Holes Outperform Electrons in Group IV Semiconductor Materials

A record‐high mobility of holes, reaching 4.3 × 106 cm2 V−1 s−1 at 300 mK in an epitaxial strained germanium (s‐Ge) semiconductor, grown on a standard silicon wafer, is reported. This major breakthrough is achieved due to the development of state‐of‐the‐art epitaxial growth technology culminating in superior monocrystalline quality of the s‐Ge material platform with a very low density of background impurities and other imperfections. As a consequence, the hole mobility in s‐Ge appears to be ≈2 times higher than the highest electron mobility in strained silicon. In addition to the record mobility, this material platform reveals a unique combination of properties, which are a very large and tuneable effective g*‐factor (>18), a very low percolation density (5 × 109 cm−2) and a small effective mass (0.054 m 0). This long‐sought combination of parameters in one material system is important for the research and development of low‐temperature electronics with reduced Joule heating and for quantum‐electronics circuits based on spin qubits.


Introduction
The semiconductor industry is one of the largest industries in the world, with global sales of $555 billion in 2021. [1] Over 99% of all semiconductor devices are made of or on silicon (Si) wafers. Novel group IV semiconductor epitaxial structures composed of silicon (Si), germanium (Ge), carbon and diamond (C), or DOI: 10.1002/smsc.202200094 A record-high mobility of holes, reaching 4.3 Â 10 6 cm 2 V À1 s À1 at 300 mK in an epitaxial strained germanium (s-Ge) semiconductor, grown on a standard silicon wafer, is reported. This major breakthrough is achieved due to the development of state-of-the-art epitaxial growth technology culminating in superior monocrystalline quality of the s-Ge material platform with a very low density of background impurities and other imperfections. As a consequence, the hole mobility in s-Ge appears to be %2 times higher than the highest electron mobility in strained silicon. In addition to the record mobility, this material platform reveals a unique combination of properties, which are a very large and tuneable effective g*-factor (>18), a very low percolation density (5 Â 10 9 cm À2 ) and a small effective mass (0.054 m 0 ). This long-sought combination of parameters in one material system is important for the research and development of lowtemperature electronics with reduced Joule heating and for quantum-electronics circuits based on spin qubits. and applications. Holes in strained Ge (s-Ge) possess additional unique properties compared to all semiconductors, including p-Si and p-GaAs. In particular, they have much smaller effective mass, which can be engineered down to 0.035 m 0 , [7] and a large electric-field tuneable effective g*-factor of up to 15. [8] The small effective mass and large g*-factor are essential properties for simplifying fabrication technology for quantum devices since larger orbital quantization and spin splitting energies enable larger lithographic features and higher-operation temperatures. These properties are also attractive for the development of topological quantum devices employing Majorana fermions. [9] One more significant property should be emphasized: Ge is a centrosymmetric elemental semiconductor, and therefore does not inherit the large bulk Dresselhaus component of SOI that would arise from bulk inversion asymmetry. [10] The Dresselhaus bulk SOI is very strong in all III-V compound semiconductors causing additional spin-orbit (SO)-mediated decoherence processes that are harmful for spin qubit devices. In contrast, SO is a useful interaction mechanism if it could be controlled. This is exactly the case for s-Ge devices with structural inversion asymmetry in the growth direction. Indeed, only the Rashba SO component remains active in s-Ge structures, which can be engineered and controlled by a gate voltage. For example, it can be completely switched off or quickly turned on at desired moments for spin manipulations and quantum-information processing. [10,11] This is a very advantageous property of Rashba-type SOI compared to the Dresselhaus coupling, because the coupling parameter describing strength of the Dresselhaus spin-orbit forces is a bulk material constant (tensor) and cannot be tuned by external electric fields.
