Spin-Resolved Angle-Resolved Photoemission Spectroscopy:A Brief Overview

Austin de St Croix, Sydney Dufresne, Sebastian Gitt, Ryan Roemer

For PHYS 502, Dr. Mona Berciu, UBC

Submitted November 23rd 2018

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1 Introduction

Angular-resolved photoemission spectroscopy(ARPES) is an experimental technique that allowsfor the determination of the electronic band struc-ture of a material. This is typically done by illumi-nating the sample with ultraviolet light and measur-ing the angle of the emitted photoelectrons, which isproportional to their crystal momentum. The mostcommon sources of high energy light for ARPES aresynchrotron radiation and vacuum ultraviolet lasers,each of which have unique advantages and disadvan-tages. By polarizing the light from these sources,spin-dependent probing can be achieved in materi-als to investigate whether bands are spin-degenerate.In materials that lack inversion symmetry, the va-lence bands can split due to spin-orbit coupling,which can be detected with spin-resolved ARPES(SR-ARPES). This technique is an effective toolwhich can probe exotic materials such as topologi-cal insulators, which have nontrivial spin-polarizedsurface states.

2 Theory

2.1 ARPES

Photoemission spectroscopy (PES) experiments arebased on the photoelectric effect, in which electronsare ejected from a sample using high energy radia-tion. This technique enables one to probe the elec-tronic structure of materials by relating the energyand momentum of the photoemitted electron to theenergy and momentum the electron had in the sam-ple. Suppose an electron is excited by a photon withenergy hν into a state with momentum ~k. The elec-tron is ejected from the sample, and measured invacuum as having energy εk and momentum ~K atan angle θ (Figure 1) relative to the surface normal.The kinetic energy of the photoemitted electron sat-isfies εk = hν−φ−|EB| where φ is the work functionof the material, and EB is the binding energy relativeto the Fermi level EF (EB = E(~k)− EF ).

Conservation of momentum applies to the par-allel components of momentum due to translationalsymmetry at the surface of the material [1] (see Ap-pendix for discussion of perpendicular components).

k|| = K|| =

√2mεkh

sin(θ) (1)

Figure 1: Schematic of APRES geometry. Emittedphotoelectron direction is specified by (θ, φ) [2].

In experiments, the photoemitted electron inten-sity (or photocurrent) for a given range of energiesand angular distributions is the quantity being mea-sured by ARPES. Starting with a N-electron system,incident radiation will excite an electron in a single-particle state φi to a state φf , where we separate

the remaining (N-1)-electron states before (Ψ(N−1)i )

and after (Ψ(N−1)f ) removal of the electron. The to-

tal transition probability per unit time, associatedwith the time-dependent perturbation Hint ∝ ~A · ~p,is given by Fermi’s Golden Rule (see Appendix forderivation)

I = Σi,fwi→f (2)

where

wi→f ∝ Σs|〈Ψ(N−1)f,s |Ψ(N−1)

i 〉 |2|Mi,f |2δ(Ef − Ei − hν)(3)

where |Mi,f |= |〈φf (εk,~k)| (~ε · ~r) |φi(~ki)〉 | is theARPES matrix element [1]. The final (N-1)-electron state may contain a range of electron and

quasiparticle excitations (Ψ(N−1)f,s ) depending on the

strength of electron-electron, electron-phonon, etc.,interactions in the system. All of these interac-tions are described by a continuous function knownas the spectral function A(~k, ω) (related to the

|〈Ψ(N−1)f,s |Ψ(N−1)

i 〉 | term in Eq. 3)

A(~k, ω) =1

π

Im(Σ(~k, ω))

(hω − ε0k − Re(Σ(~k, ω)))2 + Im(Σ(~k, ω))2

(4)

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Σ(~k, ω) is known as the self-energy. The self-energyis a complex function that has real and imaginaryparts corresponding to a shift in the electron en-ergy in the band ε0k due to interactions, and the life-time of the quasi-particle, respectively [3]. In thecase of non-interacting systems, the self-energy iszero and the spectral function reduces to a deltafunction centered at the band energy ε0k. In theinteracting case close to the Fermi surface (whereε0k − µ >> |Im(Σ(~k, ω))|, as in the case of a Fermiliquid), one can then expand the spectral intensity(Eq. 13) about the poles. The singular and non-singular terms of this expansion correspond to thecoherent (main line, quasiparticle peak) and inco-herent (satellite parts of the spectral intensity), re-spectively.

