Reversal of the superconducting diode effect

The recently discovered superconducting diode effect requires a magnetic field to break the time reversal symmetry. Our experiments show that, beyond a certain magnetic field threshold, the sign of the effect can abruptly change. A simple model links such reversal to the elusive 0-π-like transition.
Published in Physics
Reversal of the superconducting diode effect

Transistors, the building blocks of computer's CPUs, dissipate heat. This is because a transistor is a current switch, thus some current must pass through the very material of the transistor itself. Since this is resistive, the process is dissipative. Indeed, there are special transistors which do not dissipate heat, the so-called Josephson junction field effect transistors. They are based on the Josephson junction, a sandwich between two superconductors and a normal conductor, which still carries a zero-resistance current. After its discovery by B. Josephson (awarded by the Nobel Prize in 1973), Josephson junctions quickly found applications in various fields such as medicine, metrology, and astrophysics.  More recently, they became key components of quantum computers, since they are at the heart of transmons, the qubit-carrying devices in superconducting quantum processors. The initial announcement of the first superconducting diode based on a Josephson junction [1], following the discovery of the same effect in thin films [2], caused significant excitement. This enthusiasm arises from the potential for superconducting diodes to serve as the fundamental building blocks for new types of superconducting transistors, similar to how semiconductor diodes were the main component of the first transistors.

The distinctive property of an ordinary semiconducting diode is its asymmetry: its resistance can be very high or very low depending on which of its two terminals is connected to the cathode and which of on the anode of your battery. This asymmetry leads to the diode's most important property: current rectification. Instead, a superconducting diode exhibits no resistance, so its working principle must be different. In a superconducting diode zero-resistance current (a supercurrent) can indeed flow in both directions . What distinguishes the two polarities is the critical current value –the current threshold above which the junction switches to the resistive state. The graph in Fig. 1c shows the voltage (dissipation) plotted as a function of the current applied. The color indicates the two directions of the (super)current: green is the “+” direction, orange is the “-” (but I could have as well called them left/right, up/down, Joe/Donald directions).  For sufficiently low current, both the “+” curve and the “-“ one show no dissipation. At a certain threshold value (labelled as Ic), the device switches to the dissipative state. The working principle of a superconducting diode is based on the difference between Ic+ and Ic-, which gives rise to a window of current where the device can carry a zero-resistance current only in one direction –a diode-like behavior. The question is: what determines this direction [3]?

Figure 1
Figure 1. a, (Left) reverse and (right) forward biased semiconducting diode (pn junction). (Bottom) Ideal resistance versus voltage curve. Negative voltages correspond to reverse bias. b, (left) current-biased SNS Josephson junction where electric field (leading to spin-orbit interaction) E, magnetic field B, and current vector form a left handed set of vectors. If the polar electric field gives rise to strong spin-orbit interaction, then strong magnetochiral anisotropy effects arise. (Right) corresponding right-handed case. (Bottom) Inductance L versus current I., in the presence of magnetochiral anisotropy. Positive currents correspond to the left-handed case. c, Voltage-current characteristics for a SNS junction with magnetochiral anisotropy. The positive direction is arbitrarily chosen to be that of the left-handed case. The diode effect consists in a difference between the critical current for positive and negative bias. Such difference gives rise to a finite window (gray dotted lines) where dissipation-free current can only flow in one direction. d, 3D sketch of a SNS Josephson junction, with the indication of the E, B, and I vectors.

The direction for which the critical current is larger is determined by applying an external magnetic field. Here's how it works: electrons within superconducting diodes are confined to quasi-2D systems and subjected to an electric field perpendicular to the 2D plane. This electric field couples to the electron spin via the so-called spin-orbit interaction. It turns out that if the supercurrent, the applied in-plane magnetic field, and  such build-in electric field form a right handed  trio of vectors (as the usual right handed Cartesian coordinates,  see right side of Fig. 1b) then that direction of supercurrent has larger critical current (and label it as “+”). This is the reason why the superconducting diode effect originated in this way is called a magnetochiral anisotropy effect.

