167978-1-4799-5296-0/14/$31.00 © 2014 IEEE

PROC. 29th INTERNATIONAL CONFERENCE ON MICROELECTRONICS (MIEL 2014), BELGRADE, SERBIA, 12-14 MAY, 2014

Analytical Modeling of Thermal Processesin Gas Microsensors Operating at High Temperature

A.G. Kozlov, A.N. Udod

Abstract – The analytical method is presented to determine the temperature distribution in gas microsensors operating at high temperatures. The 2D structure of a gas microsensor is devided into regions, the temperature distribution in which is found by the eigenfunction method. To take into account the temperature dependencies parameters the iteration procedure is used. The values of the parameters are determined using the weighted mean temperature in each region. The application of this method is considered on the example of the gas microsensor based on proton conducting solid electrolyte.

I. INTRODUCTION

The most of gas microsensors operate at high temperature. For example, operating temperatures of gas sensors based on semiconductor oxides can reach 400 0C[1]; gas sensors based on solid state electrolytes have operating temperatures up to 1000 0C [2]. In this case, the key element of these sensors is a heater. The structure of a microsensor and power supply conditions of a heater influence on the values of operating temperature.

In designing the gas microsensors, it is important to choose such values of the design parameters and the power supply conditions for which the normal operating temperature is ensured and the temperature distribution in the region where the gas-sensitive element is placed is uniform. The experimental investigation of the temperature distribution in the structure of gas microsensors presents a difficult problem because of its small dimensions and low power consumption. The only possible way to solve this problem is the mathematical modeling.

The aim of this paper is to present the analytical method allowing one to model the temperature distribution in gas microsensors operating at high temperatures.

II. METHOD FOR DETERMINING THE TEMPERATURE DISTRIBUTION IN GAS MICROSENSORS

The temperature distribution in the structure of gas microsensors is found using the analytical method proposed in [3] for determining the steady-state temperature distribution in the various structures of thermal microsensors. This method has the following algorithm.

1. The 2D structure of a thermal microsensor is

divided into rectangular regions depending on the composition of layers and heat-generating conditions. Each region is replaced by an equivalent region with the homogeneous parameters.

2. For each region, the heat exchange conditions with the environment, the adjacent regions and the support bulk silicon frame are determined.

3. For each region, the steady-state heat deferential equation is defined and then it is solved by eigenfunction method. The heat flux densities between the regions are presented as the sums of orthogonal functions with unknown weighting coefficients.

4. The unknown weighting coefficients are determined using the adjoint boundary conditions between the regions.

For determining the temperature distribution in gas microsensors operating at high temperatures, the method must be modified to take into account the temperature dependencies of a number of parameters: the thermal conductivity of the materials of the layers and the ambient air and the convective coefficient. The modification is to use the iteration procedure that includes the following steps:

1. For each region of the gas microsensor, the temperature distribution is determined, the values of the temperature dependant parameters being equal to the values at the environment temperature and the total surface heat transfer coefficient being determined at the environment temperature.

2. The weighted mean temperature of each region, wmjT , is determined:

jj

l b

jjjjj

j bl

dydxyxTT

j j

0 0wm

,, (1)

where jT is the temperature of the region j ; jl and jbare the length and width of the region j , respectively.

3. The average temperature of the ambient air, airjT ,

over each region is determined:

2

wmenair j

jTT

T

, (2)

where enT is the environment temperature. 4. For each region of the gas microsensor, the values

of the temperature dependant parameters are determined: the parameters of the ambient air are found at the

A.G. Kozlov, A.N. Udod are with the Physical Faculty, Omsk State University, Pr. Mira 55a, 644077, Omsk, Rassia, E-mail: kozlov1407@gmail.com

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temperature equal to airjT ; the thermal conductivities of the

materials of the layers and the convective coefficient, fcjh ,

are found at the temperature equal to wmjT . The expression

for fcjh is obtained by using the similarity theory [4] and

has the following form

15.0air4.0

2.0en

25.0airair75.0airfc 394.0

jj

jjjjj

b

TTgch

, (3)

where airj , air

jc , airj , air

j are the thermal conductivity, the specific heat, the density, and the kinematic viscosity of ambient air over the region j , respectively; is the thermal coefficient of volumetric expansion of ambient air, g is the gravity acceleration. The ambient air parameters are the temperature dependence parameters. Their temperature dependencies and the value of the thermal coefficient of volumetric expansion are given in [5].

