Angle-resolved photoemission studies of quantum materials

The physics of quantum materials is dictated by many-body interactions and mathematical concepts such as symmetry and topology that have transformed our understanding of matter. Angle-resolved photoemission spectroscopy (ARPES), which directly probes the electronic structure in momentum space, has played a central role in the discovery, characterization, and understanding of quantum materials ranging from strongly correlated states of matter to those exhibiting nontrivial topology. Over the past two decades, ARPES as a technique has matured dramatically with ever-improving resolution and continued expansion into the space, time, and spin domains. Simultaneously, the capability to synthesize new materials and apply nonthermal tuning parameters in situ has unlocked new dimensions in the study of all quantum materials. These developments are reviewed, and the scientific contributions they have enabled in contemporary quantum materials research are surveyed.

Figure 1

Top-left panel: schematic of angle-resolved photoemission spectroscopy (ARPES). Photons of energy $h\nu$ are used to photoemit electrons into vacuum, where their kinetic energies and emission angles are resolved. Other panels: the scope of quantum materials studied by ARPES, corresponding to sections in this review (heavy fermion materials are representative of the materials discussed in Sec. 8).

Figure 2

Schematic depiction of microscopic interactions and their distinct signatures in an ARPES spectrum. Starting with a crystal potential $H_0$, the ARPES spectrum reflects the band structure of independent electrons. Electron-phonon coupling $H_{e\text{−}\mathrm{ph}}$ introduces a “kink” in the band dispersion, with an increased peak width below the kink energy. Electron-electron coupling $H_{e\text{−}e}$ leads to an energy-dependent peak width and a renormalized band velocity. Finally, spin-orbit coupling $H_{\mathrm{SOC}}$ can split spin or orbital degeneracy. Together these interactions give rise to the physics of quantum materials, including topological materials (with spin-polarized surface states), superconductors (with superconducting gap ${\mathrm{\Delta }}_{\mathrm{SC}}$), charge density waves (with folding wave vector $q_{\mathrm{CDW}}$), and Mott insulators (with Mott gap ${\mathrm{\Delta }}_{\text{Mott}}$).

Figure 3

State of the art in ARPES over time. (a)–(d) Improvement in energy and momentum resolution, as exhibited by the spectral function along diagonal $\mathrm{Cu}\text{−}\mathrm{Cu}$ direction in cuprates. Data in (d) additionally benefited from an improvement in spatial resolution to overcome surface mosaicity. Adapted from [825], [923], [482], and [389]. (e) Approximate number of publications per year utilizing photoemission as determined by the Web of Science. Note the steady rise with the introduction of new materials and techniques over the past 30 years. The discontinuity around 1990 coincides with the development of third generation synchrotrons but may also contain changes in record keeping methodology. (f) Energy resolution sampled from representative publications showing an overall near-exponential improvement enabled in large part by new light sources. Different light sources are color coded differently.

Figure 4

Kinematics of the photoemission process in the three-step model. An electron is excited from an initial state with binding energy $|E_B|$ below $E_{\mathrm{F}}$ to a final state above $E_{\mathrm{vac}}$, with ${\mathbf{k}}_{\perp }$ conserved. The final state is often approximated by a free-electron dispersion offset by an inner potential $V_0$, shown here in a reduced zone scheme. After transmission through the surface barrier, the photoelectron has kinetic energy $E_{\mathrm{kin}}=h\nu -\varphi -|E_B|$, where $\varphi$ is the sample work function. ${\mathbf{k}}_{\perp }$ is not conserved during transmission through the sample surface, while ${\mathbf{k}}_{||}$ is conserved throughout the entire photoemission process. Note that the detected kinetic energy $E_{\mathrm{kin}}^A$ is referenced to the vacuum level in the analyzer, which is determined by the analyzer work function ${\varphi }_A$.

Figure 5

Working flowchart from detector image to energy-momentum spectrum for modern ARPES with multiplexing detectors. Electron events are amplified in a multichannel plate (MCP) detector and imaged on a phosphor screen using a camera. In many implementations, a wire mesh is used to establish a uniform electric field but leaves an imprint on the raw data that must be removed. Various methods exist for removing the grid pattern during acquisition (so-called dithering or swept modes) or during postprocessing. In this flowchart, the effect of the mesh and detector inhomogeneity are exaggerated for clarity; see the text for more details of the subsequent analysis steps.

