www.nature.com/npjqi ARTICLE OPEN Valley dependent anisotropic spin splitting in silicon quantum dots 1 2 2 2,3 4 4 4 4 Rifat Ferdous , Erika Kawakami , Pasquale Scarlino , Michał P. Nowak ,D.R.Ward , D. E. Savage , M. G. Lagally , S. N. Coppersmith , 4 4 2 1 Mark Friesen , Mark A. Eriksson , Lieven M. K. Vandersypen and Rajib Rahman 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 micro-magnets 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 micro-magnet and an external magnetic ﬁeld. We show by atomistic calculations that the spin-orbit 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 atomic-scale 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. npj Quantum Information (2018) 4:26 ; doi:10.1038/s41534-018-0075-1 INTRODUCTION bulk silicon has six-fold degenerate conduction band minima, in quantum wells or dots, electric ﬁelds and often in-plane strain in How microscopic electronic spins in solids are affected by the addition to vertical conﬁnement results in only two low lying crystal and interfacial symmetries has been a topic of great interest over the past few decades and has found potential valley states (labeled as v and v in Fig. 1b) split by an energy − + 1–7 applications in spin-based electronics and computation. While gap known as the valley splitting. SOI enables the control of spin the coupling between spin and orbital degrees of freedom has resonance frequencies of the valley states by gate voltage, an 16,25 been extensively studied, the interplay between spin and the effect measured in refs. . However, the ESR frequencies and momentum space valley degree of freedom is a topic of recent their Stark shifts were found to be different for the two valley interest. This spin-valley interaction is observed in the exotic class states. In another work, an inhomogeneous magnetic ﬁeld, 8–10 of newly found two-dimensional materials, in carbon nano- created by integrated micro-magnets in a Si/SiGe quantum dot 11 12–14 tubes and in silicon —the old friend of the electronics 15 device, was used to electrically drive ESR. Magnetic ﬁeld industry. gradients generated in this way act as an extrinsic spin-orbit Progress in silicon qubits in the last few years has come with the coupling and thus can affect the ESR frequency. Remarkably, demonstrations of various types of qubits with exceptionally long although SOI is a fundamental effect arising from the crystalline 15,16 coherence times, such as single spin up/down qubits, two- structure, the ESR frequency differences between the valley states 17,18 19 electron singlet-triplet qubits, three-electron exchange-only 15,25 20 21 observed in refs. have different signs when the external ﬁelds and hybrid spin-charge qubits and also hole spin qubits are oriented in the same direction with respect to the crystal axes. realized in silicon (Si) quantum dots (QDs). The presence of the In this work we will show that the atomic scale details of the Si valley degree of freedom has enabled valley based qubit interface determine these signs. proposals as well, which have potential for noise immunity. To To understand and achieve control over the coupled behavior harness the advantages of different qubit schemes, quantum between spin and valley degrees of freedom, several key gates for information encoded in different bases are 9,23,24 questions need to be addressed, such as (1) What causes the required. A controlled coherent interaction between multiple device-to-device variability?, (2) Can an artiﬁcial source of degrees of freedom, like valley and spin, might offer a building interaction, like inhomogeneous B-ﬁeld, completely overpower block for promising hybrid systems. the SOI effects of the intrinsic material?, (3) What knobs and An interesting interplay between spin and valley degrees of freedom, which gives rise to a valley dependent spin splitting, has device designs can be utilized to engineer the valley dependent 15,25–27 been observed in Si QDs in recent experiments. Although spin splittings? 1 2 Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA; QuTech and Kavli Institute of Nanoscience, TU Delft, Lorentzweg 1, 2628 CJ Delft, The 3 4 Netherlands; AGH University of Science and Technology, Academic Centre for Materials and Nanotechnology, al. Mickiewicza 30, 30-059 Krakow, Poland and University of Wisconsin-Madison, Madison, WI 53706, USA Correspondence: Rifat Ferdous (email@example.com) Received: 19 June 2017 Revised: 20 April 2018 Accepted: 14 May 2018 Published in partnership with The University of New South Wales Valley dependent anisotropic spin splitting in silicon quantum dots R Ferdous et al. Fig. 1 Valley dependent anisotropic ESR in a Si QD with integrated micro-magnets. a False-color image of the experimental device showing the estimated location of the quantum dot (magenta colored circle) and two Co micro-magnets (green semi-transparent rectangles). The external magnetic ﬁeld (B ) was rotated clockwise in-plane, from the  (θ = 0°) crystal