Josephson effect

Josephson junction array chip developed by the National Bureau of Standards as a standard volt

The Josephson effect is the phenomenon of supercurrent—i.e. a current that flows indefinitely long without any voltage applied—across a device known as a Josephson junction (JJ), which consists of two superconductors coupled by a weak link. The weak link can consist of a thin insulating barrier (known as a superconductor–insulator–superconductor junction, or S-I-S), a short section of non-superconducting metal (S-N-S), or a physical constriction that weakens the superconductivity at the point of contact (S-s-S).

The Josephson effect is an example of a macroscopic quantum phenomenon. It is named after the British physicist Brian David Josephson, who predicted in 1962 the mathematical relationships for the current and voltage across the weak link.[1][2] The DC Josephson effect had been seen in experiments prior to 1962,[3] but had been attributed to "super-shorts" or breaches in the insulating barrier leading to the direct conduction of electrons between the superconductors. The first paper to claim the discovery of Josephson's effect, and to make the requisite experimental checks, was that of Philip Anderson and John Rowell.[4] These authors were awarded patents on the effects that were never enforced, but never challenged.

Before Josephson's prediction, it was only known that normal (i.e. non-superconducting) electrons can flow through an insulating barrier, by means of quantum tunneling. Josephson was the first to predict the tunneling of superconducting Cooper pairs. For this work, Josephson received the Nobel prize in physics in 1973.[5] Josephson junctions have important applications in quantum-mechanical circuits, such as SQUIDs, superconducting qubits, and RSFQ digital electronics. The NIST standard for one volt is achieved by an array of 19,000 Josephson junctions in series.[6]

Applications

The electrical symbol for a Josephson junction

Types of Josephson junction include the pi Josephson junction, varphi Josephson junction, long Josephson junction, and Superconducting tunnel junction. A Dayem bridge is a thin-film variant of the Josephson junction in which the weak link consists of a superconducting wire with dimensions on the scale of a few micrometres or less.[7][8] The Josephson junction count of a device is used as a benchmark for its complexity.The Josephson effect has found wide usage, for example in the following areas:

The effect

Diagram of a single Josephson junction. A and B represent superconductors, and C the weak link between them.

The basic equations governing the dynamics of the Josephson effect are[14]

U(t) = \frac{\hbar}{2 e} \frac{\partial \phi}{\partial t} (superconducting phase evolution equation)
\frac{}{} I(t) = I_c \sin (\phi (t)) (Josephson or weak-link current-phase relation)

where U(t) and I(t) are the voltage and current across the Josephson junction, \phi(t) is the "phase difference" across the junction (i.e., the difference in phase factor, or equivalently, argument, between the Ginzburg–Landau complex order parameter of the two superconductors composing the junction), and Ic is a constant, the critical current of the junction. The critical current is an important phenomenological parameter of the device that can be affected by temperature as well as by an applied magnetic field. The physical constant \frac{h}{2 e} is the magnetic flux quantum, the inverse of which is the Josephson constant.

Typical I-V characteristic of a superconducting tunnel junction, a common kind of Josephson junction. The scale of the vertical axis is 50 μA and that of the horizontal one is 1 mV. The bar at \scriptstyle U = 0 represents the DC Josephson effect, while the current at large values of \scriptstyle |U| is due to the finite value of the superconductor bandgap and not reproduced by the above equations.

