Electrically driven electron spin resonance mediated by spin-valley-orbit coupling in a silicon quantum dot

The ability to manipulate electron spins with voltage-dependent electric fields is key to the operation of quantum spintronics devices, such as spin-based semiconductor qubits. A natural approach to electrical spin control exploits the spin-orbit coupling (SOC) inherently present in all materials. So far, this approach could not be applied to electrons in silicon, due to their extremely weak SOC. Here we report an experimental realization of electrically driven electron-spin resonance in a silicon-on-insulator (SOI) nanowire quantum dot device. The underlying driving mechanism results from an interplay between SOC and the multi-valley structure of the silicon conduction band, which is enhanced in the investigated nanowire geometry. We present a simple model capturing the essential physics and use tight-binding simulations for a more quantitative analysis. We discuss the relevance of our findings to the development of compact and scalable electron-spin qubits in silicon.

duction of micromagnets, generating local magnetic field gradients and hence an artificial SOC [6][7][8][9], or on the use of hole spins [10], for which SOC is strong. In both cases, relatively fast coherent spin rotations could be achieved through resonant radio-frequency (RF) modulation of a control gate voltage. While the actual scalability of these two solutions remains to be investigated, other valuable opportunities may emerge from the rich physics of electrons in silicon nanostructures [1, [11][12][13][14][15][16].
Silicon is indeed an indirect band-gap semiconductor with six degenerate conduction-band valleys. This degeneracy is lifted in quantum dots (QD) where quantum confinement leaves only two low-lying valleys that can be coupled by potential steps at Si/SiO 2 interfaces. The resulting valley eigenstates, which we label v 1 and v 2 , are separated by an energy splitting ∆ ranging from a few tens of µeV to a few meV [17][18][19][20]. ∆ depends on the confinement potential, and can hence be tuned by externally applied electric fields [4,21,22]. Even if weak [23], SOC can couple valley and spin degrees of freedom when, following the application of a magnetic field, E Z ∼ ∆, where E Z is the Zeeman energy splitting. It has been shown that this operating regime can result in enhanced spin relaxation [3,22,24].
Here we demonstrate that it can be exploited to perform electric-dipole spin resonance (EDSR) [25][26][27][28]. This functionality is enabled by the use of a quantum dot with a lowsymmetry confinement potential. We discuss the implications of these results for the development of silicon spin qubits.

II. RESULTS
The experiment is carried out on a silicon nanowire device fabricated on a 300 mm diameter silicon-on-insulator wafer using an industrial-scale fabrication line [10]. The device, shown in the schematic of fig. 1a and in the scanning electron micrograph of fig. 1b, consists of an undoped, 30 nm-wide and 12 nm-thick silicon channel oriented along [110], with ndoped contacts. Two 35 nm-wide top-gates (gate 1 and gate 2), spaced by 30 nm, partially cover the channel. An additional gate (gate 3) is located on the opposite side at a distance of 50 nm from the nanowire. Electron transport measurements were performed in a dilution refrigerator with a base temperature T = 15 mK. At this temperature, two QDs in series, labeled as QD1 and QD2, can be defined by the accumulation voltages V g1 and V g2 applied to gate 1 and gate 2, respectively. The two QDs are confined against the nanowire edge I/V I ds 10  covered by the gates, forming so-called "corner" dots [29,30], as confirmed by tight-binding simulations of the lowest energy states, whose wave-functions are shown in fig. 1c. We tune the electron filling of QD1 and QD2 down to relatively small occupation numbers n 1 and n 2 , respectively (n 1 , n 2 < 10, as inferred from the threshold voltage at room temperature and the charging energy [10]). The side gate is set to a negative V g3 = −0.28 V in order to further push the QD wave-functions against the opposite nanowire edges.
In the limit of vanishing inter-dot coupling and odd occupation numbers, both QD1 and QD2 have a spin-1/2 ground state. At finite magnetic field, B, the respective spin degeneracies are lifted by the Zeeman energy E Z = gµ B B, where µ B is the Bohr magneton and g is the Landé g-factor, which is close to the bare electron value (g 2) for electrons in silicon [31]. In essence, our experiment consists in measuring electron transport through the double dot while driving EDSR in QD2. The polarized spin in QD1 acts as an effective "spin filter" regulating the current flow as a function of the spin admixture induced by EDSR in QD2. This Pauli blockade regime can be achieved only when the double dot is biased in a charge/spin configuration where inter-dot tunneling is forbidden by spin conservation [32].
