The Piezoelectric Effect - an Indispensable Solid State Effect for Contemporary Actuator and Sensor Technologies - IOPscience
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Journal of Physics: Conference Series
PAPER • OPEN ACCESS
The Piezoelectric Effect – an Indispensable Solid State Effect for
Contemporary Actuator and Sensor Technologies
To cite this article: Rüdiger G. Ballas 2021 J. Phys.: Conf. Ser. 1775 012012
View the article online for updates and enhancements.
This content was downloaded from IP address 46.4.80.155 on 15/05/2021 at 01:30ICOEO 2020 IOP Publishing
Journal of Physics: Conference Series 1775 (2021) 012012 doi:10.1088/1742-6596/1775/1/012012
The Piezoelectric Effect – an Indispensable Solid
State Effect for Contemporary Actuator and Sensor
Technologies
Rüdiger G. Ballas
Professor for Electrical Engineering, Wilhelm Büchner Hochschule – Mobile University of
Technology, Department of Engineering Sciences, D 64295 Darmstadt, Germany
E-mail: ruediger.ballasl@wb-fernstudium.de
Abstract. The piezoelectric effect, which was discovered for the first time by the brothers
Pierre and Jacques Curie, combines electrical with mechanical quantities and vice versa. If
piezoelectric materials (e.g. quartz, turmaline) are subjected to electrical signals along certain
crystal orientations, deformations along well-defined crystal orientations appear. Contrary, a
mechanical deformation results in a generation of polarization charges. Even if there exist
numerous publications on this so-called direct and reciprocal piezoelectric effect, the aim of
this paper is to convey a clear and easy understanding of this essential solid body effect in
particular for the non-specialist, since a large number of publications is rather superficial and
unfortunately sometimes incorrect. A variety of ionic crystals show the direct and reciprocal
piezoelectric effect. In this paper, an illustrative representation of both effects is given by the
molecular structure of alpha-quartz, a stable modification of the silicon dioxide, the second most
common mineral of the earth’s crust. Both effects always involve an important physical quantity,
the so-called electrical polarization, which represents an Euclidean vector being defined as the
quotient of the total dipole moment resulting from the deformation of the hexagonal unit cell
of alpha-quartz and the volume of the unit cell. Based on the physical explanation of the dipole
moment, it is shown how the directions of the electrical polarization can be calculated in a simple
manner. This finally enables the physical understanding of both effects that are nowadays used
in numerous technical applications in the broad field of sensor and actuator technologies.
1. Introduction
The common characteristic of all piezoelectric crystals is the existence of one or more polar axes.
In crystallography, a polar axis is characterized by the fact that its front and rear ends are not
equivalent, i. e. a rotation around an axis perpendicular to the polar axis by an angle of 180◦
does not match the original position of the respective crystal [1]. Figure 1 illustrates this fact on
the macroscopic scale by means of the typical crystal form of quartz, also called low-quartz or
alpha-quartz. In its low-temperature modification occurring below 573 °C (hence the term low-
quartz ), the quartz crystal belongs to the trigonal crystal system, one of seven crystal systems
in crystallography serving for the three-dimensional classification of crystals [2]. There are three
polar axes, which are marked with X1 , X2 and X3 . Each axis connects two opposing edges of
the hexagonal prism. However, the opposing edges are not equivalent, which becomes obvious
by the fact, that for example the small surfaces marked with a and b are present at the rear edge
which is assigned to the axis X2 , but these are missing at the opposite edge. The axis marked
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Published under licence by IOP Publishing Ltd 1ICOEO 2020 IOP Publishing
Journal of Physics: Conference Series 1775 (2021) 012012 doi:10.1088/1742-6596/1775/1/012012
Figure 1. Alpha-quartz crystal (low-quartz) and corresponding crystal axes.
with Z represents the so-called crystallographic main axis (often referred to as the optical axis)
and is non-polar. This is because after a 180◦ rotation around one of the X-axes, the quartz
crystal matches its original position [3].
