Module 9 : Experimental probes of Superconductivity Lecture 1 : Experimental probes of Superconductivity - I Among the various experimental methods used to probe the properties of superconductors, there are methods which sample the bulk properties such as x-ray diffraction, magnetisation, heat capacity, electrical conductivity, thermal conductivity, etc. These give information about the bulk of the sample in the sense that it is often not possible to unambiguously determine the separate contributions making up the measured quantity. For example, the magnetisation will have both spin and orbital contributions. The orbital part will comprise of a Van Vleck term as also a core diamagnetism term. There will also be a Landau diamagnetism term in case of metallic systems. The spin part, likewise, will have a Pauli paramagnetic contribution (for metallic systems) and a Curie-like contribution (for localised spins). In case of impurities, there will be such contributions from the impurity phases as well. Local probes such as nuclear magnetic resonance, muon spin rotation, neutron diffraction and scattering often provide much more detailed information pertaining to the normal state as also the superconducting state in terms of the symmetry of the order parameter, vortex structure, fluctuations, etc. Electrical resitivity The measurement of electrical resistivity as a function of temprature is historically the first method used for detection of superconductivity. Typically, four contacts are made on a bar-shaped sample and leads connected to them. A constant current source is connected to two of the leads to pass a current. The voltage thus generated is measured with a sensitive voltmeter. The sample with these lead connections is placed in a cryostat equipped with a temperature sensor. An arrangement of cryogenic liquids/gases along with a temperature controller enable resistivity measurements as a function of temperature. The is determined as the temperature below which no perceptible voltage is developed in the sample inspite of a current flowing through it. In addition, these measurements can be done in the presence of an applied magnetic field. This enables one to determine the decrease of the due to a magnetic field and hence infer the critical field. A schematic of the arrangement is shown in the figure. Schematic for resistivity measurement These measurements can be done in the presence of an applied magnetic field. The resulting data enable to track the decrease of with magnetic field and hence determine or infer the critical field. Typical variation of data are shown in the accompanying figure.
Schematic variation of the resitivity with temperature for a superconductor The advantage of this technique is that is provides a very quick method to check for superconductivity. The drawback however is that it can give ``false positive'' results in the sense that even in samples where only a small fraction is superconducting, a zero resistivity might be observed. Magnetic suseptibility Among the methods to measure the magnetisation or susceptibility, extraction method and the vibrating sample method are most popular. In the extraction method, a sample placed in a static magnetic field (generated by a solenoid) is extracted through secondary pick-up coils. The changing flux induces an emf in the pick-up coils which is sensed with a SQUID. Schematic of coil arrangement for extraction method. In contrast, in a vibrating sample magnetometer, the sample is vibrated in a static field. This gives rise to a changing flux in a pair of pick-up coils which after suitable calibration gives the magnetic moment of the sample.
Schematic for VSM A yet another method measures the susceptibility of a sample. The scheme involves a primary coil to which an current is fed. There is a pair of identical secondary coils wound on top of the primary coil and co-axial with it. The secondary coils are wound counter to each other. The sample located on a sample holder at the end of a sample rod is placed in one of the secondary coils. Ideally, the emf induced in the secondary coils (which are counterwound and in series) is zero in the absence of a sample. When a sample is mounted on the sample rod, an emf proportional to the susceptibility of the sample is induced. Schematic of ac susceptibility measurement
Once again, this arrangement is in a cryostat to enable controlled temperature variation. The above figure is a schematic for variable temperature measurements. The sample is mounted on a sample holder and inserted in a cryostat shown on the left. The cryostat is equipped with a heat exchanger and allows for flow of cold gas from a dewar (see on the right side) via a transfer tube to the sample space. The gas flow and the heater power can be controlled enabling one to stabilize the temperature and then carry out the necessary measurements. For the detection of superconductivity, one needs to check for shielding (diamagnetism) and the Meissner effect. For the former, the sample is cooled in zero applied field to the lowest temperature. At this point a small magnetic field (~ 50 Oe) is applied and the sample magnetic moment is measured in this field as a function of temperature upon warming. In a superconductor, in a small magnetic field one expects a significant diamagnetism below and the magnetisation sharply goes to (nearly) zero at and stays zero at higher temperatures. In the second type of measurement, the sample is cooled from room temperature to the lowest temperature when a field is present (~ 50 Oe). After cooling the sample in a field, sample magnetic moment is measured on warming. Here again, one expects a diamagnetic response below while the magnetic moment is negligible above with a sharp change at. This demonstrates the Meissner effect. Typical variation of susceptibility for a superconductor is given in the figure. The figure illustrates the response of a Type II Superconductor to a small magnetic field. When the sample is cooled in zero field & then the susceptibility measured on warning the lower curve (ZFC) results. This is called shielding. On the other hand when the sample is cooled in a field (FC), the upper curve results. This is called Meissner effect. Trapped flux can give rise to incomplete shielding. Nowadays, many laboratories are housed with commercial equipment with different attachments which enable automated measurements of magnetisation, heat capacity, electrical and thermal transport, etc. in a wide range of temperatures and magnetic fields.
