On second-order self-energy corrections and plasmon resonances in silicon

On second-order self-energy corrections and plasmon

resonances in silicon

Citation for published version (APA):

Daling, R. (1991). On second-order self-energy corrections and plasmon resonances in silicon. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR353576

DOI:

10.6100/IR353576

Document status and date: Published: 01/01/1991

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On second-order self-energy corrections

and

plasmon resonances in silicon

On second-order self-energy corrections

and

plasmon resonances in silicon

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof. dr. J.H. van Lint, voor een commissie aangewezen door het College van Dekanen in het

openbaar te verdedigen op vrijdag 17 mei 1991 te 16.00 uur

door

ROELOF DALING

geboren te Zwolle.

Dit proefschrift is goedgekeurd door de promotor

prof. dr. W. van Haeringen

..t.<>p<>ow u{iw uvv

..t.<>pva uhw uvv

Contents

1 Introduction

2 Some basic diagrammatics 2.1 Abstract . . .

2.2 The pictures . . . . . . . 2.3 The formulas . . . . . .

2.4

2.3.1 The Coulomb interaction . 2.3.2 The Green's function . . .

2.3.3 The interaction and Green's function put together. 2.3.4 The Coulomb interaction vertex

Summary . . . .

3 The self-energy in silicon 3.1 Abstract . . . . 3.2 Introduction . . . . 3.3 Diagrams to be calculated

3.3.l The screened versus the bare interaction 3.3.2 The relevant second-order diagrams . . . 3.3.3 Metal versus Semiconductor, an infinite difference 3.4 The physics bebind the second-order diagrams

3.5 Basic starting points . . . 3.5.1 Choice of basis functions .. . 3.5.2 Choice of a Hamiltonian .. . 3.6 Numerical evaluation of the diagrams 3.6.l Expressions . . . . 3.6.2 Numerical procedure . . . . . 3.6.3 Convergence of integration procedure 3. 7 Results . . . . . . . . . . . . . . . 1 9 9 9 17 18 19 22 26 27 29 29 30 32 32 33 35 38 41 41 42 42 43 45 49 52

Il 3.8 3.7.l 3.7.2 3.7.3 3.7.4

The first-order diagrams The second-order diagrams . The validity of GW . . . Comparison with other work. Conclusions . . . .

4 The dielectric matrix 4.1

4.2 4.3

Abstract . . . . Introduction. . . . . The RPA dielectric matrix .. 4.3.l Fotmulation . . . 4.3.2 Formulas . . . . 4.3.3 Analytic properties 4.3.4 The

k --.

i5

limit . .

4.3.5 Connection with experiment 4.3.6 Previous work .

4.4 Method of calculation. . . . . 4.4.1 The genera! idea . . . 4.4.2 The analytic continuation 4.5 Numerical Results . . . . . 4.6 Physical results . . . .

4.6.1 The energy loss function 4.6.2 Plasmon resonances . 4.7 Summary and conclusions. A Silicon in a nutshell B Fourier transforms C Symmetry relations Bibliography Samenvatting CONTENTS 53 54 57 58 59 61 61 61 64 64 67 69 70 71 72 75 75 76 78 86 86 91 101 105 109 111 113 118

Chapter 1

Introduction

In classica! as well as in quantum mechanics the problem of dealing with many inter-acting particles is a longstanding and difficult one. In both cases the theory provides us with an equation goveming the particles behaviour, Newtons equation in the classi-ca! case and the many-particle Schrödinger equation in the quant urn mechanica! case. The quantum mechanica! description is complemented with a rule stating that the total many-particle wavefunction has to be symmetrie or antisymmetric under permu-tation of any two of its arguments, an argument being the total set of labels referring to one-particle, e.g. space coordinates together with intern al degrees of freedom such as spin. The antisymmetric case applies if the involved particles have half-integral spin (fermions) and the symmetrie case if they have integral spin (bosons).

Although easy to write down for a many-particle system with specified inter-particle interactions, the Schrödinger equation as well as Newtons equation can only be solved relatively easy in case of two particles, either analytically or numerically depending on the specific interaction, the three particle case already being very com-plicated. If a large number of particles is involved, the problem can only be solved if the interaction between the particles is disregarded since in that case the many -particle equation reduces to many identical one-particle equations.

In quantum mechanics, the theory to which we will confine our attention from now on, the solution of such a one-particle Schrödinger equation leads to a set of eigenstates and eigenvalues, the eigenvalue being the energy of a particle that is in the corresponding eigenstate. The many-particle ground state is constructed by placing the particles in the lowest possible eigenstates, thereby paying attention to the above mentioned requirement of symmetry or antisymmetry. In the fermion spin ~ case, e.g. electrons in a solid or nucleons in a nucleus, the requirement of antisymmetry

2 Chapter 1. lntroduction

(the Pauli principle) allows the presence of only one particle in each eigenstate or, if spin degrees of freedom are disregarded in the wavefunction, two particles in each eigenstate. The excited states of the non-interacting system are simply obtained by taking one or more particles out of an occupied state and placing them in a state that is unoccupied in the ground state.

The simplest way to account approximately for the inter-particle interaction is to add to the one-particle Schrödinger equation a potential which accounts for the presence of all other particles in some average way. In this way the simple picture of independent particles moving in an external potential remains intact and the most el-ementary excitations of the system can still be described in terms of one particle being moved from an occupied to an unoccupied state. Although the quantitative agree-ment between the experiagree-mentally determined excitation energies and the excitation energies following from a one-particle description need not always be good, it turns out that in many cases the experimentally observed excited states of a many-particle system can very well be associated with the simple excitations obtained theoretically as the solutions of a one-particle equation and this fact makes it possible to under-stand a very complicated system at least partially in a much simpler one-particle picture. This picture emerges for example in atomie physics where it is possible to identify many of the observed excited states of many-electron atoms with hydrogen-like orbits of a single electron moving in the mean field of the nucleus and the other electrons (see for example (Con57]). A more striking example is provided by the shell model of nuclear physics in which many of the observed nuclear levels can be understood approximately by considering the nucleons to be moving in an external potential well which is set up by the nucleons themselves (see for example [Eis87]). More relevant in the context of the present work is the fact that the concept of an electron bandstructure, which finds its origin in an independent electron picture, is compatible with the experimental data concerning the excitation structure of many soli ds.

