Subharmonic gap Structures Yanir Schwartz

Introduction BCS Theory Andreev Reflection BKT Model Subharmonic Gap Structures Multiple Andreev Reflection OBTK Model and Later Octavio’s Corrections

BCS An electron moving through a conductor will attract nearby positive charges in the lattice Local deformation of the lattice causes another electron, with opposite spin, to move into the region of higher positive charge density The two electrons then become correlated and form what known as a Cooper pair

BCS Cooper pair illustration New quasiparicle is formed which can condense due to its integer spin

BCS Conclusion: At low enough temperatures, a band gap is formed.

BCS The theory was proposed by John Bardeen, Leon Cooper, and John Robert Schrieffer in 1957 They received the Nobel Prize in Physics for this theory in 1972. 1, Bardeen, 2 Cooper, 3 Schrieffer

Andreev REflection In the Normal-metal-Superconductor interface: Due to the band gap in the superconductor described in BCS’s theory, an electron coming from the normal metal side has zero probability to cross over to the superconductor side – there are simply no vacant states It does, however, have a positive probability to form a cooper pair in the S side, so that even at low energies Current can be conducted

Andreev REflection The formation of cooper pairs in the N/S interface was explained by Andreev: An electron meeting the N/S Interface produces a Cooper Pair in the superconductor, while a hole is reflected in the normal metal

BTK Theory Models the point-contact junction considers the transmission, reflection, and Andreev reflection of a lone electron at the boundary between a normal metal and a superconductor Developed by Blonder, Tinkham & Klapwijk in 1982. Uses Andreev reflection to explain model the interface while is ignoring Josephson effect for the simplicity of the theory

BTK Theory - Assumptions We take the potential to be a δ-function barrier of amplitude H at the interface Energy gap of a constant ∆ inside the superconductor The incident wave is a 1D plane wave traveling from the normal metal into the superconductor

BTK Theory - Formalism BTK Used the Bogoliubov–de Gennes (BdG) equations to generalize the BCS formalism to treat superconductors with spatially varying pairing strength ∆(x), chemical potential μ(x), potential V(x). The excitation is defined with the following annihilation-creation operators 𝛾: Ukgamma = creation of electron vgammad = annihilation of hole

BTK Theory- Formalism A new state can be defined as: Where 𝑢 𝑘 and 𝑣 𝑘 satisfy the BdG equations:

BTK Theory- Formalism Deep in the superconducting electrode where ∆(x), μ(x), and V (x) are constants, the solutions to the equation above are time-independent plane waves: Solving the above equations yield the following results: בעומק על המוליך, כאשר אין תלות מרחבית בפרמטרי הבעיה, פתרונות המשוואות הם גלים מישוריים קבועים בזמן

BTK Theory - Formalism For each energy 𝐸 𝑘 , there are four corresponding k values, ±k±, where 𝑘 + labels the electron-like quasiparticles, and 𝑘 − labels the hole-like quasiparticles

BTK Theory - Formalism Since 𝑢 𝑘 and 𝑣 𝑘 are paired through ∆ the relations between them is: And the new state can be written as

BTK Theory - Formalism Similarly, in the N side, far away from the interface, V(x) = ∆(x) = 0 we get a similar expression for the Energies: The wavefunctions are: המשוואות לא מצומדות על ידי דלתא כאן ניתן לראות שיש רק אלקטרון או רק חור. שני מצבים, בניגוד למוליך על בו יש קוואזי חלקיק

BTK Theory- Formalism To model the N-S interface, the interface is represented by a repulsive 𝛿-function potential V(x) = H We restrict our discussion to elastic tunneling processes The constraint that, for an incident particle with a positive group velocity only transmitted particles with positive group velocities and reflected ones with negative group velocities can be generated.

BTK Theory - Formalism Given an incident electron from the normal electrode, it can be either: Andreev-reflected as an hole Reflected normally as an electron – scattered Transmitted as an electron-like quasiparticle, or transmitted as a hole-like quasiparticle

BTK Theory - Formalism For electrons traveling from N to S, the wavefunctions of the normal and superconducting electrodes consist of: A,b,c,d הסתברויות לפיזור או מעבר A,b בצד המתכת C,d בצד על מוליך

BTK Theory - Formalism Schematic diagram of energy vs. momentum at N-S interface The figure illustrates the four allowed processes for an incident Electron: the Andreev-reflected hole (A), the normal-reflected electron (B), the transmitted electron-like quasiparticle (C) and the transmitted hole-like quasiparticle (D)

BTK Theory - Formalism Schematic diagram of energy vs. momentum at N-S interface Normal-Reflected Electron The figure illustrates the four allowed processes for an incident Electron: the Andreev-reflected hole (A), the normal-reflected electron (B), the transmitted electron-like quasiparticle (C) and the transmitted hole-like quasiparticle (D) Transmitted Quasiparticles C – electron-like D – hole-like Andreev-reflected hole

