Abstract
Spin qubits hosted in silicon (Si) quantum dots (QD) are attractive due to their exceptionally long coherence times and compatibility with the silicon transistor platform. To achieve electrical control of spins for qubit scalability, recent experiments have utilized gradient magnetic fields from integrated micromagnets to produce an extrinsic coupling between spin and charge, thereby electrically driving electron spin resonance (ESR). However, spins in silicon QDs experience a complex interplay between spin, charge, and valley degrees of freedom, influenced by the atomic scale details of the confining interface. Here, we report experimental observation of a valley dependent anisotropic spin splitting in a Si QD with an integrated micromagnet and an external magnetic field. We show by atomistic calculations that the spinorbit interaction (SOI), which is often ignored in bulk silicon, plays a major role in the measured anisotropy. Moreover, inhomogeneities such as interface steps strongly affect the spin splittings and their valley dependence. This atomicscale understanding of the intrinsic and extrinsic factors controlling the valley dependent spin properties is a key requirement for successful manipulation of quantum information in Si QDs.
Introduction
How microscopic electronic spins in solids are affected by the crystal and interfacial symmetries has been a topic of great interest over the past few decades and has found potential applications in spinbased electronics and computation.^{1,2,3,4,5,6,7} While the coupling between spin and orbital degrees of freedom has been extensively studied, the interplay between spin and the momentum space valley degree of freedom is a topic of recent interest. This spinvalley interaction is observed in the exotic class of newly found twodimensional materials,^{8,9,10} in carbon nanotubes^{11} and in silicon^{12,13,14}—the old friend of the electronics industry.
Progress in silicon qubits in the last few years has come with the demonstrations of various types of qubits with exceptionally long coherence times, such as single spin up/down qubits,^{15,16} twoelectron singlettriplet qubits,^{17,18} threeelectron exchangeonly^{19} and hybrid spincharge qubits^{20} and also hole spin qubits^{21} realized in silicon (Si) quantum dots (QDs). The presence of the valley degree of freedom has enabled valley based qubit proposals^{22} as well, which have potential for noise immunity. To harness the advantages of different qubit schemes, quantum gates for information encoded in different bases are required.^{9,23,24} A controlled coherent interaction between multiple degrees of freedom, like valley and spin, might offer a building block for promising hybrid systems.
An interesting interplay between spin and valley degrees of freedom, which gives rise to a valley dependent spin splitting, has been observed in Si QDs in recent experiments.^{15,25,26,27} Although bulk silicon has sixfold degenerate conduction band minima, in quantum wells or dots, electric fields and often inplane strain in addition to vertical confinement results in only two low lying valley states (labeled as v_{−} and v_{+} in Fig. 1b) split by an energy gap known as the valley splitting. SOI enables the control of spin resonance frequencies of the valley states by gate voltage, an effect measured in refs.^{16,25}. However, the ESR frequencies and their Stark shifts were found to be different for the two valley states.^{25} In another work, an inhomogeneous magnetic field, created by integrated micromagnets in a Si/SiGe quantum dot device, was used to electrically drive ESR.^{15} Magnetic field gradients generated in this way act as an extrinsic spinorbit coupling and thus can affect the ESR frequency.^{28} Remarkably, although SOI is a fundamental effect arising from the crystalline structure, the ESR frequency differences between the valley states observed in refs.^{15,25} have different signs when the external fields are oriented in the same direction with respect to the crystal axes. In this work we will show that the atomic scale details of the Si interface determine these signs.
To understand and achieve control over the coupled behavior between spin and valley degrees of freedom, several key questions need to be addressed, such as (1) What causes the devicetodevice variability?, (2) Can an artificial source of interaction, like inhomogeneous Bfield, completely overpower the SOI effects of the intrinsic material?, (3) What knobs and device designs can be utilized to engineer the valley dependent spin splittings?
