Table of Contents

Simulation output

For each simulation run, a new output folder is created in the simulation output folder. The created folder has the name of the input file.

In addition, date-time can be added to the folder name by the nextnanomat setting (Tools → Options → Expert settings before Aug 2021, Tools → Options → View/Output since Aug 2021). For nextnanomat before Aug 2021, this option is recommended in order to avoid overwriting existing output data. For nextnanomat after Aug 2021, simulation output is per default not overwritten and instead enumerated unless you manually check the option “Overwrite existing simulation…”.

The created output folder contains:

'Input' folder

The folder Input/ contains all the simulation input such as material parameters as a function of position.

Strain

If the strain option is activated, a folder Strain/ is created.

If the crystal has not been rotated, above files contain identical values.

Piezo and pyroelectric polarization

The folder Polarization/ contains the piezoelectric and pyroelectric polarization if these options are activated.

Initial electronic states

The folder Init_Electr_Modes\ contains 3 different folders corresponding to 3 different sets of basis states. They are calculated at the first step of the calculation, before the NEGF calculation. These 3 sets of states are basis of the reduced Hilbert space obtained after applying the energy cut-off <Energy_Range_Axial>.

These states are displayed for a default voltage of <Energy_Range_Axial>/2. This voltage at which the states are visualized can be modified by the input file command: 

<Simulation_Parameter>
  ...
  <Bias_for_initial_Electronic_Modes unit="meV">54</Bias_for_initial_Electronic_Modes>
  ...
</Simulation_Parameter>

'Reduced Real Space' modes

The 'reduced real space' modes are eigenstates of the position operator in the reduced Hilbert space (i.e. after the energy cut-off). Because of the energy cut-off, these states are spatially extended instead of being $\delta$ functions. This basis set is the one which is used in the NEGF calculation. It does not depend on the applied voltage. However, this basis has generally little use in terms of physical interpretation.

The folder Init_Electr_Modes\ReducedRealSpace\ contains:

$$\Psi_i(z) = f^{\text{c}}_i(z)u_{\text{c}}(z) + f^{\text{LH}}_i(z)u_{\text{LH}}(z) + f^{\text{SO}}_i(z)u_{\text{SO}}(z)$$

'Tight-binding' states

The Tight-binding\ folder contains data only if one or several <Analysis_Separator> are defined in the input file. The tight-binding basis corresponds to piecewise solution of the Schrödinger equation between these separators.

Wannier-Stark states

The Wannier-Stark states correspond to the eigenstates of the Schrödinger equation without accounting for Poisson equation (i.e. electrostatic mean-field).
It contains:

The expression in the single-band case is: $$ d_{ij} = \int dz f_i(z) ~ z ~ f_j(z) $$ In the multiband case: $$ d_{ij} = \sum_{\mu} \int dz f^{(\mu)}_i(z) ~ z ~ f^{(\mu)}_j(z) $$

Oscillator strength

The oscillator strength is calculated from the formula $$ f_{\alpha \beta} = \frac{2 \vert p_{\alpha \beta}\vert^2}{m_0 (E_{\beta} - E_{\alpha})} $$ Note that the electron mass $m_0$ entering the above formula is the bare electron mass.

This oscillator strength (which is sometimes referred as the unnormalized one), differs from the usual definition in the single band case by the ratio $m^*/m_0$, i.e. $\frac{m^*}{m_0} f_{\alpha \beta}$ is called the normalized oscillator strength.

The advantage of this unnormalized definition is that it is general enough to be applied to the multiband case.

Note that in the parabolic single-band case, the usual sum-rule is retrieved by using the normalized definition $$ \sum_{\beta \neq \alpha} \frac{m^*}{m_0} f_{\alpha \beta} = 1 $$

In-plane discretization

The file Lateral_spectrum.dat gives the energy discretization for the states used to describe the 2-Dimensional (2D) motion in the directions (x,y) perpendicular to the heterostructure. The lateral motion is discretized using cylindrical boundary conditions, and the corresponding eigenstates are Bessel funcitons.
$x$ axis: Lateral state index
$y$ axis: order of Bessel (zero index)-1 of Bessel Relative Energy (meV).

Simulation output for each voltage/temperature step

For each voltage or temperature step, the following files are produced as a result of the NEGF calculation:

$$ T_{\text{eff}} = \sum_{i} ~ p_{i} ~ E_{\parallel}(i) ~ / ~ k_b $$ where $p_{i}$ is the fraction (i.e. population normalized to 1) of occupation in the in-plane state $i$, $E_{\parallel}(i)$ is the in-plane energy for the in-plane state $i$, and k_b the Boltzmann constant.

