Impact of dislocations on the thermal conductivity of gallium nitride studied by time-domain thermoreflectance

Gallium nitride (GaN) semiconductors are of great interest in photonics and electronics. In solid-state lighting, (In)GaN-based light emitting diodes (LEDs) revolutionized general lighting and are driven with power densities exceeding 100 W/cm².¹ In emerging wireless networks, (Al)GaN/GaN high electron mobility transistors (HEMTs) are leading in 5G efforts, and their power densities are projected to reach 60 W/mm.² GaN-based devices can sustain such high-power densities thanks to GaN's high thermal stability (<500 °C)³ and high thermal conductivity (theoretical upper bound being >336 W/m K).^4,5 Yet, GaN thermal conductivity κ_GaN is reported to range from 110 to 269 W/m K, suggesting strong dependence on not only growth methods and conditions but also measurement techniques and assumptions.^6–12 For instance, freestanding GaN substrates grown by hydride vapor phase epitaxy (HVPE) and the ammonothermal method reportedly have higher thermal conductivities.^9,10,12 However, most GaN layers are grown on non-native substrates for reduced cost, high scalability, and integrated functionality, and a complete study of such heterogeneously-grown GaN is lacking.¹³ It is thus important to explore thermal conductivities of such GaN-on-foreign substrates so that LED and HEMT researchers can create accurate thermoelectrical modeling for thermal management studies.^14,15

Here, we report a time-domain thermoreflectance (TDTR) study of thermal conductivities of four types of c-plane GaN: (1) 350-μm-thick freestanding GaN grown by hydride vapor phase epitaxy (HVPE), (2) 350-μm-thick freestanding GaN grown by high nitride pressure (HNP), (3) 4.5-μm-thick GaN grown on the sapphire substrate by metal-organic chemical vapor phase deposition (MOCVD), and (4) 5-μm-thick GaN grown on the Si(111) substrate by MOCVD with step-graded Al_xGa_1-xN and AlN buffer layers in between [i.e., 5-μm-thick GaN/400-nm-thick Al_0.33Ga_0.67N/290-nm-thick Al_0.60Ga_0.40N/200-nm-thick Al_0.82Ga_0.18N/240-nm-thick AlN/Si(111)].¹⁶ 

TDTR setup uses a mode-locked Ti:sapphire laser that generates 783 nm laser pulses at an 80 MHz repetition rate as the light source. The laser pulses are split into pump and probe beams. The pump laser beam is modulated by an electro-optical modulator and passes through a mechanical delay stage where its arrival time at the sample is controlled. The laser modulation frequency is set to 11 MHz, and the mechanical delay stage introduces up to 3.6 ns of delay with respect to the probe laser beam. The probe laser beam is modulated by a chopper that operates at 200 Hz to suppress coherent laser signal pickup. A fast-response photodiode detector coupled with a radio-frequency (RF) lock-in amplifier is then used to pick up the reflected laser signal. The RF lock-in amplifier has outputs of an in-phase V_in signal and an out-of-phase V_out signal at the laser modulation frequency. The ratio –V_in/V_out is fit to the one-dimensional thermal transport model from an analytical solution for heat flow in a multilayered structure.¹⁷ Fitting is done by minimizing the sum of the squares of error between the thermal transport model and measurement data while sweeping through a range of κ_GaN and Al/GaN thermal boundary resistance (TBR).¹⁸ Throughout the analyses in this work, the sensitivities of the fitting parameters are closely monitored to ensure the fittings are valid.¹⁹ A laser spot size (1/e² radius) of 10.6 μm and a modulation frequency of 11 MHz create a thermal penetration depth (∼1.4 μm)—much smaller than the GaN layer thicknesses (∼4.5 μm) and the ratio of the lateral heat spread and thermal penetration depth is large enough (>4), allowing the use of the one-dimensional thermal transport model,^20,21 and ignoring any thermal conductivity anisotropy.^22,23

A thin layer of aluminum (Al) is deposited by dc magnetron sputtering on each sample to serve as a transducer.²⁴ Al film thicknesses (h_Al) (∼100 nm) are obtained from the picosecond acoustics echo observed at the short time delay (<100 ps) part of the TDTR measurement while assuming a longitudinal speed of sound v_l = 6.42 nm/ps and a 3-nm-thick native oxide layer accordingly: h_Al = 6.42 × t_echo/2 + 3, where t_echo is the arrival time of the first echo. Al film thicknesses are also verified using X-ray reflectivity (XRR) measurements and simulations. Thermal conductivities of the deposited Al films are obtained using a four-point probe measurement. Volumetric heat capacity of Al²⁵ and GaN^11,26 is taken from the literature as C_P,Al = 2.43 J/cm³K and C_P,GaN = 2.64 J/cm³K, respectively.