The simplest case involves the inter-dot charge transition (n 1 = 1, n 2 = 1) → (n 1 = 0, n 2 = 2), where one electron tunnels from QD1 into QD2. The two electrons may indeed form singlet (S) or triplet (T) states. While the singlet S(1, 1) and triplet T (1, 1) states are only weakly split by exchange interations and magnetic field and may both be loaded, the triplet T (0, 2) states remain typically out of reach because they must involve some orbital excitation of QD2. The system may hence be trapped for long times in the T (1, 1) states since tunneling from T (1, 1) to the S(0, 2) ground-state is forbidden by Pauli exclusion principle [32]. This scenario can be generalized to the (n 1 , n 2 ) → (n 1 − 1, n 2 + 1) transitions where n 1 and n 2 are odd integers. The current is strongly suppressed unless EDSR mixes T (1, 1) and S(1, 1) by rotating the spin in QD2.
Because the opposite (0, 2) → (1, 1) transition (or, more generically, (n 1 − 1, n 2 + 1) → (n 1 , n 2 )) is never blocked (there is always a (1,1) spin singlet to tunnel to), the Pauli blockade regime can be revealed by source-drain current rectification [33]. fig. 2 presents measurements of the source-drain current, I ds , as a function of (V g1 , V g2 ) in a charge configuration exhibiting Pauli rectification. fig. 2a corresponds to a source-drain bias voltage V ds = −2.5 mV and a magnetic field B = 0.7 T. Current flows within characteristic triangular regions [32] where the electrochemical potential of dot 1, µ 1 (n 1 , n 2 ), is lower than the electrochemical potential of dot 2, µ 2 (n 1 −1, n 2 +1). The energy detuning ε between the two electrochemical potentials increases when moving along the red arrow. Current contains contributions from both elastic (i.e. resonant) and inelastic inter-dot tunneling. fig. 2b shows that reversing the bias voltage (i.e. V ds = 2.5 mV) yields the desired Pauli rectification characterized by truncated current triangles (In Supplementary Note 1 we discuss the presence of a concomitant valley-blockade effect similar to the one shown by Hao et al. [12]).
The extent of the spin-blockade region measured along the detuning axis corresponds to the energy splitting, ∆ ST , between singlet and triplet states in the (n 1 − 1, n 2 + 1) charge configuration (which is equivalent to (0, 2)), basically the singlet-triplet splitting in QD2 1 .
We find ∆ ST = 1.9 meV. figures 2-c) and 2-d) show I ds as a function of B and ε for negative and positive V ds , respectively. As expected [33], in the non-spin blocked polarity ( fig. 2c) I ds shows essentially no dependence on B. In the opposite polarity, spin blockade is lifted at low field (B 0.1 T), due to spin-flip cotunneling [34,35], as well as at B = 0.31 T. This unexpected feature will be discussed later.
We now focus on the spin resonance experiment. To manipulate the spin electrically, we set V g1 and V g2 in the spin blocked region and apply a microwave excitation of frequency ν on gate 2. fig. 3a displays I ds as a function of B and ν at constant power at the microwave source (The power at the sample depends on ν and is estimated to be −53 dBm 0.7 mV peak at ν ≈ 9.6 GHz). Several lines of increased current are visible in this plot, highlighting resonances along which Pauli spin blockade is lifted. They are labeled A, B, C and V. In the simplest case, spin resonance occurs when the microwave photon energy matches the Zeeman splitting between the two spin states of a doublet, i.e. when hν = E Z = gµ B B.
We assign such a resonance to line A, because line A extrapolates to the origin (B = 0, ν = 0). Its slope gives g A = 1.980 ± 0.005, which is compatible with the g-factor expected for electrons in silicon [31]. Also line C extrapolates to the origin but with approximately half the slope (i.e. g C = 0.96 ± 0.01). We attribute this line to a second-harmonic driving process [36].
We now focus on resonances B and V. The slope of line B is also compatible with the electron g-factor (g B = 2.00 ± 0.01). However, line B crosses zero-frequency at B V = 0.314 ± 0.001 T, corresponding exactly to the magnetic field at which the non-dispersive resonance V appears. Consequently, line B can be assigned to transitions between spin states associated with two distinct orbitals. When these spin states cross at B V , Pauli spin blockade is lifted independently of the microwave excitation leading to the non-dispersive resonance V (see Supplementary Note 1 for details on the lifting of spin blockade at B V ).
