A TTOSECOND SPECTROSCOPY IN THE X- RAY REGIME IN COMPLEX SYSTEMS .

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A TTOSECOND SPECTROSCOPY IN THE X- RAY REGIME IN COMPLEX SYSTEMS .

ADOCTORAL THESIS BY

G

IOVANNI

C

ONSALVO

C

ISTARO

SUPERVISED BY

A

NTONIO

P

ICÓN

A

LVAREZ

F

ERNANDO

M

ARTÍN

G

ARCÍA

DEPARTAMENTO DEQUÍMICA

FACULTAD DECIENCIAS

UNIVERSIDADAUTÓNOMA DEMADRID

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Abstract

The ultrafast motion of electrons is a driving force for chemical reactions and the possibility to have control of it makes it a highly desirable avenue for study. This thesis uses the quantum mechanics of extended systems to simulate pump-probe experiments like ATAS in which the electrons can be promoted from inner shells and then are free to move through bands that are responsible for the physical properties of materials. The main tool that will be used is the resolution of the Von Neumann equation for the density matrix in crystal momentum representation, and so there will be the possibility to keep track of the real-time movement of electrons.

The main subjects of this thesis are hexagonal lattices in 2D, graphene and hBN, as well as their extension in 3D, graphite. The simulations will allow to promote electrons from the core levels (K edge) to bands that are close to the Fermi energy, and all the spectra that will be observed are shown to be sensitive to the main properties of the materials; in case of injection of electrons from the valence band, also, the ATAS observed keep track of the interference felt by the electrons.

Also, many computational methods related to high-performance computing will be investigated for the resolution of the Von Neumann equations, to speed up the simulations, as they are very demanding from the numerical point of view.

The methodologies used and the results obtained pave the way for a deeper understanding of the physics of materials, from the electron coherence to the control of their quantum states.

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5

Resumen

El movimiento ultrarrápido de los electrones es una fuerza impulsora de las reac- ciones químicas y la posibilidad de controlarlo lo convierte en una vía de estudio muy deseable. Esta tesis utiliza la mecánica cuántica de sistemas extendidos para simular experimentos de pump-probe como ATAS en los que los electrones pueden ser promovidos desde bandas internas y luego son libres de moverse a través de bandas que son responsables de las propiedades físicas de los materiales. La her- ramienta principal que se utilizará es la resolución de la ecuación de Von Neumann para la matriz de densidad en la representación del momento del cristal, por lo que habrá la posibilidad de realizar un seguimiento del movimiento de los electrones en tiempo real.

Los temas principales de esta tesis son los solidos hexagonales en 2D, grafeno y hBN, así como su extensión en 3D, grafito. Las simulaciones permitirán promover electrones desde los niveles del núcleo (K edge) hacia bandas cercanas a la energía de Fermi, y todos los espectros que se observarán se muestran sensibles a las principales propiedades de los materiales; en caso de inyección de electrones desde la banda de valencia, además, el ATAS observado realiza un seguimiento de la inter- ferencia que sienten los electrones.

Además, se investigarán muchos métodos computacionales relacionados con la com- putación de alto rendimiento para la resolución de las ecuaciones de Von Neumann, para agilizar las simulaciones, ya que son muy exigentes desde el punto de vista numérico.

Las metodologías utilizadas y los resultados obtenidos abren el camino para una comprensión más profunda de la física de los materiales, desde la coherencia elec- trónica hasta el control de sus estados cuánticos.

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List of publications

Cistaro, G., Plaja, L., Martín, F., Picón, A. (2021). Attosecond x-ray transient absorption spectroscopy in graphene. Physical Review Research, 3(1), 013144.

https://doi.org/10.1103/PhysRevResearch.3.013144

Cistaro, G., Malakhov, M., Esteve-Paredes, J.J., Uría-Álvarez, A.J., Silva, R.E.F., Martín, F., Palacios, J.J, Picón, A. (2022) A theoretical approach for electron dynamics and ultrafast spectroscopy

https://doi.org/10.48550/arXiv.2207.00249

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I

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13

Introduction

The observation and the understanding of the real motion of electrons in quantum systems, from atomic and molecular systems to solids, is now a reality. This is mainly for the development of novel technologies from the experimental side, but also the development of more sophisticated numerical tools and increasingly high- performance computers.

Since the discovery of quantum mechanics, many illustrious scientists started to develop theories whose purpose was the description of condensed-matter systems, exploiting the Hilbert space of both electrons and ions inside the materials and giv- ing rise to the understanding of band structures and experimental results. The Bloch theorem was a cornerstone in this context. The well-known Bloch theorem supplies a basis for the space of quantum electrons subject to a periodic potential, linking the good quantum numbers of extended systems, i.e. the quasi-momentum, to others that were already known from basic quantum mechanics. However, the Bloch eigenstates can cause non-trivial problems in the definition of many operators. In the early stages, there were two important initiatives. On one hand, Blount [8] and Adams [1] were deriving the properties of position, velocity, and all the other operators for Bloch electrons. On the other hand, Wannier developed the theory for a new basis, known nowadays as the Wannier basis, which under the constraint of being well localized in space can “smooth” the Bloch functions making it analytic [81].