Figure 1 shows actual TDTR measurement data (open symbols) and thermal transport model calculations along with key experimental parameters for (a) HVPE GaN, (b) HNP GaN, (c) GaN/sapphire, and (d) GaN/Si samples. A good fit is achieved with the coefficient of determination (i.e., R²) >0.97, and Al/GaN interfaces have consistent TBRs (∼10 m²K/GW) across samples. The thermal transport model calculation with κ_GaN± 10% is also plotted (dashed lines). Most of the actual TDTR measurement data points are within the κ_GaN ± 10% curves throughout the entire time delay, indicating that the uncertainties of the measurements are relatively small and that the sensitivities of the κ_GaN data are sufficiently large, ensuring a valid fitting. Multiple TDTR measurements are done on each sample to obtain average κ_GaN. The results are tabulated in Table I. HVPE and HNP GaN exhibit a relatively high κ_GaN of 204.7 (±4.6) and 206.6 (±6.8) W/m K, respectively, GaN/sapphire exhibits a moderate κ_GaN of 191.5 (±10.5) W/m K, and GaN/Si exhibits the lowest κ_GaN of 164.4 (±3.2) W/m K. κ_GaN of HVPE and HNP GaN are statistically indistinguishable; yet compared to that of the GaN/Si sample, they are larger by ∼25%.

To explore the origins of κ_GaN differences among the samples, dislocation densities (σ_D) are measured using cathodoluminescence (CL). Figure 2 shows CL images of (a) HVPE GaN, (b) HNP GaN, (c) GaN/sapphire, and (d) GaN/Si samples, taken in a panchromatic view with an acceleration voltage of 5 kV. As non-radiative recombination centers, dislocations appear as dark spots, which are then counted for density calculations.²⁷ Five measurements per sample are collected. HVPE GaN, HNP GaN, GaN/sapphire, and GaN/Si samples have an average σ_D of 4.80 (±0.42) × 10⁵, 3.81 (±0.08) × 10⁶, 2.43 (±0.20) × 10⁸, and 1.10 (±0.10) × 10⁹ cm⁻², respectively (Table I). Additionally, X-ray diffraction (XRD) (0002) symmetric and (⁠$101¯2$⁠) asymmetric ω scans are performed to verify σ_D. Based on the full-width-at-half-maximum (FWHM) of the (0002) and (⁠$101¯2$⁠) ω scans, screw-type and edge-type σ_D are calculated^16,28 (see the supplementary material) and tabulated in Table I. For the GaN/sapphire and GaN/Si samples, both techniques reveal similar σ_D. For the HVPE and HNP GaN samples, XRD estimates a higher σ_D than the CL technique. We believe that the discrepancy in this case is most likely due to the limitations of the FWHM-based XRD σ_D calculation model in which it is assumed that the crystal forms a mosaic structure consisting of similar-sized blocks.²⁹ For samples with low σ_D, the dislocations form in clusters making it difficult to assume that there exists a constant lateral coherence length between dislocations.^27,30

A secondary ion mass spectroscopy (SIMS) is performed to estimate the impurity concentrations in the GaN samples. The concentrations of Si, H, C, and O are measured and are then used to quantify the “scattering strength” $Γ$ through Klemens's model $Γ=∑ifi[1−Mi/M¯]2$⁠, where f_i is the fractional concentration of the ith impurity atom, M_i is the atomic mass of the ith impurity atom, and $M¯$ is the average atomic mass. Both measured impurity concentrations and calculated $Γ$ values are tabulated in Table I. The largest $Γ$ obtained from the GaN/Si sample is still a factor of five times smaller than the scattering due to naturally occurring isotopes (⁠$Γisotope$ = 2.744 × 10⁻⁴), suggesting that impurity concentrations of these samples in this work will not limit κ_GaN (see the supplementary material for SIMS measurement data and discussion).