2. Macroscopic Generation of Piezoelectricity
Figure 2a shows a hexagonal prism, which can be assumed to be cut out from the quartz crystal
parallel to the plane of the polar axes X1 , X2 and X3 shown in Fig. 2. Now, a mechanical
pressure load is applied to the prism along its X1 -axis (see Fig. 2b). In case of an applied
mechanical pressure load it can be observed, that electric charges of equal magnitude and
opposite sign occur at the ends of the respective polar axes, in other words, piezoelectricity
is generated. The mechanical pressure load does not necessarily have to act directly along the
Figure 2. Generation of piezoelectricity: (a) unloaded hexagonal quartz prism; (b) occurrence
of equal electric charges of opposite sign at the edges of the quartz prism generated by a
mechanical pressure load.
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polar axis direction. An existing pressure load component acting along the direction of the polar
axis is already sufficient. In order to be able to explain physically the piezoelectric phenomenon
occurring on the prism, it is necessary to discuss the structure of alpha-quartz and the chemical
bonding properties at the molecular level.
3. Molecular Structure of Alpha-Quartz
Alpha-quartz represents the most important modification of silicon dioxide (SiO2), the second
most common mineral of the earth’s crust. At the molecular level, alpha quartz consists of a
network of continuously connected [SiO4]4– tetrahedra [4]. In silicates, including alpha-quartz,
[SiO4]4– tetrahedra are the most important structuring coordination polyhedra. As illustrated
in Fig. 3a, the silicon atom is surrounded by four oxygen atoms. With regard to the Si–O bond
and thus to the bonding properties within the [SiO4]4– tetrahedron, neither a pure covalent bond
(see Fig. 3b) nor a pure ionic bond is present (see Fig. 3c). The truth about the real bonding
properties is somewhere in between.
2
O O
2 4+ 2
O Si O O Si O
2
O O
Figure 3. Structuring coordination polyhedron of alpha-quartz and its bonding boundaries
(valence electron pairs are indicated by dashes as usual): (a) [SiO4]4– tetrahedron; (b) pure
covalent bond (atomic bond); (c) pure ionic bond.
By means of the electronegativities of silicon (χSi = 1,8) and oxygen (χO = 3,5) – the
electronegativity of a chemical element is a measure of its ability to gain electrons in a molecule –
the difference of electronegativity results in ∆χ = 1,7. For such a value, a partial ionic character
of 50 % can be assumed for the Si–O bond [5][6]. In the following, it is assumed that there
is a pure ionic bond within the [SiO4]4– tetrahedron. This assumption allows for a very clear
explanation of the piezoelectric effect.
In a first approximation, the alpha-quartz-specific spatial cross-linking of [SiO4]4– tetrahedra
results in a hexagonal unit cell, as shown in Fig. 4. The unit cell is composed of three formula
units SiO2. The assumption of ionic bonding properties is associated with the fact that the silicon
ion (cation) occupies a smaller volume of space than the oxygen ion (anion). By representing
the different sizes of the silicon and oxygen ions in Fig. 4, this fact is taken into account. The
crystallographic main axis Z is perpendicular to the plane defined by the polar axes X1 , X2
and X3 and points into the drawing plane. With respect to the drawing plane, the silicon ion
1 is located over the silicon ion 2 and this again over the silicon ion 3. The arrangement of the
oxygen ions a, b and c respectively turned by 60◦ with respect to the polar axes is equivalent
[7]. Thus, it appears that the position of the silicon ions and the oxygen ions continues helically
in counterclockwise direction into the drawing plane. The direction of rotation of the spiral
corresponds to the direction of optical rotation of the present crystal structure — thus, it is a
left-handed alpha-quartz crystal. The unit cell of alpha-quartz according to Fig. 4 is electrically
neutral towards the outside. Each of the three silicon ions carries 4 positive and each of the six
oxygen ions carries 2 negative unit charges, therefore all charges are saturated each other [1].