Output in basis sets (ReducedRealSpace, WannierStark, TightBinding)

3 folders are created to output physical quantities in the 3 different basis sets (Reduced Real Space, Wannier-Stark, and Tight-Binding).

For each basis set, the folder contains:

$$ \langle E_i \rangle = \frac{ \sum_{k} ~ p_{i,k} ~ E_{\parallel}(k)}{\sum_{k} ~ p_{i,k}}, $$ where $E_{\parallel}(k)$ is the in-plane kinetic energy.

$$ T^{\text{eff}}_i = \langle E_i \rangle / ~ k_b $$

2D plots

The folder 2D_plots\ contains 2D color maps as a function of position [nm] (horizontal axis) and energy [eV] (vertical axis). Note that these 2D plots show 2 QCL periods although only 1 period is simulated.

For different extensions of 2D outputs, please also see advanced settings in the input file.

Gain

The folder Gain\ contains one- and two-dimensional plots of the intensity gain simulated. A negative value of gain corresponds to absorption.

2D color maps show the gain $G(z,E_{\rm ph})$ [cm-1 nm-1], where the horizontal axis is position $z$ [nm] and the vertical axis is photon energy $E_\rm{ph}$ in units of either energy [meV] or frequency [THz]. Note that the units of gain in the nextnano.MSB code are [eV-1 cm-1]. Also note that these 2D plots show 2 QCL periods although only 1 period is simulated.

1D plots show the gain $G(E_\rm{ph})$ [cm-1] against photon energy [meV], frequency [THz], and wavelength [micron].

Note that the gain output is only done for the voltages specified in the input file. 

<!-- Calculate gain only between the following values of
     potential drop per period in order to save CPU time -->
     <Vmin unit="mV"> 160 </Vmin>
     <Vmax unit="mV"> 400 </Vmax>

Green's functions

The folder GreenFunctions/ contains information on the Green's functions.

The electron density $n(x,E_x)$ is related to the lesser Green's function $\mathbf{G}^<$ (“G lesser”): $$n(x,E_x) = - \frac{{\rm i}}{2\pi} \mathbf{G}(x,x^\prime=x,E_x)$$

The local density of states $\rho(x,E_x)$ is related to the spectral function $\mathbf{A}$: $$\rho(x,E_x) = \frac{1}{2\pi} \mathbf{A}(x,x^\prime=x,E_x)$$ $\mathbf{A}$ is defined as $\mathbf{A} = {\rm i} (\mathbf{G}^{\rm R} - \mathbf{G}^{{\rm R}\dagger}) = - 2 {\rm Im}(\mathbf{G}^{\rm R})$. $\mathbf{G}^{\rm R}$ is the retarded Green's function.

Density matrix

The folder DensityMatrix/ contains the density matrix $\rho$ which is a complex quantity and it is dimensionless. The trace of the density matrix equals 1. In our case, the trace is 1 if we sum over one period. The state labels (state $i$, period $j$) are specified in the complex density matrix. $$\rho(i,j) = \rho({\rm state},{\rm period})$$

Output files for voltage sweep

If you sweep voltage, the following files are generated.

Combined temperature-voltage sweep

a combined temperature-voltage sweep can be done using the keyword Temperature-Voltage in the field <SweepType> of <SweepParameters> (see the example of code below). In this case, the simulation can be parallelized. <Threads> defines the number of parallel threads. Its optimal value should be the number of CPU cores available (if the available memory is sufficient). Within each parallel temperature sweep, a serial voltage sweep is performed. 

<SweepParameters>   
    <SweepType>Temperature-Voltage</SweepType>
    <MinV> 50</MinV> 
    <MaxV> 60</MaxV> 	
    <DeltaV> 2</DeltaV> 	

    <MinT> 25</MinT> 
    <MaxT> 300</MaxT> 	
    <DeltaT> 25</DeltaT> 

    <Threads>12</Threads> <!-- Parallelization for Temperature-Voltage sweep -->
</SweepParameters>

Note that for such voltage-temperature sweep, <Maximum_Number_of_Threads> in <Simulation_Parameter> should be set to 1. (A combined parallelization will result in lower performances.) 

<Simulation_Parameter>
  ...
  <Maximum_Number_of_Threads>1</Maximum_Number_of_Threads>
</Simulation_Parameter>

At the end of the simulation, current and gain maps can be displayed. Gain_map.fld gives the maximum gain at each (voltage,temperature) point. Max_Gain_frequency.fld gives the map of the corresponding photon energy for which the gain is maximum.

Folder view:

Gain map (V,T):