Understanding the pairing mechanism that gives rise to high-temperature superconductivity is one of the longest-standing problems of condensed-matter physics. Almost three decades after its discovery, even the question of whether or not phonons are involved remains a point of contention to some. Here we describe a technique for determining the spectra of bosons generated during the formation of Cooper pairs on recombination of hot electrons as they tunnel between the layers of a cuprate superconductor. The results obtained indicate that the bosons that mediate pairing decay over micrometre-scale distances and picosecond timescales, implying that they propagate at a speed of around 106 m s−1. This value is more than two orders of magnitude greater than the phonon propagation speed but close to Fermi velocity for electrons, suggesting that the pairing mechanism is mediated by unconventional repulsive electron–electron, rather than attractive electron–phonon, interactions.
Superconductivity is caused by pairing of electrons resulting from virtual exchange of bosons. In low-temperature superconductors Cooper pairing is mediated by phonons, but for high-temperature superconductors the pairing interaction is not yet confidently known1,2. There are arguments both in favour of conventional electron–phonon2,3,4,5,6,7 and unconventional electron–electron2,8,9,10,11,12,13,14,15,16 coupling mechanisms. The electron–phonon coupling in cuprates can be strong because of their ionic structure with high polarizability17 and poor screening3,4. However, it is difficult to reconcile this with a d-wave symmetry of the order parameter, which more naturally arises from repulsive electron–electron interactions, for example, via antiferromagnetic magnons8 or plasmons10,11,12,13.
In this work, we perform a new type of non-equilibrium boson generation–detection spectroscopy, which allows us to probe the pairing boson. The idea of the experiment is illustrated in Fig. 1a. When non-equilibrium quasiparticles (QPs) are injected into a superconductor through a tunnel junction, they relax to the ground state eventually recombining into Cooper pairs. This inelastic process is accompanied by the emission of bosons that are mediating in pairing. Therefore, identification of recombination bosons provides an unambiguous clue about the pairing glue18,19. This has been convincingly shown by similar experiments on low-Tc superconductors20,21,22,23, in which phonon emission has been detected, reaffirming the electron–phonon pairing mechanism. Here we perform such an experiment on a layered Bi2Sr2CaCu2O8+δ (Bi-2212) cuprate, using natural atomic-scale intrinsic Josephson junctions24, both for generation and detection of non-equilibrium bosons. In contrast to the conventional tunnelling spectroscopy, we probe not the single electron current into the sample, but hunt down the emission of non-equilibrium bosons that are responsible for Cooper pairing. We observe that recombination bosons carry a clear spectroscopic information about the superconducting energy gap Δ, which reassures the concept of electron–boson coupling with a well-defined pairing glue for cuprates. Analysis of the bosonic decay length yields the boson propagation speed ∼106 m s−1 close to the electronic Fermi velocity vF. This provides an evidence for the unconventional electron–electron coupling mechanism of high-Tc superconductivity in cuprates.
Figure 1b,c represents the top view and a sketch of the studied sample. It consists of ten small mesa structures of different sizes with attached Au electrodes micro/nanofabricated on top of a slightly underdoped Bi-2212 single crystal with Tc≃81 K. Mesas contained N=12±1 stacked intrinsic Josephson junctions. Each mesa can be biased independently using three or four terminal configuration18 and can be used either as a generator or a detector. Figure 2a represents a set of current–voltage (I–V) characteristics of the (generator) mesa 4a at different temperatures. Kinks in the I–Vs at T<Tc represent sum-gap singularities at eV=±2NΔ, see ref. 25.
Generation–detection with unbiased detector
Figure 2b shows dc-voltages Vdet of unbiased detectors 4b (close to 4a) and 2a (∼22 μm from 4a) as a function of dc-voltage Vgen in the generator 4a, at T=10 K. It is seen that a small negative Vdet appears at finite Vgen. Note that Vdet does not depend on the bias direction of Vgen. Therefore, there is no direct current injection from the generator into detectors. The detector signal is solely due to reabsorption of uncharged non-equilibrium bosons, which leads to depairing and appearance of excess QPs, as sketched in Fig. 1a. Tunnelling of excess QPs leads to a capacitive charging of unbiased detector junctions, which is measured in the experiment26. In this case, the amount of excess QPs and their charge do not depend on the bias direction in the generator. The negative Vdet corresponds to positive charge of carriers, as expected for hole-doped Bi-2212. Note that Vdet is not due to a thermoelectric effect in the base crystal, because the Seebeck coefficient in the superconducting state is zero. Rather, it is similar to a photo-effect in p–n diodes.
