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nnp:piezo:piezoelectricity_in_wurtzite [2019/10/11 14:42]
takuma.sato [Piezoelectric effect (first-order)]
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-===== 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. 
- 
-==== Specify crystal orientation ==== 
-<figure wurtzite>​ 
-{{:​nnp:​piezo:​romanov_wurtzite4.jpg?​direct&​500}} 
-<​caption>​Rotation of a wurtzite structure. The blue plane is parallel to the interface.</​caption>​ 
-</​figure>​ 
- 
-nextnano software treats the rotation of crystal orientation by the **Miller-Bravais indices** in the input file. 
-The setup of our system is as follows: the x-axis of the simulation coordinate system (hereafter **x'​**-axis) is taken to the normal vector of the interface. The z-axis of the simulation system (**z'​**) is normal to the (-1 2 -1 0) plane of the crystal, i.e. it is along **a<​sub>​2</​sub>​** direction in Figure {{ref>​wurtzite}}. The rotation axis indicated with red line is along **z'​**-axis,​ and the interface is shown as the blue plane. The inclination angle $\theta$ is defined as the angle between the c-axis [0001] and the normal vector of the blue plane, which is **x'​**-axis. Then the crystal orientation is specified in nextnano++ input file as 
-<​code>​ 
-crystal_wz{ 
-     x-hkl = [ 1, 0, l(theta)] 
-     z_hkl = [-1, 2, 0] 
-} 
-</​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)). ​ 
- 
-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}}. 
- 
-<figure crosssection>​ 
-{{:​nnp:​piezo:​romanov_crosssection.jpg?​direct&​500}} 
-<​caption>​Cross-section of the wurtzite lattice. The dashed blue line indicates the **x'​**-direction,​ which is normal to the interface (solid blue line).</​caption>​ 
-</​figure>​ 
- 
-  * 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. 
-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: 
-<​code>​ 
-x_hkl = [floor(100*sin(theta)),​ 0, floor(100*2c*cos(theta)/​sqrt(3)a)] ​ 
-</​code>​ 
- 
-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 ==== 
-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>​ 
-{{ :​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>​ 
-</​figure>​ 
- 
-<figure strain_cry>​ 
-{{ :​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>​ 
-</​figure>​ 
- 
- 
-==== Piezoelectric effect (first-order) ==== 
-The piezoelectric effect is at first instance described by a linear response against strain. In crystal coordinate system, 
-$$ 
-P_i^{(1)}=\sum_{j=1}^6 e_{ij}\epsilon_j,​ 
-$$ 
-where the strain tensor read 
-$$ 
-\begin{pmatrix} 
-\epsilon_{1} \\ \epsilon_{2} \\ \epsilon_{3} \\ \epsilon_{4} \\ \epsilon_{5} \\ \epsilon_{6} 
-\end{pmatrix} 
-=\begin{pmatrix} 
-\epsilon_{xx} \\ \epsilon_{yy} \\ \epsilon_{zz} \\ 2\epsilon_{yz} \\ 2\epsilon_{xz} \\ 2\epsilon_{xy} 
-\end{pmatrix}. 
-$$  
-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 $e_{ij}$: 
-$$ 
-\mathbf{P}^{(1)}=\begin{pmatrix}P_x^{(1)} \\ P_y^{(1)} \\ P_z^{(1)} \end{pmatrix} 
-= \begin{pmatrix}2e_{15}\epsilon_{xz} \\ 2e_{15}\epsilon_{yz} \\ e_{31}(\epsilon_{xx}+\epsilon_{yy})+e_{33}\epsilon_{zz} \end{pmatrix} 
-$$ 
- 
-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 components, piezoelectric polarization vector in crystal- and simulation- coordinate 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). ​ 
- 
- 
-Figure {{ref>​comparison}} compares the results of nextnano software, [Romanov2006] and S. Schulz and O. Marquardt, Phys. Rev. Appl. **3**, 064020 (2015). The analytical expression Eq.(18) in [Romanov2006] misses the factor $2$ before the parameter $e_{15}$ and contains a typo. As shown in [Shulz2015],​ the correct result is derived by multiplying rotation matrix to the polarization vector and strain tensor: 
-$$ 
-P_{z'​}^{(1)}= 
-e_{31}\cos\theta\epsilon_{x'​x'​} 
-+\left(e_{31}\cos^3\theta+\frac{e_{33}-2e_{15}}{2}\sin\theta\sin 2\theta \right) \epsilon_{y'​y'​} \\ 
-+\left(\frac{e_{31}+2e_{15}}{2}\sin\theta\sin 2\theta+e_{33}\cos^3\theta \right) \epsilon_{z'​z'​} \\  
-+[(e_{31}-e_{33})\cos\theta\sin 2\theta+2e_{15}\sin\theta\cos 2\theta] \epsilon_{y'​z'​} 
-$$ 
-The analytical results in Figure {{ref>​comparison}} are the plot of this equation, with an interchange of **x'​**- and **z'​**-axes. ​ 
- 
-<figure comparison>​ 
-{{ :​nnp:​piezo:​romanov_ingan_comparison2.png?​direct&​600 |}} 
-<​caption>​Piezoelectric polarization as a function of inclination angle. The gray curve is the analytical result Eq.(18) in [Romanov2006].</​caption>​ 
-</​figure>​ 
- 
-From the result 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 ==== 
-<figure alloy> 
-{{ :​nnp:​piezo:​romanov_ingan_piezo_nnp_alloy.png|direct&​400}} 
-<​caption>​Alloy content dependence of the piezoelectric polarization.</​caption>​ 
-</​figure>​ 
- 
-==== Piezoelectric effect (second-order) ==== 
  
nnp/piezo/piezoelectricity_in_wurtzite.1570804945.txt.gz ยท Last modified: 2019/10/11 14:42 by takuma.sato