1. Introduction to Josephson Tunneling and Macroscopic Quantum Tunneling
Marc Manheimer
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2. Outline Review of Josephson Tunneling.
Derivation of tilted washboard potential.
Thermal lifetime (Fulton & Dunkleberger).
Macroscopic Quantum Tunneling (Voss&Webb).
More recent work.
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3. Basic Tunnel Junction i v
NIN Tunneling
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4. NIS Tunneling
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5. SIS Tunneling
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6. SIS Tunneling
No current flows at T=0 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. (D1+D2 per electron)
The tunneling current is given by:
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7. 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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8. 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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9. Simple Derivation… Couple two superconductors…
Separate real and imaginary…
Impose a voltage between the two superconductors…
We get Josephson’s relationships with:
Substitute the pair density…
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11. 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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12. RSJ Model Tilted Washboard Potential I + i v Icsinq R C _
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13. Tilted Washboard Potential II
The Potential
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14. Mechanical Analogue Tilted Washboard Potential III q G mg
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17. 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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18. Fulton & Dunkleberger
H(K)
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20. 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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21. Voss & Webb Verify thermal switching at high T
As T®0, the switching rate becomes dominated by quantum tunneling.
Caldeira and Leggett fix the parameters, at T=0.
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22. Misc Parameters For Voss & Webb: Ic=1.6mA Ic=160nA 2x1011sec-1
3.2x10-3eV
~35K
3.2x10-4eV
~3.5K
For Fulton & Dunkleberger:
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23. Voss & Webb An interesting aside, is that V&W write the barrier as:
Also, V&W determined x=I/Ic by fitting to the exponential.
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24. Voss & Webb w/o zero point subtraction Incl zero point subtraction
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25. Voss & Webb
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26. Note: Curves change with T in MQT regime, as Ic continues to change.
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29. Finite Temperature MQT
Subsequnt to V&W, 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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30. WWV&F
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31. WWV&F
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