## Abstract

We propose a new class of optoelectronic devices in which the optical properties of the active material is enhanced by strain generated from micromechanical structures. As a concrete example, we modeled the emission efficiency of strained germanium supported by a cantilever-like platform. Our simulations indicate that net optical gain is obtainable even in indirect germanium under a substrate biaxial tensile strain of about 1.5% with an electron-hole injection concentration of 9×10^{18} cm^{-3} while direct bandgap germanium becomes available at a strain of 2%. A large wavelength tuning span of 400 nm in the mid-IR range also opens up the possibility of a tunable on-chip germanium biomedical light source.

©2009 Optical Society of America

## 1. Introduction

Crystalline germanium (Ge)-based [1–6] emitters are the subject of growing interest because of their CMOS compatibility and their C band direct bandgap emission spectrum. One major disadvantage of Ge as an emission medium is its indirect nature, which limits direct optical transitions by way of electronic scattering into the indirect valleys. Such losses can be suppressed using heavy n-doping [2,5,6] and the emission rate enhanced by the Purcell effect in highly confined optical cavities [4]. The small 0.14 eV separation between the direct Γ and the indirect L valley electrons can be further minimized in tensile-strained Ge grown on germanium-silicon-tin (GeSiSn) alloys [1]. A biaxial tensile strain of approximately 2% [7] is thereby sufficient to transform Ge into a direct bandgap material and an efficient luminescence source.

Despite the potential these methods hold, some issues remain unresolved. Firstly, Purcell effect does not directly address the large intervalley electronic scattering which is the most serious hurdle for efficient Ge light emission. In the case of the n-doped Ge light emitter, the near ohmic carrier concentration in the active layer causes unwanted current flow during electrical operation. Forward biasing of the device is needed to introduce minority holes into the active medium but will simultaneously inject significant number of electrons from the electron-rich active layer into neighbouring non-active regions where their energy will be dissipated. For Sn-based group IV alloys, buffer deposition is constrained by the thermal budget of downstream CMOS processes, some of which, such as dopant annealing, require temperatures of about 800°C. As a comparison, liquid phases in Ge1-xSnx mixtures exist at equilibrium for temperatures greater than 232°C for x>1.2% and x is restricted to below 0.7% for solid alloys at a temperature of 800°C [8].

There is therefore sufficient motivation to explore alternatives for enhancing Ge light emission. Strain engineering of Ge appears promising since the direct to indirect bandedge offset of Ge can be significantly reduced or even eliminated under tension. Johansson *et al*. [9] first reported large uniaxial tensile strains of almost 6% in Si micromechanical cantilevers and the use of mechanically applied strain to enhance transistor performance is well-known [10]. We believe that applying a similar approach in Si/Ge photonics will reap similar benefits. The electroabsorption contrast of a Ge-based Franz-Keldysh modulator is limited by the L valley optical absorption [11], which is reducible by tensile strain. The linear electro-optical effect has also been observed in strained Si-based photonic crystals where the inversion symmetry is broken [12]. Further, Liu *et al*. [2] estimated that at a biaxial tensile strain of 2%, the Ge direct bandgap will narrow down to 0.5 eV or a wavelength of 2.5 µm. Such small bandgaps corresponds to the mid-IR range around 3 µm, which is the “fingerprint” spectrum region in biochemical sensing.

While tensile strained Ge is advantageous for many applications, present techniques to realize this are limited. Ultrathin layers of compressive Ge have been pseudomorphically grown on Si whose lattice mismatch is as large as 4.2%. Compression is however, unsuitable for light emission work since the absorptive indirect band now penetrates deeper into the direct bandgap of the material. By exploiting the difference in thermal expansion coefficients of Si and Ge, Ishikawa *et al*. [13] achieved a 0.2% biaxial tensile strain in bulk Ge on Si deposited by a 2-step chemical vapor deposion (CVD) process [14]. Post-growth silicidation on the Si wafer backside can increase biaxial tensile strain to 0.24% through substrate warping [15] while the larger 1.2% biaxial tensile strain on free standing SiGe/Si sandwich structures supported by SiO2 pedestals exists in the Si layer [16]. Thus, strain obtainable by these methods remains insufficient to create a direct bandgap in Ge.

