Thin Solid Films, 161 (1988) 111 ELECTRONICS AND OPTICS
THE SCATTERING OF ELECTRONS ON GRAIN BOUNDARIES: ELECTRICAL CONDUCTIVITY F. WARKUSZ Institute of Physics, Technical University of Wroctaw, Wybrze~e Wyspiahskiego 27, 50370 W¢octaw (Poland) (Received November I0, 1986; revised October 8, 1987; accepted January 4, 1988)
The electrical conductivity of granular metal films was calculated, the boundaries of grains being treated as potential barriers. It was shown that the electrical conductivity decreases with the width and height of the potential barrier. For an electron energy higher than potential barrier height, the electrical conductivity oscillates and its mean value is smaller than that of metal without grain boundaries.
1. INTRODUCTION In many cases metal films have a distinct granular structure and the electrons are subject to an additional scattering on the grain boundaries. The width of the grain boundary is, in general, equal to several Angstr6ms. Both the description of the electrical conductivity and the interpretation of the results of its measurement in granular metals have been so far concentrated on the assumption that electron scattering by the grain boundaries is described by introducing either the parameter R for the reflection of electrons from the grain boundaries 1'2 or the parameter t for electron transmission through the grain boundaries 35. It should be noted here that in the sofar published papers the reflection parameter R, introduced by Mayadas and Shatzkes 2, and the transmission parameter t mentioned by Tellier and Tosser 5 and Warkusz 3'4 do not depend on the width and height of the potential barrier existing on the grain boundary. The present paper deals with the calculation of the electrical conductivity in granular films. The main attention was focused on the nature of the grains, i.e. the grain boundaries are treated as potential barriers of suitable widths and heights. Reflection of electrons from a barrier, expressed by the coefficient R, depends on the width and the height of barrier. The sum of the reflection coefficient R, and the transmission coefficient T of electrons through the grain boundary potential barrier is R + T = 1. In this paper we shall not be interested in the problem of external surfaces and electron scattering on these surfaces, since they are well known from the papers by Fuchs 6 and Sondheimer 7 and from more recent literature sl°. Therefore it may be assumed that the results obtained by us concern either thick films or such films in which the coefficients of electron reflections from external surfaces are close to unity. 00406090/88/$3.50
© Elsevier Sequoia/Printed in The Netherlands
2
F. W A R K U S Z
2. THEORY
2.1. Boltzmann equation and effective relaxation time Let a metal film of a granular structure be given in the plane xy. Let us assume that the grain boundaries are perpendicular to the direction of the electric field g and to the direction x (Fig. 1). Let the metal film be sufficiently thick and its external surface be smooth so that the essential contribution to the electrical conductivity is solely due to electron scattering on the grain boundaries.
I
\
I
//
Fig. 1. G e o m e t r y of electron scattering on potential barriers.
In this case the linear Boltzmann equation for the electron distribution function has the form
IP fl(k) e o~ v,, Of°(k) 0E = . J (k,k') {fl(k)f~(k')} d3k ' + z
(1)
wherefdk) = f(k)fo(k) is the deviation of the electron distribution function from its equilibrium valuef0(k), e is the electron charge, E is the electron energy, vx is the x component of the electron velocity in the direction of the applied electric field and P(k, k') is the probability of electronic transition from k to k' states in the vicinity of the grain boundaries. In order to find the integral of the probability P(k,k') in eqn. (1), we show that, as far as the scattering potentials are concerned, the assumption that the effect of this scattering can be represented by a grain boundary relaxation time zg is very reasonable. The latter is a function of the electron velocity in the direction x, the sizes D of the crystalline grains and the electron scattering parameter R ( T = 1  R). In the following we suppose that
fp(k,k,){f~(k)_fl(k,)} dak, = vOx R(a,~) T(a, f~(k)
(2)
where R(a, U) is the coefficient of electron reflection from the grain boundary and T(a, U) is the coefficient of electron transmission through the grain boundaries. If no electrons are travelling through the grain boundary then R(a, U) = 1 (T(a, U) = O) and the integral (2) is infinite; the grain boundary relaxation time zg is zero. If all electrons pass the grain boundary, then R(a, U) = 0 and the integral (2) is also zero;
