### Dark Current

**Semiconductor diodes can pass current even in the dark,** giving rise to dark current that provides a background present in any measurement. This current comes primarily from the thermally generated diffusion of minority carriers out of the n and p regions into the depleted junction region, where they recombine. The current-voltage equation for a pn diode (in the dark) is:

where IS is the saturation current that flows at large back bias (V large and negative). This equation represents the current that passes through any biased pn junction. Photodiodes use pn junctions reverse biased (V < 0) to avoid large leakage current.

**Here P is the ideality factor,** which varies from 1 to 2, depending on the diode structure. In a metal-semiconductor junction (Schottky barrier) or an ideal pn junction in which the only current in the dark is due to minority carriers that diffuse from the p and n regions, then P = 1. However, if there is thermal generation and recombination of carriers in the space-charge region, then P tends toward the value 2. This is more likely to occur in long-wavelength detectors.

**The saturation current IS is proportional to the area A of the diode in an ideal junction:**

where Dn, Dp are diffusion constants, Ln, Lp are diffusion lengths, and np0, pn0 are equilibrium minority carrier densities, all of electrons and holes, respectively. The saturation current IS can be related to the diode resistance measured in the dark when V = 0. Defining

then:

The dark resistance is inversely proportional to the saturation current, and therefore to the area of the diode.

**The diffusion current in Eq. (101)** has two components that are of opposite sign in a forward-biased diode: a forward current IS exp (eV/pkT) and a backward current -IS. Each of these components is statistically independent, coming from diffusive contributions to the forward current and backward current, respectively. This fact is important in understanding the noise properties of photodiodes.

**In photodiodes, V < 0.** For clarity, write V = -V’ and use V as a positive quantity in the equations that follow. For a reverse-biased diode in the dark, diffusion current flows as a negative dark current, with a magnitude given by

The negative dark current flows opposite to the current flow in a forward-biased diode. Holes move toward the p region and electrons move toward the n region; both currents are negative and add. This dark current adds to the negative photocurrent. The hole current must be thermally generated because there are no free holes in the n region to feed into the p region. By the same token, the electron current must be thermally generated since there are no free electrons in the p region to move toward the n region. The dark current at large reverse bias voltage is due to thermally generated currents.

**Using Eq. (104)** and assumingthe negative dark current equals the saturation current:

It can be seen that the dark current increases linearly with temperature and is independent of (large enough) reverse bias. Trap-assisted thermal generation current increases P; in this process, carriers trapped in impurity levels can be thermally elevated to the conduction band. The temperature of photodiodes should be kept moderate in order to avoid excess dark current.

**When light is present in a photodiode,** the photocurrent is negative, in the direction of the applied voltage, and adds to the negative dark current. The net effect of carrier motion will be to tend to screen the internal field. Defining the magnitude of the photocurrent as then the total current is negative:

### Noise in Photodiodes

**Successful fiber-optic communication systems** depend on a large signal-to-noise ratio. This requires photodiodes with high sensitivity and low noise. Background noise comes from shot noise due to the discrete process of photon detection, from thermal processes in the load resistor (Johnson noise), and from generation-recombination noise due to carriers within the semiconductor. When used with a field-effect transistor (FET) amplifier, there will also be shot noise from the amplifier and 1/f noise in the drain current.

**Shot Noise**. Shot noise is fundamental to all photodiodes and is due to the discrete nature of the conversion of photons to free carriers. The shot noise current is a statistical process. If N photonsare detected in a time interval At, Poisson noise statistics cause the uncertainty in N to beUsing the fact that N electron-hole pairs create a current I throughthen the signal-to-noise ratio (SNR) isWriting the frequency bandwidth in terms of the time interval throughgives:

**The root mean square (rms) photon noise, given by****creates an rms shot noise current of:**

Shot noise depends on the average current I; therefore, for a given photodiode, it depends on the details of the current voltage characteristic. Expressed in terms of PS, the optical signal power (when the dark current is small enough to be neglected), the rms shot noise current is

where X is the responsivity (or sensitivity), given in units of amps per watt.

**The shot noise can be expressed directly** in terms of the properties of the diode when all sources of noise are included. Since they are statistically independent, the contributions to the noise current will be additive. Noise currents can exist in both the forward and backward directions, and these contributions must add, along with the photocurrent contribution. The entire noise current squared becomes:

Clearly, noise is reduced by increasing the reverse bias. When the voltage is large, the shot noise current squared becomes:

The dark current adds linearly to the photocurrent in calculating the shot noise.

