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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 8x10-4 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, ?, can be expressed by

V=?I2πr=?I2π1S1+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 profile and found to be 0.116 and 0.104 respectively. These two values differ by approximately 10% which is reasonable considering that they are from the same sample.

The resistivity profiles of the samples from batch two can be seen below in figure 21. No profiles of sample 1 could be obtained as it broke while being affixed to the probe due to its thinness.

Each of the profiles shown in figure 21 displays linear behaviour in the normal state region as predicted. As stated before, lines of linear regression were fitted to the data in the normal state after the transition and reduced chi-squared tests were carried out all resulting in values between 6x10-5 and 2x10-5. Normally this might signify an over estimation of errors used with data in the trend. However, as these values were calculated without errors to begin with it merely shows that the sheer number of data points dements the result of any reasonable statistical test measuring goodness of fit. Adjusted R2 values were found to 0.999 or better with respect to a linear fit.

All of the transition temperatures from the profiles in figure 21 and their indicators of sample purity were calculated in exactly the same way as before and can be seen in figure 22

Sample

TC

Error

PI

2

88.1

0.3

0.079

88.1

0.4

0.067

89.1

0.6

0.090

3

88.3

0.1

0.070

87

1

0.109

4

87.9

0.4

0.073

88.0

0.2

0.085

88.1

0.4

0.075

85.97

0.04

0.043

It can be seen in figure 21 that of the profiles from sample 2, two follow an almost identical path while in the normal state the third and final profile remains approximately 0.014Ωmm greater than the previous two at all times. This difference is also reflected in figure 22, where the transition temperature of the final profile can be seen to be 1K greater than the other two. This could be attributed to flaking contacts on the four point probe. Due to thermal shock over time in the form of the cryostat heating and cooling, the silver paint which held the four point probe contacts in place on the sample would sometimes flake slightly, thus increasing the perceived resistance and hence the resistivity of the sample. This theory is further confirmed by what look like rogue data points on the normal state side of the transition giving the graph a set of what look like spikes away from a linear fit.

The first profile from sample 3 cut out early due to a technical fault. It can also be seen that the differences between the first and second profiles are similar to those in sample 2. This is also likely to have been caused by loose connections or flaking lending a greater resistance to the second profile. This is confirmed again by a spiking character towards the higher temperature end of the normal state trend.

The profiles of sample 4 group well both in the linear, normal state region and, as shown in figure 22, in their calculated value of the transition temperature. There is, however, a large discrepancy between the resistivity of sample 4 and that of samples 2 and 3; almost double at all points in the normal region. One possible explanation for this could be the known difference in the values of resistivity along the a or b and the c axis of the unit cell of YBCO. This leads to the idea that one sample may have unit cells aligned in a different orientation to the other two when current is passed through the sample. Literature values of this ratio range from 30 to 150[xvi] rendering this hypothesis highly unlikely.

A large source of error was that of the separations of the contacts of the four point probe through to the calculation of the resistivity. This enormous error on the value of r (in equation 8) gives each value of the resistivity an error of 20% of its own value (these have not been included in figure 21 as due to the large number of data points all of the errors from each profile blend into one another making it extremely difficult to determine which set of errors belong to which data set). If this error was not included then the error on the resistivity, just as that on the resistance, would be negligibly small. 20% seems far too large and could be a result of various factors. The first of these is the large spread and small number of measurements taken of the separations which itself results from two factors. The first being that the separations were measured by eye using vernier callipers; had a travelling microscope been used then much more accurate measurements could have been taken. The second is that as the contact wires themselves were not straight, a 'linear' distance between them was difficult to establish (photo in figure 23). It could also have been that the separation of the wires was measured from the wrong points on the sample. This is explained using figure 23.

Figure 23: A photo of a sample and four point probe with a simplified cross section of it to demonstrate current movement through the sample both in the superconducting state (green) and the normal state (red).

The separations of the contacts of the four point probe were measured from the centres of the wires. It can be seen in figure 23 that this is a reasonable approximation while the sample is in the superconducting state; the current will work to minimise the distance it travels through the conductive silver paint due to its higher conductivity. However, in the normal state this is reversed and the current will work to minimise the distance travelled through the sample. Therefore it may have been a better idea to measure the contact separations from the edge of the silver paint. This effect could be negated if thin film materials were used. However, these have different physical properties to bulk materials and so would change the purpose of the investigation altogether.

