Yambo Tutorial: Silicon wire

Si(111) hydrogenated wire
[Shortcut to examples 01 02 03 04 05 06 07 and exercises ]
by Andrea Marini & Maurizia Palummo

The material

The wire is along the x direction.
32 atoms in the cell (74 electrons)
Plane waves cutoff 8 Hartree (10500 RL vectors)

Introduction

Nanostructuring of semiconductors is an alternative means of developing new electronic and opto-electronic devices. The huge efforts made towards matter manipulation at the nanometer scale have been motivated by the fact that desirable properties can be generated just by changing the system dimension and shape. In particular, the possibility of tuning the optical response of nanosized materials by modifying their size has become one of the most challenging aspects of recent semiconductor research. Among the different nanostructures, nanowires have recently attracted a lot of interest. Being one-dimensional structures, they seem potentially useful as well as the Carbon-nanotubes and probably more, due to the possibility to tailor their chemistry and to be used to create nano-sized lasers. They exhibit extreme quantum confinement effects such that charge carriers are free to move only along the wire. The knowledge of the electronic and optical properties of Silicon nanowires is of particular interest due to their natural compatibility with silicon based technologies, and due to the discovery of photoluminescence in the visible range in Porous Silicon and in quantum matrices of Silicon. Here we aim to examine how the quantum-confinement and how local-fields and excitonic effects influence the optical spectrum of a very thin (D=4 A) Silicon nanowire, grown along the 111 crystallographic direction.

Considerations similar to the other sections of this tutorial, about convergence parameters, have to be done. Infact a small kinetic energy-cutoff, a sparse k-points mesh, and a small number of unoccupied bands are used here, in order to speed up the calculations. Nevertheless the main physical features will be captured, such as: the importance of local-field effects for the light polarization perpendicular to the wire axis [1], and the presence of strongly bound excitons about 2 eV [2].

Although this tutorial is not dedicated to the quasi-particle electronic structure calculations within the GW method, it is important to mention that in these 1-D subnanometer systems, one of the recent findings (see refs. [2],[3], for more details) is that the self-energy corrections to the DFT-KS eigenvalues increase as the wire size is decreased. In particular for the wire studied here a self-energy correction more than 3 times larger than the value obtained in the Silicon bulk is found[3]. Furthermore before starting the simulation we have to underline that for nanowires the comparison with the experiments is, unfortunately, not so straightforward. In fact, from an experimental point of view, it is very difficult to grow monosized wires, with a well characterized morphology and surface reconstruction. The main interest should be to study the trend of the electronic and optical properties by changing the size, the orientation and the morphology of the nanowire. This cannot clearly be done here and you will not find any experimental curve in the following... sorry!!!

Lateral and top view of the Si(111) wire. Large red spheres represent Si atoms; small blue spheres are hydrogen atoms used to saturate the dangling bonds.

[01] Initialization: 01_init(yambo -i -V RL)

To run this example enter the 1D_Si_wire/yambo directory and type

 localhost:>yambo -F Inputs/01_init -J 01_init

[02-03] Random-Phase approximation: 02_RPA_no_LF_par(yambo -o c), 02_RPA_no_LF_perp(yambo -o c), 03_RPA_LF_par(yambo -o c), 03_RPA_LF_perp(yambo -o c), 03_RPA_LF_QP_par(yambo -o c -V all)

As we have learned in the other sections of this Tutorial, the easiest way to calculate the optical properties, is to perform a Random Phase Approximation (RPA) calculation without the inclusion of LFE. We will calculate here the two components of the polarizability tensor of the wire, for light polarized along the wire axis and perpendicular to it. We will see that, already at this independent particle level, the optical response is strongly anisotropic. Furthermore we will find that the onset is blue shifted with respect to the RPA optical spectrum of the bulk Silicon and this is actually due to Quantum Confinement effect.

The Inputs/02_RPA_no_LF_par and Inputs/02_RPA_no_LF_perp are the inputs for simple RPA response function calculations without the inclusion of Local Field Effects (LFE) ,for light polarized along and perpendicular to the wire's axis, with

 NGsBlkXd=1     RL      #       (Xd)     Response block size

To run these two examples, type:

 localhost:>yambo -F Inputs/02_RPA_no_LF_par -J 02_RPA_no_LF_par

 localhost:>yambo -F Inputs/02_RPA_no_LF_perp -J 02_RPA_no_LF_perp

After running, you should obtain the following curves:

We can now see, the effect of including more RL components in the response function (in other words, by including Local Field Effects) for the polarizability for light polarized along the wire axis (Inputs/03_RPA_LF_par) and perpendicular to it (Inputs/03_RPA_LF_perp).

