a hysteresis model for a vanadium dioxide transition-edge microbolometer

1030 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 4, AUGUST 2001

A Hysteresis Model for a Vanadium DioxideTransition-Edge Microbolometer

Luiz Alberto Luz de Almeida, Member, IEEE, Gurdip Singh Deep, Senior Member, IEEE,Antonio Marcus Nogueira Lima, Member, IEEE, Helmut Franz Neff, and Raimundo Carlos Silvério Freire

Abstract—This paper presents the adaptation of the Preisachmodel, originally developed for magnetic hysteresis, to describemathematically the hysteresis in the resistance-temperature char-acteristics of vanadium–dioxide (VO2) thin film radiation sensors.The necessary and sufficient conditions for the applicability of thePreisach model to a VO2 film sensor are experimentally verified.Experimentally measured characteristics are compared with thosegiven by the model for minor and major loops.

Index Terms—Bolometer, hysteresis modeling, metal-insulatortransition, Preisach model, thermal radiation, vanadium–dioxide.

I. INTRODUCTION

T HE vanadium-dioxide (VO) thin film sensor has been em-ployed recently as a bolometer in the nonhysteretic part of

the resistance-temperature ( ) characteristics. The oper-ating point was set at about 25C using external cooling by aPeltier element [1]. However, this sensor can also be used as abolometer within the transition region between 40C to 60 C,thus exploiting its high resistance-temperature coefficient by ad-equately biasing the sensor and adjusting the operation point byJoule heating.

VO is characterized by a temperature-driven, metal-insu-lator phase transition [2] from a low temperature semiconductorto a high temperature metallic phase. Fig. 1 shows the experi-mental characteristic curve of VO, when the film temper-ature is raised from 20C to 80 C and subsequently reduced to20 C. This structural phase transition exhibits a crytallographictransformation that is accompanied by significant change of theelectrical and optical film properties [2].

There are some difficulties in biasing and operating the sensorin the hysteretic portion of the curve. The properties ofmajor hysteresis loops of VOthin film have been reported inthe literature [3], but there is very little information availableabout minor loops, and the influence of the thermal history onthe hysteresis trajectories [4]. Toward a theoretical evaluationof the VO bolometer performance, operating in the hysteretictransition region, a valid mathematical model is required to fullydescribe the major, as well as the minor hysteresis loops.

Manuscript received May 4, 2000; revised May 23, 2001.L. A. L. de Almeida is with the Departamento de Engenharia Elétrica,

Universidade Federal da Bahia and with the Departamento de EngenhariaElétrica, Universidade Federal da Paraíba, Campina Grande, PB, Brazil(e-mail: [email protected]).

G. S. Deep, A. M. N. Lima, H. F. Neff, and R. C. S. Freire are with the De-partamento de Engenharia Elétrica, Universidade Federal da Paraíba, CampinaGrande, PB, Brazil (e-mail: [email protected]; [email protected];[email protected]; [email protected]).

Publisher Item Identifier S 0018-9456(01)07891-3.

Fig. 1. R � T characteristic of a VOfilm.

There exist various physical and mathematical models to de-scribe hysteretic phenomena. Some of these models are derivedfor magnetic hysteresis, like the Preisach model [5]. Others,such as the Jiles model [6], explain magnetization in terms ofthe movement of magnetic domain walls. However, due to itsmathematical generality, the Preisach model is widely acceptedas a suitable tool to formally describe hysteresis phenomena.

In the following, we propose the adaptation of the Preisachmodel to describe the thermal hysteresis of a VOthin filmthat applies for a bolometric sensor. The original Preisach tri-angle and relay operator are redefined, along with adapted in-terpretations of its properties. Eventually, the experimentallyobtained VO hysteresis curves were verified for the applica-bility of the Preisach model. A surface is proposed to fit a set ofexperimental first order descending (FOD) curves, in order toderive the Preisach distribution or weighting function. Finally,the model performance is evaluated by comparing experimentalminor and major loops with those obtained from the model.

