The structural, electronic and vibrational properties of InN under pressures up to 20 GPa have been investigated using the pseudo-potential plane wave method (PP-PW). The generalized-gradient approximation (GGA) in the frame of density functional theory (DFT) approach has been adopted. It is found that the transition from wurtzite (B4) to rocksalt (B1) phase occurs at a pressure of approximately 12.7 GPa. In addition, a change from a direct to an indirect band gap is observed. The mechanism of these changes is discussed. The phonon frequencies and densities of states (DOS) are derived using the linear response approach and density functional perturbation theory (DFPT). The properties of phonons are described by the harmonic approximation method. Our results show that phonons play an important role in the mechanism of phase transition and in the instability of B4 (wurtzite) just before the pressure of transition. At zero pressure our data agree well with recently reported experimental results.

Introduction

The III-nitrides group AlN, GaN and InN and their alloys have become an important class of semiconductor materials for optoelectronic applications, such as blue light emitting diodes (LEDs) and lasers. Amongst the III-nitrides InN is the least explored, due to difficulties in synthesizing high quality single crystal. Very recently, these problems have been overcome and some of the key band parameters have been conclusively determined [1,2]. However, currently one of the most controversially discussed parameter is the nature of the band-gap energy of InN.

The behavior of lattice vibrations under high pressure provides useful information concerning structural instability; phase transformations, phonon–electron interactions, bonding properties and carriers transport properties. In addition, many thermodynamic phenomena such as Raman and neutron diffraction spectra behavior, thermal expansion, specific heats and heat conduction can be investigated. It is worth pointing out that the study of electronic and vibrational structure of InN under pressure is a highly challenging task in terms of fundamental physics and high-pressure experiments.

The aim of this work is therefore to determine the properties of the wurtzite (B4) and rocksalt (B1) phases for a wide range of pressures up to 20 GPa.

Computational methods

The calculations reported in this work were performed using the Cambridge Serial Total Energy Package (CASTEP) code [3,4]. In this package, the density functional theory [5] and the Kohn–Sham approach were used to calculate the fundamental eigenvalues [6]. In order to reduce the basis set of plane wave (PW) functions used to describe the real electronic functions, the pseudo-potential (PP) approximation was introduced, where the nucleus and the core electrons were substituted for an effective potential. These approaches, which are very important in terms of memory usage and computing time, are essential factors in first principles methods.

The exchange-correlation effects were treated within the generalized gradient approximation (GGA-PBE) [7]. Coulomb potential energy produced by electron–ion interaction is described using Vanderbilt's ultra-soft pseudo-potentials [8], in which the orbitals of In (4d105s25p1), and N (2s22p3) are treated as valence electrons. The main parameter in first principles calculations is the ground state energy [9], which is strongly affected by the energy cut off and the Brillouin zone sampling [4]. The coordinate optimization in the strained lattice is realized using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization algorithm by taking into account the total energy as well as the gradients to relax the atomic positions [10,11].

The cut-off energy for the plane wave expansion is 380 eV. The Brillouin zone sampling was carried out using the 9×9×6 and 5×5×5 set of Monkhorst–Pack mesh [12] division of the reciprocal unit cell for hexagonal and cubic structures, respectively. The sampling k points with distances between grind points are less than 0.04 Å−1. These values ensure a good convergence, and consequently provide reliable results. Throughout this study, the maximum tolerance on the energy and the force was less than 5.10−6 eV/atom and 10−2 eV/Å, respectively.

The calculated total energies and pressures as a function of the unit cell volume for InN are used to determine the structural properties, the bulk modulus and its pressure derivative by fitting the data to a third-order Birch–Murnaghan equation of states (EOSs) [13].

