Superdense Hexagonal BP and AlP with Quartz Topology: Crystal Chemistry and DFT Study

Abstract

The superdense hexagonal phosphides BP and AlP, whose structures are formed by distorted tetrahedra and characterized by quartz-derived (qtz) topology, were predicted from crystal chemistry and first principles as potential high-pressure phases. From full geometry structure relaxations and ground state energy calculations based on quantum density functional theory (DFT), qtz BP and AlP were found to be less cohesive than the corresponding cubic zinc-blende (zb) phases with diamond-like (dia) topology, but were confirmed to be mechanically (elastic constants) and dynamically (phonons) stable. From the energy–volume equations of state, qtz phases were found to be energetically favorable at small volumes (high pressures), with zb-to-qtz transition pressures of 144 GPa for BP and 28 GPa for AlP. According to the electronic band structures and the site projected density of states, both phosphides exhibit larger band gaps of the zinc-blende phases compared to the qtz phases; the smaller values for the latter result from the smaller volumes per formula unit, leading to increased covalence.

1. Introduction

III–V compound semiconductors composed of III^A-group electropositive elements (B, Al, Ga, In) combined with V^A-group electronegative elements (N, P, As, Sb) have long been considered promising material bases for a number of well-established commercial technologies, as well as for new and advanced classes of electronic and optoelectronic devices [1,2,3,4]. Examples include heterostructure and high-electron-mobility bipolar transistors, light-emitting diodes, diode lasers, photodetectors, and electro-optic modulators. The physical properties of the constituent materials are critical to the operating characteristics of these devices.

Under normal conditions, most III-V semiconductor compounds crystallize in the cubic zinc-blende (zb) structure (space group $F\overline{4}3m$, No. 216), and these compounds can be considered as a single series in which there is a gradual change of chemical bonding from the covalent type to the iono-covalent type, while the bonding energy is virtually determined by the first tetrahedral coordination sphere.

For III-V compounds of light elements, while zb-BN is characterized by a large band gap (E_g = 6.36(3) eV [5]), a value three times smaller is reported for zb-BP: E_g ≈ 2.1 eV [6]. In fact, boron nitride is polar covalent with significant charge transfer from B to N as opposed to BP, where less charge transfer is expected. The difference in behavior has a chemical origin derived from the electronegativity (χ) values according to Pauling: χB= 2.04; χN = 3.04 with Δχ(BN) = 1. Conversely, with χP = 2.19, Δχ(BP) = 0.15, and BP is considered as a covalent semiconductor (cf. [7]). As for Al belonging to the 2nd period, χAl = 1.50, and for AlP Δχ(_Al-P) = |0.69|, i.e., more than four times greater than in the case of BP.

Recently, we found a significant densification (with increased hardness) for BN and SiC when adopting a quartz-derived structure, i.e., qtz topology [8]. The aim of the present work is to extend the model approach to equiatomic monophosphides of III^A-group elements and to consider the possibility of their phase transitions at high pressure into superdense quartz-based hexagonal phases. Such high-pressure structures have been identified for zinc chalcogenides in addition to the cubic zinc-blende polymorphs adopted by the other II–VI compounds [9].

As in previous works [7,8], our investigations are based on calculations of energies, structures, and derived properties within the density functional theory (DFT) quantum mechanics framework [10,11]. Specifically, superdense qtz BP and AlP were found to be cohesive, albeit with smaller magnitudes than those known for zinc-blende BP and AlP with dia topology (vide infra, Section 3.1), and were identified as both mechanically (elastic constants) and dynamically (phonon band structures) stable. From the establishment of the respective equations of states (EOS), the trends of zb-BP and zb-AlP transformations to qtz high-pressure forms maintaining tetrahedral coordination have been identified. The electronic band structures of the new qtz phases show reduced band gaps compared to zinc-blende polymorphs.

