Scattering mechanisms in shallow undoped Si/SiGe quantum wells

Over the past three decades, advances in epitaxial growth and a better understanding of scattering mechanisms have led to a tremendous mobility enhancement in Si/Si_xGe_1−x two-dimensional electron gases (2DEGs). The implementation of Ge-graded buffer layers,¹ the use of thicker buffers and spacers,^2,3 and the optimization of the channel thickness and of the Ge content of the buffer layer⁴ have resulted in an increase of doped Si/SiGe electron mobilities from ∼2 × 10³ cm²/(V ⋅ s) at 2 K and a density of ∼3 × 10¹² cm⁻² in 1985⁵ to ∼5 × 10⁵ cm²/(V ⋅ s) at 0.35 K and a density of ∼7 × 10¹¹ cm⁻² in 1995.⁴ The development of atomic-layer-deposited (ALD) Al₂O₃ allowed for the fabrication of undoped heterostructure field-effect transistors (HFET)⁶ in which the mobility of Si/SiGe quantum wells was further improved to ∼2 × 10⁶ cm²/(V ⋅ s) at 0.3 K and a density of ∼1.4 × 10¹¹ cm⁻².⁷ This massive mobility increase opens up the possibility to fabricate and improve upon Si/SiGe nano-devices such as quantum wires^8,9 and quantum dots.^10–13 

When patterning nano-structures, a balance must be attained between high mobility, usually achieved in channels buried deep underneath the surface (∼ 500 nm), and a sharp confinement potential, which is sharper the shallower the channel is. Sharp confinement allows for fabrication of smaller and better defined nano-structures. On the other hand, higher mobility facilitates the exploration of the clean limit in nano-structures where the electron mean free path is much greater than their dimensions. Phenomena arising from electron phase coherence are also more easily observable in high mobility material due to the longer electron phase coherence length and thermal length.¹⁴ Low-temperature and high-frequency experiments on nano-devices also benefit from improved mobility as the electrical resistance, and thus Joule heating and RC delays, are reduced in these systems. As such, a better understanding of scattering mechanisms in shallow undoped Si/SiGe HFET would help optimize heterostructures for nano-structures by balancing the need of high mobility and shallow 2DEGs.

The study of scattering mechanisms in 2DEGs is a mature field of research where a tremendous amount of work has been performed. A common approach to experimentally determine the dominant scattering mechanisms in two-dimensional systems is to extract the power-law exponent from the density dependence of the mobility. Several experimental and theoretical studies have used similar techniques to analyze disorder in GaAs/AlGaAs structures,^15–21 Si MOSFETs,²² and doped^4,23–25 and undoped^26–28 Si/SiGe heterostructures.

With this in mind, we present in this work a systematic study of the depth dependence of scattering mechanisms in undoped shallow Si/SiGe quantum wells with channel depth ranging from 100 nm to only 10 nm away from the surface, the shallowest Si/SiGe device reported to date. The peak mobility of the devices decreases with increasing shallowness, from 1.35 × 10⁶ cm²/(V ⋅ s) for a 100 nm deep channel down to 9.9 × 10³ cm²/(V ⋅ s) for a 10 nm deep channel. For all devices, clear regimes where the mobility varies as a power-law function of the density (μ ∝ n^α) are observed, making comparison with theoretical predictions and identification of the dominant scattering mechanisms readily available.

The heterostructures used in this study were grown in an ultra-high-vacuum chemical-vapor-deposition system with SiH₄ and GeH₄ as precursors at a temperature of 550 ^∘C, similar to previous reports.^6,7 Prior to the growth, the 10 Ω ⋅ cm p-type Si (100) substrate was dipped in 10% HF solution and the growth chamber was pre-baked and coated with Si. A graded SiGe virtual substrate was first grown, reaching a 14% Ge composition and ∼1.4 μm in thickness. Subsequently, a ∼3 μm thick relaxed Si_0.86Ge_0.14 spacer layer was grown. Following this, a 20 nm thick strained Si quantum well was grown. Finally, a Si_0.86Ge_0.14 spacer layer and a 2 nm thick Si cap were grown on top of the structure. The purpose of the Si cap is to protect the SiGe spacer layer from oxidization. Exposure to ambient air partly oxidizes the Si cap, effectively creating a thin SiO₂ layer on the surface of the heterostructure. Four different spacer thicknesses were selected, creating a series of quantum wells buried ∼ 10 nm, 25 nm, 50 nm and 100 nm under the surface of the heterostructure. Cross-sectional transmission electron microscope (XTEM) images of all 4 heterostructures are presented in Fig. 1(a)-1(d), showing the actual width and depth of the quantum well and of the spacer layer.