Figure 2c shows Vdet normalized by the generator power Pgen=IgenVgen for the same data. It is seen that the detector response carries a clear spectroscopic information: it peaks at eVgen=2NΔ and shows a secondary dip/upturn at approximately twice the sum-gap voltage eVgen≃4NΔ (ref. 18; a slight deviation from this equality is caused by slightly different energy gaps in the generator and the detector, see Supplementary Note 1). With increasing the distance from the generator to the detector, the detector response smears out and decays in amplitude. However, spectroscopic features remain recognizable at x>20 μm from the generator.
To understand the output of our boson generation–detection experiment, we performed numerical simulations of nonlinear kinetic balance equations for QPs and bosons in stacked Josephson junctions, together with the self-consistency equation for the superconducting gap. Simulations are made for a stack of N=2 junctions made of ordinary superconductors with an s-wave symmetry of the gap at T=0.5 Tc. The formalism used is suitable for any type of electron–boson interaction with a well-defined bosonic glue2.
Figure 2d shows calculated spectra of generated non-equilibrium bosons for different Vgen (here Δ0 is the equilibrium value of the gap, which is slightly larger than the actual Δ, suppressed by the current injection19). The corresponding I–V of the generator is shown in the inset of Fig. 2e. At eVgen>2NΔ, a large number of non-equilibrium QPs is injected at an energy eVgen/N−2Δ above the gap. Non-equilibrium QP relaxation usually follows a two-step process18,19,20,21,22 as sketched in Fig. 1a; first, QPs relax to the edge of the gap, emitting Ω≤eVgen/N−2Δ relaxation bosons, then two QPs form a pair with emission of Ω≥2Δ recombination bosons. The corresponding two bosonic bands are seen in Fig. 2d. At eVgen/N=4Δ, the two bands overlap. Thus, at eVgen/N<4Δ only recombination bosons have enough energy for pairbreaking, but at eVgen/N>4Δ an additional depairing will be caused by the high-energy part of relaxation bosons with 2Δ≤Ω≤eVgen/N−2Δ.
We assume that the detector signal is due to pairbreaking by non-equilibrium Ω≥2Δ bosons, which causes excess QP population in the detector and leads to capacitive charging of the detector, similar to a photo-effect in p–n diodes. In this case, the detector response is proportional to the number of bosons with Ω≥2Δ and does not depend on the direction of current in the generator, consistent with our experimental results in Fig. 2b. The main panel of Fig. 2e shows the number of generated non-equilibrium bosons δN with Ω=2Δ as a function of the generator voltage. Figure 2f shows the calculated detector response, normalized by the generator power. It shows a sharp peak at eVgen=2NΔ and a secondary dip/upturn at eVgen=4NΔ. A sharp increase at eVgen→2NΔ is caused by the sharp onset of recombination boson generation, as shown in Fig. 2e. A sharp drop at eVgen>2NΔ is caused by a rapid redistribution of the boson spectrum. As seen from Fig. 2d, at eVgen>2NΔ a significant part of emitted bosons is in the relaxation band with Ω<2Δ. Those low-energy bosons do not cause depairing and are not detected. This leads to the drop in the detection efficiency. However, at eVgen≥4NΔ the upper edge of the relaxation band exceeds 2Δ and high-energy relaxation bosons start to contribute to depairing18,20,21,22, leading to a secondary upturn in the detector response. The simulated detector responses from Fig. 2e,f can be explicitly compared with experimental data from Fig. 2b,c, correspondingly. A larger smearing of experimental features is likely to be due to the d-wave symmetry of the order parameter in cuprates (see Supplementary Note 2 and Supplementary Fig. S1). Otherwise, there is a good overall agreement between measured and calculated detector responses.
