Lehmann Representations

−iL0(1,¯3⁰,1,3;¯ ω) iΞ(¯3,4¯,¯3⁰,4¯⁰) −iL(¯4⁰,2,¯4,2;ω). (2.23) Inserting the explicit form ofΞin Eq. 2.21, we obtain

−iL(1,2;ω) =

−iL0(1,2;ω) +

−iL0(1,3¯⁰,1,3;¯ ω) δ(¯3,3¯⁰)δ(4¯,¯4⁰)v_C(3¯,4) −δ(3¯,¯4⁰)δ(3¯⁰,¯4)W(3¯⁰,3¯)

×

×

−iL(¯4⁰,2,4¯,2;ω).

(2.24)

This equation is the final result of this Chapter, the Bethe-Salpeter equation for the polariz-ability. The dielectric properties, especially the macroscopic dielectric tensor, are obtained from the solutions of Eq. 2.24.

2.2.2 Lehmann Representations

Equation 2.19 shows that the polarizability χ is obtained by a contraction of the electron-hole correlation functionL. Lehmann representations ofL0andχ similar to the ones known

2Here, we introduce an additional notation for non-local functions, where (1,2;ω) = (r1,r2;ω). This

2 X-Ray and Optical Absorption Spectroscopy

for the one- and two-particle Green’s function [63, 125] can be derived. These Lehmann representations are employed in the next chapters to connect observables in absorption and scattering spectroscopy to quantities determined within many-body perturbation theory.

A fundamental ambiguity in the Lehmann representation exists for time-ordered quantities that only depend on time differencest1−t2, since the Fourier transformations with respect to either time differencesτ⁽¹⁾ = t1−t2orτ⁽²⁾ =t2 −t1 yield distinctly different Lehmann representations. While physical observables are unaffected by the choice of Lehmann rep-resentation, the implementation of absorption and scattering spectroscopy differ depending on the choice of time direction in the Fourier transformation. For the polarizabilityχ(1,2), we obtain the Lehmann representation χ_τ⁽¹⁾(r¹,r²,ω) with respect toτ⁽¹⁾ as

χ_τ(1)(r¹,r²,ω) = X

N,0

h0|ψˆ^†(r¹)ψˆ(r¹)|NihN|ψˆ^†(r²)ψˆ(r²)|0i ω−E_N +iη

−h0|ψˆ^†(r²)ψˆ(r²)|NihN|ψˆ^†(r¹)ψˆ(r¹)|0i ω+E_N −iη ,

(2.25)

where the sum includes all excited many-body states|Niwith total energyE_N. Without loss of generality, the total energy of the ground state is set to zero,i.e.E⁰ = 0. The polarizability is given as the sum of two terms, where the first, called the resonant contribution, has poles at the excitation energiesω = E_N, while the second, the anti-resonant contribution, has poles at negative excitation energies,ω = −E_N. Matrix elements of the polarizability χ_ijk,i⁰_j⁰_k⁰(q)in an arbitrary single-particle basis{ψ_ik(r)}are given as

χ_ijk,i⁰_j⁰k⁰(q,ω) =X

N,0

h0|cˆ_ikcˆ_j^†_(k+q)|NihN|cˆ_i^†0(k⁰+q)cˆ_j⁰_k⁰|0i

ω−E_N +iη + h0|cˆ_ikcˆ_j^†_(k+q)|NihN|cˆ_i^†0(k⁰+q)cˆ_j⁰_k⁰|0i ω +E_N −iη .

