LESSONS OF EASTER ISLAND
March 17th, 2016
The concept of a fermi level was introduced in Chap. 4 to describe the energy distribution of electrons and holes in thermal equilibrium. In nonequilibrium, quasifermi levels or imrefs provide a useful tool for semiconductor device analysis as outlined below. Formally, they correspond to the electrochemical potentials of nonequilibrium thermodynamics.
A system in thermal equilibrium has a single, constantvalued fermi level. The case of pn junction in thermal equilibrium is shown in Fig. E.1. For nondegenerate conditions (n « Nc, p « Nv):
n = NCexp [(EfEC)/kT] (E.1)
p = NV exp [(,Еу – Ef)/ kT (E.2) Fig. E.1: PN junction in thermal equilibrium. 
When voltage is applied and/or the pn junction is illuminated, the concept of a fermi level no longer applies. However, "quasifermi" levels allow the analysis of semiconductors away from equilibrium. Separate electron and hole quasifermi levels, Efn and Epp, are defined by the following equations [again for nondegenerate conditions, see Eq. (5.28) of text for the more general form]:
n = Nc exp[( Efn – Ec )/kT ] (E.3)
p = Nv exp[( Ev – Epp )/kT ] (E.4)
np = NcNv exp[(Eg + EfN – Efp )/kT (E.5)
Note that:
np = n2 exp[( EfN – Efp )/kT ] (E.6)
The latter equation suggests that quasifermi levels are a measure of the level of disturbance from equilibrium (np = щ2 at equilibrium). Also, from Eqns. (5.29) and (5.30) of the text:
Jn = (E.7)
d
Jp =VppddP (E.8)
Currents are determined by the product of carrier concentration and the quasi – fermi level gradients – a second useful attribute. As an aside, note that substituting Eq. (E.3) into Eq. (E.7) gives:
The first term is just the drift component of current since the electric field, F, is given by:
F _ 1 dEc
q dx









E.5 NonEquilibrium PN Junction
If a voltage, Va, is applied to a pn junction in the dark, the energyband diagram will take the form of Fig. E.3. Applying a voltage between the metal contacts gives an energy separation between the quasifermi levels in the two contacts equal to qVa as shown. Across the interfaces, the majority carrier quasifermi levels are essentially continuous with the respective metal fermi level, while the minority carrier quasifermi levels need not be continuous.
In the bulk quasineutral regions, the majority carrier quasifermi levels will be approximately constant. This is because the majority carrier concentrations are large here and small quasifermi level gradients will support large current flows. Combined with the constancy across the metal interface, it follows that Efn – Epp ~ qVa in the junction region.
■T” 

efn 

___ 

F 

P N 
Fig. E.3: Quasifermi levels in a nonequilibrium pn junction. 
Hence, near the junction:
np = nt2 exp (qVJkT) (E.12)
This is an alternative derivation from that in more elementary texts (e. g., Martin A. Green, "SOLAR CELLS: Operating Principles, Technology and System Applications", PrenticeHall, New Jersey, 1982). The assumptions required in the derivation are (i) no change in majority carrier quasifermi level across the contact/semiconductor interface; and (ii) constancy of quasifermi levels across the depletion region. In a more elementary derivation, the presentation usually manages to "sidestep" the first assumption. The second is equivalent to the argument that drift and diffusion components are both large and opposing in the depletion region, with nett current flow being due to a small imbalance.
The quasifermi level formulation gives a procedure for checking the validity of these assumptions and refining initial estimates. Assuming constancy, first estimates of carrier concentrations and current flows could be calculated. Then, the current density Eqs. (E.7) and (E.8) could be used to calculate the variation of
the quasifermi levels and hence allow the initial solutions to be refined.
Quasifermi levels are a valuable tool in analyzing semiconductor devices. They allow energy band diagrams in nonequilibrium situations to be sketched which can then form the basis for more detailed analysis. Attributes and key properties include:
(i) Contact regions easily handled. Differences between metal and semiconductor quasifermi levels provide the "driving force" for carrier transport across the contact. The majority carrier quasifermi level is usually aligned to the metal fermi level (a discontinuity between these two levels corresponds to "contact resistance").
(ii) Quasifermi levels vary only slowly with position in the semiconductor with no discontinuities except at interfaces.
(iii) In bulk quasineutral regions in low injection, the majority carrier quasi – fermi level is separated from the band edge by nearly the same energy as in equilibrium. Electric fields in these regions will produce a gradient in the band edge (qF = dEcy/dx) which will largely transfer to the majority imref.
(iv) Imrefs are approximately constant across depletion regions.
As an example of their use, an energy band diagram of a onedimensional (1 D) model of a pnp bipolar transistor will be constructed with a positive bias on the first junction and a negative on the second.
Step 1:
Sketch in imrefs where values are known, namely at the contacts as shown in Fig. E.4. (Only two contacts can be shown on a 1D model. The centre contact is modelled by forcing the majority carrier imref to a known value at a point in the n – type base region).
P N P
Fig. E.4: Majority carrier quasifermi levels at the contacts of a pnp transistor structure.
Step 2:
Join up the quasifermi levels as shown in Fig. E.5 noting that the minority and majority carriers will tend to come together at the emitter and collector contacts, if these are assumed to have a high recombination velocity (also at the base contact, although this contact is not really shown in the above diagram – it is some distance either out of, or into, the page).
P N
Fig. E.5: Linking quasifermi levels.
Step 3:
Add the energy band diagram in quasineutral regions as shown in Fig. E.6, noting bands lie in the same relationship to majority carrier imrefs as in equilibrium.
P
N
Fig. E.6 : Addition of conduction and valence band edges.
Step 4:
Complete the energy band diagram by joining up bands across depletion regions as shown in Fig. E.7.
Fig. E.7: Linking of conduction and valence band edges. Step 5:
Analyze the device. Several properties are already apparent from the above figure (e. g., np ~ щ2 exp [(EmB – Emp)/kT] near the emitter junction and np ~ щ2 exp [(EmB – Emc)/kT] near the collector junction, Jp is likely to have same direction in emitter and collector, and so on). The extension to less familiar devices such as 4layer structures is straightforward. High injection causes the majority carrier imref to move closer to the edge of the majority carrier band than in equilibrium. However, the relative insensitivity of imref energy to such effects allow sketches even in this case. Graded bandgap problems, and more complicated contact models, are also quite easily handled.
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