ARTICLE IN PRESS

0921-4526/$ - se

doi:10.1016/j.ph

�Correspondifax: +9111 276

E-mail add

agni@physics.d

Physica B 362 (2005) 158–166

www.elsevier.com/locate/physb

Occurrence of hysteresis-like behavior of resistance of Sb2Te3film in the dynamical measurement of heating–cooling cycle

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

Department of Physics and Astrophysics, Thin Film Lab., University of Delhi, Delhi 110 007, India

Received 30 July 2004; received in revised form 4 February 2005; accepted 4 February 2005

Abstract

Experimental observations of a peculiar behavior observed in the dynamical measurement of resistance of Sb2Te3films in heating and cooling cycle at different heating and cooling rates are described. The film regained its original

resistance, forming a closed loop, on the completion of the heating–cooling cycle which was reproducible for identical

conditions of heating and cooling over many such cycles. The area enclosed by the loop was found to depend on (i) the

thickness of the film, (ii) the heating rate, (iii) the maximum temperature to which film was heated and (iv) the cooling

rate. The observations are explained on basis of model which considers the film to be a resultant of parallel resistances.

The film’s finite thermal conductivity gives rise to a temperature gradient along the thickness of the film, due to which

the parallel combination of resistance changes with temperature in accordance with the temperature coefficient of

resistance of the material. Difference in heating and cooling rates give different temperature gradient, which explains

the observed hysteresis. This kind of dynamical measurement of resistance, exhibiting hysteresis could be quite useful in

estimating the thermal conductivity of the material.

r 2005 Elsevier B.V. All rights reserved.

PACS: 73.61.Le

Keywords: Thin films; Chalcogenides and layered compounds

1. Introduction

The electrical conductivity measurement is quitean important characterization technique for thin

e front matter r 2005 Elsevier B.V. All rights reserve

ysb.2005.02.006

ng authors. Tel.: +9111 26468401;

67061.

resses: arunp92@physics.du.ac.in (P. Arun),

u.ac.in (A.G. Vedeshwar).

films and are routinely carried out for variousmaterials ranging from metals to high-resistivesemiconductors. The temperature dependence ofresistivity yields information about intrinsicband gap of material, activation energy forconduction in polycrystalline films (grain bound-ary barrier height), impurity activation energy,etc. In most of these measurements data aretaken either in heating or cooling direction of

d.

ARTICLE IN PRESS

P. Arun et al. / Physica B 362 (2005) 158–166 159

temperature variation. If the system is homoge-neous and the rate of change of temperature isconstant, no hysteresis can be expected in hea-ting–cooling cycle. However, considerable hyster-esis has been observed when an amorphous filmwas heated above crystalline transition tempera-ture and cooled back which can be understood asdue to structural changes [1–3]. Such hysteresishave been observed in Bi films even withoutstructural changes [4]. A hysteresis behavior wasalso observed in Pd film which was ascribed toabsorption and desorption of hydrogen duringheating–cooling cycle [5]. In most cases themeasurement details are not available in theliterature. The measurements under isothermalconditions are very much essential for the accuratedetermination of resistance and its temperaturedependence which maybe expected to give iden-tical results in heating or cooling processes.However, such routine measurements (mostlyusing computer-interfaced data acquisition) aremade continuously during heating or cooling ofthe sample. Data of both temperature andresistance are acquired simultaneously for deter-mining temperature dependence. We call this asdynamical measurement. Therefore, it motivatedus to investigate more thoroughly such dynamicalmeasurement of the temperature dependence ofelectrical resistance in heating–cooling cycles forSb2Te3 film in vacuum. We report here theinteresting observations of hysteretic behavior inrepeated cycles of heating–cooling. The results canbe explained by the parallel resistor model [6].

