## 1. Introduction

Recombination of excess (nonequilibrium) electrons and holes in semiconductors through impurity recombination centers (traps) referred to in this chapter as trap-assisted (Shockley-Read-Hall) recombination is in many cases the dominant process [1-9]. Hall [10] and Shockley and Read [11] have proposed the theory of trap-assisted recombination as early as 1952. Further, the theory of trap-assisted recombination has been developed in many aspects and details [1-5, 12-14]. At the same time, due attention was not paid to study dependences of lifetimes of excess electrons

That reasonable understanding is incompletely adequate to reality. As shown below, lifetimes of excess electrons and holes (see our definition of

In this chapter, we generalize the theory and give systematic mathematical and detailed physical analysis of dependences

## 2. Preliminaries

Let’s consider the case of small deviation of free charge carriers’ concentrations from equilibrium values. This situation occurs often in semiconductors used for registration of low-level signals, for example, optical signals. It will be shown that both

It is reasonable to expect that the growth of lifetimes

where

It was first reported in [20] that vanishing

In our case,

Below, in case of single recombination level, we consider in detail the impact of photoinduced space charge

Also, ignoring approximation of quasi-neutrality, we study, at low-level illumination, the effectiveness of band-to-band photoexcitation of charge carriers and photo-emf

Analyzing above-mentioned problems, we do not use conventional (Shockley-Read) expression-based form [1-5, 9, 11-14] of generation-recombination rate. This form does not allow to express explicitly dependences

These difficulties are dramatized by the fact that under certain conditions,

## 3. Model and basic relations

Consider nondegenerated semiconductor doped by shallow fully ionized single type impurity (for definition donors) with concentration

Let either band-to-band excitation (Figure 5b) or injection on the contacts produce excess electrons and holes. Then, in stationary case, equation

determines the charge state of recombination impurity atoms.

Recombination-generation rates of electrons

Here,

For small deviation of charge carriers’ and their capture centers’ concentrations

we obtain

where

High-performance photoconductors operate with extremely low-level illumination. Therefore, linear for

In view of the above provision, we can write expressions for the density of photocurrent components as follows:

where

must satisfy continuity equations:

and also

Let limit voltage be applied to sample by value

that allows to neglect by the dependence of

## 4. Lifetime of excess charge carriers

Using distribution function of electrons over acceptor level states [12, 27], we can write neutrality equation

for nondegenerated semiconductor at thermodynamic equilibrium as follows:

where

From expressions (8), (9), (20), and (23), it follows that

Expressions (20), (24), and (25) determine dependences

### 4.1. Mathematical analysis of hole lifetime

The analysis of equation

which determines extremum points of dependence

To determine minimum point, let us set out equation (26) into the form

where absolute value of function

(29) |

is much less than unity at

It means that the first root of equation (26)

It follows from this formula that ratio

To determine maximum point of dependence

where absolute value of function

is much less than unity at

It means that the second root of equation (26)

In particular

It follows from (22), (25), (27), (30), and (34) that

(37) |

Relation (37) shows that function

increases with decreasing recombination level energy

The dependence of

where

### 4.2. Mathematical analysis of electron lifetime

The analysis of equation

which determines extremum points of dependence

To determine minimum point let us set out equation (40) into the form

where absolute value of function

(43) |

is much less than unity at

It means that the first root of equation (40)

It follows, from this formula, that ratio

We can transform equation (40) to form (32) where

(46) |

At value

where

Value

increases, as for

### 4.3. Physical interpretation

Let’s explain physical mechanisms of the above regularities.

#### 4.3.1. Hole lifetime

Reciprocal hole lifetime

consists of three partial components.

First component

corresponds to the change of capture rate of holes

Second component

corresponds to the change of capture rate of holes

Third component

corresponds to the change of thermal emission rate of holes from impurity level states into valence band

Lifetime

If conditions (27) are fulfilled and

When

As shown above, minimum point

For the same reasons, non-monotonic dependence

Non-monotonic character of dependence

Maximal value of ratio

Note that with increasing energy

The character of dependence

Values

Value

#### 4.3.2. Electron lifetime

By analogy with hole lifetime, reciprocal electron lifetime consists of three partial components:

First component

corresponds to the change of electron capture rate

Second component

corresponds to the change of electron capture rate

Third component

corresponds to the change of thermal emission rate of electrons from impurity level into valence band

Times

Value

At values

As shown above, in contrast to dependence

Similar to the behavior of hole lifetime, non-monotonic dependence

Due to decreasing

## 5. Relation between concentrations of photoholes and photoelectrons outside approximation of quasi-neutrality

Note first, from formulas (10), (11), (20), and (23) follow

Differentiating (12) with respect to

Recall that

From equation (16) and formula (5), we find

Eliminating

where

Eliminating

where

In quasi-neutrality approximation

## 6. Derivation of equation for distribution function of photoexcited charge carriers’ concentration outside quasi-neutrality

From expressions (12)–(14), it follows that

Plugging this expression for intensity

where

Plugging relation (66) between

(72) |

where

Formulas (73) and (74) are, none other than, well-known (in quasi-neutrality approximation) expressions for ambipolar diffusion constant

is much less than unity (see below). On the other hand, as it follows from equations (63) and (64),

Equating (72) and (76), we obtain equation

(77) |

Because

then

where

Therefore, we can rewrite equation (77) as follows:

where

Equation (81) is the desired equation. It depicts adequately the continuity of electron, hole, and total currents [see (15)-(17)].

## 7. Solitary Illuminated Sample

In this section, we will consider opportunities for improving photoexcitation of charge carriers and photo-emf

### 7.1. Preliminaries: Basic relations

We will call the sample as solitary, if it is not in external electric field and external electrical circuit is open.

