X-ray methods in correlated electron ­science

The relevance for correlated-electron materials of ps dynamics at the nanoscale, together with strong interaction of X-ray photons with all four of the C-S-O-L degrees of freedom, promises important applications of the SwissFEL. To begin the discussion of relevant X-ray techniques, we consider the Hamiltonian describing an X-ray photon field interacting with the electrons in the sample [6, 7]. For the moment, we treat a single free electron, without spin:
H = \frac{(\bar{p} - e\bar{A})^2}{2m} + H_{radiation}
where H_{radiation} = \sum_{\epsilon, k} \hbar \omega_k
(a^{\dagger}_{\epsilon \bar{k}} a_{\epsilon \bar{k}} + \frac{1}{2}) describes photons in the radiation field, with wavevector and polarization ε. Expanding the expression for H, we obtain:
H =H_{electron} + H_{interaction} + H_{radiation}
H_{electron} = \frac{p^2}{2m}
H_{interaction}  = \frac{e^2 A^2}{2m} - \frac{e\bar{A} \cdot
\bar{p}}{m} \equiv H_1 + H_2
The interaction Hamiltonian, Hinteraction, is responsible for producing transitions from an initial state to a final state of the combined system of X-rays and sample. According to Fermi’s Golden Rule, the transition rate is given by:
w = \frac{2 \pi}{\hbar} \left| \langle f|H_1|i \rangle + \sum_n
\frac{\langle f | H_2 | n \rangle \langle n | H_2 | i \rangle}{E_0 -
E_n + \hbar \omega_k} \right|^2 \delta \left((\hbar \omega_k - \hbar
\omega_{k'}) - (E_f - E_0)\right)
Here we are interested in “photon-in” (k) to “photon-out” (k’) transitions – thus we include only those terms which are quadratic in the vector potential A. We disregard the linear terms, which are responsible for photoemission. The transition rate is proportional to the square of the “scattering factor”, w \propto |S(\bar{Q}. \omega)|^2, which, in turn, is a function of the momentum and energy transfers \bar{Q} = \bar{k}' - \bar{k} and \omega = \omega_{k'} - \omega_k.

Consider now the following cases:

Hard X-ray diffraction

If the photon energy is much larger than the excitation energies En-E0 of the system, we need only consider the first matrix element in the expression for w:
\langle f|H_1|i \rangle \propto \sum_j \langle0 |e^{i \bar{Q}
\cdot \bar{r_j}}|0 \rangle = S(\bar{Q}, 0) = F(\bar{Q})
where is the ground-state of the system, and where we now sum over all the electrons, with coordinates. For a crystalline sample, the scattering factor F(\bar{Q}), is written
F(\bar{Q}) = \sum_{l.m.n} e^{i \bar{Q} \cdot \bar{R}_{lmn}}
\sum_{i=1}^I e^{i \bar{Q} \cdot \bar{r_i}} f_i(\bar{Q})
i.e., as a lattice sum over the unit cells and a “structure-factor” sum over the I atoms per unit cell. The quantity f_i(\bar{Q}) is known as the atomic scattering factor. For a perfect crystal, the lattice sum dictates that F is non-zero only if \bar{Q} = \bar{G}_{hkl}, a reciprocal lattice vector. This is the condition for Bragg scattering, which, using the relations Q = \frac{4 \pi \sin \theta}{\lambda} and G = \frac{2 \pi}{d}, can be expressed as Bragg’s law: \lambda= = 2 d \sin \theta. We thus see that diffraction at the Bragg angle θ is possible when the X-ray wavelength λ is shorter than twice the lattice-plane spacing (2d). The scattered intensity at the Bragg condition is proportional to |F(\bar{Q})|^2. The sensitivity of the Bragg condition to the lattice parameters implies that the diffraction of short X-ray pulses can be used to directly observe lattice phonons.

