Symmetry and Piezoelectricity: Evaluation of $\alpha$-Quartz coefficients

TL;DR

This paper evaluates the piezoelectric coefficients of α-Quartz using three symmetry-based methods, highlighting Royer-Dieulesaint as the most efficient and elegant approach among them.

Contribution

It compares three different symmetry-based methods for deriving piezoelectric coefficients of α-Quartz, emphasizing the efficiency of Royer-Dieulesaint.

Findings

Royer-Dieulesaint method is the most elegant and efficient

Fumi method is tedious

Landau-Lifshitz method requires additional physical principles

Abstract

Piezoelectric coefficients of $\alpha$-Quartz are derived from symmetry arguments based on Neumann's Principle with three different methods: Fumi, Landau-Lifshitz and Royer-Dieulesaint. While Fumi method is tedious and Landau-Lifshitz requires additional physical principles to evaluate the piezoelectric coefficients, Royer-Dieulesaint is the most elegant and most efficient of the three techniques.

Tables

Table 1. Table 1: Point symmetry groups in the Schoenflies (S) and Hermann-Mauguin (H-M) nomenclature with corresponding symmetry operations and order N 𝑁 N . The primed and double primed operations such as C 3 ′ subscript superscript 𝐶 ′ 3 C^{{}^{\prime}}_{3} and C 3 ′′ subscript superscript 𝐶 ′′ 3 C^{{}^{\prime\prime}}_{3} correspond to 3-fold rotation by 2 ​ π / 3 2 𝜋 3 2\pi/3 with respect to axes other than the standard axis (usually z 𝑧 z ). σ v subscript 𝜎 𝑣 \sigma_{v} is a reflection operation with respect to a vertical plane (containing the z 𝑧 z axis) whereas σ h subscript 𝜎 ℎ \sigma_{h} is reflection with respect to a horizontal plane and σ d subscript 𝜎 𝑑 \sigma_{d} is reflection with respect to a diagonal plane. The primed and double primed operations such as σ ′ superscript 𝜎 ′ \sigma^{\prime} and σ ′′ superscript 𝜎 ′′ \sigma^{\prime\prime} correspond to reflection operations with respect to planes other than the standard plane.

(S)	(H-M)	Symmetry operation	$N$
Cubic systems
$T$	23	$E$, 4$C_3$, 4$C_3^2$, 3$C_2$	12
$T_h$	m3	$E$, 8$C_3$, 3$C_2$, 3${\sigma }_v$, $i$, 8$S_6$	24
$O$	432	$E$, 6$C_4$, 8$C_3$, 3$C_2$, 6$C_2^{{}^{\prime }}$	24
$T_d$	43m	$E$, 8$C_3$, 3$C_2$, 6$S_4$, 6${\sigma }_d$	24
$O_h$	m3m	$E$, 8$C_3$, 6$C_2$, 6$C_4$, 3$C_2^{{}^{\prime }}$, $i$,
		6$S_4$, 8$S_6$, 3${\sigma }_h$, 6${\sigma }_d$	48
Tetragonal systems
$C_4$	4	$E$, $C_4$, $C_2$, $C_4^3$	4
$S_4$	4	$E$, $S_4$, $C_2$, $S_4^3$	4
$C_{4h}$	4/m	$E$, $C_4$, $C_2$, $C_4^3$, $i$, $S_4^3$, ${\sigma }_h$, $S_4$	8
$D_4$	422	$E$, 2$C_4$, 2$C_2$, 2$C_2^{{}^{\prime }}$, 2$C_2^{{}^{\prime \prime }}$	8
$C_{4v}$	4mm	$E$, 2$S_4$, $C_2$, 2$C_2^{{}^{\prime }}$, 2${\sigma }_d$	8
$D_{2d}$	42m	$E$, 2$S_4$, $C_2$, 2$C_2^{{}^{\prime }}$, 2${\sigma }_d$	8
$D_{4h}$	4/mmm	$E$, 2$C_4$, $C_2$, 2$C_2^{{}^{\prime }}$, 2$C_2^{{}^{\prime \prime }}$, $i$,
		2$S_4$, ${\sigma }_h$, 2${\sigma }_v$, 2${\sigma }_d$	16
Orthorhombic systems
$D_2$	222	$E$, $C_2$, $C_2^{{}^{\prime }}$, $C_2^{{}^{\prime \prime }}$	4
$C_{2v}$	mm2	$E$, $C_2$, ${\sigma }_v$, ${\sigma }_v^{\prime }$	4
$D_{2h}$	mmm	$E$, $C_2$, $C_2^{{}^{\prime }}$, $C_2^{{}^{\prime \prime }}$, $i$, $\sigma$, ${\sigma }^{\prime }$, ${\sigma }^{\prime \prime }$	8
Monoclinic systems
$C_2$	2	$E$, $C_2$	2
$C_3$	m or 2	$E$, ${\sigma }_h$	2
$C_{2h}$	2/m	$E$, $C_2$, $i$, ${\sigma }_h$	4
Triclinic systems
$C_1$	1	$E$	1
$C_i$	1	$E$, $i$	2
Trigonal systems
$C_3$	3	$E$, $C_3$, $C_3^2$	3
$S_6$	3	$E$, $C_3$, $C_3^2$, $i$, $S_6^5$, $S_6$	6
$D_3$	32	$E$, 2$C_3$, 3$C_2$	6
$C_{3v}$	3m	$E$, 2$C_3$, 3${\sigma }_v$	6
$D_{3d}$	3m	$E$, 2$C_3$, 3$C_2$, $i$, 2$S_5$, 3${\sigma }_d$	12
Hexagonal systems
$C_6$	6	$E$, $C_6$, $C_3$, $C_2$, $C_3^2$, $C_6^5$	6
$C_{3h}$ ($S_3$)	6 or 3/m	$E$, $C_3$, $C_3^2$, ${\sigma }_h$, $S_3$, $S_3^5$	6
$C_{6h}$	6/m	$E$, $C_6$, $C_3$, $C_2$, $C_3^2$, $C_6^5$,
		$i$, $S_3^5$, $S_6^5$, ${\sigma }_h$, $S_6$, $S_3$	12
$D_6$	622	$E$, 2$C_6$, 2$C_3$, $C_2$, 3$C_2^{{}^{\prime }}$, 3$C_2^{{}^{\prime \prime }}$	12
$C_{6v}$	6mm	$E$, 2$C_6$, 2$C_3$, $C_2$, 3${\sigma }_v$, 3${\sigma }_d$	12
$D_{3h}$	6m2	$E$, 2$C_3$, 3$C_2$, ${\sigma }_h$, 2$S_3$, 3${\sigma }_v$	12
$D_{6h}$	6/mmm	$E$, 2$C_6$, 2$C_5$, $C_2$, 3$C_2^{{}^{\prime }}$, 3$C_2^{{}^{\prime \prime }}$,
		$i$, 2$S_3$, 2$S_6$, ${\sigma }_h$, 3${\sigma }_d$ , 3${\sigma }_v$	24

