# Quasi-Fermi Levels E.1 Introduction

The concept of a fermi level was introduced in Chap. 4 to describe the energy distribution of electrons and holes in thermal equilibrium. In non-equilibrium, quasi-fermi levels or imrefs provide a useful tool for semiconductor device analysis as outlined below. Formally, they correspond to the electrochemical potentials of non-equilibrium thermodynamics.

E.2 Thermal Equilibrium

A system in thermal equilibrium has a single, constant-valued fermi level. The case of p-n junction in thermal equilibrium is shown in Fig. E.1. For non­degenerate conditions (n « Nc, p « Nv):

n = NCexp [(Ef-EC)/kT] (E.1)

 p = NV exp [(,Еу – Ef)/ kT (E.2) Fig. E.1: P-N junction in thermal equilibrium.

E.3 Non-Equilibrium

When voltage is applied and/or the p-n junction is illuminated, the concept of a fermi level no longer applies. However, "quasi-fermi" levels allow the analysis of semiconductors away from equilibrium. Separate electron and hole quasi-fermi 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 quasi-fermi 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.4 Interfaces At interfaces between different materials, quasi-fermi levels can be discontinuous as shown in Fig. E.2 for a metal/semiconductor interface. The electron and hole currents across the interface depend on the corresponding carrier concentration near the interface and the size of the discontinuity between Efm and the corresponding quasi-fermi level, Efn or Efp. For the above case of contact to p-type material (Fig. E.2), the hole concentration near the interface is large. Quite small differences between EFm and Efp will support large current flows, i. e., Efp would be closely aligned to the metal fermi level, Efm. The extend of the misalignment between Efn and Efm for the contact to p-type material depends on the surface recombination velocity of the interface. High surface recombination velocity will enforce conditions similar to thermal equilibrium at the interface, i. e., Efm ~ Efn ~ Efp – Low surface recombination velocity will allow large separation between Efn and Efp with dEFN/dx ~ 0 near the interface, for p-type material.

 C

 E FN

 EF P Ev

 P Fig. E.2 : Idealized interfacial conditions for the case where both electrons and holes flow from semiconductor to metal.

E.5 Non-Equilibrium P-N Junction

If a voltage, Va, is applied to a p-n junction in the dark, the energy-band diagram will take the form of Fig. E.3. Applying a voltage between the metal contacts gives an energy separation between the quasi-fermi levels in the two contacts equal to qVa as shown. Across the interfaces, the majority carrier quasi-fermi levels are essentially continuous with the respective metal fermi level, while the minority carrier quasi-fermi levels need not be continuous.

In the bulk quasi-neutral regions, the majority carrier quasi-fermi levels will be approximately constant. This is because the majority carrier concentrations are large here and small quasi-fermi 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: Quasi-fermi levels in a non-equilibrium p-n 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", Prentice-Hall, New Jersey, 1982). The assumptions required in the derivation are (i) no change in majority carrier quasi-fermi level across the contact/semiconductor interface; and (ii) constancy of quasi-fermi levels across the depletion region. In a more elementary derivation, the presentation usually manages to "side-step" 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 quasi-fermi 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 quasi-fermi levels and hence allow the initial solutions to be refined.

E.6 Use Of Quasi-Fermi Levels

Quasi-fermi levels are a valuable tool in analyzing semiconductor devices. They allow energy band diagrams in non-equilibrium 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 quasi-fermi levels provide the "driving force" for carrier trans­port across the contact. The majority carrier quasi-fermi level is usually aligned to the metal fermi level (a discontinuity between these two levels corresponds to "contact resistance").

(ii) Quasi-fermi levels vary only slowly with position in the semiconductor with no discontinuities except at interfaces.

(iii) In bulk quasi-neutral 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 one-dimensional (1- D) model of a p-n-p 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 1-D 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 quasi-fermi levels at the contacts of a p-n-p transistor structure.

Step 2:

Join up the quasi-fermi 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

Step 3:

Add the energy band diagram in quasi-neutral 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 4-layer 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.