Semiconductor heterostructures have a built-in strain that is induced by the mismatch of the crystal lattices of the composed materials. It is an essential parameter used for energy band structure engineering of the heterostructures based on various semiconductors, including Si and Ge. Development of highmobility-strained Si, SiGe, and Ge quantum-well (QW) heterostructures requires special growth techniques-molecular beam epitaxy (MBE) and chemical vapor deposition (CVD)-combined with epitaxial technologies to overcome various challenges in heteroepitaxy of these materials. Over the past decades, progress in epitaxial technologies have led to achieving record-high 2D hole gas (2DHG) mobilities in s-Ge QW modulation-doped (MOD) heterostructures, epitaxially grown on a standard Si(001) substrate, at both cryogenic and room temperatures. [4,[12][13][14][15] The mobility range of 2DHGs at room temperature was extended by 50% to 4500 cm 2 V À1 s À1 . [13] This value outperforms the roomtemperature 2DHG mobility of any known epitaxially grown semiconductors with nonzero bandgap, including III-V, and single-layer 2D materials. At lower temperatures, that is, below 10 K, much higher 2DHG mobilities in the range of 100 000-1 300 000 cm 2 V À1 s À1 have been demonstrated. [7,12,[15][16][17][18][19][20][21] It is interesting to note that these high 2DHG mobilities at both low and room temperatures were not predicted theoretically.
The historic evolution of 2DHG mobility in the group IV semiconductors at low temperatures is shown in Figure 1. [22] The first MOD-strained Si 1Àx Ge x (s-Si 1Àx Ge x ) QW heterostructures reported in 1984 were grown by solid-source MBE (SS-MBE) and had a 2DHG mobility of just 3300 cm 2 V À1 s À1 . [23,24] By 1996, this value was improved up to 16 800 cm 2 V À1 s À1 in a low-Ge-content Si 0.87 Ge 0.13 QW. [25] The large lattice mismatch between Ge and Si (4.17% at 293 K) made it impossible to grow QW heterostructure with high Ge content coherently on a Si substrate. For this reason, relaxed Si 1Àx Ge x buffers were developed and utilized to produce high-quality s-Ge structures on Si. By 1993, the highest 2DHG mobility in s-Ge QW heterostructure was 55 000 cm 2 V À1 s À1 and the material was grown by SS-MBE. [25] The next improvement was achieved by using the low-energy plasma-enhanced CVD (LEPE-CVD). This growth technology was able to enhance hole mobility by a factor of %2 up to %100 000 cm 2 V À1 s À1 , by 2002. [26] Development of reduced-pressure CVD (RP-CVD) provided a breakthrough in 2012, resulted in a Â10 enhancement of 2DHG mobility up to %1 000 000 cm 2 V À1 s À1 . [12,27] Finally, 10 years later and 38 years since the invention of the Si 1Àx Ge x QW heterostructures, the results reported here present another long-awaited major breakthrough in achieving the next record of hole mobility in s-Ge QW of 4.3 Â 10 6 cm 2 V À1 s À1 . This is over %3 times higher than the previously reported record of %1.3 Â 10 6 cm 2 V À1 s À1 in s-Ge QW. [7] This remarkable breakthrough in mobility can be viewed as an emergence of new class of quantum materials based on s-Ge with unique spin properties reported here. For comparison, in Figure 1, we also show the highest mobility of a 2D electron gas (2DEG) in tensile-strained Si (s-Si) QW. [28] This is the first time that electrons are outperformed by holes in a group IV semiconductor material at low temperatures. Currently, the record hole mobility is twice that of electrons for this material. We are unaware of a similar situation for any other semiconductor. Electron mobility has always been higher than that of holes in other semiconductors.
