Theory of Nonlinear Interactions Between X Rays and Optical Radiation in Crystals

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 provides 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. In our derivation, the , = electromagnetic interaction described by Hint is treated as where the current density operator [27] is given by jop(x t ) − , = p + A , a perturbation, where e is the electron charge, A(x t ) jop(x) jop(x t ). The first term being the momentum current

033133-2 THEORY OF NONLINEAR INTERACTIONS BETWEEN … PHYSICAL REVIEW RESEARCH 1, 033133 (2019)

p = 1 { , } and denote the input x-ray beam, the output x-ray beam, and density jop(x) 2m Dop(x) p and the second term is the A , = e , the optical beam as the pump, signal, and idler respectively. so called gauge current density jop(x t ) m Dop(x)A(x t ). We expressed these operators by the charge density operator, Since the wavelengths of the x rays are also on the order of the distance between the atomic planes, the reciprocal lattice which we define as Dop(x) =−e|xx|. The current density can be expressed as a power series in the field A vector is used to comply with the requirement for momentum conservation (phase matching), given by the equation kp + (n) j(x, t ) = j G = ks + kid , where G is the reciprocal lattice vector, and n kp, ks, kid stand for the pump, signal and idler wave vectors respectively. Thus the measured intensity is proportional to = j(0)(x) + j(1)(x, t ) + j(2)(x, t ) +··· , (6) the Fourier coefficient that corresponds to the selected recip- where n signifies the order in which the current density de- rocal lattice vector. pends on the field. Following Eq. (4), we find that the second- It is possible to write the Fourier component of the second- , , order current density is j(2)(x, t ) =j(2 p) +j(2 A), where order nonlinear conductivity that corresponds to the reciprocal (2,p) = ρ(AA) p j tr( jop(x)) is the second-order current den- lattice vector G of the signal as a sum of two terms that are sity that arises from the momentum part and j(2,A) = originated from the gauge part and from the momentum part ρ(A) A , of the current density operator: tr( jop(x t )) is the second-order current density that arises from the gauge part of the current density operator. The tem- e σ (ω ; G) = D (−ω ; G)δ poral part of the second-order current density can be Fourier ijk s imω k id ij expended p e3 1 − ω +ω + B (ω , k ; G), (10) (2) (2) i( l l )t ijk id id j (x, t ) = j (x; ωl + ωl )e . (7) Vm3ω ω 2 p s l,l Here, ω ,ω , and ω are the frequencies of the signal, idler, Each Fourier coefficient in the direction of the generated field, s id p and pump, respectively, and δij is the Kronecker delta. is related to the conductivity [28] by the following expression: As described in Sec. II B, in our procedure, we expand (2) j (x; ωl + ωl ) · eî the density matrix in the unperturbed basis using the Bloch states. In this work, we are interested in referring the nonlinear 3 (2) current density to microscopic and inter unit cell information, = σ (x; ω + ω )(ε(ω ) · eˆ )(ε(ω ) · eˆ ), (8) ijk l l l j l k hence we use the Wannier basis, which contains this infor- j,k=1 mation. The |Wn is a Wannier function of band n.Weusea wheree î is a unit vector in the ith direction and {i, j, k} standard relation between the Bloch and Wannier basis, which indicates Cartesian coordinates. Since the unperturbed system is given by is periodic in the lattice spacing, we Fourier expand spatial 1 part of the nonlinear conductivity, |n q=√ eiq·RT R |W , ( ) n (11) (2) (2) · N R σ ω + ω = σ ω + ω iG x, ijk (x; l l ) ijk ( l l ; G)e G −ip·R where T (R) = e h¯ is the translation operator. To further (2) 1 (2) − · simplify our formulas, we assume that the Wannier functions σ ω + ω = σ ω + ω iG x. ijk ( l l ; G) dx ijk (x; l l )e (9) V V are very localized (there is no overlap between neighboring functions at different sites), such that the overlap between where G is the reciprocal lattice vector and V is the crystal nearest-neighbor Wannier functions is very weak as in the volume. The spatial Fourier component of the conductivity case of insulators and semiconductors. By making this as- can be selected via the phase-matching condition relevant to sumption, we can separate the contribution of the intermolec- the process of choice. ular interactions and the band structure. Since the optical response is comprised of a sum over all quantum numbers, the D. Second-order conductivity for spontaneous parametric interaction dependence on the matrix elements and the band down-conversion of x rays into UV/visible radiation structure is intertwined. However, by assuming that the Here we derive the expression for the conductivity relevant Wannier functions are very localized, the summation over the to the process of SPDC of x rays into UV/visible radiation. matrix elements is reduced to a sum over only the band num- In this case, we simplify the expressions by assuming that the bers. The rest of the terms depend on all quantum numbers x-ray wavelengths are far above any electronic transitions and hence have the meaning of the spectral dependence of the use the dipole approximation for the optical beam (but not for interaction. Following these steps we find that Dk (−ωid ; G) the x rays). We adopt here the notation that is used for SPDC and Bijk(ωid , kid ; G)aregivenby

