Quantum computing, as a core frontier in today's technological realm, is receiving global attention. Due to their scalability and compatibility with traditional CMOS (complementary Metal oxide semiconductor) processes, silicon-based quantum bits have become an essential approach to constructing large-scale integrated quantum bit systems^[1−3]. In this domain, strained Ge/Si_1−xGe_x heterostructure two-dimensional hole gas quantum wells (2DHG QW), owing to their unique advantages, emerge as extraordinary material platforms for spin-hole quantum bits. The primary research directions of these quantum wells include:

(1). Lightly-strained quantum wells (QWs): Characterized by in-plane strain ε_∥ < −0.5% and ΔE_(HH−LH) < 40 meV (ΔE_(HH−LH) is defined as energy difference between LH and HH subbands at Γ point). With their minor lattice mismatch, these wells exhibit smoother growth interfaces and superior crystal quality. By increasing the depth of the quantum well (the thickness of the upper barrier), it is possible to overcome the impact of low barrier height susceptible to surface remote impurity scattering, thus achieving higher material mobility. Quantum dots fabricated from deep well lightly-strained materials possess faster response speeds and longer operation times. They are a significant pathway for fabricating quantum dots utilizing spin−orbit coupling (SOC) effects^[4−6].

(2). Standard-strained quantum wells (QWs): These wells have an in-plane strain of −0.5% < ε_∥ < −0.8% and ΔE_(HH−LH) of 30 meV < ΔE_(HH−LH) < 60 meV. With a moderate barrier height and relatively shallow quantum well depth, they form the fundamental material for gate-controlled quantum dots^[7−9]. The optimized intrinsic standard strained quantum well mobility has reached two million, and a 4-bit quantum processor based on this material has been developed, becoming the preferred option for the expansion of strained Ge quantum bits^[10−14].

(3). Heavily-strained quantum wells (QWs): These wells are defined by an in-plane strain ε_∥ < −0.8% and ΔE_(HH−LH) < 60 meV. They have the deepest barrier height, providing the strongest longitudinal confinement capability for quantum dots. Additionally, the two-dimensional hole gas is less affected by remote impurity scattering. Ultra-shallow heavily-strained quantum wells significantly reduce the complexity of microwave cavity coupling^[15−17]. Furthermore, their notable light−heavy hole subband separation serves as an important basis for studying single-triplet quantum bits^{[18, 19]}.

In summary, strained Ge/Si_1−xGe_x heterostructure two-dimensional hole gas quantum wells demonstrate tremendous potential for application in the field of quantum computing. The growth of complex and variously strained quantum wells is a current hotspot in quantum bit preparation.

The diverse developmental prospects pose distinct challenges in the design and growth of three types of quantum well structures. Dedicated studies have been conducted by various research teams on complex quantum effects in different strained quantum wells, focusing on specific structural parameters. Berkutov et al. and Laroche et al. independently discovered that varying quantum well depths lead to different transport phenomena under a magnetic field^{[20, 21]}. Su et al. investigated the impact of quantum tunneling on transport properties by altering the depth of the quantum wells^[22]. These studies reveal the influence of quantum effects of two-dimensional hole gas in heterostructures. The material properties of two-dimensional holes in heterostructures are influenced by multiple factors, including the strain induced barrier height, the dimension size of quantum well and the gate voltage, but systematic research on the growth of quantum well materials remains insufficient. This study systematically designed and grew quantum wells with different strains and examined the effects of strain and structural parameters on the magnetic transport of two-dimensional hole gas. Additionally, combined with the first-principle simulations, we developed a percolation density model for quantum well structures and proposed a light−heavy hole subband mixed transport model in response to variations in effective mass and strain. This research aims to address the growth challenges of strained quantum well structures in various applications and is expected to set a strategy in the growth of germanium-based quantum well.

In this study, we utilized reduced pressure chemical vapor deposition (RP-CVD) system to grow heterostructure intrinsic Ge quantum wells on 8-inch silicon wafers^[23]. Germane mixed with hydrogen (GeH₄ in H₂) and dichlorosilane (SiH₂Cl₂) were used as precursors to fabricate a germanium layer of approximately 1.7 μm thickness, serving as a virtual substrate. Subsequently, a high-temperature annealing process was performed at 820 °C for 20 min, aiming to reduce the density of threading dislocations, thus enhancing the quality of the germanium epitaxial layer. Then, at 800 °C, a reverse-graded Si_1−xGe_x (0.75 < x < 0.85) layer was grown by maintaining a constant flow of GeH₄ while gradually increasing the flow of SiH₂Cl₂^[24]. This was followed by the growth of a constant composition, fully relaxed Si_1−xGe_x (0.75 < x < 0.85) layer at lower temperatures. This step is designed to minimize remote impurity scattering on the sample surface while providing the necessary compressive stress for the quantum well. Finally, a silicon capping layer was grown, providing an excellent dielectric interface for device fabrication.