Enhanced electron and hole mobilities are not only important for developing novel electronic devices, but also open opportunities for new research in nanostructures that may lead to discoveries of new quantum phenomena. Moreover, very low-effective mass of 2DHG in s-Ge QW [7] facilitates the realization of laterally gated quantum devices such as quantum-point contacts (QPC), 1D wires, and quantum-dot (QD) devices, [29] operating at milli-Kelvin or up to 4.2 K. [4] High mobility at small carrier www.advancedsciencenews.com www.small-science-journal.com densities along with small effective mass and large effective g*-factor of the s-Ge-based materials enables fabrication of reproducible arrays of quantum devices with designer properties that can be reliably predicted. Moreover, the fact is that the devices based on the planar 2DHG structures with relatively small effective mass have relatively large lateral dimensions and, therefore, possess pure 2D characteristics determined by the hole energy spectrum originated from quantization in the vertical, that is, growth, direction of the heterostructure means that such properties can be engineered. [5] 2. Results and Discussions

Epitaxial Growth of s-Ge QW Heterostructures
For the presented research, an undoped s-Ge QW heterostructure was grown by RP-CVD on a relaxed Si 0.15 Ge 0.85 buffer on a standard Si(001) wafer of 150 mm diameter. [7,27] A schematic cross section of the heterostructure and fabricated gated Hall-bar device, with its source (S) and drain (D) Ohmic contacts and gate (G) stack, is shown in Figure 2a. [7,27] Epitaxy of the relaxed buffer layer and active region of a 15 nm-thick s-Ge QW were carefully optimized to substantially improve the material quality. In particular, the growth temperature of the s-Ge QW region was below 500°C to suppress the Si and Ge interdiffusion, Ge segregation and strain relaxation in the s-Ge QW. Both Si and Ge precursors were additionally purified to suppress background contamination. All epilayers were intentionally undoped. Accumulated by the negative gate voltage, holes are confined in the 15 nm-thick undoped QW, positioned 100 nm below the Si 0.15 Ge 0.85 cap epilayer and %1 nm-thick Ge cap interface. A thick Si 0.15 Ge 0.85 cap epilayer was intentionally selected to minimize impact of the remote ionized impurities located at the Ge cap/Al 2 O 3 gate dielectric and within the gate dielectric. [8] A %1 nm Ge cap epilayer thickness is expected to remain after the surface cleaning of an initially thicker Ge cap, prior to the deposition of the gate dielectric. Absence of a boron MOD impurity layer eliminates the presence of ionized doping impurities in close vicinity to the 2DHG confined in the s-Ge QW. [22] Omitting the in situ doping process also eliminates any unintentional boron doping due to either diffusion of impurities or their segregation in the case of inverted MOD. [7] Scattering on both remote and background ionized impurities is responsible for limiting 2DHG mobility at low temperatures. In our heterostructure, we achieved a very small level of ionized background impurities, resulting in the new benchmark record of 4.3 Â 10 6 cm 2 V À1 s À1 for hole mobility in the group IV semiconductor material.

Gated Hall-Bar Devices Microfabrication
For transport characterization of 2DHG of varied density, gated Hall-bars were fabricated using standard UV lithography, dry etching, and thin-film deposition techniques. Figure 2b shows a typical optical microscope image of a Hall bar with its channel oriented along the 〈110〉 in-plane crystallographic direction [7] and defined by the mesa structure etched in Cl 2 /Ar plasma. The Hall-bar's channel width is 100 μm and the distance between the nearest potential contacts is 200 μm. The alloyed AlSiGe Ohmic contacts are prepared by evaporating a 120 nm-thick Al film and then annealing at %275°C in N 2 ambient for 30 min. The contacts show low resistivity and excellent linear Ohmic behavior at cryogenic temperatures. The top accumulation gate is made of 20 nm Ti followed by 200 nm Au on a 50 nm-thick Al 2 O 3 dielectric layer deposited by atomic layer deposition (ALD) at 200°C. Figure 2c shows how the variation of applied gate voltage from approximately À0.6 up to À0.85 V at 290 mK induces changes to the 2DHG density and sheet conductivity from %0.32 Â 10 11 to %1.4 Â 10 11 cm À2 and from %10 to %90 mS sq À1 , respectively. The linear dependence of the hole density versus gate voltage has no noticeable hysteresis that indicates a relatively good quality Al 2 O 3 dielectric and an interface with a low density of rechargeable traps.

Magnetotransport Characterization
The gated Hall-bar was mounted in a 3 He cryostat equipped with a superconducting solenoid. For the low-temperature transport www.advancedsciencenews.com www.small-science-journal.com measurements, we employed a standard lock-in technique with an excitation current, I, of 20 nA limited by a 1 MΩ in-series resistor. We used to two synchronized lock-ins at 88 Hz to measure longitudinal (U xx ) and transverse or Hall (U xy ) voltages simultaneously. Zero magnetic-field resistivity, ρ 0 = wU xx /IL, and the Hall voltage in a small non-quantizing magnetic field are used to calculate carrier density p ¼ B=eR xy of 2DHG and carrier-transport mobility, μ ¼ R xy ðBÞ=Bρ xx ð0Þ. A gate voltage applied between the gate G and the drain D, in Figure 2b, is used to control the 2DHG density and conductance of the s-Ge QW channel.

