diff --git "a/6dE4T4oBgHgl3EQfcQxJ/content/tmp_files/load_file.txt" "b/6dE4T4oBgHgl3EQfcQxJ/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/6dE4T4oBgHgl3EQfcQxJ/content/tmp_files/load_file.txt" @@ -0,0 +1,680 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf,len=679 +page_content='Reconfigurable magnetic-field-free superconducting diode effect in multi-terminal Josephson junctions Fan Zhang,1 Mostafa Tanhayi Ahari,2 Asmaul Smitha Rashid,3 George J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' de Coster,4 Takashi Taniguchi,5 Kenji Watanabe,6 Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Gilbert,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content='7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content='2 Nitin Samarth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content='1∗ Morteza Kayyalha3∗ 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' PA 16802,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' USA 2Materials Research Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' The Grainger College of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' University of Illinois,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' IL 61801,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' USA 3Department of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' PA 16802,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' USA 4DEVCOM Army Research Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 2800 Powder Mill Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Adelphi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' MD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 20783,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' USA 5International Center for Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Nanoarchitectonics National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 1-1 Namiki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Tsukuba 305-0044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Japan 6Research Center for Functional Materials National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 1-1 Namiki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Tsukuba 305-0044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Japan 7Department of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' University of Illinois,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' IL 61801,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' USA ∗Correspondding author: nsamarth@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' ∗Correspondding author: mzk463@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' The superconducting diode effect (SDE) has attracted growing interest in re- cent years as it potentially enables dissipationless and directional charge trans- port for applications in superconducting quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Here, we demon- strate a materials-agnostic and magnetic-field-free approach based on four- terminal Josephson junctions (JJs) to engineer a superconducting diode with 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content='05081v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content='supr-con] 12 Jan 2023 a record-high efficiency (∼ 100%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We show that the SDE is reconfigurable by applying control currents to different terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We attribute the observed SDE to the asymmetry of the effective current-phase relation (CPR), which we derive from a circuit-network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Our findings demonstrate the emergence of a new form of the CPR in multi-terminal JJs that can emulate macroscopic transport signatures of superconducting systems with broken inversion and time-reversal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Introduction In linear electrical networks, the concept of reciprocity implies a symmetric relationship be- tween the applied current and measured voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' In other words, the voltage magnitude remains the same if the polarity of the current source is reversed from positive to negative (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Vio- lating this fundamental symmetry in semiconductor technology has led to a plethora of new devices including diodes, transistors, rectifiers, and photodetectors (2,3,4,5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' In superconduc- tors, engineering non-reciprocity requires simultaneous breaking of time-reversal and inversion symmetries, known collectively as chiral symmetry (6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' The superconducting diode effect (SDE) is defined as an asymmetry in the critical current when the current sweep direction is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Macroscopic transport signatures of the SDE are determined by the current-phase relation (CPR), which typically depends on the band structure and microscopic details of the underlying material system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Therefore, careful engineering of the interplay between spin-orbit interactions, topological phases, and magnetic fields can lead to the observation of the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' To this point, the SDE has been experimentally reported in a multitude of systems including but not limited to: non-centrosymmetric superconductors with the magneto-chiral anisotropy (17, 18, 19, 20, 21), Josephson junctions (JJs) based on Dirac semimetals with finite-momentum Cooper pairing 2 (22), two-dimensional (2D) van der Waals heterostructures (23, 21, 24, 25, 26), three-terminal JJs based on InAs in the presence of a magnetic field (27), and a network of graphene JJs (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' In this work, we develop a reconfigurable, materials-agnostic, and magnetic-field-free method to engineer a synthetic CPR that emulates macroscopic transport signatures of systems with bro- ken inversion and time-reversal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We consider a multi-terminal Josephson junction (MTJJ) fabricated in graphene, which has a symmetric Fermi surface with no spin-orbit cou- pling (both inversion and time-reversal symmetries are preserved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We show that the MTJJ can emulate the SDE, which is a macroscopic transport signature predicted to emerge in sys- tems with broken inversion and time-reversal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We further show that the SDE is reconfigurable: the MTJJ exhibits the typical Josephson effect with no SDE under a symmetric current-bias configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' However, it manifests the SDE under an asymmetric current-bias configuration (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 1 and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' S1 for details of asymmetric and symmetric bias configura- tions, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' The observed SDE is also magnetic-field-free and reversible with efficien- cies as large as ∼ 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We explain the experimental