Thermal conductivity of Sc_xAl_1−xN and Y_xAl_1−xN alloys Open Access

Alloys of AlN with transition metals Sc and Y have recently attracted significant research interest as new materials for high-frequency (HF) electronic and acoustic devices.^1,2 Due to the large deviation from the ideal wurtzite crystal lattice in Sc_xAl_1−xN and Y_xAl_1−xN (x  $≤$ 0.5), their spontaneous polarization, $P S P$(x), and piezoelectric coefficient, $e 33$(x), increase as compared to the respective values for AlN. The enhancement of the piezoelectric response in Sc_xAl_1−xN and Y_xAl_1−xN alloys has been demonstrated both theoretically^2–4 and experimentally.^5,6 An increase in spontaneous polarization has also been predicted.^2,7,8

The large piezoelectric coefficient in Sc_xAl_1−xN and Y_xAl_1−xN results in a high electromechanical coupling,^9,10 which enabled the development of acoustic resonators for fifth-generation (5G) mobile communication technology. Sc_xAl_1−xN-based surface acoustic wave (SAW) resonators^11,12 and film bulk acoustic resonators (FBARs)¹³ with a high-quality factor and operating up to 3.5 GHz have been demonstrated.

Due to the enhanced spontaneous and piezoelectric polarizations, a higher two-dimensional electron gas (2DEG) density can be achieved in (Sc,Y) $x$Al_1−xN/GaN high mobility transistor (HEMT) heterostructures compared to the common Al_xGa_1−xN/GaN HEMTs.¹⁴ Furthermore, the power figure-of-merit of Sc_0.18Al_0.82N (lattice matched to GaN) has been estimated to be 5–7 times higher than that of GaN and Al_xGa_1−xN (x  $≤$ 0.3),¹⁵ which is promising for implementation in high-power (HP) devices. Recently, lattice-matched Sc_0.07Al_0.38Ga_0.55N/GaN HEMT with an output power of 5.77 W/mm (at a drain bias of 20 V) and a cutoff frequency of >70 GHz has been demonstrated.¹⁶ 

Efficient heat dissipation is crucial for HF device performance. For example, overheating in SAW resonators and FBARs constrains the maximum transmission rate¹⁷ and decreases the quality factor.^18,19 In the case of HEMTs operating at high currents and high voltages, Joule heat generation becomes a severe problem for the device's performance and reliability. The self-heating leads to an increase in junction temperature, which not only reduces the electron mobility and saturation velocity but also causes thermal degradation.^20–22 Heat dissipation in device structures depends mainly on the thermal conductivity of the constituent materials. Therefore, thermal conductivity is a key parameter for the design and optimization of high-performance devices.

Despite the strong interest in (Sc,Y) $x$Al_1−xN alloys, their thermal properties are not well studied. There are only two reports on the thermal conductivity of Sc_xAl_1−xN.^23,24 No experimental data for thermal conductivity of Y_xAl_1−xN alloys are available.

In this work, we experimentally determine the thermal conductivity of Sc_xAl_1−xN and Y_xAl_1−xN layers with composition in the range 0  $≤$x  $≤ 0.22$⁠. Experimental data are measured at variable temperatures from 100 to 400 K and analyzed by a modified Callaway's model. The dominant phonon-scattering processes that contribute to the lattice thermal conductivity are identified and discussed.

Wurtzite Sc_xAl_1−xN and Y_xAl_1−xN layers with (0001) crystal orientation were grown by reactive direct current (DC) magnetron sputter epitaxy on Si(001) and sapphire substrates. Details of the growth method and layer properties can be found elsewhere.^25–28 Alloy compositions were determined by time of flight energy elastic recoil detection analysis (ToF-EERDA). The dislocation density was determined by high-resolution x-ray diffraction (HR-XRD) using Empyrean diffractometer (PANalytical). Rocking curves (⁠ $ω$-scans) were measured using a hybrid monochromator with 2-bounce Ge(220) as incident optics and a parallel plate collimator 0.27° on the detection side. The density of screw- and edge-type dislocations (⁠ $D S$ and $D E$⁠) was determined from HR-XRD rocking curves of symmetric and asymmetric reflections, respectively,²⁹ using $D i$ = $α i 2 / 4.35 b i 2$ (i = S, E), where $α S$ and $α E$ are the tilt and twist angles, respectively, and $b S$ and $b E$ are the Burger vectors for screw and edge dislocations, respectively. The total dislocation density in the studied layers was found to be about $1 × 10 11$ cm⁻². The thickness of the layers was determined from ultraviolet-visible spectroscopic ellipsometry to be 300 nm for all samples.

