# Photodetector Performance Parameters

This is a continuation from the previous tutorial - photodetector noise.

Several parameters are commonly used to define the performance characteristics of photodetectors. These parameters can be considered as the figures of merit of a photodetector. They are used for comparing one photodetector with another and for determining the suitability of a photodetector for a particular application.

## Spectral Response

Because the response of a photon detector is wavelength dependent, a given photodetector is responsive only within a finite, specific range of the optical spectrum.

The spectral range of response for a photodetector is determined by the material, the structure, and the packaging of the detector.

The spectral response of a photodetector is usually specified in terms of the spectral responsivity or the spectral detectivity of the detector.

In choosing a photodetector for an application, the match between. the spectral content of the optical signal and the spectral response of the detector is the first thing to be verified.

## Quantum Efficiency

Quantum efficiency is the probability of generating a charge carrier in a photodetector for each photon that is incident on the detector.

Similarly to the external quantum efficiency, $$\eta_\text{e}$$, of an LED or a semiconductor laser, the external quantum efficiency of a photodetector is reduced from its internal quantum efficiency, $$\eta_\text{i}$$, by the transmission efficiency, $$\eta_\text{t}$$, of the incident optical beam into the active region of the detector and by the collection efficiency, $$\eta_\text{coll}$$, of the photogenerated electrical carriers into a photocurrent.

Thus, we can express the external quantum efficiency of a photodetector as

$\tag{14-35}\eta_\text{e}=\eta_\text{coll}\eta_\text{t}\eta_\text{i}$

Comparing this relation with that in (13-67) [refer to the light-emitting diodes (LEDs) tutorial] for an LED and that in (13-127) [refer to the semiconductor laser characteristics tutorial] for a semiconductor laser, we find that the carrier collection efficiency $$\eta_\text{coll}$$ of a photodetector is equivalent to the carrier injection efficiency $$\eta_\text{inj}$$ of an LED or a laser, and that the optical transmission efficiency $$\eta_\text{t}$$ of a photodetector is equivalent to the photon extraction efficiency $$\eta_\text{t}$$ of an LED or a laser.

As expressed in (14-13) the external quantum efficiency can be defined as the ratio of the number of photogenerated charge carriers, in the form of either photoelectrons or electron-hole pairs, that actually contribute to the photocurrent to the number of incident photons: $$\eta_\text{e}=\mathcal{N}/\mathcal{S}$$.

According to (14-14), the external quantum efficiency of a detector can then be expressed in terms of the incident optical power and the photocurrent as

$\tag{14-36}\eta_\text{e}=\frac{i_\text{ph}/e}{P_\text{s}/h\nu}=\frac{h\nu{i_\text{ph}}}{eP_\text{s}}$

The quantum efficiency of a photodetector is a function of the wavelength of the incident photons because of the spectral response of the detector.

Its wavelength dependence arises not only from its explicit dependence on the optical frequency $$\nu$$ seen in (14-36) but also from the wavelength dependence of the ratio $$i_\text{ph}/P_\text{s}$$ defined below as the responsivity of the detector.

Example 14-2

A Si photodetector responds to an optical signal at 850 nm of 1 mW power with a photocurrent of 500 μA. What is its external quantum efficiency?

At $$\lambda=850\text{ nm}$$, we have

$\frac{h\nu}{e}=\frac{1239.8}{850}\text{ V}$

Therefore, for $$i_\text{ph}=500$$ μA in response to $$P_\text{s}=1\text{ mW}$$, we find from (14-36) the following external quantum efficiency for this detector.

$\eta_\text{e}=\frac{h\nu{i_\text{ph}}}{eP_\text{s}}=\frac{1239.8}{850}\times\frac{500\times10^{-6}}{1\times10^{-3}}=72.9\%$

## Responsivity

Responsivity is an important parameter for a photodetector. It allows one to determine the available output signal of a detector for a given input optical signal.

The responsivity of a photodetector is defined as the ratio of the output current or voltage signal to the power of the input optical signal.

