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Superconducting Transition Temperature Determination

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Published: Fri, 23 Feb 2018

Fabrication of YBa2Cu3O7-δ and Determination of its Superconducting Transition Temperature

A superconducting material is one which below a certain critical temperature exhibits, amongst other remarkable traits; a total lack of resistivity, perfect diamagnetism and a change in the character of the specific heat capacity. The BCS theory describes perfectly the phenomenon of superconductivity in low temperature superconductors, but cannot explain the interaction mechanism in high temperature superconductors. In order to determine the superconducting transition temperature of two laboratory fabricated batches of YBCO their resistivity and specific heat capacity were measured as functions of temperature. From resistivity measurements the two batches were found to have transition temperatures of 86.8(±0.8)K and 87.8(±0.4)K respectively which were used to infer their oxygen contents of 6.82(±0.01) and 6.83(±0.01) atoms per molecule respectively. These agreed with XRD data and the literature upper value of the transition temperature of 95K (with an oxygen content of 6.95). Specific heat capacity measurements of the first batch gave questionable confirmation of these results, but could not be performed on the second batch due to time constraints.

19 January 2010Page 14 of 14Josephine Butler College

I. Introduction and Theory

A superconducting material is defined as one in which a finite fraction of the electrons are condensed into a ‘superfluid’, which extends over the entire volume of the system and is capable of motion as a whole. At zero temperature the condensation is complete and all of the electrons participate in the forming of the superfluid. As the temperature of the material approaches the superconducting transition temperature (or critical temperature, given by Tc) the fraction of electrons within the superfluid tends to zero and the system undergoes a second order phase transition from a superconducting to a normal state.[i]

The phenomenon of superconductivity was first observed by Kamerlingh Onnes in Leiden in 1911 during an electrical analysis of mercury at low temperatures. He found that at a temperature around 4K the resistance of mercury fell abruptly to a value which could not be distinguished from zero.[iii]

The next great leap in experimental superconductivity came in 1986 when Müller and Bednorz fabricated the first cuprate superconductor[v].

After its lack of resistivity one of the most striking features of a superconductor is that it exhibits perfect diamagnetism. First seen in 1933 by Meissner and Ochsenfeld, diamagnetism in superconductors manifests itself in two ways. The first manifestation occurs when a superconducting material in the normal state is cooled past the critical temperature and then placed in a magnetic field which will then be excluded from the superconductor. The second appears when a superconductor (in its normal state) is placed in a magnetic field and the flux is allowed to penetrate. If it is then cooled past the critical temperature it will expel the magnetic flux in a phenomenon know as the Meissner effect.[vi] This can be seen qualitatively in figure 1.

In 1957, Bardeen, Cooper and Schrieffer managed to construct a wave function in which electrons are paired. Know as the BCS theory of superconductivity it is used as a complete microscopic theory for superconductivity in metals. One of the key features of the BCS theory is the prediction of an energy gap, the consequences of which are the thermal and most of the electromagnetic properties of superconducting materials. The key conceptual element to this theory is the formation of Cooper pairs close to the Fermi level.

Although direct electrostatic interactions between electrons are repulsive it is possible for the distortion of the positively charged ionic lattice by the electron to attract other electrons. Thus, screening by ionic motion can yield a net, attractive interaction between electrons (as long as they have energies which are separated by less than the energy of a typical phonon) causing them to pair up, albeit over long distances.

Given that these electrons can experience a net attraction it is not unreasonable that the electrons might form ‘bound’ pairs, effectively forming composite bosons with integer spin of either 0 or 1. This is made even more likely by the influence of the remaining electrons on the interacting pair. The BCS theory takes this idea one step further and constructs a ground state in which all of the electrons form ‘bound’ pairs.

This electron-phonon interaction invariably leads to one of the three experimental proofs of the BCS theory. A piece of theory known as the isotope effect provided a crucial key to the development of the BCS theory. It was found that for a given element the super conducting transition temperature, TC, was inversely proportional to the square root of the isotope mass, M (equation 1).

