We report the design, the fabrication, and the magneto-transport study of an electron bilayer system embedded in an undoped Si/SiGe double-quantum-well heterostructure. Combined Hall densities (nHall) ranging from 2.6 × 1010 cm−2 to 2.7 × 1011 cm−2 were achieved, yielding a maximal combined Hall mobility (μHall) of 7.7 × 105 cm2/(V ⋅ s) at the highest density. Simultaneous electron population of both quantum wells is clearly observed through a Hall mobility drop as the Hall density is increased to nHall > 3.3 × 1010 cm−2, consistent with Schrödinger-Poisson simulations. The integer and fractional quantum Hall effects are observed in the device, and single-layer behavior is observed when both layers have comparable densities, either due to spontaneous interlayer coherence or to the symmetric-antisymmetric gap.
In recent years, much interest has been shown in the development and improvement of Si/SixGe1−x heterostructures. Stemming from the possibility of combining band-structure engineering with Si technology compatibility,1 increasing efforts have been made to improve the electron and hole mobility in Si/SiGe heterostructures, ultimately attempting to improve upon the standard Si/SiO2-based devices.2 Recently, mobilities above 1.6 × 106 cm2/(V ⋅ s) have been reported in undoped Si/SiGe heterostructure field-effect transistors (HFETs),3,4 which have been used to study the fractional quantum Hall effect of Si electrons.5
Despite the rapid improvement in Si/SiGe heterostructures quality and the increasing number of experimental studies on the subject, little attention has been paid to Si/SiGe bilayer systems. Two-dimensional (2D) holes in Si/SiGe double quantum wells have been previously investigated,6 principally in attempt to improve the quality of room-temperature metal-oxide-semiconductor-field-effect transistors7–9 and to study g-factor tuning.10,11 Notwithstanding, high mobility electrons in Si/SiGe bilayers have been largely ignored, due to difficulty in material growth, leaving the fields of strongly correlated bilayers systems12–14 and quantum computing in bilayer quantum Hall droplets15,16 uncharted in this material.
With this in mind, we present in this letter the design of an undoped Si/SiGe heterostructure for hosting an electron bilayer system, and magneto-transport measurements performed in a HFET fabricated on this bilayer. Hall densities ranging from 2.6 × 1010 cm−2 to 2.7 × 1011 cm−2 were achieved in this device, with a maximal Hall mobility of 7.7 × 105 cm2/(V ⋅ s) reached at the largest electron density. The crossover density (ncrossover), the electronic density required to populate the top quantum well of the heterostructure, was determined to be ∼3.3 × 1010 cm−2, matching the value obtained from Schrödinger-Poisson (SP) simulations quite well. Also, the high quality of this bilayer structure allowed a clear observation of the integer and fractional quantum Hall effects.
While a simple modulation-doping scheme produces single-layer Si electron systems consistently, an electron bilayer Si/SiGe structure is difficult to grow with the same technique, due to surface segregation of the intentional dopants intended for supplying electrons to the bottom quantum well. Undoped heterostructures do not require intentional doping, but rely on a gate to capacitively induce carriers, and hence, allow us to work around this technical difficulty. However, since the gating is asymmetric, with the gate only active from the top side, the structure has to be carefully engineered to allow for the population of both quantum wells simultaneously.
The simplest case of two quantum wells with equal width does not host an electron bilayer. The ground state energies of the two wells are the same and the top quantum well screens the bottom one preventing it from being populated. Therefore, the two quantum wells need to be asymmetric, with the top quantum well narrower than the bottom one, so that the electrons first fill the bottom quantum well until the ground state energy of the top well drops below the Fermi energy and electrons begin to fill the top well. The SiGe interlayer barrier also needs to be thin so that the voltage drop across the barrier does not cause the energy crossover to occur too early, leaving a density of electrons in the bottom well too low for transport experiments. This behavior can be seen clearly in Figs. 1(a)–1(c), which corresponds to the simulated band diagram and electron density distribution before the crossover, right after it, and at comparable layer densities for the heterostructure design selected in this work. Additional details about the heterostructure and the simulations will be presented later.
