- Email: [email protected]
Note on polar scattering of conduction electrons in regular crystals
Meijer, Polder,
H. J..G. D.
Physica
XIX
255-264
19.53
NOTE
ON POLAR ELECTRONS
SCATTERING IN REGULAR
OF CONDUCTION CRYSTALS
by H. J. G. ME1 JER and D. POLDER Philips
Research
Laboratories,
W.V.Philips’
Gloeilanlpenfabricken,
*) Eindhovcn,
~edrrlnd
Synopsis The reciprocal relaxation time of conduction electrons due to polar interaction with piezoelectrically active acoustical vibration modes is calculated for two-atomic regular crystals. It is found to be proportional to T’k It is shown for the case of ‘ZnS (sphalerite) that below 150°K the numerical value of this reciprocal relaxation time is comparable with the reciprocal relaxation time due to polar interaction with optical vibration modes. In crystals with sodium chloride structure the mass difference of the ions can in principle be responsible for polar scattering by acoustical modes. The corresponding reciprocal relaxation time is proportional to T3/2. In the special case of LiF it becomes equal to polar scattering by optical modes at about 200°K.
1. Ilztroduction. A theory of the thermal scattering of conduction electrons in thermal equilibrium with a polar lattice with sodium chloride structure was given by F r 6 h 1 i c h and M o t t ‘) who based their considerations on an earlier article by F r ij h 1 i c h “). In such a structure in general only the optical vibration modes are effective in scattering electrons by polar interaction. It is the purpose of this note to show that in crystals with lower symmetry than the alkali halides, as e.g. ZnS, there is another source of polar scattering, namely the piezoelectrically active am&id modes. It is seen that in the case of ZnS (sphalerite) they become relatively important below 150°K. It will be shown also that even in crystals with sodium chloride structure the acoustical vibration modes can in principle be active in polar scattering because of the mass difference of the ions. In the special case of LiF this effect becomes equal to polar scattering by optical modes at about 200°K. *) H. H. Wills’
Physical
Laboratories,
-
University
255 -
of Bristol,
Bristol,
England.
256
H.
J. G. MEIJER
AND
D.
POLDER
For crystals of both structures there will also be non-polar scattering by acoustical vibration modes. Unfortunately only very rough estimates can be made of its importance by assuming that it is of the same order of magnitude as in non polar insulating crystal9). Thus for the case of ZnS at about 100°K it is presumably of the same order of magnitude as the effect treated here but, having a different dependence on temperature it becomes smaller at very low temperatures. For the case of LiF the non-polar scattering is presumably always much larger. The theories of Frohlich and of Frohlich and Mott have been partly extended and partly reformulated by C a 11 e n “). We shall take some of his results as the starting point for our investigations. Thus in the second paragraph Callens expression for the transition probability of an electron due to polar interaction with optical modes of vibration is rewritten in a slightly generalized form so as to cover also the case of polar interaction with acoustical modes of vibration. In the third paragraph the relaxation time following from this transition probability is written down. It is shown that in the case of optical modes of vibrations it reduces to an expression nearly equal to that of Frijhlich and Mott. The relaxation time for acoustical modes can be calculated if the polarization in the direction of propagation of an acoustical wave is known. This is calculated in the next paragraph. In this calculation the influence of the electrostatic energy due to the non-vanishing polarisation is taken into account. A final formula for the reciprocal relaxation time due to acoustical modes in a regular piezoelectric crystal is derived. Its numerical value as a function of temperature is given in a table and compared with the same quantity due to optical modes for the special case of ZnS (sphalerite). In the last paragraph a formula is given for the reciprocal relaxation time due to acoustical modes for the case of a crystal with sodium chloride structure. Its numerical value is compared with that of the “optical” relaxation time for the special case of LiF. 2. Transition probability due to polar interaction. As is well known the general motion of the ions of a two-atomic crystal can be described by a superposition of plane waves (wavelength A) with wave vector a(lol = l/A). The N possible independent values of Q lie in the first Brillouin zone, N being equal to the number of unit cells
XOTE
ON
POLAR
SCATTERING
OF
CONDUCTION
ELECTRONS
257
within the elementary cube, used in formulating the Born von Karman periodic boundary conditions. To any CJthere belong six independent modes. For waves of long wavelength each is characterized by its direction of polarization (one longitudinal and two transverse directions) and by the sign of the ratio of the deviations from equilibrium of the two sorts of ions. This is positive for an “acoustical” vibration and negative for an “optical” one. We thus can write for the deviations from equilibrium of the two sorts of ions respectively : Ul w = %i Ul,u,i (4 (1)