For time-dependent theories of Bloch electrons, other interesting aspects emerged, in which the description of the dynamics was related to the well-known gauge transformations of the electromagnetic field. Houston [31] and later Krieger [35] managed to find relations that stress the similarities of quasi-momenta with the properties of the eigenstates of continuum electrons in isolated systems, defining the canonical quasi-momentum that takes into account a displacement with the vector potential.

Hence, a complete quantum-mechanical framework has been built up for extended systems. Using this framework enables now to face novel challenges arising in the context of light-induced dynamics.

Recently, innovative technologies involving the production of intense petahertz (femtosecond, 10⁻¹⁵ s) laser fields have enabled the possibility to excite and drive fast currents in dielectrics, semiconductors and modern materials [68,72, 46,22,7]

and to produce high harmonics photons from the driving field, the high-harmonic generation (HHG). HHG is a non-linear non-perturbative process that can be used in gas-phase atomic systems to produce attosecond (10⁻¹⁸ s) laser pulses. Those short pulses are very special. An attosecond is the natural time scale of an electron.

Hence, the production of pulses in this time regime opens the door to manipulating and observing the electron motion. In the context of extended systems, as the lattice is not able to respond fast enough to those short pulses, quantum coherence can be exploited, allowing for unique applications beyond classical electronics [29, 60, 26,27]. With the available laser technology, several methods have been constructed to infer the electron dynamics at the attosecond time scale. HHG is typically studied in gas-phase systems, but in recent years HHG has been extended to solids [23].

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This emerging field has witnessed great progress. HHG enables the tomography of energy bands [30, 43] and is sensitive to both Berry phase [39, 42, 75, 15] and valley pseudospin [83,84,37]. In these schemes based on HHG, the same laser is used to both excite the system and probe the electron dynamics. More sophisticated methods can be used using two pulses. In general, one pulse induces the electronic motion while the other one probes the induced movement. Among those methodologies, streaking and RABBIT (reconstruction of attosecond beatings by the interference of two-photon transitions) have been used to investigate transport properties [14,74,52] and photoionization time delays [40,76]. The method primarily studied in this thesis is based on transient absorption, which offers wide flexibility to investigate the light-induced dynamics at different times. In this method, an IR pulse excites the system at energies close to the Fermi level, while the second laser excites electrons from inner bands to valence and conduction bands. Changing the delay between the two pulses provides a measurement that contains information on the motion of electrons in real-time, similar to taking snapshots for constructing a movie. When the probe pulse is an attosecond pulse, then this method is called attosecond transient absorption spectroscopy (ATAS). One of the main advantages of ATAS and that makes it unique is that one can achieve a good resolution both in energy and in time. Early experiments in condensed-matter systems have shown that ATAS has the potential to track the carrier injection from the valence to the conduction band and follow the electron-hole dynamics [71,46,38,41,86,49,69,79].

Here it is important to note that if the probe laser excites electrons from bands that have already significant properties, i.e. they are involved in the chemical bonds, all the dynamics can be blurred out and the extracted information may be distorted.

However, the attosecond technology permits the production of pulses with XUV frequencies [49,79]. Hence, electrons that are promoted because of the probe pulse come from inner/core levels, which are well-known to be flat energy bands and do not have any significant feature in reciprocal space. The dynamical features of ATAS should be the consequence of the Fermi levels properties, which are the ones that play an important role in the physics and chemistry of the materials. Furthermore, the current technology enables to extend ATAS to the X-ray regime, i.e. the attosecond probe pulse can be generated with X-ray frequencies [77,58,11,32,73]. This is very important because one can reach deep core levels that are well localized in real space and control the site specificity.

There is a need to simulate the transient state dynamics of condensed-matter systems under laser excitation, both for understanding the underlying mechanism of out-of-equilibrium properties and for correlating the microscopic electron dynamics with the macroscopic measurements, observables such as current and absorption. The modeling of an ATAS experiment shows many complexities, coming from the fact that different frequencies, one in the IR and the other in the XUV regime, are combined; hence, many electronic states are taken into account in the propa- gation to obtain an absorption spectrum at a particular time delay. Numerically, this means that a lot of resources are required, but also that the program needs to be optimized to speed up the calculations. Moreover, the typical intensities used in these experiments excite non-linear dynamics, making it impossible to use perturba- tion theory (and therefore the Kubo formula),. Last but not least, in all non-metallic two-dimensional materials known to date, the optical response is dominated by the excitonic effect, i.e. by the coherence that exists between an electron that has been excited and the hole that lives in its previous state.

