Band structure of germanium carbides for direct bandgap silicon photonics Free

Germanium and its alloys have recently gathered attention as laser materials for silicon photonics. Not only is Ge compatible with silicon IC fabrication, with dislocation density below 10⁵ cm⁻²,¹ but the direct (Γ) conduction band (CB) valley of Ge is only 140 meV higher in energy than its indirect (L) valley. Adding strain or alloying with the appropriate material can reduce the energy of the Γ valley, allowing increased light emission and even a direct bandgap. Lasers have been demonstrated using GeSn² and tensile-strained Ge.^3,4 This paper investigates another potential direct bandgap group IV alloy, Ge_1−xC_x.

Ge_1−xC_x is a highly mismatched alloy (HMA) similar to dilute nitrides such as InGaAs:N. Carbon and N are much more electronegative than their respective hosts Ge and As: 2.55 and 3.04 vs. 2.01 and 2.18 Pauling units, respectively. This difference introduces a strong local perturbation in the host crystal. In the dilute nitrides, the result is a splitting of the conduction band at Γ leading to a reduced CB minimum and decrease in bandgap, and a large electron effective mass. These effects are often described by a band anticrossing (BAC) model in which N introduces a localized state within the CB that interacts with and repels the nearby CB states to lower energies.⁵ 

However, like other HMAs, Ge_1−xC_x can readily form defects, especially from typical growth techniques. C-C bonds are stronger than C-Ge bonds,⁶ forming CC_Ge split interstitial defects and other carbon clusters, all of which form trap states within the bandgap.⁷ C triplets are the most common molecular species evaporated thermally from graphite sources,⁸ with heavier clusters up to C₁₄ detected.⁹ Atomic sources of C⁹ offer no panacea because typical Ge epitaxial techniques favor surface segregation of C,¹⁰ leading to C-C bonds as well. Indeed, Park et al. showed it would be virtually impossible to achieve fully substitutional growth of Ge_1−xC_x alloys by conventional molecular beam epitaxy (MBE) techniques.¹⁰ 

The tendency to C-C bonds and carbon clusters has led to a large range of reported characteristics, particularly in reports on the effect of C on the bandgap. Theoretical analysis has been inconclusive, with some techniques reporting a linear increase in bandgap with %C,¹¹ others claiming band bowing¹² to various extents, and a few predicting a composition range with a direct bandgap.⁸ 

This work seeks to resolve these discrepancies using improved ab initio modeling and growth techniques to extract the intrinsic properties of Ge_1−xC_x in the absence of C-C defects. Hybrid density functionals are used for numerical modeling with high accuracy despite the large difference in atom size and electronegativity between Ge and C. These results are coupled with material growth using a technique that preferentially incorporates substitutional C atoms rather than clustering and interstitial defects. All alloy percentages below are given as mole fractions (at. %).

Density functional theory (DFT) calculations were performed using the Vienna ab initio Simulation Package (VASP),^13–16 which implements DFT using plane waves. The projector-augmented wave (PAW) core electron method was used with the generalized gradient approximation (GGA) and Perdew-Burke-Ernzerhof (PBE) functional,^17–20 with a high cutoff energy of 400 eV to account for the "hard" C atom core. The PBE functional alone underestimates the Ge bandgap to the point of predicting it to be a semimetal, so the Heyd-Scuseria-Ernzerhof (HSE06) range separated hybrid exchange-correlation functional was also used in order to give a bandgap close to experimental values.²¹ When feasible, spin-orbit coupling (SOC) was also included in order to accurately represent the splitting of the valence bands. Ion locations were first relaxed within the GGA, then the lattice constant was varied using fixed fractional coordinates using the HSE06 potential to minimize system energy to a tolerance of 10⁻⁵ eV. Due to computational limits and the large size of the supercells used here (128 atoms), inclusion of PBE0 hybrid functionals²² and d electrons were impractical, although these made only minor shifts to the band structures of small Ge supercells. Further simulation details can be found in Ref. 22.

DFT of supercells larger than the primitive crystal unit cell leads to folded band structures, as discussed below. Fig. 1 shows the folded band structure for Ge_0.9922C_0.0078 using HSE06 and spin-orbit coupling. No attempt was made to fit salient features such as bandgaps to experimental data for Ge_1−xC_x alloys due to the lack of consistent data to fit to. However, pure Ge offers a first order correction since its properties are well known, and HSE slightly underestimates the direct bandgap. By decreasing the lattice constant by 1.33% and changing the fraction of exact exchange (mixing parameter) in the hybrid functional from 25% to 18%, the band alignments and bandgaps of Ge can be accurately reproduced.^23,24 Fig. 2 shows the Ge band structure with and without these corrections.

