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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 Pubhshmg Company
117
STUDY OF INTERFACE AND SURFACE ELASTIC WAVES IN P I E Z O E L E C T R I C MATERIALS BY USING THE SURFACE GI~EEN FUNCTION M A T C H I N G (SGFM) M E T H O D V.R. VELASCO lnstttuto de Fiswa del Estado Sohdo (CSIC-UAM). Umversldad A utonoma, Madrid-34, Spare Recelxed 19 January 1983
It as shown that the surface Green function matclung method can be readily apphed to plezoelecmc materials by deflmng a four-&mens~onal Green function wbach includes both the elasttc and the electrostauc fields. For an interface system tlus reduces to one half (with respect to the usual treatments) the order of the secular determinant, w~th consequent saxang in computation As an illustration an application is made to the study of shear horizontal waves for surfaces and interfaces w~th 6ram symmetry.
1. Introduction
As it is known, piezoelectricity, an interdependence of elastic and electric properties m 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 [3,4,6-9] 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 [11]. This method has proved very useful in the study of 0039-6028/83/0000-0000/$03.00 © 1983 North-Holland
118
V.R. Velasco / Study of interface and surface elastw waves
surface [ 12-15] and interface [ 13,16] dynamical and thermodynarmcal 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 S G F M 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 S G F M 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 [11], or spectral functions of interest [14,15]. The S G F M 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 [11]. 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 m 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 3 m s - i ) and those propagating with electromagneuc velocities (VE~ t -------10 5 VA). Thus it appears that, even for strongly piezoelectric materials, the interaction between the three elasUc 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") [18].
2.1. Statement of the problem In the statzc-field approxtmation the elastic wave equations for the elastic displacements u: along the coordinate axes xj are intercoupled to the aniso-
V R Velasco/ Study of interface and surface elasnc waves
119
tropic Laplace equation for the potential ~ by the piezoelectric tensor euk. Thus the particle displacements and the potential must satisfy in each medium the following set of four equations: 02% p ~~t
a2u,
a2~ =0, - c,;~l ax, axl - ek,; ~x,axA
a2uk e,k t Ox,~xt
a2ep - =0, e,~ Ox,~xk
(2.1a)
(2.1b)
z,j,k,l=l,2,3,
where p is the density of the medium concerned, c,;kt is the elastic stiffness tensor measured at constant electric field, and e,A is the dielectric permittivity tensor measured at constant electric field. Then we must supplement the equations of motion with adequate boundary conditions. We shall consider the plane x 2 = 0 as the interface between the two piezoelectric media. Medium l occupies the half-space x 2 < 0 and medium 2 occupies the remaining half-space x 2 > 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 u = c u ~ , S x , - e ~ u E k,
(2.2a)
D, = e,~ E~ + e,AtS~,,
(2.2b)
with T representing the stress tensor, S the strain tensor, E the electric field and D the electric d~splacement. Thus the normal component of D m either of the anisotropic, piezoelectric media is expressed in terms of the potential and the mechanical displacements by D 2 = e 2 x t ( a u k / O x t ) - e2~ ( 3 e p / a x , ) ,
(2.3)
since SAt = ½(au/,/ax t + a u l / ~ x k ) ,
E~ = - Oep/~x/,.
Similarly the traction stresses which enter into the boundary conditions are of the form:
T2,=
C2;~,(3u~/Sxt) + e k 2 , ( a e p / a x k ) .
(2.4)
In order to gwe a concise version of the equations involved it is convenient to introduce the following tetravector: P ' = ( u , , u 2, u 3, ( e / e ) O ) ,
(2.5)
where u, (t = l, 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 interface and surface elasnc waves
the equations of motion in a compact form given by
L-
V = 0,
(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, t') = a ( r - - r') 8 ( t - - t') /,
(2.7)
where / is the unit matrix in a four-dimensional space. After Fourier transforming (2.7), the Green function can be obtained as
6(k; ~ ) = L-'(k; ~ ) .