Equations

$$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}\cos \varphi & -\sin \varphi & 0\\ \sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$$

$$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}\cos \varphi & \sin \varphi & 0\\ -\sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$$

$$\left(\begin{array}{cccccc}{d}_{11}& d_{12}& d_{13}& d_{14}& d_{15}& d_{16}\\ d_{21}& d_{22}& d_{23}& d_{24}& d_{25}& d_{26}\\ d_{31}& d_{32}& d_{33}& d_{34}& d_{35}& d_{36}\end{array}\right)$$

$$x^{\prime }=x,y^{\prime }=-y,z^{\prime }=-z$$

$$x^{\prime }{x}^{\prime }{x}^{\prime }=xxx$$

$$y^{\prime }{x}^{\prime }{x}^{\prime }=(-y)xx$$

$$\left(\begin{array}{cccccc}{d}_{11}& d_{12}& d_{13}& d_{14}& 0& 0\\ 0& 0& 0& 0& d_{25}& d_{26}\\ 0& 0& 0& 0& d_{35}& d_{36}\end{array}\right)$$

$$x^{\prime }=-\frac{1}{2}x-\frac{\sqrt{3}}{2}y,y^{\prime }=\frac{\sqrt{3}}{2}x-\frac{1}{2}y,z^{\prime }=z$$

$$x^{\prime }{x}^{\prime }{x}^{\prime }$$

$$=$$

$$d_{11}^{\prime }=-\frac{1}{8}{d}_{11}-\frac{3}{8}{d}_{12}-\frac{3}{8}{d}_{26}\equiv d_{11}$$

$$x^{\prime }{y}^{\prime }{y}^{\prime }$$

$$=$$

$$d_{12}^{\prime }=-\frac{3}{8}{d}_{11}-\frac{1}{8}{d}_{12}+\frac{3}{8}{d}_{26}\equiv d_{12}$$

$$x^{\prime }{z}^{\prime }{z}^{\prime }$$

$$=$$

$$y^{\prime }{x}^{\prime }{z}^{\prime }$$

$$=$$

$$z^{\prime }{x}^{\prime }{z}^{\prime }$$

$$z^{\prime }{x}^{\prime }{y}^{\prime }$$

$$=$$

$$d_{312}^{\prime }=\frac{1}{4}{d}_{312}-\frac{3}{4}{d}_{321}\equiv d_{312}$$

$$\left(\begin{array}{cccccc}{d}_{11}& -d_{11}& 0& d_{14}& 0& 0\\ 0& 0& 0& 0& -d_{14}& -2d_{11}\\ 0& 0& 0& 0& 0& 0\end{array}\right)$$

$$\left(\begin{array}{cccccc}b& -b& 0& a& 0& 0\\ 0& 0& 0& 0& -a& -b\\ 0& 0& 0& 0& 0& 0\end{array}\right)$$

