Lattice dynamics properties of zinc-blende and Nickel arsenide phases of AlP

Physics Letters A 372 (2008) 5340–5345

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Lattice dynamics properties of zinc-blende and Nickel arsenide phases of AlP S. Aouadi a,∗ , P. Rodriguez-Hernandez b , K. Kassali c , A. Muñoz b a b c

Institut of Science and Technology, University of Souk-ahras, Souk-ahras 41000, Algeria Departamento de Física Fundamental II, Universidad de La Laguna, La Laguna E-38205, Tenerife, Spain Institut of Physics, University Ferhat Abbes Setif, Setif 19000, Algeria

a r t i c l e

i n f o

Article history: Received 23 April 2008 Accepted 9 June 2008 Available online 12 June 2008 Communicated by A.R. Bishop PACS: 61.50.Ah 61.50.Ks 62.50.-p 63.20.D-

a b s t r a c t Ab initio calculations, based on norm-conserving non-local pseudopotentials and the density functional theory (DFT), have been performed to investigate the behaviour under hydrostatic pressure of the structural, electronic, elastic and dynamical properties of AlP, in both zinc-blende and nickel arsenide phases. Our calculated structural and electronic properties are in good agreement with previous theoretical and experimental results. The phonon dispersion curves, the elastic constants, Born effective charge, etc., were calculated with the local density approximation and the density functional perturbation theory (DFPT). Our results in the pressure behaviour of the elastic and dynamical properties of both phases are in agreement with the experimental data when available, in other case they can be considered as predictions. © 2008 Elsevier B.V. All rights reserved.

Keywords: III–V compound Elastic constants Phonons High frequency constant Born effective charges Grüneisen parameter

1. Introduction The development of computer simulations has opened up many interesting and exciting possibilities in the study of the solid. The most important results obtainable from first-principles studies are crystal structures, structural and electronic properties under pressure, phonon spectra, elastic and dynamical properties, etc. Today it is possible to obtain results in very good agreement with experimental data and to make good predictions of the properties of a solid which were previously inaccessible to experiment. Due to the improvement in the resolution of the experimental methods, the development of high-pressure cells, and the efficiency and accuracy of the theoretical first-principles calculations, the study of the high pressure phases of many semiconductor compounds have revealed the existence of new and unexpected high pressure phases [1]. Like many III–V semiconductor compounds, at normal condition AlP crystallizes in the cubic zinc-blende (ZB) structure. High pressure experiments of AlP are difficult because AlP is unstable in air, emitting poisonous phosphine gas, which explains the experi-

*

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mental difficulties encountered in characterizing the high pressure phases map. The low pressure phase is destabilized under hydrostatic pressure and a phase transition appears, Wanagel et al. [2] (1976) using resistivity measurements in an anvil device, determined that AlP under goes a sluggish transition to a metallic phase at pressure slightly lower than that observed for ZnS, the transition pressure in AlP was reported at 14 GPa. In a theoretical study, Froyen and Cohen [3] (1983), postulated a NiAs phase of AlP, similar to the one that appear in AlAs under pressure. The experimental confirmation of such a phase in AlAs as well as AlP came much later by Green et al. [4]. The qualitative high pressure behaviour of AlP, AlAs and AlSb having a transition to a NiAs phase were reported in Refs. [5–7]. Although considerable progress has been made in theoretical description of structural and electronic properties of AlP under pressure [8,9], many of their vibrational and elastic properties are still not well established. An accurate description of the structural, electronic, elastic and dynamical properties of AlP under hydrostatic pressure in both phases, can play an important role in determining some properties related with phase transition, electronphonon interaction, etc. The aim of this work is to investigate the above properties by employing the pseudopotential method, the density functional the-

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Table 1 Calculated equilibrium lattice constants (a, c ), bulk modulus ( B 0 ) and first derivative of bulk modulus with pressure ( B 0 ) for AlP in the ZB and the NiAs structures compared to other theoretical calculations and available experimental data Structure Zinc blende

NiAs

This work Ref. [8] Ref. [19] Experiment [20] This work Ref. [21]

a (Å)

c (Å)

B 0 (GPa)