Figure 6

Dispersion extraction and self-energy analysis in ARPES. (a) Cut showing the photoemission intensity as a function of energy and momentum. The momentum distribution curve (MDC) and energy distribution curve (EDC) at the red and blue dashed lines are shown above and to the right, respectively. The green line is an MDC fitted dispersion. The black dashed line is the noninteracting bare-band dispersion used in the simulation. (b) Using MDC (shown here) or EDC fitting analysis, it is possible to extract ${\mathrm{\Sigma }}^{\prime }$ and ${\mathrm{\Sigma }}^{\prime \prime }$ (green lines on the top and bottom panels, respectively). From a modeling of their energy dependence, they may be further decomposed into contributions such as those from electron-electron ($e\text{−}e$) and electron-phonon ($e$-ph) scattering (dashed lines).

Figure 7

Light sources for ARPES and their main characteristics and typical photon energy ranges.

Figure 8

Specialized ARPES techniques. (a) Spatially resolved ARPES with a $0.6\mu \mathrm{m}$ spot size. (i) Optical image and (ii) schematic cross section of an exfoliated ${\mathrm{WSe}}_2$ flake with monolayer, bilayer, and bulk regions capped with monolayer graphene. (iii) EDCs from the three spatial regions and (iv) spatial map of the peak energy demonstrating the ability of $\mu$-ARPES to map heterostructured samples. Adapted from [973]. (b) Spin-resolved ARPES spectra from the Au(111) surface state demonstrating the efficiency of a state-of-the-art 2D imaging-type spin detector. From [913]. (c) Fermi surface maps for strained and unstrained ${\mathrm{Ca}}_{2-x}{\mathrm{Pr}}_x{\mathrm{RuO}}_4$ demonstrating an insulator-to-metal transition via nonthermal tuning knobs. From [762]. (d) ARPES cuts at the $K$ point of graphene using in situ electrostatic gating to shift the chemical potential. Adapted from [654]. (e) Measurement of a second topological surface state above $E_{\mathrm{F}}$ in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ demonstrating the ability of trARPES to measure unoccupied states. Adapted from [850].

Figure 9

Photoelectron spectrometers for ARPES. (a) Simplified depiction of lens modes for a modern analyzer. In angle-dispersive mode, electrons of the same emission angle (denoted by color) arrive at the same spatial position on the detector plane. In imaging mode, the real-space position of the electrons is mapped onto the detector, independent of emission angle. See [965] for quantitative calculations. (b) Hemispherical analyzer. For ARPES, the lens column is operated in angle-dispersive mode to map the angular distribution of electrons onto the entrance slit of a hemispherical deflector. A two-dimensional detector records the distribution of electrons as a function of emission angle ${\vartheta }_y$ and kinetic energy $E_{\mathrm{kin}}$. (c) Angle-resolved time-of-flight detector. The lens column images the angular distribution of electrons onto a two-dimensional detector with time resolution, which records the emission angles $({\vartheta }_x,{\vartheta }_y)$ and photoelectron flight times.

Figure 10

Common spin polarimetry techniques. (a) Mott scattering employs the spin-orbit interaction between the incoming electrons and heavy nuclei in the scattering target. In the geometry shown here, detectors placed at $\pm x$ will detect contrast attributed to the $y$ component of the electron spin. A pair of detectors can be placed at $\pm y$ to simultaneously detect the $x$ component (not shown). (b) VLEED scattering employs the exchange interaction between the incoming electrons and a ferromagnetic scattering target. The measurement is sensitive to the spin component parallel to the magnetization of the target. Contrast is obtained by flipping the magnetization using Helmholtz coils.

Figure 11

(a) Schematic of a trARPES experiment. A pump pulse optically excites the sample, and a time-delayed probe pulse generates photoelectrons that are collected by a photoelectron analyzer. (b)–(d) Sketches of the main classes of excitation phenomena, including (b) direct optical transitions and subsequent population relaxation dynamics, (c) destruction of an ordered phase followed by its recovery, (d) excitation of a coherent phonon mode associated with oscillatory binding energies, and (e) dressing of the electrons by a periodic field forming Floquet-Bloch states.

