# Avalanche Photodiodes

This is a continuation from the previous tutorial - * junction photodiodes*.

The avalanche photodiode (APD) is the solid-state counterpart of the PMT (photomultiplier tube) [refer to the photoemissive detectors tutorial]. An APD versus an ordinary junction photodiode [refer to the junction photodiodes tutorial] is similar to a PMT versus a vacuum photodiode [refer to the photoemissive detectors tutorial].

However, the high-gain and low-noise characteristics of PMTs are difficult for conventional APDs to match.

An internal gain is built into an APD to multiply the photogenerated electrons and holes. The physical process responsible for the internal gain in an APD is ** avalanche multiplication** of charge carriers through

**, as illustrated in Figure 14-16.**

*impact ionization*

In the impact ionization process, an electron or hole of a sufficiently high kinetic energy can create a secondary electron-hole pair by transferring its kinetic energy to the excitation of the secondary carriers through collision with the lattice.

In the presence of a high electric field, the newly generated electron and hole can be accelerated to gain sufficient kinetic energies for impact ionization to generate more electron-hole pairs. A cascade of these events leads to avalanche multiplication of the photogenerated carriers.

This process does not take place in an ordinary photodiode.

The spectral response of an APD is similar to that of an ordinary photodiode with a threshold photon energy of \(E_\text{th}=E_\text{g}\) determined by the bandgap of the absorption region where electron-hole pairs are photogenerated.

The threshold kinetic energies for an electron or hole to initiate impact ionization in a semiconductor of a bandgap \(E_\text{g}\) fall between \(E_\text{g}\) and \(2E_\text{g}\), depending on the effective electron and hole masses and the details of the band structure.

These kinetic threshold energies are much higher than the kinetic energies of electron and holes at their respective saturation velocities. Therefore, no avalanche multiplication takes place in an ordinary photodiode even when the photogenerated electrons and holes in the device are accelerated to reach their respective saturation velocities, such as in a high-speed p-i-n photodiode.

In an APD, the average drift velocities of electrons and holes remain at the saturation velocities, but high-energy carriers at the tail of the energy distribution can have kinetic energies higher than the threshold energies for impact ionization.

The impact ionization process is characterized quantitatively by the ionization coefficients, \(\alpha_\text{e}\) for electrons and \(\alpha_\text{h}\) for holes (quoted per meter, but also often quoted per centimeter).

The ionization coefficient for an electron or hole represents the probability for an electron or hole that travels a unit distance to create an electron-hole pair through impact ionization.

Both \(\alpha_\text{e}\) and \(\alpha_\text{h}\) are characteristics of a semiconductor and are strong functions of both electric field strength and temperature. They increase rapidly with an increasing electric field strength but decrease with increasing temperature.

Their ratio, known as the ** ionization ratio**, is defined as

\[\tag{14-104}k=\frac{\alpha_\text{h}}{\alpha_\text{e}}\]

The ionization ratio is a function of field strength and temperature. It also varies among different semiconductors. When \(k\lt1\), impact ionization by electrons dominates. When \(k\gt1\), impact ionization by holes dominates.

For Si, \(k\lt1\), and the value of \(k\) can be as small as 0.01, depending on the field strength. Therefore, impact ionization in Si is completely dominated by electrons.

For Ge and InP, \(k\gt1\), but the value of \(k\) is not large. For GaAs, \(k\approx1\).

As we shall see below, to maximize the avalanche gain and minimize the excess noise, an ideal APD must have only electrons initiating impact ionization, thus \(k\ll1\), or only holes initiating impact ionization, thus \(k\gg1\). A \(k\) value close to unity is not desirable because it limits the avalanche gain due to a large excess noise.

The total current gain, \(G=i_\text{s}/i_\text{ph}\) as defined in (14-23) [refer to the photodetector noise tutorial], of an APD is the ** avalanche multiplication factor** of photogenerated carriers. It depends on the thickness and the structure of the avalanche region in the APD, as well as on the reverse voltage applied to the APD.