**In addition to shot noise due to the random variations in the detection process**, the random thermal motion of charge carriers contributes to a thermal noise current, often called Johnson or Nyquist noise. It can be calculated by assuming thermal equilibrium withso that Eq. (109) becomes:

This is just Johnson noise in the resistance of the diode. The noise appears as a fluctuating voltage, independent of bias level.

**Johnson Noise from External Circuit. An additional noise component will be from the load resistor RL and resistance from the input to the preamplifier, Ri:**

Note that the resistances add in parallel as they contribute to noise current.

**Noise Equivalent Power.** The ability to detect a signal requires having a photocurrent equal to or higher than the noise current. The amount of noise that detectors produce is often characterized by the noise equivalent power (NEP), which is the amount of optical power required to produce a photocurrent just equal to the noise current. Define the noise equivalent pho-tocurrent INE, which is set equal to the noise current iSH. When the dark current is negligible,

Thus, the noise equivalent current isand depends only on the bandwidth Af. The noise equivalent power can now be expressed in terms of the noise equivalent current:

The second equality assumes the absence of dark current. In this case, the NEP can be decreased only by increasing the quantum efficiency (for a fixed bandwidth). In terms of sensitivity (amps per watt):

This expression is usually valid for photodetectors used in optical communication systems, which have small dark currents.

When dark current is dominantso that:

**This is often the case in infrared detectors such as germanium**. Note that the dark-current-limited noise equivalent power is proportional to the square root of the area of the detector because the dark current is proportional to the detector area. The NEP is also proportional to the square root of the bandwidth Af. Thus, in photodetectors whose noise is dominated by dark current, NEP divided by the square root of area times bandwidth should be a constant. The inverse of this quantity has been called the detectivity D* and is often used to describe infrared detectors. In photodiodes used for communications, dark current usually does not dominate and it is better to use Eq. (114), an expression which is independent of area, but depends linearly on bandwidth.

## AVALANCHE PHOTODIODES, MSM DETECTORS, AND SCHOTTKY DIODES

### Avalanche Detectors

**When large voltages are applied to photodiodes**, the avalanche process produces gain, but at the cost of excess noise and slower speed. In fiber telecommunications applications, where speed and signal-to-noise are of the essence, avalanche photodiodes (APDs) are frequently at a disadvantage. Nonetheless, in long-haul systems at 2488 Mb/s, APDs may provide up to 10 dB greater sensitivity in receivers limited by amplifier noise. While APDs are inherently complex and costly to manufacture, they are less expensive than optical amplifiers and may be used when signals are weak.

**Gain (Multiplication).** When a diode is subject to a high reverse-bias field, the process of impact ionization makes it possible for a single electron to gain sufficient kinetic energy to knock another electron from the valence to the conduction band, creating another electron-hole pair. This enables the quantum efficiency to be >1. This internal multiplication of pho-tocurrent could be compared to the gain in photomultiplier tubes. The gain (or multiplication) M of an APD is the ratio of the photocurrent divided by that which would give unity quantum efficiency. Multiplication comes with a penalty of an excess noise factor, which multiplies shot noise. This excess noise is function of both the gain and the ratio of impact ionization rates between electrons and holes.

**Phenomenologically, the low-frequency multiplication factor is:**

where the parameter n varies between 3 and 6, depending on the semiconductor, and VB is the breakdown voltage. Gains of M > 100 can be achieved in silicon APDs, while they are more typically 10 to 20 for longer-wavelength detectors, before multiplied noise begins to exceed multiplied signal. A typical voltage will be 75 V in InGaAs APDs, while in silicon it can be 400 V.

**The avalanche process involves using an electric field** high enough to cause carriers to gain enough energy to accelerate them into ionizing collisions with the lattice, producing electron-hole pairs. Then, both the original carriers and the newly generated carriers can be accelerated to produce further ionizing collisions. The result is an avalanche process.

**In an i layer (where the electric field is uniform) of width W,** the gain relates to the fundamental avalanche process through M = 1/(1 – aW;), where a is the impact ionization coefficient, which is the number of ionizing collisions per unit length. When aWt ^ 1, the gain becomes infinity and the diode breaks down. This means that avalanche multiplication appears in the regime before the probability of an ionizing collision is 100 percent. The gain is a strong function of voltage, and these diodes must be used very carefully. The total current will be the sum of avalanching electron current and avalanching hole current.

