Pressure and Strain Effects on the Structural, Electronic, and Optical Properties of K₄ Phosphorus

1 Introduction

As known, at ambient conditions, white phosphorus, orthorhombic black phosphorus, and various forms of red phosphorus are the stable allotropes of phosphorus. Many allotropes of phosphorus have been synthesised; meanwhile, there are still many allotropes of phosphorus that are in the prediction phase [1], [2], [3]. Two-dimensional (2D) few-layer black phosphorus [4], [5], [6] has been successfully fabricated, and researchers have found that this material is chemically inert and has remarkable transport properties. It was reported that 2D few-layer black phosphorus has a carrier mobility up to 1000 cm²/V ⋅ s, and an on/off ratio up to 10⁴ was achieved for the phosphorene transistors at room temperature [4], [5]. Researchers have been exploring its structural features [7], electronic properties [8], unique mechanical features [9], and so on. Some new phases have been proposed [10], [11]. After the identification of K₄ geometry in mathematics [12], researchers proposed a boron K₄ crystal that is stable under ambient pressure [13]. Moreover, under a pressure >110 GPa, researchers have successfully synthesised the nitrogen K₄ crystal from molecular nitrogen using a laser-heated diamond cell [14]. As known, nitrogen, as the element of the fifth main group of the periodic table, is similar in some respects to phosphorus. Recently, inspired by the unique geometry of the K₄ nitrogen structure, researchers proposed a K₄ phosphorus structure [15], which is a semiconductor with indirect band gap and many unique properties. Pressure usually has a great impact on the structural properties of the semiconductor [16], [17]. Electron characteristics are a very important property of semiconductor nanostructures, and strain has been a very common method to regulate the electronic properties of materials [18], [19]. The study found that nanostructures can still maintain their integrity under great strain [20], [21], which is of great significance to the strength of the extended strain to nanostructures [22], [23], [24]. These studies motivated us to study the pressure and strain effects on the properties of K₄ phosphorus. In this work, at first, we checked the stability of K₄ phosphorus under a hydrostatic pressure. Above 7 GPa, the mechanical stability criteria cannot be satisfied. We calculated the band structure and density of state (DOS) of K₄ phosphorus at 0 GPa, and compared the elastic anisotropies at 0 and 7 GPa. We calculated the ideal tensile stress-strain curves to check the stable range under tensile strain, and studied the influence of tensile strain and pressure on the optical properties and band gap of K₄ phosphorus.

2 Theoretical Method and Computational Details

Our calculations were performed based on the density functional theory (DFT) [25], [26], as implemented in the Cambridge Serial Total Energy Package (CASTEP) code [27]. We employed the Vanderbilt ultrasoft pseudopotentials to describe the electronic interactions in the calculations. The exchange correlation energy was described in the generalised gradient approximation (GGA) using the Perdew-Burke-Enzerhof (PBE) functional [28]. The equilibrium crystal structures were achieved by utilising geometry optimisation in the Broyden-Fletcher-Goldfarb-Shanno method [29]. In our calculations, the electronic wave functions were extended in a plane-wave basis set with energy cut-off of 440 eV. The special k-point method proposed by Monkhorst and Pack [30] was used to characterise energy integration in the first irreducible Brillouin zone, and the k-point mesh was taken as 10 × 10 × 10. Bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio were estimated by using Voigt-Reuss-Hill approximation [31]. In addition, we recalculated the elastic constant calculations by using the Vienna Ab-initio Simulation Package (VASP) [32] at GGA-PBE level. The electron-ion interaction was described by the frozen-core all-electron projector augmented wave method [33]. The energy cut-off of 500 eV and a 10 × 10 × 10 k-point grid were used in the calculations. The Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional [34] was used for the high accuracy of electronic structure calculations and the optical properties calculations.

3 Results and Discussion

K₄ phosphorus has an I2₁3 symmetry and belongs to the space group No. 199. We have established the structure of K₄ phosphorus and optimised the lattice parameters in a conventional cell and optimised the cell, as listed in Table 1. The optimised structures of K₄ phosphorus at 0 GPa are shown in Figure 1. We can see that the lattice parameters we calculated are in an excellent agreement with previous results, so our calculations are reliable. The optimised structure of K₄ phosphorus is body-centred cubic, and in the conventional cell of K₄ phosphorus, there are eight atoms located at the 8a (0.204, 0.204, 0.204) Wyckoff position. The central phosphorus atom is sp³-hybridised, which is connected to its three neighbouring atoms with equal bond lengths and band angles. The bond lengths at 0 GPa become 2.23 Å with the bond angles of 102.3°. As the pressure increases to 7 GPa, they are 2.21 Å with the bond angle of 99.5°. Both the bond lengths and bond angles decrease when the pressure increases.

