Cross-plane thermal conductivity of GaN/AlN superlattices

The tunable bandgap, high thermal stability, high electric field strength, high electron mobility, and the presence of a spontaneous and piezoelectric polarization in wurtzite III-nitrides led to the integration of GaN and its alloys in state-of-the-art electronics technology.¹ Current III-nitride-based devices range from light emitting diodes in the ultraviolet^2,3 and visible range^4,5 to high-electron mobility transistors^6–8 and biosensors.⁹ Another research field involving III-nitrides is opened up by doping them with transition metals and by studying the emergent spin-related phenomena.^10–13 Generally, devices experience efficiency losses due to internal heating^14,15 and low thermal conductivity materials are needed as the basis for thermoelectric devices^16,17 or for on-device cooling.¹⁸ Thus, a detailed understanding of the thermal properties of thin films and nitride-based heterostructures is mandatory in the perspective of efficient thermal management.

Previous works have investigated the thermal conductivity of III-nitrides in bulk¹⁹ and in thin films²⁰ in detail. Furthermore, several groups studied the decrease in thermal conductivity with increasing Al concentration in Al_xGa_1−xN bulk²¹ and thin films.^20,22–25 A decrease in the bulk thermal conductivity of III-nitrides due to impurity doping^26–29 was also found, while an enhanced conductivity is reported for isotopically enriched GaN.³⁰ For AlN/GaN stacks, significant changes in the phonon properties due to interface effects^31–34 and strain³⁵ were reported, while for compositionally graded interfaces, it was found that alloy scattering is dominant over mismatch scattering³⁶ of phonons.

In order to minimize the thermal conductivity of a heterostructure, either the thickness of the single layers must be reduced or scattering centers are introduced by interdiffusion. For instance, in the case of Al_0.4Ga_0.6N, the cross-plane (perpendicular to the sample's surface) thermal conductivity $κ⊥$ is theoretically predicted to decrease to 1 W/(m·K) (Ref. 22) when reducing the layer thickness down to 2 nm. Due to the anisotropy of the thermal conductivity tensor, the in-plane component $κ∥$ of the thermal conductivity is expected to exhibit a different heat conduction than the cross-plane one.³⁷ The interface resistance affects the interface crossing of phonons and leads to a quenching of the thermal conductivity, as reported for Ge/Si superlattices^38–40 and for up to eight pairs of AlN/GaN multilayers.⁴¹ 

In this work, the thermal conductivity of heterostructures consisting of up to 50 GaN/AlN pairs with different layer thicknesses is measured by the differential $3ω-$method^42,43 and simulated with ab initio calculations.

The investigated samples are grown by metal organic vapor phase epitaxy (MOVPE) according to the procedure detailed in the supplementary material. In particular, all samples are grown on a c-sapphire substrate, upon deposition of a low temperature nucleation layer followed by a 1.5 μm thick GaN buffer layer and a GaN/AlN superlattice (SL). The considered samples together with their relevant parameters are listed in Table I. The SL heterostructures consist of 50 GaN/AlN pairs. The thicknesses of each GaN and AlN layer are varied between 4 nm and 16 nm over the sample series, covering typical thicknesses for GaN/AlN quantum wells⁴⁴ and the number of layers needed for high-reflectivity bands of distributed Bragg mirrors.^45,46 A sketch of the SL structure is given in the inset to Fig. 1(a), and in the following, the SLs are indicated by SL4, SL8, SL12, and SL16, where the numbers refer to the thickness of the single layers in nm. Additionally, two reference samples, namely, a clean sapphire substrate and a GaN buffer, are also included in the study.

The crystallographic properties are analyzed by high-resolution x-ray diffraction (XRD) and transmission electron microscopy (TEM). In Fig. 1(a), the $ω−2θ$ scan of the SL4 around the GaN (0002) reflection is plotted and the signal simulated with the AMASS software from PANalytical is given. The satellite peaks of the main GaN reflection arise due to the interference in the diffraction of the different layers of the multilayer structure. The individual layers of the SL are evidenced in the high-angle annular dark-field image (HAADF), given for sample SL4 in Fig. 1(b). The upward arrow indicates the growth direction. Due to interdiffusion, the interfaces between the GaN and the AlN layers is smeared out in an Al_xGa_1−xN layer,⁴⁷ as evidenced by the energy dispersive x-ray measurements shown in Figs. S3(a) and S3(b) of the supplementary material and marked by the arrow in Fig. 1(c). The interdiffusion at the interface can suppress the coherent nature of phonons, favor diffusive scattering, and influence the cross-plane heat conduction.^37,41,48,49 The detailed analysis of the layers' thickness with TEM and XRD can be found in the supplementary material, where the TEM images in Figs. S2(a)–S2(d) display V-shaped defects, which are also reported in previous works.⁵⁰ 

For the thermal conductivity measurements, a 60-nm-thin insulating AlO_x layer is deposited onto the specimens, followed by a metal structure consisting of a 120-nm-thick and 10-μm-wide Au layer, which is used for heating the underlying material and for determining the corresponding temperature rise. The fabrication steps and geometry of the contacts are given in Sec. II C of the supplementary material.

