This article is an adaptation of my Masters thesis report ( second semesters ), where I worked on the modelling and simuation of Josephson Junctions.\nAll the codes and data used in this article can be found in the github repository<\/a>.\nThe previous article talked about the the fabrications of Josephson Junctions and their charecterisation ( first semester ).\nAt the beginning of this thesis, we introduce basic theoretical concepts\nthat underlay the device we are trying to study, i.e. the Josephson\nJunction (JJ<\/span>). The theorey of superconductivity was breifly described in the previous article. We first look at the Josephson effect, which describes the physics of a\nSuperconductor-Insulator-Superconductor sandwich and then look at a\npopular model of a realistic Josephson junction, namely the RCSJ<\/span> model.\nLater we will try to understand various aspects of fabrication and\ncharacterisation of such\u00a0devices.<\/p>\n Prior to 1962, researchers were familiar with quantum mechanical\ntunneling of normal electrons through a weak barrier; however, the\nprobability of tunneling of a cooper pair was thought to be\ninsignificant given that the pair as a whole would have to tunnel\nthrough the barrier. In 1962 Brian David Josephson showed that this\ntunneling probability is not low as previously thought. He predicted\ntheoretically that two superconductors that are coupled (are in close\nproximity) by a weak link, which link may be made of a normal metal, an\ninsulator, or a constriction of superconductivity, can still let the\nsuper current flow through them (Josephson 1962). This macroscopic\nphenomenon was given the name Josephson\u00a0effect.<\/p>\n Josephson demonstrated that, for a short junction, the current that\nflows through the junction when no voltage bias is applied, and the\nphase difference \\(\\phi\\)<\/span> across the junction, which is the difference in\nthe phase factor between the order parameter of the two superconductors,\nare related through the\u00a0relation:\n<\/p>\n Here, \\(I_{c}\\)<\/span> is the super\ncurrent amplitude and \\(\\delta=\\phi_{1}-\\phi_{2}\\)<\/span> , where \\(\\phi_{i}\\)<\/span> is\nthe phase of each superconductor. This phenomenon is known as the DC<\/span>\nJosephson effect. Josephson also showed AC<\/span> Josephson effect where an\napplied constant voltage bias V on the junction leads to sinusoidal\noscillations in the junction current and is governed by the\u00a0equation:\n<\/p>\n where\n\\(\\Phi_{0}\\approx2\\times10^{-15}\\)<\/span> Weber is the flux\u00a0quantum.<\/p>\n The DC<\/span> Josephson effect is explained by a process known as Andreev\nreflection (Schrieffer and Tinkham 1999). A.F.<\/span>Andreev explained the\nphenomenon in 1964 establishing the concept of the so-called Andreev\nreflection This reflection occurs at the interfaces between the\nsuperconductor S and a normal metal N. Andreev suggested that an\nelectron that approaches the interface from the normal metal side can\ntravel through the superconductor side by the formation of a Cooper pair\nwith another electron with opposite momentum and spin on the\nsuperconductor side. At the same time, reflect a hole inside the normal\nmetal region thus balancing the charge. As a result of this cycle, a\npair of correlated electrons is transferred from one superconductor to\nanother, creating a super current flow across the junction. It explains\nhow a normal current in the normal metal side becomes a super current in\nthe superconductor side. The AC<\/span> Josephson relation in essence suggests\nthat a Josephson junction can be a perfect voltage-to-frequency\nconverter. The inverse is also possible by using a microwave frequency\nto induce a DC<\/span> voltage in a Josephson junction, this phenomena is known\nas inverse AC<\/span> Josephson\u00a0effect.