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Study of interface and surface elastic waves in piezoelectric materials by using the surface Green function matching (SGFM) method
Surface Science 128 (1983) 117-127 North-Holland Publishing Company
STUDY
OF INTERFACE
PIEZOELECTRIC FUNCTION
117
AND SURFACE
MATERIALS
MATCHING
ELASTIC WAVES
IN
BY USING THE SURFACE GREEN
(SGFM)
METHOD
V.R. VELASCO Irutrwto de Fisica del Esrodo Solido (CSICReceived
19 January
[/AM),
Universidad Autonoma,
Madrid
- 34, Spain
1983
It is shown that the surface Green function matching method can be readily applied to piezoelectric materials by defining a four-dimensional Green function which includes both the elastic and the electrostatic fields. For an interface system this reduces to one half (with respect to the usual treatments) the order of the secular determinant, with consequent saving in computation. As an illustration an application is made to the study of shear horizontal waves for surfaces and interfaces with 6mm symmetry.
1. Introduction
As it is known, piezoelectricity, an interdependence of elastic and electric properties in certain materials, is intimately related to the study of elastic waves [ 1,2]. Most transducers involve direct or inverse piezoelectric effects. The wide utilization of the remarkable features of certain materials in electronics, for filtering, relies on piezoelectricity. Electromechanical resonators are directly inserted into the circuits, the vibration being maintained by the electric field. These are some of the possible applications of piezoelectric crystals. But also on the fundamental side piezoelectric materials exhibit quite interesting features. In the case of surface elastic wave propagation in piezoelectric solids, the existence of the Bleustein-Gulyaev wave [3,4] has no analogue in the case of non-piezoelectric crystals. The same is true for the case of an interface in which a wave of the Bleustein-Gulyaev type exists on piezoelectric materials of the class 6mm [5]. Studies of elastic (long) wave propagation on surfaces [3,4,6-91 and interfaces [5,10] of piezoelectric crystals are usually based on solving for the elastic displacements and electric potential. In these calculations crystals of different symmetries are considered with several combinations of surfaces and propagation directions (mainly high symmetry cases in order to reduce computation). We propose here as an alternative procedure for solving the problems the use of the SGFM method [ 111. This method has proved very useful in the study of 0039-6028/83/0000-0000/$03.00
0 1983 North-Holland
118
V.R. Velasco/
Study
of rnrerjaceand surface elastic
WCICPS
surface [ 12- 151 and interface [ 13,161 dynamical and thermodynamical properties of elastic crystals. There is no reason why this method should not work equally well for piezoelectric crystals. One of the main advantages of the SGFM method in the case of interface waves is that the order of the secular determinant is halved with respect to that obtained by using the usual approach [17]. It will be seen here that this is also true in the case of piezoelectric crystals, and clearly this provides a fair saving of computing time in the most complicated cases. Besides, by using the full formal capabilities of the SGFM method, we can obtain not only the dispersion relation of the surface and interface modes but also the displacements (and potentials) of these waves and the surface and interface contributions to the thermodynamical functions [ 111, or spectral functions of interest [ 14,151. The SGFM method relies on the matching at the surface or interface of a given linear field and some linear combination of this field and its normal derivatives [ll]. This can be done in the case of the piezoelectric crystals in a very simple way and then we can use all the formulae of the method to obtain the required physical information. In section 2 we shall explain the modifications needed to study the piezoelectric crystals. In order to show the capability of the method a simple but non-trivial case is worked out in detail. Conclusions are presented in section 3.
2. SGFM method applied to piezoelectric crystals For piezoelectric media the problem. involves not only the elastic particle displacements but also the electric and magnetic fields, in such a way that the applicable equations are combinations of the elastic equations of motion and of Maxwell’s equations, intercoupled by the piezoelectric tensors of the media. The intercoupling is usually weak enough that the solutions of the equations can be divided into two classes, those propagating with acoustic velocities (VA = 10” m SK’) and those propagating with electromagnetic velocities (VEX, = lo5 VA). Thus it appears that, even for strongly piezoelectric materials. the interaction between the three elastic waves and the two electromagnetic waves is weak, as their velocities are very different. Therefore, the two types of propagation can be considered independently. We shall be interested in elastic wave propagation and with this restriction the magnetic fields can be neglected and the electric fields derived from a scalar potential (‘static-field approximation”) [ 181. 2.1. Statement
of the problem
In the static-field approximation the elastic wave equations displacements u, along the coordinate axes x, are intercoupled
for the elastic to the aniso-
V. R. Velasco / Study of interface and surface elastic waves
119
tropic Laplace equation for the potential C#Jby the piezoelectric tensor erJA. Thus the particle displacements and the potential must satisfy in each medium the following set of four equations:
a2uk p;$-
- ‘Wax,
-
e
(2.la)
-=
3
(2.lb)
-=
9
j, j, k, I = 1, 2, 3, where p is the density of the medium concerned, c,,~, is the elastic stiffness tensor measured at constant electric field, and E,~ is the dielectric permittivity tensor measured at constant electric field. Then we must supplement the equations of motion with adequate boundary conditions. between the two We shall consider the plane x2 = 0 as the interface piezoelectric media. Medium 1 occupies the half-space x2 < 0 and medium 2 occupies the remaining half-space x2 > 0. The constitutive equations of the media, from which the equations of motion were derived by substitution into Newton’s second law and div D = 0 for the charge-free dielectrics, are T, ‘= c,,&,
(2.2a)
- e,,,&,
D, =: E,~E,: -I erklSk,,
(2.2b)
with T representing the stress tensor, S the strain tensor, E the electric field and D the electric displacement. Thus the normal component of D in either of the anisotropic, piezoelectric media is expressed in terms of the potential and the mechanical displacements by D, =: e,,,(au,/ax,)
-~&+/a+),
(2.3)
since s,, =: i(auh/ax,
+ au,/axk),
Similarly the traction the form:
E, =
-a+/ax,.
