Introduction to Josephson Tunneling and Macroscopic Quantum Tunneling

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Introduction to Josephson Tunneling and
Macroscopic Quantum Tunneling

• Marc Manheimer
• November 5, 1999

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Outline

• Review of Josephson Tunneling.
• Derivation of tilted washboard potential.
• Thermal lifetime (Fulton Dunkleberger).
• Macroscopic Quantum Tunneling (VossWebb).
• More recent work.

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Basic Tunnel Junction
NIN Tunneling
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NIS Tunneling
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SIS Tunneling
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SIS Tunneling

• No current flows at T0 until the gap voltage is
  exceeded.
• It takes 2D1 to break a Cooper pair, and leave it
  at the Fermi level, and another 2D2 to bring it
  to the conduction band in the second metal.
  (D1D2 per electron)
• The tunneling current is given by

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The Wavefunction

• The superconducting condensate is described by a
  Schrodinger equation, with wavefunction
• The phase of the wavefunction plays an important
  role in Josephson tunneling.

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Josephson Tunneling

• In 1962, Josephson predicted...
• A zero voltage super current
• An evolving phase difference, if a voltage is
  maintained across a junction

Oxide barrier
Metal 1
Metal 2
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Simple Derivation
Couple two superconductors
Separate real and imaginary
Impose a voltage between the two superconductors
We get Josephsons relationships with
Substitute the pair density
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Josephson Energy

• One can derive the coupling free energy stored in
  the junction by integrating the electrical work
  done by a current source in changing the phase
• With a convenient reference for f

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RSJ Model
Tilted Washboard Potential I

Icsinq
i
v
R
C
_
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The Potential
Tilted Washboard Potential II
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Mechanical Analogue
Tilted Washboard Potential III
q
G
mg
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Fulton Dunkleberger
Measured the effect of thermal noise on the
lifetime of the zero voltage state. They scanned
junction current, lowering the potential barrier,
until the junction made the transition into the
finite voltage state. The thermal lifetime is
given by The probability of switching to the
finite voltage state is
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Fulton Dunkleberger
H(K)
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Desired System Properties for QMT

• Metastable state separted from a continuum.
• Two macroscopically distinguishable states.
• Frequency of small oscillations high enough that
• Barrier height variable.
• Experimentally describable in classical terms.

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Voss Webb

• Verify thermal switching at high T
• As T0, the switching rate becomes dominated by
  quantum tunneling.

Caldeira and Leggett fix the parameters, at T0.
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Misc Parameters
For Voss Webb
Ic1.6mA
Ic160nA
2x1011sec-1
7x1010sec-1
3.2x10-4eV 3.5K
3.2x10-3eV 35K
For Fulton Dunkleberger
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Voss Webb
An interesting aside, is that VW write the
barrier as
Also, VW determined xI/Ic by fitting to the
exponential.
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Voss Webb
w/o zero point subtraction
Incl zero point subtraction
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Voss Webb
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Note Curves change with T in MQT regime, as Ic
continues to change.
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Finite Temperature MQT

• Subsequnt to VW, several groups developed a
  finite T model.
• MQT increases with T.
• Washburn, Webb, Voss Faris, published a
  follow-on which verifies predictions. PRL54,
  p2712 (1985).
• Groups at Berkeley and SUNY/SB also verified
  predictions.

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WWVF
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WWVF