Figure 12

Schematic temperature-doping phase diagram of electron- and hole-doped cuprate superconductors. Top-left inset: lattice arrangement for one unit cell of electron-doped cuprate $(\mathrm{La}/\mathrm{Nd},\mathrm{Ce}{)}_2{\mathrm{CuO}}_4$ ($T_c^{\max }\sim 30\mathrm{K}$). Top-right inset: lattice arrangement for one half unit cell of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ ($T_c^{\max }\sim 96\mathrm{K}$). Middle inset: top view of the ${\mathrm{CuO}}_2^{n-}$ plane in real space. Red circle, copper; gray circle, oxygen; blue circle, electron; white circle, doped hole; purple star, the critical doping that separates two different metallic regimes.

Figure 13

Orbital contents of low-energy bands in overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$. (a) $d_{x^2-y^2}$ component selected with in-plane polarized light. (b) $d_{3r^2-z^2}$ component selected with out-of-plane polarized light. (c) DFT calculation of $d$-orbital contents in the low-energy bands. Adapted from [602]. (d) $k_z$ dispersion along nodal in-plane momenta. (e) $k_z$ dispersion along antinodal in-plane momenta. (f) Schematic plot of the three-dimensional Brillouin zone of LSCO. Adapted from [356].

Figure 14

Schematic evolution of the electronic structure in hole-doped cuprate. (a)–(c) Normal-state spectral function evolution with hole-doping along high-symmetry directions. Solid black lines, $t\text{−}J$ model or tight-binding model band calculations. The unoccupied side is only qualitatively sketched based on Hubbard model calculations. (d) Fermi surface in underdoped and overdoped cuprates. The dashed diamond is the AFM zone boundary. (e) $M$ point $(\pi ,0)$ and (f) nodal ${\mathbf{k}}_{\mathrm{F}}$ near $(\pi /2,\pi /2)$ EDC evolution with hole doping. Circles, high-energy spectral feature associated with Mott physics; triangles, low-energy spectral features that are, or will evolve into, quasiparticles; black lines, half filling (HF); red lines, underdoped (UD); blue lines, overdoped (OD).

Figure 15

Spectral characters of the pseudogap [red squares in (c),(d)] and its relation to superconducting quasiparticle peaks [blue dots in (c),(d)]. Temperature-dependent, Fermi-function divided antinodal spectra for (c) near-optimally doped and (d) heavily overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$. $T_{\mathrm{AN}}$ is the gap-closing temperature at the antinode. (a),(b) Spectral weight evolution within [0,70] meV binding energy with temperature, normalized by total spectral weight over [0,250] meV binding energy. Adapted from [314]. (e) Momentum- and doping-dependent spectral weight competition between superconducting coherent peak (down arrow) and pseudogap (up arrow) in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CuO}}_{6+\delta }$. Graduated shades correspond to the marked direction of doping. Adapted from [471]. (f) Particle-hole asymmetry for the pseudogap in optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CuO}}_{6+\delta }$. Adapted from [312].

Figure 16

Emergence of quasiparticles in different parts of momentum space with hole doping. (a),(b) Nodal quasiparticles [the straight segment of the dispersions in (a) and the purple shade in (b)] are created in ${\mathrm{Ca}}_2{\mathrm{CuO}}_2{\mathrm{Cl}}_2$ immediately upon Na doping on Ca sites. Adapted from [822]. (c) Doping-dependent antinodal spectra in the normal state across the critical doping in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$. Black dashed lines indicate the chemical potentials for each doping. Adapted from [925], and [330].

Figure 17

Electronic correlation related energy gaps. (a) Momentum dependence of the gap near the node in underdoped YBCO. Adapted from [678]. (b) Energy-momentum spectra (inset) and doping dependence for the AFM gap at the AFM zone boundary in electron-doped cuprate ${\mathrm{Nd}}_{2-x}{\mathrm{Ce}}_x{\mathrm{CuO}}_4$. Adapted from [323].

