^{1,a)}, J. A. Rowlands

^{1}, S. D. Baranovskii

^{2}and Kenkichi Tanioka

^{3}

### Abstract

The review of avalanche multiplication experiments clearly confirms the existence of the impact ionization effect in this class of semiconductors. The semilogarithmic plot of the impact ionization coefficient versus the reciprocal field for holes in and electrons in and places the avalanche multiplication phenomena in amorphous semiconductors at much higher fields than those typically reported for crystalline semiconductors with comparable bandgaps. Furthermore, in contrast to well established concepts for crystalline semiconductors, the impact ionization coefficient in increases with increasing temperature. The McKenzie and Burt [S. McKenzie and M. G. Burt, J. Phys. C19, 1959 (1986)] version of Ridley’s lucky drift model [B. K. Ridley, J. Phys. C16, 3373 (1988)] has been applied to impact ionization coefficient versus field data for holes and electrons in and electrons in . We have extracted the electron impact ionization coefficient versus field ( vs ) data for from the multiplication versus and photocurrent versus data recently reported by M. Akiyama, M. Hanada, H. Takao, K. Sawada, and M. Ishida, Jpn. J. Appl. Phys.41, 2552 (2002). Provided that one accepts the basic assumption of the Ridley model that the momentum relaxation rate is faster than the energy relaxation rate, the model can satisfactorily account for impact ionization in amorphous semiconductors even with ionizing excitation across the bandgap,. If is the mean free path associated with momentum relaxing collisions and is the energy relaxation length associated with energy relaxing collisions, than the model requires . The application of the model with energy and field *independent* to leads to ionization threshold energies that are quite small, less than , and requires the possible but improbable ionization of localized states. By making energy and field dependent, we were able to obtain excellent fits to vs data for both holes and electrons in for both and . In the former case, one expects occupied localized states at to be ionized and in the second case, one expects excitation across the bandgap. We propose that ionization excitation to localized tail states very close to the transport band can explain the thermally activated since the release time for the observed activation energies is much shorter than the typical transit times at avalanche fields. For the case, leads to the following conclusions: (a) For holes, has negligibly little field dependence but increases with energy. At the ionization threshold energy . (b) For electrons, increases with energy and the field with at the ionization threshold and at impact ionization fields. For electron impact ionization in , the data can be readily interpreted in terms of near bandgapionization and a that decreases with increasing field, and having very little energy dependence. The energy relaxation length has opposite tendencies in and , which probably reflects the distinctly different types of behavior of hot carriers in the transport band in these two amorphous semiconductors.

I. INTRODUCTION: AVALANCHE MULTIPLICATION IN AMORPHOUS SEMICONDUCTORS

II. EXPERIMENTAL EVIDENCE

III. IMPACT IONIZATION BY LUCKY DRIFT

IV. APPLICATION OF THE LUCKY DRIFT IMPACT IONIZATION MODEL TO AMORPHOUS SEMICONDUCTORS

A. Holes in

B. Electrons in

C. Electrons in

V. CONCLUSIONS

ACKNOWLEDGMENTS

### Key Topics

- Ionization
- 87.0
- Avalanche photodiodes
- 46.0
- Amorphous semiconductors
- 29.0
- Image sensors
- 21.0
- Semiconductor device modeling
- 20.0

## Figures

Comparison of multiplication versus field for (holes) (see Ref. 8) and (electrons) (see Ref. 25).

Comparison of multiplication versus field for (holes) (see Ref. 8) and (electrons) (see Ref. 25).

The semilogarithmic plot of the dependence of impact ionization coefficient on the reciprocal field for not only and , but, for comparison, also for various crystalline semiconductors. for holes and for electrons. The axis is a base-10 logarithm of in which is in . Data for from Akiyama *et al.* (see Ref. 25); electrons and holes from Tsuji *et al.* (see Ref. 12); holes from Juska and Araluskas (see Ref. 9) from which was obtained by reanalyzing their multiplication data. Data for crystalline semiconductors are for , Logan and White (see Ref. 38); , Ghin *et al.* (see Ref. 39); , Plimmer *et al.* (see Ref. 40); , Cook *et al.* (see Ref. 41); , Lee *et al.* (see Ref. 42); (calculation only), Bulutay (see Ref. 43). Electrons in (Tsuji *et al.* ^{12}); electrons in (Akiyama *et al.* ^{25}); and holes in (Tsuji *et al.* ^{12}); holes in (Juska and Arlauskas^{9}).