From the considerations discussed above, the double-quantum well structure was designed to contain a 5 nm top quantum well, a 2 nm Si0.86Ge0.14 barrier, and a 15 nm bottom quantum well. Similarly to previous work,3,4 the heterostructure was grown in an ultra-high-vacuum chemical-vapor-deposition system with SiH4 and GeH4 as precursors at a temperature of 550 °C. 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 Si0.86Ge0.14 spacer layer was grown. Following this, the (strained) double quantum well structure was grown. Finally, a ∼100 nm thick Si0.86Ge0.14 spacer layer and a 2 nm Si cap were grown on top of the structure. Figure 2(a) shows a cross-sectional transmission electron miscroscope (XTEM) picture of the grown heterostructure. The measured thickness of the top (bottom) quantum well is 5.7 nm (19.7 nm) and the SiGe barrier is 2.0 nm thick, as designed. The difference between the designed and measured thicknesses of the silicon quantum wells could be attributed to the effect of temperature deviation on the growth rate.
In order to electrically measure the bilayer, a HFET was fabricated following a process flow similar to previous reports.3,4 Ohmic contacts were defined using an Au/Sb/Au (150 Å/40 Å/4000 Å) metal stack deposited through e-beam evaporation and annealed at 450 °C for 5 min in forming gas, following standard photolithography procedures. Subsequently, 1000 cycles of atomic-layer-deposited 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, following standard photolithography procedures. A schematic cross-section of the completed device is depicted in Fig. 2(b).
Based on the device parameters and the measured layer thicknesses, an iterative self-consistent SP simulation was performed. At low voltage bias and low temperature, it was previously shown that electrons do not accumulate at the oxide/semiconductor interface since the charge distribution never reaches thermal equilibrium.17 To simulate this non-equilibrium behavior with an equilibrium SP simulation, the bandgap of the upper half of the SiGe top spacer was artificially increased, thus preventing electron accumulation at the oxide/semiconductor interface. The simulated top well, bottom well, and total electron densities as a function of gate voltage are shown in Fig. 3(a). The gate voltage (Vg) values used in the simulation were shifted by a constant offset such that the starting point of the experimental data, to be presented below, and of the simulation coincide. The crossover density from simulation is ncrossover ∼4.2 × 1010 cm–2, which is an experimentally accessible density and should be detectable through magneto-transport measurements.
All the measurements reported in this letter were performed in a 3He cryostat with a base temperature of 400 mK using low-frequency (13 Hz) standard lock-in measurement techniques in a standard four-terminal geometry with a constant 80 nA excitation current. The HFET device was operated in enhancement mode, with a positive voltage applied to the gate. Prior to the initial operation of the device, a large negative bias (∼−5 V) was applied to evacuate negative charges trapped at the dielectric layer interface, improving the uniformity. Four contacts available in a Hall configuration were used for both mobility and density measurements throughout the experiment.
The Hall density (nHall) of the device is determined using the slope of the low magnetic field Hall resistance, and its dependence upon gate voltage is also presented in Fig. 3(a). If the mobility in both layers is significantly different, nHall can depart from the sum of both well densities (ntotal = ntop + nbottom).18 For this device, however, the Hall density matches the simulated total density, up to a constant gate voltage offset, and varies linearly with gate voltage until Vg = 1.40 V, where the Hall density of the device saturates at nHall = 2.7 × 1011 cm−2 (not shown in Fig. 3(a)). From the simulation, it is found that the wider (bottom) quantum well populates first until it reaches a saturation density of n = 4.2 × 1010 cm−2 at Vg = 0.73 V. As the gate voltage bias keeps increasing, the narrower (top) well starts to populate, eventually passing the bottom well density at Vg = 0.85 V.
Fig. 3(b) shows our main result, the density dependence of the HFET's Hall mobility (μHall, calculated using the van der Pauw method). At the lowest densities, the mobility increases with increasing Hall density until nHall ∼3.3 × 1010 cm−2 (corresponding to Vg = 0.67 V). At slightly higher densities, the mobility starts decreasing with increasing density. Such a mobility drop is characteristic of inter-layer (or inter-subband) scattering,18 and is a clear indication that the second well of the bilayer structure is starting to populate with electrons. While the ncrossover (taken to be the density at the mobility peak) slightly differs from the simulated results, it, nonetheless, confirms that a bilayer system is created in the device. The mobility drops from 2.1 × 105 cm2/(V ⋅ s) at the peak to 1.7 × 105 cm2/(V ⋅ s) at its minimum. This change of ∼20% is comparable to what was observed in GaAs/AlGaAs.18 At nHall ≳ 4.5 × 1010 cm−2, the mobility starts increasing again with increasing Hall mobility and peaks at μHall ∼ 7.7 × 105 cm2/(V ⋅ s) for the maximal achievable density.