2. Experimental details

Thin films of Sb2Te3 were grown by thermalevaporation using a molybdenum boat on micro-scope slides glass substrates at room temperatureat a vacuum better than 10�6 Torr: The startingmaterial was 99.99% pure stoichiometric ingotsupplied by Aldrich (USA). Films were grown onthe pre-deposited indium contacts using a maskfor the four probe configuration of resistivitymeasurements. The electrical contacts were con-firmed to be ohmic at room temperature for all thesamples studied. We have checked the nature of

the contacts up to 120 �C for all samples whichdoes not show any considerable deviation fromohmic nature. However, a small change (0.02%) inresistance was observed at 120 �C on voltagereversal. Since this is very much less than theoverall observed change in resistance due totemperature, we neglect this contribution andreasonably assume the contacts to be ohmic. Thissmall change in the interfacial property may wellbe due to the temperature dependence of thechemical potential of the semiconductor used. Thetemperature of the film was measured by athermocouple placed very near to the sample onthe substrate. The structure, chemical compositionand morphology of the films were determined byX-ray diffraction (Philip PW1840 X-ray diffract-ometer), photo-electron spectroscopy (Shimadzu’sESCA 750) and scanning electron microscope(JOEL-840), respectively. The film thickness wasmonitored during its growth by quartz crystalthickness monitor and was subsequently confirmedby Dektak IIA surface profiler, which uses themethod of a mechanical stylus movement onthe surface. The movement of the stylus acrossthe edge of the film determines the step height orthe film thickness. All the films were found to bestoichiometric and micro-crystalline after theywere allowed to age in vacuum for few weeks [7].Various films of thickness between 100 and 500 nmwere used during the experiment. All the heating–cooling cycles for resistivity measurements werecarried out in vacuum of about 10�6 Torr: Thiswas done to prevent any oxidation of the filmswhile heating, which would result in an irreversiblechange in the film resistance. The samples wereheated by placing them on a copper block which isheated by a heating coil embedded in it.The heating rate was varied by the voltage

applied to the heating coil. The temperature andthe resistance of the film was measured as afunction of time. The heater was switched off forcooling the film. The cooling of the film from itselevated temperature was very slow as the mea-surements were done in vacuum. Therefore, whilesome control of the heating rate was maintainedby controlling the voltage applied to the heatingcoil, the cooling rate was essentially determined bythe maximum temperature at which the heater was

ARTICLE IN PRESS

Fig. 1. Variation of temperature with time during the

heating–cooling cycle. Region ‘‘AB’’ and ‘‘BC’’ are indicated.

P. Arun et al. / Physica B 362 (2005) 158–166160

switched off. The heating and cooling rates hence,were largely different.Fig. 1 shows the variation of temperature with

time for both heating and cooling cycles. Thetemperature increases with time as indicated insegment ‘‘AB’’. The heater was switched off onattaining the pre-determined temperature. Point‘‘B’’ corresponds to the instant the heater wasswitched off. This is followed by a period ofconstant temperature ‘‘BC’’. The onset of coolingprocess starts at point ‘‘C’’. The cooling of thefilms in vacuum could be either by radiation lossesor by conduction through the substrate side orboth. After the heater is switched off, since theprocess is in vacuum, the substrate temperatureremains constant for an appreciably long timebefore it starts falling. From Fig. 1, we find thatthe fall in substrate temperature commences 200 slater after the heater is switched off. Beyond point‘‘C’’ the cooling process starts. However, asexplained earlier, since the cooling process takesplace in vacuum, it is very slow, hence the figuredoes not show the attainment of room temperatureon cooling. Data of region ‘‘AB’’ is best fitted bythe equation

T subðtÞ ¼ Tmaxð1� e�QtÞ þ T rt. (1)

The equation shows saturation of temperaturewith time when the heating due to the heater iscompensated by cooling. T rt is the room tempera-ture. The magnitude of Tmax and ‘‘Q’’ are relatedin a complex manner to the heater voltage, cooling

mechanism, etc. An estimate of these factors aremade by fitting Eq. (1) to the experimental datashown in Fig. 1. The value of coefficients Tmax;‘‘Q’’ and T rt were estimated as 360 �C; 0:00039 s�1

and 14:5 �C; respectively. From Eq. (1), the rate ofheating during the heating cycle is given by

dT sub

dt¼ TmaxQe�Qt. (2)

Similarly, the cooling rate can be found by fittingdata of the cooling region by the equation