SCATTERING OF ELECTRONS ON GRAIN BOUNDARIES
3
the grain boundary relaxation time zg is infinite. Using eqns. (2) and (1) we obtain
efo
vx Rta, t:) ~ +fl_
(3)
where z is the relaxation time for an electron in the metal without the grain boundaries and responsible for the background scattering of electrons. Assuming a spherical system (v, q~,O) in velocity space with c= = v cos/9 and introducing an effective relaxation time z* given by 1 _ "~*
~cos tk sin 0 + 1 "/7
(4)
l"
where 
2R DT
(5)
is the scattering parameter 1'2 and 2 is the mean free path in the metal without grain boundaries, we can write eqn. (3) in the following form: e 8 v cos ~bsin 0~~  flz,
(6)
From the above equation it may be seen that the difference between the distribution function f l and function for the electron distribution in a bulk body without grain boundaries is that instead of z the effective relaxation time (eqn. (4)) was introduced. The first term on the righthand side of eqn. (4) is responsible solely for the electron scattering on the grain boundaries. Equation (4) may be also presented in the form 1
1
l"*
"Cg
4
1
(7)
where T
zg  ~ cos ~bsin 0
(8)
and zg is the relaxation time resulting from the introduction of the granular structure. From eqns. (5) and (8) it follows that the greater is the ratio D/2 of the grain diameter D to the mean free path 2 of electrons, and the greater is the ratio of the transmission coefficient T to the coefficient R of electron reflection from the boundaries separating the grains, the longer is the relaxation time z,. It should be noted that in our model the grain boundaries are perpendicular to the direction x (Fig. 1); therefore in this direction zg had a minimum value zg = z/~ whereas, in the y or z direction, z s = oo. Taking account of eqn. (7) we see that the effective relaxation time z* in directions in which there are no grain boundaries is z (see Appendix A).
2.2. Potential barrier on the grain boundary Crystalline grains in a metal are separated from one another by a space, called
4
F. WARKUSZ
the grain boundary. In this space there is a potential barrier characterized by its width a and its height U (see Fig. 1)1'2. The Schr6dinger equation needs to be solved for the problem of a particle transmission (transition) through the potential barrier 1x for the cases when electron energy E is (1) higher than the potential barrier height U, (2) equal to U and (3) smaller than U. If the coefficient R of electron reflection at the potential barrier is defined by the ratio of the square of the reflected wave amplitude to the square of the incident wave amplitude, i.e. R = IBd2/IAll 2, and the transmission coefficient T is defined by the ratio of the square of the passing wave amplitude to the square of the incident wave amplitude, i.e. T = IA312/IA~I2, the following expression for R will be obtained:
{ 4E(UE) }1 R = U2sinh2[(a/h){2m(U_E)}~12] ~1
fore < U
(9a)
tr 2h 2 "k 1 g = tm~~ + 1)
for E = U
(9b)
4 E ( E  U) t1 R = [U2sinE[(a/h){2m(E_U)}~12] 41_
forE > U
(9c)
where a is the width of the potential barrier, U is the energy height of the potential barrier, E is the electron energy, m is the electron mass and h is Planck's constant. For the reflection coefficient R and transmission coefficient T the following relation holds: RWT= 1
(10)
Then from eqns. (9), (10) and (5) the scattering parameter 0c can be expressed as follows:
~ = D 4E(U  E) smh2
2m(U  E)
for E < U
(1 la)
forE = U
(llb)
for E > U
(1 lc)
)~ a e
~  D 2h 2mE c~ = D 4E(E  U) sin2
2re(E U)
2.3. Electrical conductivity of granular films The electrical conductivity for a metal film when electron scattering on the grain boundaries is also additionally included is determined from the expression 13 or, = ~nSJiV~Ei
~
(12)
where z* is given by formula (7),
(m'~ 1/2 IVy
=
\~h~)
E'i:
(13)
SCATTERING OF ELECTRONS ON GRAIN BOUNDARIES
5
and
E dSr = ~2m  v sin 0 dO d~
(14)
Taking into account eqns. (7), (13), (14) and (12) we obtain /'2~ a g _ 3 4n Jo ao
/',
cos2~bsin30
d4~Jod01 + a cos ~bsin 0
(15)
where ~ is given by formulae (11). Using the known methods 2,3 we compute the integral (15) and obtain trg 3 3 3 ao  20~ ~2 I~ln(1 +~)
(16)
where ~ro is the conductivity of the bulk metal without grain boundaries: cro 
8,~e2m2TvF 3 3h 3
(17)
The expression (16) can be written in approximate forms as follows: O'_gg
= 1~IX+~O~2½(X3+...
