# Photoconductive Detectors

This is a continuation from the previous tutorial - **photoemissive detectors**.

Photoconductive detectors are based on the phenomenon of ** photoconductivity**. The conductivity of a photoconductor, which can be an insulator but is usually a semiconductor, increases with optical illumination due to photogeneration of free carriers.

The conductivity of a semiconductor that has electron and hole concentrations of \(n\) and \(p\), respectively, is

\[\tag{14-63}\sigma=e(\mu_\text{e}n+\mu_\text{h}p)\]

where \(e\) is the electronic charge and \(\mu_\text{e}\) and \(\mu_\text{h}\) are the electron and hole mobilities, respectively.

In the absence of optical illumination, the conductivity, known as the dark conductivity, \(\sigma_0=e(\mu_\text{e}n_0+\mu_\text{h}p_0)\) because the electron and hole concentrations in this situation are the equilibrium concentrations, \(n_0\) and \(p_0\), respectively.

When a semiconductor is illuminated with light of a sufficient photon energy, carriers in excess of the equilibrium concentrations are generated. The photoconductivity is the additional conductivity contributed by these photogenerated excess carriers:

\[\tag{14-64}\Delta\sigma=\sigma-\sigma_0=e(\mu_\text{e}\Delta{n}+\mu_\text{h}\Delta{p})\]

where \(\Delta{n}=n-n _0\) and \(\Delta{p}=p-p_0\) are the photogenerated excess electron and hole concentrations, respectively.

Similarly to photoemission, photoconductivity also has a threshold photon energy, \(E_\text{th}\), and a corresponding threshold wavelength, \(\lambda_\text{th}\), that are characteristic of a given photoconductor.

Together with the spectral dependence of the absorption coefficient, they determine the spectral response of a photoconductor.

Depending on the processes involved in the photogeneration of free carriers, there are two principal types of photoconductivity.

The ** intrinsic photoconductivity** is contributed by the excess electrons and holes that are generated by band-to-band absorption of incident photons, as shown in Figure 14-9(a).

The threshold photon energy of intrinsic photoconductivity is clearly the bandgap energy of the photoconductor:

\[\tag{14-65}E_\text{th}=E_\text{g}\]

The ** extrinsic photoconductivity** is contributed by carriers that are generated by optical transitions associated with impurity levels within the bandgap of an extrinsic semiconductor.

In an n-type extrinsic semiconductor, the impurity levels have an energy \(E_\text{i}=E_\text{d}\) below the conduction-band edge; electrons are excited from these donor levels to the conduction band, as shown in Figure 14-9(b).

In a p-type extrinsic semiconductor, the impurity levels have an energy \(E_\text{i}=E_\text{a}\) above the valence-band edge; electrons are excited from the valence band to these acceptor levels, as shown in Figure 14-9(c).

Thus, the threshold photon energy of extrinsic photoconductivity for either n-type or p-type photoconductors is

\[\tag{14-66}E_\text{th}=E_\text{i}\]

Photoconductors cover a broad spectral range from the ultraviolet to the far infrared. In particular, there are many sensitive photoconductors in the infrared region beyond 1.2 μm wavelength where no photoemissive detectors exist.

Both direct-gap and indirect-gap semiconductors can be used for photoconductors. All of the semiconductors discussed in the introduction to semiconductors tutorial, including the group IV semiconductors, the III-V and II-VI compounds, and the IV-VI compounds, can be used for intrinsic photoconductors.

Among them, intrinsic silicon photoconductors are the most important photoconductive detectors in the visible and near infrared spectral regions at wavelengths shorter than 1.1 μm, while intrinsic germanium photoconductors are the most important photoconductive detectors in the near infrared region at wavelengths up to 1.8 μm.

In the mid infrared region between 2 and 7 μm wavelengths, one finds intrinsic photoconductors based on InAs, InSb, PbS, and PbSe. Extrinsic photoconductors are available for these mid infrared wavelengths as well as for longer wavelengths well into the far infrared region.

The most important extrinsic photoconductive detectors are p-type germanium photoconductors such as Ge:Au, Ge:Hg, Ge:Cd, Ge:Cu, and Ge:Zn.

Figure 14-10 shows the specific detectivity of representative photoconductive detectors as a function of optical wavelength.

A sensitive photoconductor must have a low dark conductivity so that the photoconductivity caused by optical illumination amounts to a significant change in its total conductivity.

