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Pyroelectricity and piezoelectricity are not true volume effects
~
So]id State Communications, Vol.40, pp.971-974. Pergamon Press Ltd. 1981. Printed in Great Britain.
0038-1098/81/470971-04502.00/0
PYROELECTRICITY AND PIEZOELECTRICITY ARE NOT T R U E VOLUME EFFECTS Roll Landauer IBM Thomas J. Watson Research Center P. O. Box 218 Yorktown Heights, New York 10598
(Received 23 September 1981 by G. Burns)
The total polarization of a crystal is sensitive to the state of the surface, even for very large volumes. This point was emphasized by Larmor and later became the basis of more detailed discussions of pyroelectricity and piezoelectricity, by Woo and by the author, questioning the usual textbook theorems. The earlier arguments are summarized and supplemented by a new model: a discussion of electronic redistribution in a one-dimensional ordered periodic array of two types of potential wells.
The occurrence of pyroelectricity and piezoelectricity, in crystals, is usually related 1 to the symmetry of the unit cell rather than to the symmetry of the actual crystal, including its surfaces and electrodes. Dipole moments, however, are not ordinary bulk quantities. Dipole moments remain critically sensitive to the state of the surface, even if we keep increasing the volume. 2 This difficulty was emphasized by Sir Joseph Larmor, in 1921, in a paper3 which seems to have been largely ignored. As Larmor made clear, our widely accepted symmetry theorems originated at a time when the distinction between atoms, molecules, and unit cells was still murky, and the freedom available for structuring the surface not yet appreciated. In a 1960 paper 4 on pyroelectricity 1 rediscovered Larmor's point, without awareness of Ref. 3. Later Woo 5 broadened the discussion to include piezoelectricity, and provided a specific calculation for piezoelectricity under hydrostatic pressure, in the zinc-blende structure. This work was in turn criticized by Martin, 6 and a reply to that critique also published. 7 It is important to note that Martin's critique 6 is, in fact, a partial acceptance of Refs. 3-5. Martin states: Conventional
the zinc-blende structure, that the detailed consideration of surface effects still gives the classically expected conclusion: no piezoelectricity under hydrostatic compression. Martin's conclusion must, therefore, be dependent on the details of his calculation. At this point it is also appropriate to mention a "review" paper by Aizu, 8 which contains no citations, but clearly emphasizes the role of surfaces and surface composition. Aizu distinguishes between "ordinary" boundaries and "ordinary" sampies, and others. Despite a strong similarity to Refs. 3-5, this author is not sure that he understands the terminology of Ref. 8 adequately, to be able to claim that it agrees with Refs. 3-5. The reader may find the discussion of Aizu's work, in a recent text, 9 more accessible than the original discussion. Fig. l a illustrates our basic point via a one dimensional chain of equally spaced positive and negative charges. This could, for example, be, interpreted to represent a succession of (1,1,1) planes in the NaCI structure. Alternatively, it can just be considered to be a one dimensional example. Clearly a large net static polarization, as shown in Fig. la, cannot exist in static equilibrium. The divergence of polarization must be compensated by charges attracted to the ends of the sample, or else by charges in electrodes deposited onto the ends of the crystal. In pyroelectricity and piezoelectricity, however, we are not concerned with a net static polarization, but only with small polarization changes. In a highly ionic crystal an unsymmetrical structure, as suggested in Fig. la, may not be easy to achieve, but does not seem impossible. Crystals
treatments o f piezoelectricity tacitly assume that the polarization produced by strain is purely a bulk effect whose symmetry properties are determined solely by the symmetry o f the environment o f the atoms in an infinite crystal. However, Woo and others have correctly pointed out that a complete description o f the polarization produced by a strain in a finite crystal must include possible effects o f the surfaces, i.e., effects o f the specific termination o f the infinite crystal to produce the given finite sample. Martin then goes on, however, to argue, in the specific case of 971
972
PYROELECTRICITY AND PIEZOELECTRICITY
-+-+-+-+ (a) F+
,.j_
"7"
,
-F-- T---r
t...___jL___.J.__
_j
+
(b) + 1--1 -- -- "1" j__ r" i ~+ 1 - - - - : ~ I.