$$\left(\begin{array}{cccccc}b& -b& 0& 2a& 0& 0\\ 0& 0& 0& 0& -2a& -2b\\ 0& 0& 0& 0& 0& 0\end{array}\right)$$

$$A=\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}& \frac{i}{\sqrt{2}}& 0\\ \frac{i}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\ 0& 0& 1\end{array}\right)$$

$${\eta }_{ijk}={\lambda }_i{\lambda }_j{\lambda }_k{\eta }_{ijk}$$

$$\left(\begin{array}{cccccc}-2.3& 2.3& 0& -0.67& 0& 0\\ 0& 0& 0& 0& 0.67& 4.6\\ 0& 0& 0& 0& 0& 0\end{array}\right)$$

Full text

Piezoelectricity, piezoelectric constants, piezoelectric materials

I Introduction and motivation

Physics students are exposed to various types of symmetry Gross and conservation laws in Graduate/Undergraduate Mechanics and Electromagnetism with Lorentz transformation and Gauge symmetries, in Graduate/Undergraduate Quantum Mechanics during the study of Atoms and Molecules. In undergraduate courses such as Special Relativity, Lorentz Transformation is used to unify symmetries between Mechanics and Electromagnetism.

In Graduate High Energy Physics, the CPT theorem where C denotes charge conjugation $(Q\rightarrow-Q)$, P is parity $(\bm{r}\rightarrow\bm{-r})$ and T is time reversal $(t\rightarrow-t)$ as well as Gauge symmetry ($A_{i}\rightarrow A_{i}+\partial_{i}\chi)$ provide an important insight into the role of symmetry in the building blocks of matter and unification of fundamental forces and interaction between particles.

Graduate/undergraduate Solid State Physics provide a direct illustration of how Crystal Symmetry plays a fundamental role in the determination of physical constants and transport coefficients as well as conservation and simplification of physical laws. The relation between symmetry and dispersion relations through Kramers theorem (T symmetry) is another example of the power of symmetry in Solid State physics. In Graduate/undergraduate Statistical Physics students are exposed to the role of symmetry and its breaking in phase transitions with existence of different phases while possessing different symmetries are each characterized by an order parameter that controls the behaviour of the corresponding free energy.

The emergence on symmetry in physical systems in not obvious, however a good starting point to understand this particular point is through crystal symmetry paradigm simply illustrated with ice formation by slowly cooling liquid water.

This work about the role of crystalline symmetries and their role in the determination of piezoelectric coefficients $d$ of $\alpha$-Quartz on the basis of three different methods. It could be used to illustrate the role of symmetry and its implications in an undergraduate or graduate Solid State Physics, Statistical Physics or Materials Science course.

It is organized as follows. After reviewing $\alpha$-Quartz properties and symmetries, we tackle the evaluation of $d$ coefficients with symmetry on the basis of Fumi method. In section 3 we treat the same problem by Landau-Lifshitz method that contains a more physical approach than Fumi and finally in Section 4 we tackle it with a special method, the Dieulesaint-Royer procedure that combines both previous approaches. The appendix contains detailed information about Point Groups and Symmetry operations.

II $\alpha$-Quartz symmetries

Quartz is a very important material from the technological point of view since it is an essential component of all oscillators (clocks) used in consumer electronics devices (watches, computers, resonators, cameras, ovens…). Quartz is the second most important material after Silicon. Its formula is silicon dioxide SiO2 and its solid state unit cell is shown in fig.1.

$\alpha$-Quartz has a trigonal structure (rhombohedral Landau ) belonging to ${\bm{D}}_{3}$ point symmetry group (Schoenflies classification) or 32 (Hermann-Mauguin or International classification).

Quartz exists in two varieties: left-handed and right-handed that are mirror images of each other as displayed in fig.2. Handedness or Chirality implies that the two varieties have the same lattice energy (crystal energy of formation) and lack of center symmetry within each variety indicates that they belong to non-centro-symmetric groups as explained in the appendix.

The technological importance of Quartz originates from the values of its quality factor $Q$ that indicates the sharpness of resonance and electro-mechanical coupling coefficient $K$ that determines the conversion efficiency of mechanical into electrical energy and vice versa as compared with other materials as seen in fig.3.

Piezoelectricity is a fundamental property of Quartz and is found in non-centrosymmetric crystals that occur in two types of point symmetry groups (PSG) (see fig. 5). There are ten PSG called polar groups (possessing a special direction) associated with pyroelectric and piezoelectric materials (possessing spontaneous polarization along the special direction) and ten other PSG that are piezoelectric only (their polarization being induced by mechanical deformation).