Of course there are also cases in which the simple one-particle picture breaks down and in which there exist correlations between the particles that cannot be described in a simple one-particle picture. This may concern the ground state as well as the excited states of a many-particle system. As an example of a system in which the ground state is governed by strong correlations we mention a ferromagnet. In a ferromagnet all the individual electrons have their spin-magnetic moments pointed more or less in the same direction and this gives rise to a macroscopie magnetic moment. The so called exchange interaction between the electron spins giving rise to

3

this ferromagnetic ordering has its origin in the spin independent Coulomb interaction in combination with the Pauli exclusion principle. An other well known example of a phenomenon which is incomprehensible in a one-particle theory is the occurence of superconductivity. Excited states sucb as the vibrational and rotational levels in a atomie nucleus or the plasma oscillations in a solid are examples of collective excitations in which all particles partake. The plasma oscillations, or plasmons as tbey are called, will be the subject of Chapter 4. It has to be noted that these collective excitations exist side by side with single-particle excitations.

The self-energy

As mentioned above many properties of an interacting particle systern can be de-scribed in terms of independent particles moving in a fixed single-particle potential, which accounts in an approximate way for the presence of all particles. The one-particle potential which gives an optimum description of the many-one-particle system depends of course on the system but also on the specific properties that are to be cal-culated. Questions concerning the ground state properties, such as the total energy, of a semiconductor can very well be answered by using the so called Local Density Approximation (LDA) ([Koh65]) to account for the Coulomb interaction between the electrons ([Zun77], [Yin82], [Den85]). This same LDA, however, is inadequate to de-scribe the excitation properties of semiconductors and obtaining accurate values for the excitation energies requires the use of another single-particle Hamiltonian in the Schrödinger equation.

An example of a very simple one-particle Hamiltonian that gives accurate values for of the excitation energies of semiconductors is given in the so called Empirica! Pseudo potential Method (EPM) ([Coh70]). This Hamiltonian contains only a few parameters which can be fitted so as to obtain a bandstructure which is in good agreement with experiment. Although this demonstrates that certain excitations in a solid can be understood in a very simple one-particle picture, the EPM Hamiltonian does not give insight in the role of the inter-electron Coulomb interaction. A one-particle potential that is based directly on the physics of the interaction bet ween the electrons is the Hartree potential. In the Hartree approximation the electron-electron Coulomb interaction, which acts between any pair of electrons, is replaced by the Hartree potential which describes the Coulomb interaction of an electron with the distributed charge density of all other electrons together. If the requirement of an antisymmetric many-particle wave function is implemented in the Hartree potential the well known Hartree-Fock potentialis obtained. This potential, however, does not

4 Chapter 1. Introduction

account in suffi.cient detail for the effects of the Coulomb interaction between electrons in semiconductors since it turns out that Hartree-Fock bandstructure calculations for these systems yield bandgaps which are about a factor of two larger than the experimental bandgaps ([Dov80],[Gyg86],[Lin86]). Apparently a more sophisticated potential is needed.

This more sophisticated potential can be obtained in the framework of many-particle perturbation theory. In this context the effective potential is commonly referred to as the self-energy operator. This self-energy operator is given as a pertur-bation series in powers of the electron-electron Coulomb interaction and the Hartree-Fock potential corresponds in fact to the two first-order terms of this series.

Due to the long range nature of the Coulomb interaction one of the second-order terms in the series either diverges (metals) or gives rise to very strong potentials (semiconductors) and in actual calculations it is therefore necessary to screen the Coulomb interaction and to reformulate the self-energy perturbation series in powers of the resulting screened interaction (W). Calculations in which the self-energy oper-ator is approximated by the first-order term of the reformulated perturbation series have been very successful in predicting semiconductor bandstructures without using experimental input. This particular approximation is known as the GW approxima-tion and it was introduced by Hedin in [Hed65]. Examples of these GW calculations can be found in [Str82], [Wan84],(Hyb86],[Lin88],[God88] and [Ham90].

The question which naturally arises is whether the higher order terms in the expansion for the self-energy operator can be neglected. From a pragmatical point of view an affirmative answer can be given since the first-order GW approximation bandstructure is in good agreement with experiment. A more principal solution of this problem, however, requires the explicit demonstration that the higher order self-energy contributions give small corrections on for example the first-order bandgap. This approach is followed in [Ben78] where a crude estimate is given of the magnitude of the second-order self-energy contribution to the direct gap of silicon. The Penn model ([Pen62]) is used to model the semiconductor electron bandstructure. The second-order self-energy contribution to the direct gap is estimated to be of the order of 10% of the first-order GW contribution.

In Chapter 3 we investigate the validity of the GW approximation in Si by calcu-lating the relevant second-order self-energy contributions using a real bandstructure instead of modelling it as in [Ben78].

5

The dielectric function

Since the introduction of the dielectric constant by Farada.y some 150 years ago the concept of dielectric response has been playing an important röle in the field of con-densed matter physics. Initially the dielectric constant (.::) was introduced to express the fact that if a material is placed in an external statie electric field (..Ö) the total internal field (.Ê) that arises inside the material is weaker than the external field. This phenomenon of screening as it is called, is expressed by Ê

=

fJ / .::. From this relation we see that it is actually the inverse of the dielectric constant which determines the response of the material to an externally applied field. In the study of the interaction of light with matter it became clear that the response of a material to an external electric field depends on the frequency w of the field. The exact frequency dependence depends on the specific material hut has characteristic features for materials belong-ing to a particular class of materials, for example metals or semiconductors. In later developments it turned out that the response to an external field a]so depends on the wavelength .À of the field or, alternatively, on its wavenumber k

=

2r. /.À. In this way the concept of dielectric response evolved from the simple dielectric constant.:: to the notion of a complicated dielectric function .::(k,w). [Kel89] is a recent book which handles in detail the concept of dielectric response in condensed matter systems.

In [Noz58) Nozières and Pines introduced the dielectric function in the many-body problem. It has to be noted that the dielectric function which features in their formulation is the so called longitudinal dielectric function which determines the response of the system to a longitudinal electric field such as the field of an external charge. This longitudinal dielectric function bas to be distinguished from the transverse dielectric function which determines the reponse to a transverse electric field (light). As shown in [Noz58) the longitudinal dielectric function can be used to give an in principle exact expression for the ground state energy of a many-body system. The longitudinal dielectric function is also the quantity which is needed to cakulate the energies of the above mentioned collective plasmon oscillations wh.ich occur in an electron gas. Moreover, in the G\V bandstructure calculations a ma.jor role is played by the longitudinal dielectric function because the screened interaction

Wis related to the Coulomb interaction v according to W

=

v/t:

The basic quantity which has to be calculated in order to obtain the dielectric func-tion is the so called irreducible polarization function P (see Chapter 4). This function describes the polarizability of the system in terms of the creation of electron-hole pairs out of the ground state. In many-particle perturbation theory this function is given

6 Chapter 1. Introduction

as a series in powers of the electron-electron Coulomb interaction which provides us with possible approximations for its actual calculation. The one-particle properties of the system as described by quantum mechanics are the ingredients entering in these calculations.

