Abstract
Within the linear response theory, a local bondpolarization model based on the displacement–displacement Green’s function and the Born potential including central and noncentral 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 [001] direction emptycolumn 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 highestenergy 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.
Keywords:
Raman scattering; Phonons; Germanium nanostructuresIntroduction
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 Sibased 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 [1]. Recently, porous Ge (pGe) [24] 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 highperformance 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 nondestructive 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 noncentral 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
Raman scattering analysis is a very powerful tool for studying the composition, bonding, and microstructure of a solid. However, the elementary excitation processes involved are complicated to describe theoretically. In general, the Raman response depends on the local polarization of bonds due to the atomic motions. Considering the model of the polarizability tensor developed by Alben et al. [7], in which the local bond polarizabilities α(j)] are supposed to be linear with the atomic displacements u_{μ}(j), i.e., c_{μ}(j) = ∂α(j)/∂u_{μ}(j) alternates only in sign from site to site in a single crystal with diamond structure, the Raman response R(ω)] at zero temperature could be expressed within the linear response theory as [8,9].
where μ, μ′ = x,y, orzi and j are the index of atoms, and G_{μ, μ′} (i,j, ω) is the displacement–displacement Green’s function determined by the Dyson equation as
where M is the atomic mass of Ge, I stands for the identity matrix, and Φ is the dynamical matrix, whose elements are given by
Within the Born model, the interaction potential (V_{ij}) between nearestneighbor atoms i and j can be written as [10].
where u(i) is the displacement of atom i with respect to its equilibrium position, α and β are, respectively, central and noncentral 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.
Results
In order to determine the parameters of the Born model for Ge, we have performed a calculation of the phonon band structure for crystalline Ge (cGe) using α = 0.957 N cm^{−1} and β = 0.244 N cm^{−1}, and the results are shown in Fig. 1a. Notice that the optical phonon bands are reasonably reproduced in comparison with the experimental data [11], since these optical phonon modes are responsible for the Raman scattering. It is worth mentioning that these parameter values are very close to those used in a generalized Born model for cGe [12]. The Raman response of cGe obtained from Eq. 1 is shown in Fig. 1b. Observe that the Raman peak is located at ω_{0} = 300.16 cm^{−1}[13,14], which corresponds to the highestfrequency of optical modes with phonon wave vector q = 0, since the q of the visible light is much smaller than the first Brillouin zone and then the momentum conservation law only allows the participation of vibrational modes around the Γ point.
Figure 1. (a) Calculated phonon dispersion relations (solid line) compared with experimental data (open circle). (b) Raman response of cGe obtained from a primitive unitary cell, as illustrated in the inset
The pGe is modeled by means of the supercell technique, in which columns of Ge atoms are removed along the [001] direction [15]. In Fig. 2, the highestfrequency Raman shift (ω_{R}) is plotted as a function of the porosity for square pores, increasing the size of supercells and maintaining the thickness of two atomic layers in the skeleton. The porosity is defined as the ratio of the removed Geatom number over the original number of atoms in the supercell. In Fig. 2, we have removed 18, 50, 98, 162, 242, and 338 atoms from supercells of 32, 70, 128, 200, 288, and 392 atoms, respectively. Observe that the results of ω_{R} are close to 270 cm^{−1}, instead of 300.16 cm^{−1} for cGe, due to the phonon confinement originated by extra nodes in the wavefunctions at the boundaries of pores. However, this confinement is only partial since the phonons still have extended wave functions, and the Raman shifts in Fig. 2 are mainly determined by the degree of this partial confinement. The inset of Fig. 2 illustrates the highestfrequency Raman peak and the corresponding pGe structure with a porosity of 56.25%.
Figure 2. Variation of Raman peaks as a function of porosity for the squarepore case. Inset: The main Raman peak for pGe with a porosity of 56.25%, which corresponds to a supercell of 32 Ge atoms, removing 18 of them
Another way to produce pores consists in removing different number of atoms from a fix large supercell. In this work, we start from a cGe supercell of 648 atoms formed by joining 81 eightatom cubic supercells in the x–y plane. Columnar pores with rhombic crosssection are produced by removing 4, 9, 25, 49, 81, 121, 169, 225, and 289 atoms, as schematically illustrated in the upper inset of Fig. 3 for a pore of 121 atoms. The results of ω_{R} are shown in Fig. 3 as a function of porosity. In the lower inset of Fig. 3, we present the variation of ω_{R} with respect to its crystalline Raman peak ω_{0}, i.e., Δω ≡ ω_{0} − ω_{R}, as a function of the inverse of partial confinement distance between pore boundaries (d) in a log–log plot. Observe that for the highporosity regime (small d) the slope tends to two, similar to the electronic case [16].
Figure 3. The Raman shift as a function of the porosity for a fixed supercell of 648 atoms. Inset: Δω = ω_{0} − ω_{R} versus the inverse of partial confinement distance (d), which is illustrated in the upper inset
For modeling GeNW, we start from a cubic supercell with eight Ge atoms, and take the periodic boundary condition along zdirection and free boundary conditions in x and y directions. For GeNW with larger crosssections, Ge atomic layers are added in x and y directions to obtain GeNW with different shapes of crosssection. We have performed the calculation of the Raman response for GeNW, whose supercells containing from 8 to 648 Ge atoms. In Fig. 4, ω_{R} is plotted as a function of the length (L) of crosssections with square (open squares), rhombic (open rhombus), and octagonal (open circles) forms. These results are compared with experimental data (solid square) obtained from Ref. [14], observing a good tendency agreement. The inset shows Δω ≡ ω_{0} − ω_{R} as a function of 1/L. Observe that with ν is 1.4–2.0 when L → 0. This result is in agreement with the effective mass theory, i.e., 2L is the longest wavelength in x and y directions accessible for a GeNW of width L, and then the highestphonon frequency of the system can be approximately determined by evaluating the frequency of optical mode at π/L.
Figure 4. For Ge nanowires, ω_{R} is plotted versus the length (L) of crosssections with square (open squares), rhombic (open rhombus), and octagonal (open circles) form, in comparison to experimental data (solid square) obtained from Ref. [14]. Inset: Δω = ω_{0} − ω_{R}as a function of 1/Lis shown in a log–log plot
In Fig. 5, the calculated Raman response spectrum of a GeNW with L = 2.11 nm is compared with the experimental one [5]. The theoretical results include an imaginary part of energy η = 13 cm^{−1}, in order to take into account the thermal and size distribution effects, and a weight function proportional to exp(−ω − ω_{R}/8). The inclusion of this weight function is to preserve basic ideas of the momentum selection rule, since in principle only Γpoint or infinitewavelength optical modes are active during the Raman scattering and for a GeNW there are only finitewavelength modes in x and y directions. In other words, if the Raman selection rule is visualized as a δfunction at Γpoint, it should be broadened for finitesize systems due to the Heisenberg uncertainty principle, i.e., optical modes with a longer wavelength should have a larger participation in the Raman response.
Conclusions
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 pGe, contrary to the crystallite approach, the supercell model emphasizes the interconnection of the system, which could be relevant for longrange correlated phenomena, such as the Raman scattering. The results show a clear phonon confinement effect on the values of ω_{R}, 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. [5]. 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.
Appendix A
For tetrahedral structures, the positions of four nearestneighbor atoms around a central atom located at (0,0,0) are , , , and , where a = 5.65 Å.
From Eq. 3 in “Modeling Raman Scattering”, the interaction potential between central atom 0 and atom 1 is
where and then, the element xx of the first interaction matrix is given by
In a similar way, one can obtain other elements of the matrix. Therefore, the four interaction matrices ϕ_{i}, bounding the central atom to its nearestneighbor atom i, can be written as
Fig.A1 The positions of four tetrahedral nearest neighbors around a central atom.
and
These four interaction matrices ϕ_{1}, ϕ_{2}, ϕ_{3}, and ϕ_{4} are indicated in the inset of Fig. 1b. Due to the tetrahedral symmetry it is easy to prove that
where I is the identity matrix.
Within the supercell model, the equilibrium positions of atoms i and j can be, respectively, written as and , being , the coordinates of unit cell and , the positions inside the cell. For an eightatom supercell, the Fourier transform of Φ can be written as
Hence, Eq. 2 can be rewritten as
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,Nbeing the number of atoms in the supercell.
Acknowledgments
This work was partially supported by projects 58938 from CONACyT, 2007045 from SIPIPN, IN100305 and IN114008 from PAPIITUNAM. The supercomputing facilities of DGSCAUNAM are fully acknowledged.
References