(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 x 3 direction and motion independent of this coordinate. In this case we have a decoupling between the xt, x 2 d~rections and the x 3 direction, due to the symmetry [3]. We shall be concerned here only w~th the piezoelectric wave which has the electric potential 4' coupled to the elasuc displacement u 3. In this case the (4 x 4) operator ~s reduced to a (2 x 2) one given by
Ell
=
--C44 (a2//ax~ 4- a2/ax22 ) 4-10 a2/at 2,
(2.9a)
L,~= -(e,%/~,,)(a~/a~, ~ + a~/ax]),
(2.9b)
L~, = e,~(a~/ax~ + a~/axg),
(2.9c)
L~ = - e,~(a~/a~, ~ + a ~ / a ~ ) .
(2.9d)
After Fourier transforming eq. (2.9) it is easy to obtain by matrix inversion:
a,,(k,, k~; o:)=~,,/N,
(2.10a)
Gl2(kl, k2; co2) = - e l s / N ,
(2.10b)
G2,( k,, k2; 6o2) = e,,/N,
(2.10c)
G22(k,, k2; ~2)= [(C44k2 -
p~o2)e,t/e,5]/kZN,
(2.10d)
where N -- Ck 2 - el,p~o 2,
C = C 4 : l , + e~5-
Of course N = 0 gives the dispersion relation for the bulk piezoelectric (transverse) wave
~2 = et2k 2,
et 2 = (C44/p)( 1 4- ef5/EiiC44 ).
Now the boundary conditions are of two kinds:
V R. Velasco/ Study of interfaceand surfaceelasncwaves
121
Mechamcal. Continuity of the displacements at the interface u l " = u l 2',
1=1.2.3.
(2.11a)
Continuity of the normal stresses at the interface T~]'= T ~ ' .
j = 1, 2, 3.
(2.1 lb)
Electrical. Continuity of the potentml at the interface ~(i)= cb(2).
(2.1 lc)
Continuity of the normal component of the electrical displacement at the interface D~' ' = D2~2'.
(2.1 ld)
These boundary conditions can be easily expressed in terms of the tetravectot V defined m (2,6) and consequently of the Green function defined m (2.8). The continuity of the elastic displacements and the electric potential amounts to the continuity of (V l, V2, V3, (e/e)V4) a t the interface. The continuity of the normal stresses and the normal component of the electrical d~splacement 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 V. Proceedmg as in ref. [11], we have (OG~.)
A2o
e
= C22~1(aG~ +e -'-~-xl )
A4, = e2~l( --~-xl )
-
e(
0G4a/
e[ OG4~
,22 e , ax, ) .
e2~ e ( OG4c' axA ]I,
(2.12a)
(2.12b)
(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
~ll ~ ax3 ] A2;
= _ eu ( aGij
(2.13a)
( aG2j
--~-x3)+e,5, a x 3 ) .
(2.13b)
Having set out the problem, the SGFM analysis proceeds in the same way as for acoustic waves [I 1]. The difference is that now we have four equations of
122
V R Velasco / Study of interface and surface elastJc waves
motion, including u and q,, or the tetravector V, instead of three for u as m 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 q, (i.e. of V) at the interface is a boundary condition of the first kind [11] and can be expressed as ~l)~,)=,,, -,, ~)~2), e(l~O~l)= ~(1) ~4a ":a
(2.14a)
e (2) ~2)~2) ~(2) ~O4a ")a '
(2.14b)
t = 1, 2, 3, a = 1, 2, 3,4, where ~('~ stand for fictitious stimuli introduced m the S G F M method [11] and ~ stand for the surface projected Green function of each medium, defined as
~(K, ~z) = lim
1
~o
0 _--- f el'2n {~(k, 0~2) dk2, n - 2,rr j _ oo
with k = (k I, k 2, k3) = (1¢, k2) the wavevector and ~o2 the etgenvalue, 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 ~, form a boundary condition of the second kind [11], and can be expressed by
(2.15)
=
where ~ +) ~s the surface projected linear operator A gwing the boundary condiuon of the second kind m terms of the Green function, and including therefore a linear combination of the surface projected Green function and its normal derivatives ' ~ +)[11] defined as '~' +'(K, ~ 2 ) = 2 - ~ limn_o
ik 2 G(k, ~2) dk2,
the signs + and - being related to the sense in which we approach to the interface [ 11 ]. In this way all the formal study prevaously done [11] ~s valid m our case and it wdl not be repeated here. We shall state only the S G F M formula for the dispersion relation of the interface modes (2.16a)
d e t l ~ '1 = 0, with -~-)-~;',
(2.16b)
123
V R Velasco / Study of interface and surface elasttc waves
where ~ - l stand for the inverse of a projected Green function defined as
~,,~=
t= 1,2,3,
o~= 1, 2, 3, 4.
