Abstract
Thermal conductivity in nonmetallic crystalline materials results from cumulative contributions of phonons that have a broad range of mean free paths. Here we use high frequency surface temperature modulation that generates nondiffusive phonon transport to probe the phonon mean free path spectra of GaAs, GaN, AlN and 4HSiC at temperatures near 80 K, 150 K, 300 K and 400 K. We find that phonons with MFPs greater than 230 ± 120 nm, 1000 ± 200 nm, 2500 ± 800 nm and 4200 ± 850 nm contribute 50% of the bulk thermal conductivity of GaAs, GaN, AlN and 4HSiC near room temperature. By nondimensionalizing the data based on Umklapp scattering rates of phonons, we identified a universal phonon mean free path spectrum in small unit cell crystalline semiconductors at high temperature.
Introduction
The physics of heat transport in condensed matter is critical to thermal management in a diverse range of technologies as well as thermoelectric energy conversion. The dominant carriers of heat in nonmetallic crystalline materials are phonons, defined as quantized lattice vibrations. Specific heat, phonon group velocity and phonon mean free path (MFP)–the average distance a phonon travels between scattering events–determine a material's thermal conductivity. Recent studies have shown that it is possible to experimentally measure the MFP dependent contributions of phonons to thermal conductivity–the phonon MFP spectrum^{1,2,3,4,5}. In this work we present experimental evidence of a universal phonon MFP spectrum in several crystalline semiconductors. Broadband frequency domain thermoreflectance (BBFDTR)^{1}, an optical pumpprobe technique, was used to measure the integrated phonon MFP spectrum, referred to as the thermal conductivity accumulation function (k_{accum}) of single crystal (100) gallium arsenide (GaAs), (0001) gallium nitride (GaN), (0001) aluminum nitride (AlN) and (0001) 4Hsilicon carbide (SiC) at temperatures (T) near 80 K, 150 K, 300 K and 400 K. By nondimensionalizing the data based on Umklapp scattering rates of phonons, which is the dominant resistive scattering mechanism at high temperature (herein meaning high relative to the temperature of peak thermal conductivity), we discovered a universal thermal conductivity accumulation function (k_{universal}).
GaAs, GaN, AlN and SiC are critical to the high power electronics and optoelectronics industries^{6}. Primary applications of these materials include high electron mobility transistors^{7}, multijunction solar cells^{8} and light emitting diodes^{9,10}, where poor heat dissipation leads to electrical and optical inefficiencies and shorter lifetimes^{11}. The bulk thermal conductivities of GaAs, GaN, AlN and SiC (4H and 6H) at T = 300 K are 50^{12,13}, 230^{14,15}, 285^{16} and 490 Wm^{−1}K^{−1}^{17,18}, respectively. First principles calculations have been used to evaluate k_{accum} of GaAs, though the predictions have not been experimentally confirmed^{19}. Thin film architectures and nanostructures possess reduced thermal conductivities compared to bulk materials due to boundary scattering, which limits phonon MFPs^{20,21,22}. Heat sources with dimensions smaller than phonon MFPs also perceive a locally suppressed thermal conductivity^{3,23}. Therefore, experimental measurements of the intrinsic k_{accum} are required to understand these effects, which are prevalent in GaAs, GaN, AlN and SiC technologies.
Kinetic theory can be used to derive an approximate expression for thermal conductivity, k, as , where C is the volumetric heat capacity, v_{RMS} is the rootmeansquared velocity and is the average MFP that a particle travels between scattering events. Though this gray approximation for the MFP of particles provides accurate predictions of thermal conductivity in gases, it does not accurately predict thermal conductivity in solids, where phonons exhibit a broad distribution of MFPs^{24}. To accurately describe thermal conductivity in a nonmetallic crystal, a summation weighted by the modedependent phonon properties is required. While the heat capacity and phonon group velocity can be predicted and measured, experimental determination of the phonon MFP spectrum has remained a challenge. To resolve thermal conductivity as a function of phonon MFP, the thermal conductivity accumulation function was defined as^{24},
where l is the phonon MFP, v is the phonon group velocity, C_{MFP} is the volumetric heat capacity per unit phonon MFP and s indexes the polarization of phonons. Since the integral is taken from 0 to l*, k_{accum} quantifies the contribution to bulk thermal conductivity of phonons with a MFP less than or equal to l*.