2 ihe¯ −iG·x D (−ω ; G) = W e W W p · eˆ W I , (ε , k ), (12) k id mV n2 n1 n1 k n2 n2 n1 id id n1 n2 ω , = − · −iG·x · − · · ε , , Bijk( id kid ; G) h¯(kid G) Wn2 e [ˆei(p eˆ j ) eˆ j (p eˆi )] Wn1 Wn1 p eˆk Wn2 In2,n1 ( id kid ) (13)

n1 n2

033133-3 R. COHEN AND S. SHWARTZ PHYSICAL REVIEW RESEARCH 1, 033133 (2019)

− ε + − ε V f0 n1 (q kid ) f0 n2 (q) I , (ε , k ) = dq , (14) n1 n2 id id π 3 ε ε + − ε − ε + γ (2 ) B.Z. id n1 (q kid ) n2 (q) id ih¯ n1 q+kid ,n2 q where we denote the spectral dependence of the interaction to and their expression is given by ε , be In1,n2 ( id kid ). The summation indices n1 and n2 stands for g , (ε, ±k ) band numbers, and the idler energy is given by εid = h¯ωid . 2 1 id I±(εid , kid ) = v dε , (16) As we show in the Appendix, the function Dk (−ωid ; G) εid [(ε ∓ εid ) ± ih¯γ ] is the induced charge density tensor Fourier component. Each term has a different polarization dependence; while the where the integration is performed over states between the two first term is symmetric with respect to (i ↔ j), the tensor bands separated by a wave vector difference equals to kid as illustrated in Fig. 2, v is the volume of the primitive cell, and Bijk(ωid , kid ; G) is antisymmetric with respect to (i ↔ j). This polarization dependence implies that the first term is the weight function in the integral is the joint density of states, ε, = 1 dS which is defined as gn ,n ( k) 3 nonvanishing when the polarization of the signal is parallel 1 2 (2π ) S(ε) |∇q ( εn ,n (q,k))| ε , = ε + − ε 1 2 to the pump, and the second term is nonvanishing when where n1,n2 (q k) n1 (q k) n2 (q). Note that for they are orthogonal to each other. The different polariza- k = 0 the expression reduces to the standard definition of the tion dependence of the two terms can be used to probe the joint-density of states [30], and can be considered as such for contributions of the induced charge density, band transitions a wave vector much smaller than a typical size of the first | | 2π dependence, and intermolecular interactions matrix elements. Brillouin zone ( k a ). From Eq. (16) and from Fig. 2,it ε The functions Bijk(ωid , kid ; G) and Dk (−ωid ; G) contain the is clear that id acts as a resonance in the integral form of information on the interaction dependence on the idler mode the function I+(εid , kid ), and therefore is more sensitive to (the long wavelength), the band structure, and intermolecular the joint density of states than I−(εid , kid ). This sensitivity interactions matrix elements. is related to the contribution that arises from the number of The information about the intermolecular interactions is energy transitions at the idler energy. encoded in the matrix elements and the interaction de- To illustrate this, we consider the following simple energy pendence on the band structure is given by the function dispersion: In ,n (εid , kid ). It is a measure of the number of band transi- 1 2 ε k = V |g k |, tions (n1 ↔ n2) with separation of kid and population differ- 1( ) ss 1( ) ε + − ε ence of f0( n1 (q kid )) f0( n2 (q)) that are closely related ε2(k) = εgap + Vss(2 −|g1(k)|), (17) to the idler energy εid . Moreover, it is important to note that for maximally localized Wannier functions (as assumed here) where εgap is the gap energy, Vss is the width of each band, there will only be contributions from interband transitions, and g1(k) is the band structure form function we used for this when the dipole approximation is assumed with respect to the example [31]. The dependence of the functions I±(εid , kid ∼ idler mode. This is because under this assumption the Wannier 0) ≡ I±(εid ) on small idler energies is evaluated numerically functions are real [29].

