Symmetry Breaking in Superconducting Phase Transitions

Transcription

1 Symmetry Breaking in Superconducting Phase Transitions Ewan Marshall H.H. Wills Physics Laboratory November 26, Introduction Since the beginning of the universe matter has had to undergo phase changes to transform, from a quark gluon plasma as seen in the big bang into the complex solids and fluids that make up the world around us. As we travel down the energy scale from the big bang new phenomena appear at lower and lower energies. The emergence of these phenomena is triggered by a breaking of some form of symmetry. The emergence of the new phenomena at decreasing energies is directly related to a phase transition that will be due to a loss of symmetry in the system. Symmetry is the property of a system that allows a transformation to occur and the system to appear unchanged after completing the transformation. This could be a familiar form of symmetry such as rotational symmetry or mirror symmetry or something more complicated such as local gauge symmetry. The breaking of symmetry can be best explained using the example of rotational symmetry as it is extremely visual. Take a circle, if we rotate this shape by any amount it will always look the same therefore it has infinite rotational symmetry. Compare this to a square, here the shape will only look identical after a rotation of π/2, π or 2π thus there is less rotational symmetry in this case. Symmetry breaking is the loss of a certain type of symmetry after a system undergoes a change taking it past some critical temperature. In the example of a liquid to solid transition both rotational and translational symmetry are broken. In discussing this transition the particles making up the substance have to be considered. A liquid is seen as an average of its particles movements not as a random snapshot of their positions, as the particles in a liquid are randomly moving at all times this results in Figure 1: Schematic showing emergence of Phenomena down the energy scale. Image taken from [1] both rotational and translational symmetry. After either of these transitions the average movement of the particles will not have changed. On passing through the critical temperature the particles align and form a solid in this case the substance is a lattice and so a translation or rotation of the lattice will not always be symmetric thus the symmetry is broken and the new phenomena of a crystalline solid is created. New phenomena are not only found via a breaking of symmetry. As is seen in the case of the quark gluon plasma no symmetry is broken as it becomes a sea of particles but the quarks order, meaning that there is a loss of states available to them. These particles at even lower energies will bind into Hydrogen atoms for this phase change again there is no breaking of symmetry but there is an ordering. Each instance of ordering can be assigned an order parameter this defines the amount of order present in the system thus as energy low- 1

2 Figure 2: Graph showing the action of magnetisation with temperature notice the Curie temperature (Tc) where the transition from paramagnet to ferromagnet occurs ( 290 C). As temperature drops the magnetisation grows as a greater proportion of the spins align. Image taken from [3] ers matter becomes more ordered and symmetry is sometimes broken this is illustrated in figure 1. 2 Symmetry Breaking Down the Temperature Scale 2.1 Magnetism As shown in the above figure (1) there are three main cases of symmetry breaking in condensed matter physics as discussed before there is the formation of a crystal structure from a liquid or gas. At a much lower temperature than the appearance of solids the highest temperature magnets appear (AlNiCo is magnetic at 1130 K. This is a type of magnet with one of the the highest known curie temperatures [2]) A ferromagnet has a critical temperature at which it will become a paramagnet due to the spins becoming randomly orientated so there is no net magnetic moment. This is due to the domination of the entropy in the free energy at high temperatures. This can be seen in the free energy equation ( T = H T S). At low temperatures the entropy term is not as important as the enthalpy and so the spins start to orientate with their neighbours to reduce the internal energy, this occurs when spins are pointing in the same direction, thus giving a magnetic field. This transition into magnetism occurs at T c and as the temperature lowers more spins orientate to point the same way and a net magnetisation is reached [4] through this transition (shown in figure 2) the rotational symmetry of the randomly orientated magnetic moments is lost and an arbitrary direction of the total magnetic moment is chosen. As this transition is 2nd order (continuous) we can model this phase change in more detail i.e. around the point T c by expanding the free energy as a power series and specifying the magnetisation as the order parameter this is shown in equation 1 G(P, T, E) = EP + g 0 + g 2 2 P 2 + g 4 4 P G(M, T ) = g 0 + g 2 2 M 2 + g 4 4 M 4 + g 6 6 M G(M, T ) = g 0 + g 2 2 M 2 + g 4 4 M 4 (1) Here G n is dependant upon temperature and P is the order parameter we specify M (magnetisation) as the order parameter for this particular transition and assume there is no applied field, as shown in the second line. In the phase change we realise that g 2 must change sign in order to distort the graph (Figure 3) into having a net magnetisation i.e. g 2 = γ(t T c ) where γ is a positive constant. As we know the magnetic phase change is continuous and if we take g 4 as positive we can see that the G 6 term can be neglected as it will not add anything new to the equation [5]. We now end up with the last line of equation 1 that describes the action of magnetisation around T c. Landau theory can be used to describe any phase change of any order by using slightly different assumptions in simplifying the power law expansion. 2.2 Superconductivity As temperature drops still we enter the realm of high T c superconductors (currently the record at ambient pressure stands at 135 K [6]) These are complex iron or copper based compounds doped with various other elements that below T c show a remarkable transition into a state with interesting properties. Simple superconductors are materials with zero electrical resistance and a property 2