stresses which enter into the boundary
conditions
are of
(2.4)
T2,=C2,h~(auk/ax!)+ek2,(a~/axk).
In order to give a concise version of the equations introduce the following tetravector: v=
(ulr u2, u3, (e/e)+).
involved
it is convenient
to
(2.5)
where U, (i = 1, 2, 3) are the components of the elastic displacements, E is one of the non-vanishing elements of the dielectric permittivity tensor of the medium and e one of the non-vanishing elements of the piezoelectric tensor. In this way V has the dimensions of an elastic displacement. Now we can write
120
V. R. Velasco / Study of mterface and surface elastic waces
the equations
of motion
in a compact
form given by
L. V=O,
(2.6)
where L is a (4 X 4) partial differential operator of second order, which can be obtained from (2.1). It is now possible to find the Green function of this problem. The Green function will be the solution of L . G( r, r’; t, 1’) = 6( r - r’) 6( f - t’) 1.
(2.7)
where I is the unit matrix in a four-dimensional ming (2.7) the Green function can be obtained G(k;
w’) = L-r@;
space. After Fourier as
transfor-
w’).
(2.8)
It is interesting to illustrate this situation with a practical case. Let us choose, to simplify the calculations, a piezoelectric crystal of the 6mm class with its C axis parallel to the x3 direction and motion independent of this coordinate. In this case we have a decoupling between the x,, x2 directions and the x3 direction, due to the symmetry [3]. We shall be concerned here only with the piezoelectric wave which has the electric potential $ coupled to the elastic displacement ug. In this case the (4 x 4) operator is reduced to a (2 x 2) one given by L,, = -
c,,(a*/a
x: + a*/ax;)
L,, = - ( e:,/E,,)(a2/ax: L,, =
e,,(a*/ax:
+ p a*/at2.
(2.9a)
+ a*/ax;),
(2.9b)
+ a*/ax;),
L,, =
-e,,(a2/a$
After
Fourier
(2.9~)
+ a*/ax;). transforming
(2.9d)
eq. (2.9) it is easy to obtain
by matrix
inversion:
k,; 0’) = &,,/‘N,
(2.10a)
G,,( k,, k,; 0’) = -e,,/N,
(2.10b)
G,,( k,, k,; 0’) = e,,/N,
(2.1Oc)
G,,(k,,
G,,(k,,
k,; a’) = [( C’,k’
(2.10d)
- po’)e,,/e,,]/kZN,
where N = Ck* - E,,Pw’,
C = Cb4q, + e&.
Of course N = 0 gives the dispersion verse) wave w2 = 2;2k*,
c? = (C&p)(l
Now the boundary
conditions
relation
for the bulk piezoelectric
+ &/E,lC44). are of two kinds:
(trans-
K R. Yel~sco / Study of interface and surface elastic waues
Mechanical.
Continuity
Uo), ,
i = I, 2, 3.
U(2) < *
~ontinujty
of the displacements
121
at the interface (2.11a)
of the normal stresses at the interface
7”(l)= 77’2’ j= 21 ’ z,/ Electrical.
(2.11b)
1,2,3.
Continuity
of the potential
at the interface
#“__+,12’_
(2.1 lc)
Continuity interface
of the
L)il) = D’J’ 2
.
