Physica B 322 (2002) 289–296

Large grain size dependence of resistance ofpolycrystalline films

P. Arun*, Pankaj Tyagi, A.G. Vedeshwar

Department of Physics and Astrophysics, University of Delhi, Delhi 110 007, India

Received 28 January 2002; received in revised form 21 March 2002

Abstract

A qualitative behavior of grain size dependent resistance of polycrystalline films has been worked out by extending

the earlier model (Volger’s model) for polycrystalline films. Growth of grain size is considered to be accompanied with a

decrease in the number of grains present. The variation of the number of grains is restricted along one direction at a

time, assuming it to be constant along the other two directions, to simplify the problem. Combining the results along

film thickness and film length, the calculated resistance versus grain size shows a family of curves. These curves can be

used to know the growth direction by comparing the measured grain size dependence of the resistance. r 2002 Elsevier

Science B.V. All rights reserved.

PACS: 73.61; 73.61.G; 81.40.C

Keywords: Thin films; Chalcogenides

1. Introduction

The studies on physical properties of polycrys-talline films of various materials have been thesubject of interest for many years in basic as wellas applied research. It is a well-established factthat physical properties of polycrystalline filmsdepend on film parameters and differ significantlyfrom those of respective single crystals or bulkmaterial. The existence of grain boundaries andthe surface effects of grains could considerablyaffect transport and optical properties of the film

[1–4]. Obviously, these effects could be realized inthe manifestation of grain size dependence ofphysical properties of the polycrystalline films. Itcan be noted that exact theories dealing with thisproblem are hard to conceive because of thecomplex and statistical nature of grain sizedistribution, and hence, grain boundary structure.However, one can explain qualitative behaviorusing simplified model based calculations. Thenumber of experimental reports show grain sizedependent transport and optical properties in alimited range of grain size. Different models wereproposed in the literature, even though quitescattered, to explain the variation of Hall coeffi-cient of polycrystalline films in contrast to theirsingle-crystal counterpart [4–6]. Still there areno concrete results explaining the grain size

*Corresponding author. Tel.: +91-11-6468-401; fax: +91-

11-7667-061.

E-mail addresses: arunp92@physics.du.ac.in, arunp92@ya-

hoo.co.in (P. Arun), agni@physics.du.ac.in (A.G. Vedeshwar).

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 1 9 5 - X

dependent physical properties in the literature.This problem of grain size dependence shouldconsider the variations in grain size, inter-graindistances and grain density. Therefore, we find itinteresting and worth while to investigate thisproblem by extending a simple model of Volger [7].

2. The model

In this model, grains are assumed to be cubicalwith the edge size a: Further the grains areassumed to be arranged in an ordered manner,as shown in Fig. 1, with equal spacing between theneighboring grains. The inter-grain distances aredifferent along x, y and z directions and are samealong any one direction. The choice of cubicalnature of grains allows us to take the same grainresistance Rg along the three axes. The resistivityof the grains is considered to be the same as that ofa single crystal, and hence, can be written as

Rg ¼ rg=a:

Considering the film of thickness d; length l andwidth w; the number of grains q can be imagined tobe arranged regularly at equal inter-grain spacing

tx along the length l: Similarly, r and p grains arearranged along the width and thickness of the filmwith inter-grain spacing ty and tz; respectively.Then the total number of grains present in thefilm is

n ¼ pqr: ð1Þ

Also, from the assumed arrangement of grainsas shown in Fig. 1 we have

d ¼ paþ ðp� 1Þtz;

l ¼ qaþ ðq� 1Þtx;

w ¼ raþ ðr� 1Þty: ð2Þ

Further, we assume the arrangement of grains tobe unaltered even as grains grow in size. Since thefilm dimensions do not change, the new grainsformed have to be confined within the volume (V )of the film and leads to the condition

na3 ¼ V � Vvoid ¼ constant: ð3Þ

It should be noted that as a result of graingrowth, a decrease in the number of grains occurs.This result follows from the assumption that thedensity of a grain remains constant even as itgrows in size. The resistance along the length ofthe film can be measured by taking the contactsacross the cross section in the yz plane. Thus, thepoints A–B, C–D, etc. shown in Fig. 1 are at equalpotential. Seto [8] made a similar simplificationstep by assuming the problem to be that of onedimension. The equivalent DC circuit of thisarrangement of measurement would be as shownin Fig. 2, where Rb represents the high resistanceof the inter-grain voids which is a function of txwhich in turn would depend on the mechanism bywhich charge carriers would cross the inter-grainboundary. Many suggestions have been made for

Fig. 1. An idealized assumption of how cubic grains are

arranged along the dimensions of the film.

