The modulations in reciprocal space

In the following paragraph, I would like to illustrate the modulations in reciprocal space. The easiest method for this is to calculate the power spectral density (PSD) in the periodogram approach. For details please see Appendix B.2. Fig. 5.8.5 shows the square root of the power spectral density (equal to the structure factor) for the two-layer Bi₂Sr₂CaCu₂O_8+δ. The power spectral density can be practically seen as both the Fourier-transformed autocovariance and the squared absolute value of a Fourier transformation of the original STM pattern. Therefore,

5.8.5: Square root of the PSD of the measurement of optimally doped Bi2Sr2CaCu2O8+δ with TC=92 K, (20x20)nm², RT, as shown in Fig. 5.8.1. The PSD was evaluated by the periodogram technique.

5.8.6: Same figure as Fig. 5.8.5, but with marks to explain the features found in the square root of the PSD. For details see text.

a modulation reveals them as peaks in the PSD. The position of the peaks will indicate its wavevector and the height of its oscillation strength. To increase the visibility of peaks with different intensities, the square root of the PSD can be used. What cannot be seen due to the normalization is the intensity of the constant STM background which corresponds to the undisturbed potential of the crystal. The constant signal is given as a broadened Kronecker delta at (0,0) whose width represents the uncertainty of the constant signal due to the finite probed area.

I would first like to discuss the straightening which must always be done in STM measurements because of the unequal attenuation of the piezos, for example. During this discussion, I will already highlight some features that are visible in the square root of the PSD. After that, I will calculate ’by hand’ the transformation of the 2d modulation from real-space to reciprocal-space to explain some high-order features. This knowledge will be required in the next section.

The straightening was done by the points B and D. The reason for using these points will be explained in the following. To achieve this, the original data as depicted in Fig. 5.8.1 was rotated counterclockwise by 42 degrees and then sheared by the matrix:

S = 1 0.11

0 1

!

(5.5) Here (1,0), is the horizontal direction. The points at A in Fig. 5.8.6 were assumed to be the spots produced by the Bi surface lattice. Here, a nearly quadratic Bi surface lattice might be assumed. It should be noted that the spots are not perfectly on their position and the spots at (±2π/a,±2π/b) are not visible. Instead, point G is clearly visible. Perhaps this is not accidental but has physical meaning. However, please note also that the spots of this lattice are distorted

5.2Comparison to the 2d modulations in Pb-free Bi2212

by the one-dimensional superstructure which results in replicas of the original lattice points at the length of about 1/5×2π/b. The point denoted as B is the result of a commensurate 2×a superstructure visible for Bi2201 and Bi2212 in diffuse x-ray diffraction [218]. It is, of course, also visible in LEED and for ARPES it produces the ’shadow band’ [219]. In this work, for the one-layer Bi_1.44Pb_0.42Sr_1.44La_0.50Cu_1.20O_6+δ, the visibility of this feature can be verified by LEED as shown in Fig. 5.8. There, the extra spot is visible at (1,0) while it is absent at (0,1).

Point D in Fig. 5.8.6 represents the one-dimensional ≈ 1×5 superstructure in its first order and can be attributed to a wave vector of q_SS=0.21×2π/b (see, e.g., [220, 157]). Therefore, the straightening was done by points B and D, because the exact position for them is confirmed by the citations above.

Figure 5.8: LEED pattern for Bi_1.44 -Pb_0.42Sr_1.44La_0.50Cu_1.20O_6+δ at elec-tron energies of 54 eV. Please note the spot at (1,0), which is absent at (0,1).

The spot (1,0) is the same as spot B in Fig. 5.8.6.

The points at C represent the intersection of the Bi sur-face Brillouin zone with the bulk Brillouin zone. This is a typical phenomenon in surface condensed matter physics.

The Bi-surface lattice produces the spots A, the bulk Bril-louin zone produces the C-spots. Please compare also with Fig. 3.4 in Chapter 3. For the general description of an intersection of a surface lattice with a bulk Brillouin zone see, e.g, [221]. If the inequality in the symmetry of the points C and C? is confirmed, it may also represent the orthorhombicity. Point E1 is the combination of the first order of the wavevector q_E1 of the two-dimensional mod-ulation and the second order of the≈1×5 superstructure.

That is why the intensity is very high and comparable to the first order of the one-dimensional superstructure.

This combination of the two modulations is also the rea-son why the peak looks broadened or doubled. Point E2 is then the second wave-vector of the two-dimensional mod-ulation.

For the preceding, I will show in the following that E1 and E2 are the positions of the reciprocal lattice vectors.

As mentioned above, the 2d lattice vectors are~r₁= 0.8~a +2.3~b and ~r₂=-7.2~a and span the area of 17 ~a~b. Their

Here,~a^∗ and~b^∗ are the reciprocal lattice vectors of the atomic lattice. In Fig. 5.8.6, there are also the points F. Although these points are of minor importance for the modulation structure, they must be addressed because of their significance to later shown findings. These points are

produced by a combination of the present modulation wave-vectors

q_F =±2q_E2±q_SS =±0.27~a^∗±(0.094±0.21)~b^∗ (5.6)

5.2.2 Comparison to other work concerning 2d modulations in Bi2212