Thermal conductivity measurements were performed using transient thermoreflectance (TTR). A detailed description of this technique has been given in our recent publications.^30–32 In the measurements, transients of the reflectance from a 200 nm thick Au transducer layer deposited on the sample surface are acquired at different pump laser powers. The spot size of the probe laser is chosen to be much smaller than that of the pump laser, ensuring that only the heat transport along the direction perpendicular to the sample surface is probed. The transients are fitted by least-square minimization using one-dimensional heat transport equations with two fitting variables—thermal conductivity (k) of the layer and thermal resistance at the Au/layer interface. In the fitting procedure, the thickness and specific heat capacity (⁠ $C p$⁠) of Au transducer, layers, and substrates, and the thermal conductivity of Au and substrates are used as input parameters. The thermal conductivity of Si and sapphire substrates was initially measured. For the specific heat capacity of Au, Si, and sapphire and the thermal conductivity of Au, the available literature data are used.^33–35 Due to the lack of data for the specific heat capacity of Sc_xAl_1−xN and Y_xAl_1−xN alloys, the value for the AlN³⁶ is used. This is a reasonable assumption having in mind that the layers under investigation are with low Y and Sc compositions (x $≤$ 0.22). We also note that the sensitivity of the measured thermal conductivity vs the specific heat capacity is found to be very small, $S C p k$ = $δ k / δ$ C_p  $≈ 2 ×$ 10⁻⁸ m²/s. Therefore, the assumption for the specific heat capacity in alloys gives a minor effect on the fitted thermal conductivity.

In order to calculate the phonon dispersion, the alloys are modeled by ordered structures such that one of the metal ions in wurtzite AlN is replaced by Sc or Y. Such a simplified approach allowed us to obtain approximated phonon dispersion and to extract parameters needed for thermal conductivity calculations. The electronic structure calculations here were done using density functional theory implemented into the Quantum Espresso package⁴³ with local density functional approximation (LDA) and 70 Ry energy-cutoff.

Sc_xAl_1−xN and Y_xAl_1−xN alloys crystallize in wurtzite (wz) or rock salt (rs) structure depending on the composition x. The transition from wz phase to rs phase occurs at composition of x = 0.55 for Sc_xAl_1−xN²⁵ and x = 0.75 for Y_xAl_1−xN.²⁶ Here, we examine only wurtzite alloys with x  $≤ 0.5$⁠. The calculated structural parameters for the relaxed crystal lattice of wz-AlN, wz-Sc_0.5Al_0.5N, and wz-Y_0.5Al_0.5N are listed in Table I together with the available literature data. Note that our results are in good agreement with the generalized gradient approximation (GGA) calculations. The phonon dispersion was calculated utilizing the finite displacement method implemented into Phonopy⁵¹ using (3 × 3 × 3) supercells of the model structures and (6 × 6 × 6) Monkhorst–Pack k-point mesh. Figure 1 shows the phonon dispersion in wz-AlN and ordered wz-Y_0.5Al_0.5N and wz-Sc_0.5Al_0.5N.

From the calculated phonon dispersion, $v s$⁠, $θ D s$⁠, and $γ s$ parameters are extracted. We limit our consideration to the phonon dispersion in the $Γ$-A direction of the Brillouin zone, where TA1 and TA2 modes are degenerated, since k is measured only along the [0001] (c-axis). The values of $θ D s$ for TA and LA modes are obtained from the phonon frequency at A-point of the Brillouin zone. Recently, we have shown that mainly the low-frequency phonons (i.e., those with a long mean-free-path) contribute to the lattice thermal conductivity of semiconductor alloys.³¹ Then, the sound velocities are determined from the phonon dispersion by $v s = δ ω s ( q ) δ q | q → 0$⁠. The phonon-mode Grüneisen parameters as a function of the phonon frequency are calculated using Eq. (2). Since the scattering time for all resistive processes depends on the average value $⟨ γ s 2 ⟩$⁠, the mode-averaged Grüneisen parameters, $⟨ γ s 2 ⟩$⁠, are evaluated and used.³⁸ The obtained $θ D s$⁠, $v s$⁠, and $⟨ γ s 2 ⟩$ for wz-AlN, wz-Y_0.5Al_0.5N, and wz-Sc_0.5Al_0.5N are listed in Table II. In the modeling of thermal conductivity, these material parameters are assumed to vary linearly between x = 0 and x = 0.5.

Figure 2 shows the experimentally determined room-temperature thermal conductivity of all studied Sc_xAl_1−xN and Y_xAl_1−xN layers. For comparison, experimental data reported for 160-nm-thick Sc_xAl_1−xN layers grown by reactive sputtering²³ and 400-nm-thick Sc_xAl_1−xN layers grown by plasma-enhanced molecular beam epitaxy²⁴ are also provided. It is seen that the thermal conductivity decreases rapidly with increasing Sc(Y) composition. Similar behavior has been observed in other semiconductor alloys.^32,53–55 The factor driving the thermal conductivity variation with alloy composition is the A-scattering, which dominates the heat transport in the alloys. The thermal conductivity of Y_xAl_1−xN is found to be lower than that of Sc_xAl_1−xN for all compositions studied. We also note that for both alloys, almost the same values of the thermal conductivity of layers grown on Si(001) and sapphire substrates are measured.

