# Tag Archives: Josephson Junctions

## Precision in Many-Body Systems

Measurements of the quantum Hall effect give a precise conductance in units of $e^2/h$. Measurements of the frequency of the AC current in a Josephson junction give us a frequency of $2e/h$ times the applied voltage. Hydrodynamic circulation in liquid 4He is quantized in units of $h/m_{4He}$. These measurements (and similar ones like flux quantization) are remarkable. They yield fundamental constants to a great degree of accuracy in a condensed matter setting– a setting which Murray Gell-Mann once referred to as “squalid state” systems. How is this possible?

At first sight, it is stunning that physics of the solid or liquid state could yield a measurement so precise. When we consider the defects, impurities, surfaces and other imperfections in a macroscopic system, these results become even more astounding.

So where does this precision come from? It turns out that in all cases, one is measuring a quantity that is dependent on the single-valued nature of the (appropriately defined) complex scalar  wavefunction. The aforementioned quantities are measured in integer units, $n$, usually referred to as the winding number. Because the winding number is a topological quantity, in the sense that it arises in a multiply-connected space, these measurements do not particularly care about the small differences that occur in its surroundings.

For instance, the leads used to measure the quantum Hall effect can be placed virtually anywhere on the sample, as long as the wires don’t cross each other. The samples can be any (two-dimensional) geometry, i.e. a square, a circle or some complicated corrugated object. In the Josephson case, the weak links can be constrictions, an insulating oxide layer, a metal, etc. Imprecision of experimental setup is not detrimental, as long as the experimental geometry remains the same.

Another ingredient that is required for this precision is a large number of particles. This can seem counter-intuitive, since one expects quantization on a microscopic rather than at a macroscopic level, but the large number of particles makes these effects possible. For instance, both the Josephson effect and the hydrodynamic circulation in 4He depend on the existence of a macroscopic complex scalar wavefunction or order parameter. In fact, if the superconductor becomes too small, effects like the Josephson effect, flux quantization and persistent currents all start to get washed out. There is a gigantic energy barrier preventing the decay from the $n=1$ current-carrying state to the $n=0$ current non-carrying state due to the large number of particles involved (i.e. the higher winding number state is meta-stable). As one decreases the number of particles, the energy barrier is lowered and the system can start to tunnel from the higher winding number state to the lower winding number state.

In the quantum Hall effect, the samples need to be macroscopically large to prevent the boundaries from interacting with each other. Once the states on the edges are able to do that, they may hybridize and the conductance quantization gets washed out. This has been visualized in the context of 3D topological insulators using angle-resolved photoemission spectroscopy, in this well-known paper. Again, a large sample is needed to observe the effect.

It is interesting to think about where else such a robust quantization may arise in condensed matter physics. I suspect that there exist similar kinds of effects in different settings that have yet to be uncovered.

Aside: If you are skeptical about the multiply-connected nature of the quantum Hall effect, you can read about Laughlin’s gauge argument in his Nobel lecture here. His argument critically depends on a multiply-connected geometry.

## Transport Signatures in Charge Density Wave Systems

This post is inspired in part by Inna’s observation that a Josephson junction can act as a DC-AC converter. It turns out that CDWs can also act in a similar manner.

Sometimes I feel like quasi-1D charge density waves (CDWs) are like the lonely neglected child compared to superconductors, the popular, all-star athlete older sibling. Of course this is so because superconductors carry dissipationless current and exhibit perfect diamagnetism. However, quasi-1D CDWs can themselves exhibit pretty stunning transport signatures associated with the CDW condensate. Note that these spectacular properties are associated with incommensurate CDWs, as they break the translational symmetry of the crystal.

To make a comparison with superconductivity (even though no likes to be compared to their older sibling), here is a cartoon of the frequency-dependent conductivity (taken from G. Gruner’s Review of Modern Physics entitled Dynamics of Charge Density Waves):

Frequency-dependent conductivity for (a) a superconductor and (b) an incommensurate CDW

In the superconducting case, there is a delta function at zero frequency, indicative of dissipationless transport. For the CDW, there is also a collective charge transport mode, but in this case it is at finite energy (as it is pinned by impurities), and it is dissipative (indicated by the finite width).

This collective charge transport mode can be “depinned” and results in a nonlinear conductivity known as  a sliding CDW. This is evidenced below in the I-V characteristics. Below a threshold electric field/voltage, usual Ohmic characteristics are observed, associated with the “normal” non-condensed electrons. However, above the threshold electric field/voltage the collective mode is depinned and contributes to the I-V characteristics.

