nextnano³ - Tutorial

next generation 3D nano device simulator

2D Tutorial

Exciton energy in quantum wires

Author: Stefan Birner

-> 2Dwire_excitonCdTe_10x10nm.in  -   10 nm x 10 nm CdTe quantum wire
   2Dwire_excitonCdTe_14x14nm.in  -   14 nm x 14 nm CdTe quantum wire
   2Dwire_excitonCdTe_20x20nm.in  -   20 nm x 20 nm CdTe quantum wire
   2Dwire_excitonCdTe_30x30nm.in  -   30 nm x 30 nm CdTe quantum wire
   2Dwire_excitonCdTe_40x40nm.in  -   40 nm x 40 nm CdTe quantum wire
   2Dwire_excitonCdTe_50x50nm.in  -   50 nm x 50 nm CdTe quantum wire
   2Dwire_excitonCdTe_70x70nm.in  -   70 nm x 70 nm CdTe quantum wire
-> 2Dwire_excitonCdTe_1x30nm.in   -    1 nm x 30 nm CdTe quantum wire
   2Dwire_excitonCdTe_2x30nm.in   -    2 nm x 30 nm CdTe quantum wire
   2Dwire_excitonCdTe_3x30nm.in   -    3 nm x 30 nm CdTe quantum wire
   2Dwire_excitonCdTe_4x30nm.in   -    4 nm x 30 nm CdTe quantum wire
   2Dwire_excitonCdTe_6x30nm.in   -    6 nm x 30 nm CdTe quantum wire
   2Dwire_excitonCdTe_8x30nm.in   -    8 nm x 30 nm CdTe quantum wire
   2Dwire_excitonCdTe_10x30nm.in  -   10 nm x 30 nm CdTe quantum wire
   2Dwire_excitonCdTe_14x30nm.in  -   14 nm x 30 nm CdTe quantum wire
   2Dwire_excitonCdTe_20x30nm.in  -   20 nm x 30 nm CdTe quantum wire
 

Exciton energy in quantum wires

This tutorial aims to calculate the exciton binding energy for 2D quantum wires (QWRs).

In order to correlate the calculated optical transition energies of a 2D quantum wire (QWR) to experimental data, one has to include the exciton (electron-hole pair) corrections. In this tutorial we study the exciton correction of the electron ground state to the heavy hole ground state (e1 - hh1).

In Paul Harrison's book "Quantum Wells, Wires and Dots" (Section 6.5 "The two-dimensional and three-dimensional limits"), the exciton energy has been calculated for CdTe quantum wells, see 1D tutorial "Exciton energy in quantum wells".

Here, we calculate the exciton energy in quantum wires where we also try to approach the limits of "bulk" and "quantum well".

• Bulk
  
  The 3D bulk exciton binding energy can be calculated analytically
    E_ex,b = - µ e⁴ / ( 32 pi² h_bar² e_r² e₀²) = - µ / (m₀ e_r²) x 13.61 eV
         where µ is the reduced mass of the electron-hole pair: 1/µ = 1/m_e + 1/m_h
                 GaAs: 1/µ = 1 /  0.067 + 1 / 0.5 ==> µ = 0.0591
                 CdTe:  1/µ = 1 /  0.096 + 1 / 0.6 ==> µ = 0.0828
         h_bar is Planck's constant divided by 2pi
         e is the electron charge
         e_r is the dielectric constant (GaAs: 12.93, CdTe: 10.6)
         e₀ is the vacuum permittivity
         m₀ is the rest mass of the electron and
         13.61 eV is the Rydberg energy.
    In GaAs, the 3D bulk exciton binding energy is equal to -4.8 meV with a Bohr radius of lambda = 11.6 nm.
    In CdTe it is equal to -10.0 meV with a Bohr radius of 6.8 nm.
    Thus the energy of the exciton, i.e. band gap transition, reads
    GaAs:  E_ex = E_gap + E_ex,b = 1.519 eV - 0.005 eV = 1.514 eV.
    CdTe:  E_ex = E_gap + E_ex,b = 1.606 eV - 0.010 eV = 1.596 eV.
  