Experimental measurements of k_{accum} have been reported using BBFDTR, time domain thermoreflectance (TDTR) and transient grating techniques^{1,2,3,4,25}. Based on observations of suppressed thermal conductivity in semiconductor alloys, Koh and Cahill^{2} hypothesized that the thermal penetration depth, , limited the diffusive phonons interrogated by TDTR to those having a MFP less than L_{P}, where f was the modulation frequency of the pump laser. Further studies using TDTR have equated l* to the laser spot size on Si^{3} and the dimensions of nanopatterned heaters on sapphire^{26}. More recently, transient grating experiments have varied the grating period to study related ballistic phonon transport effects in Si membranes^{4}.
The BBFDTR apparatus used to measure k_{accum} is shown in Fig. 1(a). The wavelengths of the continuous wave pump and probe lasers in BBFDTR are 488 nm and 532 nm, respectively. The pump laser is intensity modulated by an electrooptic modulator (EOM) at a frequency f_{1}, which upon absorption in the Au transducer film induces a temperature response in the sample at frequency f_{1}. The continuous wave probe laser beam then reflects off the sample with a modulated intensity at frequency f_{1}, as a result of Au's thermoreflectance. The reflected pump and probe have identical frequencies with a phase difference representing the phaselag of temperature to heat flux at the sample surface. The pump and probe signals are then heterodyned by a second EOM at frequency f_{2} to reduce coherent and ambient noise as described in the Supplementary Information. To obtain a plot of accumulated thermal conductivity as a function of phonon MFP, phaselag data from BBFDTR was fit to an analytical solution of the heat diffusion equation in a layered medium^{27}.
BBFDTR and TDTR measurements have demonstrated that a constant value of thermal conductivity over the range of measured heating frequencies (0.2 ≤ f_{1} ≤ 200 MHz) accurately identifies the expected bulk values of SiO_{2} and platinum samples with little deviation between the measured data and analytical fits^{1,2}. This result suggests that a constant value of thermal conductivity is appropriate when the MFPs of energy carriers are much shorter than L_{P}. On the other hand, a constant thermal conductivity over all heating frequencies was found to under predict thermal conductivity in single crystal Si^{1} and likewise in the GaAs, GaN, AlN and 4HSiC crystals studied here. As an example, the measured data and constant thermal conductivity fit, shown in Fig. 1(b), yield a bulk thermal conductivity value of 194 Wm^{−1}K^{−1} for 4HSiC at T = 304 K (40% of k_{bulk}). This suppression is expected when L_{P} is shorter than the MFPs of some phonons^{1,2}. Phaselag data was instead divided into overlapping windows of 13 data points and the thermal conductivity of each window was individually fit. The thermal conductivity of 4HSiC extracted from each window, k_{j}, is plotted in Fig. 1(c) as a function of the windows' median frequency and decreases from 294 Wm^{−1}K^{−1} to 74 Wm^{−1}K^{−1} over the frequency range (200 kHz to 200 MHz).
To generate an accumulation function, the thermal conductivity was plotted against , where f_{j} was the median frequency of the j^{th} window. At low f, phonon transport was primarily diffusive, while at high f, phonons with a MFP > L_{P} traveled ballistically as illustrated in Fig. 1(d). Consistent with prior studies, we assumed that ballistic phonons did not contribute to the measured value of thermal conductivity^{1,2,3,4,26}. This convention equates l* to L_{P} and provides a reasonable comparison between experimental k_{accum} and theoretical predictions^{1,2}, though it has not been rigorously proven and neglects spot size effects that have influenced TDTR measurements in Si^{3}.
