nextnano³ - Tutorial

next generation 3D nano device simulator

1D Tutorial

Growth of layers on strained (or stressed) substrates (biaxial and uniaxial)

Authors: Stefan Birner, Michael Povolotskyi

-> 1D_GaAs_biaxial_on_InAs001.in
-> 1D_InAs_on_biaxially_strainedGaAs001.in
-> 1D_InAs_on_uniaxially_stressedGaAs001.in
-> 1D_InAs_on_uniaxially_stressedGaAs001_input_stress.in
-> 1D_InAs_on_biaxially_stressedGaAs001.in
-> 1D_InAs_on_biaxially_stressedGaAs001_input_stress.in

These input files are included in the latest version.

Growth of layers on strained (or stressed) substrates (biaxial and uniaxial)

In this tutorial we show how to input a fixed strain tensor for a semiconductor starting layer in addition to its bulk lattice constants.
All other semiconductor layers that are grown on top are pseudomorphically strained with respect to the strained substrate layer.
In other words, the strain tensor of one specific layer is kept fixed whereas in all other layers the elastic energy is going to be minimized.

Material parameters of GaAs and InAs

Lattice constants
GaAs: a = 0.565325 nm
InAs: a = 0.60583  nm

Elastic constants
GaAs: c11 = 122.1  GPa      c12 = 56.6  GPa
InAs: c11 =  83.29 GPa      c12 = 45.26 GPa

a) Biaxially strained GaAs on InAs(001) substrate

-> 1D_GaAs_biaxial_on_InAs001.in

If one assumes that GaAs is grown pseudomorphically with respect to an unstrained InAs(001) substrate along the z direction, then the following strain tensor components are obtained:

e_xx = e_yy = e_|| = ( a_substrate - a_layer ) / a_layer = 0.071649051

e_zz = e_{_|_} = - D⁰⁰¹ e_|| = - 2 (c₁₂/c₁₁) e_|| = - 0.066426474

e_hydro = Tr(e) = e_xx + e_yy + e_zz = 0.076871628

Here, GaAs is biaxially strained and the overall strain is tensile (i.e. the hydrostatic strain ( Tr(e_ij) ) is positive) because GaAs has a smaller lattice constant than InAs.

b) InAs on biaxially strained GaAs(001) substrate

-> 1D_InAs_on_biaxially_strainedGaAs001.in

We now grow InAs pseudomorphically on a strained GaAs substrate where the GaAs substrate has the following strain tensor components:

$strain-minimization-model
 ...
 input-substrate-strain      = yes             ! 'yes' = read in strain tensor components of substrate

 strain-epsilon-substrate-xx =  0.071649051d0  ! e_||   of a)
 strain-epsilon-substrate-yy =  0.071649051d0  ! e_||    of a)
 strain-epsilon-substrate-zz = -0.066426474d0  ! e_{_|_} of a)
 strain-epsilon-substrate-xy =  0d0            ! e_xy
 strain-epsilon-substrate-xz =  0d0            ! e_xz
 strain-epsilon-substrate-yz =  0d0            ! e_yz

The GaAs strain tensor components have been chosen in such a way, as if GaAs is biaxially strained with respect to an InAs(001) substrate, pseudomorphically grown along the z growth direction.

If one minimizes the elastic energy with respect to the strained GaAs substrate, then InAs is calculated to be unstrained, as one would expect.
Note that the GaAs substrate's lattice constants are kept fixed.

The same result would be obtained if strain-epsilon-substrate-zz takes an arbitrary value, as only the in-plane strain tensor components e_|| are relevant.

Obviously, the substrate can have any arbitrary strain tensor components, e.g. hydrostatic, biaxial, uniaxial or shear strain.

c) InAs on uniaxially and biaxially strained GaAs(001) substrate

Again, the material system is an InAs layer grown on a GaAs(001) substrate that itself is subject to a variable uniaxial or biaxial stress.
We first apply a tensile uniaxial stress of 1 GPa ( = 10 kbar) on the GaAs substrate followed by a tensile biaxial stress of 1 GPa ( = 10 kbar).
Since nextnano³ accepts as inputs the strain tensor components of the substrate (and not the stress), we first have to calculate these components based on uniaxial and biaxial stresses of 1 GPa.

• Tensile, uniaxial stress of 1 GPa: (sigma_xx = 1 GPa, all other stress tensor components of sigma are zero.)
  
  a) Reading in the strain tensor
  
-> 1D_InAs_on_uniaxially_stressedGaAs001.in

To calculate the strain tensor analytically, we use the following equation:
  
     sigma_i = E_ij . epsilon_j         (i,j = 1...6)         (matrix-vector notation of Hooke's law, Voigt notation, E_ij is the elasticity matrix)
  
  In this particular example (uniaxial for zinblende) it holds for the three unknows e_xx, e_yy, e_zz:
  (Note that all shear components are zero for this case.)
  
     sigma_x = sigma_xx = c₁₁ * epsilon₁ +  c₁₂ * epsilon₂ + c₁₂ * epsilon₃ =
                               = c₁₁ * e_xx         +  c₁₂ * e_yy        + c₁₂ * e_zz = 1 GPa
  
     sigma_y = sigma_yy = c₁₂ * epsilon₁ +  c₁₁ * epsilon₂ + c₁₂ * epsilon₃ =
                               = c₁₂ * e_xx         +  c₁₁ * e_yy        + c₁₂ * e_zz = 0
  
     sigma_z = sigma_zz = c₁₂ * epsilon₁ +  c₁₂ * epsilon₂ + c₁₁ * epsilon₃ =
                               = c₁₂ * e_xx         +  c₁₂ * e_yy        + c₁₁ * e_zz = 0
  