(2.26) Equation 2.25 can be simplified by applying both the quasiparticle approximation (Eq. 1.58) and the Tamm-Dancoff approximation (Eq. 1.60). As such, the field operatorsψˆ(r) = Pikϕ_ik(r)cˆ_ikare expanded in quasiparticle wavefunctionsϕ_ik, and we obtain

χ_τ(1)(r¹,r²,ω) = X

q

X

cvk

X

c⁰v⁰k⁰

ϕ_vk^∗ (r¹)ϕ_c(k+q)(r¹)χ_cvk,c^R,1 0v⁰k⁰(q,ω)ϕ_c^∗0(k⁰+q)(r²)ϕ_vk⁰(r²) +ϕ_vk(r¹)ϕ_c^∗_(k+q)(r¹)χ_cvk,c^A,1 0v⁰k⁰(q,ω)ϕ_c⁰_(k⁰_+q)(r²)ϕ_v^∗0k⁰(r²).

(2.27)

Here, we have used the translational symmetry of the non-local polarizability, i.e. χ(r+ R,r⁰+R;ω) = χ(r,r⁰;ω) for any lattice vectorR, such that we can write the polarizability as χ =P

qχ(q). As in Eq. 2.27, we find resonant and anti-resonant contributions, and the

28

Bethe-Salpeter Equation Formalism 2.2

corresponding matrix elements of the polarizability are given by χ_cvk,c^{R1 0}_v⁰_k⁰(q,ω) =X

N,0

h0|cˆ_vk^† cˆ_c(k+q)|NihN|cˆ_c^†0(k⁰+q)cˆ_v⁰k⁰|0i

ω−E_N +iη , (2.28)

and

χ_vck,v^A1 0c⁰k⁰(q,ω) =X

N,0

h0|cˆ_v^†0k⁰cˆ_c⁰_(k⁰_+q)|NihN|cˆ_c^†_(k+q)cˆ_vk|0i

ω+E_N −iη . (2.29)

Alternatively, a Lehmann representation χ_τ(2)(r¹,r²,ω) with respect to τ⁽²⁾ = t2 −t1 is obtained as

χ_τ(2)(r¹,r²,ω) = X

N,0

h0|ψˆ^†(r²)ψˆ(r²)|NihN|ψˆ^†(r¹)ψˆ(r¹)|0i

ω−E_N +iη −h0|ψˆ^†(r¹)ψˆ(r¹)|NihN|ψˆ^†(r²)ψˆ(r²)|0i ω+E_N −iη ,

(2.30) which, in quasiparticle and Tamm-Dancoff approximation, becomes

χ_τ(2)(r¹,r²,ω) =X

q

X

cvk

X

c⁰v⁰k⁰

ϕ_vk(r¹)ϕ_c^∗_(k+q)(r¹)χ_cvk,c^R,2 0v⁰k⁰(q,ω)ϕ_c⁰_(k⁰_+q)(r²)ϕ_vk^{∗ 0}(r²)

+ϕ_vk^∗ (r¹)ϕ_c(k+q)(r¹)χ_cvk,c^A,2 0v⁰k⁰(q,ω)ϕ_c^∗0(k⁰+q)(r²)ϕv⁰k⁰(r²).

(2.31) The matrix element in this representation are defined as

χ_cvk,c^R2 0v⁰k⁰(q,ω) =X

N,0

h0|cˆ_v^†0k⁰cˆ_c⁰_(k⁰_+q)|NihN|cˆ_c^†_(k+q)cˆ_vk|0i

ω−E_N +iη , (2.32)

and

χ_vck,v^{A2 0}_c⁰_k⁰(q,ω) =X

N,0

h0|cˆ_vk^† 0cˆ_c(k+q)|NihN|cˆ_c^†0(k⁰+q)cˆ_v⁰_k⁰|0i

ω+E_N −iη . (2.33)

It is apparent, that the matrix elementsχ^R1 in Eq. 2.28 and χ^R2in Eq. 2.32, as well as their antiresonant counterparts in Eqs. 2.29 and 2.33, are closely related. The denominator is identical in the two representations, while the nominator in the second one is complex con-jugated with respect to the first one. Although the difference appears miniscule, we will demonstrate that it leads to different expressions for the dielectric function and other spec-troscopic observables in the two different representations. More details on the Lehmann representations, and their relationship to the retarded polarizability are provided in Ap-pendix B.

