Incoherent-to-coherent crossover in thermal transport through III–V alloy superlattices

Thermal transport in superlattices (SLs) is not yet fully understood despite numerous experimental and theoretical investigations.^1–8 Given that the thermal management of thermoelectric energy converters, quantum cascade lasers (QCLs), and other optoelectronic devices based on SLs is vital for their optimal performance, it is essential to understand thermal transport in these systems. For example, in a SL system such as the active region of mid-infrared-emitting QCLs grown on InP, which consists of hundreds of layers of alternating ternary materials InGaAs and InAlAs, thermal transport will depend on multiple parameters such as, periodicity, total sample thickness, interface properties, layer composition, strain, and temperature. The problem is complex because thermal transport is inherently broadband, with phonons of vastly different wavelengths (thus interacting strongly with largely different spatial scales of disorder) contributing to thermal conductivity. On the one end are short-wavelength phonons, whose transport is largely incoherent and well described by a particle-like picture and the semiclassical Boltzmann equation. The SL period is larger than the mean free path for intralayer scattering due to alloy disorder and three-phonon scattering, and scattering from random rough interfaces is a phase-breaking process. On the other end are long-wavelength phonons, capable of traveling across multiple periods without undergoing a phase-breaking process, and their transport is largely coherent. The Simkin–Mahan⁹ model for thermal transport in SLs, which notably does not incorporate diffuse interface scattering, predicts a minimum in the thermal conductivity as a function of period. This minimum indicates a crossover between predominantly incoherent transport at larger periods (lower interface densities, ID) and coherent transport at smaller periods (higher IDs). Various experimental and computational studies^5,6,10–13 have shown that the presence of a minimum thermal conductivity in SLs depends on interface quality; when interfaces are rough, the minimum in the thermal conductivity may not be observed. However, first principles-calculations by Garg and Chen¹⁴ showed that a minimum in the thermal conductivity might be reached even with the presence of roughness at the interfaces. Indeed, experimental measurements of the thermal conductivity using time-domain thermoreflectance (TDTR) on high-quality perovskite SLs of SrTiO₃/CaTiO₃ and SrTiO₃/BaTiO₃ with atomically sharp interfaces showed a decrease and subsequent increase in thermal conductivity as a function of ID,¹⁵ which is evidence of a crossover from incoherent to coherent phonon transport in high-quality perovskite SLs. A minimum in the thermal conductivity as a function of the period thickness was also observed in a lattice-matched TiN/(Al,Sc)N metal/semiconductor SL¹⁶ and a TiNiSn/HfNiSn half-Heusler SL.¹⁷ 

With regard to semiconductor SLs, numerous experimental and computational papers have reported on the thermal conductivity of group-IV SLs, whereas the work on III–Vs has largely focused on the SLs between two binaries.^{2–6,18–21} In contrast, there have been few reports on the thermal conductivity of III–V alloy/alloy SLs, especially for the InGaAs/InAlAs material system, which is of great importance in optoelectronics.²² Sood et al.²³ studied cross-plane thermal conduction using TDTR in lattice-matched In_0.53Ga_0.47As/In_0.52Al_0.48As SLs and investigated the dependence of thermal conductivity on the total period thickness and the layer-to-period thickness ratio. They found that the effective thermal conductivity increased monotonically with increasing fraction of In_0.53Ga_0.47As and decreased with increasing fractions of In_0.52Al_0.48As, as expected based on the thermal conductivity values of the bulk alloys.²⁴ When the thickness ratio was fixed, the conductivity did not change appreciably with increasing period thickness. These trends were in good agreement with a semiclassical Boltzmann transport model that employed specular interface conditions, pointing to the interfaces of high quality and low interface thermal resistance. Mei and Knezevic²⁵ presented a semiclassical model describing the full thermal conductivity tensor of III–V compound SL structures and applied it to III-arsenide systems. The thermal-conductivity calculation for SLs involved the conductivity of each layer, the effect of partially diffuse interface scattering, and thermal boundary resistance. This model agreed with the in-plane and cross-plane thermal conductivity of III-arsenide SLs from several experimental studies. Jaffe et al.²⁴ studied the effect of strain on the thermal resistivity of InGaAs/InAlAs SLs. The resistivity of the strain-balanced SLs as a function of the layer thickness deviated from the series resistance model at approximately 4-nm layer thickness and flattened out at smaller layer thicknesses (i.e., larger interface densities). This flattening of the thermal conductivity vs interface density called into question the limits of validity of the purely semiclassical transport model for phonon dynamics in SLs. Given the existence of a minimum in the thermal conductivity vs interface density for perovskites, metal/semiconductor, and half-Heusler SLs, a natural question arises whether such a crossover between incoherent and coherent transport is also present in III–V alloy/alloy SLs and how robust its experimental signatures might be as the temperature increases. This question is of both fundamental importance for understanding the influence of interface and alloy disorder on phonon dynamics in SLs, and of practical interest for optimizing the heat flow and device performance of quantum cascade lasers and other heterojunction-based emitters, detectors, and sensors.