normal
component
of the electrical
displacement
at
the
(2.1 Id)
These boundary conditions can be easily expressed in terms of the tetravector Y defined in (2.6) and consequently of the Green function defined in (2.8). The continuity of the elastic displacements and the electric potential amounts to the continuity of (V,, VI, V,, (e/e)V,) at the interface. The continuity of the normal stresses and the normal component of the electrical displacement at the interface amounts to the continuity at the interface of a linear differential operator A given by (2.3) and (2.4) expressed in terms of I-’ Proceeding as in ref. [ 111, we have (2.12a)
(2.12b)
(2.12c)
(2.12d) where Latin symbols run from 1 to 3 and greek symbols from 1 to 4. It would be instructive to consider the particular case of a piezoelectric crystal belonging to the 6mm class studied before. In this case we have (2.13a)
(2.13b) Having set out the problem, the SGFM analysis proceeds in the same way as for acoustic waves [ 1 I J. The difference is that now we have four equations of
122
V.R. Velasco / Studv of interface
and
surfaceelastic
ti’aoes
motion, including u and (p, or the tetravector V, instead of three for u as in the elastic case, and eight boundary conditions instead of six (in the problem of the interface) as for the purely elastic case. But, as we have seen, we have obtained a super-Green function G solution of the coupled system of four equations of motion. Also, there is no formal difference between the new boundary conditions and those of the purely elastic case. The continuity of u and (p (i.e. of V) at the interface is a boundary condition of the first kind [ 1 l] and can be expressed as (2.14a) (2.14b) i = 1, 2, 3, (Y= 1, 2, 3, 4, where %,(I) stand for fictitious stimuli introduced in the SGFM method [ 1l] and C?istand for the surface projected Green function of each medium, defined as 1
m elh2” @(k, a’) dk,, s 27l --Cc
C+(K, 02) = lim,,,-
with k = (k,, k,, k3) = (K, k2) the wavevector and o2 the eigenvalue, in our case the squared angular frequency. The continuity of the normal stresses and the normal component of the electric displacements, which include normal derivatives of 8, form a boundary condition of the second kind [ 111, and can be expressed by &-(“.$ I
=&J-‘.C$ 2
(2.15)
2’
where &’ *) is the surface projected linear operator A giving the boundary condition of the second kind in terms of the Green function, and including therefore a linear combination of the surface projected Green function and its normal derivatives ‘!$ ‘) [ 1 l] defined as 1 ‘@(t)(K, 02) =-llm,,, 27r
e+‘l‘l” ik, G(k,
a’) dk2,
the signs + and - being related to the sense in which we approach to the interface [ 1I]. In this way all the formal study previously done [ 1l] is valid in our case and it will not be repeated here. We shall state only the SGFM formula for the dispersion relation of the interface modes (2.16a)
det]g51 ’1= 0, with g- ’ = ,y 3
. lj; ’ - @‘.
&y 1,
(2.16b)
123
V. R. Velusco / Study of interface and surface elastic woes
where
B- ’
stand
for the inverse
of a projected
Green
function
defined
as
Q , c3~ks =
rn
i=
1,2,3,
ff = 1, 2, 3, 4.
1 (e/e>%,3 The expressions (2.16) are valid for the general case of an interface between two piezoelectric crystals. Different particular cases will be considered later. In order to illustrate these different situations we shall consider the case of an interface between two piezoelectric crystals of the class 6mm. The plane x2 I= 0 will be the interface and medium 1 extends from - CC to the interface and medium 2 occupies the other half-space. Remembering the definition of the surface projected Green function and its normal derivatives given before, it is easy to obtain = q,/xP,
(2.17a)
w’) = -e,,/2CP,
(2.17b)
g,] ( k, ; J) Blz(kI;
(2.17~)
%,( k, w’) = &u/2@, e,,( k,; Q’) = (2e,,)-‘(k;’
(2.17d)
- e&/CP),
where p = (kf - E,,pw2/c)“2. Then 2
45
P(k,:J)=2k,
$ i
t-3 i
-e15
+-&
(2.18)
,
cl5
Tel5
+&II 'fg':'(k,;
II 2
+E
+
-11
(2.19)
.
C-e;,
e15
I
I
The linear operators &’ ’ ) expressing the boundary conditions in terms of the surface projected Green function and its normal derivatives are given by
a\;‘=
C44'8i,"+
&
_e
2J
15
qi,+, I/
(2.20a)
(ef',/e,,)'G$.f', 15
VJ(k, 2,
(2.20b)
.
It is easy to verify that these operations
essential
in the general
arguments
satisfy the property
of the SGFM
method
[ 111. The dispersion
124
V. R. Velasco
relation
/ Study
of interface
and surface
ehstic
wuces
follows from (2.16) and in our case we obtain:
(2.21)
where e, = e15,,.
i = 1, 2, and < = C,/E,. From (2.21) it follows that
E, = Ebb,,, 2
(&,fF2)
i
c,&p+c2$-~ I
’
which after some handling
2
+(e,+e$=0.
(2.22)
1
can be cast as (2.23)
which is the expression given in ref. [S], obtained by the usual procedure. We can pass to consider now several particular cases of this general expression. 2.1. I. Metullized interface In this case we assume that the interface is completely coated with an infinitesimally thin, perfectly conducting electrode which is grounded. The boundary conditions are now u,(‘)=uj2),
$(‘)=O,
The dispersion
@J(~)=O,
relation
TYj)= T,‘:),
i.
jz
+ c1P2/kl
= e:/q
which is the expression
T’z’=O, 2J
We are assuming crystal.
(2.24) from @.and &’ considering of the electric potentials at is easy to see, in the case of geometry given above. that (2.25)
+ ef/e,,
given in ref. [5].