. . .
.t
. . . .
t
_
+ "T'II__ +If| _
_.I.
_
_ _ j
(e) F I G U R E 1 Periodic chain of equidistant charges with a net dipole moment. In (b) the unit cell is chosen with the maximum potential symmetry, whereas in (c) the unit cell is chosen so that the entire crystal can be included in an integral number of cells. do not, generally, grow out from their center, and almost inevitably, are subject to some symmetry breaking process during their growth. One can also imagine, at least in principle, a crystal grown by deposition from the vapor, or by molecular beam epitaxy, onto an electrode made of the metallic constituent of the crystal. To avoid questions of this sort, about the actual realizability of a crystal with two different end surfaces, Refs. 4 and 5 concentrated on the zinc-blende structure. In that case a crystal, with the minimum number of broken bonds at its surfaces, has the required polarization. To emphasize the central conceptual point, however, and permit simple one-dimensional illustrations, we shall continue, here, with the model of Fig. la. The conventional view divides the array in Fig. la, as shown in Fig. lb, each cell having the maxium allowed symmetry. Now consider some modulation parameter, such as temperature or hydrostatic pressure, which controls the degree of ionicity of the crystal, and causes the effective charges in Fig. 1 to be changed. A simple inspection of Fig. la would lead one to expect a change in the total dipole moment. The conventional theory, based instead on Fig. 1b, will assume that the symmetry in each cell remains unbroken, as a result of the scalar modulation, and then goes on to ignore the left-over ends. Alternatively, we can, as advocated in Refs. 4,5, and 7, view the crystal as suggested in Fig. lc, i.e. as a sequence of polarized entities, in which the symmetry breaking introduced by the surface is respected. Calculations of piezoelectric coefficients, from first principles, typically concentrate on charge motion in the infinite, unlimited crystal, and then at best do handwaving to extrapolate to the case of the terminated crystal. We can, here, cite only a few such calculations. ~o The actual crystal, on which
11
Vol. 40, No.
measurements are done, is likely not only to be terminated, but to have electrodes. No calculation, known to me. even remotely does justice to this. The "handwaving" involved in Ref. 5, and implicitly invoked in our discussion of Fig. lc, assumes that the changes in effective charge, calculated for the infinite crystal, applies to each ion in Fig. 1, including the end ions. The "handwaving" invoked, instead, in Ref. 6, and consistent with our discussion of Fig. I b, is that the flux of charge, along each bond separating a + and - charge, is the same as in the infinite crystal. Thus the leftmost ion in Fig. 1 sees only half the change in effective charge of the other negative ions, in the interior of the sample. Neither of the two " h a n d w a v i n g " assumptions is likely to be satisfied exactly. Note, however, that the "handwaving" assumption associated with Fig. l b must be satisfied exactly to lead to an absence of a change in the total dipole moment; any compromise b et w een the two assumptions will lead to a change in dipole moment.t~ If, in Fig. lb, the interior cells remain unpolarized as the effective charge is modulated, and if we furthermore assume that the end charges are modulated, almost as effectively as those in the interior, then there must be a transfer of charge from the left hand end to the right hand end. (If, instead, the needed charge change, at each end, comes from a nearby electrode, then we cannot expect a measurable charge flow in the external circuit.) This has led to an assertion, in many oral debates of this viewpoint, and also in Ref. 6, that such a long range transport of charge is possible only in a conductor, and that therefore the discussion of Refs. 4, 5, and 7 does not apply to insulators. To put this question to rest, and to provide an additional simple example of the symmetry breaking role of the termination of a periodic array, we shall discuss modulation of a full band in the potential shown in Fig. 2.
