What Determines the Fermi Wave Vector of Composite Fermions?

Composite fermions (CFs), exotic particles formed by pairing an even number of flux quanta to each electron, provide a fascinating description of phenomena exhibited by interacting two-dimensional electrons at high magnetic fields. At and near Landau level filling $\nu =1/2$, CFs occupy a Fermi sea and exhibit commensurability effects when subjected to a periodic potential modulation. We observe a pronounced asymmetry in the magnetic field positions of the commensurability resistance minima of CFs with respect to the field at $\nu =1/2$. This unexpected asymmetry is consistent with the CFs’ Fermi wave vector being determined by the minority carriers in the lowest Landau level, and suggests a breaking of the particle-hole symmetry for CFs near $\nu =1/2$.

Figure 1

(a) Magnetoresistance trace for a 2DES with density $n=1.74\times 10^{11}{\mathrm{cm}}^{-2}$ and subjected to a periodic potential modulation, exhibiting strong CF commensurability oscillations near $\nu =1/2$. The inset schematically shows the commensurability condition of the quasiclassical CF cyclotron orbits, marked as $i=1$, 2, and 3, with a periodic potential modulation. Dotted vertical lines mark the expected positions of the FQHSs, based on the 2D electron density. (b) The CF commensurability oscillations are shown in greater detail. Vertical solid lines mark the expected positions of the resistance minima when the CF density ($n^{*}$) is assumed to be equal to the electron density; these positions are symmetric about $\nu =1/2$. If $n^{*}$ equals the minority density, then the expected positions for the $B<B_{1/2}$ side are those shown with dashed vertical lines. The schematic insets indicate the basis of the CF minority density model which assumes that CFs are formed by the minority carriers in the LLL (hatched parts of the broadened level).

Figure 2

Magnetoresistance traces near $\nu =1/2$ for four 2D electron samples, normalized to the resistance value $R_{1/2}$ at $B_{1/2}$. Each trace is accompanied by a box whose top row gives the 2D electron density in units of $10^{11}{\mathrm{cm}}^{-2}$, while its bottom row contains the QW width $W$ and the modulation period $a$ in nm. The expected positions of the minima when $n^{*}=n$ are marked with vertical solid lines which are symmetric about $\nu =1/2$. The dashed lines represent the expected positions from the minority CF density model.

Figure 3

Magnetoresistance traces, normalized to the resistance value $R_{1/2}$ at $B_{1/2}$ in the vicinity of $\nu =1/2$ for three 2D hole samples with $W=17.5\mathrm{nm}$, density $\simeq 1.5\times 10^{11}{\mathrm{cm}}^{-2}$, and different modulation periods. The expected positions of resistance minima based on $n^{*}=n$ model are indicated with vertical solid lines which are symmetric about $\nu =1/2$. The dashed lines represent the expected positions from the minority CF density model.

Figure 4

Summary of the positions of the observed CF commensurability minima for $i=1$, labeled $B_{1,>}^{*}$ (open symbols) for $B>B_{1/2}$ and $B_{1,<}^{*}$ (closed symbols) for $B<B_{1/2}$. Short horizontal lines mark the expected positions $B_1^{*\prime }$ of the minima for $B<B_{1/2}$ based on the minority CF density model. All these fields ($B_{1,>}^{*}$, $B_{1,<}^{*}$, and $B_1^{*\prime }$) are normalized to the expected values $B_1^{*}$ based on the commensurability condition with $n^{*}=n$. Data for 2DHSs are shown in (a) and are grouped based on the modulation period $a$. The 2D hole density in each group increases from left to right. Data for 2DESs are shown in (b); for all 2DES samples $a=200\mathrm{nm}$ except for the last $W=40\mathrm{nm}$ QW, where $a=400\mathrm{nm}$.