Flat coils based TDO for measuring the in-plane penetration depth

investigate the relative temperature variations in London penetration depth. The fact that it is a resonant method, combined with the great sensitivity of frequency counters, implies that minute changes in penetration depth can be detected with reported values as small as 1 Å [24, 101].

The resolution of the method is dependent of a number of factors. Looking at the expression in Eq. 3.53, we can see that larger values of the effective dimension of the sample will results in larger frequency variations. Recalling the expression for from Eq. 2.22, considerable increase in ( ) measurement resolution can be obtained considering large surface samples. Moreover, from the expression for in Eq. 3.54, we can see that the same effect can be achieved by considering thinner samples which would increase the demagnetization factor. Consequently, the use of flat slab shaped samples can significantly improve the resolution of the technique. Just so it happens most HTC

superconductors are layered structures which can be cleaved to result in thin specimens, which is also the case of our samples, as we will mention in more detail in Chapter IV. These layered superconductors are anisotropic meaning that the field penetration is dependent on orientation in respect to the sample’s crystalline structure. We mentioned the effects of field orientation in Chapter II where we showed that, depending on the direction of the applied field, the effective susceptibility can be the result of a

contribution from the in-plane London penetration depth and the out of plane correspondent. If the applied field is perpendicular to the surface of the sample (layer planes), the measured susceptibility changes are related only to the in-plane penetration depth .

Apart from the sample geometry, we can we improve the resolution of the TDO technique for London penetration depth measurements in flat specimens by adapting the geometry of the inductor to increase the sample-coil filling factor ⁄ . Coils with volume close to the sample volume should promote larger measured frequency shifts. However, in relating the frequency variation to the in-plane penetration depth changes alone, the excitation field produced by the coil should be perpendicular to the sample surface. Moreover, simple analytical expressions like the one in Eq. 3.53 can only be used if the probing field of the coil is uniform in the sample region.

Solenoid inductors are most commonly used since they will accomplish these conditions

however, for flat samples the filling factor is very small. An example is illustrated in Fig. 3.8(a) where we show the magnetic field density distribution for a disk shaped sample inside a solenoid inductor. The left side of Fig. 3.8 depicts the field distribution for the case when the sample has zero susceptibility (empty coil) while the right panel shows the case of a strong diamagnetic specimen.

Figure 3.8 COMSOL simulations of magnetic field lines and flux density distribution in a cross section of an axially symmetric sample-coil geometry for the case of (a) solenoid inductor, (b) planar inductor, and (c) parallel pair of planar inductors. The disk shaped sample has zero susceptibility (left) and diamagnetic susceptibility close to unity (right).

It is easy to see that the field is indeed perpendicular to the surface of the sample; however, uniformity is achieved if the sample is constrained to a small central region of the coil or if long solenoids are used. In either case, the filling factor is considerably reduced for thin slab shaped samples.

A significant increase in TDO resolution for flat specimens can be achieved by making use of planar inductors like the ones suggested in [132, 133]. An example is illustrated in Fig. 3.8(b) where a disk shaped sample is placed near a flat spiral coil. We can see from the right panel of Fig. 3.8 (b) that, when the material is diamagnetic, the field distribution around the coil is significantly modified by the sample. Consequently, the inductance value shifts caused by the sample will be significant. Moreover, since spiral coils have significantly lower inductance values compared to solenoids, the relative

inductance changes are higher. However, the complicated field distribution for an empty single flat coil makes it difficult to extract quantitative information. It is easy to see from the left side of Fig. 3.8 (b) that the field is neither uniform nor perpendicular to the sample. Moreover, for anisotropic superconductive samples, the resulting TDO frequency changes will be a result of both and contributions to the

susceptibility.

For measuring the in-plane London penetration depth of our flat-like samples, we considered

using pairs of planar inductors in a parallel configuration like in the case illustrated in Fig. 3.8(c). Looking at the left panel, we can see that, by using an additional flat circular coil, in a symmetric geometry with respect to reflection across the center plane of the sample, the field of the empty inductor is perpendicular to the sample surface and uniform within a significantly large central region of the coil volume.

We have shown in the previous section that the TDO frequency changes are directly proportional to the changes in the in-plane London penetration depth. For flat like specimens, the proportionality relation in Eq. 3.53 was obtained considering a uniform excitation field perpendicular to the sample surface in normal state which, as we have shown, stands true regardless of the geometry of the inductor creating the field. To make use of the increased filling factor provided by planar inductors, hence the sensitivity of our measurements, while providing a uniform and perpendicular field in the region of the sample in normal state, we used the pair configuration of planar inductors for our TDO setup.