u2w
=
%J,i
*2,e,r
w
The indes ,i can take six different values indicating resp. longitudinal acoustical, transverse acoustical etc. It is convenient to choose for the normal modes u,,=,* (r) and u~,~,~ (r) the following expressions :
u,,u.i(r) = ( ~)l’sbu,i{R}
WI,U,~
e2rriur) (2)
u2,0,j
(4
=
( $)il’bn,i{
L}
t”2,e,i
e2niar)
are vectors, ‘with complex component, proportional to the amplitude of the deviations of the two sorts of ions. They are supposed to be normalized according to Here
ul,ts,i
and
u2,a,i
I”l,fJ,i12
nrl
+
I”2,tJ,i12
“‘2
=
l
(3)
where N, and ?jz2are the masses of the two sorts of ions. The symbol I means, that if the imaginary part is adopted for the mode Q, R I I the real part should be adopted for the mode - Q. The b,,; are normal coordinates. If Ul,,,i and U2,0,i are unequal there is a nonvanishing relative displacement ~,,~,~(r) -u2,a,i(r) of the two sorts of ions due to the mode Q, i, causing a periodically varying polarization of the crystal (@Zarization wave *) po,i
An expression -__.___ *) We polarization Physica
thus
=
(F)“‘&
for the probability
{i>
(4)
ehiur)
cD,,~ that an electron
do ml limit the USC of the term polarization caused by an oplicnl mode of vibration.
SIS
(P,,;
wave
to a periodically
with wave wr!iw 17
258
H.
J. G. MEIJER
AND
D.
POLDER
number K undergoes a transition into a state K’ due to the interaction with a polarization wave Q, i is easily obtained, e.g. by a slight generalization of Callen’s derivation of his formula (44) of reference 4). This is a calculation based on ordinary time dependent perturbation theory using the electrostatic potential due to the polarization wave as the perturbation. We thus find: cD,,~ = (e’/c? NTz~~,~) Is * PJ2 iln,,i 6(n&
n,i-1)
+ f-Q;+
6(K-
K’f
1) 6 ($,i, ng,i + 111 * (a/V
IJ, 0) (sin2 Et/E’).
(5)
where e is the electronic charge, s the unit vector in the direction of a, cob i EF 2xvo ,i the angular frequency of the mode Q, i and Q the average number of the corresponding vibrational quanta; %o,i= [exp fhw,ifkT)
--1-j-’
; 2tzt=
(h”/2WZ*) (K2-K2)+
A mm,<
where m* is the effective mass of a conduction electro-n.The probability that an electron with wave number K absorbs a quantum Q, i (@z,J or emits such a quantum (@bi) is given by the terms of (5) proportional to n,, i and (n,j + 1) respectively. Our formula (5) goes over into Callen’s expression (441, which is valid only for longitudinal optical modes of long wavelength, if ( s * P,,;I is changed into e*/2a31/M, where e* is an “effective charge” (see e.g. ref. 3), M is the reduced mass of the ions and 2a3 E A is the volume of the smallest unit cell of the crystal. 3. General expression for the relaxation time. The reciprocal relaxation time is given by 5) 1/q = c, (-
a/K) cos a @t,i + C, (-
a/K) cos a @‘“,,i
(6)
where a is the angle between K and Q and the sums are taken over all modes of sort i appropriate with the process of absorption and emission respectively. If @“,,i and @pd,;depend only on the absolute value of cr and not on its direction, which condition is fulfilled in the case of optical modes of long wavelength, the above sums, after having been transformed into integrals can easily be evaluated according to standard methods of the theory conduction. In the case of acoustical modes (even those of infinite wavelength) however, @Gand @i do depend on the direction of Q through the quantities js * P,j and wQ. In order to avoid lengthy calculations we now replace the factors in !D:,r, which depend on the direction of Q by
NOTE
ON POLAR
SCATTERING
OF CONDUCTION
259
ELECTRONS
their average values over all directions of Q *). We indicate a quantity averaged in this way by a bar. The integration over a then yields the expression :
Here ufr,mm,,naxand ~9r,mtn,,nnxare the integration limits for the modes of sort i appropriate to the absorption and the emission process respectively. In the case of qbtical longitudinal vibration modes the following approximations are usually made: IsP,l = e*/AdM and 232 Y,,~= =u+,=const. for all CL If the temperature T
+ K2 + K and
- K = .\/(2m*,,/2~+ so that we find for the relaxation time due to the optical u&nin
1/to = (8n(e*e)22/m*/3
d/2 ‘A((tzc0~)~‘2M) e?““:kT
modes: (8)
This expression becomes equal to that derived by Frohlich and Mott if e* is taken equal to the “normal” charge of the ions and m* is taken equal to the mass of a free electron. In the case of acoustical vibration modes the limites of integration are given by: 4,ttitr
= 0, e,J,,a, = 2K
(9)
This means that for not too high temperatures in the case of nondegenerate thermal electrons (which is treated here) only modes of long wavelength need to be considered. As the behaviour of a piezoelectric crystal as regards acoustical modes of long wavelength can completely be described by macroscopic phenomenological equations we can avoid the use of any microscopic picture. This is satisfying because as yet no microscopic theory of piezoelectric behaviour does exist, which is satisfactory from a quantitative point of view. *) This is B somewhat arbitrary procedure; affect the final conclusions in B qualitative
it is believed, way.
