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nnp:piezo:piezoelectricity_in_wurtzite [2019/10/11 14:42]
takuma.sato [Piezoelectric effect (first-order)]
nnp:piezo:piezoelectricity_in_wurtzite [2019/10/21 13:54]
takuma.sato removed
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 ===== Piezoelectricity in wurtzite ===== ===== Piezoelectricity in wurtzite =====
-nextnano<​sup>​3</​sup> ​and nextnano++ ​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 contribute ​as an additional component to the total charge density profile. The important consequence of this analysis is that the piezoelectric polarization normal to the interface **becomes zero at a nontrivial angle** ​around 40 degrees. 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. ​Therefore, ​the efficiency of the devices may be optimized ​by tuning ​the growth direction of the crystal.+Author: Takuma Sato, nextnano GmbH 
 + 
 +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 === 
 +  * A.E. Romanov //et al//., Journal of Applied Physics **100**, 023522 (2006) 
 +  * S. Schulz and O. Marquardt, Phys. Rev. Appl. **3**, 064020 (2015) 
 +  * S.K. Patra and S. Schulz, Phys. Rev. B **96**, 155307 (2017) 
 +The corresponding input files are: 
 +  * Romanov_InGaN_theta_nnp.in 
 +  * Romanov_AlGaN_theta_nnp.in 
 +  * Romanov_InGaN_theta_nnp_2nd.in 
 +  * Romanov_InGaN_theta_nn3.in 
 +  * Romanov_InGaN_theta_nn3_2nd.in
  
 ==== Specify crystal orientation ==== ==== Specify crystal orientation ====
Line 12: Line 25:
 <​code>​ <​code>​
 crystal_wz{ crystal_wz{
-     x-hkl = [ 1, 0, l(theta)] +     x_hkl = [ 1, 0, l(theta)] ​# x axis perpendicular to (hkl) plane = (hkil) plane 
-     z_hkl = [-1, 2, 0]+     z_hkl = [-1, 2, 0]        # z axis perpendicular to (hkl) plane = (hkil) plane
 } }
 </​code>​ </​code>​
-where $l(\theta)$ is an integer determined by the inclination angle. This statement means //the **x'​**-axis is normal to the (1 0 -1 $l(\theta)$) plane of the crystal, whereas **z'​**-axis is normal to the (-1 2 -1 0) plane// (note that nextnano++ does not require the third entry in Miller-Bravais notation (hkil)). ​+where $l(\theta)$ is an integer determined by the inclination angle. This statement means //the **x'​**-axis is normal to the (1 0 -1 $l(\theta)$) plane of the crystal, whereas **z'​**-axis is normal to the (-1 2 -1 0) plane//(Note that nextnano++ does not require the third entry, i.e. the letter ''​i'', ​in Miller-Bravais notation (hkil) ​because ''​i=-(h+k)''​.)
  
 The index $l(\theta)$ is deduced from a simple geometry consideration. Figure {{ref>​crosssection}} shows the cross-section of a wurtzite lattice that is perpendicular to the rotation axis in Figure {{ref>​wurtzite}}. The index $l(\theta)$ is deduced from a simple geometry consideration. Figure {{ref>​crosssection}} shows the cross-section of a wurtzite lattice that is perpendicular to the rotation axis in Figure {{ref>​wurtzite}}.
Line 26: Line 39:
  
