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1 Synthesis of an open-framework allotrope of silicon Duck Young Kim, Stevce Stefanoski, Oleksandr O. Kurakevych, Timothy A. Strobel Electronic structure calculations Electronic structure calculations and ionic relaxation were performed using density functional theory (DFT) [1, 2] with the generalized gradient approximation (GGA) and Perdew, Burke, and Ernzerhof (PBE) exchange-correlation functional [3, 4], as implemented in the Quantum ESPRESSO software [5]. We used a plane-wave basis set cutoff of 60 Ry and a Brillouin-zone integration grid of a 16x16x16 k-points. Crystal structure Table S1. Crystallographic data for Si 24. Si 24 full profile refinement, = Å Space group Cmcm (No. 63) a (Å) (14) b (Å) (4) c (Å) (5) Atomic coordinates x y z Occupancy U iso Si1 (8f) (2) (2) (7) Si2 (8f) (2) (2) (6) Si3 (8f) (2) (2) (6) Refinement statistics χ 2 = wrp = (-Bknd) Rp = (-Bknd) NATURE MATERIALS 1

2 Table S2. Crystallographic data for Si 24 (DFT, PBE) Si 24 DFT, PBE calculations, 1 atm Space group Cmcm (No. 63) a (Å) b (Å) c (Å) Atomic coordinates x y z Occupancy Si1 (8f) Si2 (8f) Si3 (8f) Figure S1. Unit cell of Si 24 and unique silicon atoms. Three different atomic positions of Si are represented with different colors. Crystal fragments are shown on the right side with the most deviated angle from the perfect tetrahedral angle (109.5 ). Angles are from DFT optimization. Band-gap and Absorption calculations We calculated band gaps for d-si and Si 24 using several computational approaches to make it clear that Si 24 is a quasidirect band gap semiconductor. It is well-known issue that standard DFT (PBE here) underestimates the band gap of materials. GW (where G means the single-particle Green s function and W the screened Coulomb potential) calculations were performed to correct the PBE band gap values and the Bethe-Salpeter equation (BSE) [6, 7] was used to compute the Coulomb correlation between the photoexcited electrons and holes using the ABINIT software [8]. We conducted GW 0 calculations with the cutoff dielectric matrix of 5 Hartree, which was tested to various semiconductors and insulators successfully [9]. We applied BSE calculations to d-si for testing convergence of the calculations and then calculated Si 24. For BSE calculations, we used a cutoff of 3.0 Hartree for the dielectric matrix. The GW approximation was applied to the self-energy (the proper exchange-correlation potential acting on an excited electron or hole), which can be written as the product of the one-electron Green s function times the screened Coulomb interaction =igw. In our calculations, we have used both single shot GW(G 0 W 0 ) and partially self-consistent GW 0. It is 2 NATURE MATERIALS

3 SUPPLEMENTARY INFORMATION worth noting that full correction to both G and W (GW) on d-si overestimated the band gap significantly [13]. As shown in Table S3, for d-si, G 0 W 0 and GW 0 give excellent agreement with the experimental band gap for d-si (1.17 ev) and in the main text, we used the G 0 W 0 results. The Heyd-Scuseria-Ernzershof (HSE) exchange-correlation functional [14] was also tested by us, which is more accurate for large band gap materials. Table S3. Calculated band gaps for d-si and Si 24 using various functionals. Units are in ev. d-si, indirect Si 24, indirect (direct) PBE 0.62 b 0.53 (0.57) HSE a 1.41 (1.45) G 0 W (1.34) GW b 1.43 (1.46) a Ref [15], b Ref [9] Lattice parameter change with respect to sodium concentration We calculated the lattice parameters, a, b, and c for Na x Si 24 (0 x 4) at different values of x. Supercells of of Si 24 unit cells were constructed with only one sodium atom: 1x1x1(x=1), 2x1x1 (x=0.5), 3x1x1 (x=0.333). Atomic positions were relaxed to determine the influence of Na content on the lattice parameters. Theoretical optimizations are in excellent agreement with experimental data for Na 4 Si 24 and Si 24 (Figure S2). 13 Lattice Parameters (Å) Na content, x, in Na x Si 24 Figure S2. Lattice parameters of Na x Si 24 with respect to sodium concentration x. NATURE MATERIALS 3

4 Metal-Insulator transition in Na x Si 24 Computationally, we checked if Na x Si 24 becomes a semiconductor at low values of x. Figure S3 shows the electronic density of states of Na 1 Si 24, Na 0.5 Si 24, Na Si 24, and Na Si 24. By lowering sodium concentraion, the system remains metallic with a decrease in the number of conduction electrons at the Fermi level. Figure S3. Electronic density of states for Na x Si 24. Phonon dispersion relations Phonon calculations were performed using Density Functional Perturbation Theory [16], as implemented in the Quantum-espresso package. The electronic wave function was expanded with a kinetic energy cutoff of 60 Ry. A Uniform with a 6x6x6 q-point mesh of phonon momentum has been calculated with a 12x12x12 k-point mesh. The dynamical stability of Si 24 was examined at ambient pressure and at 10 GPa. Figure S4 shows the evolution of the phonon dispersion relations along high symmetry lines and the corresponding phonon density of states. One can see that Si 24 is dynamically stable to 10 GPa. Above this pressure, it becomes destablized, indicating a structural transformation. 4 NATURE MATERIALS

5 SUPPLEMENTARY INFORMATION Figure S4. Phonon dispersion relations of Si 24 at ambient pressure (a) and 10 GPa (b). Raman Scattering Raman scattering data were collected from degassed Si 24 and pristine Na 4 Si 24 samples. A 532nm diode laser was used as an excitation source and focused onto the sample using a 20 long working distance objective lens. The power at the sample was approximately 10 mw. Scattered radiation was collected in the 180 back-scatter geometry and focused onto a 50 m confocal pinhole, which served as a spatial filter. This light was then passed through two narrowband notch filters (Ondax, SureBlock) and focused onto the entrance slit of a spectrograph (Princeton Instruments, SP2750). Light was dispersed off of an 1800 gr/mm grating and recorded using a liquid nitrogen-cooled charge-coupled device detector (Princeton Instruments, Plyon). The Raman actives mode were calcualted using density functional perturbation theory [17]. A Brlliouin zone sampling grid with Å -1 was used with a plane basis set cutoff of 500 ev. The ionic positions were carefully relaxed at ambient pressure. Figure S5 compares experimental and theortical Raman data for Si 24. Figure S5 shows an excellent agreement for peak positions between experiment and calculations with some deviation in intensity. The calculations were performed at zero temperature with the harmonic approximation and some intensity differences could be due to the frequency dependence of the polarizability derivatives and vibrational anharmonicity [18]. In addition, no attempt was made to control the polarization NATURE MATERIALS 5

6 condition during the measurements and crystallite orientation anistropy may also contribute to intensity variation. A g,b 2g A g A g B 1g B 3g B 2g B 3g A g 10 A g A g B 1g DFT B 1g Experiment Raman Shift (cm -1 ) Figure S5. Raman spectrum of Si 24. Experimental (bottom) and calculated (top). Calculated spectra are represented by Lorentzian peaks. The symmetry of each mode is indicated. Figure S6 compares Raman spectra for Si 24 with Na 4 Si 24 and calculated Raman mode frequencies. Significant differences are observed between the vacant and filled structures, while calculated frequencies are in generally good agreement for both cases. These results provide additional support for the removal of Na from the silicon framework and support the synthesis of pure Si NATURE MATERIALS

7 SUPPLEMENTARY INFORMATION Relative Intensity (arb. units) Na 4 Si 24 Si Raman Shift (cm -1 ) Figure S6. Raman spectrum of Si 24 and Na 4 Si 24. Experimental spectra are shown and compared with calculated Raman active frequencies (tick marks). Energy-dispervive X-ray spectroscopy (EDXS) The absence of Na from the Si 24 structure was verified using EDXS analyses with a JEOL JSM-6500F microscope and Aztec software. While point analyses demonstrated the absence of Na within the detection limits of the instrument (Fig. 2, main text), larger sample areas were mapped to confirm chemical homogeneity. Figure S7 shows EDXS maps for Si+Na+C and for Na and Ne. The synthesized Si 24 specimens are chemically homogeneous and consist of nearly pure silicon with minor surface carbon contamination (surface organic carbon was also observed on pure single crystal d-si sample standards). However, Na does appear to be present when mapped as a single element (Fig. S7a). This mapping analysis assumes a perfectly flat specimen and therefore, under this assumption, any background noise for a particular element would be uniformly distributed throughout the mapping area. As the specimens we have analyzed do have topographical features (Fig. S7), the apparent distribution of Na is not uniform (Fig. S7b). To confirm that Na is truly absent from the mapping area and only originates from background noise in the fitting routine, we have also performed mapping analysis on Ne (Fig. S7c). Ne was included for two main reasons: (i) it has close, but no overlapping energy peaks with those of Na and, (ii) it is positively not present in the structure. From Figures S7b and S7c one can see that the topographical distribution of Na and Ne is identical and, therefore, is a result of the background noise in the Bremsstrahlung radiation in the peak fitting routine. NATURE MATERIALS 7

8 A B C Figure S7. EDXS mapping. (A) EDXS mapping analysis showing Si+Na+C; (B) Na mapping; (C) Ne mapping. Thermal Stability of Si 24 In order to assess the thermal stability of Si 24 in air, a specimen with this composition was heated in a tube furnace in air at temperatures ranging from 423 K to 773 K. The specimen was held at each temperature for 30 minutes, the temperature was quenched, and X-ray diffraction patterns were subsequently collected at room temperature (Figure S8). From Figure S8 the decomposition temperature of Si 24 was estimated to be approximately 750 K, which is comparable to 797 K for SrSi 6 [19], 737 K in CaSi 6 [20], and 777 K in EuSi 6 [21]. 8 NATURE MATERIALS

9 SUPPLEMENTARY INFORMATION 773 K Intensity (arb. units) 723 K 673 K 623 K 573 K 523 K 473 K q (degrees), Cu K a Figure S8. Thermal decomposition of Si 24 by XRD. XRD patterns of Si 24 pieces measured at room temperature after heating in air up to 773 K for 30 minutes at each temperature. 423 K start Uniaxial compression effect on the band gap Strictly speaking, Si 24 is an indirect band gap material, however, electronic dispersion relations show nearly flat bands along the to Z direction. Due to the small difference between E d and E i, we examined band gap changes during uniaxial compression of Si 24 along c-axis. The difference between the direct and indirect band gaps (E d E i ) is shown in Figure S9. An indirectto-direct band gap transition occurs when the initial lattice constant, c 0, is reduced by ~2%. NATURE MATERIALS 9

10 Figure S9. Difference between E g and E i during uniaxial compression along c. References [1] Hohenberg, P. & Kohn, Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 136, B864-B871 (1964). [2] Kohn, W. & Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 140, A1133-A1138 (1965). [3] Perdew, J. P. et al. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B 46, (1992). [4] Perdew, J. P., Burke, K.& Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, (1996). 10 NATURE MATERIALS

11 SUPPLEMENTARY INFORMATION [5] Giannozzi P. et al., QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. J. Phys. Condens. Matter 21, (2009). [6] Salpeter, E. E. & Bethe, H. A. A Relativistic Equation for Bound-State Problems. Phys. Rev. 84, (1951). [7] Albrecht, S., Reining, L., Sol, R. D. & Onida, G. Ab Initio Calculation of Excitonic Effects in the Optical Spectra of Semiconductors. Phys. Rev. Lett. 80, (1998). [8] Gonze, X., et al. ABINIT: First-principles approaches to materials and nanosystem properties. Computer Phys. Commun. 180, (2009). [9] Tran, F. & Blaha, P., Accurate Band Gaps of Semiconductors and Insulators with a Semilocal Exchange-Correlation Potential. Phys. Rev. Lett. 102, (2009). [10] Hedin, L. New Method for Calculating the One-Particle Green s Function with Application to the Electron-Gas Problem. Phys. Rev. 139, A796 (1965). [11] Hedin, L.& Lundquist, S. Effect of Electron-Electron and Electron-Phonon Interactions on the One-Electron States of Solids in Solid State Physics 1 pp. 23 (Ehrenreich, H., Seitz, F.& Turnbull D. Eds., Academic, New York, 1969). [12] Onida, G., Reining, L. & Rubio, A., Electronic excitations: density-functional versus manybody Green s-function approaches. Rev. Mod. Phys. 74, (2002). [13] Schöne, W. D.& Eguiluz, A. G. Self-Consistent Calculations of Quasiparticle States in Metals and Semiconductors. Phys. Rev. Lett., 81, (1998). [14] Heyd, J., Peralta, J. E., Scuseria, G. E.& Martin, R. L. Energy band gaps and lattice parameters evaluated with the Heyd-Scuseria-Ernzerhof screened hybrid functional. J. Chem. Phys. 123, (2005). NATURE MATERIALS 11

12 [15] Shishkin, M.& Kresse, G. Self-consistent GW calculations for semiconductors and insulators. Phys. Rev. B 75, (2007). [16] Baroni, S., Gironcoli, S., Corso, A. D.& Giannozzi, P. Phonons and related crystal properties from density-functional perturbation theory. Rev. Mod. Phys. 73, (2001). [17] Refson, K., Tulip, P. R.& Clark, S. J. Variational density-functional perturbation theory for dielectrics and lattice dynamics. Phys. Rev. B 73, (2006). [18] Halls, M. D. & Schlegel, B. Comparison study of the prediction of Raman intensities using electronic structure methods. J. Chem. Phys. 111, (1999). [19] Wosylus, A., Prots, Y., Burkhardt, U., Schnelle, W., Schwarz, U. & Grin, Y. High-pressure synthesis of strontium hexasilicide. Z. Naturforsch 61, (2006). [20] Wosylus, A., Prots, Y., Burkhardt, U., Schnelle, W., Schwarz, U. High-pressure synthesis of the electron-excess compound CaSi 6. Sci. Technol. Adv. Mater. 8, (2007). [21] Wosylus, A., Prots, Y., Burkhardt, U., Schnelle, W., Schwarz, U. & Grin, Y. Breaking the Zintl rule: High-pressure synthesis of binary EuSi 6 and its ternary derivative EuSi 6-x Ga x (0 x 0.6). Solid State Sciences 8, (2006). 12 NATURE MATERIALS