TC�M-12 (1)[vii]

This same relationship holds for characteristic vibrational frequencies of atoms in a crystal lattice and therefore proves that the phenomenon of superconductivity in metals is related to the vibrations of the lattice through which the electrons move. However this only holds true for low temperature superconductors (a fact which will be discussed in more detail at a later stage in this section).

Both of the two further experimental proofs of BCS theory come from the energy gap in the superconducting material. The first proof is in the fact that it was predicted and actually exists (figure 2) and the second lies in its temperature dependence. From band theory, energy bands are a consequence of a static lattice structure. However, in a superconducting material, the energy gap is much smaller and results from the attractive force between the electrons within the lattice. This gap occurs Δ either side of the Fermi level, EF, and in conventional superconductors arises only below TC and varies with temperature (as shown in figure 3).

Figure 2: Dependence of the superconducting and normal density of states, DS and Dn respectively. From ‘Superconductivity’, Poole, C.P., Academic Press (2005), page164

At zero Kelvin all of the electrons in the material are accommodated below the energy gap and a minimum energy of 2Δ must be supplied in order to excite them across the gap. BCS theory predicts equation 2 which has since been experimentally proven,

ΔT=0=CkBTC (2) [viii]

where theoretically the constant C is 1.76 although experimentally in real superconductors it can vary between 1.75 and 2.45.

Figure 3: Temperature dependence of the BCS gap function, Δ. Adapted from ‘The Superconducting State’, A.D.C. Grassie, Sussex University Press (1975), page43

As before stated it has been found that the first of these ‘BCS proofs’ does not hold for high temperature superconductors. In these materials it has been found that in the relation stated as equation 1, the exponential tends towards zero as opposed to minus one half. This indicates that for high temperature superconductors it is not the electron-phonon interaction that gives rise to the superconducting state. Numerous interactions have been explored in an attempt to try and determine the interaction responsible for high temperature superconductivity but so far none have been successful.

Figure 4: A plot of TC against TF derived from penetration depth measurements. Taken from ‘Magnetic-field penetration depth in K3C60 measured by muon spin relaxation’, Uemura Y.J. et al. Nature (1991) 352, page 607.

In figure 4 it can be seen that the superconducting elements constrained by BCS theory lie far from the vast majority of ‘new’ high temperature superconducting materials which appear to lie on a line parallel to TF, the Fermi temperature and TB, the Bose-Einstein condensation temperature, indicating a different interaction method.

One of the most extensively studied properties of the superconductor is its specific heat capacity and how its behaviour changes with temperature (seen in figure 5). It is known that above the transition temperature the normal state specific heat of a material, Cn, can be given by equation 3 (below) which consists of a linear term from the conduction electrons and a cubic phonon term (the addition Schottky contribution has been ignored in this case and γ and A are constants).

Cn=γT+AT3 (3)[ix]

Due to the aforementioned energy gap it is also predicted by BCS theory that at the superconducting transition temperature there will be a discontinuity in the specific heat capacity of the material of the order 1.43 as seen in equation 4 (where CS is the superconducting state heat capacity) and figure 5.

CS-γTCγTC=1.43 (4)[x]

However for high temperature superconductors this ratio is likely to be much smaller due to a large contribution from the phonon term in the normal state specific heat capacity.

Figure 5: Heat Capacity of Nb in the normal and superconducting states showing the sharp discontinuity at TC. Taken from ‘The Solid State Third Edition’, H.M Rosenberg, Oxford University Press (1988), page 245

Now that the concept of the high temperature superconductor has been explained this report can return to one of the initial concepts of how the behaviour of resistivity changes with temperature. A low temperature superconductor is likely to obey the T5 Bloch law at low temperatures and so its resistivity will fall to zero in a non-linear region. In contrast the resistivity of a high temperature superconductor should fall to zero before it leaves the linear region. The resistivity profile of a high temperature superconductor can also be used to determine its purity. By comparing the range of temperatures over which the transition occurs with the transition temperature itself an indicator of purity can be determined (equation 5, where PI is the purity indicator and ΔT the magnitude of the region over which the transition occurs). In this case a value of zero would indicate a perfectly pure sample.

