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Topological Superconductivity in Twisted Cuprate Double Layers
In the last of the prepared presentations, Marcel Franz of the University of British Columbia spoke on topological superconductivity in twisted cuprate double layers. However, he said, to avoid much of the complexity that inevitably accompanies oxides, he would focus on some very simple physics in order to bring the oxides down to a level of simplicity that is comparable to what is seen in graphene. That is, he said, there is a very simple low-energy description that he would use in order to understand and predict some interesting new phenomenon. Some of what he was going to describe had already been published, he said, and he offered the references for those papers (Can et al. 2021a; Tummuru et al. 2021).
ENGINEERING A BILAYER TOPOLOGICAL SUPERCONDUCTOR
The basic idea behind his approach, Franz explained, is to engineer a high-Tc cuprate bilayer into a topological superconductor. Most monolayer cuprates, such as Bi2Sr2CaCu2O8+δ (BSCCO), are d-wave (dx2–y2) superconductors, he said. The d-wave superconducting order parameter is typically pictured as a cloverleaf (see Figure 7-1), and the order parameter is such that when the sample is rotated by 90 degrees, the order parameter changes, which is represented by the two different colors used for the lobes in the figure.
Next, one adds a second layer of this cuprate superconductor but rotated by a large angle, close to 45 degrees, to form a bilayer system and let the electrons tunnel between the two layers. In the presence of weak interlayer coupling, this setup has the potential to produce what is known as a dx2–y2 + idxy superconductor if there
SOURCE: Marcel Franz, University of British Columbia, presentation to the workshop, May 19, 2021.
is a complex phase between the two order parameters. It is known, Franz said, that this is a fully gapped topological superconductor with protected chiral edge modes and it exhibits spontaneously broken time-reversal symmetry. Such materials have been discussed in the literature, he said, but it has been mostly theoretical discussions as there are no actual physical examples of this type of material.
Thus, he continued, his goal with his presentation was to show that the bilayer construction he described, if realized, will likely form this exotic d + idxy topological superconductor. He would also, he said, explain what that means for those who are not used to thinking about topology.
The basic design for such a topological superconductor, he said, would be a strip consisting of two layers of cuprate superconductor, with one of them twisted relative to the other at an angle close to 45 degrees. Away from the edges, this would be a fully gapped superconductor with no excitations at low energies, but at the edges there would be topologically protected chiral edge modes.
This theoretical exercise “is not a pure fantasy” Franz said, pointing to a 2019 paper by Yu et al. After many years of trying, those researchers were able to isolate single layers of the cuprate superconductor Bi2Sr2CaCu2O8+δ and show that such a single layer becomes superconducting at about 90 K, or within a few percentage points of the critical temperature of the bulk material. For three decades, he said, researchers had debated whether a single copper-oxygen plane from a high-temperature superconductor would itself be superconducting, and the answer is yes. Furthermore, the researchers were able to perform various measurements on the single layers and show that they behave like the bulk superconductor in various other ways as well. One benefit of working with a single layer, he said, is that it is
straightforward to adjust the amount of oxygen in the layer via annealing, whereas with the bulk material one must produce many different samples in order to see how properties vary with changes in the amount of oxygen.
“So what I’m going to talk about is basically grounded in reality,” Franz said. “What I’m proposing here to take two such monolayers and assemble them into a bilayer with a twist to realize this topological superconductor.”
THEORETICAL UNDERPINNINGS
After explaining his goal of using two monolayers of Bi2Sr2CaCu2O8+δ with a twist of about 45 degrees to create a topological superconductor, Franz sketched out the theoretical calculations that indicate that it should work. He began by explaining topological superconductivity and, specifically, what it means to add the two order parameters with a complex phase.
