Figure 1

(a) Local-mode displacements, {${\mathbf{u}}_i$}, of the cells centered on the $y=6$th $B$ plane of the $12\times 12\times 12$ dot under zero electric field. The arrows give the direction of these displacements, projected on the $xz$ plane, and the arrow length indicates the projected magnitude. (b) The same as (a), but under a field of $E=5\times {10}^8\mathrm{V}/\mathrm{m}$ applied along the $z$ axis. (c) The same as (a), but under a field of $E={10}^9\mathrm{V}/\mathrm{m}$ applied along the $z$ axis. (d) Local-mode displacements {${\mathbf{u}}_i$} of the cells centered on the $y=12$th $B$ plane of the $24\times 24\times 24$ dot. (e) Illustrations of four specific configurations of two dipoles with fixed centers but different orientations. Note that the dipole-dipole electrostatic energy increases from (i) to (iv). The dipoles axis (dotted line) is defined as the straight line connecting the centers of two dipoles. (For enlarged figure, see 25.)Reuse & Permissions

Figure 2

(a) Response of the net mode-average $⟨{\mathbf{u}}_z⟩$ per unit cell, in unit of bulk lattice constant, as a function of electric field in quantum wires of different lengths. Four curves, from bottom to top, correspond to $L_z=2.4$, 4.8, 9.6, and 14.4 nm, respectively. The $x$-axis and $y$-axis lengths are both fixed at 4.8 nm. (b) Dependences of the poling field ($E_{pf}$) and the dielectric susceptibility (of stage I) on the wire length $L_z$. This susceptibility is different from the calculated bulk value of $\sim 124$, since the latter does not correspond to dipole flipping. (c) Response of the strain ${\eta }_3$ to the applied electric fields in four different wires considered in (a). (d) Piezoelectric coefficient $d_{33}$ (of stage II) as a function of the wire length. Symbols in this figure show the direct results of the simulations, while lines are guides for eyes [except in (b), where the analytically fitted result for the poling field is shown].Reuse & Permissions