured interface resistivity much closer to model predictions than conventional Fourier analysis. The remaining discrepancy between theory and experiment $(3–4 \text{nK-}{\mathrm{m}}^2/\text{K})$ is well within reach of correction terms associated with electron-phonon coupling in the metal and atomic roughness of the interface.