5. For each region of the gas microsensor, the total surface heat transfer coefficient at the temperature equal to the weighted mean temperature is determined;

6. For each region of the gas microsensor, the temperature distribution is determined, the values of the temperature dependant parameters and the total surface heat transfer coefficient being equal to the values which have been determined at step 4 and 5, respectively.

Further, the iteration procedure is repeated beginning with step 2. The number of the cycles depends on the required precision.

III. NUMERICAL RESULTS

The present method was used to calculate the temperature distribution in the gas microsensor based on proton conducting solid electrolyte. Figure 1 shows the structure of the microsensor. The microsensor is placed on the suspended thin plate made by anisotropic etching of bulk silicon. The suspended plate has the rectangular shape and is attached to the bulk silicon frame with the help of the four thin narrow beams. On the surface of the suspended plate are placed in series the layer of solid electrolyte, the heater layers and the conductive layers. In this microsensor structure, the heating is done using two heaters located on the two longitudinal sides of the solid electrolyte layer. The layers of the heaters are electric contacts to the solid electrolyte layer to form together with this layer the structure of the resistive gas sensor. Such design allows one to obtain the more uniform temperature distribution in the domain where the gas sensitive layer of solid electrolyte is placed.

The above-mentioned structure of the gas microsensor has the following values of the design parameters: (a) dimensions of the rectangular suspended plate: length - 950

m; width - 100 m; thickness - 1.0 m; (b) dimensions of the beams: length - 200 m; width - 25 m; thickness - 1.0 m; (c) materials of the suspended plate and the beams - silicon dioxide; (d) thickness of the layer of solid electrolyte - 0.15 m; material of the layer of solid electrolyte – BaCeO3; (e) thickness of the resistive layer - 0.15 m; material of the resistive layer - platinum; (f) thickness of the conductive layer - 0.3 m; material of the conductive layer – gold.

Taking into account the mirror symmetry of the structure of the gas microsensor and supposing the homogeneous heat dissipation in the heater the domain of modeling can be mark out, which contains only the one fourth part of the structure. This domain of modeling for the gas microsensor is shown in Fig. 2. In its turn, the domain should be divided into the following regions (Fig. 2):

1) region occupied by the layer of solid electrolyte (region 1);

2) region not occupied by layers (region 2); 3) region occupied by the layer of solid electrolyte

and the resistive layer (region 3), this region 3 is the heat-generating one;

4) region occupied by the resistive and conductive layers (region 4);

5) region occupied by the conductive layer (region 5);

Fig. 1. Structure of the gas microsensor based on proton conducting solid electrolyte: (1) silicon base; (2) etched cavity; (3) thermally isolated structure based on the plate suspended by four bridges; (4) thin-film heater; (5) proton conducting solid electrolyte layer; (6) thin-film conductor; (7) contact pad.

1 2 3 4

4 5 6 7

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In accordance with [3], the temperature distribution in the equivalent regions of the structure can be present as follows. For the regions 1-4 having the Neumann boundary conditions on all boundaries, the temperature distribution is

,coscos

4+

+cos2+

+cos2+

+

1 12

22,

12

2,0

12

20,

20,0

2

j

j

j

j

k= m

jjj

jmk

jjj

j

j

m=

jj

jm

jjj

j

j

k=

jj

jk

jjj

jjjj

j

jjj

jj

bym

lxk

pbm

lk

Dbl

bym

pbm

Dbl

lxk

plk

Dbl

pbl

D

pd

qT

(4)

where

;1

1),(),(),(),(),(),(

),(),(,

vjk

vjujm

ujtjk

tjm

sjm

sjkjmkD

(5)

j is the thermal conductivity of material of the region j ;

jx , jy are the coordinates of the region j ; tjk

, , s,jm

are the weighting coefficients determining the heat flux densities on the boundaries between regions j and t and

between regions j and s , respectively; ),( sj is the coefficient equal to the ratio of the thicknesses of the regions j and s ; k , m are the summation indices on xand y coordinates, respectively.

