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4.8 Conclusions and Proposed Future Experiments
In this chapter I have shown that the short-ranged CDW present in Bi2Sr2CaCu2O8+δ
and NaxCa2−xCuO2Cl2 has a predominantly d-symmetry form factor. While predicted
for some time, this is the first CDW with a d-symmetry form factor to be observed in nature. Similar observations have also been made in YBa2Cu3O6+x by x-ray scat-
tering, which is in contrast to the predominantly s’-symmetry form factor observed in La2−xBaxCuO4+δ [114–116].
A d-symmetry form factor arises naturally in spin-fermion “hot-spot” models of the cuprates, where it has a deep connection with the d-wave superconductivity because they arise out of the same fermions and the same effective interaction [94–97, 101, 103, 104]. Ad-symmetry form factor is also a natural partner tod-wave superconductivity in
the sense that it acts to protect the coherence of nodal quasi-particles against scattering from the disordered CDW in so called “cold-spots” [118].
Another possible explanation for the observedd-symmetry form factor is the Coulomb repulsion between holes on the oxygen sites of the CuO2plane, which acts to favour their
unequal occupation within the unit cell, i.e ad-symmetry form factor [98, 102, 105, 106]. The observation of a short ranged CDW in the electron doped cuprate Nd2−xCexCuO4,
where carriers are first doped into bands derived from the Cu 3dx2−y2 orbitals, provides a way to falsify this hypothesis [131]. If measurements sensitive to the CDW form factor reveal that it is predominantly d-symmetry in electron doped cuprates the hypothesis cannot be true.
I also showed in this chapter that the magnitude of the incommensurate CDW wave- vector in Bi2Sr2CaCu2O8+δ decreases with increasing hole concentration, as also ob-
served by x-ray studies of Bi2Sr2CaCu2O8+δ, Bi2−yPbySr2−xLaxCuO6+δ, HgBa2CuO4+δ
and YBa2Cu3O6+x but in contrast to the doping dependence in La2CuO4 derived
cuprates which shows the opposite trend [58]. This phenomenology clearly suggests a role for the Fermi surface geometry in determining the CDW wave-vector.
This chapter demonstrates that the charge order in Bi2Sr2CaCu2O8+δ and
NaxCa2−xCuO2Cl2consists of nanoscale domains of unidirectionald-symmetry form fac-
tor CDW. Recent high-magnetic field measurements in YBa2Cu3O6+xreveal a magnetic-
field induced CDW that is completely unidirectional [55, 56]. This is suggestive that the CuO2 plane has an intrinsic instability to unidirectional ordering. A cautionary
caveat is that YBa2Cu3O6+x is structurally orthorhombic, and so posseses no symmetry
between Cu-O directions. I propose that measurements of magnetic field induced CDW in cuprates with a structural symmetry between the Cu-O directions be performed with the aim of directly demonstrating this unidirectional tendency of the CuO2 plane.
The Scanned Josephson
Tunnelling Microscope
The main results of this chapter are reported in “Direct Detection of a Cooper-pair Density Wave in Bi2Sr2CaCu2O8+δ”,Nature,532, 343 (2016).
This chapter details the development of a scanned Josephson tunnelling mi- croscope. Motivated by the search for a pair density wave in cuprates, we employ a high temperature d-wave superconducting STM tip operating at temperatures < 50mK [132].
Scanned Josephson tunnelling microscopy (SJTM) is a direct probe of the supercon- ducting order parameter on the nanometer scale [133]. It utilises the tunnelling of Cooper-pairs between a superconducting STM tip and a superconducting sample to measure the Cooper-pair condensate directly. In contrast, traditional STM measures the spectrum of single-particle excitations. Prior to the development of SJTM, super- conductivity could be probed on the nanometer scale only indirectly using STM (by measuring the single-particle tunnelling gap) or with atomic site sensitivity using NMR orµSR.
There are many cases where the gap in the single-particle tunnelling spectrum is not sufficient to characterise spatial variations in superconductivity. Firstly, it is known that at magnetic impurities ins-wave superconductors the superconducting order parameter, Ψ, is suppressed whereas the single-particle gap, ∆, is unperturbed [134–137]. Similarly,
there is also a gap at the centre of superconducting vortices in cuprate superconductors where Ψ → 0 [138] . Finally, there are pair density wave states which we would not expect to exhibit a modulation in ∆ and thus cannot be unambiguously detected by STM.
In the face of these adversities a handful of groups have initiated efforts to develop SJTM as a direct probe of the superconducting order parameter [139–145]. In our group we have focussed on an implementation of SJTM using high temperatured-wave superconducting tips to search for a pair density wave in cuprate superconductors [132].
