Theory of Nonlinear Interactions Between X Rays and Optical Radiation in Crystals
Total Page：16
File Type：pdf, Size：1020Kb
PHYSICAL REVIEW RESEARCH 1, 033133 (2019) Theory of nonlinear interactions between x rays and optical radiation in crystals R. Cohen and S. Shwartz Physics Department and Institute of Nanotechnology, Bar-Ilan University, Ramat Gan 52900, Israel (Received 17 December 2018; published 27 November 2019) We show that the nonlinear interactions between x rays and longer wavelengths in crystals depend strongly on the band structure and related properties. Consequently, these types of interactions can be used as a powerful probe for fundamental properties of crystalline bulk materials. In contrast to previous work that highlighted that these types of nonlinear interactions can provide microscopic information on the valence electrons at the atomic scale resolution, we show that these interactions also contain information that is related to the periodic potential of the crystal. We explain how it is possible to distinguish between the two contributions. Our work indicates on the possibility for the development of novel multidimensional pump-probe metrology techniques that will provide spectroscopic information combined with structural information including ultrafast dynamics at the atomic scale. DOI: 10.1103/PhysRevResearch.1.033133 I. INTRODUCTION matching). The use of the reciprocal lattice vector also pro- vides the atomic scale resolution, which can be achieved by The possibility to utilize nonlinear interactions between measuring the efficiencies of the effect for many reciprocal x rays and radiation at wavelengths ranging from infrared lattice vectors and by using Fourier analysis [3]. Since the (IR) to ultraviolet (UV) as an atomic-scale probe for inter- goal of the measurements is to probe microscopic informa- molecular interactions and for properties of valence electrons tion, the use of crystals introduces a new challenge for the has been discussed in several publications [1–9]. This probe interpretation of the results since the measured signal depends can provide a new insight into atomic-scale processes, such not just on the atomic or inter unit cell interactions between as charge transfer between atoms in molecules and micro- the valence electrons but also on the periodic potential of the scopic redistribution of charges in response to illumination. materials. It is therefore essential to develop a formalism that The technique relies on the atomic scale wavelengths of the enables the separation of the two contributions. x-rays, which provide the high resolution, while the longer We note that other approaches such as inelastic x-ray scat- wavelengths (UV/optical) are used to enhance the interactions tering (IXS) are very useful for the investigation of properties with the valence electrons, which are usually weak for x rays. of valence electrons including, for instance, valence charge The strong wavelength dependence of spontaneous para- distribution, density of states, and the macroscopic linear metric down conversion (SPDC) of x rays into UV, which response [12–14]. In resonant inelastic scattering (RIXS), has been observed in recent experiments [3,6,8–11] sug- the core electrons are excited to valance states and decay gests that nonlinear interactions between x rays and longer by nonradiative scattering processes to lower valence states. wavelengths can be used also as new spectroscopy tools for In the last step of the process, the electrons return to the the investigation of phenomena that traditionally are probed original core state and radiate at lower photon energy with an by using long-wavelength radiation with the advantage of energy that corresponds to the energy difference between the providing microscopic atomic-scale information. Moreover, x two intermediate valence states. Other IXS processes that do rays penetrate into materials more than electrons and therefore not require the excitation of core electrons such as Compton provide bulk properties in contrast to methods that are relied scattering [15], can be used to probe the ground-state electron on electron or ion scattering. momentum density (EMD) and the electron charge density of The main challenge in measuring x ray and optical/UV the investigated sample [16]. However, the spatial resolution nonlinear interactions is the weakness of the effects. To over- of the measurement is limited by the spatial resolution of the come this challenge, experiments are performed with crystals detectors and by the beam size, whereas in nonlinear x-ray where the periodic structure is used to enhance the signal in processes the use of the reciprocal lattice vector for momen- analogy to Bragg scattering, in which the reciprocal lattice tum conservation provides the possibility to utilize Fourier vector is used to achieve momentum conservation (phase synthesis to measure atomic scale structural information in contrast to conventional IXS. An interesting analogy between between SPDC and IXS (in the low-energy transfer limit) was introduced by Freund and Levine [17], by describing the Published by the American Physical Society under the terms of the interaction as Thomson scattering of x rays from optically Creative Commons Attribution 4.0 International license. Further induced valence charges. In this work, we indeed show that distribution of this work must maintain attribution to the author(s) the nonlinearity depends on the induced charge at the visible and the published article’s title, journal citation, and DOI. frequency, but only to a certain part of the nonlinearity. 