The influence of anisotropy of InP on its elasticity and phonon properties

Xingyuan Xu; Hongzhi Fu

doi:10.1515/phys-2025-0251

Article Open Access

The influence of anisotropy of InP on its elasticity and phonon properties

Xingyuan Xu and Hongzhi Fu

Published/Copyright: December 31, 2025

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From the journal Open Physics Volume 23 Issue 1

Abstract

Using Bond matrix model and tensor theory, the 3D structures are investigated for Zener and Every anisotropies, stress-strain, heavy hole and light hole, and Phillips ionicity in InP. The lattice wave propagation, phonon focusing and phonon distribution in InP are studied in detail based on lattice dynamic theory. The lattice waves have a mixture of longitudinal and transverse modes, and the anisotropy of lattice waves is completely consistent with the crystal symmetry. The Gaussian curvature plays a decisive role in phonon focus, phonon image and caustics. The study of phonon images reveals the distribution of phonons, which not only maintains the symmetry of the crystal itself, but also maintains the anisotropy of the crystal itself.

Keywords: alloys; lattice dynamics; anisotropy; phonon distribution

1 Introduction

III–V semiconductors are a class of materials with unique optical properties; which were of considerable interest for both fundamental research and technological applications [1], [2], [3], [4], [5]. In contrast to silicon, most materials in the III–V series had direct band gaps. This allowed them to be used in electroluminescent devices [6]. Among III–V compound semiconductors, InP was one of the most promising optoelectronic materials in III–V compound semiconductor with n-type behavior. It has a face centered cubic “Zinc blende” structure and is also an interesting material for developing devices that operate at 1.3 and 1.55 μm wavelengths [7]. They are mostly manufactured on InP substrates and usually require the growth of different InP epitaxial layers because the losses of propagation in the fiber is lowest at these wavelengths [7]. InP is a key material for optoelectronics and is used in different optical and microwave devices. As a substrate material, InP provided good thermal stability and crystal perfection in semiconductor technology [8]. InP had been extensively investigated recently for various scientific and technological aspects in order to excess new class of fundamental material for high speed semiconductor technology [8].

Experimentally, a new Cu /n-InP Schottky junction with cytosine interlayer had been formed by drop casting process. The current-voltage (I–V) and capacitance-voltage (C–V) characteristics of Cu / cytosine / n-InP structure were investigated at room temperature [9]. The electrical and current transport properties of prepared Ti / a-amylase / p-InP metal / polymer / semiconductor (MPS) junction by current–voltage (I–V) approach had been demonstrated [10]. Recently, the effects of ligand electric resonance [11] on the luminescence properties of InP/ZnS quantum dots [12] and interface defects on the optical properties of InP/ZnS quantum dots [13] had been experimentally studied through low-temperature synthesis of InP. The experimental results of photoluminescence spectrum were also shown that the emission wavelength of the new quantum dots was increased by about 1–3 nm, depending on the electrical resonance properties of the bidentate dithiocarbamate ligands. The improvement of electronic properties such as current density and brightness of the device was observed [14]. The nanowires of InP [15] and nonlinear optical properties of InP / ZnS core–shell quantum dots toluene solution [16] were investigated by Z-scan and transient absorption technique with femtosecond pulses and nanosecond pulses at 532 nm wavelength, respectively. The results shown that InP / ZnS core–shell quantum dots exhibit saturated absorption under femtosecond pulse excitation, and the conversion from saturated absorption to reverse saturated absorption was observed under nanosecond pulse excitation. The mechanism of the switch was excited state absorption.