Figure 2: Spectral distribution peaks as a functionof wavevector ~k and energy E near the Fermi energy.The in coherent part of the spectrum broadens, whilethe quasiparticle lifetime grows as the Fermi energyis approached [2].

Spectral intensity (I(~k, ω) ∝ Σi,f,s|Mi,f |2A(~k, ω))

measured as a function of momentum ~k is known asan energy distribution curve (EDC). Peaks in thesecurves correspond to high photoelectron densitiesindicating the center of an electron band, while thewidths of these peaks correspond to the lifetimes ofquasiparticles, as shown in Figure 2.

2.2 Spin Polarization

SR-ARPES experiments take advantage of the spin-dependence in scattering experiments to characterizethe spin-polarization P (~k, ω) of electrons with mo-

mentum ~k measured with the spectrometer:

P (~k, ω) =I↑(~k, ω)− I↓(~k, ω)

I↑(~k, ω) + I↓(~k, ω)(5)

where I↑↓(~k, ω) is the SR-ARPES spectrum corre-sponding to a particular spin axis [4].

3 Light Source

Synchrotron radiation is a common source of light forSR-ARPES experiments. Beams can be produced bythe cyclotron-motion of electrons about a constantmagnetic field along a circular track. Another com-mon technique is the use of undulators, which areperiodically spaced magnets with fields in opposingdirections, causing the electrons to oscillate as theytravel down a path [5]. Diagrams of the two dif-ferent types can be seen in Figure 3. In both cases,this acceleration causes the electron to emit photons,which are sent down beam lines to be used for ex-periments. Synchrotrons produce photons with anenergy range of approximately 20-200 eV, but a grat-ing monochromator can be used to select narrowerbands of frequencies [6]. Any arbitrary polarizationcan be achieved from synchrotrons using wave-plates[7].

Figure 3: Diagrams of the bending magnet (top)and wiggler (bottom) synchrotron designs, as wellas their corresponding output [8].

Another common source of light for SR-ARPESare solid-state vacuum ultraviolet lasers, which arecapable of providing photon energies of around 5-7eV, exceeding the work function of most materials[9]. Typically, frequency quadrupling of Ti:sapphirelasers (through non-linear effects in crystals such asβ-barium borate) are used to produce photon ener-gies of 6 eV [10]. Unlike synchrotron sources, these

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lasers are not tunable, but they do have a muchhigher energy (momentum) resolution due to theirlower photon energy. Laser-based ARPES systemshave been demonstrated with photon energy resolu-tion as low as 0.26 meV, while state-of-the-art syn-chrotron sources only reach as low as 4 meV [11][9].Another advantage of using lower energy photons isthat it restricts emitted electrons to the first Bril-louin zone, reducing background noise due to scat-tering.

Furthermore, lasers are able to provide photonfluxes on the order of 1015 photons/s, which is atleast two orders of magnitude higher than fluxes pro-vided by modern synchrotron sources. This greatlyimproves the signal-to-noise ratio [11]. Both laserand synchrotron sources are capable of achievingspot sizes down to 0.2 mm [9].

If too many photoelectrons are ejected simulta-neously, they will experience a Coulombic repulsionknown as the space charge effect, introducing anuncertainty into the final momentum measurement[13]. For this reason, Ti:sapphire lasers use pulses ofa few picoseconds, as opposed to tens of femtosec-onds, which is a limitation in studying time-resolvedelectron dynamics. Systems using synchrotrons ex-perience much greater space charge effects, whichcan be attributed mostly to their larger energy band-width [9].

Figure 4: Inelastic mean free path vs. electron en-ergy relation that is approximately valid for mostmaterials [12].

Another important consideration is the inelas-tic mean free path (IMFP) of the emitted elec-tron, which describes how far an electron will travelthrough a solid before losing its kinetic energy dueto collisions, resulting in the formation of electron-hole pairs, phonons, and plasmons [14]. The mean

free path is approximately the same for different ma-terials, instead varying primarily due to the kineticenergy of the electron [12]. As seen in Figure 4, themean free path is the smallest for energies in the 20-100 eV range, which is the main operating range ofsynchrotron sources.

Ultraviolet lasers are therefore capable of pene-trating further into the material, with depths up to3-10 nm, while synchrotrons are only go to depthsof 0.5-2 nm, limiting them to surface states [9].

