Density dependence of valley polarization energy for composite fermions

In two-dimensional electron systems confined to wide AlAs quantum wells, composite fermions around the filling factor $\nu =3/2$ are fully spin polarized but possess a valley degree of freedom. Here we measure the energy needed to completely valley polarize these composite fermions as a function of electron density. Comparing our results to the existing theory, we find overall good quantitative agreement but there is an unexpected trend: the measured composite fermion valley polarization energy, normalized to the Coulomb energy, decreases with decreasing density.

Figure 1

(Color online) (a) Experimental setup showing the sample glued to a piezo and a strain gauge glued to the opposite face. (b) Simple energy fan diagram in the CF model for a two-valley, single-spin system. The [100] and [010] valleys are degenerate in energy at $ϵ=0$ and the degeneracy is lifted as strain is applied. (c) Magnetoresistance traces taken at $T=300\text{mK}$, at different values of strain (shown on the right in units of ${10}^{-4}$) for $n=5.1\times {10}^{11}{\text{cm}}^{-2}$. The traces are offset vertically for clarity.

Figure 2

(Color online) Magnetoresistance traces at $T=50\text{mK}$ for sample A taken at different values of strain for $n=5.0\times {10}^{11}{\text{cm}}^{-2}$.

Figure 3

(Color online) Piezoresistance traces taken for sample A at $T=300\text{mK}$ [panel (a)] and 50 mK [panel (b)] for different $\nu$. The values of $n$ are given in units of ${10}^{11}{\text{cm}}^{-2}$.

Figure 4

(Color online) (a) Piezoresistance traces taken for sample B at $T=50\text{mK}$ for different $\nu$. (b) Piezoresistance for sample B at $B=0$. The values of $n$ are given in units of ${10}^{11}{\text{cm}}^{-2}$.

Figure 5

(Color online) Valley polarization energy in units of Coulomb energy as a function of $1/p$. Data are shown for both samples at comparable $n$. Theoretical phase boundaries (originally worked out for two-spin system) based on the microscopic CF theory (black circles) and $m_p$ model (solid line) from Ref. 18 are also shown.

Figure 6

(Color online) [(a) and (b)] Valley polarization energy in units of Coulomb energy plotted as a function of $n$ for samples A and B. The dotted horizontal lines are theoretical predictions based on the $m_p$ model for the various $\nu$ as indicated. When not shown, error bars are comparable to the symbol size.