Theoretical Background

2.1 Schottky barrier photodiodes

Schottky barrier photodiodes have been studied quite extensively and have also found application as ultraviolet detectors [1]. These devices reveal some advantages over p-n junction photodiodes: fabrication simplicity, absence of high-temperature diffusion processes, and high speed of response.

Current transport processes

The current transport in metal-semiconductor contacts is due mainly to majority carriers, in contrast to p-n junctions, where current transport is due mainly to minority carriers. The current can be transported in various ways under forward bias conditions as shown in Fig. 2.1. The four processes are: [2]

(a) Emission of electrons from the semiconductor over the top of the barrier into the metal,

(b) Quantum mechanical tunneling through the barrier, (c) Recombination in the space-charge region,

(d) Recombination in the neutral region (equivalent hole injection from the metal to the semiconductor).

Fig. 2.1 Four basic transport processes in forward-biased Schottky barrier on an n-type semiconductor.

The transports of electrons over the potential barrier have been described by various theories, namely: diffusion, thermionic emission, and unified thermionic emission diffusion. It is now widely accepted that, for high-mobility semiconductors with impurity concentrations of practical interest, the thermionic emission theory appears to explain qualitatively the experimentally observed I –V characteristics.

Some workers have also included in the simple thermionic theory the quantum effects (i.e., quantum mechanical reflection and tunneling of carriers through the barrier) and have tried to obtain modified analytical expressions for the current–voltage relation.

This, however, has essentially led to a lowering of the barrier height and a rounding off of the top.

The thermionic emission theory by Bethe is derived from the assumptions that the barrier height is much larger than kT, thermal equilibrium is established at the plane that determines emission, and the existence of a net current flow does not affect this equilibrium. Bethe’s criterion for the slope of the barrier is that the barrier must decrease by more than kT over a distance equal to the scattering length. The resulting current flow will depend only on the barrier height and not on the width, and the saturation current is not dependent on the applied bias. Then the current density of majority carriers from the semiconductor over the potential barrier into the metal is expressed as

where saturation current density

) and is the Richardson constant, m* is the effective electron mass, and n is the ideality factor. Equation is similar to the transport equation for p-n

3

* 2

* 4 qk m /h

A =

π

junctions. However, the expression for the saturation current densities is quite different.

2.2 Quantum Efficiency

The quantum Efficiency η (0≤η≤1) is of a photodetector is defined as the probability that a single photon incident on the device generates a photocarrier pair that contributes to the detector current. When many photons are incident, as is almost always the case, η is the ratio of the flux of generated electron-hole pairs that contribute to the detector current to the flux of incident photons [2,4].

The quantum Efficiency is defined as

)]

exp(

1 [ ) 1

( R

ζ α

η

where R is the optical power reflectance at the surface, ζ is the fraction of electron-hole pairs that contribute successfully to the detector current, α the absorption coefficient of the material (cm^-1), and d the photodetector depth.

The first factor (1-R) represents the effect of reflection at the surface of the device. Reflection can be reduced by the use of antireflection coatings. The second factor ζ is the fraction of electron-hole pairs that successfully avoid recombination at the material surface and contribute to the useful photocurrent. The third factor represents the fraction of the photon flux absorbed in the bulk of the material. The device should have a sufficiently large value of d to maximize this factor.

2.3 Responsivity

The responsivity related the electric current flowing in the device to the incident optical power. If every photon were to generate a single photoelectron, a photon flux Φ (photons per second) would produce an electron flux Φ, corresponding to a short circuit photocurrent Iphoto= e Φ. An optical power Popt= h ν Φ (watts) at frequency ν would then give rise to an photocurrent Iphoto= e Popt / h ν .Since the fraction of photons producing detected photoelectrons is η rather than unity, the photocurrent is

opt

The proportionality factor R, between the photocurrent and the optical power, is defined as the responsivity R of the device. R has units of A/W and is given by

m

R increases with λo because photoelectric detectors are responsive to the photon flux rather than to the optical power. As λo increases, a given optical power is carried by more photons, which, in turn, produce more electrons. The region over which R increases with λo is limited, however, since the wavelength dependence of η comes into play for both long and short wavelengths.

