Model of Schottky Barrier Hot-Electron-Mode Photodetection

V. E. Vickers

Air Force Cambridge Research Laboratories, Air Force Systems Command, Bedford, Massachusetts 01730. Received 2 April 1971.

The use of Schottky barrier diodes as photodetectors operating in the hot electron mode has received recent interest.1 This mode offers extended wavelength photoresponse as well as very broad bandwidth. For this purpose, there is a need for a more precise quantitative theory to predict the external quantum yield η in collected electrons per incident photon. Archer et al.2

extended the usual Fowler relationship for photoemission3 to provide a quantitative theory for Η, taking into account optical absorptance and hot carrier scattering, assuming a thick metal film. In order to explain the experimental observation2,4 of greatly enhanced quantum yields for thin metal films, the theory must be further extended, taking into account multiple reflections of the excited electrons from the surfaces of the metal film, in addition to collisions with phonons, imperfections, and cold electrons. Assuming a thin metal film, we develop a phenomeno-logical, semiclassical, ballistic transport model for the effects of the scattering mechanisms, resulting in a multiplicative yield factor A3, a function of metal film thickness normalized to an effective hot-electron mean free path.

We begin by assuming that the time-average number of elec­trons excited in the metal layer of the device at a distance z from the metal-semiconductor junction is proportional to R e {L · S } , where S is the complex Poynting vector, in general a function of z, obtainable from the standard optical theory.5-7 For thin metal layers, L-S is approximately constant, so that when we normalize to the number of incident photons, the number of excited elec­trons per unit thickness is simply (P/d), where P is the optical absorptance of the metal layer and d is its thickness. (P is a complex but well-known6,7 and readily programmable function of the thicknesses and optical parameters of all the layers of the device.) We assume plane-parallel geometry on the macroscopic level throughout the model.

Next, we asume that the excited electrons are distributed in energy E from a few kT below the Fermi level EF to a few kT above (EF + hv), where hv is the incident photon energy. If we were to assume this distribution approximately uniform, we could write the normalized distribution N(E), in excited elec­trons per incident photon per unit thickness per unit energy range, as

where ƒ(E) is the Fermi distribution function and

We assume that an excited electron located in the vicinity of the junction will be captured if and only if its momentum orientation is within a solid angle ΔΩ determined by E and the large-scale properties of the potential barrier at the metal-semiconductor interface, and we assume isotropic distribution of the momentum orientations. We also assume that ΔΩ·As is much less than 4π, throughout the energy distribution. (Note tha t these last assumptions imply that collected electrons must have travelled from their last collision along a path approximately normal to the over-all plane of the interface.)

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At this point we pause in developing the model to note some relations to previous work. We define the junction capture efficiency ηj as

We need a functional relationship between ΔΩ and E; assuming an ideal metal, we can use8,9

and set ΔΩ zero for excitation (E − Ep) below the barrier height ψms. If we expand Eq. (4) in powers of (E − EF)/EF, retain only the first term, and substitute into Eq. (3), using only the first two terms of Eq. (1), we obtain an expression incorporating Fowler's temperature dependence3 into the zero-Kelvin yield expression of Archer et al.2

We also note that, if desired, Eq. (4) could be multiplied by a suitably averaged transmission coefficient without disturbing the following argument.

Resuming, we assume that excited electrons located in the metal layer, but not at the junction, will travel ballistically, with constant energy and constant momentum, until they undergo a collision, which can be any of three types: with another elec­tron, with a phonon (we will here include the effects of imper­fections), or with one of the two boundary surfaces of the metal layer. We assume that the probability of collision with a cold electron is described by a mean free path Le, and tha t the energy loss in such an event will be so great tha t the excited electron can no longer get over the barrier. We assume that the prob­ability of collision with a phonon is described by a mean free path Lp; that the mean energy loss ΔEP is small; and that the mean resulting change in orientation of the momentum vector is large compared with (ΔΩ/π)y.10 And we assume that on col­lision with one of the boundary surfaces (provided tha t the col­lision is not within ΔΩ at the junction boundary) the energy loss ΔEb is zero, while the direction of momentum is independent of its previous orientation.11

Now, of the initial distribution N(E) of excited electrons located at a distance z from the junction, ΔΩ/4π are oriented so as to be captured; of these, exp(− [1/LP + 1/Le]z) will arrive without being deflected en route. We define a reduced path length

and integrate over z to obtain the accumulated probability α that such electrons will arrive at the junction:

Next, the fraction of the originally excited electrons which are oriented within ΔΩ after one reflection were oriented with equal probability over the entire half-sphere prior to reflection, so that their accumulated probability β of arrival at the reflection boundary is given by

where we recognize the exponential integrals E2 and E. For multiple reflections, we can similarly calculate the prob­

ability δ that ultimately capturable electrons can get from one boundary surface to another for additional reflection: we find

Then the total contribution dη0 to the quantum yield from the energy E prior to any scattering events in the bulk is given by

We define the function

so tha t

we discover that

within 10% for d/L* ≥ 0.20, within 5 % for d/L* ≥ 0.35. The exact function and its approximation are plotted in Fig. 1.