In order to electrically measure the devices, HFETs were fabricated following a process flow similar to previous reports,^6,7 with the notable difference that the Ohmics contacts were defined using ion implantation of phosphorus, followed by an anneal at 625 ^∘C. Subsequently, 300 cycles of ALD Al₂O₃ were deposited at 200 ^∘C. Finally, a Ti/Au (100 Å/ 2000 Å) gate in the shape of a 300 μm wide Hall bar was deposited through e-beam evaporation. A schematic cross-section of the completed device is depicted in Fig. 1(e).

All devices were measured in a ³He cryostat at a base temperature of 300 mK using low-frequency (10-100 Hz) lock-in measurement techniques in a standard four-terminal geometry with a constant excitation current ranging from 10 to 50 nA. The HFETs operated in enhancement mode, with a positive voltage applied to the gate to induce a 2DEG. After a Hall measurement was complete for one density, the gate voltage was incremented, and a settling time of ∼ 45 s was waited out between each measurement. The mobility μ of the devices and their Hall density n are determined from the slope of the low-field Hall resistance and the Hall bar longitudinal resistance R_xx.

The 2DEG’s mobilities are calculated over the whole range of available densities in each device, and are presented as a log-log plot in Fig. 2. A key feature of this set of data is the strong mobility decrease with increasing shallowness of the 2DEGs. Indeed, a maximal mobility of ∼1.35 × 10⁶ cm²/(V ⋅ s) is achieved in the 2DEG buried 100 nm underneath the surface (blue curve) while the maximal mobility of the 2DEG located 10 nm beneath the surface is only 9.9 × 10³ cm²/(V ⋅ s) (black curve). For each curve in Fig. 2, the mobility behaves as a power-law function of the density past a density threshold, as can be observed from the linearity of the log-log plot and highlighted by dotted lines. This density threshold occurs at different densities for each device, occurring at higher densities for shallower quantum wells and preventing us to study the low density regime in shallower 2DEGs. Below the density threshold, the mobility decreases more rapidly with decreasing density than above it. This is a regime dominated by the interplay of strong disorder and electron-electron interactions,²⁹ and studying this regime goes beyond the scope of this work. Above the transition threshold, each device other than the shallowest one exhibits two different power-law regimes, denoted by the gray and orange (or green for the deepest 2DEG) dotted lines in Fig. 2. The power-law exponents α were extracted in all regimes and are summarized in Table I.

It is well established that, at sufficiently low temperatures (T ≲ 4.2K) so that phonon scattering is negligible, and when a single subband is populated in the 2DEG, the two dominant scattering mechanisms are scattering from charge centers and interface roughness scattering. Interface roughness scattering causes the mobility to decrease with increasing density^7,30–32 and is thus extremely unlikely to be the dominant scattering mechanism in our devices, except near the highest achievable densities in the 100 nm deep device (green dotted line). Scattering from charge centers could arise from many sources in our devices: background impurities in the Si quantum well, background impurities in the SiGe barriers, trapped charges at the Si/SiGe (quantum well / spacer) interfaces, trapped charges at the cap/spacer interface, trapped charges at the amorphous SiO₂/Si interface, charges trapped at the SiO₂/Al₂O₃ interface, or charges trapped at the gate/insulator interface.

Since we observe a very strong dependence of the 2DEG’s mobility as a function of its depth, we can rule out all charged impurity sources whose strength remains constant regardless of 2DEG depth. Therefore, background impurities in the Si and the SiGe layers, as well as charges trapped at the quantum well/spacer interfaces are unlikely to be the dominant source of scattering in our devices. In addition, scattering from impurities inside or close to the quantum well would lead to a weak mobility dependence on the density, μ ∝ n^α, α ≤ 0.3,^30,31,33,34 which is not what we experimentally observe in our devices. Charges accumulating directly underneath the metal gate are also unlikely to be the dominant source of scattering in this device due to the strong shielding the metal gate itself would induce on such charges.