Figure 18

High-resolution spectra near the node. (a) Fermi-function divided near-nodal cuts in optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ taken at the bulk $T_c$. Note the visible upper Bogoliubov quasiparticle dispersion above $E_{\mathrm{F}}$. (b),(c) Symmetrized energy distribution curves from node to off-node direction at 10 K and the bulk $T_c=92\mathrm{K}$. Note the persistent $d$-wave gap at $T_c$ due to superconducting fluctuations. Adapted from [472]. (d) High-resolution measurement examining the $d$-wave form of the energy gaps on both the bonding and antibonding bands in overdoped Bi-2212 measured with VUV laser ARPES. AB, antibonding band; BB, bonding band; SS, superstructure; gray line, $d$-wave fit. Adapted from [5].

Figure 19

Momentum- and doping-dependent superconducting gaps in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$. Near-node gap $\mathrm{\Delta }(\mathbf{k})$ measurements in (a) underdoped and (b) overdoped samples. Assuming a negligible pseudogap contribution near the node, the nodal gap velocity $v_{\mathrm{\Delta }}$ may be equated with the superconducting gap at the antinode. Adapted from [934]. (c) Doping-dependent $d$-wave superconducting gap determined by fitting $v_{\mathrm{\Delta }}$. Superconducting $T_c$ dome (gray crescent) is plotted against hole doping according to the empirical parabolic relation with maximum $T_c$ between 91 and 98 K ([726]) and is converted to energy by the weak coupling $d$-wave gap-to-$T_c$ ratio. Data points were compiled from [934], [20], [472], and [330], [332]. The variance in the underdoped region is a combined result of deviation from simple $d$-wave gap form and different fitting momentum ranges for $v_{\mathrm{\Delta }}$.

Figure 20

Cuprate Fermi surface evolution with carrier doping. (a),(b) Schematics and experimental data on the electron-doped Fermi surface in $(\mathrm{Nd},\mathrm{Ce}{)}_2{\mathrm{CuO}}_4$ with a coherent nodal hole pocket (solid black) and a less coherent antinodal electron pocket (gray shadow). The dashed lines are Fermi surface sheets before AFM folding. The thin solid diamond is the AFM zone boundary. Adapted from [852]. (c),(d) Expected tight-binding Fermi surface and the actual Fermi surface close to half filling in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CuO}}_{6+\delta }$. Adapted from [311]. (e),(f) Fermi “arc” in underdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ with a coherent node and incoherent antinode. Adapted from [752]. (g),(h) Above a certain critical doping $p_c$ the entire Fermi surface recovers coherence and connects to a single large hole pocket ([331]). The dashed and solid lines represent the antibonding and bonding bands in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$, respectively. (i),(j) Further hole doping passing the van Hove point at $M$, undergoing a holelike to electronlike Fermi surface Lifshitz transition at $p_L$. Adapted from [205]. The right panels are enlargements of the lower-right quadrants of the full BZs on the left.

Figure 21

Signatures of bosonic mode coupling on the nodal spectra in cuprates. (a) Mode-coupling effects on both the BB and AB in overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ at the node and near node. (b) Extracted antibonding band dispersions from nodal to near-antinodal region. Adapted from [19]. (c),(d) Doping dependence of the nodal dispersion in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CuO}}_6$, with the low-energy dispersion anomalies enlarged in (c). Adapted from [473].

Figure 22

Rapid change of the antinodal mode coupling across the critical doping. Parallel momentum-integrated antinodal spectra for (a) optimally doped and (b) heavily overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ at different temperatures. The integration window is indicated by the red bar in the inset schematic Fermi surface. (c) Doping dependence of the spectral weight of the antinodal “dip” feature, reflecting the mode-coupling strength. Insets: false color plots of the antinodal spectra in the superconducting state for (a) optimally doped and (b) heavily overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$. Adapted from [330].

Figure 23

(a) Schematic temperature-doping phase diagram in K-doped Ba-122 FePn system. Adapted from [434]. Inset: crystal structure of parent compound Ba-122. DQMO, double-$Q$ magnetic order. Adapted from [773]. (b) Top view of the Fe sublattice after both the nematic order and the collinear antiferromagnetic order develop. The gray shaded diamond indicates the two-Fe unit cell when considering the As atoms alternately positioned above and below the Fe plane. The red shaded rectangle is the simplified one-Fe unit cell that ignores the As atoms. (c) Low-energy band structure along the Fe—As bond direction in a three-orbital model of a typical FeSC. (d) Typical Brillouin zone (BZ) of a FeSC projected to the FeAs plane. Symmetry labels in parentheses are projected from $k_z=\pi /c$. Red, one-Fe BZ; gray, two-Fe BZ. Schematics for (e) normal-state orbital-selective band renormalization, (f) nematicity and AFM-induced band reconstruction (neglecting $d_{xy}$ band), (g) unconventional superconductivity with interpocket pairing. and (h) electron-phonon-coupling-induced electron shakeoff band. Adapted from [1020], and [1025].