5.43 5.40 5.48 5.46 3.55 3.466

– – – – 5.71 5.571

90.46 90 88

ory and the density functional perturbation theory to study the linear response of this compound under pressure. The calculated structural parameters and electronic properties are compared with previous experimental and theoretical studies. The evolution under pressure of the calculated phonon dispersion, the elastic constants, the Born effective charges, the dielectric tensor ε (∞) and Grüneisen parameter γ of both phases are also compared with the available experimental data. To our knowledge many of the present results have been not previously reported and we hope that this work will stimulate new experimental studies in the future. The Letter is organised as follows. In Section 2 we describe the method of calculation. Results and discussions concerning the various properties are presented in Section 3. In Section 4, we summarize our conclusions.

B 0 3.72 – – – 4.21 –

– 111.5 –

Table 2 Direct and indirect band gap energy of AlP in the ZB structure at equilibrium volume

This work E g (eV) [23] E g (eV) [26] E g (eV) [27] E g (eV) (theory) * ** ***

EΓ −X

E Γ −Γ

1.41 1.49 2.50 1.635 2.17* /1.45**

3.11 – 3.62 3.073 3.26***

Ref. [28]. Ref. [29]. Ref. [30].

2. Method of calculation We have performed first-principles total-energy calculations using the pseudopotential method based in the density functional theory [10,11] (DFT), within the local density approximation (LDA). We employed norm-conserving non-local Troulliers–Martins pseudopotentials [12]. The exchange-correlation energy of electrons is described in the local density approximation with the Ceperley Alder prescription [13]. The calculations were carried out using the ABINIT [14] code. The Kohn–Sham (KS) single particle functions were expanded in a plane wave basis set. Self consistent solutions of the (KS) equations were obtained by sampling the irreducible Brillouin zone with the special k-point method. Well converged results were obtained using a kinetic energy cutoff of 80 Ry and with a set of 28 k-special points for the ZB structure and 76 special points for the NiAs structure, which correspond to 6 × 6 × 6 and 12 × 12 × 8 k-points mesh in the Monkhorst– Pack notation [15]. Having obtained self consistent solutions of the (KS) equations, phonon frequencies and elastic constants are obtained using the self-consistent density functional perturbation theory [16,17] (DFPT), which avoids the use of supercells and allows the calculation of the dynamical matrix at arbitrary q vectors. The DFPT also allows the calculation of the high frequency dielectric tensor ε (∞) and the Born effective tensor charges Z B for each inequivalent atom in the unit cell. These quantities are necessary to compute the non-analytic part of the dynamical matrix. We ensure the convergence of the phonon frequencies to within 2–3 cm−1 . 3. Results and discussions 3.1. Structural and electronic properties The equilibrium lattice parameters have been calculated by minimizing the crystal total energy obtained for different volumes and fitted with the Murnaghan’s equation of state [18]. In Table 1, we report the calculated equilibrium lattice constants (a, c ), bulk modulus ( B 0 ) and first pressure derivative of bulk modulus ( B 0 ) for AlP in the ZB and the NiAs phases. Our results compare quite

Fig. 1. Enthalpy vs. pressure curves for the ZB and the NiAs phases of AlP.

well with the available experimental results and with other theoretical calculations. We analyze our results of the calculated enthalpies as a function of pressure for AlP in the ZB and the NiAs phases (see Fig. 1). We found that the pressure induced phase transition from the ZB structure to a NiAs structure occurs at a transition pressure of 6.78 GPa, which is in reasonable good agreement with the available experimental data and other theoretical studies [1]. Fig. 2(a) and (b) displays our calculated electronic band structure of AlP in the ZB and the NiAs phases respectively, along the high-symmetry directions. Previous tight-binding band structure calculations [22] for the ZB structure are in good agreement with our valence band structure, both in width and shape. The conduction bands are also similar. The valence band maximum is located at Γ and the conduction band minimum is located at X resulting in an indirect energy band gap of 1.407 eV. A comparison of our results with the experimental and other theoretical data is given in Table 2. We note that some values of the previous theoretical calculations of the energy gap are adjusted to coincide with experimental ones. Huang and Ching [22] have used correction for the band gaps beyond the LDA. Reshak et al. [23] does not use correction for the band gaps yet. Our calculated energy gaps are roughly 40% smaller than the experimental as expected from an LDA cal-

