Time-resolved ARPES studies of quantum materials

Angle-resolved photoemission spectroscopy (ARPES), with its exceptional sensitivity to both the binding energy and the momentum of valence electrons in solids, provides unparalleled insight into the electronic structure of quantum materials. Over the past two decades, the advent of femtosecond lasers, which can deliver ultrashort and coherent light pulses, has ushered the ARPES technique into the time domain. Currently time-resolved ARPES (TR-ARPES) can probe ultrafast electron dynamics and the out-of-equilibrium electronic structure, providing a wealth of information that is otherwise unattainable in conventional ARPES experiments. This review begins with an introduction to the theoretical underpinnings of TR-ARPES followed by a description of recent advances in state-of-the-art ultrafast sources and optical excitation schemes. It then reviews paradigmatic phenomena investigated by TR-ARPES thus far, such as out-of-equilibrium electronic states and their spin dynamics, Floquet-Volkov states, photoinduced phase transitions, electron-phonon coupling, and surface photovoltage effects. Each section highlights TR-ARPES data from diverse classes of quantum materials, including semiconductors, charge-ordered systems, topological materials, excitonic insulators, Van der Waals materials, and unconventional superconductors. These examples demonstrate how TR-ARPES has played a critical role in unraveling the complex dynamical properties of quantum materials. The conclusion outlines possible future directions and opportunities.

Figure 1

The rapid adoption of time- and angle-resolved photoemission spectroscopy. Left panel: increase of the approximate number of annual TR-ARPES publications over the past three decades. Exponential technological progress (milestone advancements in cyan arrows) and interest in new materials (blue stars marking the initial TR-ARPES publications for various classes of materials) have fueled the growth of the TR-ARPES technique over the past two decades. Right illustration: key classes of quantum materials extensively studied via TR-ARPES.

Figure 2

Working principles of TR-ARPES. (a) Illustration of the pump-probe approach in TR-ARPES using two ultrafast laser pulses separated by time interval $\tau >0$. The pump pulse drives the system out of equilibrium, and the probe pulse triggers the photoemission process. (b) Simplified sketch of the pump-induced evolution of the TR-ARPES signal for a Dirac cone dispersion. Top row: TR-ARPES maps for different pump-probe delays $\tau$ (${\tau }_{\mathrm{ee}}$ defines the electron-electron thermalization time). The red dashed arrow indicates the pump photon energy and related vertical optical transition from occupied into unoccupied states. Bottom row: differential TR-ARPES maps highlight the pump-induced redistributions of spectral weight.

Figure 3

Schematic configuration of an ARPES experiment. Light in the UV to XUV spectral range prompts the photoemission of electrons from the sample. Photoelectrons are then collected by an electron analyzer (a hemispherical analyzer in the sketch), which discriminates them using both the kinetic energy and the angle of emission, thus generating the 2D ARPES spectrum.

Figure 4

Different components that constitute the transient photoemission intensity, with an illustration of how their intertwined contributions may be reflected in TR-ARPES data. (a) Diagram of the different contributions, and the corresponding encoded information, that characterize the transient photoemission intensity $I(\mathbf{k},\omega ,\tau )$. (b) Hypothetical temporal evolution of the spectral broadening $\mathrm{\Gamma }$, electronic temperature $T_{\mathrm{e}}$, and matrix element $|M|$. $\mathrm{\Gamma }$ is assumed to have a Fermi-liquid form $\mathrm{\Gamma }(\tau )={\mathrm{\Gamma }}_{\mathrm{imp}}+\beta [{\omega }^2+{\pi }^2{k}_B^2{T}_{\mathrm{e}}^2(\tau )]$, where ${\mathrm{\Gamma }}_{\mathrm{imp}}=14\mathrm{meV}$ and $\beta =8$. The temporal evolution of $\mathrm{\Gamma }$ is displayed for $\omega =0$. (c) Simulated differential ARPES spectrum at $\tau =3\mathrm{ps}$ of a linear dispersion (dashed black line) assuming the transient parameters displayed in (b) and accounting for the $\omega$ dependence of $\mathrm{\Gamma }$. (d) Comparison between the transient evolution of the photoemission intensity at three different binding energies from (c) (black dashed lines, right axis) and the evolution of the electronic distributions given solely by $T_{\mathrm{e}}(\tau )$ in (b) (gray solid lines, left axis) highlighting the discrepancy in their dynamics.

Figure 5

Toy model depicting some of the challenges associated with the interpretation of the transient photoemission intensity for the case of a superconductor with gap $\mathrm{\Delta }$ with a strong electron-phonon coupling that renormalizes the electronic dispersion (also known as the kink). Multiple effects may occur, such as direct optical transitions, transient modifications of the bare dispersion from the unperturbed ${ϵ}_{\mathrm{bare}}$ to ${ϵ}_{\mathrm{pump}}$ after excitation (dashed blue and red lines, respectively), enhancement of the electronic temperature, filling of the superconducting gap, and softening of the electron-phonon band-structure renormalization. These effects are often intertwined, which complicates the interpretation of the TR-ARPES signal.

