Probing topological Floquet states in graphene with ultrafast terahertz scanning tunneling microscopy

TL;DR

The paper tackles the challenge of obtaining direct, real-space, energy-resolved access to Floquet-engineered topological states in solids, focusing on graphene. It develops a nonequilibrium Green's-function framework and a Floquet extension of STM theory to describe ultrafast tunneling currents under circular driving, and demonstrates how THz-STM can reveal Floquet-induced bulk gaps, edge states, and their dispersions. The results show that time-averaged and pump–probe measurements can map dynamical gaps with $oldsymbol{ riangle}_1, oldsymbol{ riangle}_2$ and reconstruct edge dispersions via Floquet QPI, while circular-pump LDOS dichroism offers a direct probe of edge-state chirality. Collectively, THz-STM provides a spatially resolved complement to ARPES and ultrafast transport for exploring Floquet topologies, with potential extensions to Floquet–cavity engineering and other quantum materials.

Abstract

Floquet control of band topology is a central theme in ultrafast quantum materials science. Established experimental probes of light-induced topological states include ultrafast transport and time- and angle-resolved photoemission spectroscopy, each with important strengths but also well-known limitations. Here we propose ultrafast terahertz scanning tunneling microscopy (THz-STM) as a real space energy-resolved probe of Floquet physics. We show that THz-STM enables direct local detection of bulk Floquet gaps and distinct Floquet edge state signatures. We derive a nonequilibrium Green's-function formalism for time-dependent tunneling that directly extends standard STM theory and provides an intuitive interpretation of rectified ultrafast tunneling currents. We apply the approach to bulk graphene and graphene nanoribbons of variable width. For the bulk, we show that THz-STM provides direct spectroscopic access to Floquet-induced gap openings, and we contrast pulsed pump-probe protocols with the continuous-wave Floquet steady-state limit. For finite ribbons, we demonstrate time- and space-resolved imaging of Floquet-induced topological edge states and identify the ribbon-width scale below which edge state protection breaks down. We further show how band structures of graphene nanoribbons and Floquet chiral edge modes can be reconstructed via Floquet quasiparticle interference. Finally we demonstrate that chiral impurities that break time-reversal symmetry induce characteristic spatial THz-STM signatures that can be used as a direct probe of Floquet edge state chirality.

Figures (6)

