# strain{}¶

The documentation for this keyword is also available here (old documentation layout).

Below are various variables and functions that can be used inside strain{}.

## Specifying the strain model¶

pseudomorphic_strain{}

Homogeneous strain for 1D layer structures (analytical calculation).

This feature also works in 2D or 3D but the user must be sure that the model makes sense from a physical point of view (i.e. the 2D/3D structure should consist of different layers along the growth direction whereas the layers must be homogenous along the two perpendicular directions).

minimized_strain{}

Minimization of the elastic energy for 2D and 3D geometries (numerical calculation).

It can also be used for 1D simulations. In this case, the results will be equivalent to the analytical model pseudomorphic_strain{}.

import_strain{}
import_from
value

string

example

"strain_tensor"

Reference to imported data in import{}.

The data being imported must have exactly 6 components. The expected order of strain tensor components is: $$\varepsilon_{xx}\ \varepsilon_{yy}\ \varepsilon_{zz}\ \varepsilon_{yz}\ \varepsilon_{zx}\ \varepsilon_{xy}$$

coordinate_system
value

string

default

simulation

example

crystal

The imported strain tensor is with respect to the simulation or crystal coordinate system (optional parameter).

Note

In order to calculate the strain, one has to provide a substrate with respect to which the layers are strained. This can be done with the keyword global{} ==> substrate{…}.

## Piezo- and pyroelectric charges¶

piezo_density
value

yes or no

default

yes

Calculate piezoelectric charge density and take it into account while solving the Poisson equation.

If no strain is solved, this flag is ignored.

pyro_density
value

yes or no

default

yes

Calculate pyroelectric charge density and take it into account while solving the Poisson equation.

If material system is not wurtzite, this flag is ignored. The pyroelectric charge density due to spontaneous polarization applies to wurtzite only. In order to obtain pyroelectric charges, it is not necessary to calculate strain. Pyroelectric charges are only present in wurtzite materials but not in zinc blende.

second_order_piezo
value

yes or no

default

no

Include 2nd order piezoelectric coefficients in the calculation

Note

Not fully implemented for wurtzite, only “standard growth directions” supported for wurtzite as the most general formula was not known to us at the time of implementation.

## Specify growth direction (for pseudomorphic strain model)¶

growth_direction
example

growth_direction = [1, 0, 0]

• Vector in crystal coordinate system

• Can be specified in a 2D or 3D simulation but not in a 1D simulation (x axis is taken by default in 1D)

• If not set, x axis of simulation coordinate system is taken by default.

## Solver definitions¶

linear_solver{}
iterations
value

integer

default

10000

Number of iterations for linear equation solver in strain algorithm

abs_accuracy
value

double

default
• 1e-8 [GPa] (1D)

• 1e-8 [GPa nm] (2D)

• 1e-8 [GPa nm2] (3D)

rel_accuracy
range

[0.0,0.01]

default

1e-12 (dimensionless)

use_cscg
value

yes or no

Composite step conjugate gradient solver (try this one if standard solver fails to converge)

## Debugging¶

debuglevel
options

-1 0 1 2 3

default

2

The higher this integer number, the more information on the numerical solver is printed to the screen output

## Output definitions¶

output_strain_tensor{}

output (symmetric) strain tensor : $$\varepsilon_{ij} = (u_{ij} + u_{ji})/2$$ [dimensionless]

crystal_system
value

yes or no

default

no

output strain tensor in crystal coordinate system

simulation_system
value

yes or no

default

yes

output strain tensor in simulation coordinate system (useful if simulation coordinate system differs from crystal coordinate system)

Note

The ordering of the strain tensor components is: $$\varepsilon_{xx},\ \varepsilon_{yy},\ \varepsilon_{zz},\ \varepsilon_{xy},\ \varepsilon_{xz},\ \varepsilon_{yz}$$

boxes
value

yes or no

For each grid point, in 1D two points are printed out to mimic abrupt discontinuities at interfaces (in 2D four points, in 3D eight points)

output_stress_tensor{}

output (symmetric) stress tensor : $$\ \sigma_{ij} = C_{ijkl}\ \varepsilon_{kl}$$ [GPa]

crystal_system
value

yes or no

default

no

output stress tensor in crystal coordinate system

simulation_system
value

yes or no

default

yes

output stress tensor in simulation coordinate system (useful if simulation coordinate system differs from crystal coordinate system)