Results
Experiment and theory
Here we report experimentally measured anisotropy in the ESR frequencies of the valley states \(f_{v_  }\) and \(f_{v_ + }\) and their differences \(f_{v_  }  f_{v_ + }\), as a function of the direction (θ) of the external magnetic field (B_{ext}) in a quantum dot formed at a Si/SiGe heterostructure with integrated micromagnets. At specific angles of the external Bfield, we also measure the spin splittings of the two valley states as a function of the Bfield magnitude (B_{ext}). By performing spinresolved atomistic tight binding (TB) calculations of the quantum dots confined at ideal versus nonideal interfaces, we evaluate the contribution of the intrinsic SOI with and without the spatially varying Bfields from the micromagnets to the spin splittings, thereby relating these quantities to the microscopic nature of the interface and elucidating how spin, orbital and valley degrees of freedom are intertwined in these devices. Finally, by combining all the effects together, we explain the experimental measurements and address the key questions raised in the introduction.
We show that the SOI and micromagnetic fields all make essential contributions to the dependence of the spin splitting on the magnetic field orientation. We also show that physically realistic choices for the interface condition and of the vertical electric field yield quantitative agreement with the experimental measurements. We show that a Dresselhauslike SOI makes \(f_{v_ \pm }\) anisotropic in a Si QD, even without any micromagnetic field. The valley dependence of the Dresselhaus coefficient makes \(f_{v_  } \ne f_{v_ + }\) and \(f_{v_  }  f_{v_ + }\) anisotropic. This Dresselhaus SOI, missing in bulk Si, results due to the interface inversion asymmetry. Consequently, the details of the interface, like the presence of monoatomic steps, control both the sign and magnitude of the Dresselhaus SOI and \(f_{v_  }  f_{v_ + }\). The micromagnetic fields can be separated into two parts, a homogeneous (spatial average, \({\bf{B}}_{{\mathrm{micro}}}^\theta\)) and an inhomogeneous (spatially varying, ΔB^{θ}) magnetic field. Both of these fields depend on the direction of B_{ext}, but not on its magnitude (see Methods and Supplementary Section S4), hence the superscript θ. The homogeneous component vectorially adds to B_{ext}, modifies \(f_{v_ \pm }\) and makes them anisotropic. Interface steps can cause the spatial distribution of the valley states to be nonidentical. Therefore a spatially varying magnetic field can contribute to \(f_{v_  }\) and \(f_{v_ + }\) differently. The inhomogeneous micromagnetic field, in a similar fashion as its homogeneous counterpart, adds to the anisotropy of \(f_{v_  }  f_{v_ + }\). The contributions of the different components can be distinguished because the contributions arising from the homogeneous and inhomogeneous magnetic fields are independent of B_{ext}, while the contribution arising from SOI is proportional to B_{ext}.
First we show that the experimental measurements (anisotropy of \(f_{v_  }\) and \(f_{v_  }  f_{v_ + }\) in Fig. 1 and B_{ext} dependence in Fig. 2) agree well with the theoretical calculations including all the components: SOI and the homogeneous and inhomogeneous micromagnetic fields. Then we discuss the effects of the different components separately (SOI in Fig. 3 and the inhomogeneous micromagnetic field in Fig. 4) in detail.
Anisotropy
The external magnetic field in the experimental device is swept from the [110] to \([1\bar 10]\) crystal orientation. Details of the device (shown in Fig. 1a) and the measurement technique of the spin resonance frequency can be found in ref.^{15}. A schematic of the energy levels of interest is shown in Fig. 1b depicting the v_{−} and v_{+} valley states with different spin splittings, where v_{−} is defined as the ground state. In the experiment, the lowest valleyorbit excitation is well below the next excitation, justifying this fourlevel schematic in the energy range of interest.