The temperature distribution in the region 5 is

.2

12sincos

212

14+

+2

12sin

212

12

5

5

5

5

1 1 25

2

5

2

5

)5,4()5,4(

555

5

5

1 25

2

5

)5,4(0

)5,4(

5555

bym

lxk

pb

mlkbl

bym

pb

mblT

k= m

km

m=

m

(6)

The Eqs. (4)-(6) contain the weighting coefficients the values of which are unknown. To find the values of the weighting coefficients one should use the temperature equality conditions on the boundaries between the regions. Combining the equations obtained from these conditions

Fig. 2. Marking out the domain of the modeling from the structure of the gas microsensor: (1) etched cavity; (2) suspended plate; (3) cantilever beams; (4) domain of the modeling.

1 2 3

4

region 4

region 1 region 2

region 3 region 5

Fig. 3. Temperature distribution in the domain of the modeling of the gas microsensor.

170

gives the generalized system of the linear equations for the unknown weighting coefficients which can be conveniently written in matrix representation

ΦM , (7) where M is the matrix of the coefficients; is the vector of the unknown weighting coefficients; Φ is the vector of the right parts. The solution of the system (7) allows one to determine the values of the weighting coefficients. Using these values one can find the temperature distribution in the regions 1-5 (Eqs. (1)-(6)).

Figure 3 shows the temperature distribution in the domain of the modeling of the gas microsensor for the supply current equal to 15 mA. In the present temperature distribution important part is the temperature distribution in region 1, since there the sensitive layer of the proton conducting solid electrolyte is placed. It is desirable to ensure the temperature distribution in the region 1 as uniform as possible. As seen from Fig. 3 the region 1 has a large area with a small temperature gradient.

As an integral characteristic of the temperature distribution in the region 1 one can be used the weighted mean temperature (Eq. (1)). Figure 4 shows the dependence of the weighted mean temperature in the region 1 on the length of the supported beams. The present dependence is nonlinear. With increasing the length of the beams the weighted mean temperature increase in the region 1 is

slowed. Figure 5 shows dependencies of the weighted mean temperature in the region 1 and the heater power on the supply current of the heater. The comparison of these dependencies shows that above the certain value of the supply current the heater power is growing faster than the weighted mean temperature of the region 1.

IV. CONCLUSION

The present method allows one to exactly model thermal processes in gas microsensors operating at high temperature. The method can be used to analyze the influence of design parameters, environment conditions and power supply conditions on the operation parameters of the gas microsensors. The main advantage of the method is to use simple analytical expressions to determine the weighted mean temperatures in regions of gas microsensors. The use of the iteration procedure in the present method of determining the temperature distribution gives the additional advantage which is due to the fact that, for each region of the microsensor, this procedure is independently conducted. This makes it possible to raise the accuracy of the temperature distribution in the microsensors operating at high temperatures.

REFERENCES

[1] I. Simon, N. Bârsan, M. Bauer, U. Weimar, “Micromachined metal oxide gas sensors: opportunities to improve sensor performance”, Sensors Actuators B. Chemical, 2001, vol. 73, pp. 1-26.

[2] A. Dubbe, “Fundamentals of solid state ionic micro gas sensors”, Sensors Actuators B. Chemical, 2003, vol. 88, pp. 138-148.

[3] A. G. Kozlov, “Analytical modeling of steady-state temperature distribution in thermal microsensors using Fourier method. Part 1. Theory,” Sensors Actuators A. Physical, 2002, vol. 101, pp. 283-298.

[4] Y. Jaluria, Natural convection. Heat and Mass Transfer,Oxford: Pergamon Press, 1980.

[5] A. G. Kozlov, “Optimization of structure and power supply conditions of catalytic gas sensor,” Sensors Actuators B. Chemical, 2002, vol. 82, pp. 24-33.

Fig. 5. Dependencies of the weighted mean temperature in the region 1 and the heater power on the supply current of the heater.

0 4 8 12 16 20I (mA)

0

200

400

600

800

1000

T wm

(0 C)

0

5

10

15

20

25

P (m

W)

Fig. 4. Dependence of the weighted mean temperature in the region 1 on the length of the supported beams.

0 100 200 300 400 500Lbeams (m)

630

640

650

660

670

680

690

T wm

(0 C)