T subðtÞ ¼ T finale�St þ T rt, (3)

where T final is the final temperature attained beforethe heater is switched off. In the present workT finaloTmax: The cooling rate is governed by ‘‘S’’and in our case since cooling was occurringnaturally in vacuum (10�6 Torr), ‘‘S’’ was verylow �0:00021 s�1:

3. Results and discussions

The resistance of as grown film was found to behigh and decreases with time saturating in a fewweeks [7]. No further change in resistance wasobserved with time thereafter. The as grownamorphous films were found to age into apolycrystalline state, with peak positions matchingthose given in ASTM card 15-874. The filmresistance follows a hysteresis path with heating–cooling cycle which was reproducible underidentical conditions. Reproducibility was testedas many as 20 times in some samples. The filmswere heated to various temperatures below 120 �C(T final) in vacuum except one sample which washeated to 250 �C to demonstrate the dependence ofhysteresis on T final: It was ensured no change inmaterial properties like structural, compositionalor morphological took place by repeated char-acterizations after each cycle or few cycles. Thiswas quite crucial in ruling out the contributionsfrom these parametric changes to irreversibility orhysteretic behavior of resistance with temperature.Fig. 2 shows the X-ray diffractograms of a 210 nm,film after repeated heating–cooling cycles. As canbe seen from the figure, even after the seventh cycleof heating and cooling, there was no change in the

ARTICLE IN PRESS

Fig. 2. X-ray diffractograms of Sb2Te3 film after (a) one, (b)

two, (c) six and (d) seven heating–cooling cycles.

Fig. 3. Variation of resistance with temperature during the

heating and cooling cycles for films of thickness (a) 1300 A (b)

1800 A (c) 2100 A (d) 3800 A and (e) 5000 A. The maximum

temperature to which the films were heated and the heating rate

was same in both cases. Regions ‘‘AB’’, ‘‘BC’’ and ‘‘CD’’ are

indicated.

P. Arun et al. / Physica B 362 (2005) 158–166 161

crystal structure or any improvement in the grainsize, etc. Physical and chemical changes hence canbe ruled out as to be occurring due to theheating–cooling cycles. For further comprehensiveunderstanding of the hysteretic behavior, we havecarried out experiments varying only one para-meter and keeping others same as described below.

3.1. Film thickness dependence

Films of different thickness were heated at thesame heating rate to reach identical maximumtemperature (T final). Fig. 3 shows the hysteresispaths taken during the heating and coolingprocesses for various film thicknesses. We havemarked three different regions as ‘‘AB’’, ‘‘BC’’ and‘‘CD’’ on the hysteresis loop. The resistancedecreases with increasing temperature (region‘‘AB’’) indicating the semiconducting nature ofthe Sb2Te3 film, which is a p-type narrow band gapmaterial [8]. Point ‘‘B’’ marks the point where theheater is switched off. Even though the heater wasswitched off, the temperature of the sample does

not decrease immediately as the measurementswere done in vacuum (see Fig. 1). However, theresistance continues to fall till point ‘‘C’’ at thealmost constant temperature within some span oftime. Thus, variation in resistance in region ‘‘BC’’is with time. Hence, in Fig. 4, we show thevariation of resistance with time. As can be seen,the heater is switched off at ‘‘B’’ and the resistanceof the sample continues to decrease with time tillthe onset of the cooling process marked by point‘‘C’’. It may be noted that the measured tempera-ture in the region ‘‘BC’’ was constant as shown inFig. 3 and the decrease in resistance looks to bevery steep in this region. Fig. 4 depicts the samefact, that is, increasing resistance with decreasingtemperature in cooling process beyond ‘‘C’’ as seenin Fig. 3. The increase in resistance during coolingin Fig. 3 (between ‘‘CD’’) is almost parallel to thedecrease in segment ‘‘AB’’. At point ‘‘D’’, the filmreaches room temperature and the film resistance

ARTICLE IN PRESS

Fig. 4. Variation of resistance with time during the heating–

cooling cycle. Regions ‘‘AB’’ and ‘‘BC’’ are indicated.

