Carrier cooling in direct bandgap hexagonal silicon-germanium nanowires

The monolithic integration of a silicon-based laser onto a silicon electronic chip has been a longstanding research subject. Unfortunately, silicon has an indirect bandgap, impeding efficient light emission and lasing. To circumvent the lack of a silicon laser, a lot of work has been directed toward integrating direct bandgap group III–V materials on silicon.¹ An emerging technology is to exploit metastable, direct bandgap group IV materials, such as strained germanium (Ge),² germanium tin (GeSn),^3–9 or hexagonal (lonsdaleite) silicon-germanium (hex-SiGe),¹⁰ as the light-emitting active material. However, efficient light emission and lasing also require efficient carrier cooling toward the band edges by inelastic carrier-optical phonon scattering.^9,11

In polar III/V semiconductors, like gallium arsenide, carrier relaxation is dominated by the Fröhlich interaction through emission of small wave vector optical phonons, as displayed in Fig. 1(a). The Fröhlich interaction drives carrier cooling within a conduction band (CB) or valence band (VB) valley, referred to as intravalley scattering. It has been studied extensively in group III/V semiconductors and is found to be a very efficient carrier cooling process.^12–14 Conventional group IV semiconductors like cubic germanium (Ge) are nonpolar.^11,15,16 As such, due to the absence of Fröhlich interaction, carrier relaxation in the CB is primarily governed by intervalley deformation potential scattering (intervalley DPS), as displayed in Fig. 1(b).¹² After the electrons have scattered from the Γ-valley to the indirect X- and L-valleys, large wavevector intervalley DPS between the different X- and L-valleys results in efficient electron cooling on a timescale of hundreds of femtoseconds to a few picoseconds,^17–20 where the hole temperature follows the electron temperature by fast electron–hole scattering.^13,17 For indirect bandgap semiconductors, the small amount of intravalley carrier relaxation within the Γ-valley has been overshadowed by ultrafast intervalley DPS and could not be experimentally assessed.

In direct bandgap group IV semiconductors like GeSn and hex-SiGe, as displayed in Fig. 1(c), the carriers are first very rapidly cooled by intervalley scattering by the emission of zone edge phonons. In these semiconductors, intervalley scattering is however no longer adequate for carrier cooling toward the bottom of the CB. Here, intravalley scattering within an isolated Γ-valley is the only option to cool down from the minimum of the lowest indirect valley toward the global minimum in the Γ-valley, and likewise in the VB.²¹ The intravalley cooling is expected to occur by emission of long-wavelength optical phonons. In hex-SiGe alloys, the phonon energies have been investigated using Raman spectroscopy, resulting in phonon energies between 35 and 57 meV for hex-Ge and hex-Si, respectively.²² Carrier cooling in such an isolated Γ-valley in a group IV semiconductor has not been studied yet. Hex-SiGe is unique as a direct bandgap group IV semiconductor with a large directness (ΔE_Γ-U), which is defined as the energy separation between the lowest indirect minimum and the global minimum. As an added advantage, its directness can be tuned by the alloy composition (up to 315 meV for hex-Ge), without using external strain.¹⁰ 

Experimental study on the carrier relaxation dynamics in group IV semiconductor nanowires emitting in the mid-infrared requires measurements of the hot carrier distributions as a function of time. This can be performed using either two-wavelength time-resolved pump–probe techniques or picosecond time-resolved measurement of the temporal evolution of the complete photoluminescence (PL) spectrum.^20,23 Although PL lifetime measurements on GeSn have been reported as a function of emission energy,^9,24,25 simultaneous measurements of the time-resolved and spectrally resolved PL, allowing to measure the carrier cooling time, are currently lacking.