V
A
B
A
B
A
B
A
B
A
F I G U R E 2 One-dimensional model of an ordered array of A and B atoms. Fig. 2 shows a periodic potential, terminated in such a way that the overall symmetry is broken by the "walls" at the end of the array. Each period of the potential has two wells, suggesting a sequence of alternating atoms. Let us assume a filled band, so that we are dealing with an insulator. In a given band we can expect a distribution between A and B wells; in general the electrons in a band will
Vol.
40, No.
II
PYROELECTRICITY
not occupy the two types of wells with equal probability. In the tight binding a p p r o x i m a t i o n we can, in fact, expect the electrons to be p r e d o m i n a n t l y in one of the two types of well, unless the A and B well have almost c o i n c i d e n t energy levels. N o w if we m o d u l a t e the potential and change it in some way, without affecting its symmetry, we can expect some electronic redistribution. If the m o d u l a t i o n is slow, then the original band states will be mapped adiabatically into the new band states. If the m o d u lation is applied quickly, we can expect transitions to higher lying bands, but eventually will return to the ground state in which the lowest lying bands are filled. Such a redistribution of charge, as a result o f a change in the lattice potential, has no relationship to the existence of a d c c o n d u c t i v i t y , and occurs just as much in a filled band, as it does in a conductor. In considering the redistribution of electronic charge, our s u b s e q u e n t arguments in this note will be based on the same " h a n d w a v i n g " a s s u m p t i o n made before. We will give intuitive a r g u m e n t s clearly applicable to an infinite crystal, or to one with periodic b o u n d a r y conditions, and will then assume that in the t e r m i n a t e d crystal the charges in each well of Fig. 2 will change similarly. Clearly there is room for a better calculation. As before, h o w e v e r , any change in net polarization, as the potential is modulated, will defeat the t e x t b o o k theorems, the change does not have to be exactly the one predicted from our " h a n d w a v i n g . " The redistribution of charge can be induced, for e x a m p l e , by d e e p e n i n g one of the two wells. This will not disrupt the existing local s y m m e t r y , and is therefore a " s c a l a r " peturbation, for our purposes, similar to a t e m p e r a t u r e variation. T o make the scalar nature of this modulation entirely clear, consider the case where only one of the two wells is attractive, and the other is repulsive. T h e n let the c o e f f i c i e n t multiplying this p o t e n t i a l be the c o n trolled scalar variable. If we change the sign of this scalar, we can certainly e x p e c t an electronic redistribution, In case the reader is dissatisfied with this example, we can proceed in the slightly more complex fashion described in the next paragraph. In fact, almost any r e a r r a n g e m e n t of the potential will induce some shift b e t w e e n A and B wells. The theory of one-dimensional band structures ~2 shows that for suitably adjusted p o t e n tials the forbidden gap b e t w e e n a d j a c e n t bands can be made to vanish. This basic fact may not be widely appreciated; for the two most widely investigated one d i m e n s i o n a l periodic p o t e n t i a l s , the K r o n i g - P e n n e y m o d e l and the M a t h i e u e q u a t i o n , such " a c c i d e n t a l " band edge crossings do not occur.
