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uniform E_F in the steady state. Considering the 1D p-n junction when all donor and acceptor atoms are ionized, Poisson’s equation for electrostatic potential ψ and unique space charge distribution is given as follows [10,23,24],

(1.11)

The above situation is well represented in Figure 1.9(a) in the energy band diagram of an abrupt junction in the steady state. There is a unique space charge distribution at the semiconductor junction. At distances far away from the barrier, net hole density is equal to the net electron density such that the total space charge density is zero maintaining the charge neutrality. In this case, from equation (1.11) [10,11,23,24],

      (1.12)

      and,

(1.13)

In case of a p-type neutral region, N_D = 0 and p >> n. Now, setting N_D = n = 0 in equation (1.13), we get, p = N_A and putting it in equation (1.3) [10,11,23,24],

      (1.14)

Figure 1.9 The above figure (a) displays band diagram of an abrupt junction in steady state and (b) displays an approximation of space charge distribution.

      Similarly, in case of n-type neutral region, N_A = 0 and n ≫ p Now, setting N_A = p = 0 in equation (1.13), we get, n = N_D and putting in equation (1.3) [10,11,23,24],

      (1.15)

      In the steady state, the total electrostatic potential difference between p-type and n-type neutral regions is defined as the built-in-potential (V_bi) and is given as follows [10,11,23,24],

      (1.16)

In between the neutral regions and semiconductor barrier, a narrow transition region exists which has a width smaller in comparison to the width of the barrier or the depletion region. This is true for regions existing on both sides of the depletion region. On neglecting the transition regions in comparison to the depletion region, a nearby rectangular space charge distribution is obtained as shown in Figure 1.9(b) [10,23,24]. Here, x_p and x_n are the widths of the depletion layer of p- and n-type blocks. In case of completely depleted region, the amount of p and n dopants will be zero, and then from equation (1.11) [10,24],

(1.11)

(1.17)

The physics of Poisson’s equation lies in the fact that distribution of impurities can be performed in the form of shallow diffusion or low energy ion implantation or in the form of deep diffusions or high-energy ion implantations [10,23,24]. The type of ion implantation describes the doping profile according to the energy dose applied [30]. Shallow diffusion or low-energy ion implantation introduces foreign atoms at low depths to form abrupt p-n junction as shown in Figure 1.10(a). In this case, the doping concentration profile shows a rapid changeover between the p-type and n-type regions. In case of high-energy ion implantations, distribution of doping profiles can be approximated almost linearly across the barrier called as linearly graded junction as shown in Figure 1.10(b).

In case of an abrupt junction, the free charge carriers are completely depleted such that under the condition, -x_p ≤ x < 0, equation (1.17) becomes [10],

(1.18)

      In case of an abrupt junction, the free charge carriers are completely depleted such that under the condition, 0 < x ≤ x_n, equation (1.17) becomes [10],

(1.19)

Figure 1.10 Approximation of foreign atom doping profiles in semiconductor forming (a) abrupt junction due to shallow diffusion and (b) linearly graded junction as a result of deep diffusion.

      For the space charge neutrality of the semiconductor as a whole,

      (1.20)

      The total depletion layer width is given as follows,

      (1.21)

The electric field distributed in the barrier can be obtained by integrating equations (1.18) & (1.19) [10,11,23,24].

      In case of -x_p ≤ x < 0,

(1.22)

      In case of 0 < x ≤ x_n,

(1.23)