Fig. 2. An equivalent circuit of a polycrystalline film based on

simplified assumptions.

P. Arun et al. / Physica B 322 (2002) 289–296290

explaining the cross over, such as ohmic conduc-tion, tunneling or thermonic emission [9–11]. Itmay also be a combination of these, depending onthe actual grain boundary structure.

As can be seen in Fig. 2, the whole film can beconsidered to be a parallel combination of pr

resistive elements, where resistance of each elementis given by

R1 ¼ qRg þ ðq� 1ÞRb:

Thus, the net resistance along the length of the filmbetween the two contacts would be given as

Rnet ¼qRg þ ðq� 1ÞRb

pr: ð4Þ

Substituting modified Eq. (2) in Eq. (4), we get

Rnet ¼

l þ tx

aþ tx

� �raþ

l þ tx

aþ tx� 1

� �Rb

d þ tz

aþ tz

wþ ty

aþ ty

on rearranging the above equation, we have

Rnet ¼aþ ty

wþ ty

� �aþ tz

d þ tz

� �

�ðl þ txÞrg þ aðl � aÞRb

aðaþ txÞ

� �: ð5Þ

Under normal conditions of resistance measure-ments the length of the film is far greater than thethickness (lbd). The growth of grain size is limitedby thickness along z and the width along y

direction. In any case, it is quite reasonable toapply the condition lba and lbtx above. Then,Eq. (5) reduces to

Rnet ¼l2

naðaþ txÞ2ðrg þ aRbÞ:

Substituting for n in terms of a; the grain size inthis equation, we have

Rnet ¼a2l2

ðV � VvoidÞðaþ txÞ2ðrg þ aRbÞ: ð6Þ

Both in ohmic and tunneling mechanisms ofcharge transfer between the grains, Rb stronglydepends on tx [12]. Eq. (6) reduces to single-crystalresult

Rnet ¼l2rgV

¼rgl

wd

when tx ¼ 0; Vvoid ¼ 0 and hence Rb ¼ 0: Ingeneral, the inter-grain resistance Rb is quitehigher than the grain resistance Rg: Rb may bewritten as rgf ðtxÞtx=a

2; where rg is a very largequantity. The function f ðtxÞ; would depend on themechanism of charge transfer between the twoneighboring grains. While it would be an expo-nentially falling function, expð�tx=aÞ; for tunnel-ing process, it is unity for ohmic conduction.Therefore, Eq. (6) may be written as

Rnet ¼a2l2

ðV � VvoidÞðaþ txÞ2

rg þ rgf ðtxÞtx

a

� �:

For the condition tx5a; for both mechanisms,tunneling and ohmic, f ðtxÞ may be consideredas unity. Substituting rg=rb as k we may writeEq. (6) as

V � Vvoid

rgl2Rnet ¼

a2

ðaþ txÞ2

1þ ktx

a

� �: ð7Þ

Writing the above equation in terms of thevariable x ¼ tx=a; we get

R0 ¼V � Vvoid

rgl2Rnet ¼

1þ kx

ð1þ xÞ2: ð8Þ

Now, we require a relationship between tx and a

for understanding the variation of resistance withchanging a and tx:

3. Variation of tx with grain size

The problem reduces to finding a functionaldependence of tx on the grain size a: However,as the grains grow in all three directions,constraint within the initial volume (we assumethat at all time grains maintain their cubical natureand retain their arrangement as given in theprevious section), simultaneously the grain num-ber p; q and r and in turn n decrease. We can easilyrealize this change by simply rewriting Eq. (1)using Eq. (3) as

n ¼V � Vvoid

a3: ð9Þ

Thus, tx is not only related to a but also to thegrain numbers (p; q and r) in a complex manner.Therefore, we try to correlate tx with the grain size

