Largely reduced cross-plane thermal conductivity of nanoporous In_0.1Ga_0.9N thin films directly grown by metal organic chemical vapor deposition

Abstract

In recent year, nanoporous Si thin films have been widely studied for their potential applications in thermoelectrics, in which high thermoelectric performance can be obtained by combining both the dramatically reduced lattice thermal conductivity and bulk-like electrical properties. Along this line, a high thermoelectric figure of merit (ZT) is also anticipated for other nanoporous thin films, whose bulk counterparts possess superior electrical properties but also high lattice thermal conductivities. Numerous thermoelectric studies have been carried out on Si-based nanoporous thin films, whereas cost-effective nitrides and oxides are not systematically studied for similar thermoelectric benefits. In this work, the cross-plane thermal conductivities of nanoporous In_0.1Ga_0.9N thin films with varied porous patterns were measured with the time-domain thermoreflectance technique. These alloys are suggested to have better electrical properties than conventional Si_xGe_1−x alloys; however, a high ZT is hindered by their intrinsically high lattice thermal conductivity, which can be addressed by introducing nanopores to scatter phonons. In contrast to previous studies using dry-etched nanopores with amorphous pore edges, the measured nanoporous thin films of this work are directly grown on a patterned sapphire substrate to minimize the structural damage by dry etching. This removes the uncertainty in the phonon transport analysis due to amorphous pore edges. Based on the measurement results, remarkable phonon size effects can be found for a thin film with periodic 300-nm-diameter pores of different patterns. This indicates that a significant amount of heat inside these alloys is still carried by phonons with ~300 nm or longer mean free paths. Our studies provide important guidance for ZT enhancement in alloys of nitrides and similar oxides.

Keywords

nanoporous film / thermoelectrics / phonon / mean free path / diffusive scattering

Introduction

Gallium Nitride (GaN) high electron mobility transistors (HEMTs) are widely used for high-power and/or high-frequency applications in power electronics, microwave communications, and optoelectronics. In a GaN HEMT, two-dimensional electron gas (2DEG) is formed on the interface between GaN and its ternary alloy, such as AlGaN. This 2DEG has a high carrier mobility up to 2000 cm²/V and a carrier density larger than 10¹³ cm⁻², leading to excellent electron transport for power electronics [¹]. However, the superior performance of these devices is often restricted by the significant overheating within the device, which would dramatically reduce the charge carrier mobility and thus lower the output power [²]. In addition to the degraded performance, the strong overheating will also shorten the lifetime of GaN devices, which largely affects their long-term applications [³]. Along this line, various cooling technologies have been proposed, including microchannel cooling within the substrate [⁴,⁵], enhanced heat spreading by coating the device with an ultra-high-thermal-conductivity layer [⁶,⁷]. In addition to these efforts, on-chip thermoelectric (TE) cooling can be more effective because heat can be directly and effectively removed from the hot spot. To integrate such TE coolers with a GaN HEMT, the selected TE materials should have a thermal expansion coefficient similar to that of GaN for better compatibility. Correspondingly, GaN-based alloys are recommended for such TE devices [⁸]. Beyond cooling of a GaN HEMT, TE devices using GaN alloys can also be widely used for high-temperature waste heat recovery and other applications, which benefits from the superior thermal stability of these alloys [⁹–¹²].

Despite many promising applications of a GaN-based TE device, the TE performance of GaN alloys is still relatively low. In physics, the effectiveness of a TE material is evaluated by its dimensionless TE figure of merit (ZT), defined as ZT= S²σT/k, where S, σ, k, and T represent the Seebeck coefficient, electrical conductivity, thermal conductivity, and absolute temperature, respectively [¹³]. Here k can be further split into two parts, the lattice (phonon) contribution k_L and the electronic contribution k_E. In principle, good TE materials should have a high power factor S²σ but a low k, which is challenging to be balanced within the same material. For GaN and its alloys, the power factor S²σ can be better than those for the state-of-the-art high-temperature TE materials such as Si_xGe_1−x alloys. Similar to the case of Si-based materials for TE applications, a high ZT is hindered by the intrinsically high k_L of GaN. This high k_L can be dramatically suppressed in GaN alloys with strong point-defect scattering of short-wavelength phonons, without deteriorating the power factor. In general, the ZT of wide-bandgap GaN alloys rapidly increases with elevated temperatures, which is due to decreased k, together with maintained or sometimes improved S²σ. In calculations, ZT of 0.85 at 1200 K was estimated for In_0.1Ga_0.9N at an optimized carrier density [¹¹]. In experiments, ZT was found to be 0.08 at 300 K and reached 0.23 at 450 K for In_0.36Ga_0.64N alloy, which was comparable to that for SiGe alloys [¹⁴]. Furthermore, ZT of In_0.17Ga_0.83N would reach a ZT of 0.34 at 875 K [¹⁵].

