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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 [2019/10/16 16:14] (current)
takuma.sato [Piezoelectric effect (second-order)]
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 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. 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.
  
 +=== 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)
 ==== Specify crystal orientation ==== ==== Specify crystal orientation ====
 <figure wurtzite>​ <figure wurtzite>​
Line 12: Line 16:
 <​code>​ <​code>​
 crystal_wz{ crystal_wz{
-     x-hkl = [ 1, 0, l(theta)]+     x_hkl = [ 1, 0, l(theta)]
      z_hkl = [-1, 2, 0]      z_hkl = [-1, 2, 0]
 } }
Line 51: Line 55:
 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 notation
 $$ $$
 \begin{pmatrix} \begin{pmatrix}
Line 62: Line 66:
 \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. 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} 
 +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} = \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 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).  +This theory is implemented both in nextnano<​sup>​3</​sup>​ and nextnano++, 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/​nextnano3/​input_parser/​keywords/​output-strain.htm|nextnano3]] and [[https://​www.nextnano.com/​nextnanoplus/​software_documentation/​input_file/​strain.htm|nextnano++]] documentation).  
 +<​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 the consistency,​ in the input file we have rewritten the material parameters with the values used in [Romanov2006]. ​
  
-Figure {{ref>​comparison}} compares the results of nextnano software, ​[Romanov2006and SSchulz and O. MarquardtPhys. RevAppl. **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:+Analytical expression is derived as follows ​[Schulz2015]. Since we are interested in the polarization normal to the interfaceit is useful to switch to the simulation coordinate system $(x', y', z')$This can be done by multiplying rotation matrix to the polarization vector and strain tensor 
 +$$ 
 +P_{\mu'​}^{(1)}=\sum_{\mu=1}^R_{\mu'​\mu} P_\mu^{(1)},\ \  
 +\epsilon_{\mu'​\nu'​}=\sum_{\mu,​\nu=1}^3 R_{\mu'​\mu}R_{\nu'​\nu}\epsilon_{\mu\nu},​ 
 +$$ 
 +where $3\times3$ matrix $R$ accounts for a rotation of angle $\theta$. Prime denotes the axes in simulation coordinate system. The second equation can be expressed in the six-dimensional notation as  
 +$$ 
 +\epsilon_{i}=\sum_{j'​=1}^6 S_{ij'​}\epsilon_{j'​},​ 
 +$$  
 +where $S$ is a $6\times6$ matrix, whose elements depend on $\theta$. This transformation is given in Eq.(13) in [Romanov2006]. The piezoelectric polarization vector in simulation coordinate system is therefore given as 
 +$$ 
 +\begin{pmatrix} 
 +P_{x'​}^{(1)} \\ P_{y'​}^{(1)} \\ P_{z'​}^{(1)} 
 +\end{pmatrix} 
 +=R(\theta) 
 +\begin{pmatrix} 
 +0 & 0 & 0 & 0 & e_{15} ​& 0 \\ 
 +0 & 0 & 0 & e_{15} & 0 & 0 \\ 
 +e_{31} & e_{31} & e_{33} & 0 & 0 & 0 
 +\end{pmatrix} 
 +S(\theta) 
 +\begin{pmatrix} 
 +\epsilon_{x'​x'​} \\ \epsilon_{y'​y'​} \\ \epsilon_{z'​z'​} \\ 2\epsilon_{y'​z'​} \\ 2\epsilon_{x'​z'​} \\ 2\epsilon_{x'​y'​} 
 +\end{pmatrix} 
 +$
 +The **z'​**-component ​is explicitly
 $$ $$
 P_{z'​}^{(1)}= P_{z'​}^{(1)}=
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 +\left(e_{31}\cos^3\theta+\frac{e_{33}-2e_{15}}{2}\sin\theta\sin 2\theta \right) \epsilon_{y'​y'​} \\ +\left(e_{31}\cos^3\theta+\frac{e_{33}-2e_{15}}{2}\sin\theta\sin 2\theta \right) \epsilon_{y'​y'​} \\
 +\left(\frac{e_{31}+2e_{15}}{2}\sin\theta\sin 2\theta+e_{33}\cos^3\theta \right) \epsilon_{z'​z'​} \\  +\left(\frac{e_{31}+2e_{15}}{2}\sin\theta\sin 2\theta+e_{33}\cos^3\theta \right) \epsilon_{z'​z'​} \\ 
-+[(e_{31}-e_{33})\cos\theta\sin 2\theta+2e_{15}\sin\theta\cos 2\theta] \epsilon_{y'​z'​}++[(e_{31}-e_{33})\cos\theta\sin 2\theta+2e_{15}\sin\theta\cos 2\theta] \epsilon_{y'​z'​}.
 $$ $$
-The analytical results in Figure {{ref>​comparison}} are the plot of this equation, with an interchange of **x'​**- and **z'​**-axes. ​+The analytical expression Eq.(18) in [Romanov2006] misses the factor $2$ and contains a typo. Figure {{ref>​comparison}} compares the results of nextnano software, [Romanov2006] and [Shulz2015]. ​The analytical results in Figure {{ref>​comparison}} are the plot of this equation, with an interchange of **x'​**- and **z'​**-axes. ​
  
 <figure comparison>​ <figure comparison>​
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 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. 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 ==== ==== Alloy content dependence ====
 +One can also sweep the alloy content $x$. The following result corresponds to Figure 7(a) in [Romanov2006]. One can see that the zero point is universal for different alloy content.
 <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.</​caption>​
 </​figure>​ </​figure>​
  
 +==== AlGaN ====
 +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.</​caption>​
 +</​figure>​
 ==== Piezoelectric effect (second-order) ==== ==== Piezoelectric effect (second-order) ====
 +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 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>​
 +
 +
  
nnp/piezo/piezoelectricity_in_wurtzite.1570804945.txt.gz · Last modified: 2019/10/11 14:42 by takuma.sato