Abstract
Although silicon is a promising material for quantum computation, the degeneracy of the conduction band minima (valleys) must be lifted with a splitting sufficient to ensure the formation of welldefined and longlived spin qubits. Here we demonstrate that valley separation can be accurately tuned via electrostatic gate control in a metal–oxide–semiconductor quantum dot, providing splittings spanning 0.3–0.8 meV. The splitting varies linearly with applied electric field, with a ratio in agreement with atomistic tightbinding predictions. We demonstrate singleshot spin readout and measure the spin relaxation for different valley configurations and dot occupancies, finding oneelectron lifetimes exceeding 2 s. Spin relaxation occurs via phonon emission due to spin–orbit coupling between the valley states, a process not previously anticipated for silicon quantum dots. An analytical theory describes the magnetic field dependence of the relaxation rate, including the presence of a dramatic rate enhancement (or hotspot) when Zeeman and valley splittings coincide.
Introduction
Silicon is at the heart of all modern microelectronics. Its properties have allowed the semiconductor industry to follow Moore’s law for nearly half a century, delivering nowadays billions of nanometrescale transistors per chip. Remarkably, silicon is also an ideal material to manipulate quantum information encoded in individual electron spins^{1,2,3}. This is a consequence of the weak spin–orbit coupling and the existence of an abundant spinzero isotope, which can be further enriched to obtain a ‘semiconductor vacuum’ in which an electron spin can preserve a coherent quantum superposition state for exceptionally long times^{4}.
In order to define a robust spin1/2 qubit Hilbert space, it is necessary that the energy scale of the twolevel system is well separated from higher excitations. In this respect, a major challenge for the use of silicon is represented by the multivalley nature of its conduction band. In a bulk silicon crystal the conduction band minima are sixfold degenerate, but in a twodimensional electron gas (2DEG), the degeneracy is broken^{5} into a twofold degenerate ground state (Γ valleys) and a fourfold degenerate excited state (Δ valleys), owing to vertical confinement of electrons with different effective mass along the longitudinal and transverse directions, respectively. Furthermore, the Γ valley degeneracy is generally lifted by a sharp perpendicular potential^{6,7,8,9} and the relevant energy separation is termed valley splitting (VS).
The VS depends on physics at the atomic scale^{10,11,12} (for example, roughness, alloy and interface disorder), and so it is not surprising that experiments have revealed a large variability of splittings among devices, ranging from hundreds of μeV^{5,13,14,15} up to tens of meV in exceptional cases^{16}. At present, the lack of a reliable experimental strategy to achieve control over the VS is driving an intense research effort for the development of devices that can assure robust electron spin qubits by minimizing multivalley detrimental effects^{17,18}, or even exploit the valley degree of freedom^{19,20} for new types of qubits.
Another crucial parameter to assess the suitability of a physical system to encode spinbased qubits is the relaxation time of spin excited states (T_{1}). Spin lifetimes have been measured for gatedefined Si quantum dots (QDs)^{21}, Si/SiGe QDs^{22,23} and donors in Si (ref. 24), reporting values that span from a few milliseconds to a few seconds. Furthermore, the dependence of the spin relaxation rate () on an externally applied magnetic field (B) has been investigated. Different mechanisms apply to donors and QDs, accounting for the observed B^{5} and B^{7} dependencies^{25}, respectively. In principle, (B) depends on the valley configuration and the details of the excited states above the spin ground state. However, until now, no experimental observation of the effects of a variable VS on the relaxation rate has been reported.
Here, we demonstrate for the first time that the VS in a silicon QD can be finely tuned by direct control of an electrostatic gate potential. We find that the dependence of the VS on vertical electric field at the Si/SiO_{2} interface is strikingly linear, and show that its tunability is in excellent agreement with atomistic tightbinding predictions. We demonstrate accurate control of the VS over a range of about 500 μeV and use it to explore the physics of spin relaxation for different QD occupancies (N=1, 2, 3). We probe both the regime where the VS is much larger than the Zeeman splitting at all magnetic fields and that where the valley and spin splittings are comparable. We observe a dramatic enhancement of the spin decay rate (relaxation hotspot) when spin and valley splittings coincide. To our knowledge, such hotspots have been predicted for relaxation involving orbital states^{26,27} (not valley states), but these are yet to be observed. We develop an analytic theory that explains the Bfield dependence of the relaxation rates and the details of the relaxation hotspot in terms of admixing of spinvalley states. This mechanism is seen to be significantly more prominent than the conventional spin–orbit hybridization^{28}.
