Electrically tunable hole $g$ factor of an optically active quantum dot for fast spin rotations

We report a large $g$ factor tunability of a single hole spin in an InGaAs quantum dot via an electric field. The magnetic field lies in the in-plane direction $x$, the direction required for a coherent hole spin. The electrical field lies along the growth direction $z$ and is changed over a large range, 100 kV/cm. Both electron and hole $g$ factors are determined by high resolution laser spectroscopy with resonance fluorescence detection. This, along with the low electrical-noise environment, gives very high quality experimental results. The hole $g$ factor $g_h^x$ depends linearly on the electric field $F_z,d{g}_h^x/d{F}_z=(8.3\pm 1.2)\times {10}^{-4}$ cm/kV, whereas the electron $g$ factor $g_e^x$ is independent of electric field $d{g}_e^x/d{F}_z=(0.1\pm 0.3)\times {10}^{-4}$ cm/kV (results averaged over a number of quantum dots). The dependence of $g_h^x$ on $F_z$ is well reproduced by a $4\times 4 k·p$ model demonstrating that the electric field sensitivity arises from a combination of soft hole confining potential, an In concentration gradient, and a strong dependence of material parameters on In concentration. The electric field sensitivity of the hole spin can be exploited for electrically driven hole spin rotations via the $g$ tensor modulation technique and based on these results, a hole spin coupling as large as $\sim 1$ GHz can be envisaged.

Figure 1

(a) Layer sequence of the device. On top of the substrate a short-period superlattice (SPS) of AlAs/GaAs periods is grown, followed by an $n^{+}$-doped layer, an intrinsic region with a second SPS, the quantum dot layer and a third SPS, completed with a $p^{+}$-doped layer on top. On the sample surface a semitransparent electrode is fabricated, with a hemispherical solid-immersion lens (SIL) positioned on top. (b) Contour plot of the resonance fluorescence (RF) signal as a function of the applied voltage. The charged exciton $X^{1-}$ in an in-plane magnetic field exhibits four transitions. The four energy versus voltage dependencies are sketched with the solid lines. The dashed line represents the resonant laser frequency. RF peaks are assigned to the energy transitions with two numbers. The first number indicates the excited transition and the second the recombination channel for the emission. The transitions are indicated in (c). Red and black correspond to the diagonal and vertical transitions, respectively. The color scale is a linear representation of the CCD camera output from background counts (white) to maximum counts 1200 cts/s (red). (c) The quantum states of a single electron spin in an in-plane magnetic field. $\uparrow ,\downarrow$ indicate an electron spin, $\Uparrow ,\Downarrow$ a hole spin. (d) Schematic of the sample orientation in the microscope and of the applied fields.

Figure 2

(a)–(d) Resonance fluorescence (RF) spectra of the exciton transitions of QD1 at four fixed resonant laser energies. Detuning is achieved by tuning the voltage applied to the device. The four spectra are taken at different electric fields spanning the working area of the device and at an in-plane magnetic field of 3.00 T. Each of the four peaks corresponds to one of the four transitions in the quantum system [1–4, Fig. 1]. With the help of a quadruple Lorentzian function (red) the peak positions can be determined with a precision of $\pm 0.2 \mu \mathrm{eV}$. (e) All RF peak positions (resonant excitation energy) versus electric field. The energies can be described in each case by a quadratic function of electric field consistent with the dc-Stark effect. The insets, both with a field extent of 9 kV/cm and an energy extent of $200 \mu \mathrm{eV}$, highlight the electric field dependence of the energy splittings. [The colors match the respective RF peaks (a)–(d).]

Figure 3

In-plane electron $g_e^x$ and hole $g_h^x g$ factors of two QDs as a function of a perpendicular electric field $F_z$. The electron $g$ factor (green and olive, $E_1-E_3$ and $E_2-E_4$), with a mean value of $g_e^x=-0.39\pm 0.03$ is not influenced by the external electric field. The in-plane hole $g$ factor (red and black, $E_1-E_2$ and $E_3-E_4$) can be tuned with voltage at a rate of $d{g}_h^x/d{F}_z=(8.3\pm 0.1)\times {10}^{-4}$ cm/kV for QD1 (circle) and $d{g}_h^x/d{F}_z=(7.9\pm 0.9)\times {10}^{-4}$ cm/kV for QD2 (triangle).

Figure 4

(a) The in-plane hole $g$ factor for QD2 at 3.00 T. For a vertical hole confinement length of 2.4 nm the theoretical description matches the experimental data (solid green line). (b) Theoretical analysis of the electric field dependence of the in-plane hole $g$ factor for different values of the vertical hole confinement length.