Theoretically, the structural, electronic, and mechanical properties and phonon dispersion curves of InP were studied using the framework of density functional theory and density functional perturbation theory (DFPT) implemented in the Quantum ESPRESSO package [17]. Latter, the properties of first bandgap were explored in phonon polariton curves for InP crystal driving by tunable ultrasonic [18]. The effective frequency and width of the band gap were obtained. These results can be used for terahertz frequency filtering and will have a profound impact on noise attenuation in terahertz communication field [18]. The Numerical modelings were presented theoretically for the electronic and optical properties of InAs / InP quantum dot with a dome cross-section [19]. The electrons, heavy hole energy and their interband transition energies were also studied. The linear, third-order nonlinear, interband light absorption coefficient and refractive index were investigated as a function of height point (H) and incident light intensity (I) with the density matrix formalism [19]. The effect of a magnetic field had been investigated on the excitonic and optical properties of an InP / ZnS core /shell nanodot (type I) as well as the influence of the geometrical confinement effect [20]. The investigations cover the exciton binding energy, oscillator strength, linear and nonlinear electronic, optical absorption coefficients and electronic dipole moment; and the optical gain were also studied for different ratios of the core / shell dot radii with and without the application of a magnetic field.

The main aim of this work is to critically discuss and to evaluate the anisotropy and phonon properties of InP. In the following chapters, firstly, the 3D anisotropy of InP are studied using Bond matrix method. Secondly, the 3D stress – strain structures of InP are studied theoretically by continuum tensor theory. Finally, based on the sound field and wave theory, the three-dimensional structure of the phonons was mainly studied.

2 Methodology

The wave equation combines the stress-strain and momentum conservation equations. Then, based on the properties of the medium and the direction of propagation, measurable quantities such as phase velocity and energy velocity can be obtained from the dispersion equation. The “eigenequation” (the Kelvin–Christoffel equation) [21] becomes,

(1) Γ − ρ v 2 I ⋅ u = 0

with the eigenvalues ρ v 2 j and eigenvectors/displacement u _j, j = 1, 2, 3, where I is 3 × 3 unit matrix, ρ is density of matter, v is the phase velocity. The dispersion equation is given by,

(2) det Γ − ρ v 2 I = 0

In explicit form, the components of the Kelvin–Christoffel matrix are,

(3) Γ 11 = c 11 l 1 2 + c 66 l 2 2 + c 55 l 3 2 + 2 c 56 l 2 l 3 + 2 c 15 l 3 l 1 + 2 c 16 l 1 l 2 , Γ 22 = c 66 l 1 2 + c 22 l 2 2 + c 44 l 3 2 + 2 c 24 l 2 l 3 + 2 c 46 l 3 l 1 + 2 c 26 l 1 l 2 , Γ 33 = c 55 l 1 2 + c 44 l 2 2 + c 33 l 3 2 + 2 c 34 l 2 l 3 + 2 c 35 l 3 l 1 + 2 c 45 l 1 l 2 , Γ 12 = c 16 l 1 2 + c 26 l 2 2 + c 45 l 3 2 + c 46 + c 25 l 2 l 3 + c 14 + c 56 l 3 l 1 + c 12 + c 66 l 1 l 2 , Γ 13 = c 15 l 1 2 + c 46 l 2 2 + c 35 l 3 2 + c 45 + c 36 l 2 l 3 + c 13 + c 55 l 3 l 1 + c 14 + c 56 l 1 l 2 , Γ 23 = c 56 l 1 2 + c 24 l 2 2 + c 34 l 3 2 + c 44 + c 23 l 2 l 3 + c 36 + c 45 l 3 l 1 + c 25 + c 46 l 1 l 2 ⋅

where c _ij are elastic constants and l 1 , l 2 , l 3 are direction cosine of the wave vector.

The dispersion equation for orthorhombic media has the form

(4) v 6 − a 2 v 4 + a 1 v 2 − a 0 = 0 ,

where

(5) a 1 = Γ 22 Γ 33 + Γ 11 Γ 33 + Γ 22 Γ 11 − Γ 23 2 − Γ 13 2 − Γ 12 2 , a 2 = Γ 11 + Γ 22 + Γ 33 , a 0 = Γ 22 Γ 33 + Γ 11 Γ 33 + Γ 22 Γ 11 − Γ 23 2 − Γ 13 2 − Γ 12 2 ,

with

(6) Γ 11 = c 11 l 1 2 + c 66 l 2 2 + c 55 l 3 2 , Γ 22 = c 66 l 1 2 + c 22 l 2 2 + c 44 l 3 2 , Γ 33 = c 55 l 1 2 + c 44 l 2 2 + c 33 l 3 2 , Γ 12 = ( c 12 + c 16 ) l 1 l 2 , Γ 13 = ( c 13 + c 55 ) l 1 l 3 , Γ 23 = ( c 44 + c 23 ) l 2 l 3 ⋅

The group velocity V _g can be found from the phase velocity v by V g = d ω d k . Substituting k = ω/v into this equation yields,

(7) V g = d ω d ω v p − 1 = d ω d ω v p − ω d v p v p 2 − 1 = v p 2 v p − ω d v p d ω − 1 = v p 2 v p − f d v p d f − 1

with ω = 2πf.

