Tuesday 03.11 Angle-Resolved Photoemission Spectroscopy Monday 9.11 Read: Superconductivity Chapter in Kittel Experimental Surve

Tasks

Tuesday 03.11 Angle-Resolved Photoemission Spectroscopy Prepare: Spectral function Monday 9.11 Read: Superconductivity chapter in Kittel Experimental survey + London & Landau Theory Tuesday 10.11 Student Presentation: Reza – Magnetic excitations David – Spin liquids Monday 16.11 Read: Superconductivity chapter in Kittel Josephson Junctions Tuesday 17.11 Student Presentation: Jens – Vortex, Charge order & SC Lorena – Skyrmions Monday 23.11 Unconventional superconductivity Tuesday 24.11 Student Presentation: Ron – Magic angle graphene & SC Wojtek – Room temperature SC

TODAY Lecture 1: TODAY Lecture 2: (1) Photo-emission spectroscopy (1) Meisner effect and dissipationless transport (2) Spectral function (2) Specific heat (3) Self-energy (3) London Theory Tasks

Magnetic Excitations Literature Spin Liquid Literature DOI: 10.1038/NPHYS4077 doi:10.1038/nature08917 PRL 105, 247001 (2010) https://doi.org/10.1038/s41535-019-0151-6 Organization – Dec

Monday 30.11 Guest Lecture: Fabian Natterer

Tuesday 01.12 Guest Lecture: Thomas Greber Student Presentation: Chuang – Fermi liquids Haunlong – Quantum Oscillations Monday 07.12 Guest Lecture: Marta Gibert Tuesday 08.12 Guest Lecture: Marc Janoschek Student Presentation: Jianfei – Quasi crystals Yoel – Human brain MRI Monday 14.12 Student Presentation: Ugur – SC cubits Lebin – Majorana Fermions Pascal – Eder – Tuesday 15.12 Recap + Exam Prep.

(1) Quantum Oscillations (3) Superconductivity without phonons B. Ramshaw et al., Science 348, 317 (2015) Nature 394, 39 (1998) – Pressure induced superconductivity (2) Unconventional superconductivity Nature 450, 1177 (2007) – Review article Science 336, 1554-1557 (2012) – Penetration depth @ QCP (4) What is the evidence p-wave superconductivity? Starting point: Rev. Mod. Phys. 75, 657 (2003) Nature Physics 11, 17–20 (2015) – SC fluctuations in URu2Si2 Angle-resolved photoemission electron spectroscopy (ARPES)

Electron Spectroscopy course: Spring Semester Photoelectric effect

In 1905, Albert Einstein solved this apparent paradox by describing light as composed of discrete quanta, now called photons, rather than continuous waves. Based upon Max Planck's theory of black-body radiation, Einstein theorized that the energy in each quantum of light was equal to the frequency multiplied by a constant, later called Planck's constant. A photon above a threshold frequency has the required energy to eject a single electron, creating the observed effect. This discovery led to the quantum revolution in physics and earned Einstein the Nobel Prize in Physics in 1921. From wikipedia Angle-resolved photoemission electron spectroscopy (ARPES) ARPES and inverse ARPES ARPES Instrumentation

Laboratory: Helium lamps 20 & 40 eV Lasers 7-9 eV

Synchrotrons: Tunable photon energy 20-1000 eV. Former UZH students

Dr. Moritz Hoesch Prof. Felix Baumberger DESY (Hamburg) Uni-Genf Photon – Energy Spectrum “Two-dimensional” crystal structures

NbSe2 structure Graphene Typical layered oxide structure Damascelli, Hussain, and Shen: Photoemission studies of the cuprate superconductors 477 typically used on ARPES systems equipped with a gas- should be noted that ٌ A might become important at Ϫ1 • discharge lamp) it is only 0.5% (0.008 Å ). If, on the the surface, where the electromagnetic fields may have a other hand, the photon momentum is not negligible, the strong spatial dependence, giving rise to a significant in- photoemission process does not involve vertical transi- tensity for indirect transitions. This surface photoemis- tions, and ␬ must be explicitly taken into account in Eq. sion contribution, which is proportional to (␧Ϫ1) where (2). For example, for 1487-eV photons (the Al K␣ line Ϫ1 ␧ is the medium dielectric function, can interfere with commonly used in x-ray photoemission) ␬Ӎ0.76 Å , the bulk contribution, resulting in asymmetric line which corresponds to 50% of the zone size. shapes for the bulk direct-transition peaks.3 At this A major drawback of working at low photon energies point, a more rigorous approach is to proceed with the is the extreme surface sensitivity. The mean free path for so-called one-step model in which photon absorption, unscattered photoelectrons is characterized by a mini- electron removal, and electron detection are treated as a mum of approximately 5 Å at 20–100 eV kinetic ener- single coherent process.4 In this case bulk, surface, and gies (Seah and Dench, 1979), which are typical values in vacuum have to be included in the Hamiltonian describ- ARPES experiments. This means that a considerable ing the crystal, which implies that not only bulk states fraction of the total photoemission intensity will be rep- have to be considered, but also surface and evanescent resentative of the topmost surface layer, especially on states, as well as surface resonances. However, due to systems characterized by a large structural/electronic an- the complexity of the one-step model, photoemission isotropy and, in particular, by relatively large c-axis lat- data are usually discussed within the three-step model, tice parameters, such as the cuprates. Therefore, in or- which, although purely phenomenological, has proven to der to learn about the bulk electronic structure, ARPES be rather successful (Fan, 1945; Berglund and Spicer, experiments have to be performed on atomically clean 1964; Feibelman and Eastman, 1974). Within this ap- and well-ordered systems, which implies that fresh and proach,ARPES: Three the photoemission- processstep model is subdivided into flat surfaces have to be prepared immediately prior to three independent and sequential steps: the experiment in ultrahigh-vacuum conditions (typically at pressures lower than 5ϫ10Ϫ11 torr). So far, the (i) Optical excitation of the electron in the bulk. best ARPES results on copper oxide superconductors (ii) Travel of the excited electron to the surface. have been obtained on samples cleaved in situ, which, (iii) Escape of the photoelectron into vacuum. however, requires a natural cleavage plane for the material under investigation and explains why not all the The total photoemission intensity is then given by the cuprates are suitable for ARPES experiments. product of three independent terms: the total probability for the optical transition, the scattering probability for the traveling electrons, and the transmission prob- B. Three-step model and sudden approximation ability through the surface potential barrier. Step (i) contains all the information about the intrinsic elec- To develop a formal description of the photoemission tronic structure of the material and will be discussed in process, one has to calculate the transition probability detail below. Step (ii) can be described in terms of an wfi for an optical excitation between the N-electron effective mean free path, proportional to the probability N N ground state ⌿i and one of the possible final states ⌿f . that the excited electron will reach the surface without This can be approximated by Fermi’s golden rule: scattering (i.e., with no change in energy and momentum). The inelastic-scattering processes, which deter- 2␲ N N 2 N N mine the surface sensitivity of photoemission (as dis- wfiϭ ͦ͗⌿ ͉Hint͉⌿ ͦ͘ ␦͑E ϪE Ϫh␯͒, (4) ប f i f i cussed in the previous section), also give rise to a N NϪ1 k N NϪ1 continuous background in the spectra which is usually where Ei ϭEi ϪEB and Ef ϭEf ϩEkin are the initial- and final-state energies of the N-particle system ignored or subtracted. Step (iii) is described by a trans- k mission probability through the surface, which depends (E is the binding energy of the photoelectron with ki- B on the energy of the excited electron as well as the ma- netic energy E and momentum k). The interaction kin terial work function ␾. with the photon is treated as a perturbation given by In evaluating step (i), and therefore the photoemis- e e sion intensity in terms of the transition probability wfi , H ϭϪ ͑A"pϩp"A͒ϭϪ A"p, (5) int 2mc mc it would be convenient to factorize the wave functions in Eq. (4) into photoelectron and (NϪ1)-electron terms, where p is the electronic momentum operator and A is the electromagnetic vector potential (note that the gauge ⌽ϭ0 was chosen for the scalar potential ⌽, and 3 the quadratic term in A was dropped because in the For more details on the surface photoemission effects see for example Feuerbacher et al. (1978); Miller et al. (1996); Hansen linear optical regime it is typically negligible with re- et al. (1997a, 1997b). spect to the linear terms). In Eq. (5) we also made use of 4See, for example, Mitchell (1934); Makinson (1949); Buck- ;(the commutator relation ͓p,A͔ϭϪiបٌ•A and dipole ingham (1950); Mahan (1970); Schaich and Ashcroft (1971 approximation (i.e., A constant over atomic dimensions Feibelman and Eastman (1974); Pendry (1975, 1976); Liebsch and therefore ٌ•Aϭ0, which holds in the ultraviolet). (1976, 1978); Bansil and Lindroos (1995, 1998, 1999); Lindroos Although this is a routinely used approximation, it and Bansil (1995, 1996).