Figures 1c, d show how the SOI and both the micromagnetic fields come into play to explain the experimentally measured anisotropic spin splittings. The atomistic calculation with SOI alone (labeled “B_{ext} (TB)”) for a QD at a specifically chosen, as discussed below, nonideal interface and vertical electric field (E_{ z }) qualitatively captures the experimental trend of \(f_{v_  }  f_{v_ + }\) in Fig. 1c, but fails to reproduce the anisotropy of the measured \(f_{v_  }\) in Fig. 1d in the larger GHz scale. The differences between the experimental data and the SOIonly calculations in both figures arise from the micromagnets present in the experiment. The inclusion of the homogeneous part of the micromagnetic field creates an anisotropy in the total magnetic field (Supplementary Fig. S7), which captures the anisotropy of \(f_{v_  }\) in Fig. 1d very well (\(f_{v_  } \approx g\mu \left {{\bf{B}}_{{\mathrm{ext}}} + {\bf{B}}_{{\mathrm{micro}}}^\theta } \right{\mathrm{/}}h\), where g is the Landé gfactor, μ is the Bohr magneton and h is the Planck constant), but quantitative match with the experimental data in Fig. 1c is not obtained. Next, we also incorporate the inhomogeneous part of the micromagnetic field, and witness a close quantitative agreement in the anisotropy of \(f_{v_  }  f_{v_ + }\), while the anisotropy of \(f_{v_  }\) is unaffected. This experimenttheory agreement of Fig. 1c is achieved for a specific choice of interface condition and E_{ z }, whose influence will be discussed later. Here, we conclude that mainly the intrinsic SOI and the extrinsic inhomogeneous Bfield govern the anisotropy of \(f_{v_  }  f_{v_ + }\) on the MHz scale, while the anisotropy in the total homogeneous magnetic field introduced by the micromagnet dictates the anisotropy of \(f_{v_  }\) (and \(f_{v_ + }\)) on the larger GHz scale.
Magnetic field dependence
In Fig. 2, we show that the measurements of the spin splittings as a function of B_{ext} can not be quantitatively explained as well, without the inclusion of all three components: SOI and both the homogeneous and inhomogeneous applied magnetic fields. The bottom panels show \(f_{v_  }  f_{v_ + }\) (Fig. 2a) and \(f_{v_  }\) (Fig. 2b) for B_{ext} along [110] (θ = 0°), whereas the top panels correspond to the Bfield along \([1\bar 10]\) (θ = 90°). In Fig. 2b, \(f_{v_  }\) depends on B_{ext} through g_{−}μB_{tot}/h, with B_{tot} = B_{ext} + B_{micro}. The addition of B_{micro} causes a change in B_{tot} and shifts \(f_{v_  }\) to coincide with the experimental data. The contributions of ΔB and SOI are negligible here in the GHz scale.
On the other hand, comparing the calculated \(f_{v_  }  f_{v_ + }\) from SOI alone (labeled “B_{ext} (TB)”) with the experimental data (in both the top and bottom panels of Fig. 2a), it is clear that the experimental Bfield dependence of \(f_{v_  }  f_{v_ + }\) (the slope, \(\frac{{d\left( {f_{v_  }  f_{v_ + }} \right)}}{{dB_{{\mathrm{ext}}}}}\)) is captured from the effect of intrinsic SOI. However there is a shift between the SOI curve and the experimental data (different shift for θ = 0° and θ = 90°). The addition of B_{micro} alone does not result in the necessary shift to match the experiment. Only after adding ΔB can a quantitative match with the experiment be achieved. Again the experimenttheory agreement is conditional on the interface condition and E_{ z }. Moreover, we see that the addition of ΔB does not change the dependency on B_{ext}. Therefore, to properly explain the observed experimental behavior, we can ignore neither the SOI, which is responsible for the change in \(f_{v_  }  f_{v_ + }\) with B_{ext}, nor the inhomogeneous Bfield which shifts \(f_{v_  }  f_{v_ + }\) regardless of B_{ext}.
