∫ ∫ ∫

Forward Bias (Center of Figure 2.4) If the potential of the p-region is raised with respect to the n-region, then the voltage across the pn-junction falls by the amount of the externally applied voltage VF, and the electric field at the junction is reduced. According to Formula (2.12), the width of the depletion layer W is adjusted as follows:

V n-p E x( ) xd xp-

∫– qN Aεs----------– x xp+( )⋅⎝ ⎠⎛ ⎞ xd xp-

∫– qNDεs---------- x xn-( )⋅⎝ ⎠⎛ ⎞ xd 0

∫–= =

⋅

⋅+=

W xnND N A

------------ xn+ xn ND N A+

N A ---------------------⋅= =

V n-p q

2εs ------- N Axp

ND N A+( )2 ----------------------------+⎝ ⎠⎜ ⎟

⎛ ⎞ ⋅=

xp W xn-W W N A

ND N A+ ---------------------– W

ND ND N A+ ---------------------= = =

W 2εs q

-------

N A ND+ N AND

--------------------- V n-p⋅ ⋅=

W 2εs q

-------

N A ND+ N AND

--------------------- V bi V F-( )⋅ ⋅=

Reverse Bias (Bottom of Figure 2.4) If we apply a negative voltage VR to the pn-junction, we reduce the potential of the p-region with respect to the n-region. As a result, the energy barrier at the pn-junction is raised to q(Vbi+VR).The electric field becomes stronger, and the width of the depletion layer increases to

(2.13)

Depletion Layer Capacitance Any change in voltage across a pn-junction results in an adjustment of the depletion layer width, and hence, in the displacement of electric charge. Therefore, a pn-junction exhibits a capacitance Cj, given by

W 2εs q

-------

N A ND+ N AND

--------------------- V bi V R+( )⋅ ⋅=

(2.14)

dQ is the displaced charge for a voltage change of dV. According to Poisson's law, a modification of the depletion layer width in the

p-region by dxp alters the electric field by

Note that dxp · qNA represents the displaced charge dQp. The displacement of dQp changes the voltage drop across the p-side by

Analog to this, we obtain for the n-side

Due to the requirement of overall neutrality, we have dQp = dQn = dQ. Hence, the change of voltage across the pn-junction is given by

We enter the last result into Equation (2.14) and obtain:

Substituting W by Equation (2.13) leads us to the depletion capacitance of a pn-junction:

(2.15)

Derivation of the Current-Voltage Characteristics of the pn-Junction Figure 2.5 illustrates how the flow of current in a pn-junction is established. Because of their thermal movement, holes from the p-side and electrons from the n-side continually enter the depletion layer of the pn-junction. The electric field in the depletion layer exerts a retarding force on the carriers, forcing them into the reverse direction.