Density-functional theory (DFT) is the workhorse of computational modeling for materials at equilibrium. However, the out-of-equilibrium dynamic is beyond the

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Contents 15

scope of DFT and there are three main alternatives. The first one is time-dependent DFT (TDDFT) [64,44, 85], which consists in solving a time-dependent Kohn-Sham equation. There are several TDDFT codes implemented in real-time and in real- space ideal for condensed-matter systems interacting with laser pulses, see for example Refs. [3, 50]. To account for excitonic interactions in TDDFT, a long-range nonlocal exchange functional is needed and the numerical implementation thus implies a high computational cost [59]. The second one is based on many-body pertur- bation theory (MBPT). Starting from the Kohn-Sham DFT electronic structure, the well-known Bethe-Salpeter equation (BSE) is solved and the energy and wavefunctions of excitons are obtained, the latter expressed as a superposition of single particle excitations [63,2]. BSE provides accurate energies and it is ideal for spectroscopy calculations. However, BSE is not a time-domain framework, it cannot describe real- time non-equilibrium dynamics and ultrafast spectroscopy experiments. In recent years there have been several theoretical approaches to extend BSE in the time domain [5, 62, 66]. This consists in resolving the Kadanoff-Baym equations based on the nonequilibrium Green’s function theory [70].

The third method is similar to solving the Kadanoff-Baym equations for the Green function, but starting instead from a second quantization formalism and evolv- ing the reduced density matrix. Within this formalism the well-known semiconduc- tor Bloch equations can be derived [25, 34], which evolve the density matrix of the system in reciprocal space. Our approach is based on this theoretical framework. In this work, it is shown how the equations of motion for the density matrix can be ef- ficiently implemented. We model relevant physical scenarios that illustrate the flexibility of our approach through either simple tight-binding (TB) models or within the Kohn-Sham DFT scheme, both with localized orbitals or Wannier basis, the description of excitonic effects in optical absorption spectra, and the feasibility to model ATAS using attosecond X-ray pulses both in 2D and 3D materials. In summary, this thesis has a strong component of numerical development and theoretical modeling of light-induced electron dynamics in extended systems. The thesis covers the main theoretical framework and numerical implementation of the equations of motion to describe the transient electron system interacting with laser pulses. The developed code is quite flexible, enabling us to use Wannier or atomic basis, include excitonic interactions, and describe several laser pulses interacting with the system, with frequencies ranging from THz/IR to X rays. As a main result, the ATAS of graphene is calculated using realistic parameters for a possible experiment. The ATAS contains information on both the local band dispersion around Van Hove singularities and the (gauge invariant) phase related to the Berry connection. Moreover, it is shown how the code can reproduce ATAS starting from more realistic DFT calculations, and some results using the Coulomb interaction that also is consistent with what is now present in literature.

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II

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17

Chapter 1 Main ideas in condensed matter physics

This chapter is an introduction to the basic theoretical framework that is needed to describe condensed matter systems. Starting from the definition of a geometrical periodic system, the form of Bloch functions ψ_nk(r)will be derived, together with their basic properties. These wavefunctions will form the basis on which all our theory will be developed.

1.1 Bravais lattices

Crystals are materials in the solid phase in which atoms are arranged in a specific way and a set of symmetries are respected on a macroscopic scale. While some electrons move almost freely inside the solid, the atomic nuclei (with other tightly bound electrons) form symmetric structures in space and, being heavier, are quite stable in their positions.

All bulk properties of crystals can be derived by considering that the nuclei structure is repeated throughout the whole space, with no limits; what is lost is everything related to the boundary information of the extended system.

The geometrical object that is used to describe a crystal is known as Bravais lattice. A Bravais lattice is a discrete and infinite set of points in space, such that the structure appears the same as seen from each point of the lattice [4]. We can define the lattice vectors{a_i}_i₌_1,2,3 as the ones that span the grid: given a point in the Bravais lattice, every translation by a vector

R=n₁a₁+n₂a₂+n₃a₃ n₁, n₂, n₃ ∈_Z (1.1) leads to another point of the structure. Note that n₁, n2 and n3are integer numbers.

The volume delimited by the three lattice vectors is known as unit cell and represents the unity of volume repeated throughout the whole space.

There are infinite ways of defining the three crystal vectors. The lattice vectors with the smallest length are called primitive vectors and the unit cell represented by them is known as primitive cell. It is important and it has a unique name because different Bravais lattices have different primitive cells. Moreover, the primitive cell contains only one point of the lattice and there is no distinction between points of the lattice and R vectors.

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1.2 Bloch theorem

In this section, it is introduced one of the most celebrated results in Quantum Me- chanics for extended periodic systems, the Bloch theorem. This theorem provides a unique way to express the eigenfunctions of the Hamiltonian under a quantum number, the quasi-momentum k, that is directly connected to the translational sym- metry of the system.

