Quantum Transport, Lecture 14: Josephson effects

they we continue in the field of superconductivity focusing on this family of effects proposed theoretically by Josephson and these are not just physical phenomena it is a very natural for our course to study this because it all happens in a device the building block is the Josephson Junction a junction between two superconductors and pretty much any effect you can imagine related to this concept is another device so many interesting effects and in subsequent lectures we will see more devices based on superconductors and Josephson effect including quantum bits at the end so last week we focused on a piece of superconductor and I asked you to think about the superconductor in a very simple way as a wave function this one piece of superconductor all the Cooper pairs are is a complicated many-body state but they are all condensed in a bose-einstein condensate in the ground state protected by the superconducting gap from excitations and all of those Cooper pairs are in exactly the same state described by one complex number with an amplitude which is proportional to the square root of density and a phase it's by a whole piece of superconductor would be determined by this one complex number and today we are going to study systems where tune pieces of superconductor are in close proximity to each other very close proximity with a little gap which is called a barrier and this barrier can be vacuum can be an dial trick insulator can be a metal can be a semiconductor and pretty much in all these situations the barrier is small enough you can see the Josephson effects the Josephson effect is a flow of super current between two distinct disconnected pieces of superconductor that's what Josephson effect is if you think about it it's quite remarkable because in this region in between there is no conditions for super conductivity there is no attractive interaction between the two electrons there are no there is no condensate none of that is there yet somehow through this piece of dielectric or a metal current can flow without dissipation the super current can flow Oh even though it's a you put this piece of let's say let's stick a piece of copper in between put a piece of copper and measure it separately on your table it will always have resistance no matter the temperature at the lowest temperature it will still have finite resistance put it in between two super conductors and you can pass a current without any resistance so that is very remarkable and that's called the Josephson effect and the notion behind this effect why this works is because you can think of this wave function on the left overlapping with the wave function on the right because they are close so like wave functions of simple electrons not Cooper pairs overlapping over a tunneling barrier and allowing electrons to go from one side to the other same way wave functions of Cooper pairs overlap and form through a single wave function over the barrier something like this here I show superconductor on the Left superconductor on the right a tunneling barrier in between a constant wave function here and a constant wave function here inside the superconductor and outside the superconductor I allow them to slowly decay or maybe they decay fast it depends on the materials I motivated it to you in the last lecture when I mentioned to you the proximity effect right so this this is possible if you imagine that this barrier is a metal the electrons can just fly into here and for a while they will remember each other's phases so they will think they're a Cooper pair for an evanescent time also works with a dielectric or vacuum actually the quantum mechanical wave function can go in in under the barrier and if the two are close enough there will be an overlap and then the dashed line I plot the sum of the two wave function nothing more well approximately so going from left to right this wave function also called the order parameter exhibits a a kind of a weak spot the amplitude is reduced but if it's not reduced to zero Cooper pairs can go from left to right and they can carry super current so Johson effect can be thought of as a tunneling of Cooper pairs one thing is not plotted here in this graph is phase well or you could say the phase of the left superconductor is the same as the phase of the right superconductor but as the two pieces are distinct disconnected in principle the two phases don't have to be the same in general they can be different and the phase difference across the junction is the key parameter to everything we're going to talk about today it determines all the properties of Josephson junctions so the phase drop across the tunnel barrier v i1 minus Phi 2 Delta Phi and for a large fraction of the talk I will just forget to write this Delta I will just write I just faith and I will call it phase because of course phase is additive I can just subtract a constant from somewhere in the face and but it's the phase difference yeah I want you to keep in mind that whenever I'll be talking about phase it will always be the phase difference between the 2 pieces of superconductor that matters it affects the physical properties and why that quantity matters it's because this macroscopically quantum coherence state and this microscopically quantum coherent state form 1 coupled state and in order for super current to flow we have to maintain phase coherence if we lose phase coherence and this quantity does not matter then Cooper pairs cannot travel from one superconductor to the other over a piece of copper and still create no dissipation still remain in a condensed state so phase coherence is important if you ever get to measure a Josephson Junction you would get an IV characteristic which may look like this in this case what they well this is a cartoon but what you would have to do to measure this kind of characteristics you have to sweep the current and up to some critical current there will be no voltage across the junction so after the critical current the junction often rapidly switches as a jump into the finite voltage state after which it behaves kind of like a resistor the line that goes through here extrapolates to zero so this region is often called a normal state of a Josephson Junction and this is the super current state you can see that this is by no means a linear IV especially clearly not down here so if you put make a circuit that includes the Josephson Junction as an element this will not be a linear circuit it will be distinctly different from circuits that have resistors capacitors inductors those circuits we understand very well but this will be nonlinear and so you would have