For a photodetector that has an output current signal, the responsivity is defined as

$\tag{14-37}\mathcal{R}=\frac{i_\text{s}}{P_\text{s}}\qquad(\text{A W}^{-1})$

For a photodetector that has an output voltage signal, the responsivity is defined as

$\tag{14-38}\mathcal{R}=\frac{v_\text{s}}{P_\text{s}}\qquad(\text{V W}^{-1})$

Because most of the commonly used photodetectors have output current signals, we consider in further detail the responsivity of such photodetectors in the following. Similar concepts can be extended to photodetectors that have output voltage signals.

For a photodetector without an internal gain, the signal current is simply the photocurrent, $$i_\text{s}=i_\text{ph}$$. By using (14-36), we find the following expression for its responsivity:

$\tag{14-39}\mathcal{R}=\frac{i_\text{ph}}{P_\text{s}}=\eta_\text{e}\frac{e}{h\nu}$

For a photodetector with an internal gain, however, the signal current is amplified by the gain, $$i_\text{s}=Gi_\text{ph}$$, and the responsivity is

$\tag{14-40}\mathcal{R}=\frac{Gi_\text{ph}}{P_\text{s}}=G\eta_\text{e}\frac{e}{h\nu}=G\mathcal{R}_0$

where $$\mathcal{R}_0$$ is the intrinsic responsivity of the detector defined as

$\tag{14-41}\mathcal{R}_0=\frac{i_\text{ph}}{P_\text{s}}=\eta_\text{e}\frac{e}{h\nu}$

The responsivity of a photodetector without an internal gain is simply its intrinsic responsivity, $$\mathcal{R}=\mathcal{R}_0$$, whereas one with an internal gain has a responsivity $$\mathcal{R}=G\mathcal{R}_0$$.

The spectral response of a photodetector is usually characterized by the responsivity of the detector as a function of optical wavelength, $$\mathcal{R}(\lambda)$$, which is known as the spectral responsivity.

In addition, the responsivity of a photodetector is also a function of signal frequency $$f$$. Its frequency dependence, $$\mathcal{R}(f)$$, characterizes the frequency response of the detector, as discussed laster.

Example 14-3

Find the responsivity at 850 nm for the Si photodetector described in Example 14-2.

From (14-39), the responsivity of this detector at 850 nm is simply

$\mathcal{R}=\frac{i_\text{ph}}{P_\text{s}}=\frac{500\times10^{-6}}{1\times10^{-3}}\text{ A W}^{-1}=0.5\text{ A W}^{-1}$

## Noise Equivalent Power

The noise equivalent power (NEP) of a photodetector is defined as the input power required of the optical signal for the signal-to-noise ratio to be unity, SNR = 1, at the detector output.

Then, using the relations in (14-32) and (14-33) [refer to the photodetector noise tutorial], the NEP for a photodetector, with or without an internal gain, that has an output current signal can be defined as

$\tag{14-42}\text{NEP}=\frac{\overline{i_\text{n}^2}^{1/2}}{\mathcal{R}}=\frac{\text{rms}(i_\text{n})}{\mathcal{R}}\qquad(\text{W})$

where $$\overline{i_\text{n}^2}$$ is the mean square noise current at an input optical power level for SNR = 1 and $$\mathcal{R}$$ is the responsivity defined in (14-37).

Using the relation in (14-34) [refer to the photodetector noise tutorial], the NEP for a photodetector that has an output voltage signal can be defined as

$\tag{14-43}\text{NEP}=\frac{\overline{v_\text{n}^2}^{1/2}}{\mathcal{R}}=\frac{\text{rms}(v_\text{n})}{\mathcal{R}}\qquad(\text{W})$

where $$\overline{v_\text{n}^2}$$ is the mean square noise voltage at an input optical power level for SNR =1 and $$\mathcal{R}$$ is the responsivity defined in (14-38).

For most detection systems at the small input signal level for SNR =1, the shot noise contributed by the input optical signal is negligible compared to both the shot noise from other sources and the thermal noise of the detector.