In order to understand the experimental EDSR spectrum of fig. 3a, we neglect in a first approximation the hybridization between the two QDs and consider only QD2 filled with one electron. We have developed a model that accounts for the mixing between spin and valley states due to SOC. In our silicon nanowire geometry, the confinement is strongest along the z direction (normal to the SOI substrate), so that the low-energy levels belong to the ∆ ±z  The RF magnetic field associated with the microwave excitation on gate 2 is too weak to drive conventional ESR [25]. Since a pure electric field cannot couple opposite spin states, SOC must be involved in the observed EDSR. The atomistic spin-orbit Hamiltonian primarily couples the different p orbitals of silicon [38]; the ∆ ±z states are, however, linear combinations of s and p z orbitals with little admixture of p x and p y , which explains why the SOC matrix elements are weak in the conduction band of silicon. Yet the mixing between |v 1 , ↑ and |v 2 , ↓ by "inter-valley" SOC can be strongly enhanced when the splitting between these two states is small enough. We can capture the main physics and identify the relevant parameters using the simplest perturbation theory in the limit B B V . The states |⇓ ≡ |v 1 , ↓ and |⇑ ≡ |v 1 , ↑ indeed read to first order in the spin-orbit Hamiltonian where: Therefore, |v 1 , ↑ admixes a significant fraction of |v 2 , ↓ when the splitting ∆ − gµ B B between these two states decreases. As |v 2 , ↓ can be coupled to |v 1 , ↓ by the RF electric field, this allows for Rabi oscillations between |⇑ and |⇓ . Along line A, the Rabi frequency at resonance (hν = g A µ B B) reads: where δV g2 is the amplitude of the microwave modulation on gate 2, D(r) = ∂V t (r)/∂V g2 is the derivative of the total potential V t (r) in the device with respect to the gate potential V g2 , and: is the matrix element of D(r) between valleys v 1 and v 2 . The gate-induced electric field essentially drives motion in the (yz) plane. D v 1 v 2 is small yet non negligible in SOI nanowire devices because the v 1 and v 2 wavefunctions show out of phase oscillations along z, and can hence be coupled by the vertical electric field. The field along y does not result in a sizable D v 1 v 2 unless surface roughness disorder couples the motions along z and in the (xy) plane [39,40]. Although C v 1 v 2 is weak in silicon, SOC opens a path for an electrically driven spin resonance |⇓ → |⇑ through a virtual transition from |v 1 , ↓ to |v 2 , ↓ , mediated by the microwave field, and then from |v 2 , ↓ to |v 1 , ↑ , mediated by SOC. Note, however, that the where |2 and |3 have a mixed |v 1 , ↑ /|v 2 , ↓ character. The maximum Rabi frequency . The Rabi frequency remains however sizable over a few ∆B FWHM . We point out that C v 1 v 2 and D v 1 v 2 may depend on the actual roughness at the Si/SiO 2 interface.
The calculated Rabi frequencies compare well against those reported for alternative silicon based systems. For example, the expected Rabi frequency is around 4.2 MHz for B = 0.35 T, close to anticrossing field B V , and for a microwave excitation amplitude δV g2 = 0.7 mV, close to the experimental value (see fig. 4b). This frequency is comparable with those achieved with coplanar antennas [3,42] and in some experiments with micromagnets [7,9].
One of the most salient fingerprint of the above EDSR mechanism is the dependence of f on the magnetic field orientation. Indeed, it must be realized that C v 1 v 2 may vary with the orientation of the magnetic field (as the spin is quantized along B in Eq. (2)).
Actually, symmetry considerations supported by TB calculations show that C v 1 v 2 and hence f are almost zero when B is aligned with the nanowire axis, due to the existence of a (yz) mirror plane perpendicular to that axis (see Supplementary Note 4). As a simple hint of this result, we may consider a generic Rashba SOC Hamiltonian of the form where E is the electric field, p the momentum, and σ the Pauli matrices. Symmetric atoms on each side of the (yz) plane contribute to C v 1 v 2 with opposite E x and p x components.