Figure 24

Family dependence and the multiorbital nature of FeSCs. (a) Schematic temperature-doping phase diagrams for different FeSC families. Black, nematic order; gray, SDW order; blue, superconductivity; orange shade, more metallic transport. (b) Compilation of superconducting gap vs $T_c$ for representative FeSCs (colored markers), single-layer cuprate Bi-2201 (increasing hole doping clockwise), ${\mathrm{MgB}}_2$ ($\sigma$, $\pi$, and surface bands), and $(\mathrm{Ba},\mathrm{K})\mathrm{Bi}{\mathrm{O}}_3$. For FeSCs, square and diamond (circle) markers represent superconducting gaps on the hole (electron) pockets. Dashed lines are reference $2\mathrm{\Delta }/k_B{T}_c$ values at 3.5 ($s$-wave limit) and 7.5. For definitions of the compound acronyms see [1025]. (c) Coulomb interaction $U$ and Hund’s coupling $J_H$ cooperatively generate new phenomena from single-band to multiband systems.

Figure 25

Low-energy electronic band structure, Fermi surface, and correlation effects in the normal state of FeSCs. (a),(c) Schematic and (b),(d) measured Fermi surfaces in the normal state of NaFeAs. Different experimental geometries are employed to highlight different orbital components based on the dipole-transition matrix element effect. Light gray indicates where all intensities are suppressed. Adapted from [1074]. (e) Normal-state $d_{yz}$ (green) and $d_{xy}$ (blue) orbital-dominated bands across different families with different correlation strengths. The bands are sketched according to experimental data. The numbers represent the renormalization factors on top of local-density approximation calculated band dispersions. Adapted from [1025]. (f) Orbital-selective quasiparticle decoherence at high temperatures in ${\mathrm{FeTe}}_{0.56}{\mathrm{Se}}_{0.44}$. Adapted from [1030]. (g) The $d_{yz}$ orbital renormalization factor positively correlates with the iron chalcogen or iron pnictogen bond lengths. (h) Relative renormalization strength between $d_{xy}$ and $d_{yz}$ orbital-dominated bands as a function of the iron pnictogen bond angles of different FeSCs. The acronyms represent different FeSC families and are fully indexed by [1025].

Figure 26

Band reconstructions associated with the nematic order and collinear antiferromagnetic order in underdoped FeSCs. (a) Nematic-order-induced band shift along two high-symmetry cuts in ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ at 80 and 160 K ($T_N\sim 138\mathrm{K}$). Adapted from [1027]. (b) Energy-momentum cut along $\mathrm{\Gamma }\text{−}M$ in the SDW state. Note the electron pocket folded from the zone corner (green arrows). Adapted from ([476]). (c) Nematic band shift energy scale and the SDW gap size at the $M$ point plotted against their respective transition temperatures across various FeSCs. Adapted from [892], and [1025].

Figure 27

Superconducting gap symmetry and anisotropy across different families of FeSCs. (a) Extended temperature-doping phase diagram with corresponding Fermi surfaces and the gap structures in various FeSCs. The thin red circle represents hole pockets, and thick blue represents electron pockets. Polar plots of the gap amplitude on the hole and electron Fermi surface sheet obey the same color coding, respectively. The dashed circles are grid lines for size references. Red shading indicates substantial $k_z$ dependence of the gap when projected onto the $x\text{−}y$ plane. (b) Proposed interpocket pairing mechanisms for (left panel) $s^{\pm }$-wave and (right panel) $d$-wave symmetry. Colors indicate different signs of the order parameter. Adapted from [348], and [349]. (c) Pairing amplitude computation based on coexisting electron and hole pockets. Different colors represent different symmetries as a function of nearest-neighbor and next-nearest-neighbor exchange couplings $J_1$ and $J_2$. Adapted from [837]. (d) Dependence of superconducting transition temperature $T_c$ on the $d_{yz}$ orbital renormalization strength. The same doping series of materials are represented by markers of the same color.