As can be seen in figure 22, all of the samples from batch two had similar levels of purity (at least all of the same order of magnitude).

The transition temperatures for batch two were averaged and found to be 87.8(±0.4)K. This agreed with the transition temperature calculated for batch one which is logical as the XRD data indicates that they are extremely similar compounds. Although this might be the case it would make no sense to average the transition temperatures of both batches even by weight due to the current inability to ensure that both batches contain the same oxygen content (the overriding factor determining transition temperature).

In an attempt to determine the oxygen content of the two batches of YBCO a phase diagram such as the one shown in figure 24 was used. In this way the oxygen contents of batches one and two were determined to be 6.82(±0.01) and 6.83(±0.01) respectively. This again supports the XRD data plotted in figure 14 in confirming the similarity in compound and structure.

When trying to compare the two transition temperatures calculated (86.8(±0.8)K and 87.8(±0.4)K) to literature data two major problems are encountered. The first is that this report has determined the oxygen content of the samples using their calculated transition temperature. Therefore it seems somewhat counterintuitive to find a reference transition temperature based upon an oxygen content which has been determined from the transition temperature itself. It is, however, widely established that the upper bound of the superconducting transition temperature of YBCO is 95K which the data determined here fits.

IV.I Specific Heat Capacity Methods

In order to obtain a value of the superconducting transition temperature independent of the electrical properties of the samples, two similar methods of measuring how its specific heat capacity varied as a function of temperature were employed.

For both methods a sample of YBCO was suspended at the base of a cryostat probe using thermally and conductively insulating dental floss. Two opposing sides of the sample were then coated in a thermally conductive heat sink paste. On one of these sides a strain gauge was placed and on the other a platinum thermometer. These were secured using a layer of thermally insulating varnish. Both gauge and thermometer were attached to an ammeter, voltmeter (Keithley 2000 DMMs) and current source using four point resistance set ups as can be seen below in figure 25.

Figure 25: Cross sectional view of a sample prepared for specific heat capacity measurements. The sample is shown in dark grey, the heat sink paste in light grey, varnish in brown, platinum thermometer in blue and the strain gauge in red and yellow.

The first method involved placing this probe in an airtight sample space within a vat of liquid nitrogen. The space was filled with helium until the sample had reached the same temperature as the liquid nitrogen and then vacuumed to a pressure of 2.20x10-6 Torr . A constant current was then applied to the strain gauge causing it to heat. All of the variables were recorded using a LabView program in appendix 2. Two assumptions were made during this experiment. The first was that all of the electrical power going into the strain gauge was transferred into thermal energy, and the second was that all of this thermal energy was then conducted through the sample and detected by the platinum thermometer (equation 9 where Q is the energy supplied, tf is the final time, m is the mass of the sample, ΔT is the change in temperature and the rest of the variables are as before in this report).

Q=0tfIVdt=mCVΔT (9)[xvii]

The second method took the sample set up as shown in figure 25 and placed it in a cryostat in which the temperature could be controlled. Random bursts of power were applied to the strain gauge (as opposed to a continuous flow) at a range of different temperatures and the corresponding rise in temperature was recorded with the same program as was used for the first method. The LabView program used to record data for this experiment for this experiment can also be found in appendix 2.

IV.II Specific Heat Capacity Results, Analysis and Interpretation

The results of the first of the two experiments to measure the heat capacity of the sample from the first batch can be seen below in figure 26. An initial static temperature of approximately 74K was recorded which at first appeared to be an equipment failure as liquid nitrogen boils at approximately 77K (at atmospheric pressure). This could have been explained if the liquid nitrogen had been kept under pressure within the vat. However, considering that the sample space was able to be inserted into the top of the vat freely this was not the case; the liquid nitrogen must have been at atmospheric pressure. The platinum thermometer was even retested within a standard cryostat and agreed with equation 8.

Due to only the cumulative energy and temperature being recorded a small but not unreasonable assumption had to be made when calculating the specific heat capacity. It was assumed that at any given point the total energy supplied up until that point would result in the temperature measured whether the sample had been cooled between measurements or heated continuously.

The initial drop in heat capacity can be attributed to the time taken for the initial thermal energy supplied by the strain gauge to travel through the bulk of the sample to the platinum thermometer. This trend can be seen to continue until approximately 76K.