2. Computational Methodology

The identification of the ground state structures corresponding to the energy minima and the prediction of their mechanical and dynamical properties were carried out by DFT-based calculations using the Vienna Ab initio Simulation Package (VASP) code [12,13] and the projector augmented wave (PAW) method [13,14] for the atomic potentials. Exchange correlation (XC) effects were considered using the generalized gradient functional approximation (GGA) [15]. The relaxation of the atoms to the ground state structures was performed with the conjugate gradient algorithm according to Press et al. [16]. The Blöchl tetrahedron method [17], with corrections according to the Methfessel and Paxton scheme [18], was used for geometry optimization and energy calculations, respectively. Brillouin zone (BZ) integrals were approximated by a special k-point sampling according to Monkhorst and Pack [19]. Structural parameters were optimized until the atomic forces were below 0.02 eV/Å and all stress components were <0.003 eV/ų. The calculations were converged at an energy cutoff of 400 eV for the plane-wave basis set in terms of the k-point integration in the reciprocal space, from k_x(6) × k_y(6) × k_z(4) up to k_x(12) × k_y(12) × k_z(8) for the final convergence and relaxation to zero strains. The plane wave energy cutoff implicit of the PAW potentials was 300 eV. In the post-processing of the ground state electronic structures, charge density projections were operated on the lattice sites.

The mechanical stability and hardness were obtained from the elastic constant calculations. The analysis of the elastic tensors was carried out using the ELATE software [20], which provides the bulk (B), shear (G), and Young’s (E) moduli along different averaging methods; Voigt’s method [16] was chosen here. Vickers hardness (H_V), according to elastic properties, was evaluated using the empirical Mazhnik–Oganov model [21].

The dynamic stabilities of the considered phases were confirmed by the positive phonon magnitudes. The corresponding phonon band structures were obtained from a high resolution of the Brillouin zone, according to Togo and Tanaka [22]. The electronic band structures were obtained using the all-electron DFT-based ASW method [23] and GGA XC functional [15]. The VESTA (Visualization for Electronic and STructural Analysis) [24] program was used to visualize the crystal structures and charge densities.

3. Results and Discussion

3.1. Crystal Structure

With the aim of classifying structures into families, a topological representation of crystal types has been proposed using the TopCryst system [25]. Cubic diamond has dia topology, and its rare hexagonal form, called “lonsdaleite”, has lon topology. These two topological types are characteristic of many crystal systems. For example, zinc-blende phases have dia topology and wurtzite phases have lon topology. Recently, a hexagonal tricarbon C₃ allotrope was claimed by ab initio studies to be superdense and superhard with an assigned qtz topology [26]. Later, we proposed qtz C₆, which is characterized by an ultra-hardness comparable (or even superior) to that of diamond [27]. For the sake of completeness, we note that the qtz topology is also adopted by trigonal binary phases [8].

The III–V compounds are found to crystallize:

• In tetrahedral coordination as zinc-blende ZnS (space group $F\overline{4}3m$, No. 216; dia topology) and wurtzite ZnO (space group P6₃mc, No. 186; lon topology);
• In octahedral coordination as rock salt NaCl (space group $Fm\overline{3}m$, No. 225; pcu topology) and NiAs (space group P6₃/mmc, No. 194; seh topology).

Knowing that a tetrahedral void is larger than an octahedral void, one might expect denser NaCl and NiAs-type structures to prevail at high pressures. However, an interesting feature of the topology of qtz systems, i.e., very high density, is used in the present work to propose new, hypothetical high-pressure phases with tetrahedral topology for the well-known III–V phosphides, BP and AlP, based on crystal chemistry and DFT studies. Gallium and indium phosphides are not considered here due to their dynamic instability (see Section 3.5 for details).

The ground state structures of zinc-blende BP and AlP belong to the high-symmetry space group $F\overline{4}3m$, where B/Al atoms occupy the 4a position and P atoms occupy the 4c position. The corresponding structures presented in Figure 1 show a perfectly regular arrangement of tetrahedra with a characteristic tetrahedral angle of ∠109.47°. The calculated structure parameters are very close to the experimental values [28,29] (cf.