AND p IEZOELECTRICITX
973
B a n d crossings were discussed by S h o c k l e y 13 in c o n n e c t i o n with the existence of surface states, in o n e - d i m e n s i o n a l models of m o n a t o m i c lattices. If we c o n s i d e r modulating the relative shapes of the A and B wells, then at the point where the two wells b e c o m e equal the unit cell is doubled. At that point one half of the f o r b i d d e n bands, those resulting f r o m the " s u p e r l a t t i c e " structure, must disappear. More generally, w h e n e v e r we m o d u l a t e the potential of Fig. 2 in such a way that a level of the A well passes a level of the B well, we can e x p e c t that s o m e w h e r e in that n e i g h b o r h o o d there must be a vanishing gap. N e a r such a vanishing gap, i.e. w h e n the gap is present but small, the wave function in a band will have a particularly s t r o n g admixture of states from both wells. Let us assume we are in such a band, say one which is primarily an A band, but with a strong admixture of B. N o w assume the lattice is dilated, in such a way that the A and B wells keep their shape, but the t u n n e l i n g regions b e t w e e n the wells are lengthened. Thus the A and B bands b e c o m e more strongly decoupled, and in the case we have considered, can expect transfer of charge from the B wells to the A wells. The viewpoint a d v o c a t e d in Refs. 3-5 and 7, and r e p e a t e d here, has r e c e i v e d scant a t t e n t i o n . The author is not aware of any m o d e r n text or review paper that discusses it, with the solitary e x c e p tion of Ref. 9. Several points deserve to be made in that c o n n e c t i o n . First of all, it is c o n c e i v a b l e that the ordinary t e x t b o o k t h e o r e m s are right, or at least a p p r o x i m a t e l y correct, in a large p e r c e n t a g e of cases, as a result of microscopic surface considerations, not yet properly stated. I consider this alternative unlikely, but cannot rule it out. A more significant point: large p i e z o e l e c t r i c and p y r o e l e c t r i c c o e f f i cients, that are t e c h n o l o g i c a l l y useful, and easily m e a s u r e d , are u n d o u b t e d l y associated with a t o m i c d i s p l a c e m e n t s , and not just with the t r a n s f e r of electronic charges. Such polar m o t i o n c a n n o t occur, say in Fig. 1, without disrupting the existing local symmetry. Thus our v i e w p o i n t b e c o m e s irrelevant and the t e x t b o o k t h e o r e m s apply. Finally, if we want to be c o m p l e t e l y rigorous, we must admit that if the surfaces break the s y m m e t r y , as illustrated in Fig. la, then this s y m m e t r y breaking may have an effect, albeit i m m e a s u r a b l e , on the atomic positions. T h e n one can assert that the ordinary t h e o r e m s still apply, but that the atomic a r r a n g e m e n t in the interior of real crystals may have a s y m m e t r y which is lower than that ascertained in the prevailing measurements.
974
PYROELECTRICITY
~ND PIEZOELECTRICITY
Vol. 40, No.
REFERENCES 1.
2.
3.
4. 5. 6.
G. Burns, Introduction to G r o u p Theory with Applications (Academic, New York 1977) Chapt. IV; H. J. Juretschke, Crystal Physics (W. A. Benjamin, Reading 1974) Chapt. XII, but also see the reservations expressed briefly on pg. 49. The extent to which this is explicit, in elementary texts, varies. F o r a particularly careful discussion see R. W. P. King, Fundamental Electromagnetic Theory (Dover, New York 1963) Chapt. I J. Larmor, Proc. R. Soc. London 99, 1 (1921). R. Landauer, J. Chem. Phys. 32, 1784 (1960). J. W. F. Woo, Phys. Rev. B4, 1218 (1971). R. M. Martin, Phys. Rev. B6, 4874 (1972).
7. 8.
9.
10.
11. 12.
13.
J. W. F. Woo and R. Landauer, Phys. Rev. B6, 4876 (1972). K. Aizu, Rev. Mod. Phys. 34, 550 (1962). J. C. Burfoot and G. W. Taylor, Polar Dielectrics and Their Applications (U. Calif. Press, Berkeley 1979) Section 8.2. G. Arlt and P. Quadflieg, Phys. Status Solidi 25, 323 (1968); R. M. Martin, Phys. Rev. BS, 1607 (1972); W. A. Harrison, Phys. Rev. B I 0 , 767 (1974); T. Hidaka, Phys. Rev. BIS, 4426 (1978). See footnote 3, Ref. 4., for more precise language. G. Allen, Phys. Rev. 91, 531 (1953); B. F. Buxton and M. V. Berry, Philos. Trans. R. Soc. London 2 8 2 , 4 8 5 (1976). W. Shockley, Phys. Rev. 5 6 , 3 1 7 (1939).