• Figure 1: (a) Experimental geometry for measuring topological Floquet states in graphene by THz-STM. A circular pump pulse $E(t)$ excites a graphene nanoribbon under normal incidence. The in-plane drive creates a topological Floquet state with chiral edge modes on opposite edges that have opposite propagation directions. The state is probed using an ultrafast THz bias voltage $V_\mathrm{THz}(t)$ coupled to the STM tip. (b) Floquet band structure around the $K$ point and (c) corresponding LDOS in the bulk under circularly polarized driving. The electric field strength is $E = 350~\mathrm{kV\,cm^{-1}}$ and the photon energy is $\hbar\Omega = 0.4~\mathrm{eV}$. The band structure comprises Floquet replica bands shifted by integer multiples of the photon energy, with the spectral intensity shown along $k_x$ in the 3D plot. Hybridization gaps appear at the edges of the Floquet Brillouin zone at $\varepsilon = \hbar\Omega/2$ and a Haldane gap opens at the Dirac point.
• Figure 2: Measurement protocols for lightwave-driven tunneling spectroscopy of Floquet states. (a) Idealized configuration with a continuous-wave (CW) drive of amplitude $E_0$ inducing a Floquet steady state. The time-averaged current $\bar{I}$ is measured as a function of the static bias $V_\mathrm{DC}$. (b) Experimental configuration with a multi-cycle driving pulse with peak field strength $E_0$ inducing the transient Floquet state. The DC bias is modulated by a single-cycle THz probe pulse $V_\mathrm{THz}$. The rectified current $Q_\mathrm{rect}$ is recorded as a function of the peak THz bias $V_{\mathrm{pk}}$. The upper panels of a) and b) illustrate the measurement protocols; the middle and lower panels show $\bar{I}$ [$Q_\mathrm{rect}]$ and tunneling conductances $\mathrm{d} \bar{I}/\mathrm{d}V_{\mathrm{DC}}$ [$\mathrm{d}Q_\mathrm{rect}/\mathrm{d}V_{\mathrm{pk}}$] as a function of $V_\mathrm{DC}$ [$V_\mathrm{pk}$] for driven and undriven cases, respectively. Dynamical band gaps appear as minima in the tunneling conductances. Simulations were performed for $E = 350~\mathrm{kV\,cm^{-1}}$ and $\hbar\Omega = 0.4~\mathrm{eV}$.
• Figure 3: Floquet edge states in a graphene nanoribbon. (a) Real-space schematic: Zigzag ribbon with counter-propagating edge modes as indicated by red arrows. (b) Quasienergy spectrum of a semi-infinite graphene ribbon with zigzag edges along the $x$ direction, projected onto the first 30 unit cells measured from the edge. The driving parameters are $E_0 = 10~\mathrm{MV\,cm^{-1}}$ and $\hbar\Omega = 1.5~\mathrm{eV}$, which are chosen to create large enough gap to ensure sufficient real-space localization within our computational limits. The edge state bridges the dynamical gap at $\varepsilon = 0.75~\mathrm{eV}$. The blue curves shows the THz probe pulse with variable amplitude as scanned for the simulations in (c). (c) Simulated generalized differential conductance $\mathrm{d}Q_\mathrm{rect}/\mathrm{d}V_\mathrm{pk}$ along the $y$ direction for the same driving parameters as in (b) and varying ribbon width of $N = 70, 50, 30, 10$ unit cells. The $\mathrm{d}Q_\mathrm{rect}/\mathrm{d}V_\mathrm{pk}$ values are obtained by sweeping $V_\mathrm{pk}$ and numerical differentiation of the $Q_\mathrm{rect}-V_\mathrm{pk}$ curve. The profiles are vertically offset for clarity and referenced to the zero level defined by $\mathrm{d}Q_\mathrm{rect}/\mathrm{d}V_\mathrm{pk}=0$ (dashed line). Upon sweeping $V_\mathrm{pk}$ through the bulk gap, the edge states manifest as peaks at the ribbon edges, which decays towards the bulk for ribbon sizes down to $N = 30$ unit cells.
• Figure 4: Edge state interferometry in a narrow nanoribbon. (a) Spectral function for $N = 30$ and $E_0 = 10~\mathrm{MVcm}^{-1}$, $\hbar \Omega = 1.5~\mathrm{eV}$. Due to inter-edge coupling, two edge states are visible. The dashed line marks the energy used for the LDOS calculation in (c). (b) Sketch of the device architecture for edge state interferometry: A narrow ribbon allowing a finite hybridization of edge states in the bulk is restricted from both sides leading to Floquet QPI patterns. (c) LDOS variation due to hard-wall potential at $\varepsilon = 0.7 ~ \mathrm {eV}$. (d) QPI map as defined in the main text, showing the scattering channels that are allowed between the states plotted in (a). (e) Zoom into the QPI map region marked by the rectangle in (d), showing an image of the unperturbed ribbon band structure. The mean QPI intensity, corresponding to a peak at $q_x = 0$, is subtracted to improve clarity.
• Figure 5: Polarization-dependent Floquet LDOS near a chiral defect. The chiral defect is implemented by Haldane-like complex next-nearest neighbor hopping amplitudes $\mathrm i \gamma'$ with $\gamma' = 0.7~\mathrm{eV}$ as illustrated in the real-space sketch of the graphene lattice (left panel). The right plots show the LDOS variation $\delta \bar{\mathcal{A}}_{\vec{r}}(\varepsilon)$ due to the presence of the defect (marked in yellow). (a) and (b) highlight the difference upon switching from right- to left-handed circular polarization, leading to an enhancement or a suppression of the LDOS. The LDOS is computed for a semi-infinite ribbon with zigzag edges and driving parameters $E_0 = 10~\mathrm{MVcm^{-1}}$ and $\hbar \Omega = 1.5~\mathrm{eV}$ in the middle of the gap at $\varepsilon = 0.75~\mathrm{eV}$.