With or without these semi-empirical corrections, the change in the direct bandgap energy with the addition of dilute C is expected to be correct. We find the Γ conduction band valley decreases by approximately 170 ± 50 meV/%C for the first percent C, which is very close to the observed change in dilute nitrides (150–200 meV/%N for the first percent N).^25,26

Modeling of dilute alloys requires large supercell sizes, which leads to a folding of the Brillouin zone (BZ) and apparent band structure. The supercell used here consisted of 128 atoms: the 2 atom Ge primitive unit cell repeated four times along each of its basis vectors ½[110], ½[101], and ½[011]. Without unfolding, the bands are folded 4× in each direction, which overlays the Γ, X, and L points. Unfolding helps clarify changes in the band structure away from the Brillouin zone center at Γ. Unfolding is essential when attempting to distinguish between multiple minima and maxima in a single band, such as L and Γ valleys in the Ge CB, to establish whether the bandgap is direct or indirect.

To validate the interpretation of band splitting and effective masses from the folded band structures, bands were unfolded using the method described by Tomić et al.²⁷ Fig. 3 shows the unfolded band structure for Ge_0.9922C_0.0078 without spin-orbit coupling, but including the semiempirical corrections from Section II B. For comparison, the band structure without these corrections is plotted in Fig. 4, i.e., the same uncorrected computational lattice constant as in Figure 1, and a mixing parameter of 25%. The splitting of the CB minimum at Γ into E⁺ and E⁻ bands is clearly apparent, and the direct gap is clearly the lowest CB minimum. The origin of several noisy data points near ²/₃⟨111⟩ remains unclear. These points do not appear to significantly affect the bands at Γ or L, and they do not appear to be attached to any band, so we speculate that they result from either insufficient k-points during the band structure extraction or insufficient convergence. Band structure calculations with additional k-points are currently underway.

As mentioned earlier, in order to grow high quality Ge_1−xC_x, the C must be sourced by a method that preferentially incorporates the atoms in substitutional sites. This can be accomplished through gas chemistry, by building a molecule that has one carbon atom bonded to four Ge atoms.²⁸ When coupled with kinetically limited growth, this prevents C atoms bonding with other C atoms as well as C incorporating in interstitial sites.

Preliminary growth was completed using a hybrid gas/solid-source molecular beam epitaxy (MBE) chamber using tetrakis(germyl)methane (4GeMe) as the carbon source, diluted by solid-source Ge. A smooth epitaxial GaAs surface was prepared on a GaAs(100) wafer in a Veeco Gen 930 III-V MBE chamber by desorbing the native oxide, then growing a buffer of 100 nm GaAs, 50 nm AlAs for a back barrier, and 20 nm GaAs on top. The wafer was then transferred under ultrahigh vacuum to an Intevac Mod Gen II hybrid source MBE with a background pressure of 6 × 10⁻¹⁰ Torr without liquid nitrogen. Next, 10 nm of Ge was deposited at a 60 nm/h growth rate using solid-source Ge at a wafer temperature of 400 °C, calibrated by pyrometer. The Ge shutter remained open for the remaining growth.

Ge_1−xC_x quantum well (QW) growth began by adding 4GeMe in cycles as follows. To maximize 4GeMe incorporation efficiency, the gate valve was closed to the MBE turbomolecular pump. 4GeMe was injected from a nozzle ∼10 cm from the wafer surface at a foreline reading of 30 mTorr for 20 s until the growth chamber pressure reached 5 × 10⁻⁵ Torr. 4GeMe gas flow was stopped for a 40 s pause to permit additional residence time in the chamber. Then the gate valve was reopened for 20 s to pump out residual gases. The cycle was repeated for 20 min or 15 cycles, corresponding to 20 nm of equivalent solid-source Ge growth. Then the wafer temperature was increased by 25 °C, and a 50 nm Ge barrier layer was grown. This process was then repeated for 5 more wells at successively higher temperatures, up to 525 °C.

