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I will now go on to discuss how the objects shown in figure 4.9 (a) are tiled in space to fill a large field of view. Although elements indistinguishable from figure 4.9 (a) can be seen in figures 4.10 (a) and (b), no long-range order is obvious. In Fourier space this manifests itself as the broad peaks in figures 4.10 (c) and (d). These correspond to a spatial correlation length of≈24˚A(6.3a0 or∼1.6 CDW wavelengths).

To explore the spatial arrangements of the CDW I introduce the complex functions

Dx(~r) and Dy(~r), which are a spatially-resolved measure of the d-symmetry form fac-

tor amplitude. D(~q) is Fourier filtered around the wave-vectors Q~x,y to determine the

complex-valued Dx,y(~r), as detailed in section 2.8. Summarising, we take take the full

2nm x y T = 4.2 K < Tc Z(r,150mV) x y 2nm F < -0.3 F > +0.3 qx qy x y 2nm 2nm x y T = 45 K > Tc Z(r,150mV) +1.0 -1.0 High Low qx qy

Figure 4.10: (a) Measured Z(~r,150meV) at T < Tc in the superconducting phase

ofp∼8% hole-doped Bi2Sr2CaCu2O8+δ (T =4.2 K).(b)Measured Z(~r,150meV) at

T > Tc in the pseudogap phase of p ∼8% hole-doped Bi2Sr2CaCu2O8+δ (T ∼45K).

The CDW phenomena are essentially indistinguishable from observations at T < Tc. (c)Thed-symmetry form factor power spectral density|DZ_(~q,|E|)|2_{=|( ˜O}Z

x(~q,|E|)−

˜

O_yZ(~q,|E|))/2|2 _{determined from sub-lattice segregated analysis of data in (a). (d)} |DZ(~q,|E|)|2 _{determined from sub-lattice segregated analysis of data in (b). The ~q-}

space structure of the CDW phenomenology is identical in the pseudogap phase and in the superconducting phase (apart from a slightly larger s’-symmetry form factor component). (e,f ) Using only the regions within the dashed circles in (c) and (d) the ~r-space amplitudes of the CDW in (a) and (b) are calculated for modulations along Q~x andQ~y. Then using F(~r) = (|Dx(~r)| − |Dy(~r)|)/(|Dx(~r) +|Dy(~r)|) regions

primarily modulating along y-axis with−1.0< F(~r)<−0.3 are shaded blue. Regions primarily modulating along x-axis with +0.3 < F(~r)<+1.0 are shaded orange. (g)

Configuration of unidirectionald-symmertry form factor CDW modulations contained in (a) atT < Tc. The unidirectionality colour scale forF(~r) demonstrated in (e) and (f)

is overlaid on the data in (a). The dashed circle shows the~r-space radius equivalent to the~q-space filter used to generate theDx,y(~r) images by Fourier filtering. (h)Domain

configuration of unidirectionald-symmetry form factor CDW modulations contained in figure at T > Tc. The unidirectionality colour scale forF(~r) demonstrated in (e) and

of d-symmetry Form Factor CDW amplitude at wave-vectorsQ~x,y, Dx(~r) = 2 (2π)2 Z d~qei~q·~rD˜(~q)e− (~q−_Qx~ )2 2Λ2 (4.11) Dy(~r) = 2 (2π)2 Z d~qei~q·~rD˜(~q)e− (~q−_Qy~ )2 2Λ2 , (4.12)

where Λ−1 is the characteristic length scale over which variations inDx,y(~r) can be re-

solved and is set by the filter width in Fourier space. The magnitudes of these functions,

|Dx(~r)|= p (ReDx(~r))2+ImDx(~r))2 (4.13) |Dy(~r)|= q (ReDy(~r))2+ImDy(~r))2 , (4.14)

represent the local amplitude of d-symmetry form factor CDW modulations along Q~x

andQ~y respectively. As such, we may visualise the local directionality of CDW domains

through

F(~r) = |Dx(~r| − |Dy(~r|

|Dx(~r|+|Dy(~r|

. (4.15)

Through its sign, this function will identify regions where the CDW modulation is pri- marily along the x or y axis.