2.1.2. Non-metallized surface In this case the boundary conditions +(i)=+(2),
3.
is now given by
det@i+‘.~,’ - $;-‘.‘8;‘1= 0. . where & and g-’ are the (3 x 3) matrices obtained only the indices i, j = 1, 2, 3, due to the vanishing the interface. From the relations obtained above it the piezoelectric crystals of the class 6mm with the the dispersion relation is given by C,P,/k,
1, 2,
are
D,‘“+2’.
that medium
1 is the vacuum
and medium
2 the piezoelectric
V.R. Velasco / Study of interface and surface elastic waves
125
In the vacuum there is no elastic displacement and we have only the scalar electric potential. So for the vacuum our Green function is a scalar quantity but we can obtain the dispersion relation for the surface modes from the general tensorial form in the following way: det]e,k,6,,
- Q:-‘.@;‘]
where Ed is the dielectric for the vacuum: G(
K,
(2.26)
= 0, permittivity
of the vacuum.
This is due to the fact that
k,; w’) = - l/q,k’,
and then
In the case o/the
($1-f&_ i
C2G
crystals
of the 6mm class we obtain
as dispersion
relation
‘/2
I
(e2b212 =
(l/E0
+
(2.27)
1/E2) ’
and finally
(e2/9j4 Cj( l/&g
(2.28)
+ 1/E2)2
which is the dispersion relation for the Bleustein-Gulyaev wave [3,4], for a non-metallized surface. This dispersion relation can also be obtained from (2.23) considering medium 1 as the vacuum, i.e.. Ci4 = e:5 = 0, E:, = eO.
1.1.3. Merallized surface In this case the boundary #‘)=.O,
9’2’=0,
conditions
are given by
T:,“LO.
Due to the vanishing of the electric potentials analogous case to that of the metallized interface given by
at the surface we are in an and the dispersion relation is
(2.29)
det)-&:-‘.~;‘l=O. In the case of the piezoelectric following dispersion relation @/k
crystals
of the 6mm
class we obtain
the
(2.30)
, = e2/e,
which can be put as w’(C,/p)k,(l
- e4/G2e2),
(2.31)
which is the relation given in ref. [3] for this case and obtained by the usual method. All these particular cases must be considered in detail in the formal analysis in order to obtain the Green function G, [ 1 l] of the system with a surface or an interface, but the dispersion relations can be deduced from the general formulae (2.16) by taking into account the physical facts embodied in the different boundary conditions for these particular situations. Thus we have shown in a simple (but not trivial) and well known case that the SGFM method gives the dispersion relation for all the possible cases. either from the general relation (2.16) or from the restrictions of this general formula given above. 3. Conclusions We have seen that the SGFM method, which has proved to work very well in purely elastic problems for surfaces and interfaces, can be easily modified to study the same kind of problems for piezoelectric crystals. Since the method gives readily a great deal of physical information, besides the dispersion relation, it is interesting to extend the formalism in order to include more complicated cases. Though the method has a greater formal complication than the usual procedures employed to obtain the dispersion relations in piezoelectric crystals, we have seen that it lends itself quite easily to obtain the desired dispersion relations with the advantage of providing a general formula from which particular cases follow at once, without need for an independent treatment of each particular situation as ‘is the case with the customary treatment of this problems. Besides, our method halves the size of the secular determinant, thus reducing the computation required and enabling us to consider more complicated situations. Studies of propagation in crystals of other symmetries and in general directions of propagation are in progress. Acknowledgements I am grateful to Professor F. Garcia-Moliner for critical reading of the manuscript. It is also a pleasure to thank Professor V. Heine (TCM) and Dr. R.F. Willis (PCS) for their kind invitation to the Cavendish Laboratory (Cambridge), where this work was done.
References [l] E. Dieulesaint and D. Royer, Elastic Waves in Solids (Wiley. New York 1980). [2] J. Sapriel, Acousto-Optics (Wiley. New York. 1979). [3] J.L. Bleustein, Appl. Phys. Letters 13 (1968) 412.
K R. Velasco / Study o/ interface and surface elasrlc waues
[4] [5] [6] (71 (8) [9] [IO]
[ ll] [ 121 [ 131 [ 14) [15] [16] [17] [ 1S]
127
Yu.V. Gulyaev, Soviet Phys. -JETP Letters 9 (1969) 37. C. Maerfeld and P. Tournois, Appl. Phys. Letters 19 (1971) 117. C.C. Tseng and R.M. White, J. Appl. Phys. 38 (1967) 4274. C.C. Tseng, J. Appl. Phys. 41 (1970) 2270. C.C. Tseng, Appl. Phys. Letters 16 (1970) 253. G.W. Farnell, in: Physical Acoustics, Vol. 6. Eds. W.P. Mason and R.N. Thurston (Academic Press, New York, 1970). G.W. Farnell and E.L. Adler, in: Physical Acoustics, Vol. 9. Eds. W.P. Mason and R.N. Thurston (Academic Press, New York. 1972). F. Garcia-Moliner. Ann. Physique 2 (1977) 177. V.R. Velasco and F. Garcia-Moliner, Surface Sci. 67 (1977) 555. V.R. Velasco and F. Garcia-Moliner, Surface Sci. 83 (1979) 376. V.R. Velasco and F. Garcia-Moliner, J. Phys. Cl 3 (1980) 2237. V.R. Velasco and F. Garcia-Moliner. Solid State Commun. 33 (1980) 1. V.R. Velasco. Phys. Status Solidi (a) 60 (1980) K61. R. Stoneley. Proc. Roy. Sot. (London) A106 (1924) 416. H.F. Tiersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969).
