nextnano - Software for
semiconductor nanodevices

We apply the Contact Block Reduction (CBR) method to a simple GaAs nanowire of cuboidal shape. The corresponding tutorial for nextnano³ is here.

We consider a GaAs cuboidal tube of dimensions 10 nm$\times$10 nm$\times$20 nm. Two leads of 10 nm$\times$10 nm each are attached to the edge of the device. The grid spacing is 1 nm in all directions. The effective electron mass is assumed to be constant throughout the device and equal to $0.067m_0$.

To simulate 3D (or 2D) system with CBR method in nextnano++ correctly, The quantum regions have to be appropriately specified in the input file.

quantum{
	region{
		name = "lead_1"
		x    = [-6,6]
		y    = [-6,6]
		z    = [-0.1,0.1]
		boundary{ x=dirichlet y=dirichlet z=cbr }
                Gamma{ num_ev = $num_eigenstates_device }
	}
}

The perpendicular directions, i.e. x- and y-directions, of the system are elongated by one grid due to the treatment of edge points in nextano++. Since the simulation is three dimensional, the lead region specified here has to be two dimensional. The number $\pm0.1$ is chosen to be smaller than the grid spacing, so that the region “lead_1” becomes a 2D sheet (Note: this is slightly different in nextnano³ input). CBR boundary condition has to be imposed in the propagation direction, i.e. z-direction, whereas Dirichlet boundary condition is set for perpendicular directions.

cbr{
         name = "device"
         lead{ name = "lead_1" }
         lead{ name = "lead_2" }
         delta_energy   = $delta_energy
         abs_min_energy = $E_min
         abs_max_energy = $E_max
         options        = [1, 0, 0]
}

Here we specify the device region and leads attached to the device. The program calculates transmission through the region “device”, from “lead_1” to “lead_2”. The resolution, minimum and maximum of the energy axis can be also tuned here.

The biggest advantage of the CBR method is that it can correctly predict the spectrum without calculating all eigenmodes of the 3D device. That means that, for low energies, one can significantly reduce the simulation load to calculate the transmission spectrum [Birner2009]. To demonstrate it we perform three different simulations, where the number of modes considered in the calculation is sweeped.

 $CBR_case   = 1                                             # (ListOfValues:1,2,3)

 $CBR_light  = iszero($CBR_case-1)  
 $CBR_medium = iszero($CBR_case-2)     
 $CBR_heavy  = iszero($CBR_case-3)                                           

#if $CBR_light $num_eigenstates_device    =   200            #  5.6% of all device modes  
#if $CBR_light $num_eigenstates_lead      =    30            # 17.8% of all lead modes
                                          
#if $CBR_medium $num_eigenstates_device   =   400            # 11.3% of all device modes  
#if $CBR_medium $num_eigenstates_lead     =    50            # 30.0% of all lead modes

#if $CBR_heavy $num_eigenstates_device    =   600            # 16.9% of all device modes  
#if $CBR_heavy $num_eigenstates_lead      =    80            # 47.3% of all lead modes

The following figure shows the calculated transmission coefficient as a function of energy. The result of nextnano³ is shown for reference.

<figure transmission> <caption>Transmission coefficient of a GaAs 3D nanowire. Arrows indicate the cut-off energies, namely the eigenenergy of the highest device eigenmode considered in each simulation.</caption> </figure>