Results
QD addition spectrum
Our device is fabricated using a multilevel gated metal–oxide–semiconductor (MOS) technology^{29}, and its architecture is depicted in Fig. 1a–d. A quantum dot is formed under gate P by applying a positive bias voltage to induce an electron accumulation layer. Strong planar confinement for the dot’s potential well is achieved by negatively biasing gates B, C1 and C2. A 2DEG reservoir is also induced by positively biasing gates R1 and R2, and the QD occupancy can be modified by inducing electrons to tunnel between this reservoir and the dot. The remaining gates, namely, SB1, SB2 and ST, are employed to define a singleelectron transistor (SET), capacitively coupled to the QD and used as a readout device. The high flexibility of our design would allow us to use the same device also as a (singlelead) doubledot structure by rearranging the gate bias (for example, dots can be formed under gates B and C2). However, in this work, we only present results relevant to the singledot configuration.
In order to characterize the addition spectrum of the QD, we make use of a technique previously developed for GaAsbased systems, which combines charge detection and gate pulsing^{30}. There is no direct transport through the singlelead QD and, therefore, addition/removal of charge is only detected via modifications in the SET current. In particular, charge transitions are detected as current peaks in the SET signal whenever the QD energy eigenstates come into resonance with the reservoir’s Fermi level. Note that the SET–QD coupling is merely capacitive (via C_{cpl}) and electrons do not tunnel between them. In order to maximize charge sensitivity in the detector, we employ dynamic voltage compensation^{31} on different gates, which makes our readout signal virtually unaffected by slow charge drifts and random charge rearrangements. A comprehensive discussion of the charge stability measurements can be found in Supplementary Note 1.
Figure 1e illustrates the addition energy spectrum for the first 14 electron additions to the QD. There is very little variation of charging energy (E_{C}) for high occupancies (E_{C}≈11 meV for N>9). However, by decreasing the electron number, the charging energy steadily increases, as expected when the dot size is significantly affected by the electron number. This evidently indicates that the fewelectron regime has been achieved. Most interestingly, the energy spectrum shows peaks for the addition of the fifth and thirteenth electrons. The extra addition energy needed for those transitions can be attributed to complete filling of the first and second orbital shells. As illustrated in Fig. 1f, this is consistent with the energy spectrum of twovalley 2D Fock–Darwin states^{32}, where the first and second orbital shells hold four and eight electrons, respectively. This confirms that we can probe the occupancy until the last electron. To our knowledge, such a clear manifestation of twodimensional shell structure has been observed before only in InGaAs dots^{33} and in Si/SiGe dots^{34}.
Spinvalley lifetimes
In order to measure the spin state of individual electrons in the QD, we use an energyselective readout technique^{35}. The readout protocol consists of three phases clocked by a threelevel pulsed voltage applied to gate P, which directly controls the dot’s electrochemical potential (see Fig. 2a). First, an electron of unknown spin is loaded into the dot, causing a sudden decrease in the sensor current. Next, the potential of the dot is lifted so that the Fermi level of the reservoir lies between the spinup and spindown states of the dot, meaning that a spinup electron can tunnel off the dot while a spindown electron is blocked. This is the read phase, during which the presence of a spinup state would be signalled by a current transient (spinup tunnels out and then spindown tunnels in) whereas a spindown electron would lead to no current modification. Finally, the dot’s potential is further lifted to allow the electron to tunnel off, regardless of its spin orientation. In Fig. 2b, singleshot traces for both spinup (in blue) and spindown (in green) detection are plotted. The longer the system is held in the load phase before performing a read operation, the more likely it is for the spinup excited state to decay to the spindown ground state. Thus, by varying the length of the load phase and monitoring the probability of detecting a spinup electron, we can determine^{35,24} the spin lifetime, T_{1}. In our experiments the B field is directed along the [110] crystallographic axis. A comprehensive discussion of both the spinup fraction measurements and the fitting procedure to evaluate T_{1} is included in the Supplementary Note 2. As shown in Fig. 2d, we observe a wide range of spin lifetimes as a function of magnetic field, with lifetimes as long as 2.6 s at the lowest fields studied, B=1.25 T. These are some of the longest lifetimes observed to date in silicon quantum dots^{25}.