In this situation, the NEP of a photodetector with no internal gain that has an output current signal can be expressed as

$\tag{14-44}\text{NEP}=\frac{(2e\overline{i_\text{b}}+2e\overline{i_\text{d}}+4k_\text{B}T/R)^{1/2}}{\mathcal{R}}B^{1/2}$

The most fundamental limit of a photodetector is the noise contributed by the ubiquitous blackbody radiation in the background. This background radiation sets the absolute minimum of NEP for a photodetector.

It is often the limitation for photodetectors in mid- and far-infrared spectral regions, but it is normally not important for photodetectors in visible and ultraviolet spectral regions.

For most photodetectors responding to optical wavelengths shorter than 3 μm, the noise from background blackbody radiation is dominated by that from the dark current or that from resistive thermal noise, or both.

For such a photodetector, the intrinsic NEP is that defined by its dark current by assuming that the load resistance is sufficiently large if the detector generates a photocurrent signal, or sufficiently small if it generates a photovoltage signal.

However, in order to reduce its RC time constant, a high-speed photodetector that has a current signal normally has a small area, thus a small dark current, but requires a small load resistance, thus a large thermal noise.

Therefore, the NEP of a high-speed photodetector is usually limited by the thermal noise from its external load resistance rather than by the shot noise from its internal dark current.

Because the mean square noise of a detector is proportional to the detector bandwidth, $$\overline{i_\text{n}^2}\propto{B}$$ and $$\overline{v_\text{n}^2}\propto{B}$$, the NEP of a photodetector is proportional to the square root of the detector bandwidth: $$\text{NEP}\propto{B}^{1/2}$$.

Therefore, the NEP of a photodetector is often specified in terms of the NEP for a bandwidth of a 1 Hz as $$\text{NEP}/B^{1/2}$$, in the unit of $$\text{W Hz}^{-1/2}$$.

Example 14-4

The Si photodetector considered in Examples 14-2 and 14-3 has an active area of $$\mathcal{A}=5\text{ mm}^2$$, a bandwidth of $$B=100\text{ MHz}$$, and a dark current of $$i_\text{d}=10\text{ nA}$$.

(a) Find its shot-noise-limited NEP, its thermal-noise-limited NEP, and its total NEP, all for a bandwidth of 1 Hz.

(b) Find its shot-noise-limited NEP, its thermal-noise-limited NEP, and its total NEP, all for its entire bandwidth.

(a)

The shot noise from the dark current is

\begin{align}\overline{i_\text{n,sh}^2}&=2eB\overline{i_\text{d}}=2\times1.6\times10^{-19}\times10\times10^{-9}\times{B}\qquad(\text{ A}^2\text{ Hz}^{-1})\\&=3.2\times10^{-27}B\qquad(\text{ A}^2\text{ Hz}^{-1})\end{align}

The thermal noise is

\begin{align}\overline{i_\text{n,th}^2}&=\frac{4k_\text{B}TB}{R}=\frac{4\times25.9\times10^{-3}\times1.6\times10^{-19}}{50}\times{B}\qquad(\text{ A}^2\text{ Hz}^{-1})\\&=3.32\times10^{-22}B\qquad(\text{ A}^2\text{ Hz}^{-1})\end{align}

The total noise $$\overline{i_\text{n}^2}=\overline{i_\text{n,sh}^2}+\overline{i_\text{n,th}^2}=3.32\times10^{-22}B\qquad(\text{ A}^2\text{ Hz}^{-1})$$, which is completely dominated by thermal noise.

