Spin polarization of composite fermions and particle-hole symmetry breaking

We study the critical spin-polarization energy $\left({\alpha }_{\mathrm{C}}\right)$ above which fractional quantum Hall states in two-dimensional electron systems confined to symmetric GaAs quantum wells become fully spin polarized. We find a significant decrease of ${\alpha }_{\mathrm{C}}$ as we increase the well width. In systems with comparable electron layer thickness, ${\alpha }_{\mathrm{C}}$ for fractional states near Landau level filling $\nu =3/2$ is about twice larger than those near $\nu =1/2$, suggesting a broken particle-hole symmetry. Theoretical calculations, which incorporate Landau level mixing through an effective three-body interaction, and finite layer thickness capture certain qualitative features of the experimental results.

Figure 1

$R_{xx}$ traces for 2D electrons confined to a 65-nm-wide, GaAs QW near $\nu =3/2$. The density for each trace is indicated (in units of ${10}^{11}$ ${\mathrm{cm}}^{-2})$, and traces are shifted vertically for clarity. Top inset shows the $\left(B=0\right)$ calculated charge distributions at $n=0.73$, 1.40, and $1.94\times {10}^{11}$ ${\mathrm{cm}}^{-2}$. Spin transitions of FQHSs, seen as a weakening or disappearing of $R_{xx}$ minima, are marked with arrows.

Figure 2

Summary of the measured $n_C$ above which the FQHSs become fully spin polarized. The dotted lines, drawn as guides to the eye, represent the phase boundaries that separate the partially polarized (below) from the fully polarized states (above). Data are shown for FQHSs near $\nu =3/2$ (closed symbols) and near $\nu =1/2$ (open symbols). The error bar for $n_C$ is about $\pm 5\%$ for FQHSs near $\nu =3/2$ and $\pm 10\%$ for those near $\nu =1/2$.

Figure 3

Summary of ${\alpha }_{\mathrm{C}}$, the critical ratio of the Zeeman to Coulomb energies, needed to fully spin polarize FQHSs at different $\nu$ and in different samples. Dotted lines represent the phase boundaries that separate the partially (below) and fully polarized (above) states. Closed (open) symbols are data measured near $\nu =3/2$ $\left(\nu =1/2\right)$.

Figure 4

Measured critical polarization energy ${\alpha }_{\mathrm{C}}$ as a function of the layer thickness, parametrized by $\lambda /l_B$, at filling factors $\nu =7/5$ (solid symbols) and its particle-hole counterpart at $\nu =3/5$ (open symbols). $\lambda$ is the standard deviation of the electron position, deduced from self-consistent charge distribution calculations, examples of which are shown in the panel above the figure. The dashed curve is a guide to the eye. The solid curves represent the theoretically calculated ${\alpha }_{\mathrm{C}}$ when $\kappa =0$ (black) and 0.6 (red). In this plot, we also include the $\kappa$ value (in blue) of experimental points.