The role of pressure on lattice thermal conductivity and its related thermodynamical parameters in In_0.53Ga_0.47As nanofilms

Abstract

Lattice thermal conductivity (LTC) for In_0.53Ga_0.47As alloy films, with thicknesses ranging from 10 nm to 1.4 μm, was investigated under pressures of up to 11 GPa and temperatures between 1 and 450 K, utilizing the modified Debye–Callaway model. The effects of structural and thermodynamical parameters, as well as phonon interactions, on LTC were examined. The Clapeyron, Murnaghan, and Post equations were applied to determine the pressure dependence of the melting temperature, lattice volume, and Debye temperature, respectively. A novel derivative form of the bulk modulus, suitable for nanomaterials, has been introduced. It was found that decreasing the film thickness increases the Gruneisen parameter, while increasing pressure decreases it. The LTC of nanofilms is significantly affected by their thickness and pressure strength; notably, under 11 GPa, films with a thickness of 10 nm exhibit a substantial decrease in LTC. LTC_max declines due to the greater influence of boundary scattering compared to dislocations. These findings suggest potential applications in managing nanofilm temperature and size, which are key to advancing nanomaterials and enhancing the efficiency of electronic devices.

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1. Introduction

In recent decades, researchers have focused on the thermal conductivity of nanostructured materials^1,2 due to its direct relation to the efficiency of thermal dissipation. Accurate calculation of lattice thermal conductivity is crucial for designing high-quality nano-electronic³ power devices,^4,5 generating transistors,⁶ and thermoelectric generators. Hydrostatic pressure serves as a principal thermodynamic variable used to accelerate reactions, design novel structures, and define the properties of nanostructures.^7,8 Pressure is a significant factor in semiconductor behavior, meriting further study.⁹ The alloy In_0.53Ga_0.47As, formed from InAs and GaAs, has a zinc blende structure with a lattice parameter a(∞) of 5.868 Ź⁰ and a direct band gap, making it suitable for high-performance photodiodes.¹¹ Its nanofilms are versatile, and used in devices for their radiation resistance and durability.¹² The alloy's applications now include lasers,¹³ photodetectors,¹⁴ transistors,^5,15 and photovoltaics,^5,15 with transistors being prevalent in modern electronics like smartphones and computers.

Pressure's impact on thermal conductivity¹⁶ is key for understanding phase transitions and material properties, and is studied both theoretically and experimentally. The process is vital in fields such as astrophysics and geosciences. Many authors have predicted theoretical calculations of the pressure effect on size-dependent LTC in InAs nanowires, Si nanofilms, and multilayer hexagonal boron nitride.^17–19 Theoretical models predict how pressure affects the lattice thermal conductivity (LTC) in various nanostructures, aiding in the design and analysis of their physical and thermal properties.⁷ For In_0.53Ga_0.47As nanofilms, such analysis is crucial as it allows for precise LTC calculations under pressure. However, for validating these results, experimental data exist only for the zero pressure LTC of In_0.53Ga_0.47As films (10 nm–1.4 μm),^5,20 making these findings useful for overcoming experimental challenges in other pressure ranges of up to 11 GPa, where the challenges could range from technical and financial constraints. The Clapeyron model, utilized in this work, is preferred due to its broad application in analyzing phase transitions, such as melting, because it describes how pressure affects the melting temperature. It is now also applicable at the nanoscale.^17–19 In contrast, other used models, e.g. the DNA melting model, are specifically designed for DNA's thermal denaturation, considering nucleotide sequences and hydrogen bonds. The Clapeyron model predicts shifts in melting temperature with changes in pressure along phase boundaries, while the DNA model's temperature is determined by the genetic composition and stability of the DNA strands.^21–24