2.4 Gain

The formulas presented above are predicated on the assumption that each carrier produces a charge e in the detector circuit. However, many devices produce a charge q in the circuit that differs from e [4]. Such devices are said to exhibit gain. The gain G is the average number of circuit electros generated per photocurrent pair. G should be distinguished from η, which is the probability that an incident photon produces a detectable photocurrent pair. The gain, which is defined as

e

G= q _(2.6)

can be either greater than or less than unity. Therefore, more general expressions for the photocurrent and responsivity are

ν

The responsivity of a photoconductor is given by (2.8). The device exhibits an internal gain which, simply viewed, comes about because the recombination lifetime and transit time generally differ. Suppose that electrons travel faster than holes and that the recombination lifetime is very long. As the electron and hole are transported to opposite sides of the photoconductor, the electron completes its trip sooner than the hole. The requirement of current continuity forces the external circuit to provide another electron immediately, which enters the device from the wire at the left. This new electron moves quickly toward the right, again completing its trip before the hole reaches the left edge. This process continues until the electron recombines with the hole. A single photon absorption can therefore result in an electron passing

through the external circuit many times. The expected number of trips that the electron makes before the process terminates is [3]

e

G

τ

=

τ

Where τ is the excess-carrier recombination lifetime and τe is the electron transit time across the sample. The charge delivers to the circuit by a single electron-hole pair in this case is q = G e＞e so that the device exhibits gain.

2.5 Response time and Bandwidth

The frequency response of a photodiode may be determined by basically three effects: [2]

(1) the time of carrier diffusion to the junction depletion region, τd; (2) the transit time of carrier drift across the depletion region, τe;

(3) the RC time constant, τRC associated with circuit parameters including the junction capacitance C and the parallel combination of diode resistance and external load (the series resistance is neglected).

The response time of photoconductor detectors is, of course, constrained by the transit time and RC time-constant. The carrier-transport response time is approximately equal to the recombination time τ, so the carrier-transport bandwidth B is inversely proportional to τ. Since the gain G is proportional to τ in accordance with (2.9), increasing τ increases the gain, which is desirable, but it also decreases the bandwidth, which is undesirable. Thus the gain-bandwidth product GB is roughly independent of τ.

As a final point, we mention that photodetectors of a givens material and structure often exhibit a fixed gain-bandwidth product. Increasing the gain results in a decrease of the bandwidth, and vice versa. This trade-off between sensitivity and

frequency response is associated with the time required for the gain process to take place.

2.6 Detectivity

The specific detectivity D* for a photodetector, a figure of merit used to characterize performance, is equal to

kT A R h

e _o

ν

η

where e is the electronic charge, ηis the quantum efficiency, h is Planck’s constant, νis the frequency of the radiation, Ro is the dynamic resistance at zero bias, A is the detector area, k is Boltzman’s constant, and T is the absolute temperature [5].

2.7 References

1. David Wood, “Optoelectronic Semiconductor Devices”, (Prentice Hall, New York, 1994).

2. M. Razeghi and A. Rogalski, J. Appl. Phys. 79, 7433 (1996).

3. S. M. Sze, Physics of Semiconductor Devices, 2nd ed. (Wiley, New York, 1981), pp.743-770.

4. Bahaa E. A. Saleh and Malvin Carl Teich, ”Fundamentals of Photonics”, (Wiley-Interscience Publication, New Yok, 1991).

5. C. K. Wang, T. K. Ko, C. S. Chang, S. J. Chang, Y. K. Su, T. C. Wen, C. H. Kuo, and Y. Z. Chiou, IEEE Photon. Technol. Lett. 17, 2161 (2005).