In this Letter, we report on the TDTR measurement of the thermal conductivity as a function of ID (approximately 0.0374–2.19 nm⁻¹) at different temperatures (80, 135, and 295 K) and the thermal conductivity as a function of temperature of strain-balanced In_0.63Ga_0.37As/In_0.37Al_0.63As SLs relevant to QCLs. In addition, we used a three-dimensional finite difference time-domain (FDTD) solver for the elastic wave equation to calculate the thermal conductivity of In_0.63Ga_0.37As/In_0.37Al_0.63As SLs. In these simulations, we varied the interface roughness and correlation length at the heterointerfaces within the SLs to investigate the sensitivity of the model and the data to interfacial mixing. Machine learning was used to determine the interface roughness and correlation length. The FDTD simulations are in good agreement with measurements below 295 K. We observe a minimum in the thermal conductivity as a function of ID for In₀.₆₃Ga_0.37As/In_0.37Al_0.63As SLs, which is evidence of an incoherent-to-coherent crossover in phonon transport in alloy/alloy SLs and also corroborated via FDTD modeling.

We employed TDTR to measure the thermal conductivity of nine In_0.63Ga_0.37As/In_0.37Al_0.63As SL samples. TDTR is a noncontact optical pump–probe technique mainly used to measure the thermal conductivity^26–29 and thermal conductance of solid/solid and solid/liquid interfaces.^26,29–33 A schematic of the experimental setup is presented in Fig. 1. A train of <120 fs pulses at a repetition rate of 76 MHz was generated by a FLINT femtosecond oscillator with a wavelength centered on 1030 nm. A polarizing beam splitter is used to split the beam into a pump and probe path. The pump beam passed through a piece of bismuth triborate crystal (BIBO;Newlight), in which frequency doubled the pump beam to 515 nm. After passing through the BIBO crystal, an electro-optic modulator (Thorlabs) modulated the pump beam at a frequency of 7.0 MHz, which served as a reference for lock-in detection. The sample was thermally excited by the modulated pump beam pulses. The probe beam was delayed relative to the pump beam via a linear stage (Newport). The probe beam was focused on the sample surface after it passed through a 10× objective lens (Mitutoyo). A photodiode sensor (Thorlabs) is used to detect changes in the intensity of the reflected probe beam. The small signal was detected from the background noise using a radio frequency lock-in amplifier (Zurich UHFLI). The data acquired by the lock-in amplifier were related to surface temperature changes in the transducer at various pump–probe delay times. A thermal model^34–36 was used to determine the thermal properties of the material underneath the transducer, which are linked to temperature changes. The probe and pump radii were approximately 5 and 7 μm, respectively. We used a cryostat (Oxford Instruments MicrostatN) to control the sample temperature. The cross-plane thermal conductivity measurements of the SLs were obtained at 80, 135, and 295 K and an ID from 0.0374 to 2.19 nm⁻¹. The cross-plane thermal conductivity was the average of four measurements at each temperature, and the experimental uncertainty was calculated by taking into account the aluminum (Al) film thickness (used as a transducer) and the standard deviation of the four measurements.