These PSG are classified as polar and non polar:

• •

Pyroelectric and piezoelectric (Polar groups displaying spontaneous polarization along a special direction):

• –

Triclinic system ${\bm{C}}_{1}$

• –

Monoclinic system ${\bm{C}}_{s}$, ${\bm{C}}_{2}$

• –

Orthorhombic system ${\bm{C}}_{2v}$

• –

Tetragonal system ${\bm{C}}_{4}$, ${\bm{C}}_{4v}$

• –

Trigonal (Rhombohedric) system ${\bm{C}}_{3}$, ${\bm{C}}_{3v}$

• –

Hexagonal system ${\bm{C}}_{6}$, ${\bm{C}}_{6v}$

• •

Piezoelectric only (Non polar groups characterized by a polarization induced by mechanical deformation):

• –

Orthorhombic system ${\bm{D}}_{2}$

• –

Tetragonal system ${\bm{D}}_{4}$, ${\bm{D}}_{2d}$, ${\bm{S}}_{4}$

• –

Trigonal (Rhombohedric) system ${\bm{D}}_{3}$

• –

Hexagonal system ${\bm{D}}_{6}$, ${\bm{C}}_{3h}$, ${\bm{D}}_{3h}$

• –

Cubic system ${\bm{T}}$, ${\bm{T}}_{d}$

Quartz belongs to ${\bm{D}}_{3}$ group that possesses an order 3 rotation symmetry axis ($2\pi/3$ angle) that we might take along ${\bm{z}}$ axis. This axis has the ${\cal R}({\bm{z}},2\pi/3)$ rotation symmetry operation as well as three order 2 axes ($\pi$ rotation symmetry) in the $xy$ plane. The coordinate system we use is cartesian with basis vectors $\bm{e}_{1},\bm{e}_{2},\bm{e}_{3}$ such that $\bm{e}_{i}\cdot\bm{e}_{j}=\delta_{ij}$ and any vector is expressed in this basis as: $\bm{r}=x\bm{e}_{1}+y\bm{e}_{2}+z\bm{e}_{3}$.

Neumann’s principle states that ”Symmetry elements of any physical property of a crystal must include the symmetry elements of the point group of the crystal” implying that crystal physical quantities are preserved after performing point group symmetry operations on them.

A symmetry operation such as a rotation by an angle $\phi$ about the ${\bm{z}}$ axis denoted by ${\cal R}({\bm{z}},\phi)$ and represented by:

[TABLE]

transforms $\bm{r}=x\bm{e}_{1}+y\bm{e}_{2}+z\bm{e}_{3}$ into $\bm{r^{\prime}}=x^{\prime}\bm{e}_{1}+y^{\prime}\bm{e}_{2}+z^{\prime}\bm{e}_{3}$. This is different from the case of rotation with basis change implying that the transformed vector $\bm{r^{\prime}}=x^{\prime}\bm{e^{\prime}}_{1}+y^{\prime}\bm{e^{\prime}}_{2}+z^{\prime}\bm{e^{\prime}}_{3}$ is expressed in the rotated basis $(\bm{e^{\prime}}_{1},\bm{e^{\prime}}_{2},\bm{e^{\prime}}_{3})$ such that:

[TABLE]

We examine below symmetry operations and implications of Neumann’s principle in order to simplify the piezoelectric coefficients.

III Evaluation of piezoelectric coefficients by Fumi method

Piezoelectric coefficients are represented by $d_{i,jk}$ a rank 3 tensor with indices $i,j,k=1,2,3$ corresponding to $x\rightarrow 1,y\rightarrow 2,z\rightarrow 3$. They originate from the relation ${\bm{P}}_{i}=d_{i,jk}\sigma_{jk}$ linking polarization vector ${\bm{P}}$ to stress tensor $\sigma_{jk}$.

In total, we have 27 $d_{i,jk}$ coefficients since $i,j,k=1,2,3$, however writing $d_{i,jk}$ means index $i$ must be treated separately from indices $j,k$ since $i$ relates to polarization ${\bm{P}}$ whereas $j,k$ indices relate to the symmetric stress tensor $\sigma$ i.e. $\sigma_{jk}$=$\sigma_{kj}$.

The $j,k$ symmetry is exploited with Voigt notation that amounts to replace two indices by a single one according to the recipe: when $j=k,(j,k)\rightarrow j$ and when $j\neq k,(j,k)\rightarrow 9-(j+k)$. More specifically, we have six possibilities: $11\rightarrow 1,22\rightarrow 2,33\rightarrow 3,23\rightarrow 4,31\rightarrow 5,12\rightarrow 6$.

The total number of $d_{i,jk}$ coefficients is 18 since $i=1,2,3$ and Voigt index has six possibilities.

As a result, the Voigt piezoelectric matrix is $6\times 3$ with the explicit entries:

[TABLE]

where elements whose Voigt index is 4,5,6 are given by: $d_{i4}=d_{i23}+d_{i32}$, $d_{i5}=d_{i31}+d_{i13}$, $d_{i6}=d_{i12}+d_{i21}$ for $i=1,2,3$ as a result of symmetry.

