eresis-likeependalculated

ndy in the

Physics Letters A 313 (2003) 126–131

www.elsevier.com/locate/pla

Hysteresis-like behavior of resistivity of thin filmsin heating–cooling cycle

P. Arun, A.G. Vedeshwar

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

Received 18 December 2002; received in revised form 9 April 2003; accepted 17 April 2003

Communicated by R. Wu

Abstract

The temperature profile along the film thickness is analyzed and presented for understanding the reproducible hystbehavior in resistivity of polycrystalline films in heating–cooling cycle. The temperature profile was found to crucially don the thermal parameters of the film, nature of the heating source and rate of heating/cooling. The film resistance cin heating–cooling cycle using the temperature profile shows hysteresis-like loops. The present analysis could be haqualitative estimation of thermal conductivity of the film from resistivity measurements in heating–cooling cycles. 2003 Elsevier Science B.V. All rights reserved.

PACS: 73.61; 73.61.G; 81.40.C

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n-ti-ssis-hileledsis-

ture

ult-w-ob-

ol-behenduenotsis-

ro-x-ityraliornd

er-be-

1. Introduction

Among the transport properties the electrical retivity has been quite important and common in chacterizing conductors or semiconductors in both band thin film form. The temperature dependenceresistivity yields information about the materials itrinsic band gap, impurity activation energy, the acvation energy for conduction in polycrystalline filmdue to grain boundary barrier height and so on. Retance measurements are normally made either wthe material is being heated or while it is being coofrom an elevated temperature. One expects the retance of a material to depend only on the tempera

E-mail addresses: arunp92@physics.du.ac.in,arunp92@yahoo.co.in (P. Arun), agni@physics.du.ac.in(A.G. Vedeshwar).

0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rigdoi:10.1016/S0375-9601(03)00650-9

and not on the rate of change of temperature, resing the heating and cooling cycles to coincide. Hoever, the temperature dependence of resistivity wasserved to follow different paths in heating and coing cycles [1–4]. This kind of difference can easilyexpected and understood for amorphous films wheated above crystalline transition temperature asto structural changes [1–3]. In such cases films doregain their initial resistance and hence the hysterelike loop does not form completely and is not repducible. Surprisingly, some polycrystalline films ehibit reproducible hysteresis-like loops in resistivduring heating–cooling cycles without any structuchanges, regaining its initial value. Such a behavwas observed in bismuth films [4] when heating acooling rates were kept different. This is quite intesting and since no attempt is made to explain thishavior, we attempt to do the same in this Letter.

hts reserved.

P. Arun, A.G. Vedeshwar / Physics Letters A 313 (2003) 126–131 127

romin

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2. Theory

We consider the heating process to proceed fsubstrate side to the surface of the film as shownFig. 1 indicating the initial conditions. The spatial atemporal variation of the temperature along the fithickness in this process can be given by [5]

(1)cv∂T

∂t= λ

∂2T

∂x2,

where λ is the thermal conductivity andcv is thespecific heat of the film. The solution of this partdifferential equation crucially depends on the initand boundary conditions of the problem. Dependon the initial and boundary conditions the solutiwould be quite different [6]. The initial condition fothe film of thicknessd being heated from the substraside would be given as

(2)T (x = 0, t = 0) = Tsub,

(3)T (x = d, t = 0) = Tsur.

After a long time the film would be uniformly heatewith the surface attaining the same temperature asof the substrate, i.e.,

(4)T (x,∞) = Tsub.

From the given conditions, we search for a solutionthe form

(5)T (x, t) = g(x)h(t).

Or in other words, we use the method of separavariables. The solution of Eq. (1) can be expressed

(6)T (x, t) = a + b sin

(πx

2d

)e− π2Dt

4d2 ,

where D (= λ/cv) is the thermal diffusivity. Onapplying the conditions stated in (2) and (3) tsolution may be written as

(7)T (x, t) = Tsub− (Tsub− Tsur)sin

(πx

2d

)e− π2Dt

4d2 ,

where d is the film thickness. Eq. (7) can be usto calculate the spatial and temporal variationtemperature along the film thickness for the givsubstrate temperature. For numerical calculationsheating process,T (x, t) and the resulting newTsurhave to be calculated for each changing value

Fig. 1. Direction of heat flow and initial condition of temperatureboth surfaces of the film.

the substrate temperature. Therefore, the variaof substrate temperature (source) with time isimportant parameter (rate) in the measuremencan be determined by the nature of heating/coosources employed in the measurement. We take mcommonly employed source of heating which isresistive heating. The measured variation ofTsub withtime was found to be following the equation

(8)Tsub(t) = Tsat(1− e−Qt

) + Tr.

HereTr is the room temperature,Tsat is the maximumtemperature attainable (which saturates) when hefor a long time andQ is the rate of change. In resistivheating Tsat can be varied by changing the dc/voltage applied to it. The rate of change of tempera(Q) depends onTsat. Thus, we can investigate theffect ofQ in the resistivity measurements.