The current on line A is, to a first approximation, proportional to f 2 [43,44]. The TB f 2 is plotted in fig. 4c as a function of the angle θ between an in-plane magnetic field B ⊥ z and the nanowire axis x. It shows the ∝ sin 2 θ dependence expected from the above considerations. The experimental I ds , also plotted in fig. 4c, shows the same behavior, supporting our interpretation. The fact that I ds remains finite for B x may be explained by the fact that the (yz) symmetry plane is mildly broken by disorder and voltage biasing.
In a recent work, Huang et al. [45] proposed a mechanism for EDSR based on electrically induced oscillations of an electron across an atomic step at a Si/SiO 2 or a Si/SiGe hetero-

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding authors upon reasonable request.  Supplementary notes for "Electrically driven electron spin resonance mediated by spin-valley-orbit coupling in a silicon quantum dot"

SUPPLEMENTARY NOTE 1: VALLEYS AND SPIN-VALLEY BLOCKADE
In the main text, we have discussed the nature of the A, B, C and V line in a one-particle picture. In this supplementary note, we introduce a two-particle picture for the blockade, is hence suppressed. At reverse source-drain bias, the current flows through the sequence of charge configurations (2n + 1, 2m + 1) → (2n, 2m + 1) → (2n, 2m + 2) → (2n + 1, 2m + 1)..., which can not be spin-blocked, giving rise to current rectification (see Fig. 2 of main text).

The observation of inter-valleys resonances suggests that m is even (otherwise only tran-
sitions between v 2 states would be observed in dot 2). We assume from now on that n is also even. As a matter of fact, the absence of visible bias triangles for lower gate voltages suggests that m = n = 0, though we can not exclude the existence of extra triangles with currents below the detection limit.
We discuss below the role of valley blockade in the present experiments. We assume the valley splitting is much larger in dot 1 (∆ 1 ) than in dot 2 (∆ 2 = 36 µeV) due to disorder and bias conditions. The valley splitting in dot 1 is actually beyond the bandwidth of the EDSR setup. This reflects the stochastic variations from one dot to an other, as confirmed by tight-binding simulations.

S1
(1,1) Similar states can be constructed in the (0, 2) charge configuration. The S v 1 v 1 (0, 2) and S v 1 v 1 (1, 1) are detuned by the bias on gates 1 and 2. We focus on detunings smaller than the orbital singlet-triplet splitting ∆ ST = 1.9 meV, so that neither the v 1 v 1 nor the v 2 v 2 triplets can be reached from the (1, 1) states.
Given the small ∆ 2 = 36 µeV extracted from spin resonance, we need, however, to recon-S2 sider the mechanisms for current rectification. Indeed, the system must be spin-and-valley blocked [S3] since the detuning is typically much larger than ∆ 2 so that T v 1 v 2 (0, 2) states are accessible in the bias window. Assuming that both spin and valley are conserved during tunneling, the spin and valley blocked (1, 1) states are actually are, in principle, also spin and valley blocked, they may be mixed with the nearly degenerate S v 1 v 1 and S v 2 v 2 states by, e.g., spin-orbit coupling (SOC) and nuclear spin disorder, [S4, S5] and be therefore practically unblocked.
We states. These transitions give rise to the same spectrum as in the one-particle picture.
Line V is independent on the microwave frequency and also appears when no microwaves are applied. At the magnetic field B V ∆ 2 /(gµ B ), the states We consider a silicon QD with strongest confinement along the z direction so that the low-lying conduction band levels belong to the ∆ ±z valleys. This dot is controlled by a gate with potential V g . In the absence of valley and spin-orbit coupling, the ground-state is fourfold degenerate (twice for spins and twice for valleys). Valley coupling[S6-S10] splits this fourfold degenerate level into two spin-degenerate states |v 1 , σ and |v 2 , σ with energies E 1 and E 2 , separated by the valley splitting energy ∆ = E 2 − E 1 (σ = ↑, ↓ is the spin index). In the simplest approximation, |v 1 , σ and |v 2 , σ are bonding and anti-bonding combinations of the ∆ ±z states.
The remaining spin degeneracy can be lifted by a static magnetic field B. The energy of state |v n , σ is then E n,σ = E n ± 1 2 gµ B B (+ for up states, − for down states, the spin being quantized along B). Here g is the gyro-magnetic factor of the electrons, which is expected to be close to g 0 = 2.0023 in silicon. [S11] We may neglect the effects of the magnetic field on the orbital motion of the electrons in a first approximation. The wave functions ϕ n,σ (r) = r|v n , σ can then be chosen real.