to apply its own logic it has its own logic there are other examples of nonlinear elements are tunnel junctions diodes transistors etc they also come with their own logic so when Josephson theoretically envisioned this effect everyone said that he was crazy she turned out to be true later on but for a different reason but the reason why they said he was crazy was because to think about this tunneling people broke it down into tunneling of single electrons and the probability of two electrons tunneling and being a Cooper pair here and then forming a Cooper pair again here was thought to be very small and then so Josephson understood that that amplitude is actually not small if you think about Cooper pairs tunneling and when he derived his original equations he came up with this formula which connects super current flowing in the junction so not the critical current but the super current at the given moment of time to the phase drop across the junction so a critical current then is just a constant and very simple formula super current with an amplitude of the critical current is proportional to sign of the phase difference and that is often called the first Josephson relation and he came up with two we will talk about the second one later so coming back to this experiment if we slowly ramp up the current and we are still in this super current state right we increase the current and according to this equation we are winding the phase we are creating a phase difference across the junction because the two are connected right so what would be the phase difference at the critical pouran point at the critical current value so just a sign becomes one we would be at the phase difference of Pi over two so as we go from here to here we will be going from zero right for zero current phase difference is zero or 2pi or 4pi and as we increase we will be going from zero to pi over two over a sinusoidal curve and then we exceed the critical current and something else happens which we will discuss later and it all ok a couple words about the most important examples of Josephson junctions at least for this course these are on technology you already seen this method before when I talked about tunnel junctions this is the shadow technique where you create a suspended mask over your substrate and then you put it in the evaporator and atoms in the ultra-high vacuum electron beam evaporator come down in straight lines so you can deposit them at one angle then at another angle and so you will form overlaps between the two layers and in between the two layers if you put a bunch of oxygen in this chamber you will oxidize the surface of the bottom layer and therefore here you will create a tunnel Junction well if you oxidize for a kind of a long time this tunnel barrier will be quite thick it will be a very good tunnel Junction and then there will be no super current because the tunneling barrier is too high so if you oxidize a little bit less and the two super conductors in this case this works most often with aluminum the two layers of aluminum are a little bit closer then Cooper pairs can tunnel and so with the same technique you can create good tunnel junctions also excellent Josephson super conductor insulator super conductor junctions this all relies on the fact that this aluminum oxide is an excellent oxide it grows very uniformly without any holes and with a very thin layer that it can be very well controlled with the thickness of that layer is so these junctions are very often used in quantum transport experiments and for most of the superconducting qubits for example here's an example of a qubit I think this is a sample that was fabricated in Germany in a group of Alex Haley's Tina and here he made a little loop with three overlap areas which are these tiny Josephson junctions of aluminum aluminum oxide so this is how it might look this scale bar is one micron so these are much smaller than a micron and you have to use electron beam lithography to define these patterns and you also see shadows in this picture so because you first deposit this angle and then at that angle everything is doubled and so you can see this shape and then offset from it another shape and it's the overlap of the double shapes that they control with angles of their position and create these junctions so this is a prototype of a something that is called a three josephson junction flux qubit we will discuss it in next week I think next yeah this is another technology that is very mature also very reliable and it is called the niobium trilayer junctions the is a bunch of layers but the important ones are they deposit a layer of niobium which is a much stronger superconductor than aluminum then they cover it with aluminum then they oxidize like in the previous step they oxidize and then they put a layer of niobium on top so they make a sandwich of niobium aluminum aluminum oxide niobium and then they chop this sandwich up and make circuits out of it so this whole bunch of layers is a is a process which is a factory quality process developed by this company Hyper's and they call it niobium I see if you see at the top there niobium integrated circuit process this is from their website you can download the manual for how to design circuits for rules for all these layers and submit it to them and they will give you chips with your circuits and this is used by several research groups around the world for magnetic sensors and things like that based on Josephson junctions also by this company d-wave i believe in in canada i think at least at some point they purchased their chips from from high price maybe not anymore but they used to so they would order a circuit and and it will come also by a community which tried to create the computers based on Yodas injunction so not on based on silicon semiconductors but on something called rapid single flux quantum RS FQ logic this is an example of such sir let's hear the analogy to integrated circuits is quite applicable because this is also a computing circuit where classical not quantum logic operations are done with magnetic fields created in the loops of Josephson junctions and to hear a bunch of different Josephson junctions some are bigger some are smaller other elements include inductors and the reason why you have so many layers in addition to this important one is because you want to connect different junctions with with superconducting interconnects and also maybe create shunt resistors etc so some of these are shunt resistor layers some are interconnects and some are dielectrics to separate different layers so this is a very highly controlled technology niobium Josephson junctions is something