For reasons of simplicity, in a next step the oxygen ions located below the anions a and b and the
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Figure 4. Unit cell of alpha-quartz with corresponding polar axes (the crystallographic main
axis Z points perpendicularly into the drawing plane) [according to [7]].
oxygen ion located above the anion c are combined into a single oxygen ion – each consisting of
two O2– ions – with 4 unit charges. This results in the simplified unit cell of alpha-quartz shown
in Fig. 5. It shows a regular hexagonal basic structure, where the negatively charged O2– and the
positively charged Si4+ ions are arranged at its corners. The polar axes X1 , X2 and X3 represent
2-fold axes of rotation, the crystallographic main axis Z, which points perpendicularly into the
drawing plane, represents a 3-fold axis of rotation. In crystallography, an axis is called an n-fold
axis of rotation, if a crystal lattice already results in a configuration which does not differ from
the initial configuration of the crystal lattice after a rotation around 360◦ /n. If the structure cell
as shown in Fig.5b is rotated for example around the polar axis X1 , a configuration is obtained
which is identical to the initial configuration when rotated by 180◦ . Thus, the rotation by 180◦
is a symmetry operation and the 2-fold axis of rotation (360◦ /2 = 180◦ ) represents the symmetry
element. If this symmetry operation is executed twice in succession, all the ions of the unit cell
are in the original position [8]. In the same way, the 3-fold characteristics of the crystallographic
main axis Z can be explained.
Figure 5. Simplified unit cell of alpha-quartz with corresponding polar axes: (a) Arrangement
of the Si4+ and O2– ions in a regular hexagon; (b) 2 and 3-fold axes of rotation as symmetry
elements.
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4. The direct piezoelectric effect
If the simplified unit cell is subjected to a mechanical pressure load along the direction of the
polar axis X1 (see Fig. 6a), the upper silicon ion shifts between the two upper oxygen ions and the
lower oxygen ion shifts between the two lower silicon ions. The displacement of the positively and
negatively charged ions of the structure cell against each another causes an electric polarization
P along the polar axis X1 and thus along the direction of the mechanical pressure load. As
a consequence, positive charges are induced on the upper electrode A and positive charges are
induced on the lower electrode A′ resulting in an external electric polarization voltage. This
effect is referred to as a direct longitudinal piezoelectric effect, which explains the piezoelectric
phenomenon along the polar axis X1 of the quartz prism in Fig. 2b at the macroscopic level.
Figure 6. Direct piezoelectric effect within the simplified unit cell of alpha-quartz. (a)
longitudinal piezoelectric effect; (b) transversal piezoelectric effect.
If, on the other hand, the unit cell is subjected to a mechanical pressure load acting
perpendicularly to the polar axis X1 , the silicon and oxygen ions on the left and right sides of
the structure cell shift uniformly inwards and their charges cancel each other. Thus, no charges
are influenced on both external electrodes B and B′ (see Fig. 6b). However, the upper silicon
and lower oxygen ion of the unit cell are shifted outwards, thus an electric polarization P is
generated along the direction of the polar axis X1 , but rotated by 180◦ and is now perpendicular
to the mechanical pressure load. On the upper electrode A positive charges and on the lower
electrode A′ negative charges are induced resulting in an external electric polarization voltage
with reversed sign. This effect is called direct transversal piezoelectric effect.
