Patterson, Bailey - Solid State Physics Introduction to theory

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What are surface states? A solid surface is a solid terminated at a twodimensional surface. The effect on charge carriers is modeled by using a surface potential barrier. This can cause surface states with energy levels in the forbidden gap. The name “surface states” is used because the corresponding wave function is localized near the surface. Further comments about surface states are found in Chap. 11.

Surface states can have interesting effects, which we will illustrate with an example. Let us consider a p-type semiconductor (bulk) with surface states that are donors. The situation before and after equilibrium is shown in Fig. 6.20. For the equilibrium case (b), we assume that all donor states have given up their electrons, and hence, are positively charged. Thus, the Fermi energy is less than the donorlevel energy. A particularly interesting case occurs when the Fermi level is pinned at the surface donor level. This occurs when there are so many donor states on the surface that not all of them can be ionized. In that case (b), the Fermi level would be drawn on the same level as the donor level.

One can calculate the amount of band bending by a straightforward calculation. The band bending is caused by the electrons flowing from the donor states at the surface to the acceptor states in the bulk. For the depletion region, we assume,

If nd is the number of donors per unit area, the surface charge density is σ = end. The boundary condition at the surface is then

Integrating a second time, we find

This band bending is caused entirely by the assumed ionized donor surface states. We have already mentioned that surface states can complicate the analysis of metal-semiconductor junctions.

Fig. 6.21. p-type semiconductor under bias voltage (energies in each figure are relative)

6.3.7 Surfaces Under Bias Voltage (EE)

Let us consider a p-type surface under three kinds of voltage shown in Fig. 6.21:

(a) a negative bias voltage, (b) a positive bias voltage, and then (c) a very strong, positive bias voltage.

In case (a), the bands bend upward, holes are attracted to the surface, and thus, an accumulation layer of holes is founded. In (b), holes are repelled from the surface forming the depletion layer. In (c) the bands are bent sufficiently such that the conduction band bottom is below the Fermi energy and the semiconductor becomes n-type, forming an inversion region. In all these cases, we are essentially considering a capacitor with the semiconductor forming one plate. These ideas have been further developed into the MOSFET (metal-oxide semiconductor fieldeffect transistor, see Sect. 6.3.10).

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6.3.8 Inhomogeneous Semiconductors Not in Equilibrium (EE)

Here we will discuss pn-junctions under bias and how this leads to electron and hole injection. We will start with a qualitative treatment and then do a more quantitative analysis. The study of pn-junctions is fundamental for the study of transistors.

Fig. 6.22. The pn-junction under bias V: (a) Forward bias, (b) Reverse bias. (Only relative shift is shown)

We start by looking at a pn-junction in equilibrium where there are two types of electron flow that balance in equilibrium (as well as two types of hole flow which also balance in equilibrium). See also, e.g., Kittel [6.17, p. 572] or Ashcroft and Mermin [6.2, p. 600].

From the n-side to the p-side, there is an electron recombination (r) or diffusion current (Jnr) where n denotes electrons. This is due to the majority carrier electrons, which have enough energy to surmount the potential barrier. This current is very sensitive to a bias field that would change the potential barrier. On the p-side, there are thermally generated electrons, which in the space-charge region may be swiftly swept downhill into the n-region. This causes the thermal generation (g) or drift current (Jng). Electrons produced farther than a diffusion

length (to be defined) recombine before being swept across. As mentioned, in the absence of potential, the electron currents balance and we have

where the 0 in Jnr(0), etc. means zero bias voltage. Similarly, for holes, denoted by p,

We set the notation that forward bias (V > 0) is when the p-side is higher in potential than the n-side. See Fig. 6.22. Since the barrier responds exponentially to the bias voltage, we might expect the electron injection current, from n to p, to be given by

Adding everything up, we get the Shockley diode equation for a pn-junction under bias

where J0 = Jnr(0) + Jpr(0).

We now give a more detailed derivation, in which the exponential term is more carefully argued, and J0 is calculated. We assume that both electrons and holes recombine (due to various processes) with characteristic recombination times τn and τp. The usual assumption is, that as far as net recombination goes with no flow,

340 6 Semiconductors

and

where r denotes recombination. Assuming no external generation of electrons or holes, the continuity equation with flow and recombination can be written (in one dimension):

Fig. 6.23. Schematic of pn-junction (p region for x < 0 and n region for x > 0). Ln and Lp are n and p diffusion lengths

342 6 Semiconductors

which means we only need to find the minority carrier concentrations. In the steady state, neglecting carrier drift currents, we have

Diffusion lengths measure the distance a carrier goes before recombining. The solutions obeying appropriate boundary conditions can be written

6.3 Semiconductor Device Physics 343

To finish the calculation, we need expressions for np(−dp) − np0 and pn(−dn) − pn0, which are determined by the injected minority carrier densities.

Across the depletion region, even with applied bias, Jn and Jp are very small compared to individual drift and diffusion currents of electrons and holes (which nearly cancel). Therefore, we can assume Jn 0 and Jp 0 across the depletion regions. Using the Einstein relations, as well as the definition of drift and diffusion currents, we have

If φ is the built-in potential and φb is the bias voltage with the conventional sign

and

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Substituting, we can find the total current, as given by the Shockley diode equation

Reverse Bias Breakdown (EE)

The Shockley diode equation indicates that the current attains a constant value of −J0 when the reverse bias is sufficiently strong. Actually, under large reverse bias, the Shockley diode equation is no longer valid and the current becomes arbitrarily large and negative. There are two mechanisms for this reverse current breakdown, as we discuss below (which may or may not destroy the device).

One is called the Zener breakdown. This is due to quantum-mechanical interband tunneling and involves a breakdown of the quasiclassical approximation. It can occur at lower voltages in narrow junctions with high doping. At higher voltages, another mechanism for reverse bias breakdown is dominant. This is the avalanche mechanism. The electric field in the junction accelerates electrons in the electric field. When the electron gains kinetic energy equal to the gap energy, then the electron can create an electron–hole pair (e− → e− + e− + h). If the sample is wide enough to allow further accelerations and/or if the electrons themselves retain sufficient energy, then further electron–hole pairs can form, etc. Since a very narrow junction is required for tunneling, avalanching is usually the mode by which reverse bias breakdown occurs.

6.3.9 Solar Cells (EE)

One of the most important applications of pn-junctions is for obtaining energy of the sun. Compare, e.g., Sze, [6.42, p. 473]. The photovoltaic effect is the appearance of a forward voltage across an illuminated junction. By use of the photovoltaic effect, the energy of the sun, as received at the earth, can be converted directly into electrical power. When the light is absorbed, mobile electron–hole pairs are created, and they may diffuse to the pn-junction region if they are created nearby (within a diffusion length). Once in this region, the large built-in electric field acts on electrons on the p-side, and holes on the n-side to produce a voltage that drives a current in the external circuit.