We performed time-resolved micro-PL measurements on the as-grown core shell hex-SiGe nanowire arrays. Specifically, we study the carrier cooling time in a hex-Si_0.32Ge_0.68 nanowire (NW) sample with a NW diameter (length) of 2.4 μm (8.4 μm) and in a hex-Si_0.2Ge_0.8 NW sample with a NW diameter (length) of 1.3 μm (9.2 μm). The nanowires are spaced with a pitch of $4$μm in a square pattern, as shown in the Scanning Electron Microscopy (SEM) images shown in the supplementary material A. The samples are grown using the crystal transfer method with wurtzite (WZ) GaAs NWs with a diameter of 180 nm, as a template by metal-organic vapor phase epitaxy (MOVPE).^26,27 The details of the gold-catalyzed WZ-GaAs NW growth are given in Fadaly et al.¹⁰ The hex-SiGe shells are grown around the core NWs by MOVPE at a substrate temperature of 595 °C. The diameters of both nanowire samples are suitable to strongly confine the fundamental modes, as shown in Appendix E, resulting in a similar absorbed power density per area for both samples.

The time-resolved micro-PL setup combines a Fourier Transform Infrared (FTIR) spectrometer with a superconducting nanowire single photon detector (SNSPD). This method allows for experimental observation of the time evolution of the complete PL spectrum, which is essential to measure the carrier cooling dynamics. Our experimental setup thus features the same functionality as a streak camera but in the experimentally difficult-to-access near-to-mid-infrared (NMIR) spectral region. An NKT photonics Origami pulsed fiber laser at 1030 nm (1.2 eV), with a pulse length of 200 fs at a repetition frequency of 40 MHz, is used as an optical pump source. A reflective Cassegrain objective (36x, NA = 0.52) focuses the laser beam on the sample to an excitation spot with a diameter of approximately 4 μm, exciting only a single standing nanowire from the top. The objective is also used for PL collection.

In this equation, $IPL(E)$ is the PL spectrum, $Eg$ is the bandgap energy, $χ$ is a scaling factor, and $Δμ$ is the chemical potential describing the shift in quasi-Fermi levels by the photoexcited carrier densities. By fitting the LSW model to the time-resolved PL spectra, as shown by the red dashed curves in Figs. 3(a) and 3(b), we obtain the carrier temperature $TC$ as a function of the elapsed time after optical excitation. Since we are interested in the carrier cooling dynamics after excitation, we define the initial time t = 0 at the maximum of the integrated PL-decay curve.

We focus on two different hex-SiGe alloy compositions, featuring different ΔE_Γ-U. A schematic of the approximate band structure and possible carrier cooling pathways is shown in Fig. 3(c), showing the three lowest CB minima, at the Γ-, M-, and U-points, for both alloy compositions, based on Borlido et al.³² The hex-Si_0.2Ge_0.8 alloy (blue) has a large directness of ∼140 meV (Ref. 32) and has a bandgap of 0.56 eV. Intervalley scattering will initially cool down the carriers toward (1 optical phonon energy above) the bottom of the U-valley.^17–20 This relaxation is too fast to be measured in our setup and thus takes place before the onset of our measurement. The carriers are subsequently scattered by an optical phonon emission toward high energy Γ-band states (yellow arrow) at a net excess energy of approximately ΔE_Γ-U above the CB minimum. The carrier distribution within the Γ-valley is experimentally observed by the broad initial PL emission spectrum in Fig. 3(a). Its width of >300 meV corresponds to a carrier temperature of 680 K.

The hex-Si_0.32Ge_0.68 alloy (green) has a low directness of approximately 70 meV (Ref. 32) and a bandgap of 0.61 eV. Likewise, the carriers are rapidly cooled down by intervalley scattering between the indirect valleys.^17–20 The fast intervalley scattering processes [Fig. 3(c), yellow arrow] are expected to subsequently scatter the carriers into the central Γ-valley with a net energy ΔE_Γ-U, only slightly above the CB minimum, requiring a smaller amount of intra-Γ-valley cooling. This is experimentally confirmed by the much lower initial carrier temperature $Tc(0)$ of 280 K, as shown in Fig. 3(b). We thus observe a large difference in the initial $Tc(0)$ of roughly 400 K between the different hex-SiGe alloys.

