The theory of planar ballistic SNS junctions

Abstract The paper presents the theory of planar ballistic SNS junctions with equal Fermi velocities and effective masses in all layers. The theory takes into account phase gradients in superconducting layers commonly ignored in the past. At $$T=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , the current-phase relation was derived for any thickness L of the normal layer in the model of the steplike pairing potential model analytically. The obtained current-phase relation is essentially different from that in theory neglecting phase gradients, especially in the limit $$L\rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> (short junction). The analysis resolves the problem with the charge conservation law in the steplike pairing potential model. The current-phase relation at temperatures exceeding the energy spacing between Andreev levels but less than the critical temperature was also calculated numerically. The current at these temperatures is temperature independent and decreases with growing L as $$1/L^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> </mml:math> . The previous theory predicted the current exponentially decreasing with growing T and L . Possible implications of the analysis for planar junctions with non-equal Fermi velocities and for non-planar junctions (narrow normal bridge between two bulk superconductors) are also discussed.