Research Article...

Hindawi Publishing CorporationAdvances in OptoElectronicsVolume 2011, Article ID 896962, 4 pagesdoi:10.1155/2011/896962

Research Article

Modeling of Photoconductivity of Porous Silicon

L. S. Monastyrskii, B. S. Sokolovskii, M. R. Pavlyk, and P. P. Parandii

Ivan Franko National University of L’viv, 50 Dragomanov Street, 79005 Lviv, Ukraine

Correspondence should be addressed to L. S. Monastyrskii, [email protected]

Received 13 December 2010; Accepted 12 May 2011

Academic Editor: Samir K. Mondal

Copyright © 2011 L. S. Monastyrskii et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The paper investigates a model of the photoconductivity of macroporous silicon in the conditions of homogeneous generationof photocarriers. By the finite element method, the stationary photoconductivity and the time evolution of photoconductivityafter instantaneous shutdown of light are calculated. Dependences of the stationary photoconductivity and relaxation time ofphotoconductivity on the velocity of recombination of nonequilibrium carriers at the surfaces of pores, radius of pores, andaverage distance between them are analyzed.

1. Introduction

One of urgent today’s problems is the control of toxicand other harmful substances in air atmosphere, drinkingwater, foods, and so forth. In this connection, there isa necessity for development of microelectronic systemsof detection of gases for industrial, office, and housingapartments which can be components of integral systemsof protection of human’s life and health. Of special interestis the sensors based on nanoporous semiconductors, inparticular porous silicon. Gas sensors of such type aredescribed in a number of papers [1–4]. Among suchsensors, one can distinguish the structures from poroussilicon, based on the change of photoconductivity and itskinetics under the action of different gas environmentswhich can be used as effective gas analyzers. For designingsuch sensors, it is necessary to know dependences ofphotoconductivity of porous silicon on the state of surface,depending on a gas environment where it is, as well ason the degree of porosity of a material and geometry ofpores.

In this paper, for the case of macroporous silicon withcylindrical pores, modeling of stationary photoconductivityand process of its relaxation occurring after instantaneousshutdown of illumination is carried out.

2. Formulation of the Problem

Let us consider a plate of macroporous silicon with arrangedparallel cylindrical pores of radius r0, which are perodicallydistributed in the semiconductor volume with the averagedistance between their centers being equal to 2R. These poresform a quadratic lattice in the xOy plane lying perpendicularto the axis of the cylindrical pores. At calculating, we shallsuppose the origin of coordinates is on axis of one ofthe pores. This semiconductor, for concreteness of p-typeconductivity, is illuminated by light from the region of fun-damental absorption for which the function of generation ofphotocarriers G, that is, the amount of photocarriers whichare generated for one second in unit volume does not dependon the coordinate. It should be noted that the condition ofhomogeneous generation of photocarriers is satisfied in thecase of weak absorption of light.

The generated photocarriers recombine both in thesemiconductor volume and at the surfaces of pores. Atconstant illumination, it has been established such a nonuni-form spatial distribution of photocarrier concentration atwhich balance between the processes of generation andrecombination of nonequilibrium charge carriers is ensured.

In the case of diffusion transport of photocarriers, thedependence of the photocarrier concentration Δn(x, y, t) on

2 Advances in OptoElectronics

the coordinates x, y, and time t can be determined from theequation

τnL2n

∂Δn

∂t= ∂2Δn

∂x2+∂2Δn

∂y2− Δn

L2n

+τnG f (t)

L2n

, (1)

where Ln is the diffusion length of electrons, τn is theirlifetime, and f (t) is the function of photocarrier generationnormalized on its maximum value G.