Soft X-ray resonant diffraction

If the incoming photon energy lies close to an atomic absorption edge, it is the second-order contribution from H2 which dominates the scattering rate. The case of elastic scattering, \omega_{k'} - \omega_k = 0, is treated via an energy-dependent correction to the atomic scattering factor:
f_i(\bar{Q} \Rightarrow f_i(\bar{Q}, \omega) = f_i (\bar{Q}) +
\triangle f_i (\bar{Q}, \omega)
where the correction term obeys:
\triangle f_i \propto \sum_n \frac{\langle 0| \bar{\epsilon} \cdot
\bar{r} e^{i \bar{k} \cdot \bar{r}} | n \rangle \langle n |
\bar{\epsilon'} \cdot e^{i \bar{k} \cdot \bar{r}} | 0 \rangle}{\hbar
\omega_k - (E_n - E_0) - i \Gamma}

We see that the scattering is sensitive to the electronic structure of the ground- and intermediate states and to the polarization of the incoming and outgoing photons (see Fig. V.6 a). The existence of a “core-hole” in the intermediate state is responsible for introducing the linewidth parameter Γ, representing the lifetime of the state |n>.

For soft X-rays, where the photon wave-vector k is significantly larger than the atomic dimensions, one can expand the exponentials into dipole, quadrupole and octupole terms:
\langle 0 |\bar{\epsilon} \cdot \bar{r} e^{i \bar{k} \cdot \bar{r}}|n \rangle \approx \langle 0 | \bar{\epsilon} \cdot \bar{r} | n \rangle + i \langle 0 | (\bar{\epsilon} \cdot \bar{r})(\bar{k} \cdot \bar{r}) | n \rangle - \langle0 | (\bar{\epsilon} \cdot \bar{r})(\bar{k} \cdot \bar{r})^2 | n \rangle
The product of two such matrix elements yields a hierarchy of tensorial terms: dipole-dipole (rank 2), dipolequadrupole (rank 3), quadrupole-quadrupole (rank 4), etc. These tensorial components can be enhanced by a suitable choice of polarizations and scattering vector. In TMOs, interesting resonances are the L2 and L3 transition-metal edges, which connect filled 2p and unfilled 3d states. The corresponding photon wavelength, approximately one nm, allows the observation of resonant soft-X-ray diffraction, and the dependence of the matrix elements on valence-band electronic structure produces superstructure Bragg reflections, corresponding, for example, to orbital-ordering in the correlatedelectron material. And when circularly-polarized X-rays are used, XMCD-contrast (see Chapter I) makes magnetic order visible. Finally, a charge-dependent shift of the initial core level shifts the energy-dependent resonant scattering profile (see Fig. V.6 b), providing a chargeorder contrast. One should note that soft-X-ray resonant diffraction is not easy: it requires scanning of the incoming photon energy, a multi-axis diffractometer in vacuum, and perhaps polarization-analysis of the scattered beam. With the SwissFEL, a pump-probe resonant diffraction experiment can follow, for example, the melting of orbital order by a laser pump pulse and its recovery at later times. Its shor t pulses and flexible energy tuning, par ticularly near 1 nm wavelength, make the SwissFEL an ideal source for such investigations of TMO correlated electron materials. A par ticularly interesting application of pump-probe resonant elastic scattering at the SwissFEL would be the time-resolved study, at the nanometer scale, of so-called “orbitons”, wave-like excitations of the orbitally-ordered phase, in manganites (see Fig. V.7).

Resonant inelastic X-ray scattering (RIXS)

