Modeling of Photoconductivity of Porous Silicon

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


Introduction
One of urgent today's problems is the control of toxic and other harmful substances in air atmosphere, drinking water, foods, and so forth.In this connection, there is a necessity for development of microelectronic systems of detection of gases for industrial, office, and housing apartments which can be components of integral systems of protection of human's life and health.Of special interest is the sensors based on nanoporous semiconductors, in particular porous silicon.Gas sensors of such type are described in a number of papers [1][2][3][4].Among such sensors, one can distinguish the structures from porous silicon, based on the change of photoconductivity and its kinetics under the action of different gas environments which can be used as effective gas analyzers.For designing such sensors, it is necessary to know dependences of photoconductivity of porous silicon on the state of surface, depending on a gas environment where it is, as well as on the degree of porosity of a material and geometry of pores.
In this paper, for the case of macroporous silicon with cylindrical pores, modeling of stationary photoconductivity and process of its relaxation occurring after instantaneous shutdown of illumination is carried out.

Formulation of the Problem
Let us consider a plate of macroporous silicon with arranged parallel cylindrical pores of radius r 0 , which are perodically distributed in the semiconductor volume with the average distance between their centers being equal to 2R.These pores form a quadratic lattice in the xOy plane lying perpendicular to the axis of the cylindrical pores.At calculating, we shall suppose the origin of coordinates is on axis of one of the pores.This semiconductor, for concreteness of p-type conductivity, is illuminated by light from the region of fundamental absorption for which the function of generation of photocarriers G, that is, the amount of photocarriers which are generated for one second in unit volume does not depend on the coordinate.It should be noted that the condition of homogeneous generation of photocarriers is satisfied in the case of weak absorption of light.
The generated photocarriers recombine both in the semiconductor volume and at the surfaces of pores.At constant illumination, it has been established such a nonuniform spatial distribution of photocarrier concentration at which balance between the processes of generation and recombination of nonequilibrium charge carriers is ensured.
In the case of diffusion transport of photocarriers, the dependence of the photocarrier concentration Δn(x, y, t) on the coordinates x, y, and time t can be determined from the equation where L n is the diffusion length of electrons, τ n is their lifetime, and f (t) is the function of photocarrier generation normalized on its maximum value G. Equation ( 1) should be complemented by the boundary conditions Equation ( 2) corresponds to recombination of photocarriers on the surface of cylindrical pore with the surface recombination velocity S which depends on the state of surface of pores, in particular, on the kind and of the adsorbed molecules of a gas.Equations ( 3) and ( 4) follow from the symmetry conditions of the problem which give rise to the fact that, at the faces of "elementary cell" (x = ±R, y = ±R), the photocarrier concentration reaches its maximum value.The initial condition for (1) can have different forms, particularly, in the case of instantaneous shutdown of illumination, it takes the following form: where Δn 0 (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 where integration is over the region with area A represented as a square with side 2R with a central circular cutout of radius r 0 .It should be noted that the total number of photocarriers is proportional to the photoconductivity of porous silicon.
For convenience, we further pass to the following dimensionless values:

Calculation Results and Discussion
3.1.Stationary Photoconductivity.In the stationary case, the left-hand side of ( 1) is equal to zero and, on the righthand 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 systems with complicated geometric configuration.The calculated dependences of the total number of photocarriers on the radius of pores and the ratio of distances between pores to their radius for different values of the surface recombination velocity are shown in Figure 1.
It is seen from Figure 1 the photoconductivity of porous semiconductor at fixed nonzero value of the velocity of surface recombination decreases with increasing the pore's radius and increases with rising the average distance between them.Increase of the velocity of surface recombination at the fixed values of pore's radius is accompanied by decreasing the photoconductivity which is saturated at very large values of the surface recombination velocity.
Along with the numerical modeling, we have carried out analytical calculation for the case of cylindrical symmetry of the problem which takes place when R L n .In this case with using the independent variable r = x 2 + y 2 , (1)-( 4) reduce to the following ones: whose solution via the dimensionless values defined by expressions (7) is where I n , K n are the Bessel's functions of imaginary argument [6].
Comparison of the results of numerical and analytical calculations shows that, at > 2L n , the latter coincide with an error less than 1%.When approaching R to 0.5 L n the error increases to 30%, but using the correction coefficients introduced by us in [7], the error amounts to 10-15%.

Nonstationary Case of Photoconductivity.
We shall analyze the nonstationary photoconductivity for the case of instantaneous shutdown of light falling on the semiconductor plate, that is, when the function f (t) featured on the right-hand side of (1) has the following form: where H(t) is the Heaviside's function whose value is zero for negative argument and one for positive argument.The total number of photocarriers N * was calculated by the finite element method for different values of surface recombination velocity and porous silicon geometrical parameters.It has been obtained that the increase of S * leads to the decrease of both the stationary value N * and the time of photoconductivity relaxation, with at lower values of the distance between the pores these changes being more significant.
The calculated dependences N * (t) allow us to determine the relaxation time τ rel as a slope of these dependences presented in the semilogarithmic scale.The obtained results presented in Figures 2(a) and 2(b) show that τ rel nonlinearly decreases with increasing S * and the most significant decrease of τ rel takes place when S * value is of the order of one.It turns out that, at the expense of changing the surface recombination velocity, one can reach decreasing the relaxation time by almost one order of magnitude.The latter occurs in the materials with high level of porosity when the area of pore's surface per unit volume is large.The values of variation of the relaxation time realized in porous silicon at the surface recombination velocity exceeding the velocity of photocarrier diffusive motion (L n /τ n ) are quite acceptable to be registered by simple measurement techniques.

Conclusions
In this paper, we have calculated by the finite element method the stationary photoconductivity and the time of photoconductivity relaxation for the case of macroporous silicon with cylindrical pores.It has been shown that both stationary photoconductivity and time of photoconductivity relaxation significantly depend on the surface recombination velocity of photocarriers, radius of pores, and average distance between pores what can be used for manufacturing gas sensors.Photoconductivity of macroporous silicon at fixed nonzero value of the velocity of surface recombination decreases with increasing the pore's radius and increases with rising the average distance between them.Photoconductivity relaxation time nonlinearly decreases with increasing the velocity of surface recombination, and the most significant decrease of relaxation time takes place when value of the velocity of surface recombination is of the order of one.