Discussion
The only knobs we have to adjust to obtain a quantitative agreement between the experiment and the atomistic TB calculations, are (1) E_{ z } and (2) interfacial geometry, i.e., how many atomic steps at the interface lie inside the dot and where they are located relative to the dot center. These adjustments have to be done iteratively since the steps and E_{ z } not only affect the intrinsic SOI but also the influence of the inhomogeneous Bfield. It is easy to separate out the contribution of the SOI from the micromagnet in the B_{ext} dependence of \(f_{v_  }  f_{v_ + }\). It will be shown in Figs. 3 and 4 that, the slope, \(\frac{{d\left( {f_{v_  }  f_{v_ + }} \right)}}{{dB_{{\mathrm{ext}}}}}\) originates from the SOI, while the micromagnetic field shifts \(f_{v_  }  f_{v_ + }\) independent of B_{ext}. First we individually match the experimental “slope” from the SOI and the “shift” from the contribution of the micromagnet for some combinations of the two knobs. Finally both effects together quantitatively match the experiment for E_{ z } = 6.77 MVm^{−1}, and an interface with four evenly spaced monoatomic steps at −24.7, −2.9, 18.7, 40.4 nm from the dot center along the x ([100]) direction. This combination also predicts a valley splitting of 34.4 μeV in close agreement with the experimental value, given by 29 μeV.^{27} To describe the QD, a 2D simple harmonic (parabolic confinement) potential was used with orbital energy splittings of 0.55 and 9.4 meV characterizing the x and y ([010]) confinement respectively. As the interface steps are parallel to y direction, the orbital energy splitting along y has negligible effects, but the strong y confinement significantly reduces simulation time.
To further our understanding, we have complemented the atomistic calculations with an effective mass (EM) based analytic model with Rashba and Dresselhauslike SOI terms (Supplementary Section S1), as used in earlier works.^{25,29,30,31} We have also developed an analytic model to capture the effects of the inhomogeneous magnetic field (Supplementary Section S2). Although our largescale atomistic tightbinding simulation enables us to quantitatively capture the effects of the SOI and the atomicscale details of the interface automatically, they are computationally expensive. The EM model, benchmarked with our atomistic results, allows us to get quick insight with the help of a set of fitting parameters. The contributions of the SOI and ΔB on \(f_{v_  }  f_{v_ + }\) obtained from these models are shown in Eqs. (1) and (2), respectively.
Here, α_{±} and β_{±} are the Rashba and Dresselhauslike coefficients respectively, l_{ z } is the spread of the electron wave function along z, \(\left\langle {x_ \pm } \right\rangle\) and \(\left\langle {y_ \pm } \right\rangle\) are the intravalley dipole matrix elements, ϕ is the angle of the external magnetic field with respect to the [100] crystal orientation and \(\frac{{dB_i^\phi }}{{dj}}\) are the magnetic field gradients along different directions (i, j = x, y, z) for a specific angle ϕ. It is clear from these expressions that to match \(f_{v_  }  f_{v_ + }\) the difference in SOI and dipole moment parameters between the valley states are relevant (but not their absolute values). The parameters used to match the experiment are β_{−} − β_{+} = −2.5370 × 10^{−15} eV · m, α_{−} − α_{+} = 9.4564 × 10^{−19} eV · m, \(\left\langle {x_  } \right\rangle  \left\langle {x_ + } \right\rangle\) = −0.169 nm, \(\left\langle {y_  } \right\rangle  \left\langle {y_ + } \right\rangle = 0\) nm and l_{ z } = 2.792 nm. These fitting parameters in the EM calculations enable us to obtain an even better match with the experimental data compared to TB in Figs. 1c and 2a (cyan solid lines). Here we want to point out that the accuracy of the numerically calculated micromagnetic field values depends on our estimation of the dot location. But as we calculate (β_{−} − β_{+}) and (α_{−} − α_{−}) independently by comparing the measured \(\frac{{d\left( {f_{v_  }  f_{v_ + }} \right)}}{{dB_{{\mathrm{ext}}}}}\) for [110] and \([1\bar 10]\) with Eq. (1) (Supplementary Section S5), any uncertainty in the estimated dot location or the micromagnetic field values does not effect the extracted SOI parameters.
