# Junction Photodiodes

This is a continuation from the previous tutorial - photoconductive detectors.

Every junction diode has a photoresponse that can be utilized for optical detection. Junction photodiodes are the most commonly used photodetectors in the photonics industry.

They can take many forms, including semiconductor homojunctions, semiconductor heterojunctions, and metal-semiconductor junctions.

Similarly to that of a photoconductor, the photoresponse of a photodiode results from the photo generation of electron-hole pairs.

In contrast to photoconductors, which can be of either intrinsic or extrinsic type, a photodiode is normally of intrinsic type, in which electron-hole pairs are generated through band-to-band optical absorption.

Therefore, the threshold photon energy of a semiconductor photodiode is the bandgap energy of its active region:

$\tag{14-83}E_\text{th}=E_\text{g}$

Junction photodiodes cover a wide spectral range from ultraviolet to infrared. All of the semiconductor materials used for intrinsic photoconductors discussed in the photoconductive detectors tutorial can be used for photodiodes with similar spectral characteristics.

Figure 14-15 shows the spectral responsivity of representative photodiodes as a function of optical wavelength at 300 K.

All junction photodiodes share some basic principles and characteristics. Therefore, we first consider a simple p-n homojunction photodiode for a general discussion of the common principles and characteristics. Specific characteristics of photodiodes with different structures are discussed later in this tutorial.

The general characteristics of a semiconductor p-n homojunction in the absence of optical illumination are thoroughly discussed in the semiconductor junctions tutorial

In a semiconductor photodiode, generation of electron-hole pairs by optical absorption can take place in any of the different regions: the depletion layer, the diffusion regions, and the homogeneous regions.

In the depletion layer of a diode, the immobile space charges create an internal electric field with a polarity from the n side to the p side, resulting in an electron energy-band gradient shown in Figure 14-16.

When an electron-hole pair is generated in the depletion layer by photoexcitation, the internal field sweeps the electron to the n side and the hole to the p side, as illustrated in Figure 14-16. This process results in a drift current that flows in the reverse direction from the cathode on the n side to the anode on the p side.

If a photoexcited electron-hole pair is generated within one of the diffusion regions at the edges of the depletion layer, the minority carrier, which is the electron in the p-side diffusion region or the hole in the n-side diffusion region, can reach the depletion layer by diffusion and then be swept to the other side by the internal field, as also illustrated in Figure 14-16. This process results in a diffusion current that also flows in the reverse direction.

For an electron-hole pair generated by absorption of a photon in the p or n homogeneous region, no current is generated because there is no internal field to separate the charges and a minority carrier generated in a homogeneous region cannot diffuse to the depletion layer before recombining with a majority carrier.

Because photons absorbed in the homogeneous regions do not generate any photocurrent, the active region of a photodiode consists of only the depletion layer and the diffusion regions. For a high-performance photodiode, the diffusion current is undesirable and is minimized. Therefore, the active region mainly consists of the depletion layer where a drift photocurrent is generated.

The external quantum efficiency, $$\eta_\text{e}$$, of a photodiode is the fraction of total incident photons absorbed in the active region that actually contribute to the photocurrent.

For a vertically illuminated photodetector, in which the optical signal reaches the active region in a direction perpendicular to the junction plane, the external quantum efficiency can be expressed as

$\tag{14-84}\eta_\text{e}=\eta_\text{coll}\eta_\text{t}\eta_\text{i}=\eta_\text{coll}(1-R)T_\text{h}(1-\text{e}^{-\alpha{W}})$

where $$\eta_\text{coll}$$ is the collection efficiency of the photogenerated carriers, $$\eta_\text{t}=(1-R)T_\text{h}$$, and $$\eta_\text{i}=1-\text{e}^{-\alpha{W}}$$.

Here, $$R$$ is the reflectivity of the incident surface of the photodiode, $$T_\text{h}$$ is the transmittance of the homogeneous region between the incident surface and the active region, $$\alpha$$ is the absorption coefficient of the active region, and $$W$$ is the width of the depletion layer that defines the active region.

To improve the quantum efficiency, the surface reflectivity can be reduced by antireflection coating. Besides, the homogeneous region through which the optical signal enters must be made thin to reduce absorption of the optical signal in this region.

For a p-n photodiode that has the incident surface on the p side, the p region has to be very thin and heavily doped so that the depletion layer extends mostly into the thick and lightly doped n region.

Ultimately, the quantum efficiency of a photodiode is determined by the absorption coefficient $$\alpha$$ and the depletion layer thickness $$W$$.

Clearly, there are two contributions to the photocurrent in a junction photodiode: a drift current from photogeneration in the depletion layer and a diffusion current from photogeneration in the diffusion regions.

The homogeneous regions on the two ends of the diode act like blocking layers for the photogenerated carriers because carriers neither drift nor diffuse through these regions.

Consequently, a junction photodiode acts like a photoconductor with two blocking contacts, which is discussed in the photoconductive detectors tutorial. It has a unity gain, $$G=1$$, with the external signal current simply being equal to the photocurrent:

$\tag{14-85}i_\text{s}=i_\text{ph}=\eta_\text{e}\frac{eP_\text{s}}{h\nu}$

This photocurrent is a reverse current that depends only on the power of the optical signal.

When a bias voltage is applied to the photodiode, the total current of the photodiode is the combination of the diode current given in (12-117) [refer to the semiconductor junctions tutorial] and the photocurrent:

$\tag{14-86}i(V,P_\text{s})=I_0(\text{e}^{eV/ak_\text{B}T}-1)-i_\text{s}=I_0(\text{e}^{eV/ak_\text{B}T}-1)-\eta_\text{e}\frac{eP_\text{s}}{h\nu}$

which is a function of both the bias voltage $$V$$ and the optical signal power $$P_\text{s}$$.