ΔTTC=PI (5)[xi]

Other than for scientific purposes, within the laboratory, the biggest application of superconductors at the moment is to produce to the large, stable magnetic fields required for magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR). Due to the costliness of high temperature superconductors the magnets used in these applications are usually low temperature superconductors. It is for this same reason that the commercial applications of high temperature superconductors are still extremely limited (that and the fact that all high temperature superconducting materials discovered so far are brittle ceramics which cannot be shaped into anything useful e.g. wires).

Yttrium barium copper oxide (or YBCO) is just one of the aforementioned high temperature, cuprate superconductors. Its crystal structure consists of two CuO2 planes, held apart by a single atom of yttrium, either side of which sits a BaO plane followed by Cu-O chains. This can be seen in greater detail in figure 6.

Figure 6: The orthorhombic structure of YBCO required for superconductivity. Adapted from ‘High-Temperature Superconductivity in Curpates’, A. Mourachkine, Kluwer Academic Publishers (2002), page 40

If the structure only has 6 atoms of oxygen per unit cell then the Cu-O chains do not exist and the compound behaves as an antiferromagnetic insulator. In order to create the Cu-O chains and for the compound to change to a superconductor at low temperatures it has to be doped gradually with oxygen. The superconducting state has been found to exist in compounds with oxygen content anywhere from 6.4 to 7 with optimal doping being found to occur at an oxygen content of about 6.95.[xii]

This report intends to determine the superconducting transition temperature of a laboratory fabricated sample of YBCO. This will be achieved by measuring how both its resistivity and specific heat capacity vary as a function of temperature.

II.I Fabrication and Calibration Methods

To ensure an even firing of the sample within the furnace and to find out where in the furnace the heating profile was closest to that of the actual heating program, three temperature profiles of the furnace were taken while heating. The length of the furnace was measured with a metre ruler and found to be 35±1cm. Four k-type thermocouples were then evenly spaced (every 11.5±0.5cm) along the length of it, as can be seen in figure 7 below.

Figure 7: Transverse section of the furnace. Thermocouples are numbered 1 to 4 and the length of the furnace surrounded by heating coils is shown in green, blocked at either end by a radiation shield.

Temperature profiles were taken for each of the temperature programs displayed in figure 8; all started at room temperature and were left to run until the temperature displayed by the thermocouples had stopped increasing.

Target Temp

(°C)

Heating Rate

(°Cmin-1)

Elapsed time between data (s)

350

10

180

650

15

180

950

10

300

Figure 8: Details of furnace programs used to obtain the temperature profiles shown in section III.

While this was being done samples of YBCO were fabricated. The chemical equation for the fabrication of YBCO is as follows in equation 6 and the amounts of the reactants required to fabricate 0.025 mol are displayed in figure 9

Y2O3+4BaCO3+6CuIIO→2YBa2Cu3O7-δ (6)

Reactant

Mol

RMM (gmol-1)

Mass (g)

Y2O3

0.0125

225.81

2.8226

BaCO3

0.050

197.34

9.8675

CuIIO

0.075

79.54

5.9655

Figure 9: Quantities of reactants required to fabricate 0.025 mol YBCO. Relative molecular masses (RMMs) calculated using relative atomic masses

The procedure for fabrication can be seen in figure 10 and using this technique two batches of YBCO were fabricated, the first yielded just one pellet and the second batch yielded four.

Figure 10: Describes the steps taken during fabrication of superconducting YBCO samples.

In order to obtain a more accurate value of the temperature within the sample space of the cryostat the resistance of a platinum thermometer was measured as a function of temperature. In order to do this a Pt100 platinum thermometer was varnished to one side of a cryostat probe and connected via a four point probe to a power source (as can be seen in figure 11), an ammeter and a voltmeter (Keithley 2000 DMMs). The ammeter and the voltmeter were connected to a computer in order that live data could be fed straight into a LabView program (appendix 2) which would record the data to both a much great accuracy and precision than could be done by a human. Although a stable and constant current was used it was felt, in the interest of good practise, necessary to add the live feed ammeter into the LabView program as tiny fluctuations in current could have potentially changed results which would not have been noticed otherwise.