Following the existing work, an even higher ZT can be achieved in nanoporous In_xGa_1−xN alloys, where phonons with long wavelengths and mean free paths (MFPs) can be further scattered by the nanopore edges as the classical phonon size effect. For Si, their periodic nanoporous films have been widely studied in recent years as low-cost and earth-abundant TE materials [¹⁶,¹⁷]. By introducing periodic nanopores, the lattice thermal conductivity k_L can be largely suppressed in these films but bulk-like S²σ can still be conserved, leading to enhanced TE performance [¹⁸–²⁰]. At 300 K, ZT~0.4 was measured for nanoporous Si films with pitch of 55 nm and 35% porosity, in comparison to ZT~0.01 for solid Si films [²⁰]. With alloy atoms and nanopores, phonon transport can be largely suppressed for different wavelength regimes in nanoporous GaN alloys. Besides phonon scattering by these atomic to nanoscale features, the phonon dispersion can also be largely modified to lower k_L when the nanoporous structure sizes are decreased and become comparable to the wavelength of dominant phonons, i.e., a few nanometers at 300 K [²¹]. Known as “phononic effects,” this phenomenon considers the wave nature of phonons and is caused by coherent phonon transport within a periodic nanostructure, such as periodic nanoporous materials. The demonstration of strong phononic effects can be very challenging due to the small nano-features required to match the dominant phonon wavelength. However, a very high ZT is anticipated if such phononic effects can be combined with alloying to minimize k_L [²²]. As another mechanism, amorphization and oxidation of pore edges should also be considered for k_L reduction. This mechanism was first proposed in one early study on microporous Si films [²³] and later considered in molecular dynamics simulations for Si films with ~10 nm pores [²⁴]. Such amorphization can often be found for nanofabricated pores due to the structural damage during the fabrication process, as observed in the transmission electron microscopy (TEM) image of nanoporous Si films [²⁰]. In a simple treatment, amorphous pore edges can expand the effective pore diameter. For measured k_L of a Si film with a 34 nm period and 10–16 nm pore diameters [¹⁹], agreement between the simulations and measurements are found after slightly expanding the pore diameter [²⁵].

In this work, nanoporous In_0.1Ga_0.9N thin films with varied porous patterns are studied for their cross-plane k. Different from previous cross-plane k studies on nanoporous Si films [²⁶,²⁷], our measured In_0.1Ga_0.9N thin films are directly grown on a sapphire substrate to avoid the use of deep reactive ion etching (DRIE) for pore drilling. This eliminates the amorphous pore edges introduced by DRIE and thus simplifies the data analysis. Strong phonon size effects are observed even with sub-micrometer porous patterns. These results suggest remarkable k_L contribution by phonons with sub-micron or longer MFPs though phonons in these alloys are suggested to be scattered by nanometer-scale compositional inhomogeneities [²⁸]. This conclusion is consistent with the observation of strong k_L contribution by long-MFP phonons in alloyed nanostructures such as SiGe nanowires [²⁹]. Our work provides important guidance for ZT enhancement in general TE nitrides and similar oxides.

Sample preparation and measurements

In previous measurements on nanoporous Si films, nanopores through the film were mostly fabricated by defining the nanopatterns with various lithography techniques and then drilling the pores with reactive ion etching (RIE) or DRIE [¹⁸–²⁰,²³,²⁶,³⁰–³³]. In one recent work, a focused ion beam was also employed to directly drill holes across a Si thin film [³⁴]. For these studies, phononic effects were negligible because the size of nanoporous features was much larger than the phonon wavelengths in Si, as 1–10 nm at 300 K [²⁵,³⁵] . In addition to diffusive phonon scattering by pore edges, the structural damage during the pore-drilling process also affected the phonon transport. The divergence between theoretical predictions and measurements was thus attributed to the expanded effective pore diameter due to the amorphization and oxidation on pore edges [²⁵]. To avoid such pore-edge defects introduced by nanofabrication, in this work the studied nanoporous films were directly grown on a patterned substrate using metal organic chemical vapor deposition (MOCVD). An array of vertical SiO₂ nanopillars was fabricated on the sapphire substrate as masks to prevent local growth of GaN or GaN alloys. In the literature, a mask with the inversed pattern (i.e., 40-nm-thick nanoporous SiO₂ film) was employed to grow vertical micrometer-length, 50-nm-diameter GaN nanowires with minimized surface defects [³⁶]. Similarly, the use of a SiO₂ mask for direct MOCVD growth also minimized pore-edge defects and eliminated the corresponding uncertainties in data analysis.

