Lecture Notes Attempt to Give a Pedagogical Account of the Basic Physics of Majorana Bound States and the Topological Superconductors Which Host Them
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Advanced Quasiparticle Interference Imaging for Complex Superconductors
ADVANCED QUASIPARTICLE INTERFERENCE IMAGING FOR COMPLEX SUPERCONDUCTORS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Rahul Sharma May 2020 © 2020 Rahul Sharma ALL RIGHTS RESERVED ADVANCED QUASIPARTICLE INTERFERENCE IMAGING FOR COMPLEX SUPERCONDUCTORS Rahul Sharma, Ph.D. Cornell University 2020 State-of-the-art low-temperature spectroscopic imaging scanning tunneling microscopy offers a powerful tool to study materials with unprecedented spatial (subangstrom) and energy (micro-electronvolts) resolution. Imaging the quasi- particle interference of eigenstates in a material, offers a window into the un- derlying Hamiltonian. In this thesis, we performed quasiparticle interference imaging under extreme (milliKelvin temperatures) conditions and developed novel analysis techniques to address important contemporary problems in su- perconductivity which have defied complete understanding due to their com- plex multiband nature and small energy scales. Discovered almost 25 years ago, the momentum space gap structure of Sr2RuO4 has remained a mystery despite being an intensely researched topic due to possibilities of correlated and topological superconductivity. The multi- band nature of Sr2RuO4 makes the problem complicated because usual thermo- dynamical probes cannot directly reveal which bands the subgap quasiparti- cles are coming from. Our first advanced approach addresses this problem by applying milliKelvin spectroscopic imaging scanning tunneling microscopy (SI- STM) to visualize the Bogoliubov quasiparticle interference (BQPI) pattern deep within the superconducting gap at T=90 mK. From T-matrix modeling of the subgap scattering, we are able to conclude that the major gap lies on α : β bands and the minima/nodes lie along (±1; ±1) directions. -
Three-Dimensional Majorana Fermions in Chiral Superconductors
Three-dimensional Majorana fermions in chiral superconductors The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Kozii, V., J. W. F. Venderbos, and L. Fu. “Three-Dimensional Majorana Fermions in Chiral Superconductors.” Science Advances 2, no. 12 (December 7, 2016): e1601835–e1601835. As Published http://dx.doi.org/10.1126/sciadv.1601835 Publisher American Association for the Advancement of Science (AAAS) Version Final published version Citable link http://hdl.handle.net/1721.1/109450 Terms of Use Creative Commons Attribution-NonCommercial 4.0 International Detailed Terms http://creativecommons.org/licenses/by-nc/4.0/ SCIENCE ADVANCES | RESEARCH ARTICLE THEORETICAL PHYSICS 2016 © The Authors, some rights reserved; Three-dimensional Majorana fermions in exclusive licensee American Association chiral superconductors for the Advancement of Science. Distributed under a Creative Vladyslav Kozii, Jörn W. F. Venderbos,* Liang Fu* Commons Attribution NonCommercial Using a systematic symmetry and topology analysis, we establish that three-dimensional chiral superconductors License 4.0 (CC BY-NC). with strong spin-orbit coupling and odd-parity pairing generically host low-energy nodal quasiparticles that are spin-nondegenerate and realize Majorana fermions in three dimensions. By examining all types of chiral Cooper pairs with total angular momentum J formed by Bloch electrons with angular momentum j in crystals, we obtain a comprehensive classification of gapless Majorana quasiparticles in terms of energy-momentum relation and location on the Fermi surface. We show that the existence of bulk Majorana fermions in the vicinity of spin- selective point nodes is rooted in the nonunitary nature of chiral pairing in spin-orbit–coupled superconductors. -
3 Fermions and Bosons