The probe was then placed in the sample space which was subsequently vacuumed (to a pressure of 8×10-4 Torr) and flushed with helium twice. The sample space was then left full of helium due to its high thermal conductivity. The cryostat was cooled with liquid nitrogen to a temperature of approximately 77K and the LabView program left to record the change in the resistance of the platinum thermometer (using Ohm’s law, V=IR) and it’s corresponding temperature (from the intelligent temperature controller or ITC) while the cryostat heated up naturally. The temperature increase function of the program was not used as leaving the cryostat to heat up as slowly as possible allowed data to be gathered over a much greater period of time which lead to a relationship with less error. This relationship was plotted in order that the temperature dependant resistance profile of the platinum thermometer could be incorporated into the LabView program for use in future experiments to determine more accurately the temperature of the sample space.

While this was being done the dimensions of the cut samples were measured using vernier callipers and weighed in order to determine a density for YBCO. Each dimension was measured six times (to reduce random error) by two different people (to reduce systematic error). The off cuts of each batch of YBCO were then sent off for X-ray diffraction analysis in order to determine the chemical composition of the fabricated samples. The diffraction was carried out using a wavelength of 1.54184Ǻ.

II.II Fabrication and Calibration Results, Analysis and Interpretation

The three temperature profiles of the furnace can be seen below in figure 12. The results are slightly skewed due to one end of the furnace having been left open in order to allow the thermocouples to sit inside the furnace. This can be seen back in figure 7.

The measurements were taken by eye over a 10 second time period. It was therefore decided that the errors on the time should be ±5 seconds and the error on the temperature ±1K, both of which are unfortunately too small to be seen on the profiles. The data points were fitted to cubic curves as this best matched the physical behaviour of the heating.

Figure 12: Temperature profiles of the furnace. The temperature of the program is shown in black crosses and the temperatures of thermocouples 1, 2, 3 and 4 are shown in yellow, red, green and blue respectively.

It can immediately be seen from figure 12 that, during the initial stages of heating, the temperatures of all of the thermocouples lag behind that of the furnace program, specifically those of the thermocouples at the open end of the furnace (1 and 2). This can be accounted for due to poor thermal insulation at the open end of the furnace.

It can also be seen that as the furnace reaches its required temperature and begins its dwell time the temperatures of the thermocouples continue to rise for a short duration before also levelling out. The most likely reason for this is that once the furnace reaches its required temperature the program will instantaneously cut the current to the heating coils. They will still however have thermal energy in them which will leach through the ceramic inner of the furnace into the firing space itself. Another striking feature of the profiles that can be seen is that the longer the furnace has to reach the required temperature, the more linear the increase in temperature is throughout the furnace.

It was therefore deduced that had the furnace been sealed at both ends with radiation rods and covers, then the centre of the furnace would be that which had a temperature profile closest to that of the furnace program. It was also decided that in order to ensure a steady, linear rate of heating, a slower increase in temperature would be used.

The masses of the batches before and after calcinations were compared and were found to have decreased by an average of 2.44(±0.01)% of their initial masses. This was expected as one of the by-products created during the calcination of BaCO3 is CO2 which would have been removed from the furnace during this heating period therefore reducing the mass of the compound.

The weights of the samples from batch two before and after annealing were compared and it was found that each of the samples of YBCO had increased in mass by an average of 3.51(±0.03)% of their initial masses. This was unexpected as during the annealing process the compound is reduced and so should lose mass. One possible explanation for this could be a simultaneous reduction and oxygen doping of the compound in order to try and fill the copper and oxygen chains shown in figure 6.

The densities of both batches of YBCO were calculated by weighing each of the samples from that batch and dividing their masses by their measured volumes. The densities of batches one and two were found to be 5.25(±0.04)gcm-3 and 3.5(±0.1)gcm-3 respectively. The greater error stated with the value of the density of the second batch of YBCO is a result of an error on the mean being taken whereas the error on the density of the first batch is merely propagated from those of its volume and mass as there was only one sample.