  More details on bulk excitons can be found in Section 6.1 "Excitons in bulk" (p. 181) of Paul Harrison's book "Quantum Wells, Wires and Dots".
   
• Quantum well (type-I)
  
  A 1D quantum well for a type I structure has two exciton limits for the ground state transition (e1-hh1):
  - infinitely thin quantum well (2D limit):                     E_ex,qw = 4E_ex                                   lambda_ex,qw = lambda_ex / 2
  - infinitely thick quantum well (3D bulk exciton limit): E_ex,qw =   E_ex                                  lambda_ex,qw = lambda_ex
  Between these limits, the exciton correction which depends on the well width has to be calculated numerically, not only for the ground state but also for excited states (e.g. e2 - hh2, e1 - lh1).
  
  See 1D tutorial "Exciton energy in quantum wells" for more details.
   
• CdTe quantum wire with infinite barriers
  
  In this tutorial we study the exciton binding energy of CdTe quantum wires (with infinite barriers) as a function of well width.
  
  The material parameters used are the following:
  
   !-----------------------------------------------------------!
 ! Here we are overwriting the database entries for CdTe.    !
 !-----------------------------------------------------------!
 $binary-zb-default !
  binary-type                 = CdTe-zb-default              !
  apply-to-material-numbers   = 1 2                          !
  conduction-band-masses      = 0.096d0  0.096d0  0.096d0    ! Gamma      [m0]
                                ...

  valence-band-masses         = 0.6d0    0.6d0    0.6d0      ! heavy hole [m0]
                                ...
  static-dielectric-constants = 10.6d0   10.6d0   10.6d0     !

  To keep things simple, we chose infinite barriers, although the code itself is more general, i.e. it can take into account finite barriers and different effective mass tensors for the well and barrier materials.
   
• The following figure shows the exciton binding energy in an infinitely deep CdTe quantum wire as a function of size.
  
  For the black curve, the "well width along x" is varied whereas the well width along y is kept constant, i.e. 30 nm (i.e. a rectangular N x 30 nm QWR).
  For the red curve, the wire width along x and y is varied, i.e. a square N x N nm quantum wire.
  
  
  
  Similar to the 1D tutorial "Exciton energy in quantum wells", one can see that with such a variational approach the 3D bulk exciton limit cannot be modeled correctly. For the rectangular QWR that approaches a QW that is infinitely thin, the QW exciton limit could be reached, in principle. However, this is not feasible with respect to our numerical approach because one side of the wire must be infinitely thin whereas the other must be infinitely thick.
  One can also see that for small square QWRs, the exciton binding energy diverges - in agreement with theory.
  
  The following figure shows the same data but this time on a linear scale.
  
  
   
• Our numerical approach is described in detail in:
  
         Electronic and optical properties of [N11] grown nanostructures
         M. Povolotskyi, A. Di Carlo, S. Birner
         physica status solidi (c) 1 (6), 1511 (2004)
  
  The exciton binding energy is minimized with respect to variational parameters.
  We use a separable wave function:
    g(z) = [ 2 / (a² pi) ]^1/4 exp (- z² / a²)
  where z = z_e - z_h is the distance between particles in the "non-quantized" direction and a is one of the variational parameters.
  Thus the 3D limit is not reproduced correctly in our approach (not shown in the figure).
  To obtain the 3D limit, a nonseparable wave function has to be used: psi (r,z_e,z_h)
  
  Thus, our 2D exciton code is only suited for quantum wires, i.e. we suppose that the particle motion is really confined in two directions and is free in the third direction. Therefore, the attempt to use this particular part of the code for a "QW-like structure" or for "bulk" leads to results that we cannot trust. Of course, our approach may be generalized for a structure of any dimensinality.
   