Measurements of the cross-plane thermal conductivity $κ⊥$ are performed with the differential $3ω$-method⁴² in the temperature range between 160 K and 455 K. First, the thermal conductivity $κ⊥$ of the sapphire substrate and the GaN buffer layer is measured. The obtained values serve as references for the thermal conductivity for the SLs as their influence on the measurement have to be deducted. The data evaluation is carried out by solving the heat diffusion equation in a matrix formalism and by fitting the measured temperature oscillations as a function of the heater frequency.⁵¹ The whole SL is treated as a single layer.

The measured cross-plane thermal conductivity of the clean substrate and the buffer deposited onto the substrate as a function of temperature T in the interval (160–455) K is plotted as symbols in Fig. 2(a) and the values for room temperature are included in Table I. The simulated conductivity vs temperature of the reference samples is also given in Fig. 2(a). The theoretical calculations for the sapphire substrate are performed following the procedure by Burghartz and Schulz⁵² with a sapphire thickness of 330 μm and a temperature-dependent specific heat capacity taken from Ref. 53. The values of the measured sapphire substrate follow the trend of the simulated conductivity up to 200 K. The measured GaN film thermal conductivity slightly increases for decreasing temperatures and does not feature the steep slope below 200 K, which would be expected for bulk GaN.

The measured cross-plane thermal conductivity as a function of T of the four considered samples containing SLs is reported in Fig. 2(b), and the obtained values at room temperature are collected in Table I. For the SL4 with the thinnest individual layers, also the lowest thermal conductivity of 5.9 W/(m·K) at 300 K is measured, while for the SL16, the thermal conductivity reaches 10.1 W/(m·K) at the same temperature. The measured signal is mainly influenced by the SL and by the substrate, while the contribution of the buffer layer is negligible for the investigated samples, leading to an error on the measurements lower than 1.5% over the whole temperature range. A decrease in thermal conductivity with decreasing temperature is found. In addition to the experiments, theoretical simulation accounting for phonon scattering mechanisms is performed and included as lines in Fig. 2(b).

The calculations of the bulk GaN and GaN/AlN SL thermal conductivity are performed with the almaBTE software package⁵⁴ in the relaxation time approximation (RTA) and with a uniform wavevector grid of 24³. Further details about the program can be found in Sec. III of the supplementary material. First, the temperature-dependent thermal conductivity parallel to the c-axis of GaN is analyzed. The simulated results are plotted in Fig. 3, where the topmost curve is the calculated conductivity for bulk GaN. The same simulation results are included for comparison in Fig. 2(a) (solid line). The simulations are repeated for several layer thicknesses ranging from 0.01 μm to 100 μm. The thickness of the MOVPE grown GaN buffer is 1.5 μm; therefore, a simulation of a GaN layer with the same thickness is included as a line combined with reverse triangles in Fig. 3 and as a dashed line in Fig. 2(a). The $κ⊥$ conductivity of GaN reaches a maximum at 100 K for a film thickness of 1.5 μm, and for thicker layers, this maximum is shifted to lower temperatures, namely around 15 K for bulk GaN. Inyushkin et al.¹⁹ reported a maximum of 3770 W/(m·K) around 28 K. No sharp maximum is reached for films with thicknesses of 1 μm and lower, for which, instead, the conductivity saturates at temperatures above 100 K. The highest achieved thermal conductivity decreases with the decreasing layer thickness due to the significant effect of interface- and boundary-scattering of phonons.

Two limit regimes of heat conduction are relevant for crystalline films. In particular, in films with thicknesses much lower than the cross-plane mean free path (MFP) $λdom$²² of phonons, which depends on the cross-plane bulk thermal conductivity and on the mean free path of the cross-plane projection, $κ⊥$ is predominantly ballistic. In contrast, for films with the film thickness much larger than $λdom$⁠, the heat conduction is quasi-diffusive and, therefore, no distinct dependence on the film thickness is found. The film thicknesses considered in this work are smaller than the characteristic lengths²² for the transition to the ballistic regime; thus, a distinct change of $κ⊥$ with different individual layer thicknesses is expected.

In order to gain further insights into the effect of alloy scattering, first a heterostructure is simulated, which assumes a well-defined stoichiometry of the GaN/AlN layers without intermixing and without a substrate. This ideal structure has a concentration x of Al of either 0 or 1. The SL crystal structure is built with the superlattice package of the almaBTE software, following the procedure reported in Ref. 38. In Fig. 4(a), the computed temperature-dependent cross-plane thermal conductivity of the SL4 for one to 15 pairs of GaN/AlN is given. The thermal conductivity increases over the whole temperature range with the increasing number of SL pairs. For completeness, also an inverse structure AlN/GaN (not shown) is calculated for five pairs, and the same thermal conductivity as in the case of the GaN/AlN sequence is found. In Fig. 4(b), the simulated $κ⊥$ values of one SL pair for all four thicknesses of the individual layer, i.e., 4 nm, 8 nm, 12 nm, and 16 nm, are reported over temperature. The effect of scattering of phonons at interfaces is less pronounced with increasing layer thickness, leading to an increase in thermal conductivity,^55,56 as evidenced in Fig. 4(b).