you should be able to relate to because I've been talking about these materials for a large fraction of the course Josephson junctions based on a semiconductor nano wires carbon nanotubes graphene etc also many new materials like topological insulators oxide 2 legs etc call them hybrid the junctions because they came a little bit after the development of those other types of junctions like aluminum and niobium and they couple semiconductors to superconductors so this was a long challenge for technical reasons and therefore when when all of this became possible with high quality in these new materials see the papers got very a lot of attention and got into big journals but actually the physics that they study it goes back to the 1960s to the original work of Josephson after the fact the devices appear to be very simple you take for example a semiconductor nano wire and you put two aluminium electrodes on it and so the contact is good enough then the current flows from one electrode to the other without dissipation so it's a super current here is a current on this axis and they sweep the current and they get no voltage until at some point it switches to this other state in this case the junction is actually hysteretic so the red trace is going in this direction and then when they sweep it back it stays in the resistive state for longer and it switches back to superconducting state at a different point so at I R it's called a reach wrapping current critical current and REE trapping current so if the junction is retract back into the superconducting state we will discuss why that happens in later slides but this shows you that this is not simply a an element where up to some parameter it's a zero resistor and after another parameter when it's parameter is exceeded its a finite resistor it's much more than that it's it's there's something strange nonlinear going on in this Junction here for a carbon nanotube a little bit hard to see the beautiful fabry-perot data you should look at this paper if you want to study it but what the data shows is a critical current now which is a tunable by gate so these are values are for different settings of a back gate a VG underneath the nanotube so it seems that in a semiconductor you can affect the value of the critical current by sweeping a gate just like you could affect conductance in a quantum dot or a quantum point contact by a gate you can also affect the super current you can do all these same experiments in graphene now with those hybrid semiconductor systems in mind let's go back to Andreev reflection and try to understand from this perspective what gives rise to super current what gives rise to super current is something called Andre of bound States first let's look at at one side of this Junction and remind ourselves what and rave reflection is on where reflection is a mechanism through which electrons can enter the superconductor right electron comes to the superconductor and it is reflected as a whole with the opposite energy so a Cooper pair then goes this way into the superconductor and charge is conserved because we gave one electron and we extracted a hole so we transfer the charge of TUI per and drave reflection into the superconductor and the energy works because this one is plus e this one is minus E and Cooper pair has zero so energy is conserved charge is conserved so this is allowed and it does happen now let's think what happens if there is another superconductor on this side well so if you come with an electron do this have a reflection here you go back you do another entry reflection and you send a Cooper pair this way maybe and then you can go back so you can kind of go in a loop in this process so this is now at zero bias so different from multiple Andre reflection where at each bounce the two electrons were gaining energy from bias at zero bias this will just go in an infinite loop and actually it is not so different from a quantum dot if you think about it because this electron hole particle which goes as an electron comes back as a hole and this does this loop this particle is bound in between the two super conductors it is not bound exactly like an electron in a box because at every reflection a Cooper pair is kind of nucleates and goes into the superconductor but that's okay because the number of Cooper pairs in the superconductor is not conserved the bosons so you can throw as many of them as you like in but it is like a quantum dot because these states form a discrete spectrum due to the confinement between the two superconductors so the condition for them to form this spectrum is very simple it's like wave matching solution like like in a particle in the box or an interferometer you have to make sure that as electron goes and comes back the phase accumulated in this entire process is equal to 2 pi so like like Bohr Sommerfeld model in an atom right you have to fit the number of wavelengths inside the trajectory on a on an orbital the same way very simple and just that simple condition would give you a spectrum of these andreev bound states for each Junction here are some extremely simple calculations the phase difference can come from just the phase difference between the two superconductors Phi 1 minus Phi 2 that's the quantity we discussed in the first slide then there could be a phase difference accumulated between and halt if there is no perfect symmetry between them and that that would scale with length of the junction and then there's this term comes from the confinement energy scale the gap because the the height of the potential barrier is the gap so if we are in a fairly short Junction and this term is irrelevant or if we have perfect particle-hole symmetry we can also put this to zero and extract energy from here we will get that the energies of the end rave bound states in this kind of simplified situation are proportional to the cosine of phase over two and again remember this file is phase it's a phase difference between the 2 superconductors a phase drop across the junction so once again this phase drop plays a role in the energy spectrum of under a bound states in in this superconductor normal metal superconductor Junction if it's a very long Junction and this term dominates then we actually get a something that's very simple energy rules linearly with phase and then another important thing about these states is that they always come in pairs that's due to particle-hole symmetry but the reason the easy way to understand this plus minus sign is the following remember I'm showing you that electron starts from here and becomes a hole there is another state where a hole starts from here and becomes an electron so