From Example 14-3, we have $$\mathcal{R}=0.5\text{ A W}^{-1}$$ for this detector. Thus, the shot-noise-limited NEP for a bandwidth of 1 Hz is

$\frac{(\text{NEP})_\text{sh}}{B^{1/2}}=\frac{\overline{i_\text{n,sh}^2}^{1/2}}{B^{1/2}\mathcal{R}}=\frac{(3.2\times10^{-27})^{1/2}}{0.5}\text{ W Hz}^{-1/2}=113\text{ fW Hz}^{-1/2}$

The thermal-noise-limited NEP for a bandwidth of 1 Hz is

$\frac{(\text{NEP})_\text{th}}{B^{1/2}}=\frac{\overline{i_\text{n,th}^2}^{1/2}}{B^{1/2}\mathcal{R}}=\frac{(3.32\times10^{-22})^{1/2}}{0.5}\text{ W Hz}^{-1/2}=36.4\text{ pW Hz}^{-1/2}$

The total NEP for a bandwidth of 1 Hz is

$\frac{\text{NEP}}{B^{1/2}}=\frac{\overline{i_\text{n}^2}^{1/2}}{B^{1/2}\mathcal{R}}=\frac{(3.32\times10^{-22})^{1/2}}{0.5}\text{ W Hz}^{-1/2}=36.4\text{ pW Hz}^{-1/2}$

(b)

For $$B=100\text{ MHz}$$, we find that the shot-noise-limited NEP for the entire bandwidth is

$(\text{NEP})_\text{sh}=113\times10^{-15}\times(100\times10^6)^{1/2}\text{ W}=1.13\text{ nW}$

The thermal-noise-limited NEP for the entire bandwidth is

$(\text{NEP})_\text{th}=36.4\times10^{-12}\times(100\times10^6)^{1/2}\text{ W}=364\text{ nW}$

The total NEP for the entire bandwidth is

$\text{NEP}=36.4\times10^{-12}\times(100\times10^6)^{1/2}\text{ W}=364\text{ nW}$

We see that this detector is completely limited by the thermal noise of its load resistance.

## Detectivity

The detectivity characterizes the ability of a photodetector to detect a small optical signal. It is defined as the inverse of the NEP of the detector:

$\tag{14-45}D=\frac{1}{\text{NEP}}\qquad(\text{W}^{-1})$

As discussed above, $$\text{NEP}\propto{B}^{1/2}$$ when the shot noise contributed by the input optical signal is negligibly small compared to the noise from other sources.

In addition, the background radiation current, $$i_\text{b}$$, and the dark current, $$i_\text{d}$$, are often proportional to the surface area, $$\mathcal{A}$$, of a photodetector. Therefore, when $$i_\text{b}$$ and $$i_\text{d}$$ are the dominant sources of noise for a photodetector, the intrinsic noise characteristics of the detector can be better quantified by normalizing $$\text{NEP}$$ to $$(\mathcal{A}B)^{1/2}$$.

A useful intrinsic parameter of a photodetector is the specific detectivity, $$D^*$$, defined as

$\tag{14-46}D^*=\frac{(\mathcal{A}B)^{1/2}}{\text{NEP}}\qquad(\text{cm Hz}^{1/2}\text{ W}^{-1})$

Then, for a dark-current-limited photodetector without an internal gain, we have

$\tag{14-47}D^*\approx\frac{\mathcal{A}^{1/2}\mathcal{R}}{(2e\overline{i_\text{d}})^{1/2}}$

The specific detectivity $$D^*$$ is independent of the area of the detector. It is a measure of the intrinsic detection capability of the material and the structure of the detector.

The detectivity of a photodetector is a function of the wavelength of the optical signal. The spectral characteristics of the detectivity, given as $$D(\lambda)$$ or $$D^*(\lambda)$$, reflect the spectral response of a photodetector.

The detectivity is also a function of the modulation signal frequency $$f$$ carried by the optical beam.

Example 14-5

Find the detectivity and the specific detectivity of the Si photodetector considered in Example 14-4 for the following two situations: (a) when the detector is shot-noise limited by its dark current with a large load resistance and (b) when the detector ha a 50 Ω load resistance.