STUDY
OF INTERFACE
PIEZOELECTRIC FUNCTION
117
AND SURFACE
MATERIALS
MATCHING
ELASTIC WAVES
IN
BY USING THE SURFACE GREEN
(SGFM)
METHOD
V.R. VELASCO Irutrwto de Fisica del Esrodo Solido (CSICReceived
19 January
[/AM),
Universidad Autonoma,
Madrid
- 34, Spain
1983
It is shown that the surface Green function matching method can be readily applied to piezoelectric materials by defining a four-dimensional Green function which includes both the elastic and the electrostatic fields. For an interface system this reduces to one half (with respect to the usual treatments) the order of the secular determinant, with consequent saving in computation. As an illustration an application is made to the study of shear horizontal waves for surfaces and interfaces with 6mm symmetry.
1. Introduction
As it is known, piezoelectricity, an interdependence of elastic and electric properties in certain materials, is intimately related to the study of elastic waves [ 1,2]. Most transducers involve direct or inverse piezoelectric effects. The wide utilization of the remarkable features of certain materials in electronics, for filtering, relies on piezoelectricity. Electromechanical resonators are directly inserted into the circuits, the vibration being maintained by the electric field. These are some of the possible applications of piezoelectric crystals. But also on the fundamental side piezoelectric materials exhibit quite interesting features. In the case of surface elastic wave propagation in piezoelectric solids, the existence of the Bleustein-Gulyaev wave [3,4] has no analogue in the case of non-piezoelectric crystals. The same is true for the case of an interface in which a wave of the Bleustein-Gulyaev type exists on piezoelectric materials of the class 6mm [5]. Studies of elastic (long) wave propagation on surfaces [3,4,6-91 and interfaces [5,10] of piezoelectric crystals are usually based on solving for the elastic displacements and electric potential. In these calculations crystals of different symmetries are considered with several combinations of surfaces and propagation directions (mainly high symmetry cases in order to reduce computation). We propose here as an alternative procedure for solving the problems the use of the SGFM method [ 111. This method has proved very useful in the study of 0039-6028/83/0000-0000/$03.00
0 1983 North-Holland
118
V.R. Velasco/
Study
of rnrerjaceand surface elastic
WCICPS
surface [ 12- 151 and interface [ 13,161 dynamical and thermodynamical properties of elastic crystals. There is no reason why this method should not work equally well for piezoelectric crystals. One of the main advantages of the SGFM method in the case of interface waves is that the order of the secular determinant is halved with respect to that obtained by using the usual approach [17]. It will be seen here that this is also true in the case of piezoelectric crystals, and clearly this provides a fair saving of computing time in the most complicated cases. Besides, by using the full formal capabilities of the SGFM method, we can obtain not only the dispersion relation of the surface and interface modes but also the displacements (and potentials) of these waves and the surface and interface contributions to the thermodynamical functions [ 111, or spectral functions of interest [ 14,151. The SGFM method relies on the matching at the surface or interface of a given linear field and some linear combination of this field and its normal derivatives [ll]. This can be done in the case of the piezoelectric crystals in a very simple way and then we can use all the formulae of the method to obtain the required physical information. In section 2 we shall explain the modifications needed to study the piezoelectric crystals. In order to show the capability of the method a simple but non-trivial case is worked out in detail. Conclusions are presented in section 3.
2. SGFM method applied to piezoelectric crystals For piezoelectric media the problem. involves not only the elastic particle displacements but also the electric and magnetic fields, in such a way that the applicable equations are combinations of the elastic equations of motion and of Maxwell’s equations, intercoupled by the piezoelectric tensors of the media. The intercoupling is usually weak enough that the solutions of the equations can be divided into two classes, those propagating with acoustic velocities (VA = 10” m SK’) and those propagating with electromagnetic velocities (VEX, = lo5 VA). Thus it appears that, even for strongly piezoelectric materials. the interaction between the three elastic waves and the two electromagnetic waves is weak, as their velocities are very different. Therefore, the two types of propagation can be considered independently. We shall be interested in elastic wave propagation and with this restriction the magnetic fields can be neglected and the electric fields derived from a scalar potential (‘static-field approximation”) [ 181. 2.1. Statement
of the problem
In the static-field approximation the elastic wave equations displacements u, along the coordinate axes x, are intercoupled
for the elastic to the aniso-
V. R. Velasco / Study of interface and surface elastic waves
119
tropic Laplace equation for the potential C#Jby the piezoelectric tensor erJA. Thus the particle displacements and the potential must satisfy in each medium the following set of four equations:
a2uk p;$-
- ‘Wax,
-
e
(2.la)
-=
3
(2.lb)
-=
9
j, j, k, I = 1, 2, 3, where p is the density of the medium concerned, c,,~, is the elastic stiffness tensor measured at constant electric field, and E,~ is the dielectric permittivity tensor measured at constant electric field. Then we must supplement the equations of motion with adequate boundary conditions. between the two We shall consider the plane x2 = 0 as the interface piezoelectric media. Medium 1 occupies the half-space x2 < 0 and medium 2 occupies the remaining half-space x2 > 0. The constitutive equations of the media, from which the equations of motion were derived by substitution into Newton’s second law and div D = 0 for the charge-free dielectrics, are T, ‘= c,,&,
(2.2a)
- e,,,&,
D, =: E,~E,: -I erklSk,,
(2.2b)
with T representing the stress tensor, S the strain tensor, E the electric field and D the electric displacement. Thus the normal component of D in either of the anisotropic, piezoelectric media is expressed in terms of the potential and the mechanical displacements by D, =: e,,,(au,/ax,)
-~&+/a+),
(2.3)
since s,, =: i(auh/ax,
+ au,/axk),
Similarly the traction the form:
E, =
-a+/ax,.