however,
that
it does not
260
H. J. G. MEIJER
AND
D. POLDER
4. Calculation of s * P and cod for acoustical modes with simple directions of propagation. The equations of motion for free acoustical
vibrations
of an e&tic pii,
medium
with density g are :
= &,/ax + &rypy + ax.r,,la.v (x = x, y, z)
(10)
Here ii, denotes the second time derivative of the displacement component u, and the xX,, denote the components of the stress tensor (x, y, z are assumed to be along the crystal axis). These stresses can be derived from the phenomenological expression for the internal energy density T/lr expressed in terms of the components of the deformation tensor uX,, z au,/ay and the components of the electric polarization p. Under the condition that within the dielectric no macroscopic electric field E is present we get as phenomenological expression for the energy density in the case of a regular piezoelectric crystal : Jko
= $(4nl(&-
1)) PZ--e,,PW-
+ cl2 zlgy c + 4 [c44 + 4n/(&
P&L,,,.. + u,J + Qcll & + - 1)) &I (uys+ ~4~)~ + (~1.) (11) 1))
where E is the dielectric constant at constant strain, e,4 is the piezoelectric constant (px/(uYZ + uZr) at constant E) -and the cik are the elastic constants at constant field strength E 6). In our problem the condition E = 0 does not hold so that we have to add to W,=, an expression taking into account the electrostatic energy density due to the presence of a polarisation wave p of the form given by (4) *). It is easily seen that for a wave of long wavelength this energy is given by 2n(p.=s, + p,,s,, + pZs,J2. We thus get for the total energy density : w = 274PA + Pysy+ PA) + JVE=, (14 If we neglect inertia effects in the polarization and keep in mind that there is no electric field due to external sources, we may write awjap,
= 0
(x = x, y, 2)
(13) With (13) we can eliminate p.r, p,,, p, from (12). Calling the resulting expression (which does not contain the components of p anymore) *) We thus assume from the beginning deformation waves which by piezoelectric form (4).
that a solution of (10) cm be built up from coupling induce polarization waves of the
NOTE
ON POLAR
SCATTERING
W’ we find the components
OF CONDUCTION
ELECTRONS
of the stress tensor according to : ;r,, = a w’lau.,,
(14
%y = awyau.ry Making placement
use of equations u is of the form:
261
(IO)-(
14) and assuming
that the dis-
(15) where U is assumed to be normalized according to IU12 (m, + m2) = 1 the equations of motion become, after some straightforward algebra ec* + ~1, s; + c,,b,” + $1 + (16~ ef&) (~~4~1 + + u,ic,, + c44 + (16ne?,/4 sf) v,+ WC,, + c44+ + (16~ ey4/e) .s$ s,sZ = 0
v,[-
Here c is the velocity From
Qu = 2rKrC(D) (13), (15) and (4) it follows that P . s) is given by (P - s) =
(16)
of sound, defined by
(47q4ld
vJzsysz + q&s,
+ U,s,s,J
(17) (18)
From (16) the components of U and the velocity of sound c can be found easily for some simple directions of propagation. The results for s = (1, 0, 0); s = (l/d/2) (1, 1, 0) and s = (l/d/3) (1, 1, 1) are given in table I. TABLE 1B.s and Direction of Propagation
Longitudinal
1, 1, 1
some simple
directions )
of propagation ‘rransverse
(I)
Transverse
(2)
ect
1, 0, 0 1, 1, 0
c* for
I
Cl1
Hc**
+ C,? +
2c,,) r,,--c,,+c,, 3
5. Final formula for the acoalstical relaxation time in a piezoelectric regular crystal. As is well known, in our case of non degenerate thermal electrons, the acoustical modes are such that lio/kT
0
262
H.
J. G. MEIJER
AND
D.
