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Phonon thermal transport properties of GaN with symmetry-breaking and lattice deformation induced by the electric field

Electric fields commonly exist in semiconductor structures of electronics, bringing to bear on phonon thermal transport. Also, it is a popular method to tune thermal transport in solids. In this work, phonon and thermal transport properties of GaN with wurtzite and zincblende structures at finite electric field are investigated using first principles calculations from perspectives of symmetry breaking and lattice deformation. Effects of electric field on phonon transport properties including phonon dispersion and thermal conductivity from response of electron density distribution only and response from lattice changes are studied in zincblende GaN. It is found that the former has a small but qualitative impact on phonon dispersion relations, i.e., splitting of phonon branches, since it breaks symmetry of zincblende lattice. While the latter affects both lattice symmetry and size, causing significant changes of phonon properties and increase of thermal conductivity. In wurtzite GaN, space-group-conserved lattice changes at finite electric field are studied with lattice deformation only, where thermal conductivity decreases at electric fields significantly with increase of anisotropy, much different from the changes in zincblende GaN. This work provides a comprehensive understanding on phonon thermal transport properties in GaN at finite electric field, which promises to benefit phonon transport tuning and provide reference for thermal management in GaN-based information and power electronics.


I. Introduction
Generally, electric field exists in semiconductor structures of electronics since it plays a key role in manipulating electrons. Specifically, in GaN-based high electron mobility transistors (HEMTs), it is supposed that the high density of electrons at heterostructure results from strong polarization at AlGaN/GaN interface [1,2]. As one of the most important semiconductor materials, thermal transport in GaN has received great attention in both scientific researches and industrial applications [3][4][5][6][7][8]. However, the effects of electric fields on lattice thermal transport in GaN are still unclear, which is critical for thermal management in GaN-based transistors [4]. Besides, applying an external electric field is an effective way to tune the electrical and thermal transport properties of dielectric materials [9][10][11][12][13][14][15]. Theoretically, the response of lattice to the finite homogenous electric field can be classified as three parts [16,17]: (i) response of electron wave function (density distribution), which can be represented by Born effective charges and dielectric function, as well as interatomic force constants in phonon calculations; (ii) changes in atomic coordinates; and (iii) lattice strain, including electronic part and lattice part. Hence, for materials of which electronic or lattice response to electric field is strong, their thermal conductivity promises to be tuned by external electric field.
By performing first-principles calculations, several studies have been carried out for two-dimensional and layered materials with out-of-plane external electric fields. In detail, external electric field is used to tune electronic structures of two-dimensional BN, C2N-h2D, graphene, and layered germanane [18][19][20][21], as well as phonon properties and lattice thermal conductivity of layered graphene [22], silicene [23], and borophene [24]. In Qin's work [23], the ultra-low thermal conductivity of silicene was obtained with external electric fields which induced phonon renormalization by affecting interatomic force constants. With similar mechanisms, anisotropy of thermal transport in borophene is manipulated by out-of-plane electric fields [24]. Besides affecting the magnitude and distribution of electronic charges, out-of-plane external electric fields can also break the inversion symmetry of layered structures, e.g., layered graphene, mixing in-plane optical phonon branches [22].
Compared with responses of two-dimensional materials, responses of threedimensional periodic solids are much more abundant. External electric fields not only affect magnitudes of lattice constants and interatomic force constants, but also leave qualitative changes for lattice vibration and thermal transport by inducing structure phase transition and symmetry-breaking. However, investigations of thermal transport properties in three-dimensional periodic lattice under finite electric field are quite limited, since calculating the responses of three-dimensional periodic lattice structure to finite electric field from first principles is not an easy work. The main difficulty is that the scalar potential "-Ε· r" (Ε is the electric field and r is the position vector) is nonperiodic and unbounded from below [25]. Under the framework of the modern theory of polarization, Souza et al. proposed an appropriate variance method based on the minimization of electric enthalpy functional [25]. Then, methods for calculating total energy of periodic solids as well as forces and stress, Born effective charges, dielectric function, and phonon properties from first principles were proposed and implemented in first-principles calculation software [16,17,26,27]. Later, an efficient approach to determine the optimized structure in finite electric field was developed based on the above understanding by modifying free energy and Hellmann-Feynman forces with Born effective charges and polarization, where the response of electronic part to electric field is ignored with approximations [26]. In cases where the response of electronic part can be ignored, this method has been applied well to study the lattice thermal transport accompanied by structure phase transition. As a result, it is now feasible to investigate the thermal transport properties of three-dimensional periodic solids [12][13][14]28]. Bagnall et al. [28] investigated the variance of optical phonons in wurtzite GaN induced by the changes in atomic coordinates from electric fields, and explained the shift of Raman peak positions accurately. In Liu's work [13], a larger thermal conductivity switch ratio in Barium Titanate was realized with external electric fields which can manipulate structure phase transition. Also, domain wall response to electric fields was introduced in ferroelectric materials, besides the responses mentioned above, which further enhances bidirectional tuning of thermal conductivity [12,14].
While the wurtzite structure GaN is thermodynamically stable and widely used in information and power electronics, the zincblende structure performs better in several other aspects, e.g., it is more suitable for light-emitting devices due to the absence of built-in piezoelectric fields and spontaneous polarization effects [29,30]. In this work, we perform first-principles calculations on GaN with these two lattice structures at external electric fields for two main purposes. One is to understand the response of lattice thermal transport of wurtzite GaN to external electric fields, and the other is to analyze different responses of lattice thermal transport to external electric fields in different structures of GaN, i.e., symmetry-breaking and symmetry-conserved cases. It is found that both positive and negative electric fields decrease the thermal conductivity of wurtzite GaN, and the responses of thermal transport in wurtzite and zincblende GaN to electric fields are significantly different.