stresses which enter into the boundary
conditions
are of
(2.4)
T2,=C2,h~(auk/ax!)+ek2,(a~/axk).
In order to give a concise version of the equations introduce the following tetravector: v=
(ulr u2, u3, (e/e)+).
involved
it is convenient
to
(2.5)
where U, (i = 1, 2, 3) are the components of the elastic displacements, E is one of the non-vanishing elements of the dielectric permittivity tensor of the medium and e one of the non-vanishing elements of the piezoelectric tensor. In this way V has the dimensions of an elastic displacement. Now we can write
120
V. R. Velasco / Study of mterface and surface elastic waces
the equations
of motion
in a compact
form given by
L. V=O,
(2.6)
where L is a (4 X 4) partial differential operator of second order, which can be obtained from (2.1). It is now possible to find the Green function of this problem. The Green function will be the solution of L . G( r, r’; t, 1’) = 6( r - r’) 6( f - t’) 1.
(2.7)
where I is the unit matrix in a four-dimensional ming (2.7) the Green function can be obtained G(k;
w’) = L-r@;
space. After Fourier as
transfor-
w’).
(2.8)
It is interesting to illustrate this situation with a practical case. Let us choose, to simplify the calculations, a piezoelectric crystal of the 6mm class with its C axis parallel to the x3 direction and motion independent of this coordinate. In this case we have a decoupling between the x,, x2 directions and the x3 direction, due to the symmetry [3]. We shall be concerned here only with the piezoelectric wave which has the electric potential $ coupled to the elastic displacement ug. In this case the (4 x 4) operator is reduced to a (2 x 2) one given by L,, = -
c,,(a*/a
x: + a*/ax;)
L,, = - ( e:,/E,,)(a2/ax: L,, =
e,,(a*/ax:
+ p a*/at2.
(2.9a)
+ a*/ax;),
(2.9b)
+ a*/ax;),
L,, =
-e,,(a2/a$
After
Fourier
(2.9~)
+ a*/ax;). transforming
(2.9d)
eq. (2.9) it is easy to obtain
by matrix
inversion:
k,; 0’) = &,,/‘N,
(2.10a)
G,,( k,, k,; 0’) = -e,,/N,
(2.10b)
G,,( k,, k,; 0’) = e,,/N,
(2.1Oc)
G,,(k,,
G,,(k,,
k,; a’) = [( C’,k’
(2.10d)
- po’)e,,/e,,]/kZN,
where N = Ck* - E,,Pw’,
C = Cb4q, + e&.
Of course N = 0 gives the dispersion verse) wave w2 = 2;2k*,
c? = (C&p)(l
Now the boundary
conditions
relation
for the bulk piezoelectric
+ &/E,lC44). are of two kinds:
(trans-
K R. Yel~sco / Study of interface and surface elastic waues
Mechanical.
Continuity
Uo), ,
i = I, 2, 3.
U(2) < *
~ontinujty
of the displacements
121
at the interface (2.11a)
of the normal stresses at the interface
7”(l)= 77’2’ j= 21 ’ z,/ Electrical.
(2.11b)
1,2,3.
Continuity
of the potential
at the interface
#“__+,12’_
(2.1 lc)
Continuity interface
of the
L)il) = D’J’ 2
.