CHAPTER 3

Inelastic X-ray and Electron Scattering Spectroscopy

In a general x-ray or electron scattering experiment, a collimated beam of monochromatic particles, either x-ray photons or electron, is focussed on a sample, and the scattered beam is recorded at a certain solid angle Ω². The scattered beam is then analyzed spectrally with an energy resolutiond~ω2. The probability of scattering into the solid angle element Ω²,Ω²+dΩ²

within an energy range

~ω2,~ω2+d~ω2

is given by thedouble differen-tial cross sectiond²σ/dΩ²dω². Quantum-mechanically, the many-body systems is scattered from an initial many-body state|i,K¹λ1i, which consists of the initial many-electron state

|iiand the state|K¹λ1iof the incident particle, whereK¹is the momentum of the incoming particle andλ¹ is a generic quantum number. The final state|f,K²λ²iis a superposition of the final (excited) electronic many-body state|fiand the state|K²λ2iof the scattered par-ticle. For photons,λ1 andλ2characterize the polarization states of the incoming and scat-tered photon, respectively, which can be either two linear polarization states or left- and right-hand circular polarization states. Generally, the scattering rated²σ is given by [169, 170]

d²σ = j2(r,K2,λ2)lrˆ ²dΩ²dE1

j1

, (3.1)

wherej¹is the incoming current, and the numerator describes the number of scattered par-ticles, sincej2(r,K²,λ2) is the current of scattered particles in state|K²,λ2iin the direction ofr, and lˆ = r/r is the direction vector along r. In the quantum-mechanical scattering theory, the numerator of Eq. 3.1 is given by the transition probabilityw(i,K¹,λ¹;f,K²,λ²) as

j2(r,K2,λ2)lrˆ ²dΩ2dE1 = N0

X

f

X

K2λ²

w(i,K1,λ1;f,K2,λ2)D(K2)d³K2, (3.2)

whereN¹is the number of probing particles that impinge on the sample andD(K²) =

V 2π

is the density of states of the scattered particle. We write D(K²)d³K2 = 2^VπK²dΩ². The

3 Inelastic X-ray and Electron Scattering Spectroscopy

double differential cross section is given as d²σ

The initial current j1 and the dispersionE2(K2) differ depending on the probing particle, and the transition probabilityw depends on the interaction of the probing particle with the sample.

In the following chapter, an expression for the double differential cross section, known as the generalized Kramers-Heisenberg formula, is derived from a perturbative treatment of the electron-photon interaction. We will then show that two distinctive scattering pro-cesses occur depending on the energy of the probing particle, known as the resonant and non-resonant scattering. For the non-resonant scattering, we will derive a simplified ex-pression that connects the cross section to the dielectric tensor introduced in Chapter 2.

Finally, we discuss the process of resonant inelastic scattering.

3.1 Generalized Kramers-Heisenberg Formula

In Eq. 1.2, the Hamiltonian of a system of interacting electrons in the electrostatic potential of the nuclei has been introduced. For a system of interacting electrons in a quantized elec-tromagnetic field, a more general Hamiltonian is needed, which also includes the Hamilto-nian of the photons and the interaction between electrons and photons. The most general form of this Hamiltonian is given by [170]

H =H⁰+α²

whereAis the electromagnetic vector potential ,A˙ = ∂/∂tAits time-derivative, andV the scalar potential, as introduced in Section 2.1.2, andα = ¹_c ≈ ₁₃₇¹ is the fine-structure constant in atomic units. In the last line of Eq. 1.2, we have introduced the electronic Hamiltonian H0, the radiation-field Hamiltonian H_R, and the Hamiltonian H_i of the electron-phonon

32

Generalized Kramers-Heisenberg Formula 3.1

interaction.H⁰ gathers all terms that describe solely the electronic system,i.e.