Appendix B lists symmetry operations proper to each symmetry group. Quartz $D_{3}$ trigonal group has the symmetry operations: $E$, 2$C_{3}$, 3$C_{2}$. The 3-fold rotation by $2\pi/3$ about ${\bm{z}}$ axis is denoted ${\cal R}({\bm{z}},2\pi/3)$ and the 2-fold rotation by $\pi$ about the ${\bm{x}}$ axis is denoted ${\cal R}({\bm{x}},\pi)$.

In order to perform symmetry transformations on the $d_{i,jk}$ coefficients, we apply the Italian physicist Fausto G. Fumi Fumi rule that states they transform as $x_{i},x_{j}x_{k}$ written symbolically as $d_{i,jk}\sim x_{i},x_{j}x_{k}$ with the condition of respecting the order of the corresponding factors.

We start by considering rotational symmetry of order 2 about $x$ or ${\cal R}({\bm{x}},\pi)$ operations:

The relationship between the rotated axes and the original axes in the 2-fold rotation about ${\bm{x}}$ is given as:

[TABLE]

Let us consider the implications of this mapping on some tensor elements.

$d_{111}$ transforms as:

[TABLE]

Thus $d_{111}^{\prime}=d_{111}$ by Neumann’s Principle, implying $d_{11}\neq 0$.

Coefficient $d_{211}$ transforms as:

[TABLE]

Thus $d_{211}^{\prime}=-d_{211}=d_{211}$ by Neumann’s Principle, implying $d_{211}=0$ or $d_{21}=0$.

From this result, we infer that tensor elements with odd number of indices 2 and 3 are 0 by the 2-fold rotation (because of the transformation $y,z\rightarrow-y,-z$ ).

Hence, fourteen coefficients $d_{113},d_{131}$, $d_{112},d_{121}$, $d_{211},d_{222}$, $d_{233},d_{223}$, $d_{232},d_{311}$, $d_{322},d_{333}$, $d_{323},d_{332}$ are zero along with their ten Voigt equivalents $d_{15},d_{16},d_{21},d_{22},d_{23},d_{24},d_{31},d_{32},d_{33},d_{34}$

As a result, the piezoelectric matrix is written as:

[TABLE]

From the initial 18 coefficients only 8 coefficients $d_{11},d_{12},d_{13},d_{14},d_{25},d_{26},d_{35},d_{36}$ survive after the ${\cal R}({\bm{x}},\pi)$ symmetry.

Let us investigate the impact of rotational symmetry of order 3 about ${\bm{z}}$ axis or ${\cal R}({\bm{z}},2\pi/3)$ on these coefficients.

For the 3-fold rotation about $z$, we use eq. 1 with $\phi=\frac{2\pi}{3}$ to express the relationship between the rotated coordinates and the original ones:

[TABLE]

• •

Coefficient $d_{11}$, or $d_{111}$ transforms as:

[TABLE]

This implies: $d_{111}^{\prime}=-\frac{1}{8}d_{111}-\frac{3}{8}d_{122}-\frac{3}{8}d_{212}-\frac{3}{8}d_{221}$.

Moving on to Voigt notation and using Neumann’s Principle, we have:

[TABLE]

resulting in: $3d_{11}+d_{12}+d_{26}=0$.

• •

Coefficient $d_{12}$ or $d_{122}$ transforms as:

[TABLE]

yielding: $d_{122}^{\prime}=-\frac{3}{8}d_{111}-\frac{1}{8}d_{122}+\frac{3}{8}d_{212}+\frac{3}{8}d_{221}$.

Using Voigt notation and Neumann’s Principle, we get:

[TABLE]

implying: $d_{11}+3d_{12}-d_{26}=0$.

Combining relations $3d_{11}+d_{12}+d_{26}=0$ and $3d_{11}+d_{12}+d_{26}=0$, we get: $d_{12}=-d_{11}$ and $d_{26}=-2d_{11}$.

• •

Coefficient $d_{13}$ or $d_{133}$ transforms as:

[TABLE]

This yields $d_{133}^{\prime}=-\frac{1}{2}d_{133}-\frac{\sqrt{3}}{2}d_{233}$ and consequently $d_{133}^{\prime}=-\frac{1}{2}d_{133}$ since $d_{233}=0$. Using Neumann’s Principle, $d_{133}^{\prime}=-\frac{1}{2}d_{133}\equiv d_{133}$ implies $d_{133}=0$ and consequently $d_{13}=0$.

• •

Coefficients $d_{14}$ and $d_{25}$ :

$d_{213}$ being part of $d_{25}=d_{213}+d_{231}$ transforms as:

[TABLE]

Thus $d_{213}^{\prime}=-\frac{\sqrt{3}}{4}d_{113}-\frac{3}{4}d_{123}+\frac{1}{4}d_{213}+\frac{\sqrt{3}}{4}d_{223}$, implying $d_{213}^{\prime}=-\frac{3}{4}d_{123}+\frac{1}{4}d_{213}\equiv d_{213}$.

This leads to $d_{213}=-d_{123}$ implying $d_{25}=-d_{14}$.

