The electronic properties of cuprate high-temperature superconductors in their normal state are highly two-dimensional: transport along the crystal planes is perfectly metallic, but is insulating along the perpendicular ‘c-axis’ direction. The ratio of the in-plane to the perpendicular resistance can exceed 104 (refs 1,2,3,4). This anisotropy was identified as one of the mysteries of the cuprates early on5,6, and although widely different proposals exist for its microscopic origin7,8,9, there is little empirical information on the microscopic scale. Here, we elucidate the properties of the insulating layers with a newly developed scanning noise spectroscopy technique that can spatially map the current and its time-resolved fluctuations. We discover atomic-scale noise centres that exhibit megahertz current fluctuations 40 times the expectation from Poissonian noise, more than what has been observed in mesoscopic systems10. Such behaviour can happen only in highly polarizable insulators and represents strong evidence for trapping of charge in the charge reservoir layers. Our measurements suggest a picture of metallic layers separated by polarizable insulators within a three-dimensional superconducting state.
The difference between metals and insulators is that in the former, additional charge equilibrates in femtoseconds, whereas the latter can be statically charged. The coupling to the lattice is a necessary condition for the trapping of charge in insulators on macroscopic timescales, since electrons by themselves have too great a quantum mechanical behaviour to localize. The trapped charge is stabilized by the formation of static, localized polarons involving a reconfiguration of the atomic lattice. Strongly polarizable insulators such as SrTiO3 exemplify this process. Such trapping of charge on slow timescales has been conjectured as a cause of the anomalous, highly resistive c-axis transport in the cuprate high-temperature superconductors7, although there are alternative proposals such as incoherent transport11,12. However, for these materials, band structure theory predicts metallic transport along the c-axis due to the small but finite c-axis bandwidth of the order of 0.1 eV (ref. 13). Here, we present firm evidence that such charge trapping processes do occur in superconducting cuprates measuring atomically resolved current fluctuations.
Quite generally, fluctuations of a signal in time—the noise—can be a powerful diagnostic tool as they contain information not present in the mean value. It has historically allowed differentiation between signals carried by particles and waves or between black body radiation and the coherent radiation of a laser. More recently, noise spectroscopy has established itself as a standard method of investigating mesoscopic systems. This is usually done by looking at the noise spectral power, S(w) = < δI(t) δI(t′) > , where δI is the deviation of the current operator from the mean, and the averaging < > is both quantum mechanical and statistical. Examples where noise transport measurements led to novel insights include the study of fractional charge14, the doubling of charge in Andreev processes15, and the vanishing of noise in break junctions at the quantum conductance16. We have succeeded in bringing this technique to the atomic scale in the tunnelling regime, discovering an unanticipated phenomenon when we applied it to a cuprate high-temperature superconductor.
The flow of classical, uncorrelated charged particles between two leads is a purely Poissonian process. Its noise is independent of frequency (white) and proportional to the charge q and the flow I of the carriers, S = 2q|I|, as a direct consequence of the discreteness of the carriers17. We define the normalized noise Sn = S/2e|I|, similar to the Fano factor. Thus Sn = 1 represents Poissonian noise, where Sn < 1 and Sn > 1 refer to sub-Poissonian and super-Poissonian noise, respectively (Fig. 1a, b). For an uncorrelated electronic liquid, one expects Sn = 1; for an exotic, spatially inhomogeneous electronic liquid, this is a priori unclear. Importantly, to find Sn > 1, processes in the system are required on a frequency scale that fits into the slow (megahertz) frequency window of the noise measurements. Another property of special relevance to our data is that such noise events may be completely invisible in the mean value of the current.
Our aim is thus to measure the fluctuations in the cuprates on the atomic scale. Bringing noise measurement to the tunnelling regime comes with unique challenges which have prevented any atomic-resolution shot noise measurement thus far. The central obstacle lies in the high impedance of the tunnelling junction, which is typically 0.3 GΩ to 10 GΩ. Together with the capacitance between tip and sample and the cable capacitance, the junction acts as low-pass filter, allowing transmission of signals only in the kilohertz range where 1/f noise and mechanical resonances dominate. Possible solutions include bootstrapping of an amplifier18 or building an impedance-matching circuit19. Matching a gigaohm junction leads to considerable losses in the circuit. This is simplified when using the scanning tunnelling microscope (STM) in point contact mode or in the low-megaohm range19,20 or by using methods based on force microscopy21; however, this increases interactions between tip and sample, making it more difficult to extrapolate the sample properties. Our goal is to perform noise measurements when in the tunnelling regime, with transparencies t ~ 10−6. To accomplish this task, we build a resonance circuit with all inductors from superconducting niobium, and include a custom-built, high-mobility amplifier directly into the circuit, following the principle of devices built for noise spectroscopy measurements in mesoscopic systems22. Figure 1c shows the amplification circuit that allows us to map out noise around 3 MHz with gigaohm junction resistances. We thoroughly tested our set-up on a gold sample (see Supplementary Information).