He began with a basic representation of a d-wave superconductor, such as found in a single layer of BSCCO (see Figure 7-2, upper left line drawing). The blue circle represents a circular Fermi surface, and the orange cloverleaf represents the d-wave gap that exists on the top of that surface. “The fact it’s a d-wave is manifested by the zeros of these gap that occur along the diagonals,” he said, pointing to the
SOURCE: Marcel Franz, University of British Columbia, presentation to the workshop, May 19, 2021.
places where the cloverleaf shape intersected the circle. By symmetry, he added, the d-wave order parameter is required to change sign upon a rotation by 90 degrees, so one lobe will be positive, and the lobes on either side of it will be negative, and because the gap is represented by a continuous function, its value has to be zero between a positive lobe and a negative lobe—that is, along each diagonal. This is a pure dx2–y2 order parameter.
Rotating this by 45 degrees produces the situation shown in the upper right-hard part of the figure. This is a dxy type order parament, Franz said. It has similar point nodes, but now they occur along the x and y directions, not along the diagonal.
One can then create various composite order parameter by making linear superpositions of these two. This linear superposition can be written as dx2–y2 + eiϕdxy where ϕ is an complex number parameterizing the superposition. To illustrate what happens when the two order parameters are superposed, Franz first illustrated a real linear superposition, dx2–y2 + 0.3dxy. The result, as one can see with basically high school math, he said, is essentially just a rotation of the original order parameter (lower left-hand drawing in Figure 7-2). “These point nodes remain intact, but they rotate by some small angle that can be easily calculated.”
However, something very different happens if the two order parameters are added with a complex phase (dx2–y2 + 0.3idxy, lower right-hand drawing in Figure 7-2). “Instead of rotating, I actually open up a gap at the original nodes of the order parameter,” he said. “This is the origin of the fully gapped superconductor that can occur in this way.”
Having offered that background, Franz described in qualitative terms the results he has derived about what happens when two single layers of Bi2Sr2CaCu2O8+δ are stacked on top of each other with a twist. For small twist angles below a certain critical angle, it is energetically favorable for the two order parameters to add with just real coefficients, as in the first of the two superpositions he described, and in this case, “nothing very interesting happens.” The resulting order parameter looks like the order parameter of the original two-layer d-wave superconductor with point nodes in the momentum space that are shifted a bit by that twisting. On the other hand, if the twist is larger than a certain critical angle—which depends on the particular model parameters—then it is energetically favorable to add the two order parameters with complex phase, as in the second of the two superpositions he described, and the result is a fully gapped topological superconductor.
Time reversal symmetry is broken in this situation anytime the phase, ϕ, is different from 0 or π, Franz said. This is because under time reversal (i.e., ϕ → -ϕ, there is a different phase in the linear superposition dx2–y2 + eiϕdxy).
With that qualitative description of what happens in a twisted bilayer of BSCCO, Franz offered more detailed analyses based on Ginzburg-Landau theory for twisted d-wave bilayers and on continuum Bogoliubov–de Gennes theory.
Ginzburg-Landau Calculations
The simplest way to model the situation, he said, is to use Ginzburg-Landau theory for calculating the free energy of a superconductor near the superconducting transition. The resulting equation, based on modeling the bilayer as two complex order parameters and writing all possible terms consistent with symmetry, has five terms. Two of them describe the free energy of each of the individual layers, while the other three arise from the coupling of the two layers. Franz makes the additional assumption that the two layers are identical so that their two order parameters have the same amplitude, although there can be an arbitrary phase difference between the two.
The resulting equation for the free energy contains two terms that show how that free energy varies with the twist angle, θ, and the phase, ϕ. The twist angle terms corresponds to ordinary Josephson coupling, and because it appears in the form cos(2θ)cosϕ, the Josephson coupling term goes to zero when the twist angle is close to 45 degrees. The other term is a multiple of cos(2ϕ).
