# Computational Modeling of the Size Effects on the Optical Vibrational Modes of H-Terminated Ge Nanostructures

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## Abstract

**:**

^{−1}; however, the general effects of such confinements could still be noticed, such as the shift to lower frequencies of the highest optical mode belonging to the Ge vibrations.

## 1. Introduction

**q**) in the Brillouin zone (BZ), since the BZ is much larger than the scattered light in optical spectroscopic techniques and as consequence of momentum conservation, only the phonons on the BZ center (

**q**= 0) can contribute to their spectrum specially the Raman response. When a system is confined such as in nanowires, the phonon wave function decays near the boundary of the nanostructure, this restriction on the spatial extent of the wavefunction, via a relationship of the uncertainty principle type, leads to discrete values of wave vector

**q**, of which the smallest

**q**is π/d, and its multiples, where d is the size of the crystal. The selection rules are relaxed by these effect and some modes other than the ones at the BZ center are taken into account in the spectrum, which is reflected through an asymmetric broadening and a red-Shift of the highest peaks in the vibrational spectrum [15,16]. In the nanopore case, something similar happens, however there is only a partial confinement which is due to extra nodes in the phonon wave function at the boundary of the pores, allowing only wavelengths smaller than the separation between pores, which produces a shift of the highest optical modes towards lower frequencies [17]. Phonon confinement effects have been observed through optical spectroscopic techniques, such as Raman scattering [7]. However, there have been only a few theoretical works that attempted to characterize the vibrational properties of these materials [18,19]. In this paper, we studied the phonon density of states of Ge nanostructures with the first principles density functional perturbation theory technique using the generalized gradient approximation and norm-conserving pseudopotentials. We performed a comparison of the phonon confinement effects between porous structures and nanowires by comparing their respective density of states with similar confinement distances.

## 2. Model and Calculation Scheme

**Figure 1.**Models of Ge nanostructures. (

**a**) Top view of an 8-atom supercell of pGe with p = 12.5%, where the circle shows the confined area; (

**b**) top and side views of a GeNW grown in the [001] direction. The green and grey spheres represent Ge and H atoms, respectively.

## 3. Results and Discussion

^{−1}), the DFPT results have a better fit with the experimental results in the remaining portion of the spectrum. We thus decided to use the linear response approach for our calculations.

**Figure 2.**The phonon density of states (DOS) of crystalline Ge using two first principles approaches, finite displacement-supercell (blue line) and DFPT (red line), compared with the experimental results taken from [33].

^{−1}) is noticeably larger than the shift in the nanopore (260.5 cm

^{−1}). Along with the phonon confinement one important feature that modifies the behavior of the phonon DOS is the capillary stress due to the curved surfaces of the nanostructures, the large surface to core ratio increases the number of surface states compared to the core states as the nanowire diameter decreases, which reduces the optical frequency mode weight, these effects are accounted in [34]. To corroborate the nature of the observed effects, the 4.0 Å nanopore was compared with a nanopore with the same relative porosity but with a lower confinement distance (L = 10 Å). It can be observed that when the Ge-dihydride contributions are not present, the phonon confinement effects are more apparent because the highest optical modes in the 10 Å pore are less shifted (278.43 cm

^{−1}) compared with the 4.0 Å case (260.5 cm

^{−1}), which could also be explained in the surface states scheme as the surface to core ratio decreased thus decreasing the number of surface states. This result suggests that although the phonon confinement features always affect the vibrational properties of the nanostructures, the surface also contributes to a large extent to the vibrational properties near the highest optical modes. Furthermore, the presence of different types of bonds, even with the same element (in this case H), can substantially modify the phonon DOS of such structures.

**Figure 3.**Partial phonon density of states (PDOS) of (

**a**) GeNW with d = 4.0 Å, (

**b**) pGe with L = 4.0 Å and p = 12.5%, and (

**c**) pGe with L = 10.0 Å and p = 12.5%. The green, red and blue areas represent Ge, hydride and dihydride vibrations, respectively.

