## 3 Near-Infrared Silicon Absorption Physics
### Photoconductivity and Linear Absorption
#### 3.1.3 Internal Photoemission Absorption (IPA)  
Internal photoemission is the optical excitation of electrons in the metal to energy above the Schottky barrier and then transport of these electrons to the conduction band of the semiconductor (Figure 3). Figure 3: Energy band diagram for a metal/n-semiconductor junction. Reprinted with permission from [PERSON] _et al._[29] IOP Publishing is acknowledged. Figure 2: Absorption constant _vs._ photon energy for a silicon surface [27]. Copyright (1971) by the American Physical Society [28]. The standard theory of photoemission from a metal into a vacuum is due to [PERSON] (1957). It assumes that the density of states in the metal is described by the normal parabolic distribution associated with nearly-free electron approximation. Thus in [PERSON]'s theory, the Fermi level is several eV above the conduction band minimum when a single band is considered. In a gas of electrons obeying the Fermi-Dirac statistic, the fraction of the absorbed photons Fe which produces photoelectrons with the appropriate energy and momentum before scattering to contribute to the photocurrent is calculated as:  
\[F_{e}=\frac{\left[\left(\hbar\
u-\left(\phi_{n0}-\Delta\phi_{n}\right)\right) ^{2}+\frac{\left(k_{g}T\pi\right)^{2}}{3}-2(k_{g}T)^{2}e^{-\frac{\hbar-\left(k _{g}-\Delta\phi_{n}\right)}{k_{g}T}}\right]}{8k_{g}TE_{F}\log\left[1+e^{\frac {\hbar\
u-\left(\phi_{n0}-\Delta\phi_{n}\right)}{k_{g}T}}\right]} \tag{1}\]  
where hv is photon energy, \(\Phi_{\rm B0}\) is the potential barrier at zero bias, \(\Delta\Phi_{\rm B}\) is the lowering due to image force effect (as we will see later), k is Boltzman's constant, T is the absolute temperature and E\({}_{\rm F}\) is the metal Fermi level. [PERSON]'s theory was originally obtained without taking into account the thickness of the Schottky metal layer. In order to study the quantum efficiency 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, a phenomenological semi-classical ballistic transport model for the effects of the scattering mechanisms resulting in a multiplicative factor for quantum efficiency has been developed by [PERSON] (1957). The efficiency in collecting those electrons which have sufficient normal kinetic energy to overcome \(\Phi_{\rm B}\) will depend on their probability of collisions with cold or hot electrons and with the two boundary surfaces of the metal. Supposing that the Schottky barrier is illuminated through the front surface of the metal electrode, and that optical absorption is uniform throughout the metal thickness, we indicate with p\({}_{\rm t}\) the average probability that photoexcited electrons arrive at the semiconductor surface without collision. Then, we indicate with p\({}_{\rm r}\) the accumulated probability of arrival at the reflection boundary of those electrons which have sufficient energy to overcome \(\Phi_{\rm B}\) after one collision with cold or hot electrons, which are are oriented with equal probability over the entire half-sphere before reflection by the metal surface. Finally, for multiple reflections, we indicate with p the probability that the capturable electrons can bounce from one metal boundary to another. Therefore, the total accumulated probability that the electrons will have sufficient normal kinetic energy to overcome \(\Phi_{\rm B}\) is given by [PERSON] (1957):  
\[P_{E}=p_{{}_{t}}+p_{{}_{r}}(1+p+p^{{}^{2}}+p^{3}+...)\exp(-d\,/\,L\,\,)\cong \frac{L}{d}\Bigg{[}1-e^{-\frac{d}{L}}\Bigg{]}^{\frac{1}{2}} \tag{2}\]  
where d is the metal thickness and L the mean free path describing the probability of collision with cold or hot electrons [PERSON] (1957). It would be possible to show by plotting Equation 2 that by decreasing metal thickness d, P\({}_{\rm E}\) increases, and that a considerable gain can be obtained when d is much smaller than L (thin film).