P. Arun et al. / Physica B 322 (2002) 289–296 291

along with the grain number along each directionby assuming the grain numbers to be constantalong the other two directions. For understand-ing the same we rewrite the expression for tx ofEq. (2) as

tx ¼�qa

q� 1þ

l

q� 1: ð10Þ

We now consider the variation of tx with a

due to decreasing p; i.e. decreasing number ofgrains along the thickness, while q and r; thenumber of grains along the length and widthremain constant. It is evident from Eq. (10) thatthe variation of tx with respect to a is linear asshown in Fig. 3. However, there is a limit in thegrowth of the grains, where the maximum size towhich a grain can grow is limited by d; thethickness of the film. Thus a ¼ d; for tzB0 andpB1: Therefore, the maximum size that a graincan attain for a given qr is thus obtainablefrom modified Eq. (2) in Eq. (9). Rewriting theequation as

V � Vvoid

a3¼ qr

d þ tz

aþ tz

� �;

where qr is a constant, on rewriting the aboveequation we have

tz ¼ aqra2d � DVDV � qra3

� �; ð11Þ

where DV represents (V � Vvoid). On rearrangingwe have

tz

d¼

a=dðða2=d2Þ � ðDV=qrd3ÞÞððDV=qrd3Þ � ða3=d3ÞÞ

: ð12Þ

Since tz can either be positive or zero

l

qdX

a

dX

ffiffiffiffiffiffiffiffiffiDVqrd3

s: ð13Þ

Thus, it can be understood that the maximumsize the grain can attain is influenced by q; r; DVand the film thickness. From this condition it isevident that for the grains to grow to themaximum possible size, the film thickness, theproduct qr should take a minimum

qrminimum ¼V � Vvoid

d3: ð14Þ

Similarly, for the situation when a is growingdue to decreasing grain number along the length(q), while p and r remains constant, we write

V � Vvoid

a3¼ pr

l þ tx

aþ tx

� �

which gives

tx ¼ apra2l � DVDV � pra3

� �: ð15Þ

The dependence of tx on the grain size a; in thecase of varying grain number along the film length

Fig. 3. The variation of inter-grain distances (tx) along the

length of the film with respect to increasing grain size for

number of grains remaining constant along the length and

width of the film. The family of curves were generated with

Eq. (10), for decreasing number of grains along the length of

the film, i.e. decreasing q (q1 > q2 > q3 > q4 > q5).

Fig. 4. The variation of inter-grain distances (tx) along the

length of the film with respect to increasing grain size for

number of grains remaining constant along the width and the

thickness of the film. The family of curves were generated with

Eq. (15), for decreasing number of grains along the thickness of

the film, i.e. decreasing p (p1 > p2 > p3 > p4 > p5).

P. Arun et al. / Physica B 322 (2002) 289–296292

q with pr constant is shown in Fig. 4. Each curvewas generated for given p; and p is in decreasingorder from top to bottom of the figure. The typicalinitial values used for the computation werel ¼ 2 cm, w ¼ 2mm and d ¼ 2000 (A. Also Vvoid¼4:913� 10�12 m3 with p; q and r having values 10,105 and 104, respectively. These initial values werethe same as those used for generating Fig. 3.

Eq. (15) along with the maximum size restrictionon grain size gives

1

pX

a

dX

ffiffiffiffiffiffiffiffiffiffiffiDVprld2

s: ð16Þ

We do not consider the case of p; q constant andr varying as it does not contribute anything new inthe variation of tx with respect to a; and is similarto that expressed in Eq. (15).

