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nextnano3 - Tutorial

next generation 3D nano device simulator

1D Tutorial

Optical intersubband transitions in a quantum well - Intraband matrix elements and selection rules

Author: Stefan Birner

If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
-> 1DQW_intraband_matrixelements_infinite_nn3.in / *_nnp.in - input files for the nextnano3 and nextnano++ software
-> 1DQW_intraband_matrixelements_infinite_kp_nn3.in


Optical intersubband transitions in a 10 nm AlAs / GaAs / AlAs quantum well - Intraband matrix elements and selection rules

Eigenstates and wave functions in the quantum well

  • We consider a 10 nm GaAs quantum well embedded between AlAs barriers. The structure is assumed to be unstrained.
    We assume "infinite" AlAs barriers. (This can be achieved by chosing a band offset of 100 eV.)
    This way we can compare our results to analytical text books results.
  • -> 1DQW_intraband_matrixelements_infinite_nn3.in

     $output-1-band-schroedinger
      ...
      scale                     = 0.3d0 !
    scale the wave function psi and psi² so that it is easier to visualize them
      intraband-matrix-elements = z     !
    calculate intersubband dipole moment  | < psif* | z  | psii > |  and oscillator strength ffi
     !intraband-matrix-elements = p     !
    calculate intersubband dipole moment  | < psif* | pz | psii > |  and oscillator strength ffi
     
    complex-wave-functions    = yes   !
    to print out psi in addition to psi²

    The figure shows the six lowest eigenfunctions of the 1D GaAs quantum well. The conduction band edge of GaAs is assumed to be located at 0 eV.



    For "infinite" barriers we obtain using single-band Schrödinger effective-mass approximation (i.e. isotropic and parabolic effective masses) the following eigenvalues:

    E1 = 0.05652 eV    (0.05655)
    E2 = 0.22601 eV    (0.22618 = 2² E1)
    E3 = 0.50831 eV    (0.50891 = 3² E1)
    E4 = 0.90314 eV    (0.90473 = 4² E1)
    E5 = 1.41011 eV    (1.41365 = 5² E1)
    E6 = 2.02872 eV    (2.03565 = 6² E1)

    The analytic formula in the infinite barrier QW model reads: En = hbar²/2m* (pi n / L )² = 0.056546 n²
    where L is the width of the quantum well (L = 10 nm). The analytically calculated values are given in brackets and are in excellent agreement.

    We used an electron effective mass of 0.0665 m0 and a 0.1 nm grid resolution.
    We used:
     schroedinger-1band-ev-solv      = LAPACK-ZHBGVX ! 'LAPACK', 'LAPACK-ZHBGVX', 'arpack', 'it_jam', 'chearn'
     schroedinger-masses-anisotropic = box           ! 'yes', 'no', 'box'

 

 

Intersubband matrix elements

  • Light that propagates normal to the quantum well layers cannot be absorbed by intraband transitions.

    However, if the light propagates in the plane of the well (i.e. the electric field is oriented normal to the quantum well layers), intersubband absorption occurs.

    To understand optical intersubband (or intraband) transitions for light that travels in the plane of the QW, we have to examine the intersubband dipole moment

        | Mfi | = | integral  (psif* (z) z psii (z) dz) |

    where psi is the envelope function of the relevant state (within the same band).

    In our case, we have a symmetric quantum well with infinite barriers, thus our envelope functions are either symmetric or antisymmetric. The intersubband matrix elements will vanish if the envelope functions have the same parity, e.g. | M13 | = | M31 | = 0.
    In this simple example, the matrix elements can be calculated analytically, e.g. | M12 | = 16 / (9 pi²) L  = 1.8013 [nm].
    nextnano³ result: | M12 | = | M21 | = 1.8017 [nm]
                              | M13 | = | M31 | = 0.00000006 [nm]

    For the "infinite" QW barrier model, this matrix element is independent of the effective mass, thus the matrix elements in the conduction band are the same as in the valence bands (single-band approximation).