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 476 Damascelli, Hussain, and Shen: Photoemission studies of the cuprate superconductors

gard it has to be mentioned that several speciﬁc experimental methods for absolute three-dimensional band mapping have also been developed (see, for example, Hu¨ fner, 1995; Strocov et al., 1997, 1998). A particular case in which the uncertainty in kЌ is less relevant is that of the low-dimensional systems characterized by an anisotropic electronic structure and, in particular, a negligible dispersion along the z axis [i.e., along the surface normal; see Fig. 3(a)]. The electronic

dispersion is then almost exclusively determined by kʈ , as in the case of the 2D copper oxide superconductors which we shall focus on throughout this paper [note, however, that possible complications arising from a ﬁnite three-dimensionality of the initial and/or ﬁnal states involved in the photoemission process should always be carefully considered (Lindroos et al., 2002)]. As a result, one can map out in detail the electronic dispersion rela-

tions E(kʈ) simply by tracking, as a function of pʈ , the energy position of the peaks detected in the ARPES spectra for different takeoff angles [as in Fig. 3(b), where both direct and inverse photoemission spectra for a single band dispersing through the Fermi energy EF are shown]. As an additional bonus of the lack of z dispersion, one can directly identify the width of the photoemission peaks with the lifetime of the photohole FIG. 2. Energetics of the photoemission process. The electron et al. (iii) Binding Energyenergy distribution produced by incoming photons and mea- (Smith , 1993), which contains information on the intrinsic correlation effects of the system and is formally sured as a function of the kinetic energy Ekin of the photoelectrons (right) is more conveniently expressed in terms of the described by the imaginary part of the electron self- energy (see Sec. II.C). In contrast, in 3D systems the binding energy EB (left) when one refers to the density of linewidth contains contributions from both photohole states inside the solid (EBϭ0 at EF). From Hu¨ fner, 1995. and photoelectron lifetimes, with the latter reflecting Experimental Conditions final-state scattering processes and thus the finite prob- dicular to the sample surface are obtained from the po- ing depth; as a consequence, isolating the intrinsic many- lar (␽) and azimuthal (␸) emission angles. body effects becomes a much more complicated prob- WithinUltra the- noninteractinghigh vacuum electron picture, and by lem. taking advantageNo magnetic field of total energy and momentum conser- Before moving on to the discussion of some theoreti- vation laws (note that the photon momentum can be cal issues, it is worth pointing out that most ARPES neglected at the low photon energies typically used in experiments are performed at photon energies in the ARPES experiments), one can relate the kinetic energy h Ͻ and momentum of the photoelectron to the binding en- ultraviolet (in particular for ␯ 100 eV). The main rea- son is that by working at lower photon energies it is ergy E and crystal momentum បk inside the solid: Momentum and Energy ConservationB possible to achieve higher energy and momentum reso- Ekinϭh␯Ϫ␾Ϫ͉EB͉, (1) lution. This is easy to see for the case of the momentum resolution ⌬k which, from Eq. (2) and neglecting the ϭ ϭ mE ʈ pʈ បkʈ ͱ2 kin•sin ␽. (2) contribution due to the finite energy resolution, is

Here បkʈ is the component parallel to the surface of the 2 ⌬kʈӍͱ2mE /ប cos ␽ ⌬␽, (3) electron crystal momentum in the extended zone kin • • scheme. Upon going to larger ␽ angles, one actually where ⌬␽ corresponds to the finite acceptance angle of probes electrons with k lying in higher-order Brillouin the electron analyzer. From Eq. (3) it is clear that the zones. By subtracting the corresponding reciprocal- momentum resolution will be better at lower photon en- lattice vector G, one obtains the reduced electron crystal ergy (i.e., lower Ekin), and for larger polar angles ␽ momentum in the first Brillouin zone. Note that the per- (note that one can effectively improve the momentum pendicular component of the wave vector kЌ is not con- resolution by extending the measurements to momenta served across the sample surface due to the lack of outside the first Brillouin zone). By working at low pho- translational symmetry along the surface normal. This ton energies there are also some additional advantages: implies that, in general, even experiments performed for first, for a typical beamline it is easier to achieve high-

all kʈ (i.e., by collecting photoelectrons at all possible energy resolution (see Sec. II.E); second, one can com- angles) will not allow a complete determination of the pletely disregard the photon momentum ␬ϭ2␲/␭ in Eq. total crystal wave vector k [unless some a priori assump- (2), as for 100-eV photons the momentum is 3% tion is made for the dispersion E(k) of the electron ﬁnal (0.05 ÅϪ1) of the typical Brillouin-zone size of the cu- states involved in the photoemission process]. In this re- prates (2␲/aӍ1.6 ÅϪ1), and at 21.2 eV (the HeI␣ line

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 (ii) Surface Sensitive Technique

Sudden approximation = no scattering / interaction events ARPES intensity

Fermi Dirac Function:

Matrix Element: = =

Spectral function: Poly-crystalline Gold

Fermi Dirac Distribution

(convoluted with instrumental resolution) NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-05715-2 ARTICLE

2 a bcdCut #1 E D Cut #2 Cut #2 0.0 c pol c pol 1 –0.5 ) a / (eV) π 0 F ( E

y A - k –1.0 D E

Cut #2 –1 Cut #1 B –1.5 A C E Matrix Elements B C –2 –2 –1 0 12 482 –1 0 1 Damascelli,–1 0 Hussain, 1 Intensity and (a.u.) Shen: Photoemission studies of the cuprate superconductors Diagonal Cutkx (π/a) k|| (π /a) k|| (π /a) 482 Damascelli, Hussain,ef and Shen:Cut #1 Photoemission studiesgh of the cuprate superconductorsCut #1 Cut #2 dx 2–y 2 0.0 k p pol s pol son, 1977). In turn, this implies that (␧"x)͉␾i ͘ must be k A k E1 son, 1977). In turn, this implies that (␧"x)͉␾i ͘ must be even. In the case depicted in Fig. 5(a) where ͉␾i ͘ is also –0.5 A k even. In the case depicted in Fig. 5(a) where ͉␾i ͘ is also even, the photoemission process is symmetry allowed d 2 z (eV) E even, the photoemission process is symmetry allowed F 2 for A even or in-plane (i.e., ␧p x depends only on in- E B • for A even or in-plane (i.e., ␧p x depends only on in- - –1.0 B •