Bloch Theorem

In a Bravais lattice, the wavefunctions satisfy:

ψ_nk(r) =u_nk(r)e^ik^·^r (1.2) where the functions u_nkhave the same periodicity as the Bravais lattice:

u_nk(_r) =u_nk(_r+_R) ∀_R∈Bravais lattice (1.3) In a Bravais lattice, because of the translational symmetry, the potential operator must satisfy:

V (ˆ r) =V (^ˆ r+R) (1.4)

for every R defined by eq. (1.1). The Hamiltonian, therefore, must commute with the R-translation operators ˆT_R:

[H^ˆ, ˆT_R] =0 ∀_R∈Bravais lattice (1.5) where the operators ˆT_Rare defined from their action on a wavefunction f(r)as:

Tˆ_Rf(r) = f(r−R) (1.6) These operators satisfy the following properties:

• They commute since

Tˆ_RT^ˆ_R⁰ =T^ˆ_R₊_R⁰ (1.7)

• They are unitary, since they preserve the norm:

hα|αi = hα|T^ˆ_R^†T^ˆ_R|αi =⇒ T^ˆ_R^†T^ˆ_R= _ˆ1 (1.8) Since eq. (1.5) and (1.7) must hold, a simultaneous basis that diagonalize ˆHand all the operators ˆT_Rcan be found. The eigenfunctions of the operator ˆT₋_Rsatisfy:

ψ(r+R) =C(R)ψ(r) (1.9) It is reasonable to link the coefficient at R to the coefficients at a_i. Using eq. (1.1) and (1.7) one obtains:

C(R) =C(n₁a₁+n2a2+n3a3) =C(a₁)ⁿ¹C(a2)ⁿ²C(a3)ⁿ³ (1.10) Moreover from eq. (1.8) it follows that|C(_R)| = 1 and in particular the coefficients C(a_i)can be rewritten as a complex exponential:

C(_a_i) =e^2πikⁱ (1.11)

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1.3. First Brillouin zone 19

And finally, combining eqs. (1.10) and (1.11) the coefficient C(_R)in eq. (1.9) takes the form of a complex exponential with a scalar product:

C(_R) =e^ik^·^R (1.12)

where the quasi-momentum k is defined as

k=k₁b₁+k₂b₂+k₃b₃ (1.13) and the basis{b_i}_i₌_1,2,3are the three vectors that satisfy:

ai·_b_j =_2πδ_ij _(1.14)

In summary, there will be a wavefunction for each k of eq. (1.13), and the wavefunc- tion will satisfy

ψ_k(r+R) =e^ik^·^Rψ_k(r) (1.15) or, equivalently:

ψ_k(r) =e^ik^·^ru_k(r) with u_k(r+R) =u_k(r) (1.16) In conclusion, the Bloch theorem splits the eigenfunctions of the Hamiltonian in two terms. One term is a plane wave and resembles the momentum eigenfunctions.

The second term is a function with the periodicity of the crystal and represents an envelope for the plane wave. This means that, because of the potential of the nuclei, the electron is not anymore completely free to move (in that case the eigenfunctions will be only complex exponential). An example of Bloch function, for a 1D crystal, can be seen in fig.1.1.

1.3 First Brillouin zone

The lattice generated by the lattice vectors R of eq. (1.1) is known as direct lattice, while the space in which the quasi-momenta k live is the reciprocal space.

It is useful to define the reciprocal lattice vectors G as:

G=m₁b1+m₂b2+m₃b3 | m₁, m2, m3 ∈_Z _(1.17) The{b_i}_i₌_1,2,3vectors satisfy eq. (1.14), which implies that

e^iG^·^R=1 (1.18)

Therefore, in eq. (1.15), adding a reciprocal lattice vector to a point k, the wavefunctions will be defined by the same phase term, i.e.

eⁱ⁽^k⁺^G^)·^R=e^ik^·^R

As a consequence of this, not all the quasi-momenta are inequivalent in reciprocal space, but we can restrict to a particular volume of the reciprocal space, known as first Brillouin zone, which contains all the points that are unique. Eq. (1.17) suggests that the set of G points constitutes a Bravais lattice in reciprocal space, as it resembles eq. (1.1). The primitive cell of this lattice is the first Brillouin zone, and the primitive lattice vectors are{b₁, b2, b3}. A complete set of wavefunctions is then represented by all the eigenfunctions related to k points inside the first Brillouin zone. Once we restrict the set of quasi-momenta to the inequivalent ones, at each k point there will

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FIGURE1.1: Qualitative representation of a Bloch function for a simple 1D chain. Each dot correspond to an atomic site, and the dashed lines correspond to the plane-wave part. It must be noticed that the Bloch envelope has the periodicity of the crystal, while the amplitude

depends on the plane-wave part.

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1.4. Born Von Karman periodic boundary conditions 21

correspond more eigenfunctions and another quantum number is needed. This is known as the band index and it is an integer.

1.3.1 Volume of the Brillouin zone and volume of the unit cell

The volume of the cells play an important role in the normalization of the wavefunctions as well as in the definition of Fourier series (as it also occurs in Wannier functions).