orientation towards ½110ðÞ θ ¼ 90 . b Lowest ext energy levels of a Si QD in an external magnetic ﬁeld. The valley-split levels v and v are found to have unequal Zeeman splittings (E (v ) = − + ZS ± hf ), with ESR frequencies f ≠f . In the experiment, all the measured spin splittings are much larger than the valley splitting and are v± v vþ therefore above the anticrossing point of the spin and valley states. c Both measured (red circles) and calculated f f as a function of θ,for v v B = 0.8 T. The anisotropy in f f is governed by both internal (intrinsic SOI) and external (micro-magnetic ﬁelds) factors. The anisotropy ext v v due to the intrinsic SOI, calculated from atomistic tight binding method, for a speciﬁcally chosen (discussed later) vertical electric ﬁeld and interface step conﬁguration, is labeled as “B (TB)”. The micro-magnetic ﬁeld is separated into a homogeneous (B ) and an ext micro θ θ inhomogeneous (ΔB ) part. The inclusion of B in this case (labeled “B þ B (TB)”), shifts the curve away from the experiment. The ext micro micro θ θ θ addition of ΔB introduces additional anisotropy (labeled “B þ B þ ΔB (TB)”) and shifts the curve towards the experiment. An effective- ext micro mass calculation, with ﬁtted SOI and dipole coupling parameters, is also presented with a cyan solid line. To further clarify the labeling, we want to point out that we label the curves based on the B-ﬁeld components that are used in simulations, with the SOI included in all cases. d Both measured (red circles) and calculated f , as a function of θ, for B = 0.8 T. Calculation with the intrinsic SOI shows negligible change in v ext θ θ GHz scale, while the addition of B results in anisotropy close to the experimental data. ΔB has negligible effect on f . Hence, the micro anisotropy of f is mainly dictated by the homogeneous part of the micro-magnetic ﬁeld RESULTS inhomogeneous (spatially varying, ΔB ) magnetic ﬁeld. Both of these ﬁelds depend on the direction of B , but not on its ext Experiment and theory magnitude (see Methods and Supplementary Section S4), hence Here we report experimentally measured anisotropy in the ESR the superscript θ. The homogeneous component vectorially adds frequencies of the valley states f and f and their differences v v to B , modiﬁes f and makes them anisotropic. Interface steps ext v f f , as a function of the direction (θ) of the external magnetic v v can cause the spatial distribution of the valley states to be non- ﬁeld (B ) in a quantum dot formed at a Si/SiGe heterostructure ext identical. Therefore a spatially varying magnetic ﬁeld can with integrated micro-magnets. At speciﬁc angles of the external contribute to f and f differently. The inhomogeneous micro- v v B-ﬁeld, we also measure the spin splittings of the two valley states magnetic ﬁeld, in a similar fashion as its homogeneous counter- as a function of the B-ﬁeld magnitude (B ). By performing spin- ext part, adds to the anisotropy of f f . The contributions of the v v resolved atomistic tight binding (TB) calculations of the quantum different components can be distinguished because the contribu- dots conﬁned at ideal versus non-ideal interfaces, we evaluate the tions arising from the homogeneous and inhomogeneous contribution of the intrinsic SOI with and without the spatially magnetic ﬁelds are independent of B , while the contribution ext varying B-ﬁelds from the micro-magnets to the spin splittings, arising from SOI is proportional to B . ext thereby relating these quantities to the microscopic nature of the First we show that the experimental measurements (anisotropy interface and elucidating how spin, orbital and valley degrees of of f and f f in Fig. 1 and B dependence in Fig. 2) agree v v v ext freedom are intertwined in these devices. Finally, by combining all well with the theoretical calculations including all the compo- the effects together, we explain the experimental measurements nents: SOI and the homogeneous and inhomogeneous micro- and address the key questions raised in the introduction. magnetic ﬁelds. Then we discuss the effects of the different We show that the SOI and micro-magnetic ﬁelds all make components separately (SOI in Fig. 3 and the inhomogeneous essential contributions to the dependence of the spin splitting on micro-magnetic ﬁeld in Fig. 4) in detail. the magnetic ﬁeld orientation. We also show that physically realistic choices for the interface condition and of the vertical Anisotropy electric ﬁeld yield quantitative agreement with the experimental measurements. We show that a Dresselhaus-like SOI makes f The external magnetic ﬁeld in the experimental device is swept anisotropic in a Si QD, even without any micro-magnetic ﬁeld. The from the  to ½110 crystal orientation. Details of the device valley dependence of the Dresselhaus coefﬁcient makes f ≠f (shown in Fig. 1a) and the measurement technique of the spin v v and f f anisotropic. This Dresselhaus SOI, missing in bulk Si, resonance frequency