(e/E) The expressions (2.16) are vahd for the general case of an interface between two p~ezoelectric 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 x 2 = 0 will be the interface and medium 1 extends from - oo to the interface and medium 2 occupies the other half-space. Remembering the definition of the surface projected Green function and its normal derivatwes gwen before, it ~s easy to obtain ~,l (kl ; w2) =
e,,/2Cfl,
~12 ( k , ; w2) = _
~2,(kl 2 ) =
(2.17a)
e,5/2Cfl,
(2.17b)
Ei,/2Cfl,
(2.17c)
~22(k1: ~o2) = ( 2 e , 5 ) - ' ( k i -I -
e25/Cfl),
(2.17d)
where B =
-
I/'-
~ - ' ( k l- w 2 ) = 2 k ,
Ell
Then
e5) kl
C/~
ell
,
(2.18)
el5
--el5
~rel5
+Ell
+ C- -- e~5 el5
(2.19)
The linear operators ~ +-~ expressing the b o u n d a r y conditions in terms of the surface projected Green funcuon and its normal derivatives are gwen by (~'l; I = C,4'~'lf ' + (e?5/Ei, /~'~"-'2,-+',
(2.20a)
,. 'a~-+)~o ~ 2I-+l= j --~15 r-"lj T r"15 '~-+) 02/ "
(2.20b)
It is easy to verify that these operations satisfy the property AI +l _ Al-~
=
~,
essential in the general arguments of the S G F M m e t h o d [11]. The dispersion
V R Velasco / Study of interface and surface elasttc waves
124
relation follows from (2.16) and in our case we obtain:
(I
e 2 )+~2~2( I
C, BI kl -
(e I +
CiBi
e2 )
kl
C2~2
el
e2)
+e 2
= 0,
(2.21)
el + e2
where e, = els.,, e, = ell.,, i = I, 2, and ~ = C,/e,. From (2.21) it follows that (%+e2)(~l/31kl
e~+~/32et 2kl
e 2 ) + ( e l + e , e 2_ ) 2 = 0 ,
(2.22)
which after some handling can be cast as
c,
( ):( p,~o_____~ 2
l
c,k,
+
-P2c0 1
)/2
( e , / e , - e2/e 2 )-
=
+
(2.23)
which is the expression given in ref. [5], obtained by the usual procedure. We can pass to consider now several particular cases of this general expression. 2.1.1. Metalhzed mterface
In this case we assume that the interface ~s completely coated with an infinitesimally thin, perfectly conducting electrode which is grounded. The boundary conditions are now u ~ " = u l 2',
~("=0,
,~(2,=0,
r~:l ' = r ~ f ' ,
t,j= 1,2,3.
The dispersion relation is now given by d e t l ~ +'" ~ - ' - ~ - ' " ~2-'1 = 0,
(2 24)
where ~ and ~- i are the (3 × 3) matrices obtamed from ~ and .~- t considenng only the indices t , j - - 1, 2, 3, due to the vanishing of the electnc potentials at the interface. From the relations obtained above it is easy to see, in the case of the piezoelectric crystals of the class 6mm with the geometry given above, that the dispersion relation is gwen by C,/3,1k, + C21S2/k , = e~/e, + e ] / e 2,
(2.25)
which is the expression given in ref. [5]. 2.1.2. Non-metalhzed surface
In this case the boundary conditions are
~l~(1)--~b(2), --
T2(f)=O,
n(~)_n(2) ~t'~2
--
~ 2
•
We are assuming that medium 1 is the vacuum and medium 2 the piezoelectric crystal.
