Chapter 3b

Indirect Recombination: Trapping

In indirect materials, the probability of direct band-to-band recombination is very small recombination in these materials proceed through the assistance of recombination (or trapping) centers located within the band gap, which trap carriers of one type followed by trapping carriers of the opposite type, thus annihilating the pairs.

Fig.3.3 Energy levels of common impurities in silicon, measured from the closest band edge.
The resulting energy loss is often in the form of heat given to the lattice (instead of light emission), and, thus, these materials are not well suited for optoelectronic applications.
There are four probabilities associated with a recombination center:
(a) hole capture: when an electron from the recombination center falls to the valence band,
(b) hole emission: when an electron makes a transition from the valence band to the recombination center,
(c) electron capture: when an electron falls from the conduction band to the recombination center, and
(d) electron emission: when an electron makes a transition from the recombination center to the conduction band.
Each of these processes has their own probabilities and time constants, and the resulting analysis is significantly complicated.
This theory of recombination is known as the SHR (Shockley-Hall-Read) theory of recombination.
If process (a) follows process (c) or vice versa, recombination takes place, whereas, if process (b) follows process (a) (or vice versa) or process (c) follows process (d) (or vice versa), it is known as reemission, and the recombination center behaves like a trapping center.
Generally, centers that are located towards the middle of the band gap (e.g., Au in Si) behave like recombination centers, whereas centers located closer to the conduction or valence band behave as traps, for obvious reasons.
Alternate definition: in a center located within the band gap, if after capturing one type of carrier, the most probable next event is the capture of opposite type of carrier, then it is a recombination center, however, if the most probable next event is reexcitation, then it is a trap.
The recombination can be slow or fast, depending on the amount of time the carrier spends in the center before the capture of the opposite type of carrier happens, thus, computation of lifetime for this kind of indirect recombination is sufficiently complicated.
The decay of excess carriers can be measured by a typical photoconductive decay measurement, where light shining on a sample is suddenly switched off, and the resulting decay of current passing through the sample is measured, the rate of decay of this current gives the excess carrier lifetime.

Auger Recombination

Note: the lifetime is proportional to the inverse of the doping concentration.
However, at relatively high doping levels, the lifetime decreases at a faster rate with an increase in the doping concentration.
This is because a different recombination mechanism, called the Auger recombination becomes dominant at high doping levels.
In this recombination mechanism, the electron and hole recombine without involving trap levels, and the released energy (of the order of the energy gap) is transferred to another carrier (a hole in p-type material and an electron in n-type material).
This process is somewhat the inverse of the impact ionization process, in which an energetic carrier causes EHP generation.
Since two electrons (in n-type material) or two holes (in p-type material) are involved in this process, it is highly unlikely except in heavily doped material.
The recombination lifetime associated with the Auger recombination process is inversely proportional to the square of the majority carrier concentration, i.e., for p-type material, , and for n-type material, , where Gp and Gn are coefficients with values of and .

Surface Recombination

It is obvious that near the surface of any semiconductor device, the carrier recombination rate should be very high, due to extra defects and traps at the surface.
Thus, the diffusion flux of minority carriers at the surface is determined by the surface recombination processes.

Fig.3.4 Auger recombination: (a) n-type sample, and (b) p-type sample.

For example, when minority carriers are holes, this surface recombination can be described by:
where x = 0 corresponds to the surface of the sample, and is the surface recombination rate, with being the capture cross-section for holes, is the thermal velocity for the holes, and is the surface density of the surface states.
The capture cross-section (typically ) describes the effectiveness of the localized state in capturing a carrier.
The product may be visualized as the volume swept out per unit time by a particle with cross-section
If the surface state lies within this volume around the carrier, then the carrier gets captured by the surface state.
Note: the dimension of S is cm/sec, and, consequently, it is termed as the surface recombination velocity, even though it has nothing to do with actual velocity.

Steady State Carrier Generation: Quasi-Fermi Levels

For any temperature T, there is a thermal generation rate g(T) balanced by a recombination rate r(T).
Now, if a steady light is shone on the sample, an optical generation rate will be added to g(T), and the carrier concentrations n and p would increase to their new steady state values.
Generation/recombination rate balance equation:

For steady state recombination and no trapping,; and, under low level injection approximation

Thus, the excess carrier concentrations can be given by
In general, when trapping is present, , and and .
Note: when excess carriers are present, .
When excess carriers are present, the equilibrium Fermi level is no more meaningful; instead, the carrier concentrations are defined in terms of quasi-Fermi levels (also referred to as Imref, which is Fermi spelled backwards) as
Imref for the minority carriers deviates significantly from the equilibrium Fermi level, whereas, for majority carriers, the Imref stays very close to the equilibrium Fermi level, and the separation between these two Imrefs is a measure of how far the system is from equilibrium.
With concentrations varying with position, the Imrefs would also vary with position, thus drawing Imrefs in band diagrams clearly shows the positional variations of the carrier concentrations.

EXAMPLE 3.2

A Si sample with is illuminated by a steady light thus creating optically. Assume no trapping, and
(a) Determine the electron and hole concentrations n and p respectively, and their percentage change from the equilibrium concentrations.
(b) Comment on the magnitude of the product np.
(c) Determine the locations of the Imrefs ,and compare their locations with the equilibrium Fermi leve

SOLUTION

(a) The equilibrium hole concentration . Hence, the equilibrium electron concentration

Since there is no trapping and,therefore, the excess electron and hole concentrations can be given by

Therefore, the net electron concentration is given by

and the net hole concentration is given by

Therefore, the percentage change in the electron concentration

And the percentage change in the hole concentration

Note: with optical excitation (under the low-level injection approximation), there is a very large change in the minority carrier concentration, whereas the change in the majority carrier concentration is hardly noticeable!

(b) Note:

Whenever excess carriers are present, , and the amount of deviation quantifies the departure from equilibrium.

(c) In equilibrium,

In the presence of excess carriers, the electron Imref

and the hole Imref

Thus, the majority carrier Imref almost coincides with the equilibrium Fermi level, whereas the minority carrier Imref shows a large departure from the equilibrium value.

Photoconductive Devices

Devices which change their resistance while exposed to light.
Examples: automatic night light controllers, exposure meters in cameras, moving-object counters, burglar alarms, detectors in fiber optic communication systems, etc.
Considerations in choosing a photoconductor for a given application: sensitive wavelength range, time response, and optical sensitivity (responsivity) of the material.
The photoconductivity change while exposed to light is
Obvious that for large changes in , the carrier mobility and lifetime should be high (e.g., in , and could be used as infrared detector with high sensitivity).
Time response is limited by recombination times, degree of carrier trapping, and time required for the carriers to drift through the device in an electric field.
Dark resistance (i.e., the resistance of the device without any illumination) should be as small as possible.
Generally, all these requirements cannot be satisfied simultaneously, and some kind of optimization is required.

Diffusion of Carriers

When excess carriers are created in a semiconductor and their concentrations vary with position, then there is a net carrier motion from regions of higher concentration to regions of lower concentration.
This type of motion is called the diffusion, and it is an important charge transport mechanism in semiconductors.
Diffusion and drift are the two main current transport mechanisms.

Diffusion processes

Natural result of the random motion of individual electrons.
Electrons move randomly and experience collisions, on the average, after each mean free time .
Since the motion is truly random, an electron has equal probability of moving into or out of a volume through a boundary.

Fig.3.5 Spreading of a narrow pulse of electrons created at x = 0 at t = 0 with time

A pulse of excess electrons injected at x = 0 at time t = 0 will spread out in time due to diffusion, and eventually n(x) becomes a constant, when no more net motion takes place.