Inelastic x-ray scattering by electronic excitations under high pressure

Jean-Pascal Rueff and Abhay Shukla

Rev. Mod. Phys. 82, 847 – Published 18 March 2010

Abstract

Investigating electronic structure and excitations under extreme conditions gives access to a rich variety of phenomena. High pressure typically induces behavior such as magnetic collapse and the insulator-metal transition in $3 d$ transition-metal compounds, valence fluctuations or Kondo-like characteristics in $f$ -electron systems, and coordination and bonding changes in molecular solids and glasses. This article reviews research concerning electronic excitations in materials under extreme conditions using inelastic x-ray scattering (IXS). IXS is a spectroscopic probe of choice for this study because of its chemical and orbital selectivity and the richness of information it provides. Being an all-photon technique, IXS has a penetration depth compatible with high-pressure requirements. Electronic transitions under pressure in $3 d$ transition-metal compounds and $f$ -electron systems, most of them strongly correlated, are reviewed. Implications for geophysics are mentioned. Since the incident x-ray energy can easily be tuned to absorption edges, resonant IXS, often employed, is discussed at length. Finally studies involving local structure changes and electronic transitions under pressure in materials containing light elements are reviewed.

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DOI:https://doi.org/10.1103/RevModPhys.82.847

Authors & Affiliations

Jean-Pascal Rueff^*

Synchrotron SOLEIL, L’Orme des Merisiers, BP 48, Saint Aubin, 91192 Gif sur Yvette, France and Laboratoire de Chimie Physique-Matière et Rayonnement, Université Pierre et Marie Curie-Paris 6, CNRS-UMR7614, 11 Rue Pierre et Marie Curie, 75005 Paris, France

Abhay Shukla

Institut de Minéralogie et de Physique des Milieux Condensés, Université Pierre et Marie Curie-Paris 6, CNRS-UMR7590, 140 rue de Lourmel, 75015 Paris, France

^*jean-pascal.rueff@synchrotron-soleil.fr

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Vol. 82, Iss. 1 — January - March 2010