With this picture in mind, one would expect that the larger the magnetic field, the larger the difference between  Ic+ and Ic-, that is, the larger the diode effect. And, indeed this is what was found in early experiments, including those by our group. However, we soon found that beyond a certain threshold the effect was (i) abruptly suppressed, then (ii) it changed its sign. The change of sign was particularly visible in inductance measurements. It is well known that a Josephson junction has an inductance L associated with it, and this inductance is, in the ordinary case, a U-shaped function of the DC current I. Without magnetochiral effects, such function is symmetric around zero current. In a superconducting diode such L(I) function becomes asymmetric (see bottom part Fig.1b) and such asymmetry is relatively easy to measure. As a side note, the asymmetry of L(I) allows us to establish a more direct parallel with the semiconducting diode. We mentioned that the distinctive feature of a diode is a resistance that is a highly asymmetric function of the voltage V. The superconducting diode has no resistance for DC current, but, owing to its inductance, it has anyway a finite impedance for AC signals, and such impedence is an asymmetric function of the bias, similarly to the semiconducting diode. To quantify the sign and the magnitude of the asymmetry we could take the sign and magnitude of the current minimum for L(I), or the slope of L(I)  at zero current, which we label as L0'.

DC and AC diode effect: experiment versus theory
a, Difference between positive and negative critical current in our Josephson diode, plotted as a function of the in-plane field. (Inset) Zoom-in in a very narrow range on the vertical scale. For in-plane magnetic field above 200 mT, ΔIc changes sign, although its magnitude is at the limit of visibility. b, Results of the analytical calculations based on a minimal model developed by the group of J. Fabian at the University of Regensburg. c, Measured zero-bias slope of the L(I) curve (figure of merit of the magnetochiral anisotropy for the inductance) plotted versus in-plane field. d, Corresponding result of the calculations based on the minimal analytical model.

What our group found (studying samples grown at Purdue University my Prof. M. Manfra's group) is that the asymmetry of L(I) changed abruptly above a certain threshold magnetic field, reaching a (negative) peak value much larger in magnitude than the original positive peak. Such mysterious reversal of the diode effect was robust and reproducible. Nearly at the same time, our theory colleagues Andreas Costa and Denis Kochan in the group of Prof. Fabian at the University of Regensburg were trying to model the superconducting diode effect in Josephson junctions using an analytical model, that is, a pencil-and-paper description of the effect. In fact, the only existing descriptions were numerical simulations based on tight-binding models [4]. Such numerical description are definitely useful, but at the same time unsatisfactory, because they do not provide an insight of the relevant mechanism at work. Analytical models, on the other hand, require drastic simplifications to be treatable. In our case, the model by Andreas Costa, Denis Kochan and colleagues could be considered a minimal model, nearly the simplest possible description of the systems which still contained the essential ingredients. The result of the analytical calculations was astonishing: despite its bare simplicity, the model captured qualitatively and even semi-quantitatively, all the results of the experiments, including the mysterious sign change of the diode effect. Even more interestingly, the model linked such reversal to the so-far-elusive 0-π-like transition, a ground state change of the system induced by the magnetic field.  

[1] C. Baumgartner, L. Fuchs, A. Costa, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. E. Faria Junior, D. Kochan, J. Fabian, N. Paradiso, and C. Strunk, Supercurrent rectification and magnetochiral effects in symmetric Josephson junctions, Nature Nanotechnology 17, 39 (2022).

[2] F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota, T. Moriyama, Y. Yanase, and T. Ono, Observation of superconducting diode effect, Nature 584, 373 (2020).

[3] Here it comes the main difference with the ordinary semiconductor diode. This latter is a junction between two distinguishable semiconductors: one is doped with impurities which release electrons (n side), the other with impurities that capture electrons (p side). Also, the diode is biased with finite voltage, thus it makes clearly a difference whether I connect the positive voltage to the p side or to the n side. Instead, a superconducting diode is, in principle, completely symmetric: its two terminals are indistinguishable, at least at the macroscopic level. Moreover, a supercurrent is an equilibrium (that is, nondissipative) current. Due to the time-reversal symmetry of the laws of physics, the flow of a supercurrent in one direction must be indistinguishable from that in the opposite direction. For a dissipative flow, it is the second law of thermodynamics that breaks the time symmetry –you have to apply a voltage and electron current only flows from the negative to the positive terminal of your battery, that’s the only direction for which entropy increases and dissipation takes place.  For a Josephson junction, the time-reversal symmetry is explicitly broken by applying an external magnetic field, which couples to the spin.

[4] T. Yokoyama, M. Eto, and Y. V. Nazarov, Anomalous Josephson effect induced by spin-orbit interaction and Zeeman effect in semiconductor nanowires, Phys. Rev. B 89, 195407 (2014).

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