First-principles calculation of the piezoelectric tensor d⇊ of III–V nitrides

Abstract

We report direct first-principles density-functional calculations of the piezoelectric tensor d The piezoelectric tensor d ↔ of a polar material relates to linear order the induced polarization P to the applied stress viaIt is an especially relevant quantity in the field of III-V nitride compounds, whose piezoelectric and polarization properties are prominent [1] and unusual [2]. As implied by its definition, d↔ is relevant to electroacustic applications [3], and to the determination of polarization and electrostatic fields induced by applied stress or strain in devices or epitaxial nanostructures [4].The piezoelectric tensor e ↔ [1,5] connecting polarization and strain ǫ i viais related to the d ↔ tensor of interest here bywhere the C ij are the elastic stiffness constants at constant electric field. It is thus possible to compute d ↔ from the knowledge of elastic constants and e-piezoconstants (using e.g. the theoretical estimates of Refs.[5] and [6], see also below). On the other hand, experiments [7][8][9][10][11][12] directly access the d ij measuring the strain ǫ i caused by an applied field E via the converse piezoelectric effectThe d ij 's in Eqs. 1 and 4 are by definition identical [13], so that a direct comparison is possible, and especially interesting and useful. Here we compute directly the d ij constants in the III-V binary nitrides as derivatives of the polarization with respect to stress, calculating the Berryphase polarization of a primitive cell explicitly subject to a given external stress. This approach enables us to directly compare the calculated values with experiment, and with indirect theoretical predictions using Eq. 3, and separately calculated C ↔ and e ↔ tensors. The binary III-V nitrides AlN, GaN, and InN crystallize in the wurtzite structure, and they possess three independent components of the piezoelectric tensor, namely d 33 , d 31 (=d 32 ), and d 15 (=d 24 ). To compute these constants we run a damped Parrinello-Rahman-like dynamics [14,15] for each of the compounds considered, with the constraint that the system be subject to a given non-zero stress. The external stress is applied to the zero-stress equilibrium structure, which is identical to that reported in Ref. [5]. As a compromise between the contrasting needs for sufficiently large stresses to obtain polarizations outside the numerical noise, and for sufficiently small stresses to avoid large deformations and non-linearity, we separately apply stress components σ 1 = σ 2 = ±50 Kbar in the basal plane, and σ 3 = ±50 Kbar along the singular axis, to determine d 31 and d 33 ; for d 15 we apply a shear stress σ 5 = ±50 kBar. In the case of InN, these stresses and the ensuing deformations may lead to metallization because of the very small calculated DFT band gap. This must be avoided because macroscopic polarization cannot be defined or computed in metallic systems. We found that the maximum applied stress had to be reduced to 5 Kbar for InN (incidentally, this does not causes numerical noise problems in the polarization cal...

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“…We also plotted the depth-averaged out-of-plane strain, in-plane stress, and vertical electric field values extracted from the electro-mechanical First, the uncertainty in the out-of-plane stress values derived from the experimental measurements via Equation (26) GV/m [20], [44] or ≈20% between different published values. The in-plane stress is quite sensitive to the ratio 31 / 33 found in Equation (27), which varies as much as ≈50% between its different values of -0.44 [47], -0.48 [37], -0.56 [45] and -0.67 [20] reported in the literature. Using the values of 31 / 33 with the greatest difference of -0.44 and -0.67 to calculate the in-plane stress results in a difference of only 33% in , which is always an order of magnitude greater than the modeled value.…”

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