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Figure 1
Scattering process of a photon by an electron system (gray area).Reuse & Permissions
Figure 2
Terms in the IXS cross section: (a) nonresonant and (b), (c) resonant scattering. Wavy (straight) lines represent the photon (electron) wave functions; double lines are inner-shell core hole.Reuse & Permissions
Figure 3
$q$ dependence of the matrix element $A_{l} (q)$ in the x-ray Raman scattering cross section. The momentum transfer $q$ is given in units of $2 π ∕ a$ where $a = 3.50 Å$ is the Li unit cell parameter. From 44.Reuse & Permissions
Figure 4
One-electron picture of two types of RIXS process: (a) resonant emission and (b) direct recombination. The shaded areas are occupied states. $Δ E$ stands for the energy of one particular excited state relatively to the ground state.Reuse & Permissions
Figure 5
(Color online) Schematic explanation of the RIXS process. (a) $1 s 2 p$ -RXES process in a configuration scheme. (b) Map of the cross section in the incident vs transfer energy plane. (c) Partial and total fluorescence yield absorption spectra—inset indicates the model density of unoccupied states. (d) Comparison of RXES spectra on resonance to fluorescence regime (off resonance).Reuse & Permissions
Figure 6
(Color online) Shake-up process in the intermediate state of a RIXS process. $| i ⟩$ and $| f ⟩$ are the initial and the final state and $| m ⟩$ and $| n ⟩$ are two intermediate states. From 45.Reuse & Permissions
Figure 7
First-order Coulomb corrections (dotted line) to RIXS. In (a), an excitonic pair is formed in the intermediate state.Reuse & Permissions
Figure 8
(Color online) Johann geometry is an approximation of the Johanssonn geometry for large $Δ E$ .Reuse & Permissions
Figure 9
(Color online) Press for anodic bonding. From 28.Reuse & Permissions
Figure 10
(Color online) Diamond anvil cell equipped with a solid (left) and perforated (right) diamond. The sample shown in dark gray is contained in a gasket that is compressed between the anvil and the piston. Pressure can be realized by inflating a metallic membrane schematized by the curvy line.Reuse & Permissions
Figure 11
(Color online) Self-absorption effect in a model transition-metal sample contained in a pressure cell. From top to bottom, sample geometry (a) in-plane scattering, (b) transverse geometry, and (c) full transmission (the sample and the gasket are shown in gray); 2D emission profile; integrated intensity. The sample diameter is $100 μ m$ , and with an attenuation length of $30 μ m$ , typical of metal oxides around $8 keV$ . From 155.Reuse & Permissions
Figure 12
(Color online) Molar volume as a function of the occupation number for the $3 d$ , $4 d$ , $4 f$ , and $5 f$ series.Reuse & Permissions
Figure 13
Metal-insulator transition in correlated transition metal. From 82.Reuse & Permissions
Figure 14
(Color online) Tanabe-Tsugano diagram of a $3 d^{5}$ ion in octahedral symmetry. The diagram was computed with the CAMMAG program (32). The configuration energy $(E)$ and crystal-field strength $(Δ)$ are normalized by the Racah parameter $B$ . Thick lines indicate the ground state in weak- and high-field limits.Reuse & Permissions
Figure 15
Example of high-spin $(S = 5 ∕ 2)$ and low-spin $(S = 1 ∕ 2)$ configurations for a $3 d^{5}$ metal ion in octahedral symmetry. $10 D_{q}$ is the crystal-field strength.Reuse & Permissions
Figure 16
(Color online) Fe $K β$ emission line through a pressure-induced high-spin to low-spin transition in a $3 d^{5}$ Fe-based molecular complex. The spectra are normalized to the integral. HS-LS difference spectrum is shown in gray.Reuse & Permissions
Figure 17
(Color online) $K β$ process on a configuration level scheme in the case of a $3 d^{5}$ ion. $| g ⟩, | i ⟩, | f ⟩$ denote the ground state, intermediate, and final states. Spectroscopic terms are indicated.Reuse & Permissions
Figure 18
(Color online) Deviations of the HS fraction $γ_{HS}$ as determined from different extraction techniques in simulated XES spectra of known spin state. The spectra are constructed from a linear superposition of theoretical HS and LS spectra in a ${Fe}^{2 +}$ ion with a HS weight $γ_{HS}$ (inset). The IAD values (open circles and dots) show the best results. Adapted from 194.Reuse & Permissions
Figure 19
(Color online) HS fraction in ${Fe}^{2 +}$ . Pressure dependence of enthalpies from 188.Reuse & Permissions
Figure 20
(Color online) Calculated magnetic moment in transition-metal oxide as a function of pressure: GGA (open circles) and LDA (solid circles) results. The vertical lines denote the calculated transition pressures. From 25.Reuse & Permissions
Figure 21