E plane coordinates and is therefore even under reflection Max plane coordinates and is therefore even under reflection withIntensity (a.u.) respect to the plane) and forbidden for A odd or with respect to the plane) and forbidden for A odd or –1.5 normal to the mirror plane (i.e., ␧s•x is odd as it depends C normal to the mirror plane (i.e., ␧s•x is odd as it depends on normal-to-the-plane coordinates). For a genericC ini- Min tial state of either even or odd symmetry with respect to on normal-to-the-plane coordinates). For a generic ini- –1 0 1 the mirrorDistance plane, along thethe cut polarization (a.u.) –0.4 conditions–0.2 0 0.2 resulting 0.4 in tial state of either even or odd symmetry with respect to k ( /a) k ( /a) || π an overall even matrix element can be|| summarizedπ as the mirror plane, the polarization conditions resulting in Nat. Comm. 9:3252 (2018) Fig. 2 Nodal type-II Dirac cone in La1.77Sr0.23CuO4. a Symmetrised Fermi surface map␾k recordedeven usingϩ ϩ circularlyϩ A polarisedeven 160 eV photons. Solid black an overall even matrix element can be summarized as curves are a tight binding parametrisation of the electron-like Fermik surface. Thek arrowsi indicate͗ nodal͉ ͉ and͘ orthogonal to nodal cuts. b Nodal band ͗␾f ͉A"p͉␾i ͘ k ⇒ (22) dispersion [cut #1 in a] recorded with circular polarisation symmetrised around ͭand␾i comparedodd ͗ϩ to͉ aϪ two-band͉Ϫ͘ A (d 2 odd.2 and d 2 ) tight-binding model. The k Γ x y z ⇒ À crossing of the two bands defines the type-II Dirac cone. c Spectra going through the Dirac point in the orthogonal-to-nodal direction [cut #2 in a] and k k ␾i even ͗ϩ͉ϩ͉ϩ͘ A even symmetrised around the nodal line. As indicated by the tight-bindingIn order model, to the discuss repulsive the interaction photon leads energy to orbital dependence, hybridisation. d Energy distribution ͗␾f ͉A"p͉␾i ͘ k ⇒ (22) eikr ͭ ␾i odd ͗ϩ͉Ϫ͉Ϫ͘ A odd. curves along the cuts D and E in c. e, f Same spectra as in bfrom, but acquired Eq. (5) with and linear byp consideringand s polarisation, a plane respectively. wave Solid andfor dashed the lines indicates the ⇒ FIG. 5. Schematictight-binding representation model. of The the on/off polarization switching anddemonstrates pho- thephotoelectron even and odd mirror at symmetries the detector, of the two one bands may constituting more conve- the Dirac cone. These k 2 k ikr 2 In order to discuss the photon energy dependence, ton energy effectssymmetry in the protected photoemission properties process: are not in (a)fluenced mirror by correlationniently induced write self-energy͉M ͉ effects.ϰ͉(␧" Thek) waterfall␾ ͉e feature͉ . The indicated overlap by the inte- energy scales E1 and E2 f,i ͗ i ͘ ikr plane emission fromis discussed a dx2Ϫy brie2 orbital;fly in the (b) text. sketch Background of the subtraction optical hasgral, been applied as sketched to panels b in, c, e Fig., and 5(b),f (see Supplementary strongly depends Figs. 1, 2, andon Supplementary the Note 1). from Eq. (5) and by considering a plane wave e for the transition betweeng, atomich Intensity orbitals distributions with alongdifferent the cuts angular A–C indicated mo- indetailsb andFIG.c, respectively. of the 5. Schematicinitial-state Black bars mark wave representation the function peak positions (peak of the position polarization and pho- photoelectron at the detector, one may more conve- menta (the wave functions of the harmonic oscillator are here of theton radial energy part and effects oscillating in the character photoemission of it), and on process: (a) mirror k 2 k ikr 2 used for simplicity) and free-electron wave functions with dif- niently write ͉Mf,i͉ ϰ͉(␧"k)͗␾i ͉e ͉͘ . The overlap inte- the opposite mirror symmetry of the two bands and hencethe wavelength that the Dirac of the cone outgoing thus forms plane a weakly wave. dispersing Upon in- line along the kz ferent kinetic energies (after Hu¨ fner, 1995); (c) calculated plane emission from a dx2Ϫy2 orbital; (b) sketch of the optical crossing is indeed protected by the crystal symmetry.creasing A perpen- the photondirection. energy, Note that both the bandsEkin and appearingk increase, below 1.5 eV around the gral, as sketched in Fig. 5(b), strongly depends on the photoionization cross sections for Cu 3d and O 2p atomic k 21 dicular cut through this Dirac point is shown in Fig.and2Mc.transition AlongchangesM between point in a non-necessarily (Figs. atomic2–4) are oforbitals monotonicdxz/yz origin with fashionand different irrelevant angular in this mo- levels (after Yeh and Lindau, 1985).fi f,i details of the initial-state wave function (peak position both cuts, signi cant self-energy effects are visible. Most[see noticeable Fig.menta 5(c), (thediscussion. for the wave Cu functions 3d and the of O 2thep atomic harmonic case]. oscillator are here is the waterfall feature, indicated by the energy scales E and E in of the radial part and oscillating character of it), and on As1 a matter2 of fact, the photoionization cross section is forms for the backgroundFig. 2f. We stress function that thisB self-energy(Hu¨ fner, structure 1995), is consistentused with for simplicity) and free-electron wave functions with dif- the wavelength of the outgoing plane wave. Upon in- two are more frequentlyprevious reports used: on (i) cuprates the step-edge31–34 and other back- oxides35usually,36. ferent characterized kineticDiscussion by energies one minimum (after in free Hu¨ fner, atoms, 1995); the (c) calculated 21 so-called Cooper2 minimum (Cooper,fi 1962), and a series creasing the photon energy, both Ekin and k increase, ground, with threeAs parameters previously reported for height, in ref. energyand shown posi- in Fig. 3,photoionization the dz Dirac fermions cross are sections classi ed by for their Cu dimensionality 3d and O and 2p theatomic band has a weak but clearly detectable k dispersionof near them the in in- solidsdegree (Molodtsov to which theyet al., break 2000). Lorentz invariance (see Table 1). k tion, and width of the step-edge, which reproducesz the and Mf,i changes in a non-necessarily monotonic fashion plane zone centre. This effect translates into a weak kz dispersionBeforelevels of concluding (afterType-I Yeh Dirac this and fermionssection, Lindau, break it has 1985). Lorentz to be invariance empha- such that it is background observed all the way to EF in an unoccupied [see Fig. 5(c), for the Cu 3d and the O 2p atomic case]. the Dirac point from 1.4 eV near Γ to 1.2 eV aroundsized that Z. As the descriptionstill possible for ofE photoemissionF to intersect the based bands formingon the the Dirac point region of momentum space; (ii) the Shirley background fi La1.77Sr0.23CuO4 has body-centred tetragonal structure, the Γ and Z at only the Dirac point when considering suf ciently small As a matter of fact, the photoionization cross section is B (␻)ϰ͐␮d␻ЈP(␻Ј), which allows one to extract from suddenforms approximation for the andbackground the three-step function model,B al- (Hu¨ fner, 1995), Sh ␻ points can be probed simultaneously in constant-energythough maps artificial that regions and of oversimplified, momentum space allows about an the intuitive point. For type-II Dirac usually characterized by one minimum in free atoms, the the measured photocurrentcover first and secondI(␻) in-planeϭP(␻) zonesϩc (Fig.B (4␻j).) The d 2 dominated Sh Sh understandingz two arefermions, of more the photoemission this frequently is not possible. process. used: However, (i) the step-edge back- the contributionbandP(␻ enters) due for to binding the unscattered energies of approximately electrons 1 eV (Fig. 4d) as Three-dimensional Dirac points are characterised by linearly so-called Cooper minimum (Cooper, 1962), and a series for aground, quantitative with analysis three of the parameters ARPES spectra, for calcu- height, energy posi- (the only parameteran elongated is the constant pocket centredc ; around Shirley, the 1972). zone corner. This “cigar” dispersing bands (around the Dirac point) along all reciprocal Sh lations based on the one-step model are generally re- of them in solids (Molodtsov et al., 2000). Let us now verycontour briefly stems illustrate from the the fact effect that of the thedz2 ma-band dispersestion, faster anddirections width (kx of, ky, thekz). For step-edge, type-I, such which Dirac points reproduces have been the towards Γ = (0, 0, 0)k than2 to Z = (0, 0, 2π/c) (see Fig.quired.4k). As In the this case,fi surface discontinuity,5 multiple6,7 scat- Before concluding this section, it has to be empha- trix element term I (k,␯,A)ϰ͉M ͉ , which is respon- backgroundidenti ed observed in Na3Bi and all Cd the3As way2 . Two-dimensional to E in an Dirac unoccupied fer- binding0 energy increases,f,i this pocket grows and eventuallytering, crosses finite-lifetimemions, by effects, contrast, and have matrix linear elements dispersion forF in two reciprocal sible for the dependence of the photoemission data on sized that the description of photoemission based on the the dx2 y2 dominated band on the nodal line (i.e., theinitial- line ofregion Dirac and final-state ofdirections. momentum Graphene, crystal wave being space; a functions monolayer (ii) the are of graphite, Shirley in- has background a perfect photon energy and experimentalÀ geometry, and may ␮ points extended in kz-direction in momentumcluded space). and This accountedtwo-dimensional for by first-principles band structure. The calculations, Dirac cones found in gra- sudden approximation and the three-step model, al- even result in complete suppressionfi of the intensity (Go- BSh(␻)ϰ͐␻d␻ЈP(␻Ј), which allows one to extract from happens rst at 1.2 eV in the second zone near Zas (Fig. we4f) shall and discussphene inare Sec. therefore IV.C purely for the two-dimensional. case of Bi2212 A three-dimensional though artificial and oversimplified, allows an intuitive beli et al., 1964;next Dietz in theetfirst al., zone 1976; in vicinity Hermanson, to Γ at 1.4 1977; eV (Fig. 4g). Thethe type-II measured photocurrent I(␻)ϭP(␻)ϩc B14 (␻) (Bansil and Lindroos,version of 1999). type-II cones has recently been uncovered in PtTeSh2 Sh. Eberhardt and Himpsel, 1980). By using the commuta- the contribution P(␻) due to the unscattered electrons understanding of the photoemission process. However, NATURE COMMUNICATIONS | (2018)9:3252 | DOI: 10.1038/s41467-018-05715-2k 2 | www.nature.com/naturecommunications 3 tion relation បp/mϭϪi͓x,H͔, we can write ͉Mf,i͉ for a quantitative analysis of the ARPES spectra, calcu- k k 2 (the only parameter is the constant cSh ; Shirley, 1972). ϰͦ͗␾f ͉␧"x͉␾i ͦ͘ , where ␧ is a unit vector along the po- E. State-of-the-art photoemission lations based on the one-step model are generally re- larization direction of the vector potential A. As in Fig. Let us now very briefly illustrate the effect of the ma- k 2 quired. In this case, surface discontinuity, multiple scat- 5(a), let us consider photoemission from a dx2Ϫy2 or- Intrix the early element stage termof theI investigation(k,␯,A)ϰ of͉M the͉ high-, which is respon- 0 f,i tering, finite-lifetime effects, and matrix elements for bital, with the detector located in the mirror plane temperaturesible for superconductors, the dependence ARPES of proved the photoemission to be data on (when the detector is out of the mirror plane, the prob- initial- and final-state crystal wave functions are in- veryphoton successful energy in detecting and dispersive experimental electronic geometry, fea- and may lem is more complicated because of the lack of an over- tures (Takahashi et al., 1988; Olson et al., 1990, 1989), cluded and accounted for by first-principles calculations, even result in complete suppression of the intensity (Go- all well-defined even/odd symmetry). In order to have the d-wave superconducting gap (Shen et al., 1993). as we shall discuss in Sec. IV.C for the case of Bi2212 nonvanishing photoemission intensity, the whole inte- Overbeli the pastet al. decade,, 1964; a great Dietz dealet of al. effort, 1976; has been Hermanson, 1977; (Bansil and Lindroos, 1999). grand in the overlap integral must be an even function investedEberhardt in further and improving Himpsel, this technique. 1980). By This using re- the commuta- under reflection with respect to the mirror plane. Be- sulted in an order-of-magnitude improvement in both k 2 cause odd-parity final states would be zero everywhere energytion and relation momentumប resolution,p/mϭϪi thus͓x,H ushering͔, we in can a write ͉Mf,i͉ on the mirror plane and therefore also at the detector, new eraϰ in␾ electronk ␧ x ␾ spectroscopyk 2, where and␧ allowingis a unit a detailed vector along the po- k ͦ͗ f ͉ " ͉ i ͦ͘ E. State-of-the-art photoemission the final-state wave function ␾f itself must be even. In comparisonlarization between direction theory and of the experiment. vector The potential rea- A. As in Fig. particular, at the detector the photoelectron is described sons for this progress are twofold: the availability of by an even-parity plane-wave state eikr with momentum dedicated5(a), photoemission let us consider beamlines photoemission on high-flux second- from a dx2Ϫy2 or- In the early stage of the investigation of the high- in the mirror plane and fronts orthogonal to it (Herman- andbital, third-generation with the synchrotron detector facilities located (for in a de- the mirror plane temperature superconductors, ARPES proved to be (when the detector is out of the mirror plane, the prob- very successful in detecting dispersive electronic fea- Rev. Mod. Phys., Vol. 75, No. 2, April 2003 lem is more complicated because of the lack of an over- tures (Takahashi et al., 1988; Olson et al., 1990, 1989), all well-defined even/odd symmetry). In order to have the d-wave superconducting gap (Shen et al., 1993). nonvanishing photoemission intensity, the whole inte- Over the past decade, a great deal of effort has been grand in the overlap integral must be an even function invested in further improving this technique. This re- under reflection with respect to the mirror plane. Be- sulted in an order-of-magnitude improvement in both cause odd-parity final states would be zero everywhere energy and momentum resolution, thus ushering in a on the mirror plane and therefore also at the detector, new era in electron spectroscopy and allowing a detailed k the final-state wave function ␾f itself must be even. In comparison between theory and experiment. The rea- particular, at the detector the photoelectron is described sons for this progress are twofold: the availability of by an even-parity plane-wave state eikr with momentum dedicated photoemission beamlines on high-flux second- in the mirror plane and fronts orthogonal to it (Herman- and third-generation synchrotron facilities (for a de-