We first investigate with this new scanning noise microscopy instrument the cuprate high-temperature superconductors23 with the hope of finding signs of the slow, glassy fluctuations associated with charge- or current loop order that have been claimed to show up in the noise24,25,26. These are not present, but instead we found a surprise that we will now explain.
As a sample material, we decided to use the slightly overdoped bi-layer cuprate (Pb,Bi)2Sr2CaCu2O8+δ with Tc = 79 K. The substitution of Pb for Bi has the advantage of supressing the characteristic supermodulation seen in many Bi-based cuprates, simplifying the interpretation and making higher-voltage measurements possible. We cleave the samples in ultrahigh vacuum at pressures below 10−10 mbar, and directly insert them into our STM head at 3.2 K. The single-crystal sample is held at a constant voltage Vbias, which drops over the approximately gigaohm junction between the tip and the uppermost CuO2 layer (see Supplementary Information for details). The tunnelling process starts with electrons originating from the CuO2 layer which then tunnel through the SrO and BiO layers27. Tunnelling through these charge reservoir layers does not have much effect on the STM signal, except for some spatial filtering27,28. Figure 2a shows a topographic image revealing atomic resolution. The square Bi-lattice is clearly resolved with some bright protrusions induced by Pb substitutions for Bi atoms (see Supplementary Information).
On recording the noise as a function of spatial location with atomic resolution at bias voltages of ± 0.1 eV we find homogeneously Poissonian noise (see Supplementary Information), with no sign of the fluctuating orders. However, this changes markedly when increasing the bias, as shown in Fig. 2b, c. Although most locations still exhibit Poissonian noise, a few atomic locations reveal striking enhancements of the noise. These super-Poissonian noise centres show noise values up to 40 times the expectation from Poissonian processes—more than anything that has been observed in mesoscopic systems10. We emphasize that the increase in noise is extremely localized, the width of the peaks in space being around 0.5 nm for most centres. The density of super-Poissonian noise centres is roughly 0.3%, referenced to the number of Cu atoms, and they are evenly distributed throughout the sample.
A key insight comes from the energy dependence of the noise. Noise spectra are shown in Fig. 2d–f. Below a certain threshold, the noise in the tunnelling current is purely Poissonian, Sn = 1. But above around 1 eV, or below around −650 meV, the normalized noise rises rapidly. Surprisingly, all noise centres appear to be highly asymmetric in energy: the noise enhancements at positive bias do not spatially correlate with the noise enhancement seen at negative energy. Although different locations show different strengths of noise enhancement, the onset energy seems to be independent of the noise centre, roughly −0.8 eV for the negative energy noise centres, and roughly +1 eV for the positive ones. This indicates a common mechanism and turns out to be an important diagnostic tool, as discussed below.
To understand these observations, it is worth taking a step back to look at engineered and natural systems that exhibit non-Poissonian noise, Sn ≠ 1. In electronic systems this is most often sub-Poissonian noise10,12,16,18, usually due to sequential tunnelling in quantum dots or Pauli exclusion effects. However, in the tunnelling regime considered here (t ~ 10−6), the latter are minimal. Super-Poissonian noise on the other hand is a much rarer occurrence, as it always has to include some sort of interaction10. Experimentally, super-Poissonian noise was first observed in semiconductor double-wells and later in quantum dots29. Examples of super-Poissonian noise include bi-stable systems that lead to random telegraph noise in the transport, and coupling to inelastic modes30. Our data allow us to exclude all these mechanisms as the observed noise is bias dependent and asymmetric.
Instead, our data suggest the following scenario known from multi-level quantum dots. Two tunnelling processes are present, one fast, accounting for almost all the tunnelling current, and one slow, acting as a switch for the first process. This switching mechanism is usually based on Coulomb interaction. For example, if the state of the slow process is occupied, it raises the energy level of the state necessary for the fast process and effectively blocks it, as shown in Fig. 3. This leads to an effective switching, producing bunching of electrons that causes super-Poissonian noise. Such mechanisms have been discussed in detail for mesoscopic systems10,12,31 and confirmed by experiments in multi-level quantum dots32. We note that this scenario predicts a clear threshold energy after which the noise increases, is asymmetric with respect to energy, and is localized on the atomic scale. The key insight that follows from our observations is that some slow process is involved, indicating a form of charge trapping that is known from strongly polarizable insulators, but not from a metal.
One expects these noise centres to correspond to some form of polarons being localized at defects in the crystal lattice, in contrast to more itinerant polarons at other locations. To shed light on their precise nature, we searched for impurity states, following earlier work33 but now taking fully normalized (dI/dV) / (I/V) density of states maps over a large energy range to be able to differentiate impurity states that overlap in energy. The energy-dependent (dI/dV) / (I/V) maps reveal the spatial distribution of the different impurity states. Such states have been identified as apical oxygen vacancies, Sr(Ca)-site impurities, interstitial oxygen dopants and Pb impurities. Figure 4 shows signatures of these specific states; details of which can be found in the Supplementary Information.