(Color online) Magnetic collapse in monoxides observed by XES. (a)–(d) $K β$ -XES spectra measured in MnO, FeO, CoO, and NiO in both low- (close circles) and high-pressure (open circles) phases. * indicates spectra obtained after of-line laser heating. (e)–(h) Calculated spectra at ambient pressures (thick lines) and high pressures (thin lines). Ticks represent the multiplet states, before broadening. From 124.Reuse & Permissions
Figure 22
(Color online) Phase diagram of the magnetic collapse in the transition-metal oxides. (a) The point coordinates refer to calculated values of the crystal-field splitting $10 D_{q}$ and ligand bandwidth $W (O − 2 p)$ , in both the HS (open symbols) and LS (closed symbols) states, as obtained from comparison with the experimental spectra. (b) Solid lines mark the calculated HS-LS transition boundary for CoO for different values of $V_{e_{g}}$ . From 124.Reuse & Permissions
Figure 23
(Color online) Competition between structure and magnetism in hematite. (a) Fe $K β$ emission in ${Fe}_{2} O_{3}$ as a function of pressure. (b) Diffraction patterns through the metastable phase transition and corresponding XES spectra. From 12.Reuse & Permissions
Figure 24
Pressure-induced closing of the correlation gap. (a) RIXS in NiO at ambient pressure; the incident energy is tuned to the quadrupolar prepeak in the absorption spectrum (arrow in inset). Inset: PFY-XAS spectrum of the Ni $K$ edge (solid line) with an arrow showing the choice of incident energy and constant-final-state scan (dashed line) at an energy loss of $5 eV$ . (b) Dependence of the RIXS spectra as a function of pressure; the dashed line is the nonresonant background. From 167.Reuse & Permissions
Figure 25
(Color online) RIXS in CoO as a function of pressure (open circles); lines are three-point average of the data. The incident energy is tuned to the quadrupolar prepeak in the absorption spectrum (arrow in inset). (R) indicates pressure release.Reuse & Permissions
Figure 26
Total energy (left scale) and magnetic moment (right scale) of Fe in the fixed spin moment method. From 130.Reuse & Permissions
Figure 27
Integrated absolute difference calculated from the XES spectra in the Fe-Ni Invar. The open, half-filled, and solid circles represent IAD values for the consecutive series of measurements. The right scale is deduced from the pure Fe XES data. Horizontal lines emphasize the three magnetic states (high spin, low spin, and nonmagnetic) of the Fe atom. From 158.Reuse & Permissions
Figure 28
Integrated absolute difference calculated from the XES spectra (cross) in pure Fe compared to the $α$ -phase fraction determined by Mössbauer spectroscopy (185) (solid circles) across the transition. The diffraction pattern measured in both phases, shown in the inset, confirm the structural change. The spin state is indicated. From 154.Reuse & Permissions
Figure 29
Fe $K β$ emission line in FeS as a function of pressure. At low pressure, the satellite at $7045.5 eV$ is characteristic of the HS state. The decrease in the satellite at high pressure denotes the transition to the LS state (dotted lines). From 153.Reuse & Permissions
Figure 30
(Color online) Magnetic collapse and temperature effects. (a) Fe x-ray emission spectra measured in magnesiowüstite solid solution $({Mg}_{0.83}, {Fe}_{0.17}) O$ . From 11. (b) Computed LS fraction $n$ in the $(P, T)$ space. The arrows correspond to the transition pressure range as obtained from XES measurements, the full line is a lower mantle geotherm. From 188.Reuse & Permissions
Figure 31
(Color online) Successive magnetic transitions in Fe-perovskite. (a) Fe x-ray emission spectra measured in $({Mg}_{0.9}, {Fe}_{0.1}) Si O_{3}$ between 20 and $145 GPa$ . (b) XES spectra in model compounds (solid lines) are superimposed on to three spectra representative of the three different spin states. From 13.Reuse & Permissions
Figure 32
Pressure vs temperature effects in Co. (a) Evolution of the $K β$ emission line in $La Co O_{3}$ as a function of pressure. (b) Pressure $(T = 300 K)$ and temperature (at ambient pressure) dependence of IAD values derived from XES (left scale) and estimated Co spin state (right scale). From 195.Reuse & Permissions
Figure 33
(Color online) High pressure, low temperature Co spin state. (a) Evolution of the Co $K β$ emission line in ${La}_{1 - x} {Sr}_{x} Co O_{3}$ $(x = 0.18)$ as a function of pressure at $T = 34 K$ . (b) Pressure dependence at 34 and $300 K$ of the spin state (IAD values) derived from XES. From 108. Dashed lines are guides to the eye.Reuse & Permissions
Figure 34
Electron hopping mechanism in the ${Co}^{3 +} ∕ {Co}^{4 +}$ lattice in ${La}_{1 - x} {Sr}_{x} Co O_{3}$ . From 107.Reuse & Permissions
Figure 35
Equation of state of several $4 f$ elements. From 125.Reuse & Permissions
Figure 36