For this reason, it is necessary to minimize the thermal equilibrium concentrations, \(n_0\) and \(p_0\), of free electrons and free holes in a photoconductor. According to the law of mass action given in (12-31) [refer to the electron and hole concentrations in semiconductors tutorial], \(n_0p_0=n_\text{i}^2(T)\).

In an intrinsic semiconductor, both electron and hole concentration in the dark can be reduced by lowering the temperature because \(n_0=p_0=n_\text{i}(T)\).

Because \(n_\text{i}\) depends exponentially on \(-E_\text{g}/2k_\text{B}T\), as seen in (12-29) [refer to the electron and hole concentrations tutorial], the dark electron and hole concentrations can be significant for a semiconductor that has a small bandgap energy.

Reduction of the dark free carrier concentrations by lowering temperature is particularly important for intrinsic photoconductors of small bandgap energies, such as InSb and HgCdTe. For this reason, such small-bandgap photoconductors are normally operated at the liquid nitrogen temperature of 77 K or lower.

In an extrinsic semiconductor, conductivity is predominantly contributed by the majority carriers because the majority carrier concentration is much higher than both \(n_\text{i}\) and the minority carrier concentration.

It is therefore important for the functioning of an extrinsic photoconductor that most of the free majority carriers be photogenerated rather than thermally generated. This condition requires that the donors in an n-type photoconductor and the acceptors in a p-type photoconductor not be ionized when an extrinsic photoconductor is not optically illuminated. Ideally, they should be ionized only optically when the photoconductor is illuminated.

Because the value of \(E_\text{i}\) for an extrinsic photoconductor is small, it is normally necessary to operate an extrinsic photoconductor at a low temperature to reduce the dark concentration of the majority free carriers. Some extrinsic photoconductors, such as Ge:Au, are operated below 77 K. Some, such as Ge:Cu and Ge:Zn, are often operated at the liquid helium temperature of 4 K.

From these discussions, it is clear that a photoconductor of a small threshold photon energy, thus a long threshold wavelength, is required to operate at a low temperature irrespective of whether it is an intrinsic or an extrinsic type.

As a rule, the operating temperature for a detector with a threshold energy \(E_\text{th}\) has to be \(T\lt{E}_\text{th}/25k_\text{B}\approx460E_\text{th}(\text{eV})\). A photoconductor for the mid infrared normally requires an operating temperature of 77 K, and one for the far infrared requires an even lower operating temperature often down to 4 K.

The operation of a photoconductor requires that a voltage be applied to the device. A photoconductor has a photoconductive gain that depends on many parameters of the photoconductor and on the properties of the electrical contacts.

To facilitate quantitative discussions, we consider a simple photoconductor of a length \(l\) between its electrodes, a width \(w\), and a thickness \(d\), as shown in Figure 14-11.

Thus, the optically illuminated area is \(\mathcal{A}=lw\), but the cross-sectional area between the electrodes is \(wd\). A voltage \(V\) is applied across the length \(l\) while the photoconductor is uniformly illuminated with an optical beam of a power \(P_\text{s}\).

The external quantum efficiency of the photoconductor, which is illuminated on surface \(\mathcal{A}\), can be expressed as

\[\tag{14-67}\eta_\text{e}=\eta_\text{coll}\eta_\text{t}\eta_\text{i}=\eta_\text{coll}(1-R)(1-\text{e}^{-\alpha{d}})\]

where \(\eta_\text{coll}\) is the collection efficiency of the photogenerated carriers, \(\eta_\text{e}=1-R\) with \(R\) being the reflectivity of the incident surface, and \(\eta_\text{i}=1-\text{e}^{-\alpha{d}}\) with \(\alpha\) being the absorption coefficient of the photoconductor.

To improve the external quantum efficiency, the electrodes have to be carefully designed to maximize the collection efficiency, and the surface reflectivity can be reduced by antireflection coating.

In practice, both \(\eta_\text{coll}\) and \(\eta_\text{t}\) can be made close to unity. Then \(\eta_\text{e}\) can be made very close to 100% by increasing the thickness of the photoconductor for \(\eta_\text{i}=1\) with \(d\gg\alpha^{-1}\).

However, the practically important performance indicator of a photodetector is not its external quantum efficiency but rather its detectivity, or its specific detectivity.