the state that goes in the opposite direction and that will have the kind of like the opposite energy so then we talk about hole like and drave bound states and electron like and drave bound States and they will always be symmetric with respect to zero energy drove bound states have been measured directly and indirectly many times I want to show you a recent a beautiful experiment from France where single and drave bound states were detected in a carbon nanotube this is their device and it is a carbon nanotube I don't know if you can see but it's this hair in the picture over here and they've made an aluminum device around it using the same shadow technique like for tunnel junctions but they they didn't really accid eyes anything but the reason why they use the shadow technique is to create this different electrode in the middle so the Josephson junction is between the green and the green and this guy is a tunneling probe on top it's on top of the nanotube so what they the difference between them is that when they before putting the green layer they clean the bit I think the nanotubes so they made a very good contact here and for this one they didn't so it works as a tunneling probe so it's a it's a higher potential barrier to get in so what they have is a way to do spectroscopy on this Josephson Junction from a tunneling probe and what they also introduced here is this loop we will talk about loops in the next lecture but just to motivate it for you for now this loop allows them to adjust the phase difference between the fork between the two green electrodes the way that works is they put magnetic flux into this loop and this flux creates a phase difference between the two superconductors similar to our own of bomb interferometers any other interferometers so I realized I did not touch on this in this course but just for now I want you to think about loop as a phase control mechanism create a phase difference so then when they sweep the phase they just sweep actually current for some coil to change the flux in this loop they see in their tunneling characteristics these lines that that wiggle around so the measurement is send a voltage bias from here to the superconductor so current goes here into the nanotube splits around and goes goes out but from the red to the nanotube it has to tunnel and so we are doing a tunneling spectroscopy experiment like scanning tunneling microscope or tunneling from a normal lead onto a quantum dot so if we have a quantum level aligned with the voltage bias that we apply we will observe a resonance in transmission so it's a similar concept as in quantum dot lectures so they identify these resonances as Andreev bound States because they had this characteristic dependence on phase this is a line cut one of the states is a superconducting gap they assign it to Delta so remember from last lecture at the edge of the superconducting gap we have an anomaly so that will show up as a resonance in transmission if you tunnel from a normal metal to the superconductor there will be a resonance at the gap and below the gap these phase dependent resonances are enjoy a bound States and it looks I guess I think you will all agree that it looks like they have some kind of a sinusoidal Oracle's cosinusoidal dependence on the face and that is very typical and remember the Josephson equation was a sine Phi term so pretty much in most of the junctions that we study it will be a sine Phi or a cosine Phi one way or another what they also could do in this device is of course sweep or gate the the gate in this case is a bad gate it's a layer underneath the substrate this one and they saw that these resonances also change as a function of gate voltage there is some periodicity related to the period of Coulomb Peaks when they add another electron on the quantum dot on a nanotube so this is also in agreement with what they found earlier when they saw that the super current depends on the gate so then if spectrum of andreev bound states which carries super current from left to right depends on the gate and also the super current will depend on the gate okay so once again in a superconductor normal metal superconductor Junction confined by the superconducting gab on two sides electrons will form and drave bound states which go like this electron goes into a hole and in one cycle of this Cooper pair is transferred from left to right and because phase coherence is maintained energy of these states is even dependent on phase that is why the two condensates on the left and on the right are coupled and the current can flow without dissipation this is like one wavefunction going through this dre of stage and it's perfectly fine to also think about this process as just the overlap of the wave functions on the left and on the right and drave method is very powerful gives a very good predictions as I'll show you in a later slides and very useful for many of the new experiments to think about it in terms of andreev States now let's think about what happens if you have not a superconductor normal metal where electrons can fly but the super conductor insulator structures well if you just have two wave functions here and here they will just have to overlap through the insulator and allow for the tunneling process to occur you can go continuously from one regime to the other keeping in mind the notion of and rave bound States so I want to to show you this as an example let's start here and let's make a little break here like a little mirror that reflects these states with some small probability then it turns out that the formula for a bound States from this converts into this so the difference is this towel is a transmission of an Andre bound state so if an Andre bound state has a final transmission from here to here you have to add this towel and I'm not deriving the formulas here because they're fairly basic and I want to go forward so if you are curious how is derived I look it up now if we increase this gap more and more we can think about two sets of Andre bound states one is on the left side and one is on the right side we can even infinitely shrink this this guy and then what you have to think about is Cooper pairs undergo andre reflection here then the electrons tunnel another in a reflection here and they get out this is how you could continuously go in your mind from and rave States to tunneling you just have to introduce tunnelling between two pairs of and rebound States now what happens in the spectrum is that when you introduce this tunneling between and rave bound States you hybridize the whole like and the electron like branches of this process because on one side you start with an electron