Table 1. Crystal structure parameters of BP and AlP polymorphs.

	zb-BP (Z = 4) $\mathit{F}\overline{4}3\mathit{m}$ (No. 216)	qtz BP (Z = 3) P6422 (No. 181)	zb-AlP (Z = 4) $\mathit{F}\overline{4}3\mathit{m}$ (No. 216)	qtz AlP (Z = 3) P6422 (No. 181)
a, Å	4.541 (4.538 [28])	3.274	5.501 (5.463 [29])	3.854
c, Å	–	6.983	–	8.658
Vcell, Å3	93.64	64.82	166.50	111.25
V/FU, Å3	23.41	21.61 ΔVzb-qtz/FU = 1.80	41.63	37.08 ΔVzb-qtz/FU = 4.55
Shortest bond, Å	1.97	2.00	2.38	2.40
Angles (deg.)	109.47	109.17/90.2	109.47	106.39/92.20
Atomic positions	B (4a) 0, 0, 0 P (4c) ¼, ¼, ¼	B (3c) ½, 0, 0 P (3d) ½, 0, ½	Al (4a) 0, 0, 0 P (4c) ¼, ¼, ¼	Al (3c) ½, 0, 0 P (3d) ½, 0, ½
Etotal, eV Ecoh/atom, eV Etotal/FU, eV	−51.59 −1.16 −12.89	−35.73 −0.67 −11.91 ΔE/FU = −0.98 eV	−41.47 −0.88 −10.36	−29.80 −0.66 −9.94 ΔE/FU = −0.427 eV

).

The alternative description of the qtz C₆ structure (space group P6₅22, No. 179 with unique (6a) sites [27]) is possible in space group P6₄22, No. 181, with two distinct Wyckoff positions, namely, (3c) and (3d), as shown in Table 1. Such a configuration corresponds to the binary equiatomic phosphides BP and AlP. The lattice parameters of the ground state structures are given in Table 1. The corresponding crystal structures are shown in Figure 1 with ball-and-stick and tetrahedral representations, with the latter featuring corner-sharing irregular tetrahedra (vide infra). The atoms occupy the Wyckoff positions (3c) ½, 0, 0 and (3d) ½, 0, ½. Differences can be observed in the volumes (total and atom-averaged) and the interatomic distances, which increase due to the significant increase in the atomic radius from B to Al: 0.83 Å versus 1.83 Å; r(P) = 1.10 Å. Looking at the angles related to the constituent tetrahedra, they differ significantly from the regular tetrahedral type (∠109.47°), thus showing the specificity of the qtz topology. Energies in eV are given in the last row of Table 1. The total energy was used to extract the atom-weighted cohesive energy (E_coh/atom) obtained by subtracting the energy of each atom, a process conducted in a large box, i.e., E(B) = −5.33, E(Al) = −3.37, and E(P) = −5.24. At negative E_coh/atom values, all four phases were cohesive, albeit with larger magnitudes for the ground state zinc-blende phases than for the less cohesive qtz structures. The other energy-relevant result is shown for the total energies per formula unit (FU): the qtz phases had higher energies than the zinc-blende ones, which are known as ground state structures under normal conditions. This observation, together with the smaller qtz volumes per FU, suggests possible phase transitions at high pressures (vide infra). Note that the ΔE/FU of AlP, which is two times smaller than that of BP, should allow a phase transition at a lower pressure. This will be discussed in Section 3.4.

3.2. Projections of the Charge Densities

For a qualitative assessment, Figure 2 shows the projected charge densities (yellow volumes) that were found to be centered on the electronegative element P, with a shape difference between BP and AlP. In BP, a tetrahedral shape of the charge density on P can be observed, with a skew towards B. In contrast, in AlP, a charge centered on P is observed.

