Electrostatically defined Quantum Dots in a Si/SiGe Heterostructure

We present an electrostatically defined few-electron double quantum dot (QD) realized in a molecular beam epitaxy grown Si/SiGe heterostructure. Transport and charge spectroscopy with an additional QD as well as pulsed-gate measurements are demonstrated. We discuss technological challenges specific for silicon-based heterostructures and the effect of a comparably large effective electron mass on transport properties and tunability of the double QD. Charge noise, which might be intrinsically induced due to strain-engineering is proven not to affect the stable operation of our device as a spin qubit. Our results promise the suitability of electrostatically defined QDs in Si/SiGe heterostructures for quantum information processing.


I. INTRODUCTION
Electrostatically defined quantum dot (QD) structures are attracting increasing interest as building blocks for solid state based quantum information processing. In such structures, the electron spin decoherence time is crucial for coherent manipulation of spin qubits. Electron spin phenomena have already been investigated in QD structures in AlGaAs/GaAs heterostructures 1 . The hyperfine interaction of the electrons confined in such QDs with roughly 10 5 thermally fluctuating nuclear spins has been identified as a limiting decoherence mechanism for electron spin qubits in GaAs [2][3][4] . This problem can be addressed by manipulating nuclear spins in GaAs [5][6][7] or by choosing an alternative host material. Silicon (Si) as a host material, offers a promising path towards extending the electron spin coherence time compared to GaAs based qubits, because naturally composed Si-crystals contain only a fraction of about 4.7% of nuclear spin carrying isotopes 8 compared to 100% in GaAs. Since the hyperfine interaction strength is roughly proportional to the fraction of nuclear spin carrying isotopes, much longer coherence times are expected for Si. Furthermore, Si has a weaker spin-orbit interaction 9,10 and is not piezo-electric 11 .
In this emerging research field, open questions remain, such as the influence of Si/SiGe specific material properties on device performance and tunablility. In this contribution, we present a Si/SiGe heterostructure whose material properties can be precisely controlled in molecular beam epitaxy (MBE). The heterostructure contains a strain-induced high mobility two-dimensional electron system (2DES) and is equipped with metallic top gates. In the resulting device, we implement a double QD combined with a single electron transistor (SET) as a charge sensor, both tunable by the field effect. An important fundamental difference of Si-to GaAs-based structures is the considerably larger effective electron mass in the 2DES (m * e,Si = 0.19 · m e ≈ 3 · m * e,GaAs ). We discuss the direct consequences of a high electron mass which can be observed e.g. in form of a small Fermi-energy of the 2DES and low tunneling rates of electrons across electrostatic barriers. In our measurements, charge noise strongly affects a large scale stability diagram, but is a minor issue as long as gate voltages are changed only slightly. Thereby, we demonstrate stable operation of our double QD device and its suitability as a spin qubit.