In the final phase of our study, we successfully synthesized three representative samples: LS (lightly-strained quantum well), SS (standard-strained quantum well), and HS (heavily-strained quantum well). To achieve the desired strain effects, specifically the composition of the barrier beneath the quantum well, we maintained a constant gradient buffer layer growth rate of 10% μm⁻¹while varying the growth time (ramp time)^{[25, 26]}. This approach enabled us to obtain fully relaxed constant barrier Si_1−xGe_x layers with altered Ge compositions x between 0.75 and 0.85. The strained Ge quantum wells grown on these fully relaxed Si_1−xGe_x (0.75 < x < 0.85) compositions achieved our targeted range of strain values. The grown quantum well structure is shown in Fig. 1(a), where the Si cap and Si_1−xGe_x top barrier layers determine the depth of the quantum well, and the thickness of the sGe QW corresponds to the quantum well thickness. Given that lightly-strained quantum wells are more susceptible to surface scattering, whereas heavily-strained wells benefit from deeper potential barriers and are less affected, we adjusted the thickness of the upper barrier layers accordingly. For lightly-strained quantum wells, we increased the thickness of the top barrier layer and well to reduce the tunneling density and shield against high-angle surface scattering. Conversely, for heavily strained quantum wells, we implemented an opposite strategy to capitalize on their confinement advantages. The strain levels in the three types of grown samples correspond to the three defined regions calculated using the VSAP (Vienna Ab-initio simulation package) method, as depicted in Fig. 2.

In this study, high resolution X-ray diffraction (HRXRD) and scanning transmission electron microscopy (STEM) techniques were employed for the detailed characterization of the thickness and strain of each film layer in the quantum well structure^{[7, 27]}. Atomic force microscopy (AFM) was utilized to characterize the morphology of the sample, aiming to evaluate the quality of the QW’s top-layer interface.

Fig. 3(a) shows the reverse space mapping (RSM) image of the (−2 −2 4) diffraction direction of the lightly-strained QW obtained by high-resolution X-ray diffraction (HRXRD) measurement^[28]. The silicon substrate peak is located at the upper left, and the Ge virtual substrate peak is located at the lower right of the image. The Ge quantum well (QW) is located directly below the fully relaxed constant composition SiGe barrier layer, indicating that the Ge QW is subjected to full compressive strain from the constant composition layer below, as shown by the green line in the figure. When the quantum well strain gradually increases, that is, the LS−SS−HS changes, the green line gradually moves to the left and approaches the substrate silicon peak. The strain values calculated by Matlab peak search are −0.43%, −0.61%, and −0.19%^[19], which are consistent with experimental expectations.

Fig. 3(b) shows the surface morphology of the lightly strained QW. Its regular and evenly distributed cross-hatch indicates that no 3D island deposition growth occurs, and it has good surface roughness (R_q= 2.03 nm)^[29].

To explore the low-temperature transport characteristics and evaluate the performance of our heterostructure, we fabricated electrically isolated Hall bar structures with different strain samples, whose aspect ratio was L/w = 3. The mesa was patterned using a 385-nm laser writing system, followed by reactive ion etching (RIE) with a SF₆/O₂ gas mixture. Phosphorus ions were implanted at an energy of 19 keV with a dose of 2 × 10¹⁵ cm⁻², and the sample underwent an activation anneal at 700 °C to establish a robust ohmic contact for the source and drain regions, effective even at 250 mK. A 30 nm thick Al₂O₃ gate oxide layer was deposited using an atomic layer deposition (ALD) system (model ALD−100A), ensuring precise gate control. The top gate is composed of a 10/70 nm Ti/Au bilayer, and the test terminals and source and drain are composed of 80 nm Al, deposited using an AdNaNotek JEB-3 electron beam evaporator. Wire bonding is performed to establish connections to the low temperature test equipment. Fig. 1 (b) shows the device structure and test principle of the Hall bar. Low-temperature transport measurements were carried out in a He³ cryostat system on LS, SS, and HS samples^{[30, 31]}. The device structure and test process are the same as Ref. [7]. The data are collected at temperature ranged from 250 to 650 mK, maintaining consistency in device fabrication and bonding parameters. As indicated in Table 1, the magnetic transport results revealed that the magnetic transport characteristics are closely associated with the strain and structural parameters of the materials. The LS QW exhibited the lowest percolation density, the smallest effective mass, and a high mobility of up to 7.301 × 10⁵ cm²∙V⁻¹∙s⁻¹, while the HS QW displayed the opposite characteristics. These differences are primarily attributed to the variations in the thickness, depth, and strain of the quantum wells.

