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nextnano3 - Tutorial

next generation 3D nano device simulator

1D Tutorial

Strained 001 and 110 AlAs/AlGaAs QW - Crossover of ground states

Author: Stefan Birner

If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
-> 1DAlAs_QW_110_DasguptaAPL2008.in
-> 1DAlAs_QW_110_DasguptaAPL2008_SchroedingerPoisson.in
 


Strained 001 and 110 AlAs/AlGaAs QW - Crossover of ground states

-> 1DAlAs_QW_110_DasguptaAPL2008.in

Note: Please also have a look at this tutorial: AlAs QW crossover I

 

This tutorial is based on the following paper:

Single valley high-mobility (110) AlAs quantum wells with anisotropic mass
S. Dasgupta, S. Birner, C. Knaak, M. Bichler, A. Fontcuberta i Morral, G. Abstreiter, M. Grayson
submitted to Applied Physics Letters (2008)

We investigate the ground state energies of the X band edges in an AlAs quantum well as a function of quantum well width.
The AlAs quantum well is embedded by Al0.45Ga0.55As barriers that are strained with respect to the GaAs substrate.
The AlAs quantum well is assumed to be strained with respect to the GaAs substrate.
For that reason, the three X band edges are split due to strain.
This splitting depends on the surface orientation of the quantum well plane, which is determined by the orientation of the GaAs substrate, i.e. on the growth direction.
We solve the single-band Schrödinger equation for the quantum well, i.e. for each X band edge, taking into account automatically substrate orientation, strain, deformation potentials, rotation of mass tensors and different effective masses for the well and barrier materials.

 

Lattice constants

The 300 K lattice constants are given by:

AlAs: lattice-constants            = 0.56611d0  0.56611d0  0.56611d0  ! [nm] 300 K
GaAs: lattice-constants            = 0.565325d0 0.565325d0 0.565325d0 ! [nm] 300 K

 

The temperature coefficients for the lattice constants are given by:

AlAs: lattice-constants-temp-coeff = 2.90d-6    2.90d-6    2.90d-6    ! [nm/K]
GaAs: lattice-constants-temp-coeff = 3.88d-6    3.88d-6    3.88d-6    ! [nm/K]

 

This yields the following values for the lattice constants of GaAs and AlAs at T = 4 K:

AlAs: 0.56611  nm + 2.90d-6 nm/K (T - 300 K) = 0.562516   nm
GaAs: 0.565325 nm + 3.88d-6 nm/K (T - 300 K) = 0.56417652 nm

 

The calculated lattice constants for Al0.45Ga0.55As and GaAs can be found in the following output file:
lattice_constants1D.dat

 

Orientation

We distinguish between
- the crystal coordinate system (xcr = [100], ycr = [010], z = [001]) and
 [Dasgupta2] "crystal-basis x"
- the simulation coordinate system (x = [1-10], y = [110], z=[001]).
 [Dasgupta2] "growth-basis a":   a = [1-10], b = [110], z=[001])

For (110) surface orientation we are using:
  $domain-coordinates
   ...
   hkl-x-direction-zb     = 1 -1  0 !
for (110) surface ! along the [1-10] direction in the crystal coordinate system
   hkl-y-direction-zb     = 1  1  0 !
for (110) surface ! along the [110]  direction in the crystal coordinate system
 ! hkl-z-direction-zb     = 0  0  1 !              along the [001]  direction in the crystal coordinate system

   growth-coordinate-axis = 0 1 0 ! growth coordinate axis is y axis
  
Our growth direction is the y=[110] direction ([Dasgupta2] b=[110] direction).

For (010) surface orientation we are using:
   hkl-x-direction-zb     = 1  0  0 ! for (010) surface ! along the [100]  direction in the crystal coordinate system
   hkl-y-direction-zb     = 0  1  0 !
for (010) surface ! along the [010]  direction in the crystal coordinate system
 ! hkl-z-direction-zb     = 0  0  1 !              along the [001]  direction in the crystal coordinate system
Note: For (010) surface orientation, both crystal and simulation coordinate systems are identical.

 

Strain

AlAs is strained biaxially with respect to the GaAs substrate's lattice constant, and is thus subject to a compressive strain.

The elastic constants of AlAs are given by:
 elastic-constants = 125.0d0 53.4d0 54.2d0 ! [GPa] c11,c12,c44

 

The strain tensor components are thus given by:

   Biaxial strain (in plane of interface, i.e. in (x,z) plane):
   For (110) and (010) surface orientation:
      AlAs: e|| = exx,sim = ezz,sim = ( asubstrate - alayer ) / alayer = ( aGaAs - aAlAs ) / aAlAs = -0.00190195   (-0.19 % lattice mismatch)

   Uniaxial strain (perpendicular to interface, i.e. along y direction):

   For (110) surface orientation:
      AlAs: e_|_ = eyy,sim = - D110 e|| = - (c11+3c12-2c44) / (c11+c12+2c44) e|| = +0.0011725 (+0.12 % lattice mismatch)
      [D110 = 0.616457461]