3 Discussions

3.1 The anisotropy properties of InP

Anisotropy is widespread in materials physics, which is manifested by the anisotropy of atomic arrangement and electronic structure. The 3D anisotropic structure plays an important role in the study of physical and chemical properties of materials. Anisotropy factor is a quantitative method to characterize anisotropy. Unfortunately, the anisotropy factor is expressed differently depending on different crystal systems [22]. Even for cubic crystals (high symmetry), there is no unified expression for the anisotropy factor. For example, the Zener anisotropy [23], (A _z = 2c ₄₄ /(c ₁₁-c ₁₂)) and Every anisotropy [24] (A _E = (c ₁₁-c ₁₂-2c ₄₄)/(c ₁₁-c ₄₄)) are commonly used for cubic crystals. Obviously, anisotropy has different expressions, that is, A _z and A _E cannot represent each other. The A _z = 1 and A _E = 0 represented isotropic crystals, respectively. The degree of deviation from these limits indicates the magnitude of crystal anisotropy.

In order to study the three-dimensional structures of the following crystal properties, the most effective coordinate transformation is used through the Bond matrices [25],

N = α 11 2 α 12 2 α 13 2 α 12 α 13 α 13 α 11 α 11 α 12 α 21 2 α 22 2 α 23 2 α 22 α 23 α 23 α 21 α 21 α 22 α 31 2 α 32 2 α 33 2 α 32 α 33 α 33 α 31 α 31 α 32 2 α 21 α 31 2 α 22 α 32 2 α 23 α 33 α 22 α 33 + α 23 α 32 α 21 α 33 + α 23 α 31 α 22 α 31 + α 21 α 32 2 α 31 α 11 2 α 32 α 12 2 α 33 α 13 α 12 α 33 + α 13 α 32 α 13 α 31 + α 11 α 33 α 11 α 32 + α 12 α 31 2 α 11 α 21 2 α 12 α 22 2 α 13 α 23 α 12 α 23 + α 13 α 22 α 13 α 21 + α 11 α 23 α 11 α 22 + α 12 α 21

and

(8) M = α 11 2 α 12 2 α 13 2 2 α 12 α 13 2 α 13 α 11 2 α 11 α 12 α 21 2 α 22 2 α 23 2 2 α 22 α 23 2 α 23 α 21 2 α 21 α 22 α 31 2 α 32 2 α 33 2 2 α 32 α 33 2 α 33 α 31 2 α 31 α 32 α 21 α 31 α 22 α 32 α 23 α 33 α 22 α 33 + α 23 α 32 α 21 α 33 + α 23 α 31 α 22 α 31 + α 21 α 32 α 31 α 11 α 32 α 12 α 33 α 13 α 12 α 33 + α 13 α 32 α 13 α 31 + α 11 α 33 α 11 α 32 + α 12 α 31 α 11 α 21 α 12 α 22 α 13 α 23 α 12 α 23 + α 13 α 22 α 13 α 21 + α 11 α 23 α 11 α 22 + α 12 α 21

C ′ = M ⋅ C ⋅ M T and S ′ = N ⋅ S ⋅ N T

where α _ij are the directional cosine between the x _i-axis and x′_p-axis; the [C] and [S] are stiffness-constant and compliance-constant matrix; [M]^T and [N]^T are the transpose matrices of [M] and [N]_, respectively. The Bond matrices are the strict tensors operation applicable to various crystal structures without any approximations, unlike the perturbation method for elastic properties which requires isotropic approximations [26].