4 Detector

Compared to traditional ARPES, spin resolved mea-surements must sacrifice some energy and momen-tum resolution to resolve the photoelectron’s spin inaddition to ~K and E. This is due to the more com-plex electron detector which typically involves theelectrons undergoing an additional scattering eventbefore detection.

The following discussion pertains to both tradi-tional and spin resolved ARPES. In this low energyphoton-electron scattering there is no biasing be-tween the lab frame and center of momentum frame[15]. Because of this, the detector is usually rotatedrelative to the sample at a fixed radius. A sketch ofthe experimental setup is shown in Figure 5, wherethe electron detector measures the scattering angleθ in the mirror plane. It is important that the sam-ple be properly aligned with respect to the incomingphotons such that φ and θ properly reflect the scat-tering angles. If not, the components of K|| will berotated and erroneous in magnitude (|K|||∝ sin θ).

Figure 5: A SIMION simulation showing how elec-trons with different energy behave in the half-sphereelectron detector. The outer shell is at negative po-tential while the inner shell is positively charged.Taken from [16].

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spa

Figure 6: Left: The incoming electrons scatter to either side depending on their spin coupling with theinduced ~B field. Right: The Mott effect is dominant at θ ≈ 120o in the differential cross section, thisexample for 50 keV electrons impinging on gold. Image adapted from [18].

Upon entering the detector the electrons are first col-limated using a series of electrostatic lenses (analo-gous to focusing light)1. A half sphere electron de-flector is then used to select the energy of the photo-electrons. The outer and inner spheres (with radiusRout and Rin) are held at potential difference ∆V .Highly energetic electrons will not be sufficiently de-flected while low energy electrons are attracted tothe inner sphere (at positive potential). The energyof surviving electrons E, and energy resolution ∆Eare given by the formulas below, where the slit widtha = Rout−Rin and ∆θ is the polar acceptance angleof the detector (with respect to the sample) [17].

E =e∆V

RinRout− Rout

Rin

∆E = E

[2a

Rin +Rout+ (

∆θ

2)2

]Before entering the deflector, a retarding potentialcan be applied to lower E and improve the energyresolution. The expression above for ∆E is for aperfectly monochromatic light source, but in realitythe bandwidth of the laser must be convolved withthe detector’s resolution to give a total energy reso-lution. The momentum resolution is approximately

given by ∆k|| ≈√

2mE/h2 cos θ∆θ. In ARPES acontinuous dynode or multi-channel electron multi-plier (electron counter) will be placed at the exitingend of the deflector to count the electrons.

In spin resolved measurements an additionalcomplexity is required before counting electrons.

The most common technique2 uses Mott scattering,a spin-orbit interaction, to resolve spins. After pass-ing through the deflector described above the elec-trons are accelerated to ∼ 100keV before impingingupon a thin film of high Z material (typically a noblemetal, such as gold). The electrons penetrate to themetal’s nucleus and see a net positive charge whichinduces a magnetic field [18].

~B = −1

c~v × ~E =

Ze

cr3~r × ~v =

Ze

mcr3~L (6)

Where ~E = Zer3~r was used. The electron’s spin will

couple with this magnetic field through the dipoleinteraction:

VLS = −~µ · ~B =Ze2

2m2c2r3~L · ~S (7)

This spin-orbit potential will pull the electron to oneside or another depending on the spin, as seen inFigure 6. However this effect is weak and cannotbe easily resolved at all scattering angles or ener-gies. The effect becomes more prominent at higherenergies and when the electrons back-scatter off thefoil at about θ ≈ 120o. At low energy the electroncloud shields the nucleus reducing the Mott effect,and for softer scattering angles the electron is likelyto multi-scatter. As θ approaches 180o ~L decreases,also weakening the spin-orbit potential VLS . At veryhigh energies brehmsstrahlung and other effects oc-cur, so these detectors must operate within theselimits to minimize other non-Mott scattering effects.

1It is interesting to note that the particle density of the electron beam in the phase space (~r, ~p) is conserved by Liouville’stheorem. Performing collimation will increase the momentum spread but the product of momenta and displacement for thebeam population will remain constant

2Very Low Energy Electron Diffraction, VLEED, is becoming more popular in the field. It utilizes spin exchange inter-actions with spin-polarized copper-oxides to resolve spin. It works with much lower voltage and has a better signal to noiseratio compared to Mott detectors.