For an APD that has a uniform field across its avalanche multiplication region of thickness \(d_\text{m}\), the field-dependent parameters \(\alpha_\text{e}\), \(\alpha_\text{h}\), and \(k\) have spatially independent, constant values over the thickness \(d_\text{m}\).

In this ideal situation, the avalanche multiplication gain for electron or hole injection into the avalanche region can be expressed as

\[\tag{14-105}G=\frac{1-k}{\text{e}^{-(1-k)\alpha_\text{e}d_\text{m}}-k}=\frac{1-1/k}{\text{e}^{-(1-1/k)\alpha_\text{h}d_\text{m}}-1/k}\]

When \(k=1\),

\[\tag{14-106}G=\frac{1}{1-\alpha_\text{e}d_\text{m}}=\frac{1}{1-\alpha_\text{h}d_\text{m}}\]

We see from the above two relations that the multiplication gain \(G\) increases nonlinearly with an increase in the value of \(\alpha_\text{e}d_\text{m}\), with a corresponding increase in that of \(\alpha_\text{h}d_\text{m}\), for any given value of \(k\).

At a certain value of \(\alpha_\text{e}d_\text{m}\) and its corresponding value of \(\alpha_\text{h}d_\text{m}\) for a given \(k\), however, \(G\) increases quickly to approach infinity. The consequence is an instability leading to avalanche breakdown.

In practice, the gain of an APD is often expressed emipircally as

\[\tag{14-107}G=\frac{1}{1-(V_\text{r}/V_\text{br})^n}\]

where \(V_\text{r}\) is the reverse voltage on the APD, \(V_\text{br}\) is the avalanche breakdown voltage, and \(n\) is an empirically fitted parameter typically in the range of \(3-6\). The values of \(V_\text{br}\) and \(n\) depend strongly on the device structure and operating temperature.

The gain of an APD is very sensitive to both reverse bias voltage and temperature. Voltage and temperature stabilization is often required for the operation of an APD at a constant gain.

In normal operation, an APD is biased at a fixed voltage below, but close to, the breakdown voltage. Typical gains range from 10 to 20 for Ge and InGaAs APDs, and from 50 to 200 for Si APDs.

Because of the internal gain, the responsivity of an APD is \(\mathcal{R}=G\mathcal{R}_0\), where \(\mathcal{R}_0\) is the intrinsic responsivity of an equivalent photodiode without an internal gain.

**Example 14-14**

A superlattice InGaAs/InP APD, which is described in further detail in Example 14-16 below, has an avalanche multiplication region that consists of an InAlGaAs/InAlAs superlattice layer of \(d_\text{m}=231\text{ nm}\). It has an ionization ratio of \(k=0.25\). When an average electric field of \(E_\text{m}=63\text{ MV m}^{-1}\) is established by a reverse bias voltage in this avalanche multiplication layer, the electron ionization coefficient is \(\alpha_\text{e}=6.5\times10^6\text{ m}^{-1}\).

(a) Find the avalanche multiplication gain in this condition.

(b) If the device has a breakdown voltage of \(V_\text{br}=20\text{ V}\), what is the reverse bias voltage?

**Solution:**

**(a)**

With the given parameters, we have \(\alpha_\text{e}d_\text{m}=6.5\times10^6\times231\times10^{-9}=1.5\). The multiplication gain is obtained by using (14-105)

\[G=\frac{1-k}{\text{e}^{-(1-k)\alpha_\text{e}d_\text{m}}-k}=\frac{1-0.25}{\text{e}^{-(1-0.25)\times1.5}-0.25}=10\]

**(b)**

We can estimate the reverse bias voltage by using (14-107). With \(V_\text{br}=20\text{ V}\) and \(G=10\), we have

\[V_\text{r}=\left(1-\frac{1}{G}\right)^{1/n}V_\text{br}=0.9^{1/n}\times20\text{ V}\]

Because we do not have the information on the parameter \(n\), we can only calculate the limits of the reverse bias voltage to be \(19.31\text{ V}\le{V_\text{r}}\le19.65\text{ V}\) by assuming that \(3\le{n}\le6\).