The optimum thickness of a photoconductor for a maximum value of \(D^*\) is \(d=1.256\alpha^{-1}\) for \(\eta_\text{i}=71.5\%\). Because the value of \(D^*\) varies very slowly with \(d\) around this optimum thickness, there is much freedom to choose the thickness of a photoconductor in the range of \(\alpha^{-1}\le{d}\le1.5\alpha^{-1}\) for \(D^*\) to have a value that is more than 99% of its maximum value.

We first consider intrinsic photoconductivity under the conditions leading to the relations in (12-55) and (12-56) [refer to the carrier recombination in semiconductors tutorial] so that the photogenerated electrons and holes have the same concentration and the same lifetime: \(\Delta{n}=\Delta{p}=N\) and \(\tau_\text{e}=\tau_\text{h}=\tau_\text{s}\).

We also assume that the contacts of the electrodes are ohmic contacts that allow electrons and holes to be freely removed from or injected into the semiconductor.

With a quantum efficiency of \(\eta_\text{e}\), the photogeneration rate of free carriers in the semiconductor is \(\eta_\text{e}P_\text{s}/h\nu\), which is equal to \(lwd\cdot{N}/\tau_\text{s}\) because the generation rate equals the recombination rate in the steady state.

Therefore, the total number of photogenerated free carriers in the photoconductor is

\[\tag{14-68}\mathcal{N}=lwd\cdot{N}=\eta_\text{e}\frac{P_\text{s}}{h\nu}\tau_\text{s}\]

Because the carriers have a lifetime of \(\tau_\text{s}\), the photocurrent resulting from the photogeneration of these carriers is

\[\tag{14-69}i_\text{ph}=\frac{e\mathcal{N}}{\tau_\text{s}}=\eta_\text{e}\frac{eP_\text{s}}{h\nu}\]

which is exactly the relation given in (14-14) [refer to the photodetector noise tutorial], as expected.

This is not the external signal current of the photoconductor, however. The external signal current, \(i_\text{s}\), is generated by the applied voltage \(V\) on the photoconductance, which is \(\Delta\sigma{wd}/l\) for the photoconductor of the geometry shown in Figure 14-11.

Therefore, we find that

\[\tag{14-70}i_\text{s}=V\frac{\Delta\sigma{wd}}{l}=\frac{e\mathcal{N}}{\tau_\text{tr}^\text{e}}+\frac{e\mathcal{N}}{\tau_\text{tr}^\text{h}}\]

where

\[\tag{14-71}\tau_\text{tr}^\text{e}=\frac{l}{\mu_\text{e}E}=\frac{l^2}{\mu_\text{e}V}\qquad\text{and}\qquad\tau_\text{tr}^\text{h}=\frac{l}{\mu_\text{h}E}=\frac{l^2}{\mu_\text{h}V}\]

are the ** transit times** of electrons and holes in the photoconductor, respectively.

The transit time of an electron or hole is the time it takes for the charge carrier to cross the length \(l\) of the semiconductor at its drift velocity of \(v=\mu{E}\) under an applied field of \(E=V/l\).

Under these results, we find the following photoconductive gain

\[\tag{14-72}G=\frac{i_\text{s}}{i_\text{ph}}=\frac{\tau_\text{s}}{\tau_\text{tr}^\text{e}}+\frac{\tau_\text{s}}{\tau_\text{tr}^\text{h}}=\frac{\tau_\text{s}}{\tau_\text{tr}^\text{e}}\left(1+\frac{\mu_\text{h}}{\mu_\text{e}}\right)\]

The photoconductive gain in (14-72) is obtained for an intrinsic photoconductor with ohmic contacts on both electrodes. It is not valid if the photogenerated electrons and holes do not have the same concentration, as is the case in an extrinsic photoconductor, or if one or both contacts are not ohmic.

It is also clearly not valid for all values of the applied voltage because (14-71) leads to an unphysical conclusion that the gain can be made arbitrarily large simply by increasing the voltage \(V\) to reduce the transit times.

Nevertheless, the photoconductive gain can be generally expressed as

\[\tag{14-73}G=\frac{\tau}{\tau_\text{r}}\]

where \(\tau\) is a ** carrier lifetime** that can take different forms in different situations and \(\tau_\text{r}\) is a

**that depends on the properties of the photoconductor, the contacts, and the applied voltage.**

*relaxation time constant*The values of \(\tau\) and \(\tau_\text{r}\) in this relation depend on the properties and the operating conditions of a photoconductor, as discussed below.