Figure 1:

Perspective view of the conventional unit cell (a) and crystal structure viewed from the [001] direction (b) of K₄ phosphorus at 0 GPa.

Elastic properties have always been a very important part of the many properties of materials. According to the elastic constant, we can obtain the stability, hardness, and anisotropy information of the materials. In a cubic phase structure, the mechanical stability criteria for K₄ phosphorus at 0 GPa are as follows [35]:

Under isotropic pressure, the criteria of mechanical stability are provided in [36]:

where

where P is the isotropic pressure.

The calculated crystal elastic constants C_ij of K₄ phosphorus under various pressures are shown in Table 2. We can see that the calculated elastic constants at 0 GPa are much smaller than those in [15]. Then we recalculated the elastic constants by using the strain-stress method [37], which is implemented in VASP code, and the results are also listed in Table 2. Clearly, the results calculated by using VASP are in a good agreement with those calculated via CASTEP. Thus, we think our results are reliable. A possible explanation for the discrepancy with [15] is as follows: the results reported in [15] might correspond to the elastic tensor in the rigid ion approximation only, not including the contribution due to the relaxation of the internal degrees of freedom, which makes a material stiffer. From Table 2, we can also see that K₄ phosphorus is mechanically stable up to 7 GPa. In addition, to ensure its dynamical stability at 7 GPa, we calculated the phonon spectra. As Figure 2a shows, there is no imaginary frequency in the whole Brillouin zone, indicating that K₄ phosphorus is dynamically stable up to 7 GPa. For comparison, the elastic constants and bulk modulus of black phosphorus are also calculated. It can be seen that our calculated results are in good agreement with those in [38] and the Materials Project [39]. The bulk modulus of K₄ phosphorus (20.1 GPa) is lower than that of black phosphorus (43.9 GPa), which means K₄ phosphorus has larger stiffness than black phosphorus.

Figure 2:

Phonon spectra of K₄ phosphorus at 7 GPa (a) and tensile strength of K₄ phosphorus (b).

In order to better characterise the intrinsic hardness of K₄ phosphorus, the ideal stress-strain curves for large strains were calculated. The ideal strength of a material generally means the stress that is applied to a perfect crystal that makes the crystal mechanically unstable. This stress sets an upper limit for material strength. For cubic phase, we only need to consider the a-direction as the tension direction. As seen from Figure 2b, the ideal tensile strength is 8.5 GPa with the strain of 0.3, which means that K₄ phosphorus would cleave if the strain is >0.3.

In crystal physics and engineering science, the bulk modulus B denotes the resistance to fracture, and the shear modulus G represents the resistance to plastic deformation. According to the Voigt-Reuss-Hill approximations, for cubic phase [31], [40], [41]:

we can obtain the bulk modulus B and shear modulus G:

The ratio of bulk to shear modulus (B/G) proposed by Pugh [42] is an indication of a ductile or brittle character. A larger B/G ratio is associated with a more facile ductility, whereas a smaller B/G ratio corresponds to a brittle nature. If B/G > 1.75, we consider that the material is ductile [43]; otherwise, the material is brittle. From Table 2, we can see that the B/G ratio of K₄ phosphorus is 1.44 at 0 GPa and 2.37 at 7 GPa; thereby, as pressure increases from 0 to 7 GPa, K₄ phosphorus changes from being brittle to being ductile.

To obtain more information about the elastic properties, Young’s modulus E and Poisson’s ratio ν were calculated. As a measure of the stiffness of a solid material, Young’s modulus E is defined as the ratio between stress and strain. When a material receives a tension or compression in one direction, the absolute value of the ratio of transverse contraction strain to longitudinal extension strain is called Poisson’s ratio, ν. They are given by [31], [44], [45]:

From Table 2, the Young’s modulus E of K₄ phosphorus at 7 GPa is smaller than that at 0 GPa. As known, the larger the value of E, the stiffer the material. We can see that at 0 GPa, K₄ phosphorus possesses the largest stiffness, and when the pressure increases, the stiffness gets smaller. The typical value of ν is 0.1 for covalent materials and 0.33 for metallic materials [46]. We can see that the Poisson’s ratios of K₄ phosphorus increases when pressure increases, indicating that the directionality degree of covalent bonding becomes weaker when pressure increases.