can be found in ref. . A schematic of the v v results due to the interface inversion asymmetry. Consequently, energy levels of interest is shown in Fig. 1b depicting the v and the details of the interface, like the presence of monoatomic steps, v valley states with different spin splittings, where v is deﬁned + − control both the sign and magnitude of the Dresselhaus SOI and as the ground state. In the experiment, the lowest valley-orbit f f . The micro-magnetic ﬁelds can be separated into two excitation is well below the next excitation, justifying this four- v v parts, a homogeneous (spatial average, B ) and an level schematic in the energy range of interest. micro npj Quantum Information (2018) 26 Published in partnership with The University of New South Wales 1234567890():,; Valley dependent anisotropic spin splitting in silicon quantum dots R Ferdous et al. bottom panels show f f (Fig. 2a) and f (Fig. 2b) for B v v v ext along  (θ = 0°), whereas the top panels correspond to the B- ﬁeld along ½110 (θ = 90°). In Fig. 2b, f depends on B through ext g μB /h, with B = |B + B |. The addition of B causes a − tot tot ext micro micro change in B and shifts f to coincide with the experimental tot data. The contributions of ΔB and SOI are negligible here in the GHz scale. On the other hand, comparing the calculated f f from SOI v v alone (labeled “B (TB)”) with the experimental data (in both the ext top and bottom panels of Fig. 2a), it is clear that the experimental df f ðÞ v v B-ﬁeld dependence of f f (the slope, ) is captured v v dB ext 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 alone does not result in the micro necessary shift to match the experiment. Only after adding ΔB can a quantitative match with the experiment be achieved. Again the experiment-theory agreement is conditional on the interface condition and E . Moreover, we see that the addition of ΔB does not change the dependency on B . Therefore, to properly explain ext the observed experimental behavior, we can ignore neither the Fig. 2 Measured ESR frequencies, (f ) and their differences for the SOI, which is responsible for the change in f f with B , nor v v ext two valley states as a function of the external B-ﬁeld magnitude B þ ext the inhomogeneous B-ﬁeld which shifts f f regardless of B . along two crystal directions, and comparison with theoretical v vþ ext calculations. a f f and b f with B along  (θ = 0°) v v v ext (bottom panel) and ½110ðÞ θ ¼ 90 (top panel). As in Fig. 1c,d, the calculations progressively include SOI (labeled “B (TB)”), homo- ext DISCUSSION geneous (labeled “B + B (TB)”), and gradient (labeled “B + ext micro ext The only knobs we have to adjust to obtain a quantitative B + ΔB (TB)”)B-ﬁeld of the micro-magnet. The cyan solid lines micro agreement between the experiment and the atomistic TB are the effective mass calculations and the red circles are the calculations, are (1) E and (2) interfacial geometry, i.e., how many experimental data. The dependence (slope) of f f on B in v v ext atomic steps at the interface lie inside the dot and where they are (a) comes from the SOI, while the micro-magnetic ﬁelds provide a shift independent of B located relative to the dot center. These adjustments have to be ext done iteratively since the steps and E not only affect the intrinsic SOI but also the inﬂuence of the inhomogeneous B-ﬁeld. It is easy to separate out the contribution of the SOI from the micro-magnet Figures 1c, d show how the SOI and both the micro-magnetic in the B dependence of f f . It will be shown in Figs. 3 and ﬁelds come into play to explain the experimentally measured ext v v df f ðÞ v v anisotropic spin splittings. The atomistic calculation with SOI alone 4 that, the slope, originates from the SOI, while the micro- dB ext (labeled “B (TB)”) for a QD at a speciﬁcally chosen, as discussed ext magnetic ﬁeld shifts f f independent of B . First we v v ext below, non-ideal interface and vertical electric ﬁeld (E ) qualita- individually match the experimental “slope” from the SOI and tively captures the experimental trend of f f in Fig. 1c, but v v the “shift” from the contribution of the micro-magnet for some fails to reproduce the anisotropy of the measured f in Fig. 1din combinations of the two knobs. Finally both effects together the larger GHz scale. The differences between the experimental −1 quantitatively match the experiment for E = 6.77 MVm , and an data and the SOI-only calculations in both ﬁgures arise from z interface with four evenly spaced monoatomic steps at −24.7, the micro-magnets present in the experiment. The inclusion of −2.9, 18.7, 40.4 nm from the dot center along the x () the homogeneous part of the micro-magnetic ﬁeld creates direction. This combination also predicts a valley splitting of 34.4 an anisotropy in the total magnetic ﬁeld (Supplementary μeV in close agreement with the experimental value, given by 29 Fig. S7), which captures the anisotropy of f in Fig. 1d very well μeV. To