As shown in Figs. 1 and 2, three physical attributes play a key role in explaining the experimental data, 1) SOI, 2) B_{micro}, and 3) ΔB. Each of these contribute to \(f_{v_ \pm }\), and only their sum can accurately reproduce the experimental data for a specific interface condition and vertical electric field, the two knobs mentioned in earlier paragraph. In Figs. 3 and 4, we show separately the effects of (1) and (3) respectively. We show how the contributions of SOI and ΔB are modulated by the nature of the confining interface (knob 2). The influence of E_{ z } (knob 1) on the effects of SOI and ΔB are shown in the Supplementary Figs. S3 and S4 respectively. We also show how B_{micro} modifies the total homogeneous Bfield in the Supplementary Fig. S7.
Spinorbit interaction in a Si QD and valley dependent spinsplitting
The intrinsic SOI in a Si QD makes \(f_{v_ \pm }\) anisotropic. Figure 3a shows the angular dependence of \(f_{v_ \pm }\) for a Si QD with a smooth interface, without any micromagnetic field, calculated from TB. Both \(f_{v_  }\) and \(f_{v_ + }\) show a 180° periodicity but they are 90° out of phase. From analytic effective mass study (Supplementary Section S1), we understand that the anisotropic contribution from the Dresselhauslike interaction, caused by interface inversion asymmetry,^{31} results in this angular dependence in \(f_{v_ \pm }\). Moreover, the different signs of the Dresselhaus coefficients β_{±} for the valley states, give rise to a 90° phase shift between \(f_{v_  }\) and \(f_{v_ + }\). It is important to notice that the change in \(f_{v_ \pm }\) is in MHz range. So, in GHz scale, like the blue curve (diamond markers) in Fig. 1d, this change is not visible. However, if we compare \(f_{v_  }\) and \(f_{v_ + }\) for this ideal interface case, we see \(f_{v_  } > f_{v_ + }\) at θ = 0° and \(f_{v_  } < f_{v_ + }\) at θ = 90°, which does not explain the experimentally measured anisotropy. We now discuss the remaining physical parameters needed to obtain a complete understanding of the experiment.
The atomicscale details of a Si QD interface actually define the Dresselhaus SOI. It is wellknown that the interface between Si/SiGe or Si/SiO_{2} has atomicscale disorder, with monolayer atomic steps being a common form of disorder.^{32} To understand how such nonideal interfaces can affect SOI, we first introduce a monolayer atomic step as shown in Fig. 3b and vary the dot position laterally relative to the step, as defined by the variable x_{0}. By fitting the EM solutions to the TB results (Supplementary Section S5), we have extracted β_{±} and plotted them in Fig. 3c as a function of x_{0}. It is seen that β_{±} changes sign as the dot moves from the left to the right of the step edge. Both the sign and magnitude of β_{±} depends on the distribution of the wave function between the neighboring regions with one atomic layer shift between them, as shown in Fig. 3b. To understand this behavior, we have to understand the atomic arrangements in a Si crystal, where the nearest neighbors of a Si atom lie either in the [110] or [1\(\bar 1\)0] planes. A monoatomic shift of the vertical position of the interface results in a 90° rotation of the atomic arrangements about the [001] axis, which results in a sign inversion of the Dresselhaus coefficient of that region^{31} (Supplementary Section S6). So whenever there is a monoatomic step at the interface, β changes sign between the two sides of the step. A dot wave function spread over a monoatomic step therefore samples out a weighted average of two βs with opposite signs.^{29,30} Thus the interface condition of a Si QD determines both the sign and strength of the effective Dresselhaus coefficients.