    A useful quantity is the oscillator strength ffi which is defined as follows:

        ffi = 2m* / hbar² (Ef - Ei)  | Mfi
        ffi = - fif

    f21 for our simple infinite barrier example is given by
       f21 = 256 / (27 pi²) = 0.9607
    and is independent of the well width.
    The nextnano³ result is:  f21 = 0.9603 = - f12

    We can also see that this is a strong transition because all transitions from state '1' to state 'f' must add up to unity (so-called "f-sum rule"):
        sumf (ff1) = 1.0            (Thomas-Kuhn sum rule for constant effective mass m*.)
    Thus all other transitions are much weaker.

    It is interesting to look at the transitions starting from the second level i = 2. The lowest oscillator strength f12 = - 0.96 is negative, but the sum over all ff2 must still give unity, thus oscillator strengths larger than 1 are possible, e.g. f32 = 1.87.
  • The intersubband dipole moments and the oscillator strenghts are contained in this file:
      Schroedinger_1band/intraband_z1D_cb001_qc001_sg001_deg001_dir.txt - Gamma conduction band

    For each transition, the transition energy is given.

    The effective masses that have been used for the calculation of the oscillator strengths are also indicated. They are calculated by building an average of the parallel effective masses for each grid point, weighted by the square of the wave function on each grid point. In this particular example, the effective masses are constant and do not vary with position (m|| = 0.0665 m0).
    (Assuming that the masses are isotropic, it is fine to use the parallel effective masses.)


    -------------------------------------------------------------------------------
    Intersubband transitions
     => Gamma conduction band
    -------------------------------------------------------------------------------
    Electric field in z-direction [kV/cm]: 0.0000000E+00
    -------------------------------------------------------------------------------

    -------------------------------------------------------------------------------
                       Intersubband dipole moment  | < psi_f* | z | psi_i > |  [Angstrom]
    ------------------|------------------------------------------------------------
                                      Oscillator strength []
    ------------------|--------------|---------------------------------------------
                                                     Energy of transition [eV]
    ------------------|--------------|--------------|------------------------------
                                                                m* [m_0]
    ------------------|--------------|--------------|----------|-------------------
    <psi001*|z|psi001>  249.0000
    <psi002*|z|psi00118.01673       0.9602799      0.1694912  6.6500001E-02
    <psi003*|z|psi001>  6.1430171E-07  2.9757722E-15  0.4517909  6.6500001E-02
    (same parity: symmetric)
    <psi004*|z|psi001>  1.441336       3.0698571E-02  0.8466209  6.6500001E-02
    <psi005*|z|psi001>  1.6007220E-07  6.0536645E-16  1.353592   6.6500001E-02
    (same parity: symmetric)
    <psi006*|z|psi001>  0.3971010      5.4281605E-03  1.972205   6.6500001E-02
    <psi007*|z|psi001>  5.1874160E-08  1.2690011E-16  2.701849   6.6500001E-02
    (same parity: symmetric)
    <psi008*|z|psi001>  0.1634139      1.6508275E-03  3.541806   6.6500001E-02
     ...
    <psi020*|z|psi001>  1.0178176E-02  3.9451432E-05  21.81846   6.6500001E-02
    Sum rule of oscillator strength: f_psi001 = 0.9994023

    <psi001*|z|psi00218.01673      -0.9602799     -0.1694912  6.6500001E-02
    <psi002*|z|psi002>  249.0000
    <psi003*|z|psi00219.45806       1.865556       0.2822997  6.6500001E-02
    <psi004*|z|psi002>  2.0636767E-06  5.0333130E-14  0.6771297  6.6500001E-02
    (same parity: antisymmetric)
    <psi005*|z|psi002>  1.838436       6.9852911E-02  1.184101   6.6500001E-02
    <psi006*|z|psi002>  1.4976163E-08  7.0571038E-18  1.802713   6.6500001E-02
    (same parity: antisymmetric)
    <psi007*|z|psi002>  0.5605143      1.3886644E-02  2.532358   6.6500001E-02
    <psi008*|z|psi002>  8.7380023E-08  4.4941879E-16  3.372315   6.6500001E-02
    (same parity: antisymmetric)
    <psi009*|z|psi002>  0.2461317      4.5697703E-03  4.321757   6.6500001E-02
    <psi010*|z|psi002>  8.3240280E-07  6.5062044E-14  5.379748   6.6500001E-02
    (same parity: antisymmetric)
    <psi011*|z|psi002>  0.1302904      1.9393204E-03  6.545245   6.6500001E-02
     ...
    <psi020*|z|psi002>  2.7233656E-07  2.8025147E-14  21.64897   6.6500001E-02
    Sum rule of oscillator strength: f_psi002 = 0.9975320