describe the QD, a 2D simple harmonic (parabolic (f gμ B þ B =h, where g is the Landé g-factor, μ is the v ext micro conﬁnement) potential was used with orbital energy splittings of Bohr magneton and h is the Planck constant), but quantitative match with the experimental data in Fig. 1c is not obtained. Next, 0.55 and 9.4 meV characterizing the x and y () conﬁnement we also incorporate the inhomogeneous part of the micro- respectively. As the interface steps are parallel to y direction, the magnetic ﬁeld, and witness a close quantitative agreement in the orbital energy splitting along y has negligible effects, but the anisotropy of f f , while the anisotropy of f is unaffected. v v v strong y conﬁnement signiﬁcantly reduces simulation time. This experiment-theory agreement of Fig. 1c is achieved for a To further our understanding, we have complemented the speciﬁc choice of interface condition and E , whose inﬂuence will atomistic calculations with an effective mass (EM) based analytic be discussed later. Here, we conclude that mainly the intrinsic SOI model with Rashba and Dresselhaus-like SOI terms (Supplemen- 25,29–31 and the extrinsic inhomogeneous B-ﬁeld govern the anisotropy of tary Section S1), as used in earlier works. We have also f f on the MHz scale, while the anisotropy in the total v vþ developed an analytic model to capture the effects of the homogeneous magnetic ﬁeld introduced by the micro-magnet inhomogeneous magnetic ﬁeld (Supplementary Section S2). dictates the anisotropy of f (and f ) on the larger GHz scale. v v þ Although our large-scale atomistic tight-binding simulation enables us to quantitatively capture the effects of the SOI and Magnetic ﬁeld dependence the atomic-scale details of the interface automatically, they are computationally expensive. The EM model, benchmarked with our In Fig. 2, we show that the measurements of the spin splittings as atomistic results, allows us to get quick insight with the help of a a function of B can not be quantitatively explained as well, ext without the inclusion of all three components: SOI and both the set of ﬁtting parameters. The contributions of the SOI and ΔB on homogeneous and inhomogeneous applied magnetic ﬁelds. The f f obtained from these models are shown in Eqs. (1) and (2), v v Published in partnership with The University of New South Wales npj Quantum Information (2018) 26 Valley dependent anisotropic spin splitting in silicon quantum dots R Ferdous et al. Fig. 3 Effect of the intrinsic SOI on f in a Si QD. a Calculated f as a function of θ, in a QD with ideal (ﬂat) interface, for B = 0.8 T, without v v ± ± ext any micro-magnet. The anisotropies in these curves are in the MHz range and will appear ﬂat on a GHz scale, like the SOI line (labeled "B ext (TB)") of Fig. 1d. b Schematic of a QD wave function near a monoatomic step at the interface. The distance between the dot center and the step edge is denoted by x . c Computed Dresselhaus parameters β as a function of x . β changes sign between the two sides of the step. 0 ± 0 ± d QD wave functions subjected to multiple interface steps. Four different cases are shown (c1(5), c2(3), c3(4), c4(4)) that are used in Figs. 3e, f and also in Figs. 4c, d. The number in parentheses is the total number of steps within the QD. Though c3 and c4 has the same number of steps, the location of the steps are different. c3 is the step conﬁguration used in Figs. 1 and 2. e Calculated f f as a function of θ, for different v v interface conditions, for B = 0.8 T. Interface steps affect both the magnitude and sign of f f . f f f with respect to B along  ext v v v v ext þ þ (θ = 0°)/ ½110 (θ = 180°) (bottom panel) and ½110 (θ = 90°)/½110 (θ = 270°) (top panel). f f for c3 (red lines with circular marker), in both v v Figs. 3e, f, corresponds to the SOI lines (blue dashed lines with diamond marker) of Figs. 1c and 2a. The parabolic conﬁnement and E used here are the same as that of Figs. 1 and 2, except for Fig. 3c, where a smaller dot (with a parabolic conﬁnement in both x and y corresponding to orbital energy splitting of 9.4 meV) is used to accommodate for large variation in dot location respectively. nm and l = 2.792 nm. These ﬁtting parameters in the EM calculations enable us to obtain an even better match with the 4πjj e l SOI z (1) Δ f f B β β sin2ϕ ðÞ α α experimental data compared to TB in Figs. 1cand 2a (cyan solid v v ext þ þ lines). Here we want to point out that the accuracy of the numerically calculated micro-magnetic ﬁeld values depends on our ΔB gμ dB Δ f f cosϕðÞ hi x hi x v v þ h dx estimation of the dot location. But as we calculate (β − β )and − + df f ðÞ v v dB þ þðÞ hi y hi y (α − α ) independently by comparing the measured for − − dy dB ext (2)  and ½110 with Eq. (1) (Supplementary Section S5), any dB þsinϕðÞ hi x hi x dx uncertainty in the estimated dot location or the micro-magnetic ﬁeld values does not effect the extracted SOI parameters. dB þðÞ hi y hi y dy As shown in Figs. 1 and 2, three