Generation–detection with ac-biased detector
The measurement accuracy can be improved using a lock-in technique. In this case, we send a small ac current through the detector mesa and measure the decrease in ac resistance caused by excess QP population. High sensitivity of this method is due to a strong temperature dependence of the resistance R(T) (ref. 25) shown in Fig. 3a, which can be used for probing the effective electronic temperature of the detector27. Figure 3b shows measured ac resistances of different detector mesas as a function of the power in the generator mesa 6a at T=61 K. Resistances are normalized to the corresponding equilibrium values at Pgen=0. The decrease of Rdet(Pgen) indicates the increase of non-equilibrium QP population. The detector response decays rapidly with increasing the distance from the generator. It is largest for the nearest mesa 6b, and smallest for the farthermost mesa 1b. The curves for detector mesas 4a and 4b, which have different sizes but are at the same distance from the generator 6a, collapse in one, indicating that the response does not depend on the geometry of the detector but solely on the distance from the generator. As in case of dc measurements in Fig. 2, the ac response does not reduce to a trivial self-heating, but carries a clear spectroscopic information. This is demonstrated in Fig. 3c, which shows the response of the detector 4b to the nearby generator 4a. The arrow indicates a kink in Rdet(Pgen), which is not present in R(T). The kink appears when the generator reaches the sum-gap voltage.
Figure 3d represents a set of dI/dV curves at different T for the generator 4a. The sum-gap singularities are seen as sharp peaks in conductances25. Figure 3e,f represents the corresponding detector responses dRdet/dPgen, normalized by the equilibrium value Rdet(Pgen=0), as a function of Vgen (4a), for the detector mesas 4b and 2a. It is seen that detector responses peak at eVgen=±2NΔ, consistent with dc measurements and simulations in Fig. 2c,f. Thus, the detector signal carries a clear spectroscopic information about the superconducting gap. It yields Δ0≃33 meV, consistent with previous studies5,15,25. This is an important observation, because it proves that pairing bosons have well-defined energies, that is, the ratio of the width of boson level to its energy is small. In the opposite case, the pairing glue would be ill defined23. This reassures applicability of the conventional concept of electron–boson coupling to cuprates.
From Figs 2b,c and 3b,e,f, it is seen that the detector response decreases rapidly with increasing the distance from the generator. In Fig. 4a,b we show distance dependence of ac responsivities at zero bias (Vgen→0) and at the sum-gap peak (eVgen=2NΔ) for different temperatures and generator/detector configurations. In all cases, the detector responsivity decays approximately exponentially with distance at a characteristic decay length xτ of several microns.
We want to emphasize that such an exponential decay is not expected in case of simple self-heating27,28, that is, in case when the power is simply spread out from the generator without time decay. We demonstrate this in Fig. 4c, which shows calculated temperature distributions at the surface of the crystal for the case of heat diffusion (solid lines) and for partly ballistic phonon transport with a mean free path of 5 μm (dashed lines). Calculations are based on the corresponding analytic expressions (equations 3 and 4 in ref. 28 and equation 3 in ref. 27) for a 3-dimensional T-distribution from circular mesas with radii a=5 (thick) and 1 μm (thin lines) on top of a bulk crystal. Despite some differences, at large distances all of them approach a universal δTeff∝1/r dependence, characteristic for heat diffusion from a point source. It is much slower than the observed exponential decay. This indicates that the recombination bosons do not only spread out in the base crystal but also decay with time δNbos∝exp(t/τ), where τ is the boson lifetime. This leads to exponential decay of the detector signal with distance, owing to longer time-of-flight to the farthermost mesas. It follows that bosons propagate ballistically at an approximately micrometre scale. A certain scattering does take place at larger distances >10 μm, as seen from smearing out of the spectroscopic peak in the furthermost detector in Fig. 2c.