For comparison purpose, seven different patterns were defined with SiO₂ nanopillars on the c-plane sapphire substrate by MOCVD (Fig. 1 (a)). For all patterns, the pore diameters were fixed at 300 nm and the pores were located on either a square lattice or hexagonal lattice. All layers were grown with unintentional doping. Prior to growth, the sapphire substrate was heated in H₂ ambient at 1000°C for 3 min to remove surface contaminations. The growth of the structure began with a 50-nm-thick low temperature GaN (LT-GaN) nucleation layer grown at about 500°C for 3.5 min. Following this, a 50 nm GaN buffer layer was grown at 1060°C for 1 min. Afterwards, the growth was completed by deposition of 150 nm In_0.1Ga_0.9N layer at 805°C for 60 min. After the high-temperature MOCVD growth, SiO₂ nanopillars were removed with hydrogen fluoride to obtain the nanoporous pattern. Figure 1(a) and 1(b) show the scanning electron microscopy (SEM) images of representative nanoporous films. Periodic pores are aligned on either hexagonal or square lattices.

The cross-plane k of nanoporous thin films of all patterns were measured via the time-domain thermo reflectance (TDTR) method. TDTR is an optical-based, accurate, and robust technique applicable of probing various thermal properties, including thermal conductivity, interfacial thermal conductance, and heat capacity of sample systems ranging from thin films, bulk substrates, to nanoparticles. Prior to thermal measurements, a 55-nm-thick layer of aluminum was coated onto the whole wafer by electron beam deposition to serve as the optical transducer. The thermal conductivity was extracted by fitting the ratio of in-phase (V_in) and out-of-phase (V_out) signals from TDTR experimental data to a multilayer thermal model [³⁷]. Parameters used in the model include thickness, heat capacity, and thermal conductivity of each layer. The volumetric heat capacity was used for the porous In_xGa_1-xN alloy, which was calculated as C_InGaN = xC_InN + (1 -x)C_GaN with x being the molar percentage of InN [²⁸,³⁸,³⁹]. Further, considering that the thermal penetration depth, d = \sqrt{k/(C\pi f)}, was larger than or equivalent to the combined thickness of the nanoporous In_xGa_1−xN layer (150 nm) and the GaN layer beneath (50-nm nucleation and 50-nm buffer) for our samples, we thus treated these sub-layers as an effectively homogeneous layer of ≈250 nm in the thermal model. This will lead to an effective volumetric heat capacity depending on the porosity (j):

(1)C_{\mathrm{eff}}=(\mathrm{1}-\phi ){\displaystyle \frac{h_{\mathrm{1}}\times C_{\mathrm{InGaN}}+h_{\mathrm{2}}\times C_{\mathrm{GaN}}}{h_{\mathrm{1}}+h_{\mathrm{2}}}},

where h₁ = 150 nm and h₂ = 100 nm are the thicknesses of the In_xGa_1−xN layer and the GaN layer, respectively. The thermal conductivities of this effective layer with different patterns and porosities were extracted through the best fit of the thermal model to the TDTR experimental data. More details regarding the TDTR measurement method and data reduction can be found elsewhere [³⁷,⁴⁰,⁴¹].

Phonon modeling

Modeling cross-plane k_L of a nanoporous thin film

The cross-plane k_L of a thin film is computed based on the kinetic relationship:

(2){{\displaystyle k}}_{\mathrm{L}}={\displaystyle \frac{\mathrm{1}-\phi }{\mathrm{3}}}{\displaystyle {\sum }_{i=\mathrm{1}}^{\mathrm{3}}{\displaystyle {\int }_{\mathrm{0}}^{{{\displaystyle \omega }}_{\max },i}{{\displaystyle c}}_i}}(\omega ){{\displaystyle v}}_{g,i}(\omega ){{\displaystyle \Lambda }}_i(\omega )d\omega ,

where j is porosity, w is the phonon angular frequency, the subscript i indicates the phonon branch, c_i(ω)is the spectral volumetric phonon specific heat, v_g,i(ω)is the phonon group velocity, and {{\displaystyle \Lambda }}_i(\omega ) is the modified phonon MFP for branch i and angular frequency w. Only three acoustic branches are considered here because the optical phonon contribution is negligible due to their small group velocities. For a solid thin film, {{\displaystyle \Lambda }}_i(\omega ) is related to bulk phonon MFP {{\displaystyle \Lambda }}_{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k},i}(\omega ) by [⁴²]