Thus, the bias voltage is below, but very close to, the breakdown voltage.

Because \(V_\text{r}\) is very close to \(V_\text{br}\), the multiplication gain is very sensitive to the reverse bias voltage. For example, if we take \(n=3\) but use \(V_\text{r}=19.65\text{ V}\), which is obtained for \(n=6\), we find a gain of \(G=19.4\) instead of \(10\). This example shows that stabilization of both voltage and temperature is very important for an APD to function at a constant gain as both \(V_\text{br}\) and \(n\) vary sensitively with temperature.

The small-signal equivalent circuit of an APD is shown in Figure 14-27(a). It is similar to that of an ordinary junction photodiode [refer to the junction photodiode tutorial], except that the avalanche multiplication gain is included in the signal current \(i_\text{s}=Gi_\text{ph}\) for an APD. Figure 14-27(b) shows the noise equivalent circuit of an APD.

The shot noise of an APD has the form given in (14-25) [refer to the photodetector noise tutorial] for a photodetector that has an internal gain.

All APDs generate excess noise because of the statistical nature of the avalanche multiplication process. The excess noise factor \(F\) for an APD is a function of the avalanche multiplication gain \(G\) and the ionization ratio \(k\).

For conventional APDs, the excess noise factor for avalanche multiplication initiated by electrons can be expressed as

\[\tag{14-108}F=F_\text{e}=kG+(1-k)\left(2-\frac{1}{G}\right)\]

and that for avalanche multiplication initiated by holes can be expressed as

\[\tag{14-109}F=F_\text{h}=\frac{G}{k}+\left(1-\frac{1}{k}\right)\left(2-\frac{1}{G}\right)\]

The excess noise of an APD is minimized if \(k\lt1\) when only electrons contribute to avalanche multiplication, or if \(k\gt1\) when only holes contribute to avalanche multiplication. The theoretical minimum of the excess noise factor for an APD is \(F=2-1/G\) for \(k=0\) in (14-108) or \(k=\infty\) in (14-109).

From (14-108) and (14-109), we see that it is important to have the correct type of carriers injected into the avalanche regions in order to minimize the excess noise because injection of the wrong type of carriers will lead to a very large value of \(F\).

For this reason, an avalanche region consisting of a material with \(k\lt1\) is placed on the \(\text{n}^+\) side opposite to an absorption layer on the \(\text{p}^+\) side so that electrons are injected into the avalanche region in reverse bias, whereas an avalanche region consisting of a material with \(k\gt1\) is placed on the \(\text{p}^+\) side opposite to an absorption layer on the \(\text{n}^+\) side so that holes are injected into the avalanche region in reverse bias. This point can be clearly seen in the two structures shown later in Figure 14-29.

In practice, the excess noise factor is often expressed with the following empirically fitted formula:

\[\tag{14-110}F=G^x\]

where \(x\) is a parameter typically in the range of \(0.2-1\) obtained from fitting experimental data.

Including the thermal noise, which does not get amplified, the total current noise of an APD is

\[\tag{14-111}\overline{i_\text{n}^2}=2eBGF(\overline{i_\text{s}}+\overline{i_\text{b}}+\overline{i_\text{d}})+\frac{4k_\text{B}TB}{R_\text{L}}\]

which has the same form as that of the PMT in (14-62) [refer to the photoemissive detectors tutorial].

The SNR of an APD has the form given in (14-33) [refer to the photodetector noise tutorial] for a photodetector that has an internal gain.