5. The reciprocal piezoelectric effect
Both effects are reversible, under the influence of suitably oriented electric fields a contraction
or dilatation of the unit cell of alpha-quartz can be observed. This behavior is commonly termed
reciprocal piezoelectric effect. If an external electric DC voltage is applied between the upper
electrode A and the lower electrode A′ such that the upper electrode A is negatively and the lower
electrode A′ is positively charged, an electric field with the field strength E develops between
the electrodes. Just like the electric polarization P, the electric field strength E represents a
vector quantity. In contrast to the electric polarization vector P, the field strength vector E
points from the positive to the negative charge (see Fig. 7a). As a result, the two upper oxygen
ions are somewhat attracted towards the lower electrode A′ and the two lower silicon ions are
attracted towards the upper electrode A. Thus, the unit cell is compressed along the direction
of the polar axis X1 . The displacement of the positively and negatively charged ions causes an
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Figure 7. Reciprocal piezoelectric effect within the simplified unit cell of alpha-quartz. (a)
longitudinal piezoelectric effect; (b) transversal piezoelectric effect.
electric polarization P along the polar axis X1 and thus in parallel to the compression direction
of the unit cell. This effect is called reciprocal longitudinal piezoelectric effect.
If the voltage source is connected to the electrodes such that the upper electrode A is positively
and the lower electrode A′ is negatively charged, the direction of the electric field strength E
turns 180◦ . As a result, the two upper oxygen ions are attracted towards the upper electrode A
and the two lower silicon ions are attracted towards the lower electrode A′ . At the same time,
the silicon and oxygen ions on the left and right side of the unit cell shift uniformly inwards
(see Fig. 7b). The displacement of the positively and negatively charged ions, causes an electric
polarization P along the polar axis X1 again, but rotated by 180◦ and being perpendicular to
the compression direction of the unit cell. This effect is called reciprocal transversal piezoelectric
effect.
6. Electric Dipole and Electric Polarization
The introduced quantity electric polarization P is defined as the quotient of the total dipole
moment p resulting from the deformation of the hexagonal unit cell and the volume V of the
unit cell. In the following, the physical explanation of the dipole moment p shows the way for
the calculation of the directions of the electric polarization P shown in Fig. 6 and in Fig. 7 in
a simple manner.
An electric dipole consists of two equal charges ±Q of opposite sign separated by a distance d
(see Fig. 8a). Since the opposing charges compensate each other, a dipole is always electrically
neutral. The electric dipole is characterized by its dipole moment p:
p = Qd (1)
Thus, the dipole moment represents a vector with the magnitude |p| = Qd and is directed from
the negative charge to the positive one. If there is a system of n electrically active dipoles
within a given spatial volume V , the total dipole moment pΣ results from the vector sum of all
individual dipoles pi :
X n
pΣ = pi (2)
i=1
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Figure 8. (a) Electric dipole with dipole moment p; (b) electric polarization P defind as the
quotient of total dipole moment p and the volume V of the unit cell.
The ratio between total dipole moment pΣ and spatial volume V defines the electric polarization
P (see Fig. 8b):
n
pΣ 1 X
P= = pi (3)
V V
i=1
The defining equation (3) serves as the basis for calculating the direction of the electric
polarization P arising within the simplified unit cell of alpha-quartz in combination with the
direct or reciprocal piezoelectric effect.
First, the non-deformed configuration of the simplified unit cell of alpha-quartz is taken into
account (see Fig. 9a). For reasons of simplicity, the negative total charge of the two O2– ions
combined into a single oxygen ion and the positive total charge of the respective Si4+ ions are
represented by the quantities −Q and +Q, respectively. An illustrative representation of this
fact is shown in Fig. 9b. The respective charges ±Q of equal quantity and opposite sign,
which are arranged on the polar axes X1 , X2 and X3 , constitute an electric dipole and can be
characterized by an electric dipole moment p1 , p2 or p3 , respectively, since they are spatially
separated from each other in accordance with Fig. 9a. As shown in Fig. 9b, the respective dipole
moment vector coincides with the direction of the corresponding polar axis and is directed, by
definition, from the negative charge to the positive one.