Figure 6

Illustration of a TR-ARPES system based on a hemispherical analyzer (HA) along with a sample dataset that showcases its capabilities. (a) Illustration of the three main components of a state-of-the-art TR-ARPES system: the optical schemes for the pump excitations, the generation of the ultraviolet and/or extreme ultraviolet probe pulses, and the coupling to the ARPES system. (b) TR-ARPES data of graphite acquired with 25 eV probe and 1.2 eV pump. The high energy and momentum resolution, as well as the dynamic range, achievable with a hemispherical analyzer allowed the detection of multiple peaks corresponding to both direct transitions and photon-induced replicas. In fact, although (top panel) these peaks are not obvious in the ARPES intensity maps, (bottom panel) they stand out from the quasithermalized electron background when the EDC is integrated in momentum, at energies up to 0.6 eV above $E_F$. Adapted from [275].

Figure 7

Pulse duration and bandwidth of UV and XUV probes from different TR-ARPES studies. The black line indicates the Gaussian Fourier limit, while gray lines represent integer multiples of this fundamental limit up to $n=10$. Adapted from [129].

Figure 8

Operating principles of a time-of-flight (TOF) spectrometer. (a) Sketch of a TOF detector. Adapted from [474]. (b) One-shot constant energy cut at $-0.12\mathrm{eV}$. (c) ARPES spectrum along the $k_y$ direction acquired via TOF on ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ upon mid-IR light excitation. These maps show the appearance of Floquet-Volkov states when the impinging electric field is oriented along the $k_x$ direction (discussed in Sec. 3d) and exemplify the ability of TOF detectors to measure the photoemission intensity simultaneously over 2D momentum space, for example, $I(E,k_x,k_y)$. (b),(c) Adapted from [238].

Figure 9

Experimental configuration of a momentum microscope (MM) setup, which can locate the heterostructure in the real-space acquisition mode with micrometer resolution. (a) Illustration of a MM detector studying a ${\mathrm{WSe}}_2/{\mathrm{MoS}}_2/h\mathrm{BN}$ sample. (b) The acquired ARPES map at 0 fs shows the valence bands of ${\mathrm{WSe}}_2$, ${\mathrm{MoS}}_2$, and $h\mathrm{BN}$, as well as the bright $A$ excitons of ${\mathrm{WSe}}_2$. (c) Regions of interest in the study indicated by the red and orange $10\text{−}\mathrm{\mu }\mathrm{m}$-diameter circles. Adapted from [352].

Figure 10

TR-ARPES mapping of the unoccupied states of InSb semiconductor shows how different pump energies access distinct optical transitions. (a) Band structure of InSb with available interband optical transitions with 1.57 eV (red arrows) and 1.26 eV (blue arrows) photons. TR-ARPES spectra of $\mathrm{InSb}(110)$ acquired 50 fs after $p$-polarized (b) 1.57 eV and (c) 1.26 eV pump excitation. The solid and dashed lines represent the CB dispersion along the high-symmetry directions $\mathrm{\Gamma }\text{−}\mathit{L}$ and $\mathrm{\Gamma }\text{−}\mathit{X}$, respectively. (d) Illustration of the surface and bulk BZ for $\mathrm{InSb}(110)$. Adapted from [402].

Figure 11

Mapping of the conduction band of transition-metal dichalcogenides. (a)–(c) Sketches of the available transitions and TR-ARPES differential mapping acquired with 25 eV photons for monolayer ${\mathrm{MoS}}_2$ around $\overline{K}$ at the peak of the optical excitation using three different pump energies: (a) below the CB onset, (b) resonant to the band gap, and (c) above the CB onset. By tuning the pump photon energy, the value of the direct band gap was estimated at 1.95 eV. The solid lines in (a)–(c) mark the boundaries between the projected bulk band continuum and gaps of $\mathrm{Au}(111)$, while the dashed curves represent the equilibrium fitted dispersion. (a)–(c) Adapted from [150]. (d) Photoelectron intensity distribution as a function of parallel momentum for three binding energies at a pump-probe delay of 100 fs acquired for bulk $2H\text{−}{\mathrm{WSe}}_2$ with a 21.7 eV probe and a 3.1 eV pump. The higher pump energy allows the CB to be mapped within a wider energy range, which favors a more comprehensive comparison with an ab initio calculated band structure. The experimental data are collected in the region delimited by the dashed line; outside this region, the results of calculations are displayed, and the theoretical band dispersion along the $k_z$ direction was integrated. Adapted from [318].

Figure 12

TR-ARPES mapping of the unoccupied states acquired near the $\mathrm{\Gamma }$ point with $\sim 6\mathrm{eV}$ probe upon a 1.5 eV pump of two topological insulators: (a) ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and (b) ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. The near-IR excitation promotes carriers into the TSS and the CB. Subsequent relaxation processes lead to an accumulation of hot carriers at the bottom of the bulk CB. (a) Adapted from [380]. (b) Adapted from [154].