H0= X

The HamiltonianH0differs from the one in Eq. 1.2 only by the last term, which introduces the spin-orbit coupling. This relativistic effect is not included in the non-relativistic Hamil-tonian in Eq. 1.2. The radiation-field HamilHamil-tonianH_R is given by

H_R =X

with the photon creation and annihilation operatorsaˆ^†_Kλ andaˆ_Kλ of a phonon state (K,λ) with energyωK, respectively. Here, λ enumerates the two orthogonal polarization states of the photon field, K is the photon wavevector. The interaction between electrons and photons originates from terms proportional to the electromagnetic vector potentialAand its time derivative A˙. We recall the expression of the vector potential A(r) in terms of creation and annihilation operators of the photon field,aˆ^†_Kλ andaˆ_Kλ:

A(r) = X wheree_Kλis the polarization vector of the photon in state(Kλ). We express the interaction Hamiltonian as the sum of six terms, given by

H_i1= α²

3 Inelastic X-ray and Electron Scattering Spectroscopy

According to Fermi’s Golden Rule [171], the transition ratew is obtained from a perturba-tive treatment of the electron-photon interactionHˆ_i up to second order as

w =2π X

where |Ii, |Fi, and |Ni are initial, final, and intermediate many-body state, respectively.

We furthermore restrict our perturbation series to terms of orderα²in Eqs. 3.8-3.13,i.e.to the non-relativistic case [170, 172]. The electron-photon interaction terms in Eq. 3.8-3.13 contain the vector potentialA, and thus the photonic annihilation and creation operators through Eq. 3.7, to different orders. Not all terms contribute to the scattering, since we have restricted the possible many-body states by imposing that the initial state |i,K¹λ¹i and the final state|f,K²λ2icontain only one photon each¹. As such, in the first-order treat-ment of Eq. 3.14 only the termsH_i1 andH_i4contribute, as these contain terms of the form c(Kλ)c^†(K⁰λ⁰)that conserve the number of photons. To second order, both terms inO A andO

A²

contribute. Inserting the termsH_i1orH_i4for the second-order term in Eq. 3.14, the intermediate states|Nicontain a photon, as these terms conserve the photon number.

However, these terms are proportional toα⁴and are neglected in our non-relativistic treat-ment. Similarly, within first- and second-order perturbation theory the termsH_i5 andH_i6

only contribute for orders ofO α⁴

. Thus, only the interaction termsH_i2andH_i3remain.

Since in this work we focus on non-magnetic systems, we neglect the spdependent in-teractionsH_i3 andH_i4. Thus, we express the transition rate as

w =2πX

Combining Eqs. 3.8,3.9,3.15, and 3.7, the photonic states can be traced out, such that the expression only contains expectation values for the electronic states |ii, |ni, and |fi. We obtain

1With other boundary conditions, the transition rate of other spectroscopies can be calculated. For

ex-ample, the absorption rate is obtained with|Fi=|f,0i

34

Non-Resonant Inelastic X-ray Scattering Spectroscopy 3.2

where we have used that the energies are given byE_I = E⁰+ω¹whereE⁰ is the energy of the initial electronic system,E_N = E_n, whereE_nis the energy of the intermediate electronic system, andE_F =E_f +ω2, whereE_f is the energy of the final electronic state. Furthermore, before the scattering process the electronic system is in the groundstate, such that|ii= |0i andE_i = 0. We now employ Eqs. 3.3 to obtain the double differential cross section. For photons, the dispersion relation isE2(K2) = cK2 and the initial current isj1 = ^N_V^0c. This yields the double differential cross section as

d²σ

This expression, known as the generalized Kramers-Heisenberg formula [171, 173], con-tains the two scattering processes that occur up to second order in non-magnetic materials:

the first term represents theNon-Resonant Inelastic Scattering(NRIXS), the second term the Resonant Inelastic Scattering(RIXS).

3.2 Non-Resonant Inelastic X-ray Scattering

Im Dokument Theoretical Spectroscopy of Ga2O3 (Seite 47-55)