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Two electron counters are placed at ±120o fromthe gold foil, each one measuring a given spin po-larization. This second scattering event reduces thetotal number of electrons measured by a factor of10−3−10−4 leading to a reduced intensity. A retard-ing potential can be added to deflect multi-scattered(lower energy) electrons which allows the detectorsto be made smaller as lower potentials are needed.SR-ARPES detectors operate with energy resolu-tions on the order of 10 meV and angular resolu-tion of about one degree, although it varies greatlyfrom one system to another. This detector providesa precise method to measure the complete state ofelectrons near the surface of materials.

5 A Motivating Example

Topological states are interesting because the pre-served time reversal symmetry (the Hamiltoniancommutes with the time reversal operator) at thesurface guarantees conductance due to the elimina-tion of back-scattering. Topological states are real-ized when Kramer’s degeneracy is broken, implyinga splitting in the energies of spin states predictedby Kramer’s theorem. This theorem states that spin12 particles will have degenerate energy eigenstates.This energy splitting may be complemented by mo-mentum dependent splitting known as the Rashbaeffect, where spin-orbit coupling and asymmetry inthe crystal structure (specifically in the directionperpendicular to the proving surface) are responsi-ble.

SR-ARPES allows for the direct probing of thespin polarization of energy bands with high momen-tum resolution. Because of the net zero spin polar-ization of the system it is difficult to show any spindependence. This technique proves extremely usefulwhen considering the spin states in such materialswhere the surface or edge bands are distinct fromthat of the bulk. In these systems it is not only thelinear dispersion of the bands that is unique but theprojection of the spin. Fundamentally, this is due tothe spin orbit interaction inverting the bulk bands,which causes an energy gap.

In addition to spin-orbit coupling topologicalstates are often realized in materials with small bandgaps. Significant change to the electronic structureis brought about via the spin-orbit interaction typi-

cally introduced as a small perturbation. Thus oneshould look low on the periodic table for constituentcandidates for topological materials, where the rela-tivistic effect of spin-orbit coupling is greatest. Herewe discuss an archetypal compound, Bi2Se3 to de-velop an idea of the insight gained from SR-ARPES.

5.1 Bismuth Selenide Bi2Se3

Hexagonal Bi2Se3 with a band gap of ∼220 meVhas been suggested to host topologically non-trivialstates centered around the gamma point (center ofthe Brillouin Zone). In this region the boundary be-tween the topological trivial vacuum states (Chernnumber is equal to zero) and the non-trivial bulkstates (Chern number is non-zero) gives rise to non-trivial surface states where Kramers degeneracy islifted. A toy description of these bands can be seenin Figure 7 where a cut through the gamma pointreveals that the bulk valence band and bulk conduc-tion bands are gapped, but the topological surfacestates are continuous. These spin-split states willagain be degenerate at the Dirac point where thespin up and spin down bands overlap.

Figure 7: A toy model of the electronic structure ofBi3Se3. The bulk conduction band (BCB) and bulkvalence band are gaped where the topological sur-face state (TSS) is present within the bulk energygap. Note the m-shape of the BVB which is a telltale sign of band inversion [19]. At this point ignorethe colour of the bands.

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Figure 8: a) Spin-resolved spectra measuring the equilibrium states of Bi2Se3 taken along kx = 0 withhas no spin resolution. The color bar represents intensity of photo-emission. (b) The same spectrumas in a), but with ”pump” which optically excites electrons into previously unoccupied states above theFermi energy. (c) SR-ARPES spectra of after the optical pumping, the two dimensional colour schemerepresents the intensity where bold (faded) corresponds to high (low) intensity and colour corresponding tospin polarization asymmetry measured in the y direction following a cut through the gamma point along~kx [19].

With this picture we can imagine probing stateswith ARPES to construct such a band map through-out the Brillouin zone. Utilizing spin resolution wecan gain more information about each band. Specif-ically we will be able to determine what spin statescomprise the bands if there exist such spin polariza-tion. Thus probing energy eigenstates outside of thegamma point one expects to see an asymmetry ofspin up and spin down states.

Indeed spin polarization is experimentally ob-served. Jozwiak et al. performed SR-ARPES onBi2Se3 which confirmed a spin polarized band struc-ture of the TSS within the bulk gap. Data can beenseen in Figure 8, which confirms the existence oftopological surfaces states below EF with ARPES.This measurement allows one to picture the rela-tively high density of states of the TSS bands com-pared to the BCB.