The gate potential is modulated by a RF signal with frequency ν and amplitude δV g in order to drive EDSR between states |v 1 , ↓ and |v 1 , ↑ . At resonance hν = gµ B B, the Rabi frequency reads: where D(r) = ∂V t (r)/∂V g is the derivative of the total potential V t (r) in the device with respect to V g . 2 We discard the effects of the displacement currents (concomitant ESR), which are negligible. [S12] In the absence of SOC, hf is zero as an electric field can not couple opposite spins.
As discussed in the main text, spin-orbit couples the orbital and spin motions of the electron.[S3, S13-S17] "Intra-valley" SOC mixes spins within the ∆ +z or the ∆ −z valley, while "inter-valley" SOC mixes spins between the ∆ +z and ∆ −z valleys. Both intra-valley and inter-valley SOC may couple |v 1 , σ with the excited states of valley 1. Inter-valley SOC may also couple |v 1 , σ with all v 2 states. Its effects can be strongly enhanced if the valley splitting ∆ is small enough. Indeed, in the simplest non-degenerate perturbation theory, the states |⇓ ≡ |v 1 , ↓ and |⇑ ≡ |v 1 , ↑ read to first order in the spin-orbit Hamiltonian H SOC : with The above equalities follow from time-reversal symmetry considerations for real wave func- We have neglected all mixing beyond the four |v n , σ , because the higher-lying excited states usually lie 1 meV above E 1 and E 2 (as inferred from ∆ ST on Fig. 2, main text).
Inserting Eqs. (S2) into Eq. (S1), then expanding in powers of B yields to first order in B and H SOC : where is the electric dipole matrix element between valleys v 1 and v 2 . As expected, the Rabi frequency is proportional to δV g D v 1 v 2 , C v 1 v 2 , and to B (as the contributions from the ∝ C v 1 v 2 terms in Eqs. (S2) cancel out if time-reversal symmetry is not broken by the ∝ gµ B B terms of the denominators). It is also inversely proportional to ∆ 2 ; namely the smaller the valley splitting, the faster the rotation of the spin. R v 1 v 2 does not contribute to lowest order because it couples states with the same spin.
C v 1 v 2 and D v 1 v 2 are known to be small in the conduction band of silicon.[S3, S13, S14, S16, S17] Actually, D v 1 v 2 is zero in any approximation that completely decouples the ∆ ±z valleys (such as the simplest effective mass approximation). It is, however, finite in tightbinding[S18, S19] or advanced k · p models [S7, S9] for the conduction band of silicon. According to Eq. (S4), the Rabi frequency can be significant if E 1,↑ is close enough to E 2,↓ to S5 enhance spin-valley mixing by H SOC . This happens when ∆ is small and/or when gµ B B ≈ ∆ (see later discussion). The main path for EDSR is then the virtual transition from |v 1 , ↓ to |v 2 , ↑ (mediated by H SOC ), then from |v 2 , ↑ to |v 1 , ↑ (mediated by the RF field).
The above equations are valid only for very small magnetic fields B, as non-degenerate perturbation theory breaks down near the anti-crossing between E 1,↑ and E 2,↓ when gµ B B ≈ ∆ (see Fig. 3b of the main text). We may deal with this anti-crossing using degenerate perturbation theory in the {|v 1 , ↑ , |v 2 , ↓ } subspace, while still using Eq. (S2a) for state |⇓ . However, such a strategy would spoil the cancellations between |⇓ and |⇑ needed to achieve the proper behavior hf → 0 when B → 0. We must, therefore, treat SOC in the full {|v 1 , ↓ , |v 1 , ↑ , |v 2 , ↓ , |v 2 , ↑ } subspace.[S14, S15] The total Hamiltonian then reads: As discussed before, R v 1 v 2 is not expected to make significant contributions to the EDSR as it mixes states with the same spin. We may therefore set R v 1 v 2 = 0 for practical purposes; H then splits into two 2 × 2 blocks in the {|v 1 , ↑ , |v 2 , ↓ } and {|v 1 , ↓ , |v 2 , ↑ } subspaces.
