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^{1,a)}, Valentin Yuferev

^{2,b)}, Vassil Palankovski

^{3,c)}, Duu-Sheng Ong

^{4,d)}and Juha Kostamovaara

^{1,e)}

### Abstract

Direct measurement of the electron velocity at an extreme electric field is problematic due to impact ionization. The dependence obtained by a Monte Carlo method can be verified, however, by comparing simulated and experimental data on superfast switching in a GaAs bipolar transistor structure, in which the switching transient is very sensitive to this dependence at high electric fields (up to ). Such a comparison allows the conclusion to be made that the change from negative to positive differential mobility predicted earlier at should not happen until the electric field exceeds .

This work was generally supported by the Academy of Finland, while Vassil Palankovski received support from the Austrian Science Funds (FWF) START Project Y247-N13.

### Key Topics

- Electric fields
- 14.0
- Ionization
- 10.0
- Monte Carlo methods
- 9.0
- Conduction bands
- 6.0
- Dielectric breakdown
- 4.0

## Figures

Dependence of electron velocity on the electric field in GaAs. The scattered graphs 1–4 were obtained using Monte Carlo simulations for different doping levels : , , , . The solid lines (1–4) show the analytical dependences used in drift-diffusion simulations of the switching transient. The analytical formula was of a similar structure to that used in Ref. 13, but was modified to account for the doping dependence of the electron velocity simulated here:(1)where the exponential decay in electron velocity at high fields is determined by the parameters: ; ; ; and . The critical field (V/cm) and mobiligy values, which are dependent on the concentration , are selected as follows: : (4000, 7200), (5200, 5000), (6300, 3000), and (7900, 1400). The experimental data (Refs. 4–6) for the range are shown for the sake of comparison (scatter graph). The gray-colored curve A presents the change from negative to positive differential mobility at predicted in Ref. 11, and this dependence was accounted for in the corresponding GaAs transistor simulations by adding the term , ( and ) to formula (1). Curves B and C represent speculative situations in which the mobility sign changes at 430 and , respectively (see corresponding transistor simulations below). A difference between the simulated electron velocity and the analytical fit at is of minor importance since field values in this range are not present within or between the high-field domains. The gray-colored dashed curve 5 represents the FBMC simulation results obtained here when taking account of the valley.

Dependence of electron velocity on the electric field in GaAs. The scattered graphs 1–4 were obtained using Monte Carlo simulations for different doping levels : , , , . The solid lines (1–4) show the analytical dependences used in drift-diffusion simulations of the switching transient. The analytical formula was of a similar structure to that used in Ref. 13, but was modified to account for the doping dependence of the electron velocity simulated here:(1)where the exponential decay in electron velocity at high fields is determined by the parameters: ; ; ; and . The critical field (V/cm) and mobiligy values, which are dependent on the concentration , are selected as follows: : (4000, 7200), (5200, 5000), (6300, 3000), and (7900, 1400). The experimental data (Refs. 4–6) for the range are shown for the sake of comparison (scatter graph). The gray-colored curve A presents the change from negative to positive differential mobility at predicted in Ref. 11, and this dependence was accounted for in the corresponding GaAs transistor simulations by adding the term , ( and ) to formula (1). Curves B and C represent speculative situations in which the mobility sign changes at 430 and , respectively (see corresponding transistor simulations below). A difference between the simulated electron velocity and the analytical fit at is of minor importance since field values in this range are not present within or between the high-field domains. The gray-colored dashed curve 5 represents the FBMC simulation results obtained here when taking account of the valley.

Measured (curve 1) and simulated (curves 2, A, B, and C) collector voltages across a GaAs BJT during switching. The simulation results presented in curve 2 were obtained for the dependence represented by curves 1–4 in Fig. 1. Curves A, B, and C correspond to the curves in Fig. 1 marked with the same letters. Curve 3 presents the collector voltage simulated with the dependence , as shown by curve 5 in Fig. 1 (FBMC results obtained in this work).

Measured (curve 1) and simulated (curves 2, A, B, and C) collector voltages across a GaAs BJT during switching. The simulation results presented in curve 2 were obtained for the dependence represented by curves 1–4 in Fig. 1. Curves A, B, and C correspond to the curves in Fig. 1 marked with the same letters. Curve 3 presents the collector voltage simulated with the dependence , as shown by curve 5 in Fig. 1 (FBMC results obtained in this work).

(a) Electric field profiles across the -collector layer, simulated using two dependences : curve 1 corresponds to the NDM dependences 1–4 in Fig. 1 and curve 2 to an approximation with positive differential mobility (PDM) shown by curve A in Fig. 1. The profiles 1 and 2 correspond to instants (, in Fig. 2) when the collector voltage . Profiles with higher spatial resolution typical of a single domain are shown in (b) and (c) under the following conditions: the domains shown in (b) are simulated with NDM approximation (curves 1–4 in Fig. 1) and three domains correspond to instants when the collector voltages are 240, 135, and , respectively; the domains shown in (c) are simulated with PDM approximation (curve A in Fig. 1).

(a) Electric field profiles across the -collector layer, simulated using two dependences : curve 1 corresponds to the NDM dependences 1–4 in Fig. 1 and curve 2 to an approximation with positive differential mobility (PDM) shown by curve A in Fig. 1. The profiles 1 and 2 correspond to instants (, in Fig. 2) when the collector voltage . Profiles with higher spatial resolution typical of a single domain are shown in (b) and (c) under the following conditions: the domains shown in (b) are simulated with NDM approximation (curves 1–4 in Fig. 1) and three domains correspond to instants when the collector voltages are 240, 135, and , respectively; the domains shown in (c) are simulated with PDM approximation (curve A in Fig. 1).

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