POLDER
T > 1“K. Thus n,/~~ M kt/hac”. Furthermore the second terms square brackets in (7) can be neglected so that we get:
in
(19) where j indicates the polarization of the considered acoustical mode (longitudinal or transverse) and where P’ = P/o. In order to simplify the computations we take for the term in square brackets the weighted average over the directions of propagation given in table I, rather than the average over all directions. Because of 1/t,, =Sj .:, 1/T,,~ we finally get : 1 -=-------32n3 e2 d/m* eT4d/k 1/T h2E2 112 tll
16 __~_~. - - ~~-.-____~ 13(c,,+2c,,+4c4,+
+ 16=:,/e)
1
(20)
where x is written for E/kT = (h2/2m*) K21kT. If numerical values are inserted in this formula case of ZnS (sphalerite)
we find for the
+ ----6---13(c,, + 4ne:,l4
l/t. The evaluation
= 5.1 x 10” 1/T/dx
(21)
of (8) for the same substance gives: I/~, = 5.2 . 1013e--SWr
(22)
In table II the two reciprocal relaxation times (21) and (22) are given for an electron with E = 3kT/2 as a function of temperature. It is seen that l/r, becomes of the same order of magnitude as 1/to below 150°K and becomes quickly relatively more and more important at still lower temperatures. TABLE l/t, T in
and
I/r0
as a function
“K in
250 200 150 125 100 75
1I% 10” set-* 4.6 5.9 5.1 4.5 4.2 3.6
II of ternperaturc in
1Ito 10” set-’ 60 53 14 7.0 2.4 0.39
-
NOTE
ON
POLAR
SCATTERING
OF CONDUCTION -~__
ELECTRONS
263
6. Polar scattering by acoustical modes in LiF. In a crystal with sodium chloride structure no piezoelectric coupling does exist. A polarisation wave can be induced by an elastic wave however as a consequence of the mass difference of the ions. As this is an effect which goes beyond a description in which the solid can be considered as a continuum a microscope picture has to be used. As was indicated by B o r n ‘), the microscopic equations of motion for the ions of an ionic crystal, assuming central forces between the ions, can be solved in successive approximations by a development of the amplitudes of the displacement waves and of the square of the frequency in powers of u. The equations from which the term proportional to cr is obtained were shown by Born to be identical with the equations of motion of an elastic continuum. The equations for the second term can easily be obtained in the way indicated in the above reference. We have solved these equations for the three simple directions of propagation used in the last section for the case of a crystal with sodium chloride structure *). In this case only longitudinal acoustical modes gives rise to a non-vanishing (P * s). The results are given in table III.
c* and I’. s for a crystal Direction propagation
of
TABLE
III
with
sodium
p ?
1, 0, 0
1’. s for L’*
Cl1
1, 0
Hc,,
+
3C1?)
.llw,=
structure longitudinal ‘“, -
Cl1
.Ilo,? e*
1,
chloride
2 c,,
+
(m, 3c,c
4
+
modes nr?
m,)?G
MI, tn2 ~__ (m, + 1n*)3,'2
D~ 4n-u-
4n'n'
By proceeding in a completely analogous way as in paragraph 5 we have for the case of a crystal with sodium chloride structure using (8)
*) In this calculation the influence presence of a non vanishing polarization that followed in paragraph 4.
of the electrostatic energy density due to the was taken into account in a way analogous to
264
NOTE
ON
POLAR
SCATTERING
OF CONDUCTION
ELECTRONS
The numerical value of this expression is exceptionally large in the case of LiF because of the extremely large value of woo.For this special case we find for an electron with an energy equal to 3kT/2, taking m* = m: 1 _1 _ 4 3 x 10-a 7-2/2 e1WT (24
< i -Go
The ratio (24) is equal to 1 at about 200°K. Thanks are due to Prof. H. B. G. C a s i m i r for helpful criticism and to DrsF. A. Kroger, J. H. van Santen and H. J. V i n k for stimulating discussions. Received
12- 1 I-52.
REFERE.XCliS 1) Frii h 1 i c h, H. and 11 o t t, S. F., Proc. ray. Sot. .-I 171 (1939) F r ij h 1 i c h, H., Proc. ray. Sm. A Ifi0 (1937) 230. S e i t z, F., Phys. Rev. 73 (1948) 549. C a 11 e 11, H. B., Phys. Rev. 70 (1949) 1394. ref. 4) form. (70). 6) cf. 11 as o 11, W. P., Piezoelcctric crystals and their application D. v. Nostrand, New-York, London, 1950, p. 452. 7) Born, aI., %. Phys. 8 (1922) 390.
496.
