30-band \(\mathbf{k}\cdot\mathbf{p}\) band structure calculation

Note

This tutorial is under in development.

Attention

The 30-band model is under development in our software, therefore related syntax and available functionalities are expected to be changing. Strain effects are not included yet.

Input Files:

bulk_kp_dispersion_Si_SiGe_Ge_30band_nn3.in

Scope of the tutorial:

• Band structure of bulk Si, SiGe, and Ge within 30-band \(\mathbf{k}\cdot\mathbf{p}\)

Relevant keywords:

• bulk-kp-dispersion

• kp-parameters

• k-vectors-sample-type

Main adjustable parameters in the input file:

• set of parameters - %30band_parameters

• path of the band structure - %Bandstructure

• parameters controlling mole fraction of the modeled alloy %AlloySweepActive, %AlloySweepSize, %AlloySweepSteps

Relevant output Files:

• kp_bulk\bulk_kp30kp_dispersion*.dat (band structure)

Introduction

This tutorial shows how to calculate band structure of Si_1-xGe_x alloys within 30-band \(\mathbf{k}\cdot\mathbf{p}\) model. It answers the following questions:

• How to trigger 30-band model in nextnano³?

• How to define path on which you want to calculate the band structure?

• Which output file contain the band structure computed within the 30-band model?

The band structures presented in this tutorial are in agreement with those reported by M. Cardona, F. H. Pollak and Rideau et al. (except vicinity of the \(K\) point for the latter, which is under investigation.)

The Input File

30-band \(\mathbf{k}\cdot\mathbf{p}\) model can be called to compute bulk dispersion by specifying bulk-kp-dispersion = 30-band inside the group $output-kp-data.

$output-kp-data
bulk-kp-dispersion = 30-band
$end_output-kp-data

The path along which the band structure is computed can be specified inside the group $tighten by assigning one of available paths (use autocomplete feature in nextnanomat) to the attribute k-vectors-sample-type.

$tighten
k-vectors-sample-type = L-Gamma-X-W-K-L-W-X-K-Gamma
$end_tighten

Parameters are currently hard-coded and available only for SiGe alloys. In this tutorial two sets of parameters are used to reproduce results from M. Cardona, F. H. Pollak and Rideau et al.. Switching between the different sets of parameters is done by setting kp-parameters to kp-parameters = Rideau or to kp-parameters = Cardona-Pollak inside the group $numeric-control.

$numeric-control
kp-parameters = Rideau
$end_numeric-control

Output files

The band structure generated after running the input file bulk_kp_dispersion_Si_SiGe_Ge_30band_nn3.in can be found in a file …kp_bulkbulk_kp30kp_dispersion_BrillouinZone1_L-Gamma-X-W-K-L-W-X-K-Gamma.dat, where the first column contains indexes of following wave vectors along the path and all the following columns contain eigenvalues starting with the highest ones. Exact coordinates of wave vectors for each point of the computed band structure can be found in a related file …kp_bulkbulk_kp30kp_dispersion_BrillouinZone1_L-Gamma-X-W-K-L-W-X-K-Gamma_k_vectors.dat. There, the first column contains corresponding indexes of each k-point, the three following contain exact coordinates of each wave vector, and the last column stores the length of each wave vector.

Results

Cardona-Pollak parameters: 15 band Hamiltonian

In the paper of M. Cardona and F. H. Pollak, 15-band kp Hamiltonian is introduced to model band structures of Si and Ge. The 15-band Hamiltonian disregards spin-orbit interactions, resulting in spin degeneracy across all bands. For this tutorial, nextnano computes all 30 bands, while keeping spin-orbit couplings at zero.

To use the parameters from this paper, change kp-parameters to Cardona-Pollak

$numeric-control
kp-parameters = Cardona-Pollak
$end_numeric-control

The variable %AlloyContent in the input file can be adjusted to select between 0 (pure Germanium) and 1 (pure Silicon).

The resulting band structures for both pure Germanium and pure Silicon are illustrated in Figures Figure 3.4.2.1, and Figure 3.4.2.2, respectively.

Figure 3.4.2.1 Band structure of Germanium (Ge) computed with Cardona-Pollak set of parameters , %AlloyContent=1.0

Figure 3.4.2.2 Band structure of Silicon (Si) computed with Cardona-Pollak set of parameters, %AlloyContent=0.0

Rideau et al parameters.

In the paper of Rideau et al., the full set of parameters for 30-band Hamiltonian is given, including the spin-orbit couplings. The parameters are also proved to be valid for any alloy content in Si_1-xGe_xalloys.

Band structures of pure Germanium , Silicon-Germanium alloy and pure Silicon generated within the input file are shown in the figures Figure 3.4.2.3, Figure 3.4.2.4 and Figure 3.4.2.5, respectively.

To use the parameters from this paper, change kp-parameters to Rideau

$numeric-control
kp-parameters = Rideau
$end_numeric-control

The variable %AlloyContent should be adjusted accordingly in the input file to choose the material.

Figure 3.4.2.3 Band structure of Germanium (Ge), %AlloyContent=1.0, spin-orbit coupling included.

Figure 3.4.2.4 Band structure of Silicon Germanium (Si_0.5Ge_0.5), %AlloyContent=0.5, spin-orbit coupling included.

Figure 3.4.2.5 Band structure of Silicon (Si), %AlloyContent=0.0, , spin-orbit coupling included.

Acknowledgment

This tutorial is based on the nextnano GmbH collaboration in the scope of the SiPho-G Project aiming at development of ultrahigh-speed optical components for next-generation photonic integrated circuits, and it is funded by the European Union’s Horizon 2020 research and innovation program under grant agreement No 101017194.