^{1,a)}

### Abstract

Modeling of the electrical large-signal response of granular -type semiconductors is carried out at following three different levels: (i) simple fully analytical model, (ii) semianalytical numerical model, and (iii) numerical device simulation. The electrical transients induced by both voltage and temperature changes are calculated. The analysis is based on the dynamic electrical model of the grain-boundary (GB) region, the drift-diffusion theory, and electronic trapping in the acceptor-type electronic interface states at the GBs. The electronic trapping is described using the standard rate equation. The models are verified by performing numerical device simulations using SILVACO ATLAS. The agreement between the proposed semianalytical model and ATLAS results is excellent during the whole transient and up to rather high electric fields. Compared to ATLAS, the calculations performed with the present semianalytical model are four orders of magnitude faster on a standard PC computer. The approximative analytical formulas describing the response are valid when the voltage and temperature changes are small. The semianalytical model is also fitted to reported experimental data obtained from dc and transient measurements of ZnO powder samples. The semianalytical model fits to the data well. The current in the GB region has following three components: potential-barrier limited current,charging and discharging current, and capacitive current. The results show that the large-signal transient responses of granular semiconductors are complex, as they vary highly in both duration and magnitude. During a transient the current can change many orders of magnitude. This is mainly caused by the change in the GB trap occupancy.

Fruitful discussions with Juha Sinkkonen, Pekka Kuivalainen, Marko Yli-Koski, Antti Haarahiltunen, and Charlotta Tuovinen and the financial support from the Academy of Finland are gratefully acknowledged.

I. INTRODUCTION

II. MODEL OF A GRANULAR -TYPE SEMICONDUCTOR

A. Dynamical electric model of GB

B. dcmodel of GB

C. Electronic trapping at GB

1. Normalization of the rate equation

2. Analytical approximation of step response

3. Additional fixed electronic interface states at GB

III. RESULTS

A. dc characteristics

B. Transient characteristics

1. Temperature change

2. Voltage change

3. Comparison of analytical and numerical results

4. Temperature and voltage dependence of time constant

C. Fitting of the present model to experimental data

IV. CONCLUSIONS

### Key Topics

- Semiconductor device modeling
- 32.0
- Electrons
- 29.0
- Electric currents
- 23.0
- Semiconductors
- 20.0
- Numerical modeling
- 19.0

## Figures

1D geometrical model structure of a granular semiconductor with identical GBs. Electric current flows in the direction of the axis.

1D geometrical model structure of a granular semiconductor with identical GBs. Electric current flows in the direction of the axis.

Electronic energy bands in the GB region of a -type semiconductor (a) in the thermodynamical equilibrium and (b) when a voltage is applied across the GB region. Trapping of a conduction electron in acceptor-type interface states (energy level ) at GB is illustrated in (a). is the bottom of the conduction band, the top of the valence band, and the Fermi level. and are lengths of the depletion regions in the GB region. Regions I and II are indicated.

Electronic energy bands in the GB region of a -type semiconductor (a) in the thermodynamical equilibrium and (b) when a voltage is applied across the GB region. Trapping of a conduction electron in acceptor-type interface states (energy level ) at GB is illustrated in (a). is the bottom of the conduction band, the top of the valence band, and the Fermi level. and are lengths of the depletion regions in the GB region. Regions I and II are indicated.

dc electric current density and dc voltage across GB region (inset) plotted as a function of the dc voltage applied across the sample, calculated at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I.

dc electric current density and dc voltage across GB region (inset) plotted as a function of the dc voltage applied across the sample, calculated at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I.

Figure Steady-state trap occupancy and electron density at GB (inset) plotted as a function of the dc voltage applied across the sample, calculated at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I.

Figure Steady-state trap occupancy and electron density at GB (inset) plotted as a function of the dc voltage applied across the sample, calculated at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I.

Evolution of (a) the total current density and (b) the trap occupancy in the GB region after linearly ramping up temperature between 0 and 0.1 ms from 300 K to various values (350–500 K). The data was calculated using the present semianalytical model and ATLAS with the applied voltage and the parameter values listed in Table I and .

Evolution of (a) the total current density and (b) the trap occupancy in the GB region after linearly ramping up temperature between 0 and 0.1 ms from 300 K to various values (350–500 K). The data was calculated using the present semianalytical model and ATLAS with the applied voltage and the parameter values listed in Table I and .

Evolution of (a) the total current density and (b) the trap occupancy in the GB region after ramping down temperature from various values (350–500 K) to 300 K in 0.1 ms. The data was calculated using the present semianalytical model and ATLAS with the applied voltage and the parameter values listed in Table I and .

Evolution of (a) the total current density and (b) the trap occupancy in the GB region after ramping down temperature from various values (350–500 K) to 300 K in 0.1 ms. The data was calculated using the present semianalytical model and ATLAS with the applied voltage and the parameter values listed in Table I and .

Dynamic behavior of (a) total electric current density in the GB region and (b) trap occupancy after applying a 0.2 V voltage up-ramp between 0 and 1 ns (on-transient) calculated at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I and various values of in the range .

Dynamic behavior of (a) total electric current density in the GB region and (b) trap occupancy after applying a 0.2 V voltage up-ramp between 0 and 1 ns (on-transient) calculated at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I and various values of in the range .