   For (010) surface orientation:
      AlAs: e_|_ = eyy       = - D010 e|| = - 2 (c12/c11) e|| = +0.001625 (+0.16 % lattice mismatch)
      [D010 = 0.8544]

For the (110) orientation, the hydrostatic strain is given by:
   ehydro = Tr(eij) = exx,sim + eyy,sim + ezz,sim = 2 e|| + e_|_ = -0.00263 (-0.26 % compressive strain, decrease in volume) = exx,cr + eyy,cr + ezz,cr
For the (010) orientation, the hydrostatic strain is given by:
   ehydro = Tr(eij) = exx      + eyy       + ezz       = 2 e|| + e_|_  = -0.00218 (-0.22 % compressive strain, decrease in volume)

The off-diagonal strain tensor components are zero for both orientations with respect to the simulation coordinate system, i.e. the strain tensor is diagonal.
However, the strain tensor with respect to the crystal coordinate system has a nonzero offdiagonal component for the (110) surface orientation.
exx,cr = -0.00036474 (-0.04 % lattice mismatch)
eyy,cr = -0.00036474 (-0.04 % lattice mismatch)
ezz,cr = -0.00190195 (-0.19 % lattice mismatch)
exy,cr =  0.00153721 (-0.15 % lattice mismatch)
exz,cr =  0
eyz,cr =  0

The values for exx,cr and eyy,cr can also be obtained by rotating the strain tensor (see [Dasgupta2]).
For (110) surface orientation it holds:
exx,cr = eyy,cr = ( e|| + e_|_ ) / 2 = -0.00036474 (-0.04 % lattice mismatch)
ezz,cr = ezz,sim = e||                 = -0.00190195 (-0.19 % lattice mismatch)

The calculated strain tensor components (including hydrostatic strain) can be found in the following files:
 - strain1/strain_cr1D.dat  - strain tensor components with respect to the crystal coordinate system
 - strain1/strain_sim1D.dat - strain tensor components with respect to the simulation coordinate system
(For (010) surface orientation, both files are identical.)

 

X conduction band edges: Shift and splitting due to strain (deformation potential)

The deformation potential for the X conduction band edge in AlAs is Xi = 6.11 eV.
   uniax-cb-deformation-potentials = 0d0 ...d0 6.11d0 ! [eV] Gamma,L,X

There are three conduction band edges at the X points:

  • EX,x : longitudinal mass oriented along x=[100] direction
  • EX,y : longitudinal mass oriented along y=[010] direction
  • EX,z : longitudinal mass oriented along z=[001] direction

The X conduction band edges shift as follows:

  • (110) surface orientation:
    Delta EX,x = Xi exx,cr = 6.11 eV * (-0.00036474) =   -2.22 meV
    Delta EX,y = Xi exx,cr = 6.11 eV * (-0.00036474) =   -2.22 meV
    Delta EX,z = Xi exx,cr = 6.11 eV * (-0.00190195) = -11.62 meV
    ==> The X conduction band edge where the longitudinal mass is oriented along the z=[001] orientation is the lowest in energy.
    The other two bands have equal energy. Thus they are double-degenerate and separated by 9.4 meV from the single-degenerate band.
  • (010) surface orientation:
    Delta EX,x = Xi exx,cr = 6.11 eV * (-0.00190195) = -11.62 meV
    Delta EX,y = Xi exx,cr = 6.11 eV * (+0.001625)   =   +9.93 meV
    Delta EX,z = Xi exx,cr = 6.11 eV * (-0.00190195) = -11.62 meV
    ==> The X conduction band edges where the longitudinal masses are oriented along the x=[100] or z=[001] orientation are the lowest in energy.
    These two bands have equal energy. Thus they are double-degenerate.
    The single-degenerate band where the londitudinal mass is oriented parallel to the growth direction y=[010] is separated by 21.6 meV from the double-degenerate bands.

 

Single-band effective mass tensors for the X valleys

The mass tensors for the X valleys are ellipsoids characterized by a longitudinal and two transverse masses.
For AlAs we use the following parameters.

 conduction-band-masses = ...    ...    ...                  ! [m0] Gamma  m  m  m    (isotropic)
                          ...    ...    ...                  ! [m0] L      ml mt mt 
(anisotropic)
                          1.1d0  0.2d0  0.2d0  ! [Dasgupta2] ! [m0] X      ml mt mt 
(anisotropic)

  • mX,x : longitudinal mass oriented along x=[100] direction
  • mX,y : longitudinal mass oriented along y=[010] direction
  • mX,z : longitudinal mass oriented along z=[001] direction

 

To discretize the Schrödinger equation along the growth direction, it is necessary to rotate the inverse mass tensors into the simulation coordinate system (see [Dasgupta2]).