Mobility
The experimentally obtained 2DHG mobility as a function of the 2DHG density is plotted on log-log scale in Figure 3. The 2DHG mobility varies from 1.2 Â 10 6 cm 2 V À1 s À1 at 2DHG density of 3.2 Â 10 10 cm À2 to 4.3 Â 10 6 cm 2 V À1 s À1 at 1.8 Â 10 11 cm À2 . The mean free path of holes for these data increases from %5 μm in low density range and reaches microscopic magnitudes up to 30 μm for higher densities. The maximum mobility of 4.3 Â 10 6 cm 2 V À1 s À1 exceeds by over 3 and 4 times the previously reported highest values of %1.3 Â 10 6 cm 2 V À1 s À1 at p = 2.7 Â 10 11 cm À2 in MOD s-Ge QW heterostructure [7] and %1 Â 10 6 cm 2 V À1 s À1 at 1 Â 10 11 cm À2 in undoped gated s-Ge QW heterostructure, [30] respectively. Analysis of the mobility versus carrier density slope indicates the mobility at lower carrier density range, that is, below 1 Â 10 11 cm À2 is limited by scattering on background ionized impurities with an estimated volume density of %3 Â 10 14 cm À3 . [31] This very low background impurity density is manifested as an exceptionally high mobility in the whole range of 2DHG densities, that is, the sample show record-high 2DHG peak mobility 4.3 Â 10 6 cm 2 V À1 s À1 in the high density range of %1.8 Â 10 11 cm À2 , and also very high mobility over 1.2 Â 10 6 cm 2 V À1 s À1 in the low density range down to 3.2 Â 10 10 cm À2 , seen in Figure 3. Fitting of the experimental data in low-density range, that is, <1 Â 10 11 cm À2 shows an almost linear increase of the mobility with the density, that is, μ % p 0.85 . At higher than 1 Â 10 11 cm À2 carrier density, the mobility increase slows down, following a power-law dependence with a smaller exponent, μ % p 0.4 . This dependence indicates that hole mobility in the high-density range is limited by some additional scattering mechanisms. Most likely, they are due to remote ionized impurities at the dielectric/semiconductor interface between the Al 2 O 3 gate dielectric and the surface Ge cap layer; or/and interface roughness at the s-Ge QW and Si 0.15 Ge 0.85 barrier interface, or possibly other crystal imperfections. Defects which can limit hole mobility in our 15 nm s-Ge QW epilayer are threading dislocations originating in the relaxed buffer and propagating through the whole structure up to the surface. Our optimized relaxed buffer layer shows relatively low threading dislocations density (TDD) in the range of 10 6 cm À2 . [14,27] However, TDD may be responsible for the mobility limitations reported here. Assuming a random distribution, the previously mentioned TDD corresponds to %1 threading dislocation per 10 μm. This is comparable to the obtained transport mean free path of %30 μm. If it is indeed the case, then the only way to validate this hypothesis will require further developments of new generation of relaxed buffers with even lower TDD, that is, in the range of %10 4 -10 5 cm À2 than the current state of the art reported here.
The s-Ge QW/Si 0.15 Ge 0.85 interface roughness could also be responsible for limiting the measured maximum mobility of 4.3 Â 10 6 cm 2 V À1 s À1 at 1.8 Â 10 11 cm À2 . There are two known origins of the interface roughness in an s-Ge QW heterostructure. The first is the surface roughness of relaxed Si 0.15 Ge 0.85 / Ge buffer layer grown on the Si(001) substrate. It manifests itself by the appearance of a correlated cross-hatched pattern on the surface. Such a pattern occurs as a consequence of strain relaxation due to the inhomogeneous distribution of misfit dislocations at different planes within the buffer. It is interesting to note that the period of these cross-hatches is typically below 5 μm. [27] Though, the surface of such a buffer is very smooth with the root-mean-square (RMS) surface roughness being about of %2 nm. [27] This cross-hatched roughness may also contribute to the scattering mechanisms limiting 2DHG mobility reported here. The second origin of the interface roughness is purely due to the smoothness and heteroepitaxy fluctuations of the s-Ge QW active region. In our case, it was carefully optimized to maintain the interface to be smooth and abrupt. For a more detailed examination of the aforementioned mechanisms, we plan to conduct similar experiments on a series of QW structures with varied s-Ge QW thickness.