Under the saturation density tuned by the gate voltage, we obtained magnetic field-dependent longitudinal and transverse resistances, leading to oscillation diagrams under three strain types (as shown in Fig. 4^[7]). We refer to the transport results of ultra-shallow heavily strained quantum wells^[7] and compare the magnetic transport data of standard strained QW and lightly strained QW. The study shows that strain and hole gas carrier density jointly influence the position of the Landau plateau. The Hall conductance plateau shifts backward under the same magnetic field due to the filling of quantum states on Landau levels. The number of states per Landau level, determined by the hole gas density and the applied magnetic field, allows a hole number of eB/h. At lower magnetic fields, each Landau level hosts fewer states, enabling more Landau levels to be filled by holes, which increases the Hall conductance and, correspondingly, decreases the Hall resistance^[32]. As the magnetic field increases, the Fermi level passes through these Landau levels, and the number of states on each Landau level increases. Hence, holes initially occupying higher Landau levels transit to the additional states in lower Landau levels. This leads to a reduction in the number of Landau levels occupied by holes, thereby increasing the Hall resistance and forming higher steps. When the Landau levels are well separated, the state density at the Fermi level approaches zero. In this scenario, rapid changes in the Fermi energy result in the Fermi level being "pinned" to the broadened Landau levels. When higher Landau levels become unoccupied, the Fermi level rapidly transitions to the next level. This phenomenon is known as Shubnikov-de Haas oscillations (SdH oscillations), where the minimum oscillations of the longitudinal resistance R_xx occur at integer filling factors ν. As the magnetic field further increases, these oscillations become dispersed and amplified, and the system gradually enters the quantum Hall regime. In this regime, the Hall resistance R_xy exhibits quantized steps, i.e., R_xy remains constant at a platform value over a certain magnetic field range. At these platforms, the corresponding longitudinal resistance R_xx drops to zero, indicating that the participating hole carriers are in a dissipationless transport state. Figs. 4(a) and 4(b) display the magnetotransport curves at gate voltage V_g in the saturation region (at higher carrier densities), corresponding to transverse and longitudinal resistance, respectively. From Fig. 4(a), it is evident that as the strain increases (LS−SS−HS), the number of peaks in the transverse resistance significantly increases. This increase is due to the strain-induced separation of light and heavy hole subbands, resulting in more oscillation peaks at the same magnetic field scale. In Fig. 4(b), we observe that with the increase in strain, the scale of the Landau platforms corresponding to the same filling factor grows, and the positions of the platforms change. The nonlinear trend in platform positions is firstly due to strain altering the band structure of the 2DHG in the quantum wells. Secondly, the growth of quantum wells under different strains leads to coordinated changes in structural parameters such as quantum well thickness, depth, and barrier heights, which result in changes in carrier density in the saturation region. The carrier density directly influences the number and position of Landau platforms. Under the influence of multiple complex factors, the variations in the longitudinal resistance curves presented in Fig. 4(b) are observed.