The relation between the volume of the direct space unit cell (UC) and the volume of the reciprocal space unit cell (BZ) can be found by using the matrices

A=





| | | a₁ a2 a3

| | |



 B=





| | |

b₁ b2 b3

| | |



 (1.19)

The volume of the unit cell ΩUC and of the Brillouin zoneΩBZ can be respec- tively calculated with the determinant of the matrixA and B, because the determinant of a matrix measures the region of space delimited by the vectors contained in the columns (or, equivalently, rows) of the matrix. The two matricesA and B are connected by eq. (1.14) (in matrix form):

B=_2πA⁻^T (1.20)

Using the basic properties of determinants, it is straightforward to obtain:

ΩBZ= (2π)³ ΩUC

(1.21) The relation between the two volumes recalls the relation occurring for volumes after a Fourier transformation. This is not a case, as the Bloch theorem and all the theory that will be developed reminds some concepts that are fundamental for Fourier transformations.

1.4 Born Von Karman periodic boundary conditions

Even if the quasi-momentum k defined by the Bloch theorem is a continuous vari- able, it is usually convenient to restrict the k points to a discrete set, especially for numerical purposes. Mathematically, the discretization of this quantum number can be interpreted as the quantization of the wavefunctions in a box defined by the three lattice vectors.

If the system consists of N_iunit cells distributed along the lattice vectors a_i, then the following equation must be satisfied:

ψ_nk(r+N_ia_i) =ψ_nk(r) for i=1, 2, 3 (1.22) Now, using eq. (1.15), it is easy to reach a condition that restricts the quasi-momentum values:

k=

∑

3 i=1

m_i

N_ibi where m_i = 0, 1, . . . , Ni−₁ _(1.23) Sometimes these points have a common shifting, and the inequivalent k points will be the same in case the shift is of a quantity proportional to 1/N_i. The Brillouin zone

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is reduced to a set of discrete points, equidistant along the crystal directions. The nu- merical implementation of the k-grid defined in eq. (1.23) is known as Monkhorst- Pack (MP) grid [47]. The MP grid is well-known and well-used for numerical implementation in condensed-matter systems because it is very efficient for the calcu- lations of integrals over the k space.

1.5 From Bravais lattices to real crystals

The link that occurs between the Bravais lattice and the physical crystal can sometimes be not so obvious. Ideally one would associate to each point of the Bravais lattice R an atom of the crystal, but this cannot be done in the following situations:

• The atoms in the crystal are of a different type.

• Even if all the atoms are the same, the crystal does not have a structure that looks the same from every atom (and cannot thus be interpreted as a lone Bra- vais lattice)

When one of these conditions occur, a solution is to consider the crystal as the combi- nation of more Bravais lattices called sublattices or, equivalently, to associate to each point of the Bravais lattice more than one atom.

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Chapter 2 The Hilbert space of | _ψ _nk i

In this chapter, some basic properties of the Hilbert space spanned by the Bloch functions|ψ_nkiare discussed. Since these functions are the eigenstates of the Hamilto- nian, they represent a good basis for the quantum space of the electrons in a crystal.

Therefore, to study the physics of the system, the first step is to derive for this basis the properties of scalar products, orthogonality, operators, and gauge transformations.

2.1 Basic definition and properties

As it is discussed in the first chapter, it is sufficient to restrict the Bloch functions in the first Brillouin zone, having a set that forms a complete set of orthogonal functions. In the following, two different frameworks are discussed: infinite and finite systems.

For a finite crystal with Born Von Karman periodic boundary conditions, the number of quantum states will be given by

N₁N₂N₃N_Bands

where N_i is the total number of unit cells along the lattice vector a_i. Instead, for an infinite system, there will be a continuum of states whose quantum numbers are defined from the set

{nk | 1≤n≤ N_Bands ∧ k∈ [0, 1)b₁+ [0, 1)b₂+ [0, 1)b₃ } In Dirac notation, the Hilbert space of the system will be spanned by the kets|ψ_nki, such that

ψ_nk(_r) =hr|ψ_nki (2.1)

Similarly, the Bloch envelopes will be represented by the kets|u_nki, whose wavefunctions are

u_nk(r) =hr|u_nki (2.2)

It is then convenient to express Eq. (1.2) in form of kets:

|ψ_nki =e^ik^·^ˆr|u_nki (2.3) where ˆr is the position operator. For the purpose of this thesis, it is better that both Bloch functions and Bloch envelopes are normalized within their volumes, indepen- dently of the size of the system. In order to achieve this, the scalar products are

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defined dividing by the volume in which the integrals are calculated, i.e.

h·|·i = (₁

Ω

R

Ωd³r ifΩ<_∞

1 (2π)³

R

R³d³r ifΩ=_∞ ^(2.4)

The two possibilities can be identified with the same constant involving the number of modesN (see appendixA):

h·|·i = ¹ N_Ω_UC

Z

Ω/R³d³r (2.5)