II. Methods
First-principles calculations in this work include phonon properties, e.g., phonon dispersion relations and density of states (DOS), and lattice thermal conductivity calculations. The phonon calculations are performed based on density functional theory as implemented in ABINIT [27,31] with optimized norm-conserving (ONCV) pseudopotential, while the lattice thermal conductivity calculations are based on Vienna ab initio Simulation Package (VASP) [32] with projected augmented wave (PAW) pseudopotential [33] since several tests show that ABINIT and Phono3py [34] based calculations provide much lower thermal conductivity of wurtzite GaN compared with the data from calculations and experiments in literature [3,[5][6][7]. Generalized gradient approximation in the Perdew-Burke-Ernzerhof form [35] is adopted for the exchangecorrelation functional. The kinetic energy cutoff for plane-wave basis 1000 eV is employed with strict convergence test, and the Brillouin zones are sampled using converged 8×8×6 Gamma-centered and 10×10×10 Monkhorst-Pack k-mesh grids [36] for wurtzite and zincblende structures, respectively. In the structural optimization step, the atom positions and lattice constants are fully relaxed until the residual stress and the maximum forces acting on each atom are smaller than 10 −2 kbar and 10 −6 eV/Å, respectively, for both cases with and without external electric field. The 2nd order interatomic force constants (IFCs) are calculated using a finite-displacement supercell method implemented in Phonopy [34] with supercell size 4×4×3 and 5×5×5 for wurtzite and zincblende primitive unit cell, respectively. The Born effective charges and highfrequency dielectric constants are calculated with density functional perturbation theory to include the long-range Coulomb atomic interaction in polar materials, i.e., nonanalytical corrections for 2nd order IFCs. The finite-displacement supercell method is also applied in the calculations of 3rd order IFCs using thirdorder.py, a script in ShengBTE package [37] with supercell size 3×3×3 and 4×4×4 for wurtzite and zincblende primitive unit cell, respectively. And the interaction cutoff is up to the fifthnearest neighbors in the 3rd IFCs calculations. Lattice thermal conductivity is calculated by directly solving the phonon Boltzmann transport equation [37] with second and third interatomic force constants, where the size of q-mesh grids in the Brillouin zone are both 19×19×19 for wurtzite and zincblende structures. The parameters used in first-principles calculations as mentioned above have been carefully determined for convergence of the calculations based on literature [5][6][7][8] and our tests [3].
The case of three-dimensional solids in finite electric field corresponds to that the bulk material (or slab structure thick enough) is clamped by two electrodes with voltage difference in reality. In the relaxation process of bulk material, the absolute electric field changes with the variance of lattice constants while the voltage difference is generally constant. Hence, it is more appropriate to use the reduced electric field here which is defined as [38] ̄= ⋅ , where ai is the lattice vector and is the absolute electric field with their directions defined in Cartesian coordinates. In the following calculations of structural relaxation at finite electric field, the reduced electric field is used, instead of the absolute electric field.
In this work, forces on atoms, the total energy of the system, and lattice structure optimization in three-dimensional solids at finite electric field are calculated using open-source software ABINIT which can take into consideration the response from the electronic part, i.e., changes in electron density distribution. As mentioned above, since the combination of ABINIT and Phono3py gives a much lower thermal conductivity