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(a)

(a)

(b) Fig. 2. (a) Electronic band structure of AlP in the ZB structure at P = 0 GPa. (b) Electronic band structure of AlP in the NiAs structure at P = 8.59 GPa.

culation due to the well known band gap underestimation of the LDA. In Fig. 2(b) we plot the electronic band structure of AlP in the NiAs phase at 8.59 GPa, it is clear that AlP in the NiAs phase is metallic, in good agreement with previous first principles studies [8,9,24,25]. 3.2. Elastic properties The elastic constants of solids give interesting information about their mechanical and dynamical properties, and provide a link between dynamical and mechanical behaviour of crystals. There are different methods to obtain the elastic constants through the first principle modelling materials from their know crystal structures. For example it is possible to compute the components of the stress tensor for small strains using the method proposed by Nielsen and Martin [31], this method was used in the study of many semiconductor compounds [8,9]. One can also compute the elastic response of the system with respect to the strain perturbations, obtaining the second derivatives of the total energy with respect to all the perturbations in order to obtain the elastic constants as it is implemented in the ABINIT code. Both methods showed previous successes of such calculations for similar compound and can be used to predict the elastic properties which are not yet experimentally established. We present our results of the elastic constants for AlP in the ZB and the NiAs structures in Table 3, and compare them with experimental results for the ZB phase. In the ZB phase there is a reasonable good agreement for C 11 , C 12 and C 44 with the available experimental [19] and theoretical [9] data. To our knowledge, there are no results reported on elastic constants for the NiAs phases, so our calculated elastic constant can be considered as predictions. In Table 4, we report

(b) Fig. 3. (a) Elastic constants versus pressure for AlP in the ZB phase. (b) Elastic constants versus pressure for AlP in the NiAs phase. Table 3 Elastic constants of AlP in the ZB structure at zero pressure and in the NiAs structure at P = 8.59 GPa C 11 (GPa) C 12 (GPa) C 13 (GPa) C 33 (GPa) C 44 (GPa) C 66 (GPa) This work ZB 132.28 Ref. [19] 132 Ref. [9] 132.25 This work 229.95 NiAs

68.46 63 67.5 106.38

–

–

78.53

274.84

62.15 61.5 76.55 71.71

–

61.78

Table 4 Calculated pressure derivative of the elastic moduli for AlP

This work ZB Ref. [9] This work NiAs

∂ C 11 /∂ p

∂ C 12 /∂ p

∂ C 13 /∂ p

∂ C 33 /∂ p

∂ C 44 /∂ p

∂ C 66 /∂ p

2.73 3.59 4.136

3.30 4.19 2.514

–

–

–

3.029

2.702

0.30 1.52 2.264

0.811

the pressure derivative of the elastic constants for both phases. We note that our study is focused in the pressure stability range for each structure. In Fig. 3 we plot the pressure dependence of the elastic constants for both phases, showing clearly the linear behaviour of C i j which increases with increasing pressure. 3.3. Dielectric properties To deal with the macroscopic electric field associated with the longitudinal optical modes and the related non-analytic behaviour

S. Aouadi et al. / Physics Letters A 372 (2008) 5340–5345

of the dynamical matrix at Γ point [32], we have calculated the high frequency dielectric constant ε (∞) and the Born effective charge Z B tensors for AlP in the ZB structure, which are isotropic and scalars in cubic symmetry. The Born effective charges are the fundamental quantities that specify the leading coupling between lattice displacements and electrostatic fields in insulators [33]. In order the acoustic sum rule  B to check the quality of our −study 5 , suggesting well converged s Z s = 0 is fulfilled to within 10 calculations. Our calculated Born effective charge, Z B , and high frequency dielectric constant ε (∞) are 2.20 and 8.4, respectively. In Fig. 4, we show the pressure dependence of Born effective charges and high frequency dielectric constant ε (∞) for the ZB phase. It is clear that the Born effective charge decreases with a nearly linear behaviour with increasing pressure. While the high frequency dielectric constant decreases with increasing pressure with a quadratic behaviour. 3.4. Vibrational properties Phonon excitation plays an important role in electronic transport, non-radiative electron-relaxation processes and other proper-

Fig. 4. Born effective charge and high frequency dielectric constant of AlP versus pressure in the ZB structure.