Since they are periodic, usually for the Bloch envelopes will be involved in integrals only over the unit cell. For this purpose, we define the scalar product within a unit cell as

h·|·i_UC = ¹ ΩUC

Z

UCd³r (2.6)

2.2 Crystal momentum representation

Consider a generic single-particle operator ˆO acting on the Hilbert space spanned by a Bloch basis. Its matrix elements will be identified with the quantum numbers of the basis:

O_n0k⁰,nk = hψ_n0k⁰|O|^ˆ ψ_nki (2.7) Sometimes, it will result convenient to express the matrix elements in the basis of Bloch envelopes. Using eq. (2.3), it is possible to obtain

hψ_n⁰_k|O|^ˆ ψ_nki = hu_n⁰_k|O(^ˆ k)|u_nki (2.8) where

O(ˆ k) =e⁻îk^·^ˆrOêîk^·^ˆr (2.9) Note that the new operator Ô(k)depends on the quasi momentum and acts on the Bloch envelopes|u_nki.

When the operator ˆOis invariant under translation of a lattice vector and is local in r, it is possible to transform all the brakets involving Bloch functions in brakets involving Bloch envelopes and defined over a unit cell:

hψ_n0k⁰|O|^ˆ ψ_nki = ¹ N_Ω_UC

Z

Ω/R³d³r ψ_n^∗0k⁰(_r)O(r, . . .)ψ_nk(_r)

= ¹

N_Ω_UC

∑

R

Z

ΩUC(R)d³r ψ^∗_n0k⁰(r)O(r, . . .)ψ_nk(r)

= ¹

N_Ω_UC

∑

R

Z

ΩUC(₀)d³r ψ^∗_n0k⁰(_r+_R)O(_r+R, . . .)ψ_nk(_r+_R)

= ¹

N_Ω_UC

∑

R

e⁻ⁱ⁽^k⁰⁻^k^)·^R Z

ΩUC

d³r ψ^∗_n0k⁰(r)O(r, . . .)ψ_nk(r)

= δ(_k−_k⁰)hu_n⁰_k|O(^ˆ _k)|u_nki_UC (2.10) whereΩUC(R)is the unit cell corresponding to the lattice vector R and the change of basis used to go from second to third line is described in figure2.1.

The quantity O(r, . . .)is the operator in position representation. The dots indicates the possibility that it can depend on other variables rather than r, such as the gra-

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2.3. Orthogonality of the basis 25

dient∇_ror higher derivatives. In the last line, we used the sum over complex exponential defined in appendixA; notice that the Dirac delta must be replaced with a Kronecker delta δ_kk⁰ if the crystal is finite. To summarize, for a local and periodic operator:

hψ_n⁰_k⁰|O|^ˆ ψ_nki =δ(k−k⁰)hu_n⁰_k|O(^ˆ k)|u_nki_UC (2.11) This last equation will play an important role in the development of the theory. In particular we will use

hu_n⁰_k|O(^ˆ k)|u_nki_UC=O_n⁰_n(k) (2.12) while hψ_n⁰_k⁰|O|^ˆ ψ_nkidiverge for an infinite crystal for k⁰ = k, the integral over the unit cell is always finite. This matrix representation is valid in particular for relevant operators and it will play an important role in the time evolution of the system. For example, the unperturbed Hamiltonian, which is diagonal in k because it follows the symmetry of the lattice, is written as

H_n⁰_n(_k) = hψ_n⁰_k|H|^ˆ ψ_nki_UC (2.13)

2.3 Orthogonality of the basis

Since the Bloch functions are eigenvectors of the Hamiltonian, they must satisfy:

hψ_n0k⁰|ψ_nki_{∝ δ}(k−k⁰)δ_nn⁰ (2.14) This is true for non-degenerate eigenvalues, while in case of degeneracy it is always possible to rotate the eigenvectors of the subspace to make them orthogonal. Now, using eq. (2.11) with ˆO =1, it is possible to derive^ˆ

hψ_n0k⁰|ψ_nki =δ(k−k⁰)hu_n0k⁰|u_nki_UC (2.15) Combining eq. (2.14) and (2.15), we obtain an orthogonality relation for the Bloch envelopes (notice that it is valid only for bra and kets over the same k point):

hu_n0k⁰|u_nki_UC _{∝ δ}_nn⁰ (2.16) Until now, only the orthogonality has been discussed but nothing has been said about the norm of these basis (relations (2.14) and (2.16) are proportionality, so a constant must be fixed). It is usually convenient to impose that the Bloch envelopes are normalized within the unit cell. Thus, using the previous equations, we obtain the equalities

hψ_n0k⁰|ψ_nki = δ_nn⁰δ(k−k⁰) (2.17)

hu_n⁰_k|u_nki = δ_nn⁰ (2.18)

Anyway, in literature different normalizations can be found.