Plotting the free energy as a function of the phase, ϕ, for various values of the twist angle, θ, Franz showed that when the twist angle is small, the free energy has a single minimum at ϕ = 0. “As you would expect,” he said, “for small twist angles the two layers are in phase. There’s nothing very interesting happening.” However, as the twist angle increases and the cos(2θ) term gets smaller, there is a critical twist angle, θc–, at which point the free energy as a function of the phase reaches a critical point and has a pair of minima at non-zero values of the phase. “This signals the spontaneous time reversal symmetry breaking,” he said. “The system can choose between a plus phase or a minus phase.” Continuing to increase the twist angle beyond 45 degrees, the cos(2θ) term starts to increase again, at some second critical twist angle, θc+, the free energy again has a single minimum. Thus in the region around 45 degrees that stretches from θc– to θc+, there is time reversal symmetry breaking and topological superconductivity. This is the range, Franz said, “that I’m so excited about and that is the subject of this talk.”
Taking into account how the order parameters vary with temperature, Franz produced a phase diagram showing at what temperatures and twist angles the bilayer materials would be a topological superconductor. According to that diagram, the temperature at which the bilayer material is superconducting varies from O K (absolute zero) at θc+ or θc– up to Tc, the critical temperature of the single-layer material, when the twist angle is 45 degrees. Since single layers of materials such as BSCCO can be superconducting at temperatures as high as 90 K, this indicates that with a twist angle of 45 degrees, the bilayer material should also superconduct at around 90 K. “This would be a truly high-temperature topological superconductivity,” he said, “and this would be perhaps quite unprecedented.”
Continuum Bogoliubov–de Gennes Theory
Next, Franz described some of the microscopic theory that lies the GinsburgLandua theory, specifically continuum Bogoliubov–de Gennes theory. His goal, he said, was just to describe some key points and not go into a lot of detail.
Starting with a standard Hamiltonian for a d-wave superconductor, he briefly described some calculations that allowed him to plot the phase at which the free energy of the superconductor was a minimum as a function of the twist angle. The resulting plot matched up very closely with the plot he had calculated from Ginzburg-Landau theory, but in this case the microscopic theory gave him something additional: the size of the excitation gap in the twisted bilayer structure. “As I mentioned,” he said, “this is a fully gapped superconductor, which means there is a real energy gap now, as opposed to the untwisted superconductor.” And the calculation showed that the gap can be quite large, depending on the twist angle—up to about 20 meV. By comparison, he said, the gap in bulk Bi2Sr2CaCu2O8+δ is about 45 meV, “so this is a healthy fraction of the maximum gap in this material.” Thus it should be relatively easy to observe the gap with various spectroscopic techniques, such as scanning tunneling microscopy.
Protected Edge Modes
To explain protected edge modes, Franz described performing calculations on a long rectangular strip of the twisted bilayer material that the previous calculations had examined. It is relatively easy to calculate the spectral function A(kx, ω) inside the strip, he said. He explained that the graph of the spectral function shows the excitation spectrum but spatially resolved. The graph based on the calculations shows that the material at that point is fully gapped, which is roughly the 20-meV gap that he had just described. Thus the calculations indicate, as expected, that the material is a fully gapped superconductor on the inside of the strip.
However, when one calculates the spectral function along one of the edges, some extra states appear inside the gap, and they are all dispersing in the same direction. In the particular calculation that Franz showed, there were four of these extra states. This is a sign of topologically protected chiral edge modes, he said. It is clear that they are chiral, he explained, because if one performs the same calculation on the other edge, there is a similar group of four extra states dispersing in the opposite direction. Thus the strips have a bulk gap combined with topologically protected chiral edge modes.
Self-Consistent Theory on the Lattice
Next, Franz described another level of modeling in which he carried out a lattice analysis of a Hubbard model with nearest-neighbor attractive interactions
and on-site repulsion. In this case, he said, to make any headway one has to use commensurate twist angles. One can then solve everything with standard methods.
He showed a number of results of calculations with this approach. First, he showed a series of commensurate moiré patterns that progressively approximated a 45-degree angle, which is whether things are most interesting, he said. Then he showed a series of Fermi surfaces, first for the untwisted bilayer and then for three twisted bilayers with increasing twist angles. Next he showed a series of phase diagrams for angles getting increasingly close to 45 degrees (Can et al. 2021a). “Let me just say that near 45 degrees pretty much the whole phase diagram, wherever you land on this phase diagram, you’re bound to get something interesting,” he said. That is, close to 45 degrees, much of the phase diagram is covered by topological phases. “And this is what made us excited about this project.”