Non-linear IV characteristics indicative of collective charge transport in the CDW phase

Even more amazingly, once this CDW has been depinned, applying a DC field results in an AC response. Below is an image from a famous paper by Fleming and Grimes showing the Fourier transformed AC response with several harmonics. As the voltage is turned up, the fundamental frequency increases markedly (the voltage is highest in (a) and is decreased slowly until (e) where the CDW is no longer sliding).

AC response to a DC applied voltage in order of decreasing DC voltage in NbSe3. (a) V=5.81mV, (b) V=5.05mV, (c) 4.07mV, (d) V=3.40mV (e) V=0

The observed oscillation frequency is due to the collective mode getting depinned from its impurity site and then getting  weakly pinned successively by impurities, though this picture is debated. N.P. Ong, who did some great early work on CDW transport, has noted that the CDW “sings”. A nice cartoon of this idea is presented in the ball-and-egg-crate model shown below. One can imagine the successive “hits in the road” at periodic time intervals resulting in the AC response seen above.

Ball and egg crate model of CDW transport

Hopefully this post will help people appreciate more the shy younger sibling that is the charge density wave.

All images taken from G. Gruner RMP 60, 1129 (1988).

## Let there be (THz) light

The applications of scientific discoveries is sometimes not what you would expect, and high temperature superconductivity is no different.  When high-temperature superconductivity was discovered in copper-oxides (cuprates) in 1986, the envisioned applications were power lines, electromagnets, and maglev trains, all cooled by cheap-as-milk liquid nitrogen.  While applications involving high-temperature superconductors’ dissipationless and diamagnetic properties are slowly coming online, there are other potential technologies which most people are less aware of.  The one I want to discuss here is using the layered structure of cuprate high temperature superconductors to produce coherent THz emitters.  Creating light sources and detectors for the THz portion of the electromagnetic spectrum—the notorious THz gap—has been a pressing challenge for decades.

The Josephson effect

The Josephson effect underlies many important applications of superconductors, such as sensitive magnetometers, qubits for quantum computing, and the SI definition of the volt.  The starting point for the Josephson effect is a superconductors’ complex order parameter, $\Psi=\Psi_0 e^{\imath\varphi}$.  The amplitude, $\Psi_0$, is related to some measure of the robustness of the superconducting state–either the superfluid density or the superconducting gap.  The phase, $\varphi$, reflects that a superconductor is a phase-coherent state–a condensate.  At $T_c$, a superconductor chooses an arbitrary phase, and a current in a superconductor (a supercurrent) corresponds to a gradient in this phase.  A Josephson junction, sketched below, consists of two superconductors separated by a non-superconducting barrier.  Because each superconductor chooses an arbitrary phase and the superconducting wavefunctions can penetrate into the barrier, a phase gradient develops in the barrier region, and a supercurrent can flow.  This supercurrent is given by $I_s=I_c\sin(\delta\varphi)$, where  $I_c$ is the critical current which causes the Josephson junction to become resistive (different from the critical current which makes the superconductor resistive) and  $\delta\varphi$ is the phase difference between the two superconductors.  This is the DC Josephson effect.  In the resistive regime ($I>I_c$), one encounters the AC Josephson effect, in which the Josephson junction supports an oscillating current with AC Josephson frequency $\omega=\frac{2\pi V}{\Phi_0}$, where V is the voltage across the junction and $\Phi_0$ is the magnetic flux quantum.  The current in this regime is given by: $I(t)=I_c\sin(\delta\varphi + \frac{2\pi V}{\Phi_0}t)$

Thus, a Josephson junction can convert a DC voltage to an AC current (and vis versa).

Schematic of a Josephson junction, consisting of two superconductors with a barrier in between. The barrier may be an insulator, a metal, or a constricted piece of superconductor. Each superconducting slab has a complex wavefunction with an arbitrarily chosen phase, $\varphi_{1,2}$. Supercurrent through a Josephson junction depends on the phase difference, $\delta\varphi=\varphi_1-\varphi_2$

Schematic of IV curve of Josephson junction (solid line), from Ref [1]. For sufficiently small bias currents, a supercurrent flows through the junction and no voltage is sustained–the regime of the DC Josephson effect. At currents exceeding $I_c$, the junction becomes resistive and is able to sustain a voltage across it, even though each superconducting slab remains superconducting. This is where the AC josephson effect is realized. The dashed line is an ohmic resistance, which a Josephson junction approaches in the limit of high bias voltage.

More is better

While a Josephson junction in its resistive regime is a perfect DC to AC converter with frequency proportional to voltage, the amount of power it can output is limited by the fact that device performance (and eventually superconductivity) degrade if you crank the voltage up too high.  However, it turns out that if you have multiple Josephson junctions in series, the available power scales with the number of junctions, and if all of these junctions oscillate in phase, they can form a coherent radiation source.  This is where high temperature superconductors come in.