Fig. 5. Area enclosed under the hysteresis loop of different film

thickness.

Fig. 6. A Sb2Te3 film of thickness 2100 A was heated to two

different T final: 110 and 250 �C at same heating rate.

P. Arun et al. / Physica B 362 (2005) 158–166162

goes back to point ‘‘A’’, very slowly, over a longtime (�8–9 h). The regions ‘‘BC’’ and ‘‘DA’’ arequite puzzling, where the temperature is constantand resistance is varying with time. It is evidentthat films of different thickness films enclosedifferent area under the loop. Fig. 5 shows thevariation of the area enclosed under the loop for

various thicknesses. The graph was plotted usingthe measured area enclosed by various loops ofFig. 3.

3.2. Dependence of final temperature and cooling

rate

Samples of identical thicknesses were heatedwith the same heating rate but to different finaltemperatures by switching off the heater atdifferent temperatures (T final). The segment‘‘AB’’ in such cases coincided. However, the lengthof the segment ‘‘BC’’ varied. Fig. 6 compares twocases (for clarity only two are shown) where thesame film was heated at the same heating rate, butto two different T final: It should be noted that theexperimental error in the determination of resis-tance could be slightly larger due to the slightdeviation of ohmic nature of the contact attemperatures higher than 120 �C: Still the mainbehavior of the resistance will not be affected bythis and this is the only sample heated to such ahigh temperature for the purpose of demonstra-tion of the dependence of hysteresis of T final: Thecooling rate was not controllable in our present

ARTICLE IN PRESS

P. Arun et al. / Physica B 362 (2005) 158–166 163

experiment, as it was allowed for natural coolingin vacuum. It is clear that the cooling rate stronglydepends on T final: From the above two cases it isimplied that the area under the loop is alsodependent on the cooling rate of the films. Theexplicit dependence of cooling rate could also bestudied with a convenient and controllable coolingarrangement which is not possible in the presentstudy.

3.3. Dependence of heating rate

To understand the effect of heating rate on thearea enclosed by the loop, films of same thicknesswere heated at different heating rates, as shown inFig. 7. From Eq. (2), the heating rate also varieswith time. It’s magnitude depends on Tmax and Q,both of which can be controlled or selected by thevoltage applied to the heater. It is evident fromFig. 7, dR=dT is greater for lower heating rate.The slope is larger for lower rate of heating,resulting in smaller area enclosed. Hence, the areaenclosed increases with increasing heating rate. Inother words the rate of change of resistance, or thethermal coefficient of resistance depends on theheating rate.

Fig. 7. Variation of resistance in the heating and cooling cycle

for a film of thickness 3800 A. The heating rate was kept

different by supplying the heater different voltage. Parameter

‘‘Q’’ of Eq. (1) used were (a) 4:82� 10�4 s�1 and (b) 1�

10�5 s�1:

In summary the area under the hysteresis loophence, was found mainly to depend on thefollowing parameters of the experiment:

(a)

the film thickness, (b) the heating rate, (c) the final temperature (T final) that the sample

was heated to and,

(d) the rate of cooling.

Results of all the above experiments are quitereproducible under identical experimental condi-tions over many thermal cycles. This fact alsosupports our earlier observation of reasonablystable properties and good quality of the interfaceat the contact.Most of these observations can be explained

qualitatively using the theory in Ref. [6]. However,we briefly outline the theory and the model below.

4. Theory

We first calculate the temperature profile alongthe film thickness which need not be uniformespecially during heating/cooling process in adynamic or transient measurement. Then thecalculation of the total resistance of the film as afunction of temperature can simply be carried outby integrating across the film thickness. Thisshould be the key in explaining the observedhysteresis behavior. As described earlier the film iskept on copper block being heated. Heatingproceeds from the substrate side. Therefore, thetemperature varies along the film thickness withtime which is essentially a one-dimensional pro-blem of heating conduction across the filmthickness. The variation of temperature with timeand spatial co-ordinates is given by Carslaw andJaeger [9]

cv

qT

qt¼ l

q2Tqx2

, (4)

where l is the thermal conductivity of the film andcv is the specific heat of the film. A solution of thispartial differential equation depends on the initialand boundary conditions of the problem. Depend-ing on the initial and boundary conditions solution

ARTICLE IN PRESS

Fig. 8. Hysteresis loops formed in film resistance with the

heating–cooling cycle. The calculations were done for film

thickness of 1000 A and diffusivity (i) 5� 10�3 A2=s; (ii) 50 A2/

s, (iii) 5� 102 A2/s and (iv) 5� 103 A2/s.