We believe that the maximum mobility has still not been reached in s-Ge QW heterostructures and there is room for further improvements. Clearly, more detailed experimental and theoretical studies are required to understand microscopic mechanisms that limit hole mobility in s-Ge heterostructures. However, it is clear that the higher-quality epitaxial growth provided by RP-CVD is the key factor that contributed to obtaining %80 times higher 2DHG mobility in s-Ge QW structure grown by RP-CVD compared to the best one grown by SS-MBE. This technology also results in improved quality of interfaces and a reduction in background ionized impurities and defects, not only in s-Ge, but also in the surrounding relaxed SiGe epilayers of the heterostructure. The value of RP-CVD technology becomes even clearer considering that growth pressures are relatively high at  10-100 Torr, compared to the SS-MBE ones, which utilizes ultrahigh vacuum in the range of 10 À9 to 10 À10 Torr. This makes RP-CVD epitaxy more robust and economical and thus provides a credible path to large-scale fabrication technologies.

Percolation Density
The percolation density, p c , is another one of the important figures of merit to evaluate quality of high-mobility materials for quantum research and applications. This parameter characterizes disorder at low carrier density that is very important for the fabrication of uniform quantum devices. The percolation occurs when the Fermi level falls below the average profile of potential fluctuations and, therefore, a continuous current flow becomes very much suppressed. [32,33] In Figure 4, we present a plot of conductance at zero magnetic field, σ xx ð0Þ ¼ 1=ρ 0 , as a function of carrier density. To estimate p c , the observed data are then fitted to a percolation conductivity power-law dependence: σðpÞ ∝ ðp À p c Þ α with the fitted parameters of α ¼ 1.66, and p c ¼ 0.5 Â 10 10 cm À2 being the critical concentration, that is, the percolation density limit of conductivity. This value is 4 times smaller than the lowest published so far p c = 2.1 Â 10 10 cm À2 in an undoped s-Ge QW. [30] A detailed study is planned to acquire more physics insights in the percolation properties of this new quantum-material system at lower temperatures.

Gate Stack Characterization
As shown in Figure 3, we could not reach carrier concentrations larger than 1.8 Â 10 11 cm À2 by sweeping the accumulation gate voltage, V G . At large enough gate voltages, a shift of the percolation threshold voltage occurs, most likely due to a persistent charge accumulation at the dielectric/semiconductor interface. A very similar effect was reported earlier in gated undoped Si/SiGe heterostructures, which was explained as due to a surface tunneling from quantum well to the interface. [34] In principle, this effect can be used for the uniformity control of quantum dots in large-array circuits made from electron Si/Si 1Àx Ge x or hole Ge/Si 1Àx Ge x heterostructures. [35] More careful investigation of this phenomenon and its applications is outside of the scope of this publication. It should be mentioned that by employing this effect, we have experimentally determined an existence of a stable negative surface charge of %3 Â 10 11 cm À2 in the gated Hall-bar device after initial cool down. This charge is accumulated either at the dielectric-semiconductor interface or within the Al 2 O 3 gate dielectric. Most likely it is interface charge because we have not observed hysteresis during sweeping gate voltage in both directions in a range less than 0.25 V. This is a relatively large surface charge, comparing to Si/SiO 2 interface, that could be responsible for the mobility limitation due to scattering on remote ionized impurities. Further planned experiments will provide more insights into the origin of this interface charge and will allow us to either suppress and/or control it. For example, varying thickness of the Si 0.15 Ge 0.85 cap layer will allow us to understand impact of the dielectric/semiconductor interface on the mobility.