It is often beneficial to use an an optical pump toexcite electrons into states above EF which yieldsinsight into the dispersion of the TSS and their rel-ative intensities, data of such a measurement is pic-

tured in Figure 8. Most importantly spin resolvedARPES was used to detect the spin polarization ofeach band. As expected from a topological material,the TSS are spin split as indicated by the red-bluecolour map.

Another way to visualize c) in Figure 8 is throughthe intensity vs ~kx vector as shown in Figure 9. Herethe asymmetry in spin intensity (red(blue) trianglesindicating spin up(down) in the y basis) is preciselyshown as the difference in the two sets of data, whichcan then be related to the intensity of the dispersioncurves in Figure 8. These cuts across the gammapoint are also above the Dirac point as pictured inFigure 8 b). If the measurements are carried outwhile probing at lower energy (below the Dirac pointpictured in b)) one would observe spin polarizationopposite to that observed above the Dirac point. Inother words the positive ~kx values would be red. Asan aside the hexagonal Brillouin zone is pictured in-dicating a hexagonal Wigner-Seitz cell as expected.

The spin states of various ~k values can be de-duced from the spin intensity data along any cut

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through the gamma point i.e. if a cut is made inany direction through the gamma point in the x,yplane in any direction oppositely polarized states inthe TSS bands are observed. Thus spin locked statescan be pictured as states where the spin polarizationis orthogonal to the k vector at all points along theFermi surface. These states are commonly referredto as helical Dirac states.

The utility of SR-ARPES has been demonstratedusing a topologically insulating material which canbe used to realize new physics in many condensedmatter systems. This technique allows researchersto observe the spin helical structure caused by bandinversion which splits spin up and spin down statesinto individual bands.

Figure 9: A Spin-resolved spectra measuring the asymmetry in spin polarization along a cut along kx

through the gamma point. Spin polarization is measured in the y direction. The spin dependent statescan be deduced from the spin intensity data which demonstrates the spin helical surface states [20].

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References

[1] S. Hufner. ”Photoelectron Spectroscopy”, Springer Series in Solid State Sciences, pp 244–284 (1995)

[2] A. Damascelli. ”Probing the Electronic Structure of Complex Systems by ARPES”, Physica Scripta,61 (2004)

[3] S. Suga, A. Sekiyama. ”Photoelectron Spectroscopy”, Springer Berlin Heidelberg, pp 7–31 (2014)

[4] E. Beaurepaire, H. Bulou, F. Scheurer, J. P. Kappler. ”Magnetism: A Synchrotron Radiation Ap-proach”, Springer Berlin Heidelberg, pp 97–103 (2006)

[5] Settimio Mobilio, Federico Boscherino, Carlo Meneghini. ”Synchotron Radiation” Springer (2015 )

[6] Riccardo Comin, Andrea Damascelli. ”ARPES: A Probe of Electronic Correlations” Springer BerlinHeidelberg (2015)

[7] K.C. Westfold. ”The Polarization of Synchrotron Radiation” Astrophysical Journal 130 (1959)

[8] Giuseppe Dattoli et al. ”Linac-based free electron laser devices: oscillator and single passage operatingmodes” Charged Beam Dynamics, Particle Accelerators and Free Electron Lasers, IOP Publishing.(2017)

[9] Wentao, Zhang. ”Photoemission Spectroscopy on High Temperature Superconductor” Springer (2013)

[10] J.D. Koralek et al. ”Experimental setup for low-energy laser-based angle resolved photoemission spec-troscopy” Review of Scientific Instruments, vol. 78 (2007)

[11] Guodong Liu et al. ”Development of a vacuum ultraviolet laser-based angle-resolved photoemissionsystem with a superhigh energy resolution better than 1meV” Review of Scientific Instruments, vol79. (2008)

[12] M.P. Seah, W.A. Dench. ”Quantitative electron spectroscopy of surfaces: A standard data base forelectron inelastic mean free paths in solids” Surface and Interface Analysis (1979)

[13] X.J. Zhou et al. ”Space charge effect and mirror charge effect in photoemission spectroscopy” Journalof Electron Spectroscopy and Related Phenomena (2005)

[14] R.F. Egerton. ”Electron Energy-Loss Spectroscopy in the Electron Microscope” Springer (2011 )

[15] A de St. Croix. ”Xenon Differential Cross-Section: Kinematics” Carleton University (2017)

