As described within the previous section, the form of equations 3.4 and 5.1 are imposed on the relevant muon data such that they best predict the measured flux averages. In the case of the NS bilayer sample, the simple London profile is used to model the data with the only fit parameter being the λL value. Once the bilayer profile is known this acts as a control for the
NSIF samples and defines an upper limit for the expected standard Meissner component of the total flux profile. It also provides a baseline with which to compare the results of modelling the NSIF data. The full form in equation 5.1 is imposed on all samples containing a ferromagnetic layer. As discussed at the end of section 5.2.1, for these samples there are two free fit parameters used to describe the data: λef f andAEM, which correspond to the Meissner
and EM components respectively. By modelling the muon data associated with each oxide barrier thickness, and determining the corresponding val- ues of AEM and ASF where the latter is found through comparison to the
bilayer response, the effects of both pair breaking and EM proximity can be tracked as a function of the S-F coupling strength. The results of carrying out this analysis are presented within this section. Firstly, some represen- tative data are shown alongside the resultant best fit profiles for a subset of the measured samples. This is included simply to better illustrate the data analysis procedure. The second part of the section then focuses on the barrier thickness dependence of the two proximity amplitudes. It is found that the EM amplitude significantly outweighs the effects of pair breaking until for thicker barriers the bilayer result is recovered.
Figure 5.6 presents some example results of imposing the flux profiles on the LEµSR data. The data shown correspond to a set of samples of the form Cu(40)/Nb(50)/Y where Y = Si, Co(2.4)/Nb(3)/Si and Al2Ox(2)/Co(2.4)/
Nb(3)/Si for the NS, NSF and NSIF samples respectively. As in previous chapters the top panel presents some sample muon stopping profiles. In each case the corresponding vertical line indicates the average stopping position for muons of that energy within the sample structure shown. At the lowest measured energy the muons stop only within the Cu layer. As the probing energy is increased the stopping profiles start to spread out and shift to higher average probing depths whilst a tail remains in the Cu. For Eµ ≥
20keV the muons begin to sample the “Y” layers of each structure.
280 290 300 0 20 40 60 80 100 N/S/I/F N/S/F N/S
Average probing depth (nm)
<B > (G) Cu Nb Y 20keV 14keV 8keV 4keV p (x ) (A U )
Figure 5.6: The effect of an oxide barrier on the LEµSR results. Top panel: sample muon stopping profiles for a range of incident muon energies. The vertical lines in each case indicate the average probing depth with the pro- files forEµ≥20keV beginning to extend into the Y layers. Bottom panel:
representative results of the LEµSR results performed on the NSIF sample structures in an applied field of 302.4 Oe. The solid lines represent the results of imposing the spatial flux profile on each NSY structure where Y = Si, Co(2.4)/Nb(3)/Si and Al2Ox(2)/Co(2.4)/ Nb(3)/Si for the NS,
NSF and NSIF samples respectively. The average field values are plotted as open (closed) circles for the normal (superconducting) state data which were measured at 10K (2.5K).
The results of imposing the underlying field profiles, given by equations 3.4 and 5.1 for the bilayer and NS(I)F samples respectively, are indicated by the labelled solid lines in the bottom panel of figure 5.6. The results of the conventional averages approach are also included for comparison and are indicated by the open and closed circles for the normal and superconduct- ing state measurements respectively. All normal state measurements were conducted at a temperature of10K and the corresponding superconducting state data taken at 2.5 K. For each data point the corresponding error in the average field, typically found to be within the range(0.1−0.3)G, is also plotted but is too small to be visible on the scale of the symbol size in this case. In the normal state, for all measured samples, a constant flux density is observed which simply corresponds to the applied measurement field of
302.4Oe. From the superconducting state measurements it is clear that in all samples, across all probing energies, a flux lowering is observed. As has been previously observed and modelled within chapter 4, the NS data show the standard Meissner screening response extending across the full spatial extent of the proximitised bilayer. The best-fit cosh profile in this case cor- responds to a penetration depth of λL= 140 nm. Also modelled within the
previous chapter is the NSF trilayer result which shows a significant increase when compared with the bilayer due to the EM proximity effect. Here the best fit profile corresponds to fit parameter values of λef f = (156±1.5)nm
and AEM = (−13±0.7) G. The NSIF data show an overall enhancement
to the flux lowering. When one considers the shape of the average data it is clear this is due to a partial recovery in the normal Meissner screening, thanks to the partial decoupling between S and F, and that the EM compo- nent remains relatively constant. This is reflected in the results of the profile modelling where the best-fit result for NSIF corresponds to fit parameter values of λef f = (151±1.1) nm and AEM = (−14±0.5) G. The effective
penetration depth has started to recover and move back towards the bilayer result whilst the two EM amplitudes are within error. This naively suggests that the degree of coupling between the superconductor and ferromagnet has been altered sufficiently to affect the pair breaking but not the electro- magnetic proximity.