Density Functional Theory for Bi2Sr2CaCu2O8+δ
Franz also carried out some density functional theory calculations in hopes of better elucidating the material properties of this system. Because the crystal structure of these materials is quite complicated, the calculations are not simple, he said, but two collaborators, Ryan Day and Ilya Elfimov managed to perform full-density functional calculations for some of these commensurate angles for twisted geometries. “The principal result of the calculations is that these structures are stable,” he said, showing a graph of total energy as a function of interlayer spacing for twisted structure. The graph indicated a distinct energy minimum at about 12.6 angstroms, which is comparable to the distance between the layers in bulk crystals of the material, Franz noted.
He had hoped to abstract some parameters for his team’s tight binding models, Franz said, but the calculated band structures are extremely complicated, and “it is very difficult to model this by any sort of tight binding model with any degree of the reliability.” However, he continued, “we convinced ourselves that there should definitely be coupling between the copper–oxygen planes on the order of 10s of milli electron volts. That’s the best estimate.”
EXPERIMENTAL EFFORTS
Switching to experimental results related to these materials, Franz first discussed the work of the Ziliang Ye group at the Stewart Blusson Quantum Matter Institute at the University of British Columbia, who have been making samples of the materials and probing for time reversal symmetry breaking. He has already had some preliminary successes, Franz said. He has found, for instance, that two BSCCO flakes with a twist angle of 45 degrees superconduct at 80 K—nearly the 90 K of the bulk material.
To probe for the breaking of time reversal symmetry, Franz said, Ye is planning to measure the polar Kerr effect, which is sensitive to time reversal symmetry breaking. The experiment involves shining polarized light on the sample and measuring the rotation of the polarization in the reflected light. The angle of rotation is called the Kerr angle, and it is related to the Hall conductivity in the sample, which only can be nonzero if the time reversal symmetry is broken.
There are formulas for calculating the Hall conductivity, and Franz’s group has done those calculations for BSCCO (Can et al. 2021b). It turns out, he said, that the calculated Hall conductivity—and thus the Kerr angle—in these materials is quite high, being several orders of magnitude larger than for Sr2RuO4, which was also thought to be a chiral superconductor candidate. The difference in the magnitudes can be attributed to the much larger gaps in BSCCO, he said, because the Kerr angle is proportional to the square of the gap. But the main takeaway, Franz said, is that given the magnitude of the effect, the Kerr angle for BSCCO should definitely be measurable. “We are hopeful some results will be coming this summer,” he said, “and we are very excited about this.”
A collaboration between Franz’s group and some researchers at Harvard is studying the Josephson current that passes between two BSCCO flakes that are at an angle to one another. They are using a clever design with the electrical conducts on the bottom side of the top flake and on the top side of the bottom flake so that they see a signal primarily from the interlayer junction. The researchers in the collaboration have some very interesting results, Franz said, but since they had not been published, he could not describe them. But this is another effort at his lab that is looking for evidence of time reversal symmetry breaking.
One interesting question to ask, Franz said, is whether there is any physics in the twisted BSCCO flakes that is similar to what is seen in graphene, such as interaction effects, flat bands, and so on. “Indeed, the answer is yes,” he said, and he pointed to a preprint from a group at Rutgers University (Volkov et al. 2021). They found some interesting physics that occurs at a relatively small twist angle of around 10 degrees. According to the authors, as the twist angle is increased up to around 10 degrees, the Dirac cones from the individual layers collide, and because they have the same chirality and are protected by time reversal symmetry, they cannot gap out within a non-interacting model, so instead, at some magic angle, they form a quadratic band crossing.
