1. Excess Carriers in Semiconductors
Lecture - 10
Excess Carriers in Semiconductors
• Direct & Indirect Recombinations
• Quasi-Fermi Levels

2. Excess carriers created in a semiconductor correspondingly increase its conductivity according to :
Photoconductivity: conductivity increase due to excess carriers created from optical excitation.
Direct recombination of electrons and holes :
This recombination occurs spontaneously: the probability that an electron and a hole will recombine is constant in time !
The decay of excess carriers : the rate of decay of electrons at any time t is proportional to the number of electrons remaining at t and the number of holes, with a proportional constant of the recombination, αr.
Net rate of change in C.B. electron concentration = thermal generation – recombination rate :

3. Assume that the excess EHPs is created at t =0 by a short flash of light and the initial excess electron and hole concentrations (n and p) are equal.
Instantaneous excess carriers are also equal : δn(t) = δp(t)
If the excess carrier concentrations are small, the δn2 term can be neglected. And, if the material is p-type extrinsic, the equilibrium minority carrier term (n0) can be also neglected.
Solution is:
= recombination lifetime = decay constant = minority carrier lifetime !
Mean instantaneous excess
carrier concentration
n: their value at t = 0
where

4. Decay of Excess Electrons and Holes by Recombination
General expression for carrier lifetime of n or p-type material if the injection level is low :
Figure 4—7
Decay of excess electrons and holes by recombination, for Δn = Δp = 0.1p0, with n0 negligible, and t = 10 ns (Example 4–2). The exponential decay of n(t) is linear on this semilogarithmic graph.

5. Indirect recombination: Trapping
In column IV semiconductors and certain compounds, the probability of direct electron-hole recombination is very small so that some band gap light given off by materials such as Si and Ge during recombination is very weak in radiation.
(b) In indirect materials, most recombination events occur via recombination levels within the band gap, and the resulting energy loss by this recombining electrons is usually given up to the lattice as heat rather than by the emission of photons.
An impurity or lattice defect can serve as a recombination center if it is capable of receiving a carrier of one type and subsequently capturing the opposite type of carrier, annihilating the pair of electron and hole [EHP].
In Fig. 4-8, since Er below EF, the recombination centers are filled with electrons at equilibrium. Therefore, the first event should be hole capture step (which is equivalent to an electron at Er falling to the V.B., leaving behind an empty state in the recombination center). Thus, in hole capture, energy is given up as heat to the lattice.

6. Energy is given up when a C. B
Energy is given up when a C.B. electron subsequently falls to the empty state in Er. When both of these events have occurred, the center at Er is back to its original state (filled with an electron), but an EHP is missing. Thus, one EHP recombination has taken place, and the center is ready to participate in another recombination event by capturing a hole.
The carrier lifetime resulting from indirect recombination is somewhat more complicated than is the case for direct recombination, since it is necessary to account for unequal times required for capturing each type of carrier. In particular, the recombination is often delayed by the tendency for a captured carrier to be thermally reexcited to its original band before capture of the opposite type of carrier can occur.
When a carrier is trapped temporarily at a center and then is reexcited without recombination taking place, the process is often called temporary trapping. And, such a center of an impurity or defect is called trapping center or simply trap if after capture of one type of carrier, the most probable next event is reexcitation. However, if the most probable next event is capture of the opposite type of carrier, it is called “recombination center”.

7. Indirect Recombination: Trapping
Figure 4—8
Capture processes at a recombination level: (a) hole capture at a filled recombination center; (b) electron capture at an empty center.

8. In general, trapping levels located deep in the band gap are slower in releasing trapped carriers than are the levels located near one of the band edges. This results from the fact that more energy is required to reexcite a trapped electron from a center near the middle of the gap to the C.B. than is required to reexcite an electron from a level near the C.B to the conduction band (C.B.).
In Fig. 4-9, impurities and their energy level positions in Si are shown. In this diagram, a superscript indicates whether the impurity is positive (donor) or negative (acceptor) when ionized. Some impurities introduce multiple levels in the band gap: Zn introduces a level (Zn-) located at 0.31 eV above the V.B. edge and a second level (Zn=) near the middle of the gap.
The effects of recombination and trapping can be measured by a photoconductivity decay experiment (Fig. 4-10). A population of excess electrons and holes disappears with a decay constant characteristic of the particular recombination process. The conductivity of the sample during the decay is :
Thus, the time dependence of the carrier concentrations can be monitored by recording the sample resistance as a function of time.