Figure 14-17 shows the current-voltage characteristics of a junction photodiode at various power levels of optical illumination.

The dark characteristics for $$P_\text{s}=0$$ are simply those of an unilluminated diode described by (12-117) [refer to the semiconductor junctions tutorial].

According to (14-86), the current-voltage characteristics of an illuminated photodiode shift downward from the dark characteristics by the amount of the photocurrent, which is linearly proportional to the optical power but is independent of the bias voltage.

As shown in Figure 14-17, there are two modes of operation for a junction photodiode.

The device functions in photoconductive mode in the third quadrant of its current-voltage characteristics, including the short-circuit condition on the vertical axis for $$V=0$$.

It functions in photovoltaic mode in the fourth quadrant, including the open-circuit condition on the horizontal axis for $$i=0$$.

The mode of operation is determined by the external circuitry and the bias condition.

The circuitry for the photoconductive mode, shown in Figure 14-17(a), normally consists of a reverse bias voltage of $$V=-V_\text{r}$$ and a load resistance $$R_\text{L}$$.

In this mode of operation, it is necessary to keep the output voltage, $$v_\text{out}$$, smaller than the bias voltage, $$V_\text{r}$$, so that a reverse voltage is maintained across the photodiode.

This requirement can be fulfilled if the bias voltage is sufficiently large while the load resistance is smaller than the internal resistance of the photodiode in reverse bias, as illustrated with the load line in the third quadrant of Figure 14-17.

In the photoconductive mode under the conditions that $$R_\text{L}\lt{R}_\text{i}$$ and $$v_\text{out}\lt{V}_\text{r}$$, a photodiode has the following linear response before it saturates:

$\tag{14-87}v_\text{out}=(I_0+i_\text{s})R_\text{L}=\left(I_0+\eta_\text{e}\frac{eP_\text{s}}{h\nu}\right)R_\text{L}$

The circuitry for the photovoltaic mode, shown in Figure 14-17(b), does not require a bias voltage but requires a large load resistance.

In this mode of operation, the photovoltage appears as a forward bias voltage across the photodiode.

As illustrated with the load line in the fourth quadrant of Figure 14-17, the load resistance is required to be much larger than the internal resistance of the photodiode in forward bias, $$R_\text{L}\gg{R}_\text{i}$$, so that the current $$i$$ flowing through the diode and the load resistance is negligibly small.

In the photovoltaic mode under this condition, the response of the photodiode is not linear but is logarithmic to the optical signal:

$\tag{14-88}v_\text{out}\approx\frac{ak_\text{B}T}{e}\ln\left(1+\frac{i_\text{s}}{I_0}\right)=\frac{ak_\text{B}T}{e}\ln\left(1+\eta_\text{e}\frac{eP_\text{s}}{h\nu{I_0}}\right)$

where $$a$$ is a factor of a value between 1 and 2 in the diode equation of (12-117) [refer to the semiconductor junctions tutorial].

In photoconductive mode, electric energy supplied by the bias voltage source is delivered to the photodiode. In photovoltaic mode, electric energy generated by the optical signal can be extracted from the photodiode to the external circuit. Solar cells are basically semiconductor junction diodes operating in photovoltaic mode for converting solar energy into electricity.

Figure 14-18(a) shows the small-signal equivalent circuit of a junction photodiode.

A photodiode has an internal resistance $$R_\text{i}$$ and an internal capacitance $$C_\text{i}$$ across its junction. Both $$R_\text{i}$$ and $$C_\text{i}$$ depend on the size and the structure of the photodiode and vary with the voltage across the junction.

In photoconductive mode under a reverse voltage, the diode has a large $$R_\text{i}$$ normally on the order of 1-100 MΩ for a typical photodiode, and a small $$C_\text{i}$$ dominated by the junction capacitance $$C_\text{j}$$, as discussed in the semiconductor junctions tutorial. As the reverse voltage increases in magnitude, $$R_\text{i}$$ increases but $$C_\text{i}$$ decreases because the depletion-layer width increases with reverse voltage.

In photovoltaic mode with a forward voltage across the junction, the diode has a large $$C_\text{i}$$ dominated by the diffusion capacitance $$C_\text{d}$$, as also discussed in the semiconductor junctions tutorial. It still has a large $$R_\text{i}$$, though smaller than that in the photodiode mode, because it operates near the open-circuit condition with a very small internal current in the fourth quadrant of the current-voltage characteristics.

The series resistance $$R_\text{s}$$ takes into account both resistance in the homogeneous regions of the diode and parasitic resistance from the contacts. The external parallel capacitance $$C_\text{p}$$ is the parasitic capacitance from the contacts and the package. The series inductance $$L_\text{s}$$ is the parasitic inductance from the wire or transmission-line connections.

The values of $$R_\text{s}$$, $$C_\text{p}$$, and $$L_\text{s}$$ can be minimized with careful design, processing, and packaging of the device.

The noise of a photodiode consists of both shot noise and thermal noise. Because a junction photodiode has a unity gain, its shot noise can be expressed as

$\tag{14-89}\overline{i_\text{n,sh}^2}=2eB(\overline{i_\text{s}}+\overline{i_\text{b}}+\overline{i_\text{d}})$

where $$i_\text{s}=i_\text{ph}$$ is the photocurrent.

The thermal noise seen at the output can be expressed as

$\tag{14-90}\overline{i_\text{n,th}^2}=\frac{4k_\text{B}TB}{R_\text{eq}}$

where $$R_\text{eq}$$ is the equivalent resistance seen at the output port.