P. Arun et al. / Physica B 362 (2005) 158–166164

would be different [10]. For the given experimentalconditions the variation of temperature withspatial and time co-ordinates is given by Arunand Vedeshwar [6]. The variation in temperaturealong the film thickness with time is given as

Tðx; tÞ ¼ T sub � ðT sub � T surÞ sinpx

2d

� �e�p2Dt

4d2 , (5)

where D is the thermal diffusivity (l=cv) and d isthe film thickness. The temperature profile acrossthe film thickness can be calculated using Eq. (5).At the starting of heating, T sur in the equation cansimply be taken as room temperature with T sub

slightly hotter by few degrees. Every time a newresulting T sur is used along with the incrementedT sub: Thus, the profile can be calculated numeri-cally. T sub serves as the heat source. Obviously, thedifference between T sur and T sub would increasewith decreasing D or l of the given materialexhibiting a quite non-uniform temperature dis-tribution along the film thickness at the giveninstant of time. The time for reaching equilibriumor uniform distribution is also inversely propor-tional to l of the material because cv does not varymuch from material to material at high tempera-tures.The film can be thought of as a stack of

numerous infinitesimal identical thin layers ofsame thickness. All the layers acting as resistiveelements with the net resistance of the film as theresistance in parallel combination of these layers.Since the layers are identical, at room temperatureall of them have equal value. However, due to themetallic/semiconducting nature of the film, theresistance of these layers vary with temperature.For simplicity, the variation of resistance withtemperature is taken linear as

Rlayer ¼ R0ð1þ aTÞ (6)

where a and R0 are the temperature coefficient ofresistance (TCR) and the resistance of the identicallayers, respectively. For the case T ¼ 0 �C; thefilms resistance would be given as

1

Rfilm¼

Xi¼n

i¼1

1

R0¼

n

R0. (7)

The TCR is positive for metal while it is negativefor semiconductors. Since, spatial distribution of

temperature along thickness was calculated forvarious substrate temperatures at various instant,the films resistance can be trivially calculated as afunction of substrate temperature and time.We have calculated the film resistance as a

function of temperature as described above usingEqs. (5)–(7). We have taken 100 nm thick film as astack of 10 identical layers in parallel combinationwith each layer’s resistance of 170KO at roomtemperature and a ¼ �0:8� 10�3 �C�1: Thesenumerical values are taken from our previousstudy on Sb2Te3 films [11]. The results are plottedin Fig. 8 for varying diffusivity or mainly thethermal conductivity. The visual examination ofFig. 8 reveals a peaking behavior of hysteresis looparea with thermal conductivity of the film. Wehave, therefore, plotted the hysteresis loop areaexclusively as a function of diffusivity in Fig. 9.The loop area shows a maximum at intermediatediffusivity. This may look very surprising on theonset. However, there is a striking similaritybetween Fig. 5 and Fig. 9, that is dependence ofloop area on film thickness and thermal conduc-tivity. Fig. 9 is the direct consequence of varyingthermal conductivity as calculated by the abovemodel resulting from the temperature profileacross the film thickness. The film resistance, theparallel combination of identical resistive layerswould crucially depend on the temperature profileacross the thickness. The results could be almostsimilar for very low thermal conductivity and veryhigh thermal conductivity due to nearly uniform

ARTICLE IN PRESS

Fig. 9. The variation in the area enclosed by loops formed during the resistance variation with temperature during heating–cooling

cycles. The variation is due to the difference in the films diffusitivity.

