Phonon Excitations of Composite-Fermion Landau Levels

Phonon excitations of fractional quantum Hall states at filling factors $\nu =1/3$, $2/5$, $4/7$, $3/5$, $4/3$, and $5/3$ are experimentally shown to be based on Landau-level transitions of composite fermions. At filling factor $\nu =2/3$, however, a linear field dependence of the excitation energy in the high-field regime rather hints towards a spin transition excited by the phonons. We propose to explain this surprising observation by an only partially polarized $2/3$ ground state, making the energetically lower lying spin transition also allowed for phonon excitations.

Figure 1

(a) Schematic experimental setup with the phonon emitter on the back side and the 2DES at the front. Time resolved readout is achieved by an ultrafast digital phosphore oscilloscope (DPO). (b) Steady-state calibration of the 2DES resistance at $\nu =2/3$ for $n=0.89\times {10}^{15}{\mathrm{m}}^{-2}$. (c) Phonon signal for the same $\nu$ and $n$ after emitting $T_{\mathrm{H}}=2.09\mathrm{K}$ phonons during ${\tau }_{\mathrm{H}}=10\mathrm{n}\mathrm{s}$ at $t=0$. The 2DES was excited from a base temperature $T_0=75\mathrm{m}\mathrm{K}$. (d) 2DES temperature versus time as deduced from the raw phonon signal curve (c) using the calibration (b).

Figure 2

(a) Examples of the relative specific heat at filling factors $\nu =2/3$ and $2/5$ measured by phonon absorption for sample #5 ($n=1.21\times {10}^{15}{\mathrm{m}}^{-2}$). Data for $2/5$ are shifted for clarity by $2n$. The lines are fits according to Ref. 14 plus a constant. (b) Relative phonon energy absorbed by the 2DES, normalized to its value $E_0$ for $T_{\mathrm{H}}=2.09\mathrm{K}$, for a 10-ns pulse at filling factors $\nu =2/3$ and $2/5$ (data for $2/5$ are shifted by 0.25 for clarity). The curves are fits for an excitation across a gap ${\Delta }_{2/3}=5.7\mathrm{K}$ and ${\Delta }_{2/5}=4.9\mathrm{K}$.

Figure 3

(a) Compilation of all phonon excitation gaps $\Delta$ measured at various filling factors as a function of the magnetic field (i.e., the electron concentration for each filling factor). The dotted lines are CF Landau level transitions adjusted to our phonon data with one single mass parameter $\alpha =0.158(\pm 0.006)$. The dashed line represents the expected form of a normally forbidden spin transition at $\nu =2/3$. (b) The transport gaps are systematically lower than the phonon excitation gaps in (a).