As shown in Fig. 5(a), increased strain leads to a decrease in mobility, arising from the variance in scattering mechanisms across different strain quantum wells. At high densities in lightly strained quantum wells, the power-law index β_high = 2.18. Here, the remote impurity scattering is dominated by surface interface states. Due to the minimal separation of light and heavy hole bands in the light strain sample, this scenario is more susceptible to small-angle scattering from interface charges, with β_high > 1.5^[33]. In standard strain quantum wells under high-density conditions, the power-law index β_high = 0.40, primarily induced by internal interface roughness and background impurity scattering. The increased strain enhances interface roughness, making the system more prone to large-angle scattering effects, with β_high < 0.5. For heavily strained quantum wells at high densities, the power-law index β_high = 1.27, influenced by multiple scattering mechanisms. On one hand, increased strain elevates internal interface roughness, intensifying large-angle scattering; on the other hand, due to the shallowness of the quantum wells, significant small-angle scattering from interface state charges also plays a notable role, placing β_high between 0.5 and 1.5. With escalating strain, the percolation density gradually rises, driven mainly by two factors: 1) A reduction in quantum well thickness (from 37.0 to 24.1 nm), causing leakage of hole gas to adjacent blocking layers and thus increasing percolation density. 2) A decrease in well depth (from 22.7 to 15.1 nm), enhances the likelihood of quantum well tunneling to the surface. Additionally, heightened strain raises the quantum well barrier, strengthening the confinement of the two-dimensional hole gas, thereby exerting an opposite effect on percolation density compared to the first two factors.

To further validate the causes of percolation density and identify optimal quantum well structural design, we constructed a percolation density model based on first-principles and utilized the Schrödinger−Poisson equation to elucidate its intrinsic mechanisms. Fig. 6(a) displays a simplified structure of the quantum well transport model, while Fig. 6(b) depicts the potential energy band of the hole gas quantum well under gate-voltage-controlled modulation. The red regions in the figures intuitively represent critical components of percolation density: the density of carriers leaking from the quantum well into the blocking layer and the density of carriers tunneling to the sample surface. An integration operation on the wave functions of these two carrier components, using first-principles, established a longitudinal percolation density model. The independent variables of this model are primarily influenced by the following factors (as highlighted in green in Fig. 6 (b)): 1) QW width (QW thickness), 2) QW barrier height and strain, 3) QW depth.

First-principles simulation results at 250 mK are displayed in Fig. 7(a)^[34−36]. Fig. 7(a) reveals that with increasing strain, the barriers on either side of the quantum well rise, leading to an increase in the number of confined carriers within the quantum well, and subsequently a gradual reduction and saturation of percolation density. The calculations indicate that when the Ge component in the Si_1−xGe_x barrier layer is less than 0.7, there is no significant reduction in percolation density. Therefore, for quantum wells that require higher percolation densities, using barrier layers with a Si component over 0.3 does not effectively improve percolation density; Fig. 7(b) shows that as the thickness of the quantum well increases, the percolation density leaking into the blocking layer decreases gradually and saturates beyond a thickness of 30 nm. Fig. 7(c) indicates that when the well depth increases to 35 nm, the percolation density does not significantly decrease further, suggesting limited improvement in percolation density control for quantum wells deeper than 35 nm. Fig. 7(d) presents a contour map of quantum well depth and thickness, demonstrating a continuous decrease in percolation density with increasing well depth. Fig. 8 is extracted from our calculation on leakage current density caused by carrier tunneling. It reveals that when the quantum well depth exceeds 35 nm, changes in the energy bands initiate carrier tunneling in addition to carrier leakage, leading to an increase in leakage current density. As is well known, with the increase of quantum well depth, the influence of remote impurity scattering will continue to decrease, and the mobility will also be effectively improved. However, an increase in the depth of quantum wells can also lead to an increase in percolation density and a significant decrease in gate control capability. In order to improve the gate control capability of quantum dots and the confinement effect of quantum wells, and to find a balance point between structural parameters and conductivity, based on the previous calculations, we suggest that the maximum depth of quantum wells should not exceed 35 nm and the thickness should not exceed 30 nm. Within this range, appropriately increasing the well depth and thickness can effectively reduce the percolation density of the two-dimensional hole gas. Through thickness-depth contour map, we observe that changes in well thickness exhibit a step-like pattern in Fig. 7(d), while changes in well depth show a uniform increase, indicating that overall, well thickness has a stronger correlation with percolation density. However, when the quantum well thickness is thin (QW thickness < 20 nm), the quantum well depth shows a strong correlation with percolation density. As the quantum well thickness increases, the impact of well depth on percolation density gradually weakens. This suggests that in quantum wells with smaller thicknesses, due to the substantial leakage of carriers and increased tunneling rates to the surface, well depth becomes a key factor affecting percolation density. Therefore, in fabricating ultra-shallow quantum wells, appropriately increasing the well depth can effectively reduce percolation density. On the other hand, when the quantum well is thicker (QW thickness > 30 nm), the impact of well depth on percolation density weakens. This implies that in such cases, to reduce percolation density, the focus should be on reducing quantum well thickness.