The In_0.63Ga_0.37As/In_0.37Al_0.63As SL samples under study were grown by low-pressure metalorganic vapor phase epitaxy (MOVPE) in a close-coupled showerhead reactor. Trimethylgallium (TMGa), trimethylindium (TMIn), trimethyaluminum (TMAl), and arsine (AsH₃) were used as the precursors. The growth temperature set point was 615 °C, and all reactor chamber conditions were constant across all SL sample growths. X-ray diffraction measurements were used to quantify the SL compositions provided in Table I but do not provide information on the interfacial properties. Figure 2 depicts scanning transmission electron microscope (STEM) images of SLs with IDs of 0.151 and 0.601 nm⁻¹. The images reveal that good quality interfaces were formed for the SL with an ID of 0.151 nm⁻¹ and show intermixing at the interfaces for the SL with an ID of 0.601 nm⁻¹.

Table I lists the characteristics of each strain-balanced SL such as layer thickness, indium content, and ID. The total thickness of the SLs was kept constant at approximately 212 nm, because scattering from boundary edges (and, thus, sample size) affects thermal conductivity. The ID values ranged from 0.0374 to 2.19 nm⁻¹. The thickness of the Al film for samples 1–3 is 79 nm, the thickness of the Al film for samples 4–7 is 71 nm, and the thickness of the Al film for samples 8 and 9 is 78 nm.

Figure 3 illustrates the measured and calculated thermal conductivity values in two of the SLs (4, with ID = 0.0755 nm⁻¹, and 7, with ID = 0.601 nm⁻¹) as a function of temperature in the range 80–295 K. The error bars represent the uncertainty calculated due to the Al film thickness and the standard deviation of the four measurements. The experimental thermal conductivity values vary from 0.9–1.14 W m⁻¹ K⁻¹ at 80 K to 1.69–1.86 W m⁻¹ K⁻¹ at 295 K. In all samples, the thermal conductivity increases with increasing temperature as vibrational modes become energetically accessible, eventually saturating above approximately 295 K. The fact that the thermal conductivity approximately saturates above 295 K indicates that temperature-independent mechanisms, such as interface roughness and alloy scattering, dominate instead of three-phonon Umklapp scattering that dominates in bulk materials. The TDTR raw data and information pertaining to the quality of the model used to fit the data can be found in the supplementary material.

Thermal transport in alloy SLs is simulated by solving the elastic wave equation using the FDTD technique in the velocity-stress formulation, which we developed earlier.³⁷ FDTD is a grid-based technique, and every grid point in the alloy simulation is assigned the attributes (the Lamé parameters and the mass density) of the constituent binary materials at random, according to the alloy composition. The Lamé parameters and mass-density values of the relevant compounds are presented in the supplementary material, Table S1. We use a grid-cell size of 1 Å to achieve high simulation accuracy. The simulation domain has a width of 200 nm (2000 grid cells), a thickness of 212 nm (2120 grid cells), and a length of 1 μm (10 000 grid cells), which closely mimics the dimensions of the fabricated structures. We use a time step of 1.85 fs, which is 10% of the largest allowed time step based on the FDTD Courant stability condition for our selected materials. Interface roughness is quantified by assuming a Gaussian correlation function with a given root mean square (rms) roughness and correlation length. The surface characteristics observed in experiments on InGaAs grown for QCL devices^38,39 yield an rms roughness of approximately 0.2 nm and the correlation length close to 9 nm. To study the effect of these parameters on thermal conductivity, we performed FDTD calculations with several rms roughness values (0.1, 0.2, and 0.5 nm) and correlation lengths (5, 9, and 15 nm), and all of which are close to experimental observations.^38,39 We then used these FDTD calculations as training data for a neural-network-based machine-learning algorithm in order to identify the optimal values that would give best agreement between measurements and calculations. The optimal values identified by the machine-learning algorithm are 0.4 nm for the rms roughness and 4 nm for the correlation length. (Agreement between experiment and simulation will be discussed below, see Fig. 4.)