To measure the carbon concentration vs. depth, secondary ion mass spectroscopy (SIMS) was performed by Evans Analytical Group, shown in Fig. 5. The C composition shows two plateaus at approximately 10¹⁹ cm⁻³ (0.03%) at ∼250 nm depth and 2 × 10¹⁹ cm⁻³ (0.05%) at 100–200 nm depth. These results are an average composition over the QWs and barriers, corresponding to QW compositions of 0.1% and 0.2%, respectively. The resolution of the SIMS measurement was poor (∼100 nm) due to the large surface roughness, as shown by atomic force microscopy (AFM) in Fig. 6 and transmission electron microscopy (TEM) in Fig. 7. The higher C concentration is found closer to the surface (at a depth of 100–200 nm, compared to the concentration at 250 nm depth), suggesting increased incorporation efficiency of the 4GeMe precursor and increased C incorporation, presumably due to increased hydrogen desorption with higher growth temperature. An increase in C composition with depth beyond 300 nm is attributed to carbon contamination at the original GaAs wafer surface. Deconvolution of the As concentration using height profiles derived from AFM and TEM show that the As tail toward the surface can be largely explained by the large surface roughness; during SIMS, GaAs is exposed in the deepest valleys before the Ge and Ge_1−xC_x are consumed.

Fig. 6 shows AFM of the finished growth over a 10 × 10 μm area, with an RMS roughness of 68 nm. However, as shown in the TEM in Fig. 7, this roughness appears to originate at a specific layer, with flat valleys between the peaks.

The first few layers appear to have grown smoothly, with roughening increasing sharply at or after the fourth QW, corresponding to a growth temperature of 475 °C. These observations suggest the optimal growth temperature is close to 450 °C to allow high C incorporation but still maintain a smooth surface. Despite the rough surface, no defects or dislocations were visible by high resolution TEM (not shown), with neither twins nor stacking faults.⁸ The C fraction was too low to detect by either image contrast or by energy-dispersive x-ray spectroscopy (EDX).

Raman spectroscopy (Fig. 8) shows no sign of graphitic C clusters; however, the substitutional C percentage is too low to be directly detectable, so there is no visible peak for the Ge-C bond at 530 cm⁻¹.

Nuclear reaction analysis Rutherford backscattering spectroscopy (NRA-RBS) was performed to measure the crystallinity of the Ge_1−xC_x sample, as well as that of a control sample that consisted of a similar growth of Ge using digermane as a gas source. Fig. 9 shows very low back scattering signal along the ⟨001⟩ channels in the Ge_1−xC_x sample, but, again, the C percentage is below detectable limits to confirm the fraction of substitutional C.

In order to see the effect on the band edge, photoreflectance (PR) was performed on the Ge_1−xC_x sample. PR is an experimentally performed derivative of reflectivity, dR/R, accomplished by modulating the internal electric field while measuring reflectivity (R). The result eliminates the undesirable background reflectivity while enhancing weak features corresponding to specific electronic transitions. It is relatively insensitive to indirect gap transitions. Fig. 10 shows room temperature and 20 K PR spectra from Ge_1−xC_x and a bare Ge wafer for reference. A single resonance related to a direct optical transition at the Γ point is visible for the reference sample.

For the Ge_1−xC_x sample, a strong PR resonance followed by Franz Keldysh oscillations (FKO) is visible, rather than the single PR resonance typical of a single energy gap transition. Such a shape of PR resonance means that a built-in electric field exists in the Ge_1−xC_x layer. This field is estimated from the FKO period²⁹ to be ∼80 kV/cm. The origin of this built-in field may be due to the gradient of C concentration along the growth direction, dipping the CB deeper in layers with higher %C. Alternately, C may introduce traps whose concentration varies with %C, causing the Fermi level to vary along the growth direction, and a built-in field. The observation of a clear PR signal represents good crystal quality and a well-defined electronic band structure. But a clear redshift of the PR resonance in comparison with the reference sample is difficult to see in this case because of the presence of FKO. On the other hand, for such low C concentration, the expected red shift of the PR resonance should be very small (25 meV), making it difficult to observe.

It is worth noting that the spectral position of the direct optical transition in Ge grown on GaAs can be different than in Ge bulk due to the slight residual strain in Ge layers grown on non-native substrates. This means that Ge_1−xC_x samples with larger C concentrations will be needed for further studies of C-related changes in the electronic band structure of Ge_1−xC_x alloy. For this sample, we can estimate the energy of the direct optical transition at the Γ point from the analysis of FKO period as shown in Figure 11. The accuracy of such estimation is limited, but in this case the transition energy (0.78 eV, corresponding to a 20 meV shift in E⁻) is consistent with our computational model.

