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nextnano3 - Tutorial

next generation 3D nano device simulator

1D Tutorial

pn-junction

Note: This tutorial's copyright is owned by Stefan Birner, www.nextnano.com.

Author: Stefan Birner

If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
-> 1DGaAs_pn_junction_nn3.in     / *_nnp.in  - input file for the nextnano3 and nextnano++ software
-> 1DGaAs_pn_junction_QM_nn3.in
-> 2DGaAs_pn_junction_nn3.in     / *_nnp.in  -
input file for the nextnano3 and nextnano++ software
-> 3DGaAs_pn_junction_nn3.in     / *_nnp.in  -
input file for the nextnano3 and nextnano++ software
-> 1DGaAs_pn_junction_nn3_ForwardBias.in     -
input file for the nextnano software
-> 1DGaAs_pn_junction_nnp_ForwardBias.in     -
input file for the nextnano++ software
 


pn-junction

-> 1DGaAs_pn_junction_nn3.in / *_nnp.in  - input file for the nextnano3 and nextnano++ software

This tutorial aims to reproduce figure 3.1 (p. 51) of Joachim Piprek's book "Semiconductor Optoelectronic Devices - Introduction to Physics and Simulation" (Section 3.2 "pn-Junctions").
 

Doping concentration

  • The structure consists of 300 nm GaAs.
    At the left and right boundaries, metal contacts are connected to the GaAs semiconductor (i.e. from 0 nm to 10 nm, and from 310 nm to 320 nm).
    The structure is p-type doped from 10 nm to 160 nm and n-type doped from 160 nm to 310 nm.
  • The following figure shows the concentration of donors and acceptors of the pn-junction.
    In the p-type region between 10 nm and 160 nm, the number of acceptors NA is 0.5 x 1018 cm-3.
    In the n-type region between 160 nm and 310 nm, the number of donors ND is 2.0 x 1018 cm-3.

 

Carrier concentrations

  • The equilibrium condition for a pn-junction is achieved by a small transfer of electrons from the n region to the p region, where they recombine with holes. This leads to a depletion region (depletion width = wp + wn), i.e. the region around the pn-junction only has very few free carriers left.
  • The following figure shows the electron and hole densities and the depletion region around the pn-junction at 160 nm. Here, we assumed that all donors and acceptors are fully ionized.

 

Net charges (space charge)

  • In the depletion region, a net charge results from the ionized donors ND and ionized acceptors NA.
  • The following figure shows the net charge density of the pn-junction.

 

Electric field

  • The slope of the electric field is proportional to the net charge (Poisson equation), thus the extremum of the electric field is expected to be at the pn-junction.
  • In regions without charges, the electric field is zero.
  • The following figure shows the electric field of the pn-junction.



    The extremum of the electric field Fmax (at 160 nm) can be approximated as follows:
    Fmax = - e NA wp / (epsilon epsilon0) = - 6.997 x 1014 V/m2 wp = 387 kV/cm
            = - e ND wn / (epsilon epsilon0) = - 2.799 x 1015 V/m2 wn = 386 kV/cm

    where
       e = 1.6022 x 10-19 As
       epsilon = 12.93 (dielectric constant of GaAs)
       epsilon0 = 8.854 x 10-12 As/(Vm)
       NA = 0.5 x 1018 cm-3
       ND = 2.0 x 1018 cm-3
       wp = 55.3 nm
       wn = 13.8 nm

 

Electrostatic potential, conduction and valence band edges

  • In regions, where the electric field is zero, the electrostatic potential is constant.
  • The electrostatic potential phi determines the conduction and valence band edges:
    Ec = Ec0 - e phi
    Ev = Ev0 - e phi
  • The following figure shows the conduction and valence band edges, the electrostatic potential and the Fermi level of the pn-junction.


    Without external bias (i.e. equilibrium), the Fermi level EF is constant (EF = 0 eV).

    The built-in potential phibi was calculated by nextnano to be equal to 1.426 V.
    It can be approximated as follows:
       phibi = Fmax (wp + wn) / 2
    Assuming Fmax = 387 kV/cm, this would indicate for the depletion width: wp + wn = 73.7 nm.

    To allow for a constant chemical potential (i.e. constant Fermi level EF), a total potential difference of -e phibi is required.

 

 

Quantum mechanical calculation

-> 1DGaAs_pn_junction_QM_nn3.in

  • Here, instead of calculating the densities classically, we solve the Schroedinger equation for the electrons, light and heavy holes in the single-band approximation over the whole device. We calculate up to 300 eigenvalues for each band. Thus the electron and hole densities are calculated purely quantum mechanically.
  • The following figure shows the electron and hole concentrations for the classical and quantum mechanical calculations. For the QM calculations, different boundary conditions were used.
    - Dirichlet boundary conditions force the wave functions to be zero at the boundaries, thus the density goes to zero at the boundaries which is unphysically.
    - Neumann boundary conditions lead to unphysically large values at the boundaries.
    - Mixed boundary conditions are in between.
    For the classical calculation, the densities at the boundaries are constant.
    Nevertheless, in the interesting region around the pn-junction, all four options lead to identical densities.

  • The following figure shows the band edges of the pn-junction for the four cases:
    - classical calculation
    - quantum mechanical calculation with Dirichlet boundary conditions
    - quantum mechanical calculation with Neumann boundary conditions
    - quantum mechanical calculation with mixed boundary conditions
    For all cases the band edges are identical in the area around the pn-junction. Tiny deviations exist at the boundaries of the device.

  • This figure is a zoom into the right boundary of the conduction band edge.
    On this scale, the tiny deviations for the different boundary conditions can be clearly seen.

 

 

Nonequilibrium

  • So-called "quasi-Fermi levels" which are different for electrons (EF,n) and holes (EF,p) are used to describe nonequilibrium carrier concentrations.
    In equilibrium the quasi-Fermi levels are constant and have the same value for both electrons and holes (EF,n = EF,p = 0 eV).
    The current is proportional to the mobility and the gradient of the quasi-Fermi level EF.

    -> 1DGaAs_pn_junction_nn3_ForwardBias.in - input file for the nextnano software
    -> 1DGaAs_pn_junction_nnp_ForwardBias.in -
    input file for the nextnano++ software

 

2D/3D simulations

-> 2DGaAs_pn_junction_nn3.in / *_nnp.in  - input file for the nextnano3 and nextnano++ software
-> 3DGaAs_pn_junction_nn3.in / *_nnp.in  -
input file for the nextnano3 and nextnano++ software

Input files for the same pn junction structure as in 1D, but this time for a 2D and 3D simulation are also available.
==> 2D: rectangle of dimension 320 nm x 200 nm
==> 3D: cuboid     of dimension 320 nm x 200 nm x 100 nm

 

  • Please help us to improve our tutorial! Send comments to support [at] nextnano.com.