In this work, the size-dependent LTC for In_0.53Ga_0.47As nanofilms will be determined by using the modified Debye–Callaway model with the effects of hydrostatic pressure depending on the Murnaghan lattice parameter.²⁵ The Clapeyron equation for melting temperature T_m,^23,24 the Gruneisen parameter γ, which is derived from the Rice equation of measuring shock-waves,^26,27 the average phonon group velocity v^ave,²⁸ Debye temperature θ_D,²⁹ surface energy σ_sl²⁴ and surface stress f_sl,³⁰ and the bulk modulus B³¹ with its first derivative B′ representing new values appropriate for In_0.53Ga_0.47As nanofilms, have been determined through a fresh derivation of the nanomaterial form. This approach was initially utilized by Hofmeister in 2002 for bulk materials.³² On the assumption that the dimensions at the nanoscale do not have an impact on impurities and electronic behavior, a size-dependent density ρ³¹ is calculated for all the film's thicknesses. This work examines how the lattice thermal conductivity (LTC) of nanofilms is influenced by their thickness and the amount of pressure they can withstand. In practical terms, these insights could lead to more efficient thermal management in electronic devices, as In_0.53Ga_0.47As is known for its semiconductor properties. The potential implications include improved performance and reliability in high-tech applications, such as faster processors³³ and advanced sensors.³³

2. Methods of calculation

The Debye–Callaway model that is modified by Asen-Palmer et al.³⁴ and Morelli et al.³⁵ is used to determine the LTC in In_0.53Ga_0.47As bulk (thick) and nanofilms. The model includes both modes of one longitudinal K_L and two degenerate transverse (K_T) phonon branches, whilst acoustic phonon branches mostly contribute to heat conduction in semiconductors.^36,37 The temperature dependence of LTC in solids can be expressed as:³⁵

K = K_L + 2K_T, (1)

K_L = K_L1 + K_L2, (2)

K_T = K_T1 + K_T2, (3)

where transverse and longitudinal branches are denoted by T and L, respectively. The partial conductivity term of the modified Debye–Callaway model can be given by K_L1 and K_L2, as in the following form:³⁵The transverse contribution has two forms, which are as follows:where z(x) = x⁴e^x(e^x − 1)⁻², , ħ, k_B, T, v and ω are the Planck constant (1.05457 × 10⁻³⁴ J s), Boltzmann's constant (1.38065 × 10⁻²³ m² kg s⁻² K⁻¹), absolute temperature, acoustic phonon group velocity and phonon frequency, respectively. The combined phonon-scattering is denoted by τ_C(x),³⁸ while τ_R(x) is the resistive total scattering. Furthermore, both branches of Debye temperature are determined by θ^L(T)_D:³⁵while the dispersion of phonon branches near the zone boundary is ignored. The resistive scattering processes (τ^L(T)_R)⁻¹ are estimated by:²⁰

(τ^L(T)_R)⁻¹ = (τ^L(T)_U)⁻¹ + (τ^L(T)_M)⁻¹ + (τ^L(T)_b)⁻¹ + (τ^L(T)_ph–e)⁻¹ + (τ^L(T)_DC)⁻¹, (10)

where τ_U, τ_M, τ_b, τ_ph–e, and τ_DC are the umklapp, impurity, boundary, phonon–electron, and dislocation scattering, respectively. Accordingly, the combined phonon scattering process (τ^L(T)_C)⁻¹ is as follows:

(τ^L(T)_C)⁻¹ = (τ^L(T)_N)⁻¹ + (τ^L(T)_R)⁻¹. (11)

The detailed phonon scattering processes are as follows:

2.1 Umklapp scattering

This mechanism involves phonons colliding and transferring momentum to the crystal lattice, resulting in thermal resistance. It becomes more prominent at elevated temperatures and higher phonon frequencies, which is given by:^35,39where (γ^L(T)) represents the longitudinal (transverse) Gruneisen parameters, and their values are given by γ^L(∞) = 0.3 and γ^T(∞) = 0.22. These values were determined by fitting theoretical LTC to the experimental data from the bulk state of In_0.53Ga_0.47As.

