^{1,a)}, Ute Ebert

^{2}, Andrey Minarsky

^{3}and Igor Grekhov

^{4}

### Abstract

We present an analytical theory for impact ionization fronts in reversely biased structures. The front propagates into a depleted base with a velocity that exceeds the saturated drift velocity. The front passage generates a dense electron-hole plasma and in this way switches the structure from low to high conductivity. For a planar front we determine the concentration of the generatedplasma, the maximum electric field, the front width, and the voltage over the base as functions of front velocity and doping of the base. The theory takes into account that drift velocities and impact ionization coefficients differ between electrons and holes, and it makes quantitative predictions for any semiconductor material possible.

We are grateful to P. Ivanov for critical reading of the manuscript and helpful discussions. This work was supported by the Program of Russian Academy of Sciences, “Power semiconductor electronics and pulse technologies.” P.R. thanks A. Alekseev for his hospitality at the University of Geneva and acknowledges support from the Swiss National Science Foundation.

I. INTRODUCTION

II. MODEL

A. Basic equations in drift-diffusion approximation

B. Self-similar propagation of the ionization front

C. Relation between the current density and the front velocity

D. Final set of equations

E. Special and limiting cases

III. GENERAL PROPERTIES OF THE STATIONARY FRONT PROPAGATION

A. Ordering of scales

B. Equations in the high-field region

C. Carrier concentration just behind the ionization zone

D. Maximum electric field

E. Width of the screening region

F. Transition from high-field to low-field region

G. Parameters of the plasma region

H. Voltage over the structure

IV. ULTRAFAST FRONTS

V. NONSTATIONARY PROPAGATION

A. Adiabatic condition

B. Coupling to the external circuit

VI. SUMMARY

### Key Topics

- Ionization
- 73.0
- Electric fields
- 53.0
- Electrons
- 18.0
- Current density
- 14.0
- Doping
- 12.0

## Figures

Sketch of the electric field and total concentration of free carriers concentrations (lower panel) in the structure during the passage of the ionization front. The field corresponds to the transition from linear low-field transport to saturated drift velocities. Coordinates and correspond to stationary and comoving frames, respectively. Note relations and between the initial concentration in the depleted region, doping and plasma concentration . The relation generally holds only for and can be broken for .

Sketch of the electric field and total concentration of free carriers concentrations (lower panel) in the structure during the passage of the ionization front. The field corresponds to the transition from linear low-field transport to saturated drift velocities. Coordinates and correspond to stationary and comoving frames, respectively. Note relations and between the initial concentration in the depleted region, doping and plasma concentration . The relation generally holds only for and can be broken for .

Dependence of total carrier concentration on electric field in the traveling ionization front. See notations and comments to Fig. 1. Path A–B–C–D corresponds to piecewise linear approximation of the field profile shown in Fig. 10.

Dependence of total carrier concentration on electric field in the traveling ionization front. See notations and comments to Fig. 1. Path A–B–C–D corresponds to piecewise linear approximation of the field profile shown in Fig. 10.

Concentration just behind the ionization zone as a function of for different values of . Thick solid lines 1, 2, 3 correspond to symmetric case . Dotted lines 1d, 2d, 3d and dashed lines 1e, 2e, 3e correspond to the two limiting asymmetric cases [immobile holes, case (d)] and [immobile electrons, case(e)], respectively. Curves of first, second, and third series correspond to , , and , respectively. Thin solid line 4 shows concentration at the point of maximum electric field for the symmetric case .

Concentration just behind the ionization zone as a function of for different values of . Thick solid lines 1, 2, 3 correspond to symmetric case . Dotted lines 1d, 2d, 3d and dashed lines 1e, 2e, 3e correspond to the two limiting asymmetric cases [immobile holes, case (d)] and [immobile electrons, case(e)], respectively. Curves of first, second, and third series correspond to , , and , respectively. Thin solid line 4 shows concentration at the point of maximum electric field for the symmetric case .

Maximum electric field as a function of according to Eq. (36). Both and are normalized by . Note that when .

Maximum electric field as a function of according to Eq. (36). Both and are normalized by . Note that when .

Maximum electric field as a function of for different values of according to Eq. (36). Thick solid curves 1a, 2a, 3a correspond to the symmetric case (a) , [Eq. (37)]. Thin solid curves 1b, 2b, 3b correspond to impact ionization by electrons , [case (b), Eq. (38)]; dashed curves 1c, 2c, 3c correspond to impact ionization by holes , [case (c), Eq. (38)]. Curves of first, second, and third series correspond to , , and , respectively. The parameter corresponds to the doping level in Si.

Maximum electric field as a function of for different values of according to Eq. (36). Thick solid curves 1a, 2a, 3a correspond to the symmetric case (a) , [Eq. (37)]. Thin solid curves 1b, 2b, 3b correspond to impact ionization by electrons , [case (b), Eq. (38)]; dashed curves 1c, 2c, 3c correspond to impact ionization by holes , [case (c), Eq. (38)]. Curves of first, second, and third series correspond to , , and , respectively. The parameter corresponds to the doping level in Si.