Such observations were further quantified by treating the respective charge density output files using the AIM (atoms in molecules theory) approach developed by Bader [30]. The charge density in a chemical compound reaches a minimum between atoms, thus defining a region that separates atoms from each other. The results of the charge transfer δ were δ(BP) = 0.30 and δ(AlP) = 2.60, indicating a stronger trend from covalent BP to ionic AlP, which can be explained by the electronegativity χ difference between the constituent atoms (see Introduction).

3.3. Mechanical Properties

The analysis of the mechanical properties of the new qtz phosphides was carried out by calculating the elastic tensor through finite distortions of the lattice. The calculated sets of elastic constants C_ij (i and j correspond to directions) for cubic zinc-blende and hexagonal qtz and NiAs-type phases are given in

Table 2. Elastic constant C_ij of BP and AlP polymorphs, and qtz C₆. Bulk (B_V), shear (G_V), and Young’s (E_V) moduli calculated by the Voight averaging. All values are in GPa.

	C11	C12	C13	C33	C44	C66	BV	GV	EV
qtz C6 [27]	1186	88	64	1162	549	517	441	550	1165
zb-BP	336	77	77	336	180	180	163	160	362
qtz BP	417	45	62	380	186	199	172	186	411
zb-AlP	125	63	63	125	58	58	84	46	117
NiAs-AlP	166	81	68	166	42	46	106	47	123
qtz AlP	158	51	60	175	53	52	93	53	133

. For comparison, the values obtained for qtz C₆ [27] were added. All C_ij values were positive, suggesting mechanically stable phases. qtz C₆ had the largest C_ij values, while qtz BP and qtz AlP had significantly smaller elastic constants. The new phases can be fully described by the bulk (B), shear (G) and Young’s (E) moduli obtained by Voight’s averaging [31] of the elastic constants using ELATE software [20]. The last three columns of Table 2 show the elastic moduli, with values that follow the trends observed for C_ij. It is evident that AlP is significantly more compressible than BP, both in the case of the cubic (zb) and the hexagonal (qtz and NiAs-type) phases.

The Vickers hardness (H_V) of the new phosphides was predicted using two contemporary models: the thermodynamic (T) model [32], which is based on thermodynamic properties and crystal structure and shows perfect agreement with available experimental data [33,34,35], and the empirical Mazhnik–Oganov (MO) model [21], which is based on elastic properties. The results are summarized in

Table 3. Lattice parameters, density (ρ), and Vickers hardness (H_V) of BP and AlP polymorphs, and qtz C₆.

	Space Group	a = b (Å)	c (Å)	ρ (g/cm3)	HV (GPa)
					T †	MO ‡
qtz C6 #179 [27]	P6522	2.5975	5.5859	3.666	102	105
zb-BP #216	$F\overline{4}3m$	4.5410		2.964	30	28
qtz BP #180	P6222	3.2739	3.211	3.211	33	34
zb-AlP #216	$F\overline{4}3m$	5.4625		2.362	10	6
NiAs-AlP #194	P63/mmc	3.6084	2.967	2.967	8	6
qtz AlP #180	P6222	3.8535	2.595	2.595	11	6

^† Thermodynamic model [32]. ^‡ Mazhnik–Oganov model [21].

. Both qtz and zinc-blende BP are (super)hard phases, while the hardnesses of all AlP polymorphs are three times lower. The absence of differences in the hardnesses of AlP polymorphs calculated by the empirical MO model is apparently due to the limitations of this model.

