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nnp:piezo:piezoelectricity_in_wurtzite

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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