    <psi001*|z|psi003>  6.1430171E-07 -2.9757722E-15 -0.4517909  6.6500001E-02
    (same parity: symmetric)
    <psi002*|z|psi00319.45806      -1.865556      -0.2822997  6.6500001E-02
    <psi003*|z|psi003>  249.0000
    <psi004*|z|psi00319.85515       2.716784       0.3948300  6.6500001E-02
    <psi005*|z|psi003>  6.4708888E-07  6.5907892E-15  0.9018011  6.6500001E-02
    (same parity: symmetric)
    <psi006*|z|psi003>  2.001849       0.1063465      1.520414   6.6500001E-02
    <psi007*|z|psi003>  3.9201248E-07  6.0352080E-15  2.250058   6.6500001E-02
    (same parity: symmetric)
    <psi008*|z|psi003>  0.6432316      2.2314854E-02  3.090015   6.6500001E-02
    <psi009*|z|psi003>  2.6240454E-07  4.8547223E-15  4.039457   6.6500001E-02
    (same parity: symmetric)
     ...
    <psi020*|z|psi003>  3.1797737E-02  3.7707522E-04  21.36667   6.6500001E-02
    Sum rule of oscillator strength: f_psi003 = 0.9945912

    ...
  • The commonly used
                      Intersubband dipole moment
      | < psi_f* | z | psi_i > |  [nm]

    depends on the choice of origin for the matrix elements when f = i, thus the user might prefer to output the
                      Intersubband dipole moment  | < psi_f* | p | psi_i > |  [h_bar / nm]
    which are the intersubband dipole moments

        | Nfi | = | integral  (psif* (z) pz psii (z) dz) | = | - i hbar integral  (psif* (z) d/dz psii (z) dz) |

    and the oscillator strengths

        ffi = 2m* / hbar² (Ef - Ei)  | Mfi  = 2 / ( m* (Ef - Ei) )  | Nfi

     between all calculated states in each band from eigenvalues 'min-ev' to 'max-ev'.

    In the simple QW of this tutorial, the matrix elements can be calculated analytically, e.g. | N21 | = 8 hbar / (3 L)   = 26.66 hbar /nm.
    nextnano³ result: | N21 | = | N12 | = 2.665 hbar /Angstrom
                              | N31 | = | N13 | = 0

    Here, the definition of the oscillator strength ffi has to be adjusted slightly:

        ffi = 2m* / hbar² (Ef - Ei)  | Mfi = 2 / ( m* (Ef - Ei) )  | Nfi

        ffi = - fif

    f21 for our simple infinire barrier example is given by
       f21 = 256 / (27 pi²) = 0.9607
    and is independent of the well width.
    The nextnano³ result is:  f21 = 0.9603 = - f12


    The intersubband dipole moments and the oscillator strenghts are contained in this file:
      Schroedinger_1band/intraband_p1D_cb001_qc001_sg001_deg001_dir.txt - Gamma conduction band

    The numbers show a comparison betwenn the z and the pz matrix elements (in green).