• •

Coefficient $d_{313}$ being a part of $d_{35}$ i.e. $d_{313}+d_{331}$ transforms as:

[TABLE]

This gives $d_{313}^{\prime}=-\frac{1}{2}d_{313}-\frac{\sqrt{3}}{2}d_{323}$ resulting in $d_{313}^{\prime}=-\frac{1}{2}d_{313}$ since $d_{323}$ =0. Therefore we get with Neumann’s Principle $d_{313}=0$ and $d_{331}=0$ implying $d_{35}=0$.

• •

Coefficient $d_{312}$ being a part of $d_{36}$ or $d_{312}+d_{321}$ transforms as:

[TABLE]

thus $d_{312}^{\prime}=-\frac{\sqrt{3}}{4}d_{311}+\frac{1}{4}d_{312}-\frac{3}{4}d_{321}+\frac{\sqrt{3}}{4}d_{322}$

This yields:

[TABLE]

that is: $d_{312}=-d_{321}$ implying $d_{36}=d_{312}+d_{321}=0$.

Collecting all coefficients the piezoelectric matrix becomes:

[TABLE]

IV Evaluation of piezoelectric coefficients by Landau-Lifshitz method

Instead of working with matrices while applying Fumi Fumi recipe to the transformation of coefficients $d_{i,jk}\sim x_{i};x_{j}x_{k}$, we recall that performing rotation operations in a plane orthogonal to $z$ may be described by complex variables as done in Landau-Lifshitz book Landau :

$\xi\rightarrow\xi e^{\frac{2i\pi}{3}}$ $\eta\rightarrow\eta e^{-\frac{2i\pi}{3}}$, $z\rightarrow z$.

We can use new complex variables $\xi,\eta$ in the $xy$ plane through the variable change: $\xi=x+iy$, $\eta=\xi^{*}=x-iy$. Note that the variable sets $\xi,\eta$ as well as $xy$ are linearly independent (possessing a non-zero Wronskian) Landau .

Rotational symmetry of order 3 about ${\bm{z}}$ axis or ${\cal R}({\bm{z}},2\pi/3)$ operations:

Index separation $i,jk$ yields by Fumi Fumi rule, terms such as $({z,\xi\eta}),({\eta,z\xi}),({\xi,z\eta})$. The transformation applies in the same manner to complex phases:

$x_{i}\rightarrow x_{i}e^{i\phi_{i}},x_{j}\rightarrow x_{j}e^{i\phi_{j}},x_{k}\rightarrow x_{k}e^{i\phi_{k}}$ obtaining $(x_{i},x_{j}x_{k})\rightarrow(x_{i},x_{j}x_{k})e^{i(\phi_{i}+\phi_{j}+\phi_{k})}$ where $(x_{i},x_{j}x_{k})$ coordinates represent $({z,\xi\eta})$.

The transformation $d_{z,z\xi}\rightarrow d_{z,z\xi}e^{\frac{2i\pi}{3}}$ along with ${\bm{D}}_{3}$ symmetry (invariance with respect to rotation ${\cal R}({\bm{z}},2\pi/3)$ implies: $d_{z,z\xi}=d_{z,z\xi}e^{\frac{2i\pi}{3}}$ which results in: $d_{z,z\xi}(1-e^{\frac{2i\pi}{3}})=0$, thus $d_{z,z\xi}=0$. Similarly $d_{\xi,zz}$, $d_{\eta,zz}$ are zero since total phase would be $\pm\frac{2\pi}{3}$, same for $d_{z,\xi\xi}$, $d_{z;\eta\eta}$ for which the phase is $\pm\frac{4\pi}{3}$.

The non-zero terms invariant with respect to ${\cal R}({\bm{z}},2\pi/3)$ should contain combination of $({z,\xi\eta}),({\eta,z\xi}),({\xi,z\eta}),({\xi,\xi\xi}),({\eta,\eta\eta}),({z,zz})$ since the total phase obtained after ${\cal R}({\bm{z}},2\pi/3)$ operation is 0 or $\pm 2\pi$.

Finally the 6 non-zero terms correspond to the combination: $d_{z,\xi\eta},d_{\eta,z\xi},d_{\xi,z\eta},d_{\xi,\xi\xi},d_{\eta,\eta\eta},d_{z,zz}$. Note the existence of $d_{z,\xi\eta},d_{\eta,z\xi},d_{\xi,z\eta}$ terms with $z$ appearing only once Fumi . 2. 2.

Rotational symmetry of order 2 about $x$ or ${\cal R}({\bm{x}},\pi)$ operations:

This symmetry is carried out through the following transformations:

$x\rightarrow x,y\rightarrow-y,z\rightarrow-z$ i.e. $\xi\rightarrow\eta,\eta\rightarrow\xi,z\rightarrow-z$ .

This eliminates all terms containing an odd number of $z$ such as $d_{z,\xi\eta},d_{z,zz}$.