For (110) surface orientation, the inverse mass tensors with respect to the simulation coordinate system are given as follows

  • mX,x : longitudinal mass oriented along x=[100] direction

                (  1/m  1/mf     0    )   (1/0.3385   1/0.4889   0    )
    mX,x,simij-1 = (  1/mf    1/m   0    ) = (1/0.4889   1/0.3385   0    )
                (   0      1/m  1/mt  )   (  0          0      1/0.2  )
     
  • mX,y : longitudinal mass oriented along y=[010] direction

                (   1/m -1/m  0    )   ( 1/0.3385 -1/0.4889   0    )
    mX,y,simij-1 = ( -1/m   1/md    0    ) = (-1/0.4889  1/0.3385   0    )
                (   0       1/m 1/mt  )   (   0         0      1/0.2  )
     
  • mX,z : longitudinal mass oriented along z=[001] direction (unaffected by rotation!)

                (  1/m   0     0     )   (  1/0.2       0        0    )
    mX,z,simij-1 = (   0     1/mt    0     ) = (    0       1/0.2      0    )
                (   0       0     1/ml  )   (    0         0      1/1.1  )
     

where md=2(mlmt)/(mt+ml)=0.3385[m0] and mf=2(mlmt)/(mt-ml)=0.4889[m0].

The mass tensor components 1/myy indicated in red are the ones the enter the discretized Schrödinger equation.
Remember: The growth direction is along the y direction in the simulation coordinate system (i.e. along the [110] direction in the crystal coordinate system).

 

For (010) surface orientation, no rotation is necessary, so the mass tensors are diagonal and given by:

  • mX,x : longitudinal mass oriented along x=[100] direction

             ( 1/ml   0     0   )   ( 1/1.1   0     0   )
    mX,xij-1 = (  0    1/mt   0   ) = (  0    1/0.2   0   )
             (  0     0    1/mt )   (  0          1/0.2 )
     
  • mX,y : longitudinal mass oriented along y=[010] direction

             ( 1/mt   0     0   )   ( 1/0.2   0     0   )
    mX,yij-1 = (  0    1/ml   0   ) = (  0    1/1.1   0   )
             (  0     0    1/mt )   (  0          1/0.2 )
     
  • mX,z : longitudinal mass oriented along z=[001] direction

             ( 1/mt   0     0   )   ( 1/0.2   0     0   )
    mX,zij-1 = (  0    1/mt   0   ) = (  0    1/0.2   0   )
             (  0     0    1/ml )   (  0          1/1.1 )
     

The calculated inverse mass tensor components can be found in the following files:
 - Schroedinger_1band/mass_tensor1D_cb003_qc001_sg001_deg001.dat - contains  mX,x,sim and mX,y,sim  for (110) surface orientation
                                                         
contains  mX,x     and mX,z      for (010) surface orientation
 - Schroedinger_1band/mass_tensor1D_cb003_qc001_sg002_deg001.dat -
contains  mX,z,sim                     for (110) surface orientation
                                                          contains  mX,y                    for (010) surface orientation

 

Cross-over width Wc

The following figures show the calculated ground state energies of the two X valleys.
From these figures one can extract the cross-over width Wc for the (110) and the (010) strained AlAs QW.
The experimentally observed cross-over is 5.5 nm for the (010) QW.
For (110) strained AlAs QWs, our calculations of strain and quantum confinement predict anisotropic-mass valley occupancy for well widths greater than 5.3 nm.
Below this, double-valley occupation is predicted.
The dashed, grey lines show the calculated ground state energies for the infinite barrier model (analytical model). In that case, the cross-over width is greatly overestimated.

The respective eigenvalues can be found in these files:
  - Schroedinger_1band/ev1D_cb003_qc001_sg001_deg001_dir_Kx001_Ky001_Kz001.dat - (2fold degenerate)
  - Schroedinger_1band/ev1D_cb003_qc001_sg002_deg001_dir_Kx001_Ky001_Kz001.dat - (1fold degenerate)
Information about the numbering and the degeneracy of the Schrödinger equations can be found in this file:
  - Schroedinger_1band/sg_info.txt

 

 

Strained 001 and 110 AlAs/AlGaAs QW - self-consistent Schrödinger-Poisson calculation

-> 1DAlAs_QW_110_DasguptaAPL2008_SchroedingerPoisson.in

The following figure shows the self-consistently calculated conduction band profile of the (110) sample as presented schematically in Fig. 1 of the paper by Dasgupta et al. [Dasgupta2].
Three conduction band edges are shown:
- The Gamma conduction band (dashed-dotted line) is the highest in energy for AlAs and Al0.45Ga0.55As, but the lowest for GaAs.
- The X conduction bands are split due to strain.
  The Xz condution band edge is the lowest in energy where the longitudinal mass is oriented along the z=[001] direction.
  The growth direction is along y=[110].

The inset shows the square of the ground state wave function of the 15 nm wide, strained AlAs quantum well.

This band structure has been obtained by solving self-consistently the Schrödinger and the Poisson equation.

 

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