SUPPLEMENTARY INFORMATION

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1 Supplementary information Enhanced light emission from strained Ge M. J. Süess a,b,c,*, R. Geiger a,*, R. A. Minamisawa a,*, G. Schiefler a, J. Frigerio d, D. Chrastina d, G. Isella d, R. Spolenak b, J. Faist e, H. Sigg a a Laboratory for Micro- and Nanotechnology (LMN), Paul Scherrer Institut, CH-5232 Villigen, Switzerland b Laboratory for Nanometallurgy (LNM), Department of Material Science, ETH Zurich, CH Zürich, Switzerland c Electron Microscopy (EMEZ), ETH Zurich, CH-8093 Zürich, Switzerland d L-NESS, Dip. di Fisica del Politecnico di Milano, Polo di Como, I Como, Italy e Institute for Quantum Electronics (QOE), ETH Zurich, CH-8093 Zürich, Switzerland * These authors have equal contribution. This file contains: 1. Finite element method simulations 2. Raman measurements 3. Impact of the Si layer thickness on the strain of the Ge/SOI structures a. Raman mapping vs. FEM b. Strain uncertainty and variability 4. Micro photoluminescence spectroscopy 5. Gain modeling 6. References 7. Figures NATURE PHOTONICS 1

2 1-) Finite element method simulations Finite element method (FEM) simulations were performed using the software package COMSOL Multiphysics 4.3. All calculations assumed linear elastic deformations and an anisotropic stiffness tensor c ij oriented in [100], [010], [001] coordinates: cc!" = cc!! cc!" cc!" cc!" cc!! cc!" cc!" cc!" cc!! cc!! cc!! cc!! where c 11 = GPa, c 12 = 47.9 GPa and c 44 = 67.0 GPa are the elastic constants of Ge 1. An initial biaxial stress of GPa (corresponding to a biaxial strain of 0.15%) and the geometrical parameters A, a, B, b and L (see main text), as well as the thicknesses of the layers were determined by optical and scanning electron microscopy. With this model a full 3D distribution of each strain tensor component was calculated. 2 NATURE PHOTONICS

3 SUPPLEMENTARY INFORMATION 2-) Raman measurements (a) Raman mapping vs. FEM Raman Spectroscopy was performed in backscattering geometry on a WITec CRM200 system (100 objective lens, numerical aperture NA = 0.9) using 532 nm excitation wavelength with a 1/e 2 -penetration depth of 43 nm in Ge, and a FWHM of 360 nm at the focal point. The incoming light was polarized along the main strain direction at the constriction ([100]). In order to correlate the measured Raman shifts to the FEM simulations the calculated strain tensor components are fed into the dynamical secular equation, taking into account the unstrained phonon frequency ω 2 0 = s -2, as well as the phonon deformation potentials p/ω 2 0 = -1.45, q/ω 2 0 = and r/ω 2 0 = -1.1 of Ge 2. The Raman shift can be calculated from the eigenvalues λ i = ω 2 i - ω 2 0 (i = 1,2,3), which represent three nondegenerate Raman modes for a material with a strain tensor with non-zero shear components. ppεε!! + qq 2rrεε εε!! + εε!! λλ! ppεε + qq 2rrεε εε!" + εε λλ 2rrεε!" = 0 2rrεε!" ppεε!! + qq 2rrεε εε!! + εε!! λλ! ppεε + qq 2rrεε εε!" + εε λλ = 0 2rrεε!" 2rrεε!" ppεε!! + qq εε!! + εε!! λλ! When assuming a purely biaxial and uniaxial stress situation these phonon deformation potentials lead to strain-shift coefficients of b bi = -424 and b uni = -154, respectively, which are close to values reported in experimental works 3,4. Following the work of Akbari et al. 5, the intensity I i of each mode was calculated from:!!!!!!!! (!) II II! TT(φφ, θθ)ee!!!!! TT(φφ, θθ)ee! (!)!!!!! (!) II!!!! TT(φφ, θθ)ee!!!!!!! RR RR! ee ee!! sin θθθθθθθθθθ sin θθθθθθθθθθ RR! ee! sin θθθθθθθθθθ where T(ϕ,θ) is the microscope-objective transfer matrix, which takes into account the large NA of our setup, e (n) s, n=1,2 are the scattered polarization vectors [100] and [010], R i is the Raman polarizability tensor of the perturbed phonons and e e is the excitation polarization vector. θ m is the half opening angle of the light cone probing the medium, which can be found through Snell s law, using the NA of the objective and the refractive index of the medium 5. The final shift was then found summing all shifts weighted with their relative intensity: NATURE PHOTONICS 3