It is very important to calculate the elastic anisotropy of crystal for the study of its physical and chemical properties. For cubic symmetry, we follow Zener and Siegel [47] and use A = 2C₄₄ / (C₁₁ − C₁₂) to calculate the elastic anisotropy. The value of 1.0 indicates isotropy, and any deviation from 1.0 indicates a degree of the shear anisotropy. The calculated Zener anisotropy factor A at 0 and 7 GPa are listed in Table 2, which show an anisotropy that increases with pressure. To show the elastic anisotropy of K₄ phosphorus at 0 and 7 GPa in detail, Young’s moduli for all possible directions are shown in Figure 3. For an isotropic system, the three-dimensional directional dependence shows a spherical shape, while the deviation degree from the spherical shape represents the anisotropy [48]. At 0 GPa, the maximum of Young’s modulus E_max is 120 GPa, while the minimum of Young’s modulus E_min is 54 GPa, and the average value over all directions is 80 GPa. The ratio E_max/E_min is 2.22. At 7 GPa, the maximum of Young’s modulus E_max is 191 GPa, while the minimum of Young’s modulus E_min is 23 GPa, and the average value over all directions is 62 GPa, with a ratio E_max/E_min of 8.30. These results indicate that the anisotropy increases when the pressure increases, which is consistent with the Zener anisotropy factor calculations.

Figure 3:

Direction dependence of Young’s modulus (GPa) for K₄ phosphorus at 0 GPa (a) and 7 GPa (b).

The acoustic velocity is of great significance to the study of the chemical bonding characteristics, and the symmetry and propagation direction of the crystal determine the acoustic velocity. According to the single crystal elastic constants, Brugger [49] has successfully calculated the phase velocities of pure transverse and longitudinal modes. The cubic structure has three directions, [001], [110], and [111], for the pure transverse and longitudinal modes, and other directions are for the quasi-transverse and quasi-longitudinal waves. For a cubic phase, the acoustic velocities in the principal directions are as follows [50]:

for [100],

for [110],

for [111],

where v_l is the longitudinal acoustic velocity; v_t1 and v_t2 are the first transverse mode and the second transverse mode, respectively. The density of the structure is ρ. The acoustic velocities are calculated based on the elastic constants. Therefore, we can see the anisotropy of elasticity according to the anisotropy of the acoustic velocities. The calculated average longitudinal acoustic velocity is vlm=(B+4G/3)/ρ, and the average transverse acoustic velocity is vtm=G/ρ. The average acoustic velocity is vm=[(2/vtm3+1/vlm3)/3]−1/3.

Based on the calculation results of the average acoustic velocity, we can get the Debye temperature: ΘD=hkB[3n4π(NAρM)]1/3vm, where N_A is Avogadro’s number; h and k_B are the Planck and Boltzmann constants, respectively; n is the total number of atoms in the formula unit; M is the mean molecular weight; and ρ is the mass density.

The acoustic velocities and Debye temperatures of K₄ phosphorus at 0 and 7 GPa are listed in Table 3. We can see that the density increases when the pressure increases. As the pressure increases from 0 to 7 GPa, the average acoustic velocity decreases by 11.36% and the Debye temperature decreases by 7.34%. The strength of the covalent bond in solids is related to the Debye temperature, so the strength of the covalent bond becomes weaker for K₄ phosphorus when the pressure increases.

Electronic structure is also crucial for studying the physical and chemical properties of materials. Thus, we calculated the band structure and DOS of K₄ phosphorus at 0 GPa. The calculated electronic band structure of K₄ phosphorus in primitive cell is plotted in Figure 4a. The black dashed line represents the Fermi level (E_F). The black curve represents the results we calculated using the PBE functional. The black arrow points from the valence band maximum (VBM) to the conduction band minimum (CBM). The VBM locates at (0.3077, −0.3077, 0.3077) along the G-H direction; the CBM locates at (0.0385, −0.0385, 0.0385) along the G-H direction; and the band gap is 1.08 eV. Clearly, K₄ phosphorus is an indirect semiconductor. It is known that the band gap calculated by DFT usually should be smaller than the real values; thus, we used the more precise HSE06 functional to correct the band gap of K₄ phosphorus. As shown in Figure 4a, the red curve represents the results we calculated using the HSE06 functional. It can be seen that, using both functionals, similar band structures were obtained. The VBM and CBM calculated by the HSE06 functional locate at the same positions as those calculated by the PBE functional, and the band gap of K₄ phosphorus calculated with the HSE06 functional increases to 1.74 eV.