Effective Mass
The effective mass, m*, is another important parameter of a lowdimensional quantum system. A standard approach to determine the effective mass is to analyze temperature dependence of the Shubnikov-de Haas (SdH) oscillations of magnetoresistivity, ρ xx . [36] The SdH amplitude is conventionally described by [36,37] Δρ xx ¼ 4ρ 0 D th ðTÞexp À π ω c τ q , where τ q is the quantumscattering time. This expression assumes a Lorentzian density of states (DOS) and corresponds to a "Dingle plot" with ln(Δρ xx /ρ 0 ) to be linear in the 1/B scale with an intercept (1/B = 0) of 4. More generally, the DOS is better described by a Gaussian. If the Gaussian broadening is independent of B, the "Dingle plot" becomes proportional to 1/B 2 with a slope given by (πΓ/ħω c ) 2 . The thermal-damping term is D th = X th /sinh(X th ), where X th = 2π 2 k B T/E gap . For the simple Landau level (LL) model, without spin-splitting E gap = ħω c . Figure 5 shows the temperature-dependent traces of SdH oscillations from 290 up to 900 mK for the 2DHG density in the higher density range of 1.52 Â 10 11 cm À2 . This is a sufficiently large density, and the magnetic fields being sufficiently small, that the oscillations are dominated by the basic cyclotron gap and the standard approach described earlier can be used to estimate hole in-plane effective mass independent of the LL DOS. Collapsing all the SdH amplitude data in Figure 5 using only one adjustable parameter m* is found that the best fit occurs for an effective hole mass m* = 0.054 m 0 . This value is similar to the one reported before in an s-Ge QW of similar strain along <110> in-plane crystallographic orientation. [7]

Spin Properties and g*-Factor
Information about the spin properties of the 2DHG can be deduced from the SdH oscillations in the normal to the surface magnetic fields. Figure 6 shows SdH traces for different hole densities at the base temperature, T = 290 mK. Next, we estimate the quantum-scattering time, τ q , and quantum mobility, μ q ¼ eτ q =m Ã . For this purpose, we use the highest 2DHG density trace in Figure 6 (lower trace, p = 1.49 Â 10 11 cm À2 ) because the higher density trace exhibits standard behavior, with the even minima prevailing over the odd (spin) minima in low fields, that  is convenient for the analysis. In small magnetic fields, disorder suppresses the SdH oscillations and the amplitude varies as exp(Àπ/ω c τ q ). Using the standard Dingle-plot analysis [37,38] for p ¼ 1.49 Â 10 11 cm À2 , we obtain μ q = 4.3 Â 10 4 cm 2 V À1 s À1 and τ q = 1.3 ps. This value is two orders of magnitude smaller than the transport mobility (see Table 1) indicating that the small-angle scattering dominates in the scattering microscopic processes. The remaining small-angle scattering mechanisms are generally caused by long-range Coulomb interactions of mobile carriers with remote ionized impurities or charge dipoles in doping or buffer layers. [37,38] It is in excellent agreement with our earlier conclusions made independently from the analysis of the 2DHG mobility versus 2DHG density plotted in Figure 3. But in our case, remote impurities are located in the gate stack of the Hall-bar. Let us continue examination of the SdH oscillations shown in Figure 6 in more details starting from the bottom, the highdensity end. As the density is reduced, the critical filling factor ν c decreases and the corresponding critical magnetic field, B crit , moves to lower fields (indicted in Figure 6 for traces 1.49 and 1.30). It should be emphasized, as can be seen from the labeling in Figure 6, the dominance of the even minima disappears and at the lowest densities, the odd minima are clearly the strongest. At p = 1.02 Â 10 11 cm À2 (marked by a solid diamond in Figure 6), both odd and even minima have equal amplitudes. This is a very special situation indicating that the Zeeman spin splitting (E Z = g * μ B B) is equal to the effective cyclotron splitting (reduced by E Z , i.e., E Ã c ¼ ℏω c À E Z , that means E Z = ħω c /2). This is schematically illustrated in Figure 7b. At this special density, the effective g-factor is given by g * m*/m 0 = 1. Using m*= 0.054 m 0 , we find g* = 18.5. A similar "coincidence" method with tuneable ratio E Z /E c , but using tilted magnetic field has been used in electronic structures. For example, in Ref. [39], several coincidence points were observed, E Z /E c = r, with r = 1/2,1,3/2,2,3 and used to determine the effective electron g*-factor in a InAs QW structure. For 2DHG structures with strong 2D character, the tilted-field method does not work in principle due to the strong g-factor anisotropy of the 2D holes with the in-plane g*-factor being close to zero. [6,[40][41][42] It should be emphasized that in 2D electron and hole systems, there is a strong electron-electron (hole-hole) exchange interaction that leads to a large g*-factor enhancement. [43] In particular for materials with small effective g*-factor like GaAs, this enhancement effect dominates the Zeeman splitting. [44] However, the exchange enhancement requires a large spin polarization of the LLs, which is suppressed at low fields by the disorder damping, so the observation of the large splitting, E Z = ħω c /2, persisting to the lowest fields at which oscillations are visible suggests that the splitting is substantially intrinsic in our experiment for the trace marked by a diamond in Figure 6, and not exchange induced. In the literature, the problem of the magnitude of the exchange interaction so far is not well understood. [45] It depends simultaneously on several effects involving the LL width, B-dependent filling factor, DOS function, and temperature that requires a careful self-consistent approach.