In order to account for interdiffusion at the interfaces, a concentration-graded layer of Al_xGa_1−xN between the two ideal AlN and GaN layers is introduced in the calculations (realistic structures), while the total thickness of the structure is kept constant. In Fig. 4(c), $κ⊥$ over the temperature is given for SL4 with a different number of pairs over the samples' series. When comparing the simulated ideal structures with those including Al_xGa_1−xN regions, a significant decrease in thermal conductivity is found. In the realistic structures, additionally to the phonon interface scattering, alloy scattering is observed. As reported by Vermeersch et al.,²² the change of Al concentration from GaN toward AlN leads to a decrease in the thermal conductivity of up to one order of magnitude in bulk materials. The increase in the number of layers slightly increases the thermal conductivity of the whole structure, in accord with Ref. 57. In Fig. 4(d), the conductivity of one GaN/AlN pair with intermixing for the four thicknesses is plotted over temperature. As found in the case of ideal SLs, the increase in an individual layer thickness induces an increment of the thermal conductivity. The thermal conductivity of the realistic SLs is less influenced by a change in layer numbers than by an increase in individual layer thickness.

The simulations of five realistic GaN/AlN pairs qualitatively reproduce the shape of the measured $κ⊥$ given in Fig. 2(b). As discussed, a slight increase in the maximum value of the thermal conductivity is expected when increasing the number of pairs from five to 50. Additional changes in the thermal conductivity can arise by a variance in the Al concentration and increase when going toward an ideal SL. The simulations do not consider any kind of defects, though the V-shaped ones, present in the MOVPE grown samples, are expected to reduce the thermal conductivity. Additionally, the simulations do not contain a substrate, which is expected to influence the phonon transport at the substrate and SL boundary. Furthermore, the influence of strain, predicted to increase the thermal conductivity,³² in the layers is not treated in the simulations.

In semiconductors, thermal energy is mainly transported by phonons and the phonon frequency spectra can be used to identify the main scattering mechanism. In Fig. 5, the fraction of the normalized cumulative thermal conductivity, obtained by dividing the cumulative thermal conductivity by the bulk thermal conductivity transferred by phonons as a function of the frequency, is given at 300 K. The simulated phonon contribution for a bulk GaN is plotted as a solid line. Moreover, the four lines with symbols represent the simulations for one pair of GaN/AlN with ideal and realistic composition and five pairs of GaN/AlN with ideal and realistic composition, respectively. For realistic SLs, a major contribution to the heat conductivity originates from phonons with frequency between 0 THz and 6 THz, while phonons between 12 THz and 16 THz contribute minimally, leading to a first plateau marked by the upward arrow labeled with 1 in Fig. 5. A second plateau occurs above 19 THz, indicated by the upward arrow two, and phonons with frequencies higher than 22 THz do not contribute significantly to the thermal conductivity. The splitting in two plateaus is due to the alloy scattering of high-frequency phonons.³⁸ Due to an enhancement of low-frequency phonon scattering at the abrupt interfaces,³⁸ the conductivity of the ideal SLs is less significantly affected by phonons in the frequency range (0–4) THz. Only one prominent plateau above 8 THz is visible due to the missing effect of alloy scattering. For completeness, the simulated cumulative thermal conductivity normalized to the bulk is provided as a function of the MFP of the phonons for bulk AlN, bulk GaN, and—in both ideal and realistic cases—two superlattices with one and five GaN/AlN pairs, respectively, in Fig. S4 of the supplementary material.

In conclusion, the cross-plane thermal conductivity of sapphire, GaN, and SLs consisting of up to 50 pairs of GaN/AlN has been investigated. The temperature-dependent conductivity of various film thicknesses of GaN has been simulated, and the values are reproduced by measuring the thermal conductivity with the differential $3ω$-method. The simulated thermal conductivity of the GaN/AlN SL with a different number of pairs has been compared, and it is found that the change in absolute values is minimal. The measured values for SLs with individual layer thicknesses of 4 nm to 16 nm and 50 pairs increase at room temperature from 5.9 W/(m·K) to 10.1 W/(m·K). These results confirm the possibility of decreasing the cross-plane thermal conductivity of epitaxial crystalline GaN/AlN heterostructures to values as low as those reported for amorphous layers,⁵⁵ opening wide perspectives for heat management in III-nitride-based devices.

See the supplementary material for the details about the epitaxial growth, the in-depth TEM and XRD characterization, and the simulations.

This work was supported by the European Commission's Horizon 2020 Research and Innovation Program [Grant No. 645776 (ALMA)] and by the Austrian Science Fund (FWF) (Project Nos. P31423 and P26830). For the purpose of open access, the author has applied a CC BY public copyright license to any author accepted manuscript version arising from this submission. The authors thank Werner Ginzinger for preparing the specimens for the TEM experiments and for carrying out the related measurements, Heiko Groiss and Jesús Carrete for fruitful discussions, and Albin Schwarz for preparing the contact structures for the $3ω$ measurements.