like steak and it has to tunnel into the whole leg state on the other side so instead of this crossing here you will get an anti crossing level repulsion which is proportional to the transmission of this barrier and an interesting thing happens that these lines these and rave States are four pi periodic because it's a cosine Phi over 2 and these due to this hybridization become 2 pi periodic in practice also in this Junction what will happen if you sweep the face for example is that you will always stay in the ground state so your or your current voltage characteristics energies will be 2 pi periodic because you will go from here to the ground state here unless these are completely decoupled and you have no way of going from one to the other okay I told you sine Phi sine Phi sine Phi yes it is the case in most junctions if you put two superconductors together most likely it'll be something like sine Phi in between them but there are many important examples where it's not the case and there you have to take into account what happens in the barrier first of all over here we have a case of a point contact a quantum point contact or just the classical point contact a very clean narrow constriction it turns out in that case the current phase relation if transmission is very high it is almost linear remember like those on very bound states had a linear term KF times L well if that dominates the current phase relation will be linear so current or energy versus phase will be linear but then if you lower the transmission and the under a bound states will couple it goes into something that looks like a sine at low transmission so from ballistic regime to tunneling regime you go from linear to transmission and now this function up here you can think about it as a straight line or you could also decompose it into harmonics of a sine function if sine function is your base this these skewed functions will just simply have higher harmonics in addition to sign and these higher harmonics have an interesting interpretation you can interpret them as a higher-order processes of cooper pair tunneling if a single cooper pair tunneling gives you sine Phi term sine to Phi term is like to cooper pairs tunneling and you can make those processes is dominate in clean structures so you can observe higher-order Josephson effects in specially designed experiments here's a curious example that I spent several years of my life studying it is a sinusoidal current phase relation but it's flipped you can see that a sine function goes down as we increase the phase so it is as if the sine function is shifted by PI or if you can also say a critical current is negative whatever you concept you're more more happy with you can also say that the minimum energy of this junction is at PI but we did not introduce this energy yet and the connection of the two I think we will return to this maybe in the next lecture but this is just a another example of what can happen in a Josephson Junction and then very recently theoreticians have proposed that you just take these and very bound States and if it's really forbidden to go from the lower branch to the upper branch of this point you will just stay on this line then from two pi periodic current phase relation to the current phase relation and what protects this transition is a my Arana fermion so very often you will read in my Arana papers that a smoking gun evidence for my uranus is this four pi period in the in the Josephson effect and i think the recent understanding in this a little bit of a red herring because if you start on this lower and drave state and then go up here it's an excited state and so the the phase particle remember this is a phase difference you can fold down to this state and so you will never see the 4pi period in a real experiment but this has been proposed as a very exotic this is probably the most exotic of these all of these effects now we come to the second java simulation and this one is a has to do with a very important area what happens above the critical current and also with the time-dependent property of Josephson junctions so again without deriving it I will postulate for this lecture that if you apply voltage bias to Josephson Junction a phase difference across the junction will change in a linear fashion with that voltage it will just start winding in time so at the time zero when you turn on the voltage the phase difference will start going linear with the slope which is 2 e over H only fundamental constants it's kind of amazing you can take a aluminum Junction a nanotubes Junction graphene Junction always the same evolution of the phase so it's a very fundamental relation and it also tells you that even above the critical current it is not just a resistor I already mentioned it to you from the hysteresis but also from this you can see we are in a finite voltage state so we have some voltage on the junction and it could even be that up here Ohm's law applies pretty well it's linear change but we know that underneath this there is something happening to the phase what is happening to the phase it's changing all the time so apply to voltage between the two superconductors and the phase starts going running away from you the phase difference between the two so one very simple exercise just plug this in to here this is the first jobs of simulation the sinusoidal one you will get at the finite voltage bias a constantly changing phase and this sine function will oscillate so this coupled to this dictates to us that at the finite voltage we actually have an AC super current flowing through the junction at the frequency determined by the voltage and the AC current actually leads to radiation so people have detected Josephson radiation by detectors that were hooked up to Josephson junctions this effect is real does not show up here because this is a time average measurement it's a DC characteristic so at each point we average for many seconds so if we could see in real time what's going on at this point there will be some oscillating current creating radiation so this dynamics is what gives rise to at least in part to this normal state resistance but it is very hard to see this radiation it is fairly small in amplitude and if you do any DC measurement at time averages so people have to do sophisticated measurements to see it but it is actually very easy to see the manifestations of this second relation but you have to do the reverse experiment you have to apply microwave frequency apply high frequency and the current voltage characteristics change very dramatically so under microwave excitation current voltage characteristic from this becomes a staircase