To examine more carefully the spin-cyclotron gap coincidence transition discussed earlier, Figure 7a shows SdH oscillations for 3 selected 2DHG densities plotted versus filling factor, ν = pħ/eB. The SdH trace at high hole density of 1.485 Â 10 11 cm À2 in Figure 7a presents a normal situation of SdH oscillations when even minima prevail. However, at the lowest density of  0.654 Â 10 11 cm À2 (upper trace), the situation is clearly reversed: the odd minima corresponding to the spin gaps are much stronger than the even ones. The middle trace (p = 1.02 Â 10 11 cm À2 ) corresponds to the special "coincidence" situation when odd and even minima have the same amplitude at the whole magneticfield range. In Figure 7b, we show the qualitative energy diagrams corresponding to the three situations when the spin gap is smaller than half of the cyclotron gap, E Z < E c /2, when E Z = E c /2 (the middle trace), and when E Z > E c /2 (the upper trace). The SdH oscillations are periodic in the inverse magnetic field with minima at digital filling factors, ν = pħ/eB, as is evident in Figure 7a. At low fields, when the spin splitting is not resolved, the period of the oscillation versus the filling factor, ν, increases by 2 for each oscillation. It can be seen in the upper trace in Figure 7 in the range ν > 14. At some critical field (marked as B crit odd in Figure 5 and as v c in Figure 6), the spin splitting starts to be resolved and the period of SdH oscillations Δν is reduced to one (see Figures 5 and 6). Spin minima become more pronounced with increasing magnetic field and appear between the peaks corresponding to even filling factors. For the normal situation in Figure 6 and for p > 1.02 Â 10 11 cm À2 , the cyclotron minima, at even filling factors, remain deeper than the spin minima indicating the cyclotron gap is larger than the spin gap.
Experimentally, a linear "Dingle" plot ln Δρ xx ρ 0 vs 1/B (not shown) indicates Lorentzian broadening of the LLs, Γ % (exp(Àπ/ω c τ q ), or equivalently Gaussian with a broadening Γ that increases as B 1/2 . Frequently, however, the Dingle plot shows a quadratic dependence on 1/B indicating a broadening with constant Gaussian width Γ at least over the measurement range of fields. [46] Furthermore, while Γ may increase as B 1/2 at higher fields, the self-consistent Born approximation suggests that it also decreases as the spins become resolved. [47] This competition between the two factors means that it is not clear how the broadening actually varies with the magnetic field. A simple approximation for estimating effective g*-factor [48] is to use the two critical fields B crit odd and B crit even when the respective minima first occur and assume Ez/Γ and (ħω c -E Z )/Γ take the same value at these two fields. Assuming Γ % B 1/2 , then gives g* = E Z /μ B B = 2(m*/m 0 ) p B even /( p B even þ p B odd ). The results of this approximation are shown in Figure 7c. A more detailed analysis (to be published elsewhere) without any specific B-dependence of Γ, but assuming that it is the same for maxima and minima at each field gives similar values.  www.advancedsciencenews.com www.small-science-journal.com Estimations of the effective g*-factor at different densities can be obtained from the analysis of the LL DOS, presented in Figure 7. If we assume LLs are described by Gaussians, then the odd minima are characterized by DOS with an exp(ÀE z 2 /4Γ 2 ) dependence and the even minima being proportional to exp(À(E Z -ħω c ) 2 /Γ 2 ). The B-dependence of the minima should then reveal E Z provided the values of the LL width, Γ, are known. This is not a straightforward problem, although, it might appear that the conventional exp(Àπ/ωτ q ) amplitude dependence gives a value of Γ that apparently varies as B 1/2 . The analysis confirms the simple prediction, g* % 18, with a small downward trend with increasing density. However, there is no reason to believe that the values of effective spin gaps, E Ã Z (with accounted exchange interaction), are the same for the even and odd minima. It is well known that for exchange enhancement, [44,45] there is a strong oscillatory dependence of the effective enhanced spin splitting, E Ã Z . Nevertheless, a good gate-voltage tunability of the effective g*-factor with the gate voltage (density) is evident in Figure 7c, within a range that is more than sufficient for the quantum-computing applications. Note that an excessively large g*-factor voltage-sensitivity may result in undesirable increased spin qubit noise figures due to capacitive coupling to control gates. An apparent nonlinear dependence of g*-factor in Figure 7c may be partially due to the exchange interaction discussed therein. [43,44,49] A more detailed analysis of this complex self-consistent mechanism of the exchange interaction is outside of this paper.

