AlGaAs/GaAs RTD

This tutorial describes the nextnano.NEGF simulation of AlGaAs/GaAs resonant tunneling diode (RTD).

Sample input files are available from the sample file folder:

• RTD_Example_withScattering.xml

• RTD_Example_ballistic.xml

The first input file takes the scattering mechanisms into consideration, while the second input file ignores them.

Figure 4.4.1 Top: Local density of states calculated from RTD_Example_withScattering.xml. The structure consists of GaAs and two AlGaAs barriers. Down: Current-voltage characteristics calculated from RTD_Example_withScattering.xml.

Table of contents

Simulation input

Definition of materials

At first, the materials used in the structure have to be defined. Each material is referred by an alias, which is here “well” for GaAs and “barrier” for AlGaAs.

  <Materials>

      <Material>
          <Name>GaAs</Name>
          <Alias>well</Alias>
          <Effective_mass_from_kp_parameters>yes</Effective_mass_from_kp_parameters>
      </Material>

      <Material>
          <Name>Al(x)Ga(1-x)As</Name>
          <Alloy_Composition>0.4</Alloy_Composition>
          <Alias>barrier</Alias>
          <Effective_mass_from_kp_parameters>yes</Effective_mass_from_kp_parameters>
      </Material>

      <!-- Model nonparabolicity -->
      <NonParabolicity>yes</NonParabolicity>
      <Number_of_bands>1</Number_of_bands>

      <UseConductionBandOffset>yes</UseConductionBandOffset>

</Materials>

It is specified that the effective mass is calculated from the k.p parameters (<Effective_mass_from_kp_parameters>). Also, the effective mass is assumed as energy dependent (<NonParabolicity>) and the model used for the calculation of effective mass is the one for single-band model (<Number_of_bands>).

For the details of the syntaxes of this block, please refer to this page: Syntax of the input file -> Material definition and parameters . The models used for the calculation of the effective mass are described here: Electronic Band Structure .

Definition of layers

Next, alternating layers consisting of barrier and well have to be specified. In this tutorial, the thickness of each layer is 5.0/2.0/10.0/2.0/5.0 (nm) where AlGaAs barrier layer is in bold fonts.

<Superlattice>

    <Layer>
        <Material>well</Material>
        <Thickness unit="nm">5.0</Thickness>
    </Layer>

    <Layer>
        <Material>barrier</Material>
        <Thickness unit="nm">2.0</Thickness>
    </Layer>

    <Layer>
        <Material>well</Material>
        <Thickness unit="nm">10.0</Thickness>
    </Layer>

    <Layer>
        <Material>barrier</Material>
        <Thickness unit="nm">2.0</Thickness>
    </Layer>

    <Layer>
        <Material>well</Material>
        <Thickness unit="nm">5.0</Thickness>
    </Layer>


    <Analysis_Separator>
        <Separator_Position>6.0</Separator_Position>
        <!-- collector barrier -->
    </Analysis_Separator>

    <Analysis_Separator>
        <Separator_Position>18.0</Separator_Position>
        <!-- injector barrier -->
    </Analysis_Separator>

</Superlattice>

The resulting conduction band edge profile can be found in the file called Conduction_BandEdge.dat. This file includes the (small) band bending due to the electrostatic potential. At a bias voltage of 0 mV, it looks as follows:

Figure 4.4.2 Conduction band edge at a zero bias voltage. This is obtained from RTD_Example_withScattering.xml.

Contacts (open boundary condition)

In order to simulate a system with open boundary conditions (instead of the default field-periodic boundary condition), contacts have to be defined by adding a <Contacts> section into the input file.

In RTD_Example_withScattering.xml this section is specified as follows:

<Contacts>

    <DensityLeft unit="cm^-3">1e18</DensityLeft>
    <DensityRight unit="cm^-3">1e18</DensityRight>

    <MaterialLeft>well</MaterialLeft>
    <MaterialRight>well</MaterialRight>

    <Broadening unit="meV">10.0</Broadening>
    <Ballistic>no</Ballistic>

</Contacts>

The carrier densities in the left and right contact have to been defined using the <DensityLeft> and <DensityRight> commands, as shown above. The unit is [cm^-3].

The material of the left and right contacts needs to be defined by the command <MaterialLeft> and <MaterialRight>. The string value has to be an alias defined in the <Materials> section.

A broadening energy can be defined by the command <Broadening>. Indeed, scattering is not accounted in the contact, so that this command allows a phenomenological broadening of the density of states in the contact.

The command <Ballistic> can be used to calculate ballistic transport between the contacts (i.e. no scattering process considered) if its value is set to yes. This is the case of RTD_Example_ballistic.xml.

Note

In the current version (2020-11-19), only single band calculations are supported for open boundary conditions.

Scattering

The detailed description of the sytaxes for each scattering mechanism is described here: Syntax of the input file -> Scattering processes .

The following scattering mechanisms are included in this tutorial.