Therefore, the dominant source of scattering in our devices likely arises from charges trapped at the cap/spacer interface, the amorphous SiO₂/Si interface, or the SiO₂/Al₂O₃ interface. Experimentally speaking, since the Si cap layer is only 2 nm thick, any charges trapped at cap/spacer interface, at the amorphous SiO₂/Si interface or at the SiO₂/Al₂O₃ would all induce remote charge scattering, and would be extremely difficult to tell apart. As such, these charge centers will be lumped together and considered effectively the same for the rest of this work. The fact that experimental³⁵ and theoretical³⁶ studies have shown that a large density of charges (∼2 − 3 × 10¹² cm⁻²) gets trapped at or near the Al₂O₃ / Si interface of undoped Si/SiGe heterostructures gives further weight to this hypothesis. Within the Random-Phase-Approximation (RPA) with Thomas-Fermi screening, remote charge scattering is expected to lead to an exponent of α = 1.5.^30,37 Experimentally we observed α ∼ 2.3 in our data for the 10, 25 and 50 nm deep quantum wells. Previous experimental studies also reported similar exponents.^24,25 Low-density corrections to the RPA method have been suggested to explain this stronger dependence of the mobility upon the density. Indeed, if one goes beyond the RPA method and includes local-field corrections to the theoretical mobility calculations, exponent values of α ∼ 2.4^25,32,38 agreeing with experimental data are obtained.

In the device with the deepest 2DEG, the power-law exponent α ∼ 3 obtained at lower density (gray curve) is significantly larger than what is observed for shallower devices and what is theoretically predicted. Besides the Si/insulator interface being farther from the 2DEG, which reduces the importance of the interface, the other notable difference between this device and its shallower counterparts is the much lower density experimentally achievable. In this low density range, the RPA and scattering calculation may no longer be valid and theoretical understanding of the ground state in this regime remains controversial.²⁹ Studying this density range goes beyond the scope of this work.

The saturation of the mobility in the higher density regime (green curve in Fig. 2) for the deepest 2DEG is more readily understood. In this device, the remote charge centers are located farther away from the 2DEG, and thus do not limit its mobility as much. As the overall mobility increases, other mechanisms, such as interface roughness, may overcome remote charge centers as the dominant mobility limiting mechanism. This interpretation is consistent with experiments performed in deeper devices⁷ where interface roughness was indeed determined to be the dominant scattering mechanism. Furthermore, interface roughness scattering is characterized by a decreasing or saturating mobility with increasing density, which is observed at the highest achievable densities of this device, as shown in the inset of Fig. 2.

The transition from a regime with α ∼ 2.3 to a regime with a larger power-law exponent α ∼ 5 observed in the higher density regime of the 25 nm and the 50 nm deep 2DEGs (orange curves in Fig. 2) is not readily explained by any commonly studied scattering mechanism. In this case, rather than a change in the dominant scattering mechanism, we propose that the non-equilibrium properties of the devices are behind this abrupt increase in the density dependence of the mobility.

This concept, depicted in Fig. 3, relies on the fact that a 2nd quantum well is present in or near the Si cap layer of the heterostructure, albeit one with a much lower quality due to the presence of a large density of trapped charges nearby. Since the buried Si quantum well has a much larger width (∼ 20 nm) than the surface quantum well (< 2 nm), its ground state energy lies lower than the ground state energy of the surface quantum well. Therefore, as a positive voltage is applied to the gate, the buried well is populated by electrons first. Eventually, as the gate voltage is increased, the ground state energy of the surface quantum well drops below the Fermi energy. If the surface quantum well were disorder free, a freely flowing electron layer would immediately form at the surface, effectively shielding the buried quantum well from the gate and preventing electrons to accumulate further in it. However, due to the presence of a large amount of charges near the surface of the heterostructure, the critical density for a freely flowing surface electron layer is high and no surface 2DEG can form. Therefore, electrons keep accumulating in the buried quantum well and the system is not in thermal equilibrium with its Ohmic contacts and gate bias, although it is in a metastable state which can be probed by electron transport measurements. This non-equilibrium charge distribution was discussed in detail in Ref. 39.

With a non-equilibrium charge distribution, electrons have a tendency to tunnel across the spacer from the buried quantum well to the surface quantum well to restore equilibrium. As previously suggested,³⁹ the tunneling occurs following a direct Fowler-Nordheim process in these structures. This Fowler-Nordheim tunneling depends exponentially on the height of and the electric field across the tunnel barrier. The strong exponential dependence implies a separation of time scales. When the electric field across the SiGe spacer is higher than a critical value, which has been experimentally tabulated for a series of barriers with various barrier heights in Ref. 39, this tunneling process proceeds rapidly and brings electrons from the buried quantum well to the surface. With each electron transfer the electric field across the spacer weakens, and the tunneling slows down. A metastable state is then reached when the rate of tunneling to the surface becomes so slow that very little change in electron density occurs in a typical time scale of a transport experiment.