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 Free Electron Gas (Non-interacting Electrons )

RMP 75, 473 (2003)

= 0 Band structure: Angle resolved photoemission Band structure of Sr2RuO4

Probing the Low-Energy Electronic Structure of Complex Systems by ARPES 71

Fig. 12. ARPES spectra and corresponding intensity plot from Sr2RuO4 along (a) À-M; and (b) M-X. (c) Measured and (d) calculated [89] Fermi surface. All data were taken at 10 K on a Sr2RuO4 single crystal cleaved at 180 K (from Ref. [90]). the field [98]. This issue was conclusively resolved only by Owing to the great improvement in energy and taking advantage of the high energy and momentum momentum resolution, it has now become possible to resolution of the ‘‘new generation’’ of ARPES data: it was study by ARPES also the momentum and temperature then recognized that a surface reconstruction [99] and, in dependence of the superconducting gap on low-Tc materi- turn, the detection of several direct and folded surface bands als (until recently, experiments of this kind could been were responsible for the conflicting interpretations [90,100– performed only for the much larger d-wave gap of the high- 102]. Figure 12(a) and (b) show high resolution ARPES data Tc superconductors [11]). The data presented in Fig. 14, (ÁE 14 meV; Ák 1:5% of the zone edge) taken at 10 K which are one of the most impressive examples of ¼ ¼ with 28 eV photons on a Sr2RuO4 single crystal cleaved at combined high energy and momentum resolution in 180 K (for Sr2RuO4; as recently discovered, high-tempera- ARPES experiments on solid samples (i.e., ÁE 2:5 meV ture cleaving suppresses the reconstructed-surface contribu- and Ák 0:2 ), provide direct evidence for Fermi¼ surface ¼ tions to the photoemission signal and allows one to isolate sheet-dependent superconductivity in 2H-NbSe2 [106]. A the bulk electronic structure [90]). Many well defined superconducting gap of about 1 meV was successfully quasiparticle peaks disperse towards the Fermi energy and detected along two of the normal-state Fermi surface disappear upon crossing EF: A Fermi energy intensity map sheets, but not along the third one. In fact, the opening of (Fig. 12(c)) can then be obtained by integrating the spectra the gap is directly evidenced in Fig. 14(b) and (c) by the over a narrow energy window about EF 10 meV : As the shift to high binding energies of the 5.3 K spectra leading- ðÆ Þ spectral function (multiplied by the Fermi function) reaches edge midpoint (which is instead located at EF at 10 K, as its maximum at EF when a band crosses the Fermi energy, expected for a metal), and by the simultaneous appearance the Fermi surface is identified by the local maxima of the of a peak below EF (which reflects the piling up of the intensity map. Following this method, the three sheets of density of states due to the gap opening). This behavior is Fermi surface are clearly resolved and are in excellent absent for the inner Fermi surface pocket (Fig. 14(a)). agreement with the theoretical calculations (Fig. 12(d)). 6.3. Self energy and collective modes As discussed in Section 4, the introduction of the electron 6.2. 2H NbSe2: Superconducting gap self energy Æ k;! Æ k;! iÆ k;! is a powerful À ð Þ¼ 0ð Þþ 00ð Þ 2H-NbSe2 is an interesting quasi two-dimensional system way to account for many-body correlations in solids. Its exhibiting a charge-density wave phase transition at real and imaginary parts correspond, respectively, to the approximately 33 K, and a phonon-mediated superconducting phase transition at 7.2 K. As indicated by band structure calculations [103], the valence-band electronic structure is characterized by a manifold of dispersive bands in a 6 eV range below the Fermi energy. At low energy, three dispersive bands are expected to cross the chemical potential and define three sheets of Fermi surface in the hexagonal Brillouin zone. Both the band manyfold and the Fermi surface topology have been studied in great detail by ARPES; exception made for a weak energy renormalization, the normal-state experimental data are in extremely good agreement with the results of theoretical calculations (as shown in Fig. 13, where ARPES spectra and band structure calculations are compared for the À-K high symmetry direction). As for the low temperature charge- density wave phase, despite the intense effort no agrement Fig. 13. (a) 2H-NbSe2 ARPES spectra (measured at 20 K with 21.2 eV has been reached yet on the driving force responsible for photons), (b) corresponding image plot, and (c) band structure calcula- the transition [37,104]. tions along À-K (from Ref. [105]).

# Physica Scripta 2004 Physica Scripta T109 model as12

E(k) t 1 4cos 3ak /2 cos ak /2 4cos2 ak /2 (1) y x x

where k is the in-plane momentum, a is the lattice constant, and t is the near-neighbor hopping energy.

In Figure 1 we compare energy bands and constant energy surfaces computed using Equation 1 to

ARPES measurements applied to a single layer of graphene grown on the (0001) surface of SiC (6H polytype). The primary bands, cones centered at the K points, are surrounded by six weak replica bands; these result from the interference of the graphite and substrate lattice constants (2.4 vs. 3.07Å) and correspond to similar satellite

spots in low energy electron diffraction. 13 The primary bands are in good overall agreement with the simple model despite its having only two adjustable parameters: the hopping energy t = 2.82 eV and a 0.435 eV shift of

the Fermi energy EF above the Dirac crossing energy ED. This shift is attributed to doping of the graphene layer by depletion of the substrate’s n-type carriers near the SiC surface.

Despite this good agreement, profound deviations are observed near EF and ED. We show in Figure 2a a magnifed view of the bands measured along a line (the vertical double arrow in Figure 1b) through the K point.