Figure 13

Probing the effect of terahertz pump excitation on the topological surface state of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ in momentum space. TR-ARPES spectra along $\mathrm{\Gamma }K$ acquired (a) before the teraherz pump, (b) at the electric field node with the largest negative gradient, and (c) at the field node with the largest positive gradient. (d)–(f) Photoemission intensity of the left (blue) and right (red) branches of the Dirac cone integrated inside the blue and red boxes shown in (a)–(c). Red and blue solid lines represent simulations based on the Boltzmann equation, while the black line is the Fermi-Dirac distribution convolved with the experimental resolution. Insets in (d)–(f): illustration of the electric field waveform in which the green arrows mark the time delay of each snapshot. The observed asymmetric population (i.e., a transient momentum shift of the electrons) arising when the electric field is applied indicates the emergence of a transient surface current driven by the ultrashort terahertz pump pulse. (g) Sketch detailing how the acceleration of Dirac fermions in the topological surface state can shift the Fermi surface via an electric field. Adapted from [325].

Figure 14

Schematic illustration of the dispersion of an electron photoemitted from an exciton. Following conservation laws, an electron photoemitted from an excitonic bound state exhibits an energy-momentum dispersion that mimics that of the valence band, with a spectral intensity that diminishes when one moves away from the initial momentum location of the bound state.

Figure 15

TR-ARPES employed to map excitons in momentum space. (a) 3D rendering of the TR-ARPES mapping of the occupied and unoccupied band structures of a ${\mathrm{WSe}}_2/{\mathrm{MoS}}_2$ heterostructure with $\sim 2°$ twist angle and micrometer resolution via momentum microscopy, featuring the presence of an interlayer exciton (ILX) above $E_F$. (b) TR-ARPES spectrum along the $K\text{−}\mathrm{\Gamma }$ direction 25 ps after optical excitation, along with isoenergy maps displaying the momentum location of (left image) the top of the VB and (right image) the top of the ILX. Note the decrease of spectral weight in the VB accompanying the formation of the ILX. These maps were generated by integrating over the energy ranges specified by the blue brackets. Adapted from [192].

Figure 16

Spatially indirect excitons in 3D topological insulators. (a) TR-ARPES spectra along $\mathrm{\Gamma }\text{−}M$ of $p$-doped ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ at equilibrium and three different pump-probe delays. The energy scale refers to the position of the Dirac point in energy. (b) Sketch of the experimental observation highlighting the spin polarization and formation of the excitonic state in the intensity buildup and the topological surface state. (c) Spin-resolved spectra of the exciton intensity buildup at the exciton momentum $k_{\mathrm{EX}}$, measured at a delay time of 20 ps. Adapted from [270].

Figure 17

Emergence of Floquet-Bloch states on the surface of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ driven by a 160 meV, 300 fs light pulse. The coherent interaction between the time-periodic potential of the light pulse and the electrons in the Dirac cone results in Floquet-Bloch states that appear to be replicas in energy of the original Dirac cone. Dynamical gaps in the electronic structure open up at positions where the replica cones intersect with the original cone. The replica bands disappear once the driving field dies off. Adapted from [132], and [238].

Figure 18

Floquet-Bloch states on subcycle timescales in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. (a) Left graphic: illustration of electronic processes driven via mid-IR excitation, such as multiphoton interband excitation and intraband currents, which may also result in the emergence of Floquet-Bloch states. Right graphic: schematic experimental strategy to investigate the emergence of Floquet-Bloch states on subcycle timescales. (b) Electric field waveform from mid-IR-induced momentum streaking (black markers) and current density (red markers). The black solid line shows the analytic function of the electric field with a peak amplitude of $0.8\mathrm{MV}/\mathrm{cm}$, while the dashed red line is the electric current induced by this electric field according to the Boltzmann model ([325]). (c) Curvature-filtered TR-ARPES maps recorded at different pump-probe delays during the first half of the mid-IR driving pulse. The gray lines display the band structure calculated using density functional theory (DFT). The electronic distribution splits into multiple branches that follow the Floquet-Bloch replicas of the TSS. Adapted from [182].

Figure 19

Floquet engineering of black phosphorus. (a) Schematic of the band structure of semiconducting black phosphorus (with an electronic gap $E_g\sim 340\mathrm{meV}$) with the different mid-IR photon energies used. (b) TR-ARPES maps at zero pump-probe delay (top row) and corresponding second-derivative images (bottom row) for different excitation energies. (c) Sketch of the light-induced modifications of the electronic band structure for four different pump photon energies: below the gap, comparable to the gap, slightly above the gap, and above the gap. Adapted from [468].

Figure 20

Ultrafast melting of the CO phase upon near-IR pumping in ${\mathrm{LaTe}}_3$. (a) Top illustration: tight-binding plot of the normal-state Fermi surface of ${\mathrm{LaTe}}_3$ formed by the Te $p$ bands in the first BZ. The circle highlights the probed part of the Fermi surface and the arrow marks the CO wave vector ${\mathbf{q}}_0$. Bottom illustration: Fermi surface maps at different pump-probe delays across the photoinduced phase transition. The CO gap present before the pump arrival closes on a timescale of $\sim 100\mathrm{fs}$, whereas a full recovery of the long-range CO phase occurs only after $\sim 5.5\mathrm{ps}$. (b) Detailed time evolution of the gapped region highlighted by the orange box drawn in the $t=-1300\mathrm{fs}$ slice in (a). The white dots mark the fitted positions of the upper CO band edge. (c) Time evolution of the band structure near the Fermi level after photoexcitation acquired along the $k_{||}$ direction highlighted in (a) by the yellow line. Adapted from [476].