From the circuit shown in Figure 14-18(b), we find that

$\tag{14-91}R_\text{eq}=R_\text{L}\parallel(R_\text{i}+R_\text{s})=\frac{R_\text{L}(R_\text{i}+R_\text{s})}{R_\text{L}+R_\text{i}+R_\text{s}}$

In photoconductive mode, the photodiode has a dark current of $$i_\text{d}=I_0$$ and a relatively small load resistance. In photovoltaic mode, the dark current can be eliminated, and the load resistance is required to be very large. Therefore, a photodiode is significantly noisier in photoconductive mode under a reverse bias than in photovoltaic mode without a bias.

High-speed photodiodes are by far the most widely used photodetectors in applications requiring high-speed or broadband photodetection.

The speed of a photodiode is determined by two factors: (1) the response time of the photocurrent and (2) the time constant of its equivalent circuit shown in Figure 14-18(a).

Because a photodiode operating in photovoltaic mode has a large RC time constant due to the large internal diffusion capacitance in this mode of operation, only photodiodes operating in photoconductive mode are suitable for high-speed or broadband applications.

For this reason, we only consider the speed and the frequency response for a photodiode operating in photoconductive mode.

For a photodiode operating in photoconductive mode under a reverse bias, the response time of the photocurrent to an optical signal is determined by two factors: (1) drift of the electrons and holes that are photogenerated in the depletion layer and (2) diffusion of the electrons and holes that are photogenerated in the diffusion regions.

Drift of the carriers across the depletion layer is a fast process characterized by the transit times of the photogenerated electrons and holes across the depletion layer.

In contrast, diffusion of the carriers is a slow process that is caused by optical absorption in the diffusion regions outside of the high-field depletion region. It results in a diffusion current that can last as long as the lifetime of the carriers.

The consequence is a long tail in the impulse response of the photodiode, which translates into a low-frequency falloff in the frequency response of the device. [refer to the photodetector performance parameters tutorial].

For a high-speed photodiode, this diffusion mechanism has to be eliminated by reducing the photogeneration of carriers outside the depletion layer through design of the device structure.

When the diffusion mechanism is eliminated, the frequency response of the photocurrent is only limited by the transit times of electrons and holes.

In general, the frequency response function that is dictated by the carrier transit time depends on the details of the electric field distribution and the photogenerated carrier distribution in the depletion layer.

In a semiconductor, electrons normally have a higher mobility, thus a smaller transit time, than holes. This difference has to be considered in the detailed analysis of the response speed of a photodiode.

For a good estimate of the detector frequency response, however, the average of electron and hole transit times can be used:

$\tag{14-92}\tau_\text{tr}=\frac{1}{2}(\tau_\text{tr}^\text{e}+\tau_\text{tr}^\text{h})$

In the simple case when the process of carrier drift is dominated by a constant transit time of $$\tau_\text{tr}$$, the temporal response of the photocurrent is ideally a rectangular function of duration $$\tau_\text{tr}$$. Therefore, the power spectrum of the photocurrent frequency response can be approximately expressed by

$\tag{14-93}\mathcal{R}_\text{ph}^2(f)=\left|\frac{i_\text{ph}(f)}{P_\text{s}(f)}\right|^2\approx\mathcal{R}_\text{ph}^2(0)\left(\frac{\sin\pi{f}\tau_\text{tr}}{\pi{f}\tau_\text{tr}}\right)^2$

which has a transit-time-limited 3-dB cutoff frequency

$\tag{14-94}f_\text{3dB}^\text{ph}\approx\frac{0.443}{\tau_\text{tr}}$

The frequency response of the equivalent circuit shown in Figure 14-18(a) is determined by (1) the internal resistance $$R_\text{i}$$ and capacitance $$C_\text{i}$$ of the photodiode; (2) the parasitic effects characterized by $$R_\text{s}$$, $$C_\text{p}$$, and $$L_\text{s}$$; and (3) the load resistance $$R_\text{L}$$.

Clearly, the parasitic effects must be eliminated as much as possible because they can degrade the performance of a high-speed photodiode.

A high-speed photodiode normally operates under the condition that $$R_\text{i}\gg{R}_\text{L},R_\text{s}$$. Therefore, when parasitic inductance is eliminated, the ultimate speed of the circuit is dictated by the RC time constant $$\tau_\text{RC}=(R_\text{L}+R_\text{s})(C_\text{i}+C_\text{p})$$. Its frequency response has the following power spectrum:

$\tag{14-95}\mathcal{R}_\text{ckt}^2(f)\approx\frac{\mathcal{R}_\text{ckt}^2(0)}{1+4\pi^2f^2\tau_\text{RC}^2}$

which has an RC-time-limited 3-dB cutoff frequency

$\tag{14-96}f_\text{3dB}^\text{ckt}\approx\frac{1}{2\pi\tau_\text{RC}}=\frac{1}{2\pi(R_\text{L}+R_\text{s})(C_\text{i}+C_\text{p})}$

Combining the photocurrent response and the circuit response, the total output power spectrum of an optimized photodiode operating in photoconductive mode is

$\tag{14-97}\mathcal{R}^2(f)=\mathcal{R}_\text{ph}^2(f)\mathcal{R}_\text{ckt}^2(f)=\frac{\mathcal{R}^2(0)}{1+4\pi^2{f^2}\tau_\text{RC}^2}\left(\frac{\sin\pi{f}\tau_\text{tr}}{\pi{f}\tau_\text{tr}}\right)^2$

This total frequency response has a 3-dB cutoff frequency, $$f_\text{3dB}$$, that can be found approximately by using the following rule of the sum of squares:

$\tag{14-98}\frac{1}{f_\text{3dB}^2}=\frac{1}{(f_\text{3dB}^\text{ph})^2}+\frac{1}{(f_\text{3dB}^\text{ckt})^2}$