Kondo resonance in Ce, near the Fermi energy (set at $ω = 0 eV$ ). The photoemission and inverse photoemission data from 115 are compared to DFMT calculations. From 126.Reuse & Permissions
Figure 37
Overview of RIXS in mixed-valent systems. (a) $2 p 3 d$ -RXES process in a model mixed-valent rare-earth represented by the superposition of two valent states; $v$ stands for the valence electrons. Gray arrows indicate less probable transitions. (b) The $2 p 3 d$ -RXES in a mixed-valent ions in the incident energy $(ℏ ν_{IN})$ vs transfer energy $(ℏ ν_{T})$ plane; $F$ indicates the fluorescence contribution. From 35.Reuse & Permissions
Figure 38
Phase diagram of elemental Ce. Adapted from 52.Reuse & Permissions
Figure 39
RIXS view of the $T$ -induced valence transition in doped Ce. (a) $Ce − 2 p 3 d$ RXES spectra for ${Ce}_{0.90} {Th}_{0.10}$ at $60 K$ as a function of the transfer energy; inset shows the XAS spectra at $60 K$ and $300 K$ . (b) Variation of the spectrum measured at the $f^{2}$ resonance while changing the temperature through the transition. From 150.Reuse & Permissions
Figure 40
(Color online) $f$ states in Ce observed by XAS. (a) Experimental and (b) calculated $L_{3}$ XAS spectra in elemental Ce as a function of pressure. Ticks in (b) are the multiplet states (shown at $0 kbar$ ). From 151.Reuse & Permissions
Figure 41
(Color online) Change of Ce $f$ states under pressure from RIXS. (a) $2 p 3 d$ -RXES spectra in elemental Ce as a function of pressure; thick lines are calculated spectra. From 151. (b) Change in $f$ -occupation number $1 - n_{f}$ .Reuse & Permissions
Figure 42
(Color online) $f$ -electron delocalization in Yb under pressure. (a) PFY absorption spectra in elemental Yb as a function of pressure. From 38. (b) Change of the Yb hole number $1 - n_{h}$ .Reuse & Permissions
Figure 43
(Color online) RIXS overview of the valence change in TmTe. (a) (Top) PFY-XAS spectra measured for TmTe at the Tm $L_{3}$ edge at pressures up to $10.6 GPa$ . (Bottom) RXES spectra measured at the $2 +$ resonance (solid circles); solid lines are multiplet calculations. (b) (Top) Pressure dependence of the Tm valency in TmTe as obtained by PFY-XAS (circles) and RXES (crosses) and calculations (full squares). (Bottom) Relative volume change in TmTe from x-ray diffraction. From 85.Reuse & Permissions
Figure 44
Energy difference between divalent and trivalent $4 f$ ions calculated for rare-earth metal (open circles) and sulfides (crosses). From 174.Reuse & Permissions
Figure 45
Valence change in Yb compounds under pressure. (a) $2 p 3 d$ -RXES in $Yb {Al}_{2}$ measured at fixed incident energy set in the pre-edge region. (b), (c) PFY-XAS spectra in $Yb {Al}_{2}$ and YbS as a function of pressure. From 35 and 9.Reuse & Permissions
Figure 46
(Color online) Summary of the Yb valency as a function of pressure. Dashed lines are guide to the eye. From 35, 38 and 9.Reuse & Permissions
Figure 47
(Color online) Mixed valent Gd. (a) $2 p 3 d$ -RXES in Gd as at high pressure; (b) experimental and calculated $f^{n + 1} ∕ f^{n}$ ratio as a function a pressure. Results in pure Ce (solid squares) from 151 have been added for comparison. Adapted from 117.Reuse & Permissions
Figure 48
SmS properties across the black to golden phase transition. (a) Pressure variation of the lattice parameter $a$ of SmS at $T = 300 K$ . From 146. (b) Variation of the Sm valence in SmS obtained by RIXS (circles) and standard x-ray absorption (triangles). From 8.Reuse & Permissions
Figure 49
(Color online) PFY-XAS spectra in (a) $U {Pd}_{3}$ and (b) $U {Pd}_{2} {Al}_{3}$ as a function of pressure (open circles). Ambient PFY-XAS spectrum of $U {Pt}_{3}$ is shown (black line) for comparison purpose. From 157.Reuse & Permissions
Figure 50
(Color online) Calculated occupation number in $U {Pd}_{2} {Al}_{3}$ and $U {Pd}_{3}$ as a function of pressure. From 157.Reuse & Permissions
Figure 51
(Color online) C $K$ edge in highly oriented pyrolitic graphite as obtained by x-ray Raman scattering (XRS) from 152 and soft x-ray absorption spectroscopy (XAS) from 18. The energy scale refers to the transfer energy (incident energy) for XRS (XAS). The momentum transfer $(q)$ and polarization vector $(ϵ)$ are set either parallel (∥) or normal (⊥) to the graphite $c$ axis.Reuse & Permissions
Figure 52
C $K$ edge in graphite under pressure measured by x-ray Raman scattering (XRS). From 118.Reuse & Permissions
Figure 53
Bonding changes in BN. B (left) and N (right) $K$ edges in BN under pressure measured by x-ray Raman scattering (XRS). From 127.Reuse & Permissions
Figure 54
(Color online) O $K$ edge in water and ice phases as a function of pressure and temperature. From 20.Reuse & Permissions
Figure 55
Phase diagram in the vicinity of a quantum critical point.Reuse & Permissions

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