We now lift the restriction to elastic scattering, by allowing \omega = \omega_{k'} - \omega_k to be non-zero, requiring, in general, energy analysis of the scattered photons. A schematic of the RIXS process is shown in Figure V.8. The scattering rate is now given by:
w = \frac{2 \pi}{\hbar} \sum_f \left| \sum_n \frac{\langle f | H_2 | n \rangle \langle n | H_2 | i \rangle}{E_0 - E_n + \hbar \omega_k + i \Gamma} \right| ^2 \delta((\hbar \omega_k - \hbar \omega_{k'}) - (E_f - E_i))
Note the following features: a) The sensitivity to photon polarization and valence electronic states seen in resonant elastic scattering is also present for RIXS. b) Although there exists a (virtual) core-hole in the intermediate state, evidenced by the Γ-term in the denominator, because there is no hole in the final state, the wk’ resolution of RIXS can in principle be infinitely good – as evidenced by the energy δ-function. c) Low-energy collective excitations, such as phonons, plasmons, spinwaves, etc., of the sample are accessible with RIXS, since what is measured is the energy difference between the incoming and outgoing photons. And since two photons are involved, the dipole selection rule Δl=±1 does not apply, such that a d→d transition can be observed (see Fig. V.9 a). d) Although per formed at resonance, RIXS is a low-efficiency process: In resonant elastic scattering, the excitation of each of the N scattering atoms can be coherently added, since there is a unique final state. This results in a scattering intensity proportional to N2. For RIXS, because the vir tual excitation of different atoms leads to different final states, the contributions add incoherently, resulting in an intensity propor tional to N [7]. Performing RIXS is an extremely challenging undertaking, due to the low scattered intensity and because of the necessity of per forming an energy (and scattering-angle) analysis of the scattered radiation. For pump-probe RIXS experiments at the SwissFEL, it would therefore be par ticularly interesting to realize a single-shot mode of measuring, either in the frequency (see Infobox) or in the time (ditto) domains.

The intermediate scattering function

At the beginning of this Section, we saw how the transition rate for X-ray photon scattering, with momentum transfer \bar{Q}and energy transfer \hbar \omega, is related to the scattering function S(\bar{Q}, \omega). This function shows peaks as a function of ω corresponding, for example, to longlived oscillations (quasipar ticles), such as phonons, spin-waves, etc. But finite lifetime effects will broaden these quasipar ticle peaks, and in the limit of strong damping, it may be advantageous to observe the timedependent fluctuations of the system directly – i.e., to measure instead the so-called time-domain or intermediate scattering function [12]:
S(\bar{Q}, t) = \int_{-\infty}^{\infty} S(\bar{Q}, \omega) e^{i
\omega t} d\omega
The intermediate scattering function basically provides the correlation time (over which S(Q,t) decays to the value 1/e) for the equilibrium fluctuations of a system, as a function of the fluctuation length scale 1/Q. By monitoring the scattered intensity I(θ,t) at a par ticular scattering angle 2θ (related to the momentum-transfer by Q = 4π sinθ/λ), one has access to S(Q,t) via the intensity correlation function g2(θ,t) and the “Sieger t relation” [13]:
g_2(\theta,t) = \frac{\langle I(\theta, t)I(\theta, t + \tau)
\rangle}{\langle I(\theta, t) \rangle ^2} = 1 + |S(\bar{Q}, t)|^2
A measurement of I(θ,t) on the ultrafast time scale, pertinent to correlated electron materials, would require reading out a detector at an impossible rate of GHz -THz. Two realistic alternatives, however, which are well-suited to the characteristics of the SwissFEL, are provided by the “split-pulse” and “Mössbauer filter foil” techniques and are described in Infoboxes.

A single-shot RIXS spectrometer

When per forming resonant inelastic X-ray scattering (RIXS), the scattered intensity as a function of outgoing photon energy E_{out} = \hbar \omega_{k'} is normally acquired for par ticular settings of the incoming energy E_{in} = \hbar \omega_k by the monochromator. This is a procedure which is incompatible with single-shot operation at the Swiss- FEL. One would like to instantaneously obtain a twodimensional plot of the scattered intensity as a function of Ein, Eout. A method of per forming single-shot RIXS has been proposed by V. Strocov [25] (see Fig. V.i2). The Swiss- FEL pulse is dispersed vertically by a monochromator (upper right in the figure) and brought to a line focus on the (homogeneous) sample. Scattered light corresponding to the various incoming photon energies is then focused to a vertical line and dispersed in Eout horizontally onto a CCD detector. The result is the desired two-dimensional plot.