V.R. Velasco/ Study of interface and surface elasttc 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 tensonal form in the following way: detl~o k~8~4 - A~ -)" ~2-'1 = 0,
(2.26)
where e0 is the dmlectnc permittivity of the vacuum. This is due to the fact that
for the vacuum: G(K, k3; ~02) = - 1leo k2, and then ~(kl;w2) =-l/2eok So
G
l,
'~(+)= ~ l / 2 e o ,
(~(-+)=e0'~(-+)= ~½.
(~ +'. ~ - 1 = EOK. In the case of the crystals of the 6ram class we obtain as dispersion relation
(
P2~---~2 Gk~
1
( e2/e2)2 = (l/co+ 1/~)'
(2.27)
and finally C2 k 2 [ (e2/e 2 )4 ': = ~ ' 11 , C ~ ( l / e o + 1/~2) 2
(2.28)
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., CJ4 = eJ15--- 0, eJn = co2.1 3. Metalhzed surface
In this case the boundary conditions are given by q~ll=0,
q,(2)=0,
T~)=0.
Due to the vanishing of the electric potentials at the surface we are in an analogous case to that of the metallized interface and the dispersion relation is given by ^(--)
--I
d e t l - C2 "~2 I = 0.
(2.29)
In the case of the piezoelectric crystals of the 6mm class we obtain the following dispersion relation Cfl/k, = eZ/e,
(2.30)
which can be put as co2(C/p)k,(l - e4/C2e:),
(2.31)
126
V R Velasco / Study of interface and surface elastw waves
which is the relation given m 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 s [11] 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 m a simple (but not trivial) and well known case that the S G F M method gives the dispersion relation for all the possible cases, etther from the general relation (2.16) or from the restrictions of this general formula given above. 3. Conclusions We have seen that the S G F M 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 disperston relation, it is interesting to extend the formahsm in order to include more complicated cases. Though the method has a greater formal comphcation 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 winch 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 enabhng us to consider more complicated situations. Studies of propagatton in crystals of other symmetries and in general directions of propagation are m 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 [1] E. Dleulesamt and D. Royer, Elasttc Waves m Soltds (Wiley, New York 1980). [2] J Sapnel, Acousto-Optlcs (Wdey, New York, 1979) [3] J.L. Bleustem, Appl. Phys. Letters 13 (1968) 412
V R Velasco / Study of interface and surface elastw waves [4] [5] [6] [7] [8] {9] [10] [11] [12] [13] [141 [151 [16] [17] [18]
127
Yu V Gulyaev, Soviet Phys. -JETP Letters 9 (1969) 37. C. Maerfeld and P Tournols, 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, m: Physical Acoustics, Vol. 6, Eds. W.P Mason and R.N. Thurston (Academic Press, New York, 1970). G.W Farnell and E.L. Adler, m. Physical Acoustics, Vol. 9, Eds. W.P. Mason and R.N. Thurston (Acadermc Press, New York, 1972). F Garcia-Mohner, Ann Physique 2 (1977) 177 V.R Velasco and F. Garcia-Mohner, Surface Scl 67 (1977) 555 V.R Velasco and F. Garcia-Mohner, Surface Sct 83 (1979) 376 V.R. Velasco and F Garcia-Mohner, J. Phys. CI3 (1980) 2237. V.R. Velasco and F Garcia-Mohner, Sohd State Commun. 33 (1980) 1. V R. Velasco, Phys. Status Sohdl (a) 60 (1980) K61 R Stonel.:y, Proc. Roy Soc (London)AI06 (1924)416. H F Tlersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969).