II. PREISACH HYSTERESISMODEL FORVO

The original classical Preisach model was proposed in1935 [7]. The model was conceived on a intuitive basis [5]and is based on some plausible hypotheses about the mag-netization mechanism. Later, M. Krasnoselskii [8] showedthat the Preisach model is quite general, and it is possible toadapt it to other than magnetic hysteretic phenomena. Thereis experimental evidence that a thin VOfilm is composed

0018–9456/01$10.00 © 2001 IEEE

DE ALMEIDA et al.: HYSTERESIS MODEL FOR A VANADIUM DIOXIDE TRANSITION-EDGE MICROBOLOMETER 1031

Fig. 2. Graphical representation of the relay operator ̂ for VO .

of microcrystals, which individually exhibit a very sharphysteretic transition [9]. The overall film characteristicis associated with the existence of an ensemble of eithermetallic or semiconducting microcrystals of VO. This led usto investigate the adaptability of Preisach model for thermalhysteresis to VO thin films. The classical Preisach model forVO allows us to describe the film resistance as a sum ofsimple relay operators , representing microcrystal statesand weighted by a statistical distribution function .It is given by

(1)

Fig. 2 graphically represents the relay operator . The stateof these operators depends on sensor temperature.

A simplified graphical interpretation of (1) can be achievedin terms of the so called Preisach triangle adapted forthis sensor (Fig. 3). This right triangle is associated with onlythe hysteretic portion of the characteristics, lying be-tween 20 C and 80 C. In other words, the distribution func-tion is assumed to be zero outside triangle .This right triangle can be divided into two parts: and rep-resenting the region where the relay operators are in the states1 (semiconducting) and 0 (insulating), respectively. As an ex-ample, if the device temperature is reduced from 80C toraised to , and again reduced to , the interface line be-tween the areas and is as shown in Fig. 3. The verticesof interface line are related to the past extreme values of

. It is thus evident that and depend on the history ofthermal cycling.

Considering 1 for and 0for , then (1) can be written as

(2)

III. PREISACH MODEL REPRESENTATION

For the use of the Preisach model representation to describea given hysteretic characteristic, some conditions must befullfiled. These conditions can be stated in terms of the Preisachtriangle presented in Fig. 3. Mayergoyz showed [5] that thewiping-out property and congruency property constitute thenecessary and sufficient conditions to be satisfied for a hys-

Fig. 3. Preisach triangle adapted for VOthermal hysteresis.

Fig. 4. Experimental verification of the wiping-out property in VO: Inset: thethermal excitation waveform used in this verification.

teresis nonlinearity to be represented by the Preisach modelfor a set of piecewise monotonic excitations. Consideringthe Preisach triangle depicted in Fig. 3, we can redefine thewiping-out property, adapted for the VOcase, as:

Wiping-Out Property: Each local minimum of wipesout the vertices of , whose coordinates are above thisminimum, and each local maximum of wipes out the ver-tices of whose coordinates are below this maximum.

Consider a piecewise monotonic excitation (insetFig. 4), in an arbitrary time interval . This time intervalis composed by several subintervals , during which thetemperature excitation is monotonic. Each subinterval has alocal maximum in and a local minimum in . Now we canstate the wiping-out property as:

Each new temperature maximum at the instantwipesout any temperature maximum at an arbitrary previous in-stant when , and each new tem-perature minimum at the instant wipes out any temper-


(a)

(b)

Fig. 5. Experimental verification of the congruency property for differentminor loops: (a)R � T characteristics and (b)log(R) � T characteristics.Inset: the thermal excitation waveform used in this verification.

ature minimum at an arbitrary previous instantwhen .

The wiping-out property was experimentally verified for theVO film. Fig. 4 shows experimental curves. From thewiping-out property, we should expect that pointsand ,as well as the points and be coincident. This happens ina very close way suggesting that the wiping-out property holds.

Fig. 5(a) depicts three pairs of minor loopsand generated by corresponding three pairs of back-and-forth temperature variations and . Each pairof temperature variation has the same extremum valuesand

[inset zoom Fig. 5(a)], and occur around different temper-ature values. The variationsand produce the minor loops

and , respectively. The minor loops and are verticallyshifted in the plane due to the thermal history. The sameoccurs with the minor loop pairs and . Based onthese experimental observations, the congruency property canbe redefined for the VOfilm as:

(a)

(b)

Fig. 6. (a) Experimental FOD curve and (b) the corresponding Preisachtriangle.