So]id State Communications, Vol.40, pp.971-974. Pergamon Press Ltd. 1981. Printed in Great Britain.
0038-1098/81/470971-04502.00/0
PYROELECTRICITY AND PIEZOELECTRICITY ARE NOT T R U E VOLUME EFFECTS Roll Landauer IBM Thomas J. Watson Research Center P. O. Box 218 Yorktown Heights, New York 10598
(Received 23 September 1981 by G. Burns)
The total polarization of a crystal is sensitive to the state of the surface, even for very large volumes. This point was emphasized by Larmor and later became the basis of more detailed discussions of pyroelectricity and piezoelectricity, by Woo and by the author, questioning the usual textbook theorems. The earlier arguments are summarized and supplemented by a new model: a discussion of electronic redistribution in a one-dimensional ordered periodic array of two types of potential wells.
The occurrence of pyroelectricity and piezoelectricity, in crystals, is usually related 1 to the symmetry of the unit cell rather than to the symmetry of the actual crystal, including its surfaces and electrodes. Dipole moments, however, are not ordinary bulk quantities. Dipole moments remain critically sensitive to the state of the surface, even if we keep increasing the volume. 2 This difficulty was emphasized by Sir Joseph Larmor, in 1921, in a paper3 which seems to have been largely ignored. As Larmor made clear, our widely accepted symmetry theorems originated at a time when the distinction between atoms, molecules, and unit cells was still murky, and the freedom available for structuring the surface not yet appreciated. In a 1960 paper 4 on pyroelectricity 1 rediscovered Larmor's point, without awareness of Ref. 3. Later Woo 5 broadened the discussion to include piezoelectricity, and provided a specific calculation for piezoelectricity under hydrostatic pressure, in the zinc-blende structure. This work was in turn criticized by Martin, 6 and a reply to that critique also published. 7 It is important to note that Martin's critique 6 is, in fact, a partial acceptance of Refs. 3-5. Martin states: Conventional
the zinc-blende structure, that the detailed consideration of surface effects still gives the classically expected conclusion: no piezoelectricity under hydrostatic compression. Martin's conclusion must, therefore, be dependent on the details of his calculation. At this point it is also appropriate to mention a "review" paper by Aizu, 8 which contains no citations, but clearly emphasizes the role of surfaces and surface composition. Aizu distinguishes between "ordinary" boundaries and "ordinary" sampies, and others. Despite a strong similarity to Refs. 3-5, this author is not sure that he understands the terminology of Ref. 8 adequately, to be able to claim that it agrees with Refs. 3-5. The reader may find the discussion of Aizu's work, in a recent text, 9 more accessible than the original discussion. Fig. l a illustrates our basic point via a one dimensional chain of equally spaced positive and negative charges. This could, for example, be, interpreted to represent a succession of (1,1,1) planes in the NaCI structure. Alternatively, it can just be considered to be a one dimensional example. Clearly a large net static polarization, as shown in Fig. la, cannot exist in static equilibrium. The divergence of polarization must be compensated by charges attracted to the ends of the sample, or else by charges in electrodes deposited onto the ends of the crystal. In pyroelectricity and piezoelectricity, however, we are not concerned with a net static polarization, but only with small polarization changes. In a highly ionic crystal an unsymmetrical structure, as suggested in Fig. la, may not be easy to achieve, but does not seem impossible. Crystals
treatments o f piezoelectricity tacitly assume that the polarization produced by strain is purely a bulk effect whose symmetry properties are determined solely by the symmetry o f the environment o f the atoms in an infinite crystal. However, Woo and others have correctly pointed out that a complete description o f the polarization produced by a strain in a finite crystal must include possible effects o f the surfaces, i.e., effects o f the specific termination o f the infinite crystal to produce the given finite sample. Martin then goes on, however, to argue, in the specific case of 971
972
PYROELECTRICITY AND PIEZOELECTRICITY
-+-+-+-+ (a) F+
,.j_
"7"
,
-F-- T---r
t...___jL___.J.__
_j
+
(b) + 1--1 -- -- "1" j__ r" i ~+ 1 - - - - : ~ I.