To determine the vibrational properties the linear response method and density functional perturbation theory (DFPT) are used. DFPT is one of the most popular methods of ab initio calculation of lattice dynamics [14]. However, the applicability of the method extends beyond the study of vibrational properties. Linear response provides an analytical way of computing the second derivative of the total energy with respect to a given perturbation. The DFPT problem can be solved by minimizing the second order perturbation in the total energy. This gives the first order changes in density of states, wave functions and potential [15]. The electronic second order energy, which is minimized in this approach, and implemented in CASTEP, iswhere the superscripts 0, 1 and 2 refer to the ground state, 1st and 2nd order changes, respectively. Ionic terms have also to be included in the total energy. The pre-conditioned conjugate gradients minimization scheme can be used to find the minimum of this functional with respect to the 1st order wave functions. The dynamical matrix for a given is then evaluated from the converged 1st order wave functions and densities [16]. The total energy, which includes the electron kinetic energy, the classical electrostatic energy due to Columbian interactions, and the electronic exchange-correlation energy due to quantum mechanics, is a crucial quantity in self-consistent calculations procedure. In this study it is implemented by solving a series of one-electron Kohn–Sham equations [5] with periodic boundary conditions.The generalized gradient approximation (GGA) in the scheme of Perdew–Bueke–Ernzerhof (PBE) model [7] is applied to describe the electronic exchange-correlation energy. Coulomb potential energy caused by electron–ion interaction is described using Norm-Conserving Pseudo-potentials [17]. The plane-wave basis cut-off is set at 550 and 320 eV for B4 (wurtzite) and B1 (rocksalt) phases, respectively. Thermodynamic properties of crystals can be evaluated in a fairly straightforward way based on a knowledge of the phonon frequencies across the Brillouin zone [18,14].

The quasi-harmonic approximation has been used in this study since accurate results are reported at moderate temperatures, and only small corrections to calculated phase transition properties are required at temperatures between of 1000–2000 K [19].

A number of approximations are made in order to obtain the phonon properties due to lattice vibrations. It is assumed that (i) the mean equilibrium position of each ion i, is a Bravais lattice site, R (ii) the amplitude of atomic displacements is small compared to interatomic distances. These lead to a harmonic approximation, which is sufficiently accurate to describe most of the lattice dynamical effect of interest. However, further refinement in the form of anharmonic theory is required to explain physical properties at high temperatures when the harmonic approximation breaks down.

The N-ion harmonic crystal can be considered as 3 N independent harmonic oscillators, whose frequencies are those of 3 N classical normal modes. Thus, at each wave vector k in the Brillouin zone there are 3 N vibrational modes. There are two equivalent descriptions of this system, either in terms of normal modes or phonons. Phonons are the quanta of the ionic displacement field that describe classical sound. They are similar to photons, the quanta of the radiation field that describe classical light. The properties of phonons can be described using a harmonic approximation based on the knowledge of just one fundamental quantity, namely the force constants matrix. This is implemented in CASTEP, and is given bywhere, u refers to the displacement of a given atom, and E is the total energy in the harmonic approximation. This force constants matrix (or Hessian matrix) can also be represented in reciprocal space, and is commonly referred to as the dynamical matrix:

Classical equations of motion can be written in terms of dynamical matrices. The motion can be treated as an eigenvalue problem. Each atomic displacement is described by plane waves:where the polarization vector of each mode, , is an eigenvector with the dimension of 3 N of the eigenvalue problem:

The dependence of the frequency, ω, on the wave vector is known as the phonon dispersion. Linear response calculations seek to evaluate the dynamical matrix directly for a set of k vectors. The starting point of the linear response approach is the evaluation of the second-order change in the total energy induced by atomic displacements. The main advantage of the scheme is that there is no need to artificially increase the cell size in order to accommodate small values of k vectors, as in the frozen phonon method. A more detailed description of the linear response method can be found in Ref. [14]. The results of a linear response calculation of the phonon spectra can be used to compute energy (E), entropy (S), free energy (F) and lattice heat capacity (Cv) as functions of temperature. The CASTEP yields the total electronic energy at T=0 K.

The vibrational contributions to the thermodynamic properties are evaluated to compute E, S, F and Cv at finite temperatures as discussed below. The zero point vibrational energy is included in the formula of total energy E (T). The equations described below are based on the work by Baroni et al. [14]:

at T=0 K the total energy iswhere Ezp if the zero point vibrational energy, K is Boltzmann's constant, h is Planck's constant and F(ω) is the phonon density of states. Ezp is described by

The vibrational contribution to the free energy, F, is given by

It is important to note that the density functional perturbation theory (DFPT) is the first and the most widely used theory, which is based on the plane-wave (PW) pseudo-potential (Norm-Conserving PP's) method [20].

PWs have many attractive features: (i) simple to use, (ii) orthonormal by construction, (iii) unbiased by the atomic positions. Contrary to what happens with localized (atomic-like) basis sets, it is very simple to check for convergence by just increasing the size of the basis set and adjusting the value of the kinetic-energy cut-off. After several iterations to obtain the good convergence, a value of 550 eV for the kinetic-energy cut-off was used in this calculations try.