When literature values of the density of YBCO were consulted it was found that the compound has a variable density of anywhere from 4.4 to 5.3gcm-3.[xiii] When comparing this range to the experimentally determined values of this parameter it was found that the density of the first batch lay just inside the range whilst the density of the second batch lay well outside of the lower end of it. One possible reason for the very low value of the density of batch two could be that its samples were left in the press for less time than batch one during sintering.

All samples were checked to see whether they exhibited the Meissner effect. All did and a photograph showing this can be seen below in figure 13

The X-ray analysis of the two laboratory fabricated batches of YBCO can be seen in figure 14 below. The intensities were recorded every 0.01 degrees and then scaled appropriately using the greatest intensity in order that they could be compared to each other.

As can be seen in figure 14 when both data sets are overlaid negligible differences can be seen. This indicates that both batches have almost identical chemical compositions and structure. A reasonable amount of background noise can be seen accompanied by an offset from zero intensity which changes in magnitude as the angle of diffraction increases. This can be accounted for by two factors. The first being tiny random impurities in the batches obtained by fabrication outside of a totally clean environment. The second is that small levels of the initial reactants may have not formed the required compound during calcination and annealing.

A standard diffraction pattern of YBCO produced using the same wavelength of radiation was taken from The Chemical Database Service and can be seen below in figure 15. When this is compared to the patterns of the two laboratory fabricated samples in figure 14 all of the same intensity peaks can clearly be identified. This would indicate that YBCO had been successfully fabricated.

Figure 15: X-Ray diffraction pattern of YBCO6. Calculation of the structural parameters of YBa2Cu3O7-δ and YBa2Cu4O8 under pressure, Ludwig H. A. et al., Physica C (1992) 197, 113-122.

It was expected that the comparison of standard diffraction patterns of YBCO of different oxygen contents with those fabricated within the laboratory would allow their oxygen content to be deduced. This, however, could not be achieved as all of the standard patterns of YBCO found in journals and online databases from oxygen contents of 6 to 7 had extremely similar diffraction patterns.

The resistance of the platinum thermometer was plotted against temperature and can be seen in figure 16.

A linear relationship was fitted to the data as seen in figure 16 which produced a reduced chi squared value of 1.317 and equation 7.

T=2.4958(±0.0007)R+25.54(±0.04) (7)

The reduced chi value indicates a strong linear relationship while the equation of the line gives a resistance of 99.2(±0.2)Ω at a temperature of 273.2(±0.1)K. When compared to the technical data for this component which gives a resistance of 100.00Ω[xiv] at a temperature of 273.15K, it shows very close correspondence although not within error. A temperature of one less significant figures accuracy had to be used in this calculation due to the inability of the ITC to measure temperature to any more than one decimal place.

This slight difference between the reference and experimental values of the resistance of the Pt100 at a given temperature can be accounted for by the position of the ITC’s heat sensor. This lies just outside of the sample space and would cause the ITC’s heat sensor to detect a small increase in temperature before it was received by the Pt100 within the sample space. Thus causing the Pt100 to lag behind in temperature (even if only slightly) and would therefore cause the slightly lower resistance for the given temperature as calculated above and can be seen as a very slight systematic error.

III.I Resistivity Methods

One of the cut samples was fixed to the other side of the probe to the Pt100 with thermally insulating varnish and four copper wire contacts were painted onto it with electrically conductive silver paint. The separation of each of the four wires was measured with vernier callipers six times each by two different people for the same reasons as before and recorded for later calculation. A four point probe resistance measurement was used in order to avoid the indirect measuring of resistances other than just the sample resistance. The contact resistance and spreading resistance are also normally measured by a simple two point resistance measurement. The four point probe uses two separate contacts to carry current and two to measure the voltage (in order to set up a uniform current density across the sample) and can be seen in figure 17.

In a four point probe the current carrying probes will still be subject to the extra resistances but this will not be true for the voltage probes which should draw little to no current due to the high impedance of the voltmeter. The potential, V, at a distance ,r, from an electrode carrying a current, I, in a material of resistivity, Ï?, can be expressed by

V=Ï?I2Ï€r=Ï?I2Ï€1S1+1S3-1S1+S2-1S2+S3 (8)[xv]

where r has also been expressed in terms of the contact separations (figure 17). This can be rearranged in order to calculate the value of the resistivity of material being measured.