The so called Random Phase Approximation (RPA) is the simplest and most aften used approximation used in the calculation of the dielectric properties of a system. It corresponds to taking into account only the first term of the perturbation series for the irreducible polarization function. In this first term all interaction effects between the electron and hole partaking in the polarization process are disregarded. The RPA is most useful at small values of the wavenumber k. At larger values of k (shorter wavelenghts) the effects of the Coulomb interaction between the electrons and holes become important; the mo<lifications on the dielectric response due to this electron-hole interaction are called exchange correlation effects.

For an homogeneous electron gas the calculation of the RPA dielectric function can be done analytically, resulting in the Lindhard dielectric function ([Lind54]). The inclusion of the exchange correlation effects in the dielectric response of the electron gas has also been studied along various directions. (see for example [Sin76],[Ara82] and [Bro88])

The calculation of the <lielectric properties of a real solid has to be clone nu-merically and is very demanding with regard to the available computer power. The calculation is complicated by the presence of the ion lattice which lacks full transla-tional symmetry. Due to this lack of symmetry an external field which varies slowly on the scale of the lattice unit cell induces internal fiel<ls that vary rapidly on the scale of the unit cel!. This effect is called the local-field effect. Mathematically this local-field effect implies that the dielectric function, which is appropriate to describe the dielectric properties of a homogeneous system, becomes an infinitely sized dielectric matrix.

The actual ab initio calculations of the RPA dielectric matrix of semiconductors and insulators have mainly been concerned with the statie case w

= 0 ([Wal70],

(Bre75], (Cam76], [Ba\78], [Cam81], [Hyb87a]). The frequency dependence of the dielectric response is calculated in for example [Vec72], [Lou75], [Han80] and [Far88]. A common conclusion in the above cited works is that a proper inclusion of the local-field effects is essential for a good understanding of the dielectric properties of semiconductors and insulators. However, the mutual numerical agreement is far from perfect. For example, the calculated values for the dielectric constant of Si vary in a wide range. In [HanSO] the RPA dielectric constant is found to be .êsi

=

8.0 whereas

7

[Hyb87a] reports an RPA value of €Si= 12.2.

As we mentioned above the frequency dependent dielectric function (or matrix) is needed to calculate the plasmon energy in a many-electron system. Calculations of the dielectric response in the plasmon frequency range can be found in [Wal72], [Louï5], [Stu82] and [Tau86]. The calculation in [Wal72) is rather crude in that only a very small (too small) basisset bas been used and in that local-field effects have been neglected. In [Lou75] the local-field effects have been taken into account and the calculated plasmon energy is in reasonable agreement with e:>.."})eriment, however, the plasmon dispersion, that is the dependence of the plasmon energy on the wavevector

k

of the oscillation, bas not been considered. Whereas these last two references take in to full account the electron bandstructure in their calculation, [Stu82] obtains plasmon energies in a nearly free electron model which starts from the free electron gas and which accounts for the lattice potential in a perturbative way.

Outline

In Chapter 2 we will give a brief outline of many-particle perturbation theory. We wil! only present some elements of this theory and will not give a thorough <lerivation of the perturbation expansion. The chapter is meant to make the self-energy and polarization diagrams in Cbapters 3 and 4 more accessible to readers not acquainted with the formalism.

In Chapter 3 we address the problem of the validity of the above mentioned GW approximation for the self-energy operator in Si. In the existing literature in which this approximation is being used questions concerning its forma! validity are hardly posed. Since the GW approximation corresponds to taking into account only the first-order term of a perturbation series in powers of a screened electron-electron interaction, this question can only be answered positively if it can be shown that the sum of all higher order diagrams is small with respect to the first-order GW contribution. In Chapter 3 we make a small step towards this goal by calculating the simp lest second-order self-energy diagrams, however, because of the complexity of the screened intera.ction we have to restrict ourselves toa calculation in which we use the bare Coulomb interaction. Due to this restriction, there are two second-order self-energy diagrams one of which, the so called vertex correction diagram, bears directly on the validity of the GW approximation. We calculate the contribution of both second-order diagrams to the direct gap between the valence and conduction bands of Si at

k

=

Ö. A comparison of the contribution of the vertex correction diagram with the first-order Fock contribution to this direct gap demonstrates that the vertex

s

Chapter 1. Introduction

correction diagram can safely be neglected. The results for the other second-order diagram illustrate clearly that the expansion for the self-energy operator in powers of the Coulomb interaction is useless and that, if one insists on using the perturbation formalism for obtaining the self-energy operator, the next simple approximation which might make sense after first-order Hartree-Fock is the GW approximation.

Chapter 4 deals with the calculation of the plasmon energies in Si. In order to obtain these energies we have to calculate the dielectric matrix of Si in an energy range in which there also exist one-electron-one-hole excitations. Due to the exis-tence of these Jatter excitations the calculation of the dielectric matrix involves the integration over the lBZ ( first Brillouin zone) of functions which are singular on a surface intersecting the integration volume (lBZ). We present a new method to obtain these singular integrals. The method is based on analytic continuation and is very efficient in view of the relatively small number of integration points needed to obtain numerically converged results. We calculate the RPA dielectric matrix of Si for

k

values along the 6..- and A- axes of the lBZ and use this matrix to obtain the plasmon dispersion relations along these two directions. We present results of both EPM and LDA calculations, in the Jatter of these we have also investigated the influ-ence of the exchange correlation corrections. In all our calculations local-field effects are properly taken into account. We compare our results with earlier calculations of the dielectric properties of Si as well as with experimental data.

Plasmon excitations are observed as resonances in the so called energy loss func-tion, which can be measured in electron energy loss experiments. vVe investigate the appearance of the plasmon in this energy loss function by determining the complex zeros of the determinant of the dielectric matrix. In order to obtain the relevant zeros the analytic continuation of the dielectric matrix in to the non-physical Riemann sheet is needed. Because of our special way of calculating the dielectric matrix this analytic continuation can easily be obtained.

In Appendix A we give a short descript ion of Si and introduce some terminology. Appendix B contains our definitions of the Fourier transforms. In Appendix C we show how the symmetry operations of the Si lattice can be used to determine the number of independent matrix elements occuring in the momentum representation of for example the dielectric matrix.