In this paper we report the calculations of the frequencies of the phonons and their density of states in the B4 (wurtzite) and B1 (rocksalt) phases at P=0 and 14 GPa by means of the plane wave pseudo-potential method within the linear response technique.

Results and discussion

Structural phase transition

Under ambient conditions, InN crystallizes in the thermodynamically stable wurtzite phase with the space group P63mc. To study the stability of InN and to estimate the pressure of transition from B4 (wurtzite) to B1 (rocksalt), the enthalpies of the two phases were evaluated. The chosen stable phase is the one which has the lowest energy. In the ab-initio calculations at T=0, the enthalpy is expressed as

The enthalpy of B4 (wurtzite) and B1 (rocksalt) phases at pressures up to 20 GPa were calculated, and the transition pressure was determined at the crossing point between the two curves, i.e. when HB4=HB1. This is illustrated in Fig. 1. These theoretical results, which demonstrate that InN transform from B4 (wurtzite) to B1 (rocksalt) at nearly 12.7 GPa, tare in good agreement with the experimental findings of Ueno et al. [21] and Pinquier et al. [22] and the calculations in Refs. [23,24]. Bulk modulus B0 and its pressure derivative B0' extracted from EOS fitting are listed in Table 1. The structural parameters derived in this work agree well with the reported calculated and experimental values [25-29]. However, the pressure derivative B' of the B4 (wurtzite) phase is different from the experimental value reported in Ref. [21]. The discrepancy might be due to the problems associated with the deposition of InN in B4 (wurtzite) under high pressures. In this work, the band structures along high-symmetry lines in the first Brillouin zone were obtained by solving the Kohn–Sham equation [5] at the equilibrium lattice parameters and using Vanderbilt's ultrasoft pseudo-potentials [8]. The problem associated with the determination of the band gap in DFT associated with LDA is well-known [31]. It has been argued [16] that this problem is the underlying reason why agreement between experimental and calculated values is often unsatisfactory. To alleviate this problem a method, which involve scissors operator, was introduced by Levine and Allan [32]. This is discussed in details by Gonze and Lee [33]. The calculations reported here also use a scissors operator in order to circumvent the problem mentioned above.

Fig. 1

Transition pressure from B4 to B1.

Table 1

Calculated structural parameters for B4 (wurtzite) and B1 (rocksalt) phases compared with both experimental and theoretical data reported in the literature.

Structure	This work	Calculations	Experiment
B4 (wurtzite)
a (Å)	3.628	3.528a, 3.530b, 3.525c	3.533d, 3.545e
ca	1,617	1.618a, 1.619b, 1.613c	1.611d, 1.609e
B(GPa)	128.487	144a, 160.17b, 144c	125.5f
B0′	5,158	4.6a, 3.45b, 4.64c	12.79f
P( B4 to B1)	12.7	11.1c, 12.47g	12.1f, 11.6h
B1 (rocksalt)
a (Å)	4.604	4.641a, 4.645b, 4.636c	4.688i
B(GPa)	206	189a, 191.99b, 191c	170i
B0′	4.685	4.9a, 4.64b, 4.6c	5.0i

Ref. [26]

Ref. [27]

Ref. [24]

Ref. [28]

Ref. [29]

Ref. [22]

Ref. [25]

Ref. [23]

Ref. [30], and references therein.

The bands structure and the partial density of states (PDOS) for the B4 (wurtzite) phase are shown in Fig. 2. The top of the valence-band maximum and the bottom of conduction-band minimum occur at K(0 0 0), which indicates that B4 (wurtzite) has a direct gap around 0.68 eV. This value is very close to the experimental value determined by transmission photo-luminescence (PL) studies [27]. From the partial density of states (PDOS) data four sub-bands are clearly seen. The lowest sub-band, which has energy around −14.5 eV, is composed of two almost equal contributions from orbital In: 4d10 and N: 2s2. However, for the sub-band with energy ∼−13.5 eV there is a large contribution of orbitals In: 4d10. This is in agreement with what has been reported recently [25]. For the third sub-band between −12.95 and −10.6 eV, there is a weak hybridization between the orbitals of In: 4d10 and N: 2s2. For the fourth sub-band, which lies between −5.5 and 0 eV, there is a strong contribution of P-type orbitals for both atoms In and N. Fig. 3 shows that after the transition pressure the B1 (rocksalt) phase has a (K–L) band-gap with a value of 1.33 eV at P=14 GPa. It is also noted that the sub-bands in this phase are not as intense as in the B4 phase. However, there is still a strong contribution from In: 4d10. It is believed here that the mechanism for the changes of the band structures can be attributed to the changes in the nearest-neighbor bond lengths caused by the pressure increase. This effect affects the overlaps and the bandwidths.