Applying transformation to $d_{\eta,z\xi}$, we get $d_{\xi,-z\eta}$ or $-d_{\xi,z\eta}$ thus term $d_{\eta,z\xi}$ is not zero (resulting from changing index $\eta$ into $\xi$), whereas $d_{z,\xi\eta}$ transforms into $d_{-z,\eta\xi}$ or $-d_{z,\xi\eta}$ thus this term is zero.

Finally only 2 terms $d_{\eta,z\xi}$ and $d_{\xi,\xi\xi}$ remain since we have: $d_{\eta,z\xi}=-d_{\xi,z\eta}$ and $d_{\xi,\xi\xi}=d_{\eta,\eta\eta}$.

Going back to $x,y$ variables from $\xi,\eta$, we use energy conservation in order to avoid problems stemming from non-orthogonality of coordinate system $(z,\xi,\eta)$ in contrast to $(z,x,y)$ orthogonal system.

The energy of the system is given by $(z,\xi,\eta)$ par $-{\bm{E.P}}=-E_{i}P_{i}=-E_{i}d_{i,jk}\sigma_{jk}$, obtaining:

$-{\bm{E.P}}=-2d_{\eta,z\xi}(E_{\eta}\sigma_{z\xi}-E_{\xi}\sigma_{z\eta})-d_{\xi,\xi\xi}(E_{\xi}\sigma_{\xi\xi}+E_{\eta}\sigma_{\eta\eta})$.

We apply Fumi Fumi rule to transform energy in system $(z,x,y)$ using correspondence between indices and tensor components as follows: $E_{\xi}=E_{x}+iE_{y}$, $E_{\eta}=E_{x}-iE_{y}$. Similarly, $\xi\xi=xx-yy+2ixy$ should yield stress tensor $\sigma$ components as $\sigma_{\xi\xi}=\sigma_{xx}-\sigma_{yy}+2i\sigma_{xy}$ whereas $\xi\eta=xx+yy$ should yield: $\sigma_{\xi\eta}=\sigma_{xx}+\sigma_{yy}$ and so forth.

The energy writes: $2a(E_{y}\sigma_{zx}-E_{x}\sigma_{zy})+b[2E_{y}\sigma_{xy}-E_{x}(\sigma_{xx}-\sigma_{yy})]$, with real constants $a,b$ defined by $a=2id_{\eta,z\xi}$, and $b=2d_{\xi,\xi\xi}$. As a result, we have the remaining components $d_{x,yz}=-d_{y,zx}=a$ and $d_{y,xy}=-d_{x,xx}=d_{x,yy}=-b$.

Collecting all terms, the matrix becomes:

[TABLE]

Moving on to Voigt representation, we transform:

$d_{i,jk}\rightarrow d_{i,jj},j=1,2,3$ when $j=k$ whereas $d_{i,jk}\rightarrow 2d_{i,jk}$ for terms $j\neq k$, since we have to account for coefficients symmetry: $yz\leftrightarrow zy,xz\leftrightarrow zx,xy\leftrightarrow yx$.

Thus we obtain:

[TABLE]

that might be written in the form shown in eq. 25 which is exactly the result obtained previously by Fumi method.

V Evaluation of piezoelectric coefficients by Royer-Dieulesaint method

Royer-Dieulesaint Royer method is the most elegant. It is based on dealing with rotation matrices through their eigenvalues which classifies this method as an intermediate between Fumi and Landau-Lifshitz.

After performing 2-fold rotation about $x$ axis, we infer as before that tensor elements with odd number of indices 2 and 3 are 0 from the transformation $y,z\rightarrow-y,-z$ ).

As a result, the piezoelectric matrix is written as in eq. 7.

In order to tackle the ${\cal R}({\bm{z}},\phi)$ transformation, we start with the corresponding general rotation matrix given in eq. 1 with $\phi=\frac{2\pi}{n}$ with $n$ an integer.

The eigenvalues of this matrix are: $\lambda_{1}=e^{i\phi},\lambda_{2}=e^{-i\phi},\lambda_{3}=1$ and the corresponding eigenvectors are given by: $\bm{\xi}_{1}=\left(\frac{1}{\sqrt{2}},\frac{i}{\sqrt{2}},0\right),\bm{\xi}_{2}=\left(\frac{i}{\sqrt{2}},\frac{1}{\sqrt{2}},0\right),\bm{\xi}_{3}=(0,0,1)$.

From the eigenvectors we derive the transformation matrix that takes us from the $(\bm{\xi}_{1},\bm{\xi}_{2},\bm{\xi}_{3})$ to the initial orthonormal basis $\bm{e}_{1},\bm{e}_{2},\bm{e}_{3}$ such as:

[TABLE]

In the $(\bm{\xi}_{1},\bm{\xi}_{2},\bm{\xi}_{3})$ basis, the piezoelectric coefficient tensor is written as $\eta_{ijk}$ and the transformation from $d_{ijk}$ to $\eta_{ijk}$ is given by $d_{ijk}=A_{il}A_{jm}A_{kn}\eta_{lmn}$.