of very sharp steps so current versus voltage becomes a bunch of steps these are several curves a different amplitude of excitation at a certain frequency by the megahertz in this case I think and the highest one going all the way up here and switching has no steps that's because the excitation was at zero so this is a critical current of this Junction then we apply AC excitation at high frequency and the critical current suppresses and these steps appear so these are called shapiro steps and the reason why they appear I'll tell you in a couple of minutes but what's remarkable about them is that you can always predict the frequency the voltage at which they appear because it's related to frequency that you excite with by this relation and you can track this relation back to there so this coefficient here is the Josephson frequency coefficient and the frequency is given by the voltage so it's a rate of change of the phase so when you match the voltage bias across the junction this is the voltage bias across the junction which creates this phase dynamics with an external excitation at the same frequency you get some kind of resonances so once again phase across the junction at finite voltage evolves according to this so it has a certain frequency which is this Josephson frequency you plug it into this sinusoidal function and if we match that frequency with an external excitation we get these dramatic resonances which are called shapiro steps and that is very easy to see in an experiment so now let's introduce the Josephson energy there is an energy scale associated with the Josephson junction it's called the Josephson energy and the way you derive it is like any other energy it's a free energy stored inside the junction so we have to take I times 3 times DT in order to ramp up the current we have to apply a voltage for a given time but V times DT according to what I just said in the last slide is d Phi which is the phase difference so energy is related to super current as an integral over phase or in other words super current is a derivative of energy by phase so that's really easy to remember because if you're super current is a sine function energy will be a cosine function so for a sine function energy will be a cosine function is cosine function is a particular shape which is 1 minus cosine that's what you get from a sine Phi and that tells you that the energy difference prefers to be at zero in this Junction that is the lowest energy state another coefficient in this formula this is the Josephson energy so that the scale of energy maximum energy is given by the critical current with some again some fundamental constants so critical current of the junction is proportional to this energy scale of the Josephson junction which is called EJ now let's think a little bit more about realistic Josephson junctions I just explained to you that it would be characterized by a certain energy eg a free energy stored inside the junction from the phase evolution of the wave function of the overlap of the two wave functions into superconductors but if the junction is small enough this familiar energy scale will also play a role the charging energy because if the capacitance is small this can be quite large so if we just put these two terms together we can write a Hamiltonian for a Josephson junction which includes the capacitive term with the number of Cooper pairs hence the four here the charges - and the energy term which is proportional to cosine Phi - the integral over sine Phi for the super current so in such junctions like I showed you for small aluminum circuits with very small junctions this charging energy can actually play a big role in in the properties of these junctions and that leads us to a very simple and intuitive way to think about this phase difference I will go through these equations they're very simple but basically what we're doing here is trying to calculate the total current that flows through the junction including all of its properties the Josephson current and the capacitance and the finite resistance but what we will arrive to is a LAN is a very intuitive picture for how to think about Josephson junctions so if we want to calculate the current armed with not the Schrodinger equation but with the circuit elements with these parameters like the Josephson energy and the capacitance and the normal state resistance we use this model which is called the rsj model resistive li shunted junction model and sometimes our scj modeler sees the capacitive and it is literally this model current can flow rather than thinking that it flows from one superconductor to the other we think that it slows for a parallel connection of a Josephson element resistive element and a capacitor and a Josephson element is characterized by this Josephson relation or you have to plug in something else here if you have a non sinusoidal element but this is a nonlinear part because this current if a depends on some parameter phase so it's not an Ohm's law element this guy is related to what happens at the finite voltage right so it's related to the second Josephson relation and instead of writing v equal to some constant over phase derivative we write I times our and this R is a normal state resistance it's a constant characterizing the junction and then this is the displacement current so that this current is this one and so we know that V is the proportional to D Phi DT so DV DT for the displacement current is proportional to the second derivative of phase and we have all these three factors critical current resistance and capacitance of the Josephson junction which we just plug into this model now let's put it all together the total current is the sum of the three and it turns out that it's proportional to second derivative of phase first derivative of phase and the sinusoidal term does this equation look familiar wave equation previous semester of physics LCR or even even an earlier semester yeah I would say it's a pendulum except what is what is swinging phase difference is swinging phase difference is swinging on a pendulum so we started with something very obscure phase difference that determines all the properties of this Junction now all we need to think about is a ball in this kind of a periodic sinusoidal potential yeah because we have this sine Phi term which gives us the sinusoidal potential and this slope turns out to be an external bias current bias so if we if we sit it becomes a pendulum if we only consider this region down here or a harmonic oscillator right if we want to consider this entire landscape the proper words to say are that it is a mechanical ball inside a washboard potential or tilted