In particular, when the applied voltage is large, \(\tau_\text{r}\) is not simply determined by the carrier transit times given in (14-71).

There is a capacitance, \(C=\epsilon{wd}/l\), between the anode and the cathode of the photoconductor. A space-charge effect in the photoconductor appears when the number of charges, \(Q=CV\), supplied by the applied voltage \(V\) on this capacitance is equal to or larger than the number of the carriers in the photoconductor.

This situation takes place under the following condition:

\[\tag{14-74}V\ge{V_\text{SC}}=\frac{\sigma{l^2}}{\mu\epsilon}\]

where \(\mu\) is a mobility that can take the form of \(\mu=\mu_\text{e}+\mu_\text{h}\), \(\mu=\mu_\text{e}\), or \(\mu=\mu_\text{h}\) depending on the properties of the photoconductor and its electrode contacts, as discussed below.

In the presence of this space-charge effect, \(\tau_\text{r}=\tau_\text{d}\), where

\[\tag{14-75}\tau_\text{d}=\frac{\epsilon}{\sigma}=\frac{\epsilon}{e(\mu_\text{e}n+\mu_\text{h}p)}\]

is the ** dielectric relaxation time** of the semiconductor.

Clearly, the gain does not continue to increase with increasing voltage when the space-charge effect appears.

The following cases are of interest.

- An intrinsic photoconductor in which both electrons and holes can freely move, and both the anode and the cathode have nonblocking ohmic contacts. In this case, \(\tau=\tau_\text{s}\), which is the spontaneous carrier recombination lifetime defined in (12-56) [refer to the carrier recombination in semiconductors tutorial], and \(V_\text{SC}=\sigma{l}^2/(\mu_\text{e}+\mu_\text{h})\epsilon\). For \(V\lt{V}_\text{SC}\), \(\tau_\text{r}=\tau_\text{tr}^\text{e}(1+\mu_\text{h}/\mu_\text{e})^{-1}\). When the applied voltage is large enough that \(V\gt{V}_\text{SC}\), \(\tau_\text{r}=\tau_\text{d}\), and the gain saturates at \(G=\tau_\text{s}/\tau_\text{d}\).
- An intrinsic or extrinsic photoconductor in which only one type of carrier can freely move, and the electrodes are nonblocking ohmic contacts for such carriers. In this case, only the free-moving carriers contribute to the photocurrent. Then \(\tau\) and \(\tau_\text{r}\) are, respectively, the lifetime and transit time of such carriers. The space-charge effect is determined by \(V_\text{SC}=\sigma{l}^2/\mu_\text{e}\epsilon\) if only electrons can freely move but by \(V_\text{SC}=\sigma{l}^2/\mu_\text{h}\epsilon\) if only holes can freely move. When the space-charge effect occurs at \(V\gt{V}_\text{SC}\), \(\tau_\text{r}=\tau_\text{d}\) also in this case.
- An intrinsic photoconductor in which both electrons and holes can freely move and the cathode is ohmic but the anode is blocking holes. Then, \(\tau=\tau_\text{e}=\tau_\text{h}=\tau_\text{tr}^\text{h}\) and \(\tau_\text{r}=\tau_\text{tr}^\text{e}\). In this case, \(G=1+\mu_\text{e}/\mu_\text{h}\).
- An intrinsic or extrinsic photoconductor in which both electrons and holes can freely move but both the cathode and the anode have blocking nonohmic contacts. In this case, \(\tau_\text{r}=\tau=\tau_\text{tr}^\text{e}\) and \(G=1\). With blocking contacts on both sides, the gain is unity. This is the case of the junction photodiodes discussed in the next tutorial.

When the external quantum efficiency and the gain of a photoconductor are determined, its responsivity can be easily calculated as

\[\tag{14-76}\mathcal{R}=G\eta_\text{e}\frac{e}{h\nu}\]

Because the gain \(G\) varies with the applied voltage \(V\), the responsivity \(\mathcal{R}\) of a photoconductor is also a function of \(V\) in addition to being a function of the optical wavelength and the device parameters.

**Example 14-9**

An n-type GaAs intrinsic photoconductive detector for \(\lambda=850\text{ nm}\) has the following parameters:

\(l=w=100\) μm, \(d=1\) μm, \(\alpha=1\times10^4\text{ cm}^{-1}=1\times10^6\text{ m}^{-1}\) at 850 nm, \(\eta_\text{coll}=1\), and \(\eta_\text{t}=1\) for \(R=0\) with antireflection coating on the incident surface.

