ABSTRACT
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.