. . .
.t
. . . .
t
_
+ "T'II__ +If| _
_.I.
_
_ _ j
(e) F I G U R E 1 Periodic chain of equidistant charges with a net dipole moment. In (b) the unit cell is chosen with the maximum potential symmetry, whereas in (c) the unit cell is chosen so that the entire crystal can be included in an integral number of cells. do not, generally, grow out from their center, and almost inevitably, are subject to some symmetry breaking process during their growth. One can also imagine, at least in principle, a crystal grown by deposition from the vapor, or by molecular beam epitaxy, onto an electrode made of the metallic constituent of the crystal. To avoid questions of this sort, about the actual realizability of a crystal with two different end surfaces, Refs. 4 and 5 concentrated on the zinc-blende structure. In that case a crystal, with the minimum number of broken bonds at its surfaces, has the required polarization. To emphasize the central conceptual point, however, and permit simple one-dimensional illustrations, we shall continue, here, with the model of Fig. la. The conventional view divides the array in Fig. la, as shown in Fig. lb, each cell having the maxium allowed symmetry. Now consider some modulation parameter, such as temperature or hydrostatic pressure, which controls the degree of ionicity of the crystal, and causes the effective charges in Fig. 1 to be changed. A simple inspection of Fig. la would lead one to expect a change in the total dipole moment. The conventional theory, based instead on Fig. 1b, will assume that the symmetry in each cell remains unbroken, as a result of the scalar modulation, and then goes on to ignore the left-over ends. Alternatively, we can, as advocated in Refs. 4,5, and 7, view the crystal as suggested in Fig. lc, i.e. as a sequence of polarized entities, in which the symmetry breaking introduced by the surface is respected. Calculations of piezoelectric coefficients, from first principles, typically concentrate on charge motion in the infinite, unlimited crystal, and then at best do handwaving to extrapolate to the case of the terminated crystal. We can, here, cite only a few such calculations. ~o The actual crystal, on which
11
Vol. 40, No.
measurements are done, is likely not only to be terminated, but to have electrodes. No calculation, known to me. even remotely does justice to this. The "handwaving" involved in Ref. 5, and implicitly invoked in our discussion of Fig. lc, assumes that the changes in effective charge, calculated for the infinite crystal, applies to each ion in Fig. 1, including the end ions. The "handwaving" invoked, instead, in Ref. 6, and consistent with our discussion of Fig. I b, is that the flux of charge, along each bond separating a + and - charge, is the same as in the infinite crystal. Thus the leftmost ion in Fig. 1 sees only half the change in effective charge of the other negative ions, in the interior of the sample. Neither of the two " h a n d w a v i n g " assumptions is likely to be satisfied exactly. Note, however, that the "handwaving" assumption associated with Fig. l b must be satisfied exactly to lead to an absence of a change in the total dipole moment; any compromise b et w een the two assumptions will lead to a change in dipole moment.t~ If, in Fig. lb, the interior cells remain unpolarized as the effective charge is modulated, and if we furthermore assume that the end charges are modulated, almost as effectively as those in the interior, then there must be a transfer of charge from the left hand end to the right hand end. (If, instead, the needed charge change, at each end, comes from a nearby electrode, then we cannot expect a measurable charge flow in the external circuit.) This has led to an assertion, in many oral debates of this viewpoint, and also in Ref. 6, that such a long range transport of charge is possible only in a conductor, and that therefore the discussion of Refs. 4, 5, and 7 does not apply to insulators. To put this question to rest, and to provide an additional simple example of the symmetry breaking role of the termination of a periodic array, we shall discuss modulation of a full band in the potential shown in Fig. 2.