Next, we investigate the influence of interface steps on \(f_{v_  }  f_{v_ + }\). Figure 3e shows the anisotropy of \(f_{v_  }  f_{v_ + }\) with various step configurations shown in Fig. 3d. \(f_{v_  }  f_{v_ + }\) exhibits a 180° periodicity, with extrema at the [110], \([1\bar 10]\), \([\bar 1\bar 10]\), \([\bar 110]\) crystal orientations. Both the sign and magnitude of \(f_{v_  }  f_{v_ + }\) depends on the interface condition. Since β_{±} decreases when a QD wave function is spread over a step edge, the smooth interface case (green curve) has the highest amplitude. Figure 3f shows that the slope of \(f_{v_  }  f_{v_ + }\) with B_{ext} changes sign for a 90° rotation of B_{ext} and is strongly dependent on the step configuration. The step configuration labeled c3 in Fig. 3d is used to match the experiment in Figs. 1 and 2. So the curves for c3 in both Figs. 3e, f correspond to the SOI results of Figs. 1c and 2a. It is key to note here that, as E_{ z } also influences \(\left {f_{v_  }  f_{v_ + }} \right\) and \(\left {\frac{{d\left( {f_{v_  }  f_{v_ + }} \right)}}{{dB_{{\mathrm{ext}}}}}} \right\), shown in Supplementary Fig. S3, a different combination of interface steps and E_{ z } can also produce these same SOI results of Figs. 1c and 2a, but might not result in the necessary contribution from micromagnet to match the experiment. Now the dependence of \(f_{v_  }  f_{v_ + }\) on the interface condition will cause devicetodevice variability, while the dependence on the direction and magnitude of B_{ext} can provide control over the difference in spin splittings. These results thus give us answers to key questions 1 and 3 asked in introduction.
Inhomogeneous micromagnetic field in a Si QD and valley dependent spinsplitting
Figure 4 illustrates how the inhomogeneous magnetic field alone changes \(f_{v_  }  f_{v_ + }\) (denoted as \({\mathrm{\Delta }}\left( {f_{v_  }  f_{v_ + }} \right)^{{\mathrm{\Delta }}{\bf{B}}}\)). In the presence of interface steps the wave functions for the v_{−} and v_{+} valley states shift away from each other. This shift in an inhomogeneous magnetic field results in different \(f_{v_  }\) and \(f_{v_ + }\). This can be understood from Figs. 4a, b, and/or Eq. (2). Interface steps generate strong valleyorbit hybridization^{33,34} causing the valley states to have nonidentical wave functions (Fig. 4a), and hence different dipole moments, \(\left( {\left\langle {x_  } \right\rangle  \left\langle {x_ + } \right\rangle } \right) \ne 0\) and/or \(\left( {\left\langle {y_  } \right\rangle  \left\langle {y_ + } \right\rangle } \right) \ne 0\), as opposed to a flat interface case, which has \(\left\langle {x_ \pm } \right\rangle = \left\langle {y_ \pm } \right\rangle = 0\). Thus the spatially varying magnetic field has a different effect on the two wave functions, thereby contributing to the difference in ESR frequencies between the valley states. Figure 4b shows \({\mathrm{\Delta }}\left( {f_{v_  }  f_{v_ + }} \right)^{\Delta {\bf{B}}}\) as a function of the dot location relative to a step edge, x_{0} (as in Fig. 3b) and illustrates that ΔB has the largest contribution to \(f_{v_  }  f_{v_ + }\) when the step is in the vicinity of the dot. Since ΔB^{θ} vectorially adds to B_{ext}, an anisotropic \({\mathrm{\Delta }}\left( {f_{v_  }  f_{v_ + }} \right)^{{\mathrm{\Delta }}{\bf{B}}}\) is seen in Fig. 4c with and without the various step configurations portrayed in Fig. 3d. We also see that \({\mathrm{\Delta }}\left( {f_{v_  }  f_{v_ + }} \right)^{{\mathrm{\Delta }}{\bf{B}}}\) in Fig. 4c is negligible for a flat interface, but is significant when interface steps are present. Also, \({\mathrm{\Delta }}\left( {f_{v_  }  f_{v_ + }} \right)^{{\mathrm{\Delta }}{\bf{B}}}\) is almost independent of B_{ext}, as shown in Fig. 4d. The curves labeled c3 in both Figs. 4c, d correspond to the contribution of ΔB in Figs. 1c and 2a. Now E_{ z } also influences \(\left {{\mathrm{\Delta }}\left( {f_{v_  }  f_{v_ + }} \right)^{{\mathrm{\Delta }}{\bf{B}}}} \right\), as shown in Supplementary Fig. S4. Thus a different combination of interface steps and E_{ z } can also produce these same ΔB results of Figs. 1c and 2a, but might not result in the necessary SOI contribution to match the experiment. Therefore, only a specific combination of these two knobs results in the final allinclusive experimenttheory agreement.