For the purpose of calculating the direction of electric polarization P in general, it is necessary
to go into detail about the geometric conditions within the hexagonal basic structure of the
Figure 9. Simplified unit cell of alpha-quartz in non-deformed configuration: (a) regular
hexagonal arrangement of the Si4+ and O2– ions; (b) representation of the total charge of the
respective Si4+ and combined O2– ions by the quantities +Q and −Q, respectively, and resulting
dipole moments p1 , p2 and p3 .
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simplified unit cell in a non-deformed configuration. First, the regular hexagon according to
Fig. 9 is represented enlarged showing the geometric quantities being necessary for the later
calculation (see Fig. 10). Furthermore, the polar axes X1 , X2 and X3 are replaced by a Cartesian
coordinate system.
Figure 10. Geometric quantities within the hexagonal structure of the simplified unit cell
of alpha-quartz with corresponding distance vectors between the opposing charges ±Q (non-
deformed configuration).
The quantities ex und ey represent the unit vectors associated with the coordinate axes x
and y. Instead of applying the dipole moment vectors p1 , p2 and p3 as shown in Fig. 9b, in
Fig. 10 the respective distance vectors d1 , d2 and d3 between the opposing charges ±Q are
used taking the physical relationship according to (1) into account. The quantity a illustrated
in Fig. 10 denotes the edge length of the regular hexagon. A comparison with Fig. 9a obviously
shows that the edge length a can be represented as the sum of the radius of a Si4+ and an O2–
ion. Since the Si4+ and O2– ions are arranged concentrically around the unit cell’s corners even
in its deformed configuration —- e.g. caused by a mechanical pressure load along the x or y
axis – the edge length a can be understood as a constant quantity. The angle α and thus, the
geometric auxiliary quantities h and b depend on the deformation state of the unit cell, i.e. they
are variable quantities. On the basis of Fig. 10, following correlation for the geometric auxiliary
quantities h and b can be formulated using the edge length a:
h = a sin α, b = a cos α (4)
Using the correlations (1) and (2), the general conditional equation for the total dipole moment
pΣ within the simplified unit cell of alpha-quartz can be formulated. It can be written:
3
X 3
X 3
X
pΣ = pi = Q di = Q (dix ex + diy ey ) (5)
i=1 i=1 i=1
Subsequent evaluation of the sum results in following expression:
pΣ = Q [(d1x + d2x + d3x ) ex + (d1y + d2y + d3y ) ey ] (6)
The vector components dix and diy of the respective distance vectors di presented in (6) can
be expressed by the geometric quantities according to Fig. 10. The vector components in
x-direction result in:
d1x = 0, d2x = 2b, d3x = −2b (7)
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The vector components in y-direction result in:
d1y = a + 2h, d2y = −a, d3y = −a (8)
By inserting the equations from (4) into the obtained expressions (7) and (8), the vector
components in x- and y-directions can be reformulated as follows:
d1x = 0, d2x = 2a cos α, d3x = −2a cos α (9)
d1y = a + 2a sin α, d2y = −a, d3y = −a (10)
If the relations (9) and (10) are inserted into the (6), the calculation rule for the total dipole
moment pΣ and thus referring to the definition (3) also for the electric polarization P within
the simplified unit cell of alpha-quartz is obtained:
pΣ = Qa (2 sin α − 1) ey (11)
and
Qa
P= (2 sin α − 1) ey (12)
V
The total dipole moment pΣ or the electric polarization P are only defined by a y component
(see Fig. 6 - assuming the X1 -axis is replaced by the y-axis). Whether the y-component of the
electric polarization vector P shows the value zero or a negative or positive value results from
the fact, whether the structure cell is subjected to a mechanical pressure load, and if so, along
which coordinate axis the pressure load acts. In the following, these statements are verified
mathematically.
6.1. non-deformed configuration of the unit cell
If the unit cell shows a non-deformed configuration as illustrated in Fig. 10, the angle within
the hexagonal structure of the unit cell equals α0 = π/6. The resulting vector of the electric
polarization P yields according to the conditional equation (12)
Qa π
P= 2 sin −1 ey = 0, (13)
V | {z 6 }
=1
i.e. without any external mechanical pressure load, no electric polarization within the unit cell
of alpha-quartz can be observed.
