nextnano³ - Tutorial

next generation 3D nano device simulator

1D Tutorial

Solution of the Poisson equation for different charge density profiles

Author: Stefan Birner

If you want to obtain the input files that are used within this tutorial, please check if you can find them in the installation directory.
If you cannot find them, please submit a Support Ticket.
1) -> 1D_Poisson_dipole.in / *_nnp.in - input file for the nextnano³ and nextnano++ software (1D simulation)
2) -> 1D_Poisson_linear.in
3) -> 1D_Poisson_delta.in
 

1) Dipole: Constant charge density profile of positive and negative charge

2) Linear charge density profile of positive and negative charge

3) Delta-function like charge density profile of positive and negative charges

1) Dipole: Constant charge density profile of positive and negative charge

The following figures show a dipole charge density distribution where
- the left   region carries a constant positive  charge density (resulting from ionized donors     N_D⁺) and
- the right region carries a constant negative charge density (resulting from ionized acceptors N_A^-).

Left figure:  1Ddoping_concentration.dat
Right figure: densities/density1Dspace_charge.dat

      

We have to solve the Poisson equation: d²phi / dx² = - rho / (epsilon epsilon₀)

The following figures shows the corresponding electric field distribution (left) and the electrostatic potential (right).

Left figure:  band_structure/electric_field1D.dat
Right figure: band_structure/potential1D.dat

      

The electric field is given by E(x) = - dphi / dx and has a linear dependence (~ -x) because the electrostatic potential has a quadratic dependence (~ x²).

The maximum value of the electric field is given by:

E_max = rho / (epsilon epsilon₀) * x₀ = e * 1*10¹⁸ cm^-3 / ( 12.93 * 8.8542*10^-12 As/Vm ) * 10 nm =
        = 1.3995*10⁷ V/m = 139.95 kV/cm

where x₀ is the width of the positive (or negative) charge density region, and epsilon = 12.93 is the static dielectric constant of GaAs.

The drop of the electrostatic potential between 0 nm and 20 nm is simply given by the area that is below the graph of the electric field:

       Delta phi = 1/2 E_max * 20 nm = 139.95 mV

2) Linear charge density profile of positive and negative charge

The following figures show a linearly varying charge density distribution where
- the left   region carries a linearly decreasing positive  charge density (resulting from ionized donors     N_D⁺) and
- the right region carries a linearly increasing  negative charge density (resulting from ionized acceptors N_A^-).

Left figure:  1Ddoping_concentration.dat
Right figure: densities/density1Dspace_charge.dat

        

The following figures shows the corresponding electric field distribution (left) and the electrostatic potential (right).

Left figure:  band_structure/electric_field1D.dat
Right figure: band_structure/potential1D.dat

   

The electric field shows a quadratic dependence  (~ -x²) whereas the electrostatic potential shows a cubic dependence (~ x³).

3) Delta-function like charge density profile of positive and negative charges

The following figures show a delta-function like charge density distribution where
- in the middle       of the structure there is a constant positive  charge density   of width 1 nm         (resulting from ionized donors     N_D⁺) and
- at the boundaries of the structure there are constant negative charge densities of width 1 nm each (resulting from ionized acceptors N_A^-).

Left figure:  1Ddoping_concentration.dat
Right figure: densities/density1Dspace_charge.dat

        

The following figures shows the corresponding electric field distribution (left) and the electrostatic potential (right).

Left figure:  band_structure/electric_field1D.dat
Right figure: band_structure/potential1D.dat