117
STUDY OF INTERFACE AND SURFACE ELASTIC WAVES IN P I E Z O E L E C T R I C MATERIALS BY USING THE SURFACE GI~EEN FUNCTION M A T C H I N G (SGFM) M E T H O D V.R. VELASCO lnstttuto de Fiswa del Estado Sohdo (CSIC-UAM). Umversldad A utonoma, Madrid-34, Spare Recelxed 19 January 1983
It as shown that the surface Green function matclung method can be readily apphed to plezoelecmc materials by deflmng a four-&mens~onal Green function wbach includes both the elasttc and the electrostauc fields. For an interface system tlus reduces to one half (with respect to the usual treatments) the order of the secular determinant, w~th consequent saxang in computation As an illustration an application is made to the study of shear horizontal waves for surfaces and interfaces w~th 6ram symmetry.
1. Introduction
As it is known, piezoelectricity, an interdependence of elastic and electric properties m 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 [3,4,6-9] 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 [11]. This method has proved very useful in the study of 0039-6028/83/0000-0000/$03.00 © 1983 North-Holland
118
V.R. Velasco / Study of interface and surface elastw waves
surface [ 12-15] and interface [ 13,16] dynamical and thermodynarmcal 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 S G F M 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 S G F M 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 [11], or spectral functions of interest [14,15]. The S G F M 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 [11]. 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 m 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 3 m s - i ) and those propagating with electromagneuc velocities (VE~ t -------10 5 VA). Thus it appears that, even for strongly piezoelectric materials, the interaction between the three elasUc 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") [18].
2.1. Statement of the problem In the statzc-field approxtmation the elastic wave equations for the elastic displacements u: along the coordinate axes xj are intercoupled to the aniso-
V R Velasco/ Study of interface and surface elasnc waves
119
tropic Laplace equation for the potential ~ by the piezoelectric tensor euk. Thus the particle displacements and the potential must satisfy in each medium the following set of four equations: 02% p ~~t
a2u,
a2~ =0, - c,;~l ax, axl - ek,; ~x,axA
a2uk e,k t Ox,~xt
a2ep - =0, e,~ Ox,~xk
(2.1a)
(2.1b)
z,j,k,l=l,2,3,
where p is the density of the medium concerned, c,;kt is the elastic stiffness tensor measured at constant electric field, and e,A is the dielectric permittivity tensor measured at constant electric field. Then we must supplement the equations of motion with adequate boundary conditions. We shall consider the plane x 2 = 0 as the interface between the two piezoelectric media. Medium l occupies the half-space x 2 < 0 and medium 2 occupies the remaining half-space x 2 > 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 u = c u ~ , S x , - e ~ u E k,
(2.2a)
D, = e,~ E~ + e,AtS~,,
(2.2b)
with T representing the stress tensor, S the strain tensor, E the electric field and D the electric d~splacement. Thus the normal component of D m either of the anisotropic, piezoelectric media is expressed in terms of the potential and the mechanical displacements by D 2 = e 2 x t ( a u k / O x t ) - e2~ ( 3 e p / a x , ) ,
(2.3)
since SAt = ½(au/,/ax t + a u l / ~ x k ) ,
E~ = - Oep/~x/,.
Similarly the traction stresses which enter into the boundary conditions are of the form:
T2,=
C2;~,(3u~/Sxt) + e k 2 , ( a e p / a x k ) .
(2.4)
In order to gwe a concise version of the equations involved it is convenient to introduce the following tetravector: P ' = ( u , , u 2, u 3, ( e / e ) O ) ,
(2.5)
where u, (t = l, 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 interface and surface elasnc waves
the equations of motion in a compact form given by
L-
V = 0,
(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, t') = a ( r - - r') 8 ( t - - t') /,
(2.7)
where / is the unit matrix in a four-dimensional space. After Fourier transforming (2.7), the Green function can be obtained as
6(k; ~ ) = L-'(k; ~ ) .
