Structural and Dynamical Properties of Zincblende GaN

phys. stat. sol. (b) 209, 223 (1998)

Subject classification: 61.66.Fn; 62.20.Dc; 65.40.+g; S7.14

Structural and Dynamical Properties of Zincblende GaN

F. Benkabou (a), P. Becker (b), M. Certier (c), and H. Aourag (a)

(a) Computational Materials Science Laboratory, Physics Department,University of Sidi-Bel-Abbes, 22000 Algeria

(b) Structure Electroniques et ModeÂlisations, Ecole Centrale des Arts et Manufactures,F-92295 Chatenay Malabry Cedex, France

(c) Laboratoire de SpectromeÂtrie Optique de la MatieÁre, DeÂpartement Mesures Physiques,TechnopoÃ le Metz 2000, F-57078 Metz Cedex 3, France

(Received March 11, 1998; in revised form June 23, 1998)

The structural and dynamical properties of zincblende b-GaN are calculated within a three-bodyTersoff potential coupled with a molecular-dynamics simulation scheme for a temperature rangingfrom 300 to 900 K. A good agreement between the calculated and experimental values of the lat-tice constant, the bulk modulus and its derivative, and the cohesion energy are obtained. We havealso calculated the lattice constants, lattice thermal expansion, and specific heat. In order to eluci-date the microscopic behavior of mobile atoms with temperature, the diffusion mechanism hasbeen predicted using this scheme. The structural properties of GaN in the rocksalt structure arealso studied and compared with other works.

1. Introduction

The III±V nitrides recently gained more attention because of their application in blue-light-emitting diodes and lasers operating in the blue and ultraviolet regime. The specif-ic role of nitrogen is in the formation of short bonds which leads to smaller latticeconstants (by 20). Because these compounds have small atomic volumes, many of theirphysical properties will be similar to other wide-gap semiconductors such as diamondor BN. Under high pressure these nitride compounds undergo a structural phase trans-formation to the rocksalt structure (GaN at 47 GPa [1 to 3]) which is favored by theirhigh ionicity [4, 5]. No transition to zincblende structure is observed. GaN is provingto be particularly fascinating, largely because of the recent fabrication of a new structur-al phase of this material. It was suggested by Chelikowsky [6] that the transition pres-sure from the tetrahedrally coordinated structure (wurtzite) to the rocksalt structurechanges linearly with Philips ionicity for the same atomic volume. There are severaltheoretical [1, 3, 7 to 10] and experimental [10 to 15] studies of bulk wurtzite. The materi-al was known to crystallize only in the hexagonal (wurtzite) form, which is thermodynam-ically stable. But in the 1980s, a cubic GaN, a metastable form of the material, wasdiscovered during deposition on suitable substrates such as GaAs(001) surface [16], b-SiC(100) surface [17], and Si(100) surface [18]. The calculated lattice constant is 4.503 �Abased on the measured Ga±N bond distance in WZ-GaN, while the measured valuesrange from 4.49 to 4.55 �A [19 to 23]. The total-energy calculations verified the highstability of the wurtzite phase, which is common in column-III nitrides [24, 25]. Galliumnitride (GaN) is one of the most promising materials for future electronic and optoelec-

F. Benkabou et al.: Structural and Dynamical Properties of Zincblende GaN 223

tronic devices (for a good review on GaN and other wide-band gap semiconductors, see[26]). Experimentally very little is known about the electronic, optical, and mechanicalproperties of the cubic b-GaN. Zincblende GaN structure will be interesting partly be-cause of its potential for a higher saturated electron drift velocity and a somewhat low-er band gap than wurtzite GaN [7, 8]. So, the detailed knowledge of its structural beha-vior, especially with respect to pressure and temperature, is necessary. In order tounderstand the nature of zincblende structure at high temperature it is necessary to usesome form of microscopic modeling or simulation. The advantages of the computationalapproach is that it can provide a much clearer atomic and electronic picture of thematerials and their influences on the structures, properties, synthesis and performances.The aim of the present study is to perform molecular-dynamics (MD) simulations of b-GaN, using the three-body potential that reproduces particular experimental data asclosely as possible, within certain accepted limitations [27 to 29]. With this approach,we predict the behavior of solid b-GaN under temperature and pressure. The results ofthe simulation may also be compared to other data limits. The paper is organized asfollows. In Section 2 we give a brief description of the model used and of the calcula-tion method. Then, the results of our calculation are presented and discussed in Sec-tion 3. Finally, a conclusion is given in Section 4.