Chapter

2

Some basic diagrammatics

2.1

Abstract

In this chapter we give a short introduction to Feynman diagrams. Rather then giving a genera! discussion concerning the diagrammatic representation of the perturbation series fora given operator, which can be found in any textbook on many-body theory (e.g. [Brou63], [Mat67], [Fet71], [Ink84], [Neg88]), we will focus on the use of these diagrams in solid state physics. We do this in order to facilitate discussions concerning the properties of the diagrams which we calculate in Chapters 3 and 4 as well as to make the diagrams and corresponding expressions more accessible to the reader who is not acquainted with their application in solid state physics.

2.2

The pictures

We begin with introducing the notion of Feynman diagrams in a pictorial way, in particular we will pay attention to the diagrams that appear in the perturbation series for the so called self-energy operator. The symbols that we introduce in t.his section are defined in a sometimes heuristic way but they wil! all get their precise mathematica! meaning in the next section.

We consider N particles which are not subject to mutual interactions and construct the N-particle groundstate by placing the particles in the N energetically lowest eigen-states of the single-particle Hamiltonian that describes the behavior of each individual particle. Since the particles are assumed to be non-interacting they remain forever in the same eigenstates and nothing interesting happens. We wil! represen t this ground-state in a figure by nothing, that is by white paper. The next thing we do is to add

10 Cbapter 2. Some basic diagrammatics

(a) (b) (c)

part iele hole interaction

Figure 2.1

Graphica/ representation for an extra part iele ( a) or an extra hole (b) present in

addition to the N-particle ground state. The inter-particle interaction is represented by ( c). Time increases from the bottom to the top of the figure.

an extra particle to the system by placing it in an eigenstate which is unoccupied in the groundstate. Alternatively we can add a hole to the system by taking a particle out of an eigenstate which is occupied in the groundstate. Since the particles are still assumed to be non-interacting the added particle or hole propagates freely for

an· infinite time in the state in which it was placed and, correspondingly, the eigen-states of the single-particle Hamiltonian are said to have an infinite lifetime. The particles and holes which are present in addition to the ground state are represented graphically in Fig. 2.1 a,b by lines in addition to the graphical representation for the groundstate, so they are lines on white paper. The line representing an extra particle is distinguished by an upward arrow from the line representing an extra hole, which carries a downward directed arrow. The exact mathematica! meaning of these lines wilt become clear in section 2.3 (see in particular the comments following Eq. 2.9 in that section). In Fig. 2.1 time increases from the bottom to the top of the figure.

Later we will see that the mathematica! formalism combines the propagation of an

extra particle or an extra hole in one single quantity, the so called Green's function,

and that no definite time order exists between the endpoints of the symbol represent-ing this function. Accordingly, in the figures displaying diagrams in which this new symbol is used there is no direction of increasing time.

Henceforth we restrict ourselves to the many-particle system consisting of electrons in a semiconductor. The groundstate in a semiconductor consists of the completely filled valence bands and what we have called above an extra particle in an unoccupied

state becomes an electron in a conduction band state. Creating a hole corresponds

to taking an electron out of a valence band state.

2.2. The pictures 11

(a) (b) (c) (d)

Figure 2.2

Four different processes that can occur at an endpoint of an interaction line. Note that there is always one incoming and one outgoing line. In (a) an extra electron in (con-ductionband) state c1 scatters to (conductionband) state c2 and in (b) an extra hole in (valenceband) state v1 scatters to (valenceband) state v2 • In (c) an electron-hole pair is created and in ( d) an electron-hole pair is annihilated. Of course the sa me processes can take place at the other endpoint. Time increases in the upward direction. We will represent this Coulomb interaction by the dashed line in Fig. 2.F. The electrons in the valence bands which form the ground state interact with each other; an extra electron or hole intera.cts with the electrons making up the groundstate and furthermore an extra electron can interact with an extra hole. These ir:teractions modify the propagation of an extra electron or hole and change its energy. The ways in which the free propagation of the electron or hole is modified can be represented conveniently by Feynman diagrams, which display in a transparent way the different 'physical processes' or 'physical effects' that modify the energy of the electron or hole. We have put the pbrases 'physical ... ' between quotation marks because the actual physical process, which is experimentaJly accessible, e.g. through measurements of the bandstructure, is the 'fully interacting' propagation of an extra electron or hole through the system and this propagation is represented by the sum of all diagrams. A 'process' corresponding to an individual diagram cannot be observed isolated from all other 'processes' corresponding to the other diagrams. Nevertheless, giving a meaning and interpretation to each individuaJ diagram proves to be fruitful in that it can give a feeling for the kind of physics involved and the interpretations enable one in various cases to understand the signs of the energy correction to the electron or hole due toa given diagram (see section 3.4). Sometimes the interpretation of a diagram can be given in terms of classica] processes (charges interacting with the polarized surroundings), whereas other diagrams require a quantum mechanica] interpreta.tion (Pauli principle). This will become clear shortly when we exemplify the diagrams of Fig. 2.3a,b.

12 Chapter 2. Some basic diagrammatics

co co co

t2

t2

;o.,

_~O··

_{'·a,;o ..}

c, t, vo t, t, co Dp e co _DGs e co (a) (b) (c) Figure 2.3

( a) and (b) are two examples of diagrams representing processes that modify the

prop-agation and the energy of an extra electron. ( c) is aso ca/led unlinked diagram which should not be included when calculating the infiuence of the Coulomb interaction on

the energy of an electron in state Co· This diagram is used in the main text to

ex-plain the physical significa nee of the diagram in (b). Time increases in the upward direction.

symbols of Fig. 2.1, which represent the free propagation of electrons and holes and the

interaction between them. Since the Coulomb interaction acts between two electrons or holes both endpoints of each interaction line in a diagram always have one incoming and one outgoing line. The interpretation of what happens at the endpoint of an

interaction line depends on the particular configuration of the incoming and outgoing

line and is given in Fig. 2.2.

We will now explain how to read the diagrams of Fig. 2.3. These diagrams will play an important role in Chapter 3 and the naming of the diagrams in this figure is

chosen so as to be consistent with section3.4 in which we return to these diagrams.

The diagram labeled

D?

has to be understood as follows : An electron in a conduction

band state ( ea) moves through the semi conductor in presence of N other electrons which make up the ground state (white paper). Ata given time t1 the extra electron interacts with an electron in a valence band state and scatters to another conduction band state (ei) . The electron with which it interacts is excited toa conduction band state (c2 ) thereby leaving behind a hole (v2 ), that is leaving behind one empty state, in the completely filled valence band states. The excited electron ( c2 ) moves along

in the conduction band state, represented by the line in the loop with the upward directed arrow and the hole (v2) moves along in the valence band state, represented

by the downward running line. At a later time t2 the excited electron c2 interacts

2.2. Tbe pictures 13

so that the electron hole pair (c2 , v2 ) which was created at time t1 disappears. The

classical physical analogue to this diagram is the interaction of an extra charge with the polarization density that it induces in the medium in which it is placed. For this reason we have attributed the index P to this contribution (Dj,).

