nextnano³ - Tutorial

next generation 3D nano device simulator

1D Tutorial

Exciton energy in quantum wells

Author: Stefan Birner

-> 1DExcitonCdTe_QW.in  - CdTe quantum well with infinite barriers

Exciton energy in quantum wells

This tutorial aims to reproduce the figures 6.4 (p. 196) and 6.5 (p. 197) of Paul Harrison's excellent book "Quantum Wells, Wires and Dots" (Section 6.5 "The two-dimensional and three-dimensional limits"), thus the following description is based on the explanations made therein.
We are grateful that the book comes along with a CD so that we were able to look up the relevant material parameters and to check the results for consistency.

• In order to correlate the calculated optical transition energies of a 1D quantum well 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).
  
• 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).
  
• CdTe quantum well with infinite barriers
  
  In this tutorial we study the exciton binding energy of CdTe quantum wells (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   = 2                            !
  conduction-band-masses      = 0.096    0.096    0.096      ! Gamma      [m0]
                                ...

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

  We chose infinite barriers, in order to be able to compare the nextnano calculations with standard textbook results, originally published by G. Bastard et al. (Phys. Rev. B 26 (4), 1974 (1982)), namely the exciton binding energy of a type-I quantum well (in units of the 3D bulk exciton energy E_ex, also called effective Rydberg energy) as a function of well width (in units of the 3D bulk exciton Bohr radius lambda_ex).
  
  
• Template
  
  The following screenshot shows how to use the Template feature of nextnanomat in order to calculate the exciton binding energy as a function of the quantum well width.
  
  
  
• Open template input file (...).
• Select "List of values", select variable "QuantumWellWidth". The corresponding list of values are loaded from the template input file.
• Click on "Create input files" to create an input file for each quantum well width.
• Switch to "Run" and the jobs are executed.

Results

• The following figure shows the exciton binding energy in an infinitely deep quantum well as a function of well width.
  Both quantities are given in terms of the effective Rydberg energy and the Bohr radius for a 3D exciton in the same material.
  
  
  
  Our numerical approach is the following:
  The exciton binding energy is minimized with respect to the variational parameter lambda.
  We use a separable wave function:
    psi (r) = SQRT(2/pi) 1/lambda exp (- r / lambda)
  see e.g. S.L. Chuang, "Physics of Optoelectronic Devices", Wiley, p. 562, Eq. (13.4.27), Section 13.4.3 "Variational Method for Exciton Problem" or G. Bastard et al., PRB 26, 1974 (1982)
  
  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)
  
• The following figure shows the exciton binding energy in an infinitely deep CdTe quantum well as a function of well width.
  The nextnano³ results are in nice agreement with the Fig. 6.4 of Paul Harrison's book although we use a simpler trial wave function with only one variational parameter.
  
  
  
• The following figure shows the exciton Bohr radius lambda in an infinitely deep CdTe quantum well as a function of well width.
  The nextnano³ results for well widths smaller than 10 nm are in nice agreement with Fig. 6.5 of Paul Harrison's book.
  The discrepancy arises because he uses a different trial wave functions (i.e. a superior approach) with a second variational parameter in addition to lambda.
  The 3D bulk value of lambda in CdTe reads: lambda_ex = 6.8 nm.
  
  
  
• In order to calculate the exciton correction, the following flags have to be used:
   $numeric-control
  simulation-dimension          = 1
  calculate-exciton             = yes   ! to switch on exciton correction
  exciton-electron-state-number = 1     ! electron ground state
  exciton-hole-state-number     = 1     ! hole      ground state
  
• The output of the exciton binding energies can be found in this file:
  Schroedinger_1band/exciton_energy1D.dat

The output for the 5 nm CdTe QW looks like this:

 Exciton correction for 1D quantum wells (type-I structures)
 ===========================================================
 static dielectric constant     =  10.6000000000 []
 effective mass electron        =   0.0960000000 [m0]
 effective mass hole            =   0.6000000000 [m0]
 reduced mass                   =   0.0827586207 [m0]
 Bulk Bohr radius of 3D exciton =   6.7778780735 [nm]
 Bulk 3D exciton energy         = -10.0212560410 [meV]

 lambda [nm]        exciton energy [meV]   exciton energy [Rydberg]
 0.338893904E+001   -0.158496790E+002      0.158160603E+001
 ...
 0.421888329E+001   -0.215591082E+002      0.215133793E+001
 ...
 0.553296169E+001   -0.232757580E+002      0.232263879E+001
 -----------------------------------------------------------------
 -----------------------------------------------------------------
 Calculated lambda and exciton energy:
 0.546379967E+001   -0.232817837E+002      0.232324009E+001
 -----------------------------------------------------------------

The last iteration yields -23.28 meV for the exciton binding energy.
  "lamba" is the variational parameter which is equivalent to the exciton Bohr radius in units of [nm].