[16] M. Dogun et al. ”Double Differential Cross-Sections for Electron Impact Ionization of Atoms andMolecules” Journal of Spectroscopy ID 192917 (2013)

[17] A. Damascelli, Z. Shen, Z. Hussain. ”Angle-resolved photoemission spectroscopy of the cuprate super-conductors” (2002) arXiv:cond-mat/0208504

[18] A. Takayami. ”High-Resolution Spin-Resolved Photoemission Spectrometer and the Rashba Effect inBismuth Thin Films” Springer Theses (2015)

[19] D. Hsieh et al.”Spin-polarized surface resonances accompanying topological surface state formation”Nature (2009)

[20] Chris Jozwiak et al. ”Spin-polarized surface resonances accompanying topological surface state for-mation” Nature Communications (2016)

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Appendix

Momentum Relations

The perpendicular component of momentum can be found by accounting for electron diffraction throughthe surface barrier. Let θi be the angle of the electrons wave-vector ~kf with respect to the surface normal.The electron will be transmitted through a barrier V0, which will then decrease the overall perpendicularmomentum component K⊥, meaning the conservation of energy in the perpendicular direction gives

kf,⊥ =

√2m(Ek cos2(θ) + V0)

h2 (8)

Fermi’s Golden Rule

ARPES experiments rely on the photoelectric effect, so we will consider the effects of incident radiation~A(~r, t) on an N-electron system Ψ

(N)i , where Ψ

(N)i is an eigenstate of the unperturbed Hamiltonian. Suppose

the radiation excites one of the N electrons initially in state φi(~ki) to the state φf (εk,~k). The cumulative

initial and final N-electron states can then be written as Ψ(N−1)i φi(~ki), and Ψ

(N−1)f φf (εk,~k), respectively.

The transition probability per unit time between these states in the presence of a perturbation Hint =hemc

~A · ~p is given by Fermi’s Golden Rule as

wi→f ∝ |〈Ψ(N−1)f φf (Ekin,~k)| Hint |Ψ(N−1)

i φi(~ki)〉 |2δ(Ef − Ei − hν) (9)

where hν is the energy is the incident light [1]. In separating the photoelectrons state from the (N-1)-electron final state, we have used what is referred to as the ‘sudden approximation’. This assumes that the

photoelectron is immediately decoupled from the remaining (N-1)-electron state Ψ(N−1)f upon excitation,

before relaxation occurs. Further, we will assume Ψ(N−1)f can be expressed as the sum of all eigenstates

of the perturbed Hamiltonian Ψ(N−1)f,s with energy E

(N−1)s , i.e., Ψ

(N−1)f = ΣscsΨ

(N−1)f,s . Assuming a dipole

approximation ( ~A ∝ eikr, eikr ≈ 1), we can re-write Eq. 9 as

wi→f ∝ Σs|〈Ψ(N−1)f,s |Ψ(N−1)

i 〉 |2|〈φf (εk,~k)| (~ε · ~r) |φi(~ki)〉 |2δ(Ef − Ei − hν)

= Σs|ci,s|2|Mi,f |2δ(Ef − Ei − hν)

where ~ε is the polarization vector of the incident field, Mi,f = 〈φf (εk,~k)| (~ε · ~r) |φi(~ki)〉, and ci,s =

〈Ψ(N−1)f,s |Ψ(N−1)

i 〉. Therefore the total transition rate (often called spectral intensity or photocurrent)I = Σi,fwi→f can be written as

I ∝ Σi,f,s|Mi,f |2|ci,s|2δ(εk + E(N−1)s − E(N)

i − hν) (10)

Eq. 10 applies to atoms and molecules, but for solids, we define a continuous function A(~k, ω), called thespectral function, which accounts for all electron, phonon, and magnon excitations associated with thecreation of a hole with wavevector ~k. It is convenient to re-write Eq.10 as I(~k, ω) ∝ Σi,f,s|Mi,f |2A(~k, ω)

the physical meaning of A(~k, ω) is given by

A(~k, ω) = − 1

πIm(G(~k, ω)) (11)

where ε0k be the one-electron energy, and

G(~k, ω) =1

hω − ε0k − Σ(~k, ω)(12)

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Accounting for the self-energy, the spectral function becomes

A(~k, ω) =1

π

Im(Σ(~k, ω))

(hω − ε0k − Re(Σ(~k, ω)))2 + Im(Σ(~k, ω))2

(13)

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