(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 x 3 direction and motion independent of this coordinate. In this case we have a decoupling between the xt, x 2 d~rections and the x 3 direction, due to the symmetry [3]. We shall be concerned here only w~th the piezoelectric wave which has the electric potential 4' coupled to the elasuc displacement u 3. In this case the (4 x 4) operator ~s reduced to a (2 x 2) one given by
Ell
=
--C44 (a2//ax~ 4- a2/ax22 ) 4-10 a2/at 2,
(2.9a)
L,~= -(e,%/~,,)(a~/a~, ~ + a~/ax]),
(2.9b)
L~, = e,~(a~/ax~ + a~/axg),
(2.9c)
L~ = - e,~(a~/a~, ~ + a ~ / a ~ ) .
(2.9d)
After Fourier transforming eq. (2.9) it is easy to obtain by matrix inversion:
a,,(k,, k~; o:)=~,,/N,
(2.10a)
Gl2(kl, k2; co2) = - e l s / N ,
(2.10b)
G2,( k,, k2; 6o2) = e,,/N,
(2.10c)
G22(k,, k2; ~2)= [(C44k2 -
p~o2)e,t/e,5]/kZN,
(2.10d)
where N -- Ck 2 - el,p~o 2,
C = C 4 : l , + e~5-
Of course N = 0 gives the dispersion relation for the bulk piezoelectric (transverse) wave
~2 = et2k 2,
et 2 = (C44/p)( 1 4- ef5/EiiC44 ).
Now the boundary conditions are of two kinds:
V R. Velasco/ Study of interfaceand surfaceelasncwaves
121
Mechamcal. Continuity of the displacements at the interface u l " = u l 2',
1=1.2.3.
(2.11a)
Continuity of the normal stresses at the interface T~]'= T ~ ' .
j = 1, 2, 3.
(2.1 lb)
Electrical. Continuity of the potentml at the interface ~(i)= cb(2).
(2.1 lc)
Continuity of the normal component of the electrical displacement at the interface D~' ' = D2~2'.
(2.1 ld)
These boundary conditions can be easily expressed in terms of the tetravectot V defined m (2,6) and consequently of the Green function defined m (2.8). The continuity of the elastic displacements and the electric potential amounts to the continuity of (V l, V2, V3, (e/e)V4) a t the interface. The continuity of the normal stresses and the normal component of the electrical d~splacement 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 V. Proceedmg as in ref. [11], we have (OG~.)
A2o
e
= C22~1(aG~ +e -'-~-xl )
A4, = e2~l( --~-xl )
-
e(
0G4a/
e[ OG4~
,22 e , ax, ) .
e2~ e ( OG4c' axA ]I,
(2.12a)
(2.12b)
(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
~ll ~ ax3 ] A2;
= _ eu ( aGij
(2.13a)
( aG2j
--~-x3)+e,5, a x 3 ) .
(2.13b)
Having set out the problem, the SGFM analysis proceeds in the same way as for acoustic waves [I 1]. The difference is that now we have four equations of
122
V R Velasco / Study of interface and surface elastJc waves
motion, including u and q,, or the tetravector V, instead of three for u as m 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 q, (i.e. of V) at the interface is a boundary condition of the first kind [11] and can be expressed as ~l)~,)=,,, -,, ~)~2), e(l~O~l)= ~(1) ~4a ":a
(2.14a)
e (2) ~2)~2) ~(2) ~O4a ")a '
(2.14b)
t = 1, 2, 3, a = 1, 2, 3,4, where ~('~ stand for fictitious stimuli introduced m the S G F M method [11] and ~ stand for the surface projected Green function of each medium, defined as
~(K, ~z) = lim
1
~o
0 _--- f el'2n {~(k, 0~2) dk2, n - 2,rr j _ oo
with k = (k I, k 2, k3) = (1¢, k2) the wavevector and ~o2 the etgenvalue, 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 ~, form a boundary condition of the second kind [11], and can be expressed by
(2.15)
=
where ~ +) ~s the surface projected linear operator A gwing the boundary condiuon of the second kind m terms of the Green function, and including therefore a linear combination of the surface projected Green function and its normal derivatives ' ~ +)[11] defined as '~' +'(K, ~ 2 ) = 2 - ~ limn_o
ik 2 G(k, ~2) dk2,
the signs + and - being related to the sense in which we approach to the interface [ 11 ]. In this way all the formal study prevaously done [11] ~s valid m our case and it wdl not be repeated here. We shall state only the S G F M formula for the dispersion relation of the interface modes (2.16a)
d e t l ~ '1 = 0, with -~-)-~;',
(2.16b)
123
V R Velasco / Study of interface and surface elasttc waves
where ~ - l stand for the inverse of a projected Green function defined as
~,,~=
t= 1,2,3,
o~= 1, 2, 3, 4.
