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 )   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 . raman scattering analysis is a very powerful tool for studying the composition , bonding , and microstructure of a solid . , 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 . , in which the local bond polarizabilities (j ) ] are supposed to be linear with the atomic displacements u(j ) , i.e. , c(j ) = (j)/u(j ) alternates only in sign from site to site in a single crystal with diamond structure , the raman response r( ) ] at zero temperature could be expressed within the linear response theory as . ( 1 ) where  ,  = x , y , orzi and j are the index of atoms , and g ,  ( i , j ,  ) is the displacement  displacement green s function determined by the dyson equation as ( 2 ) where m is the atomic mass of ge , i stands for the identity matrix , and  is the dynamical matrix , whose elements are given by ( 3 ) within the born model , the interaction potential ( vij ) between nearest - neighbor atoms i and j can be written as . ( 4 ) where u(i ) is the displacement of atom i with respect to its equilibrium position ,  and  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. 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 ( c - ge ) using  = 0.957 n cm and  = 0.244 n cm , and the results are shown in fig . 1a . notice that the optical phonon bands are reasonably reproduced in comparison with the experimental data , 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 c - ge . 1b . observe that the raman peak is located at 0 = 300.16 cm , which corresponds to the highest - frequency 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  point . ( a ) calculated phonon dispersion relations ( solid line ) compared with experimental data ( open circle ) . ( b ) raman response of c - ge obtained from a primitive unitary cell , as illustrated in the inset the p - ge is modeled by means of the supercell technique , in which columns of ge atoms are removed along the   direction . in fig . 2 , the highest - frequency raman shift ( 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 ge - atom 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 r are close to 270 cm , instead of 300.16 cm for c - ge , 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 illustrates the highest - frequency raman peak and the corresponding p - ge structure with a porosity of 56.25% . variation of raman peaks as a function of porosity for the square - pore case . inset : the main raman peak for p - ge 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 c - ge supercell of 648 atoms formed by joining 81 eight - atom cubic supercells in the x  y plane . columnar pores with rhombic cross - section 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 . 3 , we present the variation of r with respect to its crystalline raman peak 0 , i.e. ,   0  r , as a function of the inverse of partial confinement distance between pore boundaries ( d ) in a log  log plot . observe that for the high - porosity regime ( small d ) the slope tends to two , similar to the electronic case . the raman shift as a function of the porosity for a fixed supercell of 648 atoms . inset :  = 0  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 z - direction and free boundary conditions in x and y directions . for genw with larger cross - sections , ge atomic layers are added in x and y directions to obtain genw with different shapes of cross - section . we have performed the calculation of the raman response for genw , whose supercells containing from 8 to 648 ge atoms . in fig . 4 , r is plotted as a function of the length ( l ) of cross - sections with square ( open squares ) , rhombic ( open rhombus ) , and octagonal ( open circles ) forms . 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 highest - phonon frequency of the system can be approximately determined by evaluating the frequency of optical mode at /l . for ge nanowires , r is plotted versus the length ( l ) of cross - sections with square ( open squares ) , rhombic ( open rhombus ) , and octagonal ( open circles ) form , in comparison to experimental data ( solid square ) obtained from ref . . inset :  = 0  ras 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 . the theoretical results include an imaginary part of energy  = 13 cm , in order to take into account the thermal and size distribution effects , and a weight function proportional to exp(|  r|/8 ) . the inclusion of this weight function is to preserve basic ideas of the momentum selection rule , since in principle only -point or infinite - wavelength optical modes are active during the raman scattering and for a genw there are only finite - wavelength modes in x and y directions . in other words , if the raman selection rule is visualized as a -function at -point , it should be broadened for finite - size systems due to the heisenberg uncertainty principle , i.e. , optical modes with a longer wavelength should have a larger participation in the raman response . raman response of a genw with l = 2.11 nm ( solid line ) compared with experimental data ( open circles ) from ref . 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 r , and the variation  is in agreement with the effective mass theory . in particular , the raman response of genw regarding to the broadening of raman peaks , an imaginary part of energy  = 13.0 cm was chosen to include inhomogenous diameters of genw , the influence of mechanical stress , as well as laser heating effects . 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 . for tetrahedral structures , the positions of four nearest - neighbor atoms around a central atom located at ( 0,0,0 ) are , 
 , , and , where a = 5.65  . from eq . 3 in  modeling raman scattering  , the interaction potential between central atom 0 and atom 1 is(a.1 ) where   and then , the element xx of the first interaction matrix is given by(a.2 ) in a similar way , one can obtain other elements of the matrix . therefore , the four interaction matrices i , bounding the central atom to its nearest - neighbor atom i , can be written as(a.3 ) fig.a1 the positions of four tetrahedral nearest neighbors around a central atom.(a.4)(a.5 ) these four interaction matrices 1 , 2 , 3 , and 4 are indicated in the inset of fig . 1b . due to the tetrahedral symmetry it is easy to prove that(a.7 ) 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 eight - atom supercell , the fourier transform of ( a.9 ) has an associate eigenvalue equation , which leads to the phonon band structure shown in fig . this work was partially supported by projects 58938 from conacyt , 2007045 from sip - ipn , in100305 and in114008 from papiit - unam .
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 [ 001 ] 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 .