## 1. Introduction

Minimal value of dark current in reverse biased ^{5} V/cm then avalanche multiplication of charge carriers occurs in ^{5} V/cm or occurs in narrow-gap layer (Osipov & Kholodnov, 1987), (Osipov &, Kholodnov, 1989). Value of electric field required to initialize avalanche multiplication of charge carriers can even exceed ^{5} V/cm. But tunnel current density ^{-6} A/cm^{2}. However, diagram does not take into account the fact that tunnel current in wide-gap multiplication layer can be much greater than in narrow-gap absorber (Osipov & Kholodnov, 1989). Therefore, total tunnel current can exceed thermal generation current.

In present chapter is done systematic analysis of interband tunnel current in avalanche heterophotodiode (AHPD) and its dependence on dopants concentrations _{} 1.35 eV and ^{8} сm^{-3} and ^{11} сm^{-3}; relative dielectric constants ^{4} сm^{-1}; specific effective masses

## 2. Formulation of the problem: Basic relations

Let’s consider

can be determined, in principal, dependences of multiplication factors

It is remarkable that responsivity

where

where

Where

## 3. Avalanche multiplication factors of charge carriers in p-n structures

### 3.1. Preliminary remarks: Avalanche breakdown field

For successful development of semiconductor devices using effects of impact ionization and avalanche multiplication of charge carriers is necessary to know dependences of avalanche multiplication factors

where

Gap ^{-3}, properly. As follows from Poisson equation, voltage value given by (10) corresponds to value of electric field at metallurgical boundary

where at

Formulas (10) and (11) cannot be used for reliable estimates of

where

It seen from expression (14) that at

### 3.2. Avalanche breakdown field

Consider

does not lead to that and other contradictions, From (17) follows that:

To determine dependences

then avalanche breakdown is controlled by

If, however,

From (17)-(21) we obtain that

Formulas (15) and (16) follow from expressions (18), (19) and requirement (23)

which means smoothness of field dependence

where

And values for

### 3.3. Avalanche breakdown voltage

It follows from expressions (6)-(9) and (14)-(16) that breakdown voltage

i.e. when diode’s base is not punch-through and

i.e. when diode base is punch-through. In expression (28)

Value of parameter

breakdown voltage of diode depends on

where

In expressions (30)-(32) ^{-3} and μm, respectively.

Avalanche breakdown voltage of double heterostructure discussed in Section 4 (Fig. 1) depends on relations between fundamental parameters of materials of

### 3.4. About correlation between impact ionization coefficients of electrons and holes

One of main goals of many experimental and theoretical studies of impact ionization phenomenon in semiconductors is to determine impact ionization coefficients of electrons

Where:

To derive relation (33) let’s consider thin

where

And, under these conditions, variation of electric field

In this case as it follows from (34) and (35)

And avalanche breakdown electric field for thin

In expressions (37) and (38) and below in this Section 3.4 concentration is measured in cm^{-3}, energy – in eV, length – in µm, electric field – in V/cm. On the other hand condition of avalanche breakdown of

takes the form

That means the same probability of impact ionization in any point of diode’s base. And relation (33) follows from expressions (38) and (40). Let’s estimate applicable electric field interval for this relation. Expression (38) will be valid when inequality (41) is satisfied both for electrons and for holes

where

From (38) and (41) we find desired interval of electric field:

Interval of electric field (43) is most often realized in experimental studies (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev, 1987). Ratio

where

### 3.5. Miller’s relation for multiplication factors of charge carriers in p-n structures

Usual way to calculate dependences of avalanche multiplication factors of charge carriers

Rewrite (33) in the form

In (45) and below in this Section 3.5 is accepted (unless otherwise specified) the following, convenient for this study, system of symbols and units (Sze, 1981): gap ^{-1}, electron charge ^{-3}; concentration gradient ^{-4}; width of SCR ^{-1}. In this section, analytical dependences

where

#### 3.5.1. Stepwise p − n junction

In this case from relations (1), (2) and (45) and Poisson equation (SI units)

follow that

where

when

when

and expression (52) was derived by standard method of integrating fast-changing functions (Zeldovich & Myshkis, 1972).