Figures 4.10 (e) and (f) shows how regions of −1.0 < F(~r) < −0.3 (shaded blue) are primarily modulating along y-axis whereas regions +0.3< F(~r)<+1.0 (shaded orange) are primarily modulating along x-axis (those with −0.3 < F(~r) < +0.3 shaded white appear at boundaries). Figures 4.10 (g) and (h) reveal the results of this analysis for the data in figures 4.10 (a) and (b) respectively.

Overall, the system is configured into spatial regions within which the CDW along only one direction is dominant. By overlaying the colour scale forF(~r) on the data in figures 4.10 (a) and (b) to create figures 4.10 (g) and (h), one can see directly these unidirectional domains of short-ranged CDW.

That the CDW in Bi2Sr2CaCu2O8+δis short-ranged (short correlation length) is quanti-

tatively consistent with x-ray scattering studies [119]. Similarly short correlation lengths (.80˚A) are observed in Bi2−yPbySr2−xLaxCuO6+δ, HgBa2CuO4+δ and YBa2Cu3O6+x

[50, 58, 120, 121]. That the CDW is not long-ranged in these materials is not surprising. A Ginzburg-Landau analysis shows that an incommensurate CDW cannot be truly long ranged ordered in a system with quenched charge disorder, which is present in all of

these materials [122]. But is disorder the factor limiting correlation lengths or are the short correlation lengths a more intrinsic property of the CuO2 plane?

The question of whether disorder is limiting the zero magnetic field CDW correlation length in YBa2Cu3O6+x was addressed by Achkaret al [123]. They used resonant x-ray

scattering to measure oxygen-ordered YBa2Cu3O6+x samples before and after disorder-

ing the oxygens in the chain layer by heating and quench-cooling. They concluded that while the additional disorder did reduce the scattering intensity in the CDW peaks, it did not significantly alter the wave-vector or correlation length. This suggests that disorder is not the factor limiting the CDW correlation length in YBa2Cu3O6+x.

That the in-plane correlation length is seen to peak and then decrease again below Tc

with decreasing temperature in YBa2Cu3O6+x suggests the role of superconductivity

in limiting the correlation length [124, 125]. Further weight is added to this by the observation in YBa2Cu3O6+x, at B=30T, of a magnetic field induced CDW with a

resolution limited in-plane correlation length ξ≥230˚A [56].

In our STM studies of both Bi2Sr2CaCu2O8+δ and NaxCa2−xCuO2Cl2 we find that

the short-range ordered CDW takes the form of small locally unidirectional domains. These observations of coexisting nanoscale unidirectional regions are in reasonable agree- ment with deductions from x-ray studies of YBa2Cu3O6+x [126]. Again, the question of

whether this unidirectionality is property of the quenched disorder or instead something intrinsic to the CuO2 plane arises [127].

Interestingly the high field CDW observed in YBa2Cu3O6+x appears to be completely

unidirectional [55, 56]. However, the orthorhombicity of YBa2Cu3O6+x means that this

is not a true breaking of rotational symmetry because the Cu-O directions were not equivalent to begin with. One could argue, though, that the symmetry breaking field provided by the orthorhombicity is sufficiently weak that the observed unidirectional- ity reflects an intrinsic property of the CuO2 plane. This is supported by theoretical

work finding that unidirectional (striped) states have a lower energy than bi-directional (chequerboard) states [95, 128].

One way to determine whether this unidirectionality is truly an intrinsic property of the CuO2 plane would be to search for a inequivalence of the magnetic field induced

equivalent Cu-O directions. The observation of any inequivalence in the field enhanced CDW amplitude between the two Cu-O directions would be a positive identification of an intrinsic tendency toward unidirectional CDW order in the CuO2 planes.

4.5

Characteristics of CDW Modulations in the Spectral