Figure 21

TR-ARPES spectroscopic signatures of the light-induced melting of the CO in various layered transition-metal dichalcogenides. Top row: illustration of the reconstructed (green lines) and original (black lines) projected Brillouin zones of three CO phases in $1T$-TMDs: (a) ${\mathrm{TaS}}_2$, (b) $1{T\text{−}\mathrm{TiSe}}_2$, and (c) $\mathrm{Rb}:1{T\text{−}\mathrm{TaS}}_2$. The original schematic Fermi surfaces (circle and ellipses) are shown along with CO wave vectors (arrows). Middle row: static ARPES dispersion of the three compounds highlighting the different characteristic energy gaps in the Brillouin zone. Bottom row: unpumped and pumped TR-ARPES spectra acquired at (a) the $\overline{M}$ point of $1{T\text{−}\mathrm{TaS}}_2$, (b) the $\overline{M}$ point of $1{T\text{−}\mathrm{TiSe}}_2$, and (c) the $\overline{\mathrm{\Gamma }}$ point of Rb: $1{T\text{−}\mathrm{TaS}}_2$. The absorbed pump energy density and temperature were set to $300\mathrm{J}{\mathrm{cm}}^{-3}$ and 110 K, respectively. Adapted from [168].

Figure 22

Light-induced 2D metastable electronic state at the surface of $1{T\text{−}\mathrm{TiSe}}_2$. (a) 3D BZ of $1{T\text{−}\mathrm{TiSe}}_2$. (b) Second-derivative ARPES spectra of the time-dependent electronic structure of $1{T\text{−}\mathrm{TiSe}}_2$ acquired along the $\mathrm{\Gamma }\text{−}\mathit{M}$ direction at 4 K [the red line in (a)]. The dashed blue, green, and red lines mark the dispersion of the Se $4p_{x,y}$ band at $\mathrm{\Gamma }$, the Se $4p_{x,y}$ band folded from point $A$, and the Ti $3d_{z^2}$ band folded from point $L$, respectively. A clear ultrafast modification of the electronic dispersion is observed, with a recovery dynamics of tens of picoseconds. (c) Schematic of the CO inversion mechanism in real space. Left graphic: The original 3D CO lattice has an order parameter $\varphi =-1$. Right graphic: Upon photon excitation, phase inversion occurs at the surface leading to an order parameter $\varphi =+1$. As a result, quasi-2D electronic states form between the inverted and original CO layers (red shaded area). (a)–(c) Adapted from [98]. (d) Time-dependent Ginzburg-Landau model as a function of the order parameter for three selected pump fluences: schematics of the energy potential before and after photoexcitation (top row) and calculated transient energy potential (bottom row). Adapted from [97].

Figure 23

Ultrafast Mott gap renormalization in $1{T\text{−}\mathrm{TaS}}_2$ upon near-IR optical excitation. (a) $\mathrm{\Gamma }$-point EDCs at different pump-probe delays measured at 30 K base temperature, highlighting the pump-induced transfer of spectral weight from the LHB to the previously gapped region. (b) Dynamic response of the Mott phase as outlined by the transient LHB intensity peak normalized to the equilibrium value. The almost-instantaneous collapse is followed by a subpicosecond recovery. (c) Simulated spectral function of the curves in (a) calculated via a single-band Hubbard model in which the electronic temperature $T_{\mathrm{e}}$ at different delays is the dominant term driving the phase transition. From [302].

Figure 24

Light-induced doublon dynamics in $1{T\text{−}\mathrm{TaS}}_2$. (a) Transient EDCs acquired at normal emission for selected pump-probe delays showing the ultrafast population of the UHB. (b) Exemplary fit of an EDC above $E_F$ consisting of a linear background and a Lorentzian peak corresponding to the UHB. (c) Comparison between the ultrafast evolution of the UHB and the slower dynamics of LHB and the underlying background supporting an electronic origin of the UHB dynamics. (d) Fluence dependence of the UHB peak width. From [227].

Figure 25

Pathway of the photoinduced MIT captured in momentum space in indium nanowires on a silicon $(111)$ surface $\mathrm{In}/\mathrm{Si}(111)$. (a)–(c) TR-ARPES spectra of $\mathrm{In}/\mathrm{Si}(111)$ at 25 K for selected time delays highlighting the transition from an insulating $(8\times 2)$ to a metallic $(4\times 1)$ phase. The solid lines in (a)–(c) are the calculated electronic band structures in the $(8\times 2)$ and $(4\times 1)$ phases. The pump-induced transition occurs in three stages: closing of the gap at $X_{(8\times 2)}$ (red arrows), downward shift of the CB at ${\mathrm{\Gamma }}_{(8\times 2)}$ (yellow arrows), and separation in momentum of the now-metallic band (blue arrows) to reach the final $(4\times 1)$ structure. (d) Experimental momentum-space distribution of excited electrons (red) and holes (blue) at the time of the pump excitation. The map is obtained by subtracting the photoemission signal measured at $-1\mathrm{ps}$ from that measured at 0 ps. Adapted from [285].