When one compares the results of the conventional averages approach with the best-fit profiles found from imposing the expected full spatial de- pendence a good correspondence between the two is observed across most
energies. As was discussed previously within section 4.3.2, the model pro- files for the NSF and NSIF samples begin to deviate from the averages at the lowest probing energy. This is most likely due to the EM compo- nent rolling off more rapidly than expected within these systems where the thickness is less than the penetration depth. In order to confirm this fur- ther measurements on thicker samples would be required. It is clear from these example data sets that the averages approach to the data analysis captures the physics of the system well, and given the right framework for interpretation, provides a good physical understanding of the system. The full spatial analysis, however, offers the opportunity to identify and extract a quantitative measure of each proximity effect and to track, given measure- ments and analysis across the full sample set, their behaviour as a function of the coupling strength as was the aim of this study. The same analysis as presented in figure 5.6 for the Cu(40)/Nb(50)/Al2Ox(2)/Co(2.4)/ Nb(3)/Si
sample was therefore performed on the muon data for all other oxide thick- nesses and the corresponding best fit values ofASF and AEM extracted for
each. The results of this analysis are shown in figure 5.7 where both prox- imity amplitudes are plotted as a function of the insulator thickness, dI,
which is used as some measure of the coupling strength.
-15 -10 0 5 0 2 4 6 8 ASF AEM ampli tude (G) AlOx thickness (nm) NS bilayer (AEM= A SF= 0) -5
Figure 5.7: The extracted proximity amplitudes as a function of dI which
corresponds to a changing S-F coupling strength. The open (closed) symbols correspond to the extractedASF (AEM) values. The NS bilayer control, for
which both amplitudes are zero, is represented by the dashed line. The light shaded region indicates the approximately constant AEM for dI ≤4 nm.
The open and closed symbols correspond the the extracted values of ASF
and AEM respectively. The plotted error bars take account of both the
error associated with the data fitting and any variation between samples in Tc−Ts; where Ts was the sample measurement temperature. This was
achieved using the known temperature dependence of the total amplitude for these sample structures as presented in appendix D. The horizontal dashed line represents the NS bilayer control result for which both AEM
and ASF are of course zero. When the niobium and cobalt are in direct
contact, for dI = 0 nm, there is both a small reduction of about 15% in
the standard Meissner screening response, which can be determined from the corresponding ASF value, and a substantial diamagnetic enhancement
(AEM) due to the EM proximity. As the degree of coupling between the
Nb and Co layers is reduced, through the insertion of the insulating oxide barriers, the direct proximity weakens and the suppression in the standard Meissner screening reduces until for dI ≥ 4 nm ASF = 0 G and the NS
bilayer result is recovered. The behaviour of the EM proximity appears to be rather different. The value of AEM appears to be approximately con-
stant for barrier thicknesses of dI ≤4 nm. This is indicated by the shaded
band shown in figure 5.7. A further increase in the thickness of the Al2Ox
layer causes the value of AEM to diminish back towards the bilayer control
response.
Through careful modelling of the LEµSR data gathered for the full range of NSIF sample structures, the dependence of both the direct and electro- magnetic proximity amplitudes on the insulating barrier thickness has been successfully extracted. Since both amplitudes return to the NS bilayer re- sponse as the barrier thickness is increased it seems the effects observed are due to the direct sampling of the ferromagnet by the condensate and are not related to any stray magnetic fields. In particular, as has already been seen in previous chapters the stray fields generated at the S/F interface ef- fect the muon depolarisation rate strongly over lengthscales closer to15nm
whereas here both proximity contributions have disappeared fordI = 8nm.
Through modelling of the muon depolarisation rate as a function of dI any
potential influence of stray fields can be explored in more detail. This is the subject of the following subsection where it is explicitly demonstrated that stray fields play little or no role in the observed physics.