In Fig. 2, the composition dependence of the thermal conductivity of Sc_xAl_1−xN and Y_xAl_1−xN calculated for 300 nm thick layers at 300 K using the model described by Eq. (1) is also shown. A very good agreement between the experimental data and the model is observed. The best-match between the experimental and calculated data is achieved for dislocation density of $5 × 10 10$ cm⁻², which is two times smaller than that estimated from XRD. Note that the layers grown by magnetron spattering epitaxy usually contain columnar microstructures oriented along the [0001] direction.^25,26,56,57 The XRD broadening is, therefore, largely determined by the lattice deviation between the domains, which is likely to overestimate the dislocation density. This also means that the phonon-grain scattering is somehow convoluted in the phonon-dislocation scattering. We do not aim to separate these two processes in the scope of this study. The lower thermal conductivity of Y_xAl_1−xN alloys is due to the stronger A-scattering as a result of the larger mass and size difference between Y and Al atoms as compared to that between Sc and Al atoms.

Figure 3 shows the thermal conductivity of Sc_xAl_1−xN and Y_xAl_1−xN layers with different compositions measured in the temperature range of 100–400 K. It is found that k increases linearly up to 250 K and then almost saturates. A similar temperature dependence of the thermal conductivity has been reported for Al_xGa_1−xN layers.⁵³ As described later, such a behavior can be explained by the interplay between the contributions of the U- and B-scattering.

In order to get insight into the effect of different phonon-scattering mechanisms on the thermal conductivity, we consider the calculated scattering time for the main resistive processes as a function of the phonon frequency for the LA mode of Y_0.22Al_0.78N [Fig. 4(a)]. It is seen that A-scattering prevails in the whole phonon frequency range. We found that the U-scattering time is significantly longer compared to other phonon-scattering time. Hence, the U-scattering has a weak effect on the thermal conductivity of Y_xAl_1−xN alloys (as well as for Sc_xAl_1−xN). The B-scattering dominates only in the low phonon frequency region (i.e., below 1 THz). The same is found for the D-scattering time. The D-scattering could become more significant only for lower Y and Sc compositions and higher dislocation densities.

The partial thermal conductivity of Y_0.22Al_0.78N considering the A-scattering coupled with one of the other resistive scattering processes is presented in Fig. 4(b). The combined (A + U) phonon-scattering contribution shows unusual temperature dependence, i.e., higher thermal conductivity is obtained at lower temperatures. This is attributed to the high contribution of the U-scattering process, particularly at frequencies lower than about 0.5 THz [Fig. 4(a)]. In this frequency range, the U- and A-scattering times are of equal magnitude. The calculations for (A + D) and (A + B) phonon-scattering combinations give smaller and almost constant thermal conductivity within the studied temperature range. It is worth noting that A-, B-, and D-scattering times are temperature independent, and, hence, the temperature variation of the thermal conductivity related to these processes is driven only by the phonon temperature distribution. From Fig. 4(b), it is seen that for temperatures above 250 K, the combined (A + B) phonon-scattering contribution matches very well with the calculated total thermal conductivity as well as the experimental data. The difference between the measured and calculated thermal conductivity for temperatures below 250 K is probably due to an underestimation of the effect of B-scattering in the model. For thin layers, the B-scattering obviously plays a significant role even at elevated temperatures and interferes with the U-scattering.

In this Letter, we experimentally determine the thermal conductivity of Sc_xAl_1−xN and Y_xAl_1−xN (0  $≤$ x  $≤ 0.22$⁠) layers grown by reactive magnetron sputtering epitaxy. For both alloys, the thermal conductivity decreases rapidly with increasing Sc and Y contents as a result of the enhanced phonon-alloy scattering. The thermal conductivity of Y_xAl_1−xN is found to be smaller than that of Sc_xAl_1−xN for all compositions. This is explained by a stronger phonon-alloy scattering in Y_xAl_1−xN stemming from the larger mass and size differences between Y and Al atoms as compared to those between Sc and Al atoms. It is also found that the thermal conductivity of both alloys increases with temperature and saturates at 250 K. All experimental data are analyzed by the modified Callaway's model based on the RTA solution of the phonon BTE. The results from this study will be useful for thermal design and optimization of acoustic and electronic devices based on Sc_xAl_1−xN and Y_xAl_1−xN alloys.

This work was performed within the competence centers for III-Nitride Technology (C3NiT-Janzén) supported by the Swedish Governmental Agency for innovation systems (VINOVA) under the Competence Center Program Grant No. 2022-03139. We further acknowledge support from the Swedish Research Council VR under Grant No. 2022-04812, the Swedish Foundation for Strategic Research under Grant No. EM16-0024, and the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University, Faculty Grant SFO Mat LiU Nos. CBET-1336464 and DMR-1506159. The first-principles computations were enabled by resources provided by the Swedish National Infrastructure of Computing (SNIC).

The authors have no conflicts to disclose.

Dat Q. Tran: Conceptualization (equal); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (lead); Writing – review & editing (equal). Ferenc Tasnadi: Methodology (supporting); Software (supporting); Validation (equal); Writing – review & editing (supporting). Agnė Žukauskaitė: Resources (equal); Validation (equal). Jens Birch: Resources (equal); Validation (equal). Vanya Darakchieva: Conceptualization (equal); Funding acquisition (lead); Methodology (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Plamen P. Paskov: Conceptualization (equal); Methodology (equal); Supervision (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.