3 Fermions and bosons exercises 3.1 and 3.4 by Dr. Peter Denteneer and Dmitry Pikulin 3.1 Second quantization Consider a system of spinless bosons in a (large) volume V . We will investigate how the diVerence between an external (one-body) potential and an interaction (two-body) potential appears when the Hamiltonian is written in terms of creation and annihilation operators (second quantization). Recall the definition of field and number operators, 1 X i~k ~r † È (~r ) e ¢ aˆ~ , nˆ(~r ) È (~r )È (~r ). Æ p k Æ V ~k (The integral over continuous ~k p~/ in infinite space, with a delta-function normalization, Æ ħ is replaced here by a sum over discrete ~k in a finite volume V .) a) In the case of an external potential u(~r ), show that the operator for the total potential energy, Z Uˆ d~r nˆ(~r )u(~r ) , Æ V can be expressed as follows in terms of creation and annihilation operators: X u~q X Uˆ aˆ† aˆ , m~ ~q m~ Æ ~q pV m~ Å where Z 1 i~q ~r u~q d~r e¡ ¢ u(~r ) ´ pV V are the Fourier components of the potential u(~r ). Let v(r ) be a short-range interaction potential, dependent only on the separation r 12 12 Æ ~r ~r . j 1 ¡ 2j b) Show that: 1 Z i(~k ~r1 ~` ~r2 m~ ~r1 ~n ~r2) v(~k~`,m~ ~n) d~r1d~r2 e¡ ¢ Å ¢ ¡ ¢ ¡ ¢ v(r12) Æ V 2 1 ±~k ~`,m~ ~n v~k m~ , Æ Å Å pV ¡ 1 where v~q are the Fourier components of v(r ). -
On the Bogoliubov-De Gennes Equations
ONTHE BOGOLIUBOV-DE GENNES EQUATIONS A. J. Leggett Department of Physics University of Illinois at Urbana-Champaign U.S.A. Support: John D. and Catherine T. MacArthur Foundation On the Bogoliubov-de Gennes equations. TQT ‐ 1 Main claims of this talk: 1. For 50 years, almost all theoretical work on inhomogeneous Fermi Superfluids, including work on topological quantum technology in (p + ip) superconductors, has been based on the Bogoliubov-de Gennes (generalized mean-field) method. 2. Consideration of some simple examples shows that results obtained by this method (at least if applied naively) may be wrong. 3. This is because in the cases of interest the response of the Cooper pairs cannot be ignored. 4. The question most relevant to TQT relates to the Berry phase: here again consideration of a simple example strongly suggests that the response of the Cooper pairs cannot be ignored, so that results obtained by (naïve application of) the BdG equations may be qualitatively misleading. 5. If so, this could be a disaster for the whole program of TQT in (p + ip) Fermi superfluids. TQT ‐ 2 The Bogoliubov-de Gennes (generalized mean-field, “BdG”) method. Original Hamiltonian: † † ∙ 2 † † , ′ ′ ′ ′ † ≡fermion creation operator(etc.) , ∗ , ′ , ′ Mean-field approximation: † † † † ′ ′ ⟹ † † † † † † ′ ′ ′ ′ c‐number operators (+ Hartree + Fock terms) MF Hamiltonian: † † ∙ 2 † † ′ Δ , ′ ′ . Δ , ′ ≡ , ′ ̂ ′ ̂ ← must eventually be chosen self-consistently TQT ‐ 3 † is bilinear in , ⟹ can be diagonalized by writing † † Ε ̂ const. where , are spinors (in ordinary spin space) satisfying the BdG equations i Δ , ′ Ε i ∗ i i Δ Ε where ≡ In simple BCS case Δ , Δ, so reduces to Δ ∗ ∗ Δ TQT ‐ 4 Some notes on the BdG method: 1. -
Arxiv:1711.00011V2 [Cond-Mat.Supr-Con] 21 Feb 2018 28 4 2
IL NUOVO CIMENTO Vol. ?, N. ? ? Majorana quasiparticles in condensed matter Ramon´ Aguado(1)(∗) (1) Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior de Investigaciones Cientficas (CSIC), Cantoblanco, 28049 Madrid, Spain Summary. | In the space of less than one decade, the search for Majorana quasi- particles in condensed matter has become one of the hottest topics in physics. The aim of this review is to provide a brief perspective of where we are with strong fo- cus on artificial implementations of one-dimensional topological superconductivity. After a self-contained introduction and some technical parts, an overview of the current experimental status is given and some of the most successful experiments of the last few years are discussed in detail. These include the novel generation of ballistic InSb nanowire devices, epitaxial Al-InAs nanowires and Majorana boxes, high frequency experiments with proximitized quantum spin Hall insulators realised in HgTe quantum wells and recent experiments on ferromagnetic atomic chains on top of superconducting surfaces. PACS --.-- {.... PACS --.-- {.... 2 1. Introduction 5 2. The Dirac and Majorana equations . 5 2 1. Dirac equation . 7 2 2. The concept of antiparticle . 8 2 3. Majorana equation . 8 2 4. Charge conjugation 11 3. Majoranas in Condensed Matter . 11 3 1. Bogoliubov quasiparticles as Majorana fermions . 13 3 2. Majorana zero modes . 14 3 3. Chiral p + ip superconductors in two dimensions . x y 16 3 4. One-dimensional p-wave superconductivity: the Kitaev model . 20 3 5. Non-Abelian braiding of Majorana zero modes 24 4. Physical implementations . 24 4 1. The Fu and Kane model . -
