SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

310 CHAPTER 7. TEMPORAL RESPONSE OF DIODES AND BIPOLAR TRANSISTORS

Cp

(a) (b)

Input ac

signal vs

Vdc

I is

Cj Cdiff

rs

Rs

Diode equivalent

circuit

Figure 7.3: (a) Ap-ndiode is biased at a dc voltageVdcand a small signal modulation is applied
to it. (b) The equivalent circuit of a forward-biased diode

whereΔ ̃p(x)is the amplitude of the ac hole concentration due to the small signal. Plugging this
back into equation 7.2.16 and simplifying, we get

jωΔ ̃p(x)=−

Δ ̃p(x)

τp

+Dp

d^2

(

Δ ̃p(x)

)

dx^2

(7.2.20)

Equation 7.2.19 allows us to solve for the ac part of the injected charge distribution. The
general solution to this equation is

Δ ̃p(x)=C 1 e−x/λ+C 2 e+x/λ (7.2.21)

whereλis the ac diffusion length and is given by

λ=

[

jω

Dp

+

1

Dpτp

]− 1 / 2

(7.2.22)

Applying appropriate boundary conditions (Δ ̃p=0whenx→∞andΔ ̃p=Δ ̃p(0)when
x=0) results in

C 2 =0 (7.2.23)

C 1 = Δ ̃p(0) (7.2.24)

and the ac injected charge distribution becomes

(Δp)ac=

[

Δ ̃p(0) exp

(

−x

√

jω

Dp

+

1

Dpτp

)]

exp (jωt) (7.2.25)