Once again, the advantage of this technique is that is provides a very quick method to check for superconductivity. The drawback as before is that it can give ``false positive'' results in the sense that even in samples where only a small fraction is superconducting, a diamagnetic susceptibility might be observed. Other than the two properties (electrical resistivity and magnetic susceptibility) mentioned above there are many other probes which provide useful information on the superconductive order parameter. Bulk methods are usually used as an integral probe of the low-energy excitation spectrum at low-temperature. Some of these are discussed below. Heat capacity The heat capacity of a substance probes the entire low-energy excitations of a solid. In particular, if the phonon contribution can be extracted, one can infer the electronic part of the heat capacity which is sensitive to the density of states at the Fermi level. In a conventional s-wave superconductor, an isotropic excitation gap opens up at the Fermi surface below. In the limit of, the heat capacity (as also thermal conductivity) vanishes exponentially and it is possible to determine the superconducting gap from a fit of the heat capacity data as shown in the figure. Typical variation of heat capacity with temperature for a superconductor is shown. There is a jump in the heat capacity at Tc. The Tc decreases as the applied field is increased. However, in case the symmetry of the order parameter is different from s-wave (such as one with line or point nodes), the existence of quasiparticle excitations in the neighbourhood of these nodes gives rise to a non-exponential behaviour of the temperature dependence of the heat capacity. In particular, a power-law dependence ( ) is expected at low temperatures. The value of the exponent provides a hint for the nodal structure of the order parameter. A behaviour is suggestive of line nodes while a behaviour is expected for point nodes of the gap function. Another test for the order-parameter symmetry arises from the magnetic field dependence of the thermal conductivity and the specific heat. In case of superconductors with nodes, the density of states in a magnetic field is dominated by contributions from extended quasiparticle states rather than the bound states associated with vortex cores. This leads to a dependence of the specific heat in case of line nodes. The heat capacity jump is an important marker of superconductivity and the magnitude of the jump might also give an idea about the fraction of the sample which is superconducting in case there is a doubt as to the origin of zero resistivity/diamagnetism from some impurity contribution. Magnetic penetration depth Temperature dependence of the magnetic penetration depth can yield information about the order-parameter symmetry. There are two techniques which enable such measurements namely surface impedance in a microwave resonator and muon spin rotation measurements ( SR) in a transverse field. Microwave measurements are made in cavity resonators applying resonance modes with frequencies in the GHz range. Such measurements probe the complex conductivity as a function of temperature and frequency. From such data, the superfluid density
as well as the properties of thermally excited quasiparticles can be deduced. The surface resistance and changes in surface reactance of the sample mounted inside the resonator are obtained from the full-width at half-maximum and the changes in of the resonance curves. In the local limit of the two-fluid model, the surface impedance and the complex conductivity are given by (1) where the real part is the surface resistance and the imaginary part is the surface reactance. The two-fluid model gives a conductivity (2) where is the superfluid density, and is the normal fluid density. In the superconducting state, is the scattering lifetime of the thermally excited quasiparticles and is their effective mass. Both constributions to the surface impedance and complex conductivity reflect that in the two-fluid model most of the current at frequencies in the microwave range will be carried as a lossless supercurrent, but there will be dissipation from the normal component for any non-zero frequency. The supercurrent density can be determined from and this will lead to the determination of the penetration depth and its temperature dependence via. Another method for infering the magnetic penetration depth is via transverse-field SR studies. Due to the presence of a flux line lattice in Type-II superconductors, an additional line broadening of the SR line is observed. If this is assumed to be a Gaussian, then the muon depolarisation rate is proportional to which is in turn proportional to. Therefore, SR measurements can give information about the value of the penetration depth, its anisotropy and temperature dependence as shown in the figure. A schematic plot of the penetration depth data obtained in high-tc superconductors via musr measurements (with an applied magnetic field in the ab plane) is shown. The data point to a non-s wave pairing symmetry associated with the order parameter. In a conventional BCS superconductor, the curve is expected to be flat at low temperature below about 0.5Tc while in the present case, it continues to rise linear as one goes to lower temperatures. In the empirical approximation of the two-fluid model, one has (3) However, the actual dependence is different in the pure local limit ( ) as compared to the pure