Majorana Bound States in D-Wave Superconductor-Based Hybrid Systems
Majorana bound states in d-wave superconductor-based hybrid systems C. C. Tsuei IBM Thomas J. Watson Research Center Yorktown Heights, NY 10598, U.S.A. Abstract For the search of Majorana fermions, we propose to employ a ramp-type Josephson edge junction between a normal metal such as Au, Ag or alloys of heavy metals and a topologically-trivial dx2-y2 –wave high-Tc superconductor YBa2Cu3O7. The success in forming the elusive zero-energy Majorana bound states (MBS) at the junction relies crucially upon the strength of the Rashba spin-orbit coupling (RSOC) at the metal/superconductor interface, the tunneling direction, and the interface quality which includes a near zero tunnel barrier height. When these junction conditions are collectively satisfied, definitive signatures of Majorana fermions will emerge. After inspecting a vast amount of published data of junctions that are close to what we have proposed, we were able to find: 1) angular resonant Andreev reflection, and 2) quantized zero bias normalized differential conductance peak (ZBCP), as possible evidence for MBS formed at the junction interface. Encouraged by these findings, we have designed a MBS- supporting d-wave superconductor based π-loop heterostructure as a qubit platform. Such π-loops are characterized by a doubly degenerate half flux quantum vortex (HQV) and are topologically protected by a parity effect arising from an odd number of sign changes of the supercurrent circulating in the loop. The topological protection is further enhanced by the formation of RSOC induced Majorana bound states at the Josephson junction interface. The proposed MBS-plus-HQV qubit platform will be useful for demonstrating non-Abelian exchange statistics and for building a prototype topological quantum computer. -
Annihilation of Colliding Bogoliubov Quasiparticles Reveals Their
Annihilation of colliding Bogoliubov quasiparticles reveals their Majorana nature C. W. J. Beenakker Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands (Dated: December 2013) The single-particle excitations of a superconductor are coherent superpositions of electrons and holes near the Fermi level, called Bogoliubov quasiparticles. They are Majorana fermions, meaning that pairs of quasiparticles can annihilate. We calculate the annihilation probability at a beam splitter for chiral quantum Hall edge states, obtaining a 1±cos φ dependence on the phase difference φ of the superconductors from which the excitations originated (with the ± sign distinguishing singlet and triplet pairing). This provides for a nonlocal measurement of the superconducting phase in the absence of any supercurrent. Condensed matter analogies of concepts from particle physics are a source of much inspiration, and many of these involve superconductors or superfluids [1]. Majo- rana’s old idea [2] that a spin-1/2 particle (such as a neu- trino) might be its own antiparticle has returned [3] in the context of low-dimensional superconductors, inspir- ing an intense theoretical and experimental search for condensed matter realizations of Majorana fermions [4]. The search has concentrated on Majorana zero-modes [5– 7] — midgap states (at the Fermi level E = 0) bound to a defect in a superconductor with broken spin-rotation and time-reversal symmetry (a so-called topological su- perconductor [8, 9]). The name Majorana zero-mode (or Majorino [10]) is preferred over Majorana fermion, since they are not fermions at all but have a non-Abelian ex- change statistics [11].