(3){\displaystyle \frac{\mathrm{1}}{{{\displaystyle \Lambda }}_i(\omega )}}={\displaystyle \frac{\mathrm{1}}{{{\displaystyle \Lambda }}_{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k},i}(\omega )}}+{\displaystyle \frac{\mathrm{4}}{\mathrm{3}t}},

in which t is the film thickness. More discussions for cross-plane k_L of a thin film can also be found elsewhere [⁴³].

When nanopores are further considered, the narrow “neck” between adjacent pores will further reduce the phonon MFP by diffusively scatter phonons. The phonon size effects for both cross-plane and in-plane geometry restrictions can be combined with Matthiessen’s rule, given as [⁴⁴]

(4){\displaystyle \frac{\mathrm{1}}{{{\displaystyle \Lambda }}_i(\omega )}}={\displaystyle \frac{\mathrm{1}}{{{\displaystyle \Lambda }}_{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k},i}(\omega )}}+{\displaystyle \frac{\mathrm{4}}{\mathrm{3}t}}+{\displaystyle \frac{\mathrm{1}}{L}},

where the characteristic length L is determined by the pore diameter d and the pitch p. Here the pitch is the center-to-center distance between adjacent pores for pores aligned on hexagonal or square lattices.

In a different viewpoint, the phonon MFP modification can be done separately for the in-plane and cross-plane directions:

(5){\displaystyle \frac{\mathrm{1}}{{{\displaystyle \Lambda }}_{i,\mathrm{0}}(\omega )}}={\displaystyle \frac{\mathrm{1}}{{{\displaystyle \Lambda }}_{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k},i}(\omega )}}+{\displaystyle \frac{\mathrm{1}}{L}},

(6){{\displaystyle \Lambda }}_i(\omega )={\displaystyle \frac{{{\displaystyle \Lambda }}_i(\omega )}{\mathrm{1}+\mathrm{4}{{\displaystyle \Lambda }}_{i,\mathrm{0}}(\omega )/\mathrm{3}t}}.

Here {{\displaystyle \Lambda }}_{i,\mathrm{0}}(\omega )first incorporates the influence of pore-edge phonon scattering and the back phonon scattering by film top and bottom surfaces is further considered in Eq. (6). It can be shown that Eq. (6) is equivalent to Eq. (4).

For two-dimensional porous structures, L can be evaluated by the mean beam length of the structure, defined as [⁴⁵]

(7)L={\displaystyle \frac{\mathrm{4}{{\displaystyle V}}_{\mathrm{S}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}}{A}}=\{\begin{array}{l}{\displaystyle \frac{\mathrm{4}{p}^{\mathrm{2}}-\pi d^{\mathrm{2}}}{\pi d}},\mathrm{square}\mathrm{}\mathrm{lattice},\\ {\displaystyle \frac{\mathrm{3}\sqrt{\mathrm{2}}{p}^{\mathrm{2}}-\pi d^{\mathrm{2}}}{\pi d}},\mathrm{hexagonal}\mathrm{}\mathrm{lattice},\end{array}

where the solid-region volume V_Solid and pore surface area A are evaluated for a periodic porous film.

For the measured k values of In_xGa_1-xN and GaN layers, the effective value k_eff can be computed based on

(8){\displaystyle \frac{{{\displaystyle t}}_{\mathrm{1}}+{{\displaystyle t}}_{\mathrm{2}}}{{{\displaystyle k}}_{\mathrm{e}\mathrm{f}\mathrm{f}}}}={\displaystyle \frac{{{\displaystyle t}}_{\mathrm{1}}}{{{\displaystyle k}}_{\mathrm{1}}}}+{\displaystyle \frac{{{\displaystyle t}}_{\mathrm{2}}}{{{\displaystyle k}}_{\mathrm{2}}}},

where the subscripts number represents individual materials. Because of the slight difference between the lattice constants of In_xGa_1-xN and GaN, the interfacial thermal resistance between the two materials are neglected in Eq. (8), as assumed in the previous study [²⁸].