In this paper, we analyze the properties of biaxial tensile strained Ge deposited on deformable micromechanical system (MEMS) Si structures and show that significant optical gains are obtainable in Ge under realistic conditions in MEMS devices.

## 2. Micromechanical strain generation

The theoretical fracture strengths of Si and Ge are large: for the crystalline <100> direction, Ruoff [17] has computed the maximum Si and Ge uniaxial tensile strain and stress to be 20.6% and 21.4 GPa as well as 18.3% and 14.7 GPa respectively, while Roundy *et al*. [18] has calculated the corresponding <111> tensile strain to be 17% at 22 GPa stress for Si and 20% at 14 GPa stress for Ge. In practice, fracture limits are significantly lower and highly dependent on sample size and crystalline quality. Namazu *et a*l. [19] noted that the average bending strength of 17.53 GPa for submicron Si beams is 38 times larger than that of millimeter-sized samples. Recently, Alan *et al*. [20] also achieved high tensile stresses of 18.2 GPa in <110> Si using cantilevers with surface passivation by methyl monolayers. These results indicate that micromechanical structures are a feasible method to approach the theoretical strain limits predicted in [17] and [18].

Biaxial Si fracture stresses was investigated by Chen *et al*. [21] in 10 mm^{2} <100>-oriented samples 230 nm to 500 nm thick. A maximum Weibull strength of 4.6 GPa was found, which, based on their assumed isotropic Young’s modulus of 170 GPa and Poisson’s constant of 0.1, would yield a biaxial tensile strain of 2.44%. We note that this strain is sufficient to transform Ge into a direct bandgap material.

Some preliminary experiments on strained Ge structures on deformed substrates have already been reported. Houghton *et al*. [22] observed type I alignment of SiGe/Si quantum wells with the application of uniaxial stress on their samples and Evans *et al*. deposited Ge hut nanostructures on thin Si membranes by molecular beam epitaxy and created significant bending under the Ge mesas [23]. However, this growth method does not lend itself to blanket deposition of bulk Ge on Si. We are unaware of any published work on crystalline Ge-based MEMS structures but low temperature deposited poly-Ge is a widely used sacrificial material in CMOS-based MEMS due to its high etching selectivity with respect to poly-SiGe and Si [24]. Franke *et al*. [25] measured the average uniaxial fracture strain in poly-Ge to be 1.7% for micron-scale structures and a corresponding poly-Si value of 1.5% is also obtained via the same method [26]. The value for poly-Si is still an order of magnitude smaller than the best reported experimental results for crystalline Si [20] and the theoretical values [17], [18]. Based on the above considerations, we shall assume that maximum biaxial tensile strain of crystalline Ge is much larger than that of poly-Ge as well and exceeds the 2% required for a direct bandgap transformation.

Figure 1 depicts an embodiment of our proposed strain-generating structure supporting a Ge-based light emitter. A square 80 µm by 80 µm <100> thin Si platform is suspended using 4 Si cantilevers which are themselves attached to the main Si chip. The entire Si structure forms a cross with each arm extending 100 µm from the platform and having a uniform thickness of 10µm. A Ge ring or disk of radius 10 µm and thickness 400 nm is sited at the center of the platform. Such Ge rings can either be fabricated by selective CVD growth [14] or by post-growth patterning. A rib-ring can be an alternative structure when the channel ring is not strained as efficiently as we expected.

