New improved capacitance–voltage model for symmetrical step junction: a way to a unified model for realistic junctions

Assia Boukredimi and Kheireddine Benchouk

Abstract  Based on the mass-action law, a new capacitance–voltage (C–V) model for symmetrical step junction is presented. Furthermore, we propose a unified C–V model for realistic junction and for any applied-voltage.

1. Introduction

    The formation of depletion regions at different interfaces in semiconductor junctions and heterostructures plays an essential role in the physics of most electron devices such as p–n junctions and Schottky contacts [1,2]. 
    C–V measurements are one of the most popular electrical techniques used to determine important parameters of semiconductor devices such as impurity concentrations, barrier heights and Fermi energy levels [3,4].
    As reported by several authors in measurements [5–8] as well as in numerical simulations and analytic models [9–11], the C–V characteristics of some p–n junction diodes show a peak in the neighborhood of the built-in potential.
  The depletion approximation model is accurate under reverse and low forward bias conditions but rapidly loses validity under moderate to strong forward bias. At forward biases, the physical mechanism of forward-capacitance is very complex and the usual C–V method needs some corrections.
    In this article, we present a new method to obtain an analytical C–V model for symmetrical step junction (SSJ) without calculating the electric potential distribution. 
    Furthermore, we propose a unified C–V model for realistic junction (RJ) and for any bias conditions by means of our recent C–V model for linearly graded junction (LGJ) [11].
    This paper is organized as follows. In Sect. 2, we describe briefly the depletion approximation model. In Sect. 3, we present the theory of SSJ by including the effects of the holes and electrons in the space charge region (SCR) via the pseudo-equilibrium approach. This approach is most usually used in the characterization of semiconductor devices. In Sect. 4, we develop “exact” and “modified” models related to the C–V characteristic for any SSJ and for any bias conditions. In Sect. 5, an analytical C–V model for RJ and for any applied voltage is proposed. Finally, we give our conclusion.

 

2. Depletion approximation model

    Both step and linearly graded p–n junctions in equilibrium were initially treated by Shockley in an ancient basic paper [12].

Shockley theory for SCR capacitance is based upon an assumption that the SCR of an abrupt p–n junction is essentially depleted of mobile charge carriers. In addition, it is also assumed that the SCR edge can be adequately approximated by a discontinuous transition between completely depleted and neutral regions.

As shown in Fig. 1, the charge density goes abruptly to zero at the depletion edge (x =±x₀) for the SSJ structure at doping concentrations N_A⁻ = N_D⁺ = N₀.

 

According to the depletion approximation model, the charge Q(V_app) and the capacitance C(V_app) of the SCR in a SSJ structure are given, respectively, by [12]:

 

where V_app is the applied voltage across the depletion layer, V_d the junction built-in potential, ε_sc the dielectric permittivity of the semiconductor and u_T the thermal voltage.
In Eqs. (1) and (2), L_Di represents the intrinsic Debye length. It is given by [11]

where and n_i is the intrinsic concentration of the semiconductor and q the elementary charge. The dimensionless parameter K₀ is defined as

In this paper, all charges and capacitances are given per unit of area.

It is well known that the built-in potential is given by [9,12]:

The above equation can be rearranged as follows:

The most important problem within the depletion approximation model as shown by Eq. (2) is the singularity that occurs at V_app = V_d characterizing an infinite capacitance (unphysical concept). For V_app > V_d, the capacitance is not defined [10]. These anomalies are essentially due to the fact that it has been assumed the absence of free carriers in the SCR.
    Although the limitations of the depletion approximation are well recognized and several improved models have been proposed to correct it, the depletion approximation itself is often misunderstood [10,13].

3. Problem formulation of SSJ

    The Poisson equation is used as an analytic model in a wide variety of fields in chemistry and physics [10–12,14,15]. The one dimensional (1-D) Poisson equation, in the form that appears in non-degenerate semiconductor device theory, does not have an analytical solution when the impurity doping concentration is not uniform [14]. The nonlinearity associated with the junction-region has made this problem difficult [16]. Therefore, numerical solution of this problem has been considered by several authors [11,14,17].

3.2 New SCR width definition

One the most important factors that affect the characteristics of p-n junctions is the presence of the SCR close to the junction. Almost all the p-n-junction characteristics, including current versus voltage, capacitance and  breakdown-voltage are affected by this SCR [13].