2643-1564/2019/1(3)/033133(7) 033133-1 Published by the American Physical Society R. COHEN AND S. SHWARTZ PHYSICAL REVIEW RESEARCH 1, 033133 (2019) To date, most theoretical models that have been consid- (a) ered for the description of nonlinear interactions between x rays and longer wavelengths have focused on the ability to ω1 observe microscopic information and on the estimation of the ω3 = ω1+ω2 ω (2) strength of the effects [2–5,7,17–20]. However, they have not 2 σ addressed the challenge of the separation of the inter unit cell information from the periodic information. We note that Freund and Levine and also Jha and colleagues [17–19,21–25] have considered the periodic structure but not the influence (b) of the electronic band structure on the nonlinear interactions, ω which can be significant for valence electrons with binding s energies that are weaker or on the order of the periodic ωp = ωs+ωid (2) potential. In a recent theoretical paper, the authors analyzed σ ωid the general case of x ray diffraction from laser driven systems using a quantum electrodynamics approach, but in that paper the main focus is on the nonperturbative regime [26]. FIG. 1. Schematic diagrams for the nonlinear processes of SFG In several recent experimental papers, the experimental ω ω results were fitted to the theory by using the band gap rather and SPDC. (a) SFG: two input photons at frequencies 1 and 2 are converted to a photon at frequency ω = ω + ω . (b) SPDC: an than the atomic binding energy, but although good agreements 3 1 2 input pump photon at frequency ω is converted to a photon pair at were obtained for transparent materials, this approach is phe- p frequencies ω and ω . nomenological and failed for energies near or above the band s id gap [6]. ε ω − ω ( l ) e i l t is the electromagnetic vector potential assumed In this paper, we illustrate how the nonlinear interaction l iωl between x rays and longer wavelengths depends on the band in the Coulomb gauge, and ε(ωl ) is the electric field envelope structure and related properties by using perturbation theory of the lth mode. The solutions of the unperturbed Hamiltonian in the density matrix formalism. As an example we describe H0 are given by the Bloch states explicitly the dependence of the nonlinear interactions on the H0|n q=εn(q)|n q, (2) joint density of states of interband transitions. In addition, we analyze the polarization dependencies of the nonlinear with eigenvalues εn(q), where n is the band number and q is interactions and predict that it is not trivial. the wave vector associated with the Bloch state. II. THEORETICAL DESCRIPTION B. Density matrix AND GENERAL FORMALISM We follow the standard procedure for the calculation of nonlinear conductivity by perturbation theory in the density We focus here on the second-order nonlinearity that con- matrix formalism [22]. The density matrix elements ρ stitutes a source term in the wave equations that describe nm evolve according to effects such as sum-frequency generation (SFG) and SPDC. Schematic diagrams for these processes are presented in ∂ρnm ih¯ = [H + H ,ρ] − ih¯γ ρ − ρ(0) , (3) Fig. 1. The nonlinear interactions can be described by the ∂t 0 int nm nm nm nm second nonlinear current density or the nonlinear conductivity. where n and m signifies the eigenstates of the unperturbed Hamiltonian H0. We denote the phenomenological damping (0) A. Hamiltonian coefficients as γnm and ρ = f0(H0 ) is the relaxed density matrix which is the normalized Fermi-Dirac distribution, such To describe the nonlinear optical interaction in the crystal, that tr(ρ(0)) = 1. The full density matrix ρ(t ), can be ex- we use the minimal electromagnetic coupling Hamiltonian of pressed as a power series in the field A an electron in a periodic potential ρ(t ) = ρ(0) + ρ(A) + ρ(AA) +··· , (4) + 2 H = H + H = (p eA) + , 0 int V (x) where the first term on the right-hand side is the relaxed 2m density matrix in the absence of the field, the second term is 2 p linear with the field amplitude, the third term is square in the H0 = + V (x), (1) 2m field amplitude and so on. e e2 H = (p · A + A · p) + A2, int 2m 2m C. Current density where the unperturbed Hamiltonian is H0, the electron mass Following the perturbation theory formalism, we calculate is m, the momentum operator is p and the potential V (x) = average current density by using V (x + R) is periodic with respect to translation by a lattice , = ρ , , j (x t ) tr( (t ) jop(x t )) (5) vector, where R is any lattice vector.