The elastic anisotropies of InP are calculated from elastic constants of reference [27], and the anisotropy factors of A _z and A _E are plotted in Figure 1. The results show that although the topological structures of A _z and A _E are different, the directions of their anisotropy are the same, that is, along [001], [111] and [110] directions (Figure 1a and b). This phenomenon shows that anisotropy is an intrinsic property of crystals, and will not change with different expressions. This phenomenon should be universal, not only for A _z and A _E. This also suggests that the 3D anisotropic structures may be an important basis for understanding different physical phenomena.

Figure 1:

Orientation dependence of elastic anisotropy in InP: (a) 3-D-A _z; (b) 3-D-A _E.

According to tensor theory of elastic medium, ε i j ′ (strain) and σ k l ′ (stress) can be expressed through compliance coefficients S ijkl ′ as ε i j ′ = ∑ k l S ijkl ′ σ k l ′ , which is Hooke’s law in tensor form in the case of coordinate transformation. For cubic symmetry, the nonzero strain components can be obtained as follows, ε x x ′ = S 11 ′ σ ′ , ε y y ′ = ε z z ′ = S 12 ′ σ ′ for the [100] uniaxial stress; ε x x ′ = ε y y ′ = ε z z ′ = S 11 ′ + S 12 ′ σ ′ 3 , ε x y ′ = ε y z ′ = ε z x ′ = S 44 ′ σ ′ 6 for the [111] uniaxial stress. The strain components are shows in Figure 2 under the [100] and [111] uniaxial stresses. It is obvious that InP has obvious anisotropy. When InP is stretched in the [100] direction, the strain ε x x ′ presents isotropic expansion in the {001} plane and anisotropic contraction in the [001] direction, accompanied by “strain valleys” with a four-degree axis of symmetry [Figure 2a]. This is because InP has 2nd and 4th -degree symmetries. The similarity also appears to ε y y ′ = ε z z ′ . It’s just that the “strain valleys” become larger [Figure 2b]. While the stress along the [111] direction, although the 4th -degree symmetry of the strain does not change, the 3D structures of ε x x ′ and ε x y ′ are significantly different from each other. In the {001} plane, the ε x x ′ is still isotropic, while the strain shows obvious anisotropy in the [001] direction, and the latter is greater than that of the former [Figure 2c]. For ε x y ′ = ε y z ′ = ε z x ′ , a perfect “four-leaf structure” begins to appear. In the {001} plane, the isotropy of the strain disappears, and the strain in the [110] direction is significantly larger than that in the [100] direction [Figure 2d].

Figure 2:

The strain of InP under uniaxial stresses. (a) The strain of ɛ _xx under the [100] uniaxial stress; (b) the strain of ɛ _yy = ɛ _zz under the [100] uniaxial stress; (c) the strain of ɛ _xx under the [111] uniaxial stress; (d) the strain of ɛ _yy = ɛ _zz under the [111] uniaxial stress.

Through the dimensionless electro -negativity X _A and X _B between A and B atoms, the Pauling ionicity f i P is defined as, f i P = 1 − exp − X A − X b 2 4 . In order to study the relationship between anisotropy and ionicity, we need to utilize the 3D structure of f _i because it is linked to the elastic coefficients C _ij. Here I would like to briefly summarize Phillips iconicity f _i. It is derived from the orbital model [28] and covalency α _c, that is, f i = 1 − α c 2 in which C 12 C 11 = 2 − α c 2 2 + 2 α c 2 . The Phillips ionicity f _i is plotted in Figure 3. It shows that the maximum Phillips ionicity is in the {002} plane, and that the minimum iconicity is along the direction [111]. It means the smaller ionic bond in the [111] direction. The dispersion relationship (E = E(k)) around the minimum value (valley) of the conduction band or around the maximum value of the valence band is usually manifested as spherical, ellipsoidal or hyperbolic surfaces. Warped constant energy surfaces are typical to valence bands and are degenerate at k = 0.The heavy h and light l mass bands are,