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ties of interest for materials characterization, devices engineering and design. The phonon dispersion is an interesting characteristic of the crystal. It determines the optical and thermal properties. In this section, we present an ab-initio study of the phonon spectra of the ZB and the NiAs structures of AlP using state-of the–art DFPT [16,17]. This method has been used for a large number of materials, including semiconductors, insulators and metals, and provides an excellent description of the vibrational properties with accuracy better than a few percent of the calculated frequencies [32]. Our calculated values of the phonon frequencies at high symmetry points for AlP in the ZB structure at zero pressure are reported in Table 5(a). For the ZB structure the TO and LO zone center phonons (ωLO (Γ ) = 494.9 cm−1 and ωTO (Γ ) = 442.6 cm−1 ) agree well with values reported by Onton [34] (ωLO (Γ ) = 501.2 cm−1 and ωTO (Γ ) = 439.4 cm−1 ) from Raman spectroscopy. To our knowledge, there are no experimental data or theoretical results available for AlP in the NiAs structure. At Γ point for AlP in the NiAs structure, there are three acoustic and nine optical phonon modes: E 2u , E 2g , A 2u , B 1g , E 1u and B 2u . Where E 1u and A 2u modes are infrared active and E 2g mode is Ra-

Fig. 5. Phonon dispersion curves of AlP in the ZB structure for zero pressure and near to the pressure of transition ( P = 6.75 GPa).

Table 5 The phonon frequencies at high symmetry points of AlP for the ZB structure at zero pressure and for the NiAs structure at P = 8.59 GPa (a) ZB Frequencies (cm−1 ) at high symmetry points

Structure

Modes

L

X

U

Zinc blende

TA

0

102.1

137.5

LA TO

0 442.6

LO

494.9

335.9 403.9 430.2 430.2

353.9 412.3 416.9 416.9

137.0 191.2 330.4 373.5 417.4 435.1

Γ

(b) NiAs Frequencies (cm−1 ) at high symmetry points

Structure

Modes

A

L

M

H

K

Nickel arsenide

TA

0

134.56

173.97

139.44

LA E 2u

0 236.31

E 2g

295.38

A 2u B 1g E 1u

310.43 348.90 378.97

B 2u

467.74

134.56 134.56 243.98 243.98 364.32 364.32 364.33 364.33 403.50 403.50

207.80 207.80 235.56 235.56 359.82 359.82 365.89 365.89 411.87 411.87

103.54 123.58 225.15 231.19 266.26 267.55 268.38 361.38 365.25 381.88 407.66 460.12

180.17 183.01 183.24 220.80 253.22 253.30 329.01 329.40 349.38 381.96 382.34 419.22

Γ

229.03 229.03 266.21 266.21 314.41 314.412 386.46 386.46 421.58 421.58

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man active. We present our calculated phonon frequencies at high symmetry points for AlP in NiAs structure in Table 5(b). The calculated phonon dispersion curves for AlP in the ZB structure is plotted along high symmetry directions in Fig. 5 at zero pressure and over the transition pressure (P = 6.75 GPa). We note that the acoustic phonon modes are well separated from optical modes in the ZB phase. From our results it is clear that under pressure the optical modes and longitudinal acoustic mode are shifted upwards whereas the transversal acoustic modes are shifted downwards. The shape of all the modes remains the same as at zero pressure, also we mention that we do not obtain any mode softening in the pressure range that we analyze. In Table 6, we report the pressure derivatives of the phonon frequencies at Γ point for both structures. From our results, it is clear that the second derivative is quite small; then we can conclude a linear behaviour. In describing the volume dependence of the phonon frequencies in solids, Barron has introduced the mode Grüneisen parameters γi [35] and defined it for the ith mode with phonon frequency ωi in the following way:



γi = −

d ln ωi d ln Ωi

 (1)

.