3 Figure 3: Graph showing the action of changing the sign of g 2 in the landau theory power law expansion of free energy here it can be seen that by taking g 2 as negative (red line) a new magnetisation moment appears known as the Miessner effect (the expulsion of an applied weak magnetic field from inside the superconductor resulting in perfect diamagnetism). This is caused by screening currents flowing around the edges of the superconductor which exactly cancel the applied magnetic field by creating one that is equal and opposite. A large magnetic field will destroy superconductivity, This value of magnetic flux density is known as H c (which is a function of temperature) at this point the action of the Miessner field breaks down and the superconductor becomes either a conductor (type I superconductor) or in the case of a type II superconductor at a H- field value H c1 perfect diamagnetism is lost and it enters a vortex state where the magnetic flux starts to penetrate the sample. Here small cylindrical pieces of material in the normal state are observed with superconducting currents orbiting them. This normal material allows the magnetic field to pass through. In this state it still possesses superconducting properties but diamagnetism breaks down until a H-field value of H c2. Where it finally becomes a conductor this is often a very much higher value than the H c value predicted and it is this form of superconductor that allows much higher values of T c to be reached. Superconductors of both types can be described by B.C.S. Theory, this links the action of lattice phonons and electrons occurring when the difference between energy states is less than the energy of the phonons hω [7]. This coupling allows the formation of cooper pairs. This phenomenon is due to the electron scattering from a state with crystal momentum k to a state k + q [8]The extra momentum has been transferred by the phonon and so plainly a phonon of momentum q has either been created or annihilated via the conservation of crystal momentum. The momentum transferred here has either come from another electron creating a phonon or, if emitted, will go on to interact with another electron [9]. This interaction of electrons via a transferred phonon is a cooper pair. This is the pairing of excited electrons that causes the total spin to change from 1 2 to an integer and thus change the pair from two fermions into a boson (and therefore no longer obey the Pauli exclusion principle). This incredibly important result allows all the conducting particles in a conductor to break global gauge symmetry and condense into a single ground state. This accounts for the unusual properties of superconductivity. This ground state can be found using the variational method so that a cooper pair consists of an electron with a set spin direction and momentum, its pair is given opposite spin and momentum this is a conventional pair resulting in an s-wave type wave function this is found in the majority of superconductors (other pairings are possible with different total spins and quantum numbers and these result in different wave types but are rarer). The wave function for this ground state can be manipulated [10] to find the superconducting energy gap k and thus give an equation (2) describing the energy of the bound quasi-particle. This is made up of the electron energy (ɛ k ) minus the chemical potential (or fermi energy) µ squared and the superconducting gap squared. E k = (ɛ u µ) k (2) This superconducting gap is created because of the slightly lower energy of a bound pair of electrons in comparison to the free electron and it is this that causes the phenomena of zero resistance when the thermal energy is less than the band gap. Electrons above ɛ f act normally and separated from them by the energy gap are the superconducting cooper pairs below ɛ f. This gap is always centred upon the fermi energy and so unlike a semiconductor does not inhibit conduction. The electrons still in the normal state above ɛ f still have resistance but are short circuited by the cooper pairs and so zero resistance is still measured. Superconductors after passing through T c break a global gauge symmetry which is the action of con- 3