The predicted, or “bare” bands in this direction are nearly perfectly linear and mirror-symmetric with respect to the K point according to Equation 1. The actual bands deviate from this prediction in two signifcant ways: frst,

at a binding energy ħωph=200 meV below EF, we observe a sharpening of the bands accompanied by a slight

kink in the bands’ dispersions. We attribute this feature to renormalization of the electron bands near EF by Band structure of graphenecoupling to phonons, as discussed later.

FigureNature Physics 1 | The bandstructure3, 36 - 40 (2007) of graphene. a The experimental energy distribution of states as a function of momentum along principal directions, together with a single-orbital model (solid lines) given by Eq. (1). b Constant energy map of the states at binding

energy corresponding to the Dirac energy (ED) together with the Brillouin Zone boundary (dashed line). The orthogonal double arrows

indicate the 2 directions over which the data in Fig. 2 were acquired. c-d Constant energy maps at the Fermi energy (EF = ED+ 0.45) 13 and ED – 1.5 eV, respectively. The faint replica bands correspond to the 6√3 × 6√3 satellite peaks in low energy electron diffraction. Non-interacting electrons Interacting electrons

RMP 75, 473 (2003)

= non-zero 480 Damascelli, Hussain, and Shen: Photoemission studies of the cuprate superconductors

