Electronic characterization of plasma-thick n-type silicon using neural networks and photoacoustic response

Abstract

In this paper, electronic semiconductor characterization using reverse-back procedure was applied to different photoacoustic responses aiming to find effective ambipolar diffusion coefficient and a bulk lifetime of the minority carriers. The main idea was to find the small fluctuations in investigated parameters due to detecting possible unwanted sample contaminations and temperature variations during the measurements. The mentioned procedure was based on the application of neural networks. Knowing that in experiments the contaminated surfaces of the sample can play a significant role in the global recombination process that we are measuring and that the unintentionally introduced defects of the sample crystal lattice could vary the carrier lifetime by several orders of magnitude, a method of PA signal adjustment by the reverse-back procedure is developed, based on the changes of the carrier electronic parameters.

Appendices

Appendix 1: Photogenerated density calculation

The Solution of the Eq. (5) can be found in the form (Markushev et al. 2018; 2019):

$$\delta \,n_{p} \left( {x,f} \right) = A_{ + } e^{{x{\kern 1pt} L^{{ - {\kern 1pt} 1}} }} + A_{ - } e^{{ - \,x{\kern 1pt} L^{{ - {\kern 1pt} 1}} }} + A_{\beta } e^{ - \beta \,x} ,$$

(15)

The $A_{\beta }$ is the solution of the inhomogeneous part of the differential equation, and $A_{ + }$ and $A_{ - }$ are the integration constants which are given in the form:

$$A_{ \pm } = \pm \frac{{A_{\beta } }}{{A_{L} }}\left[ {\left( {1 \mp \sigma_{1} } \right)\,\left( {1 + \beta {\kern 1pt} L\sigma {}_{2}} \right)e^{{ - \beta {\kern 1pt} l}} - \left( {1 \pm \sigma_{2} } \right)\left( {1 + \beta {\kern 1pt} L\sigma_{1} } \right)e^{{ \pm lL^{ - 1} }} } \right]$$

(16)

$$A_{L} = \left( {1 + \sigma_{1} } \right)\left( {1 + \sigma_{2} } \right)e^{{lL^{ - 1} }} - \left( {1 - \sigma_{1} } \right)\left( {1 - \sigma_{2} } \right)e^{{ - lL^{ - 1} }}$$

(17)

$$A_{\beta } = \frac{{I_{0} }}{{\varepsilon v_{D} }}\frac{\beta L}{{L^{ - 2} - \beta^{2} }}$$

(18)

calculated using the boundary conditions.

$$\left. {D_{p} \frac{{{\text{d}}\delta n_{p} \left( x \right)}}{{{\text{d}}x}}} \right|_{x = 0} = s_{1} \delta n_{p} \left( 0 \right)\;{\text{and}}\;\left. {D_{p} \frac{{{\text{d}}\delta n_{p} \left( x \right)}}{{{\text{d}}x}}} \right|_{x = l} = - s_{2} \delta n_{p} \left( l \right)$$

(19)

where $\sigma_{i} = D_{p} \left( {s_{m} L} \right)^{ - 1}$ is the dimensionless parameter which depends on the surface recombination speed s_m at front (m = 1) and back side (m = 2) surfaces of the sample, and $v_{D} = D{}_{p}L^{{ - {\kern 1pt} 1}}$ is the diffusion velocity of the photogenerated minority carriers.

Appendix 2: Temperature calculation

The solution of the Eq. (6) can be written as (Markushev et al. 2018; 2019):

$$T_{s} \left( {x,f} \right) = b_{1} e^{{\sigma_{i} x}} + b_{2} e^{{ - \sigma_{i} x}} + b_{3} \delta n_{p} \left( {x,f} \right) + b_{4} e^{ - \beta x} ,$$

(20)

where

$$\begin{gathered} b_{1} = \frac{{\coth \left( {\sigma_{i} l} \right) - 1}}{{2k\sigma_{i} }}\left[ {b_{3} k\left( {\frac{{{\text{d}}\delta n_{p} \left( x \right)}}{{{\text{d}}x}}} \right)_{x = 0} - e^{{ - \sigma_{i} l}} \left( {\frac{{{\text{d}}\delta n_{p} \left( x \right)}}{{{\text{d}}x}}} \right)_{x = l} } \right] + \hfill \\ \, + \varepsilon_{g} \left[ {\delta n_{p} \left( 0 \right)s_{1} + \delta n_{p} \left( l \right)s_{2} e^{{\sigma_{i} l}} } \right] + b_{4} k\beta \left[ {e^{{l\left( {\sigma_{i} - \beta } \right)}} - 1} \right] \hfill \\ \end{gathered}$$

(21)

$$\begin{gathered} b_{2} = \frac{1}{{{2}k\sigma_{i} {\text{sinh}}\left( {\sigma_{i} l} \right)}}\left\{ { - b_{3} k\left( {\frac{{{\text{d}}\delta n_{p} \left( x \right)}}{{{\text{d}}x}}} \right)_{x = l} + {\upvarepsilon }_{g} \delta n_{p} \left( l \right)s_{2} + b_{4} k\beta \, e^{ - \beta l} + } \right. \hfill \\ \, \left. { + e^{{ - \sigma_{i} l}} \left[ {b_{3} k\left( {\frac{{{\text{d}}\delta n_{p} \left( x \right)}}{{{\text{d}}x}}} \right)_{x = 0} + {\upvarepsilon }_{g} \delta n_{p} \left( 0 \right)s_{1} - b_{4} k\beta } \right]} \right\} \hfill \\ \end{gathered}$$

(22)

$$b_{3} = \frac{{\varepsilon_{g} }}{{k\tau_{p} \left( {\sigma_{i}^{2} - L^{ - 2} } \right)}}\;{\text{and}}\;b_{4} = \frac{{\beta I_{0} }}{{\varepsilon \left( {\beta^{2} - \sigma_{i}^{2} } \right)}}\left( {\frac{{b_{3} }}{{D_{p} }} - \frac{{\varepsilon - \varepsilon_{g} }}{k}} \right)$$

(23)

are constants calculated using the boundary conditions !!!!!

$$\left. { - k\frac{{{\text{d}}T_{s} \left( x \right)}}{{{\text{d}}x}}} \right|_{x = 0} = s_{1} \delta n_{p} \left( 0 \right)\varepsilon_{g} \;{\text{and}}\;\left. { - k\frac{{{\text{d}}T_{s} \left( x \right)}}{{{\text{d}}x}}} \right|_{x = l} = - s_{2} \delta n_{p} \left( l \right)\varepsilon_{g}$$

(24)

Here s₁ and s₂ are the surface recombination speeds at illuminated front (1) and nonilluminated back (2) sample surface.