Employed bulk phonon dispersion and MFPs

All input parameters for phonon modeling are listed in Table 1. To simplify, an isotropic sine-shape phonon dispersion (Born-von Karman model) is used for the In_0.1Ga_0.9N alloy. Following similar studies [²⁸,⁴⁶], only three identical acoustic phonon branches are considered. In previous studies [²⁸], a linear phonon dispersion (Debye model) was used and k_L was anticipated to be less accurate due to the over-predicted k_L contribution by phonons close to the first Brillion zone boundary.

Following the virtual crystal method, an alloy with randomly distributed atoms is viewed as an ordered virtual crystal with the virtual atomic mass \bar{M} and atomic volume V. Here the atom mass is averaged over the atomic mass M for each component, given as \overline{M}=x{{\displaystyle M}}_{\mathrm{I}\mathrm{n}\mathrm{N}}+(\mathrm{1}-x){{\displaystyle M}}_{\mathrm{G}\mathrm{a}\mathrm{N}}, in which x is the molar percentage of InN, and subscripts indicate the component. Similarly, the effective density r is computed as\rho =x{{\displaystyle \rho }}_{\mathrm{I}\mathrm{n}\mathrm{N}}+(\mathrm{1}-x){{\displaystyle \rho }}_{\mathrm{G}\mathrm{a}\mathrm{N}}. The average volume V for each atom is then computed as V=\bar{M}/\rho. This treatment is slightly different from the previous virtual-crystal models, in which the effective lattice constant d = V^1/3 is computed as the linear combination of d for InN and GaN. In practice, V=\bar{M}/\rho and V = d³give results within 1% divergence but V=\bar{M}/\rho is more consistent with the effective \bar{M} calculation. Following the V definition, the equivalent atomic distance a_D is further computed as

(9){{\displaystyle \alpha }}_{\mathrm{D}}={{\displaystyle \left({\displaystyle \frac{\pi N\mathrm{V}}{\mathrm{6}}}\right)}}^{\mathrm{1}/\mathrm{3}},

where N = 4 is the number of atoms per primitive cell in both wurtzite GaN and InN. The sine-shaped phonon dispersion is given as

(10)\omega (k)={{\displaystyle \omega }}_{\max }\sin \left({\displaystyle \frac{\pi k}{\mathrm{2}{{\displaystyle k}}_{\mathrm{0}}}}\right),

in which the maximum wave vector {{\displaystyle k}}_{\mathrm{0}}=\pi /{{\displaystyle \alpha }}_D. The maximum phonon angular frequency ω_max is related to the sound velocity v_s by

(11){{\displaystyle \omega }}_{\max }=\mathrm{2}{{\displaystyle v}}_{\mathrm{s}}/{{\displaystyle \alpha }}_{\mathrm{D}},

where v_s is averaged over v_s for the longitudinal acoustic (LA) and transverse acoustic (TA) phonons as [⁴⁷]

(12){\displaystyle \frac{\mathrm{1}}{{{\displaystyle v}}_{\mathrm{s}}^{\mathrm{2}}}}={\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}\left({\displaystyle \frac{\mathrm{1}}{{{\displaystyle v}}_{\mathrm{s},\mathrm{L}\mathrm{A}}^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{1}}{{{\displaystyle v}}_{\mathrm{s},\mathrm{T}\mathrm{A}}^{\mathrm{2}}}}\right).

Here v_s,_LA and v_s,_TA are first computed as liner combination of the corresponding values for InN and GaN, with the molar percentage as the weight. Then the effective v_s for the virtual crystal is calculated with Eq. (12). Once v_s and a_D are computed, ω_max in Eq. (11) and thus the sine-shape phonon dispersion can be determined.

For bulk phonon MFP{{\displaystyle \Lambda }}_{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k},i}(\omega ), Umklapp (U) scattering and point defect scattering within alloys are considered as resistive scattering processes. The momentum-conserved Normal scattering is not considered here to simplify the analysis, as a common practice in studies using the kinetic relationship in Eq. (2) [⁴⁷–⁴⁹]. For disordered GaN alloys, this argument is also valid because the scattering rates for Normal processes can be much smaller than those for other phonon scattering mechanisms [⁴⁶].

