Resolving Spin-Orbit- and Hyperfine-Mediated Electric Dipole Spin Resonance in a Quantum Dot

We investigate the electric manipulation of a single-electron spin in a single gate-defined quantum dot. We observe that so-far neglected differences between the hyperfine- and spin-orbit-mediated electric dipole spin resonance conditions have important consequences at high magnetic fields. In experiments using adiabatic rapid passage to invert the electron spin, we observe an unusually wide and asymmetric response as a function of the magnetic field. Simulations support the interpretation of the line shape in terms of four different resonance conditions. These findings may lead to isotope-selective control of dynamic nuclear polarization in quantum dots.

Figure 1

(a) Scanning electron micrograph of a device similar to the one measured (top) and QPC current (a background plane has been subtracted) as function of LP and RP gate voltages (bottom). $V_L=1.2V_{\mathrm{LP}}+0.6V_{\mathrm{RP}}$ and $V_R=1.4V_{\mathrm{LP}}+0.5V_{\mathrm{RP}}$. ($n$, $m$) indicate the number of electrons in the left and right dot, respectively. (b) Schematic explanation of adiabatic rapid passage (see text). Blue and red solid lines show the electron spin eigenenergies in the presence of an ac excitation transverse to the static magnetic field as a function of detuning from resonance. In the experiment, the excitation frequency was chirped in the direction of the time arrow. The insets show the effective field acting on the spin for three values of detuning ($\kappa =\hslash /g{\mu }_B$, with ${\mu }_B$ the Bohr magneton and $g$ the electron $g$ factor). (c) Electrochemical potential diagrams during the pulse cycle (see text). At the read-out stage, the electron tunnels out if and only if it is spin-down, in which case a step in the QPC current results.

Figure 2

(a) Measured spin-down probability as a function of the magnetic field for a $400\mu \mathrm{s}$ MW burst at $f_{\mathrm{MW}}=26.5\mathrm{GHz}$ (blue) and for a $400\mu \mathrm{s}$ frequency chirped MW burst with a frequency modulation (FM) depth of 40 MHz centered at 26.5 GHz (red). The double arrow shows the magnetic field span corresponding to 40 MHz. Inset: EDSR resonance frequency as a function of the magnetic field, along with a linear fit (gray curve), giving an electron $g$ factor of $-0.339\pm 0.003$. As the resonance position we took half a FM depth above the left flank of the response. (b) Similar to (a), for three different driving amplitudes. The amplitudes are estimated considering the attenuations in the MW line (traces are offset for clarity). Inset: The linewidth saturates as the power increases. (c) Similar to (a) for two different FM depths, both at a chirp rate of $150\mathrm{MHz}/\mathrm{ms}$. Arrows show the expected difference in the linewidth. (d) Measured spin-down probability at $B=5.349\mathrm{T}$ as a function of MW burst duration (FM depth 75 MHz). The gray line is an exponential fit to the data. All measurements in (a), (c), and (d) are taken with the same MW power as the red trace in (b).

Figure 3

(a) Schematic representation of the resonance condition of SOEDSR, ${\omega }_L$ (black), and HFEDSR, ${\omega }_L-{\omega }_N$, for ${}^{75}\mathrm{As}$ (blue), ${}^{69}\mathrm{Ga}$ (red), and ${}^{71}\mathrm{Ga}$ (green) nuclei. The separations are not to scale. The dashed arrow shows the direction of the magnetic field sweep in the measurements. (b) Measured spin-down probability versus the magnetic field for a 75 MHz chirp at 26.5 (black) and 36.8 GHz (red). Both measurements have been done at high MW power, where the linewidth is saturated. Arrows show the relative positions of spin-orbit- and hyperfine-mediated EDSR resonances [same color coding as in (a)].

Figure 4

Simulated spin-down probability (assuming perfect measurement fidelity) versus the magnetic field for a $500\mu \mathrm{s}$, 40 MHz chirp (blue) and a $937\mu \mathrm{s}$, 75 MHz chirp (red) for (a) ${\Omega }_{\mathrm{SO}}=1.25\mathrm{MHz}$, ${\Omega }_{\mathrm{hf}}^{{}^{75}\mathrm{As}}=0.63\mathrm{MHz}$ and (b) ${\Omega }_{\mathrm{SO}}=6.28\mathrm{MHz}$, ${\Omega }_{\mathrm{hf}}^{{}^{75}\mathrm{As}}=5.02\mathrm{MHz}$. The numbers in panel (b) express the number of resonances covered by the chirp. Arrows show spin-orbit- and hyperfine-mediated EDSR resonance positions (color coding as in Fig. 3). Double arrows show the magnetic field range corresponding to the FM depth.