E h , l k = ℏ 2 2 m 0 γ 1 k x 2 + k y 2 + k z 2 ± 2 γ 2 2 k x 4 + k y 4 + k z 4 + 3 γ 3 2 − γ 2 2 k x 2 k y 2 + k z 2 k y 2 + k z 2 k x 2 ,

where k _x, k _y, k _z are wave vector components, m ₀ is electron mass, and γ ₁, γ ₂, γ ₃ are the Luttinger parameters [29]. The plus and minus signs refer to light and heavy hole mass bands, respectively. For InP, the lack of inversion symmetry gives an additional term in the dispersion and is linear in a wave vector. The warped constant energy surfaces of light and heavy hole mass bands are plotted in Figure 4. The constant energy surface of the heavy-hole band is not spherical for γ ₃ ≠ 0. Energy contour of the heavy hole band in the region of k _x, k _y, k _z ≃ 0 are spherical but for larger values k _x, k _y and k _z the contour is warped, which shows the dispersion is linear in wave vectors (Figure 4a). It shows that the light hole is isotropic, which shows the light-hole band is almost spherical in the wide range of wave vectors (Figure 4b), while the heavy hole is clearly anisotropic. This means that the effective mass of heavy hole is larger along the [111] direction than the [100] direction [30]. These two directions are exactly the anisotropy of InP.

Figure 3:

The 3D Phillips ionicity f _i of InP.

Figure 4:

Three-dimensional constant-energy surfaces of the InP. (a) heavy-hole and (b) light-hole.

3.2 Lattice wave properties of InP

The gradient of dispersion relationship ω(k) to wave vector k is the group velocity, V _g = ∇_k ω(k). The V _g and k are usually inconsistent, except for some highly symmetrical crystal directions. The distribution of anisotropic wave vectors k in 3D space can be regarded as the propagation of incoherent energy in anisotropic media. This feature can cause other interesting phenomena in phonon focusing [31]. The transformation from spherical coordinate system to rectangular coordinate system is an effective method to study lattice waves, which is not only important in theory but also necessary in engineering. The transformation follows the following rules,

(9) V j 1 = v j ⁡ sin ⁡ θ ⁡ cos ⁡ φ + ∂ v j ∂ θ cos ⁡ θ ⁡ cos ⁡ φ − ∂ v j ∂ φ sin ⁡ φ sin ⁡ θ , V j 2 = v j ⁡ sin ⁡ θ ⁡ sin ⁡ φ + ∂ v j ∂ θ cos ⁡ θ ⁡ sin ⁡ φ + ∂ v j ∂ φ cos ⁡ φ sin ⁡ θ , V j 3 = v j ⁡ cos ⁡ θ − ∂ v j ∂ θ sin ⁡ θ , V j 1 = V j x , V j 2 = V j y , V j 3 = V j z .

The properties of slowness surfaces S θ , φ = 1 / v j θ , φ , which is usually called the reciprocal of the phase velocity, is described by Gaussian curvature [32],

(10) K = L N − M 2 E F − G 2 ,

with

E = ∂ θ S x 2 + ∂ θ S y 2 + ∂ θ S y 2 , F = ∂ θ S x ⋅ ∂ φ S x + ∂ θ S y ⋅ ∂ φ S y + ∂ θ S z ⋅ ∂ φ S z , G = ∂ φ S x 2 + ∂ φ S y 2 + ∂ φ S y 2 , L = ∂ θ , θ S x ∂ θ , θ S y ∂ θ , θ S z ∂ θ S x ∂ θ S y ∂ θ S z ∂ φ S x ∂ φ S y ∂ φ S z E ⋅ G − F 2 , M = ∂ θ , φ S x ∂ θ , φ S y ∂ θ , φ S z ∂ θ S x ∂ θ S y ∂ θ S z ∂ φ S x ∂ φ S y ∂ φ S z E ⋅ G − F 2 , N = ∂ φ , φ S x ∂ φ , φ S y ∂ φ , φ S z ∂ θ S x ∂ θ S y ∂ θ S z ∂ φ S x ∂ φ S y ∂ φ S z E ⋅ G − F 2 ,

where S x = sin ⁡ θ ⁡ sin ⁡ φ v j , S y = sin ⁡ θ ⁡ cos ⁡ φ v j , S y = cos ⁡ θ v j .

