Determinación de la moralidad del sexo, continuación

Figure 7.7:Blue dots represent the experimen- tal data of the optimally doped compound shown in Fig. 7.6a. Cyan line is the model func-

tion and is described in the text. -20 -10 0 10 20

0.2 0.4 0.6 0.8 1.0 1.2 UHmVL dI dU

compound, the gap vanishes around 30 K in agreement with literature [260]. Espe- cially the observation of a flux lattice in the Shubnikov phase of the respective system, which will be shown in the following (see Fig. 7.9), is a clear proof of the prevailing superconductivity.

Such a V-shaped gap like shown in Fig. 7.6 can be associated with a nodal pair- ing symmetry. In a report on a combined specific heat and nuclear magnetic resonance (31_{P-NMR) study, it was speculated that in case of the optimally doped SrFe}

2(As0.65P0.35)2,

a multigap exists [227]. The proposed model consists of a small nodal gap with resid- ual DOS plus additional full gaps [227]. For the present STM data shown in Fig. 7.7, we found a better agreement for a gap-equation that is closely related to the one stated in Ref. [227]. The model function, which is shown in Fig. 7.7, was created within the s + (s + d)-model, which was introduced in the previous chapter, using the following equation: ∆s+(s+d)_e = α∆0_e+√1 − α2_∆0 e √ 2cos(2θ). (7.1) In this case, ∆0

e is a mean gap value which is averaged over the reciprocal space and

which was set to 2.193 meV. The parameter α was set to 0.3 and therefore leads to nodes in the gap according to Ref. [226]. As can be seen in Fig. 7.7, the shape of the experimen- tally obtained superconducting gap is reproduced quite well. Only the quasiparticle coherence peaks are slightly smaller in the experimental data. Since the experimen- tal spectrum is an averaged spectrum, the reason for the suppressed coherence peaks could be the intrinsic impurities on the sample.

For most of the iron-based superconductors, technically, it is almost impossible to suppress superconductivity by applying a magnetic field due to their large upper criti- cal field. For SrFe2(As1−xPx)2, the upper critical field Hc2is about 60 T [229]. Neverthe-

less, the Shubnikov phase can be reached by applying a few hundred mT.

7.4 Determination of the Coherence Length

The coherence length was determined for both the optimally doped and the over- doped compound by using two different methods. The power spectral density func- tion (PSDF) was applied on a superconducting gap map and the coherence length could be extracted by analyzing vortices in the Shubnikov phase. The following two paragraphs discuss the PSDF method and the vortex method.

7 SrFe2(As1−xPx)2

7.4.1 PSDF Method

SrFe2(As1−xPx)2samples have intrinsic doping inhomogeneities due the random phos-

phorus concentration. This leads to spatial variations of the superconducting order parameter on the minimal length scale set by the coherence length [248]. As described at the beginning of this thesis, the superconducting ground state is determined by a large number of Cooper pairs where the electrons are paired over a distance of sev- eral hundred nm in the case of conventional superconductors. In this state, their wave functions overlap and the phase of each Cooper pair wave function is the same as for the superconducting ground state. The size of a single Cooper pair can be related to the coherence length ξ in the sense of BCS theory [6]. We assume now that there are spatial variations of the Cooper pairing in the sample induced by the doping inhomogeneities and that these variations are convoluted with the wave function of the Cooper pairs. The probability distribution of a Cooper pair can be used in order to estimate the size of a single Cooper pair [248, 265]. For this purpose we use the Gaussian distribution [248]

g(x, y) = 1 2πσ2e

−x2+y2

2σ2 , (7.2)

with the coherence length as the full width at half-maximum (FWHM) [248]

ξ = FWHM = 2σp2ln(2). (7.3)

In order to create our so-called superconducting gap maps, we performed spatially resolved STS measurements over an area of 30 nm×30 nm. Within this area, dI/dU - spectra where taken at each of the 256×256 points. The gap size (= order parameter) was evaluated for each spectrum. However, the gap size is not sufficient to describe the superconducting properties. Especially in the present case of a V-shaped supercon- ducting gap, the depth of the gap must be considered as well. Thus, for each spectrum, the superconducting gap area, which is sketched in Fig. 7.6a, was calculated by doing a numerical integration using the trapezoidal rule. The resulting map is shown for the optimally/overdoped sample in Fig. 7.8a/c.

As can be seen, there are variations in the intensity of these two images. Bright areas correspond to pronounced superconductivity with a larger value for the super- conducting gap area. On these superconducting gap maps, the radially resolved PSDF method was applied. The PSDF can be considered as the square of the absolute value of the Fourier transformation of a function (PSDF = |F(f (x, y))|2_{) [266, 267]. Assuming}

that the image consists of randomly distributed superconducting areas that are convo- luted with a Gaussian distribution g(x,y) representing the Cooper pairs including their coherence length [248]

|F (image)|2 = |F (random)|2

| {z }

const

∗|F (g(x, y))|2, (7.4)

the coherence length can be extracted from |F(g(x, y))|2 _{by using the relationship 7.3.}

The result is shown in Fig. 7.8b and d. In order to state the final result for the in- plane superconducting coherence length, several measurements for the optimally as

7.4 Determination of the Coherence Length gap ar e a (ar b. units ) gap ar e a (ar b. units ) small large small large a) b) c) d) rP S DF (a rb . u n it s) ξ=4.1±1.1nm Determination of ξ for P= 35% rP S DF (a rb . u n it s) distance 2π(nm-1₎ ξ=2.3±0.8nm Determination of ξ for P= 46% distance 2π(nm-1₎

Figure 7.8:a Superconducting gap map for the optimally doped compound (x=0.35, area: 30 × 30 nm2, 256 × 256 pixel). b) Blue dots correspond to the calculated radially resolved PSDF of the map shown in a. The solid line shows the applied fit. c) Superconducting gap map for the overdoped compound (x=0.46, area: 35 × 35 nm2_{, 256 × 256 pixel) d) Blue dots correspond to} the calculated radially resolved PSDF of the map shown in c). The solid line shows the applied fit. Taken from Ref. [248].