Figure 28

Coherent $A_{1g}$ phonon in FeSe. (a) The lattice displacement (top) and band energy shifts (bottom) are resolved by time-resolved x-ray diffraction and trARPES, respectively. (b) Schematic of the lattice (top) and band (bottom) modulation due to the coherently excited mode. From [271].

Figure 29

Layer and temperature dependence of the $\mathrm{FeSe}/\mathrm{SrTi}{\mathrm{O}}_3$ thin film. (a) Fermi surface of monolayer, two-layer, and three-layer FeSe films at low temperature (left to right). (b) The corresponding energy-momentum cuts through high-symmetry points: cut 1 at $\mathrm{\Gamma }$ and cut 2 at $M$. Adapted from [892]. (c) The $M$ pocket at 20 K (left) and 65 K (right). (d) Temperature dependence of the superconducting gap on the $M$ pocket. Adapted from [538].

Figure 30

Interfacial electron-phonon coupling in FeSe monolayer film. (a) $\mathrm{\Gamma }\text{−}M$ energy-momentum cut in $\mathrm{FeSe}/\mathrm{SrTi}{\mathrm{O}}_3$, with the main electron pocket and its shakeoff labeled $A$ and $A^{\prime }$, respectively. (b) Second energy-derivative-enhanced plot of (a). Adapted from [504]. (c) Superconducting gap sizes on two intercepting $M$ pockets from one K-dosed bulk FeSe (diamonds) and seven $\mathrm{FeSe}/\mathrm{SrTi}{\mathrm{O}}_3$ samples, containing ${}^{16}\mathrm{O}$ (circles) and ${}^{18}\mathrm{O}$ (squares). $\eta$ is the extracted electron-phonon interaction strength. Adapted from [854].

Figure 31

Band structure of graphene. (a) Monolayer graphene and its computed Dirac cone band structure at the Brillouin zone corner. (b) Measured isoenergy contour at the Dirac point energy $E_D$ of monolayer graphene. Adapted from [676]. (c) Band structure near the Dirac point and (d) $k_{||}-k_z$ Fermi surface maps for monolayer to four-layer graphene. Adapted from [675].

Figure 32

Interaction effects in graphene systems. (a) Low-energy dispersion anomalies in doped graphene. Adapted from [85]. (b) Substrate effect causing band renormalization in graphene. Adapted from [376]. (c) Substrate induced gap opening on the Dirac point for graphene-$6H$ SiC heterostructure. Adapted from [1085]. (d) Removal of translational symmetry and the formation of dodecagonal Dirac electron replicas in 30° twisted bilayer graphene. Adapted from [4].

Figure 33

Lattice and electronic structure of layer-dependent $2H\text{−}\mathrm{Mo}{X}_2$ ($X=\mathrm{S}$, Se). (a) Top and side views of the lattice structure. (b) PEEM image of a CVD grown flake. (c) Brillouin zone and high-symmetry points. (d) Microspot ARPES results on monolayer flake and bulk ${\mathrm{MoS}}_2$. Adapted from [404]. (e),(f) DFT calculation and experimental measurements of monolayer and (g),(h) eight-layer band structures of ${\mathrm{MoSe}}_2$. Unoccupied states are achieved via surface dosing. Adapted from [1075].

Figure 34

Layer-dependent CDW order revealed by band folding in $1T\text{−}{\mathrm{TiSe}}_2$. Single-particle spectra in (a),(c) the normal-state and (b),(d) the CDW state of $1T\text{−}{\mathrm{TiSe}}_2$. (a),(b) Monolayer thin film grown in situ on bilayer graphene. (c),(d) Bulk material. Adapted from [131].