The diagram Dbs has a somewhat more complicated interpreta.tion which cannot be associated with a classical proces but which has its origin in quantum mechanics. The line with the upward directed arrow present before time t1 again represents an extra electron in a conduction band state (eo). At time t1 two of the N electrons that form the ground state interact with each other and are excited to conduction band states (Co) and (c1 ) (the two (new) lines with the upward arrow), leaving behind two holes (va) and (v1 ) (the two lines with the downward arrows). At time t2 one

of the excited electrons (c1 ) interacts with the extra electron (eo), and both fall into the empty valence band states (va and v1 ) created at t1 • The second of the t wo

electrons excited at t1 survives and has taken the role of the extra electron in CQ. It has to be noted that in the intermediate state of this diagram (i.e. the state between times t1 and t2 ) two electrons are present in the same conduction band state

eo.

This simultaneous presence of two electrons in one and C1e same state seems to be at variance with the Pauli principle. It is, however, precisely the effect of the Pauli principle on the electron's energy which is represented by this diagram. In order to explain this seemingly contradictory statement we show in Fig. 2.3c a third second order diagram. In this diagram one extra electron is present in state Co before time

t1• As in Fig. 2.3b at time t1 two of the electrons in the ground state are excited to conduction band states Co and c1 leaving bebind two holes in va and v1. At time t2 the two extra electron hole pairs are annihilated and after t2 the electron in Co is again the

only one present. As is seen in the figure the electrons and holes in the two diagrams in Figs. 2.3b,c occupy at each instant the same set of states, and the intermediate state in Fig. 2.3c also violates the Pa.uli principle. Diagram 2.3c modifies the energy of the electron in state Co by an amount which we denote by ~Ecs(eo,c1,va,v1).

It is important to notice that this diagram consists of two unlinked parts, one part corresponding to the free propagation of an electron in state Co and the other part depicting a specific process in which only electrons take part which are present in the ground state. The energy correction ~Ees( Co, c1 , va, v1 ) is entirely due to this ground state process. The only but essential difference between the two diagrams of Fig. 2.3b and 2.3c is that in Fig. 2.3b the electron which was initially present in state Co has changed roles with an electron which was initially in the state va. In contrast to this, in Fig. 2.3c it is the same electron which occupies the initia] and

14 Cbapter 2. Some basic diagrammatics

final state

eo.

Due to this exchange of roles the energy correction on state Co due to the diagram in Fig. 2.3b equals -~Ees(Co, c1 , va, v1 ). This simple change of sign

is connected with the quantum mechanica! rule that a many-electron wavefunction has to be antisymmetric under permutation of any two of its arguments. So we see that the sum of the energy corrections of the two diagrams with these particular intermediate states cancel. We now invoke, however, the linked cluster theorem of Goldstone ([Gol57]) which states that the unlinked diagrams, of which Fig. 2.3c is a typical example, should not be incorporated in the series of diagrams representing the 'processes' inodifying the energy of an extra electron. Thus the diagram in Fig. 2.3b

gives a contribution -~Ees(

eo, c

1 , va, v1 ) to the electrons energy which is, according to the cited theorem, no longer cancelled by the diagram of Fig. 2.3c.

It is clear now that, despite its at first sight strange Pauli principle violating intermediate state, the diagram of Fig. 2.3h has a physical interpretation which we formulate as follows: Due to the Coulomb interaction there are all kind of fluctua-tions on the non-interacting ground state which modify its energy. The right part of the unlinked diagram in Fig. 2.3c is a particular example of such a fluctuation and contributes ~Ees( Co, c1 , va, v1 ) to the ground state energy. The Pauli principle allows the presence of only one electron in a given state, so part of the ground state fluctua-tions are no longer allowed if we put an extra electron in Co and as a result the extra electron suppresses the contribution ~Ees( Co, c1 , v0 , vi) to the ground state energy.

Effectively this contributes -~Ees(

eo,

c1 , va, v1 ) to the energy of the extra electron, and it is exactly this contribution which is represented by the diagram of Fig. 2.3b.

For this reason we have attributed the index GS to this contribution. We have thus arrived at the conclusion that the linked diagram with the Pauli principle violating intermediate state takes into account the consequences of the Pauli principle on the

energy of an extra electron.

It is useful to notice that corresponding to the two diagrams in Figs. 2.3•,b there are two diagrams in which

t

1

>

t2 and which can be obtained by turning the two

diagrams in Figs. 2.3•,h upside down. The interpretation of these diagrams is similar, however with the roles of electrons and holes interchanged.

Except for their very different physical interpretations there is one more dîfference between the two processes of Fîgs. 2.3•,b. As wil! be explained in section 2.3.3 the energy of the intermediate state existing between times t1 and t2 is not necessarily equal to the energy which is available for its creation at t1 , it can be aso called virtual state. The Heisenberg uncertainty relation allows the existence of such a virtual state only during a fini te time. In diagram Df., however, the possibility exists that the

2.2. The pictures 15

extra electron bas enough energy to create an electron-hole pair with conservation

of energy and the intermediate state which results is called a real state. Since this

intermediate state is energetically allowed it does not need to vanish within a finite

time and the state with the one electron in a conduction band state which existed

before t1 acquires a finite lifetime beca.use it can permanently decay to a state with

two electrons and a hole. The process depicted in D(;5 , however, cannot give rise to

lifetimes since the creation of electron-hole pairs out of the groundstate at time t1

always violates energy conservation, i.e. the intermediate state of this diagram can

never be a real state.

The self-energy operator

The above considered processes Dj, and Das both modify the energy of the extra electron. In the mathematica} formulation of many-particle physics this. energy

modi-fication is described by the so called self-energy operator

M.

This self-energy operator

can be considered as the sum of all possible processes tha.t modify the propagation

of the electron (or hole) and can be represen ted by the sum of so called self-energy

diagrams. These self-energy diagrams are obtained from the kind of diagrams we

have considered until now by deleting the external lines representing the extra elec

-tron or hole in the initial and final state. Mathematically the self-energy operator is

used as an effective potential in a quasiparticle Hamiltonian. The eigenstates of this

Hamiltonian are, by construction of

M, states

in which all kinds of interaction

pro-cesses, such as the above described creation of electron hole pairs, are incorporated. It is difficult to give the exact meaning of such an eigenstate, in fact the eigenstates

no Jonger correspond to states in which a single electron can be placed but it refers

to a specific mixture of states consisting of the extra electron with all the induced

electron-hole pairs around it. This conglomarate of electron and electron-hole pairs

is called a quasiparticle. As was explained, the quasiparticles generally have a finite

lifetime as opposed to the infinite lifetime of an electron placed in a conduction band

state of the single-electron Hamiltonian we started with.