3.4. Equations of State and Possible High-Pressure Phase Transitions

To determine energy trends when considering different crystal structures of a solid, equations of state (EOS) are required, i.e., one cannot rely solely on the quantities obtained from plane lattice optimizations, especially when comparing the energies and volumes of different phases. The underlying physics mean that the calculated total energy corresponds to the cohesion within the crystal, as discussed above. The solutions to the Kohn–Sham DFT equation indicate the energy in terms of infinitely separated electrons and nuclei. But the zero of the energy depends on the choice of pseudopotentials; then, it becomes arbitrary by its shift, not by scaling. However, the energy derivatives and the EOS remain unchanged. Therefore, it is necessary to obtain the EOS and extract the fitting parameters for an evaluation of the equilibrium values. This was carried out from a series of calculations of the total energy as a function of volume for the cubic and hexagonal phases of BP and AlP. The resulting E(V) curves, shown in Figure 3, were fitted to the third-order Birch equations of state [36]:

E(V) = E₀(V₀) + (9/8)∙V₀B₀[([(V₀)/V])^⅔ − 1]² + (9/16)∙B₀(B₀′ − 4)V₀[([(V₀)/V])^⅔ − 1]³,

where E₀, V₀, B₀, and B₀′ are the equilibrium energy; volume; bulk modulus; and its first pressure derivative, respectively (see

Table 4. Calculated and experimental properties of BP and AlP polymorphs: bulk moduli (B₀) and their first pressure derivatives (B₀′); total energies (E₀) and volumes (V₀) per formula unit; zb-to-qtz transition pressures (p_tr) and reduced volumes (V_tr/V₀).

	BP			AlP
	zb		qtz	zb		qtz
	calc.	exp.	calc.	calc.	exp.	calc.
B0 (GPa)	158	174(2) [37]	165	83	88 [38]	86.0
B0′	3.89	3.2(2) [37]	4.0	3.86	4 [38]	4.07
E0/FU (eV)	−12.90	–	−11.89	−10.37	–	−9.92
V0/FU (Å3)	23.41	23.36 [28]	21.61	41.63	40.75 [29]	37.08
Vtr/V0	0.67	–	–	0.81	–	–
ptr (GPa)	144	–	–	28	–	–

).

In the case of boron phosphide (Figure 3a), an intersection of E(V) curves of the zinc-blende and qtz phases can be observed at V/FU = 15.6 Å, which, for zb-BP, is equivalent to a reduced volume of 0.67. The corresponding pressure can be calculated by the Murnaghan equation [39]:

p = B₀/B₀′ [(V₀)/V])^B₀′ − 1]

using the experimental values of B₀ and B₀′ (Table 4), and is equal to 144 GPa. Thus, it can be assumed that the phase transformation of zb-BP into qtz BP can occur at pressures not lower than 144 GPa, which is in agreement with the experimental data on the absence of room-temperature phase transitions of zb-BP up to 110 GPa [40].

In the case of aluminum phosphide (Figure 3b), the intersection of the E(V) curves of zb-AlP and qtz AlP was observed at V/FU = 33.84 Å (V/V₀ = 0.81 for zb-AlP). The corresponding pressure calculated using the experimental B₀ and B₀^′ values of zb-AlP (Table 4) was equal to 28 GPa, i.e., qtz AlP formation could be expected at higher pressures. However, at room temperature, the phase transformation of zb-AlP into a high-pressure phase with a NiAs-type structure, which would be energetically favorable over the whole pressure range, was observed at about 10 GPa [41]. Thus, the formation of qtz AlP at sufficiently high (>28 GPa) pressures can only be possible due to alternative metastable behavior under far-from-equilibrium conditions (shock compression, etc.).