Relaxation times in cuprates τ∼1–20 ps are fairly well established via time-resolved optical experiments16,29,30,31. This time represents the inelastic QP relaxation time due to emission of bosons and the reverse process of boson decay due to absorption by QPs (see also Supplementary Note 3). Using the observed decay length xτ, we estimate the propagation velocity of recombination bosons vb∼xτ/τ∼106 m s−1. This velocity is more than two orders of magnitude larger than the phononic sound velocity 4 × 103 m s−1 (ref. 32) and is close to the electronic Fermi velocity vF (ref. 33). Thus, in contrast to low-Tc superconductors20,21,22, the detected pairing boson is not a phonon, but has a pure electronic origin. For comparison, at a similar τ∼ps phonons would decay at the length scale of a few nanometres, consistent with the observed spatial inhomogeneity of the isotope effect5. Therefore, we cannot judge about the role of electron–phonon coupling in cuprates from our data, because the corresponding phonon decay length is well beyond our spatial resolution.
What excitations can propagate at velocities close to vF? Obviously, QPs can do this. However, non-equilibrium QPs would produce Vdet that depends on the charge of injected QPs (electrons or holes), that is, on the direction of the injection current in the generator. Such non-even current response has been observed in previous works with a detector intimately connected with the generator18,34 (see also a description of the QP signal in Methods). In contrast, in our experiment the detector and the generator are well separated and the detected signal is independent from the bias direction in the generator, as shown in Fig. 2b. This clearly demonstrates that the detector signal is not due to injection of non-equilibrium charge carriers, but is due to inflow of uncharged non-equilibrium bosons. Among bosons, acoustic plasmons can have velocities close to vF (refs 10, 11, 12, 13). Collective spin waves (antiferromagnetic magnons) can also propagate at a comparable velocity (up to ∼0.5 vF), as shown by inelastic neutron scattering in cuprates35. Because of uncertainty in τ, at present we can only exclude phonons but cannot discriminate between plasmons and magnons.
The non-equilibrium boson generation–detection spectroscopy performed here is qualitatively different from conventional tunnelling spectroscopy, because it explicitly detects non-equilibrium recombination bosons emitted on QP relaxation, rather than the electronic current into the sample. This allows hunt down of bosons responsible for pairing. Therefore, our experiment provides evidence for the involvement of unconventional repulsive electron–electron coupling mechanism in high-Tc cuprates, caused by exchange of electronic bosons, such as plasmons or collective spin waves.
Sample fabrication and measurements
Small mesa structures were fabricated on top of a freshly cleaved Bi-2212 crystal. Shortly after cleaving, a gold protection layer was deposited and 6-μm-size mesas (such as mesa 3 and 5 in Fig. 1b) were patterned by means of photolithography and ion milling. Finally, some of the mesas were split in two parts and trimmed to smaller sizes by a focused ion beam. More details on sample fabrication can be found in ref. 36. Measurements were performed in a He-flow cryostat at T down to 1.6 K. The detector responses at several mesas (either dc or ac) were measured simultaneously with the generator I–V on slow variation of the dc bias in the generator.
Detection of QP signal
The response of the detector, which is even with respect to the current direction in the generator, is caused solely by uncharged non-equilibrium bosons. However, an additional signal from QPs, which changes the sign together with the bias in the generator, does exist. This non-even contribution is small at low temperatures, but successively grows as the base temperature approaches Tc. The QP contribution leads to a minor left–right asymmetry of the detector response in Figs 2 and 3. Close to Tc, the non-even QP response takes over and dominates the detector signal. The main reason for the smallness of the QP signal lies in the interlayer tunnelling nature of the c axis transport. In the studied sample, the detector and the generator are separated by an ∼20-nm-deep trench. Therefore, non-equilibrium particles on the way from the generator to the detector have to pass through more than ten layers. QPs have a difficulty to do so, because the probability of interlayer tunnelling is small. The probability of a coherent (without scattering) tunnelling through ten layers is a power ten smaller and, therefore, is very small. On the other hand, collective bosonic modes can travel freely within the crystal lattice, for example, phonons, magnons and plasmons can propagate in all directions at distances much larger that the atomic distance. This is essentially the requirement for existence of collective bosonic modes.