#### 3.5.2. Gradual (linear) p − n junction

In this case Poisson equation can be written as (SI units):

where

and therefore

In derivation of relations (55) and (56) was used known expression for voltage distribution on linear

Formula (56) differs from known formula Sze-Gibbons for avalanche breakdown voltage of linear

#### 3.5.3. Thin p + − n ( p ) − n + structure (p − i − n )

When thickness of high-resistivity region (base) of considered structure

where

we find that

where

In deriving expressions (60)-(63), multiplication of charge carriers in

we find that in zeroth-order approximation with respect to this parameter

In the case of very thin base when

electric field varies so slightly along base that probability of impact ionization is practically the same in any point of it ((Osipov & Kholodnov, 1987), (Kholodnov 1988-1), Sections 3.2 and 3.4). As a result

where

### 3.6. Discussion of the results. Comparison with computed and experimental data

#### 3.6.1. To formulas for avalanche breakdown electric field and voltage for abrupt *p*^{+} - n junction

^{+}- n

In sections 3.1-3.3 were derived approximate universal formulas for avalanche breakdown field

then formulas (25) and (26) are not true. To estimate breakdown field

If assume that in

Therefore curve 1 in Fig. 5 starts to fall significantly below unity at larger values ^{-3}. As shown on lower inset in Fig. 7, dependence

and expressed as

when

#### 3.6.2. To сorrelation between values of impact ionization coefficients of electrons and holes

In Section 3.4 is shown that there is reason to suppose existence of some correlation between values of impact ionization coefficients of electrons

#### 3.6.3. To Miller’s relation

From (48), (55) and (67) follow that, exponents in Miller’s relation (46) for multiplication factors of electrons and holes are given by

where

Then for these situations

Graphs in Fig. 4 allow comparing numerical values of exponents

where

In Fig. 15a are shown dependences

which for many semiconductors is of the order of 10^{17} cm^{-3}. At such high concentrations, as it follows from Section 3.4 and (Kholodnov, 1988-1) and relations (1) and (2), for stepwise

moreover

For comparison, in Fig. 15b are presented dependences of ^{15} cm^{-3}. At this doping avalanche breakdown in

V/V
_{BD} |
M
_{p}
| |

Experiment(Shotov, 1958) |
Theory
| |

0.65 | 1.35 | 1.30 |

0.70 | 1.50 | 1.44 |

0.75 | 1.75 | 1.65 |

0.80 | 2.10 | 1.98 |

0.85 | 2.65 | 2.55 |

0.90 | 3.70 | 3.71 |

0.95 | 7.00 | 7.30 |

Finally, it is interesting to analyze application of expressions (45) and (76) to describe avalanche process in

## 4. Tunnel currents in avalanche heterophotodiodes

### 4.1. Calculation of tunnel currents in approximation of quasi-uniform electric field and conditions of its applicability

In act of interband tunneling electron from valence band overcomes potential barrier ABC (Fig. 16a). The length of tunneling

If variation of electric field within length of tunneling

When

then true ABC barrier coincides to high degree of accuracy with triangle ABC′ to which corresponds uniform field

It follows from (83) and Poisson equation that inequalities (84) are satisfied if

at that

As shown below, due to large values of field ^{-3} and even high.

Under these conditions specific rates of charge carriers’ tunnel generation

obtained in (Kane, 1960) (see also (Burstein & Lundqvist, 1969)) for

Here

where characteristic dimensions of areas of charge carriers’ tunnel generation in layers I and II

Equation (89) is valid under conditions

These conditions mean the following. If inequalities (91) for

Fulfillment of conditions (92) is necessary at punch-through of proper layers of structure for neglecting tunneling through its hetero-interfaces which is not accounted for by formula (89). We show further, that at avalanche breakdown, inequalities (91) and (92) are valid for almost all real values of material parameters, concentrations

Breakdown fields

where

(

For many semiconductors including

From expressions (93) and (94) when relations (95) are satisfied we find the following.

then ratio

then under avalanche breakdown of proper layer of structure ratio

then length of tunneling

In expressions (96)-(99)