2. Calculations

2.1 The Tersoff potential

The Tersoff potential is based in the bond-order concept [30]. The inter-atomic poten-tial energy between two neighboring atoms i and j is written as

Vij � fc�rij� �A exp �ÿlrij� ÿ cBbij exp �ÿmrij�� ; �1�

fc�rij� �

1 ; r < RÿD

12ÿ 1

2sin

p

2�r ÿ R�=D

h i; RÿD < r < R�D

0 ; r > R�D

8>>><>>>: ; �2�

where bij is the many-body bond-order parameter describing how the bond-formationenergy (the attractive part of Vij) is affected by local atomic arrangement due to thepresence of other neighboring atoms, the k atoms. It is a many-body function of thepositions of atoms i; j and k. It has the form [31]

bij � �1� xniij �ÿ1=2ni ; �3�

xij �P

k 6�i; jfc�rij� big�qijk� ; �4�

where xij is called the effective coordination number and g�qijk� is a function of theangle between atoms and has been fitted to stabilize the tetrahedral structure. The nitridesform a specific subgroup of the III±V compounds characterized by high ionicity andvery short bond lengths. As a result of the short bonds of the nitrides these materialsshow many properties like diamond, notably exceptional hardness, high lattice thermalconductivity and a wide band gap [32]. No parameters in the literature have beenfound for the Tersoff potential which fits crystalline GaN. So, we start our fitting with

224 F. Benkabou, P. Becker, M. Certier, and H. Aourag

Structural and Dynamical Properties of Zincblende GaN 225

Ta b l e 1The adjusted Tersoff parameters for b-GaN

A (eV) 2999.9 c 10039B (eV) 179.0 d 16.217l1 ��Aÿ1� 3.6867 h ÿ0.59825l2 ��Aÿ1� 1.8532 rcut ��A� 2.395b 1.110-6 dcut ��A� 0.15n 0.72

Fig. 1. Partial pair distribution functions for b-GaN

Ta b l e 2Values of peak distances and coordinate numbers of pairs for b-GaN. Experimental val-ues in parentheses are taken from [39]

300 K 600 K 900 K

rmin ��A� h rmin ��A� h rmin ��A� h

1.939 (1.94) 3.576 (4) 1.93 3.993 1.94 4.043.165 (3.17) 11.941 (12) 3.18 11.990 3.188 24.433.717 (3.88) 11.951 (12) 3.73 12.056 4.925 18.426

the Tersoff parameters of carbon[33] as input parameters for our sim-ulation. The parameters are fittedto polytype energies [34 to 36], wehave ensured that the adjusted po-tential parameters give reasonableresults for structural and thermody-namic properties of GaN in thezincblende phase. The resultingparameters for GaN are listed inTable 1.

2.2 Computer simulations

Molecular dynamics (MD) is a di-rect simulation technique at the atomic level. Almost all the physical properties of amaterial may be determined using molecular dynamics. It requires generally an inter-particle potential. Thus molecular-dynamic computer simulations have been performedto determine the predictions of Tersoff potential for solid structure under temperatureand pressure effect. The MD cell is formed of a cube of side L with 3� 3� 3 diamondunits cells, where 216 particles are included. The periodic boundary conditions are ap-plied. The atomic structure of solid has been calculated by using a NVT molecular-dynamics simulation. The MD routine is based on a fifth-order gear-predictor-correctoralgorithm of the Newtonian equations of motion using a neighbor list technique with atime step Dt � 1:86 fs, and an efficient network cube algorithm for nearest-neighborbook keeping, details are given elsewhere [37]. After an equilibration period, a histo-gram of atomic separations is produced in order to compute the pair distribution func-tion [38] g�r� and other properties which are computed along the trajectory of the sys-tem in phase space.