III. MODEL CASE OF A SEMICONDUCTOR WITH TWO BANDS ε2(q) In this section, we focus on the nonlinear interaction in semiconductors. We begin with the spectral dependence of the nonlinear conductivity. Since we have already shown that when the Wannier functions are localized, the spectral ε , dependence can be described by the function In1,n2 ( id kid ) [Eq. (14)] it is sufficient to explore the spectral dependence Δε (q∗,k ) of this function. To get more insight we consider a simple 2,1 id example of a semiconductor at zero temperature with two energy bands, where the valence band is fully occupied and the conduction band is completely vacant and in addition, we ε1(q) assume that the damping coefficients are equal to a constant γ . Under these assumptions, the difference in the two Fermi- Dirac distributions is proportional to a delta function and therefore, the function given by Eq. (14) can be simplified by expressing it using two functions we denote as I+(εid , kid ) and ε , ∗ ∗ I−( id kid ), which satisfy the equation q q + kid

I , (ε , k ) n1 n2 id id FIG. 2. Schematic view of the energy difference between bands = δ δ ε , + δ δ ε , , ↔ ∗ n1,2 n2,1I+( id kid ) n1,1 n2,2I−( id kid ) (15) (2 1) at wave vector q , separated by kid .

033133-4 THEORY OF NONLINEAR INTERACTIONS BETWEEN … PHYSICAL REVIEW RESEARCH 1, 033133 (2019)

FIG. 4. β dependence of the nonlinear current density: The functions I˜+(˜εid ) (blue) and the difference [I˜+(˜εid ) − I˜−(˜εid )] (red) at the band gap energy (˜εid = 1). Both decreases with the parameter β.