Next we take into account contributions to the yield subsequent to scattering events in the bulk. Since the fraction removed by capture is small, our assumptions have the consequence that the scattering events are uniformly distributed over the thickness of the layer and postscattering momentum is isotropic. Hence, we see that the distribution of excited electrons immediately subsequent to their first collision in the bulk is quite like the initial distribution, save the two distinctions tha t the subsequent distribution is shifted down in energy by ΔEP and that it is only γ times as large as the initial distribution, where γ is the prob­ability that an excited electron will collide with a phonon before it collides with a cold electron. The fraction γ can be calculated as follows: the fraction undergoing a phonon col­lision over a path length dr, having travelled a total pa th length r, would be exp(− r/Lp)dr/Lp. Of these, only exp(− r/Le) would not have collided on the way with a cold electron and been removed from the process. Hence, we find

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There will be a succession of such distributions, each shifted down in energy by ΔEP from its predecessor and of amplitude γ times its predecessor.

At this point, we can complete the solution by integrating over energy and summing over the distributions. (Note that for distributions after the j th distribution, where

there will only be contributions from the tail of the Fermi func­tion and we can shortly truncate the computation.) If we have Le as a function of E, a complicated expression results. On the other hand, if we assume that Le and N(E) are constant (except for a Fermi tail on the latter) and ΔEP is negligible, we can ob­tain a simple answer: the total yield contribution dη at the energy E from all scattering processes is then given by

Using Eqs. (14), (12), and (3), we obtain for the external quantum yield

In sum, we have obtained results equivalent to those of a Monte Carlo calculation for the important special case of a thin metal film and can write the external quantum yield η for the hot-electron-mode Schottky barrier photodetector as the product of an optical term P/d, a conventional photoelectric term ηj, and a scattering term

representing the effects of scattering from the surfaces, phonons, and cold electrons. The optical absorptance is a function of the layer thicknesses and other optical parameters of the device (hence an often weak function of incident wavelength), while from Eq. (5) we see that ηj is a strong function of incident wave­length and barrier height (and less importantly of temperature).

I would like to acknowledge helpful discussions with A. C. Yang, F. D. Shepherd, Jr., and R. Schwartz. I am indebted to R. L. Phipps for graphing assistance.

References 1. F. D. Shepherd, Jr., A. C. Yang, and R. W. Taylor, Proc.

IEEE 58, 1160 (1970). 2. J. Cohen, J. Vilms, and R. J. Archer, "Investigation of Semi­

conductor Schottky Barriers for Optical Detection and Ca-thodic Emission," Final Report, AFCRL-68-0651, Dec. 1968 (Contract F19628-68-C-0090), Hewlett-Packard Labs., Palo Alto, Calif.; J. Cohen, J. Vilms, and R. J. Archer, "Investi­gation of Schottky Barriers for Optical Detection and Ca-thodic Emission," Final Report, AFCRL-69-0287, June 1969 (Contract F19628-69-C-0148), Hewlett-Packard Labs., Palo Alto, Calif.; R. J. Archer and J. Cohen, paper presented at Device Research Conference, Rochester, New York (1969).

3. R. H. Fowler, Phys. Rev. 38, 45 (1931). 4. A. C. Yang (AF Cambridge Research Labs.), private com­

munication. 5. T. C. Fry, J. Opt. Soc. Amer. 15, 137 (1927); 16, 1 (1928). 6. 0. S. Heavens, Optical Properties of Thin Solid Films (Butter-

worths, London, 1955), Chap. 4. 7. H. Wolter, in Encyclopedia of Physics, (Springer-Verlag,

Gottingen, 1956), Vol. 24, p. 461. 8. L. Apker, E. Taft, and J. Dickey, Phys. Rev. 74, 1462 (1948).

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9. S. M. Sze, "Hot Electrons in Thin Gold Films," Rept. SEL-63-032 (TR No. 1659-4), Stanford Electronics Labs., Apr. 1963.

10. J. M. Ziman, Electrons and Phonons (Clarendon Press, Ox­ford, 1960), pp. 187, 200.

11. Ref. 10, p. 456.