Basic Josephson Junctions

Lecture 11: Basic Josephson Junctions

Outline 1. Quantum Tunneling 2. Josephson Tunneling 3. Josephson current-phase and current-voltage relations 4. Basic Josephson Junction lumped element 5. AC Josephson Effect 6. DC voltage standard

October 9, 2003

Massachusetts Institute of Technology 6.763 2003 Lecture 11 The Nobel Prize in Physics 1973

"for his theoretical "for their experimental predictions of the discoveries regarding properties of a tunneling phenomena supercurrent through a in semiconductors and tunnel barrier, in particular superconductors, those phenomena which respectively" are generally known as the Josephson effects"

Leo Esaki Ivar Giaever Brian David Josephson 1/4 of the prize 1/4 of the prize 1/2 of the prize

Japan USA United Kingdom

IBM Thomas J. General Electric University of Watson Research Company Cambridge Center Schenectady, NY, Cambridge, United Yorktown Heights, USA Kingdom NY, USA Massachusetts Institute of Technology b. 1925 b. 1929 b. 1940 •http://www.nobel.se/physics/laureates (in Bergen, 6.763 2003 Lecture 11 Norway) Tunneling between a normal metal and another normal metal or a superconductor

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Massachusetts Institute of Technology http://www.nobel.se/physics/laureates/1973/giaever-lecture.pdf 6.763 2003 Lecture 11 Tunneling between two superconductors

Giaever Tunneling

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Josephson Tunneling

Massachusetts Institute of Technology 6.763 2003 Lecture 11 Tunneling Summary: We are only concerned with the Josephson Tunneling in a Basic Junction

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Massachusetts Institute of Technology 6.763 2003 Lecture 11 In the superconducting electrodes:

The Supercurrent Equations govern the electrodes,

Even in the absence of E&M fields, a gradient of the phase can cause a current and the time change of that phase can cause a voltage. For example, for

a constant current Jo, at the boundaries we find

So that the wavefunction in the electrode is

Massachusetts Institute of Technology 6.763 2003 Lecture 11 In the insulator

The current must be continuous, so it must flux through the insulating barrier; a process which is not allowed classically. But quantum mechanically the superelectrons can tunnel through the insulating barrier as a supecurrent with zero voltage. This is the Josephson current.

Because the supercurrent equation does not hold in the insulating region, the full macroscopic wave equation must be used to find Y in the insulating region, with the boundary conditions given by the wavefunction at the electrodes.

Tunneling Potential Barrier Massachusetts Institute of Technology 6.763 2003 Lecture 11 Tunneling through the Barrier

The energy of the superelectron is less than the barrier height, so that no classical particles flow.

constant

Therefore, in the insulating region

Where so that

Massachusetts Institute of Technology 6.763 2003 Lecture 11 Tunneling through the Barrier

At the boundaries.

So that

Therefore,

with

Massachusetts Institute of Technology 6.763 2003 Lecture 11 Josephson Current-Phase relation

In the presence of and electromagnetic field, the Josephson current-phase relation generalizes to

where the gauge-invariant phase is defined as

Which is invariant under , , Massachusetts Institute of Technology 6.763 2003 Lecture 11 Josephson Voltage-Phase relation

The gauge-invariant phase is

The rate of change of the gauge-invariant phase is

At the boundary in the electrodes, so that

Therefore, or

Massachusetts Institute of Technology 6.763 2003 Lecture 11 Basic Lumped Junctions Distributed element

Ic is the critical current Lumped element

Massachusetts Institute of Technology 6.763 2003 Lecture 11 Energy in a Basic Josephson Junction

The energy WJ in the basic junction is

Use the Josephson relations to write as

Therefore,

Massachusetts Institute of Technology 6.763 2003 Lecture 11 AC Josephson Effect

The voltage source is DC with v=V0 , so that

+ Vo - The resulting current is ac!

The Josephson frequency is

A dc voltage of 10 µV causes an oscillation frequency of about 5 GHz, a

Josephson microwave oscillator. But with a typical Ic of 1 mA, this oscillator delivers a very small power of the order of 10 nW. Therefore need many synchronous oscillators. Massachusetts Institute of Technology 6.763 2003 Lecture 11 AC and DC voltage drives

Let

+ Then the guage-invariant phase is v(t) -

The current is FM-like:

Use the Fourier-Bessel series to express the current as a Fourier series

A dc current will occur when 2πfJ = n ωs , that is,

Massachusetts Institute of Technology 6.763 2003 Lecture 11 Principle of the DC Voltage Standard

+ v(t) -

An ac voltage of 1 GHz applied across the junction will give a dc current, at

V0= 0 and at dc voltages of integral multiples of 2µV.

The principle of the dc Volt: Put 5000 Josephson junctions in series, and apply a fixed frequency, which can be done very accurately, and measure the interval of the resulting dc voltages that occur at precise voltage intervals.

Massachusetts Institute of Technology 6.763 2003 Lecture 11 Summary: Basic Josephson Junction (I

iθ Ψ = n e 1 Superconductor 1 1 Insulator iθ Nb 2 ~10Å, Al2O3 Ψ2 = n2e

• Josephson relations: • Behaves as a nonlinear inductor:

dI I = Ic sinϕ V = LJ , Φ dϕ dt V = 0 2π dt Φ where 0 LJ = 2π Ic cosϕ ϕ = θ 2 −θ1 Φ = flux quantum 2π 1 ρ ρ 0 − ∫ A(r,t) ⋅ dl 2 483.6 GHz / mV Φ 0 Massachusetts Institute of Technology 6.763 2003 Lecture 11