RIXS in the time-domain

A principal drawback of conventional RIXS measurements is the necessity of a fine energy-analysis of the scattered radiation, resulting in a significant loss of intensity. It has been suggested [26] that the ultrashor t pulses of the SwissFEL could be used to effectively per form RIXS in the time-, instead of the energy domain. If a suitable non-linear (NL) optical medium for soft X-rays could be developed, one could imagine performing X-ray heterodyne spectroscopy: A SwissFEL pulse is split into two pulses. One of these is scattered by the sample, causing the creation, by inelastic scattering, of multiple frequency components. This multifrequency pulse is then recombined with the unscattered reference pulse in the NL-medium, where sum and difference frequencies are generated. The difference frequencies appear at the detector as slow oscillations, corresponding to the inelastic energy loss or gain in the sample. Heterodyne spectroscopy is routinely per formed with optical pulses, using the frequency-resolved optical gating (FROG) technique (see Fig. V.i3). Realization of an X-ray FROG will require the transform-limited pulses which a seeded SwissFEL will provide.

Split-pulse XPCS

X-ray photon correlation spectroscopy measures the time-correlation function of the coherently-scattered radiation intensity from a fluctuating sample:
g_2(\tau) = \frac{\langle I(t) I(t+\tau) \rangle}{\langle
I(t)\rangle ^2}
Since this requires a detector bandwidth which exceeds that of the fluctuations, one is generally limited to times τ longer than 10 nsec. The use the SwissFEL to probe ps-dynamics will require a different approach – “split-pulse XPCS” (see Fig. V.i4) [11]. Here a single SwissFEL pulse is split and delayed, producing a pair of pulses with tunable separation τ. The 2d-detector then registers a doubleexposure speckle pattern, and the speckle contrast will decrease when τ exceeds the fluctuation correlation time τc. That one indeed can measure g(τ) with split-pulse XPCS has been demonstrated by Gutt et al. [29]: The double exposure delivers the intensity S(\tau) = I(t) + I(t+\tau) , and the speckle contrast is given by the variance:
c_2(\tau) \equiv \frac{\langle S^2(\tau)\rangle - \langle S(\tau)
\rangle ^2}{ \langle  S(\tau) \rangle ^2}
We note that:
\langle S^2 (\tau) \rangle = 2 \langle I^2 \rangle  + 2 \langle
I(t) I(t + \tau) \rangle
\langle S(\tau) \rangle ^2 = 4 \langle I \rangle ^2
and that, for a fully coherent beam,
\langle I^2 \rangle - \langle I \rangle ^2 = \langle I \rangle ^2
We thus obtain:
c_2(\tau) = \frac{\langle I(t) I(t+\tau) \rangle}{2 \langle I
\rangle^2} = \frac{g_2(\tau)}{2}.

The Mössbauer filter-foil technique

The Mössbauer filter-foil technique provides an alternate method for measuring the intermediate scattering function [30]. Foils containing a stable Mössbauer isotope are placed in front of and behind the scattering sample and, at time t=0, a resonant Swiss- FEL pulse excites the isotope. The subsequent decay is then monitored by the total transmitted counting rate, measured in the forward direction (see Fig. V.i5). Denoting by ρ(Q,t) the spatial Fourier transform of the electron density in the sample at the time t, the intermediate scattering function is given by S(Q,t) = 〈ρ(Q,0)ρ*(Q,t)〉. Further, we let |g(t)|2 denote the decay probability of the Mössbauer isotope a time t after excitation. The signal I(t) at the detector is the square of the coherent sum of the probability amplitudes for a) radiation scattered by the sample at time zero and emitted by the nuclei of the second foil at time t and b) emitted by the first foil at time t and directly scattered into the detector:
I(t) = \langle | \rho(Q,0)g(t) + g(t) \rho(Q,t)|^2 \rangle = 2
|g(t)|^2 [|\rho(Q,0)|^2 + R_e S(Q,t)]
From this expression, the intermediate scattering function can be extracted. By far the most popular Mössbauer isotope is 57Fe, with a resonant energy of 14.4 keV and an excited state lifetime of 141 ns. An upgrade option for the SwissFEL will make this photon energy accessible with the fundamental undulator radiation, as well as still higher energies using XFEL harmonics [31]. Advantages of the filter-foil method in combination with the SwissFEL are: a) extremely intense X-ray pulses with zero background during the nuclear decay, and b) measurements of S(Q,t) for times far longer than those accessible with the split-pulse XPCS method. (The maximum feasible split-pulse delay will be ns for hard X-rays and tens of ps for soft X-rays.)