The occurence of states with a finite Jifetime is mathematically expressed by the

fact that the self-energy operator, which is energy dependent, becomes non-Hermitian

at certain energies. As a result the quasiparticle Hamiltonian, in which the self-energy operator enters as an effective potential, has complex eigenvalues, the imaginary parts of which correspond to the inverse lifetimes of the quasiparticle eigenstates. If we consider a system, such as a semiconductor or insulator, with an energy gap égap

16 Chapter 2. Some basic diagrammatics

electron hole e/ectron hole

--0 --0

(a) Hartree (b) Fock

---0

selfenergy diagrarns

Figure 2.4

Hartree(a) and Fock(b) diagrams. The actual self-energy diagrams have been obtained by deleting the external lines.

only states which lie at least êgap above the lowest unoccupied state get a fini te lifetime due to the creation of electron- hole pairs. Of course the quasihole states more than êgap below the highest occupied state will also acquire a lifetime.

The examples which we have given are second order processes, that is the Coulomb interaction occured twice. There are also first order processes modifying the energy of an electron or hole, they are shown in Fig. 2.4. These diagrams are commonly referred to as the Hartree(a) and the Fock(b) diagram. The appearance of these diagrams is somewhat unnatural since the Hartree diagram contains something propagating in a closed loop and in the Fock diagram there is propagator which starts and ends at the same time. In the mathematica! formalism the closed loop in the Hartree diagram gets a clear interpretation, which is that it corresponds to the electron density in the groundstate, and the Hartree diagram represents nothing but the well known Hartree potential which approximates the effects of the inter-electron interaction by an one-electron potential due to the mean one-electron density. The Fock diagram arises from the requirement that the total many-electron wavefunction be antisymmetric and cannot be explained classically. Since it can be obtained by exchanging the role of two propagators in the Hartree diagram the Fock diagram is called the exchange version of the Hartree diagram. It is the instantaneous character of the Coulomb interaction in the diagram which causes the intermediate propagator to be instantaneous and this would not be the case if we considered a non-instantaneous interaction.

2.3. Tbe formulas 17

+

(a) (b)

Fîgure 2.5

Vertex correct ion diagram. ( a) shows the two possible timeorders of the vertex correc -tion diagram. Time increases in the upward direction. In (b) the diagrams of (a) are combined in one single diagram in which no timeorder is assumed.

feature in Chapter 3 of this thesis. In this figure we have deleted the external lines and thus the diagrams are self-energy diagrams. In Fig. 2.5° time is assumed to increase in the upward direction and the two diagrams differ only in the order between t1 and h

The intermediate states occuring in these two diagrams differ. For obvious reasons the intermediate state of the diagram in whicb t1

<

t2 is called a two-electron-one-hole

state and in the case of t1 > t2 we speak of an one-electron-two-hole intermediate

state. The two diagrams in Fig. 2.5a are the exchange versions of the self-energy diagrams that are obtained from Fig. 2.3a and 2.3b by deleting the external lines .

According to the mathematica! formulation in the next section the two diagrams of Fig. 2.5a can be combined in the one single diagram of Fig. 2.5b, which we draw horizontally in order to stress the fact that in this diagram there is no specific order between t1 and t2 . The diagram of Fig. 2.5b shows how, for example, the lower interaction line modifies the interaction vertex V of the upper interaction line, which is the reason why this diagram is generally called the 'vertex correction diagram '.

2.3

The formulas

In this section all symbols introduced in the previous section wil! be defined precisely and we wil! associate to each symbol a mathematica! expression. \Vith each Feynman diagram there is associated a unique mathematica! expression which has to be used if for example the contribution of a self-energy diagram to the self-energy operator is to

18 Chapter 2. Same basic diagrammatics

be calculated. In the same way as the diagrams of the previous section were built up from the elementary symbols in Fig. 2.1, the formulas corresponding to the diagrams consist of the basic expressions for the propagatorline and the interactionline. We wil!

show that the resulting expression for a diagram despite its complicated appearance

is almost as transparent as the diagram itself, which, however, does not mean that it can easily be calculated.

2.3.l

The Coulomb interaction

One of the basic ingredients entering in to the mathematica! expressions corresponding

to the Feynman diagrams is the instantaneous Coulomb interaction V between the

electrons which is given by:

It is convenient to have the momentum representation of the interaction at hand.

Using the definitions of the spatial Fourier transform in Appendix B we find

with e2 v(Q)

=

-1::12· to ql (2.2) (2.3)

In Eq. 2.2 the integration variable

q

extends over the whole momentum space. When

dealing with a crystalline solid it is convenient to write

q

as the sum of a lBZ (first Bril-louin zone) vector

k

and a reciprocal lattice vector

Q.

This decomposition

q

=

k

+

Q

is unique and can be used to write Eq. 2.2 in a form in which the momentum integra-tion is restricted to the lBZ and in which an additional summation over reciprocal lattice vectors occurs

"""""'1

á3

k - - "(k- Q-) ( - - ) v(r1 - r2)

=

~ - -v(k

+

Q)e' + . r i - r2 •

- IBZ (27r )3

Q

(2.4)

The symbol for the interaction in the position representation as well as in the momentum representation is shown in Fig. 2.6

2.3. The formulas 19

--?---

_q

(a) (b)

Figure 2.6

Graphical representation of the Coulomb interaction in the position representation Eq. 2.1 (

a)

and in the momentum representation Eq. 2.3 (b).

2.3.2 The Green's function

The exact one-particle Green's funtion or propagator G(r1t1 ;

f2t

2 ) is defined as the

expectation value of the time ordered product of the field operators ,,P(ri, t₁₎ and

ipt(r2,t2 ) in the N-particle groundstate

llllR.),

G(

r1

t1; fit2)

-i(w~

IT( iti(

r1,

t1 )îf

t

(r2, t2)) 1

'11~)

(2.5) ==

-iN~

IB(t1 - t2)it>(ri, t1 )îf

t (

fi,

t2) - B(t2 - t1)1/J

t ( fi,

t2)1/J(

r;,

t1

)1'11~

). The state I

wR.)

in this definition is the exact many-electron groundstate in which the Coulomb interaction bet ween the electrons bas been taken into full account. The operator 1/J

t

(r,

t) is called a creation operator since its action on a many-electron state is to add an extra electron. The operator ,,P(r, t) takes an electron out of a many-electron state and is called an annihilation operator. The second equality in Eq. 2.5 defines the time ordered product of the two ( anticommuting) operators 1jJ and 1/J

T.