   * When $\theta=0$, the interface is normal to the (0001) plane, i.e. **x'​**-axis is normal to the (0001) plane.   * When $\theta=0$, the interface is normal to the (0001) plane, i.e. **x'​**-axis is normal to the (0001) plane.
-  * When $0<​\theta<​90$ degree, definition of the index is $l(\theta):​= \frac{c}{d}$ and the following relation holds $$d=\frac{\sqrt{3}}{2}a\tan{\theta}$$ From these equations we find $$l(\theta)=\frac{2c}{\sqrt{3}a\tan{\theta}}$$ The plane to be determined can be then taken as $$(\sin\theta\ \ 0\ \ -\sin\theta\ \ \frac{2c}{\sqrt{3}a}\cos\theta)$$. 
   * When $\theta=90$ degree, the **x'​**-axis should be normal to the (1 0 -1 0) plane of the crystal.   * When $\theta=90$ degree, the **x'​**-axis should be normal to the (1 0 -1 0) plane of the crystal.
-We note that the expression in the second ​case includes the other two cases. To approximate the direction with integer entries, we multiply 100 and take the floor function:+  * When $0<​\theta<​90$ degree, definition of the index is $l(\theta):​= \frac{c}{d}$ and the following relation holds $$d=\frac{\sqrt{3}}{2}a\tan{\theta}.$$ From these equations we find $$l(\theta)=\frac{2c}{\sqrt{3}a\tan{\theta}}.$$ The plane to be determined can be then taken as $$(hkil) = (\sin\theta\ \ 0\ \ -\sin\theta\ \ \frac{2c}{\sqrt{3}a}\cos\theta).$$ 
 +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>​
-x_hkl [floor(100*sin(theta)), 0, floor(100*2c*cos(theta)/​sqrt(3)a)] ​+$h = floor(100*sin(theta)) 
 +$k = floor(100*2c*cos(theta)/​sqrt(3)a) 
 +x_hkl = [$h, 0, $l # x axis perpendicular to (hkl) plane = (hkil) plane
 </​code>​ </​code>​
 +//Since nextnano<​sup>​3</​sup>​ does not support variables in the Miller-Bravais indices, we explicitly give the indices in the input file using statements like// ''​!IF ​  ​%ORIENTATION40 ​  ​hkil-x-direction ​ =  64 0 -64 143''​.
 +
 +==== Parameter sweep of the angle using Template: Sweep over the variable theta ====
 +  * Input file: Romanov_InGaN_theta_nnp.in
 +One can make use of **'​Template'​** feature of nextnanomat to sweep the angle $\theta$ and obtain crystal orientation dependence of several physical quantities. Here, calculation is performed for every 5 degrees (Figure {{ref>​postprocessing}}).
 +<figure postprocessing>​
 +{{:​nnp:​piezo:​postprocessing.png?​direct&​700}}
 +<​caption>'​Template'​ tab in nextnanomat.</​caption>​
 +</​figure>​
 +We obtain the angle dependence using **'​Postprocessing'​** feature.
 +Here, we collect the strain tensor components $\varepsilon_{xx}$,​ $\varepsilon_{yy}$,​ $\varepsilon_{zz}$,​ $\varepsilon_{xy}$,​ $\varepsilon_{xz}$ and $\varepsilon_{yz}$ that are in columns 2, 3, 4, 5, 6, 7 of the file ''​strain_simulation.dat''​.
 +  * Select file containing values for the strain tensor components ''​strain_simulation.dat''​ by clicking on the folder icon below //​Postprocessing//​.
 +  * Select ''​1''​ for the //Maximum number of values lines//.
 +  * Select ''​2''​ for the //Number of relevant column//. //(to do: Improve nextnanomat to include all columns.)//
 +  * Click on //Create file with combined data// to generate file ''​theta_strain_simulation_Column2.dat''​.
 +  * Select ''​3''​ for the //Number of relevant column//.
 +  * Click on //Create file with combined data// to generate file ''​theta_strain_simulation_Column3.dat''​.
 +  * Select ''​4''​ for the //Number of relevant column//.
 +  * Click on //Create file with combined data// to generate file ''​theta_strain_simulation_Column4.dat''​.
 +  * Select ''​5''​ for the //Number of relevant column//.
 +  * Click on //Create file with combined data// to generate file ''​theta_strain_simulation_Column5.dat''​.
 +  * Select ''​6''​ for the //Number of relevant column//.
 +  * Click on //Create file with combined data// to generate file ''​theta_strain_simulation_Column6.dat''​.
 +  * The postprocessing results are contained in the folder ''<​name_of_input_file>​_postprocessing''​.
 +  * Finally, the plotted results of the postprocessing file can be exported to gnuplot. Add all columns to the Overlay, and then click on: //Create and Open Gnuplot (*.plt) from Items of Overlay//
  
