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Energy-band diagram

As mentioned above electrons and holes in semiconductors exhibit dedicated characteristics, like acceptance of only restricted energy levels, effective mass, etc. Some of them can be illustrated with the band structure of semiconductors and its energy-band diagram. Whereas the band structure (already mentioned above) shows important conclusions found out by quantum-mechanic analysis, generally, an energy-band diagram presents dedicated energy levels (among others the band edges and ) against the location described with a position coordinate ( , or ) of a Cartesian system. To illustrate the increase of electron energy in the energy-band diagram, usually the ordinate with an arbitrary reference level is used for that.

Before going in details, we should mention what the band model's positive features are. As we will demonstrate with examples below, in particular, the energy-band diagram is used to detect where space charges, electric fields and currents will occur within the semiconductor. Moreover, it is very useful to find out the right (internal) boundary conditions needed to calculate the potential and field distribution within the semiconductor device structures. In this way a problem of the complex contact electronics can be defused drastically for modeling purposes. Last but not least, based on it we are able to explain in a phenomenological manner the operating principle of complex semiconductor devices without a detailed analysis of the internal electrons.

Mostly, only the three energy levels (the band edges , and the Fermi-level ) as well as the potential energy of a fee unmoved electron (), and additionally, the two energy differences (the electron affinity and work function ) are sufficient to complete the energy-band model as demonstrated in the following.

From the band-structure we know that just energy levels near the band edges and are most interesting for the existence of electrons and holes in semiconductors. That is why just theses levels have to be pictured. As mentioned above, from them we find the energy gap .

If we have doped semiconductors, we still need the position of the so-called Fermi-level in the energy band diagram. The Fermi-edge is an important formal parameter which characterizes the occupation probability of available energy states in the conduction and valence band with electrons or holes (see also paragraph 1.1.3). The position of the energy level in the band model is dependent on the impurity concentration. Normally, the Fermi-level is within the forbidden band (see Figure 1.1). Such semiconductors are said to be non-degenerated. Otherwise, in heavily doped semiconductors ( or ) the level can dip into the conduction or valence band (degenerated semiconductors).

Characteristically, under thermal equilibrium (that means e.g. without currents in the semiconductor) the Fermi-edge is constant versus the position coordinate throughout the energy-band model (e.g. ). In this case, there is only one energy level for electrons and holes. However, in an unbalanced thermo-dynamic state (e.g. with current flow due to supplied voltages) the Fermi-level is split-up into a quasi-Fermi-level for electrons , and another one for holes . Normally, both levels vary differently versus the position coordinate.

It should be mentioned, that the energy equivalent of a supplied voltage has to meet the condition .

More details on how to posit in the energy-band diagram, and about and as well as their potential equivalents and will be given in the following paragraphs.

To find out locations where space charges and electric fields dominate the internal electronics in semiconductors; additionally, we need information on changing the electrostatic (macro) potential against the position coordinate. Therefore, the energy equivalent of the electrostatic (macro) potential, i.e. the level of potential energy is also pictured in the band diagram (see Figure 1.2).

Figure 1.2: Energy-band diagram [1], [2]

Because the energy difference between the energy equivalent of the macro potential and the band edges and is only dependent on the crystal characteristics, the potential and the band edges remain in parallel throughout the structure as long as the crystal's characteristics (e.g. the semiconductor material) do not change.

Regarding to , we are able to make visible two essential energy differences, namely, the electron affinity and the work function (see Figure 1.2).

The energy necessary to free an electron from the conduction band edge is called electron affinity . In other words, , where is the potential energy of a resting and free electron outside of crystals.

In particular, for modeling electronic devices in technology, we need and . These energy values are necessary to emit an electron into vacuum and silicon oxide, respectively.

The energy difference between (energy equivalent to the macro potential ) and (Fermi-level) is called work function , which is a material-specific parameter.

We know, for doped semiconductors is dependent on the impurity concentration, and consequently , too. In particular, we should distinguish between and for - and -doped semiconductors.

Based on the energy quantities pictured in the energy band diagram and their relative position to each other we are able to distill consequences out of the energy-band diagram for metals, semiconductors and insulators.

Figure 1.3: Energy band diagram: (a) metal, (b) semiconductor, (c) insulator [2]

As shown in Figure 1.3(a), metals have no forbidden band, because the valence band overlaps the conduction band. Consequently, in metals all electrons will contribute to the comparably high conductivity.