By using (14-94) for $$f_\text{3dB}^\text{ph}$$ and (14-96) for $$f_\text{3dB}^\text{ckt}$$, the 3-dB cutoff frequency of a photodiode including transit-time and circuit limitations can be expressed approximately as

$\tag{14-99}f_\text{3dB}\approx\frac{0.443}{[\tau_\text{tr}^2+(2.78\tau_\text{RC})^2]^{1/2}}=\frac{1}{2\pi[\tau_\text{RC}^2+(0.36\tau_\text{tr})^2]^{1/2}}$

Figure 14-19 shows the total frequency response given by (14-97) for a fixed value of $$\tau_\text{tr}$$ but for a few different values of $$\tau_\text{RC}$$. It is seen that the total frequency response is transit-time-limited when $$\tau_\text{tr}\gt2.78\tau_\text{RC}$$, but is RC-time-limited when $$\tau_\text{tr}\lt2.78\tau_\text{RC}$$.

The characteristics given by (14-97) and shown in Figure 14-19 represent the ultimate frequency response of a photodiode.

In practice, the frequency response of a photodiode can be substantially degraded by the presence of a significant diffusion current and by parasitic effects.

The optimum design of a high-speed photodiode requires (1) elimination of the diffusion current, (2) elimination of parasitic effects, and (3) equalization of the transit-time-limited bandwidth and the RC-time-limited bandwidth by making $$\tau_\text{tr}=2.78\tau_\text{RC}$$.

An important consideration for a high-speed photodiode is the bandwidth-efficiency product, $$\eta_\text{e}f_\text{3dB}$$, rather than the bandwidth alone because increasing the bandwidth can often result in a reduced efficiency in many device structures.

Many different approaches can be taken to optimize both the bandwidth and the efficiency for a maximum bandwidth-efficiency product. This issue is further addressed in the following discussions of various device structures.

p-i-n Photodiodes

A p-i-n photodiode consists of an intrinsic region sandwiched between heavily doped $$\text{p}^+$$ and $$\text{n}^+$$ regions. Figure 14-20 shows the comparison between a p-n junction photodiode and a p-i-n photodiode.

In a p-n photodiode, the depletion-layer width and the junction capacitance both vary with reverse voltage across the junction. The electric field in the depletion layer is not uniform.

In a p-i-n photodiode, a reverse bias voltage applied to the device drops almost entirely across the intrinsic region because of high resistivity in the intrinsic region and low resistivities in the surrounding $$\text{p}^+$$ and $$\text{n}^+$$ regions.

As a result, a p-i-n photodiode has the following two important characteristics: (1) the depletion layer is almost completely defined by the intrinsic region; (2) the electric field in the depletion layer is uniform across the intrinsic region.

In practice, the intrinsic region does not have to be truly intrinsic but only has to be highly resistive. It can be either a highly resistive p region, called a $$\pi$$ region, or a highly resistive n region, called a $$\nu$$ region.

The depletion-layer width $$W$$ in a p-i-n diode does not vary significantly with bias voltage but is pretty much fixed by the thickness, $$d_\text{i}$$, of the intrinsic region so that $$W\approx{d}_\text{i}$$.

The internal capacitance of a p-i-n diode can be predetermined in the design of the device through the choice of the thickness of the intrinsic region and the device area $$\mathcal{A}$$:

$\tag{14-100}C_\text{i}=C_\text{j}=\frac{\epsilon\mathcal{A}}{W}\approx\frac{\epsilon\mathcal{A}}{d_\text{i}}$

This capacitance is fairly independent of the bias voltage; thus it remains constant in operation.

When a reverse voltage is applied to a p-i-n diode, a uniform electric field that is linearly proportional to the reverse bias voltage exists throughout the intrinsic region:

$\tag{14-101}E\approx\frac{V_0+V_\text{r}}{W}\approx\frac{V_\text{r}}{d_\text{i}}$

for $$V_\text{r}\gg{V}_0$$.

Due to this uniform field, both electrons and holes have constant drift velocities across the depletion layer in a p-i-n photodiode.

At low and moderate fields, the drift velocities of electrons and holes both vary linearly with the electric field strength. For a p-i-n photodiode operating in this regime with a relatively low reverse bias voltage, the average carrier transit time is given by

$\tag{14-102}\tau_\text{tr}=\frac{1}{2}\left(\frac{W}{\mu_\text{e}E}+\frac{W}{\mu_\text{h}E}\right)\approx\frac{d_\text{i}^2}{2\mu{V_\text{r}}}$

where $$\mu=\mu_\text{e}\mu_\text{h}/(\mu_\text{e}+\mu_\text{h})$$.

Because the depletion-layer width in a p-i-n diode is dictated by the thickness of the intrinsic region, the transit time is inversely proportional to the bias voltage. Therefore, the response speed of a photodiode can be improved by increasing the reverse bias voltage.

At high fields, however, both electron and hole drift velocities reach their respective saturation velocities: $$v_\text{e}\approx{v}_\text{e}^\text{sat}$$ and $$v_\text{h}\approx{v}_\text{h}^\text{sat}$$, which vary little with bias voltage. For most semiconductors, this occurs at a field strength above $$100\text{ MV m}^{-1}$$ for a saturation velocity on the order of $$10^5\text{ m s}^{-1}$$.

For a p-i-n photodiode operating in this regime with a sufficiently large reverse bias voltage, electrons and holes have a constant average transit time across the depletion layer:

$\tag{14-103}\tau_\text{tr}=\frac{W}{v_\text{sat}}\approx\frac{d_\text{i}}{v_\text{sat}}$

where

$\frac{1}{v_\text{sat}}=\frac{1}{2}\left(\frac{1}{v_\text{e}^\text{sat}}+\frac{1}{v_\text{h}^\text{sat}}\right)$

So long as the reverse bias voltage is large enough to keep electrons and holes drifting at their respective saturation velocities, $$\tau_\text{tr}$$ is independent of the bias voltage and can thus be predetermined by the thickness of the intrinsic region through the design of the device.