Konle J, Presting H, Kibbel H:
Physica E. 2003, 16:596.
COI number [1:CAS:528:DC%2BD3sXhs1egsbg%3D]
Publisher Full Text 
J. Electroanal. Chem.. 2006, 589:258. Publisher Full Text

Sun D, Riley AE, Cadby AJ, Richman EK, Korlann SD, Tolbert SH:
Nature. 2006, 441:1126.
COI number [1:CAS:528:DC%2BD28Xmtlahsrw%3D]
Publisher Full Text 
Nature. 2006, 441:1122.
COI number [1:CAS:528:DC%2BD28Xmtlaiu7Y%3D]
Publisher Full Text 
Jalilian R, Sumanasekera GU, Chandrasekharan H, Sunkara MK:
Phys. Rev. B. 2006, 74:155421. Publisher Full Text

Nano Lett.. 2006, 6:1578.
COI number [1:CAS:528:DC%2BD28XlslOksLo%3D]
Publisher Full Text 
Elliott RJ, Krumhansl JA, Leath PL:
Rev. Mod. Phys.. 1974, 46:465.
COI number [1:CAS:528:DyaE2cXkvFKqsb8%3D]
Publisher Full Text 
Phys. Rev. Lett.. 1988, 61:191.
COI number [1:CAS:528:DyaL1cXltVWjurk%3D]
Publisher Full Text 
Srivastava GP: The Physics of Phonons. Adam Hilger, Bristol; 1990.

Phys. Rev. B. 1971, 3:364. Publisher Full Text

Bose G, Tripathi BB, Gupta HC:
J. Phys. Chem. Solids. 1973, 34:1867.
COI number [1:CAS:528:DyaE3sXlsFSgtLg%3D]
Publisher Full Text 
Finlayson E, AmezcuaCorrea A, Sazio PJA, Baril NF, Badding JV:
Appl. Phys. Lett.. 2007, 90:132110. Publisher Full Text

Wang X, Shakouri A, Yu B, Sun X, Meyyappan M:
J. Appl. Phys.. 2007, 102:014304. Publisher Full Text

IEEE Trans. Nanotech. 2006, 5:466. Publisher Full Text

Cruz M, Wang C, Beltrán MR, TagüeñaMartínez J:
Phys. Rev. B. 1996, 53:3827.
COI number [1:CAS:528:DyaK28Xht1ejs78%3D]
Publisher Full Text