Let us start from the trivial ⌺(k,␻)ϭ0 noninteracting time (due to ⌺Љ), and it sharpens up rapidly, thus emerg- case. The N-particle eigenfunction ⌿N is a single Slater ing from the broad incoherent component upon determinant and we always end up in a single eigenstate approaching the Fermi level, where the lifetime is infi- when removing or adding an electron with momentum nite corresponding to a well-defined quasiparticle [note k. Therefore G(k,␻)ϭ1/(␻Ϫ⑀kϮi␩) has only one pole that the coherent and incoherent part of A(k,␻) repre- for each k, and A(k,␻)ϭ␦(␻Ϫ⑀k) consists of a single sent the main line and satellite structure discussed in the line at the band energy ⑀k [as in Fig. 3(b)]. In this case, previous section and shown in Fig. 3(c), bottom right]. † Furthermore, the peak position is shifted with respect to the occupation numbers nk␴ϭck␴ck␴ are good quantum numbers and for a metallic system the momentum distri- the bare-band energy ⑀k (due to ⌺Ј): since in the interacting case the quasiparticle mass is larger than the band bution [i.e., the expectation value n(k)ϵ͗nk␴͘, quite generally independent of the spin ␴ for nonmagnetic mass because of the dressing (m*Ͼm), the total disper- systems] is characterized by a sudden drop from 1 to 0 at sion (or bandwidth) will be smaller (͉␧k͉Ͻ͉⑀k͉). Given that many of the normal-state properties of the kϭkF [Fig. 3(b), top], which defines a sharp Fermi surface. If we now switch on the electron-electron correla- cuprate superconductors do not follow the canonical tions adiabatically, so that the system remains at equilib- Fermi-liquid behavior, it is worth illustrating the rium, any particle added into a Bloch state has a certain marginal-Fermi-liquid (MFL) model, which was specifi- probability of being scattered out of it by a collision with cally proposed as a phenomenological characterization another electron, leaving the system in an excited state of the high-temperature superconductors as it will be in which additional electron-hole pairs have been cre- discussed in Sec. VIII.B.2 (Varma et al., 1989, 1990; ated. The momentum distribution n(k) will now show a Abrahams and Varma, 2000). In particular, the motiva- tion of the MFL description was to account for the discontinuity smaller than 1 at kF and a finite occupation anomalous responses observed at optimal doping, for probability for kϾkF even at Tϭ0 [Fig. 3(c), top]. As example, in electrical resistivity,week ending Raman-scattering inten- long as n(k) showsPRL 113, a256402 finite (2014) discontinuityPHYSICALZkϾ0 at REVIEWk LETTERS 19 DECEMBER 2014 ϭk , we can describe the correlated Fermi sea in terms sity, and nuclear-spin relaxation rate. The MFL assump- F (a) (b) week ending2 PRL 104, 056403 (2010) PHYSICAL REVIEWmeasurementtions LETTERS4 of areγ 8. as4–11.5 follows:mJ5 FEBRUARY=mol K 2010for (i) different there met- are momentum- of well-defined quasiparticles, i.e., electrons dressed with allic samples, respectively¼ [32]. This demonstrates that all a Κ Γ Μ level and a broad shoulder (HP) disperses to higher binding a manifold of excited states, which are characterized by low-energyenergies,independent eventually charge merging excitations withexcitations a nearly arise nondispersive from over the small most Fermi of the Brillouin zone surfacepeak (NDP) pockets centered identified at 0:8 eV above.. In LV, the Furthermore, sharp peak it strongly a pole structure similar to that of the noninteracting sys- andthat the shoulder contribute are totallyÀ suppressed and to only the spin NDP and charge polarizability suggestsremains. thatIn Figs. our1(c) ARPESand 1(d), data the PDH represent structure the is bulk electronic tem but with renormalized energy ␧k and mass m*, and structure.emphasized␹(q,␻ at different,T);k (ii)by subtracting the the latter LV from LH has a scale-invariant form as a b spectra, after normalizing both spectra at 1:5 eV. a finite lifetime ␶kϭ1/⌫k . In other words, the properties ThefunctionThe symmetry role of of correlations the of orbitals frequency probed in by theÀ photoemission low-energy and temperature, excitations namely, Im␹ candepends be estimated on the beam polarization, from the as dictated renormalization by selection of the Fermi of a FermiCoherent and Incoherent liquid are similar to those ofW aeight free-electron rulesϰf [3(].␻ In/ ourT). experimental To visualize configuration, we expect the to implications of this approach, velocitydetect orbitalsvF=v evenbare withand respect the to the quasiparticle plane of incidence weight Z. While gas with damped quasiparticles. Furthermore, because vbarewithwecannot LH (here,compare this be is determineda1g and the one eg0 [ MFL4 directly]) and odd and with from LV Fermi-liquid experiment, it self energies, ne- Ca1.8Sr0.2RuO4 (c)Sr3-xLaxIr2O7 (d) NaxCo2 (here, the other e ). We overlay to our measurements in HP can often be approximatedg0 by band structure calculations the bare-electron character of the quasiparticle,NDP or pole Figs. 1(a) and 1(b) the LDA bands according to these c QP d glecting for simplicity any momentum dependence: strength (also called coherence factor),ΓK is Z ϽΓ1M and the withinparities. the Clearly, LDA. the slope In of Fig. both the4(b) QP andwe the thus HP compare the k experimentaldispersions correspond quasiparticle to that of the dispersiona1g band [see for also2 x 0.065 with2 Fig. 2]. The⌺ questionFL͑ arises␻͒ϭ as to␣ why␻ thereϩ isi␤ a ‘‘break’’͓␻ ϩ in ͑¼␲kBT͒ ͔; (17) total spectral weight must be conserved-1 [see Eq. (19)], -1 aLDA SO VCA calculation for the same doping kF=0.52 Å k =0.6 Å the a1g dispersion, giving rise to the QP and HP parts. F þ we can separate G(k,␻) and A(k,␻) into a coherent andAlong relaxedÀM, a large atomic− hybridization positions. gap is predicted Using between averaged Fermi -1 a1g and eg0 , which seems to be able to produce such a k=0.78 Å k=0.75 Å-1 velocities from all Fermi surfacex crossings␲ we find pole part and an incoherent smooth part without poles situation, as⌺ proposed͑ before␻͒ [ϭ7]. We␭ note,␻ however,ln thatϪi x . (18) -1 the position expectedMFL for e at k ( 0:7 eV) is much closer k=1.1 Å -1 vF=vbare 0.5 2 .Theerrorbarisestimatedfromaslightg0 F k=0.97 Å À to the NDP¼ than toð theÞ HP [ 0:25 eVͫ, see Fig. 1(d)␻]. The 2 ͬ c (Pines and Nozie`res, 1966): k=1.34 Å-1 uncertainty in the experimental Fermi velocity due to -1 À k=1.2 Å main problem with this explanation is that no hybridization finitegap is predicted resolution along ÀK effects, whereas we and measure the almost high the sensitivity of -1.5 -1.0 -0.5 0.0 -1.5 -1.0 -0.5 0.0 (E-E )(eV) Here xϷmax(͉␻͉,T), ␮ϭ0, ␻ is an ultraviolet cutoff, F (E-E )(eV) same PDH in the two directions. Even assuming that the c Zk F the calculated dispersion to the rotation angle of the PRL 104, 056403 (2010) hybridization gap could be larger along ÀK than in the Sutter et al. 2018G͑k,␻͒ϭ PRL 113, 256402 (2014)ϩGinchFIG. 1, (color online). (a), (b) ARPES intensity (15) plots along octahedra.calculation,and ␭ it We seemsis further a highly coupling unlikely estimate that it could constant the produce quasiparticle a (which residue could in principle be FIG. 4 (color online).ÀK Luttinger(left) and ÀM volume(right) with and LH (a) quasiparticle and LV (b) polariza- ␻Ϫ␧kϩi⌫k Z nearlydirectly identical by dispersion analyzing of a1g thenear E coherentF as that along weight.ÀM To this end residue. (a) Luttinger volumetions. The of LDA the dispersion experimental of even bands Fermi are superimposed surface to [Fig.momentum2(a)]. Therefore, we propose dependent). the alternative explana- From Eqs. (17) and (18) at T FIG. 4. Temperature dependence of QP spectral weight. (a-d)the ARPESLH image and spectra of the alongodd band S– to thefor LV temperatures image [4]. The a as indicated. (e,f,g) 1g wetion subtract that the PDH a smooth is an intrinsic background structure of froma due the to raw EDCs and Background subtracted (as in Ref.(blue),37 and compared38) EDCs with of the thecharacter↵-and nominal is indicated-band Luttinger by at theT size=1 volume of.3K the markers.(cyan) of 3 (c),andx= (d)2 30K Top: (black)ϭ and0 we fixed see that, while1g in a Fermi liquid the quasiparti- ⌫k /␲ Spectra at k in LH (large circles, light gray/red) and LV (small fitcorrelation the resulting effects. spectraThe similarity along between the the entire two direc- Fermi surface with momenta indicated by dashed verticalarising lines from in (a) the and stoichiometry (d). (f) is aF zoom of nearSr1−xELaF xof3 theIr2O EDC7 if displayed every in (e).tions The is then shaded natural, as the bandwidth is quite similar in the area in (e) andA͑ (f)k represents,␻͒ϭZ incoherentk and coherent spectralϩcircles,A weight black).inch modelled Bottom:,ð Difference by fitsÞ spectra to an (LH-LV), exponentially (16) in blue, at modifiedtwocles peaks Gaussian are representing well defined the coherent because quasiparticle⌺Љ(␻ and)/⌺Ј(␻) vanishes as lanthanum atom2 dopes2 the one indicated itinerantk values, electron fitted with a (redLorentzian line). cut by The the Fermi two directions. and a Lorentzian function truncated͑␻ byϪ the␧ Fermi-Dirack͒ ϩ⌫ distribution, respectively. The dashed line indicates the sum. (h) doping evolution of thek function conduction for the QP band and an minimum asymmetric function is shown for the inHP [21]. incoherentSurprisingly, hump there is [see no clear Fig.eg0 dispersions4(c)]. While detected this analysis is Normalized spectral weight, integrated within the magenta () and blue (↵) boxes shown in (a-d), versus T . ␻ for ␻ 0 and Zk is finite at kϭkF , in a marginal black on the right axis. (b) Band dispersion along M together somewhatin these measurements. model dependent This is particularly and tends clear in to LV underestimate Z, Ϫ1 Γ [Fig. 1(b)], where only→ the broad NDP is observed. This Z ϭ Ϫ Theϭ singleZ crystalsϩ were prepared by a standard flux Fermi liquid ⌺ (␻)/⌺ (␻)ϰ1/ln␻ is only marginally sin- ␻)SO −,VCA␧k calculationk(⑀ (kU ⌺0,Јx),0. and065) along⌫k itproblem clearly is analogous shows to a the significant well-knownЉ absence coherentЈ of e weight Zץ/⌺ aЈ LDAץwhere k (1 with method and characterized by transport and magnetic mea- g0 1 þ 1 ¼ ¼ pockets at E in Na CoO [6,7]. The e bands seem to ¼ the QPϭZ residue⌺ Z ,[1 and@! the(0,⌃(! 0,)]− self-energy2=(1π=c) – (02)π=ais,surements 0, and−ment2π [=c5]. with) its In corresponding Fig. our derivatives1, we DMFT show ARPES resultsto the measurementsΓ atM areTΓ = 390 K.25 (seegular–0. Supple-5 alongF for thex ␻ 2 entire0, andFermig0 there surface. are This no indicates Fermi-liquid-like qua- k͉ Љ͉ ⌘ < independent of ! and T . A FLdirection is therefore of the 2D expected surfacetaken Brillouin atmentary the APE zone. Fig. beam The2). line The hole of ELETTRA QP pocket excitations found [15], with of a thebe↵-band shifted away thus from EF, which may be due to a larger acrystal-field Fermi-liquid-like splitting between→ statea and withe thanZ assumed≈ vF=v inbare and thus a to displayevaluated (1) a linear at QP␻ dispersion,ϭin␧ thisk . calculation (2)It a should line (see width theSCIENTA also Supplementalfulfil, SES2002 be in the emphasized analyzer, Material most strict an[38] angular sense,) has resolution all thatbeen criteria of of asiparticles FL. because1g g0 Zk vanishes as 1/ln␻ at the Fermi 0.2 , and an energy resolution of 20 meV. The tempera- weaklythe calculation momentum (this splitting dependent changes from self-energy,10 meV to in stark con- that scales as !2, (3) a QP peaksuppressed amplitude for proportional clarity and the electron pocket at M was shifted À the Fermi-liquid description isture valid was 20Resistivity K,only the photon in and energy proximity specific 86 eV, and heat the measurements, beam to was trast300surface to meV, however, lightly depending doped (in on the method turn, cuprates of calculation for where [k16 stronglyϭ]). AskF the momentum corresponding Green’s to Z, independent of ! and T inand energy (4) a to FL match cut-o↵ theen-linearly experimentaldisplay polarized, FL either Fermi behavior in the wave plane for vectors. ofT< incidence1 K (c), [Linear only anda muchresult of heav- a higher splitting, eg0 bands could essentially dependentcontribute to interaction the NDP. It is also effects possible lead that eg0 tobandsZ ≪ are v =v for the ergythe scale ! FermiFL below which surfaceZ(d)⌃(! Quasiparticle) and< ! [46 rests]. Using residueHorizontal onZier the QPalong (LH)] masses or condition perpendicularly the [15 Fermi]. Reconciliation [Linear surfaces␧ VerticalkϪ at␮ is (LV)] reachedand by analysis spectral functions are entirelyF bare incoherent). As for a ⌃(!)=! (1 1/Z), this criterion|= can| be rewritten as(the plane of incidence is defined by the incoming beam nodalweak, quasiparticles especially at this photon[17,36] energy,. and somewhat ӷ ⌺ for small (π␻=2Ϫ; π=␮2 )(circles and and (and triangles,Ϫ theof sample’s the). respectively) extremely surface Furthermore, normal). dressed estimated The sample-band was from⌫ QPaligned stateshiddenMFL around by some the description amorphous background of also the present inhigh-T superconductors, note < ⌃(!) ͉< Љ⌃͉(!)/(1 Z). ForÆð an unsaturatedÞ FL, wek kF k c 2 the weight2 of the coherentby LEED, quasiparticles.S-point. and this These alignment QP was amplitudes, confirmed by the proportional peri- theThe NDP. to roleZ In,are both of cases, correlations the subtraction of in Fig. Sr13willIr2O be7 a is thus rather refer|= to| QPs< for which Z has ! or temperature depen- ϰ͓(␲kBT) ϩ(␧kϪ␮) ͔ for a Fermi-liquidodicityaccessed over two through Brillouin system Zones. EDC’s. Additional In in contrast two measure- to theintricate.verythat↵-band, efficient On from the way one to reveal Eq. hand, the (18) true theya1g PDH are one line crucial obtains shape, for driving a contribution the linear in T dence. This implies a nFL self-energy, i.e. non-linearmentsQP at the peak SIS amplitude beam line of of the the Swiss-band Light exhibits Source awhich pronounced we now investigate. or more dimensions (Lutinger,and 1961; the CASSIOPEE Pines beam and line of Nozie SOLEIL were`res, used to insulatingtoA standard the ground way electrical to estimate state the in strength the resistivity parent of the inter- compound (i.e., via␻ϭ a0), consistent with ⌃(!) for ! 0. suppression with increased T (Fig. 4e–g). A box integra- < ! complement this study. Figure 1 shows that the spectra are substantialactions is to orbital calculate the dependent effective mass shiftmÃ, through and deformation of We have extracted the Luttinger volume in Sr1 xLax Ir2O7 1966), although additionalvery logarithmic differenttion of under spectral LH (top weight panel) corrections− and around LV (middle3 the respective panel) theexperimentskF renormalization’s reveals of the Fermi at optimal velocity V =V doping. Furthermore, as we will Examination of the ↵-band,from with pure extensivedxz, dyz fitschar- to the experimental Fermið surface.Þ As the bare band structure [10,11]. OnF theLDA ¼ other hand, they acter, reveals an essentially T -independent QP ampli-polarizations.the same In LH trends [also large (Fig. circles4h). spectra As both(light gray/ the ↵-m andLDA=m-bandsÃ. The dispersion can be obtained either by fitting should be included in the two-dimensionalred) in bottom panels], a sharp case peak (QP) (Hodges crosses the Fermi appearthesee difference to in play Sec. spectra a minor at VIII.B.2, fixed rolek in(EDC the low-energy for the Energy MFL physics self-energy once has been used tude (Fig. 4g). The QP dispersionshown is approximately in Fig. 4(a), lin- the valuesare measured are fully simultaneously, consistent with this e a↵ect is not a result of 056403-2 ear "et↵ al.v ,k 1971).k ,implying Whensmall⌃(!)=(1 Fermi we compare surfacevb /v ) !, of volumesurface the degradation. electron3x=2, where We removalx areis thus the led tometallicity concludefor the that is induced line-shape by light electron analysis doping of with the La. ARPES spectra from k ⇡ F| F| < F F with v and vb being dressedindependently and bare Fermi measured veloci- Lathe concentration.dxy dominated The-band factor states of display non-saturatedIn summary, we investigated the doping evolution of the F F Bi Sr CaCu O (Bi2212; Abrahams and Varma, tiesand [47]. Assuming addition an isotropic spectra FL andfor using a Fermivb = liquidFL behavior. of Furthermore,quasiparticles the ratio betweencorrelated coherent2 2 insulator2 Sr Ir8ϩO␦. Our ARPES data show 3=2 arises from theF stoichiometry of Sr1−xLax 3Ir2O7 and 3 2 7 ˚ b and incoherent spectral weight (see Fig. 4e) indicates that 2.34with eVA, the those QP residue of yields a noninteractingZ = vF/vF =0.26(4), electronð systemÞ (in the coherent,2000). ungapped As a quasiparticles last remark, with it a large should and nearly be emphasized that the indicates that every La ionZ contributes1 around theone S-point, electron in accordance to the with the DMFT consistent with DMFT that findsFermiZxz sea.=0 The.23. interpretation Anal- ⌧ of our data within a conven- momentum independent spectral weight forming a closed lattice periodic potential), the effectvalue Z ofxy the0.05. We self-energy have thus demonstratedscale-invariant that the low-energy excitation spectrum assumed ysis of the MDC linewidth (HWHM) at T = 30 K ⇡ 2 tional Fermi liquid˚ 1 pictureQP with mass small renormalization Fermi surfaces and FL is QP breakdownsmall Fermi are or- surface, which can be accounted for by yieldscorrections(!)=0 + ⌘! becomeswith 0 =0 evident.020(2) A and [see Figs. 3(c) and (b), re- in the MFL model is characteristic of fluctuations asso- ˚ 1 2 supported by a calculationbitally of the selective. electronic specific heat LDA SO U calculations. This phenomenology is in ⌘ = 10.6(6) A eV being constants. This is docu- þ þ mentedspectively]. by plotting ⌃ The(!)=(from quasiparticle(!) the ARPES) vb versus data. peak! UsingIn the now summary, experimental has we a have Fermi finite presented surface life- a combinedstarkciated contrast ARPES, withto lightly a dopedTϭ cuprates0 quantum[16–18] and critical surface point, as also dis- = 0 (Fig. 3). By comparing ⌃(!)volume/(1 Z)A andFS and⌃(! quasiparticle), we DFT, velocities and DMFT for studyx of0.065 Ca1.8weSr0.2RuOdoped4.Ourresults Sr2IrO4 [14], suggesting that the properties of estimate a ! 80meV< Fermi liquid cut-o↵ =– in agree- revealed the2 complete¼ low-energy electronic structure. FL calculate cyclotron masses mÃ ℏ AFS=2π ϵ 3.1 9 Sr3Ir2O7 are unique among doped correlated insulators ⇠ ¼ ∂ ∂ ¼ ð Þ Rev. Mod. Phys., Vol.me 75,for No. each 2, Fermi April 2003 surface lens. The doping dependence and not suitable for the engineering of cupratelike high-Tc of mÃ is negligible within the current accuracy of our superconductivity. It remains an open question how our experiment. Assuming two dimensionality, these masses findings relate to recent reports of strongly correlated states correspond to a Sommerfeld coefficient γARPES 9 3 mJ= in doped single layer iridates [14,15] and Ru-substituted mol K2, in good agreement with the direct thermodynamic¼ ð Þ bilayer iridates [19].