Figure 26

Transient changes in the valence band of the excitonic insulator TNS upon a near-IR excitation. (a) Equilibrium ARPES spectra of TNS acquired around $\mathrm{\Gamma }$ at 110 K. The solid (dashed) black line is a guide for the eye for the VB dispersion before (after) the optical excitation. The red bar indicates the momentum integration interval for extracting EDCs to track the VB peak position. (b) Time-dependent energy shift of the upper VB peak at $\mathrm{\Gamma }$ for different pump fluences. Two different behaviors are observed around an empirical critical fluence $F_C\sim 0.2\mathrm{mJ}/{\mathrm{cm}}^2$. An upward shift with a 1 ps recovery to the original binding energy at low fluences is opposed to a 200-fs-delayed downward shift at higher fluences, which is followed by a settling into a lower-energy position relative to the equilibrium case. Adapted from [269].

Figure 27

Photomelting of the excitonic gap in TNS. (a) TR-ARPES spectra of TNS acquired along the Ta and Ni chains at a base temperature of 30 K showing the ultrafast emergence of a CB electronlike pocket at $\mathrm{\Gamma }$ upon a 1.77 eV excitation. (b) EDCs extracted at the top of the VB [the dashed red line in (a)] at time delay 0.3 ps as a function of the pump fluence. (c) Fluence dependence of the photoinduced change of the energy gap extracted from the EDCs in (b) (red squares). While a linear decrease in the gap size is reported at low fluences, a saturation point is reached at $F_C=0.29\mathrm{mJ}/{\mathrm{cm}}^2$, resulting in a residual gap even at high fluences. The same fluence dependence is also observed for the decay rate of the nonequilibrium electrons obtained in a 0.1 eV window above $E_F$ (green squares). Adapted from [401].

Figure 28

Tracking the light-induced broadening of the VB line shape as an indicator of the strength of electron-electron interactions in TNS. (a) TR-ARPES spectrum acquired along the $\mathrm{\Gamma }$-$X$ direction of ${\mathrm{Ta}}_2{\mathrm{NiSe}}_5$ before photoexcitation. (b) EDCs centered at $\mathrm{\Gamma }$ before and after the optical excitation showing a pump-induced broadening of the peak. (c) Transient variation of the spectral broadening of the VB at $\mathrm{\Gamma }$ for three different excitation fluences. Adapted from [143].

Figure 29

Transient evolution of the near-nodal superconducting gap of optimally doped Bi2212 ($T_c=91\mathrm{K}$). (a) TR-ARPES dispersion acquired along the gapless nodal direction ($\varphi =45°$) before and after near-IR optical pumping. For positive delays, the differential spectrum, obtained by subtracting the ARPES map at negative delay, is shown. Inset in the left panel: location of the cut in the Fermi surface. (b) Same as in (a) but along the off-nodal cut ($\varphi =31°$), where the band bending due to the presence of the gap is clearly visible. (c) Transient evolution of the superconducting gap at $\varphi =32°$ mapped via symmetrized EDCs. The spectroscopic modifications observed within the originally gapped region are described using a phenomenological model (black solid lines) that includes a single-scattering term and the superconducting gap amplitude. Adapted from [371].

Figure 30

Visualizing the filling of the superconducting gap in optimally doped Bi2212 upon near-IR pump excitation. The off-nodal TR-ARPES differential spectra at various pump-probe delays (0–10 ps range) highlight the transfer of spectral weight within the gap region without a notable energy shift of the quasiparticle peak toward $E_F$. Adapted from [299].

Figure 31

Key role of phase fluctuations in the transient filling of the superconducting gap in Bi-based cuprates. (a) Equilibrium Fermi surface and differential [$I(\tau =+0.5\mathrm{ps})-I(\tau =-0.5\mathrm{ps})$] isoenergy contour mapping at 10 meV above the Fermi level for underdoped Bi2212 with $T_c\sim 82\mathrm{K}$. (b) Off-nodal EDCs at ${\mathbf{k}}_F$ of underdoped Bi2212 with $T_c\sim 82\mathrm{K}$ normalized to momentum-integrated nodal EDCs ($\varphi =45°$) at different pump-probe delays ($\sim 8\mathrm{\mu }\mathrm{J}/{\mathrm{cm}}^2$ incident fluence). EDCs have been deconvolved from the energy-resolution broadening prior to the division. (c) Transient evolution of the superconducting gap $\mathrm{\Delta }$ and pair-breaking scattering rate ${\mathrm{\Gamma }}_p$ extracted by fitting symmetrized EDCs (left axis), and $T_e$ extracted along the nodal direction (right axis) for optimally doped Bi2212 with $T_c\sim 91\mathrm{K}$. The solid lines are phenomenological double-exponential decay fits to the data. (d) Comparison between the normalized transient variation of the quantities in (c). The faster and disentangled dynamics characterizing ${\mathrm{\Gamma }}_p$ attest to the nonthermal origin of the ultrafast enhancement of the phase fluctuations. (e) Sketch of the transient collapse of the condensate driven by a loss of phase coherence. The top panels show a schematic diagram of the energetics of the process and the related real-space condensate phase coherence, the middle panels display the spectral function at $k=k_F$ when phase fluctuations are induced, and the bottom panels show the temporal evolution of the pairing strength ($\mathrm{\Delta }$, black line) and of ${\mathrm{\Gamma }}_p$ (blue line). While the pairing is defined by $T_{\mathrm{e}}$ (red spheres and dashed lines), the superconductivity and the macroscopic $T_c$ are determined by the onset of phase coherence $T_{\mathrm{\Theta }}$ (green spheres and dashed lines), which is proportional to the superfluid density. Adapted from [33], and [477].