Time reversal symmetry does not allow the formation of isolated flat bands as happens in graphene, Franz explained, but the equivalent of the magic angle in graphene is the angle at which the quadratic band crossing forms, and the authors argue that although interactions are perturbatively irrelevant at the Dirac point, they become relevant at the quadratic band crossing. Then infinitesimal interactions can actually break time reversal symmetry and gap out the system at that magic angle. The authors proposed a phase diagram in which the mate-
rial has time reversal symmetry broken in a region around the magic angle and close to 0 K.
In response to that work, Franz’s team carried out some calculations of its own, and he showed a phase diagram that they calculated for the material assuming interaction-induced s-wave instability. In that diagram a dx2–y2 + idxy superconducting phase appears of the sort that Franz had described earlier in his talk, and at a twist angle of 45 degrees that superconducting phase extends all the way to Tc, the critical temperature of the bulk material. This is a phase in which time reversal symmetry is broken, and it should be observable with various spectroscopies, he said.
LOOKING AHEAD
In the last part of his talk, Franz offered some forward-looking statements about the field and, in particular, about his group’s interest in Majorana fermions. It is well known, he said, that topological superconductors often host Majorana zero modes at vortices, corners, or other defects. And indeed, he added, in most papers about topological superconductivity, Majorana fermions are given as the main motivation because they can be used in schemes for topological quantum computation.
Various types of Majorana states have been reported in the literature. For instance, Majorana zero modes have been observed in semiconductor quantum wires in proximity to superconductors, and Majorana fermions have been seen in ferromagnetic atomic chains on top of a superconductor. These are both examples of a one-dimensional topological superconductor that has Majorana fermions bound to its ends, Franz said. There are also two-dimensional examples where Majorana fermions occur in the core of a superconducting vortex. This situation may arise when there is a topological insulator interfaced with an ordinary superconductor, and that interface forms a kind of topological superconductor that supports Majorana zero modes in individual vortices. This is a large field and a hot topic, he said.
So it is natural to ask whether something like this happens in the dx2–y2 + idxy topological superconductor, but the answer, he said, is unfortunately no. The reason is that this is a spin-singlet topological superconductor, and because of spin degeneracy, Majorana zero modes are not expected to appear. The next question to ask, then, is whether there is some variant on this construction that could host host Majorana zero modes, and Franz sketched out an idea of how that might work.
Instead of a d-wave superconductor, imagine that one has a p-wave superconductor that can be exfoliated to create a monolayer px superconductor. Then one could follow much the same pattern used to create the dx2–y2 + idxy bilayer superconductor. Take two monolayers of the superconductor with a twist angle of 90 degrees. By the same logic as before, this should produce a px + ipy superconductor.
This is known to host Majorana zero modes in the vortex core because this is a spin triplet superconductor with all of the symmetries. The drawback to this plan, Franz said, is that there are no such px + ipy superconductors known in nature. So, could they be engineered artificially?
“This is a somewhat farfetched idea,” Franz said, “but as theorists we can speculate.” So he offered an approach based on the semi-conductor quantum wires that have been proximitized by coating them with an ordinary superconductor. A parallel array of many such wires would form a px superconductor. Then a second such array placed on top of the first array but at an angle of 90 degrees would create a px + ipy superconductor.
Franz’s has team has analyzed such a superconductor theoretically using the same sorts of calculations they used to examine the bilayer BSCCO materials, and the calculated phase diagrams show large portions that contain the desired px + ipy superconducting phase with spontaneously broken time reversal symmetry. Furthermore, the calculations show that this type of superconductor would harbor an isolated Majorana zero mode in the vortex (Tummuru et al. 2021). This is a more speculative realization that the BSCCO bilayers, Franz acknowledged, but “I wanted to put it out there in case somebody has a more realistic idea how to leverage the spontaneous time reversal symmetry breaking to create something truly interesting.”
Finally, Franz concluded his talk by offering some questions that he and his group have been asking themselves: What is the best way to observe the topological phase experimentally? Are there any interesting uses for this novel topological superconducting phase once it has been identified? and, Are there other two-dimensional systems beyond graphene, chalcogenides, and cuprates that will produce interesting new behaviors under twist or similar geometries?
