Article

Depletion Approximation Analysis of an Exponentially Graded Semiconductor p-n Junction

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Abstract

Estimation of important properties of p-n junctions such as reverse leakage current and capacitance is greatly facilitated by the depletion approximation. Computation of the electrostatic potential and electron and hole concentrations within this approximation are commonplace in the microelectronics industry. The depletion approximation requires appropriate and accurate boundary conditions. Thus far, such reasonable boundary conditions have been widely applied only for the case in which the p-type and n-type impurity concentrations are spatially uniform. The author derives the solution of the depletion approximation with appropriate boundary conditions for the case in which the impurity concentration on one side of the diode decays exponentially with distance. He plots the potential and charge density of this exponential depletion approximation and compares these results to full numerical solution of the semiconductor equations. The agreement between the proposed approximation and the numerical solution validates this approximation scheme

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... The magnitude of built-in (diffusion) electric field in the emitter E pr can reach the values compared with the electric field strength in the DL of ð n junction. The occurrence of the field which distribution is not restricted by an immediate vicinity of metallurgical boundary of the junction, as it takes place for a stepwise impurity profile, as well as high concentration of free (majority) carriers result in characteristic features SQO, 5(2), 2002 of structure of the space-charge region [22]. Indeed, since the Debye screening length in the emitter l Dp is rather small (l Dp << x 0 ), for its main part (≈ x 0 ) the quasineutrality condition is valid: ρ(x) = 0, E pr (x) = const, where ρ(x) is the charge density. ...
... As distinct from a stepwise (quasi-uniform) impurity distribution, in the considered case besides the main DL in the vicinity of metallurgical boundary of pn junction, there is an additional DL [22] at the boundary with uniform p + region of the emitter at x e ≤ x ≤ x 0 , with the extension of W e = x e x 0 (Fig. 1). Setting the field E(x e ) = 0, and the potential ϕ(x e ) = U bi , we find for the considered DL the field distribution E(x) = E pr (x+x e )/W e , the potential ϕ(x) = U bi + [(x+x e ) 2 /(2W e )], as well as the potential difference ∆ϕ e ≡ ϕ(x e ) ϕ(x 0 ) = W e /2, where W e = λ Dp 2 [here the Debye screening length is determined by maximum value of the acceptor concentration in the emitter λ Dp = (ε 0 k B T/4πq 2 N Am ) 1/2 ]. ...
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