    -------------------------------------------------------------------------------
                       Intersubband dipole moment  | < psi_f* | z | psi_i > |  [Angstrom]
                       Intersubband dipole moment  | < psi_f* | p | psi_i > |  [h_bar / Angstrom]
    ------------------|------------------------------------------------------------
                                      Oscillator strength []
    ------------------|--------------|---------------------------------------------
                                                     Energy of transition [eV]
    ------------------|--------------|--------------|------------------------------
                                                                 m* [m_0]
    ------------------|--------------|--------------|-----------|------------------
    <psi001*|z|psi001>  249.0000  
            (matrix element <1|1> depends on choice of origin!)
    <psi001*|p|psi001>  4.3405972E-19 
    (matrix element <1|1> independent of origin)

    <psi002*|z|psi00118.01673       0.9602799      0.1694912  6.6500001E-02
    <psi002*|p|psi001>  2.6649671E-02  0.9602799      0.1694912  6.6500001E-02

    <psi003*|z|psi001>  6.1430171E-07  2.9757722E-15  0.4517909  6.6500001E-02
    (same parity: symmetric)
    <psi003*|p|psi001>  2.7325134E-18

    <psi004*|z|psi001>  1.441336       3.0698571E-02  0.8466209  6.6500001E-02
    <psi004*|p|psi001>  1.0649348E-02  3.0698579E-02  0.8466209  6.6500001E-02

    <psi005*|z|psi001>  1.6007220E-07  6.0536645E-16  1.353592   6.6500001E-02
    (same parity: symmetric)
    <psi005*|p|psi001>  6.9518724E-18

    <psi006*|z|psi001>  0.3971010      5.4281605E-03  1.972205   6.6500001E-02
    <psi006*|p|psi001>  6.8347314E-03  5.4281540E-03  1.972205   6.6500001E-02

    <psi007*|z|psi001>  5.1874160E-08  1.2690011E-16  2.701849   6.6500001E-02
    (same parity: symmetric)
    <psi007*|p|psi001>  2.8686024E-19

    <psi008*|z|psi001>  0.1634139      1.6508275E-03  3.541806   6.6500001E-02
    <psi008*|p|psi001>  5.0510615E-03  1.6508278E-03  3.541806   6.6500001E-02
     ...
    <psi020*|z|psi001>  1.0178176E-02  3.9451432E-05  21.81846   6.6500001E-02
    <psi020*|p|psi001>  1.9380626E-03  3.9452334E-05  21.81846   6.6500001E-02

    Sum rule of oscillator strength
    : f_psi001 = 0.9994023
    Sum rule of oscillator strength: f_psi001 = 0.9994023
      ...

8-band k.p calculation for k|| = (kx,ky) = 0

  • The following input file performs the same calculations as above but this time using the 8-band k.p model.
    -> 1DQW_intraband_matrixelements_infinite_kp_nn3.in
    We modified the 8-band k.p parameters and decoupled (!) the electrons from the holes (EP = 0 eV, S = 1/me). This way we have an effective single-band model and thus we are able to compare the k.p results to the single-band results in order to check for consistency.
  • The numbering of the k.p eigenstates differs slightly from the single-band eigenstates because the k.p eigenstates are two-fold spin-degenerate. The actual values for the matrix elements are identical (assuming a decoupled k.p Hamiltonian, i.e. a single-band Hamiltonian).
  • Note that the single-band definition of the oscillator strength does not really make sense for a k.p calculation where the masses usually are anisotropic, nonparabolic and are different on each grid point (due to different materials and different strain tensors).
    For the calculation of the oscillator strength in a k.p calculation, the user can specify suitable masses by overwriting the default entries:

     conduction-band-masses = 0.0665d0 0.0665d0 0.0665d0 ! Gamma band (only used for oscillator strength in k.p)
                              1.32d0   0.15d0   0.15d0   !
    L band (ignored in k.p)
                              0.97d0   0.22d0   0.22d0   !
    X band (ignored in k.p)

     valence-band-masses    = 0.500d0  0.500d0  0.500d0  !
    heavy hole (only used for oscillator strength in k.p)
                              0.068d0  0.068d0  0.068d0  !
    light hole (ignored in k.p)
                              0.172d0  0.172d0  0.172d0  !
    split-off hole (ignored in k.p)

    Of course, the masses that are used to calculate the k.p eigenstates have to be specified via the 6-band and 8-band k.p parameters.
  • The intersubband dipole moments and the oscillator strenghts are contained in this file:
      Schroedinger_kp/intraband_p1D_cb001_qc001_8x8kp_dir.txt - Gamma conduction band
               intraband_z1D_cb001_qc001_8x8kp_dir.txt -
    Gamma conduction band

    Note that the two-fold spin-degeneracy in single-band is counted explicitely in k.p.