normal
component
of the electrical
displacement
at
the
(2.1 Id)
These boundary conditions can be easily expressed in terms of the tetravector Y defined in (2.6) and consequently of the Green function defined in (2.8). The continuity of the elastic displacements and the electric potential amounts to the continuity of (V,, VI, V,, (e/e)V,) at the interface. The continuity of the normal stresses and the normal component of the electrical displacement at the interface amounts to the continuity at the interface of a linear differential operator A given by (2.3) and (2.4) expressed in terms of I-’ Proceeding as in ref. [ 111, we have (2.12a)
(2.12b)
(2.12c)
(2.12d) where Latin symbols run from 1 to 3 and greek symbols from 1 to 4. It would be instructive to consider the particular case of a piezoelectric crystal belonging to the 6mm class studied before. In this case we have (2.13a)
(2.13b) Having set out the problem, the SGFM analysis proceeds in the same way as for acoustic waves [ 1 I J. The difference is that now we have four equations of
122
V.R. Velasco / Studv of interface
and
surfaceelastic
ti’aoes
motion, including u and (p, or the tetravector V, instead of three for u as in the elastic case, and eight boundary conditions instead of six (in the problem of the interface) as for the purely elastic case. But, as we have seen, we have obtained a super-Green function G solution of the coupled system of four equations of motion. Also, there is no formal difference between the new boundary conditions and those of the purely elastic case. The continuity of u and (p (i.e. of V) at the interface is a boundary condition of the first kind [ 1 l] and can be expressed as (2.14a) (2.14b) i = 1, 2, 3, (Y= 1, 2, 3, 4, where %,(I) stand for fictitious stimuli introduced in the SGFM method [ 1l] and C?istand for the surface projected Green function of each medium, defined as 1
m elh2” @(k, a’) dk,, s 27l --Cc
C+(K, 02) = lim,,,-
with k = (k,, k,, k3) = (K, k2) the wavevector and o2 the eigenvalue, in our case the squared angular frequency. The continuity of the normal stresses and the normal component of the electric displacements, which include normal derivatives of 8, form a boundary condition of the second kind [ 111, and can be expressed by &-(“.$ I
=&J-‘.C$ 2
(2.15)
2’
where &’ *) is the surface projected linear operator A giving the boundary condition of the second kind in terms of the Green function, and including therefore a linear combination of the surface projected Green function and its normal derivatives ‘!$ ‘) [ 1 l] defined as 1 ‘@(t)(K, 02) =-llm,,, 27r
e+‘l‘l” ik, G(k,
a’) dk2,
the signs + and - being related to the sense in which we approach to the interface [ 1I]. In this way all the formal study previously done [ 1l] is valid in our case and it will not be repeated here. We shall state only the SGFM formula for the dispersion relation of the interface modes (2.16a)
det]g51 ’1= 0, with g- ’ = ,y 3
. lj; ’ - @‘.
&y 1,
(2.16b)
123
V. R. Velusco / Study of interface and surface elastic woes
where
B- ’
stand
for the inverse
of a projected
Green
function
defined
as
Q , c3~ks =
rn
i=
1,2,3,
ff = 1, 2, 3, 4.
1 (e/e>%,3 The expressions (2.16) are valid for the general case of an interface between two piezoelectric crystals. Different particular cases will be considered later. In order to illustrate these different situations we shall consider the case of an interface between two piezoelectric crystals of the class 6mm. The plane x2 I= 0 will be the interface and medium 1 extends from - CC to the interface and medium 2 occupies the other half-space. Remembering the definition of the surface projected Green function and its normal derivatives given before, it is easy to obtain = q,/xP,
(2.17a)
w’) = -e,,/2CP,
(2.17b)
g,] ( k, ; J) Blz(kI;
(2.17~)
%,( k, w’) = &u/2@, e,,( k,; Q’) = (2e,,)-‘(k;’
(2.17d)
- e&/CP),
where p = (kf - E,,pw2/c)“2. Then 2
45
P(k,:J)=2k,
$ i
t-3 i
-e15
+-&
(2.18)
,
cl5
Tel5
+&II 'fg':'(k,;
II 2
+E
+
-11
(2.19)
.
C-e;,
e15
I
I
The linear operators &’ ’ ) expressing the boundary conditions in terms of the surface projected Green function and its normal derivatives are given by
a\;‘=
C44'8i,"+
&
_e
2J
15
qi,+, I/
(2.20a)
(ef',/e,,)'G$.f', 15
VJ(k, 2,
(2.20b)
.
It is easy to verify that these operations
essential
in the general
arguments
satisfy the property
of the SGFM
method
[ 111. The dispersion
124
V. R. Velasco
relation
/ Study
of interface
and surface
ehstic
wuces
follows from (2.16) and in our case we obtain:
(2.21)
where e, = e15,,.
i = 1, 2, and < = C,/E,. From (2.21) it follows that
E, = Ebb,,, 2
(&,fF2)
i
c,&p+c2$-~ I
’
which after some handling
2
+(e,+e$=0.
(2.22)
1
can be cast as (2.23)
which is the expression given in ref. [S], obtained by the usual procedure. We can pass to consider now several particular cases of this general expression. 2.1. I. Metullized interface In this case we assume that the interface is completely coated with an infinitesimally thin, perfectly conducting electrode which is grounded. The boundary conditions are now u,(‘)=uj2),
$(‘)=O,
The dispersion
@J(~)=O,
relation
TYj)= T,‘:),
i.
jz
+ c1P2/kl
= e:/q
which is the expression
T’z’=O, 2J
We are assuming crystal.
(2.24) from @.and &’ considering of the electric potentials at is easy to see, in the case of geometry given above. that (2.25)
+ ef/e,,
given in ref. [5].
2.1.2. Non-metallized surface In this case the boundary conditions +(i)=+(2),
3.
is now given by
det@i+‘.~,’ - $;-‘.‘8;‘1= 0. . where & and g-’ are the (3 x 3) matrices obtained only the indices i, j = 1, 2, 3, due to the vanishing the interface. From the relations obtained above it the piezoelectric crystals of the class 6mm with the the dispersion relation is given by C,P,/k,
1, 2,
are
D,‘“+2’.
that medium
1 is the vacuum
and medium
2 the piezoelectric
V.R. Velasco / Study of interface and surface elastic waves
125
In the vacuum there is no elastic displacement and we have only the scalar electric potential. So for the vacuum our Green function is a scalar quantity but we can obtain the dispersion relation for the surface modes from the general tensorial form in the following way: det]e,k,6,,
- Q:-‘.@;‘]
where Ed is the dielectric for the vacuum: G(
K,
(2.26)
= 0, permittivity
of the vacuum.