For simplicity, we will characterize effectiveness of charge carriers’ photoexcitation by light-propagation-length averaged concentrations of photoelectrons

Figures 6 and 7 show calculated dependences

We have not used in study quasi-neutrality approximation [2-9, 13, 18, 21, 22, 28-31] because it can lead to unacceptable errors in calculation of dependences

Let’s consider a homogeneous semiconductor sample (Figure 5) with no voltage applied, i.e., in absence of illumination and intensity of electric field

in which

where

and function

where

Denote:

Exact solution of equation (88) is

where

In quasi-neutrality approximation, parameters

### 7.2. Effectiveness of charge carriers’ photoexcitation

We define the mean value

Let’s analyze the worst case, when recombination of excess charge carriers on illuminated

From (87), (90), and (97), it follows that

where

Thus, according to definition (96), we find

where

and

Outside quasi-neutrality approximation, expressions (20)–(25), (60), (61), (75), (80), (91)–(94), and (104)–(110) determine, in parametric form (value

Denote desired dependences in quasi-neutrality approximation as

Figure 6 shows that the effectiveness of charge carriers’ photoexcitation may grow significantly with increasing

Lengths

At

where function

This means that diffusion of photoelectrons and photoholes is independent from each other.

Therefore,

Length

At point

and

despite the fact that

When

Moreover, shielding length of photoinduced space charge

### 7.3. Effectiveness of photo-emf excitation

In view of the fact that under considered conditions

from expressions (12)–(14) and (87), we obtain that photo-emf

i.e. potential

where

It is clear from expression (121) that at infinite surface recombination rate, i.e., when conditions (97) are fulfilled, illumination produces no photo-emf. In this regard, assume that photoexcited charge carriers are not captured on surfaces and there is no charge on surfaces. In these conditions, photoexcited electric field intensity

And densities of electron

Therefore, we may write boundary conditions, in view of relation (87), as

From (85), (86), (90), (126), and (127), it follows that

where

Find consistently

where

where

If we utilize relation

then expressions (121) and (136)–(139) allow us to obtain

where

(142) |

In quasi-neutrality approximation, we get

where

We refer to the dependences (141) of photo-emf

Figures 7, 10, and 11 show these dependences and solution in quasi-neutrality approximation as well. It is clear in Figures 10 and 11 that, in quasi-neutrality approximation, maximal value

We can explain results by the fact that very long shielding length (114) of photoexcited space charge

Evidently, with thinning

Note that for impurity level energy equal to

## 8. Photoelectric gain

We will consider uniform spatial distribution of density of charge carriers’ photoexcitation rate

We will facilitate mathematical description of photoelectric gain (see Figures 1а and 2)

through utilizing small dimensionless parameter (75)

Using linearized expressions for electron (12) and hole (13) components of photocurrent

where

Therefore, expressions (4), (5), (8), and (9) with boundary conditions (1) allow us to write

where

Actual distribution

Values of parameters

Equation (152), with boundary condition (1), and relation (153) allow us to obtain the formula for photoelectric gain:

where quasi-neutral ambipolar drift length of charge carriers

effective diffusion length of charge carriers

and effective reciprocal diffusion-drift lengths

Relations (18), (20)–(22), (24), (25), (73)–(75), (83), and (153)–(157) determine, in parametric form dependence

It can be shown that equation

Let’s explain dependence

Product

determines drift length and direction in electric field of concentrational perturbation – quasi-neutral cloud of positive and negative charges [2, 31], including bound at deep impurity (here

In this case, due to the fact that

From expression (154), it follows that

where

is effective ambipolar length (156) at

Value

From expression (159), it follows that function

Increase in effective ambipolar diffusion constant

when bias voltage is applied across sample

Threshold value

## 8. Summary

Capture rate of excess charge carriers increases with increasing concentration

It seems, authors of article [25] have reported first about the availability of minimum and portion of weak growth (up to 24 %) on experimental dependence of excess charge carriers’ lifetime on recombination center concentration, which increased because of bombarding sample by high-energy electrons. Many years later, increase in lifetime, presumably, caused by increasing

The main reason for giant splash of photoresponse in semiconductors with increasing recombination center concentration

This reason is also sufficient to provide increase, in order of magnitude and more, in efficiency of charge carriers’ photoexcitation with increasing

At and about point

However, increase in orders of magnitude in charge carriers’ lifetime with increasing

In reality, recombination on contact electrodes occurs always to more or less extent [5, 9]. Therefore, under normal conditions (

Saturation in

Strict solutions of problems concerning the quantity of photoexcited electrons

It may be that

At sweep-out effect on contact electrodes, splash of

As shown in [33], when recombination impurity

As shown in [35], significant growth of charge carriers’ lifetimes with increase in concentration of recombination impurity in certain range could be stimulated by uncontrolled (background) doping by deep impurities. Even two maximums can occur.

Above-mentioned regularities occur at arbitrarily low-level photoexcitation and they become the more evident, the wider the semiconductor bandgap.

The theory of giant splash of photoresponse in semiconductors at low-level illumination with increasing concentration of recombination centers could develop further through the generalization of boundary conditions on semiconductor surfaces and current contact electrodes, accounting for nonuniformity of charge carriers’ photoexcitation along the line of current flow and fluctuation processes. The study of nonstationary (frequency domain and transient) characteristics is of particular interest.

From physical essence of considered effects, it follows that similar effects can occur in other mediums with recombination of dissociative or ion-ion type, for example, in multicomponent plasma [36]. More details about topic are given in [37-46].