Nature of Tunable Hole $g$ Factors in Quantum Dots

We report an electric-field-induced giant modulation of the hole $g$ factor in SiGe nanocrystals. The observed effect is ascribed to a so-far overlooked contribution to the $g$ factor that stems from the mixing between heavy- and light-hole wave functions. We show that the relative displacement between the confined heavy- and light-hole states, occurring upon application of the electric field, alters their mixing strength leading to a strong nonmonotonic modulation of the $g$ factor.

Figure 1

(a) Schematic cross section of a SiGe QD device. (b)–(d) Color plot of $d{I}_{sd}/d{V}_{sd}(V_{\mathrm{TG}},V_{sd})$ for $B_z=70\mathrm{mT}$, 3 T, and 5 T, respectively ($V_{\mathrm{BG}}=0$). The lines indicated by rhombi correspond to the onset of tunneling via Zeeman-split levels for $N-1$ and $N+1$ holes on the QD. The lines indicated by a star and by a circle correspond to singlet-triplet excitations for $N$ holes.

Figure 2

(a) Left: Color plots of $d{I}_{sd}/d{V}_{sd}(V_{\mathrm{BG}},V_{sd})$ for $B=70\mathrm{mT}$ and 4 T. At 4 T the Zeeman splitting is clearly visible. Right: Corresponding schematic diagram illustrating the measurement principle to extract the Zeeman energy splitting (and hence the $g$ factor) from gate-voltage sweeps at constant $V_{sd}$ (see the horizontal green line). (b) Color plots of $d{I}_{sd}/d{V}_{sd}(V_{\mathrm{BG}},V_{\mathrm{TG}})$ for a fixed $V_{sd}=2.6\mathrm{mV}$. These data sets demonstrate the modulation of $g_{\perp }$ by the top and back gates. (c) $g_{\perp }(V_{\mathrm{BG}},V_{\mathrm{TG}})$ as extracted from (b). Below $g_{\perp }\approx 0.75$ the Zeeman splitting cannot be resolved any more due to the finite broadening of the tunneling resonances.

Figure 3

Expected $g_{\perp }$ ($F$) dependence for a SiGe QD with strong compositional gradient. The numerical result (solid line) has two regimes: the dashed line shows a fit to the expression $⟨\kappa ⟩={\kappa }_{\infty }-\Delta \kappa (1+F/F_{\mathrm{intr}}{)}^{-1/3}$, derived for a triangular potential well with an intrinsic field $F_{\mathrm{intr}}$ for $z<0$ and an infinite barrier for $z>0$; the dotted line is a fit to a linear dependence, obtained for a symmetric potential well. In the latter regime, $g_{\perp }$ is most sensitive to $F$, with $d{g}_{\perp }/dF=0.41\mathrm{m}/\mathrm{MV}$. At large negative $F$, the wave function is pushed into the Si-rich region, where $g_{\perp }<0$. Inset: Energy profiles for the heavy-hole (HH), light-hole (LH), and split-off (SO) bands as calculated from Eq. (2) with $x_{\max }\text{}=0.8$. We set $x=0$ for $z<-20\mathrm{nm}$ to account for the Si substrate, and $x=0$ for $z>0$ to account for the existence of a strained, few-nm-thick Si capping layer.

Figure 4

The $g_{\perp }(F)$ dependence according to Eq. (5) (solid line) superimposed on the experimental data (squares) of Fig. 2c. Since the variations in $V_{\mathrm{BG}}$ and $V_{\mathrm{TG}}$ are proportional to each other, we take $F=a{V}_{\mathrm{TG}}+b$, with fitting parameters $a$ and $b$. The dashed line is an improved fit obtained with a model that allows for a field gradient across the SiGe nanocrystal (such a gradient is expected as a screening effect of the source and drain electrodes). For $z\in [-w,0]$, we assume an electrostatic potential of the form $V(z)=-cF(z+w/2{)}^2-Fz$, where $c$ is a fit parameter. Insets: Schematics of the HH (red) and LH (blue) wave functions at different $F$. At finite, intermediate fields (lowest insets) the two wave functions are shifted relative to each other resulting in the largest $|d{g}_{\perp }/dF|$.