The exact Green's function as defined in Eq. 2.5 describes the propagation of an electron or hole with all possible interaction effects included. In fact it describes the propagation of the quasiparticle which was introduced in section 2.2 and the whole diagrammatic expansion discussed in that section is meant to obtain this exact Green's function.

20 Chapter 2. Some basic diagrammatics

Just as the simple non-interacting propagator lines of Figs. 2.la,b were used in the series of diagrams representing the propagation of an interacting electron, the so called non-interacting Green's function is used to calculate the contribution of a given diagram to the above defined exact Green's function. This non-interacting Green's function G0(r1t1 ; ,-St2) can be defined in the same way as the exact Green's function,

however with the exact groundstate replaced by the non-interacting groundstate l<I>~), which is the state obtained by placing N electrons in the lowest N eigenstates of a single-particle Hamiltonian.

-i (

<I>~

IT( î/l( i1' t1)1f

t (

,-s, t2)) 1

<]}~)

(2.6)

-i(<I>~IB(t1

-t2)1f(i1,t1)1ft(,-s,t2) - B(t2 -

t1)1ft(f2,t2)î/J(i1,ti)

l

<I1~)

.

In order to understand the content of this unperturbed Green's function it is conve-nient to decompose the field operators î/J and 1f

t.

The creation operator î/J

t

is \vritten as the sum of two parts, one of which creates an electron in a conduction band state whereas the other part creates an electron in a valence band state. Another way of saying this is, that 1f

t

is an operator which can create an electron in a conduction band state and which can annihilate a hole in a valence band state. For the anni-hilation operator î/J the sa.me kind of decomposition can be made, showing that î/J

annihilates electrons in a conduction band state and creates holes in a valence band state. According to this decomposition the term in the unperturbed Green's function of Eq. 2.6 with t1

>

t2 describes the propagation of an extra electron in a conduction

band state which is created in addition to the groundstate at time t2 by the electron

creation part of

iP

t

and subsequently annihilated at time t1 by the electron

annihila-tion part of

if.

Similarly the term with t2

>

t1 describes the propagation of an extra

hole which is created at time t1 by the hole creation part of î/J and annihilated by the

hole annihilation part of î/J

t

.

The eigenstates </>.,. and corresponding eigenvalues e.,. of a single-electron Hamil-tonian

Ho

can be used to give an explicit expression for the unperturbed Green's function:

Go(i1t1;r"St2)

=

(2.7)

L

-iO(t1 - i2)ef>.,.(ri)</>~{r"S)e-ien(!1-t2)

+

L

iO(t2 - i1)r/>n(i1)</>~(r"S)e-Î<n(t1-t2),

n=occ. n=U7l.OCC.

with

(2.8)

In Eq. 2.7 occ. and unocc. stand for the occupied and the unoccupied states in the groundstate respectively. We can Fourier transform Eq. 2.7 from the time domain to

2.3. The formulas 21

the frequency(energy) domain (see Eq. B.l of Appendix B) yielding

(2.9)

where we have used the Bloch theorem which states that in a periodic solid the eigenstates of a single-electron Hamiltonian can be labeled by a

k

vector in the lBZ and a bandindex l which enumerates eigenstates at a given

k.

In Eq. 2.9 €F is the

energy of the highest occupied state; 1) is an infinitesemally small positive number and

the sgn-function in the numerator takes the value 1 if its argument is positive and -1

otherwise. One specific (l,

k)

combination in Eq. 2.9 describes the propagation of an extra electron in conductionband state (l, k) if l is a conductionband index and the propagation of a hole in valenceband state (1,

k)

if 1 is a vaJenceband index. These two possibilities correspond to the symbols in Fig. 2.1 a and Fig. 2.1 b respectively.

If we assume the solid to be very large the discrete summation over

k

becomes an integration over the lBZ (see Eq. B.5) so that Eq. 2.9 becomes

(2.10) The graphical symbol for the unperturbed Green's function is shown in Fig. 2. 7. As shown in this figure it combines in one symbol the propagation of an electron and of a hole.

If we insert in Eq. 2.9 the plane wave expansion for the wavefunctions which reads

,i. -(f'I __ l_ " d -(J{)ei(k+K)-r

'l'lk ' J - yTI ~ Ik ' .R

(2.11)

with

R

a reciprocal lattice vector, and use Eq. B.8 we arrive at the momentum representation of the Green's function

(2.12) The numerator of one term in the RHS of Eq. 2.12 in combination with Eq. 2.11 makes clear how the momentum representation Green's function describes the propagation of an extra electron or hole in terms of the propagation of the infinitely many different momentum components present in the wave function of state 1. This is depicted in Fig. 2.8 which shows the propagation of the component of the wavefunction with momentum

k

+

f<

2 and its scattering to the component with momentum

k

+

J?1

due

22 Chapter 2. Some basic diagrammatics

(b:'

t

l1 < l2 electron t 1 t 1 > t2 or t 1 < t2

=

,,:· t

l1 > l2 hole l2 (a) Figure 2.7

(a) Graphical representation of the Green's function defined in Eq. 2.6. No specific time order exists between t1 and t2 • In fact the single symbol in ( a) stands Jor the

symbol in (b) or for the symbol in ( c) depending on the relation between t1 and t2

to an interaction with the lattice. Due to the Bloch property the momentum can only change by a reciprocal lattice vector and the lBZ part (k) of the momentum is conserved. In the complete expression for a diagram, of which Fig. 2.8 will be apart, a summation has to be performed over all involved reciprocal lattice vectors, such as the

R

1 and

K

2 in Eq. 2.12, as far as they are not fixed by momentum conservation.

According to Eq. 2.12 the different terms in the summation will be weighted by the product of plane wave coefficients d1f(I(i)d;f(I(2).

2.3.3

The interaction and Green's function put together.

The full expression corresponding to a given diagram in the time-position repre-sentation is obtained by assigning to the propagators in the diagram the function

iGo(i1t1 ;T'2t2) with i2 t 2 the beginpoint of the propagator and i1t1 its endpoint. Fur-thermore we have to assign to each interactionline the function -iV (Eq. 2.1) with begin- and endpoint as arguments. The thus obtained expression has to be multiplied by -i(-2)NFL in which NFL is the number of closed loops in the diagram. The factor

(2)NFL accounts for the spin summation that has to be performed for each closed loop and the factor ( -1 )NFL is typical for the fermion character of the electrons. Finally we have to integrate over all internal space and time coordinates, which yields an expression depending on the external coordinates only.