V
A
B
A
B
A
B
A
B
A
F I G U R E 2 One-dimensional model of an ordered array of A and B atoms. Fig. 2 shows a periodic potential, terminated in such a way that the overall symmetry is broken by the "walls" at the end of the array. Each period of the potential has two wells, suggesting a sequence of alternating atoms. Let us assume a filled band, so that we are dealing with an insulator. In a given band we can expect a distribution between A and B wells; in general the electrons in a band will
Vol.
40, No.
II
PYROELECTRICITY
not occupy the two types of wells with equal probability. In the tight binding a p p r o x i m a t i o n we can, in fact, expect the electrons to be p r e d o m i n a n t l y in one of the two types of well, unless the A and B well have almost c o i n c i d e n t energy levels. N o w if we m o d u l a t e the potential and change it in some way, without affecting its symmetry, we can expect some electronic redistribution. If the m o d u l a t i o n is slow, then the original band states will be mapped adiabatically into the new band states. If the m o d u lation is applied quickly, we can expect transitions to higher lying bands, but eventually will return to the ground state in which the lowest lying bands are filled. Such a redistribution of charge, as a result o f a change in the lattice potential, has no relationship to the existence of a d c c o n d u c t i v i t y , and occurs just as much in a filled band, as it does in a conductor. In considering the redistribution of electronic charge, our s u b s e q u e n t arguments in this note will be based on the same " h a n d w a v i n g " a s s u m p t i o n made before. We will give intuitive a r g u m e n t s clearly applicable to an infinite crystal, or to one with periodic b o u n d a r y conditions, and will then assume that in the t e r m i n a t e d crystal the charges in each well of Fig. 2 will change similarly. Clearly there is room for a better calculation. As before, h o w e v e r , any change in net polarization, as the potential is modulated, will defeat the t e x t b o o k theorems, the change does not have to be exactly the one predicted from our " h a n d w a v i n g . " The redistribution of charge can be induced, for e x a m p l e , by d e e p e n i n g one of the two wells. This will not disrupt the existing local s y m m e t r y , and is therefore a " s c a l a r " peturbation, for our purposes, similar to a t e m p e r a t u r e variation. T o make the scalar nature of this modulation entirely clear, consider the case where only one of the two wells is attractive, and the other is repulsive. T h e n let the c o e f f i c i e n t multiplying this p o t e n t i a l be the c o n trolled scalar variable. If we change the sign of this scalar, we can certainly e x p e c t an electronic redistribution, In case the reader is dissatisfied with this example, we can proceed in the slightly more complex fashion described in the next paragraph. In fact, almost any r e a r r a n g e m e n t of the potential will induce some shift b e t w e e n A and B wells. The theory of one-dimensional band structures ~2 shows that for suitably adjusted p o t e n tials the forbidden gap b e t w e e n a d j a c e n t bands can be made to vanish. This basic fact may not be widely appreciated; for the two most widely investigated one d i m e n s i o n a l periodic p o t e n t i a l s , the K r o n i g - P e n n e y m o d e l and the M a t h i e u e q u a t i o n , such " a c c i d e n t a l " band edge crossings do not occur.