4 ΔΔΔΔ! = II! ΔΔΔΔ! II!! where Δω = ω i - ω 0 λ i /2ω 0. Additional corrections reported by de Wolf et al. 6 were performed, in order to account for beam shape, absorption and the heterogeneous strain distribution along the beam direction. For each position an average over the out-of-plane dimension z weighted with the relative absorption is calculated:! ΔΔΔΔ! xx, yy, zz ee!" ΔΔΔΔ xx, yy =! ee!" using the absorption coefficient α = cm -1 for nm (2.33 ev) wavelength 7. Additionally, the influence of the beam shape is accounted for by weighting the shift with a Gaussian beam profile, where b x = b y = 360 nm is the calculated resolution of the beam:! ωω xx!, yy! = ωω! xx, yy ee!!!!!!!!!!!!!!!!!,! ee!!!!!!!!!!!!!!!!!,! Fig. S1 shows simulated strain maps (Fig. S1A) of the strain tensor entries ε xx, ε yy, ε zz and ε xy taken near the surface. ε xz and ε yz are close to zero throughout the whole map and therefore not shown. These entries are used to calculate the Raman shift maps and linescans in the main text, as well as the Raman shift map shown in Fig. S1b, which matches very well to the measured Raman mapping (Fig. S1C). 4 NATURE PHOTONICS

5 SUPPLEMENTARY INFORMATION a xx b yy y z x Strain (%) zz c xy rel. Raman shift (cm -1 ) Figure S1: Comparison between FEM strain maps, as well as calculated and measured Raman shift maps. (A) FEM strain maps of the strain tensor entries ε xx, ε yy, ε zz and ε xy (ε xz and ε yz are not shown). (B) Calculated Raman shift map obtained by inserting the strain tensor in the secular equation (see main text). (C) Measured Raman shift map. NATURE PHOTONICS 5

6 (b) Strain uncertainty and variability Power dependent measurements assured that the laser power was sufficiently small to not cause thermal peak shift. Peak shifts were retrieved by sub-pixel fitting of the original spectra with a Lorentzian line shape. The relative statistical error of the fitting routine was 0.3%. The statistical variation from processing was tested by a random sampling over 40 structures with enhancement factor 11. The relative strain variability between structures was found to be 3.3%. 6 NATURE PHOTONICS

7 SUPPLEMENTARY INFORMATION 3-) Impact of the Si layer thickness on the strain of the Ge/SOI structures Suspended Ge structures fabricated on SOI bend due to the strain in the Ge layer, similar to a bi-layer configuration experiencing different thermal elongations (i.e. strains) upon heating. The impact of the Si layer thickness on the strain of the Ge/SOI structures is studied with FEM simulations. The strain and curvature of the Ge layers as a function of the thickness ratio between the Si and the Ge layer t Si /t Ge (A = 6 µm, a = 2 µm, B = 30 µm, b =51 µm, ε bi = 0.18) is shown in Fig. S2. Both, the curvature of the central part of the structure and the strain tensor element ε xx reach maximum values for t Si /t Ge = 0.5. For ratios below this value, both strain and curvature approach values of structures without Si layer underneath. For t Si /t Ge >> 0.5, the curvature is suppressed and the longitudinal strain will strongly decrease, as most of the relaxation will be hindered by the thick Si layer. In this work we use t Si /t Ge = 0.21, which results in a 25% additional enhancement. xx (t Si = 0) Figure S2: Systematic FEM study of the influence of the layer thickness ratio on curvature and strain ε xx at the center of the structures. NATURE PHOTONICS 7