The excess noise degrades the SNR of an APD when compared with an ordinary photodiode of the same quantum efficiency. Therefore, the use of an APD instead of an ordinary photodiode such as a p-i-n photodiode makes sense only when amplifiers are needed in the use of an ordinary photodiode for the detection of low-power optical signals.

Because of the noise from the amplifiers, an APD can have a better SNR than a photodiode-amplifier combination to justify the use of the APD. This situation occurs when detecting high-frequency signals at very low power levels because the amplifier noise dominates the detector noise at high frequencies.

The NEP (noise equivalent power) for Si APDs can be as low as 1 pW.

**Example 14-15**

Find the excess noise factor for the APD considered in Example 14-14 if electrons are injected into its avalanche region to initiate the avalanche multiplication process. What is its excess noise factor if holes are injected instead?

*Solution:*

We have \(k=0.25\) and \(G=10\) from Example 14-14. If electrons are injected, the excess noise factor is found using (14-108) to be

\[F=F_\text{e}=0.25\times10+(1-0.25)\times\left(2-\frac{1}{10}\right)=3.925\]

If holes are injected instead, we have to use (14-109) to find that

\[F=F_\text{h}=\frac{10}{0.25}+\left(1-\frac{1}{0.25}\right)\times\left(2-\frac{1}{10}\right)=34.3\]

We see that \(F_\text{h}\) is about nine times \(F_\text{e}\). Clearly, the avalanche multiplication process in this device of \(k\lt1\) has to be initiated by electrons, not by holes. in order to minimize the excess noise. If holes are injected instead, the excess noise factor would be enhanced by as much as nine times, resulting in a significant increase in the APD noise.

Like any photodiode, the response time of an APD is determined by both the response time of its signal current and the time constant of its equivalent circuit.

The speed of an APD is determined by four factors:

- The transit time \(\tau_\text{tr}\) through the absorption layer of a thickness \(d_\text{a}\).
- The diffusion time in the diffusion regions
- The avalanche buildup time \(\tau_\text{av}\) in the avalanche multiplication layer of a thickness \(d_\text{m}\)
- The circuit response time limited by the RC time constant \(\tau_\text{RC}\)

The avalanche buildup time is unique to APDs. The other three factors are common to all photodiodes, but the transit time in an APD is different from that in an ordinary photodiode.

The absorption layer of an APD is equivalent to the intrinsic region of a p-i-n photodiode. It is either intrinsic or very lightly doped and is depleted to maintain a sufficiently high field in this region for a short carrier transit time.

The avalanche multiplication layer requires an even higher field. Thus, it is also intrinsic or very lightly doped. Besides, it is much thinner than the absorption layer: \(d_\text{m}\ll{d}_\text{a}\).

Because the field strengths in these two regions are different, and their material compositions can also be different in heterostructure APDs, the carrier velocities in these two regions can be different even when they are all close to or at their respective saturation values.

A detailed analysis of the time response of an APD is very complicated because it has to take into account the spatial variations of the field strength and the carrier distribution in each region, as well as the spatial variations in \(\alpha_\text{e}\), \(\alpha_\text{h}\), and \(k\) in the avalanche region. However, by taking these parameters to be constants of their respectively spatially averaged values, a simplified analysis yields results that are very good approximations to accurate values.

In an APD where electron multiplication dominates the avalanche process, an electron generated on the \(\text{p}^+\) side of the absorption layer can generate a secondary electron-hole pair in the avalanche region located on the \(\text{n}^+\) side of the absorption layer after taking a time of \(\tau_\text{tr}^\text{e}\) to drift through the absorption layer.