Distinguishable effects of SOI and inhomogeneous magnetic field
It is important to figure out how to differentiate between the contributions of the SOI and the micromangetic fields in an experimental measurement. A comparison between Figs. 3f and 4d (also between Eqs. (1) and (2)) reveals that any dependence of \(f_{v_  }  f_{v_ + }\) on B_{ext} can only come from the SOI and not from the inhomogeneous magnetic field. This indicates that the experimental Bfield dependency in Fig. 2a can not be explained without the SOI. So the effect of the SOI cannot be ignored even in the presence of a micromagnet and this answers key question 2 raised in the introduction. However, engineering the micromagnetic field will allow us to engineer the anisotropy of \(f_{v_  }  f_{v_ + }\) (key question 3). Also, the influence of interface steps will cause additional devicetodevice variability (key question 1).
SOI vs micromagnet driven ESR in Si QDs
Now the understanding of an enhanced SOI effect compared to bulk, brings forward an important question, whether it is possible to perform electricdipole spin resonance (EDSR) without the requirement of micromagnets. Here, we predict that (Fig. 5) for similar driving amplitudes as used here the SOIonly EDSR can offer Rabi frequencies close to 1 MHz, which is around five times smaller than the micromagnet based EDSR. Moreover, the Rabi frequency of the SOIEDSR will strongly depend on the interface condition^{35} (Supplementary Section S7) and can be difficult to control or improve. On the other hand, with improved design (stronger transverse gradient field) we can gain more advantage of the micromagnets and drive even faster Rabi oscillations. However, we also predict that, both the SOI and inhomogeneous Bfield contribute to the E_{ z } dependence of \(f_{v_ \pm }\) (Supplementary Section S3) and make the qubits susceptible to charge noise.^{25} As these two have comparable contribution, both of their effects will add to the charge noise induced dephasing of the spin qubits in the presence of micromagnets.
Possible application of the spinvalley interaction in a Si QD
The coupled spin and valley behavior observed in this work may in principle enable us to simultaneously use the quantum information stored in both spin and valley degrees of freedom of a single electron. For example, a valley controlled not gate^{9} can be designed in which the spin basis can be the target qubit, while the valley information can work as a control qubit. If we choose such a direction of the external magnetic field, where the valley states have different spin splittings, an applied microwave pulse in resonance with the spin splitting of v_{−}, will rotate the spin only if the electron is in v_{−}. So we get a NOT operation of the spin quantum information controlled by the valley quantum information. Spin transitions conditional to valley degrees of freedom are also shown in ref.^{15} and an intervalley spin transition, which can entangle spin and valley degrees of freedom, is observed in ref.^{27}.
Conclusion
To conclude, we experimentally observe anisotropic behavior in the electron spin resonance frequencies for different valley states in a Si QD with integrated micromagnets. We analyze this behavior theoretically and find that intrinsic SOI introduces 180° periodicity in the difference in the ESR frequencies between the valley states, but the inhomogeneous Bfield of the micromagnet also modifies this anisotropy. Interfacial nonidealities like steps control both the sign and magnitude of this difference through both SOI and inhomogeneous Bfield. We also measure the external magnetic field dependence of the resonance frequencies. We show that the measured magnetic field dependence of the difference in resonance frequencies originates only from the SOI. We conclude that even though the SOI in bulk silicon has been typically ignored as being small, it still plays a major role in determining the valley dependent spin properties in interfacially confined Si QDs (A few works on metaloxidesemiconductor based Si QDs without any micromagnets have appeared (arXiv:1703.03840, Nat. Commun. 9, 1768 (2018)) subsequent to our submission, that validate our findings and predictions about the spinorbit interaction, its anisotropy and devicetodevice variability). These understandings help us answer the key questions from the introduction, which are crucial for proper operation of various qubit schemes based on silicon quantum dots.
