Low-threshold optically pumped lasing in highly strained germanium nanowires

The integration of efficient, miniaturized group IV lasers into CMOS architecture holds the key to the realization of fully functional photonic-integrated circuits. Despite several years of progress, however, all group IV lasers reported to date exhibit impractically high thresholds owing to their unfavourable bandstructures. Highly strained germanium with its fundamentally altered bandstructure has emerged as a potential low-threshold gain medium, but there has yet to be a successful demonstration of lasing from this seemingly promising material system. Here we demonstrate a low-threshold, compact group IV laser that employs a germanium nanowire under a 1.6% uniaxial tensile strain as the gain medium. The amplified material gain in strained germanium can sufficiently overcome optical losses at 83 K, thus allowing the observation of multimode lasing with an optical pumping threshold density of ~3.0 kW cm−2. Our demonstration opens new possibilities for group IV lasers for photonic-integrated circuits.

pre-existing film strain is set to be 0.2%. As a result of strain redistribution in the undercut structure, an amplified strain of ~2% exists within the entire gain medium uniformly. This allows us to achieve an extremely homogeneous gain medium which also plays an important role in enabling the lasing action in our highly strained germanium (Ge) lasers. 3 (CMP) step for a smooth surface for bonding, a 50-nm Al 2 O 3 sacrificial layer is deposited on the Ge surface by atomic layer deposition (ALD). Then the Ge-on-Si wafer is directly bonded at room temperature to an 8-inch Si (100) handle wafer with a 1-µm thick thermal oxide (SiO 2 ) layer, followed by a post-bonding annealing at 300 ˚C for 3 hours to enhance the bonding strength. After removing the carrier Si by grinding and selective chemical etching in Tetramethylammonium hydroxide (TMAH), the Ge layer is transferred to the handle wafer, which forms a GOI substrate. Then the Ge layer of the GOI is thinned down by CMP to the desired thickness of 220 nm with a surface roughness of < 0.2 nm. The 8-inch GOI wafer is diced into 1 × 1 cm 2 pieces for device fabrication. The Ge nanowire laser structure is defined by the electron-beam lithography (EBL), and its pattern is transferred to the Ge layer by reactive etching (RIE) using Cl 2 gas. The dry etch stops at the Al 2 O 3 layer, and the sample is wet etched in 30% KOH solvent to selectively remove the sacrificial Al 2 O 3 layer forming the undercut structure.
The releasing process causes the strain redistribution and amplifies the tensile strain in the nanowire 3,4 . Then the sample is dimmed in 100% IPA. We finalize the laser fabrication by contact drying on a hotplate, which allows the nanowire to be in contact with the SiO 2 layer rather than suspended in air, resulting in a better heat conduction in our device. In our design, we intentionally bring the germanium (Ge) layer into contact with the underlying silicon dioxide (SiO 2 ) layer during the fabrication process (Supplementary Note 2). While the SiO 2 layer can effectively confine the optical mode within the Ge layer owing to a large refractive index difference between Ge and SiO 2 , the heat accumulation by optical pumping can be significantly minimized in our architecture since the SiO 2 layer provides additional heat conducting paths towards the thick silicon   Fig. 4d). While the simulated Q factors are well above 1,000, the experimental Q factor at the threshold is measured to be ~850 and this discrepancy may be attributed to the sidewall roughness of our DBRs introduced during fabrication processes 8 . We believe that improving the cavity design and fabrication processes will allow us to further reduce the lasing threshold.  To compute the spontaneous emission coupling factor β of our germanium (Ge) nanowire, we employed the multimode laser rate equation to fit our experimental data, and obtained a β factor of 0.08. We assumed that the lasing modes (5 cavity modes between 1512 nm and 1547 nm) have the same threshold modal gain, group velocity and spontaneous emission factor 9 . The simplified rate equations describing the dynamic relation between carrier density and photon density is expressed as:

Supplementary Note 5: Curve fitting of experimental data
where N is the carrier density in the Ge nanowire, S is the photon density, P is the optical pumping power density, η eff represents the fraction of the optical pumping power absorbed by the nanowire, τ r and τ nr are the radiative and non-radiative recombination lifetimes, respectively, the Auger recombination coefficient is given as C and the number of the lasing modes is represented by m s , v g is the group velocity, g(N) is the material gain, g th is the threshold gain, Γ is the confinement factor, and β is the spontaneous emission factor.
The fitting parameters are listed in Supplementary cm -3 has been experimentally measured in a Ge-on-insulator (GOI) sample 10 . For the specific epitaxially grown Ge layer measured for ~3.12 ns carrier lifetime, the authors used a multiple hydrogen-annealing heteroepitaxy (MHAH) growth technique. The reported threading dislocation density in the Ge layer using MHAH technique 11 ranges between 1 x 10 7 cm -2 ~ 1 x 10 8 cm -2 . In our experiment, on the other hand, we employed an As-doped Ge seed layer 2 to reduce the threading dislocation density in the Ge layer to ~4.6 x 10 6 cm -2 .
Previously, the carrier lifetime has been empirically correlated to the threading dislocation density as in the following equation 12 : where τ is the minority carrier lifetime, C is a proportionality constant, ρ D is the threading dislocation density.
To make a conservative estimation of the carrier lifetime, we here take the lower bound value of the threading dislocation density 1 x 10 7 cm -2 for Ge grown by MHAH technique, and take the threading dislocation density of 4.6 x 10 6 cm -2 for the Ge layer in our present study. Since the lifetime is inversely proportional to the threading dislocation as in Supplementary Equation 3, we estimate the lifetime of our Ge layer to be ~6.78 ns by taking into account (at least) 0.46 times lower threading dislocation density in our Ge layer compared to the Ge layer reporting 3.12 ns lifetime.
In addition, lifetime is also highly dependent on doping density as extrinsic dopants act as recombination centers. For a doping density of >10 16 cm -3 , E. Gaubas et al. 13 showed that the carrier lifetime is related to the doping density as in the following equation: where n dop is the doping concentration. Since we employed a doping density of 6 × 10 18 cm -3 which is 0.6 times lower than the Ge layer with a doping density of 1 x 10 19 cm -3 which presents 3.12 ns, we finally estimate the lifetime in our Ge layer to be ~11 ns.
Although low temperature operation may also increase the lifetime quite as shown in ref 14, we did not include this temperature effect to be conservative in our lifetime estimation.
It is worth mentioning that the lifetime of ~11 ns is an order of magnitude less than commonly used 15 and also that the lifetime in bulk Ge with a doping density of 1 x 10 19 cm -3 at our measurement temperature (~83 K) can be as large as ~600 ns 14 .
For the radiative recombination lifetime, we use a direct bandgap recombination coefficient of 1.3 × 10 -10 cm 3 s -1 . By calculating the fraction of the electrons residing in the direct Г valley at a carrier injection density of 8 × 10 19 cm -3 , we obtain 28 ns for the radiative recombination lifetime. The Auger recombination coefficient is 1 x 10 -32 cm 6 s -1 for the heavily doped n-type Ge.
A linear relationship between the material gain and the carrier density is assumed, which can be expressed as The threshold gain is estimated from the threshold modal gain equation. The equation is expressed as: where Γ is the confinement factor, Γg th is the threshold modal gain at threshold, n g is the group index, k 0 is angular wavenumber and Q is the quality factor at threshold. The confinement factor and group index are estimated from our finite-difference time-domain (FDTD) simulation: Γ = 0.45, n g ~ 3.2. Near the threshold pumping density, the peak at 1529.6 nm has a FWHM of 1.80 nm, fitted to a Lorentzian function. The estimated Q factor at threshold is 850, and the corresponding threshold modal gain Γg th is calculated to be 151.2 cm -1 . Thus, we estimated the threshold gain g th to be 151.2/0.45 = 336 cm -1 .
By solving the coupled multimode laser rate equations under steady-state, we found the best fit for b is ~0.08 (plotted in Supplementary Fig. 6) and determined the lasing threshold to be ~ 3.0 kW cm -2 . The L-L curve for different b values was also plotted for comparison.  Similar emission behaviors were observed by Kurdi et al. 5 , where resonances were seen in a strained Ge microdisk at a low pumping level, but the ratio of the resonance amplitude to the background emission amplitude ceased to increase as the pump power was further increased. It is clearly stated in their manuscript that in an optical cavity with a weak value of spontaneous emission coupling factor, the cavity resonances are expected to predominate over the background when lasing action occurs. Since the resonance amplitude was not overwhelming the background emission in ref 5, they did not correlate their observation to lasing as in our low strained Ge nanowires.