The secondary hole then takes a time of \(\tau_\text{tr}^\text{h}\) to drift back to the \(\text{p}^+\) side where it is collected. Because the drift of a secondary hole follows the drift of its primary electron, the transit time in an APD is twice as long as that in an ordinary photodiode of the same intrinsic absorption-layer thickness:

\[\tag{14-112}\tau_\text{tr}=\tau_\text{tr}^\text{e}+\tau_\text{tr}^\text{h}=\frac{d_\text{a}}{v_\text{e}^\text{a}}+\frac{d_\text{a}}{v_\text{h}^\text{a}}\]

where \(v_\text{e}^\text{a}\) and \(v_\text{h}^\text{a}\) are, respectively, the electron and hole drift velocities in the absorption region.

The same transit time is obtained in an APD where hole multiplication dominates the avalanche process.

In the avalanche region, the multiplication process is not instantaneous but takes time to build up. The avalanche buildup time is a function of the gain, the ionization ratio, and the thickness of the avalanche region.

For an avalanche process initiated by electrons with \(k\lt1\), the avalanche buildup time can be approximated as

\[\tag{14-113}\tau_\text{av}\approx{Gk}\frac{d_\text{m}}{v_\text{e}^\text{m}}+\frac{d_\text{m}}{v_\text{h}^\text{m}}\]

where \(v_\text{e}^\text{m}\) and \(v_\text{h}^\text{m}\) are, respectively, the electron and hole drift velocities in the avalanche multiplication region.

For an avalanche process initiated by holes with \(k\gt1\),

\[\tag{14-114}\tau_\text{av}\approx\frac{G}{k}\frac{d_\text{m}}{v_\text{h}^\text{m}}+\frac{d_\text{m}}{v_\text{e}^\text{m}}\]

When the diffusion time of carriers in the diffusion regions is minimized, the intrinsic time constant for the signal current in an APD is the sum of the transit time and the avalanche buildup time:

\[\tag{14-115}\tau=\tau_\text{tr}+\tau_\text{av}\]

The signal-current frequency response of an APD has the form of (14-93) [refer to the junction photodiodes tutorial] but with the time constant \(\tau\) given in (14-115):

\[\tag{14-116}\mathcal{R}_\text{s}^2(f)=\left|\frac{i_\text{s}(f)}{P_\text{s}(f)}\right|^2\approx\mathcal{R}_\text{s}^2(0)\left(\frac{\sin\pi{f\tau}}{\pi{f}\tau}\right)^2\]

which has a 3-dB cutoff frequency

\[\tag{14-117}f_\text{3dB}^\text{s}\approx\frac{0.443}{\tau}\]

The circuit response of an APD is similar to that of an ordinary photodiode, with a 3-dB cutoff frequency

\[\tag{14-118}f_\text{3dB}^\text{ckt}\approx\frac{1}{2\pi\tau_\text{RC}}\]

where \(\tau_\text{RC}\) is the RC time constant of the APD equivalent circuit.

Therefore, the total frequency response of an APD can be expressed as

\[\tag{14-119}\mathcal{R}^2(f)=\mathcal{R}_\text{s}^2(f)\mathcal{R}_\text{ckt}^2(f)=\frac{\mathcal{R}^2(0)}{1+4\pi^2f^2\tau_\text{RC}^2}\left(\frac{\sin\pi{f\tau}}{\pi{f\tau}}\right)^2\]

The 3-dB cutoff frequency of an APD can be approximated by a relation similar to that given in (14-99) [refer to the junction photodiodes tutorial]:

\[\tag{14-120}f_\text{3dB}\approx\frac{0.443}{[\tau^2+(2.78\tau_\text{RC})^2]^{1/2}}=\frac{1}{2\pi[\tau_\text{RC}^2+(0.36\tau)^2]^{1/2}}\]

An important figure of merit for an APD is the gain-bandwidth product \(Gf_\text{3dB}\).

There are two modes of operation for an APD. In the normal mode of operation discussed above, the bias voltage is set at a fixed value just below the breakdown voltage. As can be seen from (14-107), the device has a fixed gain at a given operating temperature for \(V_\text{r}\lt{V}_\text{br}\).