Figure 32

Application of the MTM to the transient evolution of the effective electronic temperature of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ upon a 1.55 eV pump. Three cooling stages are observed, corresponding to different relaxation channels for the electronic and phononic bath following the optical excitation, as sketched in the inset. Adapted from [303].

Figure 33

Transient evolution of the electronic temperature and chemical potential in the topological surface state of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ after near-IR pump excitation. (a) Pump-probe TR-ARPES spectrum displayed as the difference between the signal at $+300\mathrm{fs}$ and the signal at $-500\mathrm{fs}$. Temporal evolution of (b) the effective temperature $T^{*}$ and (c) the effective chemical potential ${\mu }^{*}$ obtained from fitting a sequence of EDC curves at $k_F$. In each case, a single decay exponential function captures the observed temporal dynamics, with characteristic relaxation times equal to $\sim 2.5$ and $\sim 2.7\mathrm{ps}$, respectively. A simple thermal model based on the extracted $T^{*}(\tau )$ and ${\mu }^{*}(\tau )$ reproduces the ultrafast evolution of the nonequilibrium charge population in the topological surface state at different binding energies. Adapted from [73].

Figure 34

MTM analysis of TR-ARPES data of graphene upon (a) mid-IR and (b) near-IR excitations. Left graphic: schematics of the two different excitation schemes illustrating how only the near-IR pump pulse accesses direct optical transitions. The light gray line represents the electric field of the 300 meV pump pulse and the light red line represents the intensity of the 950 meV pump pulse. Middle panels: momentum-integrated EDCs around the $K$ point for selected pump-probe delays, along with Fermi-Dirac distribution fits. Right panels: transient electronic temperature with multiexponential decay fits. Note that while in (a) the pump-induced thermal broadening follows the conventional Fermi-Dirac distribution for all delays, in (b) EDCs at times close to zero are best described by two different Fermi-Dirac fits for electrons and holes (the dashed black lines). These time delays are indicated by red data points and the shaded area in the transient evolution of $T_{\mathrm{e}}$. Adapted from [139].

Figure 35

Disentangling the surface and bulk contributions to the light-induced coherent oscillations in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. (a) Sketch of the optical excitation and relaxation of coherent phonons in the bulk and surface bands. (b) Top panel: time-dependent shift of the TSS. Bottom panel: temporal evolution of the CB photoemission intensity integrated within the black box indicated in the inset. The black solid lines are smooth curve fits using tenth-order polynomials. (c) Fast Fourier transform (FFT) of the residuals from the fit for the TSS and CB showing the presence of distinct frequencies. While the oscillations of the CB can be captured by a single undamped cosine, the oscillations of the TSS are best fit using a beat pattern represented by the sum of two cosine functions. Adapted from [379].

Figure 36

Response to optical excitations of the $A_{1g}$ phonon mode in $\mathrm{FeSe}/{\mathrm{SrTiO}}_3$ films tracked via combined time-resolved x-ray diffraction and TR-ARPES measurements. (a) Schematic of the $A_{1g}$ phonon mode, which periodically modulates the electronic band energies. (b) Sketch of the experimental strategy. (c) Temporal evolution of the x-ray intensity of the $(004)$ Bragg peak upon near-IR pumping, which excites a coherent $A_{1g}$ phonon mode; see inset. (d) Equilibrium ARPES spectrum of FeSe along the $\mathrm{\Gamma }-X$ direction acquired at 20 K. Renormalized density functional theory calculations for the electronic band dispersions are represented by superimposed lines (solid for ARPES-detected bands and dashed for unresolved bands). The orbital characters of the bands are also indicated. (e) Relationship between (blue line) the oscillations of the Se displacement $\delta z_{\mathrm{Se}}$ probed by time-resolved x-ray scattering and (orange and green lines) the momentum-averaged binding energy shift of the electronic bands obtained using TR-ARPES. Adapted from [133].

Figure 37

Momentum-space visualization of the band selectivity of the electron-phonon coupling in $T_d\text{−}{\mathrm{WTe}}_2$ via frequency-domain ARPES. (a) Left panel: differential ARPES spectrum of $T_d\text{−}{\mathrm{WTe}}_2$ along the $\overline{\mathrm{\Gamma }}-\overline{W}$ direction acquired 120 fs after near-IR pump excitation. Right panel: transient evolution of the differential ARPES intensity around $\mathrm{\Gamma }$ showing clear oscillations of the binding energies modulated by multiple frequencies. (b) Frequency-domain ARPES maps (i.e., Fourier-transformed photoemission signals) for selected frequencies. The analysis highlights the band selectivity of the phonon excitations (the red arrow) and also applies to excited states above $E_F$ (the black arrow). From [164].