A key focus of our experiment was to electrostatically tune the valley energy separation and measure relaxation rates in different valley configurations and QD electron occupancies. As we show below, our data definitively indicate that excited valley states have a critical role in the spin relaxation processes. We develop a theory to explain how changes in the VS affect the spinvalley state mixing and leads to the observed relaxation times.
As we detail in the following section, we have attained accurate gate control of the VS, allowing us to tune it over a range of hundreds of μeV. This permits us to conduct experiments in regimes where the VS (E_{VS}) is either larger or smaller than the Zeeman spin splitting (E_{Z}), depending on the magnitude of the magnetic field (see Fig. 2c).
Figure 2d presents measurements of spin relaxation rates as a function of magnetic field for two VS values at a fixed dot population of N=1. We start by examining a configuration where the valley separation is larger than the spin splitting at all fields (green data set). In other words, we operate in a regime for which
where g is the electron gyromagnetic ratio, μ_{B} is the Bohr magneton and B is the applied inplane magnetic field.
For E_{VS}=0.75 meV (green data in Fig. 2d), we observe a monotonic increase in the rate with respect to B that becomes increasingly fast as E_{Z} approaches E_{VS}. In our experimental conditions (B field parallel to [110]), the B^{5} dependences for known bulklike mechanisms in silicon^{36,37} should not apply, while predicted^{23,25,27,38} rates B^{7} do not explain the experimental data.
By decreasing the valley separation to E_{VS}=0.33 meV, we can achieve the condition where the Zeeman splitting matches or exceeds the VS. The red data in Fig. 2d illustrate the situation where inequality (1) only holds for B<2.8 T. When E_{Z}=E_{VS} (that is, for B=2.8 T), a spike in the relaxation rate occurs. Relaxation hotspots have been previously predicted to occur for spin relaxation involving orbital states in single and coupled QDs^{26,27,39,40}. To our knowledge, this is the first experimental observation of such a phenomenon.
In order to understand the relaxation mechanisms, we have developed a model that takes into account the perturbations in pure spin states due to spin–orbit coupling (SOC), yielding eigenstates that are admixtures of spin and valley states. The four lowest spinvalley states (see Fig. 2c) are defined as , , , . These states are considered to be only very weakly affected by higher excitations, such as orbital levels that are at least 8 meV above the ground state in our device^{41}. In Supplementary Note 3 we detail how mixing to a 2plike orbital state leads to a B^{7} dependence in and is, therefore, important mainly for high B fields (above the anticrossing point). At lower fields, the prominent mechanism is the spinvalley admixing, which we now discuss in detail.
The relaxation between pure spin states is forbidden because the electron–phonon interaction does not involve spin flipping. However, in the presence of interface disorder, SOC can mix states that contain both the valley and spin degrees of freedom, thus permitting phononinduced relaxation. Indeed, in the nonideal case of QDs with a disordered interface, roughness can perturb the envelope function of both valleys (otherwise identical for ideal interfaces) and allows one to assume nonzero dipole matrix elements connecting the valley states (see Supplementary Note 3), such as , , (for ideal interfaces these are nonzero only due to a strongly suppressed Umklapp process). By means of perturbation theory, we define renormalized excited states and that can relax to the ground state , as they have an admixture of the state of the same spin projection (see Fig. 2c). The details of the SOC Hamiltonian, H_{SO}, and perturbation matrix are reported in Supplementary Note 3. The leadingorder wavefunctions are given by:
where , , and is an expression involving the detuning from the anticrossing point, δ≡E_{VS}−E_{Z}, and the energy splitting at the anticrossing:
where β_{D} (α_{R}) is the Dresselhaus (Rashba) SOC parameter, ħ is the reduced Planck’s constant and m_{t}=0.198m_{e} is the transverse effective electron mass.