Most importantly, we find a clear correspondence between positive-energy noise centres and the +1.1 V impurity state, as shown in Fig. 4c, d. This impurity state has previously been identified as an apical oxygen vacancy33 which, in the insulating oxygen materials, has various charge states stabilized by lattice distortions. This amounts to strong evidence for this signature being associated with the charge trapping process. The positive noise centres all show this signature of an impurity state in the charge reservoir layer through which tunnelling occurs but is modulated by a slow charge trapping process. Surprisingly, we do not observe a clear correspondence between a specific type of impurity resonance and the site of the negative noise centres. This might be because the state is dark due to filtering mechanisms28, or because of stronger coupling to the CuO2 layer.
In summary, we have presented direct evidence for the existence of slow charge trapping processes at defect sites in the form of the localized super-Poissonian noise signals. These are reminiscent of Coulombic impurities that occur generically in polarizable insulators, despite the three-dimensional superconducting state present in our samples. Although the atomic centres with noise enhancement are sparse, they demonstrate the polaronic nature of charge carriers tunnelling through the insulating layers. Apart from the noise centres, the polarons are probably itinerant to different degrees; a situation that at present cannot be described by theory. The c-axis physics of the cuprates thus appears to be in a literal sense similar to what is found in, for example, the Al–Al2O3–Al barriers employed for Josephson junctions: no coherent charge transport is possible for normal electrons, whereas the virtual tunnelling of Cooper pairs suffices for a coherent Josephson contact. In the cuprates, this is manifested in the form of the c-axis Josephson plasmons observed in the optical conductivity34, emerging from a completely overdamped charge transport in the normal state3. The surprise is that apparently an oxidic layer that is only two atoms thick living in a metallic environment suffices for the polaronic trapping of charge.
The role of the ‘c-axis phenomena’ in the mechanism of high-temperature superconductivity is a long-standing question5,6,34, as is the unusual nature of the coupling of the polar insulator phonons to the electrons5,6,8,9,35. This acquired new impetus recently with the discovery that when a single layer of FeSe is removed from bulk and put on an polarizable insulator, the critical temperature increases by a factor of four36, with evidence reported suggesting that the coupling to the polarizable insulating substrate may indeed play a critical role37. Further, interfaces of the polarizable insulators SrTiO3 and LaAlO3 host two-dimensional superconductors with the highest Tc per charge density ratio35. So much is clear from our findings, that even for the atomically thin insulating layers, the polar electron–phonon interactions are severe enough to slow down electronic motions to macroscopic timescales. Although a great challenge for established theory, this conundrum deserves further close consideration.
The slightly overdoped bi-layer cuprate (Pb,Bi)2Sr2CaCu2O8+δ crystals were flux-grown in the standard manner with a Pb:Bi ratio of 0.4:1.6. Pb substitutions for Bi atoms show enhanced contrast in the atomic-resolution STM images38,39. The Pb content is highly consistent with the ratio. We characterized these sample by measuring the field-cooled magnetization superconducting transition using a DC superconducting quantum interference device (SQUID) magnetometer (MPMS XL from Quantum Design) in an applied field of 0.01 mT using the reciprocating sample option (RSO) (See Supplementary Information).
Scanning tunnelling microscopy and spectroscopy
The scanning tunnelling microscopy and spectroscopy measurements have been carried out with a modified commercial cryogenic STM (Unisoku USM 1500). The single crystals were cleaved in ultrahigh vacuum (P < 5 × 10−10 mbar) and directly transferred to the pre-cooled STM head (T ~ 3.2 K). We used mechanically sharpened Pt–Ir STM tips. The differential conductance (dI/dV) spectra were measured by a conventional lock-in technique with a voltage modulation of 50 mV and a frequency of 887 Hz. To minimize the set-up effect for LDOS maps, we normalized the dI/dV spectra with the total conductance (I/V).
Scanning noise spectroscopy
To overcome the limited bandwidth of conventional STM, we developed a low-temperature high-frequency amplifier including niobium superconducting coils and a custom-built low-noise high-mobility electron transistor40,41. The circuit is described in detail in the Supplementary Information and in ref. 4. We used a Zurich Instruments MFLI digital spectrum analyser to measure the power spectral density of the noise. The integrated noise power as a function of current on the Au (111) surface and its slope are used to compute the normalized noise (Sn). If the charge transport is purely random, Sn is equal to 1, with deviations from 1 indicating the temporal correlation between charge carriers.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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We thank C. Beenakker, A. Ben Hamida, Y. Blanter, D. Chatzopoulos, V. Cheianov, T. Klapwijk, M. Leeuwenhoek and J. van Ruitenbeek for help and valuable discussions. This project was financially supported by the European Research Council (ERC StG SpinMelt) and by the Netherlands Organisation for Scientific Research (NWO/OCW), as part of the Frontiers of Nanoscience programme, as well as through a Vidi grant (680-47-536).