Designing edge currents using mesoscopic patterning in chiral <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-wave superconductors

Chiral superconductors are topological as characterized by a finite Chern number and chiral edge modes. Direct fingerprints of chiral superconductivity are thus often taken to be spontaneous edge currents with associated magnetic signatures. However, a number of recent theoretical studies have shown that the total edge current along semi-infinite edges is greatly reduced or even vanishes in many scenarios for all pairing symmetries except chiral <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>p</a:mi></a:math> wave, thus impeding experimental detection. We demonstrate how mesoscopic finite-sized samples can be designed to give rise to a shape- and size-dependent strong enhancement of the chiral edge currents and their generated orbital magnetic moment and magnetic fields. In particular, we find that low rotational symmetry systems, such as pentagons and hexagons, give rise to the largest currents, while circular disks also generate large currents but in the opposite direction. We estimate the resulting magnetic fields to be as large as <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mn>0.01</b:mn><b:mo>–</b:mo><b:mn>0.5</b:mn></b:mrow></b:math> mT, with a magnetic moment approaching <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:msub><c:mi>μ</c:mi><c:mi mathvariant="normal">B</c:mi></c:msub><c:mo>/</c:mo><c:mn>2</c:mn></c:mrow></c:math> per Cooper pair, where <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msub><e:mi>μ</e:mi><e:mi mathvariant="normal">B</e:mi></e:msub></e:math> is the Bohr magneton. The current and magnetic signatures diverge with shrinking system sizes, eventually cut off by finite-size suppression of chiral superconductivity. We thus also extract the full phase diagram as a function of temperature and system size for different geometries, including competing superconducting orders. In geometries strongly suppressing only one of the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>d</g:mi></g:math>-wave components, we find an additional heat capacity jump, as large as <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mn>10</h:mn><h:mo>%</h:mo></h:mrow></h:math> of the bulk normal-superconducting transition, marking the transition between a chiral and a nodal <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>d</i:mi></i:math>-wave state. This further acts as an indirect signature of chiral superconductivity, measurable with nanocalorimetry. Our results are relevant for system sizes on the order of tens to hundreds of coherence lengths, and highlight mesoscopic patterning as a viable route to experimentally identify chiral <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>d</j:mi></j:math>-wave superconductivity.