2.5.2. global{ }

Global settings simulation domain.

simulate1D{}

Specify simulation dimension: 1D simulation along the x direction

global{
   simulate1D{}

simulate2D{}

Specify simulation dimension: 2D simulation in the (x,y) plane

simulate2D{}

simulate3D{}

Specify simulation dimension: 3D simulation

simulate3D{}

crystal_zb{}

crystal_wz{}

Specify orientation of crystal coordinate system with respect to simulation coordinate system. For zinc blende materials there are three-digit Miller indices. The Miller indices define a plane. There exists a vector that is perpendicular to this plane, e.g. in zinc blende blende materials, the [hkl] vector is always perpendicular to the (hkl) plane. However, for wurtzite, this is not necessarily true. For instance, although the [0001] vector is perpendicular to the (0001) plane, in general is does not hold that the vector that is perpendicular to the (hkil) plane is defined by [hkil]. Note: For a 1D simulation, the heterostructure is always grown along the x axis. For a 2D simulation, always the (x,y) plane is used.

Zinc blende

crystal_zb{
   x_hkl = [1, 0, 0]    # Specify (hkl) plane perpendicular to x axis
   y_hkl = [0, 1, 0]    # Specify (hkl) plane perpendicular to y axis

The x axis of the simulation coordinate system is perpendicular to this (hkl) plane of the crystal, here: (1 0 0). The y axis of the simulation coordinate system is perpendicular to this (hkl) plane of the crystal, here: (0 1 0). The Miller indices (here: (0 0 1)) for the z axis are determined automatically. For zinc blende it holds: The vector [hkl] is perpendicular to the (hkl) plane.

Another example:

crystal_zb{
   x_hkl = [3,  1, 1]   #
   y_hkl = [0, -1, 1]   #

x axis of simulation coordinate system is perpendicular to (3 1 1) plane of crystal coordinate system, i.e. the x axis is along [311] direction. y axis of simulation coordinate system is perpendicular to (0 -1 1 ) plane of crystal coordinate system, i.e. the y axis is along [0-11] direction. The Miller indices (here: [2, -3, -3]) for the z axis are determined automatically, i.e. (2 -3 -3) plane, i.e. the z axis is along [311] direction.

Wurtzite

Usually for wurtzite, the four-digit Miller-Bravais indices (h k i l) are used. We also use this notation but omit the ‘i’ because i = - h - k. The three integer values (Miller indices) that are given for x_hkl refer to a plane and not to a direction. The x direction is then the one that is perpendicular to this plane.

This vector along the x axis has indices that are in general not identical to the Miller indices in wurtzite.

crystal_wz{              # e.g. hexagonal [0001] axis along x axis
   x_hkl = [ 0, 0, 1]    # Specify (hkil) plane perpendicular to x axis: ( 0, 0,  0, 1)
   y_hkl = [ 1, 0, 0]    # Specify (hkil) plane perpendicular to y axis: ( 1, 0, -1, 0)

This corresponds to the four-digit Miller-Bravais indices hkil = ( 0, 0, 0, 1) that define the (hkil)=(0001) plane. Coincidently, the vector [0001] is pependicular to it. This corresponds to the four-digit Miller-Bravais indices hkil = ( 1, 0, -1, 0) that define the (hkil)=(10-10) plane. The Miller-Bravais indices for the (hkil) plane perpendicular to the z axis are determined automatically inside the code (here: (-1 2 -1 0)).

Another example:

crystal_wz{              # e.g. (10-10) plane is perpendicular to x axis
   x_hkl = [ 1, 0, 0]    # hkil = ( 1, 0, -1, 0)
   y_hkl = [-1, 2, 0]    # hkil = (-1, 2, -1, 0)

This corresponds to the four-digit Miller-Bravais indices hkil = ( 1, 0, -1, 0) that define the (hkil)=(10-10) plane. The x axis of the simulation coordinate system is perpendicular to this plane. This corresponds to the four-digit Miller-Bravais indices hkil = (-1, 2, -1, 0) that define the (hkil)=(-12-10) plane. The y axis of the simulation coordinate system is perpendicular to this plane. The Miller-Bravais indices of the (hkil) plane perpendicular to the z axis are determined automatically inside the code (here: (0001)). Coincidently, the vector [0001] is pependicular to it. In this particular case, no rotation has to be applied to the crystal (rotation matrix = identity matrix).

crystal_wz{
   ...
   rotation_c_a_ratio_use_substrate = yes  # (default: yes)
   rotation_c_a_ratio = 1.63299            # c/a ratio

In wurtzite, the c/a ratio of the two lattice constants ‘c’ and ‘a’ is important. The ideal one, \(c/a=\sqrt{8/3}=1.63299...\), is not the one present in GaN, AlN or InN, i.e. in real materials. For the rotation of the crystal coordinate system to the simulation coordinate system, a specific c/a ratio has to be assumed. By default, we use the one of the substrate material. If you want to use the ideal c/a ratio, you have to specify rotation_c_a_ratio_use_substrate = no.

Additionally, one can specify a custom value for the c/a ratio. If no customized value is specified, rotation_c_a_ratio = sqrt(8/3) (default). The actually used rotation matrix is written to the log file.

x_hkl = [ ., ., .]
y_hkl = [ ., ., .]
z_hkl = [ ., ., .]

