Composite fermion valley polarization energies: Evidence for particle-hole asymmetry

In an ideal two-component two-dimensional electron system, particle-hole symmetry dictates that the fractional quantum Hall states around $\nu =1/2$ are equivalent to those around $\nu =3/2$. We demonstrate that composite fermions (CFs) around $\nu =1/2$ in AlAs possess a valley degree of freedom like their counterparts around $\nu =3/2$. However, focusing on $\nu =2/3$ and 4/3, we find that the energy needed to completely valley polarize the CFs around $\nu =1/2$ is considerably smaller than the corresponding value for CFs around $\nu =3/2$, thus betraying a particle-hole symmetry breaking.

Figure 1

(Color online) (a) Magnetoresistance traces taken at different values of strain (indicated in units of ${10}^{-4}$) showing FQH states in the lowest Landau level. The traces are offset vertically for clarity. (b) Experimental setup to apply controllable in-plane strain. (c) Simple fan diagram showing the response of composite fermion Landau levels to strain.

Figure 2

(Color online) Piezoresistance ($R$ vs $ϵ$) traces taken at $T=300\text{mK}$ and at fixed values of filling factor around $\nu =1/2$ (left panels) and $\nu =3/2$ (right panels). $x$ axis is expressed as the strain-induced valley splitting $\left(E_v\right)$ measured in units of the Coulomb energy $\left(V_C\right)$. The values of $\nu$ and $p$ are indicated. The corresponding values of $n$ are also given in units of ${10}^{11}{\text{cm}}^{-2}$. The values of $E_{v,pol}$ are indicated by vertical arrows.

Figure 3

(Color online) Piezoresistance (right $y$ axis) and energy gaps (left $y$ axis) at $\nu =2/3$. The piezoresistance trace was taken at $B=16.6\text{T}$ $\left(n=2.67\times {10}^{11}{\text{cm}}^{-2}\right)$ at $T=250\text{mK}$. The gaps shown as (black) circles are obtained for the same $B$ from the same cooldown. The (red) triangles are gaps obtained from a separate cooldown at a comparable value of $B=15.0\text{T}$ $\left(n=2.41\times {10}^{11}{\text{cm}}^{-2}\right)$. (Inset) Arrhenius plots for two extreme values of $ϵ$.

Figure 4

(Color online) Piezoresistance at $T\cong 250\text{mK}$ for $\nu =2/3$ at different values of $B$ (corresponding $n$ are 2.0, 2.9, 3.7, and $4.6\times {10}^{11}{\text{cm}}^{-2}$). Inset illustrates that the peaks are better resolved at low temperatures for smaller $B$. The traces at $B=12.5\text{T}$ and $B=17.8\text{T}$ should be multiplied by the factors shown to obtain the actual values of resistance.

Figure 5

(Color online) Comparison of measured valley polarization energies for $\nu =2/3$ and 4/3, along with the theoretical prediction (dashed line).