In the photon-counting mode of operation, the reverse bias voltage is set above the breakdown voltage. In this situation, a single photon can trigger a constant flow of photocurrent because \(G\rightarrow\infty\) for \(V_\text{r}\gt{V}_\text{br}\), according to (14-107).

The operation of an APD in this mode is controlled by an external circuit to quench the breakdown current by reducing the voltage on the APD to below the breakdown voltage after a photon triggers the breakdown. The APD is then ready to respond to the next incoming photon.

In this mode of operation, an APD is capable of counting single photons, like a PMT (photomultiplier tube) [refer to the photoemissive detectors tutorial].

Its response speed, or time resolution, in counting successive photons is determined by the speed of the external circuit. With a passive current-quenching circuit that consists of current-limiting resistors, the time resolution is on the order of a few nanoseconds, limited by the RC time constant of the circuit. With an active current-quenching circuit consisting of a current-switching transistor, the time resolution can be as high as 20 ps, limited by the switching speed of the transistor in the circuit.

There are many different structures developed for APDs.

In principle, a p-n or p-i-n diode biased near its breakdown voltage can have an avalanche multiplication gain, thus functioning as an APD.

In practice, however, the structure of an APD is designed to optimize both the quantum efficiency and the avalanche multiplication gain of the device.

To maximize quantum efficiency, the absorption region for photogeneration of carriers has to be relatively thick.

To optimize avalanche multiplication, two conditions are required:

- The avalanche region has to be relatively thin in order to support a very high field without local breakdown.
- It is best to have a single type of carrier injected into the avalanche region rather than have both electrons and holes photogenerated throughout the region.

An ordinary p-n or p-i-n structure is not ideal for an APD because both photogeneration and avalanche multiplication of carriers take place in its depletion layer. Some Ge APDs have \(\text{n}^+-\text{p}\), \(\text{n}^+-\text{n}-\text{p}\), or \(\text{p}^+-\text{n}\) structures, which are acceptable but not optimum.

## Separate Absorption and Multiplication APD

A concept for optimizing both photogeneration and avalanche multiplication in an APD is to use a ** separate absorption and multiplication** (SAM) structure, which has separate regions for the two functions.

In such a structure, photogeneration takes place in a relatively thick region of a moderately high field to reduce the carrier transit time, whereas the ionizing carriers are injected into a thin region of a very high field for avalanche multiplication.

Figure 14-28 shows the structure and the field distribution in reverse bias of a Si SAM APD consisting of \(\text{p}^+-\pi-\text{p}-\text{n}^+\) layers.

This structure is called the ** reach-through** structure because the depletion layer under a large reverse bias voltage in the operating condition of this device reaches through the \(\pi\) and \(\text{p}\) regions from the \(\text{p}^+\) region to the \(\text{n}^+\) region.

For optimum performance of a Si APD, electron injection into he avalanche region is required because \(k\ll1\) in Si.

In the reach-through structure shown in Figure 14-28, photons are absorbed to generate electron-hole pairs mainly in the thick \(\pi\) region. The photogenerated electrons, which are minority carriers in the \(\pi\) region, are accelerated and injected into the thin \(\text{p}-\text{n}^+\) junction where avalanche multiplication takes place in the presence of a high electric field.

The photogenerated holes in the \(\pi\) region are collected in the \(\text{p}^+\) region without multiplication because of the low field in that region.

To reduce the noise caused by the leakage current at the edges of the \(\text{p}-\text{n}^+\) junction and to avoid local breakdown at these edges, a guard ring around the edges is often incorporated into a reach-through Si APD, as also shown in Figure 14-28.

Figure 14-29(a) shows the structure and the field distribution in reverse bias of a heterojunction InGaAs/InP SAM APD.

Because \(k\gt1\) in InP, hole injection, rather than electron injection, into the avalanche region for multiplication is desired in this device. Therefore, it is the \(\text{n}^--\)InP layer that is placed on the \(\text{p}^+\) side next to the \(\text{p}^+-\)InP layer.