4. The resistance as a function of grain size

As we have seen earlier, Eq. (8) contains thegrain size dependence of film resistance throughthe grain size dependence of tx: Therefore, wecalculated film resistance as a function of normal-ized grain size using the two expressions inEqs. (10) and (15) for the cases of varying grainnumbers along the thickness and along the lengthof the film, respectively. The resulting behavior isshown in Fig. 5 which contains the family ofcurves resulting from the effect of changing grain

numbers along two different directions. However,we can easily realize that the measured resistanceas a function of grain size could go along any pathenclosed by the family of curves, depending uponthe dominance of growth mechanism along aparticular direction subjected to conditions im-posed by Eqs. (13) and (16). That is, it could showan oscillatory behavior which could be prominentinitially due to unrestricted growth, which smooth-ens with decreasing amplitude and frequency asgrowth of grain size proceeds.

We have shown explicitly in Fig. 6, the case forlarger variation in p compared to q; or in otherwords when the grain number variation along thethickness is preferred over that along the length. Inthis case, as can be seen from the figure, the filmresistance should show an increasing trend withgrain size and could show few discontinuities orsteps. Similarly, Fig. 7 shows the variation of filmresistance as a function of grain size for the case oflargely varying q as compared to p; i.e., thepreferred grain growth along length as comparedto film thickness. We see on the average decreasingtrend with grain size. The discontinuities or step-like behavior is due to the small variation of p:Therefore, one can easily know the dominantdirection of grain growth from the experimentallydetermined film resistance as a function of grainsize.

However, in the literature there are not manysuch data to compare with the present model. Both

Fig. 5. The variation of the film resistance (R0) with increasing

grain size. The family of curves were generated with values from

Figs. 3 and 4.

Fig. 6. The variation of the film resistance, R0 of Eq. (8) with

increasing grain size for rapid variation in number of grains p as

compared to q: The straight line is to highlight the general

increasing trend of the film resistance.

P. Arun et al. / Physica B 322 (2002) 289–296 293

increasing and decreasing trends in film resistance(or resistivity) as a function of film thickness havebeen observed for films with thickness greater thanthe mean free path of the charge carriers. It shouldbe noted that the present model is operative forthicker films, that is, for films whose thickness isgreater than the mean free path of its chargecarriers. For thinner films, usually resistanceshows a universally decreasing trend with filmthickness due to specular scattering [13]. Usually,the scattering points are smoothened to a linearbehavior with film thickness, hence for thick filmsvariation in resistivity should be independent ofdefects and specular scattering and should dependon grain size and geometry. Literature show someattempt to correlate the variation of film proper-ties with respect to film thickness [14] and dopingconcentration [8], assuming that the grain sizevaries linearly with thickness and doping concen-tration, respectively. However, systematic data areabsent. Therefore, to establish the direct depen-dence of film resistance on grain size is almostimpossible from existing experimental data.

The above model with its initial assumptions isbest suited for understanding the variation of filmresistance with growing grain size by heat treat-ment of the films, provided surface oxidation,reduction in film thickness, etc. do not take place.Another possibility is in the cases where filmageing is observed as a result of improving

crystallinity. Such ageing in Sb2Te3 films has beenrecently reported [15]. It was observed that the as-grown amorphous Sb2Te3 films showed a drop infilm resistance with time. This was followed by asharp decrease in resistance after which the film’sresistance saturated. The duration for which thesharp decrease takes place is film thicknessdependent. Thicker the film, the sharp declinetakes place over a large time. The onset of this fallin resistance coincides with the appearance ofpeaks in the X-ray diffraction pattern, indicatinggrain formation. The films of Sb2Te3 were foundto crystallize with hexagonal structure (ASTM 15-874), cba: Fig. 8 shows this variation of resistanceof Sb2Te3 thin film with the passage of time. WhileFig. 9 shows only the boxed region of Fig. 8, whichis the region of sharp decline in resistance withtime of films with thickness 500 and 250 nm,respectively. For both films, the variation inresistance was slow enough to measure duringthe fall. Also, their thickness was enough toassume that the variation in resistivity is indepen-dent of defects and specular scattering anddepends only on grain size and geometry. Thenormalized resistance is plotted with respect to thenormalized time for the purpose of comparison. Azig–zagging behavior with a net trend of decreaseis observed, which is more prominent in the500 nm thick film. Assuming the grain growth tobe linear with time, the trend shown in Fig. 9 issimilar to that shown in Fig. 7. We can easilyrealize that the resistance of films whose crystal-lites are hexagonal structured with lattice constantc far greater than a and having c-axis alignednormal to the substrate plane should show easiergrain growth along the length and width ascompared with that along restrictive film thick-ness. As per the model q; the grain number alongthe length, would be decreasing more rapidly thanp; that along the film’s thickness, leading to ageneral trend of decrease in resistance. The factthat change in p is more probable in thicker filmswith c-axis of hexagonal crystals oriented normalto the substrate, explains the prominent zig–zagging in 500 nm films as compared to that ofthinner films. The variation in resistance duringzig–zagging is of the order of 10–50 kO. Due to thelack of significant number of X-ray peaks we