(a)

As given in Example 14-4, the detector has an active area of $$\mathcal{A}=5\text{ mm}^2=5\times10^{-2}\text{ cm}^2$$. When the detector is shot-noise limited by its dark current, it has a detectivity

$D=\frac{1}{(\text{NEP})_\text{sh}}=\frac{1}{1.13\times10^{-9}}\text{ W}^{-1}=8.85\times10^8\text{ W}^{-1}$

and a specific detectivity

\begin{align}D^*&=\frac{(\mathcal{A}B)^{1/2}}{(\text{NEP})_\text{sh}}=\frac{(5\times10^{-2}\times100\times10^6)^{1/2}}{1.13\times10^{-9}}\text{ cm Hz}^{1/2}\text{ W}^{-1}\\&=1.98\times10^{12}\text{ cm Hz}^{1/2}\text{ W}^{-1}\end{align}

(b)

When the detector has a 50 Ω load resistance, it has a detectivity

$D=\frac{1}{\text{NEP}}=\frac{1}{364\times10^{-9}}\text{ W}^{-1}=2.75\times10^6\text{ W}^{-1}$

and a specific detectivity

\begin{align}D^*&=\frac{(\mathcal{A}B)^{1/2}}{(\text{NEP})_\text{sh}}=\frac{(5\times10^{-2}\times100\times10^6)^{1/2}}{364\times10^{-9}}\text{ cm Hz}^{1/2}\text{ W}^{-1}\\&=6.14\times10^9\text{ cm Hz}^{1/2}\text{ W}^{-1}\end{align}

We find that when the photodetector is loaded with a 50 Ω resistance, its detectivity and specific detectivity are limited by the resistive thermal noise and are much lower than its intrinsic detectivity and specific detectivity, which are limited by the shot noise from its dark current.

## Linearity and Dynamic Range

Linearity of a photodetector is defined by the response of the detector being linear, meaning that its output current or voltage signal is linearly proportional to its input optical signal.

Linear response is required for a photodetector to convert the waveform of an input optical signal faithfully to an output electrical signal without distortion.

When a photodetector has a linear response, its quantum efficiency $$\eta_\text{e}$$ and responsivity $$\mathcal{R}$$ defined above are constants that are independent of the power $$P_\text{s}$$ of the input optical signal.

However, every practical photodetector only has a finite range of linear response, as shown in Figure 14-1. As the power of the input optical signal reaches a certain level, the response of a photodetector starts to saturate, thereby deviating from linearity.

The maximum input signal power acceptable is determined by the maximum deviation from the linear response of a photodetector that can be tolerated in a particular application.

Given the maximum tolerable deviation from linearity to be $$\delta$$ (for $$100\delta\%)$$, the saturation signal power, $$P_\text{s}^\text{sat}$$, for the photodetector in the application is the corresponding maximum acceptable input power.

As illustrated in Figure 14-1, the value of $$P_\text{s}^\text{sat}$$ can be found from

$\tag{14-48}\left.\frac{\text{d}i_\text{s}}{\text{d}P_\text{s}}\right|_{P_\text{s}=P_\text{s}^\text{sat}}=(1-\delta)\mathcal{R}\qquad\text{or}\qquad\left.\frac{\text{d}v_\text{s}}{\text{d}P_\text{s}}\right|_{P_\text{s}=P_\text{s}^\text{sat}}=(1-\delta)\mathcal{R}$

where $$\mathcal{R}$$ is the responsivity of the detector in the linear range.

The usefulness of a photodetector for detecting an optical signal is clearly limited by its saturation, which is quantified by $$P_\text{s}^\text{sat}$$, at the large-signal end and by its detectivity, which is determined by the $$\text{NEP}$$ of the detector, at the small-signal end.

The range of the input signal power above the $$\text{NEP}$$ but below $$P_\text{s}^\text{sat}$$ in the linear-response region is the useful range of operation for a photodetector. This range is known as the dynamic range (DR) of the detector, as indicated in Figure 14-1.

The dynamic range is usually quantified as

$\tag{14-49}\text{DR}=10\log\frac{P_\text{s}^\text{sat}}{\text{NEP}}\qquad(\text{dB})$

Alternatively, the dynamic range of a photodetector is also frequently stated in terms of the number of orders of magnitude in the input power from the $$\text{NEP}$$ to $$P_\text{s}^\text{sat}$$.

