— DEV — I–V characteristics of n-doped GaN single layer

Header

Input Files:

• IV_GaN_n_doped_1D_nnp.in

• IV_GaN_n_doped_2D_nnp.in

• IV_GaN_n_doped_3D_nnp.in

Scope of the tutorial:

• currents

• wurtzite

Main adjustable parameters in the input file:

• parameter

Relevant output files:

• IV_characteristics.dat

Introduction

This tutorial shows the accuracy of drifft-diffusion model implemented in nextnano++ on a simple example: a single layer of an n-doped GaN. We compare the I–V characteristics obtained by nextnano++ with analytical solutions.

IV characteristics of an n-doped GaN single layer

The conductivity $\sigma$ and the resistivity $\rho$ of an n-type doped GaN sample can be calculated analytically, following formulas:

$$\begin{array}{r}\begin{array}{c}\sigma =q{\mu }_{n}n,\\ \rho =d/\sigma ,\end{array}\end{array}$$

where $q$ is electron charge, $n$ is concentration of electron carriers, ${\mu }_n$ is mobility of electrons, and $d$ is thickness of the material.

This is a good check for the results obtained with nextnano++ simulations. The thickness of the GaN layer is $d=100\mathrm{nm}$.

The structure we are dealing with consists of bulk GaN that is sandwiched between two contacts. The whole structure has the following dimensions:

material	width ($\mathrm{nm}$)	doping
contact	$10$
n-GaN	$100$	$1\times {10}^{18}{\mathrm{cm}}^{-3}$
contact	$10$

As you see, the GaN is n-type doped with a donor concentration of $N_D=1\times {10}^{18}{\mathrm{cm}}^{-3}$. The energy level is chosen to be $0.01507\mathrm{eV}$ below the conduction band edge.

impurities{
    donor{ name = "Si_donor" degeneracy = 2 energy = 0.01507 }
}

This leads to the electron density of $5.2846\times {10}^{17}{\mathrm{cm}}^{-3}$. This is also equivalent to the concentration of the ionized donors. The result obtained by another commercial software is $5.355\times {10}^{17}{\mathrm{cm}}^{-3}$.

contacts{
    ohmic{ name = "left_contact" bias = 0.0 }
    ohmic{
        name = "right_contact"
        !WHEN $biassweep bias = [ $biasstart, $biasend ]
        !WHEN $biassweep steps = $biassteps
        !WHEN $nosweep   bias = $biasstart
    }
}

If $biassweep = 1, sweeping bias takes place. Otherwise, if $biassweep = 0 and $nosweep (= 1 - $biassweep) = 1, sweeping bias is not applied. Since the bias is swept from $0.00\mathrm{V}$ to $0.10\mathrm{V}$, $biasstart is set to 0.0 and $biasend is set to 0.1. In addition, $biassteps is equal to 10.

We take the GaN mobility to be constant: ${\mu }_n=100{\mathrm{cm}}^2/\mathrm{Vs}$. The mobility model that is applied is called constant and described as below.

currents{
    mobility_model = constant
    recombination_model{
        SRH = no
        Auger = no
        radiative = no
    }
    output_currents{ }
}

We sweep the voltage at the right contact and calculate the current density for $0.00\mathrm{V}$, $0.01\mathrm{V}$, $0.02\mathrm{V}$, …, $0.10\mathrm{V}$ (10 steps).

Results

1D

The current-voltage (IV) characteristic can be found in the following file: IV_characteristics.dat. Figure 2.4.55 shows the IV curve obtained by nextnano++.

Figure 2.4.55 IV curve of an n-doped GaN single layer.

The figure shows that the GaN layer is an ohmic resistor. From Figure 2.4.55, you can obtain a resistivity of the n-GaN layer of $1.1819\times {10}^{-6}\mathrm{\Omega }{\mathrm{cm}}^2$. Another commercial software results in $1.43\times {10}^{-6}\mathrm{\Omega }{\mathrm{cm}}^2$.

A good check is the analytic formula given above. From this, you can obtain:

$${\sigma }_n=e{\mu }_{n}n=1.6022\times {10}^{-19}\mathrm{As}\times 100{\mathrm{cm}}^2/\mathrm{Vs}\times 5.2846\times {10}^{17}{\mathrm{cm}}^{-3}=8.4670\mathrm{A}/\mathrm{Vcm}$$

$$\rho =d/\sigma =100\mathrm{nm}/(8.46700\mathrm{A}/\mathrm{Vcm})=1.1811\times {10}^{-6}\mathrm{\Omega }{\mathrm{cm}}^2$$

Another analytical result with the other commercial software is $1.168\times {10}^{-6}\mathrm{\Omega }{\mathrm{cm}}^2$.

Thus, you can see that the nextnano++ result agrees better with the analytical result than the result by the other commercial software.

2D

Now, we try the same structure in a 2D nextnano++ simulation to check if the 2D result agrees with the 1D one. The input file IV_GaN_n_doped_2D_nnp.in is used for this section. The width of the sample along the y direction is $200\mathrm{nm}$. The x direction is the same as in 1D.

Note that the unit for the current in a 2D simulation is $[\mathrm{A}/\mathrm{cm}]$. Dividing this two-dimensional current value by the width of the device (in our case $200\mathrm{nm}$), we obtain the current density in units of $[\mathrm{A}/{\mathrm{cm}}^2]$ which is the usual unit of a 1D simulation. As our simple 2D example structure is basically equivalent to a 1D structure, we can easily compare our 2D results with the 1D results to check for consistency.

voltage	current ($\mathrm{A}/\mathrm{cm}$) (nextnano++ 2D)	current density ($\mathrm{A}/{\mathrm{cm}}^2$) (nextnano++ 2D*)	current density ($\mathrm{A}/{\mathrm{cm}}^2$) (nextnano++ 1D)
$0$	$0$	$0$	$0$
$0.02$	$0.33845$	$16922.4$	$16922.4$
$0.04$	$0.67689$	$33844.7$	$33844.7$
$0.06$	$1.0153$	$50767.0$	$50767.0$
$0.08$	$1.3538$	$67689.2$	$67689.2$
$0.10$	$1.6922$	$84611.2$	$84611.3$

$\ast$ Here, the current density of the 2D simulation is obtained by dividing the current $[\mathrm{A}/\mathrm{cm}]$ by the width $200\mathrm{nm}$.

From the IV characteristics obtained from the 2D simulation, you can obtain a resistivity of the n-GaN layer of $1.1819\times {10}^{-6}\mathrm{\Omega }{\mathrm{cm}}^2$ which agrees very well with the 1D result (1D: $1.1819\times {10}^{-6}\mathrm{\Omega }{\mathrm{cm}}^2$).

3D

Of course, it is also possible to simulate this structure in 3D. In this case, the unit of the current is $[\mathrm{A}]$ and have to be divided by the area of the device perpendicular to the current flow direction to obtain the units of $[\mathrm{A}/{\mathrm{cm}}^2]$.

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Last update: 17/07/2024