    -------------------------------------------------------------------------------
                       Intersubband dipole moment  | < psi_f* | z | psi_i > |  [Angstrom]
                       Intersubband dipole moment  | < psi_f* | p | psi_i > |  [h_bar / Angstrom]
    ------------------|------------------------------------------------------------
                                      Oscillator strength []
    ------------------|--------------|---------------------------------------------
                                                     Energy of transition [eV]
    ------------------|--------------|--------------|------------------------------
                                                                 m* [m_0]
    ------------------|--------------|--------------|-----------|------------------
    <psi001*|z|psi001>  249.0000      
    (matrix element <1|1> depends on choice of origin!)
    <psi002*|z|psi001>  249.0000       (matrix element <2|1> depends on choice of origin!)
    <psi001*|p|psi001 1.8126842E-18 
    (matrix element <1|1> independent of origin)
    <psi002*|p|psi0011.8126842E-18 
    (matrix element <2|1> independent of origin)

    <psi003*|z|psi001>  18.01673       0.9602799      0.1694912  6.6500001E-02
    <psi004*|z|psi001>  18.01673       0.9602799      0.1694912  6.6500001E-02
    <psi003*|p|psi001>  2.6649671E-02  0.9602798      0.1694912  6.6500001E-02
    <psi004*|p|psi001>  2.6649671E-02  0.9602798      0.1694912  6.6500001E-02

    <psi005*|z|psi001>  3.5382732E-13
    <psi006*|z|psi001>  3.5382732E-13
    <psi005*|p|psi001>  2.1414240E-15
    <psi006*|p|psi001>  2.1414240E-15

    <psi007*|z|psi001>  1.441336       3.0698583E-02  0.8466209  6.6500001E-02
    <psi008*|z|psi001>  1.441336       3.0698583E-02  0.8466209  6.6500001E-02
    <psi007*|p|psi001>  1.0649348E-02  3.0698583E-02  0.8466209  6.6500001E-02
    <psi008*|p|psi001>  1.0649348E-02  3.0698583E-02  0.8466209  6.6500001E-02

    <psi009*|z|psi001>  7.2598817E-13
    <psi010*|z|psi001>  7.2598817E-13
    <psi009*|p|psi001>  1.0445775E-14
    <psi010*|p|psi001>  1.0445775E-14

    <psi011*|z|psi001>  0.3971008      5.4281550E-03  1.972205   6.6500001E-02
    <psi012*|z|psi001>  0.39710
    08      5.4281550E-03  1.972205   6.6500001E-02
    <psi011*|p|psi001>  6.8347319E-03  5.4281550E-03  1.972205   6.6500001E-02
    <psi012*|p|psi001>  6.8347319E-03  5.4281550E-03  1.972205   6.6500001E-02

     ...
    <psi039*|z|psi001>  1.0178294E-02  3.9452352E-05  21.81846   6.6500001E-02
    <psi040*|z|psi001>  1.0178
    294E-02  3.9452352E-05  21.81846   6.6500001E-02
    <psi039*|p|psi001>  1.9380630E-03  3.9452349E-05  21.81846   6.6500001E-02
    <psi040*|p|psi001>  1.9380630E-03  3.9452349E-05  21.81846   6.6500001E-02

    Sum rule of oscillator strength
    : f_psi001 = 0.9994023
    Sum rule of oscillator strength: f_psi001 = 0.9994023

    (The deviations from the single-band calculation are indicated in red.)

    We used:
     schroedinger-kp-ev-solv        = LAPACK-ZHBGVX   ! 'LAPACK', 'LAPACK-ZHBGVX', 'arpack', 'it_jam', 'chearn'
     schroedinger-kp-discretization = box-integration ! 'finite-differences', 'box-integration'
  • Please help us to improve our tutorial! Send comments to support [at] nextnano.com.