Example 14-6

With a load resistance of 50 Ω, the Si photodetector considered in the preceding examples has a saturation current of 10 mA. Find its saturation optical signal power and its dynamic range.

Because $$\mathcal{R}=0.5\text{ A W}^{-1}$$ for this detector, the saturation optical signal power corresponding to $$i_\text{s}^\text{sat}=10\text{ mA}$$ is

$P_\text{s}^\text{sat}=\frac{i_\text{s}^\text{sat}}{\mathcal{R}}=\frac{10}{0.5}\text{ mW}=20\text{ mW}$

The $$\text{NEP}$$ of this detector in the presence of a 50 Ω load resistance is 364 nW from Example 14-4. Therefore, the dynamic range of the detector is

$\text{DR}=10\log\frac{20\times10^{-3}}{364\times10^{-9}}\text{ dB}=47.4\text{ dB}$

## Speed and Frequency Response

The response speed of a photodetector is directly related to its frequency response. It determines the ability of a photodetector to follow a fast-varying optical signal.

To record an optical signal faithfully, a photodetector must have a speed higher than the fastest temporal variations in the signal or, equivalently, a frequency response that has a bandwidth covering the entire bandwidth of the signal.

In the time domain, the speed of a photodetector is characterized by the risetime, $$t_\text{r}$$, and the falltime, $$t_\text{f}$$, of its response to an impulse signal or to a square-pulse signal, as shown in Figure 14-2.

The risetime is defined as the time interval for the response to rise from 10 to 90% of its peak value, whereas the falltime is defined as the time interval for the response to decay from 90 to 10% of its peak value.

Generally, the overall speed of a photodetector is determined by both its intrinsic bandwidth and its RC circuit-limited bandwidth.

The risetime of the impulse response is determined by the intrinsic bandwidth of a photodetector, and that of the square-pulse response is determined by the RC circuit-limited bandwidth of the photodetector.

The risetime and its corresponding bandwidth have the following relation:

$\tag{14-50}t_\text{r}=\frac{0.35}{f_\text{3dB}}$

where $$f_\text{3dB}$$ is the 3-dB cutoff frequency defined below.

The frequency response, which is characterized by the frequency dependence of the responsivity $$\mathcal{R}(f)$$ at a given optical wavelength, can be obtained by simply taking the Fourier transform of the impulse response or by registering the response of the detector at one signal frequency at a time while sweeping the signal frequency.

Note that $$\mathcal{R}(f)$$ is the current or voltage response spectrum of the detector because the responsivity of a photodetector is defined as the output current or voltage signal of the detector.

The output electrical power spectrum of the detector is $$\mathcal{R}^2(f)$$, which defines a 3-dB cutoff frequency, or 3-dB bandwidth, for a photodetector as

$\tag{14-51}\mathcal{R}^2(f_\text{3dB})=\frac{1}{2}\mathcal{R}^2(0)$

Considering the rectangular time interval used to define the bandwidth $$B$$, we have the following relation between $$f_\text{3dB}$$ and $$B$$ of a photodetector:

$\tag{14-52}f_\text{3dB}=0.886B=\frac{0.443}{T}$

The 3-dB bandwidth of a photodetector is a function of the combined effect of a few different physical factors that determine the speed and the frequency response of the detector. These factors and their relative importance depend on the type of the photodetector. They are discussed in later tutorials where the physical properties of various photodetectors are addressed.

Example 14-7

Find the 3-dB cutoff frequency and the risetime in response to an impulse signal for the Si photodetector considered in preceding examples.

From Example 14-4, we find that $$B=100\text{ MHz}$$ for this detector. Therefore, its 3-dB cutoff frequency is

$f_\text{3dB}=0.886B=88.6\text{ MHz}$

The risetime of its response to an impulse signal is

$t_\text{r}=\frac{0.35}{f_\text{3dB}}=\frac{0.35}{88.6\times10^6}\text{ s}=3.95\text{ ns}$

The next tutorial covers the topic of photoemissive detectors