By fitting the TRPL spectra in Figs. 3(a) and 3(b) with the LSW model, we obtain carrier cooling curves $Tc(t)$ for both hex-Si_0.2Ge_0.8 and hex-Si_0.32Ge_0.68, which are displayed in Figs. 4(a) and 4(b). To obtain relevant carrier cooling times for a hex-SiGe nanowire laser, we used high excitation fluences of 5.31, 3.54, and 1.77 mJ/cm², reasonably above the gain threshold for typical semiconductor nanowire lasers.^33–36 At an elapsed time shorter than 0.15 ns, the emission spectrum of hex-Si_0.2Ge_0.8 shows a second peak centered around $∼$ 0.82 eV [visible at t = 0 ns in Fig. 3(a)]. As this peak affects the Fermi–Dirac shape of the high energy side of the spectrum, it results in an increase in uncertainty of $Tc$⁠, as displayed in Fig. 4(a). The energy of this peak is in the vicinity of the second valence band transition. However, the appearance of the peak is more pronounced at low excitation powers and is not evident in the spectra of hex-Si_0.32Ge_0.68. The time-resolved spectral PL maps of all measurements represented in Figs. 4(a) and 4(b) are displayed in supplementary material C.

The measured carrier cooling times (⁠ $τ0$⁠) are plotted in Fig. 4(c) as a function of the excitation fluence. For both alloys, we obtain a carrier cooling time of roughly 180 ps at an excitation density of 3.5 mJ/cm². The similar carrier cooling time observed for both alloy compositions provides evidence that we indeed measure the fundamental intravalley carrier cooling mechanism in hex-SiGe. As expected from the absence of Fröhlich interaction, the intravalley carrier relaxation in the direct Γ-valley hex-SiGe is clearly slower than in III–V semiconductors.¹⁴ 

The influence of hot phonon effects can be experimentally assessed by observing whether $τ0$ increases with excitation fluence due to the presence of a hot phonon population. This heats the carriers by phonon absorption processes and thus limits the cooling rate of the carriers.^39,40 As shown in Fig. 4(c), we only observe a very moderate increase in the carrier cooling time in our hex-SiGe nanowire alloys, suggesting that phonon bottleneck effects in hex-SiGe are less important compared to, e.g., III/V multiple-quantum well (MQW) structures where hot phonon effects are stronger.^41–43 This is consistent with our theoretical expectations that, due to the larger number of optical modes in hexagonal crystals,²² a strong phonon bottleneck effect is absent in hex-SiGe.^42,44

To establish whether the measured carrier cooling time in hex-SiGe is still acceptable for lasing, we compare our results with an InGaAs/InGaAsP MQW laser structure in Fig. 4(c) (see supplementary material D). These MQW samples are emitting in the same spectral region as our hex-SiGe samples and are measured with the same experimental setup. Importantly, we observe similar order of magnitude carrier cooling times for the InGaAs MQW reference sample as obtained for hex-SiGe. This can be understood by the large susceptibility of MQWs for hot phonon effect,^41–43 as observed in supplementary material Sec. D. The similarity of the carrier cooling time in our nonpolar hex-SiGe nanowires as compared to a conventional III/V InGaAs MQW laser structure provides evidence that, from the carrier cooling perspective, the nonpolar nature of our hex-SiGe nanowires does not provide a major obstacle for lasing.

The mechanism for carrier cooling in an isolated Γ-valley, with large ΔE_Γ-U, is still unknown. Using the current experimental method, no distinction can be made between specific scattering processes. Three potential mechanisms are postulated: (i) deformation potential scattering (DPS) in the VB, (ii) DPS in the CB, and (iii) polar optical phonon scattering due to the very small polarity of the Si-Ge bond.