QUESTION-AND-ANSWER SESSION
The question-and-answer period following Franz’s talk was moderated by Aharon Kapitulnik, who opened the session with two submitted questions: How does one find values for or estimate the interlayer coupling strengths, and what are the differences, if any, in stacking BSCCO 2201, 2212, and 2223?
Concerning the first question, Franz said that when his team started this work they assumed that after 30 years of theoretical and experimental work there should be some consensus on what the interlayer coupling constant is in, say, BSCCO 2212. But a search of the literature showed that the values from various experiments and theoretical calculations differed greatly from one another—ranging from 5–10 meV all the way up to as much as 80–100 meV—and these were all reputable articles from well-known groups. These are not simple things to estimate, he said, basically
because the band structures are so complicated. “Depending on where you look in the band structure,” he said, “you get different answers for this quantity.”
As for the second question, Franz said that many of the calculations he showed were for BSCCO 2201, which is a cuprate with just a single copper–oxygen plane per monolayer, because they are simpler to do theoretically. The phase diagrams for 2201 and 2212 are very comparable, he said. It is BSCCO 2212 that has a higher critical temperature and that has been exfoliated to monolayer limits, he said, so one should probably focus on this material. In general, though, the construction approach that he described should work for any of these materials and, in principle, for any monolayer d-wave superconductor.
Next Kapitulnik asked his own question about pseudogaps. Noting that the BSCCO materials exhibit a pseudogap, he asked what is happening in that phase and, in particular, whether time reversal symmetry might be broken already in the pseudogap phase before the superconducting transition. Franz responded that very little is known about the pseudogaps in these materials and, in particular, his group does not know how to model the pseudogap in BSCCO monolayers. “I’m trying to view this work as something to open doors to some new explorations of high-Tc cuprates in the monolayer limit,” he said, “and I expect there to be some surprises, and pseudogap might be one of the surprises.” Adding that experimental results for single monolayer show a pseudogap, he said that this question should be within reach of experiments.
Next Kapitulnik passed along a question about how Franz’s results depend on c-axis coupling. “Very good question,” Franz said. “The results are quite sensitive to this interlayer coupling strength.” In particular, while it is always that case that at a twist angle of 45 degrees in the BSCCO bilayers, the Tc of the bilayer will be equal to the critical temperature of the material, at least theoretically, but the strength of the interlayer coupling will affect how large the region is in which time reversal symmetry is broken. His group’s calculations assumed an interlayer coupling of about 10 meV, which is at the lower edge of the range in the literature, and with that coupling strength, time reversal symmetry was broken at twist angles of 45 degrees plus or minus about 7 degrees, and that represents a fairly large phase.
REFERENCES
Can, O., T. Tummuru, R.P. Day, I. Elfimov, A. Damascelli, and M. Franz. 2021a. “High-Temperature Topological Superconductivity in Twisted Double-Layer Copper Oxides.” Nature Physics 17:519–524.
Can, O., X.-X. Zhang, C. Kallin, and M. Franz. 2021b. “Probing Time Reversal Symmetry Breaking Topological Superconductivity in Twisted Double Layer Copper Oxides with Polar Kerr Effect.” Physical Review Letters 127(15):157001.
Tummuru, T., O. Can, and M. Franz. 2021. “Chiral P-Wave Superconductivity in a Twisted Array of Proximitized Quantum Wires.” Physical Review B 103(10):L100501.
Volkov, P.A., J.H. Wilson, and J.H. Pixley. 2020. “Magic Angles and Current-Induced Topology in Twisted Nodal Superconductors.” arXiv 2012:07860v1.
Yu, Y., L. Ma, P. Cai, R. Zhong, C. Ye, J. Shen, G.D. Gu, X.H. Chen, and Y. Zhang. 2019. “High-Temperature Superconductivity in Monolayer Bi2Sr2CaCu2O8+δ.” Nature 575:156–163.