If the surface quantum well were pristine, the height of the tunnel barrier would be identical across the whole 2DEG area and the density versus gate voltage (Fig. 4(a)) would sharply transition from a linear relationship to a saturation regime. The smoother transition we observe in our experiment is attributed to the trapped charges near the interface, making the height of the tunnel barrier, or in turn the rate of tunneling, spatially variable. Therefore, as the gate voltage is increased, a fraction of the 2DEG area allows fast tunneling locally while only slow tunneling occurs in the rest of the 2DEG area. This results in a more uniform tunnel barrier across the whole area after fast tunneling processes settle, and implies that the charges tunneling from the buried 2DEG to the surface quantum well effectively screen the potential landscape seen by the buried 2DEG. If enough charges were to tunnel from the buried quantum well to the surface quantum well, the density of the surface quantum well would eventually overcome the mobility edge threshold density. At this point, charges would freely flow through the Ohmic contacts from the buried 2DEG to the surface quantum well, thermal equilibrium across the device would be restored and a sharp drop in the density of the buried 2DEG should be observed. This non-equilibrium model was in fact first brought forth to explain this last phenomenon.⁴⁰ In our devices however, gate leakage prevented us to reach this last regime.

We propose that this smoothing of the potential at the surface is at the origin of the stronger power-law exponent observed in our experiment. Fig. 4(a) shows the dependence of the density upon gate voltage in the 50 nm deep and the 100 nm deep 2DEGs. Focussing on the 50 nm deep trace, the loss of linearity in the density versus gate voltage relationship, concomitant with the change in the value of the exponent α in Fig 2, gives further weight to this model. It is also a direct sign that some of the charges capacitively induced by the gate in the buried 2DEG move away from it, migrating to the surface quantum well. This curve also gives us an estimate of the density of charge that migrates to the surface quantum well by considering the departure of the measured density from its extrapolated linear dependence. We would like to point out that, while tunneling never completely stops in these devices, the tunneling rate in the metastable state is of the order of a few electrons per cm² s,³⁹ making it virtually undetectable. Any stronger tunneling events have already occurred during the settling time that is waited before each measurement. The stability of the 2DEG density can be verified from the linearity of the low-magnetic field Hall measurement, as presented in Fig. 4(b). The linearity of this trace confirms that any relaxation process still occurring after the settling time occurs on a timescale much longer than the measurement timescale.

Quantitative support that a layer of shielding charge may be responsible for the observed large α values results from simulation. We first perform Monte-Carlo simulations to calculate the mobility-density dependence with the standard formalism for scattering off remote charged impurities.³³ We consider a 1μm × 1μm area with charged impurities placed randomly at a distance d (d represents the SiGe spacer thickness) above the 2DEG area with a fixed impurity charge density. The resulting potential at the Si quantum well is then used to compute the mobility-density dependence. The results of twenty different random impurity distributions are used to obtain an average mobility and standard error bars. We have simulated the mobility-density dependence for impurity concentrations between 5 × 10¹¹ cm⁻² and 5 × 10¹² cm⁻² and obtained the best fitting with experimental data for an impurity charge density of 2 × 10¹² cm⁻². This number is of the same order of magnitude of what has been reported in the literature.^28,35,36 We obtain an exponent of α ∼ 1.6, thus confirming the validity of our numerical method within the RPA.³³ To model the non-equilibrium screening effect, we add a number of negative charges based on the difference between the extrapolated linear portion of the curve in Fig. 3(a) and the curve’s actual value. Each charge is added at a location closest to the global potential minimum. This selective placement of “shielding charges” captures the effect of electrons migrating to the surface quantum well. Once all the shielding charges have been placed, the mobility is then calculated as above. We note that simply decreasing the number of scattering centers does not lead to the observed mobility enhancement. The selective placement of charges introduces spacial correlations in the disorder potential, which was shown to be at the origin of the extremely high mobilities observed in modulation-doped GaAs/AlGaAs 2DEGs.⁴¹ 

