1 Supplementary Information: Electrically Driven Single Electron Spin Resonance in a Slanting Zeeman Field. Pioro-Ladrière, T. Obata, Y. Tokura, Y.-S. Shin, T. Kubo, K. Yoshida, T. Taniyama, S. Tarucha A. Supplementary Notes A1. Engineering the stray magnetic field to address several spins in a quantum dot array. A. Derivation of Equation 1. B. Supplementary Figures 1. ESR response below Teslas.. Power dependence of ESR peak height. 3. icro-magnet technology for scalability
A. Supplementary Notes A.1. Engineering the stray magnetic field to address several spins in a quantum dot array. We show in this note how several spins could be addressed using a common ESR gate electrode by slightly modifying the micro-magnet geometry. As in (19), we assume a linear array of gate-defined quantum dots, each holding a single electron (Fig. S3A). To perform single spin operation on any spin of the array, we employ the concept demonstrated in the main text, i.e. frequency selective voltage driven ESR. To achieve this, two large ferromagnetic strips are deposited on top of the quantum dot array. The separation between the two strips has two purposes. The first one is to produce a transverse slanting magnetic field at each dot location (Fig. S3B). The second one is to provide a strong ac electric field, which is approximately equal to the applied ac voltage between the two strips over the separation. By adjusting the frequency, phase and duration of the electric field, any single spin rotation can then be realized. The frequency selectivity is achieved by tapering the strip separation, yielding different a Zeeman field Bi B + δbi for each dot location i. This is possible because the stray field induced shift, δ, is a function of the separation in the taper. For 1% B i addressability efficiency, the difference in Larmor frequency, spins must be greater than the inverse of the intrinsic spin coherence time d i f, between two adjacent. In Fig. S3C, we present simulation results for an array of 6 spins using realistic device parameters. The 1 chosen taper yields f 1 z, which is much bigger than T for nuclei spin free systems (6). The field gradient, throughout the array. b SL, necessary for fast single spin rotation is above 1 T/µm The hybridization between the spin and charge degrees of freedom induced by the slanting magnetic field slightly modifies the exchange interaction between neighboring spins. owever, as demonstrated in (6), the CNOT gate can still be realized in such hybrid T
3 systems. Combined with the single spin addressability explained above, the proposed architecture brings all-electrical universal control of electron spins within reach. A.. Derivation of Equation 1 In this note, we derive the expression for the effective ESR field arising from both the slanting magnetic field and the spin-orbit interaction. We consider the simplified geometry depicted in Fig. 1b comprising only one of the two quantum dots, the micro-magnet and the high-frequency gate. The origin is taken at the quantum dot location and the x and are oriented along the y axis [ 11] and [ 11 ] crystal directions, respectively. Incidentally, both the external magnetic field, B r, and the dominant component of the ac electric field are aligned along the [ 11] direction in our experiment. As discussed below and in (13), the precise orientation of these two fields with respect to the crystal directions plays an important role for the spin-orbit contribution. Because of the strong vertical confinement in our structure, only the in-plane motion of the electron in the quantum dot is considered, which can be described by the twodimensional amiltonian QD px + py + V ( x, y) + eeac sin( πft)x (A..1) m where are the momentum operators associated to the in-plane coordinates, m the p x, y electron s effective mass and e its electric charge. The three terms in Eq. A..1 are, from left to right, the kinetic energy, the lateral confinement energy and the Coulomb energy originating from the ac electric field (oscillating with amplitude E at frequency f ). AC The electron spin is subject to the total magnetic field (including the external and micro-magnet s stray fields), B r, with corresponding Zeeman amiltonian