2.2 Normal scattering

Normal phonon scattering involves phonons being deflected by interactions that preserve their wave vector and net momentum. This scattering type is often overlooked compared to umklapp processes, and is computed based on:^35,39

2.3 Boundary scattering

The surfaces and interfaces of nanofilms play a crucial role in phonon scattering. Phonons can be reflected or absorbed at these boundaries, according to:

2.4 Defect scattering

The presence of defects in the lattice can scatter phonons, affecting LTC. The strength of this scattering is contingent upon the material's dispersion curves. The Rayleigh scattering theorem elucidates the interactions between high-frequency phonons and three distinct defect types: impurities (foreign atoms), isotopes, and alloy disorder parameters:^40,41(N_imp) represents the concentration of impurities, while (S) denotes the scattering factor, typically around the value of 1. The term for alloy scattering defined at which phonons scatter with the alloy is comparable to the rate of phonon-isotope scattering. The parameter (A) signifies the alloy parameter, which quantifies the intensity of the alloy's scattering effect,⁴² and is calculated by , representing the strength of the mass difference interaction, which for In_0.53Ga_0.47As, is calculated as follows:⁴³

2.5 Phonon–electron scattering

In semiconductors and other materials with free electrons, interactions between phonons and electrons can scatter phonons, influencing LTC. Zou and Balandin suggested a method for calculating the rate at which phonons scatter with electrons:⁴⁴N_e represents the density of electrons in the conduction band, E denotes the deformation potential with an approximate value of −7.79 eV,⁴⁵m* symbolizes the effective mass of an electron, which is (0.041 × m_e)⁴⁶ for In_0.53Ga_0.47As, where m_e is the resting mass of an electron, and ρ stands for the mass density.

2.6 Dislocation scattering

The rate at which phonons scatter off dislocations, considering a dimensionless parameter for short-range interactions, is determined based on ref. 40 and 41:In this context, η represents the weight factor, which is assigned a value of 0.55, and N_D denotes the total concentration of dislocations, encompassing both edge and screw types. Factors influencing phonon scattering: several factors can significantly affect phonon scattering processes in nanofilms:⁴⁷ in nanofilms, the aforementioned scattering processes can lead to a reduction in LTC compared to bulk materials. This is attributed to the increased surface-to-volume ratio, which amplifies the impact of surface scattering mechanisms. Additionally, confinement effects can alter the phonon spectrum, leading to changes in LTC. An understanding of these processes and their dependencies is crucial for the design of materials with tailored thermal properties.

2.7 Nanofilm dependencies: pressure and size factors

The aforementioned principle can be used when there is no pressure. Considering pressure, which is an important parameter and affects most properties of solids, the model can be applied to the effect of pressure on LTC in In_0.53Ga_0.47As thick and nanofilms of up to 11 GPa, with modifications of other thermophysical properties, including melting temperature. The bulk solid's melting temperature, T_m(r,P), dependent on pressure, is calculated using the Clapeyron equation:^23,24,48The Clapeyron equation, in its simplified form (first-order approximation) as a function of size and pressure, allows for easier calculations and can still provide a general understanding of phase behavior, which is useful for nanofilms, calculating the melting temperature.^23,24T_m0, using variables: molar volume V_s and V_l, for solid and liquid states, respectively. Here, V_s is the product of the zero-pressure lattice volume V and Avogadro's number N_A,²³ while V_l is 90% of V_s. Compressibility K_s, and K_l are defined as (K_s = 1/B), where B is the bulk modulus, and K_l is typically ten times K_s, and H_m is the melting enthalpy. Surface energy σ_sl and stress f_sl are critical thermodynamic parameters (Table 1), particularly for calculating T_m(P).

Table 1 Parameters are obtained through equations stated in the text for both the bulk and nanofilms of In_0.53Ga_0.47As

Material parameters	Thick films 1.4 μm (bulk)	Nanofilm thickness
		70 nm	30 nm	20 nm	10 nm
B (GPa)	58.0666	56.8123	55.1463	53.6953	49.383
V s (cm^3 mol^−1)	15.215	15.3861	15.6263	15.8417	16.5187
K s (10^−12 Pa^−1)	17.2216	17.6018	18.1335	18.6263	20.2499
B′	1.986	1.99026	1.99606	2.00129	2.01785
v ^ave (m s^−1)	3237.18	3220.51	3197.6	3176.93	3111.11
γ ^ave	0.159678	0.1617	0.1646	0.1673	0.1755
K l (10^−12 Pa^−1)	172.216	176.018	181.335	186.263	202.499
ρ (kg m^−3)	5541.07	5477.65	5393.45	5320.11	5102.06
V l (cm^3 mol^−1)	13.693	13.8475	14.0637	14.2576	14.8669