(e/E) The expressions (2.16) are vahd for the general case of an interface between two p~ezoelectric 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 x 2 = 0 will be the interface and medium 1 extends from - oo to the interface and medium 2 occupies the other half-space. Remembering the definition of the surface projected Green function and its normal derivatwes gwen before, it ~s easy to obtain ~,l (kl ; w2) =
e,,/2Cfl,
~12 ( k , ; w2) = _
~2,(kl 2 ) =
(2.17a)
e,5/2Cfl,
(2.17b)
Ei,/2Cfl,
(2.17c)
~22(k1: ~o2) = ( 2 e , 5 ) - ' ( k i -I -
e25/Cfl),
(2.17d)
where B =
-
I/'-
~ - ' ( k l- w 2 ) = 2 k ,
Ell
Then
e5) kl
C/~
ell
,
(2.18)
el5
--el5
~rel5
+Ell
+ C- -- e~5 el5
(2.19)
The linear operators ~ +-~ expressing the b o u n d a r y conditions in terms of the surface projected Green funcuon and its normal derivatives are gwen by (~'l; I = C,4'~'lf ' + (e?5/Ei, /~'~"-'2,-+',
(2.20a)
,. 'a~-+)~o ~ 2I-+l= j --~15 r-"lj T r"15 '~-+) 02/ "
(2.20b)
It is easy to verify that these operations satisfy the property AI +l _ Al-~
=
~,
essential in the general arguments of the S G F M m e t h o d [11]. The dispersion
V R Velasco / Study of interface and surface elasttc waves
124
relation follows from (2.16) and in our case we obtain:
(I
e 2 )+~2~2( I
C, BI kl -
(e I +
CiBi
e2 )
kl
C2~2
el
e2)
+e 2
= 0,
(2.21)
el + e2
where e, = els.,, e, = ell.,, i = I, 2, and ~ = C,/e,. From (2.21) it follows that (%+e2)(~l/31kl
e~+~/32et 2kl
e 2 ) + ( e l + e , e 2_ ) 2 = 0 ,
(2.22)
which after some handling can be cast as
c,
( ):( p,~o_____~ 2
l
c,k,
+
-P2c0 1
)/2
( e , / e , - e2/e 2 )-
=
+
(2.23)
which is the expression given in ref. [5], obtained by the usual procedure. We can pass to consider now several particular cases of this general expression. 2.1.1. Metalhzed mterface
In this case we assume that the interface ~s completely coated with an infinitesimally thin, perfectly conducting electrode which is grounded. The boundary conditions are now u ~ " = u l 2',
~("=0,
,~(2,=0,
r~:l ' = r ~ f ' ,
t,j= 1,2,3.
The dispersion relation is now given by d e t l ~ +'" ~ - ' - ~ - ' " ~2-'1 = 0,
(2 24)
where ~ and ~- i are the (3 × 3) matrices obtamed from ~ and .~- t considenng only the indices t , j - - 1, 2, 3, due to the vanishing of the electnc potentials at the interface. From the relations obtained above it is easy to see, in the case of the piezoelectric crystals of the class 6mm with the geometry given above, that the dispersion relation is gwen by C,/3,1k, + C21S2/k , = e~/e, + e ] / e 2,
(2.25)
which is the expression given in ref. [5]. 2.1.2. Non-metalhzed surface
In this case the boundary conditions are
~l~(1)--~b(2), --
T2(f)=O,
n(~)_n(2) ~t'~2
--
~ 2
•
We are assuming that medium 1 is the vacuum and medium 2 the piezoelectric crystal.