Dynamic behavior of total electric current density in the GB region after applying various voltage upramps between 0 and 1 ns (on-transient) calculated at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I. For the capture cross-sections the following values were used: (ATLAS), and (model).

Dynamic behavior of total electric current density in the GB region after applying various voltage upramps between 0 and 1 ns (on-transient) calculated at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I. For the capture cross-sections the following values were used: (ATLAS), and (model).

Dynamic behavior of trap occupancy after applying various voltage upramps between 0 and 1 ns (on-transient) calculated at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I. For the capture cross-sections the following values were used: (ATLAS), and (model).

Dynamic behavior of trap occupancy after applying various voltage upramps between 0 and 1 ns (on-transient) calculated at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I. For the capture cross-sections the following values were used: (ATLAS), and (model).

Electron density in the GB region calculated using ATLAS during the 1.6 V on-transient shown in Figs. 8 and 9. The GB is at .

Electron density in the GB region calculated using ATLAS during the 1.6 V on-transient shown in Figs. 8 and 9. The GB is at .

Dynamic behavior of total electric current density in the GB region after ramping down the voltage from various initial values to 0.1 V in (off-transient) calculated (a) in the ramping phase (logarithmic time scale) and (b) in the rest of the transient (linear time scale) at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I. For the capture cross-sections the following values were used: (ATLAS) and (model).

Dynamic behavior of total electric current density in the GB region after ramping down the voltage from various initial values to 0.1 V in (off-transient) calculated (a) in the ramping phase (logarithmic time scale) and (b) in the rest of the transient (linear time scale) at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I. For the capture cross-sections the following values were used: (ATLAS) and (model).

Dynamic behavior of trap occupancy after ramping down the voltage from various initial values to 0.1 V in (off-transient) calculated at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I. For the capture cross-sections the following values were used: (ATLAS), and (model).

Dynamic behavior of trap occupancy after ramping down the voltage from various initial values to 0.1 V in (off-transient) calculated at 300 K using ATLAS and the present semianalytical model with the parameter values listed in Table I. For the capture cross-sections the following values were used: (ATLAS), and (model).

Dynamic behavior of the trap occupancy after applying various voltage steps in the range 20–80 mV calculated at 300 K using the present semianalytical model and the analytical formulas [Eqs. (37)–(39)] with the parameter values listed in Table I and .

Dynamic behavior of the trap occupancy after applying various voltage steps in the range 20–80 mV calculated at 300 K using the present semianalytical model and the analytical formulas [Eqs. (37)–(39)] with the parameter values listed in Table I and .

Time constant of the trap-occupancy transient plotted as a function of temperature with various values of the trap occupancy calculated at 0 V using Eq. (39) with and the parameter values listed in Table I and . The squares indicate the temperature which corresponds to the steady-state value of . The arrows describe the evolution of during transients with increasing (, ) and decreasing temperature (, ), respectively.

Time constant of the trap-occupancy transient plotted as a function of temperature with various values of the trap occupancy calculated at 0 V using Eq. (39) with and the parameter values listed in Table I and . The squares indicate the temperature which corresponds to the steady-state value of . The arrows describe the evolution of during transients with increasing (, ) and decreasing temperature (, ), respectively.

Time constant of the trap-occupancy transient plotted as a function of the voltage applied across the GB region with various values of the trap occupancy calculated at 300 K using Eq. (39) with and the parameter values listed in Table I and . The squares indicate the value of which corresponds to the steady-state value of . The arrows describe the evolution of during an on (, ) and off-transients (, ).

Time constant of the trap-occupancy transient plotted as a function of the voltage applied across the GB region with various values of the trap occupancy calculated at 300 K using Eq. (39) with and the parameter values listed in Table I and . The squares indicate the value of which corresponds to the steady-state value of . The arrows describe the evolution of during an on (, ) and off-transients (, ).

(a) Steady-state trap occupancy and (b) time constant of the trap-occupancy transient plotted as a function of the dc voltage applied across the GB region at various temperatures calculated using (a) the present semianalytical model and (b) Eq. (39) with , , and the parameter values listed in Table I and . The curves (b) were calculated using the values of in (a).

(a) Steady-state trap occupancy and (b) time constant of the trap-occupancy transient plotted as a function of the dc voltage applied across the GB region at various temperatures calculated using (a) the present semianalytical model and (b) Eq. (39) with , , and the parameter values listed in Table I and . The curves (b) were calculated using the values of in (a).

(a) Measured (Ref. 15) and calculated dc electric current density and (b) the trap occupancy plotted as a function of the dc voltage applied across the GB region. The data was calculated using the present semianalytical model with , , , , and .

(a) Measured (Ref. 15) and calculated dc electric current density and (b) the trap occupancy plotted as a function of the dc voltage applied across the GB region. The data was calculated using the present semianalytical model with , , , , and .

(a) Measured (Ref. 15) and calculated total electric current density , (b) the voltage applied across the GB region, and (c) the trap occupancy plotted as a function of time. The density of the calculated capacitive current is shown by the dotted curve. The data was calculated using the present semianalytical model with , , , , , , and .

(a) Measured (Ref. 15) and calculated total electric current density , (b) the voltage applied across the GB region, and (c) the trap occupancy plotted as a function of time. The density of the calculated capacitive current is shown by the dotted curve. The data was calculated using the present semianalytical model with , , , , , , and .

## Tables

Parameters used in the calculations.

Parameters used in the calculations.

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