washboard potential the washboard is this thing where you can wash your clothes if you don't have a laundry machine very good device but maybe they have to come up with a new analogy because in modern times people probably don't use that anymore but we use it in the Josephson physics so it's still useful now let's look more closely at this equation I wrote it again up there so this equation describes the evolution of a phase particle which is remember the phase difference between the two superconductors across the Josephson Junction but I'll just going to call it the particle from now on in a washboard potential and so the term which is in front of the second derivative of the coordinate acceleration that's the acceleration what's in front its mass so mass is capacitance this is the analogy capacitance of the Josephson Junction is like mass first derivative that's friction friction is resistance except you get more friction if resistance is lower I have to remember that it's not like you get more resistance and you get more friction it's the other way around so it's an analogy never forget you're in an analogy it's not a son intuitive it's just simple it's just simple to remember and then the force is an external bias so individual well and its curvature is related to a plasma frequency that would be the self oscillations of the ball inside here so this is a formula you would get if you treat it as a pendulum okay so armed with that analogy we can understand a little bit the evolution of the IV curve remember in the IV curve we increase the bias and there is no current flowing oh sorry there is no voltage flowing developing we increase the current bias and voltage remain zero a what is voltage voltage is defied ET and so voltage would mean that a phase which is now just simply called X the coordinate has to change but so at low bias we are in this tilt and the ball is in one of these traps and so we are tilting it a bit but it's not coming out it's staying here so phase does not change therefore the voltage is zero so when does the voltage develop when we tilt it so much that these confinements are no longer there when the curve became flat on these parts and the ball it was sitting here and it start to go boink boink boink starts to go and as it goes we get the voltage we get D Phi DT in our equation so this is right at the critical point and then as we keep increasing the force we increase the tilt even more and then the bulges goes faster and faster and faster so this is simply considering the confinement in this potential but remember we also have friction resistance and we also have capacitance the mass the inertia so let's first talk about the regime where capacitance is small and resistance is large small also small so resistance is small we have a very viscous medium friction is large resistance a small friction is large that's when we get this simplest IV curve because I am tilting this potential and at some point the ball starts to roll and so I am just past the critical point and the ball is rolling and I tilted back and as soon as that force is gone as soon as the little traps appear again the ball is stuck again because it has so much friction and so little inertia as small capacitance and small resistance so then the IV curve is very simple a super current state that's when we are trapped inside one trap then it starts to roll but as soon as I trap it again it goes back into the super current state so this is called the overdamped Junction as opposed to the under-damped Junction the under dam junction is when you have a lot of inertia or you have a large resistance so very little friction then what happens is you tilt this washboard potential and you let the ball roll and the ball is heavy and it doesn't care about friction it just starts to go and it actually is hard to stop it you choose it back you already develop the traps but you just go over the traps because it has all this momentum so this is what's shown here you start and you don't go until you escape from the trap you switch into the finite voltage state but if you want to go back it will take you some effort to stop the ball so you have to reduce the bias way down and you get this hysteretic behavior so that's called under damped as it has a lot of inertia and you cannot stop the ball so the this analogy helps you explain very easily the hysteresis that people often see in Josephson junctions don't from the trapping current you can estimate the capacitances and parameters of the junction you can predict where this will occur from there from our rsj model sometimes this part is rounded here so that is like premature escape from the trap maybe there is a thermal excitation so you still have a wash board still has maxima and minima but they are pretty shallow and then thermal energy can kick you out but remember once you're out the next minimum is lower so you just start to go so that's why there will be some switching before you reach the critical current if the critical current is determined by the point where there is no minimum anymore in the trap so we're still here we still have minima but we can escape okay now let's think again about Shapiro steps I promised you an explanation this will get again a hand waving explanation I wish I brought a washboard with me you will have to use a little bit your imagination here but once again Shapiro steps are under the influence of external radiation we develop these resonances and what are these resonances it is that we keep increasing the voltage or we keep increasing a current but the voltage doesn't change it is a finite voltage but it remains the same so what is the voltage voltage is a rate at which this particle moves so you keep increasing the tilt but the particle moves at the same rate the rate does not increase that's what happens on a Shapiro step so this is a data from this paper without excitation they have this black curve which is a little bit hysteretic by the way so this really does happen in experiments but it doesn't have these other steps and under radiation in the range of gigahertz they start having these steps so radiation can be modeled as you have to add a force which is time dependent and it's sinusoidal it's periodic to this potential and what that does is it rocks it like that it tilts it like that so then there is a actually a classical phenomenon which is called phase locking which tells you that if the force is a certain phase then as you try to go over the well if the force always pushes you back you slow down a bit and it leads to a phase locking between the motion of this ball and this potential such that over each period of this force you