III. Results and discussion
A. Symmetry-breaking from the finite electric field In this section, it is concentrated on the changes in phonon and phonon transport properties from symmetry-breaking of lattice system at the electric field, including symmetry-breaking from the electric field with fixed lattice and atomic positions, i.e., from changes in electron density distribution only, and electric field with lattice deformation, i.e., from lattice strain and changes in atomic positions at finite electric field.
The detailed lattice structure and atomic positions are shown in Figure 1. Phonon dispersion relations contain six branches including three acoustic branches and three optical branches. The phonon degeneracies at high symmetry points are listed in Table   1, and they can also be seen from phonon dispersion relations. Since the three directions x, y, z in Cartesian coordinates are equivalent for zincblende structures, reduced electric field along z-direction is applied ( Figure 1). The breakdown electric fields (5×10 8 V/m for zincblende GaN and 3.3×10 8 V/m for wurtzite GaN) from Chow's work [40] in 1996 are generally used. These values provide reference to experimental studies for increasing breakdown electric fields, while larger values may be used in numerical and theoretical studies in an ideal GaN system, e.g., electric field 5×10 8 V/m is used for wurtzite GaN in Ref. [28]. In this study for effects of symmetry-breaking and lattice deformation induced by the electric field on phonon transport properties, we intend to apply electric field as large as possible to observe the significant response of phonon transport properties to electric fields, which can also provide information on the feasibility of thermal conductivity tuning based on electric fields. In first-principles calculations at finite electric fields, only valence band electrons are considered, which regards the system as an insulator with infinite bandgap and implies that no limitation exists for electric fields. The experimental bandgap values are used as a criterion in this work to help select high electric fields to ensure reasonability and significant responses (an estimated lower limit for bandgap can be provided to a specific electric field in In calculations with fixed lattice structure and atomic positions, only the changes in electron density distribution are considered. In this case, the symmetries of the lattice system are reduced, e.g., 2-fold rotation symmetry along x (y) axis and mirror symmetry relying on the plane not containing z-axis do not exist and equivalence among x, y, z directions is absent due to the unsymmetrical electron density distribution at finite electric field along the z-direction. It is noted here particularly that lattice symmetry is not used in phonon calculations to decrease computation costs as actual symmetry is lower than apparent lattice symmetry at finite electric fields. In Figure  ].
Due to the symmetry-breaking reflected by lattice structure and atomic positions, nondiagonal elements are not equal to zero, though they are not large compared to diagonal elements. Lattice with space group R3m belongs to the trigonal system and can be described by two approaches. The first approach is introduced here for comparison with the cubic zincblende structure. In this approach, lattice parameters a=b=c, and three  Figure 3, the contribution from high-frequency phonons is nearly zero as illustrated in Figure 5(b).
For a detailed understanding on changes in lattice thermal conductivity, phonon scattering rate, group velocity, and specific heat are calculated and plotted in Figure 6.
Based on phonon Boltzmann transport equation, lattice thermal conductivity formula is derived as [37] = 1 The with mode-specific heat V, . Scattering rate is a very important an-harmonic phonon property, depending on phonon dispersion relations and anharmonicity of atomic interaction. In Figure 6(a), it is found that the distribution of scattering rate at zero electric field is more concentrated. The differences in magnitude are not significant.
The scattering rate at the negative electric field is relatively large while that at the positive electric field is relatively small particularly at frequency 0-5 THz from which around 40% thermal conductivity is contributed. The weighted phase space for three phonon processes [44] and Grüneisen parameters to phonon frequency for zincblende GaN at different electric fields are plotted in Figures 6(b) and (c). It is found that electric fields increase the weighted space group slightly which contributes to the increase of phonon scattering, while they decrease the phonon anharmonicity significantly. The Hence, phonon group velocities can be significantly different for conditions with different electric fields though phonon dispersion relations in the low-frequency section are similar. Nearly the same specific heat is illustrated in Figure 6(e), which is not supposed to result in changes in lattice thermal conductivity.

B. Lattice deformation from the finite electric field
For wurtzite GaN, convergence for electronic calculations is difficult at the finite electric field perpendicular to the polar axis, as electric field along this direction breaks lattice symmetry, verifying robust lattice symmetry at finite electric field. In practical applications, it is more common that an electric field is applied along the polar axis In Figure 9(a), thermal conductivity with respect to a temperature ranging from 300-500 K is presented. Both positive and negative electric fields result in large decreases of thermal conductivity, opposite to the lattice thermal conductivity response to the electric field in zincblende GaN. Besides, anisotropy increases in the finite electric field while thermal conductivity is nearly isotropic at zero electric field. Here, thermal conductivity along the polar axis is regarded as out-of-plane thermal conductivity, and that in the direction perpendicular to the polar axis is in-plane one. In detail, the decrease of in-plane thermal conductivity at the positive electric field is relatively small while out-of-plane thermal conductivity decreases a lot. At negative electric field states, both in-plane and out-of-plane thermal conductivities decrease significantly, with room temperature data illustrated in Figure 9(b). Figures 9(c) and (d) show thermal conductivity contribution from each phonon band (bands 1-6) for in-plane and out-of-plane cases at room temperature. It can be seen from these data that thermal In this work, the isotope effect is not taken into consideration. While this effect is reported to be significant in first-principles calculations, it is found to be small in experiments. In calculations, the isotope effect is treated based on Tamura's mass variation approximation [46] and the Matthiessen rule. Since this type of scattering is not affected by electric fields, it only decreases the thermal conductivity of GaN at different electric fields simultaneously and will not affect the conclusions in this work.

IV. Concluding remarks
In conclusion, phonon and phonon transport properties of zincblende and wurtzite

Supplementary material
See supplementary materials for the benchmark study on AlAs, more details about lattice parameters with and without electric fields, phonon dispersion relations based on ABINIT and VASP, discussion on room temperature thermal conductivity changes concerning electric fields, optical phonon shifts, and Q-mesh convergence in thermal conductivity calculations for zincblende GaN.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.                     Figure 9. Lattice thermal conductivity of wurtzite GaN with and without lattice deformation from electric field (a) to temperature (b) at room temperature. And lattice thermal conductivity at room temperature for each low-frequency phonon band (bands 1-6) with and without the electric field for (c) in-plane and (d) out-of-plane cases.