physical attributes play a key role in explaining the experimental data, 1) SOI, 2) B , and micro Here, α and β are the Rashba and Dresselhaus-like coefﬁcients ± ± 3) ΔB. Each of these contribute to f , and only their sum can respectively, l is the spread of the electron wave function along z, accurately reproduce the experimental data for a speciﬁc interface hi x andhi y are the intra-valley dipole matrix elements, ϕ is the ± ± condition and vertical electric ﬁeld, the two knobs mentioned in angle of the external magnetic ﬁeld with respect to the  crystal ϕ earlier paragraph. In Figs. 3 and 4, we show separately the effects dB orientation and are the magnetic ﬁeld gradients along different dj of (1) and (3) respectively. We show how the contributions of SOI directions (i, j= x, y, z)for aspeciﬁcangle ϕ. It is clear from these and ΔB are modulated by the nature of the conﬁning interface expressions that to match f f the difference in SOI and dipole v v þ (knob 2). The inﬂuence of E (knob 1) on the effects of SOI and ΔB moment parameters between the valley states are relevant (but are shown in the Supplementary Figs. S3 and S4 respectively. We not their absolute values). The parameters used to match also show how B modiﬁes the total homogeneous B-ﬁeld in micro −15 the experiment are β − β =−2.5370 × 10 eV · m, α − α = − + − + the Supplementary Fig. S7. −19 9.4564 × 10 eV · m, hi x hi x =−0.169 nm, hi y hi y ¼ 0 þ þ npj Quantum Information (2018) 26 Published in partnership with The University of New South Wales Valley dependent anisotropic spin splitting in silicon quantum dots R Ferdous et al. Si/SiGe or Si/SiO has atomic-scale disorder, with monolayer atomic steps being a common form of disorder. To understand how such non-ideal 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 . By ﬁtting the EM solutions to the TB results (Supplementary Section S5), we have extracted β and plotted them in Fig. 3casa function of x . It is seen that β changes sign as the dot moves 0 ± 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  or  planes. A monoatomic shift of the vertical position of the interface results in a 90° rotation of the atomic arrangements about the  axis, which results in a sign inversion of the Dresselhaus coefﬁcient of that region (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 29,30 weighted average of two βs with opposite signs. 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 f . v v Figure 3e shows the anisotropy of f f with various step v v conﬁgurations shown in Fig. 3d. f f exhibits a 180° v v periodicity, with extrema at the , ½110, ½110, ½110 crystal orientations. Both the sign and magnitude of f f depends on v v Fig. 4 Effect of inhomogeneous magnetic ﬁeld on f f . a 1D cut v v the interface condition. Since β decreases when a QD wave þ ± of the wave functions of the two valley states close to a step edge, function is spread over a step edge, the smooth interface case highlighting their spatial differences. A large vertical E-ﬁeld, E = (green curve) has the highest amplitude. Figure 3f shows that the −1 30 MVm is used here to show a magniﬁed effect. b The change in slope of f f with B changes sign for a 90° rotation of B v v ext ext f f due to the inhomogeneous B-ﬁeld (ΔB) alone as a function v v and is strongly dependent on the step conﬁguration. The step of the distance x between the dot center and a step edge, as conﬁguration labeled c3 in Fig. 3d is used to match the ΔB deﬁned in Fig. 3b. c Angular dependence of Δ f f for the v v experiment in Figs. 1 and 2. So the curves for c3 in both Figs. various step conﬁgurations of Fig. 3d (same color code) computed 3e, f correspond to the SOI results of Figs. 1c and 2a. It is key to ΔB from atomistic TB. d Δ f f as a function of B . v v ext dfðÞ f v v ΔB ΔB note here that, as E also inﬂuences f f and , z v v Δ f f shows negligible dependence on B . Δ f f þ dB v vþ ext v vþ ext for c3 (red lines with circular marker), in Figs. 4c, d, corresponds to shown in Supplementary Fig. S3, a different combination of the contribution of ΔB (the difference between the black solid interface steps and E can also produce these same SOI results of curve/lines with circular markers and green dashed curve/lines with Figs. 1c and 2a, but might not result in the necessary contribution square markers) of Figs. 1c and 2a. The ﬁelds used in the simulations from micro-magnet to match the experiment. Now the depen- of c and d are the same as that of Figs. 1 and 2, whereas the ﬁelds dence of f f on the interface condition will cause device-to- used for b are the same as that of Fig. 3c v v device variability, while the dependence on the direction and magnitude of B can provide control over the difference in spin ext Spin-orbit interaction in a Si QD and valley dependent splittings. These results thus give us answers to key questions 1 spin-splitting and 3 asked in introduction. The intrinsic SOI in a Si QD makes f anisotropic. Figure 3a shows the angular