4 densing the many wave functions of electrons in a material into one single macroscopic wavefunction. This macroscopic manifestation of quantum mechanics is due to the constant order parameter. By choosing a gauge for the magnetic vector potential (even A = 0) a free energy cost is associated with any change in the phase therefore to minimise energy the variation of phase over the system must also be minimised. This gives a long range order of the phase and so a breaking of global gauge symmetry and a physical quantum observable. Superconductors will be focused upon in more detail later. 2.3 Superfluidity There are many similarities between superconductivity and superfluidity with a superfluid demonstrating zero viscosity (comparable to zero resistance in superconductors) and the ability to flow from a higher potential to another lower level by climbing the walls of its container in a film coating all the surfaces around it. The transition into superfluidity at tempratures of the order of 1 K Helium 3 and helium 4 both display superfluidity but have very different microscopic action at the phase transition [11] just as is seen in type I and type II superconductors for brevity s sake here only 4 He will be discussed as it is the simpler case. Unlike 4 He, 3 He is a fermion (due to its odd number of spin 1 2 particles) and so immediate links can be drawn with the superconducting state explained above as to enter this phase 3 He would have to pair in some way analogous to cooper pairs in the B.C.S. theory. This occurs between atoms rather than electrons and is mediated via spin fluctuations rather than phonons [12]. In this case instead of pairing in a simple s-wave configuration a p-wave is found, this prevents the paired atoms from ever being in the same position. In relation to the band gap seen in superconductors (leading to two sets of conducting electrons), superfluids are seen to be both viscous and non viscous at the same time. This is known as the two fluid model with helium II containing a normal and superfluid component. this is seen in two simple experiments the superflow through capillaries that shows zero viscosity and also the action of creating a rotation in the fluid that the normal component translates through the bulk via a viscosity. 4 He when undergoing the phase transition into Figure 4: The Lambda transition into superfluidity in different dimensions shows the tell tale λ shape giving the transition its name. Graph shows a plot of temperature (normalised to T c vs. heat capacity) vs. heat capacity. Image taken from [16] superconductivity the fluid goes through an asymptote in its heat capacity this is the mark of the transition between states and is when the local gauge symmetry is broken due to the macroscopic wavefunctions in the boson condensate [13]. This logarithmic asymptote is known as the λ transition this is shown in figure 4 this is due to the entropy becoming very small as the material transforms into a superfluid [14]. The gauge symmetry broken here is only the main symmetry breaking action in a superfluid that is seen at the λ point in the superfluid phase of 3 He rotational symmetry breaking can occur in the spins of the particles and so leads to multiple phases inside the superfluid phase i.e. with an applied magnetic field the specific heat jump at the λ point shows two distinct transitions implying the breaking of spin-orbit symmetry and the presence of two phase transitions [15]. 3 Superconductivity Consequences of symmetry Breaking Having discussed superconductors and the symmetry breaking mechanism in section 2.2 the consequences of this broken symmetry will be expanded upon. For any phase transition there is a set 4

