Mapping Spatial Variations in the Amplitude and Phase of LDOS Mod-

SI-STM spectroscopic maps contain a vast amount of information, with modulations oc- curring on multiple length scales and in multiple directions. Often we are only interested in the amplitude and phase of a small sub-set of these modulations. These quantities can be extracted from our data sets by filtering their complex Fourier transforms so that only wave-vectors proximal to those of interest remain. For instance, if we are interested in modulations of the spectroscopic map M(~r, E) at wave-vectorQ~ we would filter its Fourier transform ˜M(~q, E) to yield

M_Q~(~q, E) = ˜M(~q, E)e

−(~q−Q~)2

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 20 40 60 -60 -40 -20 0.2 0.3 0.4 0.5

Figure 2.13: (a) d-wave superconducting gap function measured for

Bi2Sr2CaCu2O8+δ with p = 17% using quasi-particle interference. Gap values

are plotted against β, the angle from the nodal (π, π) directions. (b) (kx, ky)

values of all points shown in (a). They trace out the underlying Fermi surface. In Bi2Sr2CaCu2O8+δ with p < 19% no quasi-particle interference is observed from

quasi-particles beyond the anti-ferromagnetic Brillouin zone (dashed lines). Red points represent the mean of 6 independentkxand ky estimates made from measured

wave-vectors~q1, ..., ~q7 using equations 2.52 and 2.53. Error bars represent the standard

deviation of these 6 independent estimates.

where Λ−1 is the characteristic length scale below which variations in the amplitude or phase of modulations atQ~ will not be resolved in this process. I will refer to this process as Fourier filtration.

Using an inverse Fourier transform one can then proceed to create the complex real-space map, M_Q~(~r, E) = 2 (2π)2 Z d~qei~q·~rM˜_Q~(~q, E) , (2.55)

that only contains modulations at wave-vectors proximal to Q~.

To make a real-space map of modulations at wave-vectorQ~ one simply takes the ampli- tude of M_Q~(~r),

A_Q~(~r, E) =

q

(ReM_Q~(~r, E))2+ (ImM_Q~(~r, E))2 . (2.56)

Similarly the phase of these modulations, φ(~r), is given by

φ(~r) = arctanImMQ~(~r, E)

ReM_Q~(~r, E)

In other cases we are interested in removing a specific Fourier component from an image because it is visually dominating and we wish to examine a smaller modulation at another wave-vector. In this case we can remove those modulations at wave-vectors±Q~0

to create the real-space map

M_±0 _~ Q0(~r, E) = 2 (2π)2 Z d~qei~q·~rhM˜(~q, E)−M˜_±0 _~ Q0(~q, E) i , (2.58) where M_±0 _~ Q(~q, E) = ˜M(~q, E)× e− (~q−Q~0)2 2Λ2 +e− (~q+Q~0)2 2Λ2 . (2.59)

Sub-Lattice Segregated SI-STM

in Cuprates

In this chapter I will detail novel methods, developed by myself and others in our group over the past 6 years, extending spectroscopic-imaging STM (SI-STM) to imaging states which have significant intra-unit-cell structure. I will a introduce a method of finding and segregating different atomic sites or sub-lattices. I will also outline the technical steps required to use the complex Fourier transform of SI-STM spectroscopic maps to probe intra- unit-cell symmetry breaking. Focussing on cuprate superconductors, these methods will be used in chapter 4 to determine the intricate intra-unit-cell structure of the charge density wave in cuprates.

SI-STM is a real-space probe and therefore in principle retains all phase information about spatial modulations, in contrast to scattering probes. One has access to the full complex Fourier transform of conductance maps, ˜g(~q, E), as opposed to just the power spectrum |g˜(~q, E)|2_.

This turns out to be incredibly useful for studying the structure of electronic states within the unit cell. Consider the CuO2 plane with one copper and two oxygens per

unit cell. In a perfectly periodic lattice, these sets of different atoms or sub-lattices will contribute to its Fourier transform with different phases. For instance, in cuprates if we measure the functionReZ˜((2_aπ

0,0), E)−Re ˜

Z((0,2_aπ

0), E), we are directly measuring only

differences in electronic structure between the two oxygen sites in the unit cell [32]. The utility of phase-resolved SI-STM lies in this sub-lattice specificity.

In order to enact such a scheme one must remove artifacts introduced in the measure- ment process that distort the lattice in SI-STM spectroscopic maps away from perfect periodicity. A procedure for correcting these distortions to give a perfect lattice with the origin of co-ordinates lying on a specific atomic site, first introduced by Lawler et al. in reference [32], will be described below.

Another advantage of mapping SI-STM data to a perfectly periodic lattice is that it enables you to easily find the positions of atoms belonging to different sub-lattices. Spectroscopic maps can then be segregated into several maps, each containing only the sites of a single sub-lattice.

All of these techniques will be heavily employed in chapter 4 where they are used to make a direct detection of a d-symmetry form factor charge density wave in cuprates, a state where a breaking of the symmetry between the oxygen sub-lattices within the unit cell is periodically modulated in space.

As with the previous chapter, the rapid and fairly specialised development of the field means that for completeness I must touch on a fairly large number of sub-topics, some of which are fairly technical; one way to use this chapter might be to back-reference to it from the main results chapters of the thesis rather than try to take in every detail on first reading.

3.1

Data Acquisition Parameters

In order to implement sub-lattice segregated, phase-resolved SI-STM one requires a large number of pixels within every unit cell. To effectively discriminate between different sub-lattices we need at least three physical measurements along each Cu-O-Cu bond; a minimum of 9 pixels per CuO2 plaquette. Measurements presented in this thesis

typically have between 9 and 25 pixels per unit cell. For Bi2Sr2CaCu2O8+δ this gives

a pixel density 0.62-1.71˚A−2 corresponding to between ≈ 400×400 and 650×650 measurement pixels in a typical 50nm ×50nm field of view.