  These are three equations with three unknowns whereas due to symmetry arguments, it additionally holds: e_yy = e_zz
  
  ==> e_xx = sigma_xx / [c₁₁ - 2  c₁₂² / (c₁₁ + c₁₂) ] = 0.011594748
  ==> e_yy = e_zz = - c₁₂ / (c₁₁ + c₁₂) * e_xx = -0.003672427
  
  Thus the following strain tensor corresponds to a strained GaAs substrate with a uniaxial stress along the x direction of 1 GPa:
  
   input-substrate-strain      = yes             ! 'yes' = read in strain tensor components of substrate

 strain-epsilon-substrate-xx =  0.011594748d0  ! e_xx
 strain-epsilon-substrate-yy = -0.003672427d0  ! e_yy
 strain-epsilon-substrate-zz = -0.003672427d0  ! e_zz
 strain-epsilon-substrate-xy =  0d0            ! e_xy
 strain-epsilon-substrate-xz =  0d0            ! e_xz
 strain-epsilon-substrate-yz =  0d0            ! e_yz

This corresponds to a tensile, hydrostatic strain in GaAs of e_hydro = 0.00424989.

Then the InAs will be strained with respect to a uniaxially stressed GaAs substrate.
  Thus the InAs has the following strain tensor components:
  
     e_xx = -0.0560392
     e_yy = -0.0702856
     e_zz =  0.0686452
  
  This corresponds to a compressive, hydrostatic strain in InAs of e_hydro = -0.0576795.



b) Reading in the stress tensor

-> 1D_InAs_on_uniaxially_stressedGaAs001_input_stress.in

An alternative way to get the same result, is to specify the stress tensor rather than the strain tensor:

 input-substrate-stress    = yes    ! 'yes' = read in stress tensor components of substrate

 stress-sigma-substrate-xx = 1d9    ! [Pa]  ! If GaAs is uniaxially stressed with sigma_xx = 1 GPa along the x direction.
 stress-sigma-substrate-yy = 0d0    ! sigma_yy = 0
 stress-sigma-substrate-zz = 0d0    ! sigma_zz = 0
 stress-sigma-substrate-xy = 0d0    ! sigma_xy = 0
 stress-sigma-substrate-xz = 0d0    ! sigma_xz = 0
 stress-sigma-substrate-yz = 0d0    ! sigma_yz = 0


 
• Tensile, biaxial stress of 1 GPa: (sigma_xx = sigma_yy = sigma_|| = 1 GPa = 10 kbar, all other stress tensor components of sigma are zero.)
  
  -> 1D_InAs_on_biaxially_stressedGaAs001.in
  
  The formulas (similar as above) lead to the following analytical equations:
  
  ==> e_xx = e_yy = e_|| = sigma_|| c₁₁ / (2  c₁₂² - c₁₁²  - c₁₁c₁₂) = 0.007922321
  ==> e_zz = e_{_|_} = - D⁰⁰¹ e_|| = - 2 (c₁₂/c₁₁) e_|| = -0.007344854
  
  Thus the following strain tensor corresponds to a strained GaAs substrate with a biaxial stress along the x and y directions of 1 GPa:
  
   input-substrate-strain      = yes             ! 'yes' = read in strain tensor components of substrate

 strain-epsilon-substrate-xx =  0.007922321d0  ! e_xx = e_||
 strain-epsilon-substrate-yy =  0.007922321d0  ! e_yy = e_||
 strain-epsilon-substrate-zz = -0.007344854d0  ! e_zz = e_{_|_}
 strain-epsilon-substrate-xy =  0d0            ! e_xy
 strain-epsilon-substrate-xz =  0d0            ! e_xz
 strain-epsilon-substrate-yz =  0d0            ! e_yz

This corresponds to a tensile, hydrostatic strain in GaAs of e_hydro = 0.00849979.

Then the InAs will be strained with respect to the biaxially stressed GaAs substrate.
  Thus the InAs has the following strain tensor components:
  
     e_xx = -0.0594660
     e_yy = -0.0594660
     e_zz =  0.0646280
  
  This corresponds to a compressive, hydrostatic strain in InAs of e_hydro = -0.0543041.
  
  
  
  b) Reading in the stress tensor

-> 1D_InAs_on_biaxially_stressedGaAs001_input_stress.in

An alternative way to get the same result, is to specify the stress tensor rather than the strain tensor:

 input-substrate-stress    = yes    ! 'yes' = read in stress tensor components of substrate
                                    !         If GaAs is biaxially stressed with
 stress-sigma-substrate-xx = 1d9    ! [Pa]    sigma_xx = 1 GPa along the x direction and with.
 stress-sigma-substrate-yy = 1d9    ! [Pa]    sigma_yy = 1 GPa along the y direction (sigma_xx = sigma_yy).
 stress-sigma-substrate-zz = 0d0    ! [Pa]    sigma_zz = 0
 stress-sigma-substrate-xy = 0d0    ! [Pa]    sigma_xy = 0
 stress-sigma-substrate-xz = 0d0    ! [Pa]    sigma_xz = 0
 stress-sigma-substrate-yz = 0d0    ! [Pa]    sigma_yz = 0

The features presented in this tutorial can be used to model stresses that often occur in the case of thermal expansion mismatches for III-V films grown on sapphire or silicon substrates.
The selective growth on pre-patterned mesas where local stress/strain variations obviously play a crucial role in the resulting electronic/optical properties is another possible application.