forct < 1
(18a)
for ~ = 1
(18b)
for ~ > 1
(18c)
O"o
a~ = 3 (In 2  ½) = 0.5793 go %_ ao
3 2~
3 3 + ~2 ~4
3 1  ÷ 2~5 ~6
""
If we compare eqn. (16) in the present paper with eqn. (1.355) of Tellier and Tosser 5 then we shall see that these equations are identical, assuming that
1
= 
V
(19)
where D v  2 In(l/t)
(20)
v being defined in earlier papers 35. In refs. 35 the parameter t appearing in formula (20) was also termed the "coefficient of electron transmission through the grain boundaries". The grain boundary is treated there as a plane perpendicular to the electric field direction (direction x) and the parameter t determines the probability of electron transmission through the planes (the layer model). In the present paper the parameter Tis the probability of electron transmission through the potential barrier that appears on the grain boundary. Such a potential barrier is defined by its thickness a and height U. This parameter will thus depend on a and U as well on the energy of the incident electrons. Both t and T determine the electron transmission through the grain boundary, but these relationships are neither identical nor linear.
6
F. W A R K U S Z
In the abovementioned papers 3 5 the parameter t is defined as follows: t = exp
2gCOS~bsinO
(21)
where 2g is the mean free path of electrons, resulting from the granular structure. Taking account of eqns. (5), (lO), (19) and (20) we obtain 1 r
1
texp(~)orTl_ln
t
(22)
The function T = f ( t ) illustrated in Fig. 2 is not a linear dependence T = t, but for t > 0.02 the function T = 0.8t +0.2 gives a good approximation. From Fig. 2 it may be seen that T = t will be true only for t > 0.8, i.e. for small coefficients of electron reflection from the grain boundaries. 10

T
zz
~
/~//
06
02
/
#
/#
02
i
0)~
Fig. 2. T r a n s m i s s i o n
3.
I
06
I
08
~0 t c o e f f i c i e n t s T a n d t.
DISCUSSION AND CONCLUSIONS
We shall focus our attention on three problems, namely electrical conductivity vs. potential barrier width, electrical conductivity vs. the difference between the electron energy and the potential barrier height, and electrical conductivity vs. the diameter of the crystalline grains. The dependence of the conductivity ag/ao on the potential barrier width is plotted in Fig. 3. The calculations have been performed for an average electron energy E = 5 eV, a reduced grain diameter D/2 = 10 and potential barrier heights U = 3, 4, 5, 6, 7 and 10 eV. The width of the potential barrier varied from 0 to 0.5 nm. If E > U (in our case E = 5 eV) then the oscillation of the electrical conductivity depends on the width a of potential barrier and its height U. This problem will be dealt with in more detail while discussing Fig. 4. For E ~< U oscillation of the conductivity does not occur. From Fig. 3 we see that the conductivity ag/ao decreases with increasing width of the potential barrier and it decreases by more as the potential barrier becomes higher. In practice at an electron energy equal to 5 eV the fading of the electrical
SCATTERING OF ELECTRONS ON GRAIN BOUNDARIES
1.0
EUZ[eV]
1 . 0 '  
O~B
%1~o
0.6
0.8
/ 0.6
O.Z
I
3
2
~.
7
\~ooo
/.... \
I
5 QIll
o [;q
Fig. 3. ag/a o as a function of potential barrier width a. Fig. 4. Oscillation in the conductivity ag/a o, for E > U, vs. the potential barrier width.
conductivity occurs for a potential barrier height of 10 eV for a potential barrier width equal to 0.4 nm. The oscillation of the electrical conductivity ag/t~ o for the case E > U is shown in Fig. 4. Both the period and the amplitude of oscillation depend on the difference E   U , being greater for smaller differences. Oscillation is a static process (time independent). F o r E = U the oscillation of ag/t~ o does not occur. We m a y see in Fig. 4 that for our data the largest period and the highest amplitude occur for E  U = 0.25 eV, i.e. E = 5 eV, U = 4.75 eV. The mean values of ag/a o for these oscillations were also calculated and are shown in this figure by the broken horizontal lines. The electrical conductivity oscillates a r o u n d these mean values, and for E  U = 2 eV, 1 eV, 0.5 eV and 0.25 eV the respective mean values of ~g/~o are the following: 0.987, 0.972, 0.935 and 0.875. It must be added that the amplitude of the conductivity in these intervals depends on the grain diameters. The greater D/2 is, the smaller the amplitude is, since it results also from eqns. (11) and (16). In our calculations we have assumed D/2 = 10. The m a x i m u m conductivity ~g/ao occurs for a~
n~h
{2m(E v)} 1/2
and the m i n i m u m for (2n + 1)~h a = 2 { 2 r e ( E  U ) } 1/2
n = 0,1,2,3 ....