7 SrFe2(As1−xPx)2

well as for the overdoped compound were averaged by applying the above-mentioned method on different regions of the surface. For the optimally doped compound, the coherence length has a value of ξx=0.35= 4.1 ± 1.1 nm and ξx=0.35= 2.3 ± 0.8 nm for the

overdoped compound.

In the following paragraph, a second method for the determination of the coherence length will be presented in order verify the results just mentioned. This method will be referred to as vortex method.

7.4.2 Vortex Method

In order to apply this method, the sample has to be driven in the Shubnikov phase. Therefore, a magnetic field of 1 T was applied. By taking dI/dU or d2_I/dU2 _{maps, the}

vortex lattice can be resolved. Since the contrast is better in the case of d2_I/dU2 _maps,

the latter will be presented. In order to measure d2_I/dU2 _{maps, the bias voltage was}

set to 2/1.2 mV for the optimally/overdoped compound because peaks are visible at this energy in the second derivative of the tunneling current. These peaks correspond the local maximum of the slope of the superconducting gap in the first derivative of the tunneling current. By means of the vortex lattice, superconducting and normally conducting areas can be distinguished. In Fig. 7.9a and c, such a vortex lattice is shown for the optimally and the overdoped compound.

In these images, the dark almost circular areas correspond to the normally conduct- ing areas, where the superconducting order parameter (superconducting gap) is sup- pressed, and hence, there is no peak at 2/1.2 mV. The superconducting order parame- ter can be described by Ψ(r) = |Ψ(r)|eiθ_{. For an isolated vortex, the Ginzburg-Landau}

equation is solved by Ψ(r) = Ψ∞tanh(√r_2ξ). The distance from the vortex core is labeled

with r and the value of the superconducting order parameter in absence of a magnetic field with Ψ∞[6, 268]. In Fig. 7.9a and c, one vortex is marked with a green rectangle.

Along a line across such a vortex, dI/dU spectra were measured in order to determine the variation of the superconducting energy gap, i.e., |Ψ(r)SC| = |Ψ∞tanh(√r_2ξ)|Thus,

the coherence peak separation can be measured, which reflects the width of the super- conducting energy gap. The corresponding values, normalized to the value in absence of a magnetic field |Ψ(r)SC|/|Ψ∞|, are shown as dots in Fig. 7.9b and (d). The coherence

length can now be extracted by fitting these data with a function f (r) = a · tanh(_√r 2ξ).

For the optimally/overdoped compound, this method gives a value of the coherence length of ξ = 5.0 ± 1.0 nm /ξ = 2.9 ± 0.6 nm. This agreement with the values ob- tained by the PSDF method is rather good for both compounds. Nevertheless, when the PSDF method and the vortex method are compared, it becomes obvious that the former is more accurate since a higher number of local spectra is taken into account. Furthermore, when taking dI/dU spectra along a line through a vortex as in the vortex method, the vortex should not move during the measurement. However, the vortices were not well pinned in the case of the optimally doped compound. As can be seen in Fig. 7.9a, they turned out to be mobile even during scanning, which made an accurate linegrid measurement through a vortex quite challenging. Of course, the position of the vortex was checked before and after taking the linegrid by taking a scan. But it is

7.4 Determination of the Coherence Length r (nm) r (nm) |ψ SC |/|ψ ∞ | Δ/Δ ∞ |ψ SC |/|ψ ∞ | r (nm) ξ = 5.0 ± 1.0 nm ξ = 2.9 ± 0.6 nm a) b) c) d) 20nm 20nm x=0.35 x=0.46 x=0.35 x=0.46

Figure 7.9:a) d2I/dU2 map measured at U=2 mV for the optimally doped compound. b) Dots represent the normalized width of the superconducting energy gap. Each dot corresponds to a value obtained from an individual tunneling spectrum recorded along a line through a vortex. The corresponding vortex is marked with a green rectangle in a). c) d2I/dU2map measured at U=1.2 mV for the overdoped compound. d) Dots represent the normalized width of the super- conducting energy gap. Each dot corresponds to a value obtained from an individual tunneling spectrum recorded along a line through a vortex. The corresponding vortex is marked with a green rectangle in c). Taken from Ref. [248].

7 SrFe2(As1−xPx)2

still difficult to determine the exact position of the vortex.

Besides the two methods of determining the coherence length that have just been presented, a theoretical estimation was made by using the relationship for the upper critical field Hc2 = _2πξ∆02 [31]. As already mentioned, Hc2 is about 60 T for the optimally

doped compound [229]. As a result, the theoretical coherence length would be ξtheo ≈

2.34 nm. This is in good agreement with the previous results. Additionally, the com- parison of coherence lengths of similar systems gives results within the same order of magnitude [269].