Calculation of electrical levels 

Electrical levels 

In the persuit of improved electronics, a significant problem is the engineering of materials that satisfiy the need for semiconducting materials with efficient doping. Most semiconducting material is deliberately chosen to be impure, mainly because one wishes to have material that has an excess of one type of charge carrier: electrons and holes. The simplest dopants substitute for host atoms but have either one more or one less valence electron. A species with an excess of electrons is a donor (they donate these electrons to the aid conduction) and are "negative" in charge: this produces ntype material. Similarly, a deficit of electrons creates an acceptor (the species will accept electrons from the host) leaving a positviely charged valencehole: this produces ptype material. Important devices (such as diodes) can be manufactured by the interfacing of n and ptype semiconductors. At zerotemperature, however, all these electrons and holes will remain on the dopants and the conductivity will be zero. As the temperature increases the fraction of donors or acceptors that are ionized (give up a carrier to the material) increases exponentially, the ionization fraction depends on the depth of the level from the semiconductor bandedge (see figure.1): the shallower the level, the more effective the dopant. 

Formation energies 


Donor and acceptor levels may lie anywhere in the bandgap. Technically, the question which we wish to answer is what are the range of the chemical potential of the electrons for which a given charge state is thermodynamically most stable? This question can be tackled by attempting to calculate the formation energy for each charge state: The terms in this expression for system X in charge state q, in order, are the total energy taken from some calculation, a sum over the atomic chemicalpotentials, an electroniv term (in braces) which accounts for the valence band top and chemical potential of electrons, and finally a term that accounts for a number of artifacts in the calculation of the first term arising due to, for instance, boundary conditions. We calculate the formation energies for all possible charge states, and find which is the lowest energy for all values across the bandgap, as illustrated in figure 2. 
Problems 

However, using standard densityfunctionaltheory (DFT) methods there are problems which make this calculation particularly challenging:
 
A potential solution: The Marker Method 

One way around these problems is to assume that the errors are similar for comparable systems, and then take the bold step to simply cancel them: this is the markermethod. This method is very simple in that one takes two like systems, X and Y, for one of which the electrical levels are already known (Y), and this allows one to calculate the difference in the (say) acceptor levels: Here the calculation only requires total energy difference involving the same charge state. Even if the bandgap is underestimated by 50%, it is in error by the same amount for X and Y, so the effect (largely) cancels. However, for the errors to cancel to the best possible degree one requires:


Selected References