8 4-) Micro photoluminescence spectroscopy µ-photoluminescence (µpl) measurements were performed using a diode-pumped continuous-wave solid-state laser emitting at 532 nm as the excitation source. The laser beam was focused using a 50 microscope objective (effective NA = 0.2) to a spot size of ~ 0.8 µm. The emission signal was collected with the same objective (using the full NA = 0.8), transmitted via a 400 µm diameter optical fiber and analyzed with a Fourier transform infrared spectrometer to which a liquid nitrogen cooled InSb detector with a cold short pass filter for suppression of dark currents is attached. The filter causes a cut-off of the sensitivity at about 0.56 ev not very far from the onset of the PL signal for a constriction with 3.1% strain. To confirm the position of the PL peaks of Fig. S3a, measurements were repeated with a home-made infrared (IR) microfocus set-up equipped with all-metal optics and an unfiltered InSb detector. The same laser has been used for excitation, but with a spot-size of 1.8 µm. The spectra were recorded in step-scan mode with typically a resolution of 100 cm -1 (12.4 mev). Good agreement is obtained between the spectra taken on a constriction with a strain of 3.1 % with the two microscope set-ups. One of the key concerns for PL measurements in Ge is the impact of heating as the band gap decreases with increasing temperature. In order to quantify the role of heating, we have performed power dependent measurements using the broad band IR detection set-up and compared the results to heat transfer FEM simulations. Fig. S3a displays the normalized PL spectra of a 3.1% strained Ge structure measured at different excitation powers. The signals below 0.4 ev are attributed to blackbody (BB) emission. Increasing the power density from 0.10 to 0.77 MW/cm 2 results in a shift of the PL onset of about 20 mev as is seen in the inset of Fig. S3a. The intensity was estimated assuming a flat top beam profile with the aforementioned spot diameter and the accurately measured integrated power. The maximum power in this experiment was 19.5 mw. The temperature profile along a constricted structure obtained from FEM simulations is shown in Fig. S3b. It indicates that the temperature in the constriction is increased by ~100 K at 0.77 kw/cm 2. For unstrained Ge 8 this temperature induces a band gap narrowing at Γ of ~20 mev, which nicely confirms our experimental data. Fig. S4c shows that the integrated PL intensity strongly increases with increasing the excitation power following a power law with an exponent of NATURE PHOTONICS

9 SUPPLEMENTARY INFORMATION M M M M a Normalized PL Intensity (a. u.) Integrated PL Int. (a. u.) c Exp. b Figure S3: (A) Power dependent PL spectra of a 3.1% strained Ge structure excited in the center of the constriction by a 532 nm wavelength laser with spot size of about 1.8 µm. The data are normalized to the peak intensity of the PL in order to highlight the impact of the excitation power on the PL on-set. The signal starting at 0.2 ev is attributed to blackbody emission; its relative contribution decreases for higher power. The inset displays a zoom of the PL on-set. (B) FEM simulation of the temperature profile along one constricted structure. The position zero corresponds to the center of the constriction, where the incident laser of spot size 1.8 µm is positioned. The maximum temperature enhancement reached at 0.77 kw/cm 2 is about 100 K and is found to fall off quickly towards the outer end of the constriction (shown by the dotted line at 3 µm). (C) Integrated PL intensity for different excitation powers. The integrated intensity strongly increases, which can be described by a power law with exponent 1.6. NATURE PHOTONICS 9