256402-4 480 Damascelli, Hussain, and Shen: Photoemission studies of the cuprate superconductors

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 Fermi Liquid Theory

KORDYUK et al. PHYSICAL REVIEW B 71, 214513 ͑2005͒ KORDYUK et al. PHYSICAL REVIEW B 71, 214513 ͑2005͒

by mainly one parameter, the high-energy cutoff ␻c. This by mainly one parameter, the high-energy cutoff ␻c. This gives us the way to solve the whole problem, examining a gives us the way to solve the whole problem, examining a wider energy range of the ARPES data. wider energy range of the ARPES data. B. Quadratic dispersion B. Quadratic dispersion One more complication should be addressed here: in the One more complication should be addressed here: in the wider energy range a deviation of the bare dispersion from a wider energy range a deviation of the bare dispersion from a line should be taken into account. Along the nodal direction line should be taken into account. Along the nodal direction the TB band in the occupied part can be well approximated the TB band in the occupied part can be well approximated by a simple parabola ␧͑k͒=␻ ͑1−k2 /k2 ͒,9 for which we still by a simple parabola ␧ k =␻ 1−k2 /k2 ,9 for which we still 0 F ͑ ͒ 0͑ F͒ need one energy scale parameter: the bottom of the bare band need one energy scale parameter: the bottom of the bare band ␻ or the bare Fermi velocity v =−2␻ /k . Using this dis- ␻ or the bare Fermi velocity v =−2␻ /k . Using this dis- 0 F 0 F 0 F 0 F persion in Eq. ͑1͒, one can ﬁnally modify Eqs. ͑2͒ and ͑3͒ to persion in Eq. ͑1͒, one can ﬁnally modify Eqs. ͑2͒ and ͑3͒ to

vF 2 2 vF 2 2 ⌺ ␻ = k ␻ − k + ␻, 4 ⌺ ␻ = k ␻ − k + ␻, 4 Ј͑ ͒ ͓ m͑ ͒ F͔ ͑ ͒ Ј͑ ͒ ͓ m͑ ͒ F͔ ͑ ͒ 2kF 2kF FIG. 1. Color online. Bare band dispersion solid line and FIG. 1. ͑Color online.͒ Bare band dispersion ͑solid line͒ and ͑ ͒ ͑ ͒ vF 2 2 renormalized dispersionvF ͑points2͒ on top2 of the spectral weight of ͱ renormalized dispersion ͑points͒ on top of the spectral weight of ⌺Љ͑␻͒ = − W͑␻͒ͱk ͑␻͒ − W ͑␻͒. ͑5͒ ⌺Љ͑␻͒ = − W͑␻͒ km͑␻͒ − W ͑␻͒. ͑5͒ interacting electrons. Thoughm intended to be general, this sketch kF interacting electrons. Though intended to be general, this sketch � = kF = ∗ = represents the nodal direction of an underdoped Bi-2212. represents the nodal direction of an underdoped Bi-2212. C. Fitting procedure C. Fitting procedure represented by the solid line in Fig. 1. When interactions are represented by the solid line in Fig. 1. When interactions are present, the self-energy leads to a shifting and broadening of Inpresent, short, the the fitting self-energy machinery leads is to based a shifting on Eqs. and͑4 broadening͒, ͑5͒, of In short, the fitting machinery is based on Eqs. ͑4͒, ͑5͒, the noninteracting spectral function. The resulting picture is and ͑A1the͒. noninteracting One can define spectral three steps function. here. The In the resulting two first picture is and ͑A1͒. One can define three steps here. In the two first steps, the real part of the self-energy, for given ␻ , ␻ , and n essentially that which is measured in ARPES ͑the blurred steps,essentially the real part that of the which self-energy, is measured for given in ARPES␻0, ␻c, and͑then blurred 0 c region in Fig. 1 illustrates this͒. If one neglects the momen- ͑whichregion characterizes in Fig. 1 the illustrates tails, see this below͒. If one͒, is neglects calculated the in momen- ͑which characterizes the tails, see below͒, is calculated in ⌺ ⌺ tum dependence of the self-energy, then, from Eq. ͑1͒, the two waystum͑ dependencei͒ ⌺dispЈ by Eq. of͑ the4͒, self-energy,͑ii͒ ⌺KKЈ by Eq. then,͑5͒ fromwith Eq. sub-͑1͒, the two ways ͑i͒ dispЈ by Eq. ͑4͒, ͑ii͒ KKЈ by Eq. ͑5͒ with sub- sequent KK transform A1 . Then, in step iii , the param- sequent KK transform ͑A1͒. Then, in step ͑iii͒, the param- momentum distribution curves ͓MDC͑k͒=A͑k͒␻=const͔ have momentum distribution͑ ͒ curves ͓MDC͑ ͑k͒͒=A͑k͒␻=const͔ have eters ␻ , ␻ , and n ͓see Eq. ͑A11͔͒ are varied until ⌺Ј ͑␻͒ eters ␻ , ␻ , and n ͓see Eq. ͑A11͔͒ are varied until ⌺Ј ͑␻͒ maxima at km͑␻͒ determined by ␻−␧͑km͒−⌺Ј͑␻͒=0 for a maxima0 c at km͑␻͒ determined by ␻−␧͑km͒−⌺Јdisp͑␻͒=0 for a 0 c disp and ⌺Ј ͑␻͒ coincide. In practice, we fit the difference and ⌺Ј ͑␻͒ coincide. In practice, we fit the difference given ␻. In other words, ⌺Ј͑␻͒=␻−␧͑km͒, is that which is givenKK ␻. In other words, ⌺Ј͑␻͒=␻−␧͑km͒, is that which is KK illustrated in Fig. 1 by the double-headed arrow. In the region ⌺dispЈ −illustrated⌺KKЈ to a in small Fig. contribution 1 by the double-headed of experimental arrow. resolu- In the region ⌺dispЈ −⌺KKЈ to a small contribution of experimental resolu- where the bare dispersion can be considered as linear ͑with a tion. Thewhere details the bare of the dispersion procedure can are be given considered in Secs. as 3 linear and 4͑with a tion. The details of the procedure are given in Secs. 3 and 4 of the Appendix. of the Appendix. slope vF͒ one can write slope vF͒ one can write

