$g$-factors in ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ quantum dots

We investigate the $g$-factors of individual electron states in gate-defined quantum dots fabricated from ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ heterostructures. We consider both the case of effective positive charging energy ($U>0$) where single electrons are added upon increasing the local gate voltage, and the case of $U<0$ where electron pairing is observed. The $g$-factors are extracted from the field dependence of the quantum dot addition spectrum. Tunnel couplings and confinement are tunable by the gate voltages and in the regime of weakest coupling, we find $g$-factors close to 2 due to quenching of the orbital magnetic moment. For stronger coupling, $g$-factors are anisotropic and exhibit values up to $\sim 4.5$ in the out-of-plane orientation. We further show examples of the sequential addition of electrons with the same spin as a consequence of exchange interactions.

Figure 1

(a) Free energy as a function of gate voltage for three successive charge states in the case of $U>0$ (orange curves) and $U<0$ (blue curves). Solid (dashed) lines correspond to $B=0$ ($B>0$). The odd-$N$ parabolas exhibit Zeemann splitting and the intersection points between even and odd charge states (circles), where zero-bias transport is allowed, shift linearly along $V_{\mathrm{sg}}$ (circles). The inset shows a top view scanning electron microscopy image of device S1. The scale bar is $100\mathrm{nm}$. (b), (d) $G$ vs $V_{\mathrm{sg}}$ and $V_{\mathrm{sd}}$ for samples S1 and S2, respectively. Dashed lines in (b) indicate the Coulomb blockade diamonds. (c), (e) Conductance vs $V_{\mathrm{sg}}$ and $B$ for $U>0$ (S1) and $U<0$ (S2), respectively. The circles indicate the linear shift of a pair of Coulomb peaks [cf. (a)].

Figure 2

Field dependence of the $V_{\mathrm{sg}}$ separation of zero-bias CB peaks. The values have been converted into energy by scaling with the lever arm $\alpha$. (a) and (b) show results for devices S1 and S2 with effective $U>0$ and $U<0$, respectively, and symbols correspond to those in Figs. 1 and 1. The corresponding $g$-factors are found from linear fits (solid lines).

Figure 3

(a) Bias spectroscopy of sample S2 in a regime of higher $V_{\mathrm{sg}}$. (b) Zero-bias conductance vs $V_{\mathrm{sg}}$ for equidistant values of magnetic field magnitude from $-1$ to $1\mathrm{T}$; curves are offset by $(B+1)e^2/h$ with $B$ in tesla. The three panels show measurements for different orientations of $B$ as indicated. For each trace, the peak positions indicated by markers are found by fitting the trace to a single- or double-peak profile and the pairing field is identified. (d)–(f) Anisotropy of $g_e$ extracted from $B_{\mathrm{p}}$ (see text).

Figure 4

(a) Zero-bias conductance vs $V_{\mathrm{sg}}$ and $B_{\mathrm{z}}$ for sample S1 in a different gate regime. (b) The field dependence of the peak separations offset to coincide at $B=0$. Solid lines show linear fits and the corresponding $g$-factors are indicated. Field-independent peak spacings (solid symbols) result from the successive addition of electrons to the QD with the same spin.