At a temperature of just over 86K the specific heat capacity of the sample seemed to gain an almost exponential character. It has been suggested that this was due to ineffective thermal insulation of the sample in the sample space, and instead of the thermal energy being transferred through the sample it was transferred to the liquid nitrogen through convection/radiation. Convection to the sample space wall seems unlikely considering the very low pressure within the sample space and radiation would not be an efficient enough transfer process to account for this large and constant drain of thermal energy from the experimental system.

Enlarging the area of the graph around the previously found transition temperature a very small discontinuity can clearly be seen just before the large amount of noise in the upper, normal state, heating regions. This can be seen in figure 27.

When compared to figure 5 in section I it can be seen that this discontinuity, although not as sharp, matches the shape expected in theory; a sharp jump followed by a change in the character of the heat capacity. Even with errors set by the trend described by equation 7 (±0.04K), this discontinuity still distorts the general trend to an extent that could not be explained by a singular anomaly.

The peak of this sharp discontinuity in heat capacity was found to occur at a temperature of 86.1(±0.1)K, which agrees with the transition temperature for the first batch found in the resistivity experiment of 86.8(±0.8)K, albeit at the extremes of the errors.

Due to time constraints only two further experiments were run in an attempt to try and replicate these initial results but both failed to do so. The first was repeated with a current within error of that used in the first experiment and yielded the results seen in figure 28.

Although it can be seen that this experiment, like the first, gives rise to an almost linear rise in temperature at around 86K it peaks just before whereas the first experiment peaked just after 86K. Also, a discontinuity can be seen within this curve, similar to the first experiment. The main differences start to occur when the discontinuity is seen in more detail. Instead of a continuing change in specific heat capacity such as that seen in figure 27, this discontinuity has a shape that is more reminiscent of a few rogue data points, possibly due to a loose wire in the platinum thermometer electronics. This can be seen more clearly in figure 29 below.

The second of these two repeat experiments can be seen below in figure 30. Conducted with a slightly higher current, this trend continues past 86K and starts to rise just after 88K. There is no discontinuity in this trend, for several possible reasons. The first is that its specific heat capacity never reaches 17.30(±0.03)Jg-1K-1 (the value of the specific heat capacity at the discontinuity found in figure 27). This is primarily a consequence of the experiment cutting out early due to a technical fault. The second is possibly that the discontinuity found in figure 27 is an anomaly and should not normally occur.

The minimum points on each of figures 24, 26 and 28 were determined and found to be 2.31(±0.03)Jg-1K-1, 2.13(±0.03)Jg-1K-1 and 2.82(±0.03)Jg-1K-1 respectively. Although not agreeing within error, these values do lie in the same order of magnitude and so could be different due to a slight change in environmental conditions (e.g. concentration of helium/air inside the sample space).

The only reference data found on this topic combined the normal and superconducting heat capacities using equation 4. This assumes that the phonon contribution to the normal state heat capacity is equal to zero, which is clearly not the case in the high temperature superconductors and so cannot be compared here.

Only one set of data was able to be taken for the second heat capacity experiment and the results of this can be seen below in figure 31. Although the initial heating spike seen in each of the previous heat capacity experiments results can be seen it lacks any other similar characteristics. The trend moves straight past the region in which the transition temperature was thought to lie with no visible effect on the results. This could be due to several reasons. The first could be that not enough data points have been collected and any discontinuity was simply missed. The second and much more likely reason could be that in an attempt to hold the cryostat at a constant higher temperature all of the liquid nitrogen boiled off and thermal energy from the samples surrounding caused it to heat instead of the energy supplied from the strain gauge. This was a problem with the design of the experiment

It is for the very reason stated above that the minimum specific heat capacity achieved in the second heat capacity experiment could not be measured. External heating simply made the results obtained from this method far too unreliable to use in any reliable data analysis.

V Conclusions

The calibration of the furnace gave results instrumental in the successful fabrication of YBCO and the data obtained from the calibration of the Pt100 allowed the temperature of the sample space to be measured to a much greater accuracy during experiments. The relationship of the Pt100 platinum thermometers resistance and its temperature was also shown to differ from reference data very little.