observations by modeling our system using a circuit network of coupled resistively-shunted junctions (RSJs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We find that the semi- classical RSJ model accurately captures the non-reciprocal transport effect in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We calculate an effective non-sinusoidal CPR arising from circuit network effects among the super- conducting terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We show that this effective CPR is not symmetric under sign-reversal of the superconducting phase, resulting in the observation of the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Our combined experimental and theoretical findings establish a new materials-agnostic platform based on MTJJs to engineer novel forms of CPRs and non-reciprocal superconducting properties for potential applications in superconducting cryogenics and quantum technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 3 Results and Discussion We fabricate four-terminal JJs on hBN/graphene/hBN van der Waals heterostructures which are edge-contacted by Ti(10 nm)/Al(100 nm) superconducting electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Figure 1A shows an atomic force microscope (AFM) image of a representative four-terminal JJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We characterize the junction in two asymmetric bias-current configurations (see our prior work (29) and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' S1 in the Supplementary Materials for results obtained from symmetric configurations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' In configuration 1 (2), we ground terminal 2 (3) and apply a constant control current to terminal 3 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' In both configurations, we measure the SDE by sweeping the current I1 of terminal 1 and applying a control current I4 to terminal 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Figure 1B shows V12 vs I1 measured in Config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 1 at I3 = 0 nA, I4 = 10 nA, Vg = 30 V, B = 0 G, and T = 12 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We observe that the critical current (I+ c ) for positive sweep direction, marked by red arrows, is around 4 nA, whereas the critical current (I− c ) for the negative sweep direction, marked by black arrows, is around −24 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We also observe different critical currents (I+ c or I− c ) and return currents (I+ r or I− r ) for each sweep direction, likely due to the Joule heating effect (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' We note that increasing the temperature reduces the impact of Joule heating, thereby resulting in similar critical and return currents (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' S4 in the Supplementary Materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' To elucidate the origin of the observed SDE, we consider a circuit-network model of coupled RSJs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 1C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' The RSJ model represents individual junctions by a two-fluid system in which the total junction current is the sum of a pair current ip jk(t) and a dissipative quasiparticle current iq jk(t) (31,32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Here, we assume diffusive transport in which the pair current is given by the first Josephson relation ip jk(t) = Ijk c sin(φjk(t)), where Ijk c is the critical current between terminals j and k and φjk(t) ≡ φj(t) − φk(t) is the gauge-invariant phase difference satisfying the sec- ond Josephson relation dφjk(t)/dt = (2e/ℏ)Vjk(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' The quasiparticle current is due to a finite voltage Vjk(t) across the junction, iq jk(t) = GjkVjk(t), where Gjk is a constant phenomenolog- 4 ical conductance tensor (see the Supplementary Materials for the RSJ parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' A circuit network model of JJs typically includes a parallel capacitance as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' In our graphene-based JJs, however, the junction capacitance is negligible and, hence, we only consider the resistance of the junction (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Imposing current conservation (Kirchhoff’s current law) at terminal j, we obtain Ij = � k (ip jk + iq jk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' (1) In this study, we are interested in the emergent non-reciprocal superconducting properties in the four-terminal JJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Therefore, we only consider small bias currents such that no quasiparticle current flows between the terminals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=', iq jk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' In this case, starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 1 and assuming I3 = 0 nA for Config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 1, we may analytically obtain expressions for the other terminal currents as I1 = I14 c sin φ14 + I13 c sin φ13 + I12 c sin φ1, I4 = I41 c sin φ41 + I43 c sin φ43 + I42 c sin φ4, 0 = I32 c sin φ3 + I31 c sin φ31 + I34 c sin φ34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' (2) For fixed (φ1, I4) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 2, we find φ3, φ4, and, consequently, the following effective CPR: I1(φ1, I4) = � n an sin nφ1 + bn cos nφ1, (3) where n is an integer and (an, bn) are the amplitudes of the nth harmonic for the sine and cosine functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' Comparing our numerical simulation to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 3, we find that b0, a1, and a2 are the only dominant factors in our device (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' This leads to I1(φ1, I4) ≈ b0 + a1 sin φ1 + a2 sin 2φ1, (4) where b0 ∝ I4 and (a1, a2) are independent of I4 (see Supplementary Materials for more de- tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 5 The symmetry of this effective CPR depends on the current-bias configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' For ex- ample, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 4 is symmetric under simultaneous sign reversal of φ1 and I4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=', I1(φ1, I4) = −I1(−φ1, −I4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' However, for a fixed I4 ̸= 0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' 4 represents a CPR which is asymmetric under sign reversal of φ1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=', I1(φ1, I4) ̸= −I1(−φ1, I4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' This is because for fixed I4 ̸= 0, b0 does not change sign when φ1 → −φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' In general, the non-reciprocity arises from an asymme- try in the free energy of the system for opposite current directions ±I1 and fixed I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE4T4oBgHgl3EQfcQxJ/content/2301.05081v1.pdf'} +page_content=' To see this, we consider the expression for the free energy of the system F(I1, I4, φ1, φ3, φ4) = (ℏ/2e) � I1φ1 + I4φ4 + � j