2) 3) 4) 5)
to ultrasonirs,
H. J..G. D.
Physica
XIX
255-264
19.53
NOTE
ON POLAR ELECTRONS
SCATTERING IN REGULAR
OF CONDUCTION CRYSTALS
by H. J. G. ME1 JER and D. POLDER Philips
Research
Laboratories,
W.V.Philips’
Gloeilanlpenfabricken,
*) Eindhovcn,
~edrrlnd
Synopsis The reciprocal relaxation time of conduction electrons due to polar interaction with piezoelectrically active acoustical vibration modes is calculated for two-atomic regular crystals. It is found to be proportional to T’k It is shown for the case of ‘ZnS (sphalerite) that below 150°K the numerical value of this reciprocal relaxation time is comparable with the reciprocal relaxation time due to polar interaction with optical vibration modes. In crystals with sodium chloride structure the mass difference of the ions can in principle be responsible for polar scattering by acoustical modes. The corresponding reciprocal relaxation time is proportional to T3/2. In the special case of LiF it becomes equal to polar scattering by optical modes at about 200°K.
1. Ilztroduction. A theory of the thermal scattering of conduction electrons in thermal equilibrium with a polar lattice with sodium chloride structure was given by F r 6 h 1 i c h and M o t t ‘) who based their considerations on an earlier article by F r ij h 1 i c h “). In such a structure in general only the optical vibration modes are effective in scattering electrons by polar interaction. It is the purpose of this note to show that in crystals with lower symmetry than the alkali halides, as e.g. ZnS, there is another source of polar scattering, namely the piezoelectrically active am&id modes. It is seen that in the case of ZnS (sphalerite) they become relatively important below 150°K. It will be shown also that even in crystals with sodium chloride structure the acoustical vibration modes can in principle be active in polar scattering because of the mass difference of the ions. In the special case of LiF this effect becomes equal to polar scattering by optical modes at about 200°K. *) H. H. Wills’
Physical
Laboratories,
-
University
255 -
of Bristol,
Bristol,
England.
256
H.
J. G. MEIJER
AND
D.
POLDER
For crystals of both structures there will also be non-polar scattering by acoustical vibration modes. Unfortunately only very rough estimates can be made of its importance by assuming that it is of the same order of magnitude as in non polar insulating crystal9). Thus for the case of ZnS at about 100°K it is presumably of the same order of magnitude as the effect treated here but, having a different dependence on temperature it becomes smaller at very low temperatures. For the case of LiF the non-polar scattering is presumably always much larger. The theories of Frohlich and of Frohlich and Mott have been partly extended and partly reformulated by C a 11 e n “). We shall take some of his results as the starting point for our investigations. Thus in the second paragraph Callens expression for the transition probability of an electron due to polar interaction with optical modes of vibration is rewritten in a slightly generalized form so as to cover also the case of polar interaction with acoustical modes of vibration. In the third paragraph the relaxation time following from this transition probability is written down. It is shown that in the case of optical modes of vibrations it reduces to an expression nearly equal to that of Frijhlich and Mott. The relaxation time for acoustical modes can be calculated if the polarization in the direction of propagation of an acoustical wave is known. This is calculated in the next paragraph. In this calculation the influence of the electrostatic energy due to the non-vanishing polarisation is taken into account. A final formula for the reciprocal relaxation time due to acoustical modes in a regular piezoelectric crystal is derived. Its numerical value as a function of temperature is given in a table and compared with the same quantity due to optical modes for the special case of ZnS (sphalerite). In the last paragraph a formula is given for the reciprocal relaxation time due to acoustical modes for the case of a crystal with sodium chloride structure. Its numerical value is compared with that of the “optical” relaxation time for the special case of LiF. 2. Transition probability due to polar interaction. As is well known the general motion of the ions of a two-atomic crystal can be described by a superposition of plane waves (wavelength A) with wave vector a(lol = l/A). The N possible independent values of Q lie in the first Brillouin zone, N being equal to the number of unit cells
XOTE
ON
POLAR
SCATTERING
OF
CONDUCTION
ELECTRONS
257
within the elementary cube, used in formulating the Born von Karman periodic boundary conditions. To any CJthere belong six independent modes. For waves of long wavelength each is characterized by its direction of polarization (one longitudinal and two transverse directions) and by the sign of the ratio of the deviations from equilibrium of the two sorts of ions. This is positive for an “acoustical” vibration and negative for an “optical” one. We thus can write for the deviations from equilibrium of the two sorts of ions respectively : Ul w = %i Ul,u,i (4 (1)