• Interface roughness scattering

<Interface_Roughness>
<Amplitude_in_Z unit="nm">0.1</Amplitude_in_Z>
    <InterfaceAutoCorrelationType>0</InterfaceAutoCorrelationType> <!-- Correlation type: 0=Exponential, 1=Gaussian -->
<Correlation_Length_in_XY unit="nm">8</Correlation_Length_in_XY>
</Interface_Roughness>

• Acoustic phonon scattering

<!-- Acoustic phonons -->
<Acoustic_Phonon_Scattering>no</Acoustic_Phonon_Scattering> <!-- Comment: Acoustic phonons are in general not efficient - can be neglected in most cases -->
<AcousticPhonon_Scattering_EnergyMax unit="meV">3.0</AcousticPhonon_Scattering_EnergyMax> <!-- Maximum acoustic phonon energy -->

• Charged impurity scattering

<!-- Charged impurities -->

<!-- Effective temperature of the electrons involved in electrostatic screeening: 3 models available -->
<!-- model #1: Teff = T + Toffset * exp(-T/Toffset) with Toffset specified as Temperature_Offset_parameter  -->
<!-- model #2: self-consistent calculation (requires several iterations of the all calculation). The ccuracy specified by Accuracy_Self_consistent_Electron_Temperature -->
<!-- model #3: Teff is directly specified by Electron_Temperature_for_Screening-->
<Model_Temperature_for_Screening>1</Model_Temperature_for_Screening> <!-- integer 1,2 or 3 is required accordingly to the desired model -->

<Temperature_Offset_parameter>150</Temperature_Offset_parameter> <!-- enter Toffset for model#1 only such as Teff = T + Toffset * exp(-T/Toffset) -->
<Accuracy_Self_consistent_Electron_Temperature>0.05</Accuracy_Self_consistent_Electron_Temperature>  <!-- for model #2 only: self-consistent calculation until the effective temperature convergences below the desired accuracy-->
<Electron_Temperature_for_Screening>200</Electron_Temperature_for_Screening> <!-- for model#3 only: the effective temperature is directly specified -->


<ImpurityScattering_Strength>1</ImpurityScattering_Strength> <!-- 1.0 is the normal physical calculation. Other values may be used for testing the importance of impurity scattering. -->

• Electron-electron scattering

<Electron_Electron_Scattering>yes</Electron_Electron_Scattering>

• Alloy scattering

<Alloy_scattering>yes</Alloy_scattering>

(Advanced) In this sample input file, the Coulomb scatterers (ionized impurities and other charge carriers) are assumed to be homogeneously distributed in order to speed up the calculation.

<Homogeneous_Coulomb>yes</Homogeneous_Coulomb>

Note

LO-phonon scattering is applied by default as all materials have intrinsic parameters controlling the electron-phonon interaction. We can modify this default behavior by this command.

Poisson equation

<Poisson>yes</Poisson>

If yes, the Poisson equation is included in the program flow. Then the electrostatic mean-field interactions (electron-electron and electron-impurities interactions) can be taken into account.

Lateral motion

<Lateral_motion>
    <Material_for_lateral_motion>well</Material_for_lateral_motion>
    <!-- Lateral energy spacing  -->
    <Value unit="meV">10</Value>
</Lateral_motion>

In this sample file, the parameters for the lateral motion (i.e. the two-dimensional free motion in the directions perpendicular to the growth axis) are taken from the material of well, i.e. GaAs.

Also, the discretization energy for this lateral motion is specified as 10 meV.

Simulation output

Electron eigen states

The electron eigenstates calculated for the whole region biased for each voltage is written in WannierStark/WannierStark_states.dat. These states are used as the basis states of the Green’s function.

Figure 4.4.3 Electron eigenstates for the whole region at the bias of 100 mV.

Note

There is another output WannierStark/TightBinding_states.dat that describes the electron eigenstates confined in each GaAs region. These eigenstates are calculated from the Schrödinger equations for each region separated by <Analysis_Separator> in <Superlattice> section.

For the detailed explanation, please refer to here .

Local density of states

The following figures show the local density of states (LDOS) for the bias of 100 mV written in 2D_plots/DOS_energy_resolved.vtr. Please note that the scaling of the colormap is different in the two figures. The gnuplot file which generates the gif animation is also available in the top directory as Animation_DensityOfStates.plt.

The LDOS tells us where and at which energy electronic states are available that the charge carriers can occupy. The LDOS is shown for \(\mathbf{k}_\parallel=0\), i.e. there are also electronic states available for \(\mathbf{k}_\parallel\neq0\). But they are not shown in this plot because then the picture could not show the minimum energy of each subband , which is at \(\mathbf{k}_\parallel=0\), so nicely.

Figure 4.4.4 Local density of states calculated with scattering at the bias of 100 mV.

Figure 4.4.5 Local density of states calculated with the ballistic condition at the bias of 100 mV.

Note

The gnuplot files that generate the above figures are DOS_energy_resolved.plt. The file explorer in nextnanomat does not show .plt file so please access through your default file explorer.

Electron density

The following figures show the energy resolved electron density \(n(x,E)\) for the bias of 100 mV written in 2D_plots/CarrierDensity_energy_resolved.vtr. Please note that the scaling of the colormap is different in the two figures. The gnuplot file which generates the gif animation is also available in the top directory as Animation_CarrierDensity.plt.

The electron density is obtained from occupying the LDOS (for both \(\mathbf{k}_\parallel=0\) and \(\mathbf{k}_\parallel\neq0\)) with charge carriers. It is a non-equilibrium occupation that is not described by a Fermi distribution.

Figure 4.4.6 Electron density calculated with scattering at the bias of 100 mV.

Figure 4.4.7 Electron density calculated with the ballistic condition at the bias of 100 mV.

Current density

The following figures show the energy resolved current density \(j(x,E)\) for the bias of 100 mV written in 2D_plots/CurrentDensity_energy_resolved.vtr. Please note that the scaling of the colormap is different in the two figures. The gnuplot file which generates the gif animation is also available as Animation_CurrentDensity.plt.

Figure 4.4.8 Current density calculated with scattering at the bias of 100 mV.

Figure 4.4.9 Current density calculated with the ballistic condition at the bias of 100 mV.

Current-voltage characteristics

The following figure shows the current-voltage characteristics calculated both with and without scattering. These results are taken from Current_vs_Voltage.dat.

Figure 4.4.10 Current-voltage calculated with and without scattering.