Figure 35

Surface 2D electron gas in ${\mathrm{SrTiO}}_3$. (a) 2DEG carrier density vs UV irradiation dosage on the ${\mathrm{SrTiO}}_3$ $(001)$ surface. Adapted from [607]. (b) Electronic structure of the 2DEG on the ${\mathrm{SrTiO}}_3$ $(001)$ surface. Note the high in-plane dispersion and the lack of $k_z$ dispersion. (c) Strong polaronic shakeoff on the 2DEG surface state at low carrier concentrations. (d) Maximum curvature plot of (c) to highlight the shakeoff bands. (e) EDC fitting of the shakeoff band consisting of multiple phonon sidebands. Adapted from [964].

Figure 36

Evidence for spin-charge separation in the 1D chain compound ${\mathrm{SrCuO}}_2$. (a) Raw energy distribution curves for the bifurcated single-particle excitation consisting of spinon (red) and holon (blue) branches. (b) Fitted dispersions collapsed from different perpendicular momenta. The dashed lines are band theory calculations, while the red and blue solid lines are analytical fits to the spinon and holon dispersions. Adapted from [446].

Figure 37

A simplified view of the band structure evolution in topological states of matter studied by ARPES. A trivial insulator has a finite band gap between a conduction band (CB) and a valence band (VB). Closure of the gap can produce a Dirac semimetal with linear dispersion. Continued evolution of the gap results in band inversion, which can produce a topological insulator with surface states (SSs) bridging the gap. The Dirac semimetal becomes a Weyl semimetal if either time-reversal or spatial-inversion symmetries are broken, whereas the topological insulator exhibits exotic phenomena when coupled with superconductivity and magnetism.

Figure 38

${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ as the first 3D topological insulator. (a)–(c) Electronic structure for pure Bi, the alloy, and pure Sb, respectively. The band arrangement in pure Bi is trivial, while the bands at the $L$ point of pure Sb are inverted. Within a critical doping range, the alloy becomes a direct band gap semiconductor while retaining the topologically nontrivial band inversion. Adapted from [309]. (d) Sketch and (e) ARPES measurement of the surface state dispersion of ${\mathrm{Bi}}_{0.91}{\mathrm{Sb}}_{0.09}$ showing five Fermi-level crossings. The arrows denote the spin polarization of the bands, as verified by spin-resolved ARPES measurements. Adapted from [362].

Figure 39

ARPES measurements of the 3D topological insulators (a) ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and (b)–(d) ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. (a) Adapted from [983]. The topological surface state is observed linearly dispersing across the bulk band gap. (c) Fermi surface of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ showing the anisotropic dispersion of the surface state. (d) MDCs of the cut shown in (b). Adapted from [143].

Figure 40

Deviations from an ideal helical spin texture in 3D TIs. The ideal texture, shown in (a), consists of 100% polarized in-plane spins tangentially oriented along an isotropic Dirac cone. Top row: circular-dichroism ARPES on ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. (b) Difference in photoemission intensity between left-hand and right-hand circularly polarized light. (c) By modeling matrix elements in a spin-orbit coupled system, all three components of the spin polarization can be calculated from the data. (d) Summary of the deduced spin polarization, including an out-of-plane component associated with hexagonal warping. Adapted from [960]. Bottom row: Spin-orbital texture of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. (e) Spin polarization measured by spin-resolved ARPES reverses sign when the light polarization is rotated. Adapted from [417]. (f) This is because the eigenstates are linear combinations of spin (arrows) and orbital (blue and green shapes) components. The experiment measures the spin polarization associated with the orbital component photoemitted by the incident light polarization. Adapted from [1057].

Figure 41

(a) Series of three 2DEG states on a ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ surface contaminated with adsorbates. The lowest subband exhibits Rashba splitting due to strong spin-orbit coupling. (b) Model for the formation of 2DEGs. The conduction band energy $E_c$ is subject to a band-bending potential near the surface, leading to quantum confinement of the bulk wave function. Adapted from [461].

Figure 42

(a) ARPES spectra of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ thin films synthesized to the thickness shown (QL, quintuple layer). There is no Dirac cone observed for sufficiently thin samples. (b) Gap at the Dirac point extracted as a function of film thickness. The gapless topological surface state recovers for thicknesses of more than six QLs. Adapted from [1072].

Figure 43

Magnetically doped topological insulator. (a) Ungapped Dirac cone of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. (b) Gapped Dirac cone of Fe-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. Adapted from [144].

