Spin and the Coulomb gap in the half-filled lowest Landau level

The Coulomb gap observed in tunneling between parallel two-dimensional electron systems, each at half-filling of the lowest Landau level, is found to depend sensitively on the presence of an in-plane magnetic field. Especially at low electron density, the width of the Coulomb gap at first increases sharply with in-plane field, but then abruptly levels off. This behavior appears to coincide with the known transition from partial to complete spin polarization of the half-filled lowest Landau level. The tunneling gap therefore opens a window onto the spin configuration of two-dimensional electron systems at high magnetic field.

Figure 1

(a) Typical tunneling $IV$ characteristic at $\nu =1/2$. Data taken at $T=50$ mK with $B_{\perp }=4.09$ T and $B_{\left|\right|}=0$. $I_{\text{max}}$ and $V_{\text{max}}$ denote location of maximum tunneling, while $\mathrm{\Delta }$ is voltage where tunneling current has reached $0.02I_{\text{max}}$. (b) Density dependence of $V_{\text{max}}$ at $B_{\left|\right|}=0$. Solid line is an unweighted least-squares fit of $V_{\text{max}}$ versus $n^{1/2}$; the fit extrapolates to $V_{\text{ex}}=-2.0$ mV at $n=0$.

Figure 2

Effect of an in-plane magnetic field on the tunneling $IV$ curve at $\nu =1/2$. (a) Maximum tunneling current versus $q\ell$ at various densities. Dashed line is theory prediction described in text. (b) Normalized $\nu =1/2$ tunneling $IV$ curves at $n=4.24\times {10}^{10}{\mathrm{cm}}^{-2}$ for $B_{\left|\right|}=0$ and $B_{\left|\right|}=2.9$ T. Upward arrows near lower left show change in $\mathrm{\Delta }$ induced by the in-plane field. Data taken at $T=50$ mK.

Figure 3

Effect of in-plane magnetic field on $\nu =1/2$ tunneling critical points, $V_{\text{max}}$ and $\mathrm{\Delta }$, at three different densities, plotted versus total magnetic field $B_{\text{tot}}$. For each density, the perpendicular field $B_{\perp }$ is fixed while the in-plane field $B_{\left|\right|}$ is varied. The leftmost data point of each set corresponds to $B_{\left|\right|}=0$ (tilt angle $\theta =0)$. The arrows indicate our assignment of the “knee” in the $\mathrm{\Delta }$ vs $B_{\text{tot}}$ data.

Figure 4

Solid red circles: Critical normalized Zeeman energy determined from knee in Coulomb gap $\mathrm{\Delta }$ data vs 2DES density $n$. Open circle: CF spin transition point found by Tracy et al. [30]. Dashed line: Theoretical estimate of Park and Jain [34].