where I˜±(˜εid ) are dimensionless, and will be used for our analysis. It is interesting to note that the difference function I+(εid ) − I−(εid ) can be related to the induced charge density when the dipole approximation is assumed with respect to the idler mode. In this case, we find that Dk (−ωid ; G) = (k) ε − ε (k) f2,1 (G)(I+( id ) I−( id )) where f2,1 (G) encapsulates the inter-molecular interactions with respect to bands 2, 1 and the reciprocal lattice vector G. I+(εid ) − I−(εid ) reflects the spectral dependence of the induced charge density on the joint density of states. FIG. 3. Band structure dependence of the nonlinear current den- We show the dependence of I˜±(˜εid ) and of their difference ˜ ε sity. (a) The term I+(˜id ) (green) shows a clear peak at the band gap in Fig. 3. The prominent peak in I˜+(˜εid ) near the band gap ˜ ε energy while the term I−(˜id ) (red) decreases monotonically with the energy is due to its strong dependence on the large number of relative idler energyε ˜ . (b) The difference between the I˜+(˜ε )and id id interband transitions at the band gap, while I˜−(˜εid ) decreases the I˜−(˜ε ) term. The vertical purple line indicate band gap energy. id monotonically. The difference I˜+(˜εid ) − I˜−(˜εid ) also exhibits a very distinctive peak near the band gap energy. It is possible under the assumption that the width of the levels are very to measure this peak if the spectral resolution of the detector is narrow and therefore the damping coefficient γ is neglected. higher than the FWHM of the peak profile. This enhancement near the band gap energy demonstrates the sensitivity of the Furthermore, in order to see how I±(εid ) scale with the band gap energy and to investigate the dependence of the nonlinear response and of the induced charge to the joint density of states, and is also in agreement with a recent choice of bandwidth Vss, we define dimensionless energy parameters experiment, showing an enhancement just at the band gap energy [6]. ε ε = id ,β= 2Vss , To investigate how the bandwidth effects the model, we ˜id ˜ ε β = /ε εgap εgap evaluated I±(˜id ) for various values of 2Vss gap.The functions I˜±(˜ε ) display the same behavior for all β differing ε (k) − ε (k) id ε k = 2 1 = + β −|g k | , only by a factor. This difference is displayed in Fig. 4 where ( ) ε 1 (1 1( ) ) (18) gap the peaks of the functions decrease with growing values of β. This decrease is due to the broadening of the joint density of ε β ε where ˜id , , and (k) are the idler energy, bandwidth en- states, as β grows, which leads to a reduction in the density ergy, and interband transition energy relative to the band gap of transitions about the energy gap. The spectral dependence ε energy gap respectively. With these definitions, the functions of the optical response on β, can be used to probe changes ε I±( id ) can be expressed by dimensionless functions times a in the band structure due to an external field, and also the scale factor (k) corresponding changes in f2,1 (G). I˜±(˜εid ) After we have described the spectral dependence of the I±(ε ) = , id ε2 nonlinear effects, we estimate the contribution of the inter- gap molecular interactions in the case of two bands by working in v 1 a two-dimensional subspace spanned by the Wannier states of I˜±(˜ε ) = dk , (19) · id π 3 iG x (2 ) B.Z. ε˜id ( ε(k) ∓ ε˜id ) each band. Since the operator e shares the same eigenstates