The 3D structures of the three slowness surfaces of InP are plotted in Figure 5. It shows the following characteristics: the outermost is the slow shear lattice wave (ST), the middle is the fast shear lattice wave (FT), and the innermost is the fastest longitudinal wave (L). For ST and FT, they intersect only in the highly symmetric [100] direction of InP [Figure 5a]. This means that FT > ST is not always true, and they often have degenerate properties at high symmetry points. Again, the L-waves and S-waves are not pure L-waves and S -waves, but mixtures of them, so they are called quasi- L -wave and quasi- S -wave [33]. In the three lattice waves, the magnitude of their anisotropy follows the following relation, ST > FT > L [Figure 5a]. Then, the structure of ST with the Gaussian curvature (GC) is also plotted in [Figure 5b]. It shows that the slowness surface of ST is divided into the three sections. The first part, GC > 0, represents the divergence properties of lattice waves. The second part, GC < 0, represents the convergence properties of lattice wave. In the middle of them is the third part, GC = 0, which is represented by the red curve. The Gaussian curvature is divided into positive and negative parts. This red line represents the infinite density of phonon states, which leads to the phenomenon of phonon focusing and cuspidal edges of group velocities discussed below. On the one hand, group velocity is the gradient of dispersion relation, on the other hand, it also represents the direction of energy transmission. In the case of our discussion, k and V _g are no collinear. The one-to-one correspondence existed in isotropic media is no longer exists. The 3D structures of group velocity for L, ST and FT are plotted in Figure 6. It has been shown that the group velocity is always perpendicular to the slowness surface [34], 35]. It may be noted that for group velocity the ST is highly anisotropic than that of FT and L [Figure 6a–d], which is the same as the order of the phase velocity. That is, the longitudinal wave is almost isotropic, with ST anisotropy being the greatest, followed by FT anisotropy. For L, cuspidal edges don’t appear at all [Figure 6a], which means that L has little effect on phonon focusing. For FT [Figure 6c], cuspidal edges begin to appear, but not obvious, indicating that FT is beginning to play a small role in phonon focusing. Not like FT, the arrangement of cuspidal edges in ST is considerably more complex [Figure 6b and d], which play a key role in phonon focusing. The cuspidal edges appear in the [100], [001] and [111] directions [Figure 6b and c]. This indicates that one-to-one correspondence between phase velocity vector and group velocity vector does not exist in these directions. That is, the phonon is focused along the direction. This indicates that the phase velocity vectors in these directions are associated with multiple group velocity vectors, resulting in highly anisotropic wave propagation along these paths. These features are also reflected on the slowness surface. One can expect very interesting phenomena such as conical refraction, phonon amplification, etc., to occur in these directions [36], [37], [38].

Figure 5:

(color on line) 3-D representations of three slowness surfaces of InP. (a) The combination of the L, ST and FT. (b) ST with Gaussian curvature.

Figure 6:

(color on line) Three-dimensional representations of the three sheets of the group velocity surface of InP. (a) The longitudinal (L) surface; (b) the slow transverse (ST) surface; (c) the fast transverse (FT); (d) the combination of three group velocity.

3.3 Phonon properties of InP

The complete solution of phonon eigenproblem is connected with the phonon dispersion relation ω = ω q , j and eigenvectors e = e j q which are labeled by wave vector q and branch index j = 1, 2, 3. Atomic vibrations corresponding to acoustic or optical, can be longitudinal or transverse, or a mixture of them. In anisotropic crystals, there is no clear relationship between e and q (wave vector), except in the direction of high symmetry. The scheme for calculating the phonon density of state g(ω) is based on the Dirac function, g ω = V 8 π 3 ∑ q , j δ ω − ω q , j . The phonon number N and phonon energy E [39] are mathematically relevant,

(11) N = ∑ q s n ω q , j , T = ∫ n ω ( q , j ) , T × g ω ( q , j ) d ω , E = ∑ q , s E j = ∫ ℏ ω q , j g ω n ω , T d ω = ∑ q s ℏ ω ( q , j ) 2 + ℏ ω ( q , j ) e ℏ ω K BT − 1 , g ω q , j = V 2 π 3 ∫ δ ω − ω ( q , j ) d k , n ω q , j , T = 1 e ℏ ω q , j K B T − 1 ,

where n ω , T is Bose- Einstein distribution.