**In most pin diodes the i region is really low n-doped.** This means that the field is not exactly constant, and an integration of the avalanche process across the layer must be performed to determine a. The result depends on the relative ionization coefficients; in III-V materials they are approximately equal. In this case, aWt is just the integral of the ionizing coefficient that varies rapidly with electric field.

**Separate Absorber and Multiplication (SAM) APDs.** In this design the long-wavelength infrared light is absorbed in an intrinsic narrow-bandgap InGaAs layer and photocarriers move to a separate, more highly n-doped InP layer that supports a much higher field. This layer is designed to provide avalanche gain in a separate region without excessive dark currents from tunneling processes. This layer typically contains the pn junction, which traditionally has been diffused. Fabrication procedures such as etching a mesa, burying it, and introducing a guard ring electrode are all required to reduce noise and dark current. All-epitaxial structures provide low-cost batch-processed devices with high performance characteristics.68

**Speed.** When the gain is low, the speed is limited by the RC time constant. As the gain increases, the avalanche buildup time limits the speed, and for modulated signals the multiplication factor decreases. The multiplication factor as a function of modulation frequency is:

wherewhere t is the multiplication-region transit time and p is a number that changes from 2 to M as the gain changes from 1 to 1000. The gain decreases from its low-frequency value whenIt can be seen that it is the gain-bandwidth product that describes the characteristics of an avalanche photodiode in a communication system.

**Noise**. The shot noise in an APD is that of a pin diode multiplied by M2 times an excess noise factor Fe:

where

**In this expression,** P is the ratio of the ionization coefficient of the opposite type divided by the ionization coefficient of the carrier type that initiates multiplication. In the limit of equal ion-ization coefficients of electrons and holes (usually the case in III-V semiconductors), Fe = M and Fh = 1. Typical numerical values for enhanced APD sensitivity are given in Vol. I, Chap. 17, Fig. 15.

**Dark Current.** In an APD, dark current is the sum of the unmultiplied current Idu, mainly due to surface leakage, and the bulk dark current experiencing multiplication Idm, multiplied by the gain:

The shot noise from dark (leakage) current id:

The proper use of APDs requires choosing the proper design, carefully controlling the voltage, and using the APD in a suitably designed system, since the noise is so large.

### MSM Detectors

*Either a pn junction or bulk semiconductor material* can reside under the interdigitated fingers. The MSM geometry has the advantage of lower capacitance for a given cross-sectional area, but the transit times may be longer, limited by the lithographic ability to produce very fine lines. Typically, MSM detectors are photoconductive. Volume I, Chap. 17, Fig. 17 shows the geometry of highspeed interdigitated photoconductors. These are simple to fabricate and can be integrated in a straightforward way onto MESFET preamplifiers.

**Consider parallel electrodes deposited on** the surface of a photoconductive semiconductor with a distance L between them. Under illumination, the photocarriers will travel laterally to the electrodes. The photocurrent in the presence of Ps input optical flux at photon energy hv is:

**The photoconductive gain G is the ratio of the carrier lifetime t to the carrier transit time Ttr:**

Decreasing the carrier lifetime increases the speed but decreases the sensitivity.

**The output signal is due to the time-varying resistance that results from the time-varying photoinduced carrier density N(t):**

where is the sum of the electron and hole mobilities, w is the length along the electrodes excited by light, and de is the effective absorption depth into the semiconductor.

Usually, MSM detectors are not the design of choice for high-quality communication systems. Nonetheless, their ease of fabrication and integration with other components makes them desirable for some low-cost applicationsâ€”for example, when there are a number of parallel channels and dense integration is required.

### Schottky Photodiodes

**A Schottky photodiode uses a metal-semiconductor** junction rather than a pin junction. An abrupt contact between metal and semiconductor can produce a space-charge region. Absorption of light in this region causes photocurrent that can be detected in an external circuit. Because metal-semiconductor diodes are majority carrier devices they may be faster than pin diodes (they rely on drift currents only, there is no minority carrier diffusion). Up to 100 GHz modulation has been reported in a 5- x 5-|im area detector with a 0.3-^m thin drift region using a semitransparent platinum film 10 nm thick to provide the abrupt Schottky contact. Resonance enhancement of the light has been used to improve sensitivity.