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The bulk (thick film in this work) structure surface energy σ_sl in eqn (20) is calculated according to the Gibbs–Thomson equation:²⁴, and for nanosized films, it is modified to:

where (∞) and (r) denote both bulk state (the thick film) and nanofilms, h is the first solid surface monolayer and calculated from the relation h = 1.429d_mean(∞),⁴⁹ and S_vib is the material's vibrational entropy obtained from S_vib = S_m − R.²⁹d_mean(∞) is the mean bond length, and it is dependent on the thickness, where S_m(∞) and R are the melting entropy and the gas constant, respectively. S_m = H_m/T_m,²⁹ where H_m is the melting enthalpy, and for a tetrahedral compound semiconductor it is calculated from this relation:⁵⁰H_m(∞) = −10⁻⁵T_m²(∞) + 0.059T_m(∞) − 21.33. Its modification for nanosize-dependent solids can be expressed as:

H_m(r) = −10⁻⁵T_m²(r) + 0.059T_m(r) − 21.33. (22)

The intrinsic surface stress f_sl in eqn (20) is the pressure effect on the surface creating a lattice contraction and it is a reversible work per unit area, which is calculated according to:³⁰, for nanofilms, the form is modified to:The bulk modulus B for the bulk value is calculated from B(∞) = v_ave²(∞) ρ(∞), and its nanosize-dependent modification is:³¹

B(r) = v_ave²(r) ρ(r) (24)

Additionally, the average phonon group velocity for the bulk state is ,⁵¹ and its nanosized modification is as in the following:where v^L and v^T are the longitudinal and transverse phonon group velocities, respectively. At the nanoscale, the deflection or reflection of phonons depends strongly on the size. The mean bond length d_mean(∞) at high temperature is an average distance for a phonon. Its change according to the nanosize scale Δd_mean(r) is given by:⁴⁹where r is half of the nanofilm thickness, and r_c is the critical radius at which the material melts at zero Kelvin and calculated by r_c = (3 − D)h, where D is 0, 1, 2 and 3 for nanoparticles, nanowires, nanofilms and the bulk state (thick films), respectively. For nanofilms, r_c = h, Δd_mean(r_c) in eqn (26), which is the maximum increase in the mean bond length calculated from:⁴⁹

Δd_mean(r_c) = h − d_mean(∞), and then for nanofilms, the size-dependent mean bond length d_mean(r) is calculated according to the relation:

d_mean(r) = h − Δd_mean(r). (27)

The nanosize-dependent lattice parameter a(r) is calculated from the relation: .⁴⁹ Accordingly, the lattice volumes for both bulk and nanosize dependence of cubic zinc blende shape structures including In_0.53Ga_0.47As are calculated from relations and , respectively.^52,53 The nanostructure mass density ρ(r) is calculated from the size-dependent lattice volume V(r) of a nanofilm according to ,³¹ where ρ(∞) and V(∞) are the mass density and lattice volume for the bulk value. The bulk density is given by , where the average atomic mass M for In_0.53Ga_0.47As can be calculated as:⁴³Hence, Y = 1 and X = 0.47, and the average atomic masses for In, Ga, and As are abbreviated as M_In, M_Ga, and M_As, respectively. These are calculated from the relation where c_i is the natural abundance of the isotope composition and M_i is the ith isotope atomic mass, with values of 60.107%, 39.89%, 4.33%, 95.67%, and 100% for ⁶⁹Ga, ⁷¹Ga, ¹¹³In, ¹¹⁵In, and ⁷⁵As, respectively.³⁵ Accordingly, their average atomic mass is equal to 139.95 × 10⁻²⁷ kg. The size-dependent melting temperature T_m0(r) is given by:where for In_0.53Ga_0.47As, T_m0(∞) is 1373.15 K.⁵⁴ According to eqn (20), the dependence of both bulk state, the thick film in this case, and nanosize melting temperature on pressure is calculated as shown in Fig. 3 (Fig. 3–11 in Appendix A).

