Strain Engineering of Germanium Nanobeams by Electrostatic Actuation

Germanium (Ge) is a promising material for the development of a light source compatible with the silicon microfabrication technology, even though it is an indirect-bandgap material in its bulk form. Among various techniques suggested to boost the light emission efficiency of Ge, the strain induction is capable of providing the wavelength tunability if the strain is applied via an external force. Here, we introduce a method to control the amount of the axial strain, and therefore the emission wavelength, on a suspended Ge nanobeam by an applied voltage. We demonstrate, based on mechanical and electrical simulations, that axial strains over 4% can be achieved without experiencing any mechanical and/or electrical failure. We also show that the non-uniform strain distribution on the Ge nanobeam as a result of the applied voltage enhances light emission over 6 folds as compared to a Ge nanobeam with a uniform strain distribution. We anticipate that electrostatic actuation of Ge nanobeams provides a suitable platform for the realization of the on-chip tunable-wavelength infrared light sources that can be monolithically integrated on Si chips.

. The red symbols indicate the voltage levels required to achieve 2% tensile strain in Ge nanobeam for various Ge nanobeam widths for a fixed thickness and length of 50 nm and 1 m, respectively. Solid line is the guide to the eye. Dashed line indicates the required voltage to achieve 2% strain calculated using a 2D model.         The calculated photoluminescence spectra of the deflected and uniformly strained Ge nanobeams for 1%, 2%, 3% and 4% axial strains. (c) The ratio of cumulative radiative recombination in a deflected beam to that in a uniformly strained beam as a function of uniform carrier generation rates for various axial strains. Figure S12. The deflection along the nanobeam simulated by FEM and analytically calculated using Equations (S20) and (S21) 1 . The length and thickness of the Ge nanobeam are 2 m and 30 nm, respectively. The applied voltage is 10 V. The gap between the Ge nanobeam and Si substrate is 100 nm.

Section 1: Comparison of 2D and 3D Ge nanobeams
The results presented in this paper, except for the ones in the initial strain induction, are obtained via 2D simulations. In a 2D geometry (i.e. when the nanobeam is very wide), electric field forms between the Ge nanobeam and Si substrate as straight lines. However, for relatively narrower nanobeams, the electric field lines between the Si substrate and the top and side surfaces of the nanobeam should be taken into account. These additional electric field lines are called fringing fields. Fringing fields provide an additional force on the Ge nanobeam and reduce the required voltage significantly with decreasing nanobeam width as shown in Figure S2. As a result, this paper presents upper limits of the required voltages for 3D Ge nanobeams. 1. The deflection is much smaller than the thickness of the nanobeam. Therefore, the geometric nonlinearity effects such as stress stiffening and stretching can be ignored.

Section 2: Small Deflection Theory
2. The force caused by the fringing fields are negligible.
3. The edges are fixed. 4. The deflection is much smaller than the gap distance so that, the electric force does not alter with deflection.
Under the same set of assumptions, a model to estimate the required voltage and the deflection to reach a predetermined strain can be developed. Based on the small deflection theory, the state of uniaxial stress can be assumed, where, around the symmetry axis, upper portions of the beam undergo negative strain and stress, while the lower portions are in tension. This implies that there exists a neutral axis, where strain and stress are zero. Assuming a local curvature 'Rcv', the longitudinal strain (εxx) at a distance of z from the neutral axis can be calculated as: According to Saint-Venant's principle, Equation (S1) is valid at locations sufficiently far from the supports, where localized distortions are present 2 . Since the nanobeam is prismatic, the neutral axis lie at the mid-plane in the z axis. Thus, the maximum strain at the plane of symmetry can be written where t is the thickness of the nanobeam. Since the first derivative of the radius of curvature is zero at the plane of symmetry, the expression becomes where w(x) denotes the deflection profile of the nanobeam. If the deflection profile is taken as a fourth order polynomial as found by Choi et al. 1 , where wmax is the maximum deflection and L is the length of the nanobeam, then the maximum strain at the plane of symmetry is found as For the calculation of the maximum deflection, linear stiffness assumption is made. For a fixed-edge prismatic beam, the spring constant under uniform load is (S10) When the condition of wmax = g/3 is applied to Equations (S9) and (S10) and they are rearranged to provide required voltage and the deflection to reach a predetermined strain value, Equations (S9) and (S10) become (S12) Equations (S11) and (S12) predict that the required deflection depends linearly on the maximum strain at the symmetry plane and the required voltage varies with 1.5 th power of the maximum strain at the symmetry plane, respectively.