^{−1}, there is a red area that decreases its width while the nanowire diameter increases, which correspond to bending vibrational modes in the Ge-dihydrides. The gradual narrowing of this area is due to the lower number of dihydride bonds relative to single hydride bonds, also due to the lower hybridization between dihydride phonon modes as the nanowire corners are separated when the diameter increases. This feature allows for a clearer phonon confinement signature on the larger diameter nanowires compared with the smaller ones since the shift of the highest optical modes to lower frequencies is not masked by the dihydride vibrations. A second region can be observed in the nanowire case (at approximately 500 to 750 cm

^{−1}), which is comprised of dihydride and hydride vibrations and no contribution from the Ge; this region consists of bending modes. As the hydride/dihydride ratio increases, the hydrogen bending region becomes mostly dominated by single hydride vibrations, which can be expected due to the large number of single hydrides. Finally, a fourth region can be identified (around 2000 cm

^{−1}), which is composed of stretching vibrations of single hydrides and dihydrides. It can be observed that even when the nanowire diameter increases the single hydride and dihydride contributions are clearly distinguished, which could be important to the characterization of such nanowires with the different spectroscopy techniques. As for the nanopore case, only three regions could be identified (expected due to the absence of the dihydrides at the surface): the Ge dominated vibration region (0 to 300 cm

^{−1}), the single hydride bending interval (400 to 750 cm

^{−1}), and the H stretching (2000 cm

^{−1}) modes. It is important to note that the results shown in Figure 4c,d have the same relationship as those depicted in Figure 3a,b but with L≈d of 8 Å. When comparing the Ge contribution (0–300 cm

^{−1}) in the cases of Figure 4c,d the effects of the dihydrides are reduced due to the increased concentrations of Ge-hydride bonds, hence greater shift of the highest optical modes in the case of the nanowire was observed, according to quantum confinement scheme. Other possible explanation of the decreased contribution of the dihydrides to the DOS around 300 cm

^{−1}in the 8 Å NW case, could be due to the separation between the dihydrides. In the lower diameter NWs the corners are too close hence a phonon hybridization could take place, thus shifting to lower energies the dihydride rocking modes (normally at 500 cm

^{−1}for Ge dihydride bonds [35]); as the nanowire diameter increases these modes become less hybridized shifting to their normal interval of frequencies (400 to 500 cm

^{−1}). Regardless the dihydride behavior it can be observed that when comparing this set of nanowires with nanopores in a 32 atom supercell; the effect of the phonon confinement is observed since the last highest optical modes with Ge contribution of the GeNW were shifted to a lower frequency compared with that of the pGe.

**Figure 4.**PDOS of GeNWs with a d of (

**a**) 4 Å, (

**b**) 6 Å, (

**c**) 8 Å, and pGe with a p, L of (

**d**) 28.125%, 8 Å, (

**e**) 12.5%, 10 Å and (

**f**) 3.125%, 12 Å. The green, blue and red areas represent Ge, hydride, and dihydride contributions, respectively.

**Figure 5.**GeNWs vibration modes of (

**a**) single hydride bending mode (ω

_{B-H}), (b) dihydride scissor-bending mode (ω

_{sci-2H}) and (c) dihydride asymmetrical stretching mode (ω

_{as}).

## 4. Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Trejo, A.; Cruz-Irisson, M.
Computational Modeling of the Size Effects on the Optical Vibrational Modes of H-Terminated Ge Nanostructures. *Molecules* **2013**, *18*, 4776-4785.
https://doi.org/10.3390/molecules18044776

**AMA Style**

Trejo A, Cruz-Irisson M.
Computational Modeling of the Size Effects on the Optical Vibrational Modes of H-Terminated Ge Nanostructures. *Molecules*. 2013; 18(4):4776-4785.
https://doi.org/10.3390/molecules18044776

**Chicago/Turabian Style**

Trejo, Alejandro, and Miguel Cruz-Irisson.
2013. "Computational Modeling of the Size Effects on the Optical Vibrational Modes of H-Terminated Ge Nanostructures" *Molecules* 18, no. 4: 4776-4785.
https://doi.org/10.3390/molecules18044776