AND p IEZOELECTRICITX
973
B a n d crossings were discussed by S h o c k l e y 13 in c o n n e c t i o n with the existence of surface states, in o n e - d i m e n s i o n a l models of m o n a t o m i c lattices. If we c o n s i d e r modulating the relative shapes of the A and B wells, then at the point where the two wells b e c o m e equal the unit cell is doubled. At that point one half of the f o r b i d d e n bands, those resulting f r o m the " s u p e r l a t t i c e " structure, must disappear. More generally, w h e n e v e r we m o d u l a t e the potential of Fig. 2 in such a way that a level of the A well passes a level of the B well, we can e x p e c t that s o m e w h e r e in that n e i g h b o r h o o d there must be a vanishing gap. N e a r such a vanishing gap, i.e. w h e n the gap is present but small, the wave function in a band will have a particularly s t r o n g admixture of states from both wells. Let us assume we are in such a band, say one which is primarily an A band, but with a strong admixture of B. N o w assume the lattice is dilated, in such a way that the A and B wells keep their shape, but the t u n n e l i n g regions b e t w e e n the wells are lengthened. Thus the A and B bands b e c o m e more strongly decoupled, and in the case we have considered, can expect transfer of charge from the B wells to the A wells. The viewpoint a d v o c a t e d in Refs. 3-5 and 7, and r e p e a t e d here, has r e c e i v e d scant a t t e n t i o n . The author is not aware of any m o d e r n text or review paper that discusses it, with the solitary e x c e p tion of Ref. 9. Several points deserve to be made in that c o n n e c t i o n . First of all, it is c o n c e i v a b l e that the ordinary t e x t b o o k t h e o r e m s are right, or at least a p p r o x i m a t e l y correct, in a large p e r c e n t a g e of cases, as a result of microscopic surface considerations, not yet properly stated. I consider this alternative unlikely, but cannot rule it out. A more significant point: large p i e z o e l e c t r i c and p y r o e l e c t r i c c o e f f i cients, that are t e c h n o l o g i c a l l y useful, and easily m e a s u r e d , are u n d o u b t e d l y associated with a t o m i c d i s p l a c e m e n t s , and not just with the t r a n s f e r of electronic charges. Such polar m o t i o n c a n n o t occur, say in Fig. 1, without disrupting the existing local symmetry. Thus our v i e w p o i n t b e c o m e s irrelevant and the t e x t b o o k t h e o r e m s apply. Finally, if we want to be c o m p l e t e l y rigorous, we must admit that if the surfaces break the s y m m e t r y , as illustrated in Fig. la, then this s y m m e t r y breaking may have an effect, albeit i m m e a s u r a b l e , on the atomic positions. T h e n one can assert that the ordinary t h e o r e m s still apply, but that the atomic a r r a n g e m e n t in the interior of real crystals may have a s y m m e t r y which is lower than that ascertained in the prevailing measurements.
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REFERENCES 1.
2.
3.
4. 5. 6.
G. Burns, Introduction to G r o u p Theory with Applications (Academic, New York 1977) Chapt. IV; H. J. Juretschke, Crystal Physics (W. A. Benjamin, Reading 1974) Chapt. XII, but also see the reservations expressed briefly on pg. 49. The extent to which this is explicit, in elementary texts, varies. F o r a particularly careful discussion see R. W. P. King, Fundamental Electromagnetic Theory (Dover, New York 1963) Chapt. I J. Larmor, Proc. R. Soc. London 99, 1 (1921). R. Landauer, J. Chem. Phys. 32, 1784 (1960). J. W. F. Woo, Phys. Rev. B4, 1218 (1971). R. M. Martin, Phys. Rev. B6, 4874 (1972).
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J. W. F. Woo and R. Landauer, Phys. Rev. B6, 4876 (1972). K. Aizu, Rev. Mod. Phys. 34, 550 (1962). J. C. Burfoot and G. W. Taylor, Polar Dielectrics and Their Applications (U. Calif. Press, Berkeley 1979) Section 8.2. G. Arlt and P. Quadflieg, Phys. Status Solidi 25, 323 (1968); R. M. Martin, Phys. Rev. BS, 1607 (1972); W. A. Harrison, Phys. Rev. B I 0 , 767 (1974); T. Hidaka, Phys. Rev. BIS, 4426 (1978). See footnote 3, Ref. 4., for more precise language. G. Allen, Phys. Rev. 91, 531 (1953); B. F. Buxton and M. V. Berry, Philos. Trans. R. Soc. London 2 8 2 , 4 8 5 (1976). W. Shockley, Phys. Rev. 5 6 , 3 1 7 (1939).