1. Subharmonic gap Structures
Yanir Schwartz

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

3. 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

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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

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

13. 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

14. 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

15. 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

16. 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

17. 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.

18. 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

19. 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

20. 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)

21. 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

22. 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:

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

24. 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

25. 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.

26. 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

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

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

29. 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.

30. 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

31. 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.

32. 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

33. OBTK Theory

34. 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

35. 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

36. 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

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

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

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

40. 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

41. OBTK Theory - Calculations
Normalized Resistance vs Normalized Voltage

42. OBTK Theory - Calculations

43. THE EnD
THANK YOU
Yanir Schwartz