5.1
Fundamentals of SJTM Operation
The purpose of the scanned Josephson tunnelling microscope is to directly visualise spatial variations in the superconducting order parameter on the nanometer scale. To achieve this we employ a superconducting STM tip, as shown in figure 5.1 (a). In contrast to normal STM, this allows the tunnelling of Cooper pairs between the tip and a superconducting sample, otherwise known as the Josephson effect. As will be detailed in section 5.3, this tunnelling of Cooper pairs leads to a peak in the tunnelling current near zero voltage bias. The magnitude of this peak current can be used as a direct measure of the superconducting order parameter.
The superconducting tip also allows superconductor-insulator-superconductor (SIS) single-particle tunnelling between the tip and sample, as discussed in section 2.1.4. Thus, the same tip-sample junction can be used to measure both the superconducting order parameter and properties of the single-particle excitation spectrum in the sample, such as the superconducting gap energy or the energy of an impurity resonance.
SJTM operates like an enhanced version of the spectroscopic-imaging STM (SI-STM) that was discussed in chapter 2. As with SI-STM, junctions between the tip and sam- ple will be established with set-point voltage and current, Vs and Is, on a fine two-
dimensional grid of points {~ri}. At each of these points the STM feedback loop will
be turned off upon attaining the set-point current and the tip-sample separation, zi,
x y
x
y y x
20
Figure 5.1: (a)A high temperatured-wave scanned Josephson tunnelling microscope
(SJTM). A nanometer sized piece of Bi2Sr2CaCu2O8+δ is attached to a tungsten STM
tip to form a superconducting STM tip. (b) A typical I(VB) characteristic for tun-
nelling between a Bi2Sr2CaCu2O8+δ sample and our Bi2Sr2CaCu2O8+δ nano-flake tip.
For|VB|>1mV the current is carried almost entirely by single-particle tunnelling. (c)
g(VB) =dI/dVB corresponding to (b). (d)Enlargement of (b) near VB = 0 showing
a region with far greater conductance than the single-particle channel that terminates in a peak current Ic. In this region the current is carried predominantly by Cooper
pair tunnelling. (e)Ic(~r) map visualising variations in the superconducting order pa-
rameter. It is derived fromI(VB) curves of the type shown in (d), measured on a fine
grid of points in the field of view. (f ) g(~r,−20mV) for the same field of view as (e), simultaneously visualising variations in the single-particle excitation spectrum.
The bias voltage, VB1, is then ramped and the current and differential conductance,
I(VB) and g(VB) = dI/dVB, recorded. For each point on the surface of the sample,
I(VB) andg(VB) such as those shown in figures 5.1 (b) and (c) are recorded. Combining
the measurements from all points in the field of view yields the familiar spectroscopic maps I(~r, E=eVB) andg(~r, E =eVB).
For |VB|,|Vs| > 1mV, T(~r), I(~r, E = eVB) and g(~r, E = eVB) are entirely equivalent
to those obtained using normal SI-STM, albeit with a more complicated tip density of states. This is because, in this voltage range, the current is carried almost entirely by single particles. Conversely, for|VB|.10−100µV the current is carried predominantly
by Cooper pairs. This allows us to extract another spectroscopic map, Ic(~r), which
directly measures spatial variations in the superconducting order parameter.
In section 5.3 I will show that the expected signature of Cooper pair tunnelling is a steep quasi-linear increase in current away forVB = 0, reaching a maximum currentIc, before
dropping sharply. Figure 5.1 (d) shows an enlarged portion of figure 5.1 (b) nearVB= 0
that exhibits just this phenomenology. On increasing VB beyond the point where the
current drops sharply, the current once again increases but with a much smaller gradient. Here the current is predominantly carried by single particles.
The Cooper pair tunnelling is confined to a small range of voltages around VB = 0
and exhibits a much larger differential conductance than the single-particle tunnelling channel (steeper gradient nearVB= 0 in figure 5.1 (d)). This allows us to effectively sep-
arate single-particle and Cooper pair tunnelling contributions and hence simultaneously visualise superconducting order parameter and local density of states variations using the spectroscopic maps Ic(~r) and g(~r, E = eVB) respectively. Ic(~r) and g(~r, E =eVB)
measured in the same field of view are shown in figures 5.1 (e) and (f). This abil- ity to simultaneously and directly visualise both the ground state condensate and its single-particle excitation spectrum lies at the heart of SJTM’s utility.
Having briefly described the utility and operation of SJTM I will now discuss how the Josephson effect may be used as a probe of the superconducting order parameter and how, for our ultra-small junctions, it leads toI(VB) characteristics of the form shown in
figure 5.1 (d). 1
In this chapter the bias voltage output by the STM electronic control unit will be denoted VB to