Contrarily, an insulator (see Figure 1.3(c)), is characterized by a large energy gap. Therefore, a relatively high energy is necessary to lift valence band electrons into the conduction band. That is why the thermal generation of carriers is extremely weak and at room temperature pretty unlikely, i.e. no mobile charges are available within the insulator.

The semiconductor (see Figure 1.3(b)) is neither as a good conductor as the metals are, nor an insulator. The particularity exploited for electronic devices is due to its small intrinsic conductivity, which can be prepared goal-oriented by doping.

To meet the expectations and to illustrate the explanation given above, the energy-band diagram of a - and -doped semiconductor under thermal equilibrium will be presented at first (see Figure 1.4).

Obviously, the main differences between these energy-band diagrams concern the relative position of the Fermi-level within the forbidden band, and consequently, the work function of the -and -type semiconductor.

Figure 1.4: Energy band diagram of a - and -doped semiconductor [1], [2]

If we physically implement a - and -type semiconductor layer in the same crystal, a so-called -junction is formed. To construct its energy-band model we have to bring together the band-diagrams given in Figure 1.4(a) and (b), carefully. In doing so we should consider that is constant throughout the structure. In fact, there are no interface and dipole charges at the stoichiometric junction where the conducting type is technologically changed from -type to -type or vice verse. That is why the electrostatic potential within the structure may not exhibit any discontinuities. As mentioned above, the energy equivalent to the electrostatic potential has to stay in parallel to the band edges. Consequently, careful combining the energy-band diagrams of a - and -type semiconductor layer results in energy-band diagram of a -junction as shown in Figure 1.5.

Figure 1.5: Energy-band diagram of a -junction [1], [2]

Without any analytic calculations we can learn from it, that within a small region around the junction from a - to an -type semiconductor () the electrostatic potential varies with the -coordinate. Below (e.g. see paragraph 1.2) we will show, that this fact signalizes an electric field due to space charges in this region. Moreover, caused by this electric field an internal (built-in) potential difference occurs, which has to be considered for determining the potential and field distribution within the structure correctly.

Another important basic structure of semiconductor devices is the MOS-structure shown in Figure 1.6 for a -doped semiconductor.

Figure 1.6: Energy-band diagram of a MOS-structure under thermal equilibrium

In our example we assume, that the gate is made from aluminum .

To come up with the energy band diagram of such a MOS-structure we need the band model for the gate metal () and the -doped semiconductor. Simply, the metal's energy-band model is given with the Fermi-level and its work function . Separated by a very thin layer a -doped semiconductor completes the MOS-structure where the semiconductor's band model is well-known already.

Once again, we have to make sure that under thermal equilibrium conditions is constant throughout the complete structure (see Figure 1.6).

Due to the difference in the work function of () and () we get a linear increase of the potential (potential energy) within the insulator layer. Supposed there are no interface charges located at the interface between the insulator and semiconductor, the electrostatic potential and its slope (the slope corresponds to the electric field as discussed in paragraph 1.2), may not exhibit any kind of discontinuities at the interface. Moreover, deep in the semiconductor, i.e. in a sufficient distance from the interface between oxide and semiconductor, the MOS-structure's energy-band diagram has to agree with that of a -doped semiconductor. Therefore, we have to accept the band bending as shown in Figure 1.6. Finally, gives the energy difference to , wherewith the energy-band diagram of the MOS structure can easily be completed, supposed all relevant rules given above are considered carefully.

To sum up what we can learn from this example, we should recognize the following:

Already under thermal equilibrium (i.e. without supplied voltages) we can identify a space charge at the interface between and . Roughly speaking, this space charge layer ( with the layer width ) is due to complex phenomena of contact electronics and leads to a electric field built-in the oxide and also in the semiconductor. Consequently, a drop of potential across the oxide () as well as the semiconductor () will occur. The sum of these voltages is the so-called contact potential , which plays an important role to define the correct boundary conditions for a detailed analysis of the potential and field distribution within the MOS structure with power supply.

Next: Energy and potential dependency Up: Charge carriers in semiconductors Previous: Electrons and holes in   Contents
Christoph Clauss, Karl-Heinz Diener, Peter Trappe 2005-01-21