-One can make use of **'​template'​ feature** of nextnanomat to sweep the angle $\theta$ and obtain crystal orientation dependence of several physical quantities. Here, calculation is performed for every 5 degrees. Since nextnano3 does not support variables in the Miller-Bravais indices, we explicitly give the indices in the input file. 
 ==== Strain ==== ==== Strain ====
-Figures {{ref>​strain_sim}} and {{ref>​strain_cry}} are the strain tensor elements as a function of inclination angle $\theta$, with respect to simulationand crystalcoordinate systems, respectively. One can confirm that they reproduce correctly Figure 5 and 6 in [Romanov2006]. Please note that Romanov takes **z'​**-axis as growth direction, while we take **x'​**-axis. Therefore **x'​**- and **z'​**-axes are interchanged from [Romanov2006].+Figures {{ref>​strain_sim}} and {{ref>​strain_cry}} are the strain tensor elements as a function of inclination angle $\theta$, with respect to **simulation** and **crystal** coordinate systems, respectively. One can confirm that they reproduce correctly Figure 5 and 6 in [Romanov2006]. Please note that Romanov takes **z'​**-axis as growth direction, while we take **x'​**-axis. Therefore **x'​**- and **z'​**-axes are interchanged from [Romanov2006].
  
 <figure strain_sim>​ <figure strain_sim>​
 {{ :​nnp:​piezo:​romanov_ingan_strain_sim.png?​direct&​500 |}} {{ :​nnp:​piezo:​romanov_ingan_strain_sim.png?​direct&​500 |}}
-<​caption>​Elastic ​strains ​as a function of c-axis inclination angle $\theta$ in simulation coordinate system.</​caption>​+<​caption>​Elastic ​strain tensor components ​as a function of c-axis inclination angle $\theta$ in **simulation** coordinate system.</​caption>​
 </​figure>​ </​figure>​
  
 <figure strain_cry>​ <figure strain_cry>​
 {{ :​nnp:​piezo:​romanov_ingan_strain_cry.png?​direct&​500 |}} {{ :​nnp:​piezo:​romanov_ingan_strain_cry.png?​direct&​500 |}}
-<​caption>​Elastic ​strains as a function of c-axis inclination angle $\theta$ in crystal coordinate system.</​caption>​+<​caption>​Elastic ​strain tensor componentsas ​a function of c-axis inclination angle $\theta$ in **crystal** coordinate system.</​caption>​
 </​figure>​ </​figure>​
  
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 The piezoelectric effect is at first instance described by a linear response against strain. In crystal coordinate system, The piezoelectric effect is at first instance described by a linear response against strain. In crystal coordinate system,
 $$ $$
-P_i^{(1)}=\sum_{j=1}^6 e_{ij}\epsilon_j,​+P_\mu^{(1)}=\sum_{j=1}^6 e_{\mu j}\epsilon_j,​
 $$ $$
-where the strain tensor ​read+where $\mu=1,2,3$ and the strain tensor ​is expressed in six-dimensional Voigt notation
 $$ $$
 \begin{pmatrix} \begin{pmatrix}
Line 62: Line 101:
 \end{pmatrix}. \end{pmatrix}.
 $$  $$ 
-Please note that the indices $x, y, z$ without prime refer to the axes of the crystal coordinate system. For the symmetry of the wurtzite structure, only three components ​remain in the piezoelectric coefficient tensor $e_{ij}$:+Please note that the indices $x, y, z$ without prime refer to the axes of the crystal coordinate system. The superscript $^{(1)}$ indicates first-order piezoeffect. For the symmetry of the wurtzite structure, only three parameters ​remain in the piezoelectric coefficient tensor $e_{ij}$ ​
 $$ $$
-\mathbf{P}^{(1)}=\begin{pmatrix}P_x^{(1)} \\ P_y^{(1)} \\ P_z^{(1)} \end{pmatrix} +\begin{pmatrix} 
-= \begin{pmatrix}2e_{15}\epsilon_{xz} \\ 2e_{15}\epsilon_{yz} \\ e_{31}(\epsilon_{xx}+\epsilon_{yy})+e_{33}\epsilon_{zz} \end{pmatrix}+P_x^{(1)} \\ P_y^{(1)} \\ P_z^{(1)} ​\\  
 +\end{pmatrix} 
 +
 +\begin{pmatrix} 
 +0 & 0 & 0 & 0 & e_{15} & 0 \\ 
 +0 & 0 & 0 & e_{15} & 0 & 0 \\ 
 +e_{31} & e_{31} & e_{33} & 0 & 0 & 0 
 +\end{pmatrix} 
 +\begin{pmatrix} 
 +\epsilon_{xx} \\ \epsilon_{yy} \\ \epsilon_{zz} \\ 2\epsilon_{yz} \\ 2\epsilon_{xz} \\ 2\epsilon_{xy} 
 +\end{pmatrix} 
 += \begin{pmatrix}2e_{15}\epsilon_{xz} \\ 2e_{15}\epsilon_{yz} \\ e_{31}(\epsilon_{xx}+\epsilon_{yy})+e_{33}\epsilon_{zz} \end{pmatrix},
 $$ $$
 +cf. Eq. (4) in [Schulz2015]. __**Note that Eq. (14) in [Romanov2006] misses the factor 2 for off-diagonal elements of the strain tensor.**__
  