10 5-) Gain modeling In order to obtain the band edge energies and the effective masses m* of uniaxially stressed Ge structures, the band structure was calculated employing the deformation potential and 8- band k p models implemented in the nextnano3 software package 9. The parameters used in the simulation and the respective references are displayed in Table S1. The 6-band Dresselhaus parameters L, M and N for Ge were calculated ( from the experimental Lüttinger parameters γ 1, γ 2 and γ 3 obtained by Hensel and Suzuki 10 : " % L$ 2 ' = γ # 2m 1 4γ & " % M$ 2 ' = ( 2γ # 2m 2 γ 1 1) 0 & " % N$ 2 ' = 6γ # 2m 3 0 & ( ) We scaled the 6-band parameters to the respective 8-band parameters ( using the dipole matrix element E p and the direct band gap of relaxed Ge E gap at 300K 11. The matrix element E p = 26 ev was obtained by fitting the absorption coefficient e 5 2 & 2m α(hν) = ( r 6πε 0 nm 0 c ' 2 ) + * 3 2 E p hν hν E gap to experimental data published elsewhere 12. Here, ε 0 is the electrical permittivity, n the refractive index, m 0 the free electron mass, c the vacuum speed of light, hν the photon energy, e the elemental charge, ħ the reduced Planck s constant and m r is the reduced mass of electrons and holes at the Γ point of the Brillouin zone. Excitonic effects were not considered in the model. Furthermore, the inversion symmetry parameter B was set to 0, whereas the Kane parameter S was calculated ( using E p, E gap, the unstrained electron mass m e at k=0 and the spin-orbit coupling Δ SO. 10 NATURE PHOTONICS

11 SUPPLEMENTARY INFORMATION Table S1. Parameters used in the 8 band k p simulations. Deformation potentials a v = ev (Ref. 13 ) a c (Γ) = ev (Ref. 13 ) a c (L) = ev (Ref. 13 ) b = ev (Ref. 14 ) d = ev (Ref. 14 ) Ξ u (Γ) = 0 ev Ξ u (L) = ev (Ref. 15 ) Elastic constants C 11 = GPa (Ref. 1 ) k p parameters Effective masses in units of m 0 C 12 = 47.9 GPa (Ref. 1 ) C 44 = 67.0 GPa (Ref. 1 ) L (ħ2/2m 0 ) = L + E p /E gap = 1.16 ( M (ħ2/2m 0 ) = M = -5.9 ( N (ħ2/2m 0 )= N + E p /E gap = ( γ 1 = (Ref. 10 ) γ 2 = 4.24 (Ref. 10 ) γ 3 = 5.69 (Ref. 10 ) E p = 26 ev E gap = 0.80 ev (Ref. 11 ) B = 0 S = 1 m e E p E gap Δ SO = ( E gap + Δ SO E gap m e = [17] ( ) Δ SO = ev (Ref. 16 ) m l Γ L (Ref. 17 ) (Ref. 17 ) VB VB VB m t At 3.1% strain, the simulation predicts a separation of 46 mev between the Γ and L valleys. The quasi-fermi levels µ e (for electrons) and µ h (for holes) were obtained by inverting NATURE PHOTONICS 11

12 and nn!" = nn!" =! eedd!! EE, mm ff! EE, TT, μμ! dddd!!!! eedd!! EE, mm ff! EE, TT, μμ! dddd!!! for a given electron density n el and hole density n h, respectively. Here, D 3D (E,m*) is the three dimensional density of states (DOS), f e/h (E,T,µ e/h ) the Fermi distribution for electrons and holes and T the temperature. The simulations were performed for a temperature of 300K. For the conduction band DOS we considered Γ and the L valleys and for the valence band DOS heavy hole-, light hole- and spin-orbit split-off bands which do mix under the applied strain. In Table S1 the effective masses for 3.1% strain used for the DOS calculations are listed. VB1, VB2 and VB3 denote the valence bands, with the highest band being referred to as VB1. For Γ and the valence bands, m l corresponds to the [100] direction and m t to the effective mass along the [010] and [001] directions, whereas for L, m l and m t correspond to the effective masses along the [111] and [ 110] directions, respectively. The L effective masses are assumed to be unaltered by strain. The gain spectrum g(hν) was calculated from Fermi s golden rule using g(hν) = e 2 η E p 6πε 0 nm 0 c hν k t dk t m l { ( ) f ( E k,v ( k t ), µ h,t)} f E 2 k k,c ( k t ),µ e,t l where m l is the longitudinal reduced effective mass of electrons and holes near the Γ-point, and k t and k l the transversal and the longitudinal wavevector, respectively. The model takes into account that the transitions take place between the spherical s-like conduction band and the uppermost valence band, and thus E p is scaled by η=3/4. Transitions in the second valence band are neglected because its position at Γ is 130 mev lower in energy at the strain of 3.1% than the uppermost valence band states. To achieve net gain, the material s losses due to free carrier absorption and the intraband transitions between the strongly dispersive and strain split valence band levels have to be overcome. Such absorption cross sections were provided recently by Carroll et al. 18 who determined for biaxially strained Ge (0.15%) at the energy of 775 mev a cross section of σ h = cm 2 for the holes and estimated that σ e = cm 2. The absorption increases as the energy decreases by α tot (hν) = (N e,tot σ e + N h,tot σ h ) { (0.775 ev hν)} (as extracted from Carroll et al. 18, Fig. 2a), where N e,tot and N h,tot correspond to the total electron and hole densities, respectively. 12 NATURE PHOTONICS