Compared to a p-n photodiode, in which the depletion-layer width varies with bias voltage, a p-i-n photodiode has a number of advantages because its depletion-layer width is determined by the thickness of the intrinsic region and is independent of the bias voltage.

Both the quantum efficiency and the frequency response of a p-i-n photodiode can be optimized by the geometric design of the device, whereas those of a p-n photodiode depend strongly on the bias voltage.

From the above discussions, it is clear that the transit time, the RC time constant, and the internal quantum efficiency of a vertically illuminated p-i-n photodiode, shown in Figure 14-21(a), all depend on the thickness $$d_\text{i}$$ of the intrinsic region: $$\tau_\text{tr}\propto{d}_\text{i}$$, $$C_\text{i}\propto{d}_\text{i}^{-1}$$, and $$\eta_\text{i}=1-\text{e}^{-\alpha{d_\text{i}}}$$.

For a high quantum efficiency, the thickness $$d_\text{i}$$ of the intrinsic region can be chosen to be larger than the absorption length: $$d_\text{i}\gt1/\alpha$$.

To optimize the speed of a p-i-n photodiode, both the thickness of the intrinsic region and the area of the device have to be properly chosen.

To reduce the diffusion current, $$d_\text{i}$$ can be chosen to be larger than the electron diffusion length in the $$\text{p}^+$$ region and the hole diffusion length in the $$\text{n}^+$$ region: $$d_\text{i}\gg{L}_\text{e},L_\text{h}$$.

A large $$d_\text{i}$$ reduces the RC time constant of the device by reducing $$C_\text{i}$$, but it increases the transit time $$\tau_\text{tr}$$. Because the electric field is relatively constant throughout the active region of a p-i-n photodiode, the transit time can be optimized with a chosen $$d_\text{i}$$.

Because $$C_\text{i}$$ can be reduced by reducing the device area, a p-i-n photodiode normally has an intrinsic region that has a thickness chosen to optimize the quantum efficiency and the transit time. For a high-speed p-i-n photodiode, the device area is made small enough that the RC time constant is not a limiting factor of its frequency response.

One major limitation of p-i-n photodiodes that are made of indirect-gap semiconductors, such as Si and Ge, is the small absorption coefficients of these semiconductors in the spectral regions where only indirect absorption takes place in such semiconductors.

For example, at $$\lambda=850\text{ nm}$$, the absorption coefficient at 300 K is only about $$7\times10^4\text{ m}^{-1}$$ for Si but is about $$1\times10^6\text{ m}^{-1}$$ for GaAs though 850 nm is farther away from the bandgap wavelength of 1.11 μm for Si than from that of 871 nm for GaAs.

This results in a low quantum efficiency, thus a small responsivity, for a Si or Ge p-i-n photodiode of even just a moderate speed because of the conflicting requirements on the thickness $$d_\text{i}$$ for reducing $$\tau_\text{tr}$$ and increasing $$\eta_\text{i}$$ in a vertical p-i-n photodiode shown in Figure 14-21(a).

One solution to this problem is provided by the lateral p-i-n geometry shown in Figure 14-21(b).

In a lateral p-i-n, both $$\tau_\text{tr}$$ and $$C_\text{i}$$ still depend on $$d_\text{i}$$ in the same manner as in a vertical p-i-n, but the internal quantum efficiency is not a function of $$d_\text{i}$$ but is a function of the trench depth $$d$$ as $$\eta_\text{i}=1-\text{e}^{-\alpha{d}}$$. Thus, $$f_\text{3dB}$$ and $$\eta_\text{i}$$ can be independently optimized by properly choosing a value of $$d_\text{i}$$ to optimize $$\tau_\text{tr}$$ and $$C_\text{i}$$ for a large $$f_\text{3dB}$$ while making a deep enough trench for a high value of $$\eta_\text{i}$$.

One additional advantage of a lateral p-i-n photodiode is that the incident optical signal does not have to pass through the homogeneous $$\text{p}^+$$ or $$\text{n}^+$$ region before it reaches the active intrinsic region, thus improving the external quantum efficiency.

This feature is significant for a homojunction p-i-n used for optical detection at short optical wavelengths, such as a Si p-i-n for blue or ultraviolet wavelengths, where the absorption coefficient is very high and the optical penetration depth is very small.

Example 14-12

A vertically illuminated InGaAs/InP p-i-n photodiode for $$\lambda=1.3$$ μm consists of a lightly doped $$\text{n}^-$$-InGaAs layer of a thickness $$d_\text{i}$$ between a thin $$\text{p}^+$$-InGaAs top layer and an $$\text{n}^+$$-InP substrate.

The device is reverse-biased at a sufficiently high bias voltage for both electrons and holes to reach their respective saturation velocities of $$v_\text{e}^\text{sat}=6.5\times10^4\text{ m s}^{-1}$$ and $$v_\text{h}^\text{sat}=4.8\times10^4\text{ m s}^{-1}$$. The absorption coefficient of InGaAs at DC and low frequencies is $$\epsilon=14.1\epsilon_0$$. Take $$R=R_\text{L}+R_\text{s}=50$$ Ω, $$C_\text{p}=0$$, and $$L_\text{s}=0$$ for this device.

This device can be designed to be either front or back illuminated and can be antireflection coated to have a high $$\eta_\text{t}$$; meanwhile, its structure can be optimized to have a high $$\eta_\text{coll}$$.