-This theory is implemented ​both in nextnano<​sup>​3</​sup> ​and nextnano++, ​with corresponding material parameters in the respective database. ​One can export the strain tensor componentspiezoelectric polarization vector in crystaland simulationcoordinate systems ​in output files (cf. [[https://​www.nextnano.com/​nextnano3/input_parser/keywords/output-strain.htm|nextnano3]] and [[https://​www.nextnano.com/​nextnanoplus/software_documentation/input_file/​strain.htm|nextnano++]] documentation). ​+These equations are implemented in both the nextnano++ and nextnano<​sup>​3</​sup> ​software ​with corresponding material parameters in the respective database. ​The following flags export the strain tensor components ​and piezoelectric polarization vector in **crystal** and **simulation** coordinate systems (cf. [[https://​www.nextnano.com/​nextnanoplus/software_documentation/input_file/​strain.htm|nextnano++]] and [[https://​www.nextnano.com/​nextnano3/input_parser/keywords/output-strain.htm|nextnano3]] documentation). The piezoelectric polarization vector with respect to the simulation coordinate system can be found in the file ''​Strain\piezoelectric_polarization_vector_simulation.dat''​. 
 +<​code>​ 
 +# nextnano++ 
 +strain{ 
 +   ​output_strain_tensor{ 
 +      crystal_system ​   = yes 
 +      simulation_system = yes 
 +   } 
 +    
 +   ​output_polarization_vector{ 
 +      crystal_system ​   = yes 
 +      simulation_system = yes 
 +   } 
 +
 +</​code>​ 
 +<​code>​ 
 +! nextnano3 
 +$output-strain 
 +   ​strain-simulation-system = yes 
 +   ​strain-crystal-system ​   = yes 
 +   ​polarization-vector ​     = yes 
 +$end_output-strain 
 +</​code>​ 
 +For consistency,​ we have used the same material parameters as [Romanov2006],​ i.e. we have overwritten our default material parameters of the database with the values specified in the input file
  