II. MATERIAL AND SAMPLE DEVELOPMENT
Our double QD is electrostatically formed within a 2DES in a strained-Si quantum well (QW) of a MBE grown Si/Si 1−x Ge x heterostructure with x = 24 %. The heterostructure layout and composition is shown in figure 1(a). A layer doped by phosphorus gives rise to a maxium 2DES density of about 3.5 × 10 11 cm −2 and an electron mobility of 1.1 × 10 5 cm 2 (Vs) −1 in this wafer at the temperature T = 1.4 K. The biaxial tensile strain in the Si QW lifts the sixfold valley degeneracy of bulk Si. The energy of the two valleys in [0 0 1] growth direction is lowered by 230 meV below the conduction band edge of the surrounding Si 0.76 Ge 0.24 layers. From a one-dimensional self-consistent band structure calculation with nextnano++ 27,28 , we obtain an intravalley subbband spacing between the first two subbands on the order of 8 meV which is large compared to the Fermi energy of E F = 1.1 meV. This small Fermienergy, compared to typical GaAs heterostructures, is a consequence of the high effective electron mass in Si and the two-fold valley degeneracy.
The double QD is defined in a mesa fabricated by wet-chemical etching. Ohmic contacts are formed by diffusing Sb/Au into the heterostructure. Electric top-gates are fabricated by electron beam lithography and palladium (Pd) evaporation 29 . The Pd on the device surface pins the Fermi energy at about 750 meV 30,31 below the conduction band edge. This strong pinning is as consequence of surface states at the Pd-Si interface. The surface states bind most of the electrons otherwise remaining at the doping layer. Together with the high work function of Pd, this results in a large Schottky barrier. The latter helps to minimize leakage currents from biased gates into the heterostructure. Our double QD gate design has been adapted from comparable GaAs-based structures 32 . A nominally identical device to the one investigated in this work is shown in an AFM micrograph in figure 1(b). High frequency coaxial cables lead to the gates bL and bR on the sample surface while all other gates are connected via low-frequency wires. After cool-down to T 2DES ≈ 100 mK, samples from the studied wafer require weak illumination with a red LED in order to populate the 2DES. We find that even at zero applied bias, the mere presence of Pd on top of the Si cap layer completely depletes the 2DES underneath. This behavior has been observed before 33-35 and is mainly a consequence of the saturation of Si dangling bonds 36 and the related Fermi level pinning at the Pd-Si interface 37 at low doping concentrations. Consequently, positive voltages are typically applied to all gates in order to drive currents from ohmic contacts III or V to IV. An unintended electrical short between gates PL, bC, PR and bR forces these gates to be on the same electrical potential. We will refer to this potential as V bR in the following. As a consequence of the short, the inter-dot barrier, the energy levels and the tunnel barriers from both dots to the leads cannot be tuned independently.
Based on a 3D self-consistent band structure calculation performed with nextnano++, where we take into account the dot capacitances for the given gate geometry, we estimate the double QD occupation of The disappearance of I DQD below V bR ≤ 205 mV and the overall large effective resistance V SD /I DQD ≥ 60 MΩ for the stability diagram is in part caused by the large effective electron mass m * in Si-based 2DES since tunneling rates are exponentially suppressed as the mass of the tunneling particle is increased. However, in our device, the low I DQD is furthermore a consequence of the short between gates PL, bC, PR and bR. This short not only results in a strong capacitive coupling of V bR to the right, but also to the left QD and in a strong suppressing effect of V bR on the QD-lead tunneling rates. Furthermore, V bR can be expected to asymmetrically influence the tunneling rates of the left and right QD to its leads. We can observe the effect of asymmetric QD-lead tunneling rates in figure 2 in a larger current value along the charging lines of the right QD (marked by ⊲ in figure 2) compared to the charging lines of the left QD (marked by ⊳ in figure 2). Away from the triple points, current along the charging lines of the right (left) QD involves a first order tunneling process and a second order co-tunneling process in series. The current is roughly given by Here, Γ L and Γ R are the respective QD-lead tunneling rates and Γ iD is the inter-dot tunneling rate. The asymmetry energy ∆ separates the energies of the localized states with the electron being either in the left or in the right QD. As we observe I ⊲ DQD > I ⊳ DQD (where we use the realistic assumption ∆ < Γ iD ), which corresponds to Γ R < Γ L , the tunnel coupling between the right QD and its lead is weaker than the tunneling coupling between the left QD and its lead.  Electrons can tunnel resonantly from the right lead IV into the right QD followed by an elastic co-tunneling process via the Coulomb-blocked left QD into the left lead III. Note that inelastic co-tunneling processes are also possible, but do not change our qualitative argument. Dotted lines in figure 4(b) mark the single particle excitation spectrum of the right QD. The observed current steps and transconductance oscillations in figure 4(a) for V SD > 0 imply that the excited states of the right QD contribute separately to I DQD as depicted by arrows in figure 4(b). This also implies that the energy relaxation rate Γ E within the right QD is slow compared to the co-tunneling rates between the right QD and the left lead III. In figure 4(a), we resolve two excited states with a characteristic excitation energy of approximately 200 µeV. For V SD < 0, no excited states are observed along the charging lines of the right QD in figure 3(a). Here a co-tunneling process is followed by resonant tunneling from the right QD to the right lead IV as sketched in figure 4(c). However, energy relaxation in the right QD is fast compared to the slow tunneling rate Γ R . Hence, the missing excitation spectrum of the right QD for V SD < 0 not only confirms the previous finding Γ R < Γ L , but furthermore suggests Γ R ≪ Γ L ,Γ iD .

