Frequency dependence of the thermal conductivity of semiconductor alloys

Yee Kan Koh and David G. Cahill

Phys. Rev. B 76, 075207 – Published 23 August 2007

Abstract

The distribution of phonons that carry heat in crystals has typically been studied through measurements of the thermal conductivity $Λ$ as a function of temperature or sample size. We find that $Λ$ of semiconductor alloys also depends on the frequency of the oscillating temperature field used in the measurement and hence demonstrate a novel and experimentally convenient probe of the phonon distribution. We report the frequency dependent $Λ$ of ${In}_{0.49} {Ga}_{0.51} P$ , ${In}_{0.53} {Ga}_{0.47} As$ , and ${Si}_{0.4} {Ge}_{0.6}$ as measured by time-domain thermoreflectance over a wide range of modulation frequencies $0.1 < f < 10 MHz$ and temperatures $88 < T < 300 K$ . The reduction in $Λ$ at high frequencies is consistent with a model calculation that assumes that phonons with mean free paths larger than the thermal penetration depth do not contribute to the thermal conductivity measured in the experiments.

Received 14 May 2007

DOI:https://doi.org/10.1103/PhysRevB.76.075207

Authors & Affiliations

Yee Kan Koh and David G. Cahill

Department of Materials Science and Engineering and Frederick Seitz Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801, USA

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Issue

Vol. 76, Iss. 7 — 15 August 2007

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Images

Figure 1
Fits of the thermal model (solid lines) to the TDTR measurements (open circles) used to determine the thermal conductivity of a $2010 nm$ thick InGaP layer at various frequencies. In these fits, the thermal conductance of the $Al ∕ In Ga P$ interface is fixed for all three curves at $G = 51 MW m^{2} K^{- 1}$ . The curves are labeled by the modulation frequency of the pump laser, $V_{in}$ and $V_{out}$ are the in-phase and out-of-phase signals of the rf lock-in amplifier that detects the small changes in the intensity of the reflected probe beam produced by the pump, and $t$ is the time delay between the pump and probe optical pulses.Reuse & Permissions
Figure 2
Frequency dependence of the out-of-phase signal normalized by the jump in the in-phase signal. Measured $V_{out} ∕ Δ V_{in}$ of Si (open squares), $Si O_{2}$ (solid squares), GaAs (open diamonds), InP (solid diamonds), InGaP (open circles), InGaAs (solid circles), and ${Si}_{0.4} {Ge}_{0.6}$ (open triangles) are compared to the respective calculations from the thermal model (solid lines), from top to bottom, of $Si O_{2}$ , semiconductor alloys, GaAs, InP, and Si. For calculation of semiconductor alloys, we assume $Λ = 3 W m^{- 1} K^{- 1}$ , $C = 1.69 J {cm}^{- 3} K^{- 1}$ , and Al film thickness of $93 nm$ . Only measurements of semiconductor alloys deviate from the calculations.Reuse & Permissions
Figure 3
Room temperature thermal conductivity of single crystals of Si, InP, and GaAs; a $1 μ m$ thick layer of amorphous $Si O_{2}$ ; and epitaxial layers of semiconductor alloys as a function of the modulation frequency used in the measurement. Data for $2010 nm$ thick InGaP, $3330 nm$ thick InGaAs, and $6000 nm$ thick ${Si}_{0.4} {Ge}_{0.6}$ are shown as open circles, filled circles, and open triangles, respectively.Reuse & Permissions
Figure 4
Comparison of the frequency and thickness dependences of the room temperature thermal conductivity of III-V semiconductor alloys. Data for $2010 nm$ InGaP (triangles) and $3330 nm$ InGaAs (circles) acquired at different frequencies (open symbols) are plotted as a function of penetration depth, $d = \sqrt{Λ ∕ π C f}$ , where $Λ$ is thermal conductivity of thick layers at low frequency, $C$ is the heat capacity per unit volume, and $f$ is the modulation frequency. Also included are data for epitaxial layers of different thicknesses measured at low $f$ with $d > h$ (filled symbols) plotted as a function of the layer thickness $h$ . The dashed line is the calculated thermal conductivity using the isotropic continuum model described in the text for InGaAs that limits the mean free path of the phonons to the layer thickness.Reuse & Permissions
Figure 5
(a) Temperature dependence of the thermal conductivity of InGaP epitaxial layers of thicknesses $2010 nm$ (circles), $456 nm$ (squares), $178 nm$ (diamonds), and $70 nm$ (triangles). (b) Temperature dependence of the thermal conductivity of InGaAs epitaxial layers of thicknesses $3330 nm$ (circles), $891 nm$ (squares), $591 nm$ (diamonds), and $431 nm$ (triangles). Both plots include measurements at modulation frequencies of $10 MHz$ (solid symbols) and $0.6 MHz$ (open symbols). The data sets are labeled by either the layer thickness $h$ or the penetration depth at room temperature $d$ , whichever is smaller. The upper and lower dashed lines in each figure are thermal conductivities calculated using the isotropic continuum model described in the text that limits the mean free path of the phonons to either $1 μ m$ or $100 nm$ , respectively.Reuse & Permissions
Figure 6
Thermal conductivity distribution $κ (l)$ as a function of mean free path, $l$ . $κ (l)$ is defined by $Λ = \int κ (l) d l$ . The open symbols are $κ (l)$ converted from frequency dependent measurements. The dashed lines are calculations from the isotropic continuum model described in the text. Error bars on the data points reflect the experimental uncertainties in the determination of $κ (l)$ . The major source of experimental uncertainty is the setting of the absolute value of the phase of the reference channel of the rf lock-in amplifier and, at the lowest modulation frequencies, the correction needed to account for the optical pulses that leak through the pulse picker. The precision of the thermal conductivity measurements is approximately 1% at modulation frequencies of $10 MHz$ , 7% at $0.6 MHz$ , and 10% at $350 kHz$ .Reuse & Permissions

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