9. Energy levels of impurities in Si
Figure 4—9
Energy levels of impurities in Si. The energies are measured from the nearest band edge (Ev or Ec); donor levels are designated by a plus sign and acceptors by a minus sign.

10. Photoconductive Decay Measurement
Figure 4—10
Experimental arrangement for photoconductive decay measurements, and a typical oscilloscope trace.

11. Steady State Carrier Generation: Quasi-Fermi Levels
At equilibrium, a semiconductor experiences thermal generation of EHPs at a rate g(T) = gi, which is balanced by the recombination rate so that the equilibrium concentrations of n0 and p0 are maintained:
This equilibrium rate balance can include generation from defect centers as well as band-to-band generation.
If a steady light is shown on the sample, an optical generation rate (gop) will be added to the thermal generation, and the carrier concentrations n and p will increase to new steady state values.
The balance will be written between generation and recombination in terms of the equilibrium carrier concentrations (n0 and p0) and the deviations from the equilibrium (δn and δp):

12.  For steady state recombination and no trapping case: δn = δp
Since the thermal generation rate is
For low-level excitation case: δn2 can be neglected !
Excess carrier concentration can be written as :
Note that when trapping is not present. 

13. (Example 4-3)
Let us assume that 1013 EHP/cm3 (gop) are created optically every microsecond (μs) in a Si sample with n0 = 1014 cm-3 and τn= τp = 2μs. The steady state excess electron (or hole) concentration is then 2 x 1013 cm-3.
While the percentage change in the majority electron concentration is small, the minority carrier concentration changes
from
p0 = ni2/n0 = (1.5 x 1010)2/ = 2.25 x 106 cm-3 (equilibrium) to
p = p0 + δp = p0 + gopτp = 2.25 x 106 cm-3 + (1013 EHP/cm3-μs) x (2μs)
= 2 x 1013 cm-3 (steady state)
Note that the n0p0 =ni2 can not be used with the subscript removed: that is, np  ni2 when excess carriers are present !

14. Consider that the Fermi level (EF) is meaningful only when no excess carriers are present (i.e., at equilibrium state).
We also can write expressions for the steady state electron and hole concentrations in the same form as the equilibrium expressions by defining separate quasi-Fermi levels Fn and Fp for electrons and holes, respectively.
The quasi-Fermi levels (Fn and Fp ) are the steady state analogues of the equilibrium Fermi level EF. When the excess carriers are present, the deviations of Fn and Fp from EF indicate how far the electron and hole concentrations (n and p) are from the equilibrium concentrations (n0 and p0). And the separation of the quasi-Fermi levels (Fn - Fp) is a direct measure of the deviation from equilibrium (at equilibrium, Fn = Fp = EF).

15. 1014 cm-3 + (1013 EHP/cm3-μs) x (2μs)
From the example 4-3, the steady state electron concentration can be calculated :
Since n0 = 1014 cm-3
n = n0 + δn = n0 + gopτn =
1014 cm-3 + (1013 EHP/cm3-μs) x (2μs)
= 1014 cm x 1013 cm-3 = 1.2 x 1014 cm-3
With
1.2 x 1014 = (1.5x1010) exp[(Fn-Ei)/0.026]
Fn-Ei = eV
Similarly, Ei -Fp- = eV is obtained using
Figure 4—11
Quasi-Fermi levels Fn and Fp for a Si sample with n0 = 1014 cm23, tp = 2 μs, and gop = 1019 EHP/cm3-s (Example 4–4).

16. Photoconductive Devices
Devices which change their resistance when exposed to light !
Optical sensitivity of a photoconductor can be evaluated by examining the steady state excess carrier concentrations generated by an optical generation rate gop. If the mean time (that each carrier spends in its respective band before its capturing occurs) is τn and τp, we have :
Photoconductivity change is :
Considerations in choosing a photoconductor for a given application include:
- sensitive wavelength range
- time response
- optical sensitivity