- +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, 
-Figure ​{{ref>​comparison}} compares ​the results ​of nextnano software, [Romanov2006] and SSchulz and OMarquardtPhys. Rev. Appl. **3**, 064020 ​(2015). The analytical expression ​Eq.(18) in [Romanov2006] ​misses ​the factor $2$ before ​the parameter ​$e_{15}$ and contains a typoAs shown in [Shulz2015],​ the correct result ​is derived by multiplying rotation matrix to the polarization vector and strain tensor:+$$ 
 +P_{\mu'}^{(1)}=\sum_{\mu=1}^3 R_{\mu'​\mu} P_\mu^{(1)},​\ \  
 +\epsilon_{\mu'​\nu'​}=\sum_{\mu,​\nu=1}^3 R_{\mu'​\mu}R_{\nu'​\nu}\epsilon_{\mu\nu},​ 
 +$$ 
 +where the $3\times3$ rotation matrix $R$ accounts for a rotation ​of angle $\theta$Prime denotes the axes in simulation coordinate systemThese equations can be expressed in vector form as 
 +$$ 
 +\begin{pmatrix} 
 +P_{x}^{(1)} \\ P_{y}^{(1)} \\ P_{z}^{(1)} 
 +\end{pmatrix} 
 +=R^{-1}(\theta) ​  
 +\begin{pmatrix} 
 +P_{x'​}^{(1)} \\ P_{y'​}^{(1)} \\ P_{z'​}^{(1)} 
 +\end{pmatrix},\ \  
 +\begin{pmatrix} 
 +\epsilon_{xx} \\ \epsilon_{yy} \\ \epsilon_{zz} \\ 2\epsilon_{yz} \\ 2\epsilon_{xz} \\ 2\epsilon_{xy} 
 +\end{pmatrix} 
 +=S^{-1}(\theta) 
 +\begin{pmatrix} 
 +\epsilon_{x'​x'​} \\ \epsilon_{y'​y'​} \\ \epsilon_{z'​z'​} \\ 2\epsilon_{y'​z'​} \\ 2\epsilon_{x'​z'​} \\ 2\epsilon_{x'​y'​} 
 +\end{pmatrix} 
 +$$  
 +where $S(\theta)$ is a $6\times6$ matrix. The second transformation is given in Eq. (13) in [Romanov2006]. From equations above, we obtain ​the first-order piezoelectric effect in the simulation coordinate system 
 +$
 +\begin{pmatrix} 
 +P_{x'​}^{(1)} \\ P_{y'​}^{(1)} \\ P_{z'​}^{(1)} 
 +\end{pmatrix} 
 +=R(\theta) 
 +\begin{pmatrix} 
 +0 & 0 & 0 & 0 & e_{15} ​& 0 \\ 
 +0 & 0 & 0 & e_{15} & 0 & 0 \\ 
 +e_{31} & e_{31} & e_{33} & 0 & 0 & 0 
 +\end{pmatrix} 
 +S^{-1}(\theta) 
 +\begin{pmatrix} 
 +\epsilon_{x'​x'​} \\ \epsilon_{y'​y'​} \\ \epsilon_{z'​z'​} \\ 2\epsilon_{y'​z'​} \\ 2\epsilon_{x'​z'​} \\ 2\epsilon_{x'​y'​} 
 +\end{pmatrix}. 
 +$$ 
 +The **z'​**-component ​is explicitly
 $$ $$
 P_{z'​}^{(1)}= P_{z'​}^{(1)}=
-e_{31}\cos\theta\epsilon_{x'​x'​}+e_{31}\cos\theta\epsilon_{x'​x'​}\\
 +\left(e_{31}\cos^3\theta+\frac{e_{33}-2e_{15}}{2}\sin\theta\sin 2\theta \right) \epsilon_{y'​y'​} \\ +\left(e_{31}\cos^3\theta+\frac{e_{33}-2e_{15}}{2}\sin\theta\sin 2\theta \right) \epsilon_{y'​y'​} \\
 +\left(\frac{e_{31}+2e_{15}}{2}\sin\theta\sin 2\theta+e_{33}\cos^3\theta \right) \epsilon_{z'​z'​} \\  +\left(\frac{e_{31}+2e_{15}}{2}\sin\theta\sin 2\theta+e_{33}\cos^3\theta \right) \epsilon_{z'​z'​} \\ 
-+[(e_{31}-e_{33})\cos\theta\sin 2\theta+2e_{15}\sin\theta\cos 2\theta] \epsilon_{y'​z'​}++\left[(e_{31}-e_{33})\cos\theta\sin 2\theta+2e_{15}\sin\theta\cos 2\theta\right] \epsilon_{y'​z'​}.
 $$ $$
-The analytical results in Figure {{ref>​comparison}} are the plot of this equation, with an interchange of **x'​**- and **z'​**-axes. ​+__**Note that the corresponding analytical expression Eq. (18) in [Romanov2006] misses the factor $2$ in front of $e_{15}$ in the 2<​sup>​nd</​sup>,​ 3<​sup>​rd</​sup>​ and 4<​sup>​th</​sup>​ line, and contains a typo in the 3<​sup>​rd</​sup>​ line, i.e. $e_{33}$ has to be $e_{31}$ in the first term.**__ Figure {{ref>​comparison}} compares the results of the nextnano software with the results of [Romanov2006] and [Schulz2015],​ respectively. ​The analytical results in Figure {{ref>​comparison}} are the plot of the equation ​above, with an interchange of **x'​**- and **z'​**-axes. ​
  