In any event, its bandwidth-efficiency product is limited to $$\eta_\text{i}f_\text{3dB}$$ because $$\eta_\text{i}\ge\eta_\text{e}$$. This device is made to have a circular active area of a diameter $$2r$$.

Plot its 3-dB cutoff frequency, $$f_\text{3dB}$$, and the upper limit of its bandwidth-efficiency product, $$\eta_\text{i}f_\text{3dB}$$, as a function of the intrinsic layer thickness $$d_\text{i}$$ in the range of $$0\lt{d}_\text{i}\lt3$$ μm for the four different diameters of $$2r=10,20,40,$$ and $$80$$ μm.

The average transit time can be calculated using (14-103) with the following average saturation velocity for electrons and holes:

\begin{align}v_\text{sat}&=\left[\frac{1}{2}\left(\frac{1}{v_\text{e}^\text{sat}}+\frac{1}{v_\text{h}^\text{sat}}\right)\right]^{-1}=\left[\frac{1}{2}\left(\frac{1}{6.5\times10^4}+\frac{1}{4.8\times10^4}\right)\right]^{-1}\text{ m s}^{-1}\\&=5.52\times10^4\text{ m s}^{-1}\end{align}

The active area is $$\mathcal{A}=\pi{r}^2$$. The internal capacitance of the photodiode is $$C_\text{i}=\epsilon\mathcal{A}/d_\text{i}=\epsilon\pi{r}^2/d_\text{i}$$. Thus, the RC time constant $\tau_\text{RC}=RC_\text{i}=R\frac{\epsilon\pi{r}^2}{d_\text{i}}$ with $$R=50$$ Ω and $$\epsilon=14.1\epsilon_0$$.

From (14-99), we then have

$f_\text{3dB}\approx\frac{0.443}{[\tau_\text{tr}^2+(2.78\tau_\text{RC})^2]^{1/2}}=\frac{0.443}{\left\{(d_\text{i}/v_\text{sat})^2+[2.78R(\epsilon\pi{r^2}/d_\text{i})]^2\right\}^{1/2}}$

The values of $$f_\text{3dB}$$ in the range of $$0\lt{d}_\text{i}\lt3$$ μm are calculated using this relation for $$2r=10,20,40,$$ and $$80$$ μm. Then the bandwidth-efficiency product is calculated using

$\eta_\text{i}f_\text{3dB}=(1-\text{e}^{-\alpha{d_\text{i}}})f_\text{3dB}$

The values of both $$f_\text{3dB}$$ and $$\eta_\text{i}f_\text{3dB}$$ are plotted as a function of $$d_\text{i}$$ in Figure 14-22.

From the data shown in Figure 14-22, we see that for a given device diameter there is an optimum intrinsic layer thickness of $$d_\text{opt}$$ for a maximum value of $$f_\text{3dB}$$ and a different optimum intrinsic layer thickness of $$d_\text{opt}'$$ for a maximum value of $$\eta_\text{i}f_\text{3dB}$$. We also find that $$d_\text{opt}'\gt{d}_\text{opt}$$.

The cutoff frequency is primarily limited by $$\tau_\text{RC}$$ if $$d_\text{i}\lt{d}_\text{opt}$$, whereas it is primarily limited by $$\tau_\text{tr}$$ if $$d_\text{i}\gt{d}_\text{opt}$$.

For a given device diameter, there is one possible choice of $$d_\text{i}$$ on either side of $$d_\text{opt}$$ for a sufficiently large value of $$f_\text{3dB}$$. For a desired $$f_\text{3dB}$$, the choice of $$d_\text{i}\gt{d}_\text{opt}$$ has a larger bandwidth-efficiency product than that of $$d_\text{i}\lt{d}_\text{opt}$$.

Heterojunction Photodiodes

Heterojunction structuren offer additional flexibility in optimizing the performance of a photodiode.

In a heterojunction photodiode, the active region normally has a bandgap that is smaller than one or both of the homogeneous regions. A large-gap homogeneous region, which can be either the top $$\text{p}^+$$ region or the substrate $$\text{n}$$ region, serves as a window for the optical signal to enter.

The small bandgap of the active region determines the threshold wavelength, $$\lambda_\text{th}$$, of the detector on the long-wavelength side, while the large bandgap of the homogeneous window region sets a cutoff wavelength, $$\lambda_\text{c}$$, on the short-wavelength side.

For an optical signal that has a wavelength $$\lambda_\text{s}$$ in the range of $$\lambda_\text{th}\gt\lambda_\text{s}\gt\lambda_\text{c}$$, the quantum efficiency and the responsivity can be optimized.

A limiting factor for the speed of a heterojunction photodiode is the trapping of electrons at the conduction-band discontinuity and that of holes at the valence-band discontinuity. For high-speed applications, this limitation has to be removed by reducing the barrier height through compositional grading at the interface of the heterojunction.

Many III-V p-i-n photodiodes have heterojunction structures, which can be either symmetric with a small-bandgap active intrinsic region sandwiched between large-bandgap $$\text{p}^+$$ and $$\text{n}^+$$ regions, such as $$\text{p}^+$$-AlGaAs/GaAs/$$\text{n}^+$$-AlGaAs and $$\text{p}^+$$-InP/InGaAs/$$\text{n}^+$$-InP, or asymmetric with a large-bandgap $$\text{p}^+$$ or $$\text{n}^+$$ region on only one side, such as $$\text{p}^+$$-AlGaAs/GaAs/$$\text{n}^+$$-GaAs or $$\text{p}^+$$-InGaAs/InGaAs/$$\text{n}^+$$-InP.

Figure 14-23 shows some structures of heterojunction photodiodes.

Sophisticated heterojunction structures such as quantum wells and strained quantum wells, as well as quantum wires and quantum dots, are also used for the active region of photodiodes.