At the nanoscale, the Debye temperature, denoted as θ_D, is key to understanding atomic vibrations in lattice structures. It is calculated using Lindemann's formula: . This formula also applies to the pressure dependence of θ_D for both bulk and nanosize materials:²⁹

Fig. 4 shows the nanosize Debye temperature–pressure dependence of In_0.53Ga_0.47As. Phonon group velocity in nanofilms (v(r)) is derived from Debye temperature θ_D(r) proportionality: ,²⁸ with v(∞) being 4260 m s⁻¹ for longitudinal and 2981 m s⁻¹ for transverse branches.⁵⁵ This relationship is modified for pressure dependence (Fig. 5):These calculations are also utilized for zero-pressure size-dependent parameters for both thick (bulk state) and nanofilm thicknesses, as detailed in Tables 1 and 2.

Table 2 Calculated parameters for In_0.53Ga_0.47As in both its thick (bulk) and nanofilms

Parameters	Thick films 1.4 μm (bulk)	Nanofilm thickness
		70 nm	30 nm	20 nm	10 nm
f sl (J m^−2)	6.30066	6.14814	5.94479	5.76818	5.24622
H m (kJ mol^−1)	40.8304	40.6086	40.3077	40.0397	39.2079
σ sl (J m^−2)	1.67367	1.62881	1.56884	1.5169	1.3644
S vib (J mol^−1 K^−1)	21.42	21.1963	20.8895	20.6138	19.7435
S m (J mol^−1 K^−1)	29.734	29.5054	29.1986	28.9229	28.0526

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Lattice contraction in bulk (thick film) and nanosizes, due to pressure a^P(r,∞), is described by the Murnaghan equation:²⁵

For In_0.53Ga_0.47As semiconductor nanofilms and bulk thickness, a(r,∞) is the zero-pressure lattice parameter, and (a^P(r,∞)) is the pressured lattice parameter, while B′ determines the first derivative of the bulk modulus. This investigation uses the Murnaghan equation to find that lattice parameter density, and volume decrease with increasing pressure. Changes in pressure affect the size, shape, and bulk modulus (B) of nanoscale solids, indicating material resistance and hardness.^56,57 The formula for the first derivative of the bulk modulus for cubic materials at ambient pressure is specified:³². When pressure derivatives are equal, the first derivative of the bulk modulus is given by the free-volume equation:³², for γ_th = γ_ave. In this work, the first derivatives of the bulk moduli for bulk thickness and its corresponding nanoscales are derived as follows:Then according to eqn (32), the bulk and nanoscale lattice parameter and volume dependence on pressure are as shown in Fig. 6. The average Gruneisen parameter γ_ave in eqn (33) is influenced by nanoscale size and pressure, calculated as: .⁵¹ Rice²⁶ derived an equation for the pressure-dependent Gruneisen parameter (γ_P(∞)), which is: ; this equation accounts for pressure effects on the lattice volume of nanoscale solids.The term V_P(r) indicates the effect of pressure on nanoscale lattice volume and is estimated by: . Fig. 7 illustrates how this impacts the Gruneisen parameters for both thick (bulk) and nanofilm materials.

Based on eqn (1)–(34), pressure influences the thermodynamic and structural parameters in the LTC-modified equation for both bulk (thick) and nanofilms, as depicted in Fig. 1.

Fig. 1 LTC of In_0.53Ga_0.47As alloy nanofilms and bulk dependent on temperature under hydrostatic pressure from 0 to 11 GPa for (a) 10 nm, (b) 20 nm, (c) 30 nm, and (d) 70 nm, and (e) 1.4 μm bulk state, respectively, with experimental data for nanofilm thicknesses obtained from ref. 5, and for bulk (thick) from ref. 20.