Figure 4:

Band structure (a) and DOS (b) of K₄ phosphorus.

The DOS of K₄ phosphorus in a single atom is shown in Figure 4b. The black dashed line represents the Fermi level (E_F). As Figure 4b shows, we can see that the valence band region can be divided into two parts, the first part (−15 to 7 eV) is characterised by the contributions of s states, whereas the second part (−7 to 0 eV) originates from the contributions of p states. The conduction band region is mainly characterised by the p states. The DOS near Fermi level mainly originated from the p orbital electrons.

To study the influence of hydrostatic pressure on the energy band gap of K₄ phosphorus, we calculated the band gap as a function of pressure. Usually, the trends of change of the band gap calculated by the PBE functional and the HSE06 functional are similar [51]; thus, we used the PBE functional to calculate the band gap. The changes of band gap are shown in Figure 5. We found that, with the increase of pressure, the band gap is diminishing, and the band gap decreases almost linearly from 1.08 to 0.33 eV when the pressure increases from 0 to 7 GPa. We also calculated the band gap as a function of strain. As Figure 5 shows, when K₄ phosphorus is stretched from the strain of 0.0–0.1, the band gap decreases almost linearly from 1.08 to 0.79 eV. When it is contracted from the strain of 0.0 to −0.1, the band gap decreases from 1.08 to 0.21 eV. Thus, we can conclude that irrespective of whether K₄ phosphorus is pressurised or strained, the band gap will become smaller. As we know, the value of band gap is crucial for efficient optoelectronic devices. Thereby, we can artificially control the strain and pressure to choose any desired value.

Figure 5:

Variation of band gap versus pressure and strain of K₄ phosphorus.

We calculated the imaginary parts of dielectric function (ε₂) to study the influences of pressure and strain on the optical absorption properties. The spectral range is divided into three domains, which are infrared, visible, and ultraviolet regions, from left to right. As Figure 6a shows, in the visible region, the optical absorption of K₄ phosphorus is much stronger than that of diamond silicon, and with the pressure increase on K₄ phosphorus, the absorption coefficients have a great improvement, and the absorption coefficients of K₄ phosphorus are getting closer to the solar spectrum. When the pressure reaches 7 GPa, K₄ phosphorus has the best optical absorption properties. In ultraviolet regions, with the pressure increase, the absorption coefficients have a small improvement. For the visible region (see Fig. 6b), when the strain applied to K₄ phosphorus is from 0 to −0.08, the absorption coefficients have a great improvement, and the optical absorption is stronger than that of diamond silicon. Meanwhile, the absorption coefficients of K₄ phosphorus are getting closer to the solar spectrum. When the strain applied to K₄ phosphorus is −0.08, they have the best optical absorption properties. In the ultraviolet region, when the strain applied to K₄ phosphorus is from 0 to −0.08, the absorption coefficients also have a great improvement; however, when the energy is >4 eV, K₄ phosphorus exhibits worse optical absorption than diamond silicon. When the strain applied to K₄ phosphorus is from 0 to 0.08, the changes of absorption coefficients are tiny, and the optical absorption properties become a little worse. All the strong absorption coefficients are in the visible range of sunlight. Therefore, in the study of the absorption characteristics of the material in the visible region, K₄ phosphorus would play a crucial role in its promotion.

Figure 6:

Calculated imaginary parts of the dielectric function (ε₂) as a function of energy under pressures (a) and strains (b).

4 Conclusions

In summary, we confirmed the mechanical and dynamic stabilities of K₄ phosphorus up to 7 GPa under a hydrostatic pressure. We calculated the lattice parameters, cell volume, elastic constants, bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, average acoustic velocity, and the Debye temperature of K₄ phosphorus at 0 and 7 GPa. We found that with the pressure increasing, the stiffness become smaller, the anisotropy increases, and the average acoustic velocity and the Debye temperature decrease. The ideal tensile strength is 8.5 GPa and the strain is 0.3. By studying the band gap of K₄ phosphorus as a function of hydrostatic pressure and strain, we find that the pressure and strain will reduce the band gap. We also calculated the imaginary parts of dielectric function to study the influences of pressure and strain on the optical absorption properties, from which we found that when the pressure increases, the optical absorption properties become improved. When the strain applied to K₄ phosphorus is from 0 to −0.08, the optical absorption properties also improve; however, when the strain is from 0 to 0.08, the optical absorption properties become worse.