Such quantum structures have the advantage of high peak absorption coefficients, which lead to an improved quantum efficiency for a given thickness of the active region. They are often used for improving the bandwidth-efficiency products of high-speed photodetectors.

Schottky Photodiodes

The property of the interface between a metal and a semiconductor depends on the work functions of the metal and the semiconductor, $$e\phi_\text{m}$$ and $$e\phi_\text{s}$$, respectively, and the type of semiconductor.

The metal-semiconductor junction is an ohmic contact without a potential barrier if $$\phi_\text{s}\gt\phi_\text{m}$$ in the case of an n-type semiconductor or $$\phi_\text{s}\lt\phi_\text{m}$$ in the case of a p-type semiconductor.

A Schottky barrier of a height $$E_\text{b}=e(\phi_\text{m}-\chi)$$ for electrons to flow from the metal to the semiconductor exists at the metal-semiconductor junction if $$\phi_\text{s}\lt\phi_\text{m}$$ in the case of an n-type semiconductor, as shown in Figure 14-24(a).

A Schottky barrier of a height $$E_\text{b}=E_\text{g}-e(\phi_\text{m}-\chi)$$ for holes to flow from the metal to the semiconductor exists at the metal-semiconductor junction if $$\phi_\text{s}\gt\phi_\text{m}$$ in the case of a p-type semiconductor, as shown in Figure 14-24(b).

The general characteristics of a Schottky junction are similar to those of a p-n junction.

The characteristics of a Schottky junction formed between a metal and an n-type semiconductor can be approximated by those of a $$\text{p}^+-\text{n}$$ junction with a built-in potential of $$V_0=\phi_\text{m}-\phi_\text{s}$$, as shown in Figure 14-24(a).

Similarly, a Schottky junction between a metal and a p-type semiconductor can be considered as an $$\text{n}^+-\text{p}$$ junction with a built-in potential of $$V_0=\phi_\text{s}-\phi_\text{m}$$, as shown in Figure 14-24(b).

Therefore, the depletion-layer width $$W$$ of a Schottky junction and its dependence on bias voltage can be found by using (12-100) [refer to the semiconductor junctions tutorial] and by taking $$N_\text{a}\gg{N}_\text{d}$$ in the case of an n-type semiconductor or $$N_\text{d}\gg{N}_\text{a}$$ in the case of a p-type semiconductor.

The junction capacitance simply has the same form as that of a p-n junction given in (12-119) [refer to the semiconductor junctions tutorial].

It is also possible for a Schottky diode to function like a p-i-n diode by inserting a lightly doped $$\text{n}^--$$semiconductor layer between a metal and a heavily doped $$\text{n}^+-$$semiconductor region. In such a structure, the metal functions as a $$\text{p}^+-$$homogeneous region, and the $$\text{n}^-$$ layer functions as the intrinsic region in a p-i-n diode.

The depletion layer, which exists almost entirely the $$\text{n}^-$$ region, broadens as the reverse bias voltage increases until it reaches the metal at a voltage known as the punchthrough voltage. When the reverse bias voltage is larger than the punchthrough voltage, the depletion-layer width of such a Schottky diode becomes independent of the voltage and is simply defined by the thickness of the $$\text{n}^-$$ layer.

The characteristics and the equivalent circuit of a Schottky photodiode are similar to those of a semiconductor junction photodiode discussed above.

A Schottky photodiode can also operate in either photoconductive mode or photovoltaic mode, but it normally operates in photoconductive mode in most of its applications for the same reasons as discussed above for other junction photodiodes.

A Schottky photodiode operating in photoconductive mode can have a very high speed, particularly when an n-type semiconductor is used. Because the optical signal is absorbed in a thin layer at the junction interface, only the majority carriers, which are electrons in the case of an n-type semiconductor, have to drift across the active region. A well-designed Schottky photodiode can reach an intrinsic frequency bandwidth as high as 100 GHz.

The spectral response of a Schottky photodiode depends on whether an optical signal is absorbed by the semiconductor or by the metal.

If the optical signal is absorbed by the semiconductor, the spectral characteristics of a Schottky photodiode is the same as that of a semiconductor junction photodiode with a threshold photon energy defined by the bandgap of the absorbing semiconductor: $$h\nu\gt{E}_\text{th}=E_\text{g}$$.

This process takes place when the Schottky photodiode has a thin, semi-transparent metallic layer to allow the optical signal to enter with little attenuation before it reaches the depletion layer. This is the normal mode of operation for a high-efficiency, high-speed Schottky photodiode.

Absorption of a photon by the metal at the junction interface can also produce a photoresponse if the photon has sufficient energy to excite an electron over the Schottky barrier.

For a Schottky photodiode to operate in this mode, the metallic layer has to be thick and absorbing, but the absorption has to take place at the junction interface. The spectral response range in this mode of operation is then $$E_\text{b}\lt{h\nu}\lt{E}_\text{g}$$ for the optical signal to enter from the semiconductor side without being absorbed by the semiconductor. A Schottky photodiode in this mode is useful as an infrared detector, but its efficiency is low because a metal does not absorb light efficiently.

Example 14-13

An InGaAs/InP Schottky photodiode has a structure similar to that of the InGaAs/InP p-i-n photodiode considered in Example 14-12 above, but it has a metallic layer in place of the $$\text{p}^+$$ layer of the p-i-n photodiode.

The thickness of the $$\text{n}^-$$ layer is $$d_\text{i}=1$$ μm. The diameter of the device is $$2r=12$$ μm. It is back illuminated through the InP substrate. The device is biased above the punchthrough voltage, and the electrons have reached their saturation velocity.

(a) What is the spectral response range of this photodiode at 300 K?

(b) Find the 3-dB cutoff frequency of this photodiode if $$R=R_\text{L}+R_\text{s}=50$$ Ω and $$C_\text{p}=0$$.

