Equation (3.2.3) tells us that depletion layer is deeper on the lightly doped side and shallower on heavily doped side.
Graph 2 gives the Electric Field Profile. Electric Field(E) is Electric Flux Density (D) divided by permittivity(ϵ= ϵ 0 ϵ r ) where D is the number of Flux passing through cm 2 cross-sectional area. In Rationalized MKS units it is assumed that 1 Coulomb of Charge gives rise to 1 Coulomb of electric Flux. Hence qN D d n0 is the total positive charge within a volume of 1cm 2 cross-sectional area and length d n0 cm. Equivalently qN A d p0 is the total negative charge within a volume of 1cm 2 cross-sectional area and length d p0 cm. Electric Flux originates on +ve Charge and terminates on –ve charge. Therefore Flux Density D is maximum at metallurgical junction z = 0 and negatively directed.
Therefore at z = 0, D max = - qN D d n0 = - qN A d p0 and on the two sides D linearly decreases to zero at z = –d p0 and at z = d n0 . Hence E is maximum at z = 0 and falls linearly to zero at z = –d p0 and at z = d n0 as shown in Graph 2. From this we conclude that
Since Electric Field E is the negative of potential gradient hence potential is the area under the Electric Field profile:
Therefore:
Therefore:
Similarly
Adding Equation (3.2.7) and (3.2.8) we get:
Square rooting Equation (3.2.8):
This equation tells us that if we have one sided step junction then depletion layer will lie on lightly doped region and will be determined entirely by the lighter doping density.
Equation (3.2.9) tells us that in abrupt junctions the depletion width ‘d’ is directly proportional to the square root of the barrier potential.
In linearly graded junctions ‘d’ is directly proportional to the cube root of the barrier potential. This will be shown in the advanced version of Diode Physics.
As seen in Figure 3.3.A, there is a carrier concentration gradient in the depletion region which causes the diffusion current across the depletion region. As seen in Graph 3 of Figure 3.3. there is a potential gradient also in the depletion region. This leads to the drift of minority carriers.Drift and Diffusion together lead to detailed charge balance leading to zero current across the depletion under zero-bias condition.
3.2.1. Energy Band Diagram interpretation of Built-in Barrier Potential.
When P-Type Semiconductor and N-Type Semiconductor are alloyed together or PN-Abrupt Junction formed by diffusion then under no bias condition and under thermal equilibrium condition, Fermi-Level of P-Type and that of N-Type align themselves as shown in Figure 3.4 since Fermi-Level is a function of temperature and the two samples are at the same temperature when alloyed together. In the process there is band-bending and built-in potential gradient is created at the junction.
In Chapter 3 Advanced version of diode, it will be shown that:
From Equation 2.2.6.2.7:
Substituting Equation (3.2.11) in (3.2.10) we get:
In Equation (3.2.12), V Th is the thermal voltage kT/q= 0.026V at 300K, V G = band gap voltage of the material and for Si it is 1.12V and N C and N V are equivalent density of states= 10 19 /cc (for Si) and if N A = N D =10 16 /cc then built-in barrier potential is of 0.8V. .