Maximum electric field as a function of for different values of according to Eq. (36). Thick solid lines 1a, 2a, 3a correspond to the symmetric case , [case (a), Eq. (37)]. Thin solid lines 1b, 2b, 3b correspond to impact ionization by electrons , [case (b), Eq. (38)]. Curves of first, second, and third series correspond to , , and , respectively; . Inset shows the Townsends’s dependence for impact ionization coefficient .

Maximum electric field as a function of for different values of according to Eq. (36). Thick solid lines 1a, 2a, 3a correspond to the symmetric case , [case (a), Eq. (37)]. Thin solid lines 1b, 2b, 3b correspond to impact ionization by electrons , [case (b), Eq. (38)]. Curves of first, second, and third series correspond to , , and , respectively; . Inset shows the Townsends’s dependence for impact ionization coefficient .

Slope of electric field in the screening region normalized by the slope in the depleted region as a function of for different values of . Solid lines 1, 2, 3 correspond to the symmetric case . Dotted lines 1d, 2d, 3d and dashed lines 1e, 2e, 3e correspond to the two limiting asymmetric cases [immobile holes, case (d)] and [immobile electrons, case (e)], respectively. Curves of first, second, and third series correspond to , , and , respectively.

Slope of electric field in the screening region normalized by the slope in the depleted region as a function of for different values of . Solid lines 1, 2, 3 correspond to the symmetric case . Dotted lines 1d, 2d, 3d and dashed lines 1e, 2e, 3e correspond to the two limiting asymmetric cases [immobile holes, case (d)] and [immobile electrons, case (e)], respectively. Curves of first, second, and third series correspond to , , and , respectively.

Concentration of electron-hole plasma generated by the front passage as a function of front velocity . In panel (a) the dependence is shown for different values of . Solid curves 1, 2, 3 correspond to the case of symmetric transport , (e.g., , ). Dotted lines 1d, 2d, 3d and dashed lines 1e, 2e, 3e correspond to the limiting cases of immobile holes , [case(d)] and immobile electrons , [case (e)], and are calculated for the same values of . Curves of first, second, and third series correspond to , , and , respectively. In panel (b) the dependence is shown for different values of and and fixed value . Solid lines from 1 to 7 correspond to , , 0, 0.5, 0.8, 0.9, and 1.0, respectively. Associated dotted and dashed lines in panel (b) correspond to and , respectively, and the same value of as for the respective solid lines.

Concentration of electron-hole plasma generated by the front passage as a function of front velocity . In panel (a) the dependence is shown for different values of . Solid curves 1, 2, 3 correspond to the case of symmetric transport , (e.g., , ). Dotted lines 1d, 2d, 3d and dashed lines 1e, 2e, 3e correspond to the limiting cases of immobile holes , [case(d)] and immobile electrons , [case (e)], and are calculated for the same values of . Curves of first, second, and third series correspond to , , and , respectively. In panel (b) the dependence is shown for different values of and and fixed value . Solid lines from 1 to 7 correspond to , , 0, 0.5, 0.8, 0.9, and 1.0, respectively. Associated dotted and dashed lines in panel (b) correspond to and , respectively, and the same value of as for the respective solid lines.

Electric field in the electron-hole plasma generated by the front passage as a function of front velocity . In panel (a) the dependence is shown for different values of . Solid curves 1, 2, 3 correspond to case of symmetric transport , (e.g., , ). Dotted lines 1d, 2d, 3d and dashed lines 1e, 2e, 3e correspond to the limiting cases of immobile holes , [case(d)] and immobile electrons , [case (e)], respectively. Curves of first, second, and third series correspond to , , and , respectively. In panel (b) the dependence is shown for different values of and and fixed value . Solid lines from 1 to 7 correspond to , , 0, 0.5, 0.8, 0.9, and 1.0, respectively. Associated dotted and dashed lines correspond to and , respectively, and the same value of as for the respective solid lines.

Electric field in the electron-hole plasma generated by the front passage as a function of front velocity . In panel (a) the dependence is shown for different values of . Solid curves 1, 2, 3 correspond to case of symmetric transport , (e.g., , ). Dotted lines 1d, 2d, 3d and dashed lines 1e, 2e, 3e correspond to the limiting cases of immobile holes , [case(d)] and immobile electrons , [case (e)], respectively. Curves of first, second, and third series correspond to , , and , respectively. In panel (b) the dependence is shown for different values of and and fixed value . Solid lines from 1 to 7 correspond to , , 0, 0.5, 0.8, 0.9, and 1.0, respectively. Associated dotted and dashed lines correspond to and , respectively, and the same value of as for the respective solid lines.

Piecewise linear approximation of the field profile used to calculate the voltage across the base (Sec. III H). The respective dependence is shown by dashed line A–B–C–D in Fig. 2.

Piecewise linear approximation of the field profile used to calculate the voltage across the base (Sec. III H). The respective dependence is shown by dashed line A–B–C–D in Fig. 2.

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