<\/p>\n A Josephson junction, is typically composed of two superconducting\nelectrodes separated by weaklink which is typically insulating, thus\nsuch a junction would have some unavoidable capacitance \\(C\\)<\/span> (Just like\nthe parallel plate capacitor separated by a dielectric). If the junction\ncurrent exceeds the critical current of the junction then quasi-particle\nexcitations are generated. These quasi-particle currents are not\nsuperconducting and can be quite lossy just like a normal metal current,\nso we represent this as a normal resistor \\(R\\)<\/span>. This gives us the\nresistively and capacitivly shunted junction (RCSJ<\/span>) model. This model\nhelps us simulate the characteristics of a Josephson junction. A\nschematic representation of the same can be seen in Fig\n1<\/a>.<\/p>\n Writing out Kirchov\u2019s circuit laws for the RCSJ<\/span> model (from Fig\n1<\/a>.)we can\u00a0find<\/p>\n or\n<\/p>\n \nRearranging\u00a0as\n<\/p>\n \nwe can interpret Eq [eq:dammpedeqn]<\/a> as the dynamics of a\ndamped particle with the following physical\u00a0properties:\n<\/p>\n The dynamics of the Josephson junction phase difference in-terms of the\ndamped particle can be described as follows: (Fig\n2<\/a>)<\/p>\n The junction has a simple cosine potential when there is no junction\n current (at I=0). At this point, the pseudo-particle is caught in\n one of the cosine\u2019s wells, as shown in Fig\n 2<\/a>a. We can observe the\n effect of an extra linear term to the potential as we introduce some\n current. Because the potential now resembles a tilted washboard,\n it\u2019s known as the tilted washboard potential<\/em>. There are still\n vallies in the potential if the bias current is less than \\(I_{c}\\)<\/span>,\n and the ball remains stuck, as shown in. Fig\n 2<\/a>a. Because the\n junction phase (i.e the pseudo-particle) is stuck at a fixed value\n of \\(\\delta\\)<\/span>, the voltage is zero (as \\(V\\propto\\dot{\\delta}\\)<\/span> ). As\n demonstrated by the horizontal blue line, this is the section of the\n IV<\/span> curve where an increase in junction current does not result in an\n increase in junction voltage. Because the junction element is still\n superconducting, all current flows through the tunnel element and\n none through the resistor at this point, resulting in the\u00a0junction.<\/p>\n<\/li>\n As the current is increased past \\(I_{c}\\)<\/span>, the linear term in the\n potential begins to dominate the cosine part, and the vallies fade\n away. As demonstrated in Fig\n 2<\/a>b the junction pseudo\n particle then rolls downhill. The pseudo particle is now in a\n time-varying phase, and the junction voltage is non-zero and\n approaches a finite value, as shown by the red line in Fig\n 2<\/a>b<\/p>\n<\/li>\n Because the current I now exceeds the tunneling element\u2019s critical\n current, the tunneling element no longer acts as a superconductor.\n The formation of quasi-particles makes the connection resistant. In\n other words, the resistive element carries practically all of the\n current. Further increases in current result in a linear increase in\n voltage, identical to that of a typical metal, as indicated by\n \\(V=IR\\)<\/span>, as shown in Fig\n 2<\/a>c, and by the green\n line marked Fig\n 2<\/a>c.<\/p>\n<\/li>\n As the current is reduced, we return to the green line, as shown by\n the mark Fig\n 2<\/a>d. The rest of the\n process depends on how fast we are raising and lowering the current.\n The potential regains its cosine nature and regains vallies, as we\n lower the current below \\(I_{c}\\)<\/span>\u00a0.