BTK Theory - Formalism Still, the coefficients A,B,C,D are unknown. We apply the following boundary conditions: The wavefunction values are continuous at the interface 𝜓 𝑁 (0) = 𝜓 𝑆 (0) = 𝜓(0) The derivative of the wavefunctions satisfies the equation:

BTK Theory - Formalism The solution is then Defining 𝑍= 𝑚𝐻 ℏ 2 𝑘 𝑓 and

BTK Theory - Formalism Consequently, when a bias voltage is applied, the total current flowing from the normal to the superconducting electrode is The zero-temperature differential tunneling conductance is proportional to

BTK Theory – I-V Curves For an N/N junction, the I-V Curve is simply linear T – Matrix element, 1 if barrier = 0 N1 צפיפות המצבים במתכת אחת * הסתברות פרמי דיראק למעבר אלקטרון N2 צפיפות המצבים במתכת השניה באנרגיה שונה * 1-FD זו ההסתברות למעבר חור יש זרם הפוך מ 2 ל 1 והזרם הכולל הוא הפרש הזרמים.

BTK Theory – I-V Curves For an N/S Junction the current is given by: פתרון נומרי של המשוואות – עבור חוזק מחסום שונה עבור טמפרטורה אפס אין זרם כאשר האנרגיה שניתנת לאלקטרון קטנה מדלתא

BTK Theory – I-V Curves Differential tunneling conductance vs Bias voltage for various normalized barrier heights Z T=0

BTK Theory – I-V Curves For an S1/S2 Junction:

Weak Links A weak link is a junction between two electrodes where the current flow takes place in direct way, not via tunnel effect through an insulating layer as in the tunnel junctions Weak link צומת בין שתי אלקטרודות בינהן הזרם זורם ישירות - לא במנהור.

Subharmonic Gap Structures Measured I-V Curves showed series of current jump at eV = 2∆/𝑛 This series of current jumps is known as subharmonic gap structure due to its localization in voltages

OBTK Theory A theory which tried to explain and model the creation of the subharmonic gap structure (SGS) Based on Boltzmann equation but assuming the same scattering/transmission mechanisms Normal scattering and Multiple-Andreev-reflections is included in the model In the limit of zero scattering the technique is shown to be equivalent to the trajectory technique of KBT we discussed earlier  the Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in thermodynamic equilibrium.

OBTK Theory The problem is a generalization of the BKT model to two symmetric N/S junctions, as shown below The normal metal is the weak link between the two superconducting electrodes

OBTK Theory

OBTK Theory במבנה הספציפי הזה תתכן יותר מהחזרה אחת אלקטרון שהולך ימינה מרוויח אנרגיה חור שהולך שמאלה מרוויח גם אנרגיה מספר הפעמים שזה יכול לקרות - n והתנאי ל AR זה neV<2Delta דרך נוספת זה להסתכל על הולכה אפקטיבית של אלקטרון יחיד בעל אנרגיה neV

OBTK Theory Based on direction of motion, the electrons are separated into two subpopulations The two subpopulations are described by two different non-equilibrium distribution functions 𝑓 → (E,x) and 𝑓 ← (E,x) Neither of which is assumed to be approximated by the equilibrium Fermi function within the normal region Functions of Energy and space

OBTK Theory We write the boundary conditions for the subpopulation equations: Electrons with energy E at x =0 have energy E+eV when they arrive at the other superconductor at x =L. Similarly, electrons starting with energy E at x =L will arrive at x =0 with energy E—eV. electrons with energy E at x =0 have energy E+eV when they arrive at the other superconductor at x =L. Similarly, electrons starting with energy E at x =L will arrive at x =0 with energy E—eV. אלקטרון עם אנרגיה E ב X=0 מגיעים ל X=L עם אנרגיה E+eV באופן דומה אלקטרון שמגיע מ X=L ל X=0 עם אנריגה E מאבד אנרגיה ומגיע עם E-eV

OBTK Theory The coefficients A(E), B(E), and T(E) describe Andreev reflection, normal reflection, and transmission

OBTK Theory In the absence of scattering, B=0 יש מחזוריות של הפונקציה בקטע -Delta:Delta

OBTK Theory Eliminating dependency in 𝑓 ← (E,x) we get משוואת ההסתברות באופן כללי

OBTK Theory Calculating the current: Specifically for the special case where B=0: המקרה המנוון B=0 הוא המקרה בו אין פיזור רגיל בממשק, אלא רק Andreev reflections והעברות. המקרה הזה מדגיש את הקפיצות בזרם קרי Subharmonic Gap structure

OBTK Theory - Calculations Normalized Resistance vs Normalized Voltage

OBTK Theory - Calculations

THE EnD THANK YOU Yanir Schwartz