Usually based on the Debye model, the scattering rate for U processes has its common form as

{\displaystyle \frac{\mathrm{1}}{{{\displaystyle \tau }}_U}}\sim {\displaystyle \frac{{{\displaystyle k}}_B{\gamma }^{\mathrm{2}}{V}^{\mathrm{1}/\mathrm{3}}}{\overline{M}{{\displaystyle v}}_s^{\mathrm{3}}}}{\omega }^{\mathrm{2}}T\sim {\displaystyle \frac{{{\displaystyle \alpha }}_D{{\displaystyle k}}_B{\gamma }^{\mathrm{2}}}{\overline{M}{{\displaystyle v}}_s^{\mathrm{3}}}}{\omega }^{\mathrm{2}}T\sim {\displaystyle \frac{{\hslash \gamma }^{\mathrm{2}}}{\overline{M}{{\displaystyle v}}_s^{\mathrm{2}}{{\displaystyle \theta }}_D}}{\omega }^{\mathrm{2}}T

, where θ_D is the Debye temperature [²⁸,⁵⁰–⁵⁴]. However, there usually exists a large factor difference between different {\displaystyle \frac{\mathrm{1}}{{{\displaystyle \tau }}_U}} expressions, as discussed in one early study [⁵¹]. By analyzing the measured thermal conductivity accumulation function for bulk GaN as the x = 0 case, the pre-factor of 2 used in a recent study [⁴⁹] shows good agreements with the experimental data [⁵⁵]. The scattering rates for the U process are thus given as

(13){\displaystyle \frac{\mathrm{1}}{{{\displaystyle \tau }}_U}}=\mathrm{2}{\displaystyle \frac{{ħ\gamma }^{\mathrm{2}}}{\overline{M}{{\displaystyle v}}_s^{\mathrm{2}}{{\displaystyle \theta }}_D}}{\omega }^{\mathrm{2}}T={\displaystyle \frac{\mathrm{2}}{{({\mathrm{6}\pi }^{\mathrm{2}})}^{\mathrm{1}/\mathrm{3}}}}{\displaystyle \frac{{{\displaystyle k}}_B{\gamma }^{\mathrm{2}}{(NV)}^{\mathrm{1}/\mathrm{3}}}{\overline{M}{{\displaystyle v}}_s^{\mathrm{3}}}}{\omega }^{\mathrm{2}}T,

where g is the Grüneisen parameter. Again V and \bar{M}are atomic volume and mass, respectively. The {{\displaystyle \tau }}_U\sim N^{-\mathrm{1}/\mathrm{3}} dependence origins from {{\displaystyle \alpha }}_D={{\displaystyle \left({\displaystyle \frac{\pi NV}{\mathrm{6}}}\right)}}^{\mathrm{1}/\mathrm{3}}\sim N^{\mathrm{1}/\mathrm{3}} and indicates the reduced relaxation time for an increased number N of atoms within a primitive cell [⁵¹], which is not considered by Toberer et al. In estimation, g = 0.5 is used in all analysis, as argued in analysis of Al_xGa_1-xN alloys [⁴⁶]. Assuming U processes are dominant above 300 K, the computed accumulation function for bulk GaN is consistent with the experimental data by Freedman et al. (solid line in Fig. 2). The accumulation function at a length L₀ measures the percentage of k_L contributed by all phonons with MFP less than L₀. It is widely used to justify frequency-dependent phonon MFPs in thermal analysis [⁵⁶]. In addition, the k_L value predicted for bulk GaN is 190 W/(m·K) at 300 K, which is within 20% divergence from 234 W/(m·K) computed by first principles [⁵⁷] and up to 230 W/(m·K) by measurements [⁵⁸]. Such a divergence is reasonable without data fitting specially for GaN.

For nonlinear phonon dispersions, {{\displaystyle v}}_s^{\mathrm{3}} in Eq. (13) is further changed to {{\displaystyle v}}_g{{\displaystyle v}}_p^{\mathrm{2}}, where the group velocity v_g=dω/dk and phase velocity v_p=ω/k [⁴⁹]. However, the detailed derivation is not provided in their study. In practice, this modification leads to large divergence for both k_L accumulation and temperature-dependent k_L(dashed lines in Figs. 2). The computed bulk k_L is reduced to 142 W/(m·K) at 300 K, which is much lower than real values. Therefore, Eq. (13) as the widely used \mathrm{1}/{{\displaystyle \tau }}_Uexpression is still employed in the current analysis.

