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Now, integrating equations (1.22) & (1.23) over the barrier gives V_bi as follows,

      (1.24)

      (1.25)

Hence, area covered under the shaded triangle in Figure 1.11 corresponds to built-in-potential or V_bi. Using above equations, Vbi as a function of the total width of depletion layer is given as,

      (1.26)

The p-n junction circuit diagrams along with energy band diagrams for steady-state and in case of forward- and reverse-biased states are shown in Figure 1.12. Figure 1.12(a) shows the p-n junction circuit in the state of thermal equilibrium and the corresponding energy band diagram is shown in Figure 1.12(b) [10,11,23,24]. It is clear that the built-in-potential is V_bi across the junction formed by p- and n-blocks, while potential energy difference from p- to n-block is qV_bi. On applying positive potential V_f so that the p-n junction is forward-biased, width of the barrier is reduced due to the fact that the resultant electrostatic potential across the semiconductor junction becomes V_bi − V_f as shown in Figure 1.12(c) & (d). In case of reverse biasing the p- and n-type blocks, the p-n junction barrier is reverse-biased so that the resultant electrostatic potential at the junction becomes V_bi + V_r enhancing the overall width of the depletion layer. This almost ceases the transport of charge carriers across the junction because charge carriers cannot cross the wide barrier as shown in Figure 1.12(e) & (f).

Figure 1.11 The schematic shows the electric field distribution and the width of the depletion region or barrier. The area covered under the triangle shows the built-in-potential (V_bi). The width of the barrier is in reference to the width as shown in Figure 1.9 displaying the space charge distribution.

       Depletion Capacitance

Accumulation of holes and electrons at the junction of p- and n-type blocks or layers creates a capacitive effect at the barrier. This depletion layer capacitancse can be defined as per unit area [10,31]. Here, C is the capacitance, dq is the differential change in charge per unit area with respect to differential change in the applied voltage dV. A p-n junction with an arbitrary impurity profile is shown in Figure 1.13(a), while 1.13(b) & 1.13(c) show the profile of change in space charge as a function of change of the applied bias and the corresponding change in the electric field distribution. When the applied bias on the side of n-type block is increased by a differential voltage dV such that the total applied bias becomes V + dV, an enhancement in the region enclosed by charge carriers and the associated electric field distribution profile can be seen as given in Figure 1.13(c) within the barrier width (t). The differential change in charge dq is shown by the green coloured rectangular blocks on both sides of the barrier in Figure 1.13(b) maintaining overall charge neutrality. The differential positive change in electric field is given by . The associated differential change can be given by . Hence, capacitance per unit area due to space charge distribution of the depletion layer in the reverse-bias condition is given as follows [10,23,24, 61-63],

      (1.27)

Figure 1.12 Schematic representations of barrier width and corresponding band energy diagrams of a p-n junction in (a) steady-state equilibrium condition and (b) related band energy diagram, (c) forward biasing showing forward current (I) and voltage V_f and (d) associated band energy diagram, (e) reverse-biased diode circuit showing reverse current (I_r) and voltage (V_r) and (f) correlated energy band diagram. It is clear that the width of depletion layer decreases in forward-biased circuit in (c) so that charge-carriers cross the barrier easily, while it increases in reverse-biased circuit in (e) which makes transportation of charge-carriers across the barrier difficult.

Figure 1.13 Schematic shows an arbitrarily profile of dopants when reversed bias as shown in (a). Schematics (b) and (c) show the change in space charge distribution and electric field profile as a function of the change in applied bias.

In case of forward-biased p-n junction, width of the depletion layer decreases as a result of high force exerted by electric field acting on charge carriers making them mobile which induces diffusion capacitance as well [10,23,24]. From the discussion held above, it is clear that p-n junction acts in form of a condenser in which p- and n-type blocks or layers are the two plates, while the barrier or depletion layer itself acts as a dielectric. The junction capacitance is inversely proportional to the thickness of the depletion layer from and measured in Farads. A voltage-controlled capacitance is in which junction capacitance varies as a function of the applied potential difference. This type of device possessing a variable reactance is called variable reactor or varactor collectively [10,23,24].

1.2 Fabrication Technology of Diode

The fabrication technology of a semiconductor p-n junction diode involving different steps is displayed diagrammatically in