B. Charge Sensing
In an attempt to characterize the double QD in the few-electron regime, we use charge sensing 32 via a QD SET. The QD is located between gates bR, α R and xR. Current is measured from contact V to IV as illustrated by the dashed arrow in figure 1(b). The  The stability diagram of figure 6 shows the general tendency that the appearance of fluctuations strongly depends on V bR while V bL has almost no influence. This can be interpreted as a hint that the observed telegraph noise is not a general problem of the heterostructure, but is rather linked to the unintended short between the gates PL, bC, PR and bR, all lying on the same potential. Lateral leakage currents along the sample surface are likely to cause the short. These leakage currents can also trigger charge fluctuations which result in the observed telegraph noise.
In addition, the risk of vertical leakage currents is higher for strain-engineered Si/SiGe  figure 7(b). The distance between the solid lines is described by the function ∆E = (2∆) 2 + ( Γ id ) 2 + E C where 2∆ = (µ R − µ L ) is the asymmetry energy of the quantummechanical two-level system and E C is the classical charging energy which represents the electrostatic coupling between the two QDs 43,44 . In order to fit the charging lines in a stability diagram based on applied gate voltages, in addition a linear transformation via the lever arms α j i (compare section III A) and a rotation of the coordinate system is employed 44 . Assuming E C to be constant within a small range of applied gate voltages, we find best fits for E C ≈ 435 µeV and the tunneling rates Γ id in figure 7(b). The solid line in figure 7(b) is a fit curve based on the WKB approximation for the inter-dot tunnel coupling Γ id = Γ 0 · exp(−d m * e E B / ) ∼ exp(βV bR ), where we assume for simplicity a constant width d of the tunnel barrier and the barrier height E B = E 0 B − α B V bR and use α B V bR /E 0 B ≪ 1. The gate-barrier lever arm is defined by α B = E B /V bR . Then the scaling factor β depends on m * e , d and α B . From the fitting procedure we find β = 0.056 ± 0.023 mV −1 which corresponds to ∆V bR ≈ 40 mV that are required to change the tunneling rate by one order of magnitude.
This value is rather small compared to similar experiments with GaAs based double QDs 45,46 .
The observed strong dependence of the tunneling rate on the gate voltage can in part be attributed to the higher effective electron mass in Si.
The tendency for small tunneling rates which strongly depend on gate voltages has been independently observed for QD-lead tunneling in figure 7(a) and inter-dot tunneling in figure 7(b). It has the following two direct implications: Due to the small tunneling rates, transport spectroscopy in the few-electron regime is more difficult in Si compared to double QDs defined in GaAs because of much smaller currents. On the contrary, the strong dependence of the tunneling rates on gate voltage is a chance for experiments which require time-dependent tunnel barriers -as often the case in quantum information processing.

C. Pulsed Gate Experiments
Spin based quantum information processing requires fast initialization and manipulation of the spins in a double QD. We have combined charge sensing with pulsed gate operation 47 to demonstrate, as a first step, switching between two charge configurations. we therefore expect to find two copies of the stability diagram shifted according to the pulse direction and amplitude. This can be seen in figure 8(b) and (c) where the pulses were applied to gate bL (b) and bR (c) with amplitudes of ∆V bL = 2 mV and ∆V bR = 1.2 mV, respectively. Clearly, the charging lines split into doublets of parallel charging lines with the corresponding distance ∆V bL or ∆V bR . Due to the strong inter-dot coupling, the reconfiguration lines are rather broadened than split.
The low QD-lead tunneling rates Γ L and Γ R in the few-electron regime restricts our pulse repetition rates to no more than about 10 kHz. We have also performed pulse repetition rates up to approximately 5 MHz limited by our instruments in the regime of larger QD-lead tunneling rates.

IV. CONCLUSION
In summary, we have performed direct transport spectroscopy through a few-electron Si/SiGe double QD, charge-sensing with a remote single QD sensor and pulsed-gate measurements. We deduce material-specific implications for the implementation of double QDs and spin qubits. An important parameter influencing the transport properties of our QD devices is the comparatively large effective electron mass m * e in Si-based 2DES. It enhances the dependence of tunneling rates on gate voltage and correspondingly can cause overall low tunneling rates across electrostatic barriers. Additionally, the large m * e contributes to a small Fermi energy, together with the two-fold valley degeneracy. The combination of low tunneling rates and small Fermi-energies hampers linear response transport spectroscopy with a current flowing across a double QDs in the few-electron regime. However, these difficulties can be circumvented by smaller feature sizes in future devices. From another perspective, the relatively strong scaling of tunneling rates with gate voltage can be exploited to implement efficient tuning of tunneling rates by pulsing gate voltages with a limited amplitude.
As an alternative to transport spectroscopy, a spin qubit can also be operated at a constant overall charge of a double QD in combination with charge spectroscopy. Based on such measurements, we find QDs in our Si/SiGe heterostructure devices still exposed to more charge noise than mature GaAs-based devices. Yet, our experiments also demonstrate a promising tendency towards quiet operation of the double QD, when manipulating gate voltages only in a limited range. These results suggest a realistic path towards Si-based quantum information processing.
The key advantage for Si-based qubits is the reduced interaction of confined electron spins in Si with their volatile crystal environment that gives rise to a number of decoherence mechanisms. Phonon-mediated back-action of a remote charge sensor on a qubit, which has been observed in GaAs based QDs 48,49 , can be expected to be much weaker in Si.
Indeed, the electron-phonon coupling is reduced (e.g. no piezo-electricity) and the low Fermi energy reduces the band-width for phonon-mediated interaction 48 . Furthermore, the spin-orbit coupling is weak and the hyperfine interaction in natural Si crystals is reduced compared to GaAs. Most importantly, our results show that the presented device layout with the possibility of almost zero Overhauser field in recently realized isotopically purified 28 Si 2DES 29 makes Si-based QDs a promising candidate for spin qubits with coherence times much larger than those that can be realized in GaAs/AlGaAs heterostructures.

ACKNOWLEDGMENTS
Financial support by the Deutsche Forschungsgemeinschaft via SFB 631 and the "Nano Initiative Munich (NIM)" is gratefully acknowledged. We thank Daniela Taubert, Daniel