Fig. 10. Variation of resistance with time during the heating–

cooling cycle in Bi2Te3:

P. Arun et al. / Physica B 362 (2005) 158–166 165

temperature profile. Therefore, an increased orenhanced loop area for moderate thermal con-ductivity seems to be quite reasonable arising dueto quite non-uniform temperature profile acrossfilm thickness.The similarity between the behavior of loop area

with film thickness and thermal conductivity mayalso be expected because many physical propertieslike thermal conductivity show thickness depen-dence [12]. The exact dependence may vary frommaterial to material. In the present study, Sb2Te3films are semi-metallic and shows a linearly inverserelation of resistance with film thickness [11], inthe range of Fig. 5. Since films are semi-metallic,by Wiedemann–Franz law, we can see that thermalconductivity varies linearly with film thickness.Therefore, the experimental result of thicknessdependence of loop area shows an analogousbehavior to that predicted by the calculateddependence of loop area on thermal conductivity.However, the resistivity of Sb2Te3 films is slightlylarger due to its polycrystalline nature. Still we feelit is within the applicability of Wiedemann–Franzlaw. Also, the thermal conductivity used in themodel calculation is the total of lattice andelectron contributions. Further, we have observedhysteresis loops even in the amorphous films ofBi2Te3 as shown in Fig. 10. Similarly, thepolycrystalline InSb films also show hysteresis(not shown for brevity) although the hysteresiswidths are quite different for all three materialsunder the same experimental conditions and

parameters. This fact indicates assertively the roleof thermal conductivity in the formation ofhysteresis loops. The present model explains quitewell qualitatively the features of the hysteresisbehavior. At present we have not fitted theexperimental data with the model due to thenon-availability of few material parameters re-quired. Alternatively, one can estimate thermalconductivity of the film across its thickness byfitting the experimental data and the model. Itwould be same along parallel and perpendiculardirections of the film for isotropic materials anddiffer for anisotropic materials. We are pursuingfew other different materials in this directionalong with quantitative analysis and fitting of

ARTICLE IN PRESS

P. Arun et al. / Physica B 362 (2005) 158–166166

experimental data for the determination of ther-mal conductivity. However, the detailed analysiswill be the subject for future publication.

5. Conclusions

The routine dynamical (or non-equilibrium)measurement of temperature dependent resistanceis shown to follow different paths in heating andcooling directions forming a closed hysteresis loop.The occurrence of the hysteresis is mainly ascribedto the thermal conductivity of the material. Thisalso explains qualitatively the dependence ofhysteresis on other experimental parameter likefilm thickness, heating/cooling rate, etc. quiteconsistently. The present model and explanationsmay well be suited for the materials with in thelimits of Wiedmann–Franz law. Therefore, suchdynamical measurements may be quite useful inestimating the thermal conductivity of the materialby fitting data to the present model.

References

[1] V. Damodara Das, D. Karunakaran, Phys. Rev. B 39

(1989) 10872.

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

Commun. 106 (1998) 315.

[3] V. Damodara Das, S. Selvaraj, J. Appl. Phys. 83 (1993)

3696.

[4] K. Jayachandran, C.S. Menon, Pramana 50 (1998) 221.

[5] Y. Sakamoto, I. Takashima, J. Phys.: Condens. Matter 8

(1996) 10511.

[6] P. Arun, A.G. Vedeshwar, Phys. Lett. A 313 (2003)

126.

[7] P. Arun, P. Tyagi, A.G. Vedeshwar, Physica B 307 (2001)

105.

[8] I. Lefebvre, M. Lannoo, G. Allan, A. Ibanez, J. Fourcade,

J.C. Jumas, E. Beaurepaire, Phys. Rev. Lett. 59 (1987)

2471.

[9] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids,

Oxford University Press, Oxford, 1954.

[10] S.M. Sze, Semiconductor Devices, Physics and Technol-

ogy, Wiley, New York, 1993.

[11] P. Arun, A.G. Vedeshwar, Mater. Res. Bull. 38 (2003)

1929–1938.

[12] L.I. Maissel, R. Glang, Handbook of Thin Film Technol-

ogy, McGraw-Hill, New York, 1970.