The mobility as a function of 2DEG density is computed for depth d = 50nm, and the results from 20 random realizations are shown as a solid green line in Fig. 2. For densities above 1.8 × 10¹¹cm⁻², which marks the onset of charge migration, the power law exponent α ∼ 10 is larger than the value (∼ 5) found in the experiment. We attribute this over-estimation of α to the globally-optimal placement of each shielding charge, which results in a larger screening than a more realistic case where shielding charges get stuck in local energy minima. Below 1.8 × 10¹¹cm⁻², α ∼ 1.6 under-estimates the observed value (∼2.1), which we attribute to the limitations of the RPA method used by our model, and the omission of other effects such as temperature dependent screening and metal gate screening. Overall, the simulations give qualitative agreement with the data and support the claim that large α values may be due to non-equilibrium interface charge.

To verify the quality of the HFETs presented in this work, R_xx measurements of all devices taken near the highest achievable density are presented in Fig. 3(b). The clear observation of a zero value of resistance within 0.2% of the lock-in amplifier full scale at filling factor ν = 4 for the device with the shallowest 2DEG, and up to ν = 8 in devices with deeper 2DEGs, are a clear sign that a single high quality 2DEG has been measured in each device. Another sign of the high quality of the material is the presence of low-field Shubnikov De-Hass (SdH) oscillations, starting at 0.76 T for the shallowest device and starting between 0.25 and 0.35 T for the deeper devices. This can clearly be observed in Fig. 3(c) where a zoom-in on the low magnetic-field R_xx data for the 25 nm deep 2DEG is presented. From the onset of these oscillations, we can estimate the quantum scattering time τ_q. At this point, the disorder broadening Γ = ħ/2τ_q of the landau level is approximately equal to the cyclotron energy ħ ω_c = ħ eB_SdH/m^∗. Here, ω_c is the cyclotron frequency, e the elementary charge, B_SdH the magnetic field at which the onset of SdH oscillations is observed and m^∗ the effective electron mass.

Comparing this quantum scattering time to the transport scattering time τ_t, obtained from the mobility through μ = eτ_t/m^∗, we obtain Dingle ratios τ_t/τ_q ranging from ∼ 80 to ∼ 17 for the 3 deepest quantum wells and a Dingle ratio of ∼ 1.5 for the shallowest quantum well. The large Dingle ratios are indicative that the rate of large angle scattering, caused by background impurities located in the quantum well, is much smaller than the rate of small angle scattering caused by remote impurities,³⁰ confirming the high quality of the devices presented in this study and the hypothesis that mobility is limited by scattering from charges trapped near the heterostructure surface. The small Dingle ratio of the shallower device implies that the charge centers are in such close proximity to the 2DEG that they induce large angle scattering. Dingle rations calculated using the amplitude growth of the SdH oscillations (R_xx ∝ e^{−π/τ_qω_c})¹⁸ shows good agreement with the ratios calculated from the onset of the SdH oscillations, as shown in Table I.

In conclusion, we have performed a study of the density dependence on the mobility and of the dominant scattering mechanisms in increasingly shallow high mobility undoped Si/SiGe HFET. 2DEGs as shallow as 10 nm underneath the surface with a mobility up to 9.9 × 10³ cm²/(V ⋅ s) were successfully measured. The minimal achievable density in the HFET increases with decreasing spacer thickness. Nevertheless, regimes with mobility in excess of 1 × 10⁵ cm²/(V ⋅ s) at densities of ∼2 × 10¹¹ cm⁻² were achieved in 2DEGs 50 and 25 nm under the heterostructure surface, making such devices perfectly suited for the fabrication of nano-devices. The dominant scattering mechanism in these devices was identified to be scattering off the remote charge centers accumulating near the Si/Al₂O₃ interface, identified by the power-law exponent α ∼ 2.3. It was also shown that, at high enough density, the devices are driven out of equilibrium. This process effectively smooths out the spatial impurity energy potential and leads to a dramatic increase of the power-law exponent to α ∼ 5. Understanding of the dominant scattering mechanisms in shallow high-mobility undoped Si/SiGe quantum wells is a crucial step towards future development of Si compatible nano-devices.

This work has been supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy (DOE). This work was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. DOE, Office of Basic Energy Sciences, user facility. Sandia National Laboratories is a multi program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. DOE’s National Nuclear Security Administration under contract DE-AC04-94AL85000. The Si/SiGe heterostructures were prepared by NTU and supported by the Ministry of Science and Technology (103-2112-M-002-002-MY3 and 103-2622-E-002-031).