In these input files, the response function size is changed to 107 RL. The converged value for NGsBlkXd must be found doing several calculations with different values and checking the effect on the final spectra. After running

localhost:>yambo -F Inputs/03_RPA_LF_par -J 03_RPA_LF_par
localhost:>yambo -F Inputs/03_RPA_LF_perp -J 03_RPA_LF_perp

we will find

This Figure illustrates how the optical response of the wire is modified by taking into account the inhomogeneity of the system. Similar to other one-dimensional systems, such as carbon nanotubes, when local field effects are included [4], an important intensity reduction is observed for perpendicular light polarization which renders the wire almost transparent up to about 7 eV, while a small change of the optical spectrum for light polarized along the wire axis. This depolarization effect, which originates from the presence of microscopic electric fields due to the induced polarization charges after the application of the external field, has been actually observed experimentally both in nanotubes and in porous silicon matrices (see references [2],[4] for more details). Moreover, as explained in ref [1], the observed anisotropy can be almost entirely explained in classical terms by using Effective Medium Theory (EMT) formulas.

GW calculations [3] have shown that the self-energy corrections open the DFT gaps of about 2.3 eV. For this reason we can use here such a value as a scissor operator which simply opens rigidly the LDA gap. This is done in Inputs/03_RPA_LF_QP with the line

% XfnQP_E
 2.30000 | 1.000000 | 1.000000 |      # [EXTQP Xd] E parameters (c/v)
%

and the corresponding result (we consider here only the polarizability for light polarized along the wire axis) looks like:

[04] ALDA: 04_ALDA_g_space(yambo -o c -t a), 04_ALDA_r_space(yambo -o b -t a)

Again the first attempt to go beyond RPA is to use the TDDFT in the Adiabatic LDA approximation. As we have learned, with yambo, we can run this kind of calculation, both in the RL (Inputs/04_alda_g_space) and in the Bloch space implementation (Inputs/04_alda_r_space).

To run these two cases, type:

 localhost:>yambo -F Inputs/04_alda_g_space -J  04_alda_g_space

 localhost:>yambo -F Inputs/04_alda_r_space -J  04_alda_r_space

As you can see from the next figure, the RL implementation yields a wrong polarizability, with spurious states below the absorption gap. This anomalous effect is not present in the Bloch-space implementation, and it is due to the fact the in isolated systems (like the Si wire) there are large regions where the charge is very small. In these regions the ALDA f_xc kernel diverges and the multiplication in RL space with the non interacting response function (which vanishes) is numerically instable.

On the contrary the Bloch-space implementation properly works,inducing small but visible changes with respect to the RPA spectrum. Nevertheless the ALDA will not able to reproduce the BSE curve, the we will calculate in the next sections.

[05] The Statically screened Electron-electron interaction: 05_W(yambo -b)

As we have seen, in other sections of this tutorial, the calculation of the statically screened Electron-electron interaction is required in order to perform a Bethe-Salpeter calculation. To perform this calculation, type:

 localhost:>yambo -F Inputs/05_W

The screened interaction matrix will be written at the end of this run in the database SAVE/ndb.em1s. It is important to underline that, also at this level of the calculation, careful convergence tests in the plane-wave components and the electronic transitions involved in the screened coulomb interaction matrix, should be, in principle, required.

[06] Bethe-Salpeter equation 06_BSE(yambo -o b -y h)

We are now ready for a BSE calculation. In particular we aim to see here, how the excitonic effects influence the optical spectrum for light polarized along the wire.

To run the BSE calculation, type:

 localhost:>yambo -F Inputs/06_BSE -J 06_BSE

We should find that, also the effect of the electron-hole interaction on the optical properties of the wire, strongly depends on light polarization: for light polarized perpendicular to the wire axis, the BSE optical spectrum will be rather similar to the RPA curve when LFE are introduced, whereas for light polarized along the growth axis, a transfer of the oscillator strength to the low energy peaks is observed together with a reduction of the intensity above the electronic gap.

The BSE spectrum (blue curve) for light polarized along the wire's axis, compared with the spectra obtained at the quasi-particle (green curve) and at the ALDA (red curve) levels,looks like the following:

[07] A Bethe-Salpeter based F_xc: 07_LRC(yambo -o c -t l)

Last of all, you can try the simple Long Range Component model, as introduced in Reining using the input file Inputs/07_LRC. Here we set

LRC_alpha= -9.50000          # [TDDFT] LRC alpha factor

much larger than in the surface case (consequence of the reduced dimensionality).

Additional Exercises

Calculate the polarizability spectra like in section 02-03 increasing the polarization RL size, the polarization bands to find the converged values.
Repeat the ALDA, BSE, and LRC calculations for light polarized perpendicular to the wire axis.

References

F.Bruneval et al., Phys. Rev. Lett. 94, 219701 (2005)
M.Bruno et al., Phys. Rev. B 72, 153310 (2005)
X. Zhao et al., Phys. Rev. Lett. 92, 236805 (2004)
A.Marinopolus et al., Phys. Rev. Lett. 91, 046402 (2003)

Si(111) hydrogenated wire [Shortcut to examples 01 02 03 04 05 06 07 and exercises ] by Andrea Marini & Maurizia Palummo