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In the state of thermal equilibrium, the individual hole and electron streams passing through the barrier are ideally zero. The state of thermal equilibrium can be defined as the steady-state condition at a given temperature when no externally applied field is present. In this case, the net current density due to both drift and diffusion currents should be zero for both holes and electrons. Thus, net current density for holes is given as [10,11,23,24],

(1.1)

(1.2)

where, is the Einstein relation. Also,

The expression for hole concentration,

(1.3)

Differentiating equation (1.3) with respect to x in the equilibrium condition,

(1.4)

From equation (1.2) with the help of equation (1.4),

(1.5)

(1.6)

Similarly, net current density for electrons is given as follows,

(1.7)

(1.8)

(1.9)

Hence,

(1.10)

It is apparent from equations (1.6) and (1.10) that the Fermi level (E_F) is not dependent on x and remains uniform in whole of the semiconductor sample for zero net hole and electron densities. This is also apparent from the band diagram as shown in Figure 1.6(b). A typical space charge distribution happens at the barrier due to 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)

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].