V.R. Velasco/ Study of interface and surface elasttc 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 tensonal form in the following way: detl~o k~8~4 - A~ -)" ~2-'1 = 0,
(2.26)
where e0 is the dmlectnc permittivity of the vacuum. This is due to the fact that
for the vacuum: G(K, k3; ~02) = - 1leo k2, and then ~(kl;w2) =-l/2eok So
G
l,
'~(+)= ~ l / 2 e o ,
(~(-+)=e0'~(-+)= ~½.
(~ +'. ~ - 1 = EOK. In the case of the crystals of the 6ram class we obtain as dispersion relation
(
P2~---~2 Gk~
1
( e2/e2)2 = (l/co+ 1/~)'
(2.27)
and finally C2 k 2 [ (e2/e 2 )4 ': = ~ ' 11 , C ~ ( l / e o + 1/~2) 2
(2.28)
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., CJ4 = eJ15--- 0, eJn = co2.1 3. Metalhzed surface
In this case the boundary conditions are given by q~ll=0,
q,(2)=0,
T~)=0.
Due to the vanishing of the electric potentials at the surface we are in an analogous case to that of the metallized interface and the dispersion relation is given by ^(--)
--I
d e t l - C2 "~2 I = 0.
(2.29)
In the case of the piezoelectric crystals of the 6mm class we obtain the following dispersion relation Cfl/k, = eZ/e,
(2.30)
which can be put as co2(C/p)k,(l - e4/C2e:),
(2.31)
126
V R Velasco / Study of interface and surface elastw waves
which is the relation given m 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 s [11] 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 m a simple (but not trivial) and well known case that the S G F M method gives the dispersion relation for all the possible cases, etther from the general relation (2.16) or from the restrictions of this general formula given above. 3. Conclusions We have seen that the S G F M 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 disperston relation, it is interesting to extend the formahsm in order to include more complicated cases. Though the method has a greater formal comphcation 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 winch 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 enabhng us to consider more complicated situations. Studies of propagatton in crystals of other symmetries and in general directions of propagation are m 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 [1] E. Dleulesamt and D. Royer, Elasttc Waves m Soltds (Wiley, New York 1980). [2] J Sapnel, Acousto-Optlcs (Wdey, New York, 1979) [3] J.L. Bleustem, Appl. Phys. Letters 13 (1968) 412
V R Velasco / Study of interface and surface elastw waves [4] [5] [6] [7] [8] {9] [10] [11] [12] [13] [141 [151 [16] [17] [18]
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Yu V Gulyaev, Soviet Phys. -JETP Letters 9 (1969) 37. C. Maerfeld and P Tournols, 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, m: Physical Acoustics, Vol. 6, Eds. W.P Mason and R.N. Thurston (Academic Press, New York, 1970). G.W Farnell and E.L. Adler, m. Physical Acoustics, Vol. 9, Eds. W.P. Mason and R.N. Thurston (Acadermc Press, New York, 1972). F Garcia-Mohner, Ann Physique 2 (1977) 177 V.R Velasco and F. Garcia-Mohner, Surface Scl 67 (1977) 555 V.R Velasco and F. Garcia-Mohner, Surface Sct 83 (1979) 376 V.R. Velasco and F Garcia-Mohner, J. Phys. CI3 (1980) 2237. V.R. Velasco and F Garcia-Mohner, Sohd State Commun. 33 (1980) 1. V R. Velasco, Phys. Status Sohdl (a) 60 (1980) K61 R Stonel.:y, Proc. Roy Soc (London)AI06 (1924)416. H F Tlersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969).