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Theory of Raman Scattering by Phonons in Germanium Nanostructures
© to the authors 2007
Received: 24 September 2007
Accepted: 5 December 2007
Published: 21 December 2007
Within the linear response theory, a local bond-polarization model based on the displacement–displacement Green’s function and the Born potential including central and non-central interatomic forces is used to investigate the Raman response and the phonon band structure of Ge nanostructures. In particular, a supercell model is employed, in which along the  direction empty-column pores and nanowires are constructed preserving the crystalline Ge atomic structure. An advantage of this model is the interconnection between Ge nanocrystals in porous Ge and then, all the phonon states are delocalized. The results of both porous Ge and nanowires show a shift of the highest-energy Raman peak toward lower frequencies with respect to the Raman response of bulk crystalline Ge. This fact could be related to the confinement of phonons and is in good agreement with the experimental data. Finally, a detailed discussion of the dynamical matrix is given in the appendix section.
In comparison with silicon (Si) and III–V compounds, germanium (Ge) has a larger dielectric constant and then is particularly suitable for photonic crystal applications. Also, one can incorporate Ge islands into Si-based solar cells for more efficient light absorption. In general, the presence of many arrays of quantum dots with lower bandgap than that of the p–i–n solar cell structure in which they are embedded can lead to an enhancement of the quantum efficiency . Recently, porous Ge (p-Ge) [2–4] and Ge nanowires (GeNW) [5, 6] have been successfully produced and Raman scattering is used to study the phonon behavior in these materials. Although there are many reports about porous Si and Si nanowires, only few investigations have been carried out on Ge nanostructures. However, GeNW hold some special interest in comparison to Si ones, because Ge has, for example, a higher electron and hole mobility than Si, which would be advantageous for high-performance transistors with nanoscale gate lengths.
The reduction of crystallite sizes to nanometer scale can drastically modify the electronic, phononic, and photonic behaviors in semiconductors. Raman scattering, being sensitive to the crystal potential fluctuations and local atomic arrangement, is an excellent probe to study the nanocrystallite effects. Moreover, Raman spectroscopy is an accurate and non-destructive technique to investigate the elementary excitations as well as the details of microstructures. For example, the line position and shape of Raman spectra may give useful information of crystallinity, amorphicity, and dimensions of nanoscale Ge.
In this article, we report a theoretical study of the Raman response in Ge nanostructures by means of a local polarization model of bonds, in which the displacement–displacement Green’s function, the Born potential including central and non-central forces, and a supercell model are used. This model has the advantage of being simple and providing a direct relationship between the microscopic structure and the Raman response.
Modeling Raman Scattering
where u(i) is the displacement of atom i with respect to its equilibrium position, α and β are, respectively, central and non-central restoring force constants. The unitary vector indicates the bond direction between atoms i and j. The dynamical matrix within the Born model is described in details in Appendix A.
We have presented a microscopic theory to model the Raman scattering in Ge nanostructures. This theory has the advantage of providing a direct relationship between the microscopic structures and the measurable physical quantities. For p-Ge, contrary to the crystallite approach, the supercell model emphasizes the interconnection of the system, which could be relevant for long-range correlated phenomena, such as the Raman scattering. The results show a clear phonon confinement effect on the values of ωR, and the variation Δω is in agreement with the effective mass theory. In particular, the Raman response of GeNW is in accordance with experimental data. Regarding to the broadening of Raman peaks, an imaginary part of energy η = 13.0 cm−1 was chosen to include inhomogenous diameters of GeNW, the influence of mechanical stress, as well as laser heating effects [5, 14]. The obtained averaged width L = 2.11 nm is smaller than D = 12.0 nm estimated in Ref. . This difference could be due to a possible amorphous oxide layer surrounding the surface of GeNW. This study can be extended to other nanostructured semiconductors such as nanotubes.
where I is the identity matrix.
It is worth to mention that Eq. (A.9) has an associate eigenvalue equation, which leads to the phonon band structure shown in Fig. 1a. Furthermore, the dimension of matrixes involved in Eq. (A.9) is 3N,N being the number of atoms in the supercell.
This work was partially supported by projects 58938 from CONACyT, 2007045 from SIP-IPN, IN100305 and IN114008 from PAPIIT-UNAM. The supercomputing facilities of DGSCA-UNAM are fully acknowledged.
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