The three main effects predicted by Josephson follow from these relations:

The DC Josephson effect
The DC Josephson effect is a direct current crossing the insulator in the absence of any external electromagnetic field, owing to tunneling. This DC Josephson current is proportional to the sine of the phase difference across the insulator, and may take values between \scriptstyle  -I_c and \scriptstyle I_c.
The AC Josephson effect
With a fixed voltage \scriptstyle U_{DC} across the junctions, the phase will vary linearly with time and the current will be an AC current with amplitude \scriptstyle I_c and frequency \frac{2e}{h} U_{DC}. The complete expression for the current drive  I_\text{ext} becomes I_\text{ext} \;=\; C_J \frac{dv}{dt} \,+\, I_J \sin \phi \,+\, \frac{V}{R}. This means a Josephson junction can act as a perfect voltage-to-frequency converter.
The inverse AC Josephson effect
If the phase takes the form \scriptstyle \phi (t) \;=\;  \phi_0 \,+\, n \omega t \,+\, a \sin( \omega t), the voltage and current will be
U(t) = \frac{\hbar}{2 e} \omega ( n + a \cos( \omega t) ), \ \ \ I(t) = I_c \sum_{m \,=\, -\infty}^\infty J_m (a) \sin (\phi_0 + (n + m) \omega t).

The DC components will then be

U_{DC} = n \frac{\hbar}{2 e} \omega, \ \ \ I(t) = I_c J_{-n} (a) \sin \phi_0.

Hence, for distinct AC voltages, the junction may carry a DC current and the junction acts like a perfect frequency-to-voltage converter.

Josephson energy

The Josephson energy is the potential energy accumulated in a Josephson junction when a supercurrent flows through it. One can think of a Josephson junction as a non-linear inductance which accumulates (magnetic field) energy when a current passes through it. In contrast to real inductance, no magnetic field is created by a supercurrent in a Josephson junction — the accumulated energy is the Josephson energy.

For the simplest case the current-phase relation (CPR) is given by (aka the first Josephson relation):

I_s = I_c \sin(\phi),

where I_s\, is the supercurrent flowing through the junction, I_c\, is the critical current, and \phi\, is the Josephson phase. Imagine that initially at time t=0 the junction was in the ground state \phi=0 and finally at time t the junction has the phase \phi. The work done on the junction (so the junction energy is increased by)


U = \int_0^t I_s V\,dt
= \frac{\Phi_0}{2\pi} \int_0^t I_s \frac{d\phi}{dt}\,dt
= \frac{\Phi_0}{2\pi} \int_0^\phi I_c\sin(\phi) \,d\phi
= \frac{\Phi_0 I_c}{2\pi} (1-\cos\phi).

Here E_J = {\Phi_0 I_c}/{2\pi} sets the characteristic scale of the Josephson energy, and (1-\cos\phi) sets its dependence on the phase \phi. The energy U(\phi) accumulated inside the junction depends only on the current state of the junction, but not on history or velocities, i.e. it is a potential energy. Note, that U(\phi) has a minimum equal to zero for the ground state \phi=2\pi n, n is any integer.

Josephson inductance

Imagine that the Josephson phase across the junction is \phi_0\, and the supercurrent flowing through the junction is

I_0 = I_c \sin\phi_0\,

(This is the same equation as above, except now we will look at small variations in I_s\, and \phi\, around the values I_0\, and \phi_0\,.)

Imagine that we add little extra current (dc or ac) \delta I(t)\ll I_c through JJ, and want to see how the junction reacts. The phase across the junction changes to become \phi=\phi_0+\delta\phi\,. One can write:

I_0+\delta I = I_c \sin(\phi_0+\delta\phi)\,

Assuming that \delta\phi\, is small, we make a Taylor expansion in the right hand side to arrive at

\delta I = I_c \cos(\phi_0) \delta\phi\,

The voltage across the junction (we use the 2nd Josephson relation) is


V = \frac{\Phi_0}{2\pi}\dot{\phi} 
= \frac{\Phi_0}{2\pi}(\underbrace{\dot{\phi_0}}_{=0} + \dot{\delta\phi})
= \frac{\Phi_0}{2\pi} \frac{\dot{\delta I}}{I_c \cos(\phi_0)}.

If we compare this expression with the expression for voltage across the conventional inductance


 V = L \frac{\partial I}{\partial t}
,

we can define the so-called Josephson inductance


  L_J(\phi_0) = \frac{\Phi_0}{2\pi I_c \cos(\phi_0)}
  = \frac{L_J(0)}{\cos(\phi_0)}.