By evaluating the relaxation rate via the electron–phonon deformation potentials (proportional to the deformation potential constants, ), we obtain the rate below the anticrossing as:
where the purevalley relaxation rates are (for longitudinal and transverse phonons):
where ρ is the silicon mass density, v_{σ} is the speed of sound in silicon, , are the angular integrals, and ΔE_{v′v} and r are the energy difference and the dipole matrix element relevant to the transition, respectively (see also Supplementary Note 3). The experimental condition for which the hotspot occurs (that is, E_{VS}=E_{Z}) is modelled as an anticrossing point for the mixed states and . At that point, spin relaxation is maximized and approaches the valley relaxation rate, as δ→0 in equation (5).
Above the anticrossing (that is, E_{VS}<E_{Z}), the relevant relaxation transitions are and (the subsequent decay is in the form of a fast intervalley transition, and is therefore neglected). The analytical formulations of these contributions read:
The dashed lines in Fig. 2d show the calculated relaxation rates relevant to the two experimental values of E_{VS} discussed, also including B^{7} contribution from SOC mixing with the higher orbital state (see Supplementary Note 3). We use dipole matrix elements as a single free parameter by assuming . A leastsquare fit to the experimental data is performed by fixing the SOC strength to (β_{D}−α_{R})≈45–60 m s^{−1} (justified by the high electric field ≈3 × 10^{7} V m^{−1}, see Wilamowski et al.^{42} and Nestoklon et al.^{43}). The fit then extracts a dipole size r≈1–2 nm for both values of E_{VS}.
The good agreement between the calculations and the experiment, as well as the presence of a hotspot at the point of degeneracy between Zeeman and VS, provide strong evidence of our ansatz that the spin relaxation is predominantly due to a new mechanism: that of mixing with the excited valley states via Rashba/Dresselhauslike SOC in the presence of interface disorder.
Both the splitting at the anticrossing, equation (4), and the intervalley relaxation, equation (6), depend crucially on the size of the dipole matrix element, r, predicting a fast phonon relaxation of ≈10^{7}–10^{8} s^{−1} for r=1–3 nm, at the hotspot of Fig. 2d. This confirms our core findings that when spinvalley states anticross, the intervalley rates are fast for these samples, with the only available relaxation mechanism being the intervalley decay. We point out that these relaxation rates are expected to be sample/materialdependent, given the effect of interface disorder on valley mixing.
We now examine the case where N=2 electrons, and investigate the dependence of the relaxation rate on the magnetic field at a fixed VS (E_{VS}=0.58 meV). We note that the energy levels accessible for loading the second electron in the dot, when the N=1 spindown ground state is already occupied, are either the singlet (S) or the two lower triplets (T_{−}, T_{0}), whereas the higher triplet (T_{+}) would require a spin flip and is, therefore, not readily accessible (see Fig. 3a). In general, for triplet states, the antisymmetry of the twoelectron wavefunction requires one electron to occupy a higher energy state. For our multivalley QD (Fig. 1f), this requirement is fulfilled when the two electrons occupy different valley states (see Fig. 3a). For low fields, the ground state is S and the triplets have higher energies. This results in excited states (triplets) that extend over two valleys and relax to a singlevalley ground state (singlet).
As the magnetic field is increased, S and T_{−} undergo an avoided crossing (B≡B_{ST}), and then T_{−} becomes the ground state. We adjust the levels of our pulsed readout protocol so that during the load phase only S and T_{−} are below the reservoir’s Fermi energy, whereas, during the read phase, the Fermi energy is positioned within the singlet–triplet (ST) energy gap. As a consequence, for B<B_{ST} (B>B_{ST}) a T_{−} (S) state would be signalled with a current transient, and relaxation rates can be extracted as for the N=1 occupancy. The experimental relaxation rates in Fig. 3b show a strongly nonmonotonic behaviour, approaching an absolute minimum at the anticrossing point (B_{ST}=5 T). The trend is strikingly symmetric, as can be appreciated when is plotted against the detuning energy, as shown in the right inset. This symmetry is reflected in the QD energy spectrum (Fig. 3a), as far as the detuning δ is concerned. For B<B_{ST}, the ST energy gap decreases with increasing B, resulting in slower relaxation rates. By contrast, for B>B_{ST}, the ST energy gap increases with increasing field, and so does the relaxation rate.