Equation (1) should be complemented by the boundaryconditions

1√x2 + y2

[x∂Δn

∂x+ y

∂Δn

∂y

]∣∣∣∣∣x2+y2=r2

0

= SτnL2nΔn

∣∣∣∣∣x2+y2=r2

0

,

(2)

∂Δn

∂x

∣∣∣∣x=±R

= 0, (3)

∂Δn

∂y

∣∣∣∣∣y=±R

= 0. (4)

Equation (2) corresponds to recombination of photocarrierson the surface of cylindrical pore with the surface recombina-tion velocity S which depends on the state of surface of pores,in particular, on the kind and of the adsorbed moleculesof a gas. Equations (3) and (4) follow from the symmetryconditions of the problem which give rise to the fact that,at the faces of “elementary cell” (x = ±R, y = ±R), thephotocarrier concentration reaches its maximum value.

The initial condition for (1) can have different forms,particularly, in the case of instantaneous shutdown ofillumination, it takes the following form:

Δn(x, y, t

)∣∣t=0 = Δn0

(x, y

), (5)

where Δn0(x, y) is the stationary solution of (1).The total number of photocarriers N in the plate per its

unit thickness can be calculated as

N =∫∫

Δn(x, y, t

)dx dy, (6)

where integration is over the region with area A representedas a square with side 2R with a central circular cutoutof radius r0. It should be noted that the total number ofphotocarriers is proportional to the photoconductivity ofporous silicon.

For convenience, we further pass to the following dimen-sionless values:

x∗ = x

Ln, y∗ = y

Ln, t∗ = t

τn, r∗0 =

r0

Ln,

R∗ = R

Ln, S∗ = Sτn

Ln, N∗ = N

GτnA.

(7)

3. Calculation Results and Discussion

3.1. Stationary Photoconductivity. In the stationary case, theleft-hand side of (1) is equal to zero and, on the right-hand side, f (t) = 1. The transformed equations (1)–(5)were solved numerically by the finite element method [5]which is especially effective in the calculation of systemswith complicated geometric configuration. The calculateddependences of the total number of photocarriers on theradius of pores and the ratio of distances between pores totheir radius for different values of the surface recombinationvelocity are shown in Figure 1.

It is seen from Figure 1 the photoconductivity of poroussemiconductor at fixed nonzero value of the velocity ofsurface recombination decreases with increasing the pore’sradius and increases with rising the average distance betweenthem. Increase of the velocity of surface recombination at thefixed values of pore’s radius is accompanied by decreasing thephotoconductivity which is saturated at very large values ofthe surface recombination velocity.

Along with the numerical modeling, we have carried outanalytical calculation for the case of cylindrical symmetry ofthe problem which takes place when R � Ln. In this case

with using the independent variable r =√x2 + y2, (1)–(4)

reduce to the following ones:

1r

d

dr

(rdΔn

dr

)− Δn

L2n= −Gτn

L2n

,

dΔn(r0)dr

= SτnL2nΔn(r0),

dΔn(R)dr

= 0,

(8)

whose solution via the dimensionless values defined byexpressions (7) is

Δn(r∗)Gτn

= 1− S∗[I1(R∗)K0(r∗) + K1(R∗)I0(r∗)]I1(R∗)

[K1(r∗0)

+ S∗K0(r∗0)]− K1(R∗)

[I1(r∗0)− S∗I0

(r∗0)] , (9)

where In,Kn are the Bessel’s functions of imaginary argument[6].

Comparison of the results of numerical and analyticalcalculations shows that, at > 2Ln, the latter coincide

with an error less than 1%. When approaching R to 0.5Ln the error increases to 30%, but using the correctioncoefficients introduced by us in [7], the error amounts to 10–15%.

Advances in OptoElectronics 3

5040

30

20

10

0

0.2

0.4

0.6

0.8

1

0.04

0.08

0.120.16

0.2

N∗

R/r0 r0/Ln

(a)

5040

30

20

10

0 0.2

0.4

0.6

0.8

1

0.04

0.08

0.120.16

0.2

N∗

R/r0 r0/Ln

(b)

Figure 1: Dependences of total number of photocarriers on r0/Ln and R/r0. S∗ = 5 (a), S∗ = 200 (b).