This is due to the fact that
k,; w’) = - l/q,k’,
and then
In the case o/the
($1-f&_ i
C2G
crystals
of the 6mm class we obtain
as dispersion
relation
‘/2
I
(e2b212 =
(l/E0
+
(2.27)
1/E2) ’
and finally
(e2/9j4 Cj( l/&g
(2.28)
+ 1/E2)2
which is the dispersion relation for the Bleustein-Gulyaev wave [3,4], for a non-metallized surface. This dispersion relation can also be obtained from (2.23) considering medium 1 as the vacuum, i.e.. Ci4 = e:5 = 0, E:, = eO.
1.1.3. Merallized surface In this case the boundary #‘)=.O,
9’2’=0,
conditions
are given by
T:,“LO.
Due to the vanishing of the electric potentials analogous case to that of the metallized interface given by
at the surface we are in an and the dispersion relation is
(2.29)
det)-&:-‘.~;‘l=O. In the case of the piezoelectric following dispersion relation @/k
crystals
of the 6mm
class we obtain
the
(2.30)
, = e2/e,
which can be put as w’(C,/p)k,(l
- e4/G2e2),
(2.31)
which is the relation given in ref. [3] for this case and obtained by the usual method. All these particular cases must be considered in detail in the formal analysis in order to obtain the Green function G, [ 1 l] of the system with a surface or an interface, but the dispersion relations can be deduced from the general formulae (2.16) by taking into account the physical facts embodied in the different boundary conditions for these particular situations. Thus we have shown in a simple (but not trivial) and well known case that the SGFM method gives the dispersion relation for all the possible cases. either from the general relation (2.16) or from the restrictions of this general formula given above. 3. Conclusions We have seen that the SGFM method, which has proved to work very well in purely elastic problems for surfaces and interfaces, can be easily modified to study the same kind of problems for piezoelectric crystals. Since the method gives readily a great deal of physical information, besides the dispersion relation, it is interesting to extend the formalism in order to include more complicated cases. Though the method has a greater formal complication than the usual procedures employed to obtain the dispersion relations in piezoelectric crystals, we have seen that it lends itself quite easily to obtain the desired dispersion relations with the advantage of providing a general formula from which particular cases follow at once, without need for an independent treatment of each particular situation as ‘is the case with the customary treatment of this problems. Besides, our method halves the size of the secular determinant, thus reducing the computation required and enabling us to consider more complicated situations. Studies of propagation in crystals of other symmetries and in general directions of propagation are in progress. Acknowledgements I am grateful to Professor F. Garcia-Moliner for critical reading of the manuscript. It is also a pleasure to thank Professor V. Heine (TCM) and Dr. R.F. Willis (PCS) for their kind invitation to the Cavendish Laboratory (Cambridge), where this work was done.
References [l] E. Dieulesaint and D. Royer, Elastic Waves in Solids (Wiley. New York 1980). [2] J. Sapriel, Acousto-Optics (Wiley. New York. 1979). [3] J.L. Bleustein, Appl. Phys. Letters 13 (1968) 412.
K R. Velasco / Study o/ interface and surface elasrlc waues
[4] [5] [6] (71 (8) [9] [IO]
[ ll] [ 121 [ 131 [ 14) [15] [16] [17] [ 1S]
127
Yu.V. Gulyaev, Soviet Phys. -JETP Letters 9 (1969) 37. C. Maerfeld and P. Tournois, Appl. Phys. Letters 19 (1971) 117. C.C. Tseng and R.M. White, J. Appl. Phys. 38 (1967) 4274. C.C. Tseng, J. Appl. Phys. 41 (1970) 2270. C.C. Tseng, Appl. Phys. Letters 16 (1970) 253. G.W. Farnell, in: Physical Acoustics, Vol. 6. Eds. W.P. Mason and R.N. Thurston (Academic Press, New York, 1970). G.W. Farnell and E.L. Adler, in: Physical Acoustics, Vol. 9. Eds. W.P. Mason and R.N. Thurston (Academic Press, New York. 1972). F. Garcia-Moliner. Ann. Physique 2 (1977) 177. V.R. Velasco and F. Garcia-Moliner, Surface Sci. 67 (1977) 555. V.R. Velasco and F. Garcia-Moliner, Surface Sci. 83 (1979) 376. V.R. Velasco and F. Garcia-Moliner, J. Phys. Cl 3 (1980) 2237. V.R. Velasco and F. Garcia-Moliner. Solid State Commun. 33 (1980) 1. V.R. Velasco. Phys. Status Solidi (a) 60 (1980) K61. R. Stoneley. Proc. Roy. Sot. (London) A106 (1924) 416. H.F. Tiersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969).