dependence of f for a Si QD with a smooth interface, Inhomogeneous micro-magnetic ﬁeld in a Si QD and valley without any micro-magnetic ﬁeld, calculated from TB. Both f and dependent spin-splitting f show a 180° periodicity but they are 90° out of phase. Figure 4 illustrates how the inhomogeneous magnetic ﬁeld alone From analytic effective mass study (Supplementary Section S1), ΔB changes f f (denoted as Δ f f ). In the presence of we understand that the anisotropic contribution from the v v v v þ þ interface steps the wave functions for the v and v valley states Dresselhaus-like interaction, caused by interface inversion asym- − + shift away from each other. This shift in an inhomogeneous metry, results in this angular dependence in f . Moreover, the magnetic ﬁeld results in different f and f . This can be different signs of the Dresselhaus coefﬁcients β for the valley v v understood from Figs. 4a, b, and/or Eq. (2). Interface steps states, give rise to a 90° phase shift between f and f .Itis v v 33,34 generate strong valley-orbit hybridization causing the valley important to notice that the change in f is in MHz range. So, in states to have non-identical wave functions (Fig. 4a), and GHz scale, like the blue curve (diamond markers) in Fig. 1d, this hence different dipole moments,ðÞ hi x hi x ≠0 and/or change is not visible. However, if we compare f and f for this v vþ ðÞ hi y hi y ≠0, as opposed to a ﬂat interface case, which has ideal interface case, we see f >f at θ = 0° and f <f at θ = 90°, v v v v þ þ hi x ¼hi y ¼ 0. Thus the spatially varying magnetic ﬁeld has a ± ± which does not explain the experimentally measured anisotropy. different effect on the two wave functions, thereby contributing to We now discuss the remaining physical parameters needed to the difference in ESR frequencies between the valley states. Figure obtain a complete understanding of the experiment. ΔB The atomic-scale details of a Si QD interface actually deﬁne 4b shows Δ f f as a function of the dot location relative v v the Dresselhaus SOI. It is well-known that the interface between to a step edge, x (as in Fig. 3b) and illustrates that ΔB has the Published in partnership with The University of New South Wales npj Quantum Information (2018) 26 Valley dependent anisotropic spin splitting in silicon quantum dots R Ferdous et al. largest contribution to f f when the step is in the vicinity of smaller than the micro-magnet based EDSR. Moreover, the Rabi v v frequency of the SOI-EDSR will strongly depend on the interface the dot. Since ΔB vectorially adds to B , an anisotropic ext ΔB condition (Supplementary Section S7) and can be difﬁcult to Δ f f is seen in Fig. 4c with and without the various step v v control or improve. On the other hand, with improved design ΔB conﬁgurations portrayed in Fig. 3d. We also see that Δ f f v v (stronger transverse gradient ﬁeld) we can gain more advantage in Fig. 4c is negligible for a ﬂat interface, but is signiﬁcant when of the micro-magnets and drive even faster Rabi oscillations. ΔB However, we also predict that, both the SOI and inhomogeneous interface steps are present. Also, Δ f f is almost v vþ B-ﬁeld contribute to the E dependence of f (Supplementary z v independent of B , as shown in Fig. 4d. The curves labeled c3 ± ext Section S3) and make the qubits susceptible to charge noise. As in both Figs. 4c, d correspond to the contribution of ΔB in Figs. 1c these two have comparable contribution, both of their effects will ΔB and 2a. Now E also inﬂuences Δ f f , as shown in z v v add to the charge noise induced dephasing of the spin qubits in Supplementary Fig. S4. Thus a different combination of interface the presence of micro-magnets. steps and E can also produce these same ΔB results of Figs. 1c and 2a, but might not result in the necessary SOI contribution to Possible application of the spin-valley interaction in a Si QD match the experiment. Therefore, only a speciﬁc combination of The coupled spin and valley behavior observed in this work may in these two knobs results in the ﬁnal all-inclusive experiment-theory principle enable us to simultaneously use the quantum informa- agreement. tion stored in both spin and valley degrees of freedom of a single electron. For example, a valley controlled not gate can be Distinguishable effects of SOI and inhomogeneous magnetic ﬁeld 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 It is important to ﬁgure out how to differentiate between the direction of the external magnetic ﬁeld, where the valley states contributions of the SOI and the micro-mangetic ﬁelds in an have different spin splittings, an applied microwave pulse in experimental measurement. A comparison between Figs. 3f and resonance with the spin splitting of v , will rotate the spin only if 4d (also between Eqs. (1) and (2)) reveals that any dependence of − the electron is in v . So we get a NOT operation of the spin f f on B can only come from the SOI and not from the − v v ext quantum