3. Results and discussion

In LTC calculations for bulk and nanofilms, parameters such as lattice volume, Debye temperature, and group velocity are derived from eqn (20)–(34). Other parameters, like surface roughness and electron concentration, are determined by fitting to the experimental data^5,20 at 0 GPa. Excluding dislocations, the concentration of imperfections, such as impurities (N_imp) and electrons (N_e), remains constant across nanosizes and under varying pressures; these values, along with size- and pressure-dependent parameters, eqn (20)–(34), cause modifications in this model, specifically tailored to the properties of In_0.53Ga_0.47As nanofilms, where (N_e = 0.8 × 10²⁵ m⁻³) and (N_imp = 0.069 × 10²³ m⁻³), and these values are obtained from the fitting process to the experimental data^5,20 of LTC for phonon–electron and phonon-impurity scattering rates, which are inserted into (τ^L(T)_ph–e)⁻¹ and (τ^L(T)_M)⁻¹ in eqn (10).

New values for the first derivative, appropriate for In_0.53Ga_0.47As nanofilms, have been determined through a fresh derivation of the nanomaterial form (refer to Table 1). This approach was initially utilized by Hofmeister in 2002 for bulk materials.³²

Fig. 1 illustrates how hydrostatic pressure affects the LTC of In_0.53Ga_0.47As nanofilms and bulk (thick film) with a thickness range from 10 nm to 1.4 μm, highlighting phonon-based heat transfer in this semiconductor alloy.

At low temperatures, phonon wavelengths match the sample size, affecting boundary scattering. As temperature rises, LTC peaks and is influenced by imperfections like dislocations. Electron numbers and impurity concentrations remain unchanged despite size reduction. At higher temperatures, energetic lattice vibrations and short-wavelength phonons (via the umklapp process) scatter with the crystal lattices, impacting LTC, as depicted in Fig. 1a–e. Fig. 1a–e show that hydrostatic pressure decreases LTC across all temperatures for In_0.53Ga_0.47As nanofilms (10–70 nm) and bulk (1.4 μm). At low temperatures (1–30 K), phonons carry minimal thermal energy, making LTC less pressure-dependent. Boundary scattering, influenced by pressure, temperature, and size, is the key factor here. At the nanoscale, phonons are disrupted, and while their excitation energies rise with temperature, pressure diminishes this effect. Pressure effects on LTC are attributed to lattice rearrangement and reduced dislocation lines, especially in 10 nm nanofilms, lowering the LTC peak and increasing scattering, which diminishes the right-side peak of the LTC curve. Increased pressure leads to higher lattice scattering at sample boundaries, reducing phonon group velocities (Fig. 5); the effects are shown in Fig. 8(a) and (b), and both are obtained from the boundary scattering rate in the nanofilm: , where l_c is the Casimir length, and l is the sample size,⁴⁰ which at 0 GPa equals to 2.61, 2.62, 2.63, 2.64, and 2.8 μm for 10, 20, 30, 70 nm, and 1.4 μm as the bulk, respectively. Surface roughness (ε) ranges from 0 to 1, with ε = 0 for bulk thickness (complete diffusion) and ε = 1 for nanofilms (total reflection). For nanofilms of 10, 20, 30, and 70 nm, the ε values are 0.089, 0.076, 0.069, and 0.054, respectively. Phonon velocities decrease with pressure, more so in thinner films, affecting the In_0.53Ga_0.47As nanofilms’ LTC. Fig. 8(a) and (b) illustrate the inverse relationship between sample thickness and boundary scattering rate. Boundary scattering significantly affects thermal resistivity, more so in strained samples (10 nm). Hydrostatic pressure modifies this rate, reducing LTC, especially at low to intermediate temperatures. It also limits the phonon wavelengths, increasing scattering effects. Despite fewer dislocations, pressure impacts LTC reduction, as shown in Fig. 8(c). From the dislocation scattering rate formula, the N_D (dislocation concentration) values are 4.97 × 10¹⁷, 4.53 × 10¹⁷, 4.27 × 10¹⁷, 3.75 × 10¹⁷ and 1.84 × 10¹⁷ m⁻² for 10, 20, 30, 70 nm, and 1.4 μm (thick) bulk, respectively, and they are used to fit LTC to the reported experimental data. The basic assumption is that at intermediate temperatures, boundary scattering has a minor role, while lower dislocation concentration becomes the main factor in thermal energy transfer under pressure. Strained lattices scatter high-energy phonons, affecting LTC as the atom distances decrease under pressure. Smaller sizes see more impact, likely from bond weakening. The effect of pressure on smaller nanosize elucidates the influence of LTC in thinner films (Fig. 1), which is convenient with ref. 58 and 59. LTC reduction with nanosize decrease is evident in In_0.53Ga_0.47As nanofilms (Fig. 9). Despite pressure-reducing dislocations, LTC_max declines due to boundary scattering's greater influence compared to dislocations.