Note

The ordering of the stress tensor components is: $$\sigma_{xx},\ \sigma_{yy},\ \sigma_{zz},\ \sigma_{xy},\ \sigma_{xz},\ \sigma_{yz}$$

boxes
value

yes or no

output_hydrostatic_strain{}

prints out the hydrostatic strain, i.e. the trace of the strain tensor $$\mathrm{Tr}[\varepsilon_{ij}] = \varepsilon_{xx} + \varepsilon_{yy} + \varepsilon_{zz}$$ [dimensionless]

boxes
value

yes or no

Note

The hydrostatic strain output is in percent (This is different compared to nextnano³.)

output_distortion_tensor{}

output distortion tensor $$u_{ij}$$ (which can be nonsymmetric for certain growth directions) $$u_{xx}\ u_{yy}\ u_{zz}\ u_{xy}\ u_{yx}\ u_{xz}\ u_{zx}\ u_{yz}\ u_{zy}$$ [dimensionless]

crystal_system
value

yes or no

default

no

output distortion tensor in crystal coordinate system

simulation_system
value

yes or no

default

yes

output distortion tensor in crystal coordinate system

boxes
value

yes or no

output_displacement{}

output displacement vector [nm]

crystal_system
value

yes or no

default

no

output displacement vector in crystal coordinate system

simulation_system
value

yes or no

default

yes

output displacement vector in simulation coordinate system

boxes
value

yes or no

output_force_density{}

output force density vector field $$f_i$$ [nN/nm3] (at moment output may be not fully correct; not tested sufficiently)

crystal_system
value

yes or no

default

no

output force density vector field in crystal coordinate system

simulation_system
value

yes or no

default

yes

output force density vector field in simulation coordinate system

boxes
value

yes or no

output_elastic_energy_density{}

output elastic energy density ($$\frac{1}{2}\ C_{ijkl}\ \varepsilon_{ij}\ \varepsilon_{kl}$$) [eV/nm3] The integrated elastic energy is printed out in log file.

boxes
value

yes or no

output_polarization_charges{}

prints out piezo and pyroelectric charge densities [1018/cm3] in case they were calculated. Pyroelectric charges due to spontaneous polarization apply to wurtzite only. Piezoelectric charges can be calculated for both zinc blende and wurtzite in case the strain was calculated. The piezo charge density is written to: density_piezoelectric_charge.dat ($$\rho_\mathrm{pz}$$) For diamond like crystal structures that have an inversion center such a Si or Ge, piezoelectric charges do not exist.

The pyro charge density is written to: density_pyroelectric_charge.dat ($$\rho_\mathrm{py}$$) It applies to wurtzite only and is independent of strain and is due to spontaneous polarization. If both, piezo and pyroelectric charge densities were calculated, the sum of both charge densities (total polarization charge density) is written to: density_polarization_charge.dat ($$\rho_\mathrm{pol}=\rho_\mathrm{pz}+\rho_\mathrm{py}$$)

To summarize:

• zincblende: density_piezoelectric_charge.dat ($$\rho_\mathrm{pz}$$)

• wurtzite: density_piezoelectric_charge.dat ($$\rho_\mathrm{pz}$$), density_pyroelectric_charge.dat ($$\rho_\mathrm{py}$$), density_polarization_charge.dat ($$\rho_\mathrm{pol}=\rho_\mathrm{pz}+\rho_\mathrm{py}$$)

output_polarization_vector{}

prints out piezo and pyroelectric polarization vector [C/cm2]. Pyroelectric polarization due to spontaneous polarization apply to wurtzite only. The piezoelectric polarization vector depends on strain and it is zero if no strain is present.

crystal_system
value

yes or no

default

no

output polarization vector in crystal coordinate system

simulation_system
value

yes or no

default

yes

output polarization vector in simulation coordinate system

boxes
value

yes or no

output_sparse_matrix{}

output sparse matrix used in strain solver

type

enumerate desired types of sparse matrix output in .mtx format (optional).

options

values zero_nonzero zero_nonzero_absolute all

default

values

• values : output sparse matrix as it is

• zero_nonzero : output matrix containing ‘0’ and ‘1’ for zero and non-zero entries of sparse matrix

• zero_nonzero_absolute : output matrix containing ‘0’ and ‘1’ for zero and non-zero absolute values of entries of sparse matrix

• all : output all types listed above

## Output of material parameters¶

output_lattice_constants{}

Output lattice constants.

boxes
value

yes or no

output_elastic_constants

Output elastic constants.

boxes
value

yes or no

output_piezo_constants{}

Output piezoelectric constants.

boxes
value

yes or no

output_second_order_piezo_constants{}

Output 2nd order piezoelectric constants.

boxes
value

yes or no

output_pyro_constants{}

Output pyroelectric constants, i.e. spontaneous polarization constants.

boxes
value

yes or no