One can see that this inductance is not constant, but depends on the phase (\phi_0)\, across the junction. The typical value is given by L_J(0)\, and is determined only by the critical current I_c\,. Note that, according to definition, the Josephson inductance can even become infinite or negative (if \cos(\phi_0)<=0\,).

One can also calculate the change in Josephson energy


  \delta U(\phi_0) = U(\phi)-U(\phi_0) 
  = E_J (\cos(\phi_0)-\cos(\phi_0+\delta\phi)\,

Making Taylor expansion for small \delta\phi\,, we get


  \approx E_J \sin(\phi_0)\delta\phi
  = \frac{E_J \sin(\phi_0)}{I_c \cos\phi_0}\delta I

If we now compare this with the expression for increase of the inductance energy dE_L = L I \delta I\,, we again get the same expression for L\,.

Note, that although Josephson junction behaves like an inductance, there is no associated magnetic field. The corresponding energy is hidden inside the junction. The Josephson Inductance is also known as a Kinetic Inductance - the behaviour is derived from the kinetic energy of the charge carriers, not energy in a magnetic field.

See also

Wikimedia Commons has media related to Josephson effect.

References

  1. Josephson, B. D., "Possible new effects in superconductive tunnelling," Physics Letters 1, 251 (1962) doi:10.1016/0031-9163(62)91369-0
  2. Josephson, B. D. (1974). "The discovery of tunnelling supercurrents". Rev. Mod. Phys. 46 (2): 251–254. Bibcode:1974RvMP...46..251J. doi:10.1103/RevModPhys.46.251.
  3. Josephson, Brian D. (December 12, 1973). "The Discovery of Tunneling Supercurrents (Nobel Lecture)" (PDF).
  4. Anderson, P W; Rowell, J M (1963). "Probable Observation of the Josephson Tunnel Effect". Phys. Rev. Letters 10: 230. Bibcode:1963PhRvL..10..230A. doi:10.1103/PhysRevLett.10.230.
  5. The Nobel prize in physics 1973, accessed 8-18-11
  6. Steven Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion, 2003.
  7. Anderson, P. W., and Dayem, A. H., "Radio-frequency effects in superconducting thin film bridges," Physical Review Letters 13, 195 (1964), doi:10.1103/PhysRevLett.13.195
  8. Dawe, Richard (28 October 1998). "SQUIDs: A Technical Report - Part 3: SQUIDs" (website). http://rich.phekda.org. Retrieved 2011-04-21. External link in |publisher= (help)
  9. International Bureau of Weights and Measures (BIPM), SI brochure, section 2.1.: SI base units, section 2.1.1: Definitions, accessed 22 June 2015
  10. Practical realization of units for electrical quantities (SI brochure, Appendix 2). BIPM, [last updated: 20 February 2007], accessed 22 June 2015.
  11. Fulton, T.A.; et al. (1989). "Observation of Combined Josephson and Charging Effects in Small Tunnel Junction Circuits". Physical Review Letters 63 (12): 1307–1310. Bibcode:1989PhRvL..63.1307F. doi:10.1103/PhysRevLett.63.1307. PMID 10040529.
  12. Bouchiat, V.; Vion, D.; Joyez, P.; Esteve, D.; Devoret, M. H. (1998). "Quantum coherence with a single Cooper pair" (PDF). Physica Scripta T 76: 165. Bibcode:1998PhST...76..165B. doi:10.1238/Physica.Topical.076a00165.
  13. Physics Today, Superfluid helium interferometers, Y. Sato and R. Packard, October 2012, page 31
  14. Barone, A.; Paterno, G. (1982). Physics and Applications of the Josephson Effect. New York: John Wiley & Sons. ISBN 0-471-01469-9.
This article is issued from Wikipedia - version of the Friday, April 29, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.