3.5. Dynamic and Thermodynamic Properties from the Phonons

An important criterion for phase dynamic stability can be obtained from the phonon properties. Phonons are vibrational quanta whose energy is quantized using the Planck constant h in its reduced form ћ (ћ = h/2π), which gives the phonons’ energy: E = ћω (frequency: ω). The phonon band structures (red lines) obtained following the protocol presented in Section 2 are shown in Figure 4. The horizontal direction shows the main directions of the hexagonal center of the Brillouin zone, while the vertical direction shows the ω frequencies, given in units of terahertz (THz). The absence of negative frequencies indicates dynamically stable systems. The band structure includes 3N bands with 3N−3 optical modes found at higher energies than three acoustic modes starting from zero energy (ω = 0) at the Γ point, the center of the BZ, up to a few terahertz. They correspond to the lattice rigid translation modes of the crystal (two transverse and one longitudinal). While the different panels show positive frequency (ω) magnitudes, a slightly negative phonon curve can be observed along Γ-A (vertical direction), around Γ, the center of the BZ. This indicates a slight instability of the longitudinal acoustic mode, which may result from the volume of AlP (37.08 ų/FU), almost double compared to BP (21.60 ų/FU). The resulting openness of the AlP structure is consistent with the structurally dense qtz topology. This observation explains the reason why GaP and InP, characterized by large atomic radii of Ga (1.35 Å) and In (1.47 Å), could not be dynamically stabilized in qtz topology, as follows from our preliminary study (Figure 4e,f).

The thermodynamic properties of the new qtz phases were calculated from the phonon frequencies using the statistical thermodynamic approach [42] on a high-precision sampling mesh in the Brillouin zone. The temperature dependencies of the heat capacity at constant volume (C_v) and entropy (S) of qtz BP and qtz AlP are shown in Figure 5 in comparison with experimental C_p data for zb-BP [43,44] and zb-AlP [45]. It is easy to see that the heat capacity of zb-BP is higher than that of qtz BP in the entire temperature range which was studied, while the heat capacity of zb-AlP in the range of 300–1000 K practically coincides with the heat capacity of qtz AlP. Experimental data on the low-temperature heat capacity of zb-AlP are not available in the literature.

3.6. Electronic Band Structures

Using the crystal parameters in Table 1, electronic band structures were obtained using the all-electrons DFT-based augmented spherical method (ASW) [23], and are shown in Figure 6. The bands (blue lines) developed along the major directions of the hexagonal Brillouin zones. The zero energy along the vertical axis was considered with respect to E_V, at the top of the filled valence band (VB), separated from the empty conduction band (CB) by a band gap. For both phosphides, the zinc-blende phase had a larger band gap compared to the qtz phase, with the smaller magnitude for the latter resulting from the smaller volume per FU, leading to increased covalence.

The band structures are mirrored by the site-projected densities of states (DOS) shown in the right part of Figure 6. As a general feature, the P(DOS) showed higher intensity than the B(DOS) and Al(DOS) due to the lower electronegativities of boron and aluminum compared to phosphorus, as discussed above. The Fermi level is represented by a vertical line at 0 along the horizontal energy axis, while the vertical axis corresponds to the DOS expressed in eV⁻¹ units, translating the number of states per unit energy. The valence band (VB) was wider in BP than in AlP, with the characteristics of a small separation between s-prevailing states from −16 to −10 eV and the p-prevailing states from −9 eV to E_V (top of the VB) in BP, and a larger separation for s-states from −12 to −9 eV and p-states from −6 to E_V in AlP. Such a feature can be related to the larger interatomic distances in AlP vs. BP.

4. Conclusions

Based on crystal chemistry and DFT calculations, superdense hexagonal (P6₄22) BP and AlP were proposed as potential high-pressure phases with quartz-derived topology. All binary phases were found to be cohesive, with higher energies for quartz-like topologies than for ambient pressure zinc-blende phases. Subsequent energy–volume equations of states revealed that qtz BP and AlP dominate at low volumes, and the threshold pressures of the cubic-to-hexagonal phase transition are 144 and 28 GPa, respectively. At the local level, the transition mechanism is related to the distortion of the BP4/AlP4 tetrahedra that characterizes the qtz topology. The electronic band structure, reflected by the site-projected density of states (DOS), shows that smaller-volume qtz phases lead to increased covalence of both phosphides, resulting in lower band gap values compared to the corresponding zinc-blende phases.