which results from eqns. (1 lc) and (16). The changes in ~/t~ o for D/2  1, 2, 5 and 10 are shown in Fig. 5. The plots were d r a w n for two cases: E = 5 eV, U = 4 eV (E > U) and E = 5 eV, U = 5 eV (E = U). In this figure we see that these changes are distinct and that the conductivity of granular films depends substantially on the grain diameter: the smaller the diameters are, the lower the conductivity is. It seems that it will be difficult to measure experimentally the oscillation of conductivity in the case where E > U. Therefore it is better to use mean values a~/ao which in Fig. 5 are denoted by broken
8
F. WARKUSZ
horizontal lines and for E = Uand D2 = 10, 5, 2, and 1 are equal to 0.972, 0.947, 0.886 and 0.816 respectively. Despite the fact that the electron energy E exceeds the potential barrier height U, the m e a n conductivity continues to decrease as the grains in the films become smaller (see Fig. 5) and for smaller differences E  U (see Fig. 4). In the extreme case o f E = U and for E < U the conductivity decreases substantially with increasing width of the potential barrier and decreasing grain diameter. F o u r curves in Fig. 5 are obtained for E = U = 5 eV and D2 = 10, 5, 2 and 1. The lack of oscillations in fi/ao is confirmed by eqns. (1 lb) and (16). The plot of fig/fie vs. the relative value D/2 of the grain diameters at the steady values of the coefficients of electron reflection from the grain boundaries (potential barriers) is shown in Fig. 6. In this figure R = 0.444, 0.747 and 0.898 correspond to potential barrier heights U = 4, 5 and 6 eV, the electron average energy E = 5 eV and the potential barrier width a = 0.3 nm. We see that for small values o l D / 2 and high coefficients R the conductivity fig/fr0 is lOW. 10 dg/0+o ~


o.8
06
\
\ 
\ \
\
\
\
k
\
\
!P
lo[ \
\ \
,
4o8,6
\
\ o4~
0972[0 947
. . . . . . .
\ \
\ ",,
x
'
\
02 r
\
\
\ \
\
\
~o/Ooi
\ op,lo \ \ \
\ \
.,
\\
'08F!
\
'
I
/__ o (~,]
/
.
~
\.l
l
Og
1
02
I o/7,
Fig. 5. Change in conductivity ag/a o vs. grain diameter D/2 and potential barrier width. Fig. 6. ag/a o as a function of D/2 and coefficient R.
The plot of frg/fi o vs. potential barrier height is shown in Fig. 7. The calculations were performed for barrier widths a = 0.1,0.2, 0.3, 0.4 and 0.5 nm, electron energy E = 5 eV and grain diameter D/2 = 10. The conductivity decreases rapidly with increasing potential barrier height, and it decreases particularly fast when U > E. F o r U < E we observe the conductivity oscillation, which is shown more distinctly in Fig. 8. The m a x i m u m conductivity appears for
U=E
nZn2h 2 2ma 2
9
SCATTERING OF ELECTRONS ON GRAIN BOUNDARIES
10
1o
E
. . . . . . . . .
.E E xN
=
08
o"g/o'o
0.6
0.8
c f~
c ~
"~ c_
"
zz~
06
06
0.2
ESeV 10 I
2
u [eV]
~
6
U [evl
a
Fig. 7. aw/tro as a function of potential barrier height U. Fig. 8. Oscillation in the conductivity %/a o vs. U.
and the minimum for (2n + 1)2n2h2 U = E
8ma2
n = 1,2,3....
The values of the electrical conductivity of granular films obtained from calculations are of interest, and therefore further investigations devoted to this problem may bring important results concerning the structure of such films ~2 18. On the basis of many experiments, Hoffmann and coworkers 12 18 have stated that the reduction in the conductivity depends exponentially on the number of grain boundaries per mean free path. The definition of the total transmission coefficient T for the special array of grain boundaries is identical with that given by Reiss e t a l J 2, eqns. (8) and (4). The difficulties consist in the fact that we never deal with an ideal granular film and with ideal grain boundaries as potential barriers of equal width and height. In reality we have a certain distribution of barriers, which confuses theoretical results obtained for ideal structures. Nevertheless, this problem is worthy of attention, in particular that of the experimenters who at present are attempting to conduct their experiments in continually improving laboratory conditions. ACKNOWLEDGMENTS
This study was sponsored by the Institute of Physics, University of Warsaw, under Contract CPBP. Thanks are due to Professor C. Wesolowska for supervision. REFERENCES 1 2 3 4 5 6
A, F. Mayadas, M. Shatzkes and J. F. Janak, Appl. Phys. Lett., 14 (1969) 345. A.F. Mayadas and M. Shatzkes, Phys. Rev. B, 1 (1970) 1382. F. Warkusz, Electrocomponent Sci. Technol., 9 (1981) 105. F. Warkusz, Electrical and mechanical properties of thin metal films, Size effects, Prog. Surf. Sci., •0(3) (1980) 287. C.R. Tellier and A. J. Tosser, Size Effects in Thin Films, Elsevier, Amsterdam, 1982. K. Fuchs, Proc. Cambridge Philos. Soc., 34 (1938) 100.