The diagonalization of the {|v 1 , ↓ , |v 2 , ↑ } block yields energies: and eigenstates: with: and: Likewise, the diagonalization of the {|v 1 , ↑ , |v 2 , ↓ } block yields energies: and eigenstates: with: and: We can finally compute the Rabi frequencies for the resonant transitions between the groundstate |ψ − and the mixed spin and valley states |ψ ± : The Rabi frequencies for the resonant transitions between |ψ + and |ψ ± are equivalent. We can also compute the Rabi frequency between the states |ψ ± : as well as the Rabi frequency between the states |ψ ± : where We have validated the model of supplementary note 2 against tight-binding (TB) calculations.[S18, S19] TB is well suited to the description of such devices as it accounts for valley and spin-orbit coupling at the atomistic level (without the need for, e.g., extrinsic Rashba or Dresselhaus-like terms in the Hamiltonian).
We consider the prototypical device of Fig. S2a is made of a layer of SiO 2 (ε SiO 2 = 3.9) and a layer of HfO 2 (ε HfO 2 = 20). The device is embedded in Si 3 N 4 (ε Si 3 N 4 = 7.5). We did not include the lateral gate in the simulations.
All terminals are grounded except the central gate.
We compute the first four eigenstates |1 ...|4 of this device using a sp 3 d 5 s * TB model. [S20] The dangling bonds at the surface of silicon are saturated with pseudo-hydrogen atoms. We include the effects of SOC and magnetic field. The SOC Hamiltonian is written as a sum of intra-atomic terms [S21] H TB SOC = 2λ where L i is the angular momentum on atom i, S is the spin and λ is the SOC constant of silicon. The action of the magnetic field on the spin is described by the bare Zeeman Hamiltonian H z = g 0 µ B B · S, and the action of the magnetic field on the orbital motion of the electrons is accounted for by Peierl's substitution. [S22] We can then monitor the different Rabi frequencies The wave function of the ground-state |1 is plotted in Fig. S2b (V g = 0.1 V). It is, as expected for etched SOI structures, confined in a "corner dot" below the gate. Note that The energy of the first four eigenstates is plotted as a function of B y in Fig. S3a.
The reduced TB Rabi frequency F = f /δV g between |v 1 , ↓ and |v 1 , ↑ is plotted as a function of B y in Fig. S3. Hence F ≡ F − before the anti-crossing between |v 1 , ↑ and |v 2 , ↓ at B = B V , and F ≡ F + after that anti-crossing. This transition corresponds to line A of the main text. The dependence of F on B is strongly non-linear, with a prominent peak near the anti-crossing. Figure S3.  We can then switch off SOC, recompute the TB eigenstates at B = 0, extract

S9
and input these into Eqs. (S15). As shown in Fig. S3, Eqs. (S15) perfectly reproduce the TB data. There are, in particular, no virtual transitions outside the lowest four |v n , σ that contribute significantly to the Rabi frequency.
The reduced TB Rabi frequency F < = f < /δV g is also plotted on Fig. S4. This transition corresponds to line B of the main text. It also shows a peak near the B = B V , and is pretty strong.
In the present device, the TB valley splitting is much larger than the experiment (∆ 36 µeV). However, ∆ decreases once surface roughness is introduced [S9] and can range from a few tens to a few hundreds of µeV depending on the bias conditions. The TB data reported in Fig. 4 of the main text have been computed for a particular realization of surface roughness S11 disorder that reproduces the experimental ∆ 36 µeV. The surface roughness profiles used in these simulations have been generated from a Gaussian auto-correlation function with rms ∆ SR = 0.4 nm and correlation length Λ SR = 1.5 nm. [S23, S24] Note that disorder also A strategy for the control of the valley splitting in etched SOI devices will be reported elsewhere.
It is clear from Eqs. (S15) that F ± < F max , where hF max = e|D v 1 v 2 | is limited only by the dipole matrix element D v 1 v 2 . In particular, near the anti-crossing field B = B V , only depends on D v 1 v 2 . The SOC matrix element C v 1 v 2 actually controls the width of the peak around B = B V . The full width at half-maximum ∆B FWHM of this peak (F = F ac /2) indeed reads: S12

SUPPLEMENTARY NOTE 4: ROLE OF SYMMETRIES
To highlight the role of symmetries, we plot the reduced TB Rabi frequency F (corresponding to line A of the main text) and SOC matrix element |C v 1 v 2 | as a function of the orientation of the magnetic field in Fig. S5 (same bias conditions as in Fig. S3, |B| = 1.1 T).