Section 3: Non-linear effects
In Ge nanobeam analysis, since the edges are quasi-fixed, there should be an equivalent axial force to prevent the edges from moving inwards during the deflection. This axial force results in a so-called "nonlinear stretching effect" which, in turn, leads to stress stiffening. Consequently, the stiffness of the nanobeam increases super-linearly with deflection. Thus, the nanobeam undergoes a smaller deflection at an applied voltage compared to what is predicted by the small deflection theory. For nanobeams undergoing sufficiently large deflections, the nonlinear stretching effect at the plane of symmetry can result in large axial tensile strains. These large tensile strains, when superimposed on bending-induced strains (either compressive or tensile), can neutralize compressive strains and can bring about a cross-section with purely tensile strains.
While stress stiffening leads to a reduced deflection at a given voltage, strain localization (i.e. extinction of compressively strained regions) enhances strain at the same deflection when compared to that predicted by the small deflection theory. It is worth mentioning that as t/L ratio increases; the effect of shear deformation on nanobeam deflection becomes more significant, meaning that for large t/L ratios the stiffness of the nanobeam deviates from the relations obtained by disregarding shear deformation.
In addition to geometric nonlinearities, the elasticity of the silicon dioxide on which the Ge nanobeam is resting should be considered. Figure S3 shows that the deflection of and strain on the Ge nanobeam is enhanced when the elasticity of SiO2 is taken into account under the same applied voltage. Yet the elasticity of SiO2 results in an increase in the required deflection and voltage to achieve a predetermined strain as can be seen when Figure 3 and Figure

Section 4: Initial Strain Induction
When the stressed SiNx is deposited on the Ge nanobeam, the beam experience a deflection away from the Si substrate, which increases the gap distance. Consequently, when the potential difference is applied between the Si substrate and the Ge nanobeam, the electrostatic force decreases due to the increased gap distance. Nevertheless, it is possible to decrease the thickness of the SiO2 layer and achieve the predetermined strain at w=g/3. In Figure 4d, the required voltage values to reach 4% axial strain are presented when where wfinal is the deflection after electrostatic potential is applied, winitial is the deflection due to the nitride stress and tox is the oxide thickness, which is equal to the gap distance before the stressed SiNx is deposited.

Section 5: Overcoming the dielectric breakdown of SiO2
The electrical pressure (i.e. electrical force per unit area) on the Ge nanobeam for the structure shown in Figure 1 can be calculated using Equation (S7) as where is the capacitance per unit area. When a dielectric slab is deposited on the Si substrate ( Figure 5a), the electrical pressure becomes where td and ϵd are the thickness and the relative permittivity of the dielectric layer, respectively.
The pressure terms given by Equation (S14) and Equation (S15) are equal to each other when where + td/ϵd is defined as the effective gap distance, geff and is the thickness of the vacuum region after deposition of the dielectric layer.
As a result, the SiO2 layer thickness after the deposition of the dielectric layer, , becomes equal to the summation of the vacuum and dielectric layer thicknesses, which can be written in terms of geff, td and as follows.
It should be noted that the limit of the dielectric thickness, tmax, when the deflection is equal to g/3 is equal to 2 3 ⁄ .

