nnp:piezoelectricity_in_wurtzite
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nnp:piezoelectricity_in_wurtzite [2019/10/25 18:53] stefan.birner [Piezoelectricity in wurtzite] |
nnp:piezoelectricity_in_wurtzite [2020/01/27 15:40] takuma.sato [Piezoelectricity in wurtzite] |
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| === References === | === References === | ||
| - | * A.E. Romanov //et al//., Journal of Applied Physics **100**, 023522 (2006) | + | * A.E. Romanov, T.J. Baker, S. Nakamura, and J.S. Speck, Journal of Applied Physics **100**, 023522 (2006) |
| * S. Schulz and O. Marquardt, Phys. Rev. Appl. **3**, 064020 (2015) | * S. Schulz and O. Marquardt, Phys. Rev. Appl. **3**, 064020 (2015) | ||
| * S.K. Patra and S. Schulz, Phys. Rev. B **96**, 155307 (2017) | * S.K. Patra and S. Schulz, Phys. Rev. B **96**, 155307 (2017) | ||
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| We note that the expression in the third case includes the other two special cases. To approximate the direction with integer entries, we multiply 100 and take the floor function: | We note that the expression in the third case includes the other two special cases. To approximate the direction with integer entries, we multiply 100 and take the floor function: | ||
| <code> | <code> | ||
| + | $gamma = $c_InGaN / $a_InGaN # c/a ratio | ||
| + | # ideal c/a ratio in wurtzite is SQRT(8/3)=1.63299 | ||
| $h = floor(100*sin(theta)) | $h = floor(100*sin(theta)) | ||
| - | $k = floor(100*2c*cos(theta)/sqrt(3)a) | + | $l = floor(100*2*gamma*cos(theta)/sqrt(3)) |
| x_hkl = [$h, 0, $l] # x axis perpendicular to (hkl) plane = (hkil) plane | x_hkl = [$h, 0, $l] # x axis perpendicular to (hkl) plane = (hkil) plane | ||
| </code> | </code> | ||
| Line 146: | Line 148: | ||
| Analytical expression is derived as follows [Schulz2015]. Since we are interested in the polarization normal to the interface, it is useful to switch to the simulation coordinate system $(x', y', z')$. This can be done by transforming the polarization vector and the strain tensor to the simulation system, | Analytical expression is derived as follows [Schulz2015]. Since we are interested in the polarization normal to the interface, it is useful to switch to the simulation coordinate system $(x', y', z')$. This can be done by transforming the polarization vector and the strain tensor to the simulation system, | ||
| $$ | $$ | ||
| - | P_{\mu'}^{(1)}=\sum_{\mu=1}^3 R_{\mu'\mu} P_\mu^{(1)},\ \ | + | P_{\mu'}^{(1)}=\left(R P^{(1)} \right)_{\mu'}=\sum_{\mu=1}^3 R_{\mu'\mu} P_\mu^{(1)},\ \ |
| - | \epsilon_{\mu'\nu'}=\sum_{\mu,\nu=1}^3 R_{\mu'\mu}R_{\nu'\nu}\epsilon_{\mu\nu}, | + | \epsilon_{\mu'\nu'}=\left(R\epsilon R^{-1} \right)_{\mu'\nu'}=\sum_{\mu,\nu=1}^3 R_{\mu'\mu}R_{\nu'\nu}\epsilon_{\mu\nu}, |
| $$ | $$ | ||
| - | where the $3\times3$ rotation matrix $R$ accounts for a rotation of angle $\theta$. Prime denotes the axes in simulation coordinate system. These equations can be expressed in vector form as | + | where the $3\times3$ rotation matrix $R$ accounts for a rotation of angle $\theta$ and we have used the fact that the rotation matrix is orthogonal: $(R^{-1})_{\mu\nu}=R_{\nu\mu}$. Prime denotes the axes in simulation coordinate system. These equations can be expressed in vector form as |
| $$ | $$ | ||
| \begin{pmatrix} | \begin{pmatrix} | ||
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