Congruency Property:All minor loops, vertically shiftedand produced by a back-and-forth variation of temperatureexcitation between the same maximum and minimum

values are congruent.The congruency property was also experimentally verified for

the VO film. Fig. 5(a) presents the minor loops in theplane and corresponding excitation (shown in the inset).We can see that the congruency property does not hold ( ,

and ). This is primarily due to the asym-metric characteristic of the VOhysteresis curve. However, ifa plot is used, the minor loops can be seen moredistinctly over the whole range of resistance variation, and thenoncongruency effect is reduced, i.e., , , and

, as shown in Fig. 5(b).

IV. OBTAINING THE DISTRIBUTION FUNCTION

A procedure, proposed by Mayergoyz for determining thedistribution function , employs the experimentalfirst-order descending transition (FOD) curves [5], where theterm descending is related to the decreasing resistance. AnFOD curve is generated by first increasing the temperatureto its maximum value of 80C, where the resistance value isaround . Next, the temperature is monotonicallydecreased until it reaches some value with a resistance

as shown in Fig. 6(a), with the corresponding Preisachtriangle depicted in Fig. 6(b). Then, the temperature is increased


monotonically to some value where the sensor resistance is.

The increase of temperature from to removes the region[Fig. 6(b)] from and adds it to the area . This region

corresponds to the change in the sensor resistance given by

(3)

The surface integral over region can be written as the fol-lowing double integral

(4)

Considering that there is a fixed for each FOD curveand by differentiating (4) twice, the weighting function at anypoint can be determined [5] as

(5)

Applying the logarithmic transformation on the resistance data,the weighting or distribution function is defined by

(6)

V. CHOICE OF THEFOD SURFACE

Initial attempts to fit a low-order polynomial surface to theexperimentally obtained FOD curves did not yield satisfactoryresults. This is primarily due to the highly nonlinear and asym-metric characteristics of the sensor hysteresis. Polynomial sur-faces of higher order match the measured data points, but exhibithighly oscillatory behavior between the measured data points.This is in conflict with the smooth hysteresis characteristics ob-served experimentally.

Some intuitive attempts have indicated that each FOD sur-face may be adequately fitted by using a two step parameter op-timization procedure. In the first step it is assumed that thethFOD curve can be described by

C (7)

with

The upper temperature limit 80C is indicated in the Preisachtriangle shown in Fig. 3. The parameter vector for theth FODcan be determined by solving the following parameter optimiza-tion problem

(8)

where represents the experimental FOD data andis the subset of the 3 dimensional spacethat defines the validparameter search region.

At the end of the first step, we have a set oftridimensionalparameter vectors. The next step is to establish the dependenceof the components of these vectors with. To characterize thisrelationship the following functions have been used

(9)

(10)

(11)

In this case, the parameter vector

is also determined by solving the following parameter optimiza-tion problem

(12)

where , , and represent the values obtainedfrom the solution of the first optimization problem and isthe subset of the 9 dimensional space that defines the validparameter search region.

With these two steps, the FOD curves can be represented bya surface given as

(13)

whose second derivative (6) yields the estimated distributionfunction , thus concluding the hysteresis modelingprocedure. Also, considering the high temperature saturation re-sistance and , (2) can berewritten as

(14)

A. Numerical Procedure

In order to fit the experimental data, the Nelder–Mead algo-rithm [10] was employed to solve both parameter optimizationproblems described in the previous section. This is a relativelysimple nonlinear minimization algorithm that tends to convergeto a local minimum. Consequently, this algorithm requires a rel-atively good initial guess. Both parameter optimization prob-lems were solved in the MATLAB environment by using the

function that implements the Nelder–Mead method.