Calling all cuprates

The crystal structure of cuprate high temperature superconductors consists of $CuO_2$ sheet where superconductivity originates, separated by insulating layers.  While the $CuO_2$ sheets are coupled with each other, the coupling can be weak in some cuprates, such that the material behaves like an array of intrinsic Josephson junctions in series.  Thus, a structure which has to be specially manufactured for other superconductors, the cuprates give for free.

Crystal structure of the high temperature superconductor, $Bi_2Sr_2CaCu_2O_{8+\delta}$ (BSCCO) which is most commonly used to make THz emitters. The layered structure of cuprates—superconducting $CuO_2$ layers separated by insulating intervening layers– permits the material itself to be a series of Josephson junctions. Adapted from Ref. [2].

The first step to making a cuprate superconductor into an emitter of coherent THz radiation is to pattern a single crystal into a smaller structure called a mesa. The mesa behaves as a resonant cavity such that a half-integer number of wavelengths ($\lambda/2$) of radiation fit into the width, w, of the device.  The lowest-order resonance condition is met when the AC Josephson frequency is equal to the frequency of a cavity mode, $\omega_c=\frac{\pi c_0}{n w}$, where $c_0/n$ is the mode propagation velocity in the medium and n is the far-infrared refractive index.  For a given mesa width, the resonance condition is met for a specific value of applied voltage for each Josephson junction, $V_{jj}=\frac{c_0 \Phi_0}{2 w n}$.  For a stack of Josephson junctions in series, the applied voltage scales with the number of junctions (N): $V=NV_{jj}$

A schematic of such a device is shown below.  The mesa, produced by ion milling, is 1-2 microns high (corresponding to ~1000 intrinsic Josephson junctions), 40-100 microns wide (setting the resonance emission frequencies), and several hundred microns in length.  A voltage is applied along the height of the stack and THz radiation is emitted out the side of the stack.  Devices have been fabricated with emissions at frequencies between 250 GHz and 1THz.  Linewidths of ~10MHz have been achieved as have radiation powers of 80 microwatts, though it is predicted that the latter figure can be pushed to 1mW [2,3].  The emission frequency can be tuned either by fabricating a new device with a different width, or by fabricating a device shaped like a trapezoid or a stepped pyramid and varying the bias voltage [4].  The latter corresponds to different numbers of Josephson junctions in the stack oscillating coherently.

THz emitter made out of high-temperature superconducting cuprates. A ‘mesa’ is ion-milled from a single crystal of BSCCO with a restricted width dimension, w. THz radiation is emitted out of the side, with frequency depending on the width of the mesa and the applied voltage. From Refs [2-3].

Emission spectra of three devices with different widths,w, made out of high temperature superconductors operated at T~25K. Inset shows linear relationship between frequency and 1/w. From Ref [3]

Implications

Successful fabrication of coherent THz emittors out of high temperature superconductors is a relatively new achievement and there is additional progress to be made, particularly towards increasing the emitted power.  This technology is promising for filling in a portion of the THz gap outside the capabilities of quantum cascade lasers, whose lowest emission frequency is presently 1.6THz.  In the future, one can imagine a light source consisting of an array of BSCCO mesas of different dimensions producing a narrow-bandwidth lightsource which is tuneable between 250GHz and 1.5THz for security and research applications.

References

[1] J. Annett. Superconductivity, Superfluids, and Condensates, Oxford University Press (2003)

[2] U. Welp et alNature Photonics 7 702 (2013)

[3] L. Ozyuzer et al, Science 318 1291 (2007)

[4] T. M. Benseman et al, Phys. Rev. B 84 064523 (2011)

## Simple, beautiful and decisive.

Condensed matter physics has seen its fair share of landmark experiments. In the field of high temperature superconductivity, there is one series that stands out among the rest, however. These are the Josephson Interference experiments conducted in the early 90s by van Harlingen and co-workers. These investigations were technically challenging, and some hard work no doubt went into the experimental design.

To understand the impact of the experiments, a little background and historical context is needed. At the time, the field of high temperature superconductivity was approximately seven years old and the symmetry of the superconducting order parameter was an open question. It was known that the order parameter had nodes, but whether the order parameter changed sign (i.e. was d-wave or extended s-wave) was unresolved.

The Josephson Interference experiments unambiguously settled this issue. The second iteration of the experiment (PDF link!), was particularly decisive. Essentially, the expected critical current as a function of flux would look like so for an s-wave and d-wave superconductor respectively:

The authors obtained the latter pattern and the order parameter symmetry problem was solved (the symmetry was d-wave). It is not often in condensed matter physics that experiments are this clean, unambiguous and illuminating. When they are, however, they deserve to be celebrated.