32.5

21.5

10.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

50

4030

2010

0

S∗R/Ln

τ rel/τ

n

(a)

32.5

21.5

10.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

50

4030

2010

0

S∗R/Ln

τ rel/τ

n

(b)

Figure 2: Dependences of photoconductivity relaxation time on R/Ln and S∗; r0 = 0.1 (a), r0 = 0.05 (b).

3.2. Nonstationary Case of Photoconductivity. We shall ana-lyze the nonstationary photoconductivity for the case ofinstantaneous shutdown of light falling on the semiconduc-tor plate, that is, when the function f (t) featured on theright-hand side of (1) has the following form:

f (t) = H(−t), (10)

where H(t) is the Heaviside’s function whose value is zero fornegative argument and one for positive argument.

The total number of photocarriers N∗ was calculatedby the finite element method for different values of sur-face recombination velocity and porous silicon geometricalparameters. It has been obtained that the increase of S∗ leadsto the decrease of both the stationary value N∗ and thetime of photoconductivity relaxation, with at lower values

of the distance between the pores these changes being moresignificant.

The calculated dependences N∗(t) allow us to determinethe relaxation time τrel as a slope of these dependencespresented in the semilogarithmic scale. The obtained resultspresented in Figures 2(a) and 2(b) show that τrel nonlinearlydecreases with increasing S∗ and the most significantdecrease of τrel takes place when S∗ value is of the orderof one. It turns out that, at the expense of changing thesurface recombination velocity, one can reach decreasing therelaxation time by almost one order of magnitude. The latteroccurs in the materials with high level of porosity when thearea of pore’s surface per unit volume is large. The values ofvariation of the relaxation time realized in porous silicon atthe surface recombination velocity exceeding the velocity of

4 Advances in OptoElectronics

photocarrier diffusive motion (Ln/τn) are quite acceptable tobe registered by simple measurement techniques.

4. Conclusions

In this paper, we have calculated by the finite elementmethod the stationary photoconductivity and the time ofphotoconductivity relaxation for the case of macroporoussilicon with cylindrical pores. It has been shown that bothstationary photoconductivity and time of photoconductivityrelaxation significantly depend on the surface recombinationvelocity of photocarriers, radius of pores, and averagedistance between pores what can be used for manufacturinggas sensors. Photoconductivity of macroporous silicon atfixed nonzero value of the velocity of surface recombinationdecreases with increasing the pore’s radius and increases withrising the average distance between them. Photoconductivityrelaxation time nonlinearly decreases with increasing thevelocity of surface recombination, and the most significantdecrease of relaxation time takes place when value of thevelocity of surface recombination is of the order of one.

References

[1] L. T. Canham, “Silicon quantum wire array fabrication byelectrochemical and chemical dissolution of wafers,” AppliedPhysics Letters, vol. 57, no. 10, pp. 1046–1048, 1990.

[2] L. Monastyrskii, T. Lesiv, and I. Olenych, “Composition andproperties of thin solid films on porous silicon surface,” ThinSolid Films, vol. 343-344, no. 1-2, pp. 335–337, 1999.

[3] C. Baratto, G. Faglia, G. Sberveglieri et al., “Multiparametricporous silicon sensors,” Sensors, vol. 2, no. 3, pp. 121–126, 2002.

[4] S. J. Kim, J. Y. Park, S. H. Lee, and S. H. Yi, “Humidity sensorsusing porous silicon layer with mesa structure,” Journal ofPhysics D, vol. 33, no. 15, pp. 1781–1784, 2000.

[5] O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, The Finite ElementMethod: Its Basis and Fundamentals, Elsevier Butterworth-Heinemann, Oxford, UK, 2005.

[6] G. N. Watson, A Treatise on the Theory of Bessel Functions,Cambridge University Press, Cambridge, UK, 2nd edition,1995.

[7] L. S. Monastyrskii, B. S. Sokolovskii, and V. S. Vasylyshyn,“Calculation of photosensitivity of porous silicon with thecylindrical geometry of pores,” Optical Memory and NeuralNetworks, vol. 18, no. 1, pp. 55–59, 2009.

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