We have numerically determined the distributiof temperature along the thickness of 1000 Å usEqs. (7) and (8), for different thermal diffusivitiesshown in Fig. 2. The various curves in the figureeach thermal diffusivity represents the evolution oftemperature distribution (or profile) at various instaof time (indicated in the figure caption) as heating pceeds. The three thermal diffusivities chosen herefer in order of magnitudes to demonstrate the donant role of the thermal conductivity because specheat does not vary quite considerably from samplsample as compared to thermal conductivity. We hused the valuesTsat = 360◦C, Tr = 15◦C andQ =3.9×10−4 s−1 for the numerical computations. In facFig. 2 mainly illustrates the dominant role of thermconductivity of the film in the evolution of temperatudistribution along its thickness. As thermal conductity becomes poorer the non-homogeneity in tempe

128 P. Arun, A.G. Vedeshwar / Physics Letters A 313 (2003) 126–131

0 Å

d

ea-n-

re-cengfterac-ore,

timeuesay

tartdis-gm-in

cem.ally

ated to

ateing.ainhis

, thetingood

blenceing

in

isfilmthisticalk-ersm-ncena-mthethe

Fig. 2. Variation of temperature along the thickness of a 100thick film of different diffusivity (a) 5× 10−3 Å/s, (b) 5× 102 Å/sand (c) 5× 103 Å/s after (i) 0 s, (ii) 200 s, (iii) 400 s, (iv) 600 s an(v) 800 s of substrate heating.

ture distribution increases. In such situation any msured physical parameter of the film could exhibit uexpected temperature dependence.

Normally, the temperature dependence of filmsistivity is measured in situ to avoid possible surfaoxidation. When heater is switched off after attainithe desired temperature, the actual cooling starts asome time because of low heat loss by radiation in vuum or conduction towards substrate side. Theref

substrate temperature remains constant for someeven after the heater is switched off and this continto act as a constant heat source for the film. We mcall this time gap between the end of heating and sof cooling as dead time. However, the temperaturetribution along the film thickness will still be evolvindue to the temperature gradient. This evolution of teperature distribution during dead time is illustratedFig. 3 for various instant of time. Not much differencan be seen for very low thermally conducting filOn the other hand, for moderate and good thermconducting film,Tsur increases.

Once the cooling starts, the variation of substrtemperature with time can be expressed as (as fitteexperimental data)

(9)Tsub(t) = Tme−Qt + Tr,

whereTm is the maximum temperature of the substrat the end of heating and before the start of coolThe temperature distribution during cooling can agbe calculated numerically using Eqs. (7) and (9). Tis illustrated in Fig. 4 for the same values ofD as takenearlier. However, we have takenQ = 1.2 × 10−4 s−1

because of low heat losses in vacuum. Thereforein situ measurements inherently have different heaand cooling rates. Again, the moderate and gthermally conducting films show quite higherTsur thanTsub as time elapses. Therefore, it is quite reasonato expect the physical parameter like film resistato behave abnormally during in situ heating–coolcycles. Therefore, we analyze the film resistancethis context.

3. Film resistance

Since the temperature along the film thicknessnon-uniform, the temperature dependence of theresistance should be evaluated accordingly. Forpurpose the film can be considered as stack of identhin layers of same thickness. The whole film thicness can be divided into convenient number of layas suitable for the analysis. At uniform constant teperature of the film each layer has identical resistaand the total film resistance is the parallel combition of all thin layers. However, in case of non-unifortemperature distribution along the film thicknessresistance of individual layer differs depending on

P. Arun, A.G. Vedeshwar / Physics Letters A 313 (2003) 126–131 129

0 Å

,off.

ionre

0 Å

,ng.

ceallue

llic.

Fig. 3. Variation of temperature along the thickness of a 100thick film of different diffusivity (a) 5× 10−3 Å/s, (b) 5× 102 Å/sand (c) 5× 103 Å/s after (i) 0 s, (ii) 40 s, (iii) 80 s, (iv) 120 s(v) 160 s and (vi) 200 s after the source of heating was switched

temperature of that layer. Quite generally, the variatof the resistance of individual layer with temperatucan be given by

(10)R(T ) = Ro(1+ αT ),

Fig. 4. Variation of temperature along the thickness of a 100thick film of different diffusivity (a) 5× 10−3 Å/s, (b) 5× 102 Å/sand (c) 5× 103 Å/s after (i) 8000 s, (ii) 16 000 s, (iii) 24 000 s(iv) 32 000 s and (v) 40 000 s after the setting in of the films cooli

whereα is the temperature coefficient of resistan(TCR) and Ro is the resistance of each identiclayer. Ro can be taken as room temperature vaand appropriateα can be considered for the filmdepending on whether it is semiconducting or metaAt T = 0 ◦C the film resistance is simply given by

(11)1

R=

i=n∑i=1

1

Ro= n

Ro.