u2w
=
%J,i
*2,e,r
w
The indes ,i can take six different values indicating resp. longitudinal acoustical, transverse acoustical etc. It is convenient to choose for the normal modes u,,=,* (r) and u~,~,~ (r) the following expressions :
u,,u.i(r) = ( ~)l’sbu,i{R}
WI,U,~
e2rriur) (2)
u2,0,j
(4
=
( $)il’bn,i{
L}
t”2,e,i
e2niar)
are vectors, ‘with complex component, proportional to the amplitude of the deviations of the two sorts of ions. They are supposed to be normalized according to Here
ul,ts,i
and
u2,a,i
I”l,fJ,i12
nrl
+
I”2,tJ,i12
“‘2
=
l
(3)
where N, and ?jz2are the masses of the two sorts of ions. The symbol I means, that if the imaginary part is adopted for the mode Q, R I I the real part should be adopted for the mode - Q. The b,,; are normal coordinates. If Ul,,,i and U2,0,i are unequal there is a nonvanishing relative displacement ~,,~,~(r) -u2,a,i(r) of the two sorts of ions due to the mode Q, i, causing a periodically varying polarization of the crystal (@Zarization wave *) po,i
An expression -__.___ *) We polarization Physica
thus
=
(F)“‘&
for the probability
{i>
(4)
ehiur)
cD,,~ that an electron
do ml limit the USC of the term polarization caused by an oplicnl mode of vibration.
SIS
(P,,;
wave
to a periodically
with wave wr!iw 17
258
H.
J. G. MEIJER
AND
D.
POLDER
number K undergoes a transition into a state K’ due to the interaction with a polarization wave Q, i is easily obtained, e.g. by a slight generalization of Callen’s derivation of his formula (44) of reference 4). This is a calculation based on ordinary time dependent perturbation theory using the electrostatic potential due to the polarization wave as the perturbation. We thus find: cD,,~ = (e’/c? NTz~~,~) Is * PJ2 iln,,i 6(n&
n,i-1)
+ f-Q;+
6(K-
K’f
1) 6 ($,i, ng,i + 111 * (a/V
IJ, 0) (sin2 Et/E’).
(5)
where e is the electronic charge, s the unit vector in the direction of a, cob i EF 2xvo ,i the angular frequency of the mode Q, i and Q the average number of the corresponding vibrational quanta; %o,i= [exp fhw,ifkT)
--1-j-’
; 2tzt=
(h”/2WZ*) (K2-K2)+
A mm,<
where m* is the effective mass of a conduction electro-n.The probability that an electron with wave number K absorbs a quantum Q, i (@z,J or emits such a quantum (@bi) is given by the terms of (5) proportional to n,, i and (n,j + 1) respectively. Our formula (5) goes over into Callen’s expression (441, which is valid only for longitudinal optical modes of long wavelength, if ( s * P,,;I is changed into e*/2a31/M, where e* is an “effective charge” (see e.g. ref. 3), M is the reduced mass of the ions and 2a3 E A is the volume of the smallest unit cell of the crystal. 3. General expression for the relaxation time. The reciprocal relaxation time is given by 5) 1/q = c, (-
a/K) cos a @t,i + C, (-
a/K) cos a @‘“,,i
(6)
where a is the angle between K and Q and the sums are taken over all modes of sort i appropriate with the process of absorption and emission respectively. If @“,,i and @pd,;depend only on the absolute value of cr and not on its direction, which condition is fulfilled in the case of optical modes of long wavelength, the above sums, after having been transformed into integrals can easily be evaluated according to standard methods of the theory conduction. In the case of acoustical modes (even those of infinite wavelength) however, @Gand @i do depend on the direction of Q through the quantities js * P,j and wQ. In order to avoid lengthy calculations we now replace the factors in !D:,r, which depend on the direction of Q by
NOTE
ON POLAR
SCATTERING
OF CONDUCTION
259
ELECTRONS
their average values over all directions of Q *). We indicate a quantity averaged in this way by a bar. The integration over a then yields the expression :
Here ufr,mm,,naxand ~9r,mtn,,nnxare the integration limits for the modes of sort i appropriate to the absorption and the emission process respectively. In the case of qbtical longitudinal vibration modes the following approximations are usually made: IsP,l = e*/AdM and 232 Y,,~= =u+,=const. for all CL If the temperature T
+ K2 + K and
- K = .\/(2m*,,/2~+ so that we find for the relaxation time due to the optical u&nin
1/to = (8n(e*e)22/m*/3
d/2 ‘A((tzc0~)~‘2M) e?““:kT
modes: (8)
This expression becomes equal to that derived by Frohlich and Mott if e* is taken equal to the “normal” charge of the ions and m* is taken equal to the mass of a free electron. In the case of acoustical vibration modes the limites of integration are given by: 4,ttitr
= 0, e,J,,a, = 2K
(9)
This means that for not too high temperatures in the case of nondegenerate thermal electrons (which is treated here) only modes of long wavelength need to be considered. As the behaviour of a piezoelectric crystal as regards acoustical modes of long wavelength can completely be described by macroscopic phenomenological equations we can avoid the use of any microscopic picture. This is satisfying because as yet no microscopic theory of piezoelectric behaviour does exist, which is satisfactory from a quantitative point of view. *) This is B somewhat arbitrary procedure; affect the final conclusions in B qualitative
it is believed, way.