Methods
Theory
For the theoretical calculations, we use a large scale atomistic tight binding approach with spin resolved sp^{3}d^{5}s* atomic orbitals with nearest neighbor interactions.^{36} Typical simulation domains comprise of 1.5–2 million atoms to capture realistic sized dots. Spinorbit interactions are directly included in the Hamiltonian as a matrix element between porbitals following the prescription of Chadi.^{37} The advantage of this approach is that no additional fitting parameters are needed to capture various types of SOI such as Rashba and Dresselhaus SOI in contrast to k.p theory. We introduce monoatomic steps as a source of nonideality consistent with other works.^{32,34,38} We do not include strain in the calculations, in order to keep the model considered as minimal as possible; homogeneous strain adds effects similar to electric fields (Supplementary Section S8) and any inhomogeneity in strain, in the presence of atomic steps, adds effects similar to a slight change in the location of steps. The Si interface is modeled with Hydrogen passivation, without using SiGe. This interface model is sufficient to capture the SOI effects of a Si/SiGe interface discussed in refs.^{29,30,31}. We use the methodology of ref.^{39} to model the micromagnetic fields (Supplementary Section S4). Full magnetization of the micromagnet is assumed. This causes the value of the magnetization of the micromagnet to be saturated and makes it independent of B_{ext}. However, a change in the direction of B_{ext} changes the magnetization. We include the effect of inhomogeneous magnetic field perturbatively, with the perturbation matrix elements, \(\left\langle {\psi _m} \right\frac{1}{2}g\mu {\mathrm{\Delta }}{\bf{B}}^\phi \left {\psi _n} \right\rangle\) = \(\frac{1}{2}g\mu \mathop {\sum}\limits_{i,j} \left\langle {\psi _m} \right\frac{{dB_i^\phi }}{{dj}}j\left {\psi _n} \right\rangle\). Here, ψ_{ n } and ψ_{ m } are atomistic wave functions calculated with homogeneous magnetic field. For further details about the numerical techniques, see NEMO3D ref.^{36}.
Experiment
Method details about the experiment can be found in ref.^{15}. The dot location in this experiment is different from ref.^{15}. The device was electrostatically reset by shining light using an LED and all the measurements were done with a new electrostatic environment (a new gate voltage configuration). The quantum dot location is estimated by the offsets of the magnetic field created by the micromagnets extrapolated from the measurements shown in Fig. 2 and comparing to the simulation results shown in Supplementary Section S4. We also observed that the Rabi frequencies were different from ref.^{15} when applying the same microwave power to the same gate, which qualitatively indicates that the dot location is different.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported in part by ARO (W911NF120607); development and maintenance of the growth facilities used for fabricating samples is supported by DOE (DEFG0203ER46028). This research utilized NSFsupported shared facilities (MRSEC DMR1121288) at the University of WisconsinMadison. Computational resources on nanoHUB.org, funded by the NSF grant EEC0228390, were used. M.P.N. acknowledges support from ERC Synergy Grant. R.F. and R.R. acknowledge discussions with R. Ruskov, C. Tahan, and A. Dzurak.
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R.F. performed the gfactor calculations, explained the underlying physics and developed the theory with guidance from R.R. R.F., R.R., E.K., P.S. and M.P.N. analyzed the simulation results and compared with experimental data in consultation with L.M.K.V., M.F., S.N.C. and M.A.E. E.K. and P.S. performed the experiment and analyzed the measured data. D.R.W. fabricated the sample. D.E.S. and M.G.L. grew the heterostructure. R.F. and R.R. wrote the manuscript with feedback from all the authors. R.R. and L.M.K.V. initiated the project, and supervised the work with S.N.C, M.F. and M.A.E.
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Ferdous, R., Kawakami, E., Scarlino, P. et al. Valley dependent anisotropic spin splitting in silicon quantum dots. npj Quantum Inf 4, 26 (2018). https://doi.org/10.1038/s4153401800751
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DOI: https://doi.org/10.1038/s4153401800751
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