Spin susceptibility and effective mass of two-dimensional electrons in ${\text{Mg}}_x{\text{Zn}}_{1-x}\text{O}/\text{ZnO}$ heterostructures

We report measurements of the spin susceptibility and the electron effective mass for two-dimensional electrons confined at the interfaces of ${\text{Mg}}_x{\text{Zn}}_{1-x}\text{O}/\text{ZnO}$ single heterostructures ($x=0.05$, 0.08, and 0.11), grown by molecular-beam epitaxy on (0001) ZnO substrates. By tuning the built-in polarization through control of the barrier composition, the electron density was systematically varied in the range of $5.6\times {10}^{11}–1.6\times {10}^{12}{\text{cm}}^{-2}$, corresponding to a range of $3.1\le r_s\le 5.2$, where $r_s$ is the average electron spacing measured in units of the effective Bohr radius. We used the coincidence technique, where crossings of the spin-split Landau levels occur at critical tilt angles of magnetic field, to evaluate the spin susceptibility. In addition, we determined the effective mass from the temperature dependence of the Shubnikov–de Haas oscillations measured at the coincidence conditions. The susceptibility and the effective mass both gradually increase with decreasing electron density, reflecting the role of electron-electron interaction.

Figure 1

Longitudinal resistivity ${\rho }_{xx}$ and Hall resistivity ${\rho }_{xx}$ vs $B$ measured for sample A at 0.3 K. Inset depicts ${\rho }_{xx}$ at low magnetic field. The oscillations start developing at an onset of 0.7 T with minima at odd Landau filling factors $\nu$.

Figure 2

(a) Measurement configuration in tilted magnetic field. (b) Schematic fan diagram showing spin-split Landau levels as a function of tilt angle $\theta$. (c) ${\rho }_{xx}$ vs $B_{\perp }$ measured for sample A at 0.3 K. Data taken at various tilt angles are shown and shifted vertically for clarity. Coincidences at $i=1$ and 2 are emphasized as bold traces.

Figure 3

(a) Temperature dependence of ${\rho }_{xx}$ vs $1/B_{\perp }$ traces recorded for sample A at the first coincidence angle $\left(\theta =57.12°\right)$. Inset depicts the temperature dependence of the logarithmic amplitude of ${\rho }_{xx}$ minima at odd $\nu$ ranging from 15 to 35 (filled squares). Solid lines represent least-squares fits to the plots; the slopes of these lines give $m^{\ast }$. (b) The deduced $m^{\ast }/m_0$ at the first and second coincidence angles are plotted against $\nu$ as closed and open circles, respectively. The error bars are from standard deviations of the fits to $\text{ln}\left(A/T\right)$ vs $T$. Inset depicts $m^{\ast }/m_0$ plotted against perpendicular magnetic field $B_{\perp }$, where linear least-squares fits (solid lines) extrapolate to $m^{\ast }/m_0$ at $B_{\perp }=0$.

Figure 4

$g^{\ast }{m}^{\ast }/g_b{m}_b$ and $m^{\ast }/m_b$ as a function of 2D electron density, $n$. Data are evaluated for samples A–E from measurements carried out at the first coincidence angles. In addition, $g^{\ast }{m}^{\ast }/g_b{m}_b$ for samples A, B, and D can be evaluated at the second coincidence angles. The one-sided error bars for $m^{\ast }$ represent the lowest value of $m^{\ast }$ measured from the low-field range and the extrapolated $\left(B=0\right)$ value [see inset of Fig. 3b]. The normalized average interelectron spacing $r_s$ is given in the top abscissa. The horizontal line marks bulk values for $g^{\ast }{m}^{\ast }$ and $m^{\ast }$.