<\/p>\n<\/li>\n If there were no dissipative forces as is the case when we sweep\n fast enough or when whatever dissipation remains can\u2019t completely\n stop the particle, the particle would continue to roll down as it\n already has energy. Therefore, even as I is lowered below \\(I_{c}\\)<\/span> we\n still have time varying \\(\\delta\\)<\/span> and therefore still have a\n measurable voltage. This can be concluded from Fig\n 2<\/a>e and the pink line in\n Fig 2<\/a>e<\/p>\n<\/li>\n In the end, we go back to no bias case where the potential is again\n a cosine term and as we slowly sweep the voltage we slow down and\n finally stop the particle. Then as we increase the negative bias the\n process starts all over in\u00a0reverse.<\/p>\n<\/li>\n<\/ul>\n In Eq[eq:JJ1<\/span>]<\/a> we saw that he Josephson Junction\ncurrent depends on the phase difference \\(\\delta\\)<\/span> across the junction.\nWhen an external magnetic field is applied, the field influences the\nphase difference \\(\\delta\\)<\/span>, this in turn causes interesting dynamics\nbetween the Josephson Junction current and the applied external magnetic\nfields. It can be shown that in the case of a small Josephson Junction\nthis dependence follows the relation(Schrieffer and Tinkham\u00a01999):\n<\/p>\n \nhere\n\\(\\Phi_{J}=\\mu_{0}Hw(d+\\lambda_{1}+\\lambda_{2})\\text{ is the magnetic flux linked to the whole barrier }\\)<\/span>with\n\\(w\\)<\/span> being the width of the barrier, \\(d\\)<\/span> the barrier thickness,\n\\(\\lambda_{1},\\lambda_{2},\\)<\/span>the penetration depths of the two\nsuperconductors.\nThis behavior was first experimentally found by Rodwel (Rowell 1963), in\na Pb-I-Pb junction at 1.3 K. This is the standard from of the Fraunhofer\npattern \\(F(x)=I_{0}\\sin^{2}(\\pi x)\/(\\pi x)^{2}\\)<\/span>and is seen as a unique\ncharacteristic confirmation of a Josephson junctions. This\nFraunhofer-like result, which is akin to diffraction of monochromatic,\ncoherent light passing through a slit, provides a validation of the\nsinusoidal current phase relation (CPR<\/span>).\nFor a SQUID<\/span>, the critical current-magnetic field characteristic is\nsimilar to that of Josephson Junctions with the addition of SQUID<\/span>\noscillations superimposed on it. Both of these signatures are in the\nsimulations done in the later section.\nIn case of junctions with ferromagnetic behavior, at certain temperature\nand barrier width, a \\(\\pi\\)<\/span>junctions could be observed. This is due to\nexchange field-induced oscillations of the order parameter and is very\nsensitive to the temperature and the ferromagnetic layer thickness. In\nsuch materials the CPR<\/span> could be written as \\(I(\\phi)=I_{0}sin(\\phi+\\pi)\\)<\/span>,\nand doubling of periodicity in \\(I_{c}\\)<\/span>vs \\(H\\)<\/span> is observed(Frolov et al.\n2004). In materials with broken time-reversal and broken parity\nsymmetries (this can be obtained in systems with both a Zeemanfield and\na Rashba spin-orbit coupling(Assouline et al. 2019; Strambini et al.,\nn.d.)), the CPR<\/span> could take the form of \\(I(\\phi)=I_{0}sin(\\phi+\\phi_{0})\\)<\/span>\n(Assouline et al. 2019), in Josephson junctions this term leads to the\nanomalous phase shift, which could manifest as the presence of second\nharmonics in the critical current - phase relation (Stoutimore et al.\n2018). This system is also simulated as a part of the thesis, where in\nthe \\(I_{C}H\\)<\/span> behavior is replaced with the\u00a0following:<\/p>\n The characteristic signature of a Josephson junction, apart from its\ncurrent voltage relation (IV<\/span>) is the Critical current \\(I_{c}\\)<\/span>dependence\non the applied magnetic field H ( \\(I_{C}\\)<\/span>vs H or \\(I_{C}H\\)<\/span>).