It is lightly doped with \(n_0=1\times10^{12}\text{ cm}^{-3}=1\times10^{18}\text{ m}^{-3}\) and has a lifetime of \(\tau_\text{s}=100\) μs for photo generated carriers.

Both electrons and holes can freely move in the photoconductor, and both electrodes have ohmic contacts. The device is biased at \(V=2\text{ V}\) across its electrodes.

GaAs has the following characteristic parameters: \(\epsilon=13.18\epsilon_0\) at DC or low frequencies, \(\mu_\text{e}=8500\text{ cm}^2\text{ V}^{-1}\text{ s}^{-1}=0.85\text{ m}^2\text{ V}^{-1}\text{ s}^{-1}\), \(\mu_\text{h}=400\text{ cm}^2\text{ V}^{-1}\text{ s}^{-1}=0.04\text{ m}^2\text{ V}^{-1}\text{ s}^{-1}\), and \(n_\text{i}=2.33\times10^{12}\text{ m}^{-3}\) at 300 K.

(a) Find the dark conductivity. With the given bias voltage, is the device limited by a space-charge effect at any level of input optical signal?

(b) Find the external quantum efficiency. What are the gain and the responsivity of this device?

**(a)**

We have \(n_0=1\times10^{18}\text{ m}^{-3}\) and \(p_0=n_\text{i}^2/n_0=5.43\times10^6\text{ m}^{-3}\). We then find the following dark conductivity for the device at 300 K:

\[\begin{align}\sigma_0&=e(\mu_\text{e}n_0+\mu_\text{h}p_0)=1.6\times10^{-19}\times(0.85\times1\times10^{18}+0.04\times5.43\times10^6)\text{ Ω}^{-1}\text{ m}^{-1}\\&=0.136\text{ Ω}^{-1}\text{ m}^{-1}\end{align}\]

Because \(\sigma\gt\sigma_0\) at any level of input optical signal, we have

\[V_\text{SC}\gt\frac{\sigma_0l^2}{(\mu_\text{e}+\mu_\text{h})\epsilon}=\frac{0.136\times(100\times10^{-6})^2}{(0.85+0.04)\times13.18\times8.85\times10^{-12}}\text{ V}=13.1\text{ V}\]

Because \(V\lt{V}_\text{SC}\) for \(V=2\text{ V}\), the device is not limited by a space-charge effect at any level of input optical signal.

**(b)**

We find that \(\alpha{d}=1\) for this device. With \(\eta_\text{coll}=\eta_\text{t}=1\), the external quantum efficiency

\[\eta_\text{e}=\eta_\text{i}=1-\text{e}^{-\alpha{d}}=1-\text{e}^{-1}=63.2\%\]

The electron transit time at the bias voltage of \(V=2\text{ V}\) is

\[\tau_\text{tr}^\text{e}=\frac{l^2}{\mu_\text{e}V}=\frac{(100\times10^{-6})^2}{0.85\times2}\text{ s}=5.88\text{ ns}\]

Because both electrons and holes can freely move, the gain

\[G=\frac{\tau_\text{s}}{\tau_\text{tr}^\text{e}}\left(1+\frac{\mu_\text{h}}{\mu_\text{e}}\right)=\frac{100\times10^{-6}}{5.88\times10^{-9}}\left(1+\frac{0.04}{0.85}\right)=1.78\times10^4\]

At \(\lambda=850\text{ nm}\), the responsivity

\[\mathcal{R}=G\eta_\text{e}\frac{e}{h\nu}=1.78\times10^4\times0.632\times\frac{850}{1239.8}\text{ A W}^{-1}=7.71\text{ kA W}^{-1}\]

It is necessary to apply a voltage or a current to a photoconductor for measuring its photoconductivity in terms of an electrical signal. Normally a bias voltage is applied and a load resistance is used to convert the signal current into an output voltage.

Figure 14-12(a) shows the basic circuitry of a photoconductive detector that is biased with a voltage. Figure 14-12(b) shows the small-signal equivalent circuit, including the noise sources, of a photoconductive detector.

The intrinsic speed of the device is not limited by its capacitance, but the speed of the output signal at the load resistance is still influenced by the device capacitance.