(a)

The spectral response range of this back-illuminated photodiode is limited at the short-wavelength end by a cutoff wavelength $$\lambda_\text{c}$$ determined by the bandgap of the InP window layer because an optical signal has to pass through the InP substrate to reach the InGaAs active layer.

It is limited at the long-wavelength end by the threshold wavelength $$\lambda_\text{th}$$ determined by the bandgap of the InGaAs that is lattice matched to InP.

From the discussions following (12-9) [refer to the introduction to semiconductors tutorial], we find that the absorption edge of InP is at 919 nm and that of InGaAs is at 1.65 μm. Therefore, the spectral response range of this Schottky photodiode at 300 K is from $$\lambda_\text{c}=919$$ nm to $$\lambda_\text{th}=1.65$$ μm

(b)

In a Schottky photodiode, only the majority carriers, which in this case are electrons, have to drift across the active region. Thus, the transit time is simply that of the electrons. From Example 14-12, we have $$v_\text{e}^\text{sat}=6.5\times10^4\text{ m s}^{-1}$$. With $$d_\text{i}=1$$ μm, we find that

$\tau_\text{tr}=\frac{d_\text{i}}{v_\text{e}^\text{sat}}=\frac{1\times10^{-6}}{6.5\times10^4}=15.4\text{ ps}$

With $$\epsilon=14.1\epsilon_0$$ from Example 14-12, we find that the internal capacitance of the device for $$d_\text{i}=1$$ μm and $$2r=12$$ μm is

$C_\text{i}=\frac{\epsilon\pi{r^2}}{d_\text{i}}=\frac{14.1\times8.85\times10^{-12}\times\pi\times(12\times10^{-6}/2)^2}{1\times10^{-6}}\text{ F}=14.1\text{ fF}$

With $$R=R_\text{L}+R_\text{s}=50$$ Ω, the RC time constant

$\tau_\text{RC}=RC_\text{i}=50\times14.1\times10^{-15}\text{ s}=705\text{ fs}$

Therefore, the 3-dB cutoff frequency of this photodiode is

$f_\text{3dB}=\frac{0.443}{[(15.4\times10^{-12})^2+(2.78\times705\times10^{-15})^2]^{1/2}}\text{ Hz}=28.5\text{ GHz}$

Because $$\tau_\text{tr}\gg\tau_\text{RC}$$ for this device, $$f_\text{3dB}$$ is almost entirely determined by the electron transit time.

Photodiodes with Multipass Structures

A high-speed photodiode requires a thin depletion layer for a short transit time, but, according to (14-84), the quantum efficiency of the photodiode decreases as the depletion-layer width $$W$$ is reduced. Therefore, there is a trade-off between its frequency bandwidth and quantum efficiency.

To optimize both the bandwidth and the efficiency of a high-speed photodiode, a large bandwidth-efficiency product $$\eta_\text{e}f_\text{3dB}$$ is desired.

From (14-84), it can be seen that the external quantum efficiency of a photodiode can be increased without changing the depletion-layer width by (1) antireflection coating the incident surface to make $$R=0$$ and (2) using a heterostructure with a nonabsorbing large-bandgap homogeneous region for $$T_\text{h}=1$$.

Many different device structures have been developed to increase the bandwidth-efficiency product further beyond that obtained with these two simple steps. They can be divided into three basic categories:

1. Vertically illuminated photodetectors with multiple optical passes through the active region, where are discussed here;
2. Laterally illuminated photodetectors such as the lateral p-i-n photodetectors, which are discussed earlier;
3. Guided-wave photodetectors, which are discussed in a later tutorial.

Figure 14-25 shows three approaches to increasing the bandwidth-efficiency product of a photodiode by increasing its quantum efficiency without increasing the thickness of its active region.

The simple double-pass structure, shown in Figure 14-25(a), directs the optical signal to pass through the active region twice with a back reflector of reflectivity $$R_\text{b}$$, which can be simply the substrate electrode if the substrate is transparent. With this structure, the quantum efficiency can be improved by a factor close to $$1+R_\text{b}$$ if the absorbing active region has a thickness of $$W\lt\alpha^{-1}$$.

To increase the quantum efficiency further, the effective optical path length in the active region can be increased without increasing the physical thickness of the active region by using the refracting-facet structure shown in Figure 14-25(b).

In this structure, the top electrode reflects the optical signal for a second pass through the active region to keep the advantage of a double-pass structure, but the optical signal passes through the active region at an angle $$\theta$$ for a total effective path length of $$2W/\sin\theta$$. Therefore, the quantum efficiency is further increased over that of the simple double-pass structure shown in Figure 14-25(a). A bandwidth-efficiency product around 40 GHz has been obtained for refracting-facet photodiodes.

To push the quantum efficiency close to unity in a high-speed photodiode with a very thin active region, a resonant-cavity-enhanced structure shown in Figure 14-25(c) can be used.

This structure consists of both a frond and back reflector to form a resonant cavity. It functions in a manner similar to that of a VCSEL by forming a standing wave with its high-intensity crest located at the thin absorbing active region.

By using DBR reflectors of a reflectivity greater than 99% for a high-$$Q$$ cavity, a quantum efficiency greater than 90% can be achieved with this scheme. A bandwidth-efficiency product around 20 GHz has been obtained for resonant-cavity-enhanced p-i-n and Schottky photodiodes.

The resonant-cavity-enhanced structure is highly wavelength selective because of its resonance nature. This wavelength selectivity is a disadvantage for general applications because of its narrow optical bandwidth, but it is a useful feature for applications in wavelength-selective detection systems such as wavelength-division multiplexing systems.

The next tutorial covers the topic of avalanche photodiodes