Deriving Carrier Lifetime, Bandwidth, and Rise Time

Goal: derive \( \tau \), \( \Delta f \), and \( t_{\text{rise}} \) from the same first-order carrier dynamics.

Final relations

\boxed{ t_{\text{rise}}\approx2.2\tau, \qquad \tau=\frac{1}{2\pi\Delta f}, \qquad t_{\text{rise}}\approx\frac{0.35}{\Delta f} }

Carrier-rate equation

We start with the equation for the change in excess carrier concentration with respect to time, assuming a constant generation rate:

\frac{\partial \Delta n(t)}{\partial t} = G_0-\frac{\Delta n(t)}{\tau}.

Here, \(G\equiv G_0\) during illumination and the recombination rate is \(R=\Delta n(t)/\tau\). We use the initial conditions

\(\Delta n(0)=0\)
\(\Delta n(\infty)=\text{constant}\)

Solving the differential equation gives the general form

\Delta n(t)=c_1e^{-t/\tau}+c_2.

The particular solution gives

c_2=G_0\tau.

Applying \(\Delta n(0)=0\) gives the carrier-density response

\boxed{ \Delta n(t)=G_0\tau\left(1-e^{-t/\tau}\right) }.

Carrier-density response

\[ \displaystyle \Delta n(t)= \begin{cases} G_0\tau\left(1-e^{-t/\tau}\right), & t<t_0,\\[6pt] G_0\tau e^{-(t-t_0)/\tau}, & t\ge t_0. \end{cases} \]

Figure 2: Carrier density \(\Delta n(t)\) under constant generation rate \(G_0\) until time \(t_0\), followed by an exponential decay. The steady-state value during illumination is \(G_0\tau\).

The blue curve is the build-up solution

\Delta n(t)=G_0\tau\left(1-e^{-t/\tau}\right).

The red curve shows the response after the generation rate is turned off, \(G\equiv0\).

Deriving \(t_{\text{rise}}\)

First, use the blue curve to relate rise time to carrier lifetime. Define

t_{\text{rise}}\equiv t_{90}-t_{10}.

Rearranging the carrier-density equation gives

t = -\tau \ln\left( 1-\frac{\Delta n}{G_0\tau} \right).

For \(t_{90}\), set \(\Delta n=0.9G_0\tau\). For \(t_{10}\), set \(\Delta n=0.1G_0\tau\). Therefore,

t_{90} = -\tau\ln(0.1) = 2.3026\tau,

t_{10} = -\tau\ln(0.9) = 0.1054\tau.

Subtracting the two gives

\boxed{ t_{\text{rise}} = t_{90}-t_{10} = \tau\ln 9 \approx2.2\tau }.

Deriving \(\tau\) and \(\Delta f\)

Now use the red curve, where the generation rate has been turned off. If the detector had reached steady state immediately before switch-off, then

\(\Delta n(t_0)=G_0\tau\)
\(\Delta n(\infty)=0\)

The decay is therefore

\Delta n(t) = G_0\tau e^{-(t-t_0)/\tau} \theta(t-t_0),

where \(\theta(t-t_0)\) is the Heaviside step function.

Apply the Fourier transform

\mathcal{F}\{\Delta n(t)\} = \int_{-\infty}^{\infty} \Delta n(t)e^{-i\omega t}\,dt.

Using the Heaviside function to set the lower limit to \(t_0\),

\Delta n(\omega) = \int_{t_0}^{\infty} G_0\tau e^{-(t-t_0)/\tau} e^{-i\omega t}\,dt.

Evaluating the integral gives

\Delta n(\omega) = \frac{ G_0\tau^2e^{-i\omega t_0} }{ 1+i\omega\tau }.

The phase factor \(e^{-i\omega t_0}\) has unit magnitude, so

\left|\Delta n(\omega)\right| = \frac{ G_0\tau^2 }{ \sqrt{1+(\omega\tau)^2} } = \frac{ G_0\tau^2 }{ \sqrt{1+(2\pi f\tau)^2} }.

At \(f=0\),

\left|\Delta n(0)\right|=G_0\tau^2.

Define the electrical bandwidth as the \(-3\,\mathrm{dB}\) frequency, \(f_{-3\mathrm{dB}}\equiv\Delta f\). At this point, the magnitude has fallen to \(1/\sqrt{2}\) of its low-frequency value:

\left|\Delta n(2\pi\Delta f)\right| = \frac{ \left|\Delta n(0)\right| }{ \sqrt{2} }.

Substituting into the frequency-domain expression gives

\frac{1}{\sqrt{1+(2\pi\Delta f\,\tau)^2}} = \frac{1}{\sqrt{2}},

and therefore

\boxed{ \tau=\frac{1}{2\pi\Delta f} }.

Carrier lifetime \( \tau \), electrical bandwidth \( \Delta f \), and rise time \( t_{\text{rise}} \) can now be interchanged directly for this first-order model:

\boxed{ t_{\text{rise}}\approx2.2\tau, \qquad \tau=\frac{1}{2\pi\Delta f}, \qquad t_{\text{rise}} \approx \frac{0.35}{\Delta f} }.

Interpretation: the rate equation above has the same mathematical form as a first-order low-pass filter. The familiar qualitative behavior of an RC time constant therefore carries over directly: longer lifetime means slower rise time and narrower bandwidth; shorter lifetime means faster rise time and wider bandwidth.

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