• The following figure shows the exciton Bohr radius lambda in an infinitely deep CdTe quantum wire as a function of well width.
  
  (==> insert picture)
  
  The 3D bulk exciton value of lambda in CdTe reads: lambda_ex = 6.8 nm.
   

• Influence of grid resolution of the exciton binding energy:
   

  grid resolution	1.00 nm	0.50 nm	0.25 nm	0.10 nm along x
  rectangular CdTe quantum wire
  1 nm x 60 nm ("well")				-17.62 meV (0.50 nm along y)
  1 nm x 50 nm ("well")				-19.34 meV (0.50 nm along y)
  1 nm x 40 nm ("well")				-22.07 meV (0.50 nm along y)
  1 nm x 30 nm ("well")				-25.78 meV (0.50 nm along y)
  1 nm x 30 nm ("well")				-25.21 meV (0.25 nm along y)
  2 nm x 30 nm ("well")			-24.83 meV
  3 nm x 30 nm ("well")			-23.55 meV
  4 nm x 30 nm ("well")		-23.08 meV	-22.61 meV
  6 nm x 30 nm ("well")		-21.12 meV
  8 nm x 30 nm ("well")		-19.72 meV
  10 nm x 30 nm ("wire")		-18.58 meV
  14 nm x 30 nm ("wire")		-16.73 meV
  20 nm x 30 nm ("wire")		-13.92 meV
  30 nm x 30 nm ("bulk")	-12.01 meV	-11.98 meV	-11.97 meV

   

  grid resolution	1.00 nm	0.50 nm	0.25 nm	0.10 nm
  square CdTe quantum wire
  6 nm x  6 nm ("bulk")			-45.90 meV
  8 nm x  8 nm ("bulk")			-33.24 meV
  10 nm x 10 nm ("bulk")		-28.29 meV	-27.97 meV
  14 nm x 14 nm ("bulk")		-21.92 meV
  20 nm x 20 nm ("bulk")		-16.57 meV
  30 nm x 30 nm ("bulk")	-12.01 meV	-11.98 meV	-11.97 meV
  40 nm x 40 nm ("bulk")	- 9.47 meV	- 9.46 meV
  50 nm x 50 nm ("bulk")	- 7.86 meV
  70 nm x 70 nm ("bulk")	- 6.09 meV

   

• In order to calculate the exciton correction, the following flags have to be used:
   $numeric-control
  simulation-dimension                  = 2
  calculate-exciton                     = yes ! to switch on exciton correction
  exciton-electron-state-number         = 1   ! electron ground state
  exciton-hole-state-number             = 1   ! hole      ground state
  number-of-electron-states-for-exciton = 3   ! total number of electron states to be considered for expansion
  number-of-hole-states-for-exciton     = 3   ! total number of hole       states to be considered for expansion

Here, we used three electron and three holes states for the wave function expansion, i.e. taking into account two excited electron and two excited hole states in addition to the electron and hole ground states. Obviously, it it the responsibility of the user to vary this number (and also the grid resolution) in order to ensure that the results are properly converged.
  
  
  In addition, the following specifiers have to be present:
   $output-1-band-schroedinger
  ...
  complex-wave-functions = yes ! Prints out the wave functions psi for electrons and holes (and not only psi²).
  effective-mass-tensor  = yes ! Prints out the effective mass tensors m_ij for electrons and holes.
Reason:
- The wave functions psi            for electrons and holes have to be written out because they will be read in during the 2D exciton calculation.
    (This also holds for the electron and hole energies.)
- The effective mass tensors m_ij for electrons and holes have to be written out because they will be read in during the 2D exciton calculation.

This keyword is also required.
 $output-file-format
  simulation-dimension = 2
  file-format          = AVS-ASCII
  resolution           = average   ! This specifier is necessary.
 $end_output-file-format