As expected, F shows the same trends as |C v 1 v 2 | -although there are small discrepancies.
These discrepancies result from the effects of the magnetic field on the orbital motion of the electron, dismissed in the supplementary note 2. Strikingly, the SOC matrix element and Rabi frequency are almost zero when the magnetic field is aligned with the nanowire axis (x). The dependence of C v 1 v 2 on the magnetic field (or spin) orientation suggests that only the ∝ σ x term of the SOC Hamiltonian is relevant for this matrix element. This can be supported by a symmetry analysis. Now quantifying the spin along z, the TB SOC Hamiltonian [Eq. (S18)] reads: where σ α are the Pauli matrices, L α = i L i,α and L i,α is the component α = x, y, z of the angular momentum on atom i. The device of Fig. S6 is only invariant by reflection through the (yz) plane (space group C s ). The table of characters of this space group is reproduced in R, and wave functions |ψ 1 and |ψ 2 , O 12 = ψ 1 |O|ψ 2 = Rψ 1 |ROR † |Rψ 2 (S24) S13 C s E σ(yz) A 1 1 +1 A 2 1 −1 C 2v E σ(yz) σ(xz) C 2 (z)  Both |v 1 and |v 2 states belong to the irreducible representation A 1 of C s . Therefore, σ(yz)|v 1 = |v 1 and σ(yz)|v 2 = |v 2 . Also, D(r) = ∂V t (r)/∂V g is compatible with the C s symmetry, so that σ(yz)Dσ(yz) † = D. Therefore, Eq. (S24) does not set any condition on D v 1 v 2 . However, σ(yz) transforms a p x orbital into a −p x orbital, hence transforms L y into −L y , L z into −L z , but leaves L x invariant (see Table II). Eq. (S24) then imposes that L y and L z can not couple the |v 1 and |v 2 states. This is why only the ∝ L x σ x term of the atomistic SOC Hamiltonian makes a non-zero contribution. As a result, |C v 1 v 2 | and the Rabi frequency are proportional to | sin θ|, where θ is the angle between x and an in-plane magnetic field. The effective SOC Hamiltonian between |v 1 , ↓ and |v 2 , ↑ is likely dominated by Rashba-type contributions of the form H eff SOC ∝ p y,z σ x (where p y and p z are the momenta along y and z). These Rashba-type contributions may arise from the electric field of the gate along y and z, and/or from the top and lateral Si/SiO 2 interfaces bounding the corner dot.
How far is the EDSR connected with the formation of corner dots in etched SOI structures ? To answer this question, we have computed the Rabi frequency in a standard "Trigate" device where the gate surrounds the channel on three sides (see Fig. S6a). The dimensions are the same as in Fig. S2, but the central gate now covers the totality of the nanowire.
As shown in Fig. S6b, the low-lying conduction band states are still confined at the top interface, but the wave function is symmetric with respect to the (xz) plane (no "corner" effect at zero back gate voltage). It turns out that there is no EDSR whatever the orientation of the magnetic field. Actually, the SOC matrix element C v 1 v 2 is zero for all spin orientations.
Therefore the confinement in a corner dot seems to be a pre-requisite for a sizable EDSR.
As a matter of fact, the devices of Figs. S6 and S2 have different symmetries. Indeed, the device of Fig. S6 is invariant by reflections through the (yz) and (xz) planes, and by the twofold rotation around the z axis (space group C 2v ). The states |v 1 and |v 2 of this device again belong to the irreducible representation A 1 of C 2v (Table I). For the same reasons as before, Eq. (S24) does not set any condition on D v 1 v 2 . The existence of a σ(yz) mirror still imposes that only L x can couple |v 1 and |v 2 states. Yet the introduction of a σ(xz) mirror, which leaves only L y invariant, prevents such a coupling. Therefore, none of the operators L x , L y and L z can couple the |v 1 and |v 2 states. C v 1 v 2 is thus zero, and SOC-mediated EDSR is not possible.
To conclude, the symmetry must be sufficiently low in order to achieve EDSR in the conduction band of silicon. This condition is realized in corner dots where there is only one σ(yz) mirror left (yet preventing EDSR for a magnetic field along the wire axis). The design of geometries without any symmetry left would in principle maximize the opportunities for S15