2.3. The formulas Figure 2.8 electron if lis a conductionband index Jwle if lis a valenceband index 23

Representation of the Green's function in the momentum representation. The

wig-gly line represents the interaction of the electron (hole) with the ion lattice. Due to this interaction it can change its momentum by a reciprocal lattice vector. The l at-tached to the symbol for the Green's function is a band index. The two time orders in

the single symbol of Fig. 2. 7a correspond to the two indicated possibilities f or l.

As an illustration we give the expression corresponding to the vertex correction diagram of Fig. 2.5b :

Mvc(i1t1; i2t2)

=

-1

j

d3r3dt3

j

d3r4dt4Go(f3t3; ?71i1)Go(i,,t4; f3t3)Go(f2t2; i,,t4)

xV(i1t1; i,,t4)V(i2t2; ijt3) (2.13)

In practice it is far more convenient to work in the energy-momentum representation

which can be obtained by the inserting in Eq. 2.13 the Fourier expansions for the

interactions and the Green's functions. The resulting energy and momentum depen-dence of the Fouriertransform of the above expression for Mvc can be considered

separately and we will first discuss the energy dependence.

The energy denominator

Performing the integrations over the intermediate times t3 and t4 which are trivial because of the 8-functions in Eq. 2.1 and disregarding space dependencies and

in-tegrations as well as the factor -1 we are left with an expression of the following structure

(2.14)

in which the tildes are used to indicate that we consider only the time dependence of the involved quantities. Mvc is seen to depend only on the difference t2 - t 1

=

t

24 Chapter 2. Some basic diagrammatics

transform of lvlvc

Ûvc(w)

=

j

dteiwt

j ~

1

e-iw,tÓo(w1)

j ~

2

e1_""'tÓo(w2)

j ~

3

_e-1"'31_{Óo(w3). (2.15)} The time integral yields 211"8(w - w1

+

w2 - w3 ) which makes for example the w2 in tegration trivia\:

(2.16) Inserting in this equation the Green's function of Eq. 2.9 (we ooly coosider its energy dependence) the following expression is obtained:

-

J

dw1

J

dw3 Mvc(w)

=

~ ~ 1 w1 - e11 (k1)

+

i17sgn(e1, (k1 ) - t:F) 1 x - -w1

+

W3 - w - e12(k2)

+

i71sgn(e12(k2) - éF) 1 x - - ' (2.17) w3 - t:13(k3)

+

i17sgn(e13(k3) - eF)

in which we have taken in to account one(/;, k;) combination for each Green's function. Usiog Cauchy's theorem to perform the w1 and w3 integration we finally find

Üvc(w) = (2.18)

[B(êF -ê11(k1))-B(êF -ê12(k2))][ll{sgn(ó11(k1)-óF)- sgn(ó12(k2)-óF)}-B(óF -€1,(k3))]

w - ó'/1 (fi) + ê12 (k2) - ê1,(k3) + iry(sgn(i:1, (k1) - óF) - sgn(ê12(k2) - êF) + sgn(ê1, (k3) - êF ))

Since in a semiconductor or insulator all bands are either completely filled or empty the fJ functions in the numerator put no restrictions on the possible

k

values but only restrict the baodindices to certain ranges. Working out these restrictions we are left with the following two combinations of bandindices which correspond to the possible intermediate states discussed in connection with Fig. 2.5

1. 11 E CB (Conduction bands), /2 E VB (Valence bands) and 13 E CB, which

is the 2-electron-1-hole intermediate state corresponding to t2

>

t1 in Fig. 2.5.

The energy denominator is the difference between the energy parameter w and the energy e11 (ki) - e12 (k2 )

+

éz3 (k3) of the intermediate state. The numerator

in Eq. 2.18 becomes -1 and the denominator cootains an infinitesemal +i17

2. 11 E VB ,12 E CB and /3 E VB correspondîng to the 1-electron-2-hole interme -diate state occuring if t2

< t

1 in Fig. 2.5. The energy denomioator in this case îs the oegative of the difference betweeo -w and the intermediate state energy -e11(k1)

+

e12(k2) - t:13(k3) and contains an iofinitesemal -i71. The numerator

2.3. The formulas 25

In the final expression corresponding to the diagram a summation over the three bandindices bas to be performed and we see that the energy of the intermediate state can take any value allowed by the spectrum of eigenvalues. If €11 (ki) - e12

(k

2)

+

e13

(k3 )

f:.

w the intermediate state is called a virtual intermediate state and if the

energy of the intermediate state equals w we speak of a real intermediate state. The relation

- 1 -.

=

P(].) =t= i1r6(x ).

x

±

lTJ x (2.19)

shows that the occurence of a real intermediate state gives rise to an imaginary part, i.e. a non-Hermitian part, in the expression corresponding to this diagram. This explains from a mathematica! point of view the remark in section 2.2 that the possibility of a real intermediate state gives rise to lifetimes.

The momentum dependence

If we insert in Eq. 2.13 the Fourier transform Eq. 2.4 for the interactions and Eq. B.11 for Green's functions it is easily seen that we get a plane wave factor (éf·r with ij a sum of lBZ vectors and reciproca.l lattice vectors) at ea.ch internal vertex. The subsequent integration over the int.erna! space coordinates generates the requirement of momentum conservation at each internal vertex. Using Eq. B.4 to obtain the mo-mentum representation of Eq. 2.13 generates likewise the momo-mentum conservation at external vertices. The momentum representation expression of an arbitrary diagram is therefore transparent if we consider it as a graph through which an external mo-mentum fiows in such a way that the sum of all outgoing momenta at a vertex equals the sum of all inflowing momenta at the same vertex. Consider, for example, the ver-tex correct.ion diagram as it is shown in Fig. 3.6< of Chapter 3 and the corresponding expression in Eq. 3.11. The figure shows how an external momentum

k

+ !?

2 flows into the diagram and divides at the vertex into two flows. (In fact we should use

n(k

+

R

2 ) when speaking of momentum but we will ignore the factor 1i in the

follow-ing) The flow in to the Coulomb interaction line carries a momentum

k

-

q

1

+

i(2 -

Q

1

corresponding in Eq. 3.11 to the factor

l/

l

k -

q1 -

Q

1

+

!?

2

1

2 , which is the momentum

representation of the Coulomb interaction. The remaining momentum

ifi

+

Q

1 flows

in to the propagator line. Corresponding to Eq. 2.12 and Fig. 2. 7 this yields a plane wave coefficient dj191

(Q

1 ), in Eq. 3.11. Note that the complex conjugating symbol

is missing in that equation since the plane wave coefficients can be chosen real in our calculation. The momentum component ij+

Q

1 is subsequently scattered due to