Figure 38

Evaluation of the electron-phonon coupling response in an optimally doped Bi2212 cuprate via an analysis of the transient electron self-energy. (a) Nodal dispersion for three distinct pump-probe delays, showing the softening of the characteristic kink at $\sim 70\mathrm{meV}$ at $+1\mathrm{ps}$. The dashed line corresponds to the linear bare band dispersion. Insets: comparisons of MDCs before and after near-IR pumping around the kink energy. (b) Real part of the electron self-energy extracted at different delays (full circles) and at 100 K in equilibrium conditions (open circles). Arrows mark the suppression of different peaks. (c) Corresponding MDC’s full-width at half maximum (FWHM), which is proportional to the imaginary part of the self-energy. A clear pump-induced modification of both the real and imaginary parts of $\mathrm{\Sigma }$ is reported at the kink energy. Adapted from [463].

Figure 39

Momentum-resolved analysis of the quantized electron-phonon decay processes in graphite, allowing the extraction of the mode-projected electron-phonon matrix element $g_{\overline{\mathbf{k}}}$. (a) Simulation of the transient TR-ARPES intensity for a Dirac cone pumped with 1.2 eV photons, including a retarded $e$-ph interaction with a phonon of energy $\hslash {\mathrm{\Omega }}_0$. At time $t=0$, the direct transition peak (DTP) feature is observed at $E_{\mathrm{DTP}}=0.6\mathrm{eV}$; at $t={\tau }_{{\mathrm{\Omega }}_0}$, the phonon-induced replica (PIR) is observed at $E_{\mathrm{DTP}}-\hslash {\mathrm{\Omega }}_0$. (b) Calculation of $g_{\overline{\mathbf{k}}}$ for both an electron in $K$ scattering with an ${A^{\prime }}_1$ mode with a momentum $\sim K$ (green line) into a final state in $K^{\prime }$ and an electron in $K$ scattering with an $E_g^2$ mode with a momentum $\sim 0$ (blue line) into a final state in $K$. (c) Momentum-integrated EDC around the $K$ point immediately after the pump excitation. Multiple DTPs and PIRs can be identified on top of the hot-electron background based on the momentum-resolved optical joint DOS in (e). (d) Transient evolution of the amplitude of the most prominent peaks: ${\mathrm{DTP}}_1$ (${\mathrm{DTP}}_2$) in dark (light) blue and ${\mathrm{PIR}}_1$ in red. The dashed lines indicate the peak delays: ${\mathrm{DTP}}_2$ (${\mathrm{PIR}}_1$) is delayed 9 fs (47 fs) with respect to ${\mathrm{DTP}}_1$. Solid curves represent the electronic occupation of the specified states derived from a rate-equation model fit. The transfer of spectral weight from ${\mathrm{DTP}}_1$ to ${\mathrm{PIR}}_1$ is associated with an $e$-ph scattering time constant ${\tau }_{{\mathrm{\Omega }}_0}=174\pm 35\mathrm{fs}$. Adapted from [277].

Figure 40

Emergence of SPV in GaAs(110). (a) CB dynamics around the $\overline{\mathrm{\Gamma }}$ point of $p$-type $\mathrm{GaAs}(110)$ upon ultrashort mid-IR optical excitation. (b) Normalized CB EDCs at 0 (blue line), 0.5 (green line), and 5 ps (red line). (c) Temporal evolution of the half-maximum point of the high-energy edge [(HEE), squares] and low-energy edge [(LEE), circles] of the EDC peak. (d) VB $\overline{\mathrm{\Gamma }}$-EDC transient map and (e) energy shift dynamics. For both the CB and the VB the energy shift observed at positive delays can be described using double- or single-exponential fits: ${\tau }_{\mathrm{H}1}=200\pm 20\mathrm{fs}$ and ${\tau }_{\mathrm{H}2}=1.8\pm 0.3\mathrm{ps}$ for the HEE edge, ${\tau }_{\mathrm{L}}=1.4\pm 0.1\mathrm{ps}$ for the LEE edge, and ${\tau }_{\mathrm{VB}}=1.7\pm 0.1\mathrm{ps}$ for the VB. The shared $\sim 1.5\mathrm{ps}$ dynamics stem from the emergence of the pump-induced SPV, which also causes the additional upward shift of the VB at negative delays. (f) Schematic interpretation of the positive- and negative-delay dynamics in the presence of SPV, where the black lines perpendicular to the sample surface represent the SPV-induced electric field. Adapted from [447].

Figure 41

Visualization of the effects of light-induced SPV in TR-ARPES data of $p$-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The sketch on the left illustrates the topological surface state (red and blue) and bulk bands (gray) of $p$-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, where the Fermi level $E_F$ lies within the bulk band gap. The ARPES spectra at different pump-probe delays highlight the shift of the chemical potential observed before and after the optical excitation due to the SPV effect. The far right panel shows the full dynamics of the momentum-integrated intensity along $K-\mathrm{\Gamma }-K$. Adapted from [65].