Since the Bloch functions form a basis, they satisfy the completeness relation:

∑

n

Z

d³k |ψ_nkihψ_nk| =₁^ˆ (2.19) It is also convenient to define the completeness relation for a fixed k point con- structed with periodic functions. Hence, using the projection at fixed k on the sub- space of Bloch wavefunctions, ˆ1kis defined as

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FIGURE2.1: The change of coordinates from r to rINTRA+Rallows to go in the region to integrate fromΩUC(R)toΩUC(0). Each square in the

figure represent a unit cell.

∑

n

|ψ_nkihψ_nk| =

∑

n

|u_nkihu_nk| =₁^ˆ_k (2.20) Note that this means that _Z

d³k ˆ1k =₁^ˆ (2.21)

As it is expected from the orthogonality relations, while the Bloch functions can recover the whole space, the Bloch envelopes can recover the space only at a fixed quasi-momentum.

2.4 Position operator

One of the operators whose behavior is odd in the representation of Bloch functions is the position operator ˆr. The main reason is that, if the crystal is considered infi- nite, the integrals diverge as the Bloch functions cannot be normalized in the proper sense: as it has been shown in eq. (2.17), the orthogonality relation is valid only in the distribution space. Note that it is not possible to use directly (2.11), because al- though the position operator ˆr is local, it is not periodic. To find the expression for the position operator, the same procedure described by Blount [8] is followed, which

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2.5. Commutator of an operator ˆOwith position ˆr 27

is valid only in the thermodynamic limit (k is a continuous variable):

hψ_n0k⁰|ˆr|ψ_nki = hu_n0k⁰|e⁻^ik⁰^·^ˆrr e^ik^·^ˆr

|u_nki

= hu_n0k⁰|e⁻^ik⁰^·^ˆr(−i∇_ke^ik^·^ˆr)|u_nki

= −i∇_khψ_n0k⁰|ψ_nki +ihu_n0k⁰|e⁻ⁱ⁽^k⁰⁻^k^)·^ˆr|∇_ku_nki

The first integral can be identified with the orthogonality condition given by eq.

(2.17). For the second one, the sum over the unit cells is used as in eq. (2.10), see fig.

2.1, and the final result is:

hψ_n0k⁰|ˆr|ψ_nki = −i∇_kδ(k−k⁰)δ_n⁰_n+ξ_n⁰_n(k)δ(k−k⁰) (2.22) where the Berry connection ξ is defined as

ξ_n0n(_k) =ihu_n⁰_k|∇_ku_nki_UC (2.23) As it will be explained in the next chapters, the two terms found correspond to the splitting of the position in two quantities (as in fig.2.1): the intracell position operator hψ_n0k⁰|ˆrINTRA|ψ_nki =ξ_n0n(k)δ(k−k⁰) (2.24) and the lattice vector operator

hψ_n0k⁰|R^ˆ|ψ_nki = −i∇_kδ(k−k⁰)δ_n⁰_n (2.25) Note that in a more correct way (for gauge covariance reasons that will be explained within this chapter) the definition of the lattice vector operator is

hψ_n0k⁰|R^ˆ|ψ_nki = −i∇_khψ_n0k⁰|ψ_nki (2.26) The position operator is therefore:

ˆr=ˆrINTRA+R^ˆ (2.27)

2.5 Commutator of an operator ˆ O with position ˆr

The matrix elements of the commutator [ˆr, ˆO] can be calculated by means of the splitting of the position operator in intracell and lattice vector operator, see eq. (2.27), as

[ˆr, ˆO] = [R, ˆ^ˆ O] + [ˆrINTRA, ˆO] (2.28) The commutator involving the lattice vector operator can be calculated from its matrix elements (2.25) and from the completeness of the space:

[R, O]_n0k⁰,nk = −

∑

n⁰⁰

i Z

d³k⁰⁰∇_k⁰⁰δ(k⁰−k⁰⁰)δ_n⁰_n⁰⁰hψ_n⁰⁰_k”|O|^ˆ ψ_nki +

∑

n⁰⁰

i Z

d³k⁰⁰hψ_n⁰_k’|O|^ˆ ψ_n00k⁰⁰i ∇_kδ(_k⁰⁰−_k)δ_nn⁰⁰

= i h∇_k⁰ψ_n0k⁰|O|^ˆ ψ_nki + hψ_n0k⁰|O|∇^ˆ _kψ_nki

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In the same way, the commutator with the intracell operator can be calculated using its matrix elements (2.24):

[_r_INTRA_{, O}]_n0k⁰,nk = +

∑

n⁰⁰

Z

d³k⁰⁰ξ_n⁰_n⁰⁰(_k⁰⁰)δ(_k⁰−_k⁰⁰)O_n00k⁰⁰,nk

−

∑

n⁰⁰

Z

d³k⁰⁰O_n0k⁰,n⁰⁰k⁰⁰ξ_n00n(k)δ(k⁰⁰−k)

=

∑

n⁰⁰

ξ_n0n⁰⁰(k⁰)O_n00k⁰,nk−O_n0k⁰,n⁰⁰kξ_n00n(k)