As opposed to the oneelectron case, we note that the twoelectron eigenstates anticrossing leads to a minimum in the relaxation rate (coldspot), defined by a splitting at the anticrossing, , of the same order as that of equation (4) (see Fig. 3a and Supplementary Note 3). The occurrence of this minimum does not strictly depend on the nature of the states involved in the decay (spinlike, valleylike, orbitallike or admixtures). It is due to the fact that the avoided crossing takes place between the ground and the first excited state, whereas for the case N=1 it involves the first and the second excited states without affecting the ground state.
To model the twoelectron case, we build the wavefunctions for S and T_{−} from the singleparticle states by considering the Coulomb interaction as a perturbing averaged field. The corresponding states are defined as , . Next, the additional perturbation given by SOC leads to renormalized eigenstates that are admixtures of singlet and triplet:
being similar forms to equations (2) and (3). As we show in Supplementary Note 3, by evaluating the electron–phonon Hamiltonian matrix element for the transition between these states, one finds that it coincides in its form with its oneelectron counterpart for . Therefore, we can conclude that the corresponding relaxation rate, , has the same functional form as those derived in equation (8), although the matrix elements for the two cases will be different (see Supplementary Note 3). The dashed lines in Fig. 3b represent the calculated rates that are fitted to the experimental data similarly to the case where N=1. Once again, the model convincingly reproduces the main features of the experimental trend, in particular the rates for fields away from the anticrossing, together with the symmetry of the characteristics with respect to B_{ST}. Further work may be needed to improve the fit in the vicinity of the anticrossing point.
We also measured the relaxation rates for N=3 electrons. When the QD occupancy is set at N=2, the lower valley is fully occupied and for low B fields the ground state is a singlet. In this condition, the readout protocol is adjusted to probe spin relaxation within the upper valley upon loading/unloading of the third electron. By keeping E_{VS}=0.58 meV and using the same methodology described before, we measure relaxation rates for the third electron spin state. We find that there is no significant difference between the spin relaxation rates for N=3 and N=1, as shown in the left inset of Fig. 3b. Two main conclusions can be drawn from this. First, we can infer that the effect of electron–electron interactions on the multivalley spectrum may be negligible^{44}, which is plausible. Indeed, for valley threeelectron states, two electrons are just ‘spectators’, so that the remaining electron establishes the same energy level structure as in Fig. 2c, and the Coulomb corrections do not affect the VS. Second, as we report in Yang et al.^{41}, in small QDs for higher occupancies a significantly reduced energy separation between the ground state and the first excited orbital state is observed. This would introduce a nonnegligible perturbation on the relaxation if this were affected by the orbital degree of freedom. Hence, the similarities in behaviour in terms of decay rates are a further indication that for our QD the dominant relaxation mechanism resides in the degree of spinvalley admixing, as opposed to the spin–orbit admixing relevant for other semiconductor systems^{28}.
Valley splitting control
We now turn to the experimental demonstration of accurate control of the VS, E_{VS}, via electrostatic gating. To determine E_{VS}, we use two different experimental approaches. One utilizes the rapid increase in spin relaxation at the hotspot, and is applicable in the lowmagneticfield regime. The other is based on magnetospectroscopy, and is relevant for high fields.