Figure 42

SPV as a tuning knob of the Rashba spin-orbit coupling of a 2D electron gas at the surface of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. (a) ARPES dispersion at negative time delay (before pump arrival), at time zero (with pump and probe fully overlapped), and at 8 ps delay after the excitation. The differential spectra (obtained by subtracting the spectrum acquired at $-0.5\mathrm{ps}$) are also shown for the zero and positive delays. An observable time-dependent energy shifts of the QWSs at the Brillouin zone center is induced by the emergence of SPV. (b) Transient evolution of the Rashba spin-orbit coupling strength ${\alpha }_R$ in the first QWS as extracted by fitting momentum (orange) and energy (green) splitting of Rashba subbands at several time delays. Adapted from [259].

Figure 43

Critical phenomena connecting the ultrafast demagnetization in ferromagnetic Ni to the equilibrium magnetic phase transition. (a) TR-ARPES spectra of $\mathrm{Ni}(111)$ along the $\mathrm{\Gamma }-K$ direction before ($-500\mathrm{fs}$) and after (500 fs) laser excitation, highlighting the transient reduction of the exchange splitting $\mathrm{\Delta }{E}_{\mathrm{ex}}$. (b) Momentum-integrated EDCs on a logarithmic scale at different pump-probe delays at $\sim 6\mathrm{mJ}/{\mathrm{cm}}^2$ pump fluence. The electronic temperature, extracted by fitting momentum-integrated EDCs to a Fermi-Dirac distribution, reaches its maximum within $\sim 24\mathrm{fs}$ after the optical excitation. (c) Top panel: maximum electronic temperature at 24 fs pump-probe delay as a function of the pump fluence. The yellow region highlights the energy transferred to the spin bath ($\mathrm{\Delta }{F}_S$) within $\sim 20\mathrm{fs}$. Bottom panel: $\mathrm{\Delta }{E}_{\mathrm{ex}}$ at 2 ps pump-probe delay as a function of the pump fluence. The two panels report a similar critical fluence $F_c\approx 2.8\mathrm{mJ}/{\mathrm{cm}}^2$. (d) Schematic of the light-induced demagnetization process of Ni. Upon a femtosecond pump excitation, the transient electronic temperature approaches and can surpass the Curie temperature within 20 fs, which induces high-energy spin excitations, which store the magnetic energy. Demagnetization occurs later, in $\sim 175\mathrm{fs}$, driven by relaxation of nonequilibrium spins and the likely excitation of low-energy magnons. Full recovery of the spin system occurs within $\sim 0.5–100\mathrm{ps}$, depending on the laser fluence. Adapted from [407].

Figure 44

Transient mapping of the spin-resolved surface resonance state (SRS) in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. (a) Measured and calculated spin-resolved EDCs along the $\mathrm{\Gamma }-K$ direction close to the TSS. Dotted (continuous) lines indicate opposite spin-resolved photoemission intensities before (after) the optical excitation, while TSS and SRS are indicated by arrows. Adapted from [43]. (b) Spin-integrated and spin-resolved ARPES maps (left and right panels, respectively) along the $\mathrm{\Gamma }-K$ direction at 0.7 ps pump-probe delay upon near-IR excitation. Adapted from [188].

Figure 45

Selective population of the $K$ or $K^{\prime }$ valley of $2{H\text{−}\mathrm{WSe}}_2$ upon circularly polarized optical excitation. (a) TR-ARPES maps at various pump-probe delays acquired with linearly polarized pump pulses (logarithmic color scale). The dashed lines display the VB and CB dispersions obtained by DFT calculations for a bilayer compound, with the CB shifted 250 meV upward to match the experimental observations. (b) Energy distribution curves of the excited state signal at $K$ and $K^{\prime }$ 15 fs after excitation with a linearly and a circularly polarized optical pump. Note the strong valley selectivity attainable by excitation with circularly polarized light. Adapted from [25].

Figure 46

Tracking the filling of the pseudogap driven by the shortening of the spin-correlation length in the optimally doped ${\mathrm{Nd}}_{2-x}{\mathrm{Ce}}_x{\mathrm{CuO}}_4$ electron-doped cuprate. (a) Experimental Fermi surface measured with 6.2 eV at 10 K. The blue solid line is a tight-binding constant energy contour at $\omega =0\mathrm{meV}$, the red dashed line marks the antiferromagnetic zone boundary, and the violet dotted circle indicates the hot-spot (HS) position. The green and black solid cuts mark the two momentum directions explored in the study: near node and HS. (b) Transient electronic temperature extracted by fitting the Fermi edge broadening along the near-nodal direction for two pump-fluence regimes: low fluence (LF) and high fluence (HF). The gray points indicate time delays for which a pure thermal fitting is not accurate. (c) Temporal evolution for both LF and HF of the photoemission intensity at the HS for $\omega =50\pm 10\mathrm{meV}$, which represents the evolution of the pseudogap. (d) Photoemission intensity at the HS for $\omega =50\pm 10\mathrm{meV}$ as a function of the electronic temperature (the black and red circles for LF and HF, respectively). The green line and shadow represent the inverse of the spin-correlation length ${\xi }_{\mathrm{spin}}$ from neutron scattering studies. Saturation of the photoemission intensity at the HS (red shadow) marks the temperature $T^{*}$ at which a complete filling of the PG is observed. Adapted from [35].

Figure 47

Illustration of the forthcoming new frontiers of the TR-ARPES technique in the coming years.