Bias-Free Access to Orbital Angular Momentum in Two-Dimensional Quantum Materials

The demonstration of a topological band inversion constitutes the most elementary proof of a quantum spin Hall insulator (QSHI). On a fundamental level, such an inverted band gap is intrinsically related to the bulk Berry curvature, a gauge-invariant fingerprint of the wave function’s quantum geometric properties in Hilbert space. Intimately tied to orbital angular momentum (OAM), the Berry curvature can be, in principle, extracted from circular dichroism in angle-resolved photoemission spectroscopy (CD-ARPES), were it not for interfering final state photoelectron emission channels that obscure the initial state OAM signature. Here, we outline a full-experimental strategy to avoid such interference artifacts and isolate the clean OAM from the CD-ARPES response. Bench-marking this strategy for the recently discovered atomic monolayer system indenene, we demonstrate its distinct QSHI character and establish CD-ARPES as a scalable bulk probe to experimentally classify the topology of two-dimensional quantum materials with time reversal symmetry.

Figure 1

(a) ARPES dispersion at the $K$ point of $n$-doped indenene showing the two valence bands (${\mathrm{VB}}_1$, ${\mathrm{VB}}_2$) as well as a partially occupied conduction band ${\mathrm{CB}}_1$ ($\mathrm{h}\nu =21.2\mathrm{eV}$ 20 K). (b) Topological phase diagram of a $p$-electron system on the triangular lattice as a function of ISB (${\lambda }_{\mathrm{ISB}}$) and SOC strength (${\lambda }_{\mathrm{SOC}}$). The insets show the $⟨L_z⟩$ polarization (green and orange arrows) and the Berry curvature (red and blue curves) of the valence and conduction band states at $K$, calculated by DFT for the topologically trivial and nontrivial (QSHI) system. The topology of the system is reflected in both the OAM as well as the Berry curvature ordering sequence [19, 21, 22].

Figure 2

Photoelectron emission channel resolved CD-ARPES calculated according to Ref. [17] for the tight-binding model derived in Refs. [19] and [21]. Panels (a) and (b) represent calculations with the light incidence plane (red arrow) along $\mathrm{\Gamma }M$, for two experimental geometries rotated by 60° with respect to each other (${\mathrm{a}}_1$), (${\mathrm{b}}_1$). Panels (${\mathrm{a}}_2$), (${\mathrm{b}}_2$) show the $s$ channel, (${\mathrm{a}}_3$), (${\mathrm{b}}_3$) the $d$ channel, (${\mathrm{a}}_4$), (${\mathrm{b}}_4$) their interference channel and (${\mathrm{a}}_5$), (${\mathrm{b}}_5$) the total CD-ARPES dispersion around $K$ and $K^{\prime }$ along the orange path in (${\mathrm{a}}_1$) and (${\mathrm{b}}_1$), respectively. Details are given in the Supplemental Material [22].

Figure 3

Step (i)—Identification of potentially faithful photon energies: (a) 3D visualization of an STM topography scan (bias voltage: 750 mV, tunneling current: 150 pA) across a half ($\approx 5\AA$) 4H-SiC unit cell step edge, exposing two indenene domains (insets) that are mutually rotated by 180° (which in ${\mathrm{C}}_3$ symmetry corresponds to a 60° rotation and flips $K$ and $K^{\prime }$). (b) Side view ball and stick model of the step structure depicted in (a), emphasizing a 180° rotation in the 4H-SiC stacking order at every half unit cell (red path). LEED pattern of (c) single domain indenene and (d) a two domain indenene sample taken at 135 eV. (e) CD-ARPES map of the two-domain sample taken at $\mathrm{h}\nu =30\mathrm{eV}$ and 65° light incidence (red arrow) along the $M\mathrm{\Gamma }M$ direction of the hexagonal BZ (dashed) and energy-integrated over the indenene valence bands range. The inset shows the LEED pattern of this sample. (f) CD-ARPES integrated around the six valley momenta [circles in (e)] and plotted as a function of photon energy. Collective zeros in the CD-signal mark photon energies where the photoemission channel interference is potentially suppressed. These energies are candidates for faithful photon energies in step (ii).

Figure 4

Step (ii)—Confirmation and characterization of faithful photon energies: High resolution CD-ARPES measurements of a single domain indenene sample for $h\nu =80$ (a),(b) and 47 eV (c). Shown are the CD-ARPES disperion (left) as well as dichroic energy distribution curves at the valley momenta (right). The light incidence plane with respect to the hexagonal BZ of indenene is sketched in the inset. Rotation of the sample by 60° in (b) swaps TRP $K$ and $K^{\prime }$ and, consequently, inverts the dichroism signal. (c) CD-ARPES at a different faithful photon energy flips the sign of CD-ARPES as consequence of a switch of final states from $m_l=0$ to $m_l=\pm 2$. Inset: LEED images taken at 135 eV indicate the single domain phase.