Results
The k_{accum} of GaAs, GaN, AlN and 4HSiC at temperatures near 80 K, 150 K, 300 K and 400 K are displayed in Fig. 2 (offset temperatures due to laser heating are added to the nominal temperatures measured at the cryostat cold finger). Insets report the measured thermal interface conductance at each temperature. For all four materials the thermal interface conductance, G, increased with temperature in agreement with prior studies of metaldielectric interfaces^{28}. Interfaces of AlN and GaN with chromium have been measured at T = 300 K and our results are within 20% of the values^{29}. The k_{accum} data is normalized by the bulk thermal conductivity at each temperature. Bulk values of thermal conductivity are an additional source of uncertainty that are not reflected in the error bars on Fig. 2 (see Supplementary Information for justification of the chosen bulk values). Density functional theory driven simulations of k_{accum} in GaAs at T = 300 K agree with the measured data in Fig. 2(a)^{19}. BBFDTR measurements indicate that phonons with MFPs less than 230 ± 120 nm contribute 50% to the total bulk thermal conductivity of GaAs at T = 330 K. As the bulk thermal conductivity of the material increases, phonons with longer MFPs contribute more significantly to the bulk thermal conductivity. Phonons with MFPs less than 1000 ± 200 nm, 2500 ± 800 nm and 4200 ± 850 nm contribute 50% to the bulk thermal conductivity of GaN, AlN and 4HSiC at T = 309 K, 308 K and 304 K, respectively.
An isotropic model of thermal conductivity, where k_{inplane} = k_{crossplane}, was assumed in GaAs, GaN, AlN and 4HSiC across all heating frequencies. This assumption is valid for GaAs as its cubic (zincblende) crystal structure contains highorder symmetry and its thermal conductivity tensor is isotropic^{30}. Slack et al.^{16} reason that inplane and crossplane thermal conductivities of AlN varied by less than 5% at room temperature and above. GaN was assumed to exhibit similar quasiisotropic behavior to that of AlN due to the fact that GaN and AlN have identical lattice structures and similar acoustic wave velocity deviation between their inplane and crossplane velocities^{31,32}. Finally, the thermal conductivity of 6HSiC and 4HSiC along the [0001] axis has been found to be about 20–30% less than the thermal conductivity along the [1000] axis at room temperature^{17,33}. Since L_{P} ~ r (r = 2.65 ± 0.13 μm, 1/e^{2} radius laser spot) for much of the frequency range in 4HSiC, heat spreading is not purely onedimensional. Yet the data must be normalized to a single value. Therefore, we have chosen the [1000] axis bulk values because they are reported over a wide range of temperatures. To examine the effects of anisotropy in 4HSiC, a heat diffusion model considering inplane and crossplane thermal conductivities separately was used to fit phaselag data as a function of heating frequency^{34}, where k_{crossplane} = 0.8k_{inplane} was held constant. The observed thermal conductivity difference between the anisotropic and isotropic data interpretations varied by less than 11% of k_{bulk} at T = 407 K and 304 K and by less than 4% of k_{bulk} at T = 151 K and 81 K (see Supplementary Information). Therefore, data presented in Fig. 2 assumed isotropic thermal conductivity in GaAs, GaN, AlN and 4HSiC.
Figure 2(e) compares the contribution of phonons with a MFP < 1 μm to the bulk thermal conductivity in GaAs, Si^{1}, GaN, AlN and 4HSiC from 81 K to 445 K. As the bulk thermal conductivity of the material increases, phonons with a MFP < 1 μm contribute less to the total thermal conductivity. As the temperature decreases, phonon occupation and phononphonon scattering are reduced and long MFP phonons (>1 μm) become the dominant heat carriers. With the approaches from Ref. 35 the data in Fig. 2 allows for mapping of thermal conductivity suppression in nanostructured materials and devices.