3. Results

3.1 Structural properties

3.1.1 Partial pair distribution function

In order to test the stability of the zincblende structure of GaN at finite temperaturewithin the Tersoff potential model, we have calculated the pair distribution functions at300, 600 and 900 K. Fig. 1 shows the pair distribution function of b-GaN. The first peakposition at 1.93 �A for GaN represents the distance between a zincblende GaN latticepoint and its first neighboring tetrahedral site, it is in good agreement with experimen-tal values [39]. The first peak position of g�r� at 900 and 600 K is 1.94 and 1.93 �A forGaN, respectively, which is close to the distance between the first neighbor sites. Funda-


Fig. 2. Total energy of GaN as a func-tion of temperature

mentally it might be said that the structure of b-GaN approximately remains in thezincblende structure. The coordination numbers are evaluated by the equation

h � 4pr0

�r2r�r� dr ; �5�

where r0 is the number density of atoms. In Table 2 values of coordination numbersand peak positions are given. The value of h remains approximately at 4 for the threetemperatures 300, 600 and 900 K, which means that GaN has a stable zincblende struc-ture. But at 900 K, we observe a sharp nearest-neighbor peak and have an oscillatorytail about the value of 1. This behavior is confirmed by the total energy of system as afunction of temperature which shows a linear behavior as displayed in Fig. 2.

3.1.2 Ground-state properties

Ground state properties of b-GaN were calculated within the Tersoff potential modelby using MD simulation. The lattice constant and bulk moduli are obtained from thecalculation of the total energy, Etot, as a function of volume (Fig. 3) by varying thelattice parameter. These results are fitted to the Murnaghan equation [40] of state. Theresults obtained are given in Table 3.

Our results compare particularly well with those from the LMTO method in its sca-lar-relativistic form [41], in conjunction with the local density approximation (LDA) for


Fig. 3. The total energy of the system as a function of the volume at 300 K

zincblende-type semiconductors [42, 43]. Christensen et al. [44] used this method forpredicting higher-pressure phases to elaborate the role of the Ga-3d states in connec-tion with the pressure-induced structural transition.

Due to their high ionicities the nitrides are expected to transform to the rocksaltstructure when pressure is applied. In our total-energy calculations we have also exam-ined the high-pressure rocksalt structure. The results for the calculated structural param-eters are also given in Table 3. The good agreement observed suggests that the poten-tial parameters describe well the homogeneous response of b-GaN to hydrostaticcompression.

3.1.3 Elastic constants

The shear moduli require knowledge of the derivative of the energy as a function of alattice strain [45]. In the case of a cubic lattice, it is possible to choose this strain so thatthe volume of the unit cell is preserved. The strain can be chosen so that the energy isan even function of the strain, whence an expansion of the energy in powers of thestrain contains no odd powers. Thus for the calculation of the modulus C11 ±± C12 wehave used the volume-conserving orthorhombic strain tensor,

e �d 0 00 ÿd 00 0 d2=�1ÿ d2�

0@ 1A : �6�

Application of this strain changes the total energy from its unstrained value to

E�d� � E�ÿd� � E�0� � �C11 ÿ C12� Vd2 �O�d4� ; �7�


Ta b l e 3Structural properties of nitrides GaN in the zincblende and rocksalt structures atT � 300 K compared with other theoretical and experimental results

zincblende rocksalt

experiment present work other works present work other works

a ��A� 4.54 [34] 4.49 4.46 [44] 4.07 4.18 [44]4.49 [18] 4.41 [50] 4.098 [50]

B �107 Pa) 1.85 [35] 1.77 1.84 [44] 1.87 2.27 [44]1.73 [50] 2.23 [50]

B0 4.57 4.6 [44] 5.51 4.0 [44]3.64 [50] 3.69 [50]

Ecoh (eV) 4.45 [36] 4.49 4.87

Ta b l e 4Equilibrium constants for the cubic structure of GaN at T � 300 K compared with othertheoretical and experimental results

C11 �107 Pa) C12 �107 Pa)

present work 2.339 1.485experiment [51] 2.64 1.53theory [52] 2.61 1.27


Fig. 4. Mean displacements ofb-GaN at a) 300 K and b) 900 K

where V is the volume of the unit cell and E�0� is the energy of the unstrained latticeat volume V. The strain (6) can be used for any cubic lattice. For an isotropic cubiccrystal [46], the bulk modulus is given exactly by

B � 13 �C11 � 2C12� : �8�

Using (7) and (8) we obtained the values of C11 and C12. The results are displayed inTable 4 along with some data of literature. A good agreement is observed.