13 SUPPLEMENTARY INFORMATION 6-) References 1. Wortman, J.J. and Evans, R.A. Young's Modulus, Shear Modulus, and Poisson's Ratio in Silicon and Germanium. J. Appl. Phys. 36, (1965). 2. Lockwood, D.J. Light scattering in semiconductor structures and superlattices. Plenum Press 273, (1991). 3. Cerdeira, F., Buchenauer, C.J., Pollak, F.H. and Cardona, M. Stress-Induced Shifts of First- Order Raman Frequencies of Diamond- and Zinc-Blende-Type Semiconductors. Phys. Rev. B 5, (1972). 4. Pezzoli, F. et al. Strain-induced shift of phonon modes in Si1-xGex alloys. Mat. Sci. Semicond. Proc. 9, (2006). 5. Akbari, M., Buhl, S., Leinenbach, C., Spolenak, R. and Wegener, K. Thermomechanical analysis of residual stresses in brazed diamond metal joints using Raman spectroscopy and finite element simulation. Mech. Mater. 52, (2012). 6. De Wolf, I., Maes, H.E. and Jones, S.K. Stress measurements in silicon devices through Raman spectroscopy: Bridging the gap between theory and experiment. J. Appl. Phys. 79, (1996). 7. Aspnes, D.E. and Studna, A.A. Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 ev. Phys. Rev. B 27, (1983). 8. Braunstein, R., Moore, A.R. and Herman, F. Intrinsic Optical Absorption in Germanium- Silicon Alloys. Phys. Rev. 109, (1958). 9. Birner, S. et al. nextnano: General Purpose 3-D Simulations. IEEE Trans. Electron. Dev. 54, (2007). 10. Hensel, J.C. and Suzuki, K. Quantum resonances in the valence bands of germanium. II. Cyclotron resonances in uniaxially stressed crystals. Phys. Rev. B 9, (1974). 11. Schäffler, F. High-mobility Si and Ge structures. Semicond. Sci. Tech. 12, 1515 (1997). NATURE PHOTONICS 13

14 12. Seeger, K. Semiconductor physics : an introduction. Springer-Verlag (1989). 13. Wei, S.-H. and Zunger, A. Predicted band-gap pressure coefficients of all diamond and zinc-blende semiconductors: Chemical trends. Phys. Rev. B 60, (1999). 14. Chandrasekhar, M. and Pollak, F.H. Effects of uniaxial stress on the electroreflectance spectrum of Ge and GaAs. Phys. Rev. B 15, (1977). 15. Van de Walle, C.G. Band lineups and deformation potentials in the model-solid theory. Phys. Rev. B 39, (1989). 16. Madelung, O. Semiconductors: group IV elements and III-V compounds. Springer-Verlag, (1991). 17. Cardona, M. and Pollak, F.H. Energy-Band Structure of Germanium and Silicon: The k.p Method. Phys. Rev. 142, (1966). 18. Carroll, L. et al. Direct-gap gain and optical absorption in germanium correlated to the density of photoexcited carriers, doping, and strain. Phys. Rev. Lett. 109, (2012). 14 NATURE PHOTONICS