⌺Ј͑␻͒ = ␻ − vF͓km͑␻͒ − kF͔. ͑2͒ ⌺III.Ј͑␻ RESULTS͒ = ␻ − vF͓km͑␻͒ − kF͔. ͑2͒ III. RESULTS We have applied the described procedure to the experi- Assuming in addition weak k dependence of ⌺Љ along a cut Assuming in addition weak k dependence of ⌺ along a cut We have applied the described procedure to the experimental data measured along the nodal direction for theЉ fol- perpendicular to the Fermi surface ͑see discussion in Sec. 5 perpendicular to the Fermi surface ͑see discussion in Sec. 5 mental data measured along the nodal direction for the fol- 5 lowing samples: underdoped Bi͑Pb͒-2212 ͑T =77 K͒, over- of the Appendix͒, the MDCs exhibit a Lorentzian line shape of the Appendix͒, the MDCs exhibit a Lorentzianc line shape5 lowing samples: underdoped Bi͑Pb͒-2212 ͑Tc =77 K͒, over- doped ͑T =75 K͒ Bi͑Pb͒-2212, and optimally doped Bi͑La͒- with the half width at half maximum W and withc the half width at half maximum W and doped ͑Tc =75 K͒ Bi͑Pb͒-2212, and optimally doped Bi͑La͒- 2201 ͑T =32 K͒, marked in the following as UD77, OD75, c 2201 ͑Tc =32 K͒, marked in the following as UD77, OD75, and OP32, respectively. The data for UD77 and OD75 were ⌺Љ͑␻͒ = − vFW͑␻͒. ͑3͒ ⌺ ␻ = − v W ␻ . 3 and OP32, respectively. The data for UD77 and OD75 were collected at 130 K, and for OP32Љ͑ ͒ at 40F K.͑ We͒ have explored ͑ ͒ Thus, the determination of both the real and imaginary collected at 130 K, and for OP32 at 40 K. We have explored a numberThus, of excitation the determination energies in of the both range the of real 17–55 and imaginary eV a number of excitation energies in the range of 17–55 eV parts of the self-energy requires the knowledge of the bare but, asparts we of show the below, self-energy only requires at 27 eV, the at knowledge which only of the the bare ␧ 12 but, as we show below, only at 27 eV, at which only the dispersion ͑k͒͑or, in the vicinity to EF, an “energy scale,” antibonding band is visible, the described procedure can be 12 9 dispersion ␧͑k͒͑or, in the vicinity to EF, an “energy scale,” antibonding band is visible, the described procedure can be e.g., Fermi velocity vF͒. The KK transformation gives an directly applied to the bilayer Bi9 samples. The experimental e.g., Fermi velocity vF͒. The KK transformation gives an directly applied to the bilayer Bi samples. The experimental additional equation which relates these functions: ⌺Ј details can be found elsewhere.14,15 additional equation which relates these functions: ⌺Ј details can be found elsewhere.14,15 =KK⌺Љ ͓e.g., Eq. ͑A1͔͒. This opens the way to extract all Figure 2 illustrates an example of the ARPES spectrum, =KK⌺Љ ͓e.g., Eq. ͑A1͔͒. This opens the way to extract all Figure 2 illustrates an example of the ARPES spectrum, desired quantities from the experiment, but brings a new photocurrent as a function of energy and momentum, taken desired quantities from the experiment, but brings a new photocurrent as a function of energy and momentum, taken “problem of tails.” Under “tails” we mean the behavior of for UD77 Bi͑Pb͒-2212 at 130 K along the nodal direction. “problem of tails.” Under “tails” we mean the behavior of for UD77 Bi͑Pb͒-2212 at 130 K along the nodal direction. ⌺Љ͑␻͒ for energies ͉␻͉Ͼ␻m, where ␻m is a “confidence On top of it, we plot the result of the fitting procedure, the ⌺Љ͑␻͒ for energies ͉␻͉Ͼ␻ , where ␻ is a “confidence limit,” a maximal experimental binding energy to which both bare dispersion. m m On top of it, we plot the result of the fitting procedure, the limit,” a maximal experimental binding energy to which both bare dispersion. the W͑␻͒ and km͑␻͒ functions can be confidently determined. Another result of the procedure is the self-energy func- the W ␻ and k ␻ functions can be confidently determined. Fortunately, as we show in Sec. 3 of the Appendix, the tions. They͑ are͒ shownm͑ in͒ Fig. 3 for UD77 and in Fig. 4 for Another result of the procedure is the self-energy func- different but reasonable tails of ⌺Љ͑␻͒ almost do not effect OD75 andFortunately, OP32. We as remind we show that in the Sec. real 3 part of theof the Appendix, self- the tions. They are shown in Fig. 3 for UD77 and in Fig. 4 for different but reasonable tails of ⌺Љ͑␻͒ almost do not effect OD75 and OP32. We remind that the real part of the self- the low-energy behavior of ⌺Ј͑␻͒. The influence of the high- energy is represented by two functions ⌺dispЈ and ⌺KKЈ , ob- energy region on the coupling strength ͑A3͒ can be described tained,the as low-energy it is described behavior above, of from⌺Ј͑␻ the͒. The experimental influence of dis- the high- energy is represented by two functions ⌺dispЈ and ⌺KKЈ , ob- energy region on the coupling strength ͑A3͒ can be described tained, as it is described above, from the experimental dis- 214513-2 214513-2 ARPES: Theory band ground

Three step model:

*** Electron absorbs light

Event probability

*** Electron travels to the surface

Sudden approximation = no scattering / interaction events

*** Escape of the photo-electron into vacuum

Momentum and energy conservation considerations.

Electron Spectroscopy course: Spring Semester QO versus ARPES

Requirements Requirements High-purity (quality) single crystals Cleavable crystal surface Magnetic fields No magnetic fields

Capacity Capacity Bulk sensitive Surface sensitive Fermi surface area Band structure including Fermi surface Electronic mass Electronic mass Superconductivity Superconductivity – Mr. Perfect

Dissipation less charge conduction

Perfect diamagnet Meisner effect History of superconductivity

From Wikipedia High-temperature superconductivity

Georg Karl-Alex Andreas Hugo Bednorz Müller Schilling Keller

Discovery of High-Tc superconductivity Highest ambient Electron – lattice transition temperature coupling Nobel Prize in Physics 1987 Superconductivity – “Course strategy”

This course -- Experimental phenomenology

-- London Theory -- Ginzburg-Landau “Theory”

Condensed Matter Theory Course Spring 2020 – Prof. Titus Neupert Bardeen-Cooper-Schrieffer (BCS) theory Superconductivity - Overview

Step 1 Step 2 Classification of - Theory Summary Superconductors - Josephson Effect - Paraconductivity

Reading tasks Reading tasks Kittel: Kittel: Chapter: Superconductivity Chapter: Superconductivity Superconductivity – Observables Superconductivity – Specific heat

http://iopscience.iop.org/article/10.1088/0953-8984/24/5/055701 Superconducting LaPt4Ge12 Specific heat?

LaPt4Ge12

H. Pfau et al., doi:10.1209/0295-5075/82/47011 Phys. Rev. B 94, 054523 (2016) Specific heat – Phase transitions

J. Phys.: Condens. Matter 15 No 43 (5 November 2003) L643-L648