The probe was once again inserted into the cryostat and the cryostat was cooled as detailed in section II.I. Once the sample had reached a temperature equal to that of the boiling point of liquid nitrogen a LabView program was left to run which recorded the resistance of the sample and its corresponding temperature. The program used to do this can be seen in appendix 2. Although a temperature increase function was built into the program, the cryostat was left to warm up naturally for the same reason used when calibrating the platinum thermometer. The set up for this can be seen below in figure 18.

Figure 18. Schematic for the resistivity experiment. Vacuum pumps and pressure gauges have been omitted as well as the heater on the ITC as none of the bear any real relevance to the experiment. Data cables are shown in red, Pt100 in blue and sample in grey.

This was repeated for each sample of fabricated YBCO at least twice and their temperature dependant resistivity profiles can be seen in section III.II

III.II Resistivity Results, Analysis and Interpretations

The resistance profile of the sample from the first batch was measured twice and these profiles can be seen in figure 19. Unfortunately it was not possible on this occasion to measure the four point probe contact separations on this first sample before it was removed and so these profiles could not be adjusted to those of resistivity using equation 8. However, as this transformation is simply a stretch in the y-axis, it does not change the behaviour of the transition or the value of the transition temperature obtained from the profile.

It can be seen in figure 19 that although the first profile cuts out at approximately a temperature of 190 Kelvin, both profiles follow virtually the same path until that point. The first profile cuts out early due to data points being taken once every second causing the program to fail and shut down. The number of data points was then cut to one every three seconds for subsequent experiments.

With measurements being taken automatically by computer (and with the Keithley multimeter’s ability to measure currents and voltages to 7 significant figures) the errors on the resistance were negligible (±0.003% of the value of the resistance) and so can not be seen in figure 19. The same is true of the errors on the temperature. Assuming that equation 7 is correct then with a ±0.003% error on any calculated resistance, the temperature of the sample space should only have an error of ±0.04K.

Had each of the samples been perfectly pure their profiles would have a very sharp transition between the states and the transition temperature would be very clear. However as a result of the broadening of this transition due to the impurity of the samples a temperature could not be clearly defined. Had powerful enough graphing software been to hand and were the profile able to be fitted to any know curve on this software, the most reliable way to find the transition temperature would have been to plot the first derivative of resistivity with respect to temperature and then determine its maximum (corresponding to the point of inflection within the transition). This not being the case the temperature of the transition was approximated to be the temperature at the half way point in the drop between the two states.

To ascertain at which points on the profile the change in state began and ended, separate lines of linear regression were fitted to the linear data in both the normal state and the superconducting state. These two lines of regression were extended closer and closer to the transition from either side until the adjusted R2 value of the lines of best fit was 0.999, which indicated an excellent linear fit.

It was found upon inspection that the mid-point of the transition could be defined in two different ways; the mid point in resistivity and the mid point in temperature (the mid-point in resistivity obviously corresponding to slightly a different temperature than that found at the mid point of the temperature). This was due to a slight skew in the transition in the profile and so in order to clearly define the superconducting transition temperature a clearer approximation from the one stated before had to be made. It was therefore approximated that the temperature corresponding to the mid point in resistivity should be averaged with the mid point in temperature on the x-axis and the error be the temperature either side of this average value which either previous ‘mid’ value lay. This can be seen more clearly in figure 20.

Figure 20: Shows the method used to calculate the superconducting transition temperature using an expanded view of the first profile in figure 19. Lines of linear regression are shown in black either side of the area in which the transition occurs (in yellow). Both temperatures can be seen highlighted by dashed lines.

By the use of this method it was determined that the transition temperatures for both of the profiles in figure 19 were 87.6(±0.9)K and 86.0(±0.4)K for the first and second profiles respectively. Although these do not agree with each other (within the confines set by the errors) an average was taken and found to be 86.8(±0.8)K. The purity indicator was also calculated for each profil


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