 <figure comparison>​ <figure comparison>​
-{{ :nnp:piezo:romanov_ingan_comparison2.png?​direct&​600 |}} +{{ :nnp:piezo:romanov_ingan_comparison7.png?​direct&​600 |}} 
-<​caption>​Piezoelectric polarization as a function of inclination angle. The gray curve is the analytical result Eq.(18) in [Romanov2006].</​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 in [Romanov2006]. With the factor $2$ the result becomes the black curve. nextnano<​sup>​3</​sup>​ and nextnano++ reproduce the black curve.</​caption>​
 </​figure>​ </​figure>​
  
-From the result ​we can see that the piezoelectric polarization vanishes at an intermediate angle around 38 degreeand that it is maximized when inclination angle is zero.+From the results ​we can see that the piezoelectric polarization vanishes at an intermediate angle around 38 degree and that it is maximized when inclination angle is zero.
 ==== 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.
 <figure alloy> <figure alloy>
-{{ :​nnp:​piezo:​romanov_ingan_piezo_nnp_alloy.png|direct&400}} +{{ :​nnp:​piezo:​romanov_ingan_piezo_nnp_alloy.png|direct}} 
-<​caption>​Alloy content dependence of the piezoelectric polarization.</​caption>​+<​caption>​Alloy content dependence of the piezoelectric polarization ​for In<​sub>​x</​sub>​Ga<​sub>​1-x</​sub>​N/​GaN structure. In<​sub>​x</​sub>​Ga<​sub>​1-x</​sub>​N is under biaxial compressive strain with respect to GaN.</​caption>​
 </​figure>​ </​figure>​
  
 +==== AlGaN ====
 +  * Input file: Romanov_AlGaN_theta_nnp.in
 +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.
 +<figure alloy_Al>​
 +{{ :​nnp:​piezo:​romanov_algan_piezo_nnp_alloy.png|direct}}
 +<​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>​
 ==== Piezoelectric effect (second-order) ==== ==== Piezoelectric effect (second-order) ====
 +  * Input file: Romanov_InGaN_theta_nnp_2nd.in
 +Optimization of optoelectronic device design requires an accurate and detailed knowledge of the growth-direction dependence of the built-in electric field. Recently, the second order piezoelectric effect has been reported to be relevant for wurtzite III-N materials, namely GaN, AlN and InN. This potentially affects the electronic and optical properties of the devices. The piezoelectric polarization is generalized in crystal coordinate as [Patra2017]
 +$$
 +P_\mu^{\mathrm{pz}}=\sum_{j=1}^6e_{\mu j}\epsilon_j+\frac{1}{2}\sum_{j,​k=1}^6 B_{\mu jk}\epsilon_j\epsilon_k+\cdots
 +$$
 +where $e_{\mu j}$ and $B_{\mu jk}$ are first- and second-order piezoelectric coefficients,​ respectively. For binary wurtzite structure, one can show that $B_{\mu jk}$ has 8 independent components $B_{311}, B_{312}, B_{313}, B_{333}, B_{115}, B_{125}, B_{135}, B_{344}$. The explicit expression of the second-order term is given in Eq. (3) in [Patra2017],​ which is also implemented in nextnano++.
 +
 +One can turn on the second-order contribution in [[https://​www.nextnano.com/​nextnanoplus/​software_documentation/​input_file/​strain.htm|nextnano++]] as
 +<​code>​
 +# nextnano++
 +strain{
 +   ...
 +   ​second_order_piezo = yes        # default: no
 +}
 +</​code>​
 +and in [[https://​www.nextnano.com/​nextnano3/​input_parser/​keywords/​numeric-control.htm|nextnano3]]
 +<​code>​
 +! nextnano3
 +$numeric-control
 +   ...
 +   ​piezo-second-order = 2nd-order ​   ! [no/​2nd-order/​2nd-order-Tse-Pal]
 +$end_numeric-control
 +</​code> ​
 +
 +Figure {{ref>​2nd_order}} shows the result of 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 2nd_order>​
 +{{ :​nnp:​piezo:​romanov_ingan_comparison6.png|direct&​400}}
 +<​caption>​Second-order piezoelectricity. Interface planes are indicated at corresponding angles.</​caption>​
 +</​figure>​
 +
 +  * Please help us to improve our tutorial! Should you have any questions and comments, please contact support [at] nextnano.com.