The shot noise in a photoconductor is associated with the statistical nature of the generation and recombination of carriers, which leads to random fluctuations in the carrier number. This shot noise is known as the ** generation-recombination noise**.

It has contributions from the optical signal, the background radiation, and the dark current. The dark current in this case comes from the dark conductivity of the device due to thermal excitation of free carriers.

For long-wavelength infrared detectors, this dark current can be a major source of noise because of the small excitation energy. For this reason, such detectors have to be operated at low temperatures in order to minimize this noise.

Because of the gain in a photoconductor, the generation-recombination noise has the form of the amplified shot noise given in (14-25) [refer to the photodetector noise tutorial]:

\[\tag{14-77}\overline{i_\text{n,GR}^2}=2eBGF(\overline{i_\text{s}}+\overline{i_\text{b}}+\overline{i_\text{d}})\]

Because the gain \(G\) of a photoconductor is a function of the carrier lifetime \(\tau\) in the form of (14-73), the excess noise factor \(F\) is determined by the statistics of the carrier lifetime and the signal frequency as

\[\tag{14-78}F=\frac{\overline{G^2}}{\overline{G}^2}=\frac{\overline{\tau^2}}{\overline{\tau}^2}\]

For a photoconductor, in which the carrier lifetime is primarily determined by the carrier recombination process, the probability distribution of \(\tau\) is characterized by the Poisson process of a continuous random variable with the consequence that \(F=2\).

In addition to this shot noise, there is also thermal noise from the photoconductor resistance and the load resistance.

Therefore, the total noise of a photoconductor is

\[\tag{14-79}\overline{i_\text{n}^2}=4eBG(\overline{i_\text{s}}+\overline{i_\text{b}}+\overline{i_\text{d}})+\frac{4k_\text{B}TB}{R_\text{eq}}\]

where \(R_\text{eq}\) is an equivalent resistance seen at the output of the device.

Most photoconductors are shot-noise limited by their dark current. The SNR of a photoconductor has the form given in (14-33) [refer to the photodetector noise tutorial] for a photodetector with an internal gain of \(G\).

**Example 14-10**

The photoconductive detector considered in Example 14-9 is loaded with a sufficiently large resistance such that the resistive thermal noise is negligible compared to the shot noise from its dark current at the operating temperature of 300 K. The background radiation noise is also negligible.

(a) Find the dark resistance of the device. Then, find its dark current at a bias voltage of \(V=2\text{ V}\).

(b) Find the NEP of the device for a bandwidth of 1 Hz at \(\lambda=850\text{ nm}\).

(c) Find the value of \(D^*\) for the device at \(\lambda=850\text{ nm}\).

**(a)**

From Example 14-9, we have \(\sigma_0=0.136\text{ Ω}^{-1}\text{ m}^{-1}\). Thus, the dark resistance of the device is

\[R_0=\frac{l}{\sigma_0wd}=\frac{100\times10^{-6}}{0.136\times100\times10^{-6}\times1\times10^{-6}}\text{ Ω}=7.35\text{ MΩ}\]

The dark current at a bias voltage of \(V=2\text{ V}\) is

\[i_\text{d}=\frac{V}{R_0}=\frac{2}{7.35\times10^6}\text{ A}=272\text{ nA}\]

**(b)**

With \(G=1.78\times10^4\) found in Example 14-9, the noise of this photoconductor is

\[\begin{align}\overline{i_\text{n}^2}&=\overline{i_\text{n,sh}^2}=4eBG\overline{i_\text{d}}\\&=4\times1.6\times10^{-19}\times1.78\times10^4\times272\times10^{-9}\times{B}\qquad(\text{A}^2\text{ Hz}^{-1})\\&=3.1\times10^{-21}B\qquad(\text{A}^2\text{ Hz}^{-1})\end{align}\]

With \(\mathcal{R}=7.71\text{ kA W}^{-1}\) from Example 14-9, we find the following NEP for a bandwidth of 1 Hz:

\[\frac{\text{NEP}}{B^{1/2}}=\frac{(3.1\times10^{-21})^{1/2}}{7.71\times10^3}\text{ W Hz}^{-1/2}=7.22\text{ fW Hz}^{-1/2}\]