are only allowed to jump over one bump that's because and you roll here with all your momentum that you gained the potential rocks backwards and pushes you back so this this classical phase locking phenomenon leads to the fact that even though you increase the tilt the the rate at which you go is independent of that it's dependent on how fast the force is going so the force is going a little bit faster then you phase lock at a different frequency but every every period of this drive you overcome one bump then what are these other steps well that is also very simple in one period you shoot over two bumps so there's a higher-order phase locking processes two bumps three bumps etc so this is a way to think about these shapiro steps okay this dramatic picture is a 3020 Josephson junctions in a long chain and what they do with these junctions is they apply a microwave radiation like in the last slide but they have 3000 junctions for more accuracy and what they do is they measure the voltages of Shapiro's steps and this happens to be how our society sets the standard for electric volt for 1 volt because if you know the frequency with which you excite this system then the voltage difference between shapiro steps is determined by that frequency with fundamental constants like 2 e over H so this is the most accurate way we have to determine voltage knowing just the fundamental constants because we can very accurately measure and create different frequencies we can use our ability to create frequencies to very accurately measure volts so they do it at the NIST National Institute of Standards using Josephson devices and using Shapiro steps actually to measure the standard of volt in the last couple minutes I would like to play with this analogy a little bit longer I think this is a very beautiful result and this is what's driven the field of superconducting qubits for the last two decades so what do we have we have a bowl with a mass and this bowl has is described by coordinate in one dimensional space which is phase so if we make the mass of this bowl smaller and smaller and lower the temperature we might go into a quantum regime right where it should be described by the uncertainty principle between the momentum and the coordinate and I already substituted Delta Phi for the coordinate because we know what that is now what is momentum momentum is M the mass times X dot which is Phi dot which is this which is this which is this this is the charge so the uncertainty principle is actually between the charge and the phase or between the number of Cooper pairs and the phase difference across the junction so if you know the charge very well you completely don't know the phase or if you know the phase very well like you set it from a flux loop charge is completely unknown so you can you know I introduced it to you as a kind of a mental exercise let's play with the rsj model and derive this uncertainty principle but it happens to be a very fundamental property these two quantities really are connected by this uncertainty principle and you can measure it in experiments you can measure the quantum mechanical behavior of a face particle you can also make a connection between the quantum mechanical properties of the face and the fluctuating charges but today I will just show you this part and then in the next lecture we will talk about the the other quantum experiments so this is what I just described you can you can fix the face or you can fix the charge and what determines that is the ratio between the Josephson energy and the charging energy so if you have a very large Josephson energy and the charging energy is negligible that means that the charge is fixed there is no uncertainty there so you get superpositions of phase states superpositions of phase particle in this world potential and if joes and energy dump is dominated by a charging energy you can see superpositions of charges over the Josephson Junction but the phase will be fixed and there are also a continuous spectrum of devices from this extreme to this extreme the experiment I will talk about today is called macroscopic quantum tunneling and it is a treating this washboard potential as a potential barrier and trying to see the particle going from here to here quantum mechanically by tunneling not by jumping over so the way they actually do it is they bias the Josephson junction close to the tipping point and so after the particle has tunneled it just goes it runs it goes into the running state so what they had to measure is simply a voltage developing across the junction so switch into the finite voltage state so this is a very beautiful and to me very important paper that maybe one of the first demonstrated quantum properties in the Josephson junction related to the phase dynamics and what they've measured is a the distribution of switching voltages for Josephson Junction as a function of temperature and the width of that distribution so they measure a thousand times at which bias if the junction switch and that is that width of that distribution closely followed temperature for high temperatures so escape from this well to the running state was just temperature broadened events so you get energy from temperature and jump out and starts to run but then they started lowering the temperature more and more the temperature kept dropping but the escape rate the distribution of those escape rates saturated so even though it should be harder and harder to escape if you don't have energy to jump over it happened to be equally easy below certain temperature and they've attributed it to macroscopic quantum tunneling processes I encourage you to read this paper it's very easily written especially if you know the water board potential idea and to me you know the washboard itself is created by quantum mechanical processes the Josephson effect is a fundamentally quantum mechanical effect tunneling of wave functions from one super conductor to the other but now what we have done is we have defined a new degree of freedom this quantum particle which is a phase difference and we observe the quantum mechanical behavior that degree of freedom so we have layered layers of quantum of on top of each other and much of the modern condensed matter physics has to do with layer in quantum on top of quantum what was considered quantum in the 1960s we now call it classical the Josephson effect itself and quantum is the superpositions of phase particle and we will dedicate another lecture to that but for now I will stop

Channel: Sergey Frolov

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Id: -HUVGWTfaSI

Length: 78min 12sec (4692 seconds)

Published: Thu Mar 07 2013