The effective mass m* is first fitted to the background by multiply subtracting it and then fitted to the decay of the SdH oscillation using the following formula:

1ΔρxxΔρ0=Δρ/ρ0(T)Δρ/ρ0(T0)=ATAT0=Tsinh(αT0)T0sinh(αT),

where α=2πkBm*ℏeB, k_B is the Boltzmann constant, ℏ is the Planck constant, ρ₀ is the zero-field resistivity, and T₀ = 250 mK is the coldest temperature at which the oscillations were measured. The g−factor is calculated by the following expression:

2g*=2mem*×11+BS/BL,

where B_L is the magnetic field strength at the starting point of SdH, and B_S is the magnetic field strength at the starting point of Zeeman splitting. The quantum lifetime τ_q is extracted by fitting the SdH oscillation. From τt=μm*/e, we estimate the transport lifetime τt corresponding to a large Dingle ratio τt/τq^[37].

We finally obtained the effective masses (LS−0.073 m₀, SS−0.083 m₀, HS−0.091 m₀), in-plane g-factors (LS−11.09, SS−9.86, HS−8.30), and Dingle ratios (LS−20.22, SS−28.93, HS−33.33) for three types of samples from magneto-transport measurements conducted within the temperature range of 250 to 650 mK. The fitting function curve of amplitude changing with temperature is shown in Fig. 9. As summarized in Table 1, the in-plane g-factor decreases with increasing strain. Theoretically, an increase in strain should lead to larger band splitting and a rise in the g-factor^[38−41]. However, a decrease in well depth, leading to leakage of hole gas, along with increased scattering caused by strain, results in these three factors collectively producing a contrary trend of decreasing in-plane g-factor^[40]. The Dingle ratio increases with the intensification of strain, primarily due to increased surface roughness caused by the strain, leading to the accumulation of interface state charges and enhanced scattering. This intensifies the effect of large-angle scattering, as reflected in the increase in the Dingle ratio. This aligns with the power law index βhigh measured previously through the classical Hall effect, further validating the scattering mechanisms of the three structures.

The effective mass increases with strain, contradicting the band curvature model^[42] and not aligning with the traditional heavy hole single state density transport model^{[15, 43]}. Within a smaller compressive strain range, the separation of the light and heavy hole subbands is less than the energy from the Fermi level to the band top, resulting in transport carriers occupying both light and heavy hole subbands simultaneously. Therefore, when calculating the effective mass, it is necessary to consider the situation where the carrier state density occupies both light and heavy hole subbands. The following formula can be used to calculate the effective mass for this mixed transport state:

31m*=αmlh*+1−αmhh*,

where

4α=nlhnlh+nhh.

mlh* and mhh* represent the effective masses of light and heavy holes, while α represents the mixed proportion of light holes in the light and heavy hole sub bands. When only a single heavy hole state is involved in transport, the effective mass decreases with increasing strain. Considering a mixed state of light and heavy strain holes, as shown in Fig. 10, using a fixed proportion mixed state calculation model indicates that the effective mass increases with increasing strain. To more closely approximate actual transport structures, we transitioned from a fixed to a gradual mixing ratio, and the results show that within the effective strain range, the effective mass increases with increasing strain. This trend aligns with the low-temperature transport test results of our actual grown multi-strain quantum well structures. This mixed-state calculation model of light and heavy holes provides a new approach to addressing the changes in effective mass in strained quantum wells and provides a new idea for regulating the effective mass of carriers.

In summary, through systematic material growth and low-temperature transport experiments, we have comprehensively studied the effects of strain and material structure on the physical properties of quantum wells. We successfully grew quantum wells with strain ranging from −1.19% to −0.43% and mobility between 3.382 × 10⁵ and 7.301 × 10⁵ cm²∙V⁻¹∙s⁻¹. Fundamental parameters were extracted and compared across these different strained quantum wells. By constructing a quantum well structure model, we demonstrated the significant impact of well depth, well thickness, and strain on material percolation density. Additionally, we provided strategic guidance on how to correctly adjust structural parameters to grow low-percolation-density quantum wells with varying compressive strains suitable for quantum computing applications. To address the anomalous changes in effective mass with increasing strain, we proposed a new mixed heavy-hole and light-hole subband transport model. This study serves as a strategy for growing high-quality, complex strained quantum wells and offers valuable insights for the development of heterostructure 2D hole gas quantum wells with different strain levels.