The semilogarithmic plot of the dependence of impact ionization coefficient on the reciprocal field for not only and , but, for comparison, also for various crystalline semiconductors. for holes and for electrons. The axis is a base-10 logarithm of in which is in . Data for from Akiyama *et al.* (see Ref. 25); electrons and holes from Tsuji *et al.* (see Ref. 12); holes from Juska and Araluskas (see Ref. 9) from which was obtained by reanalyzing their multiplication data. Data for crystalline semiconductors are for , Logan and White (see Ref. 38); , Ghin *et al.* (see Ref. 39); , Plimmer *et al.* (see Ref. 40); , Cook *et al.* (see Ref. 41); , Lee *et al.* (see Ref. 42); (calculation only), Bulutay (see Ref. 43). Electrons in (Tsuji *et al.* ^{12}); electrons in (Akiyama *et al.* ^{25}); and holes in (Tsuji *et al.* ^{12}); holes in (Juska and Arlauskas^{9}).

The temperature dependence of the IIC for . The numbers next to the lines represent the activation energies from the slopes in eV. Data extracted from Tsuji *et al.* (see Ref. 12). Field is in : ; ; ;; ; ; . Filled in points are for electrons.

The temperature dependence of the IIC for . The numbers next to the lines represent the activation energies from the slopes in eV. Data extracted from Tsuji *et al.* (see Ref. 12). Field is in : ; ; ;; ; ; . Filled in points are for electrons.

Analysis of hole IIC versus data for . The best line represents Eq. (8) with a constant . (), (), and () curves represent typical analysis based on . (a) , , , , , or (a) , , , , . (b) , , , , . (c) , , , , .

Analysis of hole IIC versus data for . The best line represents Eq. (8) with a constant . (), (), and () curves represent typical analysis based on . (a) , , , , , or (a) , , , , . (b) , , , , . (c) , , , , .

Typical density of states for an amorphous semiconductor and proposed possible impact ionization processes (I and II).

Typical density of states for an amorphous semiconductor and proposed possible impact ionization processes (I and II).

Semilogarithmic plot of the hole IIC vs reciprocal field for , experimental data and typical curve fits. Examples of curve fits using an energy dependent analysis of hole IIC vs data for . Best curve fits based on . In all cases . , Experimental data; , (a) , , , ; , (b) , , , ; , (c) , , , ; , (d) , , , .

Semilogarithmic plot of the hole IIC vs reciprocal field for , experimental data and typical curve fits. Examples of curve fits using an energy dependent analysis of hole IIC vs data for . Best curve fits based on . In all cases . , Experimental data; , (a) , , , ; , (b) , , , ; , (c) , , , ; , (d) , , , .

Semilogarithmic plot of the hole IIC vs reciprocal field for , experimental data and two typical curve fits using two different . Energy dependent analyses of hole IIC vs data using . Curve fits are , solid curve, , , , , ; , dashed curve, , , , , .

Semilogarithmic plot of the hole IIC vs reciprocal field for , experimental data and two typical curve fits using two different . Energy dependent analyses of hole IIC vs data using . Curve fits are , solid curve, , , , , ; , dashed curve, , , , , .

Semilogarithmic plot of the electron IIC vs reciprocal field for , experimental data and examples of curve fits. , Experimental data; , (a) , field dependent , , , , , , ; , (b) , energy dependent , , , , , ; , (c) , energy and field dependent , , , , , , , , at ; , (d) , energy and field dependent , , , , , ; , at .

Semilogarithmic plot of the electron IIC vs reciprocal field for , experimental data and examples of curve fits. , Experimental data; , (a) , field dependent , , , , , , ; , (b) , energy dependent , , , , , ; , (c) , energy and field dependent , , , , , , , , at ; , (d) , energy and field dependent , , , , , ; , at .

Semilogarithmic plot of the electron IIC vs reciprocal field for , experimental data and typical examples of curve fits. , Experimental points (from : in Fig. 2); , best line (exponential dependence); , (a) , field dependent , , , , , ; , (b) , field dependent , , , , , , ; , (c) , field dependent , , , , , , .

Semilogarithmic plot of the electron IIC vs reciprocal field for , experimental data and typical examples of curve fits. , Experimental points (from : in Fig. 2); , best line (exponential dependence); , (a) , field dependent , , , , , ; , (b) , field dependent , , , , , , ; , (c) , field dependent , , , , , , .

## Tables

Hole impact ionization in . Summary of data analysis.

Hole impact ionization in . Summary of data analysis.

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