Comparison with the State of the Art
It is well known that electron g*-factor in GaAs is very small g Ã GaAs %À0.44; for holes, it is around 1.4. [50] In Si, effective g*-factor for both electrons and holes is around 2. [51,52] In III-V materials, InAs and InSb, electron g*-factor reaches relatively high values of %À15 [39] and up to À50, [53] correspondingly. Unfortunately, III-V materials are very complicated to process, very expensive, not widely abundant in the earth crust compared to Si, do not exist in isotopically pure forms, and are not compatible with the state-of-the-art Si technologies for mass production. Therefore, the obtained results make the s-Ge on Si material very attractive for further research of quantum physics and for development of novel quantum technologies.
Low-temperature 2DHG properties obtained in this work are summarized in the Table 1. For comparison, the highest 2DHG mobilities obtained in MOD and undoped s-Ge QW are shown as well. For completeness, the highest mobilities of 2DEG in s-Si and GaAs and 2DHG in GaAs are added to the table (three last rows). Material's structures along with the corresponding references are listed in the first column. The crystallographic orientation of the Hall-bar (when applicable) is specified in the second column. It was found that transport mobility strongly depends on the Hall-bar orientation in s-Ge QW structures. [7] The maximum low-temperature carrier mobility is quoted in the third column along with the corresponding concentration shown in the fourth column. The highest mobility so far is obtained for 2DEG in GaAs, 44 Â 10 6 cm 2 V À1 s À1 . We believe that hole mobility in s-Ge can reach similar values or even higher due to the record-small hole effective mass in this material system, which can be engineered using compressive biaxial strain in the Ge epilayer of very low disorder and smooth interfaces. Dingle ratio of the scattering times τ t =τ q is given in the seventh column. As expected for high-mobility samples, it is very large and reaches 100 in s-Ge in this work. The large Dingle ratio indicates a very strong dominance of small-angle scattering mechanisms that limit quantum-scattering time as discussed earlier. It will be important to study and identify exact microscopic scattering mechanisms in s-Ge QWs for further advances in quality of this material platform. Both transport-and quantum-scattering times are shown in the eighth and ninth columns, respectively. The last column shows available g*-factor data in s-Ge. Unfortunately, there are no g*-factor data in the quoted works cited in this table. Effective g*-factor values in other materials are discussed in Section 2.8.
As a last note, this work reduces the gap between the best 2DHG mobility in GaAs QW heterostructures grown on GaAs substrate, which was recently increased from 2.3 Â 10 6 cm 2 V À1 s À1 (at carrier density 6.5 Â 10 10 cm À2 ) [54] to 5.8 Â 10 6 cm 2 V À1 s À1 (1.3 Â 10 10 cm À2 ), measured at 300 mK. [55] All other known semiconductors, including III-V, II-VI, perovskites, 2D materials, etc., show substantially lower 2DHG mobility than in the s-Ge and GaAs QW structures. Moreover, it is important to emphasize that s-Ge QWs are grown on standard Si(001) wafers, which are used by the semiconductor industry to fabricate over 99% of all modern electronic devices including complementary metal oxide semiconductor (CMOS) devices. RP-CVD in particular enables epitaxial growth of these QW structures on 300 mm diameter wafers now and can be extended to larger 450 mm wafers in near future. This is an additional attractive feature of the reported s-Ge material system for large-scale applications.

Conclusions
A record-high mobility of free holes reaching 4.3 Â 10 6 cm 2 V À1 s À1 in strained germanium grown on a standard silicon wafer has been demonstrated that sets a new quality benchmark for the group IV semiconductor materials. As a consequence, electrons are outperformed by holes in the group IV semiconductor materials at low temperatures. The demonstrated hole mobility in s-Ge is twice that of the best mobility of electrons reported in state-of-the-art strained silicon. A similar situation has not been observed for any other semiconductor material. Due to the fourfold material quality improvement, it can be stated that the novel class of quantum materials for the quantum-physics research and applications has emerged. This superior material system with a combination of unique properties, which are large and tuneable effective g*-factor, strong and tuneable SOI, low percolation density, and small effective mass, will lead to new opportunities for innovative quantum-device technologies and applications in quantum as well as in classical electronics, optoelectronics, and sensors.