information controlled by the valley quantum informa- inhomogeneous magnetic ﬁeld. This indicates that the experi- tion. Spin transitions conditional to valley degrees of freedom are mental B-ﬁeld dependency in Fig. 2a can not be explained without also shown in ref. and an inter-valley spin transition, which can the SOI. So the effect of the SOI cannot be ignored even in the entangle spin and valley degrees of freedom, is observed in ref. . presence of a micro-magnet and this answers key question 2 raised in the introduction. However, engineering the micro- magnetic ﬁeld will allow us to engineer the anisotropy of f f v v CONCLUSION (key question 3). Also, the inﬂuence of interface steps will cause To conclude, we experimentally observe anisotropic behavior in additional device-to-device variability (key question 1). the electron spin resonance frequencies for different valley states in a Si QD with integrated micro-magnets. We analyze this SOI vs micro-magnet driven ESR in Si QDs behavior theoretically and ﬁnd that intrinsic SOI introduces 180° Now the understanding of an enhanced SOI effect compared to periodicity in the difference in the ESR frequencies between the bulk, brings forward an important question, whether it is possible valley states, but the inhomogeneous B-ﬁeld of the micro-magnet to perform electric-dipole spin resonance (EDSR) without the also modiﬁes this anisotropy. Interfacial non-idealities like steps requirement of micro-magnets. Here, we predict that (Fig. 5) for control both the sign and magnitude of this difference through similar driving amplitudes as used here the SOI-only EDSR can both SOI and inhomogeneous B-ﬁeld. We also measure the external magnetic ﬁeld dependence of the resonance frequencies. offer Rabi frequencies close to 1 MHz, which is around ﬁve times 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 metal-oxide-semiconductor based Si QDs without any micro-magnets have appeared (arXiv:1703.03840, Nat. Commun. 9, 1768 (2018)) subsequent to our submission, that validate our ﬁndings and predictions about the spin-orbit interaction, its anisotropy and device-to-device 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. Fig. 5 Calculated Rabi frequencies (f ) with SOI only, inhomoge- Rabi neous B-ﬁeld only and both SOI and inhomogeneous B-ﬁeld for METHODS different direction of the external magnetic ﬁeld for both v (panel a) Theory and v (panel b) valley states. Interface condition, vertical e-ﬁeld and For the theoretical calculations, we use a large scale atomistic tight binding parabolic conﬁnement for the dot used in these simulations are the 3 5 approach with spin resolved sp d s* atomic orbitals with nearest neighbor same as that used to match the experimental data in Figs. 1 and 2. The valley and orbital splittings that we get from the simulations are interactions. Typical simulation domains comprise of 1.5–2 million atoms around 34.4 μeV and 0.48 meV, respectively. The dot radius is around to capture realistic sized dots. Spin-orbit interactions are directly included 35 nm. The fastest Rabi frequencies for SOI only are around 1 MHz, in the Hamiltonian as a matrix element between p-orbitals following the which are least ﬁve times smaller compared to that of the gradient prescription of Chadi. The advantage of this approach is that no B-ﬁeld for θ = 0°. It is important to note here that the micro-magnet additional ﬁtting parameters are needed to capture various types of SOI geometry was designed to maximize the Rabi frequency at θ = 0°. such as Rashba and Dresselhaus SOI in contrast to k.p theory. We introduce The details of the f calculation are discussed in Supplementary monoatomic steps as a source of non-ideality consistent with other Rabi 32,34,38 Section S7 works. We do not include strain in the calculations, in order to keep npj Quantum Information (2018) 26 Published in partnership with The University of New South Wales Valley dependent anisotropic spin splitting in silicon quantum dots R Ferdous et al. the model considered as minimal as possible; homogeneous strain adds 3. Žutić, I., Fabian, J. & Das Sarma, S. Spintronics: fundamentals and applications. effects similar to electric ﬁelds (Supplementary Section S8) and any Rev. Mod. Phys. 76, 323 (2004). inhomogeneity in strain, in the presence of atomic steps, adds effects 4. Loss, D. & Vincenzo, D. P. Quantum computation with quantum dots. Phys. Rev. A. similar to a slight change in the location of steps. The Si interface is 57, 120 (1998). modeled with Hydrogen passivation, without using SiGe. This interface 5. Petta, J. et al. Coherent manipulation of coupled electron spins in semiconductor model is sufﬁcient to capture the SOI effects of a Si/SiGe interface quantum dots. 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Coherent singlet-triplet oscillations in a silicon-based double is different. quantum dot. Nature 481, 344–347 (2011). 