VI. EXPERIMENTAL RESULTS

Fig. 7(a)–(c) shows the fit of the functions ,and to the estimated value sets ,

, and , respectively. Theestimation of each one of the eighteen parameter vectors


(a)

(b)

Fig. 8. Comparison between the proposed model and experimental data for (a)minor loops and (b) major loop.

has employed at least 200 resistance-tempera-ture measurement pairs collected over each one of the eighteenFOD curves. A good agreement in the fit of andis visually observable in Fig. 7(b) and (c), respectively.

Fig. 7(a) shows a good agreement in the fit of fortemperatures up to 45C, but after this point, the fit is poor.Thus, the estimate of the parameters for the surface describedby (13) are the following: , , ,

, , , , ,, and .

In order to improve the fit of , an attempt has been madeto fit parameters to directly to FOD experimental data.Several other cost and penalty functions, along with differenttypes of norms, have been tested, but the simplex algorithm didnot converge to a better result.

From the estimated surface given by (13), the distributionfunction was obtained using (6). Employing a proce-dure developed in [5], the resistance can be then calculatedfor an arbitrary excitation using (14). To verify the modelquality, two minor and a major loop were experimentally ob-tained and compared with those generated from the model, as

(a)

(b)

(c)

Fig. 7. Functions (a)f(T ), (b) g(T ), and (c)h(T ) estimated from thevectors issued from the first parameter optimization stepfa ; a ; . . . ; a g,fb ; b ; . . . ; b g, andfc ; c ; . . . ; c g respectively.

we can see in Fig. 8(a) and (b), respectively. The data generatedfrom the proposed model agrees very well with experimentaldata for the minor loops [Fig. 8(a)]. Furthermore, we also ob-serve good agreement for a major loop over the range of 35Cto 80 C [Fig. 8(b)].

VII. CONCLUSIONS

The applicability of the classical Preisach model for thethermal hysteresis in VOthin films has been investigated.We focused on the hysteresis in the phase transitionregion of the sensor characteristics. For the necessary andsufficient conditions to be satisfied for the applicability ofthe Preisach model, a characteristic was foundto be more suitable than the characteristic curve,which is asymmetrical. A procedure for determining the modelparameters has been described. Model simulations have showngood agreement with experimental data for the minor loops andmajor loops over a limited temperature range. However, dueto the asymmetric behavior of the sensor hysteresis (Fig. 1),experimental data at low temperatures do not agree with themodel. The model is thus valid in the transition region wherethe resistance hysteresis is relevant.

ACKNOWLEDGMENT

The authors thank CNPq (Conselho Nacional de Desenvolvi-mento Científico e Tecnológico) and CAPES (Fundação Coor-denação de Aperfeiçoamento de Pessoal de Nível Superior) forthe award of research and study fellowship during the course ofthese investigations.


REFERENCES

[1] A. P. Gruzdeva, V. V. Zerov, and O. P. Konovalova, “Bolometric andnoise properties of elements for uncooled IR arrays based on vanadiumdioxide films,” J. Opt. Technol., vol. 64, pp. 1110–1113, Dec. 1997.

[2] H. Jerominek, F. Picard, and D. Vincent, “vanadium dioxide films for op-tical switching and detection,”Opt. Eng., vol. 32, pp. 2092–2099, Sept.1993.

[3] H. S. Choi, J. S. Ahm, J. H. Jung, T. W. Noh, and D. H. Kim, “Mid-infrared properties of a VOfilm near the metal-insulator transition,”Phys. Rev. B, Condens. Matter, vol. 54, pp. 4621–4628, Aug. 1996.

[4] J. R. Sun, G. H. Rao, and J. K. Liang, “Thermal history de-pendent electronic transport and magnetic properties in bulkLa Nd Ca MnO ,” Physica Status Solidi, vol. 163, pp.141–147, Sept. 1997.

[5] I. D. Mayergoyz, Mathematical Models of Hysteresis. New York:Springer-Verlag, 1991.

[6] D. C. Jiles and D. L. Atherton, “Ferromagnetic hysteresis,”IEEE Trans.Magn., vol. 19, pp. 2183–2185, Sept. 1983.

[7] F. Preisach, “ber Die Magnetische Nachwirkung,”Zeitschrift fr Physik,vol. 94, p. 277, 1935.