It can be concluded that YBCO was successfully fabricated within the laboratory. This was confirmed by X-ray diffraction data. If the fabrication were to be redone it would be useful to identify a way in which the actual amount of oxygen absorbed into the compound could be measured. It is by this method that the actual compound fabricated could be identified and compared to reference data as opposed to having to use experimentally determined transition temperatures to 'work backwards' using phase diagrams. It would also be beneficial to create several more samples of varying oxygen content in order to gather a wider range of profiles and transition temperatures. It would also be extremely beneficial to fabricate the compound in a much cleaner environment e.g. a clean room instead of a fume cupboard. This would not only give a much sharper transition due to a greater level of purity but would also remove much of the background noise in the XRD data making it much easier to compare to reference data.

It can also be concluded that as a result of the successful resistivity profiling of the samples, the transition temperatures of each of the batches were found to be 86.8(±0.8)K and 87.8(±0.4)K. These results were then used to determine the oxygen contents of the fabricated samples and found to be 6.82(±0.01) and 6.83(±0.01). The profiles also all showed excellent linearity in the normal states regions as predicted by theory. If this experiment was to be repeated a much more accurate way of measuring the four point probe contact separation would be determined to reduce the overly large error propagated through into the values of the resistivity.

The experiments concerning the specific heat capacity of the compound gave mixed results in terms of usable data. The primary data gathered in the first experiment seemed to confirm the transition temperature found in the resistivity experiment with a discontinuity at 86.1(±0.1)K but these results could then not be reproduced calling into question the accuracy of the experiment. However, measurements of the specific heat capacities at minimum points seemed to agree within magnitude although not within the small errors set. The second experiment failed completely to produce any sort of usable data due to poor experimental design. It was also found that the specific heat capacity of the normal state should have been recorded in order to produce any results comparable to reference literature.

With all of this taken into consideration it can still be seen that the main difficulty with these experiments was that the oxygen content of the samples was unknown. This made it almost impossible to compare the transition temperatures found to any sort of reference data.

Acknowledgements

Prof. Damian Hampshire

Mr. Mark Raine

Mr. Gary Oswald

Miss. L Falk

Mrs. S Jowitt

19 January 2010Page 14 of 14Josephine Butler College

References

[i] 'Theory of Superconductivity', J.R.Schrieffer, Perseus Books (124), page 1

[ii] 'Introduction to Solid State Physics 8th Edition', C. Kittel, John Wiley & Sons, Inc (2005), page 259

[iii] http://nobelprize.org/nobel_prizes/physics/laureates/1913

[iv] 'Possible high TC superconductivity in the Ba-La-Cu-O system', J.G. Bednorz and K.A. Müller (1986)

[v] http://nobelprize.org/nobel_prizes/physics/laureates/1987

[vi] 'Superconductivity', C. P. Poole, Jr., H. A. Farach & R. J. Creswick, Academic Press (1995), page 40

[vii] 'Superconductivity', E.A. Lynton, Methuen & Co Ltd (1962), page 75

[viii] 'Superconductivity: Volume I', R.D. Parks, Marcel Dekker Inc (1969), page 76

[ix] 'Superconductivity: Volume I', R.D. Parks, Marcel Dekker Inc (1969), page 6

[x] 'Superconductivity', C. P. Poole, Jr., H. A. Farach & R. J. Creswick, Academic Press (1995), page 96

[xi] 'Superconductivity', C. P. Poole, Jr., H. A. Farach & R. J. Creswick, Academic Press (1995), page 36

[xii] 'High-Temperature Superconductivity in Curpates', A. Mourachkine, Kluwer Academic Publishers (2002), page 40

[xiii] R. Swarup, A. K. Gupta and M. C. Bansal (1995). "Effect of sample density on magnetic penetration depth in YBaCuO ceramic superconductors". Journal of Superconductivity 8 (3): 361-364

[xiv] http://docs-europe.electrocomponents.com/webdocs/0c41/0900766b80c41b6b.pdf

[xv] 'Semiconductor material and Device Characterization', Schroder D.K., John Wiley & Sons Inc (1990), page 4

[xvi] 'Superconductivity', C. P. Poole, Jr., H. A. Farach & R. J. Creswick, Academic Press (1995), page 28-29

[xvii] 'Physics for Scientists and Engineers, Sixth Edition', Tipler P.A. & G. Mosca, W.H. Freeman and Company (2008), page 687 and 875

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