nnp:piezoelectricity_in_wurtzite
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nnp:piezoelectricity_in_wurtzite [2019/10/24 17:41] stefan.birner [Piezoelectric effect (second-order)] |
nnp:piezoelectricity_in_wurtzite [2024/01/03 17:50] stefan.birner removed |
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| ===== Piezoelectricity in wurtzite ===== | ===== Piezoelectricity in wurtzite ===== | ||
| Author: Takuma Sato, nextnano GmbH | Author: Takuma Sato, nextnano GmbH | ||
| + | |||
| + | //If you want to obtain the input files used in this tutorial, please contact support [at] nextnano.com.// | ||
| nextnano++ and nextnano<sup>3</sup> can simulate growth orientation dependence of the piezoelectric effect in heterostructures. Following A.E. Romanov //et al//., Journal of Applied Physics **100**, 023522 (2006), we consider In<sub>x</sub>Ga<sub>1-x</sub>N and Al<sub>x</sub>Ga<sub>1-x</sub>N thin layers pseudomorphically grown on GaN substrates. The c-axis of the substrate GaN is inclined by an angle $\theta$ with respect to the interface of the heterostructure. The layer is assumed to be very thin compared to substrate so that the strain is approximately homogeneous in all direction (pseudomorphic), and the ternary alloys mimic the orientation of crystallography direction. The layer material deforms such that the lattice translation vector of each layer has a common projection onto the interface. The strain in a crystal induces piezoelectric polarization, which contributes as an additional component to the total charge density profile. The important consequence of their analysis is that the piezoelectric polarization normal to the interface **becomes zero at a nontrivial angle**. The piezoelectric charge in a heterostructure in general results in an additional offset between electron and hole spatial probability distribution, thereby reducing the overlap of their wavefunctions in real space. The small overlap of electron and hole leads to an inefficient radiative recombination, i.e. lower efficiency of optoelectronic devices. The work by Romanov //et al//. paved the way to device optimization by the growth direction of the crystal. | nextnano++ and nextnano<sup>3</sup> can simulate growth orientation dependence of the piezoelectric effect in heterostructures. Following A.E. Romanov //et al//., Journal of Applied Physics **100**, 023522 (2006), we consider In<sub>x</sub>Ga<sub>1-x</sub>N and Al<sub>x</sub>Ga<sub>1-x</sub>N thin layers pseudomorphically grown on GaN substrates. The c-axis of the substrate GaN is inclined by an angle $\theta$ with respect to the interface of the heterostructure. The layer is assumed to be very thin compared to substrate so that the strain is approximately homogeneous in all direction (pseudomorphic), and the ternary alloys mimic the orientation of crystallography direction. The layer material deforms such that the lattice translation vector of each layer has a common projection onto the interface. The strain in a crystal induces piezoelectric polarization, which contributes as an additional component to the total charge density profile. The important consequence of their analysis is that the piezoelectric polarization normal to the interface **becomes zero at a nontrivial angle**. The piezoelectric charge in a heterostructure in general results in an additional offset between electron and hole spatial probability distribution, thereby reducing the overlap of their wavefunctions in real space. The small overlap of electron and hole leads to an inefficient radiative recombination, i.e. lower efficiency of optoelectronic devices. The work by Romanov //et al//. paved the way to device optimization by the growth direction of the crystal. | ||
| === 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 144: | 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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| <figure comparison> | <figure comparison> | ||
| - | {{ :nnp:romanov_ingan_comparison7.png?direct&600 |}} | + | {{ :nnp:romanov_ingan_comparison7.png |}} |
| <caption>Piezoelectric polarization as a function of inclination angle. The gray dotted curve contains a typo $e_{33}\leftrightarrow e_{31}$ and misses the factor $2$. When the first typo is fixed, the gray solid curve is obtained and looks to be consistent with Figure 7(a) in [Romanov2006]. With the factor $2$ the result becomes the black curve. nextnano<sup>3</sup> and nextnano++ reproduce the black curve and are thus consistent to [Schulz2015].</caption> | <caption>Piezoelectric polarization as a function of inclination angle. The gray dotted curve contains a typo $e_{33}\leftrightarrow e_{31}$ and misses the factor $2$. When the first typo is fixed, the gray solid curve is obtained and looks to be consistent with Figure 7(a) in [Romanov2006]. With the factor $2$ the result becomes the black curve. nextnano<sup>3</sup> and nextnano++ reproduce the black curve and are thus consistent to [Schulz2015].</caption> | ||
| </figure> | </figure> | ||
| Line 209: | Line 213: | ||
| ==== Alloy content dependence ==== | ==== Alloy content dependence ==== | ||
| One can also sweep the alloy content $x$. The following results correspond to Figure 7(a) in [Romanov2006]. One can see that the zero point is universal for different alloy contents. | One can also sweep the alloy content $x$. The following results correspond to Figure 7(a) in [Romanov2006]. One can see that the zero point is universal for different alloy contents. | ||