4 r r gµ B σ / (A..) B r where σ ( σ, σ, σ ) x y z are the Pauli spin matrices, g the g-factor and µ B the Bohr magneton. Assuming for simplicity no misalignment between the quantum dot and the micro-magnet, the magnetic field is very well approximated by the slanting form (7) r B ( B + δb b SL z)ˆ x b SL xˆ z (A..3) The linear term in z ensures that B r satisfies axwell s equations but does not play a role, again due to the strong confinement in the z direction. Notice that the sign of the gradient ( is negative in our geometry) is explicitly taken into account in the above expression. b SL The spin-orbit interaction, SO, which also influences the electron is given by SO α( p x σ y p y σ x )+ β p ( x σ x + p y σ y ) (A..4) where α and β are the Rashba and Dresselhaus spin-orbit coefficients, respectively, and σ are the momentum and spin operators in the x and y directions (along the p x, y x, y [1] and [1] crystal directions, respectively). Adding all these terms gives the total amiltonian + + (A..5) QD SO which we use to model the dynamic of the electron whose spin and charge degrees of freedom are coupled together by the slanting field (via the term xσ ) and the spin-orbit interaction (via the terms p σ ). i j x As suggested in (4), the spin-orbit coupling can be conveniently accounted up to first order in α and β by the canonical transformation: U U where
5 U Exp[in r r σ /]. The transformation leaves QD unaffected but transforms r r r + SO into + SO gµ B ( B + BSO ) σ /, resulting in a position dependent r r r correction to the magnetic field: B n B. Neglecting the terms proportional to α bsl, αδ B, bsl SO β and βδ B, the spin-orbit effective magnetic field is further simplified to n m h r B SO r r n B; m y h ( αy + βx ); n ( αx + βy ) x z ; n (A..6) Evaluating Eq. A..6 for our specific orientation, we get the simple expression r B S B xˆ z l so (A..7) where 1 1 1 l so l α lβ is defined in terms of the two spin-orbit lengths l h h ; l β (A..8) m α m β α ( h is the reduced Plank constant). Comparing Eq. A..7 with Eq. A..3, we conclude that the magnetic field correction is of identical spatial dependence as the micro-magnet s stray field. This effective slanting field, with gradient B / l so, reduces (for l so > ) or enhances (for l so < ) the total magnetic field gradient. The fit with negative slope observed in Fig. 3 implies l α < l β for our experiment. To calculate the effective ESR field, we first decompose the transformed amiltonian into a static and a time dependent part as
6 W V S S + V ( t) ; + W; px + py + V m gµ BbSLxσ z ; ( x, y) () t ee sin( πft)x AC gµ BBσ x ; (A..9) ere, B B + δb is the total in-plane magnetic field ( δ B is the stray magnetic field shift) and b SL b SL B /l so is the total field gradient including the spin-orbit correction. We then obtain the eigenfunctions of the static part of, S, by perturbation theory in W. For the unperturbed part, the Pauli spin matrices are decoupled from the quantum dot orbitals φ n ( x, y). The eigenenergies and eigenfunctions of are ε n gµ B B σ and x, y n, σ φn ( x, y) ψ σ where n 1,,..., σ ± 1 and ψ σ is the ε nσ spinor of the σ x eigenstates. The electron ground state wavefunction is spin split by the Zeeman energy, which is smaller than the orbital excitation energy: E Z gµ B B < ε ε 1. The perturbation W mixes the electron spin states with the orbital states. The two lowest energy states perturbation theory: g σ, which constitute a hybridized spin are calculated by q 1 gσ 1σ + gµ Bb SL x 1 q + E qσ (A..1) q>1 Z ( σ σ ) where σ σ and q ε q ε1. We assume symmetric confinement in the x direction such that the unperturbed states satisfy the relation qσ x qσ. The slight hybridization of the spin states with the quantum dot orbital states reduces the g-factor (6). owever, this renormalization is less than 1% and is therefore neglected in our analysis.