To exert biaxial tensile stress on the Ge emitter in the cross structure, a vertical force of 330 mN is applied at the bottom of the platform center, either using an external L-shaped hook constituting part of the external chip packaging or a MEMS-based electrostatic actuator with the silicon membrane. Locating the force applicator off-chip makes this technique amenable even to processes without the addition of MEMS capabilities. On integrated MEMS chips, on-chip movement control has the added advantage that the strain and the device parameters are tunable on the fly.

Exerting upwards pressure on the platform contorts the supporting cantilevers which, in tandem, generate a biaxial in-plane stress on the structure center. As in the case of a vertically loaded MEMS cantilever [27], the arching of the stressed structure creates a linear strain field which changes from compressive at the bottom surface to tensile at the top.

A simple cantilever-type mechanism is sufficient when a uniaxial tensile strain is required. In reference [28], Feng Zhang *et al*. predicted that uniaxial tension along <111> Ge results in a direct bandgap at a tensile stress of 6.5 GPa or a tensile strain of 4.2%. Attaining such high uniaxial strains typically involves complex selective growth processes, Ge nanowires as suggested by the authors or the optimization of the geometries and orientations of quantum dots. With cantilever structures [20], we believe that comparable uniaxial tensile strains are likewise achievable in bulk Ge.

The strain field calculated by finite element analysis (FEA) of our proposed device is shown in Fig. 1. For simplicity, we neglect the small Ge ring structure and assume that it will take on the substrate curvature and strain given the relative thicker height and substantially larger volume of the substrate. The strain distribution within the Ge layer is then linearly extrapolated from that of the Si substrate. A more exact simulation to account for the strain variation near the Ge ring will be the subject of further study. We further simplify our analysis by approximating crystalline Si as isotropic with a Young’s modulus constant of 170 GPa and a Poisson coefficient of 0.1. Chen *et al*. [21] found that this assumption does not substantially alter their FEA results as compared to a full fledged simulation with cubic elastic constants for Si.

The maximum uniaxial stress magnitude of 5 GPa is located at the outer ends of the cantilever support and is comparable to that of similar-sized cantilevers reported in [9]. The largest biaxial tensile stress, also about 5 GPa for both in-plane crystallographic axes, is located at the platform center where the Ge ring is located. This amounts to an in-plane biaxial tensile strain of 2.77% (and a corresponding out-of-plane compression strain of 2.14% and 2.08% in the silicon and germanium layers respectively) using the elastic constants given in the next section and an in-plane biaxial tensile strain of 2.64% using the isotropic assumption of the FEA simulations here. The other components of the surface stress tensor are much smaller at Ge ring. By the same axes convention in Fig. 1, the surface vertical stress σ_{zz} is negligible while the shear stress σ_{xy} and the sum total of σ_{yz} and σ_{zx} amount to 2.5% of the biaxial stress, σ_{xx} or σ_{yy} along the Ge emitter ring.

## 3. Electrooptical properties of strained Ge

The normally degenerate light hole (*lh*) and heavy hole (*hh*) bands of unstrained Ge are now lifted, resulting in a *lh* valence bandedge. Simultaneously, the energetic offset from the conductance Γ to L valleys, Δ*E*
_{ΓL} is reduced. All bandedges, move towards the bandgap center leading to a smaller direct Γ-*lh* interband transition energy *E*
_{gΓlh}.

In our simulation model, we assume that the Ge layer takes on the curvature and hence the strain of the underlying Si substrate. The Si strain tensor, and by extension the Ge strain tensor, can be calculated from the Si surface stress tensor by the generalized Hookes’ Law [29] with the appropriate elastic stiffness constants for Si, c_{11}=1.66 Mbar, c_{12}=0.639 Mbar and c_{44}=0.796 Mbar and for Ge, c_{11}=1.29 Mbar, c_{12}=0.48 Mbar and c_{44}=0.68 Mbar [30].