The absorption region in this \(\text{P}^+-\text{N}^--\nu-\text{n}^+\) heterostructure is the InGaAs \(\nu\) region, which has a smaller bandgap than the InP layers. Holes that are photogenerated in this region are injected into the avalanche region at the InP \(\text{p}^+-\text{n}^-\) junction for avalanche multiplication. Photogenerated electrons are collected in the InGaAs \(\text{n}^+\) region without multiplication.

Figure 14-29(b) shows the structure and the field distribution in reverse bias of a superlattice InGaAs/InP SAM APD.

In this structure, the avalanche region consists of either an InGaAsP/InAlAs superlattice or an InAlGaAs/InAlAs superlattice that is lattice matched to InP.

Because these superlattice materials have \(k\lt1\), electron injection, rather than hole injection, is desired. Consequently, this superlattice multiplication layer is place on the \(\text{n}^+\) side of the structure, and the InGaAs absorption layer is now on the \(\text{p}^+\) side.

There is also a thin \(\text{p}^+-\)InP buffer layer in this structure. This heavily doped buffer layer allows a sharp transition from a very high field strength in the avalanche region to a lower field in the absorption region so that relatively constant, but very different, field strengths can be maintained in both regions. Its purpose is to suppress undesirable tunneling dark current generation and avalanche multiplication in the absorption layer.

**Example 14-16**

A superlattice InGaAs/InP SAM APD designed for optical detection in the infrared spectral range covering 1.3 and 1.55 μm wavelengths has the structure shown in Figure 14-29(b).

It consists of a nearly intrinsic \(\pi-\)InGaAs absorption layer of \(d_\text{a}=1\) μm, an undoped InAlGaAs/InAlAs superlattice multiplication layer of \(d_\text{m}=231\) nm, and a heavily doped \(\text{p}^+-\)InP buffer layer of a very small thickness of 30-50 nm between these two layers.

The absorption coefficients of the InGaAs absorption layer at 1.3 and 1.55 μm wavelengths are \(\alpha=1.2\times10^6\text{ m}^{-1}\) and \(\alpha=6.6\times10^5\text{ m}^{-1}\), respectively.

In the normal operating condition of the APD, the electron and hole drift velocities are \(v_\text{e}^\text{a}=8\times10^4\text{ m s}^{-1}\) and \(v_\text{h}^\text{a}=6\times10^4\text{ m s}^{-1}\) in the InGaAs absorption layer and \(v_\text{e}^\text{m}=4.2\times10^4\text{ m s}^{-1}\) and \(v_\text{h}^\text{m}=3.2\times10^4\text{ m s}^{-1}\) in the InAlGaAs/InAlAs superlattice multiplication layer.

The impact ionization ratio is \(k=0.25\). The active area of this APD has a diameter of \(2r=40\) μm. It has a total capacitance, including its internal capacitance and parasitic capacitance, of \(C=300\) fF and a parasitic series resistance of \(R_\text{s}=10\) Ω.

Find the 3-dB cutoff frequency and the gain-bandwidth product of this APD when it operates at a multiplication gain of \(G=10\) with a load resistance of \(R_\text{L}=50\) Ω.

**Solution:**

With \(d_\text{a}=1\) μm, the transit time in the absorption layer is

\[\tau_\text{tr}=\frac{d_\text{a}}{v_\text{e}^\text{a}}+\frac{d_\text{a}}{v_\text{h}^\text{a}}=\left(\frac{1\times10^{-6}}{8\times10^4}+\frac{1\times10^{-6}}{6\times10^4}\right)\text{ s}=29\text{ ps}\]