Non-equilibrium distributions of QPs and bosons are described by a system of two coupled kinetic equations:
which describes dynamic equilibrium between injection, relaxation and escape of the corresponding particles, respectively. The formalism of electron–boson relaxation in superconductors has been developed in the seminal work by Bardeen, Cooper and Schrieffer. The QP relaxation rate can be written as:
Here, DQP(0) is the electronic density of states per spin at Fermi level in the volume of the electrode, DB(Ω) is the boson density of states per ion, ρ(E) is normalized by DQP(0) QP density of states, α2(Ω) is the electron–boson spectral function, A(E1, E2)= and B(E1, E2)= are the coherence factors, and f(E) and g(Ω) are the non-equilibrium occupation numbers for QPs and bosons, respectively. The first integral in equation (2) describes net scattering of QPs upwards with absorption of a boson, the second integral describes net relaxation with emission of a boson and the third integral describes pairbreaking and recombination.
Non-equilibrium bosons are produced on relaxation of non-equilibrium QPs, that is, there is an exact balance between boson excitation and QP relaxation. Therefore, the boson relaxation rate is opposite to the QP relaxation rate and can be written as
In the first integral, the two terms describe destruction and creation of bosons due to absorption by QPs and relaxation of QPs, respectively. In the second integral, the two terms describe destruction of bosons due to absorption by Cooper pairs (pairbreaking) and creation of recombination bosons on pairing of QPs, respectively. The factor 1/2 in front of the second integral reflects the fact that two QPs produce one boson on recombination into the Cooper pair.
To accurately describe non-equilibrium phenomena at large injection currents, it is necessary to take into account the influence of non-equilibrium QP distribution on the superconducting gap. The energy gap Δ is connected to the QP distribution f(E) (no matter equilibrium or not) via the self-consistency equation:
Here, λ is the electron–boson coupling constant and ΩD is the effective bosonic cutoff (Debye) energy. In general, this equation describes the suppression of Δ on increase of f(E); however, due to the denominator under the integral in equation (4) the gap is most sensitive to QPs at the edge of the gap E=Δ.
Non-equilibrium QPs are injected in electrodes via tunnel junctions. The QP injection rate is proportional to the tunnelling current:
where V is the bias voltage (per junction) and Rn is the tunnel (normal) resistance of the junction. Note that this equation is also nonlinear, because Δ depends on f via the self-consistency equation.
Finally, it is necessary to specify the QP and boson escape rates. Here, stacking of junctions in the generator has an important role and enhances non-equilibrium effects both in QP and boson subsystems. For QPs, the escape from inner electrodes is effectively blocked by the presence of adjacent tunnel junctions. Bosons, to the contrary, may freely travel in the crystal lattice and are thus collective for the whole stack. This leads to a cascade amplification of the non-equilibrium boson population proportional to the number of junctions in the stack18. The simulations presented in Fig. 2 were made for the minimal model of N=2 stacked Josephson junctions with one inner electrode. All the characteristics are shown for the inner electrode and thus catch the main stacking effects. In the simulations, we assumed that the QP escape rate from the middle electrode is zero, whereas the boson escape rate is just proportional to the impingement rate at the interface with a finite (50%) transmission probability.
We numerically solved the full nonlinear system of two coupled integral equations (1, 2, 3) together with the self-consistency equation (4), following the iterative finite difference procedure described in the Supplementary Material to ref. 19. It should be clarified that Fig. 1a represents only the most probable two step decay process, in which the QP first relaxes to the edge of the gap, emitting a relaxation boson, and then two QPs recombine into a Cooper pair, emitting a recombination boson. Probabilities of such processes are enhanced by the singularities in the QP density of states. However, from equations (2) and (3), it is seen that there are many other processes as well, such as reabsorption of an equilibrium boson by non-equilibrium QP, multistep QP relaxation with emission of low-energy bosons, recombination between any two QPs and creation of secondary non-equilibrium QPs due to pairbreaking. All those possibilities are taken into account on solving the nonlinear integral (equations (1, 2, 3)).
How to cite this article: Krasnov, V. M. et al. Signatures of the electronic nature of pairing in high-Tc superconductors obtained by non-equilibrium boson spectroscopy. Nat. Comm. 4:2970 doi: 10.1038/ncomms3970 (2013).
Financial support from the Swedish Research Council and technical support from the Core facility in Nanotechnology at Stockholm University are gratefully acknowledged. We are thankful to A.V. Balatsky for valuable remarks.
Supplementary Figure S1, Supplementary Notes 1-3 and Supplementary Reference