130 P. Arun, A.G. Vedeshwar / Physics Letters A 313 (2003) 126–131

at-k-

thecles.

lmoveg. 5.10

es

entg

ssestainase6

d asr-

theu-lieroes

sti-heith-

turef

nce

ut-

fs.

Rtes.ofts a.

Fig. 5. Hysteresis loops formed in film resistance with the heing–cooling cycle. The calculations were done for film thicness of 1000 Å and diffusivity (i) 5× 10−3 Å2/s, (ii) 50 Å2/s,(iii) 5 × 102 Å2/s and (iv) 5× 103 Å2/s.

Fig. 6. The variation in the area enclosed by loops formed duringresistance variation with temperature during heating–cooling cyThe variation is due to the difference in the films diffusivity.

We have numerically calculated the variation of firesistance in heating–cooling cycle using all the abnecessary details and the results are shown in FiWe have taken a 1000 Å thick film consistingresistive layers withRo = 170 k� andα = −0.80×10−3/◦C (semiconducting). These numerical valuare taken from our earlier study on Sb2Te3 films [7].The various hysteresis loops are due to differthermal diffusivities (identified in the caption) keepinall other parameters constant.

The hysteresis loop opens up and encompalarger area as thermal diffusivity increases to cervalue and the area under the hysteresis curve decrethereafter. We show this fact more explicitly in Fig.where the area under the hysteresis loop is plottea function of thermal diffusivity. The peaking at a ce

s

tain thermal diffusivity at moderate value suggestsimportance of non-uniformity in temperature distribtion along the film thickness. As we have seen eara considerable change in temperature distribution dnot occur for either very lowD or very highD duringheating–cooling cycles. Therefore, the present invegation indicates a moderate thermal diffusivity for tmaterials exhibiting resistance hysteresis loops wout phase change.

Another important consequence of the temperadistribution along the film thickness is the TCR. Ocourse, it does not matter at all for highD which canalso be shown mathematically. Writing the resistafor anyT as

(12)1

R= 1

Ro

n=d/a∫i=0

di

(1+ αT ),

where a is the thickness of individual layer. Substiting T = Tsub from Eq. (7), we get

Ro

R=

n∫0

[(1+ αTsub)

(13)− α(Tsub− Tsur)e− π2Dt

4d2 sin

(π

2ni

)]−1

di.

The solution of the above integral for highD is givenby

(14)R = Ro

n(1+ αT )

or

(15)dR

dt= Ro

n

(αPQe−Qt

)using Eq. (8). Therefore,

(16)TCRfilm = n

Ro

dR

dT= α.

The TCR of the film is simplyα the value ofthe individual layer. Therefore, TCR of the film ohigh D will be independent of heating/cooling rateHowever, for lower thermal diffusivity values, TCcan considerably depend on heating–cooling raWe illustrate this fact in Fig. 7 where two casesdifferentD are compared in (A) and (B) for differenheating rates. Finally, we have calculated TCR afunction of thermal diffusivity as shown in Fig. 8

P. Arun, A.G. Vedeshwar / Physics Letters A 313 (2003) 126–131 131

ture

e ofdate

a--

er-ter

t fit-opnd

tab-uc-te-vi-ing.

eerous

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89)

. 106

6.

ford

ogy,

es.

Fig. 7. Figure exhibits the change of resistance of with temperafor (A) a poor thermal conducting film and (D = 5×102 Å/s) (B) agood thermally conducting film (D = 5 × 103 Å/s). The heatingrates were maintained different (i) 3.6 × 10−3 ◦C/s, (ii) 72◦C/sand (iii) 216◦C/s.

It shows the importance of increasing dependencTCR on the decreasingD. Therefore, the measureTCR may not truly represent the heating/cooling rindependent value for moderate and low-D materials.

This analysis could be quite useful for the estimtion of thermal conductivity from the resistivity hysteresis loop if the specific heat is known. The thmal conductivity can be used as the fitting parameto the experimental hysteresis loop data. The besting value of thermal conductivity to the observed locould be very close to the materials value. More acareful experimental data may be required for eslishing this procedure of determining thermal condtivity. Detailed experiments are in progress for marials of both known and unknown thermal conductities. At present, it sounds encouraging and interest

Fig. 8. Computed TCR for films of different diffusivity, where thfilms are assumed to be of same thickness and made up of numlayers, with all the layers having the same TCR.

4. Conclusions

The temperature distribution along the film thicness seems to be the origin of forming hysteresis-loop of resistivity of polycrystalline films in heatingcooling cycle. The temperature profile crucially dpends on the rate of heating/cooling and so doesnature of hysteresis loop. The present analysis prethe pronounced hysteresis behavior for materials hing moderate thermal diffusivity or thermal conductity having comparable specific heats. The dependeof temperature coefficient of resistance increasesdecreasing thermal diffusivity or thermal conductiviThe qualitative estimation of thermal conductivitythe film material is quite possible from the hystereloop if specific heat is known.

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Univ. Press, Oxford, 1954.[6] S.M. Sze, Semiconductor Devices, Physics and Technol

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