The avalanche multiplication in this APD is initiated by electrons. With \(d_\text{m}=231\) nm, \(k=0.25\), and \(G=10\), the avalanche buildup time in the multiplication layer is

\[\tau_\text{av}\approx{Gk}\frac{d_\text{m}}{v_\text{e}^\text{m}}+\frac{d_\text{m}}{v_\text{h}^\text{m}}=\left(10\times0.25\times\frac{231\times10^{-9}}{4.2\times10^4}+\frac{231\times10^{-9}}{3.2\times10^4}\right)\text{ s}=21\text{ ps}\]

We find that \(\tau_\text{tr}\) is comparable to but somewhat larger than \(\tau_\text{av}\) for this APD in the given operating condition. Thus, the intrinsic time constant

\[\tau=\tau_\text{tr}+\tau_\text{av}=50\text{ ps}\]

The RC time constant

\[\tau_\text{RC}=(R_\text{s}+R_\text{L})C=(10+50)\times300\times10^{-15}\text{ s}=18\text{ ps}\]

We find that \(2.78\tau_\text{RC}=50\) ps, which is the same as \(\tau\). Thus the bandwidth of this APD in the given operating condition is equally determined by both its intrinsic time constant and its RC time constant. We have

\[f_\text{3dB}=\frac{0.443}{[\tau^2+(2.78\tau_\text{RC})^2]^{1/2}}=\frac{0.443}{[50^2+50^2]^{1/2}\times10^{-12}}\text{ Hz}=6.26\text{ GHz}\]

Therefore, with \(G=10\), the gain-bandwidth product

\[Gf_\text{3dB}=62.6\text{ GHz}\]

## Graded-Gap Staircase APD

Sophisticated heterostructures, including those using quantum wells and graded-gap layers, have been developed to improve the performance characteristics of APDs.

Figure 14-30(a) and (b) show, respectively, the unbiased and biased band diagrams of a ** staircase APD**, which consists of multiple nearly intrinsic, or lightly doped, graded-gap layers between the \(\text{p}^+\) and \(\text{n}^+\) regions.

This structure is the solid-state equivalent of the PMT (photomultiplier tube) [refer to the photoemissive detectors tutorial], with each graded-gap layer functioning as an electron multiplication state equivalent to a dynode in a PMT.

The graded gap in this structure is made by varying the composition of a semiconductor material, such as the composition \(x\) in Al_{x}Ga_{1-x}As.

The bandgap in each layer increases linearly from a small value of \(E_\text{g1}\) to a large value of \(E_\text{g2}\) with an abrupt drop back to \(E_\text{g1}\) at the end of the layer.

For typical III-V semiconductors, most of the heterostructure bandgap difference occurs in the conduction band. In reverse bias, the voltage applied to the device drops almost entirely across the nearly intrinsic multilayer graded-gap region.

At the operating bias voltage of the device, the energy band has a pattern like that shown in Figure 14-30(b). Photogenerated electrons in the \(\text{p}^+\) region are injected into successive stages of alternating low-field graded-gap regions and high-field conduction-band steps.

The energy drop at each conduction-band step is larger than the threshold impact-ionization energy for electrons. The electrons drift through a low-field region without multiplication, but they impact ionize when passing through an abrupt conduction-band step.

Holes do not contribute to avalanche multiplication but are quickly swept away because the moderately high field in the valence-band is not large enough to cause impact ionization by holes.

The value of the ionization ratio \(k\) for this structure is thus substantially reduced in comparison to a conventional structure of the same material.

The excess noise factor of a staircase APD is also much reduced due to the fact that impact ionization in this device is localized at the conduction-band steps.

At each potential step, each electron acquires only enough energy to generate one secondary electron-hole pair. The only excess noise comes from the probability that an electron, though having enough energy, may or may not impact ionize at a given potential step.

As a result, a staircase APD typically has a small excess noise factor close to unity similar to that of a PMT.

Because this device has a very small \(k\) value and a small excess noise factor, it has improved performance characteristics in terms of optimized gain, reduced noise, and increased speed.

The next tutorial covers the topic of * guided-wave photodetectors*.