<\/p>\n The PPMS<\/span> in the lab has builtin recipes only for DC<\/span> measurement and as\nsuch DC<\/span> measurements like IV<\/span> are relatively slower (1 IV<\/span> scan in 10-15\nminutes on good resolution). Thus getting data for \\(I_{c}H\\)<\/span> would\nrequire multiple IVs to be measured at a sweep of magnetic field H. This\nwould take almost a day per device on a decent resolution and thus cant\nbe done frequently. The more easier measurement would be to set and\nconstant current (say the \\(Ic\\)<\/span> at zero magnetic field) then measure the\nVoltage as a function of changing magnetic field ( V vs H or VH<\/span> )\nhowever, there is little literature regarding the characteristics of VH<\/span>\nrelation (or magneto resistance ) of a Josephson\u00a0junction.<\/p>\n Thus the main goal of the thesis is to verify the correlation between\nthe \\(I_{C}H\\)<\/span> and \\(VH\\)<\/span> signatures of a Josephson junction via simulation.\nsimulation part of the thesis is to first setup the numerical solution\nto the ODE<\/span> [eq:dammpedeqn]<\/a> , then simulate an\nI - V measurement, Iterate the IV<\/span> sweep over multiple magnetic fields\nlinearly spaced between \\(-2\\frac{\\phi}{\\phi_{0}}\\)<\/span> and\n\\(2\\frac{\\phi}{\\phi_{0}}\\)<\/span> . This would give us the data of all\npermissible sets junction current \\(I_{J},\\)<\/span>junction voltage \\(V_{J}\\)<\/span> and\nthe applied magnetic field \\(H\\)<\/span> that are the possible states of the given\nJosephson Junction. Finally, one could correlate the simulation with\nexperimental data for junctions with and without second harmonics from\n\\(I_{c}H\\)<\/span> and \\(VH\\)<\/span> measurements. First lets us try to understand the\nsystems ODE<\/span>.<\/p>\n We model the Josephson junction using the RCSJ<\/span> model as described in\nsec[subsec:RCSJ<\/span>-model]<\/a>:he total\ncurrent running through the network\u00a0is <\/p>\n Due to quasiparticle tunnelling, the resistance in\nreality relies on both the temperature and the voltage across the\njunction as described by this\u00a0equation: <\/p>\n where typically \\(R_{sg}\\gg R_{n}\\)<\/span>. The characteristic\nvoltage of the junction is accordingly defined as \\(V_{c}=I_{c}R_{n}\\)<\/span>.<\/p>\n The current-phase relation (CPR<\/span>) \\(I_{s}(t)=I_{c}\\sin(\\phi)\\)<\/span> describes\nthe supercurrent via a Josephson junction (JJ<\/span>), where \\(I_{c}\\)<\/span> is the\ncritical current and \\(\\phi\\)<\/span> is the junction phase-difference. The CPR<\/span>\ncan be stated in more broad terms as (Pal et al. 2014; Tanaka and\nKashiwaya\u00a01997)\n<\/p>\n \n, and when the first harmonic is suppressed (for example, at a 0 \u2013\\(\\pi\\)<\/span>\ntransition)(Sellier et al. 2004), the second harmonic may become\napparent. We try to determine the influence of adding a second harmonic\nCPR<\/span> in the I c H and V H behavior in this thesis by\u00a0simulation.<\/p>\n Before we try to solve the ODE<\/span>\n[eq:ODE<\/span>]<\/a>, we can try to simplify the ODE<\/span> by\nnormalising the\u00a0equation.<\/p>\n This helps in minimizing the round off errors, for instance if one\nvariable has the value 24582 (units a) the other variable could be in\nthe order 0.001861(units b). The significance of the second variable\ncould be irreversibly lost while executing any operation pertinent to\nthese variables, such as multiplication. One technique to assist limit\nthese possible losses is to normalise the variables first. The equation\nis simplified with normalized time ( unitless ) via the plasma\nfrequency, \\(\\tau=\\omega_{p}t\\)<\/span> where\n\\(\\omega_{p}=\\left(2eI_{c0}\/\\hbar C\\right)^{1\/2}\\)<\/span>:(Schrieffer and Tinkham\u00a01999)<\/p>\n \\(\\omega_{p}=\\frac{1}{\\tau_{p}}=\\frac{1}{\\sqrt{L_{c}C}}=\\sqrt{\\frac{2eI_{c}}{\\hbar C}}\\)<\/span>\nand\n\\(dt=\\frac{1}{\\omega_{p}}d\\tau\\rightarrow\\frac{d^{n}}{dt}=\\omega_{p}^{n}\\frac{d^{n}}{\\tau}\\)<\/span>,\napplying these factors we\u00a0get<\/p>\n In this eq [eq:FinalODE]<\/a> Q is the damping factor\n(or quality factor) which depends on the inherent resistive and\ncapacitive components of the RCSJ<\/span> model. This Q is identical with\n\\(\\beta_{c}^{1\/2}\\)<\/span>, where \\(\\beta_{c}\\)<\/span> is a frequently used damping\nparameter that was introduced by Stewart and McCumber. In the case of\nheavy damping \\((Q\\ll1)\\)<\/span> we see the same IV<\/span> behavior for increasing and\ndecreasing current, however in the case of under damped junction\n(\\(Q\\gg1)\\)<\/span>, while decreasing the current, we see that the junction\nremains in the non zero voltage range below \\(I_{c}\\)<\/span>. This behavior is\nalso explored in the\u00a0simulations.