Probing fractional quantum Hall effect by photoluminescence

TL;DR

This work develops a composite-fermion (CF) theory of photoluminescence (PL) in fractional quantum Hall systems, showing that while the emitted photon energy $E_g$ is not diagnostic in the SU(2) symmetric limit, the finite-temperature PL intensity carries rich information about CF excitations. By combining CF theory with CFD and exact diagonalization in spherical geometry, the authors map low-energy spectra and identify bright and dark states across Jain fillings, predicting PL intensity peaks near $ν = n/(2n±1)$ and extracting CF exciton/trion binding energies from activated temperature dependence. They find that at $\bar{ν}=1/3$ the ground state is bright, whereas at $\bar{ν}=2/5$ and $3/7$ it is dark due to a roton minimum, with finite gaps $Δ$ that govern the PL response; modest SU(2) symmetry breaking and finite well width can modify but generally preserve the qualitative PL signatures. The results connect to PL and reflectance experiments in GaAs wells and to fractional quantum anomalous Hall states in twisted TMD bilayers, suggesting PL as a practical probe to identify incompressible CF states and to measure CF bound-state energies in real materials.

Abstract

The recent discovery of fractional quantum anomalous Hall (FQAH) states - fractional quantum Hall (FQH) states realized without an external magnetic field - in twisted transition-metal dichalcogenide (TMD) bilayers represents a significant development in condensed matter physics. Notably, these states were first observed via photoluminescence (PL) spectroscopy. Surprisingly, a general theoretical understanding of PL is not available even for the standard FQH states. For an ideal two-dimensional system, the energy of the emitted photon is predicted to be independent of the correlations, but we show that the PL intensity contains valuable information. Specifically, we predict that at finite temperatures, the PL intensity peaks at the Jain fillings ν= n/(2n \pm 1), and away from these fillings, the binding energies of the composite-fermion excitons and trions can be deduced from the temperature dependence of the intensity. We discuss implications for PL experiments in semiconductor quantum wells and twisted TMD bilayers.

Figures (16)

• Figure 1: (a) The state with a hole in LL$_\downarrow$. The electron-hole recombination is associated with the emission of a photon, which produces the PL spectrum. (b) The equivalent system related by particle-hole transformation (electron $\Leftrightarrow$ hole; $\uparrow \Leftrightarrow \downarrow$, $E\rightarrow -E$). (Note that replacing $\nu$ by $1-\nu$ in LL$_{\uparrow}$ of the left panel does not produce an equivalent system.)
• Figure 2: This figure depicts the physics of photoluminescence in the FQHE in terms of CFs. We consider FQHE states in the valence band at filling factors $\bar{\nu}=n/(2n+1)$, $\bar{\nu}\gtrsim n/(2n+1)$ [shown for simplicity as the $\bar{\nu}=n/(2n+1)$ state plus a quasiparticle (QP), which is a single CF in an otherwise unoccupied $\Lambda$L] and $\bar{\nu}\lesssim n/(2n+1)$ [shown for simplicity as the $\bar{\nu}=n/(2n+1)$ state plus a quasihole (QH), which is a single missing CF in the topmost occupied $\Lambda$L]. When an electron is excited from the valence band (denoted by pseudospin-$\uparrow$) to the conduction band (denoted by pseudospin-$\downarrow$), a very complicated and highly excited state of CFs is obtained which can be written as linear combination of CF basis functions of the type shown in Column-I ("Pre-relaxation") which have $S_z=N/2-1$. Each basis function is shown in Dirac's $|\cdots \rangle$ ("bra-ket") notation, with the pseudospin-$\uparrow$ state shown schematically on the left and the pseudospin-$\downarrow$ state on the right. We make the customary assumption that, at zero temperature, prior to recombination this state relaxes to the ground state within the $S_z=N/2-1$ sector, resulting in a state of the kind shown in the Column-II ("Post-relaxation"). Depending on the filling factor, this state may contain a CF QP in $\downarrow$ sector, a CF exciton or a CF trion. The states in Column-II can have $(S, S_z)=(N/2, N/2-1)$ or $(S, S_z)=(N/2-1, N/2-1)$. Given that the state after recombination necessarily has $(S, S_z)=(N/2, N/2)$ and that the photoluminescence operator $S^{+}$ increases $S_z$ by one unit but leaves $S$ invariant, one can show, using the CF theory, that the only "bright" state is the $\bar{\nu}=1/3$ ground state; the ground states at all other $\bar{\nu}$ considered here are "dark," i.e., do not allow recombination for symmetry reasons. The lowest energy bright state (which produces the state shown in Column-III after recombination) is an excited state. At finite temperatures the lowest energy bright state has a nonzero probability of occupation proportional to $e^{-\Delta/T}$, resulting in a PL signal; here $\Delta$ is the energy difference between the lowest energy bright state relative to the ground state, estimated in the main text.
• Figure 3: Dispersion of the lowest pseudospin wave, obtained from CFD for: a) $\bar{\nu} = 1/3$, b) $\bar{\nu} = 2/5$, c) $\bar{\nu} = 3/7$, d) $\bar{\nu} = 2/3$. The energies are measured relative to the lowest bright state, marked by the red dashed circle; all other states shown are dark. The wave vector is defined as $k = L/R$, with $R=\sqrt{Q}l$. The pseudospin wave has a roton minimum for $\bar{\nu} = 2/5$ and $3/7$, producing a dark ground state.
• Figure 4: Lower panel: Plot of $\Delta$, the energy of the lowest bright state relative to the ground state, as a function of $\bar{\nu}$, the filling factor of the FQHE state in the valence band. The symbols are calculated [blue circle for the Jain $n/(2n+1)$ state, red square for the $n/(2n+1)$ state with a QP, green diamond for the $n/(2n+1)$ state with a QH, and the star at $\nu=1/2$]; the dashed lines are schematic. The inset shows the excitons (blue circle) and trions (green diamond and red square), where the two circles represent the $\uparrow$ and $\downarrow$ pseudospin sectors; for the red square, the QP and QH in the $\uparrow$ sector can annihilate leaving a single CF quasiparticle in the $\downarrow$ sector, as is the case for $\bar{\nu}=n/(2n+1)$ plus one QP in Fig. \ref{['fig:Schematic3']}. Upper panel: schematic plot of the PL intensity as a function of $\bar{\nu}$.
• Figure 5: Thermodynamic extrapolations of $\Delta$ for (a) $n/(2n+1)+{\rm QP}$, (b) $n/(2n+1)+{\rm QH}$, and (c) $1/2+\rm{QP}$.