Following the discussion in reference [31], the change in the Γ and L bandedges due to strain, ΔE_{CΓ} and ΔE_{CL} respectively are derived from the strain tensor by Eq. (7) and Eq. (17) of reference [31]. The Ge deformation potential constants (in the notation as the aforementioned reference) are taken to be *a*
^{dir}
_{C}=-8.24 eV, a^{indir}C=-1.54 eV and Ξ^{L}
_{U}=15.13 eV [31].

The corresponding behavior of hole states is more complex due to their close energetic and momentum proximity at the Γ point. Bandmixing occurs in the presence of shear strain and the new hole eigenstates are taken as a linear sum of the unstrained states by diagonalizing the Pikus-Bir Hamiltonian [32]. Since the shear components in our system are small, we shall treat the valence band carriers as *hh*-like or *lh*-like and ignore the intermixing and anti-crossings of the hole states. As with the case for the conduction band calculations, the relevant Ge parameters to determine the *lh* and *hh* bandedges, *b*=-2.55 eV, *d*=-5.5 eV, Δ_{0}=0.3 eV are extracted from reference [31]. Although shear strain lifts the degeneracy of the L band, we assume the energy of the least energetic L valley as that for all L valleys. This simplification will require a larger applied tensile strain and is therefore a more stringent criterion for a given decrease in the indirect L to direct Γ separation.

In accordance with the preceding FEA analysis, the shear components of the surface Si stress tensor are taken to be 2.5% of the in-plane biaxial hydrostatic stress value and their signs are chosen to give the least energetic L valley in the Ge layer using the proceeding method while the vertical stress components are taken to be zero. The strain tensor is calculated from the stress tensor using Hook’s law. At the Si/Ge interface, all Si strain components except for the vertical strain, εzz are transferred over to the Ge layer and the latter is subsequently calculated from the in-plane strain assuming zero vertical stress via Hook’s law. We approximate the linear Ge tensile strain field in the cantilever structure by discretizing it into 10 sublayers parallel to the Si base. For each strained sublayer, the in-plane strain changes linearly as the same rate as the Si base and the energy shifts in the density of states (DOS) of the L, Γ, *hh* and *lh* bands are calculated in accordance with the deformation potential theory.

To determine the gain, we assume dynamic equilibrium in the Ge layer and a constant number of injected electrons and holes, whose distributions are regulated by their respective pseudo-Fermi levels. For given hole and electron populations, global quasi-Fermi levels common across all sublayers were computed numerically from standard Fermi statistics. The strain-dependent change in both the conductance Γ [33] and L valleys [34], [35] are assumed small and neglected in the following computations. Biaxial tensile strain lifts the *lh* and *hh* degeneracy and introduces anisotropy in their bandedges. We approximate the modified lh and *hh* masses masses parallel and perpendicular to the strain plane as

$${m}_{\mathrm{hh}\parallel}\u2044{m}_{o}=1\u2044({\gamma}_{1}+{\gamma}_{2}),{m}_{\mathrm{hh},\mathrm{perpendicular}}\u2044{m}_{o}=1({\gamma}_{1}-2{\gamma}_{2})$$

where ${f}_{+}=\genfrac{}{}{0.1ex}{}{2x(1+\genfrac{}{}{0.1ex}{}{3}{2}(x-1+\sqrt{1+2x+9{x}^{2}}))+6{x}^{2}}{\genfrac{}{}{0.1ex}{}{3}{4}{(x-1+\sqrt{1+2x+9{x}^{2}})}^{2}+x-1+\sqrt{1+2x+9{x}^{2}}-3{x}^{2}}$ is a correction factor for the spin-orbit split-off band interaction and x=*Q*
_{ε}/Δ_{0}, whereby *Q _{ε}*=-1/2

*b*(ε

_{xx}+ε

_{yy}-2ε

_{zz}) [32]. The unstrained Ge Luttinger parameters are given by γ

_{1}=13.4, γ

_{2}=4.25 and γ

_{3}=5.69 [36]. The use of Eq. (1) assumes that the optical transitions of interest take place near the Γ point where the kinetic term E(

*k*) is smaller than the

*lh-hh*strain energy splitting (2δ

*E*

_{S}) [33]. We do not expect any gain in the low strain regime where 2δ

*E*

_{S}is small and the approximation less precise but the latter becomes more accurate towards the device operating tensile biaxial strain of about 2%.