however,
that
it does not
260
H. J. G. MEIJER
AND
D. POLDER
4. Calculation of s * P and cod for acoustical modes with simple directions of propagation. The equations of motion for free acoustical
vibrations
of an e&tic pii,
medium
with density g are :
= &,/ax + &rypy + ax.r,,la.v (x = x, y, z)
(10)
Here ii, denotes the second time derivative of the displacement component u, and the xX,, denote the components of the stress tensor (x, y, z are assumed to be along the crystal axis). These stresses can be derived from the phenomenological expression for the internal energy density T/lr expressed in terms of the components of the deformation tensor uX,, z au,/ay and the components of the electric polarization p. Under the condition that within the dielectric no macroscopic electric field E is present we get as phenomenological expression for the energy density in the case of a regular piezoelectric crystal : Jko
= $(4nl(&-
1)) PZ--e,,PW-
+ cl2 zlgy c + 4 [c44 + 4n/(&
P&L,,,.. + u,J + Qcll & + - 1)) &I (uys+ ~4~)~ + (~1.) (11) 1))
where E is the dielectric constant at constant strain, e,4 is the piezoelectric constant (px/(uYZ + uZr) at constant E) -and the cik are the elastic constants at constant field strength E 6). In our problem the condition E = 0 does not hold so that we have to add to W,=, an expression taking into account the electrostatic energy density due to the presence of a polarisation wave p of the form given by (4) *). It is easily seen that for a wave of long wavelength this energy is given by 2n(p.=s, + p,,s,, + pZs,J2. We thus get for the total energy density : w = 274PA + Pysy+ PA) + JVE=, (14 If we neglect inertia effects in the polarization and keep in mind that there is no electric field due to external sources, we may write awjap,
= 0
(x = x, y, 2)
(13) With (13) we can eliminate p.r, p,,, p, from (12). Calling the resulting expression (which does not contain the components of p anymore) *) We thus assume from the beginning deformation waves which by piezoelectric form (4).
that a solution of (10) cm be built up from coupling induce polarization waves of the
NOTE
ON POLAR
SCATTERING
W’ we find the components
OF CONDUCTION
ELECTRONS
of the stress tensor according to : ;r,, = a w’lau.,,
(14
%y = awyau.ry Making placement
use of equations u is of the form:
261
(IO)-(
14) and assuming
that the dis-
(15) where U is assumed to be normalized according to IU12 (m, + m2) = 1 the equations of motion become, after some straightforward algebra ec* + ~1, s; + c,,b,” + $1 + (16~ ef&) (~~4~1 + + u,ic,, + c44 + (16ne?,/4 sf) v,+ WC,, + c44+ + (16~ ey4/e) .s$ s,sZ = 0
v,[-
Here c is the velocity From
Qu = 2rKrC(D) (13), (15) and (4) it follows that P . s) is given by (P - s) =
(16)
of sound, defined by
(47q4ld
vJzsysz + q&s,
+ U,s,s,J
(17) (18)
From (16) the components of U and the velocity of sound c can be found easily for some simple directions of propagation. The results for s = (1, 0, 0); s = (l/d/2) (1, 1, 0) and s = (l/d/3) (1, 1, 1) are given in table I. TABLE 1B.s and Direction of Propagation
Longitudinal
1, 1, 1
some simple
directions )
of propagation ‘rransverse
(I)
Transverse
(2)
ect
1, 0, 0 1, 1, 0
c* for
I
Cl1
Hc**
+ C,? +
2c,,) r,,--c,,+c,, 3
5. Final formula for the acoalstical relaxation time in a piezoelectric regular crystal. As is well known, in our case of non degenerate thermal electrons, the acoustical modes are such that lio/kT
0
262
H.
J. G. MEIJER
AND
D.