A symmetry transformation combined with Neumann’s Principle yields:

[TABLE]

If we call $\nu_{1}$ the number of indices equal to 1 and $\nu_{2}$ the number of indices equal to 2, $\lambda_{i}\lambda_{j}\lambda_{k}=\exp[i(\nu_{1}-\nu_{2})\frac{2\pi}{n}]$ for a symmetry axis of order $n$. In the ${\cal R}({\bm{z}},2\pi/3)$ symmetry $n=3$ and $\eta_{ijk}$ components are not zero whenever $\nu_{1}-\nu_{2}$ is a multiple of $n$. This implies that $\eta_{123}$, $\eta_{213}$, $\eta_{312}$ and $\eta_{333}$ are not zero since $\nu_{1}-\nu_{2}=0$ as well as components $\eta_{111}$ and $\eta_{222}$ since $\nu_{1}-\nu_{2}=\pm 3$.

Let us consider first $d_{ijk}$ case with $i,j,k\neq 3$ such that coefficients are expressed in terms of $\eta_{111}$ and $\eta_{222}$ only.

Thus $d_{ijk}=A_{i1}A_{j1}A_{k1}\eta_{111}+A_{i2}A_{j2}A_{k2}\eta_{222}$ for $i,j,k\neq 3$.

For instance, if we want to evaluate $d_{11}$ i.e. $d_{111}$, we write: $d_{111}=A_{11}A_{11}A_{11}\eta_{111}+A_{12}A_{12}A_{12}\eta_{222}=0$. Using matrix $A$ elements given in eq. 26 we get: $d_{111}=\frac{1}{2\sqrt{2}}\eta_{111}-\frac{i}{2\sqrt{2}}\eta_{222}$.

Moving on to $d_{12}$ i.e. $d_{122}$, we obtain in the same way: $d_{122}=-\frac{1}{2\sqrt{2}}\eta_{111}+\frac{i}{2\sqrt{2}}\eta_{222}$ which implies that $d_{11}=-d_{12}$.

In the same manner we can evaluate $d_{26}$ i.e. $d_{212}$ or $d_{221}$. We obtain $d_{212}=-\frac{1}{2\sqrt{2}}\eta_{111}+\frac{i}{2\sqrt{2}}\eta_{222}$ which implies that $d_{26}=-2d_{11}$ (factor 2 originates from the fact $d_{26}$ is equivalent to $d_{212}$ or $d_{221}$ as previously done in the Landau-Lifshitz section).

Elements containing digit 3 are $d_{13}$, $d_{35}$ and $d_{36}$. They are all zero as we know from Fumi analysis. Let us retrieve this result in the case of $d_{13}$ or $d_{133}$.

In order to evaluate $d_{133}=A_{1l}A_{3m}A_{3n}\eta_{lmn}$, we use $A_{13}=A_{23}=A_{31}=A_{32}=0$, obtaining: $d_{133}=A_{11}A_{33}A_{33}\eta_{133}+A_{12}A_{33}A_{33}\eta_{233}=0$ since both $\eta_{133}$ and $\eta_{233}$ are zero.

In the $d_{35}$ case, one has to evaluate $d_{313}$ and $d_{331}$. Evaluating $d_{313}=A_{3l}A_{1m}A_{3n}\eta_{lmn}$ yields $d_{313}=A_{33}A_{11}A_{33}\eta_{313}+A_{33}A_{12}A_{33}\eta_{323}$ which is zero since both $\eta_{313}$and $\eta_{323}$ are zero.

The rest of the elements are obtained in the same fashion and the final outcome is exactly what we obtained earlier from Fumi and Landau-Lifshitz albeit in a faster and more compact form.

The piezoelectric coefficient matrix obtained is the same as Fumi and Landau-Lifshitz previous result given in eq. 25.

For right-handed $\alpha-$quartz, the actual numericalCady values are (each should be multiplied by $10^{-12}$ in order to get (Coulomb/Newton) SI units):

[TABLE]

VI Conclusion

Symmetry is illustrated in the the evaluation of piezoelectric coefficients of $\alpha$-Quartz by three distinct methods: Fumi, Landau-Lifshitz ans Royer-Dieulesaint. Fumi method is general, straightforward and tedious, Landau-Lifshitz method requires many physical concepts that must be adapted to every encountered situation whereas Royer-Dieulesaint is the most elegant while general and not requiring any additional concepts as with Landau-Lifshitz. Advanced methods to deal with symmetry are based on Group theoretical description of Tensors and Tensor fields such as described in ref. Batanouny , however they require deep knowledge of Group Theory Dresselhaus .

Appendix A Point symmetry groups and symmetry operations

We first present 3D Point Symmetry Groups in fig. 4. The classification into centro, non centrosymmetric groups as well as polar and non-polar groups is given in fig. 5. Finally symmetry operations pertaining to each group is presented in Table I.