We assume parabolic DOS for all bands, where the L and Γ band effective masses, *m*
_{L} and *m*
_{Γ}, are given by 0.22 *m*
_{0} and 0.038 *m*
_{0} respectively [30], where *m*
_{0} is the electron mass. Their *hh* and *lh* counterparts, *m*
_{x} are approximated by the DOS masses given by (*m*
_{x‖}
^{2}
*m*
_{x,perpendicular})^{1/3} where x is *lh* or *hh* respectively. The assumption of parabolic bands here neglects the anisotropy and non-parabolicity induced by strain, which will affect the correctness of physical quantities such as the DOS or the inversion population threshold. More exact modeling of the curvature of the strained carrier dispersion will be the subject of future work. At the same time, the unstrained material parameters in our model, such as the valence band Luttinger parameters as well as the conductance masses should be replaced by their strained equivalents if they become experimentally available, since the parabolic model cannot determine these parameters as a function of strain.

Tensile strain in Ge lifts the *lh* band above that of the *hh* band, resulting in stronger conduction band to *lh* transitions where TM-polarised (polarisation perpendicular to biaxial strain field) photons predominate [33]. The TM gain expression for the dichroic Ge medium can be derived from Eqs. (15) and (20) of reference [33] assuming negligible intraband relaxation time and parabolic dispersions for both the conductance and lh bands. Equations (14) and (15) in reference [33] also assume that Δ_{0} is large compared to δ*E*
_{S}. At a biaxial tensile strain of 2% in our Ge modeling, δ*E*
_{S} is approximately 0.11 eV compared to Δ_{0}=0.3 eV. To further improve the accuracy of our modeling, the spin-orbit split-off state would have to be included in the density matrix formulation of [33]. Based on these assumptions, we arrive at the following expression,

$$\times \left({f}_{c}\left(\genfrac{}{}{0.1ex}{}{{m}_{r}}{{m}_{c}}({E}_{\mathrm{photon}}-{E}_{g\Gamma \mathrm{lh}})\right)-{f}_{v}\left(\genfrac{}{}{0.1ex}{}{{m}_{r}}{{m}_{\mathrm{lh}}}({E}_{\mathrm{photon}}-{E}_{g\Gamma \mathrm{lh}})\right)\right)$$

where *E*
_{photon} is the photon energy, *E*
_{gΓlh} is the *lh* direct bandgap and (*f _{c}*-

*f*

_{v}) is the population inversion factor, which is a function of the carrier masses and

*E*

_{photon}-

*E*

_{gΓlh}.

*m*is the reduced mass given by (

_{r}*m*

_{lh}

*m*

_{Γ})/(

*m*

_{lh}+

*m*

_{Γ}). The empirical constant

*A*=2.01×10

^{4}(eV)

^{1/2}cm

^{-1}is derived from top down absorption experiments [2], [37] of 0.2% biaxial tensile strained Ge where the illuminating light polarization is parallel to the biaxial strain plane. We derive the reduced mass at 0.2% biaxial tensile strain,

*m*

_{r-0.2%strain}based on the DOS

*m*

_{lh}by Eq. (1) even though they give an overestimation of the actual values at low

*lh-hh*strain separation 2δ

*E*

_{S}. The computed gain value will consequently yield a conservative underestimation of the gain at the operating strain because of the smaller

*m*

_{r}/

*m*

_{r-0.2%strain}ratio.

*B*

_{lh}=0.54 is a correction factor reflecting the relative

*hh*and

*lh*absorption contributions as well as the TM/TE interaction strengths of the

*lh*band. It is calculated using the unstrained Ge masses (which again leads to a conservative low estimation of the

*lh*gain due to the small unstrained

*lh*DOS) as well as the dipole moments given in Eqs. (14) and (15) of reference [33].

