Supporting Information for. Design of Dipole-Allowed Direct Band Gaps in Ge/Sn. Core-Shell Nanowires

Transcription

1 Supporting Information for Design of Dipole-Allowed Direct Band Gaps in Ge/Sn Core-Shell Nanowires Elisabeth Pratidhina, Sunghyun Kim, and K. J. Chang* Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Rep. of Korea *Corresponding author. Telephone number: Fax number: Intrinsic strain induced in Ge/Sn core-shell nanowires Based on the continuum elasticity model, a methodology was proposed to determine critical dimensions for coherently strained coaxial nanowire heterostructures [1]. In [111]-oriented semiconductor core-shell nanowire structures, the equilibrium lattice strain is expressed as where and are given by = ( + ( +2 + ( + 2, (1) = ( + 2 ( +6. (2) 2( + +2 The parameters and represent the radius of the core and the thickness of the shell, respectively, and the variables and denote the equilibrium lattice constants of the core S1

2 and shell materials, respectively. In our Ge/Sn core-shell NWs, the equilibrium lattice constants and the elastic parameters,,, and, are obtained from the HSE06 calculations, as shown in Table S1. Finally, the intrinsic strain induced to the Ge core is calculated as =. (3) The intrinsic strain in Ge/Sn core-shell NWs is drawn as a color map for different core diameters and shell thicknesses in Figure S3. The results of the simple continuum model are in good agreement with the LDA calculations for small-diameter core-shell NWs, ensuring the validity of the strain model for nanowires with large diameters. S2

3 Table S1. The elastic parameters and equilibrium lattice constants obtained from the HSE06 calculations for Ge and Sn. Element (kbar (kbar (kbar (A Ge Sn Table S2. The band gaps from the LDA, GGA+U, and HSE06 calculations are compared for pure Ge n NWs consisting of n Ge layers. The LDA band gaps are improved by the GGA+U approximation for the exchange-correlation potential, in which the on-site Coulomb parameter of U = 0 ev and the on-site exchange parameter of J = 3.33 ev were chosen to reproduce the measured band gap of bulk Ge [2]. ID and D represent indirect and direct band gaps, respectively. NWs Band gap (ev) LDA HSE06 GGA+U Ge ID Ge ID Ge ID Ge ID Ge ID Ge ID Sn D Sn D Sn D Gap S3

4 Supplementary Figures Figure S1. (a) The variation of the band structure of the [100]-oriented Ge NW with the diameter of Å under uniaxial tensile strain. In the absence of strain, the valence band maximum (VBM) and conduction band minimum (CBM) states are located at the Γ and points, respectively. These states are derived from the Γ and points in the Brillouin zone of bulk Ge, respectively, whereas the conduction band edge state at the Γ point is derived from the point. The indirect gap nature remains unchanged under uniaxial strain. (b) The atomic structure of the [100]-oriented Ge NW with the diameter of Å. (c) The orbital characteristics of the VBM and the conduction band edge states at the Γ and points, which are represented by the degrees of contributions from the Γ,, and points in the bulk BZ. S4

5 Figure S2. (a) The variation of the band structure of the [110]-oriented Ge NW with the diameter of Å under uniaxial tensile strain. In the absence of strain, although the Ge NW has the direct band gap, the optical transition at the threshold energy is not dipole-allowed because the VBM and CBM states are derived from the Γ and points in the Brillouin zone of bulk Ge, respectively. Under uniaxial strain, the position of the VBM state shifts away from the Γ point, resulting in the indirect band gap. (b) The atomic structure of the [110]-oriented Ge NW with the diameter of Å. (c) The orbital characteristics of the VBM and the conduction band edge states at the Γ and points, which are represented by the degrees of contributions from the Γ and points in the bulk BZ. S5

6 Figure S3. Based on the continuum elasticity model [1], the calculated intrinsic strain induced by the Sn shell is drawn as a color map in Ge n /Sn m core-shell NWs with different core diameters. Triangles and circles represent the indirect and direct band gaps, respectively, based on the LDA calculations. For a pure Ge n=7 NW with the diameter of about 3 nm, the GGA+U (see Table S2) and LDA calculations yield the critical strains of 6.3% and 6.6%, respectively, at which the indirect-to-direct band gap transition occurs (see Figure S5). In Ge 7 /Sn m core-shell NWs, the strain induced by the Sn shell of m = 4 roughly corresponds to the critical strain. Similarly, for a pure Ge n=6 NW, the critical strains of 5.6% and 5.7%, corresponding to the induced strain by the Sn shell of m = 3, are obtained from the GGA+U and LDA calculations, respectively. S6

7 Figure S4. The LDA band structures of [111]-oriented Ge n /Sn m core-shell NWs with the diameters of Å. S7

8 Figure S5. The variations of the indirect and direct band gaps are plotted as a function of the uniaxial strain for the [111]-oriented pure Ge n NW (n = 7) with the diameter of about 3 nm. The indirect-to-direct band gap transition occurs at 6.3% strain and the band gap is estimated to be about 1.0 ev at the critical strain. In the GGA+U approximation for the exchange-correlation potential, the on-site Coulomb parameter of U = 0 ev and the on-site exchange parameter of J = 3.33 ev are used to reproduce the measured band gap of bulk Ge (see Table S2). In the LDA calculations, although the band gap is underestimated, the critical strain of 6.6% is very close to the GGA+U result. S8

9 Figure S6. (a) The LDA band gaps of pure Ge n NWs and Ge n /Sn m core-shell NWs. (b) The constant shifts of the LDA band gaps of Ge n /Sn m core-shell NWs with including the gap corrections by the GGA+U calculations for pure Ge n NWs (see Table S2). Solid lines represent the fitted exponential curves of the calculated band gaps, whereas dashed lines for Ge 6 /Sn m and Ge 7 /Sn m core-shell NWs denote the expected band gaps by assuming the same exponential behavior as that for Ge 5 /Sn m core-shell NWs. S9

10 Figure S7. The atomic structures of the Ge/Sn alloy nanowires. The alloy nanowires contain the same numbers of Ge and Sn atoms as in the Ge 4 /Sn 2 core-shell nanowire. The blue, purple, and pink balls represent Ge, Sn, and H atoms, respectively. The outermost shells are composed of only Sn atoms to avoid Ge-H bond. S10

11 References [1] Raychaudhuri, S.; Yu, E. T. Calculation of Critical Dimensions for Wurtzite and Cubic Zinc Blende Coaxial Nanowire Heterostructures. J. Vac. Sci. Technol. B 2006, 24, [2] Liu, L.; Zhang, M.; Hu, L.; Di, Z.; Zhao, S.-J. Effect of Tensile Strain on the Electronic Structure of Ge: A First-Principles Calculation. J. Appl. Phys. 2014, 116, S11