nextnano³ - Tutorial

next generation 3D nano device simulator

1D Tutorial

pn-junction

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

Author: Stefan Birner
Please send comments to nextnano3 (-at-) wsi.tum.de.

If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
-> 1DGaAs_pn_junction.in
-> 1DGaAs_pn_junction_QM.in
-> 2DGaAs_pn_junction.in
-> 3DGaAs_pn_junction.in

Note: nextnano++ input files are available for this tutorial. Please contact support [at] nextnano.de for prices.
 

pn-junction

-> 1DGaAs_pn_junction.in

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 N_A is 0.5 x 10¹⁸ cm^-3.
  In the n-type region between 160 nm and 310 nm, the number of donors N_D 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 = w_p + w_n), 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 N_D and ionized acceptors N_A.
• 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 F_max (at 160 nm) can be approximated as follows:
  F_max = - e N_A w_p / (epsilon epsilon₀) = - 6.997 x 10¹⁴ V/m² w_p = 387 kV/cm
          = - e N_D w_n / (epsilon epsilon₀) = - 2.799 x 10¹⁵ V/m² w_n = 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)
     N_A = 0.5 x 10¹⁸ cm^-3
     N_D = 2.0 x 10¹⁸ cm^-3
     w_p = 55.3 nm
     w_n = 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:
  E_c = E_c0 - e phi
  E_v = E_v0 - 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 E_F is constant (E_F = 0 eV).
  
  The built-in potential phi_bi was calculated by nextnano³ to be equal to 1.426 V.
  It can be approximated as follows:
     phi_bi = F_max (w_p + w_n) / 2
  Assuming F_max = 387 kV/cm, this would indicate for the depletion width: w_p + w_n = 73.7 nm.
  
  To allow for a constant chemical potential (i.e. constant Fermi level E_F), a total potential difference of -e phi_bi is required.

Quantum mechanical calculation

-> 1DGaAs_pn_junction_QM.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 wavefunctions 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.
  
  

Non-equilibrium

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

2D/3D simulations

-> 2DGaAs_pn_junction.in
-> 3DGaAs_pn_junction.in

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 nextnano3 (-at-) wsi.tum.de.