The first technique stems from the fact that the hotspot can be reliably detected by monitoring the spinup probability as a function of magnetic field. In Fig. 4a, we show measurements of the spinup probability performed with the same method as the one used to evaluate spin lifetimes (see Supplementary Note 2). We see that the probability of detecting a spinup electron decreases significantly at some magnetic fields. A sudden drop of the spinup fraction in a narrow range of field identifies the increase in relaxation rate associated with the hotspot. Given that valley and Zeeman splittings coincide at the hotspot, one can extract the valley separation as E_{VS}=gμ_{B}B_{HS}, where B_{HS} is defined as the field at which the hotspot is observed. For varying gatevoltage configurations, we scan B in the range 2.8 T<B<5 T, and identify B_{HS} by setting an arbitrary probability threshold (greenshaded area in Fig. 4a) below which the hotspot is assumed to occur. The use of this technique is limited to B<5 T because the lifetime drop at the hotspot can be therein confidently assessed. At higher fields the relaxation becomes increasingly fast and its enhancement at the hotspot is indistinguishable within our measurement bandwidth (≈10 kHz).
In order to evaluate E_{VS} at higher magnetic fields, we use a more conventional magnetospectroscopic approach, as shown in Fig. 4b. By employing the same gatepulsed technique used for the charge stability experiments (see Supplementary Note 1), we focus on the singlet–triplet groundstate transition as we load the second electron into the dot (that is, N=1→2 transition) in the range 5 T<B<6.5 T. This is clearly identified as the point where the S (light grey feature) and T_{−} (dark grey feature) states cross. Here, E_{VS}=gμ_{B}B_{ST}, as seen in Fig. 3a.
The data points in Fig. 4c represent the measured valley separation as a function of V_{P}, obtained by means of the aforementioned techniques. The solid line fit shows remarkable consistency between the two sets of data and reveals that E_{VS} depends linearly on the gate voltage over a range of nearly 500 μeV, with a slope of 640 μeV V^{−1}. In order to keep constant the dot’s occupancy and tunnelling rates for different V_{P}, a voltage compensation is carried out by tuning gates C1 and B accordingly. We note that we previously reported VSs of comparable magnitude (few hundreds of μeV) in devices realized with the same technology^{41,45}. However, to our knowledge, this is the first demonstration of the ability to accurately tune the VS electrostatically in a silicon device.
A linear dependence of the VS with respect to the vertical electric field has been predicted for 2DEG systems via effective mass theory^{7,9,10}. A similar dependence for MOSbased QDs^{46} has also been reported by employing atomistic tightbinding calculations^{47}. In order to compare our experimental finding with the theoretical predictions, we simulate the vertical electric field (F_{Z}) in the vicinity of the dot for the range of gate voltages used in the experiments. We employ the commercial semiconductor software ISETCAD^{48} to model the device’s electrostatic potential, and thereby the electric fields in the nanostructure. For this purpose, TCAD solves the Poisson equation with an approximation of Newton’s iterative method^{49} to obtain convergence at low temperatures.
The spatial extent of the dot is identified by regions where the calculated conduction band energy drops below the Fermi level (red area in the top inset of Fig. 4c). Note that our calculations are performed on a threedimensional geometry identical to the real device with the only free parameter being the amount of offset interface charge. This is adjusted to match the experimental threshold voltage of the device (V_{th}=0.625 V), as explained in the Supplementary Note 4.
The computed variation of interface electric field with gate voltage V_{P} is used to determine the VS according to both the atomistic^{46} and effective mass^{7} predictions. Dashed lines in Fig. 4c depict the trends for both approaches, with both exceeding by more than 1 meV of the measured values. Despite this offset, the atomistic calculations give a tunability of the VS with gate voltage, ΔE_{VS}/ΔV_{P}, in good agreement with the experiments. The calculated value of 597 μeV V^{−1} agrees with the measured value to within less than 7%. The value of 541 μeVV^{−1} calculated using the effective mass approach reveals a larger deviation (≈15%) from the experiments. The presence of an offset in the computed VS may be due to the contribution of surface roughness that is not accounted for in the models, and is thought to be responsible for a global reduction of E_{VS}^{8,10,11,12,50,51}. We emphasize, however, that the gate tunability would remain robust against this effect, which is not dependent on electric field.
Discussion
In this work, we have shown that the VS in a silicon device can be electrostatically controlled by simple tuning of the gate bias. We used this VS control, together with spin relaxation measurements, to explore the interplay between spin and valley levels in a fewelectron quantum dot.