3.1.4 Mean square displacements

The mean square displacement of atoms in a simulation can be easily computed by itsdefinition

MSD � hjr�t� ÿ r�0�j2i ; �9�where h. . .i denotes averaging over all the atoms (or all the atoms in a given subclass).The result of MSD as a function of time t obtained from the 216-atom sample is plottedin Fig. 4a and b. It shows that b-GaN has a lattice stability at 300 K, while at 900 Kthere will be a slight migration due to the diffusion effects. If the system is solid, MSDsaturates to a finite value, while if the system is liquid, MSD grows linearly with time.In this case it is useful to characterize the system behavior in terms of the slope, whichis the diffusion coefficient D,

D � hjr�t� ÿ r�0�j2i=6�t2 ÿ t1� ; �10�where r�t� means the position vector of GaN at time t. By using the Einstein relation(10) we deduce the diffusion constant at 300 K to be 0:66� 10ÿ4 cm2=s at higher densi-ties. In comparison with carbon, this value is not too much different from the value


Fig. 5. Diffusion coefficients evaluatedfrom MSDs for b-GaN

estimated by Wang et al. [47], 2:05� 10ÿ4 cm2=s. At 900 K, we found that the diffusioncoefficient is 1:28� 10ÿ4 cm2=s higher than the value at 300 K.

Fig. 5 shows the behavior of the diffusion coefficient D versus temperature as evalu-ated by equation (10). The overall behavior agrees well with results for III±V semicon-ductors in the zincblende structure.

3.2 Thermal properties

The thermal expansion coefficient a measures how the lattice constant responds to anisometric change in temperature,

a � 1a

@a

@T

� �p

: �11�

The thermal expansion coefficient at room temperature is then computed from thetemperature derivative of these lattice constants. At temperatures below the Debyetemperature (about 607 K for b-GaN [48]), quantum effects are important in determin-ing thermal expansion. Above the Debye temperature, quantum effects become unim-portant, and thermal expansion may be determined from classical calculations, such asmolecular dynamics (MD). Thermal expansion is due to anharmonic terms in the cohe-sive energy for small displacements of the atom about its equilibrium position at 0 K.The thermal expansion of b-GaN has been studied in the temperature range from 300to 900 K.

Fig. 6 shows the variation of the lattice parameter of b-GaN with temperature. Thecollected data can be fitted to polynomial functions of temperature. We deduce thethermal expansion coefficient as 4:59� 10ÿ5 Kÿ1, which is slightly smaller than the val-ue of 5:59� 10ÿ5 Kÿ1 of single crystal wurtzite (WZ) GaN [49] but in the same order


Fig. 6. The lattice parameter of b-GaNas a function of temperature

of magnitude with many other semiconductors. The specific heat of b-GaN at constantvolume �Cv� is also calculated using the Tersoff potential. The constant-volume heat capa-city measures how the internal energy responds to an isometric change in temperature,

Cv � @E

@T

� �V

: �12�

From Fig. 2 we determine the specific heat of b-GaN, which is estimated to beCv � 6:01 cal=K mol. This result is certainly in agreement with experiments at hightemperatures where the value of Cv is close to 3R, where R is the universal gas con-stant �R � 2 cal=K mol) for b-GaN. This range usually includes room temperature. Thefact that Cv is nearly equal to 3R at high temperatures regardless of the substancedescribed is called the Dulong-Petit law.

4. Conclusion

We have presented a detailed analysis of structural, dynamical and thermal propertiesof b-GaN as obtained using the Tersoff potential.

Our results of structural properties of b-GaN in both zincblende and rocksalt struc-tures in general agree well with the self-consistent linear muffin-tin-orbital band struc-ture calculations of Christensen.

The result of the thermal and dynamical properties of b-GaN are in the same orderof many other semiconductors with zincblende structure.

The diffusion coefficient of b-GaN evaluated from the relation between the meansquare displacement and time is predicted. Under temperature effects, the migrationbehavior of b-GaN solid is observed.

This work represents one of a few contributions which will surely lead to a betterunderstanding of the basic physical properties of GaN in zincblende structure.

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Structural and Dynamical Properties of Zincblende GaN