Figure 3 plots experimentally measured κ_GaN as a function of experimentally measured σ_D (open symbols). An earlier empirical model (plotted as a dotted line for reference) by Mion et al.,³¹ expressed as κ_GaN = 230 tanh^0.12(5 × 10⁶/σ_D), fails to provide a good fit to this work. A new empirical model, κ_GaN = 210 tanh^0.12(1.5 × 10⁸/σ_D), is proposed in this work (plotted as a dashed line for comparison). The two empirical models have two major differences. The first difference is the maximum κ_GaN employed in the empirical formula. In the literature, highest reported κ_GaN values are 294 (±44) and 253 (±22) W/m K (measured by the laser flash method)^32,33 and 269 W/m K (measured by the stationary heat flow method).⁸ Yet, most of κ_GaN are reportedly <230 W/m K.^{10,22,31,34–36} In close context to this work, Zheng et al.²³ recently studied high quality HVPE and ammonothermal GaN using the TDTR technique and reported a κ_GaN of ∼209 W/m K. These results suggest that the discrepancy of measured maximum κ_GaN may arise from the differences in the employed experimental technique. The second difference is the deflection point where κ_GaN starts to drop as a function of σ_D. In our modified empirical model, the σ_D level at which κ_GaN starts to drop rapidly is chosen as 1.5 × 10⁸ cm⁻²—a slightly larger value from that in Mion et al.³¹ (i.e., 5 × 10⁶ cm⁻²). With this shift, the new empirical model in this work properly captures not only our experimental data on the GaN/sapphire and GaN/Si samples but also other literature that studied GaN samples with high σ_D (>10⁸ cm⁻²).^37–39 

Based on the experimental data, we propose to modify Klemens's model to explain the findings. The original Klemens's model describes the effect of σ_D on κ_GaN by the functions^5,40

where τ_DC is the phonon scattering relaxation time associated with dislocation cores, η is the weight factor to account for the mutual orientation of the direction of the temperature gradient and the dislocation line, V₀ is the volume per atom, v is the polarization-averaged phonon velocity, v_L (v_T) is the longitudinal (transverse) sound velocity, ω is the phonon frequency, τ_S (τ_E) is the phonon relaxation time associated with screw-type (edge-type) dislocations, b_S (b_E) is the screw-type (edge-type) dislocation Burgers vector, and $γ$ is the Gruneisen parameter. Previously published literature^11,31,34 agrees that the impact of σ_D on κ_GaN is underestimated. In order to compensate for the discrepancy, a correction factor of 1000 is commonly used as a multiplier to the scattering rates to fit the experimental results.^11,34

To fit our experimental data, Klemens's model is modified accordingly: η is set as 2 × 10¹⁰ and σ_D is replaced by σ_D^0.4. The η should be considered as a correction factor rather than a factor that purely represents the effect of the relative orientation between the temperature gradient and the dislocation. These two modifications effectively increase the strength of phonon–dislocation scattering and adjust the rate of κ_GaN decrease such that κ_GaN is not vanishingly small (>35 W/m K) at σ_D > 10¹⁰ cm⁻².^6,7,31 This modified Klemens's model is plotted in Fig. 3 (solid line). The basic GaN material parameters are taken from Ref. 5. The same parameter set is used for the calculations of all four GaN samples.

In conclusion, four types of GaN samples (HVPE GaN, HNP GaN, GaN/sapphire, and GaN/Si) are studied using XRD and TDTR to determine σ_D and κ_GaN, respectively. κ_GaN of HVPE GaN, HNP GaN, GaN/sapphire, and GaN/Si are measured as 204.7 (±4.6), 206.6 (±6.8), 191.5 (±10.5), and 164.4 (±3.2) W/m K, respectively. The determining factor of κ_GaN is shown to be σ_D when σ_D> 1.5 × 10⁸ cm⁻². The new empirical model κ_GaN = 210 tanh^0.12(1.5 × 10⁸/σ_D) describes the relationship between κ_GaN and σ_D. A modified Klemens's model is proposed to increase the strength of phonon–dislocation scattering and adjust the rate of κ_GaN decrease such that it explains the experimental results. Overall, we report how κ_GaN of heteroepitaxial GaN can be estimated based on σ_D. Our work lays the foundation in thermal management of the technologically important GaN-based semiconductor devices, typically grown on foreign substrates with high σ_D.

The laser power dependent TDTR measurement results, XRD ω scans in the (0002) symmetric and (⁠$101¯2$⁠) asymmetric planes for dislocation density determination, and the SIMS measurement results are discussed in detail in the supplementary material.

This work was supported by the Air Force Office of Scientific Research (AFOSR) through Young Investigator Program Grant No. FA9550-16-1-0224 and was carried out in the Nick Holonyak, Jr. Micro and Nanotechnology Laboratory and Frederick Seitz Materials Research Laboratory Central Research Facilities, the University of Illinois at Urbana-Champaign, IL, USA.