Exactly two of these three axes have to be specified, the third one is calculated internally.

substrate{}

Specify substrate (and thus its lattice constant), i.e. the material on which all layers are grown. Consequently, all layers are strained with respect to the (unstrained) substrate. The substrate is also necessary to determine the symmetry properties of the \(\mathbf{k}_{||}\) integration.

substrate{
   name    = "GaAs"
}

or for ternary AlxGa1-xAs (Al0.3Ga0.7As)

substrate{
   name    = "Al(x)Ga(1-x)As"         # AlGaAs
   alloy_x = 0.3                      # x = 0.3
}

or for quaternary AlxGayIn1-x-yAs (Al0.3Ga0.1In0.6As)

substrate{
   name    = "Al(x)Ga(y)In(1-x-y)As"  # AlGaInAs
   alloy_x = 0.3                      # x = 0.3
   alloy_y = 0.1                      # y = 0.1
}

temperature

Specify simulation temperature. Temperature should be larger than 1 µK (T >= 1e-6 K)

temperature = 300                       # T in Kelvin (here: 300 K)

temperature_dependent_bandgap

temperature_dependent_bandgap = yes/no  # turns on/off temperature dependence of band gaps (optional, default is yes)

Note: The band gaps (and thus the conduction band edges) are adjusted when the temperature changes. This has the consequence that the conduction band offsets are temperature dependent whereas the valence band offsets are not temperature dependent. Due to Varshni formula a correction is calculated (if necessary interpolated) which is then added to the (interpolated) band gap. For instance:

\[\begin{split}\begin{aligned} E_{\rm gap}(T > 0~{\rm K}) &= E_{\rm gap}(T = 0~{\rm K}) + E_{\rm gap}^{\rm Varshni correction}(T) \\ E_{\rm gap}^{\rm Varshni correction}[{\rm A}_{x}{\rm B}_{1-x}{\rm C}] &= x \frac{-\alpha_{\rm AC} T^2}{T + \beta_{\rm AC}} + (1-x) \frac{- \alpha_{\rm BC} T^2}{T + \beta_{\rm BC}} - x(1-x) \frac{- \alpha_{\rm bow,ABC} T^2}{T + \beta_{\rm bow,ABC}} \end{aligned}\end{split}\]

temperature_dependent_lattice

temperature_dependent_lattice = yes/no  # turns on/off temperature dependence of lattice constants (optional, default is yes)

periodic{}

Note: If the grid is periodic along any of the three directions, then the Poisson and the strain equations are solved automatically with periodic boundary conditions along the appropriate directions. For the Schrödinger equation periodic boundary conditions are taken only if the quantum region extends over the whole simulation region along the relevant direction. Objects (e.g. circles, rectangles, …) extending over an edge of the simulation region will not automatically be continued on the opposite side. Periodic boundaries are not implemented for magnetic fields.

periodic{         # (optional)
   x = yes/no     # periodic along x direction (default is no)
   y = yes/no     # periodic along y direction (default is no) (2D or 3D only)
   z = yes/no     # periodic along z direction (default is no) (3D only)
}

Note: For all nextnano GmbH versions older than 2019-04-16, the periodic{} flag was specified within the grid{ } section and not within the global{ } section.

magnetic_field{ }

Specify magnetic field.

strength
value:

any float >= 0.0

default:

0.0

magnetic flux density B in units of Tesla [T] = [Vs/m²]
direction
value:

integer array of dimension 3

default:

(only 3D) orientation of magnetic field vector with respect to (x,y,z) simulation coordinate system
Note: In 2D, the specifier direction must not be present because the direction is automatically set internally to be perpendicular to the simulation plane in 2D, i.e. (x,y) plane => [0, 0, 1].
In 1D (Zeeman splitting of energy levels only), the specifier direction must not be present because the direction is parallel to the simulation direction (x).
Defining magnetic_field{} in the input file results into a different (slower) solver being called. Thus, omit this section altogether in case no magnetic field is needed.
Note: The Hall effect for the drift-diffusion current calculation is not implemented yet.
magnetic_field{
   strength  = 5.3         # magnetic flux density B in units of Tesla [T] = [Vs/m^2]
   direction = [3, 1, 1]   # (only 3D) orientation of magnetic field vector with respect
                           # to (x,y,z) simulation coordinate system, here: [311]
}

For single-band: For a normal single-band calculation, all eigenstates are internally treated to be two-fold spin degenerate (e.g. when calculating the quantum density). In contrast, if magnetic_field{} is defined, the Pauli equation is solved in all 2D and 3D single-band quantum regions instead of the single-band Schrödinger equation. So the states are no longer two-fold spin degenerate (unless the magnetic field is zero). Note that even if the magnetic field strength is explicitly set to zero, still the Pauli equation is solved in which case both spin eigenfunctions will be calculated although they have the same energy. Thus, to maximize speed in the absence of magnetic fields, leave magnetic_field{} undefined. Then the dimension of the matrix from which the eigenfunctions are calculated has only half the size (and thus only half of the eigenfunctions have to be calculated), and it is is real instead of Hermitian.

Currently, the magnetic field is input to the single-band and the multi-band k.p Hamiltonian but is not input to the drift-diffusion current equation. Thus the classical Hall effect is not taken into account. As the magnetic field modifies the eigenenergies and probability densities, and thus the quantum density, the current will be different compared to the zero-field case because the current is proportional to the density.