5 Once the system has passed into the superconducting phase there is a free energy cost associated with coming back out of the ordered state (this was mentioned in section 3.1). This relates to the Landau theory discussed in section 2.1 in relation to magnetism. Using the same method of expanding the free energy around T C but now using Ginzburg Landau theory [18] in order to take into account the fact the order parameter is complex. Equation 3 is the resulting free energy expression for the superconductor in terms of its order parameter ψ f s = f n + h2 2m ( h 2 i + 2eA)ψ(r) + a ψ(r) b ψ(r) 4 + B(r) 2 (3) Figure 5: Figure showing the jump in heat capacity associated with passing through T c (In this case 20 K. Image taken from [17] of standard consequences. These are the sudden change in behaviour at the phase transition, the rigidity of the order parameter, defects in the material due to differing order parameters over the material and new excitations associated with the order parameter. 3.1 Phase Transition The superconducting Phase transition contains a heat capacity anomaly due to the large decrease of entropy in ordering the electrons past T c. The size of the entropy change in the material on ordering gives an indication of how many electrons in the material have paired and become ordered. The entropy change between the normal state and the superconducting state (remembering that the superconductor can be brought back into the normal state at any temperature via the application of a suitably strong magnetic field) creates a free energy gap between the two states. This indicates the energy gap discussed in section 2.2. The action of specific heat in the transition to superconductivity is shown in figure Order Parameter Rigidity Taking this result, assuming no magnetic field and a 1D system just as before the free energy can be minimised (equation 4) by minimising the free energy the physical state of the superconductor can be investigated and the rigidity of the order parameter completely determines the action of the material. h2 d 2 ψ(x) 2m dx 2 + aψ(x) + bψ(x) 3 = 0 (4) In this minimisation of free energy the differential equation can be solved for specific boundary conditions. Recalling the discussion of the Miessner effect the boundary of the superconductor is extremely important as it holds the currents that expel the magnetic field. Equation 4, if solved for ψ(x = 0) = 0 (the edge of the bulk) allows us to see how the order parameter changes in this region. Equation 5 shows this solution and ξ is the coherence length, a measure of the scale upon which the order parameter changes from 0 to its bulk value ξ 0. ( ) x ψ(x) = ψ 0 tanh 2ξ(T ) ( h 2 ξ(t ) = 2m a(t ) ) 1 2 (5) looking at figure 6 we see that the magnetic field penetrates further than the change in order parameter this means that this superconductor is a type II superconductor as it is still superconducting even though it is not displaying perfect diamagetism. 5

6 Figure 6: Image showing the action of the order parametric at the boundary of the bulk (blue line) and the penetration of the magnetic field (red line) 3.3 Defects Even though there is a free energy cost associated with the surface of a superconductor, when the magnetic field is sufficiently large the sample finds that by creating a larger surface for the magnetic field to act upon the free energy is minimised. So createing rods of normal material in a triangular lattice formation (as they repel each other) inside the bulk. The superconductor is no longer in the Miessner phase and has entered the vortex phase. An STM image of these topological defects is provided in Figure 7 These defects in the material serve to pin the flux lines very strongly with superconducting currents orbiting the defects in order to screen the field as much as possible. 3.4 Excitations Excitations do occur inside a superconductor, If a vector boson travelling through the material is a transverse electromagnetic wave then an energy gap is created that will have a mass associated with it. This is due to the interaction of the gauge transformation with a conservation law [20] i.e. in this case electromagnetic waves couple to the Bose condensate. Interestingly this is analogous to the Higgs mechanism used by particle physicists to explain the generation of all mass. This is because the ground state of both of these systems are degenerate. This leads to a 3D potential shaped as a Mexican hat where the system s order parameter will spontaneously fall into one of an infinite set of states upon ordering at the phase transition. Figure 7: Image showing topological defects in a type II superconductor in its vortex phase the periodic triangular lattice formation of these defects is clear. Image taken from [19] 4 Conclusion Symmetry breaking due to phase changes have been examined throughout the energy scale of the universe with a focus upon quantum phase changes such as magnetism and the similar superconductor or superfluid transitions. These phase changes have been explained and the role of symmetry breaking and the formation of an ordered state has been shown. In the last part of the report the consequences of the superconducting phase change were focused upon and various effects were discussed in more detail. Quantum phase changes, symmetry breaking and consequently the ordering of matter is of huge importance as without these effects matter in different forms would never exist and effects that we take for granted in everyday life such as magnetism would never occur. 6

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