7 To find the effect of the ac electric field on the hybridized spin, we use the fact that only the off-diagonal matrix elements of V ( t) remain in the subspace spanned by { g σ + 1, gσ 1 }. The amiltonian of the hybridized spin is thus of the ESR form: ESR 1 gµ B B[ σ + B sin ( πft )σ z AC x ] (A..11) with the effective ESR field given by 1b x q q eeacx 1 B (A..1) AC q SL q> 1 q E Z where σ x, z are the Pauli matrices associated to the two-level system. For an explicit 1 expression, we assume parabolic confinement, V ( x, y) m ω ( x + y ), and restrict the orbital spaces to the lowest three levels q ( n,m) (,), (,1), (, 1) (characterized by the radial and angular momentum quantum numbers ( ) l orb / where h /( m ω ) ( n,m) ) with matrix elements (,±1)x, l orb. Using these matrix elements, we obtain the form B ee l b /( ) which reduces finally to Eq. 1 of the main text when AC hω >> E Z. AC orb SL E Z
8 B. Supplementary Figures a b Supplementary Figure 1. a Evolution of the ESR peak position (associated to the left dot) as the micro-magnet is magnetized by the external field B. Plotted is the value of total quantum dot Zeeman field B L B + δb L normalized over the resonant frequency f. A constant in-plane shift δ B L -6 mt is assumed. The red line is obtained using g.41. The deviation indicates variation in the stray Zeeman field by the micro-magnet magnetization process. b Corresponding peak height, normalized over the saturation current and the square of the ac electric field E. The error bar is dominated by the uncertainty in E and I. The red AC curve is a parabolic fit with small positive offset. The ESR peaks are obtained at microwave power below the onset of ESR saturation where the height is approximately proportional to I b SL E AC. The parabolic increase is consistent with a AC linear magnetization curve since b SL. The residual peak at zero external field can be attributed to ESR mediated by the hyperfine field as demonstrated in (9).
9 a b -499 I dot (fa) V R (mv) 1 5 hf -515-756 V L (mv) -74 n n 1 c Supplementary Figure. a Stability diagram obtained at the resonance condition for the left dot. Parameters are the same as in Fig. A. The circles indicate typical operation points used for power dependence. b Energy diagrams corresponding to the two operation point in a. In both diagrams, ESR lifts off the spin blockade. For the n ( n 1) case, the electron in the right dot tunnels elastically (inelastically) to the left dot. The inelastic transition is enabled by the ac potential drop across the
1 barrier separating the two dots, V b, through photon assisted tunnelling (PAT). The label n indicates the number of microwave photon absorbed in the tunnelling process. For each case, the tunnelling rate is proportional to the square of the corresponding Bessel function, ( α ) J n, where α ev b /(hf ) characterizes the strength of PAT (16). c ESR peak height as function of parameter α, obtained at frequency f.1 Gz. The purple (green) data points are taken at the n ( n 1) operation point by varying the microwave source power. Each data point is obtained by extracting the maximum dot current,, from a single external magnetic field sweep across the resonance condition. The arrow labelled the saturation point of the ESR induced spin-flip rate I dot Γ ESR α marks. The solid lines are fits of the saturated region ( α > α ) to the expected J ( α ) behaviour For α > α, does not depend on power and the current follows the Bessel function. For α < α, the current is also modulated by the power dependent n Γ ESR Γ ESR and therefore deviates from the solid curves. The two independent fits yield similar amplitudes A 611 fa and A 1 634 fa. The saturation curve shown in Fig. 3 is obtained by dividing the n data by the power-dependent saturation current I A J ( α ). The saturation field, E AC, is estimated using the value of α and the simple relation Vb EAC / d where d is the distance between the two dots, taken as 1 nm. This method allows us to estimate directly the quantum dot ac field as function of the microwave source power.
11 a icro-magnet Insulator y x d 1 d d 3 d 4 V AC B Dot array J-gate icro-magnet 5 nm b c d i S i t s δb (mt) z b SL δb y b SL (T/µm) Location index, i Supplementary Figure 3. a Quantum dot array with specially designed micro-magnet assembly. The assembly consists of two ferromagnet strips with a tapered separation d i. b Cross-sectional view at the i-th quantum dot location. The two strips are magnetized uniformly along the in-plane direction y using the external Zeeman field B. The stray field transverse component (z-component) at the dot location (shown by red arrows) is of the slanting form with gradient b SL. The in-plane component (shown in black, y-component)
1 shifts the Zeeman field by an amount δ B. c Numerical simulation of the in-plane shift and transverse gradient calculated at each spin location. The array consists of 6 quantum dots. The separation increases in increment, d d i +1 d, of 3 nm with d i1 7 nm. The i micro-magnet thickness, t, is 15 nm. A distance s of 1 nm from the quantum dot plane to the insulating layer top surface is assumed. The J-gates are used for two-spin SWAP operation as explained in (14) and demonstrated in (5).