Vibrational modes of silicon metalattices from atomistic and finite-element calculations

Transcription

1 Vibrational modes of silicon metalattices from atomistic and finite-element calculations Yihuang Xiong, Weinan hen, Ismaila Dabo Materials Science and Engineering, Penn State University October 6 th, 216

2 Intensity (a.u) Raman spectroscopy at the nanoscale Vibrational spectroscopy provides a highly sensitive probe of structural properties at the nanoscale. 1 nm Wavenumber (cm -1 ) Kuok et al., Phys. Rev. Lett. 9, 25 (23)

3 What is a metalattice? Metalattices are 3D nanoscale structures consisting of crystalline aggregates, periodically repeated on the scale of 1-1nm. 1 1 nm avity rystalline Silicon Han and respi, Phys. Rev. Lett. 86, 696 (21)

4 How to calculate acoustic modes coustic wave equation: oundary conditions: How to calculate optical modes Optical wave equation: oundary conditions: We need both acoustic and optical elastic coefficients ijkl and Dijkl. Thonhauser and Mahan, Phys. Rev. 69, (24)

5 ω (cm 1 ) Phonon spectrum of silicon from density-functional perturbation theory 5 4 TO1,2 LO TO1 TO2 LO LO TO1,2 } DFPT Expt. a Expt. b 3 L L L 2 T2 1 T1,2 T1 T1, 2 q q q X R L alculation Details: Functional = LD (local density approx.) Wavefunction utoff = 1 Ry K-Point Sampling = 24 x 24 x 24 Experiment: a Nilsson and Nelin, Phys. Rev. 6, 3777 (1972) b Dolling, Proc. Symp. Inelastic Scattering Neutrons in Solids and Liquids 2, 37 (1963)

6 ω 2 (cm 2 ) Transformation of the phonon spectrum of silicon } DFPT q 2 q 2 q 2 X R L y carrying out the transformation q q 2 and ω ω 2, all of the phonon branches become linear close to the point.

7 Linear regression of the first-principles phonon spectrum Path ranch Linear Regression Perturbation Expression [q ] q [,1] T1,T2 ω 2 = q 2 ω 2 = 44/(a 2 c 2 ρ) q 2 L ω 2 = q 2 ω 2 = 11/(a 2 c 2 ρ) q 2 TO1,TO2 ω 2 = q 2 ω 2 = ω 2 + D44/(a 2 c 2 ρ) q 2 LO ω 2 = q 2 ω 2 = ω 2 + D11/(a 2 c 2 ρ) q 2 T1 ω 2 = q 2 ω 2 = (11 12)/(a 2 c 2 ρ) q 2 Κ [q q ] q [, 3 /4] T2 ω 2 = q 2 ω 2 = 244/(a 2 c 2 ρ) q 2 L ω 2 = q 2 ω 2 = ( )/(a 2 c 2 ρ) q 2 TO1 ω 2 = q 2 ω 2 = ω 2 + (D11 D12)/(a 2 c 2 ρ) q 2 TO2 ω 2 = q 2 ω 2 = ω 2 + 2D44/(a 2 c 2 ρ) q 2 LO ω 2 = q 2 ω 2 = ω 2 + (D11 + D12 + 2D44)/(a 2 c 2 ρ) q 2 L [q q q] q [, 1 /2] T1,T2 ω 2 = q 2 ω 2 = ( )/(a 2 c 2 ρ) q 2 L ω 2 = q 2 ω 2 = ( )/(a 2 c 2 ρ) q 2 TO1,TO2 ω 2 = q 2 ω 2 = ω 2 + (D11 D12 + D44)/(a 2 c 2 ρ) q 2 LO ω 2 = q 2 ω 2 = ω 2 + (D11 + 2D12 + 4D44)/(a 2 c 2 ρ) q 2 q in unit of 2π/a, ω in cm 1, and all other quantities are in S.I. units. a = m; ρ = 237 kg/m 3 ; c = m/s.

8 Elastic constants of the acoustic and optical modes of silicon from density-functional perturbation theory We exploit the linear regression and perturbation results to obtain: LD PE PEsol SW SW a Expt. 11 (GPa) (GPa) (GPa) D11 (GPa) D12 (GPa) D44 (GPa) a owley, Phys. Rev. Lett. 6, 2379 (1988)

9 Frequency (cm 1 ) Frequency (cm 1 ) coustic vibrational modes Method = DFT-based SW Pore size = ~1.5 nm Number of atoms = 2,14 Pore volume fraction = ~52% Method = DFT-based FEM Pore size = 1.37 nm Number of nodes = 63,918 Pore volume fraction = 52%

10 Frequency (cm 1 ) Frequency (cm 1 ) coustic vibrational modes (cont d) Method = DFT-based SW Pore size = ~1.5 nm Number of atoms = 2,14 Pore volume fraction = ~52% Method = DFT-based FEM Pore size = 1.37 nm Number of nodes = 63,918 Pore volume fraction = 52%

11 Frequency (cm 1 ) Frequency (cm 1 ) coustic vibrational modes (cont d) Method = DFT-based SW Pore size = ~1.5 nm Number of atoms = 2,14 Pore volume fraction = ~52% Method = DFT-based FEM Pore size = 1.37 nm Number of nodes = 63,918 Pore volume fraction = 52%

12 Frequency (cm 1 ) Frequency (cm 1 ) Optical vibrational modes Method = DFT-based SW Pore size = ~1.5 nm Number of atoms = 2,14 Pore volume fraction = ~52% Method = DFT-based FEM Pore size = 1.37 nm Number of nodes = 63,918 Pore volume fraction = 52%

13 Frequency (cm 1 ) Frequency (cm 1 ) Optical vibrational modes (cont d) Method = DFT-based SW Pore size = ~1.5 nm Number of atoms = 2,14 Pore volume fraction = ~52% Method = DFT-based FEM Pore size = 1.37 nm Number of nodes = 63,918 Pore volume fraction = 52%

14 Frequency (cm 1 ) Frequency (cm 1 ) Optical vibrational modes (cont d) Method = DFT-based SW Pore size = ~1.5 nm Number of atoms = 2,14 Pore volume fraction = ~52% Method = DFT-based FEM Pore size = 1.37 nm Number of nodes = 63,918 Pore volume fraction = 52%

15 Summary We have calculated the vibrational modes of silicon metalattices (silicon crystal with an array of nanopores) using OMSOL. We have determined the acoustic and optical phonon dispersion spectrum of silicon metalattices in agreement with atomistic models. Those results open up the possibilities to predict the influence of nanoscale geometry on the optical response of silicon metalattices. atomistic continuum Funding: NSF-DMR 14262