Discussion
Using measured k_{accum} data a universal thermal conductivity accumulation function was identified. Umklapp scattering is the dominant resistive scattering process above the temperature of peak thermal conductivity, which typically occurs at less than 10% of the Debye temperature in bulk crystalline materials with low impurity concentration^{36}. To derive an expression for k_{accum} at high temperatures, the following form for the relaxation time, τ, of phonon scattering due to Umklapp processes was assumed^{35},
where P and C_{U} are material dependent constants that describe Umklapp scattering rates and ω is the phonon frequency. We employed the truncated Debye dispersion^{21} ω = v_{s}q for acoustic phonons, where v_{s} is the sound velocity and q is the phonon wave vector. The Debye dispersion is truncated at the Brillouin zone edge frequency, ω_{BZE} of the real dispersion relationship as not to overestimate contributions from high frequency, low group velocity acoustic phonons. An isotropic expression for k_{accum} follows from Eq. (1) for a single polarization^{24},
where is the reduced Planck constant, n is the BoseEinstein distribution and is the minimum phonon MFP determined by ω_{BZE}. Classical occupation leads to , where k_{B} is the Boltzmann constant. This assumption is accurate for phonons having , where , as derived in section 5 of the Supplementary Information. Since thermal conductivity in intrinsic crystalline semiconductors results largely from phonons with , this assumption can be valid even when T < θ_{BZE}, as it is for our materials. Therefore, the normalized k_{accum}, including acoustic longitudinal and transverse modes is (see Supplementary Information for a full derivation and tabulated values of P, C_{U}, v_{s} and ω_{BZE}),
Based upon inspection of Eq. (4) a nondimensional phonon MFP is identified as,
The assumption of classical occupation is neccesary for deriving Eqn. (4). The form of L_{p,nondimensional} (Eqn. (5)) identified in this derivation is valid, independant of occupation, when Umklapp scattering dominates thermal resistance.
The assumption of Umklapp dominated scattering is adequately met by Si, GaN, AlN and 4HSiC at T ≈ 300 K and 400 K and GaAs at T ≈ 150 K, 300 K and 400 K given that the temperatures of their peak thermal conductivity occur at ~ 25 K^{37}, ~ 55 K^{14}, ~ 60 K^{16}, ~ 50 K^{18}, ~ 20 K^{12}, respectively. The k_{accum} of GaAs, Si^{1}, GaN, AlN and 4HSiC, as normalized by k_{bulk}, are shown in Fig. 3(a). When plotted as a function of L_{P,nondimensional}, as in Fig. 3(b), k_{accum} data collapse to a single universal thermal conductivity accumulation function, k_{universal}. This collapse equivalently implies a universal phonon MFP spectrum in these materials, as the accumulation function is the integral of the MFP spectrum. To test the robustness of k_{universal} the Bornvon Karman Slack model^{24,35} was also used to find values of P and C_{U}, where an average sound velocity, and Brillouin zone edge frequency was used for the longitudinal and transverse branches. In this case, L_{P,nondimensional} is defined as,
Fig. 3(c) shows that a k_{universal} based on the Bornvon Karman Slack model also exists when using the form of L_{P,nondimensional} in Eq. (6). The scale of the L_{P,nondimensional}axis in Fig. 3(b) and Fig. 3(c) shows that approximately 90% of thermal conductivity in crystalline semiconductors results from phonons with MFPs 1–200 times the MFP of Brillouin zone edge acoustic phonons (l_{min}). For comparison the projections of k_{universal} based on truncated Debye and Bornvon Karman Slack models are shown in Figure 3b and 3c.
Some debate exists on the importance of isotope scattering in galliumbased semiconductors, though authors agree that it is diminished at high temperatures^{31,38,39,40}. The collapse of GaAs and GaN data to k_{universal} indicates that isotope scattering did not strongly affect the τ^{−1} ∝ ω^{2} scattering of long MFP phonons probed by BBFDTR.