Section 6: Fracture Analysis
To predict the possibility of any failure during bending, the transverse rupture strength of Ge 3 (=18 GPa) is set as the failure criterion, where it is assumed that material failure occurs for induced first principal stresses larger than the transverse rupture strength of Ge. In particular, stress concentration due to sharp corners is also considered. This problem is more pronounced when having initial strains as shown in Figure 4 For unstrained and undoped Ge, the concentration of the electrons residing in the conduction band of the  valley (nΓ) is much smaller than the total electron concentration in the conduction band and the nΓ/n ratio is approximately equal to 10 -4 (0.01%). This ratio increasei.es up to approximately 10 -2 (1%) for a uniaxial tensile strain of 4% 6 . Furthermore, the radiative recombination through the  valley occurs much faster compared to the transitions through L valley with approximately 5 order of magnitude difference between the radiative recombination coefficients (RL << RΓ). As a result, the radiative recombination rate is significantly higher through the  valley, specifically at strained locations, and the effect of radiative recombination through the L valley is negligible. Therefore, we implemented a single band structure featuring solely the  valley. Since the radiative recombinations through the L valley are ignored, this assumption results in a slight-underestimation of the overall radiative recombination rate for the unstrained structure. Therefore, for both the electromechanically-deflected and the uniformly-strained nanobeams, the calculated Urad enhancements compared to the unstrained structure (i.e. Urad,strained/Urad,unstrained) constitute an upper limit of what can practically be achieved. However, also note that since we disregard inter-valley transitions between the and L valleys, this upper limit is likely to be increased when the inter-valley effects are properly accounted for.
Changes in the direct bandgap (Eg,) due to uniaxial strain was calculated following Sukhdeo et al. 7 , as given in Equation S19. In this respect, Figure S10a shows the direct bandgap profile at the bottom of the electromechanically deflected nanobeams (at z = t = 30 nm, from x = 0 to x = L = 200 nm) for maximum strain values varying between 1% to 4%.
where  is the strain in percents.
In the calculation of the photoluminescence (PL) spectra, the radiative recombination rate from each mesh region is retrieved from SILVACO. We took into account the tensilely strained portions of the nanobeam only, and assumed that the emission from each tensilely strained mesh region is due to the electron-light hole radiative recombination with a Lorentzian spectral line shape. The The normalized PL spectra of the deflected and uniformly strained Ge nanobeams with 1, 2, 3, and 4% tensile strains are shown in Figure S10b. The FWHM values of the PL spectra of the deflected Ge nanobeams are significantly larger than that of uniformly strained Ge nanobeams at small strains due to non-uniform bandgap profile of the structure. However, carrier localization to the highest strain locations at the deflected Ge nanobeams become more pronounced as the strain increases. Therefore, the FWHM values of the PL spectra of the deflected Ge nanobeams approach to that of uniformly strained Ge nanobeams.
The illumination intensity can be varied in the photoluminescence measurements, and depending on the optical design of the nanobeam structure, the generation rate in the structure can vary.
Moreover, the total generation rate constitutes transitions from the valence band to both the and L valleys. Thus, the generation rate to the  valley is less than the total generation rate. Due to these factors, the calculated generation rate (G) of 10 27 cm -3 s -1 (which was calculated assuming that the generation occurs solely between the valence band and  valley) is actually subject to changes depending on the strain profile and the optical setup. Therefore, the electrical response of the nanobeams to varying G is different for the electromechanically deflected and the uniformly strained nanobeams. Auger recombination plays a significant role in the presence of elevated carrier concentrations. Due to the carrier localization mechanism in the deflected nanobeams, Auger recombination can become a dominant factor on total recombination rate. Thus, the advantage of carrier localization in the deflected nanobeams can be shadowed by the elevated Auger recombination rates, and turn to a disadvantage under high illumination levels. In this respect, Figure S10c shows the changes Uc-rad,deflected/Uc-rad,uniform under varying G. Under low to moderate illumination levels (G < 10 27 cm -3 s -1 ) and high maximum strains (>%1), the deflected beams are advantageous compared to the uniformly strained structures in terms of Uc-rad. On the other hand, for high illumination levels (G > 10 28 cm -3 s -1 ), the electromechanically deflected nanobeams can actually become disadvantageous since the carrier localization is not beneficial anymore.
In the electrical simulations, 200 nm of beam length, 30 nm of beam thickness and 9 nm fillet radius are chosen as they give the same maximum axial strain at the edges and at the plane of symmetry. This is because, obtaining the same strain is crucial for a fair comparison with the uniform strain case.

Section 8 : Validity of Simulations
To check the reliability of the simulations, the analytical equations given in the study of the Choi et. al. 1 are regenerated. In that paper, the beam which is fixed at the two edges is analyzed. In their study, when deflection is significantly lower than the gap, the deflection profile is found as where is the permittivity of the vacuum, V is the applied potential between the two terminals separated by the gap, b is the width of the beam, E is the Young Modulus of the material, I is the moment of inertia, g is the gap, t is the thickness of the beam and =L/2 is the half of the length of the beam. For the L = 2 m, g = 100nm, t = 30 nm V = 10 V and E=103 GPa, the fixed-fixed case is simulated and the results are in a good agreement with the analytic solutions as it is shown in Figure S12.