Figure 7 depict the3D phonon distribution in Center- Cut- Sphere and Center- Cut- Box for L, ST and FT with T = 300k. Due to the anisotropy of InP, the phonon distribution presents anisotropy characteristics. For L, the conical shapes of phonons appear in the [001], [010] and [100] directions (Figure 7a and b), and presents a perfect distribution of the 4th degree symmetry. This indicates that the anisotropy of the crystal can lead to a change in the phonon distribution from uniform to non-uniform. This tendency is completely determined by the anisotropy of the crystal, and there is a one-to-one correspondence between them. This means that phonon channel exists even in semiconductors. This feature can lead to the anisotropic heat transportation [38] not only in the insulators but also in the metal crystals.

Figure 7:

(color on line) 3D representations of phonon in InP. (a) Longitudinal (L) (center cut sphere); (b) longitudinal (L) (center cut box); (c) slow transverse (ST) (center cut sphere); (d) slow transverse (ST) (perspective); (e) fast transverse (FT) (center cut sphere); (f) fast transverse (FT) (center cut box).

Similar to the L, the phonon channel of ST also shows a 4th degree symmetry, but the ST structure becomes more complex [Figure 7c and d]. On the one hand, the phonon channel has become a funnel with a rectangular shape; On the other hand, small bag-phonon channels appear symmetrically between them. This phenomenon is attributed to the fact that ST has more anisotropy than L and FT. This similar phenomenon has been confirmed by experiments in fcc Ge at temperature T = 2 K [36]. In fact, the Gaussian curvature Gc = 0 is often associated with catastrophe theory, which often leads to the emergence of some strange phenomena in physics [40], such as the emergence of singularity and causticity and so on. Finally, it is about the FT which is in the middle of anisotropy [Figure 7e and f] between L and ST. On the one hand, FT retains the common features of L and ST, on the other hand, it is different from them. For example, it still has 4th symmetry and similar funnel structure. The difference is that the funnel structure is neither like L’s nor ST’s, and there is a hybrid structure between them. This structure is similar to a four-page petal with high symmetry on the diagonal of a crystal. From Figure 7, we can clearly observe that the statistics of different phonons (L, ST, and FT) vary due to their anisotropy, but they all exhibit a common characteristic, which is that they all follow the direction of anisotropy, namely the [100], [111], and [110] directions. That is to say, anisotropy has a regulatory effect on phonons. This is why phonon channel exists even in semiconductors.

4 Conclusions

The anisotropy, lattice wave and phonon properties of InP are studied based on the Bond matrix method and continuum tensor theory. Two different 3D structures of Zener and Every anisotropies give the same result of crystal anisotropy, i.e., [001], [110] and [111] directions. These directions also have the 4th degree symmetry of the crystal. The 3D structures of strain, heavy-hole and light-hole as well as the Phillips iconicity also show the same anisotropy as above. The slowness of L, ST and FT are no longer pure shear waves and longitudinal waves, but become quasi shear waves and quasi shear waves. The zero Gaussian curvature divides the lattice waves into divergent and focusing regions. The anisotropy of the three lattice waves follows the following relation: ST > FT > L, then, phonon focusing, phonon image and caustic are mainly affected by ST. The enhanced distribution of phonons in the [001], [110] and [111] directions indicates that anisotropy has an adjusting effect on phonons.

Corresponding author: Hongzhi Fu, College of Physics and Electronic Information, Luoyang Normal University, Luoyang, 471022, China, E-mail: fhzscdx@163.com

Funding information: This project is supported by the Natural Science Foundation of China (No. 51471082).
Author contribution: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-06-18

Accepted: 2025-12-07

Published Online: 2025-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2025-0251

Keywords for this article

alloys; lattice dynamics; anisotropy; phonon distribution

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