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1D Tutorial

InAs / In_0.4Ga_0.6Sb superlattice dispersion with 8x8 k.p (type-II band alignment)

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

Author: Stefan Birner, Michael Povolotskyi

If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
-> 1DInAs_InGaSb_k_zero.in
-> 1DInAs_InGaSb_k_parallel.in
-> 1DInAs_InGaSb_k_superlattice.in
 

InAs / In0.4Ga0.6Sb superlattice dispersion with 8x8 k.p (type-II band alignment)

This tutorial aims to reproduce Fig. 2(a) of

Long wavelength InAs/InGaSb infrared detectors: Optimization of carrier lifetimes
C.H. Grein, P.M. Young, M.E. Flatte, H. Ehrenreich
J. Appl. Phys. 78 (12), 7143 (1995)

Conduction and valence band edges

-> 1DInAs_InGaSb_k_zero.in

The heterostructure is a superlattice with 3.98 nm InAs and 1.5 nm In_0.4Ga_0.6Sb, where both constituents are strained with respect to the GaSb substrate.

The structure has a type-II band alignment, i.e. the electrons are confined in the InAs layer,
whereas the holes are confined in the In_0.4Ga_0.6Sb layer.

The In_0.4Ga_0.6Sb layer is strained pseudomorphically with respect to the GaSb substrate,
leading to a compressive strain (-2.5 %) which splits the degeneracy of the heavy and light hole band edges in this layer.
Thus, the heavy hole band edge lies above the light hole band edge.

The InAs layer is also strained pseudomorphically with respect to the GaSb substrate,
and is thus under slight biaxial tension (+0.6 %).
The splitting of the hole band edges is the opposite as in InGaSb, i.e. the light hole band edge is above the heavy hole band edge.

The following figure shows the electron and hole band edges.



The origin of the energy scale is set the the GaSb valence band edge energy.

Electron and hole wave function for k_|| = 0

• We simulate one period only (i.e. from 0 nm to 5.48 nm) and solve the Schrödinger equation with periodic boundary conditions to mimic an infinite superlattice.
  
  The following figure shows the conduction band edge and the heavy, light and split-off hole valence band edges in this superlattice structure
  together with the electron (c1), heavy hole (hh1) and light hole (lh1) energies and wave functions (psi²), calculated within 8x8 k.p theory.
  
  One can clearly see that the electron state (c1) is confined in the InAs layer (right part of the figure),
  whereas the heavy (hh1) and light hole (lh1) states are confined in the In_0.4Ga_0.6Sb layer (left part of the figure).
  
  
  
  We used the same material parameters as given in the above cited paper by Grein et al., apart from the k.p parameters.
  
   

Electron and hole energies for k_|| /= 0

• -> 1DInAs_InGaSb_k_parallel.in

  The following figure shows the E(k_||) dispersion of the electron ground state and the two highest hole states along two different directions in (k_x,k_y) space.
  
  
  
  This data is contained in this file: kp_data1/par1D_disp_01_00_11_hl_8x8kp_ev_min001_ev_max010.dat
  Note that the band gap is not determined by the bandgap of one individual layer. It is determined by the electron ground state in the InAs layer, and the hole ground state in the InGaSb layer. This means more freedom for band gap engineering.

Electron and hole energies for k_z /= 0

• -> 1DInAs_InGaSb_k_superlattice.in

  The right part of the following figure shows the E(k_z) superlattice dispersion of the electron ground state and the two highest hole states. k_z is the superlattice vector between 0 and 1 pi/L where L = 5.48 nm is the length of one superlattice period.
  (1 pi/L = 0.0573 1/Angstrom)
  This data is contained in this file: kp_data1/8x8kp_dispSL_hl_qc001_evmin001_evmax016.dat
  
  The left part of the figure shows the E(k_||) dispersion along [10], i.e. from (k_x,k_y) = (0,0) to (k_x,k_y) = (-0.1,0) which is shown in the figure above already.
  
  
  
  One can clearly see that these heterostructure bands are highly nonparabolic.

• The resolution of the k_z superlattice vectors is specified here:
  
   $quantum-model-electrons
  ...
  num-ks-001  = 11      ! number of superlattice vectors in [001] direction

  ...

 $quantum-model-holes
  ...
  num-ks-001  = 11      ! number of superlattice vectors in [001] direction


 
• The number of superlattice vectors to be put out is specified here:

 $output-kp-data
  ...
  cb-k-SL-min = 1       ! only for superlattices: lower bound for range of superlattice vectors (usually equal to 1)
  cb-k-SL-max = 11      ! upper bound (usually equal to num-ks-001)
  ...
  vb-k-SL-min = 1       ! only for superlattices: lower bound for range of superlattice vectors (usually equal to 1)
  vb-k-SL-max = 11      ! upper bound (usually equal to num-ks-001)
 

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