The scattering rate for point-defect phonon scattering is [⁵⁹]

(14)\mathrm{1}/{{\displaystyle \tau }}_I={\displaystyle \frac{\pi }{\mathrm{6}}}\mathrm{\Gamma }V{\omega }^{\mathrm{2}}{\displaystyle {\sum }_p{{\displaystyle D}}_p}(\omega ),

in which D_p(ω) is the phonon density of states (DOS) at an angular frequency w for branch p, the coefficient G is computed based on the averaged atomic weights for InN and GaN:

(15)\mathrm{\Gamma }=x{{\displaystyle \left({\displaystyle \frac{{{\displaystyle M}}_{\mathrm{I}\mathrm{n}\mathrm{N}}-\overline{M}}{\overline{M}}}\right)}}^{\mathrm{2}}+(\mathrm{1}-x){{\displaystyle \left({\displaystyle \frac{{{\displaystyle M}}_{\mathrm{G}\mathrm{a}\mathrm{N}}-\overline{M}}{\overline{M}}}\right)}}^{\mathrm{2}}.

Equation (10) is more accurate than the widely used τ_I expression [⁶⁰,⁶¹]

(16)\mathrm{1}/{{\displaystyle \tau }}_I={\displaystyle \frac{\mathrm{\Gamma }V}{\mathrm{4}\pi {{\displaystyle v}}_s^{\mathrm{3}}}}{\omega }^{\mathrm{4}},

where is v_s is given by Eq. (12). In the low-frequency limit (w→0), Eq. (14) will degrade to Eq. (16) as the special case of Eq. (14). The selection of Eq. (14) will better account for the point-defect scattering at high phonon frequencies.

The model described above is used to compute the k_L of In_xGa_1−xN alloys and compared with experimental data [¹⁵,²⁸,⁶³]. For TDTR measurements in Tong et al., heat is confined within a shallow layer at the top of the thin film, whose thickness is given as the thermal penetration depth d = \sqrt{k/(C\pi f)} (Fig. 3(a)). In this situation, phonons with MFPs longer than d do not contribute to the measured k value. In bulk materials, tuning f and thus d has been employed to extract the spectral lattice thermal conductivity contributed by phonon with different MFPs [⁵⁵,⁶⁴,⁶⁵]. In the previous model [²⁸], a constant d≈ 200 nm was used and a corresponding boundary scattering rate {\displaystyle \frac{\mathrm{1}}{{{\displaystyle \tau }}_B}}={\displaystyle \frac{{{\displaystyle v}}_g}{\delta }} was added to other scattering rates for phonons in k_L calculations. For alloys films with thickness t>d, this treatment did not fully exclude the contribution from phonons with bulk MFPs longer than d though their effective MFPs can be largely restricted by d. This was the case for most alloy films with 0.2<x<0.8 and a few thick films with x>0.8. To be more accurate, here the estimated d = \sqrt{k/(C\pi f)} in Fig. 3(a) is used to compute the k_L contributed only by phonons with MFPs shorter than d (solid green line in Fig. 3b). Similar phonon MFP spectral analysis can be found in the literature [⁴⁸,⁵⁶].

In general, our new model shows better agreement with experimental data (Fig. 3(b)). In addition to the treatment of d, the improved agreement also benefits from the replacement of linear phonon dispersions [²⁸] to more accurate sine-type phonon dispersions. This reduces the overestimation of k_L contributions by high-frequency phonons with reduced group velocities. The remaining divergence between the predictions and some experimental data can still be attributed to nanometer-scale compositional in homogeneities [²⁸].

Data analysis

Figure 4 presents the measured thermal conductivities of the tri-layered nanoporous film with different periodic patterns (symbols), in comparison to theoretical analysis (solid line). The uncertainties are marked in Fig. 4 to incorporate factors such as the influence of thermal penetration into the sapphire substrate. The 50-nm nucleation and 50-nm buffer GaN layers are not distinguished in the analysis. In principle, the nucleation layer may have more point defects to scatter high-frequency phonons but the overall impact is anticipated to be limited here. In addition, the contribution of electronic thermal conductivity k_E is estimated to be<0.1W/(m·K) for a film with similar compositions at 300 K [⁶⁶]. Therefore, k≈ k_L is assumed in the current analysis.

Based on Eq. (7), L=d\left({\displaystyle \frac{\mathrm{1}}{\phi }}-\mathrm{1}\right) for both pores aligned on hexagonal and square lattices. Therefore, the same k_L–j curve is observed in Fig. 4. In general, the experimental results follow the trend of the theoretical prediction. The effective k remarkably reduces for increased porosities and thus decreased neck width between adjacent pores. For a sample with pores on a square lattice (F\approxZ35%), k for the In_0.1Ga_0.9N layer is extracted as 4.2W/(m∙K) using k = 23.6 W/(m∙K) computed for the underneath double GaN layers. This value is much lower than estimated k={{\displaystyle k}}_{S\mathrm{olid}}(\mathrm{1}-\varphi )\approx8.5 W/(m∙K), where the solid-film {{\displaystyle k}}_{\mathrm{S}\mathrm{olid}}\approx13 W/(m∙K) for a In_0.1Ga_0.9N films with 233 nm thickness [²⁸]. This large k reduction is attributed to classical phonon size effects within the nanoporous film.

