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Derive an expression for responsivity of an intrinsic photo detector in terms of quantum efficiency and wavelength.
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Quantum Efficiency ɳ:

1. Quantum Efficiency is defined as the ratio of number of electrons collected to the number of incident photons.

2. It is also defined as ratio of generated current flux to input photon flux.

$ɳ_q$ = (number ofelectrons collected)/(number ofincident photons)

$ɳ_q = R_e/R_p$ = Quantum efficiency

where $R_p$ = incident photon rate (photons/sec)

$R_e$ = corresponding electrons rate (electrons/sec)

Responsivity:

1. It represents the sensitivity of a photodetector.

2. he function of photodetector is to convert the optical signal into electrical signal.

3. More photons that strike the photodetector, more change carriers will be produced. i.e greater will be the photo current I.

4. The performance of a photodiode is characterized by a term Responsivity R.

5. Photocurrent is directly proportional to incident optical power ($P_{in}$).

$$I_p \ \ α \ \ P_in$$ $$I_p = RP_{in}$$

where R is constant called Responsivity of photo detector in (A/W)

As the energy of photon E = hv, then incident photon rate $R_p = \frac{P_{in}}{hv}$

where $\ \ \$ $P_in$ = incident optical power

Electron rate $R_e = \frac{I_p}q$

Quantum efficiency $ɳ = \frac{R_e}{R_p}$

$$ɳ = \frac{{I_p}{q}}{\frac{P_{in}}{hv}}$$

$$ɳ= \frac{I_p {hv}}{_qP_{in}}$$

$$I_pP_{in} = \frac{ɳq}{hv}$$

where $R = \frac{I_p}{P_{in}}$

$R = \frac{ɳq}{hv}$

$R = \frac{ɳqλ}{hc}$

$$\boxed{R = \frac{ɳqλ}{hc}}$$

where $v = \frac{c}λ$ and $hc = E_p$(energy of photon)

From above equation, Responsivity is directly proportional to Quantum efficiency at particular wavelength