Fig. 2

Band structure and partial density of states (electrons/eV) of B4 phase at P=0 GPa.

Fig. 3

Band structure and partial density of states (electrons/eV) of B1 phase at P=14 GPa.

Phonon spectrum and density of states (DOS).

The contribution of the phonons is very significant in the mechanism of the phase transition from B4 (wurtzite) to B1 (rocksalt). There are four and two atoms per unit cell in B4 and B1, respectively. This corresponds to dispersion curves with twelve branches for B4 and six for B1. Due to the translational symmetry the phonon frequencies in the first Brillouin zone can be calculated.

Figs. 4 and 5 show the phonon dispersion curves and the density of states (DOS) along the high-symmetry points in the first Brillouin zone of B4 (wurtzite) phase at two different pressures of 0 and 12 GPa, respectively.

Acoustic and optical phonon modes of various symmetries can be observed in these figures. These are in good agreement with those determined experimentally [34,35].

Fig. 4 shows that the longitudinal acoustic (LA) and transversal optical (TO) branches are separated. The gap of ∼132 cm−1 between acoustic and optical frequencies in the DOS matches the findings reported in Ref. [35]. Along the symmetry direction K–M and K–X the LA branch is almost flat.It is observed in Fig. 4 that all the phonon modes have positive frequencies, and no modes appear to be soft. This indicates that the B4 (wurtzite) structure is mechanically stable at P=0.

Fig. 4

Phonon dispersion curves and density of states (1 cm−1) of B4 phase at P=0 GPa.

Due to the large mass difference between In and N atoms, the In atomic vibrations dominate the low phonon frequencies, whereas N vibrations dominate the high frequencies. This is shown in the DOS data, which indicates the contribution of the phonons of orbital In: 3d in the modes TA.

Fig. 5 shows the phonon dispersion curves and density of states (DOS) along the high-symmetry directions for InN in the B4 (wurtzite) structure at P=12 GPa (just before transition pressure). As can be seen the gap frequency of the phonons is about 276 cm−1. This value is very large as compared with the value at P=0. This indicates that the gap of phonons increases with the increase of the pressure. In the center of the first Brillouin zone along the K–M direction, the transverse acoustical (TA) mode near K(0 0 0) point softens to imaginary frequencies (negative value), indicating a structural instability of InN in the B4 (wurtzite) phase. An elastic instability occurs when the vibrational frequency of a mode decreases to zero and becomes imaginary (the vibrational frequencies depend on the crystal structure, and vary with stress due to changes in the structure with stress). An imaginary vibrational frequency implies that the atomic motion corresponding to that mode becomes no oscillatory, and that the imposed crystal structure represents a saddle point rather than an energy minimum [18].

Fig. 5

Phonon dispersion curves and density of states of B4 phase at P=12 GPa.

Conclusion

Using PP-PW approach based on density functional theory, within the generalized gradient approximation, the structural and electronic properties of InN under high-pressure were studied. The transition pressure, lattice constant, unit volume, bulk modulus and their derivative pressure of B4 (wurtzite) and B1 (rocksalt) phases from 0 to 20 GPa are determined.

It is found that InN undergoes a transition from B4 (wurtzite) to B1 (rocksalt) at 12.7 GPa. This is in agreement with the reported experimental data and theoretical calculations The behavior of the band-gap for both phases under high pressure were analyzed, and we found that the energy of the direct gap for B4 (wurtzite) is ∼0.86 eV at P=0, and the energy of the indirect gap for B1 (rocksalt) is ∼1.33 eV at P=14 GPa. The important role of In: 4d10 states in the nature of the bands structure for both B4 (wurtzite) and B1 (rocksalt) phases was demonstrated. Our results agree well with the experimental and theoretical studies.

The gap of the frequency of the InN phonons in B4 (wurtzite) phase at 0 and 12 GPa is ∼132 and ∼276 cm−1, respectively. The value of the gap at P=0 is in very good agreement with the experimental results reported recently. At P=12 GPa the structure is unstable. This is also in agreement with the calculations of the enthalpies, which show that the pressure of transition is ∼12.7 GPa. The important contribution of phonons in the transition phase mechanism of InN from B4 (wurtzite) to B1 (rocksalt) has been discussed.