Overview - Basics 2

Note: This documentation is nearly 10 years old and might not be related any more to the actual version of the nextnano³ source code.
It is still displayed here for historical reasons until a decent manual for nextnano³ will be available.
Interested readers should consult the diploma and PhD theses of S. Hackenbuchner, M. Sabathil, T. Zibold (nextnano++) and T. Andlauer (nextnano++) available from the publications website.

You can dowload this page as a pdf file (basics.pdf) but most of it is written in German and this file won't be updated any more. Be sure to check the online documentation for the latest version.

Contents

2       Basics 2 - Definition of k.p Hamiltonian
2.1    Wurtzite
2.1.1 Spin-orbit interaction
2.1.2 Crystal field splitting
2.1.3 Strain effects
2.1.4 Transformation to rotated coordinate system
2.1.5 Special wurtzite parameters
2.2    Zincblende
2.2.1 Spin-orbit interaction
2.2.2 Strain effects
2.2.3 Transformation to rotated coordinate system
2.3    The rotation problem
2.3.1 1D simulation
2.4    Discretization

2  Basics 2 - Definition of k.p Hamiltonian

2.1 Wurtzite

We use an eight-band model for the multi-band Schrödinger equation. Our basic Hamiltonian (without spin-orbit coupling and strain) for wurtzite is defined as follows:

The definitons of the parameters L₁, L₂, M₁, M₂, M₃, N₁, N₂ require that the z axis is the hexagonal direction of the crystal.

with

	(2.1)

One additionally has to account for the band offsets by adding the diagonal matrix:

with   k² = k_x² + k_y² + k_z².

2.1.1 Spin-orbit interaction

The spin-orbit interaction in the basis

is given by:

The basis states are expressed explicitly as follows:

2.1.2 Crystal field splitting

In wurtzite one has additionally to allow for crystal field splitting: This is accomplished by the (3 x 3)-matrix

in the subspaces

2.1.3 Strain effects

If we want to include strain effects, we have to add the following matrix to H_vv:

The strain effects on the conduction band (shift of the conduction band) is described by two absolute deformation potentials for the conduction band a_c,|| and a_{c,_|_}:

2.1.4 Transformation to rotated coordinate system

The transformation to a rotated coordinate system is formally done by the following rotation matrix:

with R^-1=R^T. The rotated Hamiltonian H' now reads:

H'(k') = RH(R^Tk')R^T

2.1.5 Special wurtzite parameters

For wurtzite usually the parameters A₁, ..., A₆ are used instead of L₁, ..., N₂. The relations among them are the following:

For the strain parameters, there are similar relations:

2.2 Zincblende

The basic Hamiltonian (without spin-orbit coupling and strain) for zincblende is defined as follows:

The definitons of the parameters L, M, N require that the x axis is the [100], the y axis the [010] and the z axis the [001] direction of the crystal.

with

	(2.22)

One additionally has to account for the band offsets by adding the diagonal matrix:

with   k² = k_x² + k_y² + k_z².

2.2.1 Spin-orbit interaction

The spin-orbit interaction in the basis

is given by:

The basis states are expressed explicitly as follows:

2.2.2 Strain effects

If we want to include strain effects, we have to add the following matrix to H_vv:

The parameters l, m, n can be derived from the absolute deformation potential a_v and the shear deformation potentials b and d via:

The strain effect on the conduction band (shift of the conduction band) is described by the absolute deformation potential for the conduction band a_c:

2.2.3 Transformation to rotated coordinate system

The transformation to a rotated coordinate system is formally done by the following rotation matrix:

with R^-1=R^T. The rotated Hamiltonian H' now reads:

H'(k') = RH(R^Tk')R^T

2.3 The rotation problem

There are two coordinate systems:

• Crystal system
  describes a vector in terms of the basis vectors in the crystal system
• Simulation system
  The simulation system is a cartesian system (x, y, z) in which the simulation variables are defined.

The rotation problem is described here.

2.3.1 1D simulation

When the simulation is carried out in 1D, there is just one variable z. This implies that the k.p Hamiltonian is discretized in z direction. It is possible, however, that the simulation direction (which is always identical to the growth direction) is the y or the x axis in the simulation system. Then, an additional rotation must be carried out, which transforms by an even permutation this simulation direction to the z axis.

In detail:

The calculation is carried out in the calculation system. So the additional rotation is defined via:

k_{calculation system} = R'.k_{simulation system}

The rotation matrix must have determinant +1 (i.e. no mirror matrix). Now we distinguish the three different cases for the simulation direction:

• simulation direction = x axis
  
  
• simulation direction = y axis
  
  
• simulation direction = z axis
  
  

2.4 Discretization

The rotated Hamiltonian can be written as:

In order to discretize the Hamiltonian, we first symmetrize the differential operators. With the position-dependent material parameters P(r) and Q(r), one thus gets:

Fig 2.1  1D grid

We start with the one-dimensional problem (homogeneous in y and z direction): The first-order derivative is approximated by:

Psi denotes the envelope wavefunction. With

(2.36)

one finally obtains:

(2.37)

For the second derivative the raltion holds:

and henceforth:

 

(2.38)

In this convention the material parameters must be given between the mesh points.

In two and three dimensions there are additionally mixed derivatives of second order:

Fig 2.2  2D grid

The discretization on the grid in picture Fig 2.2 is carried out as follows: