new low-density allotrope of silicon - nature research · 2015-01-23 · phonon dispersion...

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NMAT4140

NATURE MATERIALS | www.nature.com/naturematerials 1

Supplementary Information

New Low-Density 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 Si24.

Si24 – full profile refinement, =0.406626 Å Space group Cmcm (No. 63) a (Å) 3.82236(14) b (Å) 10.7007(4) c (Å) 12.6258(5) Atomic coordinates

x y z Occupancy Uiso Si1 (8f) 0 0.2435(2) 0.5551(2) 1 0.0119(7) Si2 (8f) 0 0.5705(2) 0.3412(2) 1 0.0119(6) Si3 (8f) 0 0.0284(2) 0.5903(2) 1 0.0073(6)

Refinement statistics χ2

= 1.247 wRp = 0.0435 (-Bknd) Rp = 0.0291 (-Bknd)

Synthesis of an open-framework allotrope of silicon

© 2014 Macmillan Publishers Limited. All rights reserved.

http://www.nature.com/doifinder/10.1038/nmat4140

2 NATURE MATERIALS | www.nature.com/naturematerials

SUPPLEMENTARY INFORMATION DOI: 10.1038/NMAT4140

Table S2. Crystallographic data for Si24 (DFT, PBE) Si24 – DFT, PBE calculations, 1 atm

Space group Cmcm (No. 63) a (Å) 3.8475 b (Å) 10.7443 c (Å) 12.7342 Atomic coordinates

x y z Occupancy Si1 (8f) 0 0.24285 0.55476 1 Si2 (8f) 0 0.57130 0.34274 1 Si3 (8f) 0 0.02862 0.59056 1

Figure S1. Unit cell of Si24 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 Si24 using several computational approaches to make it clear that Si24 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 GW0 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 Si24. 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(G0W0) and partially self-consistent GW0. It is

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, G0W0 and GW0 give excellent agreement with the experimental band gap for d-Si (1.17 eV) and in the main text, we used the G0W0 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 Si24 using various functionals. Units are in eV.

d-Si, indirect

Si24,

indirect (direct) PBE 0.62b 0.53 (0.57)

HSE06 1.28a 1.41 (1.45) G0W0 1.12 1.30 (1.34) GW0 1.2b 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 NaxSi24 (0 x 4) at different values of x. Supercells of of Si24 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 Na4Si24 and Si24 (Figure S2).

Figure S2. Lattice parameters of NaxSi24 with respect to sodium concentration x.

4.03.22.41.60.80.0

Latti

ce P

aram

eter

s (Å

)

Na content, x, in NaxSi24

4

10

11

12

13





Table S2. Crystallographic data for Si24 (DFT, PBE) Si24 – DFT, PBE calculations, 1 atm

Space group Cmcm (No. 63) a (Å) 3.8475 b (Å) 10.7443 c (Å) 12.7342 Atomic coordinates

x y z Occupancy Si1 (8f) 0 0.24285 0.55476 1 Si2 (8f) 0 0.57130 0.34274 1 Si3 (8f) 0 0.02862 0.59056 1

Figure S1. Unit cell of Si24 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 Si24 using several computational approaches to make it clear that Si24 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 GW0 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 Si24. 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(G0W0) and partially self-consistent GW0. It is

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, G0W0 and GW0 give excellent agreement with the experimental band gap for d-Si (1.17 eV) and in the main text, we used the G0W0 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 Si24 using various functionals. Units are in eV.

d-Si, indirect

Si24,

indirect (direct) PBE 0.62b 0.53 (0.57)

HSE06 1.28a 1.41 (1.45) G0W0 1.12 1.30 (1.34) GW0 1.2b 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 NaxSi24 (0 x 4) at different values of x. Supercells of of Si24 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 Na4Si24 and Si24 (Figure S2).

Figure S2. Lattice parameters of NaxSi24 with respect to sodium concentration x.

4.03.22.41.60.80.0

Latti

ce P

aram

eter

s (Å

)

Na content, x, in NaxSi24

4

10

11

12

13





Metal-Insulator transition in NaxSi24 Computationally, we checked if NaxSi24 becomes a semiconductor at low values of x.

Figure S3 shows the electronic density of states of Na1Si24, Na0.5Si24, Na0.333Si24, and Na0.125Si24. 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 NaxSi24.

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 Si24 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 Si24 is dynamically stable to 10 GPa. Above this pressure, it becomes destablized, indicating a structural transformation.

Figure S4. Phonon dispersion relations of Si24 at ambient pressure (a) and 10 GPa (b).

Raman Scattering Raman scattering data were collected from “degassed” Si24 and pristine Na4Si24 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 narrow-band 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 2× 0.04 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 Si24. 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





condition during the measurements and crystallite orientation anistropy may also contribute to intensity variation.

Figure S5. Raman spectrum of Si24. 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 Si24 with Na4Si24 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 Si24.

100 200 300 400 500

Raman Shift (cm-1)

Experiment

DFTB3gAg

B1g

B1g

Ag Ag

Ag,B2g

B1g

Ag

B3g

B2g

Ag

10

Figure S6. Raman spectrum of Si24 and Na4Si24. 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 Si24 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 Si24 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.

50 150 250 350 450 550Raman Shift (cm-1)

Rela

tive

Inte

nsity

(arb

. uni

ts)

Si24

Na4Si24





Figure S7. EDXS mapping. (A) EDXS mapping analysis showing Si+Na+C; (B) Na mapping; (C) Ne mapping. Thermal Stability of Si24

In order to assess the thermal stability of Si24 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 Si24 was estimated to be approximately 750 K, which is comparable to 797 K for SrSi6 [19], 737 K in CaSi6 [20], and 777 K in EuSi6 [21].

A

B C

Figure S8. Thermal decomposition of Si24 by XRD. XRD patterns of Si24 pieces measured at room temperature after heating in air up to 773 K for 30 minutes at each temperature. Uniaxial compression effect on the band gap

Strictly speaking, Si24 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 Ed and Ei, we examined band gap changes during uniaxial compression of Si24 along c-axis. The difference between the direct and indirect band gaps (Ed – Ei) is shown in Figure S9. An indirect-to-direct band gap transition occurs when the initial lattice constant, c0, is reduced by ~2%.

10 15 20 25 30 35 40 45 50 55

start

423 K

473 K

523 K

573 K

623 K

673 K

723 K

773 K

2q (degrees), Cu Ka

Inte

nsity

(arb

. uni

ts)





Figure S9. Difference between Eg and Ei 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, 6671-6687 (1992).

[4] Perdew, J. P., Burke, K.& Ernzerhof, M. Generalized Gradient Approximation Made Simple.

Phys. Rev. Lett. 77, 3865-3868 (1996).

[5] Giannozzi P. et al., QUANTUM ESPRESSO: a modular and open-source software project

for quantum simulations of materials. J. Phys. Condens. Matter 21, 395502 (2009).

[6] Salpeter, E. E. & Bethe, H. A. A Relativistic Equation for Bound-State Problems. Phys. Rev.

84, 1232-1242 (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, 4510-4513 (1998).

[8] Gonze, X., et al. ABINIT: First-principles approaches to materials and nanosystem properties.

Computer Phys. Commun. 180, 2582-2615 (2009).

[9] Tran, F. & Blaha, P., Accurate Band Gaps of Semiconductors and Insulators with a Semilocal

Exchange-Correlation Potential. Phys. Rev. Lett. 102, 226401 (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 many-

body Green’s-function approaches. Rev. Mod. Phys. 74, 601-659 (2002).

[13] Schöne, W. D.& Eguiluz, A. G. Self-Consistent Calculations of Quasiparticle States in

Metals and Semiconductors. Phys. Rev. Lett., 81, 1662-1665 (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, 174101 (2005).





[15] Shishkin, M.& Kresse, G. Self-consistent GW calculations for semiconductors and

insulators. Phys. Rev. B 75, 235102 (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, 515-562 (2001).

[17] Refson, K., Tulip, P. R.& Clark, S. J. Variational density-functional perturbation theory for

dielectrics and lattice dynamics. Phys. Rev. B 73, 155114 (2006).

[18] Halls, M. D. & Schlegel, B. Comparison study of the prediction of Raman intensities using electronic structure methods. J. Chem. Phys. 111, 8819-8824 (1999).

[19] Wosylus, A., Prots, Y., Burkhardt, U., Schnelle, W., Schwarz, U. & Grin, Y. High-pressure

synthesis of strontium hexasilicide. Z. Naturforsch 61, 1485-1492 (2006).

[20] Wosylus, A., Prots, Y., Burkhardt, U., Schnelle, W., Schwarz, U. High-pressure synthesis of

the electron-excess compound CaSi6. Sci. Technol. Adv. Mater. 8, 383-388 (2007).

[21] Wosylus, A., Prots, Y., Burkhardt, U., Schnelle, W., Schwarz, U. & Grin, Y. Breaking the

Zintl rule: High-pressure synthesis of binary EuSi6 and its ternary derivative EuSi6-xGax (0

x 0.6). Solid State Sciences 8, 773-781 (2006).



new low-density allotrope of silicon - nature research · 2015-01-23 · phonon dispersion...

Documents