anomalous limit ( ) or the dirty local limit ( ). In the weak coupling BCS superconductor in the clean local limit, the dependence is more like (4) with On the other hand, for line nodes in the gap, one gets Presence of impurities further changes the temperature dependence of the penetration depth. Consequently, one has to be cautious about making strong conclusions based on the temperature dependence of the penetration depth. Nuclear Magnetic Resonance Protons and neutrons possess a magnetic moment. Nuclei contain a collection of protons and neutrons and may possess a collective spin moment. In the absence of a magnetic field, there is a -fold spin degeneracy associated with nuclear energy level (see figure). This degeneracy is lifted in the presence of a static magnetic field and this is called the nuclear Zeeman effect. In thermal equilibrium, the populations of the various nuclear levels are different. The ratio of population between adjacent levels separated by an energy gap is. Transitions between the nuclear Zeeman levels can be induced by an external radio frequency field. This phenomenon is called nuclear magnetic resonance. The block diagram illustrates the NMR instrumentation. The rf synthesizer generates continuous rf output at the
designated frequency. This output is gated by a pulse programmer and amplified by the transmitter. This results in programmed pulses of an rf magnetic field which are transmitted to a coil around the sample. The axis of the coil is perpendicular to the dc magnetic field. Following the rf pulse, the nuclear magnetization undergoes precession inside the coil thereby giving rise to a time varying flux. The induced emf is then detected and amplified. The associated NMR signal can then be displayed on a computer for further analysis. For variable temperature measurements, the NMR probehead is placed inside a cryostat and a temperature controller is used to regulate the temperature. There is a hyperfine interaction between the nucleus and the electrons which effectively gives rise to an additional magnetic field (in addition to the applied field) at the nuclear site. This gives rise to a shift in the NMR resonance frequency. The fractional shift is called the Knight shift in case of metallic systems. is proportional to the magnetic spin susceptibility. Especially in situations where the spin susceptibility is temperature dependent, it is possible to determine the separate contributions to the susceptibility (spin and orbital) from a combined analysis of the NMR shift data and the bulk susceptibility data. NMR is, therefore, a sensitive probe of the static susceptibility of a compound. Information about the dynamics is also available via a measurement of the spin-lattice relaxation rate which measures the efficacy with which the nuclei which have been excited by the radio frequency field return back to the ground state after the field is switched off. Since the nuclei are ``connected'' to the electrons via the hyperfine interaction, fluctuations in the electron system give rise to de-excitation of the nuclear spins. Metallic systems are expected to follow the Korringa law which says that is temperature independent. In the case of superconductors, since a gap develops in the electron density of states below, the nuclear spin-lattice relaxation rate decreases exponentially. The superconductive gap can be determined from an analysis of the temperature dependence of of a suitable nucleus in the compound under investigation. Also, a peak in the NMR relaxation rate below (the Hebel-Slichter peak) is predicted by BCS theory. The figure shows the normalized, nuclear spin-lattice relaxation rate for a superconductor as a function of the normalized temperature. There is an increase in the rate below the superconducting transition temperature (coherence peak) which was predicted by the BCS theory.
Here T 1 is plotted for a typical superconductor as a function of inverse temperature. The region below Tc is to the right of T c /T = 1 on the x axis. The dip in T 1 will show up as a peak in 1/T 1. This is the Hebel-Slichter peak. Once again, the symmetry of the gap function and the presence or absence of nodes in it significantly affects the temperature dependence of. The temperature dependence of is often compared to theoretical predictions to throw light on the nature of the order parameter and deviations from conventional behaviour. Further, since there is flux expulsion in the superconductive state, the Knight shift should go to zero for temperatures well below. The figure shows the typical variation of Knight Shift with temperature for a conventional superconductor below its superconducting transition temperature. While it is expected that the shift go to zero at zero temperature (due to the energy gap following a spin singlet formation) as per the dashed curve, in reality it remains at a non-zero value due to the orbital contribution to the shift which is unaffected by the superconducting transition. The presence of impurities
can also affect the low-temperature behaviour. In the vortex state of Type-II superconductors, the NMR line has a characteristic shape related to the symmetry of the vortex lattice.