033133-5 R. COHEN AND S. SHWARTZ PHYSICAL REVIEW RESEARCH 1, 033133 (2019) as the position operator, we first find its matrix representation In contrast to previous publications [4,8], we have found in the eigenbasis of the position operator and then rotate that the polarization of the signal is not only in the direction this matrix to the original basis. Next, we express the matrix of the polarization of the pump. The two polarization com- elements contained in Dk (−ωid ; G) and Bijk(ωid , kid ; G)in ponents exhibits different spectral dependencies which can be terms of the position and momentum matrix elements and investigated by measuring each component. obtain We emphasize that since the nonlinear interaction we discuss is a parametric process in nature (the system does ihe¯ 2 not change its state during the nonlinear interaction) it is D (−ω ; G) =− π e−iG·a sin(G · c) k id mV k inherently ultrafast. It is therefore very likely that it would be possible to use the nonlinear x-ray and long-wavelength × (I+(ε , k ) − I−(ε , k )), (20) id id id id interaction for the study of ultrafast dynamics in solids. Our theory implies that pump-prob measurements can be used B (ω , k ; G) ijk id id to study the femtosecond and subfemotosecond dynamics of −iG·a = πke cos(G · c)[¯h(kid − G) · (êiπ j − eˆ jπi )] interesting processes. Examples include the variation of the populations between bands, ultra-fast charge transfer between × (I+(ε , k ) + I−(ε , k )), (21) id id id id electronic states, optically induced chemical potential varia- tions, ultrafast (optically induced) dynamics of band struc- where tures, and ultra-fast phase transitions. Our formalism can be combined with standard ab initio π = | · | =− | · | , i k W2 p eˆk W1 W1 p eˆk W2 (22) methods for further studies of the nonlinear x-ray and optical/UV mixing effects in many solid state systems. = | | = | | , a W1 x W1 W2 x W2 (23) We stress that nonlinear interactions can be used to reveal a very broad band spectroscopic information ranging from sub = | | = | | , c W1 x W2 W2 x W1 (24) eV to several hundred of eV and structural information of the valence electrons by using a single apparatus. By using SPDC π, , and the vectors ( a c) are real. This form of the matrix el- of x rays to long wavelengths, the energy scan can be done by ements is attained when assuming that the Wannier functions tuning the angle of the sample and the energy is selected by , = are real, and that [xi x j] 0. the detection system. Generally speaking, even in the case of a simple semiconductor the estimation of the matrix elements requires numerical calculations. However, since the Wannier functions ACKNOWLEDGMENTS are commonly constructed using the linear combination of The authors thank Emanuele G. Dalla Torre, and Efrat atomic orbital (LCAO) approach, in which Wannier functions Shimshoni for inspiring and useful discussion. This work was are a constructed by a superposition of many atomistic wave supported by the Israel Science Foundation (ISF)(IL), Grant functions [32], we can get a rough estimation of the magnitude No. 201/17. of the nonlinear conductivity by approximating the Wannier functions to be hydrogen wave functions of levels 1s and 2p. For idler and pump photon energies at 1 eV and 10 keV, APPENDIX: INDUCED CHARGE DENSITY respectively, we find that the first term of the conductivity AND THE CONDUCTIVITY ≈ −7 3 −2 is on the order of ( 10 A W ). The second term of the In this section, we derive the relation between the in- ≈ −11 3 −2 conductivity is on the order of ( 10 A W ). duced charge density tensor and the second-order conductivity which arises from the so called gauge part of the current IV. SUMMARY AND CONCLUSIONS density operator. We start with defining the average charge density to be In this work, we have shown that the contribution to the nonlinear interaction arises from both band structure proper- ρc(x, t ) = tr(ρ(t ) Dop(x)) ties and atomic-scale interactions. Our results imply that it is = ρ(n) = ρ(0) + ρ(1) , +··· , (A1) possible to extract atomic-scale information on the valence c c (x) c (x t ) electrons as predicted by previous publications [1,3,4,7,17,19] n but only if the band information can be separated from where n signifies the order of each term with respect to the the Wannier function contribution. For example, when the ρ(0) field, such that c (x) is the average charge density in the Wannier functions are localized. Consequently, the spectral ρ(1) , absence of an external field, and c (x t ) is the average dependence of the nonlinearity is essential for the construc- induced charge. The induced charge density can be Fourier tion of the microscopic information of the electronic states. expanded in the field modes, namely, Moreover, the population difference between bands or within a band also plays a role in the spectral dependence. When (1) (1) −iωl t ρ (x, t ) = ρ (x; ωl )e . considering more than two bands, there could be an effect of c c (A2) l interference between several spectral contributions including interband and intraband transitions, provided that there is a The Fourier coefficients of the induced charge density are population difference between these transitions. related to the electric field amplitudes via the induced charge

033133-6 THEORY OF NONLINEAR INTERACTIONS BETWEEN … PHYSICAL REVIEW RESEARCH 1, 033133 (2019) density tensor Finally, to relate the gauge part conductivity to the induced 3 charge density tensor, we note that ρ(1) ω = ω ε ω · . c (x; l ) D j (x; l )( ( l ) eˆ j ) (A3) (2,A) (A) A e (1) = j (x, t ) = tr(ρ j (x, t )) = ρ (x, t )A(x, t ). (A5) j 1 op m c The induced charge density tensor has the periodicity of the Using this relation along with Eqs. (7)–(9), we ﬁnd lattice and therefore can be written as · (2,A) ω = ω iG x, σ ω + ω D j (x; l ) D j ( l ; G)e ijk ( l l ; G) G e e = ω δ + ω δ . 1 D j ( l ; G) ik Dk ( l ; G) ij (A6) −iG·x imωl imωl D j (ωl ; G) = dx D j (x; ωl )e . (A4) V V