To calculate thermal conductivity using FDTD, we employed the Green–Kubo formula, where the two-time correlation function of the local heat-current vector Q is computed using FDTD. (⁠$Q$ is the product of kinetic energy and velocity, computed at every grid point, and has units of $W m$⁠. If the values of $Q$ were summed up over the whole simulation domain, then divided by volume, the result would be the familiar heat-flux, with units $W/m2$⁠.) We initialize the velocity field in the system according to a given temperature. Once equilibrium is reached, typically after 100 000 timesteps, the simulation runs for an additional 900 000 steps, during which we calculate $Q$ at every time step. We calculate the two-time correlation function by summing up the product of the cross-plane component of $Q$ at time zero and at time t over the entire simulation domain. Finally, we use the Green–Kubo formula to calculate the cross-plane thermal conductivity κ as

where $kB$ is the Boltzmann constant, T is the system temperature, $Ω$ is the system volume, $Qz$ is the cross-plane component of the heat-current vector, and $⟨Qz(0)Qz(t)⟩$ is the two-time correlation function. It should be noted that the FDTD simulation inherently does not incorporate multiphonon processes, because the underlying elastic wave equation is linear. What the FDTD simulation does well is capturing the wave nature of linear elastic waves with all the different wavelengths present at a given temperature as they interact with various types of static disorder such as alloy and interface roughness. The fact that the FDTD simulation (which, again, does not include multiphonon scattering, but accurately accounts for interface and alloy scattering) agrees so well with experiments over a wide temperature range is indirect proof that the interface and alloy scattering processes remain dominant up to high temperatures.

Figure 4(a) illustrates the thermal conductivity as a function of ID at 80, 135, and 295 K. There is a global minimum for κ as a function of ID. This minimum κ that prevails up to room temperature represents a crossover from incoherent to coherent transport of thermal carriers moving in the cross-plane direction, as has been shown within the recent decade in non-alloy SL systems (SrTiO₃/CaTiO₃) as well.¹⁵ Prior modeling work treating diffuse scattering at rough interfaces as a phase-breaking (coherence-destroying) process^9,40,41 predicted this dip at the point of crossover in SLs. Namely, when the layers are thick and the ID is lower than the ID of the minimum κ, heat carriers scatter at the rough but individually resolved interfaces, a process that randomizes the phase of the scattered waves and results in largely incoherent thermal transport. In contrast, when the layers are thin and the ID is higher than the ID of the minimum κ, heat carriers perceive the SL structure as a new, nearly uniform alloy material through which thermal transport is coherent. A typical heat carrier (whose wavelength increases with decreasing temperature and is in the 1–100 nm range for the temperatures relevant here) “averages” over all the layers contained within its wavelength, so it perceives a high-ID SL as an effective-medium alloy^9,14,40,41 and moves through it without phase breaking.

Our data in Fig. 4(a) support the idea of the incoherent-to-coherent crossover: as the ID increases from ∼0.0374 to 0.601 nm⁻¹, the thermal conductivity deviates from the thermal conductivity value measured in a quaternary alloy (denoted in dashed lines and representing the control case of no interfaces). This deviation captures the additional thermal-conductivity reduction caused by interface scattering, which goes beyond the scattering within the alloy layers. As the ID crosses ∼0.601 nm⁻¹ and approaches the maximum of 2.19 nm⁻¹, the thermal conductivity comes back up and recovers to the same magnitude from the quaternary alloy, consistent with the idea that when the thicknesses of the layers are on the order of the unit-cell size, the SL is effectively a homogeneous alloy. Similar behavior was observed in another material system: the first-principles model by Garg and Chen¹⁴ of Si/Ge SLs showed a minimum in thermal conductivity as a function of period. For short periods (1.1 nm), the calculated thermal conductivity of the SL was in excellent agreement with the measured value for a Si_0.5Ge_0.5 alloy. At this limit, the interfacial disorder permeates the entire SL making the structure essentially an alloy.¹⁴ 