The relaxation rates for a oneelectron system exhibit a dramatic hotspot enhancement when the spin Zeeman energy equals the VS, whereas for a twoelectron system the rates reach a minimum at this condition. We found that the known mechanisms for spin relaxation, such as the admixing of spin and p orbital states, were unable to explain the key features of the experimental lifetime data, and so introduced a novel approach based on admixing of valley and spin eigenstates. Our theory, which showed good agreement with experiment, implies that spin relaxation via phonon emission due to spinorbit coupling can occur in realistic quantum dot systems, most likely due to interface disorder.
Our results show that by electrical tuning of the VS in silicon quantum dots, it is possible to ensure the long lifetimes (T_{1}>1 s) required for robust spin qubit operation. Despite this, the excited valley state will generally be lower than orbital states in small quantum dots, placing an ultimate limit on the lifetimes accessible in very small dots, due to the spinvalley mixing described above.
Electrical manipulation of the valley states is also a fundamental requirement to perform coherent valley operations. However, the experimental relaxation rate at the observed hotspot was found to be fast (>1 kHz) for our devices, implying a fast intervalley relaxation rate.
Finally, in the context of realizing scalable quantum computers, these results allow us to address questions of device uniformity and reproducibility with greater optimism. Indeed, our work suggests that issues related to the wide variability of the VS observed in silicon nanostructures to date can shift from the elusive atomic level (surface roughness, strain, interface disorder) to the more accessible device level, where gate geometry and electrostatic confinement can be engineered to ensure robust qubit systems.
Methods
Device fabrication
The samples fabricated for these experiments are silicon MOS planar structures. The high purity, near intrinsic, natural isotope silicon substrate has n+ ohmic regions for source/drain contacts defined via phosphorous diffusion. High quality SiO_{2} gate oxide is 8 nm thick and is grown by dry oxidation at 800 °C. The gates are defined by electronbeam lithography, Al thermal evaporation and oxidation. Three layers of Al/Al_{2}O_{3} are stacked and used to selectively form a 2DEG at the Si/SiO_{2} interface and provide quantum confinement in all the three dimensions.
Measurement system
Measurements are carried out in a dilution refrigerator with a base temperature T_{b}≈40 mK. Flexible coaxial lines fitted with lowtemperature lowpass filters connect the device with the roomtemperature electronics. In order to reduce pickup noise, the gates are biased via batterypowered and optoisolated voltage sources. The SET current is amplified by a roomtemperature transimpedance amplifier and measured via a fastdigitizing oscilloscope and a lockin amplifier for the singleshot and energy spectrum experiments, respectively. Gate voltage pulses are produced by an arbitrary wavefunction generator and combined with a DC offset via a roomtemperature biastee.
Additional information
How to cite this article: Yang, C. H. et al. Spinvalley lifetimes in a silicon quantum dot with tunable valley splitting. Nat. Commun. 4:2069 doi: 10.1038/ncomms3069 (2013).
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Acknowledgements
The authors thank D. Culcer for insightful discussions, and F. Hudson and D. Barber for technical support. This work was supported by the Australian National Fabrication Facility, the Australian Research Council (under contract CE110001027) and by the U.S. Army Research Office (under contract W911NF1310024). The use of nanoHUB.org computational resources operated by the Network for Computational Nanotechnology, funded by the US National Science Foundation under grant EEC0228390, is gratefully acknowledged.
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C.H.Y. carried out the measurements. N.S.L. designed and fabricated the device. C.H.Y. and A.R. analysed the data. C.H.Y., A.R., S.L., G.K., A.M. and A.S.D. discussed the results. R.R. and C.T. modelled the relaxation rates. F.A.M. simulated the electric field profiles. A.S.D. conceived and planned the project. A.R. wrote the manuscript with input from all coauthors.
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Correspondence to A. Rossi.
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The authors declare no competing financial interests.
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Supplementary Figures S1S2, Supplementary Notes 14 and Supplementary Reference (PDF 367 kb)
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Yang, C., Rossi, A., Ruskov, R. et al. Spinvalley lifetimes in a silicon quantum dot with tunable valley splitting. Nat Commun 4, 2069 (2013). https://doi.org/10.1038/ncomms3069
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