Interface Hamiltonian 8-band Zinc-Blende

The Model

The interface Hamiltonian is used to model the heterostructures with atomic sharp interfaces. This is an extension of the standard 8-band $k \cdot p$ model (described here). The model, originally derived in [Kilpstein2010], includes $l = 2$ perturbation terms, which have been omitted in our implementaion, consistent with the approach in [LivnehPRB2012].

The interface Hamiltonian in the block form is given by:

\begin{array}{r} {\hat{H}}_{i n t e r f a c e} = δ (z - z_{i}) [\begin{array}{c} D_{S} & 0 & 0 & π β \\ 0 & D_{X} & π α & 0 \\ 0 & π α & D_{X} & 0 \\ π β & 0 & 0 & D_{Z} \end{array}] \end{array}

where $δ$ is the Dirac delta function, $z_{i}$ is the position of the interface, $D_{S}$ , $D_{X}$ , $D_{Z}$ , $α$ , and $β$ are the interface Hamiltonian parameters, $π$ is the parameter that can take values of $\pm 1$ . Parameter $π$ is used to distinguish different ordering of materials at the interface, for example InaAs/GaSb and GaSb/InAs interfaces would have same interface paramters, but opposite $π$ values.

Implementation and usage

The interface Hamiltonian is available only for 1D simulations. For the best results, ensure that there is a grid point at the interface position, for example like so:

grid{
    xgrid{
        ...
        line{ pos = $x_1          spacing = $x_interface }   # material1
        line{ pos = $x_interface  spacing = $x_2 }           # material2
        ...
    }
}

To add the interface Hamiltonian to the simulation, use the interface keyword in quantum{ region{ kp8_band{} } } section of input file.

Example:

quantum{
    region{
        kp_8band{
            ...
            interface{
                position = $x_interface
                D_S = 1.0
                D_X = 2.0
                D_Z = 1.5
                alpha = 0.1
                beta = 0.1
                reverse = no
            }
            ...
        }
    }
}

reverse=no corresponds to $π = 1$ , and reverse=yes corresponds to $π = - 1$ . Other interface parameters are expressed in $e V \cdot n m$ .

One can add multiple interfaces to the simulation by adding multiple interface blocks to the input file. Example below has 2 interfaces with the same paramters, but different order of materials (common case for modelling of quantum well).

quantum{
    region{
        kp_8band{
            ...
            interface{
                position = $x_interface1
                D_S = 1.0
                D_X = 2.0
                D_Z = 1.5
                alpha = 0.1
                beta = 0.1
                reverse = no
            }
            interface{
                position = $x_interface2
                D_S = 1.0
                D_X = 2.0
                D_Z = 1.5
                alpha = 0.1
                beta = 0.1
                reverse = yes
            }
        }
    }
}

To set up repeating interfaces, one can use the array_x keyword in the interface block.

quantum{
    region{
        kp_8band{
            ...
            interface{
                position = 10.0
                ... # parameters here
                array_x{
                    shift = 2.0
                    min = -2
                    max = 3
                {
            }
        }
    }
}

The above example will create 6 interfaces with the same parameters, at positions 6.0, 8.0, 10.0, 12.0, 14.0 and 16.0 nm.

Last update: 23/01/2025