Frequency Dependence of the Thermal Conductivity of Semiconductor Alloys

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

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 In0.49Ga0.51P, In0.53Ga0.47As, and Si0.4Ge0.6 as measured by time-domain thermoreflectance over a wide range of modulation frequencies, 0.11 MHz. This surprising result is consistent with a model based on the assumption that phonons with mean-free-paths greater than the thermal penetration depth, d = Λ / π Cf , where C is the heat capacity per unit volume, do not contribute to Λ measured in the experiments. Thus, by varying f and therefore d, we conveniently probe the distribution of phonon mean-free-paths. Our results also have practical importance in the design and thermal management of microelectronics: InGaP, InGaAs and SiGe, are common in high frequency devices10,11 and our finding that Λ is dependent on both the layer thickness and the time-scale of the heat transport creates new challenges for accurate modeling of temperature distributions in these devices.12

II. EXPERIMENTAL DETAILS InGaP and InGaAs samples, supplied by Epiworks Inc., were epitaxially grown on GaAs and InP substrates, respectively, by metal organic chemical vapor deposition (MOCVD). The 70 nm InGaP epitaxial layer was obtained by sequential selective etching 3

of GaAs and InGaP from a multilayer HBT InGaP/GaAs wafer, also grown by MOCVD. The Si0.4Ge0.6 sample was provided by Prof. Fitzgerald of MIT.13 The layer thickness of the samples is measured by picosecond acoustics. The longitudinal speed of sound is 5.22 nm ps-1 in InGaP and 4.25 nm ps-1 in InGaAs, derived from the average of the speed of sound in the pure crystals.14 InGaP samples (2007 nm, 178 nm and 70 nm) are undoped, except the 456 nm layer, which is lightly doped (n-type, 2.5 x1017 cm-3). The InGaAs samples have a range of dopant concentrations: n-type, 1015 cm-3 (3330 nm); p-type, 2.3x1019 cm-3 (891 nm); p-type, 2.7x1019 cm-3 (591 nm); and p-type, 9x1018 cm-3 (431 nm). The 6000 nm SiGe sample is undoped. To prepare the samples for measurements, we deposit 70-100 nm thick Al films by magnetron sputter deposition.15 We measure the thermal conductivities by time-domain thermoreflectance (TDTR).16,17 A schematic diagram of our equipment is given in Ref. 18 and our method for data analysis is described in Ref. 19. In our TDTR measurements, a laser beam from a mode-locked Ti:sapphire is split into a pump beam and a probe beam with the relative optical path being adjusted via a mechanical delay stage. The pump beam is modulated at a frequency f in the range 0.1