Figure 44

Topological phase transition driven by chemical substitution in $\mathrm{TlBi}({\mathrm{S}}_{1-x}{\mathrm{Se}}_x{)}_2$. The top row shows cuts through $k_y=0$, and the bottom row shows Fermi surface maps, with the doping level $x$ indicated. For $x\lesssim 0.6$ the material is a trivial semiconductor, while for $x\gtrsim 0.6$ the band gap becomes inverted and a topological surface states forms. Adapted from [993].

Figure 45

Topological phase transition driven by temperature in the crystalline topological insulator ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Se}$ ($x=0.23$). At a temperature between 100 and 200 K the bulk band gap inverts, leading to the formation of topological surface states. From [213].

Figure 46

Monolayer $1T^{\prime }\text{−}{\mathrm{WTe}}_2$ as a quantum spin Hall insulator. (a) Calculated band structure showing the band inversion enabled by spin-orbit coupling. (b) ARPES measurement revealing a 45 meV band gap, as indicated by the horizontal dashed lines. Adapted from [897].

Figure 47

(a) Distinction between Floquet states and Volkov states, which are dressing of electronic states in the solid and vacuum, respectively. Both states can exist simultaneously. (b) Experimental measurement of Floquet-Bloch bands on ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. Red arrows highlight avoided crossings between neighboring Floquet states. Adapted from [580].

Figure 48

Dirac semimetals. (a) Sketch of the band-inversion mechanism of Dirac semimetal formation. If two 3D bands of opposite parity are driven to invert, and are symmetry forbidden from mixing, they will necessarily form a pair of fourfold degenerate Dirac nodes that disperse linearly as a function of $k_x$, $k_y$, and $k_z$. (b) Experimental realization in ${\mathrm{Cd}}_3{\mathrm{As}}_2$. The 3D Fermi surface consists of a pair of nodal points. A cut through a node reveals a linear band dispersion (iv). Adapted from [553].

Figure 49

Weyl semimetals. (a) The simplest inversion symmetry breaking Weyl semimetal has four Weyl nodes, and the simplest time-reversal symmetry breaking Weyl semimetal has two Weyl nodes. (b) Band mapping of the inversion symmetry-breaking Weyl semimetal TaAs, with (c) a close-up of the surface Fermi arcs compared to (d) a calculation. Adapted from [1012]. (e) Fermi surface mapping of TaAs combining low-energy, surface-sensitive ARPES (green) with soft-x-ray, bulk-sensitive ARPES (orange). The soft-x-ray measurements isolate the bulk Weyl nodes, while the low-energy measurements reveal the surface Fermi arcs connecting them. (f) Schematic of this electronic structure, including the bulk Weyl nodes and surface Fermi arcs. Adapted from [998].

Figure 50

High-resolution laser ARPES on ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. (a) Fermi surface on CO passivated ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. (b) Spectral intensity on the energy-momentum cut along the red line in (a), where the band splitting is used to extract the spin-orbit coupling strength. From [890].

Figure 51

Fermi surface and development of quasiparticles in iridates. (a) Fermi surface overlaid with the anisotropic energy gap (colored dots) in ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$. (b) Emergence of quasiparticles with electron doping in ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$. Adapted from [185].

Figure 52

(a) ARPES spectrum from the heavy fermion system ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$, and (b) sketch of the spectral function as described by the periodic Anderson model. The renormalized $f$ level, or Kondo resonance, exists below $E_{\mathrm{F}}$ and hybridizes with the dispersive Rh $4d$ conduction electrons. From [178].

Figure 53

trARPES studies on the CDW in ${R\mathrm{Te}}_3$. (a) Tight-binding Fermi surface for ${\mathrm{DyTe}}_3$ consisting of the $p_x$ and $p_z$ orbitals, as indicated. Dashed lines represent the folded shadow bands. The CDW ordering vector $q_{\mathrm{CDW}}$ and gapped region are indicated. (b) Cut through the gapped region after pumping showing the gap and shadow bands. Because of photoexcitation of carriers, both upper and lower bands are visible. Markers indicate the band dispersion in a tight-binding model. (c) Spectrum in the gap region as a function of pump-probe delay. Markers indicate the peak positions of the upper and lower CDW bands, which reveal a partial closure and oscillation of the gap amplitude. Adapted from [758].