Elastic Constants and Related Properties of the Group III-Nitrides
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In this paper we present the elastic constants for cubic BN and those for both zincblende and wurtzite GaN and AIN and the relations between them. COMPUTATIONAL METHOD The method used for our calculations is the full-potential linear muffin-tin orbital method as described by Methfessel [7]. The underlying framework for the total energy calculations is the local density functional theory [8] and elastic constants are obtained as second derivatives of the total energy curve as a function of suitably chosen strain distortions. We used well converged triple n muffin-tin orbital basis sets up to d-waves in all three K values (ddd for short) for zincblende and a fdp basis set for the wurtzite with an additional s orbital included on the large empty spheres. The Brillouin zone summations were carried out with 10 special points for zincblende and 36 for wurtzite. The muffintin sphere radii were chosen to be nearly touching and kept fixed during the calculations with distortions so as not to change the volume of the interstitial region. However, for distortions along the c-axis of the wurtzite structure (c/a distortions) we allowed the small empty sphere (centered in between cation and anion in the opposite direction of the bond along the c-axis) to vary in size so as to maintain an accurate interpolation in the local interstitial region. 399 Mat. Res. Soc. Symp. Proc. Vol. 395 01996 Materials Research Society
RELATIONS BETWEEN THE ELASTIC CONSTANTS For zincblende, we calculate the bulk modulus B - (C'1 + 2C' 2)/3 directly by applying a hydrostatic strain, the tetragonal shear modulus C.,- (C'1 -Cj 2 )/2 by means of a volume conserving strain along [001], and the trigonal shear modulus C44 by a volume conserving strain along [111]. For the latter, it is necessary to relax the relative displacement of the anion and cation for each distortion as defined by the Kleinman ( parameter. The equation 1 C,'4 = C,4° - Q--[wT((a/4)]
2
,
(1)
provides a well-known relation between the elastic constants with and without internal relaxation (indicated by superscript 0), the transverse optical phonon frequency wTf and c. Here, Q is the unit-cell volume, a is the cubic lattice constant and pt is the reduced mass. From the above relations, we obtain the complete cubic elasticity tensor and the
related quantities mentioned. Similarly, we derived a relation for the wurtzite structure between the shear elastic constant corresponding to a volume conserving strain along the hexagonal c-axis,
33-
2
-
(c1 ± 6 2)/2,
(2)
the A 1 tranverse optical phonon frequency wAb, and a new internal displacement parameter = du,,mj/d(c/a), where uc = d is the bond length along the c-axis:
-
p=[(3/ wc(c/a)
(3)
The above elastic constant is essentially obtained in the process of relaxing the total energy E[Q, c/a, u] of the wurtzite crystal. For a fixed volume Q and c/a we first minimize the energy with respect to the u parameter giving E[Q, c/a, u,in]. Then we minimize the latter as a function of c/a providing E[Q, (c/a),mn,•, Umin] and finally we
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