Knowledge of k_{accum} enables mapping of thermal conductivity suppression in nanostructures and devices based on their characteristic size and is therefore critical to the thermal design of electronic, photonic and thermoelectric technologies. The existence of a material independent k_{universal} in GaAs, Si, GaN, AlN and SiC suggests that the phonon mean free path spectrum is a universal feature of intrinsic crystalline semiconductors. This is useful for the projection of k_{accum} in intrinsic crystalline semiconductors where thermal resistance is dominated by Umklapp processes based on parameters that can be obtained from historical thermal conductivity vs. temperature data. Because our data is based on small unit cell, high thermal conductivity materials, additional experiments are needed to determine whether more complex materials such as alloys and large unit cell crystals adhere to a different form of k_{univeral}.
Methods
Sample preparation
Undoped, semiconductor grade (100) GaAs, (0001) GaN, (0001) AlN and (0001) 4HSiC bulk wafer samples were purchased and provided by University Wafer Inc., Kyma Technologies Inc., HexaTech Inc. and Cree Inc., respectively. Preparation of bulk wafer samples for BBFDTR measurements included two steps. 4HSiC was dipped into an HF:H_{2}0 (1:9) solution for 10 minutes at room temperature to remove the native oxide layer and blown dry in ultra high purity nitrogen. Next, a chromium adhesion layer and a gold transducer layer of thickness reported in Supplementary Information Table S1 were deposited on GaAs, GaN, AlN and 4HSiC samples via a Perkin Elmer 6 J sputtering system. Gold was chosen as the transducer material, as it has a high absorptivity at 488 nm (pump laser) and a high coefficient of thermoreflectance at 532 nm (probe laser)^{41}. Xray reflectivity measurements were used to determine gold and chromium layer thicknesses, as detailed in Supplementary Information section S2.
Broadband frequency domain thermoreflectance
To measure a large range of k_{accum}, BBFDTR heterodynes the pump and probe signals present at f_{1} with a second modulation at frequency f_{2} producing signals at f_{2} − f_{1} and f_{1} + f_{2}. This allows for heating frequencies up to 200 MHz^{25}. This was highly advantageous compared to traditional frequency domain thermoreflectance (FDTR), where the signal is compromised by coherent and ambient noise at frequencies greater than 20 MHz. The component of the signal with frequency f_{1} + f_{2} is filtered out and the lower frequency component, f_{2} − f_{1}, is recorded using an SR830 lockin amplifier. An optical band pass filter is used to attenuate the pump or probe beam to permit for measurement of each beam individually. Frequencies f_{1} and f_{2} are concurrently swept between 200 kHz and 200 MHz while maintaining f_{2} − f_{1} = 86 kHz. A constant 1/e^{2} spot size radius of 2.65 ± 0.13 μm, measured using a knifeedge technique, was used for all data presented.
To fit thermal conductivity and thermal interface conductance two assumptions were made. (1) The thermal interface conductance between the sample and the chromiumgold layer was frequency independent. This was reasonable as the heating frequency was five orders of magnitude smaller than the relaxation rate of electrons in the gold layer at room temperature^{42}. Hence, at the interface the electrons and phonons will be in equilibrium, independent of modulation frequency. (2) The volumetric heat capacity was frequency independent. This assumption is valid, as the long MFP phonons that dominate thermal conductivity do not contribute significantly to the heat capacity of a semiconducting material^{1,43}.
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Acknowledgements
We thank Alan J. H. McGaughey and David G. Cahill for independently sharing their hypotheses that a universal phonon MFP spectrum exists, Zonghui Su for measuring the bulk thermal conductivity of AlN, the NSF GOALI for funding (NSF CBET Thermal Transport Processes award #1133394) and Casey Hansen and the NSF REU program at Carnegie Mellon University.
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J.P.F. prepared samples and performed all measurements. J.H.L., E.A.P., Z.S. and R.F.D. provided samples. J.P.F. and J.A.M. wrote the manuscript. All authors discussed the data and edited the manuscript.
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Freedman, J., Leach, J., Preble, E. et al. Universal phonon mean free path spectra in crystalline semiconductors at high temperature. Sci Rep 3, 2963 (2013). https://doi.org/10.1038/srep02963
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