[8] M. Krasnoselskii and A. Pokrovskii, Systems withHysteresis. Moscow: Nauka, 1983.

[9] I. A. Khakhaev, F. A. Ghudnovskii, and E. B. Shadrin, “Martensitic ef-fects in the metal-insulator phase transition in a vanadium dioxide film,”Phys. Solid State, vol. 36, June 1994.

[10] J. A. Nelder, “Simplex method for function minimization,”Eng.Technol. Appl. Sci., vol. 15, no. 15, pp. 12–12, 1979.

Luiz Alberto Luz de Almeida (M’00) was born October 17, 1962. He receivedthe M.Sc. degree in electrical engineering from Universidade Federal da Bahia,Salvador, Brazil, in 1995. He is currently a Ph.D. student at the Departmentof Electrical Engineering, Universidade Federal da Paraíba, Campina Grande,Brazil.

From 1986 to 1993, he worked at industry where he gained experience inthe design of analog/digital instrumentation and control systems. He joined theDepartment of Electrical Engineering, Universidade Federal da Bahia, Salvador,Brazil, in 1996, where he has been an Assistant Professor for telecommunicationsystems. His current research interests include signal processing and modelingof sensors and actuators.

Gurdip Singh Deep(M’76–SM’84) was born December 12, 1937. He receivedthe B. Tech. (Hons.) degree in electrical eng. from Indian Institute of Technology(IIT), Kharagpur, India, in 1959, the M.E. degree in power engineering (elec-trical) from the Indian Institute of Science, Bangalore, India, in 1961, and thePh.D. in electrical engineering from IIT Kanpur, India, in 1971.

From 1961 to 1965, he worked as an Assistant Professor in Guru Nanak Engi-neering College Ludhiana, and from 1965 to 1972, he was with the IIT, Kanpur,as a Lecturer/Assistant Professor. Since July 1972, he has been a titular Pro-fessor at the Centre of Science and Technology of Federal University of Paraíba,Campina Grande, Brazil. Presently, he is the coordinator of the Electronic In-strumentation and Control Laboratory of the University. He was a Consultantfor Encardio-rite Electronics (Pvt) Ltd., India, during 1969–1970. His researchinterests are electronic instrumentation and microcomputer-based process con-trol.

Antonio Marcus Nogueira Lima (S’77–M’89) was born in Recife, Pernam-buco, Brazil, in 1958. He received the B.S. and M.S. degrees in electrical en-gineering from Federal University of Paraíba, Campina Grande, Paraíba, Brazilin 1982 and 1985, respectively. He received his doctoral degree in 1989 fromInstitut National Polytechnique de Toulouse, Toulouse, France.

Since September 1983, he has been a Professor in the Electrical EngineeringDepartment, Federal University of Paraíba. His research interests are in thefields of electrical machine drives, power electronics, electronic instrumenta-tion, control systems, and system identification.

Helmut Franz Neff was born on May 10, 1948. He received the doctoral degreein physics from University of Berlin, Berlin, Germany in 1981.

He is a solid state physicist, with long standing research on micro-structuredtransition edge devices and sensors. He is presently working at VIR-TECH,Denmark, on the development of biochemical sensors, using surface plasma res-onance principles.

Raimundo Carlos Silvério Freire was born on October 10, 1955, in Poço dePedra-RN, Brazil. He received the B.S. degree in electrical engineering fromthe Federal University of Maranhão, São Luis, Brazil, in 1980, and the M.S.degree in electrical engineering from Federal University of Paraíba, CampinaGrande, Brazil, in 1982. He received his doctoral degree in electronics, au-tomation, and measurements at the National Polytechnical Institute of Lorraine,Nancy, France, in 1988.

He worked as an Electrical Engineer for Maranhão Educational Television,Brazil, from 1980 to 1983. He was a Professor of electrical engineering at Fed-eral University of Maranhão from 1982 to 1985. Since December 1989, he hasbeen on the faculty of the Electrical Engineering Department of the Federal Uni-versity of Paraíba. His research interests include electronic intrumentation andsensors, and microcomputer-based process control.

a hysteresis model for a vanadium dioxide transition-edge microbolometer

Documents