To investigate the pressure's impact on the thermal properties of the In_0.53Ga_0.47As alloy's nanosize, the LTC at the turning point temperature due to pressure, T_{tp LTC}(P), is plotted against pressure in Fig. 10. The LTC at this point is influenced by boundary and lattice dislocation at low and intermediate temperatures. As size compression increases, so does boundary scattering, which outweighs the reduction in lattice dislocations. Increased pressure significantly lowers LTC in thinner films due to a greater reduction in lattice structure and volume, with a pronounced effect in smaller nanofilms, as shown in Fig. 2 and 11.

Fig. 2 The dependence of (ΔLTC/LTC)_P on nanofilm thickness at 300 K.

4. Conclusions

(1) It is expected from this work that the concentrations of both impurities and electrons will remain the same as those in the bulk, unchanged by both effects of nanoscale dimensions and pressure.

(2) Clapeyron, Murnaghan and Post equations were successfully applied within the modified Debye–Callaway model to assess the effects of hydrostatic pressure on the film's lattice thermal conductivity (LTC), considering surface energy and stress.

(3) At a low temperature range, the reduced LTC due to pressure (ΔLTC/LTC)_P is a strongly temperature-dependent parameter that approaches its minimum value at temperatures above T = 100 K. It is also a size-dependent parameter, increasing from its minimum value at low temperatures to a maximum value at room temperature.

(4) The insights gained in this work could be crucial for optimizing the thermal properties of nano thermoelectric devices, potentially improving their functionality and efficiency.

Data availability

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

Conflicts of interest

The authors have no conflicts to disclose.

Appendix

Fig. 3–11.

Fig. 3 Melting temperature as a function of pressure for In_0.53Ga_0.47As films up to 11 GPa.

Fig. 4 (a) Longitudinal and (b) transverse modes for Debye temperature as a function of pressure for In_0.53Ga_0.47As films up to 11 GPa.

Fig. 5 Phonon group velocity dependent on pressure for In_0.53Ga_0.47As alloy films for (a) longitudinal and (b) transverse modes, for both thick (bulk) and nanofilms.

Fig. 6 Pressure-dependent (a) lattice parameter, and (b) lattice volume, for In_0.53Ga_0.47As alloy films of up to 11 GPa.

Fig. 7 Gruneisen parameter dependence on pressure for In_0.53Ga_0.47As alloy films for both modes (a) longitudinal, and (b) transverse.

Fig. 8 Δ(τ_Boundary)⁻¹ as a function of pressure in (a) and (b), Δ(τ_Dislocation)⁻¹/(τ_Dislocation)⁻¹ in (c) with different thicknesses of 10, 20, 30, and 70 nm and the bulk state for the In_0.53Ga_0.47As alloy.

Fig. 9 LTC_max as a function of pressure for In_0.53Ga_0.47As nanofilms with different thicknesses of 10, 20, 30, and 70 nm and bulk (thick).

Fig. 10 The turning point temperature for LTC as a function of pressure for In_0.53Ga_0.47As alloy films.

Fig. 11 (ΔLTC/LTC)_P as a function of temperature for In_0.53Ga_0.47As alloy with different thicknesses of (a) 10, 20, 30, and 70 nm, and (b) bulk state.

Acknowledgements

Thanks go to I. N. Qader for his assistance, and the authors would like to acknowledge the financial support under grant no. (3/1/6464-992020) from the College of Science at Salahaddin University-Erbil in Kurdistan Region, Iraq.

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