*hh*do not interact with TM photons and we neglect their influence in Eq. (2).

Finally, the modal gain of the Ge-based waveguide can be calculated from the average of the gain in the active layer weighted by the local field intensity. As a simplification, we assume that the mode overlaps uniformly with all parts of the strained germanium layer and that the net gain is simply the average of all sublayer gain coefficients.

## 4. Discussion

The relative shifts in Fig. 2(a) of the conductance and valence bands with increasing in-plane <100> biaxial tensile strain result in several consequences for light emission. Firstly, the conductance band Γ to L valley separation, ΔEΓL decreases and eventually disappears under a tensile strain of 2% when Ge becomes a direct material. Secondly, the valence band degeneracy is lifted and the *lh* optical transitions dominate that of the *hh* due to the smaller bandgap associated with the *lh* band. This allows for a more rapid change of the pseudo-Fermi levels towards inversion because of the smaller effective DOS. *lh* also have a stronger interaction with TM photons than TE photons and for subsequent discussion, we focus only on TM emission. *hh*, on the other hand does not contribute to TM gain at all.

Figure 2(b) shows typical interband gain profiles of the 1.75% strained Ge layer based on our model at injection levels from 1 to 9×10^{18} cm^{-3}. The larger Ge biaxial tensile strain further away from the Si substrate and the consequently smaller *lh* bandgap induces *e-h* pair migration to the Ge top surface, generating a stronger inversion and larger gain there. Since the active Ge layer is undoped, minority carrier injection from the adjacent p-doped regions is not suppressed unlike the case for the highly n-doped active layer proposed in reference [2]. In conjunction with SiGe cladding layers, a type I structure is also attainable for electron confinement.

Figure 2(c) plots the maximum Ge optical gain coefficient with and without free carrier absorption (FCA) as well as the wavelength at maximum net gain against the <100> biaxial tensile strain component at the same 9×10^{18} cm^{-3} injection concentration used in reference [2]. The gradual movement of the Γ point towards the bandgap center at higher tensile stress lowers the required electron concentration at inversion. Beyond a biaxial tensile strain of 1.5%, FCA is compensated by the interband gain even while Ge remains an indirect material. By varying the applied biaxial tensile strain between 1.5% and 2%, the peak wavelength is also tunable within a range of 400 nm up to a maximum wavelength of 2.6 µm. The net gain of direct bandgap Ge at 2% biaxial tensile strain can reach 1077 cm^{-1}, compared to 400 cm^{-1} predicted in n-doped Ge [2]. However, in the latter case, a value of *B*
_{lh} of 0.318 [37] is applied, assuming unstrained masses as well as polarisation independent and identical average dipole moments for both *hh* and *lh*. Substituting for this value of *B*
_{lh} yields a weaker gain of 325 cm^{-1} in our case.

## 5. Conclusion

We modeled Ge emission characteristics under biaxial tensile strain generated by micromechanical structures. Our simulations indicate that transparency is attainable even in indirect Ge at a substrate biaxial tensile strain of 1.5% assuming a carrier injection concentration of 9×10^{18} cm^{-3}. Larger biaxial tensile strains greater than 2% will transform the Ge into a direct bandgap material, resulting in an efficient mid-IR light emitter. A strain-tunable wavelength range of 400 nm also opens up the possibility of on-chip tunable light sources. Our technique can be further combined with n-doping of the Ge layer. Jifeng *et al*. [2] estimated that the net optical gain increases by a factor of 10 in 0.25% biaxial tensile strained Ge compared to unstrained Ge. By means of our proposed technique, we can substantially increase the gain of n-doped Ge on CMOS fabricated by deposition or bonding processes of Ge on wafer substrates, where the built-in strain is low.

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