| - | The zero point is different compared to [Romanov2006] as he misses the factor of 2 for the strain tensor component. As can be seen in Figure 5 shown above, this mistake is not relevant for 0 and 90 degrees. | + | The zero point is different compared to [Romanov2006] as he misses the factor of 2 for the strain tensor component. As can be seen in Figure {{ref>comparison}} shown above, this mistake is not relevant for 0 and 90 degrees. |
| <figure alloy> | <figure alloy> | ||
| {{ :nnp:romanov_ingan_piezo_nnp_alloy.png}} | {{ :nnp:romanov_ingan_piezo_nnp_alloy.png}} | ||
| Line 219: | Line 223: | ||
| Similarly, piezoelectric polarization of Al<sub>x</sub>Ga<sub>1-x</sub>N/GaN structure is calculated and shown in Figure {{ref>alloy_Al}}. | Similarly, piezoelectric polarization of Al<sub>x</sub>Ga<sub>1-x</sub>N/GaN structure is calculated and shown in Figure {{ref>alloy_Al}}. | ||
| This result corresponds to Figure 8(a) in [Romanov2006]. The piezoelectric effect vanishes at around 38 degree in this case as well. | This result corresponds to Figure 8(a) in [Romanov2006]. The piezoelectric effect vanishes at around 38 degree in this case as well. | ||
| - | Again, zhe zero point is different compared to [Romanov2006] as he misses the factor of 2 for the strain tensor component. As can be seen in Figure 5 shown above, this mistake is not relevant for 0 and 90 degrees. | + | Again, the zero point is different compared to [Romanov2006] as he misses the factor of 2 for the strain tensor component. As can be seen in Figure {{ref>comparison}} shown above, this mistake is not relevant for 0 and 90 degrees. |
| <figure alloy_Al> | <figure alloy_Al> | ||
| {{ :nnp:romanov_algan_piezo_nnp_alloy.png}} | {{ :nnp:romanov_algan_piezo_nnp_alloy.png}} | ||
| <caption>Alloy content dependence of the piezoelectric polarization for Al<sub>x</sub>Ga<sub>1-x</sub>N/GaN structure. Al<sub>x</sub>Ga<sub>1-x</sub>N is under biaxial tensile strain with respect to GaN.</caption> | <caption>Alloy content dependence of the piezoelectric polarization for Al<sub>x</sub>Ga<sub>1-x</sub>N/GaN structure. Al<sub>x</sub>Ga<sub>1-x</sub>N is under biaxial tensile strain with respect to GaN.</caption> | ||
| </figure> | </figure> | ||
| + | The sign of the piezoelectric polarization is opposite to the case of InGaN/GaN composition (Figure {{ref>alloy}}). This is due to the fact that the lattice constants of InN, GaN and AlN obey the following relation | ||
| + | $$ | ||
| + | a_{\mathrm{InN}}>a_{\mathrm{GaN}}>a_{\mathrm{AlN}} | ||
| + | $$ | ||
| + | (also for $c$). Since we take GaN as a substitute, In<sub>x</sub>Ga<sub>1-x</sub>N layer is subject to compressive strain, whereas Al<sub>x</sub>Ga<sub>1-x</sub>N is under tensile strain [Romanov2006]. | ||
| ==== Piezoelectric effect (second-order) ==== | ==== Piezoelectric effect (second-order) ==== | ||
| * Input file: Romanov_InGaN_theta_nnp_2nd.in | * Input file: Romanov_InGaN_theta_nnp_2nd.in | ||
| Line 245: | Line 254: | ||
| $numeric-control | $numeric-control | ||
| ... | ... | ||
| - | piezo-second-order = 2nd-order ! [no/2nd-order/2nd-order-Tse-Pal] | + | piezo-second-order = 2nd-order ! [no/2nd-order] |
| $end_numeric-control | $end_numeric-control | ||
| </code> | </code> | ||
| - | Figure {{ref>2nd_order}} shows the results of the nextnano software. While the second-order contribution becomes negligible between the orientation $(1 0 \bar{1} 3)$ and $(1 0 \bar{1} 2)$, it enhances the piezo effect up to 14% in other directions. This figure can be qualitatively compared to Figure 1 ( c ) in [Patra2017], but note that they consider binary InN/GaN structure there. The pink curve is different from the one in Figure {{ref>comparison}} because we employed the material parameters used in [Patra2017]. nextnano<sup>3</sup> yields slightly deviated result, as it uses different formula for the second-order effect, L. Pedesseau et al., Appl. Phys. Lett. 100, 031903 (2012). Also see [[https://www.nextnano.com/nextnano3/input_parser/keywords/numeric-control.htm|nextnano3 documentation]]. | + | Figure {{ref>2nd_order}} shows the results of the nextnano software. While the second-order contribution becomes negligible between the orientation $(1 0 \bar{1} 3)$ and $(1 0 \bar{1} 2)$, and also between 85 and 95 degrees, it enhances the piezo effect up to 14% in other directions. This figure can be qualitatively compared to Figure 1<nowiki>(c)</nowiki> in [Patra2017], but note that they consider binary InN/GaN structure there while we are using In<sub>0.2</sub>Ga<sub>0.8</sub>N/GaN. The pink curve is different from the one in Figure {{ref>comparison}} because we employed the material parameters used in [Patra2017]. nextnano<sup>3</sup> also produces a consistent result (input file: ''Romanov_InGaN_theta_nn3_2nd.in''). Within nextnano<sup>3</sup> one can also use different formulae for the second-order effect, cf. [[https://www.nextnano.com/nextnano3/input_parser/keywords/numeric-control.htm|nextnano3 documentation]]. |
| <figure 2nd_order> | <figure 2nd_order> | ||
| {{ :nnp:romanov_ingan_comparison6.png|direct&400}} | {{ :nnp:romanov_ingan_comparison6.png|direct&400}} | ||
| - | <caption>Second-order piezoelectricity. Interface planes are indicated at corresponding angles.</caption> | + | <caption>Second-order piezoelectricity. The second-order term enhances the piezoelectric polarization. nextnano<sup>3</sup> result (yellow) is consistent to the nextnano++ result (blue). Interface planes are indicated at corresponding angles.</caption> |
| </figure> | </figure> | ||
| * Please help us to improve our tutorial! Should you have any questions and comments, please send to support [at] nextnano.com. | * Please help us to improve our tutorial! Should you have any questions and comments, please send to support [at] nextnano.com. | ||
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