Having established the formation of a bilayer system, we will now present quantum Hall results in the single and the bilayer regimes. Fig. 4(a) shows the Hall resistance of the device as a function of magnetic field at a gate voltage Vg = 0.66 V, where a single layer is populated with nbottom ∼3.0 × 1010 cm−2. In this figure, we can clearly observe a quantum Hall plateau developing at ν = 1 and quantized within 2% of h/e2, as expected for the quantum Hall effect in a single-layer 2D electron system. The ν = 2 quantum Hall plateau is not clearly observed in this configuration, most likely due to the low magnetic field (B ∼ 0.6 T) at which the plateau should develop.
As the device is configured to higher densities (into the bilayer regime), additional quantum Hall plateaux become visible in the Hall resistance, as depicted in Fig. 4(b), where the Hall resistance is shown as a function of magnetic field for Vg = 0.85 V. Here, nHall ∼ 7.9 × 1010 cm−2 and the two layer densities are almost matched based on the simulation or within 50% of one another based on the experimentally determined ncrossover. The integer filling fractions ν = 1, 2, and 4 are quantized within 2% of their respective theoretical resistance and the fractional filling faction ν = 2/3 is quantized within 5% of its theoretical resistance. From the positions (in magnetic field) of these quantum Hall states, one can extract a density n = 7.9 × 1010 cm−2, virtually identical to the Hall density. This result indicates single-layer behavior rather than bilayer-behavior. Such an effect has been observed at high magnetic fields (B ≳ 0.9 T) in high-quality GaAs samples with mismatched electron19,20 and hole21–23 densities.
The electron bilayer system presented in this letter contains three kinds of degeneracy: spins, valleys, and layers. Each Landau band thus has a degeneracy of 8. The valley splitting at n = 7.9 × 1010 cm−2 is estimated to be ∼0.4 K.24 At 1.6 T (ν = 2), the spin gap is ∼2.2 K, without considering enhancement from many-body effects. The symmetric-antisymmetric gap ΔSAS, established by interlayer tunneling, is estimated to be ∼1.6 K from the SP simulation at matched densities. Since the valley splitting gap is the smallest, odd integer states arise from the valley degeneracy. The spin gap being larger than the interlayer tunneling gap, the ν = 2 state likely arises from interlayer effects. The ν = 2 state in our system thus corresponds to the ν = 1 state in GaAs bilayer systems, with either ΔSAS or spontaneous interlayer coherence at the origin of this state.25 In our device, the Coulomb energy EC = e2/ϵlB is ∼65 times larger than ΔSAS. Here, ϵ is the dielectric constant and is the magnetic length. Spontaneous interlayer coherence is expected to occur when EC ≫ ΔSAS and when the ratio of intra- to inter-layer Coulomb energy d/lB < 2.20,23 Here, d is the center-to-center separation between the two wavefunctions. Since d/lB ∼ 0.7 in our device, it appears that interlayer tunneling is limited and that the ν = 2 state arises from spontaneous interlayer coherence. However, due to the relatively large ΔSAS (more than an order of magnitude larger than in GaAs studies20,22,23), we cannot rule out that the symmetric-antisymmetric gap is at the origin of the emergence of the ν = 2 state in our device. Suppressing ΔSAS through either a thicker SiGe barrier, a larger Ge concentration in the barrier and/or back-side gating, would allow us to confirm the nature of this bilayer state.
In conclusion, we have fabricated an undoped Si bilayer system embedded in a Si/SiGe heterostructure. Using this HFET device, we directly showed the population of a second layer as the Hall density of the device was increased to ncrossover ∼3.3 × 1010 cm−2, matching simulation results. Hall densities ranging from 2.6 × 1010 cm−2 to 2.7 × 1011 cm−2 were achieved, yielding a maximal combined mobility of 7.7 × 105 cm2/(V ⋅ s) at the highest density. At comparable densities, no bilayer-like behavior is observed in the quantum Hall regime and the νtotal = 2 bilayer quantum Hall state is clearly observed, although further experiments and/or development in device fabrication and material growth will be required to assess its nature with certainty.
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, an Office of Science User Facility operated for the U.S. DOE Office of Science. 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 No. DE-AC04-94AL85000. The work at NTU was supported by the Ministry of Science and Technology (Nos. 103-2622-E-002-031 and 103-2112 -M-002-002-MY3).