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 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 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 (typi- cally 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 ma- terial 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 probabil- ity 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 momen- tum). 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