Fig. 7. The variation of the film resistance, R0 of Eq. (8) with

increasing grain size for rapid variation in number of grains q as

compared to p: The straight line is to highlight the general

increasing trend of the film resistance.

P. Arun et al. / Physica B 322 (2002) 289–296294

cannot confirm the films orientation. Presently, weare trying to establish the functional dependenceof film resistance on grain size in oriented films ofdifferent compound semiconductors.

One can predict that the films having cDaDb;growth along the thickness may not be curtailed,thus allowing p to rapidly decrease, which wouldshow an increasing resistance with grain size.Such post-deposition variation has been reportedin metallic films like silver, copper [16] andCdSexTe1�x [17,18], etc.

As discussed above, the simple ‘‘back of theenvelope like’’ model is at best useful to qualita-tively explain the influence of the growth of grainsize on the film resistance and for indicating theorientation of the grains and preferred direction ofgrowth.

References

[1] L.L. Kazmerski (Ed.), Polycrystalline and Amorphous

Thin Films and Devices, Academic Press, New York,

1980.

[2] M.E. Cowher, T.O. Sedgwick, J. Electrochem. Soc. 119

(1972) 1565.

[3] P. Rai-Choudhury, P.L. Hower, J. Electrochem. Soc. 120

(1973) 1761.

[4] T.I. Kamins, J. Appl. Phys. 42 (1971) 4357.

[5] V. Snejdar, J. Jerhot, Thin Solid Films 37 (1976) 303.

[6] J. Jerhot, V. Snejdar, Thin Solid Films 52 (1978) 379.

[7] J. Volger, Phys. Rev. 79 (1950) 1023.

[8] J.Y.W. Seto, J. Appl. Phys. 46 (1975) 5247.

[9] C.M. Hurd, J. Phys. C 18 (1985) 6487.

[10] V.D. Das, M.S.M. Sastry, M. Pattabi, Phys. Stat. Sol. A 96

(1986) 677.

[11] C.A. Neugebauer, M.B. Webb, J. Appl. Phys. 33 (1962) 74.

[12] R.L. Petritz, Phys. Rev. 104 (1956) 1508.

[13] C.R. Tellier, A.J. Tosser, C. Boutrit, Thin Solid Films 44

(1977) 201.

[14] V. Damodara Das, P. Gopal Ganesan, Solid State

Commun. 106 (1998) 315.

Fig. 9. The boxed region in Fig. 8 has been shown enlarged.

Both axes are normalized.

Fig. 8. The variation in resistance of thin film passage of time immediately after deposition shown for film thickness (a) 160 nm,

(b) 250 nm and (c) 500 nm.

P. Arun et al. / Physica B 322 (2002) 289–296 295

[15] P. Arun, P. Tyagi, A.G. Vedeshwar, V.K. Paliwal, Physica

B 307 (2001) 105.

[16] V.D. Das, M.S.M. Sastry, M. Pattabi, Phys. Stat. Sol. A 96

(1986) 677.

[17] P.J. Sebastian, V. Sivaramakrishnan, Phys. Rev. B 65

(1989) 237.

[18] P.J. Sebastian, V. Sivaramakrishnan, Phys. Stat. Sol. A

124 (1991) 505.

P. Arun et al. / Physica B 322 (2002) 289–296296