POLDER
T > 1“K. Thus n,/~~ M kt/hac”. Furthermore the second terms square brackets in (7) can be neglected so that we get:
in
(19) where j indicates the polarization of the considered acoustical mode (longitudinal or transverse) and where P’ = P/o. In order to simplify the computations we take for the term in square brackets the weighted average over the directions of propagation given in table I, rather than the average over all directions. Because of 1/t,, =Sj .:, 1/T,,~ we finally get : 1 -=-------32n3 e2 d/m* eT4d/k 1/T h2E2 112 tll
16 __~_~. - - ~~-.-____~ 13(c,,+2c,,+4c4,+
+ 16=:,/e)
1
(20)
where x is written for E/kT = (h2/2m*) K21kT. If numerical values are inserted in this formula case of ZnS (sphalerite)
we find for the
+ ----6---13(c,, + 4ne:,l4
l/t. The evaluation
= 5.1 x 10” 1/T/dx
(21)
of (8) for the same substance gives: I/~, = 5.2 . 1013e--SWr
(22)
In table II the two reciprocal relaxation times (21) and (22) are given for an electron with E = 3kT/2 as a function of temperature. It is seen that l/r, becomes of the same order of magnitude as 1/to below 150°K and becomes quickly relatively more and more important at still lower temperatures. TABLE l/t, T in
and
I/r0
as a function
“K in
250 200 150 125 100 75
1I% 10” set-* 4.6 5.9 5.1 4.5 4.2 3.6
II of ternperaturc in
1Ito 10” set-’ 60 53 14 7.0 2.4 0.39
-
NOTE
ON
POLAR
SCATTERING
OF CONDUCTION -~__
ELECTRONS
263
6. Polar scattering by acoustical modes in LiF. In a crystal with sodium chloride structure no piezoelectric coupling does exist. A polarisation wave can be induced by an elastic wave however as a consequence of the mass difference of the ions. As this is an effect which goes beyond a description in which the solid can be considered as a continuum a microscope picture has to be used. As was indicated by B o r n ‘), the microscopic equations of motion for the ions of an ionic crystal, assuming central forces between the ions, can be solved in successive approximations by a development of the amplitudes of the displacement waves and of the square of the frequency in powers of u. The equations from which the term proportional to cr is obtained were shown by Born to be identical with the equations of motion of an elastic continuum. The equations for the second term can easily be obtained in the way indicated in the above reference. We have solved these equations for the three simple directions of propagation used in the last section for the case of a crystal with sodium chloride structure *). In this case only longitudinal acoustical modes gives rise to a non-vanishing (P * s). The results are given in table III.
c* and I’. s for a crystal Direction propagation
of
TABLE
III
with
sodium
p ?
1, 0, 0
1’. s for L’*
Cl1
1, 0
Hc,,
+
3C1?)
.llw,=
structure longitudinal ‘“, -
Cl1
.Ilo,? e*
1,
chloride
2 c,,
+
(m, 3c,c
4
+
modes nr?
m,)?G
MI, tn2 ~__ (m, + 1n*)3,'2
D~ 4n-u-
4n'n'
By proceeding in a completely analogous way as in paragraph 5 we have for the case of a crystal with sodium chloride structure using (8)
*) In this calculation the influence presence of a non vanishing polarization that followed in paragraph 4.
of the electrostatic energy density due to the was taken into account in a way analogous to
264
NOTE
ON
POLAR
SCATTERING
OF CONDUCTION
ELECTRONS
The numerical value of this expression is exceptionally large in the case of LiF because of the extremely large value of woo.For this special case we find for an electron with an energy equal to 3kT/2, taking m* = m: 1 _1 _ 4 3 x 10-a 7-2/2 e1WT (24
< i -Go
The ratio (24) is equal to 1 at about 200°K. Thanks are due to Prof. H. B. G. C a s i m i r for helpful criticism and to DrsF. A. Kroger, J. H. van Santen and H. J. V i n k for stimulating discussions. Received
12- 1 I-52.
REFERE.XCliS 1) Frii h 1 i c h, H. and 11 o t t, S. F., Proc. ray. Sot. .-I 171 (1939) F r ij h 1 i c h, H., Proc. ray. Sm. A Ifi0 (1937) 230. S e i t z, F., Phys. Rev. 73 (1948) 549. C a 11 e 11, H. B., Phys. Rev. 70 (1949) 1394. ref. 4) form. (70). 6) cf. 11 as o 11, W. P., Piezoelcctric crystals and their application D. v. Nostrand, New-York, London, 1950, p. 452. 7) Born, aI., %. Phys. 8 (1922) 390.
496.
2) 3) 4) 5)
to ultrasonirs,