Field free Josephson diode effect in a d wave superconductor heterostructure Hamed Vakili 1 Moaz Ali 1 and Alexey A Kovalev1 1Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience University of Nebraska Lincoln Nebraska 68588 USA Dated June 18 2024 We study superconductor normal region superconductor S N S Josephson junction formed using superconductors with d d id and d is superconducting pairings We show that the quality factor of the Josephson diode effect and its sign can be substantially tuned by the external magnetic field gate voltage and the length of the junction for all three types of pairings We also identify the conditions under which the anomalous Josephson and Josephson diode effects can appear in the junction by analyzing appropriate symmetries In particular by breaking a rotation symmetry we show how a large field free Josephson diode effect can be realized even in the absence of spin orbit coupling We also study the role of edge states appearing in the case of chiral superconductor with d id pairing Our results demonstrate that the Josephson diode effect in a planar geometry can be used as a signature of unconventional superconducting pairings I INTRODUCTION The superconducting diode SDE and Josephson diode effects JDE have attracted considerable atten tion in recent years due to many potential applications in superconducting devices and quantum computing 1 15 The effects are characterized by non reciprocal behavior in the critical current in particular the current can be dissipationless in one direction and dissipative in the op posite direction In the absence of both time reversal and inversion symmetry Josephson junctions JJ can exhibit the JDE characterized by non reciprocal behav ior in the critical supercurrent Ic 16 26 This means c that the critical supercurrent in the direction I c differs from that in the opposite direction I This nonreciprocity in supercurrents holds promise for applications in superconducting electronics 27 28 The d wave superconductivity is very common and ap pears in cuprate superconductors The bulk of tradi tional d wave superconductors is gapless Recently chi ral d wave superconductivity has been proposed in magic angle bilayer twisted graphene 29 30 in Bi Ni bilay ers 31 and in Sn Si 111 31 It has been predicted that twisted bilayer cuprates can realize d is or d id pairings without gapless states in the bulk 32 33 how ever further research is necessary to confirm these pre dictions 34 35 With the d id pairing the bulk states are gapped and instead chiral edge states appear Prox imity induced superconductivity has been suggested in two dimensional electron gas 2DEG materials such as Bi2O2Se proximized with cuprates 36 Furthermore it has been demonstrated that a twisted bilayer cuprate ex hibits time reversal symmetry TRS breaking and JDE at specific twist angles 34 37 38 In this work we study superconductor normal re gion superconductor S N S Josephson junction formed with d wave superconductors We consider d d id and d is superconducting pairings that have been sug gested in exfoliated layers of cuprate superconductors We find that for the case of d wave pairing the finite JDE can only occur in the presence of spin orbit cou pling SOC and magnetic field The d id and d is pairings break TRS As a result in the presence of addi tional asymmetry see Fig 1 the TRS breaking results in JDE even in the absence of SOC and magnetic field According to our results for d id pairing the chiral edge modes show a clear contribution to JDE which suggests that interference effects can be used to control JDE We further confirm this by studying the distribution of the Josephson current in the junction and by identifying the Andreev bound states ABS from the local density of states LDOS Our proposal realizes a large field free JDE and demonstrates that JDE in a planar geometry can be used as a signature of unconventional supercon ducting pairings II MODEL AND METHODS We consider S N S Josephson junction JJ formed us ing 2DEG with induced d wave d id or d is pairing see Fig 1 An electrical gate Vg is applied to the normal region as shown in Fig 1 which allows for the tunability of JJ The Bogoliubov de Gennes BdG Hamiltonian of the in system written in the Nambu basis the continuous limit reads 39 cid 18 2k2 2m V SO k z h xRe k x yIm k x H cid 19 z 1 Here m is the effective electron mass is the chemical potential is the Rashba spin orbit coupling strength and h describes the Zeeman energy e g for an ex ternal magnetic field The gate voltage is defined as V VG WG 2 x The pair potential k x is 4 2 0 2 n u J 7 1 n o c r p u s t a m d n o c 1 v 7 2 1 1 1 6 0 4 2 v i X r a 2 2kF m 0 t 0 Typical material is defined as parameters we use in numerical calculations are as fol lows a 2 5nm m 0 14me SO 0 1ta 0 1eV A 0 0 01t 0 4meV and 0 1t 4meV 40 2a We calculate the conserved current inside the junction see Fig 8 a which does not require the self consistent calculation The self consistent procedure would be re quired inside superconductor to ensure the current con servation To calculate the equilibrium Josephson cur rent we use the Matsubara Green s function G i n ob tained using the analytical continuation of the retarded Green s function The local current between the sites n and m can be expressed using the standard approach 41 46 In m 2 ekBT Tr Im HnmGmn HmnGnm 4 cid 88 n along the cut in y direction I cid 80 where Gnm is the Green s function and Hnm is the Hamil tonian submatrix calculated between the sites n and m with the lattice sites taken inside the normal region The summation is performed over the fermionic Matsubara frequencies n 2n 1 kBT and the Green s functions Gnm also accounts for the self energy of the leads The total current is obtained by summing all contributions n In n ex where ex is the unit vector along the x axis see Fig 8 a The diode effect is then characterized by the quality factor Q I c are the critical current calculated by finding the maximum and minimum of current over L R 0 2 As an example in Fig 1 c we show the current phase relation CPR for d id and d is pairings The figure shows the nonvanishing anomalous Josephson effect at 0 pro vided that the two pairing lobes are rotated with respect to each other Furthermore CPR in the figure results in the quality factor Q equal to 0 2 for d id pairing and 0 14 for d is pairing c where I c and I c I c I c I We also calculate the local density of states