(i) Initially, the dominant cooling mechanism is due to intervalley scattering of electrons that cool down in the indirect valleys and subsequently scatter toward the central Γ-valley. Once the electrons have arrived in the central Γ-valley, optical phonon emission by DPS in the VB probably becomes the dominant cooling mechanism.⁴⁵ If this mechanism is dominant, electron cooling will follow the cooling of holes by electron–hole scattering. We note that this relaxation mechanism is expected to be equally valid in other direct bandgap group IV semiconductors like GeSn or SiGeSn.

(ii) For both strained Ge and GeSn, the intravalley-Γ-valley DPS scattering is forbidden due to the s-like characteristic of the CB.⁴⁵ For hex-SiGe, the intra-Γ-valley DPS is not symmetry forbidden, as the lowest CB of hex-SiGe in the Γ-valley is of hybridized sp-type character.⁴⁶ 

(iii) We cannot completely exclude polar optical phonon scattering due to the small remaining polarity in either GeSn or hex-SiGe. While the Si-Si and the Ge-Ge bonds in hex-SiGe are completely nonpolar, the randomly distributed Si-Ge bonds in the hex-SiGe alloy still give rise to a slight polarity due to their small electronegativity difference (⁠ $Δχr$⁠). The effective average electronegativity difference between the individual atoms within the alloy is $Δχr,eff$ = 0.02 and $Δχr,eff$ = 0.03 for hex-Si_0.2Ge_0.8 and hex-Si_0.32Ge_0.68, respectively.⁴⁷ So, we cannot completely exclude a small amount of polar optical phonon emission by the Fröhlich interaction. However, even though there is a difference in polarity, there is no significant difference in carrier cooling rate between the two alloys, suggesting that the Fröhlich interaction is not dominant. Consequently, nonpolar intra-Γ-valley DPS within the VB Γ-valley, or possibly also within the CB Γ-valley, is the most likely mechanism responsible for the observed carrier cooling time of 180 ps, but definite conclusions can only be drawn by a future theoretical investigation, which is beyond the scope of the present paper. Further experimental research could be performed comparing the carrier cooling behavior of hex-SiGe to other direct bandgap group IV materials to determine the effect of sp hybridization in the CB and possible effects on their technological relevance for Si photonics. To exclude the contribution of any remaining Fröhlich interaction, pure hex-Ge can be investigated, which is identical to hex-SiGe in regard to sp character and the large directness of the CB. This is presently not possible using the technique presented in this work due to limitations of the detector's measurement window.

We finally note that the intravalley scattering in hex-SiGe might be different from its cubic counterparts since the hexagonal unit cell is two times larger than the cubic unit cell. Consequently, the reciprocal unit cell of hex-SiGe is twice as small as that of its cubic counterpart, resulting in backfolding of phonon modes.⁴⁸ The backfolding introduces several additional phonon modes at the Γ-point. Furthermore, in hex-SiGe, the Si-Si, the Si-Ge, and the Ge-Ge bonds result in slightly different phonon energies (⁠ $∼$ 40 meV).²² A theoretical calculation might provide more insight into the role of these folded phonon modes for carrier cooling dynamics.

In conclusion, our hex-SiGe nanowires feature a large directness, allowing us to measure the carrier cooling dynamics in an isolated Γ-valley. We experimentally observe a hot carrier cooling time within the Γ-valley of approximately 180 ps. The slow cooling time implies that the Fröhlich interaction is either absent or strongly reduced. In conventional III/V semiconductors, carrier relaxation is efficient due to polar optical phonon scattering, mediated by the Fröhlich interaction. The absence of efficient Fröhlich and intervalley scattering within an isolated Γ-valley of a nonpolar direct bandgap semiconductor might provide a bottleneck for efficient group IV semiconductor lasers. However, we emphasize that the carrier cooling time in our hex-SiGe nanowires is of comparable magnitude to the carrier cooling time in an InGaAs multi-quantum well laser structure. We thus conclude that the near absence of Fröhlich interaction in direct bandgap hex-SiGe does not inherently limit its application potential for hex-SiGe semiconductor lasers or hex-SiGe semiconductor optical amplifiers.