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F. WARKUSZ
7 8 9 10 I1 12 13
E.H. Sondheimer, Adv. Phys., 1 (1952) 1. M.S.P. Lucas, J. Appl. Phys.,36(1965) 1632. F. Warkusz and J. Kaszubkiewicz, Mater. Sci., 8 (1982) 91. F. Warkusz, Thin Solid Films, 122(1984) 105. S. Flugge and H. Marschall, Rechenmethoden der Quantentheorie, Springer, Berlin, 1952. G. Reiss, J. Vancea and H. Hoffmann, Phys. Rev. Lett., 56 (1986) 2100. H. Hoffmann, in J. Treusch (ed.), Festk6rperprobleme: Advances in Solid State Physics, Vol. 22, Vieweg, Braunschweig, 1982, p. 255. J. Vancea, H. Hoffmann and K. Kastner, Thin Solid Films, 121 (1984) 201. J. Vancea and H. Hoffmann, Thin Solid Films, 92 (1982) 219. H. Hoffmann and J. Vancea, Thin Solid Films, 85 (1981) 147. G. Fischer, H. Hoffmann and J. Vancea, Phys. Rev. B, 22 (1980) 6065. G. Reiss, J. Vancea and H. Hoffmann, J. Phys. C, 18 (1985) L657.
14 15 16 17 18
APPENDIX A
In our considerations we employ the Baines M and Halpern A2 method. The effective time of electron relaxation, derived by us for background scattering and on the grain boundaries, has the form (eqn. (7)) 1 t'*

ct tcos q~sin 0
1 1.
l
(A1)
while the dependence given by Mayadas and Shatzkes for the same geometry, i.e. for the grain boundaries perpendicular to the direction x, is the following 1
1.*

1
+
z
ct
1
(A2)
zcos~bsin0
To find the transition probability P(k, k'), Mayadas et al. A3,A4 treat the potential
v= sX
/xx.)
n
due to all the reflecting planes located perpendicularly to the electric field o~ (grain boundaries are represented as parallel partially reflecting planes) as a perturbation • element (k] V]k' ). on the electron hamiltonian and compute the square of the matrix The planes located randomly are assumed to obey a gaussian probability distribution. These planes are in fact identified by ~ functions of strength S = Ua, the square of the strength being expressed in terms of the reflection coefficient R. Let us consider the following problems. (1) I f 0 = 0, then from eqn. (A1) we have 1.* = z while from eqn. (A2) it follows that 1.* = 0. (2) If q9 = x/2 then from eqn. (A1) we have 1.* ~ 1., while from eqn. (A2) we obtain 1.* = 0. (3) I f e = x/2 and $ = 0 then eqns. (A1) and (A2) are identical. From the assumption that there are no grain boundaries in the y and z directions it follows that z* = z, where z is the relaxation time in the metal without grain boundaries (background scattering), and not z* = 0, from eqn. (A2). Taking into account eqn.(A1) and the geometry in Fig. 1, the kinetic equations of
SCATTERING OF ELECTRONS ON GRAIN BOUNDARIES
Boltzmann m a y be written in the form
e g v~
cos ~bsin 0 = f l
cos q5 sin 0 + f l
"t
T
while in view of eqn. (A2) we obtain e ~ v ~~ cos q~ sin 0 = ~
1 fl cos q~ sin 0 ÷ z
References for Appendix A AI M . J . Baines, Proc. Cambridge Philos. Soc., 57 (1961) 606. A2 V. Halpern, J. Phys. F, 1 (1971) 608.
A3 A. F. Mayadas, M. Shatzkes and J. F. Janak, Appl. Phys. Lett., 14 (1969) 345. A4 A. F. Mayadas and M. Shatzkes, Phys. Rev. B, 1 (1970) 1382.
11