Using parameters extracted from the HSE modeling, we predict direct bandgap lasing from Ge_1−xC_x at photon energies below 0.5 eV, corresponding to wavelengths of 2.5–5 μm. A Γ valley decrease of 170 ± 50 meV/%C suggests a strongly direct bandgap for 1%C, which would increase electron population in the Γ valley for increased optical gain. It would also suppress intervalley scattering, which would increase modulation by the quantum confined Stark effect (QCSE) by increasing the exciton lifetime.³⁰ 

This is the first demonstration of Ge_1−xC_x growth in which defects are undetectable yet the bandgap has shifted. Point defects such as C-C clusters, split interstitials, and graphitic bonds were not detectable by Raman, EDX, or NRA-RBS, permitting a clear PR signal even though the C concentration itself was too low to verify Ge-C bonding. Furthermore, the lack of extended defects such as dislocations and stacking faults in both scanning and high resolution TEM indicates an absence of carbon clusters, which had been shown to nucleate extended defects.¹⁰ 

Although the 0.2% C concentration in the grown sample was insufficient for a direct bandgap, these results nevertheless show significant promise for direct bandgap devices. Strong optical transitions and the decrease in direct bandgap energy validate the DFT model results, which in turn show a direct bandgap at much smaller concentrations than previous reports. This is similar to dilute nitrides such as GaInNAsSb, in which the new E⁻ band enabled even stronger optical transitions than Ga(In)As, thus increasing laser gain and modulation.^31,32

The electron effective mass predicted by our DFT work, 0.045m₀, is 1.5× that of the Γ CB valley in Ge (0.030m₀ by DFT for a 128-atom supercell of pure Ge without semiempirical corrections). This increase is consistent with the band anticrossing model and dilute nitrides, and it suggests further increases in gain and modulation due to an increased electron-hole overlap integral in QWs.³² More importantly, it also improves on the small density of states in the Γ valley of Ge, tensile Ge, and GeSn, in which the heavy L valleys capture most of the electrons even if a direct bandgap is achieved.^33,34 Thus, Ge_1−xC_x offers a route to lasers and modulators comparable with III-V materials.

The multi-QW structure grown in this experiment and the roughness of the sample make interpretation of CV and other electrical measurements problematic. However, this work does provide a target growth temperature of 450 °C for subsequent growths of bulk Ge_1−xC_x for electrical characterization of trap energies and densities. Future efforts will focus on increasing the C concentration to demonstrate a direct bandgap and lasing in Ge_1−xC_x.

Finally, we note that the calculated bandgap of Ge_0.9922C_0.0078 is approaching the energy of the split-off valence band (Fig. 1). A slightly higher C concentration in the alloy would suppress CHHS Auger recombination if E_G < E_SOH. Likewise, the similar wavefunction character in the E⁺ and E⁻ bands (Fig. 4) suggests that free carrier absorption (FCA) from E⁻ to E⁺ may be insignificant, although such calculations are beyond the scope of this work. Suppressed Auger and intraband FCA may make Ge_1−xC_x a compelling laser source for wavelengths from 3 to 5 μm.

We have provided evidence from ab initio modeling with hybrid functionals for a direct bandgap in Ge_1−xC_x for x < 1%, supported by photoreflectance showing a 20 meV reduction in bandgap even for 0.2% C. The numerical results are generally consistent with a band anticrossing model, in which the carbon atom causes a new, localized state above the conduction band minimum. As a result, the conduction band is split at Γ into E⁺ and E⁻ bands, as it is in dilute nitrides. Because the interaction potential is stronger at Γ than L, the CB minimum at Γ decreases faster than the minimum at L, turning the alloy into a direct bandgap semiconductor.

This model is supported by experimental measurement of a strong optical transition with built-in field. Although the C mole fraction (0.2%) was too low to provide definitive measurements, no evidence of graphitic or split-interstitial C was detectable by Raman, NRA-RBS, or HRTEM, yet a 20 meV decrease in the direct bandgap was apparent in PR, consistent with our ab initio results. Lack of extended defects suggests that C surface segregation leading to C-C clustering was suppressed. From this we infer that the 4GeMe carbon precursor led to substitutional incorporation.

The experimental growth of Ge_1−xC_x without detectable carbon-carbon defects was made possible using a precursor gas that deposited C atoms on the surface pre-bonded to Ge, thus bypassing the thermodynamically favored surface segregation and carbon clusters. These are the first experimental evidence of bandgap reduction and anticrossing in Ge_1−xC_x material that is not dominated by C-C bonds or other carbon cluster defects.

Efforts to grow Ge_1−xC_x with higher concentrations of C for a strongly direct bandgap are currently underway.

This work was supported in part by the National Science Foundation under Grant No. DMR-1508646, the Extreme Science and Engineering Discovery Environment (XSEDE) supported by NSF Grant No. ACI-1053575, and a Notre Dame Energy Center postdoctoral fellowship.

The authors thank the Notre Dame Center for Research Computing for calculation time with technical assistance from Dodi Heryadi, and Vincenzo Lordi and Eoin P. O'Reilly for helpful discussions.