18. Wu, X. et al. Two-axis control of a singlet-triplet qubit with integrated micro- magnet. Proc. Natl. Acad. Sci. USA 111, 11938 (2014). Data availability 19. Eng, K. et al. Isotopically enhanced triple quantum dot qubit. Sci. Adv. 1 no. 4, The data that support the ﬁndings of this study are available from the e1500214 (2015). corresponding author upon reasonable request. 20. Kim, D. et al. Quantum control and process tomography of a semiconductor quantum dot hybrid qubit. Nature 511,70–74 (2014). 21. Maurand, R. et al. A CMOS silicon spin qubit. Nat. Commun. 7, 13575 (2016). ACKNOWLEDGEMENTS 22. Culcer, D., Sariava, A. L., Koiller, B., Hu, X. & Das Sarma, S. Valley-based noise This work was supported in part by ARO (W911NF-12-0607); development and resistant quantum computation using Si quantum dots. Phys. Rev. Lett. 108, maintenance of the growth facilities used for fabricating samples is supported by 126804 (2012). 23. Rohling, N. & Burkard, G. Universal quantum computing with spin and valley DOE (DE-FG02-03ER46028). This research utilized NSF-supported shared facilities states. New J. Phys. 14, 083008 (2012). (MRSEC DMR-1121288) at the University of Wisconsin-Madison. Computational 24. Rohling, N., Russ, M. & Burkard, G. Hybrid spin and valley quantum computing resources on nanoHUB.org, funded by the NSF grant EEC-0228390, were used. M.P.N. with singlet-triplet qubits. Phys. Rev. Lett. 113, 176801 (2014). acknowledges support from ERC Synergy Grant. R.F. and R.R. acknowledge 25. Veldhorst, M. et al. Spin-orbit coupling and operation of multivalley spin qubits. discussions with R. Ruskov, C. Tahan, and A. Dzurak. Phys. Rev. B 92, 201401(R) (2015). 26. Scarlino, P. et al. Second-harmonic coherent driving of a spin qubit in SiGe/Si quantum dot. Phys. Rev. Lett. 115, 106802 (2015). AUTHOR CONTRIBUTIONS 27. Scarlino, P. et al. Dressed photon-orbital states in a quantum dot: Inter-valley spin R.F. performed the g-factor calculations, explained the underlying physics and resonance. Phys. Rev. B 95, 165429 (2017). developed the theory with guidance from R.R. R.F., R.R., E.K., P.S. and M.P.N. analyzed 28. Tokura, Y., van der Wiel, W. G., Obata, T. & Tarucha, S. Coherent single electron the simulation results and compared with experimental data in consultation with L.M. spin control in a slanting Zeeman ﬁeld. Phys. Rev. Lett. 96, 047202 (2006). K.V., M.F., S.N.C. and M.A.E. E.K. and P.S. performed the experiment and analyzed the 29. Golub, L. E. & Ivchenko, E. L. Spin splitting in symmetrical SiGe quantum wells. measured data. D.R.W. fabricated the sample. D.E.S. and M.G.L. grew the Phys. Rev. B 69, 115333 (2004). heterostructure. R.F. and R.R. wrote the manuscript with feedback from all the 30. Nestoklon, M. O., Golub, L. E. & Ivchenko, E. L. Spin and valley-orbit splittings in authors. R.R. and L.M.K.V. initiated the project, and supervised the work with S.N.C, M. SiGe/Si heterostructures. Phys. Rev. B 73, 235334 (2006). F. and M.A.E. 31. Nestoklon, M. O., Ivchenko, E. L., Jancu, J.-M. & Voisin, P. Electric ﬁeld effect on electron spin splitting in SiGe/Si quantum wells. Phys. Rev. B 77, 155328 (2008). 32. Zandvliet, H. J. W. & Elswijk, H. B. Morphology of monatomic step edges on vicinal ADDITIONAL INFORMATION Si(001). Phys. Rev. B 48, 14269 (1993). Supplementary information accompanies the paper on the npj Quantum 33. Gamble, J. K., Eriksson, M. A., Coppersmith, S. N. & Friesen, M. Disorder-induced Information website (https://doi.org/10.1038/s41534-018-0075-1). valley-orbit hybrid states in Si quantum dots. Phys. Rev. B 88, 035310 (2013). 34. Friesen, M., Eriksson, M. A. & Coppersmith, S. N. Magnetic ﬁeld dependence of Competing interests: The authors declare no competing interests. valley splitting in realistic Si/SiGe quantum wells. Appl. Phys. Lett. 89, 202106 (2006). 35. Huang, W., Veldhorst, M., Zimmerman, N. M., Dzurak, A. S. & Culcer, D. Electrically driven spin qubit based on valley mixing. Phys. Rev. B 95, 075403 (2017). Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims 36. Klimeck, G. et al. Atomistic simulation of realistically sized nanodevices using in published maps and institutional afﬁliations. NEMO 3D: part I—models and benchmarks. IEEE Trans. Electron Dev. 54, 2079–2089 (2007). 37. Chadi, D. J. Spin-orbit splitting in crystalline and compositionally disordered REFERENCES semiconductors. Phys. Rev. B 16, 790 (1977). 1. Datta, S. & Das, B. Electronic analog of the electro-optic modulator. Appl. Phys. 38. Kharche, N., Prada, M., Boykin, T. B. & Klimeck, G. 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To view a copy of this license, visit http://creativecommons. Open Access This article is licensed under a Creative Commons org/licenses/by/4.0/. Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative © The Author(s) 2018 Commons license, and indicate if changes were made. The images or other third party npj Quantum Information (2018) 26 Published in partnership with The University of New South Wales
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