Where Ω is the volume of crystal. The mode Grüneisen parameters calculated for AlP in the ZB and the NiAs structures are presented in Table 7. Kunc et al. [36] have concluded that for transverse acoustic phonons in GaAs, the Table 6 First and second derivative of the frequencies at the Γ point with pressure obtained for AlP in the ZB and the NiAs structures

(cm−1 /GPa)

Structure

Mode

∂ω ∂P

Zinc blende

LO TO E 2u E 2g A 2u B 1g E 1u B 2u

5.56 5.86 4.73 3.36 3.06 3.89 5.74 4.47

Nickel arsenide

Table 7 Mode Grüneisen paramaters

∂2ω ∂ P2

(cm−1 /(GPa)2 )

−0.07 −0.07 −0.04 −0.02 −0.07 −0.03 −0.05 −0.04

anharmonic contributions are very important. We obtain γTA ≺ 0 for the high symmetry points L, X and U for the ZB phase and for the high symmetry points, H and M for the NiAs phase. The negative value of the mode Grüneisen parameter at low frequency imply a negative thermal-expansion coefficient at low temperatures [19]. Talwar et al. [37] found γTA = −0.64 at critical point X in the study of BP, Talwar and Vandevyer [38] note that the results for various physical properties in their study are well understood with exception of the calculated value of γTA = −3.48, which is negative, but too large. 4. Conclusion In summary, in this Letter we have investigated the structural, elastic and lattice dynamics properties of AlP under hydrostatic pressure, by employing first principles scheme, based in the plane-wave pseudopotential method, the density functional theory and the density functional perturbation theory. The calculated static properties, structural lattice constants, bulk modulus, pressure derivative of bulk modulus, are an agreement with previous experimental and theoretical available results. Our study yield and indirect band gap with the minimum of the conduction band at the X point for the ZB phase, and a metallic band structure for AlP in the NiAs phase. We have calculated the elastic constants for the low and high pressure phases of AlP. The high frequency dielectric constant, Born effective charges and phonon dispersion for the low pressure phase are obtained. We also report the phonon frequencies at selected high symmetry points for the NiAs phase. Our calculations show that all the elastic constants for AlP increases with application of hydrostatic pressure. In the other hand, the high frequency dielectric constant and Born effective charges decreases with pressure. Finally we also report the mode Grüneisen parameters for both structures. Our results are in good agreement with available data and give reliable predictions where the data are lacking. We hope that our work will stimulate future experimental efforts on the study of the behaviour of AlP under pressure.

γi at high symmetry points of AlP

(a) ZB Structure

Zinc blende

Mode Grüneisen

High symmetry points

parameter (cm−1 /Å3 )

Γ

γTA

–

γLA γTO

– 1.19

γLO

1.01

L

X

−2.33

−2.48

0.40 1.45 1.38 1.38

1.08 0.91 1.60 1.60

U

−0.83 −2.22 1.10 1.55 1.11 1.49

(b) NiAs Structure

Nickel arsenide

Mode Grüneisen

High symmetry points

parameter (cm−1 /Å3 )

Γ

A

L

γTA

–

2.31

γLA γ E 2u

– 2.42

γ E 2g

1.45

γ A 2u γ B 1g γ E 1u

2.26 1.35 1.88

γ B 2u

1.20

2.31 2.31 1.31 1.31 1.74 1.74 1.74 1.74 1.89 1.89

0.87 0.88 1.17 1.17 0.86 0.86 1.86 1.86 1.96 1.96 1.63 1.63

M

H

−1.24 −0.75 0.44 1.10 1.70 1.88 1.91 1.61 1.32 2.11 1.76 1.65

−0.40 1.22 1.22 1.93 1.93 1.44 1.44 1.75 1.75 1.72 1.72

K 0.33 0.14 0.16 2.66 0.78 0.78 2.27 2.27 1.14 1.74 1.75 1.86

S. Aouadi et al. / Physics Letters A 372 (2008) 5340–5345

Acknowledgements S. Aouadi wish to express his sincere thanks to the staff of the Departamento de Física Fundamental II of the Universidad de La Laguna for their continued interest and encouragement throughout the course of the present work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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