### 4.2. Features of interband tunnel currents in p + − n heterostructures under avalanche breakdown

Analysis of expression (89) under avalanche breakdown of

#### 4.2.1. Independent doping levels of wide-gap and narrow-gap n type layers

**I.**

In this case, at any concentration

As follows from (6)-(9), (89) and (93), if

which is fulfilled with large margin at

then tunnel current is almost independent on

If

One can see that at minimum of tunnel current, as a rule, the following inequality is valid

where

When (103) is fulfilled then

Therefore, as it follows from (6)-(9), (89), (90) and (93), concentration

where

Expression (105) is valid when inequality

As a result we find

(106) |

where

It is shown from (105) and (106) that

When

(108) |

Where

From (94), (105) and (108) follow that

Where

drops sharply, same as ^{-3}.

When concentrations

then in minimum of

Nature of above dependence

**II.**

Condition (100) is not satisfied. For example, for combination of layers

where ^{-3} and μm, respectively. Under this condition, when

avalanche breakdown is controlled by

then at

then after initial plateau

then for small enough thicknesses **I** increase **II**, increase

#### 4.2.2. Equal doping levels of wide-gap and narrow-gap n type layers

Under this condition density of tunnel current is given by expression (89), where

i.

At this relation of parameters avalanche breakdown is controlled by **I**, and is caused by the same physical grounds. The only difference is that when **I**.

This occurs because at given value

ii. Condition (117) is not satisfied.

Then, till

## 5. Basic performance of avalanche heterophotodiode

### 5.1. Responsivity

In punch-through conditions of absorber

Therefore, (Fig. 20a)

where reflection coefficient of light from illuminated surface

quantum efficiency

(120) |

where

Particularly,

Dependence

### 5.2. Noise

It was noted above that in order to achieve the best performance of SAM-APD special doping profile is formed in heterostructure which facilitates penetration of photogenerated charge carriers with higher impact ionization coefficient into multiplication layer. In this case, at given voltage bias on heterostructure, current responsivity

In

we find that value of concentration

is given by

Formulas (127) and (128) are valid when

where concentration and thicknesses, as in (127) and (128), are measured in cm^{-3} and µm, respectively.

If inequality (129) is not satisfied, then values

Allowable intervals of concentrations and thicknesses of heterostructure layers are shown in Fig. 21. As can be seen from Fig. 20b, even, when ^{1/2} corresponds to ^{2}, and value ^{1/2} corresponds to ^{2}.

### 5.3. Operational speed

Minimal possible time-of-response of this class of devices

is determined by time-of-flight of charge carriers through multiplication layer

As is evident from Fig. 20b, in

## 6. Analytical model of avalanche photodiodes operation in Geiger mode

We consider possibility to describe transient phenomena in

In present structure charge is mainly concentrated in thin near border

where

Substituting volume charge density from Poisson equation in continuity equation for

where

dielectric constant of vacuum,

Relation (135) generalizes well-known theorem of Rameau (Spinelli & Lacaita, 1997), it takes into account key feature of Geiger mode – variation over time of voltage across DL, and it is valid for any distribution profile of dopant. In our formulation of the problem (in

Because plates are very thin, then generation and recombination in them can be neglected. Now by integrating same equations with respect to thickness of plates, we find that in approximation of absence of minority carriers in

Strictly speaking, equations (139) are valid when

Relations (133)-(139) allow obtaining equations

with initial conditions

where

At delta-shaped time-evolving illumination

where

If we take

and problem is solved in quadratures. To solve in elementary functions let’s approximate exact dependence

Suppose, for simplicity

where

## 7. Conclusions

The above analysis shows that to create high performance SAM-APD (in particular, based on widely used ^{1/2} and current responsivity 10.3 A/W. Analysis shows that operational speed can be slightly increased by means of inhomogeneous doping of wide-gap multiplication layer. To ensure operational speed in picosecond range it is necessary to use as multiplication layer semiconductor layer with low tunnel current and impact ionization coefficients of electrons and holes much different from each other, for example, indirect-gap semiconductor silicon. As has long been known maximal operational speed is achieved by APD if light is absorbed in space-charge region. In this case, as it was shown in Section 6, when bias voltage