Planar rectangular spiral coils, 8 x 8 mm2 is size, were milled on a copper clad printed circuit board (PCB) with 1oz copper thickens (1.4 mils = 35µm). Using a LPKF Protomat S43 milling machine with a minimum milling tool diameter of 0.1 mm and 0.5µm translation resolution, we were able to build rectangular spiral coils with 3 turns/mm and a total of 12 turns. A magnified image of one such spiral coils is shown in Fig. 3.9. A 0.2 mm hole was drilled in the center pad of the coils to allow for lead attachment. A thin copper wire was soldered onto the center pad and the soldering joint (inset of Fig. 3.9) was trimmed to a minimum height. The coils were cleaned and covered with a thin layer of GE varnish to

prevent oxidation of the copper tracks. The typical series inductance and resistance values of the

individual coils were 0.62 µH and 0.64 Ω respectively, as measured by an Agilent 4263B LCR meter. The coils were then connected in aiding parallel, in a geometry with reflection symmetry about the halfway plane, separated by a 2.7 mm gap (as measured from the copper track surface). The choice of parallel connection, as opposed to series, was made to provide a lower total inductance of the resulting coil which would boost the relative TDO frequency changes induced by a magnetic sample. The measured series inductance and resistance values for each resulting coil pair were around 0.39 µH and 0.32 Ω

respectively.

Figure 3.9 Picture of one of the 8 x 8 mm2 flat spiral coils with 3 turns/mm milled on a copper-clad laminate 1oz PCB. Inset: soldering joint for the center lead.

Our superconductive samples for in-plane London penetration depth measurements (detailed in Chapter IV) are flat shaped single crystals with rectangular cross section and high aspect ratios (typical dimensions 2 x 2 x 0.1 mm). If the sample is positioned midway between the two planar inductors, with the ab crystallographic plane parallel to the surface of the flat coils (Fig. 3.10), considering the symmetry

of our setup and the small thickness of the samples relative to the coil gap, the probing ac field is

expected to be parallel to the c axis of the crystal. Consequently, supercurrents are only induced in the ab

plane, thus the measured changes in TDO resonant frequency are solely due to the variations in .

To test for the perpendicularity as well as the uniformity of the field in the sample region,

simulations were carried out for our specific coil-sample configuration, using the COMSOL Multiphysics software. The exact geometry used in the numerical simulations is depicted in Fig. 3.10. The coils

geometry was directly imported from the Autodesk Autocad file used by the PCB milling machine for fabricating our planar inductors. The resulting 3D geometry was imported in COMSOL where the magnetic field distribution was calculated using the Magnetic Fields interface (right side of Fig. 3.10), assuming a uniform current density through the cross section of the coil tracks (

), ⁄ -. The coil domain was assigned copper as the conductive material and air for the rest of the domains. We used the external current density approach to minimize the computational time. Since the current is oscillating at high frequencies, the current density is expected to be concentrated towards the surface of the conductor due to the skin effect. More precise calculations can be made using the frequency domain, however, the current density has to be computed before which, for 3D geometries, can take considerably more time. We have carried out simulations for both stationary and frequency domain studies and found that, for a 2D axially symmetric analog geometry, the differences in magnetic field values are negligible.

Figure 3.10 Left: Spatial arrangement of the coils and sample. The setup is symmetric with respect to reflection across the z = 0 plane. Right: COMSOL magnetic field simulations for the coil geometry

The simulated results obtained for the field lines and magnetic flux density distribution over the y

= 0 and z = 0 cross sections of the setup for the normal state case of the sample (see left side of Fig. 3.10), are illustrated in Fig. 3.11.

Figure 3.11 The simulated magnetic field distribution of our setup for the normal state of the sample. (a) Magnetic field lines and flux density distribution over the y=0 cross section of the setup (side view). (b) Flux density distribution over the z=0 cross section of the setup (top view). (c) Expanded view on the y=0 plane. (d) Expanded view on the z=0 plane. The white rectangles symbolize the domain of a 2x2x0.1 mm3 sample. The color scale corresponds to the B field magnitude relative to its value in the center of the sample (0,0,0).

In panel (a) we show the magnetic field lines and flux density distribution over the y = 0 cross section of the sample-inductor setup (side view). In panel (b) the flux density distribution over the z = 0 cross section of the setup (top view). Panel (c) and (d) show an expanded view of the y = 0 plane and the

z = 0 plane, respectively. The white rectangles symbolize the sections of a 2 x 2 x 0.1 mm3 sample. The color scale corresponds to the B field magnitude relative to its value in the center of the sample (x=0, y=0, z=0). The results confirm that the probing field from the coils is indeed perpendicular to the ab surface of the sample [Fig. 3.12(c)] and that in a central rectangular region of dimensions comparable to the sample size, the magnitude of the field is homogeneous with ∼90% uniformity [Fig. 3.12(d)] i.e. the average value of the field across the surface relative to the center value is ∼0.9. For the 1mA coil current used in the simulation, the calculated magnitude of the field in the center of the geometry is around 20 mOe.