Usually only the diagonal matrix elements in k space will be used. In this case, the matrix elements of the commutator can be rewritten, after summing the contributions above, as:

[r, O]_n⁰_n(k) =i∇_kO_n⁰_n(k) + [ξ, O]_n⁰_n(k) (2.29) Note also that this commutator can be rewritten in crystal momentum representation using the gradient of an operator, with the operator defined in eq. (2.9):

∇_kO(^ˆ _k) = ∇_ke⁻îk^·^ˆrOêîk^·^ˆr

= −iˆr ˆO(_k) +i ˆO(_k)ˆr (2.30) So it finally reads:

∇_kO(^ˆ k) = −i[ˆr, ˆO(k)] (2.31)

2.6 Velocity operator

Another important operator is the velocity operator, which can be written as ˆv = ¹

i¯h[ˆr, ˆH] (2.32)

Using eq. (2.29), the velocity operator in crystal momentum representation is v_n⁰_n(k) = ¹

¯h(∇_kH_n⁰_n(k) −i[ξ, H]_n⁰_n(k)) (2.33) Hence, the velocity matrix elements can be calculated using the Berry connection, the Hamiltonian, and the gradient of the Hamiltonian.

2.7 Quasi-momentum operator

The quasi momentum operator ˆk is defined from its action on the Bloch wavefunc- tions:

ˆk|ψ_nki =k|ψ_nki (2.34)

Because of this definition, and using the orthogonality condition (2.17), its matrix elements are

hψ_n0k⁰|ˆk|ψ_nki =k δ(k−k⁰)δ_nn⁰ (2.35) The matrix elements of the lattice operator ˆRin crystal momentum representation when n=n⁰ are exactly the same than the one of the position operator ˆr in momen- tum representation [80] (besides a constant ¯h). This suggests that the two operators

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2.8. Gauge freedom for|ψ_nki 29

are conjugate of each others and they satisfy the commutation rule:

[R^ˆ_i, ˆk_j] =i δ_ij (2.36)

2.8 Gauge freedom for | ψ

_nk

i

As it is the typical situation in quantum mechanics, a phase factor added to a state does not change the physics, and all the observables must then be invariant under a phase transformation.

The general form of a localU (1)transformation in Bloch states is [78,55]:

ψ⁰_nk

=e^iθⁿ⁽^k⁾|ψ_nki (2.37) which means:

u⁰_nk

=e^iθⁿ⁽^k⁾|u_nki (2.38) U(1)represents the group of unitary square matrices of dimension 1, i.e., complex exponentials. Note that the phase can depend on the quasi momentum. Using eq.

(2.37), one obtains the transformation O⁰_n0k⁰,nk = ψ⁰_n0k⁰

ˆOψ⁰_nk

(2.39)

= e⁻^iθⁿ⁰⁽^k⁰⁾O_n0k⁰,nke^iθⁿ⁽^k⁾

All the operators that satisfy eq. (2.39) when the basis is transformed with eq. (2.37) and (2.38) will be referred to as gauge covariant. The meaning is that, if the wavefunction changes with a phase factor, the matrix elements of the operators must change accordingly to preserve the physics.

In the next sections it is shown that the matrix elements of the ˆr and ˆv operators, defined previously in the representation of Bloch functions, are gauge covariant.

Gauge transformation of r

The position operator can be splitted in two contributions, as shown in eq. (2.27).

Here the two terms are analyzed separately when a gauge transformation defined by eq. (2.37) and (2.38) is performed.

The transformation of the intracell operator of eq. (2.24) is:

ψ_n⁰0k⁰

rINTRA

ψ⁰_nk

= ξ⁰_n0n(k)δ(k−k⁰)

= iu⁰_n0k

∇_ku⁰_nk

UCδ(_k−_k⁰)

= i e⁻ⁱ⁽^θⁿ⁰⁽^k⁾⁻^θⁿ⁽^k⁾⁾

hu_n⁰_k|∇_ku_nki_UC+i(∇_kθ_n(_k))hu_n⁰_k|u_nki_UCδ(_k−_k⁰)

= e⁻^iθⁿ⁰⁽^k⁾ξ_n⁰_n(_k)e^iθⁿ⁽^k⁾−δ_nn⁰∇_kθ_n(_k)δ(_k−_k⁰)

For n 6= n⁰, the last equation has the same structure than eq. (2.39). However, for n = n⁰, there is the contribution of the last term, and it is then clear that the transformation do not preserve gauge covariance. The lattice vector operator matrix elements of eq. (2.25) satisfy:

ψ⁰_n0k⁰

ˆR

ψ_nk⁰

= −i∇_kψ⁰_n0k⁰

ψ_nk⁰

= δ_nn⁰ −i∇_kδ(k−k⁰) +δ(k−k⁰)∇_kθ_n(k)

A TTOSECOND SPECTROSCOPY IN THE X- RAY REGIME IN COMPLEX SYSTEMS .