In measurements, phonons with MFPs longer than 1000±200 nm contribute to 50% of room-temperature k in bulk GaN [⁵⁵]. In GaN alloys, long-MFP phonons become more important due to strong point-defect scattering to suppress the contribution by high-w phonons with short MFPs. Such effects have been observed in SiGe nanowires [²⁹] and thermal analysis of alloys [²²]. In this situation, sub-micron nanoporous structures can be more effective in reducing k of alloys. Figure 5 compares the accumulated k_L for bulk GaN and In_0.1Ga_0.9N alloys. It can be observed that 50% of room-temperature k in the bulk In_0.1Ga_0.9N alloy is contributed by phonons with MFPs longer than 3 mm. When some nanometer-scale compositional inhomogeneities exist, phonons in the middle-w range can be further scattered [⁶⁷,⁶⁸] but long-MFP phonons still contribute significantly to k_L.

To refine the analysis, the phonon MFPs for pure GaN thin films are changed to the expression obtained by fitting the bulk k values [⁵⁷,⁶⁹,⁷⁰] from 4.2 to 1500 K [⁷¹].The scattering rates for point defects as isotope atoms and U processes are

(17){\displaystyle \frac{\mathrm{1}}{{{\displaystyle \tau }}_I}}=A{\omega }^{\mathrm{4}},

(18){\displaystyle \frac{\mathrm{1}}{{{\displaystyle \tau }}_U}}={{\displaystyle B}}_{\mathrm{1}}{\omega }^{\mathrm{2}}T\exp (-{{\displaystyle B}}_{\mathrm{2}}/T)\approx {{\displaystyle B}}_{\mathrm{1}}{\omega }^{\mathrm{2}}T\exp (-{{\displaystyle \theta }}_D/\mathrm{3}T).

Here Eq. (17) follows Eq. (16). Compared with Eq. (13), an additional exponential factor is added to Eq. (18) [⁷²]. This factor can be neglected at high temperatures because k_L is less sensitive to \exp (-{{\displaystyle B}}_{\mathrm{2}}/T) [⁷³,⁷⁴]. Based on the data fitting, A, B₁, and B₂ are determined as A 5.26 × 10^-45 s³, B₁ = 1.1 × 10^-19 s/K, and B₂ = 200 K, respectively. The fitted B₂ value is consistent with {{\displaystyle \theta }}_D = 600 K for bulk GaN. At both 309 and 415 K, the obtained k accumulation also agrees well with the experimental data by Freedman et al. Despite the variation in the phonon MFPs, the predicted k_L (dashed line in Fig. 4) shows no difference from the earlier prediction and the two curves overlap.

In previous studies on the cross-plane k_L of nanoporous Si films with comparable structure dimensions, it was suggested that phononic effects may play an important role in the k_L reduction [²⁶].The measured low k values cannot be fully explained with diffusive pore-edge scattering of phonons [³⁵]. In contrast, this work clearly shows that diffusive phonon scattering by pore edges alone can explain the measurement results. Our conclusion is thus aligned with the analysis that suggests negligible phononic effects for ~100 nm or larger periodic porous structures [²⁵,³²,³⁵]. The low k found for Si films may be attributed to structural damage in real films, which can be introduced by nanopore drilling with dry etching or other steps in the fabrication process. Such unintentional damage is minimized with direct MOCVD growth of nanoporous films.

Conclusions

In summary, thermal studies of nanoporous GaN-based thin films are carried out. It is found that significant k_L reduction can be achieved with periodic sub-micron pores. For thermoelectric applications, further k_L reduction can be achieved with even smaller nanoporous structures, as long as the structure sizes can still be much larger than the majority electron MFPs to preserve the electrical conductivity. Unrestricted to GaN itself, a high ZT can also be achieved in 2DEG on GaN-related junctions with an extremely high mobility to benefit the power factor. Similar concepts have been attempted in the past but the final ZT is still limited by the high k_L [¹⁰, ⁷⁵]. Along this line, nanopores or other nanostructures may be introduced for ZT enhancement.

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