Supplementary
In stark contrast, our 1.6% strained wires (Fig. 2) show predominant resonances whose amplitudes are more than one order higher than the background emission as well as the cavity resonances outside the gain bandwidth as clearly stated in our main manuscript. And this is the result of strong superlinear increase of resonance amplitudes due to the presence of optical net gain, thus presenting clear evidence of lasing.
Our additional experiments on strain-dependent emission characteristic clearly present the pivotal role of uniaxial strain for achieving low threshold lasing in Ge by showing that the resonances become larger with regards to the background emission at higher strain, and finally overwhelm the background emission at 1.6% uniaxial strain when lasing is achieved.

Detailed description on gain and loss modeling
We perform theoretical modeling to obtain the gain spectrum of uniaxial tensile strained germanium (Ge). Empirical pseudopotential method (EPM) is used to compute the bandstructure of strained Ge. EPM is an attractive approach because it allows for the computation of bandstructures with relatively small number of empirical parameters. In the calculation for bandstructure, the single electron Hamiltonian is expressed as 16 : where V loc , V nloc , and V so represent the local, nonlocal, and spin orbit contributions to the pseudopotential, respectively.
After calculating the bandstructures, the transition rates between different bands in Ge are calculated by using Fermi's golden rule. The absorption (or gain) due to band-toband transitions is calculated using: To achieve the optical net gain in Ge, the material gain should overcome the combined material loss consisting of free electron absorption (FEA) and inter-valence band absorption (IVBA) 3 . For the losses at room temperature, we take the same formalism used in ref 3, but divide the total loss by a factor of 3 assuming isotropic losses since we only consider one polarization direction for the material gain. It is required to investigate the polarization dependence of material losses to more precisely understand the gain and loss behaviour in Ge. Since IVBA is strongly dependent on temperature 17 , we take an experimentally measured IVBA at 95 K which is close to the temperature at which we observed lasing. By fitting IVBA data of Ge with p-type doping concentration of 1 × 10 19 cm -3 at 95 K 17 , we obtain: where ћω is the photon energy, N h,tot is the total hole density. α IVBA is in units of cm -1 , ћω is in units of eV, and N h,tot is in units of cm -3 . The curve for 95 K is more strongly dependent on temperature than the counterpart for 300 K, and this can be attributed to the difference in Fermi-Dirac functions at two different temperatures. We use this experimental IVBA for our modeling at 83 K, and for FEA, we employ the same formalism used for 300 K 3 since FEA is not a strong function of temperature.

Modelling results
Supplementary Figure 8a shows the separate contributions of FEA and IVBA to the combined loss for a carrier injection density of 7 × 10 19 cm -3 . At 7 x 10 19 cm -3 carrier density and at the experimental peak gain wavelength of ~1530 nm, IVBA is >214 cm -1 whereas FEA is only <96 cm -1 . Therefore, IVBA contributes to the major part of the material loss and it is critical to minimize the IVBA by lowering operating temperatures to achieve optical net gain.