**(c)**

This device has an illumination area of \(\mathcal{A}=lw=(100\times10^{-6})^2\text{ m}^2=1\times10^{-8}\text{ m}^2\). Thus, its specific detectivity at \(\lambda=850\text{ nm}\) is found to be

\[\begin{align}D^*&=\frac{(1\times10^{-8})^{1/2}}{7.22\times10^{-15}}\text{ m Hz}^{1/2}\text{ W}^{-1}=1.39\times10^{10}\text{ m Hz}^{1/2}\text{ W}^{-1}\\&=1.39\times10^{12}\text{ cm Hz}^{1/2}\text{ W}^{-1}\end{align}\]

The frequency response of a photoconductor that has a gain generally described by (14-73) is characterized by the following electrical power spectrum:

\[\tag{14-80}\mathcal{R}^2(f)=\frac{\mathcal{R}^2(0)}{1+4\pi^2f^2\tau^2}\]

which has a 3-dB cutoff frequency given by

\[\tag{14-81}f_\text{3dB}=\frac{1}{2\pi\tau}\]

Therefore, a photoconductor has the following gain-bandwidth product:

\[\tag{14-82}Gf_\text{3dB}=\frac{\tau}{\tau_\text{r}}\cdot\frac{1}{2\pi\tau}=\frac{1}{2\pi\tau_\text{r}}\]

Figure 14-13 shows the frequency response of a typical photoconductive detector.

Although the response speed of a photoconductor can be increased by reducing the carrier lifetime \(\tau\), the gain-bandwidth product is solely determined by the time constant \(\tau_\text{r}\).

As discussed above, the value of \(\tau_\text{r}\) can be reduced by increasing the applied voltage until \(V=e\mathcal{N}/C\), when \(\tau_\text{r}=\tau_\text{d}\). At this point and beyond, the gain-bandwidth product saturates at \(Gf_\text{3dB}=1/2\pi\tau_\text{d}\); thus, increasing the speed will reduce the gain, and vice versa.

We can therefore expect a high-gain photoconductor to be very slow and a high-speed photoconductor to be insensitive.

Both the gain and the speed of practical photoconductors cover a wide range, from less than 1 to over \(10^5\) for the gain and from less than 1 ps to over 1 ms for the response speed.

Figure 14-14 shows the structure of a high-speed photoconductor. It has the ** metal-semiconductor-metal** (MSM) structure with interdigitated electrodes of submicrometer features to minimize the electron transit time so that a higher gain can be obtained for a given speed.

Because its speed is limited by the carrier lifetime, the device is fabricated on a high-resistivity semi-insulating semiconductor that has a very short carrier lifetime.

For the transmission of high-frequency electrical signals, the electrodes are connected to microstrip lines.

A high-speed photoconductor is often used as a high-speed optoelectronic switch for switching an electronic circuit with an ultrashort optical pulse.

**Example 14-11**

Find the 3-dB cutoff frequency, the gain-bandwidth product, and the NEP over the entire bandwidth for the photoconductor considered in the preceding two examples.

Because \(\tau=\tau_\text{s}=100\text{ μs}\), the 3-dB cutoff frequency

\[f_\text{3dB}=\frac{1}{2\pi\times100\times10^{-6}}\text{ Hz}=1.59\text{ kHz}\]

In the operating condition under consideration in the preceding examples, we have \(G=1.78\times10^4\). Thus, the gain-bandwidth product

\[Gf_\text{3dB}=1.78\times10^4\times1.59\text{ kHz}=28.3\text{ MHz}\]

It is easily verified that \(Gf_\text{3dB}=1/2\pi\tau_\text{r}\) as expressed in (14-82) for \(\tau_\text{r}=\tau_\text{tr}^\text{e}(1+\mu_\text{h}/\mu_\text{e})^{-1}=5.63\text{ ns}\).

With \(f_\text{3dB}=1.59\text{ kHz}\), we have \(B=f_\text{3dB}/0.886=1.79\text{ kHz}\). Therefore, the NEP over the entire bandwidth is

\[\text{NEP}=7.22\times(1.79\times10^3)^{1/2}\text{ fW}=305\text{ fW}\]

We find that this detector ha a low cutoff frequency at the kilohertz level because of its large carrier lifetime of 100 μs. Because of its small bandwidth, it also has a low total NEP over its entire bandwidth.

The speed of the device can be increased by reducing its carrier lifetime, but both the gain and the total NEP will suffer if other parameters of the device remain unchanged.

The next tutorial covers the topic of ** junction photodiodes**.