<\/p>\n The \\(I_{c}\\)<\/span>in eq [eq:FinalODE]<\/a>, is the critical current\nand its dependence with magnetic field is to be incorporated separately\nin the simulation based on Eq\n[eq: IcH 2]<\/a>.<\/p>\n The ODE<\/span> [eq:FinalODE]<\/a> can be converted to a\nsystem of first order ordinary differential equations, and then it can\nbe passed on to a ODE<\/span> solver like scipy.integrate.odeint<\/em> with the\ninitial\u00a0values.<\/p>\n The result one such ODE<\/span> solve is \\(\\phi\\)<\/span> and \\(\\mathring{\\phi}\\)<\/span> as a\nfunction of time for the given initial conditions of \\(\\phi\\)<\/span> ,\n\\(\\mathring{\\phi}\\)<\/span> and \\(I\\)<\/span>. The dynamics of such a system depend vastly\non the initial condition given to the ODE<\/span>. As an example consider the\nODE<\/span> solution as a function of time (in \\(\\tau\\)<\/span>) for different initial\nconditions for \\(\\phi\\)<\/span> and \\(\\mathring{\\phi}\\)<\/span> in Fig\n3<\/a>.<\/p>\n For the first iteration of the simulation (0 current and 0 phase\ndifference) starting point for \\(\\phi\\)<\/span> and \\(\\mathring{\\phi}\\)<\/span> should be\nset to zero as any \\(\\mathring{\\phi}\\)<\/span> would have to start from the moment\na superconducting phase sets up and gradually evolves with time to reach\u00a0equilibrium.<\/p>\n After the system evolves for a certain amount of time steps (which needs\nto be adjusted depending on Q), the voltage is calculated by averaging\nover the last cycle (detected as a percentage of the entire time solve).\nIf there\u2019s no voltage cycle, the voltage gets set to zero. The initial\ncondition for the next run (next current value in the IV<\/span> seep) is the\nfinal state of that previous\u00a0one.<\/p>\n The percentage of final cycle and the number of cycles to be run is\ndetermined for each range of Q (below 0.1, below 1 below 10, below 100)\nand kept in a function called timeparams.<\/em><\/p>\n The entire process is then iterated over the given range of given\ncurrents. It must be noted that since the simulation for current depends\non the previous relaxed values for \\(\\phi\\)<\/span> and \\(\\mathring{\\phi}\\)<\/span>, the\nvoltage values for a given I depend on whether the current is a part of\nthe increasing I cycle or decreasing I cycle. Thus one could clearly\ndifferentiate between the over damped IV<\/span> and under damped IV<\/span> based on\nthe presence of re-trapping current. See fig\n4<\/a>. In the case of heavy damping Josephson Junctions\u00b6<\/a><\/h3>\n
![\"(a)](\"https:\/\/ashwinschronicles.github.io\/Photos\/a-The-superconducting-order-parameter-of-a-superconductor-S-penetrating-into-the.webp\")
\n![\"Andreev](\"https:\/\/ashwinschronicles.github.io\/Photos\/andreev_reflection.webp\")
\nRCSJ<\/span> model<\/span>\u00b6<\/a><\/h4>\n
![\"A](\"https:\/\/ashwinschronicles.github.io\/Photos\/RCSJ.webp\")
\n![\"Interpretation](\"https:\/\/ashwinschronicles.github.io\/Photos\/JJ-IV.webp\")
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Josephson Junctions in the Presence of a Magnetic Field\u00b6<\/a><\/h4>\n
Simulation Details\u00b6<\/a><\/h1>\n
Modeling the ODE<\/span>\u00b6<\/a><\/h3>\n
Simulation parameters\u00b6<\/a><\/h3>\n
![\"ODE](\"https:\/\/ashwinschronicles.github.io\/Photos\/ODE-relaxation.webp\")
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