6.2. mechanical pressure load acting along the y-axis
If the unit cell is subjected by a mechanical pressure load acting along the y-axis (i.e. along the
direction of the polar axis X1 ) (see Fig. 6a), the upper silicon ion shifts between the two upper
oxygen ions and the lower oxygen ion shifts between the two lower silicon ions, i.e the angle
within the hexagonal structure of the unit cell according to Fig. 10 yields α < π/6. Thus, the
resulting vector of the electric polarization P results in
Qa
P= 2 sin (α) −1 ey = −Py ey , (14)
V
|{z} | {z }
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6.3. mechanical pressure load acting along the x-axis
If the unit cell is subjected by a mechanical pressure load acting along the x-axis (i.e
perpendicularly to the polar axis X1 ), the silicon and oxygen ions on the left and right side
of the unit cell shift uniformly inwards, while the upper silicon and lower oxygen ions of the
unit cell shift outwards (see Fig. 6b). Thus, the angle within the hexagonal structure of the
unit cell according to Figure 10 yields α > π/6. In this case, the resulting vector of the electric
polarization P yields
Qa
P= 2 sin (α) −1 ey = +Py ey , (15)
V
|{z} | {z }
>0 >1
| {z }
>0
i.e. due to the external mechanical pressure load acting along the x-axis, an electric polarization
P is generated within the unit cell of alpha-quartz. The y-component of the polarization vector
is now positive and thus compared to the direction of the polar axis X1 , it points in the same
direction.
7. Conclusion
In this paper, the direct and reciprocal piezoelectric effect were represented illustratively by
means of the hexagonal unit cell of alpha quartz, a stable modification of the silicon dioxide and
the second most common mineral of the earth’s crust. Common to both effects is the occurring
vector of the electric polarization, which is defined as the quotient of the total dipole moment
resulting from the deformation of the hexagonal unit cell of alpha-quartz and the volume of
the unit cell. On the basis of the physical explanation of the dipole moment, it was shown
how the directions of the electric polarization occurring within the unit cell of alpha-quartz
can be calculated in a simple manner. This finally reveals the physical understanding of the
direct and reciprocal piezoelectric effect. Nowadays, both effects are used in numerous technical
applications in the broad field of sensor and actuator technologies.
About the Author
Prof. Dr.-Ing. Rüdiger G. Ballas holds the professorship for electrical engineering at Wilhelm
Büchner Hochschule - Mobile University of Technology, Darmstadt, Germany. In addition
to his activity as vice dean of the department of engineering sciences, his research activities
focus in particular on piezoelectric materials, sensors and actuators as well as on piezoelectric
energy harvesting and (micro)electromechanical systems (MEMS). Before his university career
he held leading positions in research and development at different companies concerned with
piezoelectric sensors and actuators. He earned his doctorate degree at Technical University
Darmstadt, Germany for his research work in the field of piezoelectric multilayered bending
actuators. During his pre-doctoral studies he already received several awards for outstanding
work in the field of microstructuring of quartz crystals for high-frequency applications in modern
telecommunications technologies. Furthermore, in 2017 he received the IAAM Medal due to
notable and outstanding contribution in the field of ”New Age Technology & Innovations”,
Stockholm, Sweden. In 2020 he was nominated as ”Fellow of the International Association
of Advanced Materials (FIAAM), Sweden” as well as ”VSET Fellow (Vebleo - International
Scientific Organization for Materials Science, Engineering and Technology), Sweden”. He is
author of various publications and author/co-author of several textbooks published by the
renowned publisher Springer Nature.
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[2] M. Okrusch, S. Matthes, Mineralogie: Eine Einführung in die spezielle Mineralogie, Petrologie und
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