In Fig. 4(b), we compare the experimental data to the FDTD modeling results for thermal conductivity vs ID at 80 K. The FDTD model using an interface roughness of 0.4 nm and a correlation length of 4 nm is in excellent agreement with experimental data. The optimal interface roughness and correlation length were determined using machine learning (a neural-network analysis). In real SLs, there is some atomic-scale intermixing near each interface. In our simulation, the combined effect of a smaller rms roughness obtained from experiment³⁹ and the alloy intermixing near the interfaces is accurately captured through a somewhat larger value of the effective rms roughness. When the ID increases, the SL thermal conductivity approaches the thermal conductivity of a quaternary alloy. [The resemblance of a high-ID structure to an alloy can be gleaned from the right-hand-side inset to Fig. 4(b), which depicts one such simulated structure.]

The fact that the simulation data (which does not include temperature-dependent multiphonon processes but accurately captures alloy and interface-roughness scattering) agree so well with the experimental data implies that the mechanisms captured by the simulation (alloy and interface scattering) indeed dominate thermal transport in SLs, whereas multiphonon processes are a relatively minor contributor. This conclusion—that alloy and interface scattering dominate over three-phonon scattering in alloy SLs—holds up to room temperature and is supported by the data depicted in Fig. 4(a). Namely, with increasing temperature, the curves shift up, toward higher thermal conductivities, which is consistent with increasing heat capacity (i.e., an increasing number of phonons participating in thermal transport). The minima are also expected to become shallower as the temperature increases,^9,14,40,41 which is attributed to the rapidly increasing importance of the temperature-sensitive three-phonon Umklapp scattering.^9,14,40,41 However, our experimental data do not show a flattening of the thermal-conductivity minimum all the way up to room temperature, which implies that the temperature-insensitive alloy and interface-roughness scattering in alloy SLs continue to overshadow multiphonon scattering over our entire temperature range.

In summary, we measured (using TDTR) and modeled (via FDTD) the thermal conductivity of III–V alloy SLs at different temperatures and with different IDs. We showed that alloy and interface scattering remain the dominant scattering mechanisms over Umklapp scattering up to room temperature. We observed a minimum in the thermal conductivity as a function of ID of In_0.63Ga_0.37As/In_0.37Al_0.63As SLs, which is indicative of an incoherent-to-coherent crossover as the ID increases and the structures move from well-resolved interfaces to an effective homogeneous alloy medium. This minimum in thermal conductivity persists as the temperature increases, indicative of the continued dominance of the temperature-independent interface and alloy-disorder scattering over the temperature-dependent three-phonon scattering in thermal transport through III–V alloy superlattices. Experimental data and modeling at 80 K are in excellent agreement using an interface roughness of 0.4 nm and a correlation length of 4 nm over the entire ID range. Overall, this work contributes to further understanding of the nature of thermal transport in SLs, particularly the relative importance of disorder (alloy and interface roughness) vs multiphonon processes in thermal transport through SLs, which has clear repercussions for the thermal management of electronic and optoelectronic devices.

See the supplementary material for raw data and fitting result for some samples measured by TDTR and parameters used in FDTD simulations.

The experimental work (B.M.F., L.J.M., and V.G.) was supported by AFOSR under Grant No. FA9550-19-1-0385. The theoretical work (I.K.) was supported by AFOSR under Grant Nos. FA9550-18-1-0340 and FA9550-22-1-0407.