[1] H. Danino and I. Freund, Phys.Rev.Lett.46, 1127 (1981). [14] A. Q. Baron, arXiv:1504.01098. [2] K. Tamasaku, K. Sawada, and T. Ishikawa, Phys. Rev. Lett. 103, [15] I. G. Kaplan, B. Barbiellini, and A. Bansil, Phys. Rev. B 68, 254801 (2009). 235104 (2003). [3] K. Tamasaku, K. Sawada, E. Nishibori, and T. Ishikawa, [16] M. J. Cooper, Rep. Prog. Phys. 48, 415 (1985). Nat. Phys. 7, 705 (2011). [17] I. Freund and B. Levine, Phys.Rev.Lett.25, 1241 (1970). [4] T. Glover, D. Fritz, M. Cammarata, T. Allison, S. Coh, J. [18] I. Freund and B. Levine, Opt. Commun. 3, 101 (1971). Feldkamp, H. Lemke, D. Zhu, Y. Feng, R. Coffee et al., [19] I. Freund, Chem. Phys. Lett. 12, 583 (1972). Nature (London) 488, 603 (2012). [20] K. E. Dorfman and S. Mukamel, Phys.Rev.A86, 023805 [5] B. Barbiellini, Y. Joly, and K. Tamasaku, Phys. Rev. B 92, (2012). 155119 (2015). [21] I. Freund, Phys.Rev.Lett.21, 1404 (1968). [6] A. Schori, C. Bömer, D. Borodin, S. P. Collins, B. Detlefs, M. [22] S. Jha and C. Warke, Il Nuovo Cimento B (1965-1970) 53, 120 Moretti Sala, S. Yudovich, and S. Shwartz, Phys. Rev. Lett. 119, (1968). 253902 (2017). [23] I. Freund and B. Levine, Phys.Rev.Lett.23, 854 (1969). [7] J. R. Rouxel, M. Kowalewski, K. Bennett, and S. Mukamel, [24] S. S. Jha and J. W. Woo, Phys.Rev.B5, 4210 (1972). Phys. Rev. Lett. 120, 243902 (2018). [25] S. Jha and J. Woo, Il Nuovo Cimento B (1971-1996) 10, 229 [8] D. Borodin, A. Schori, J.-P. Rueff, J. M. Ablett, and S. Shwartz, (1972). Phys. Rev. Lett. 122, 023902 (2019). [26] D. Popova-Gorelova, D. A. Reis, and R. Santra, Phys. Rev. B [9]S.Sofer,O.Seﬁ,E.Strizhevsky,S.Collins,B.Detlefs,C.J. 98, 224302 (2018). Sahle, and S. Shwartz, arXiv:1904.13146. [27] N. Bloembergen and Y. Shen, Phys. Rev. 133, A37 (1964). [10] K. Tamasaku and T. Ishikawa, Acta Cryst. Sect. A 63, 437 [28] J. Callaway, Quantum Theory of the Solid State (Academic (2007). Press, Boston, MA, 2013). [11] D. Borodin, S. Levy, and S. Shwartz, Appl. Phys. Lett. 110, [29] S. Ri and S. Ri, arXiv:1407.6824. 131101 (2017). [30] N. W. Ashcroft and N. D. Mermin, Solid State Physics (Cengage [12] S. K. Sinha, J. Phys.: Condens. Matter 13, 7511 Learning, New Delhi, India, 1976). (2001). [31] D. Chadi and M. L. Cohen, Phys. Status Solidi B 68, 405 [13] Y. J. Wang, B. Barbiellini, H. Lin, T. Das, S. Basak, P. E. (1975). Mijnarends, S. Kaprzyk, R. S. Markiewicz, and A. Bansil, [32] C. M. Zicovich-Wilson, R. Dovesi, and V. R. Saunders, Phys. Rev. B 85, 224529 (2012). J. Chem. Phys. 115, 9708 (2001).

033133-7