Superconducting RF Materials for Accelerators Dr Graeme Burt Lancaster University CI Postgraduate Lectures May 2012

Outline Fermi levels in metals Superconductivity Energy Gap and coherence length Meissner effect Two fluid model / BCS resistance Residual resistance Thermal conductivity Thermal Breakdown RF Critical fields Impurities and processing Thin films Alternate materials Stuff you will know Stuff you will have forgotten New stuff

Conductivity in metals As you all know electrons are fermions and can have +1/2 or -1/2 spin. As such due to the Pauli exclusion principle we can only have two electrons per energy level. The electrons stack up on top of each other occupying all available energy levels. The last occupied energy level is known as the Fermi level, eF. For Niobium the fermi level is at 5.32 eV, and the fermi temperature (TF=eF/kB) is 6.18x104 K and the fermi velocity (vF2=eF/2m) is 1.36x106 ms-1. Without the Pauli exclusion principle this would be much lower. eF

Superconductivity Kammerlingh-Onnes demonstrated that the dc resistance of certain materials drops to immeasurable levels when cooled below a critical temperature, Tc. Several materials have been shown to be superconducting, the two most common for RF cavities are Nb (Tc=9.2 K) and lead (Tc=7.2 K). For magnets Nb-Ti (Tc=10 K) and Nb3Sn (Tc=18 K) are used. High Tc materials are not used as they do not perform well in the presence of a magnetic field or at RF frequencies. 40 years after the discovery, the first theory of superconductivity was proposed by Bardeen, Cooper and Schrieffer. They proposed that superconductivity was based on the pairing of electrons due to the attractive potential between them due to coupling to lattice vibrations.

Cooper Pairs As an electron moves through the lattice it distorts it. A following electron feels a force due to the displaced positive charge causing a small net attraction between the electrons. The highest relaxation frequency, wd, gives the distance over which two electrons can interact (vF2p/wD) as tens of nm. The paired electrons form a new particle with twice the mass and charge of an electron. The bound state is energetically favourable and the pairing energy is given as 2D~2(1.7kBTc). This new particle, know as a cooper pair, has zero spin and is hence a boson and is no longer bound by fermi statistics. This allows the cooper pairs to all exist in the same quantum state with the same centre of mass energy (hence velocity), and can hence move coherently, this is what allows a current to flow with zero resistance. The lattice interaction is the same effect that causes resistance at high temperatures hence good superconductors tend to be bad room temperature conductors.

Coherence Length When the electrons condense into cooper pairs only those within kTc of the fermi level are involved. From this range it can be shown that the momentum is According to Heisenberg's uncertainty principle the spatial extent of the pair, ξ0, also known as the coherence length, is hence For Niobium ξ0=39 nm and for lead ξ0=83 nm.

Coherence Length and Energy gap above T= 0 K The large coherence length suggests Cooper pairs overlap and possess long range order. It also is the range over which the wavefunction decays at a normal conducting and superconducting boundary. As we increase the superconductors temperature above T= 0 K some of the electrons have enough energy to reach an excited state. The partner electron hence doesn’t have a pair and hence remains in a normal conducting electrons state. In this case the energy gap decreases very slightly until it reaches Tc when it tends to zero.

DC Electrical properties As the temperature rises above T=0 K, not all the electrons are in the superconducting state. We can consider the electrons using a two fluid model, one superconducting and the other normal conducting. Scattering of Cooper pairs can only occur if they can gain enough energy to break the pairing (as all pairs flow coherently). Scattering of current can occur at either impurities or phonons. Impurities are fixed targets and cannot change the electon energy and phonons do not have enough energy to break up the pair hence Cooper pairs cannot be scattered. In a DC case the two fluids act like two resistances in parallel, since the superconducting state has zero resistance it can carry the entire current. However an applied voltage could provide enough energy to break up the pairs. If we apply a voltage the pairs flow as one at a single velocity. The current is proportional to that velocity, given as j= -2en Dv. From the momentum calculation we can hence show that the critical current (the maximum possible current that can be carried in a superconductor) is

RF Electrical properties However in the AC case the cooper pairs still have inertia, meaning the have to be accelerated and decelerated every time the field changes. This causes a delay which allows time varying magnetic fields to penetrate into the material as the supercurrent cannot completely shield it. The time varying magnetic field in the material causes normal conducting eddy currents to flow resulting in an RF resistance. However these currents are related to the rate of change of the magnetic field it is out of phase with the supercurrent (inductive). In order to understand the effect of this we must first understand the work of London. London’s first equation is that in a perfect conductor, as there are no scattering an applied electric field will result in a current that that linearly increase such that

Miessner Effect Normal Conductor Superconductor H > 0 T > Tc

London’s 2nd Equation London realised that the Meissner effect requires that This is known as the 2nd London equation and combing this with Maxwells equations we can show that a static magnetic field will decay over a distance, lL, known as the london penetration depth. Where

RF Surface resistance If we combine the 1st London equation with Ohms law We can rearrange to get a definition of the superconducting conductivity. The normal conductivity is given by the usual definition The total current is hence

RF Surface resistance The impedance is hence defined as Considering ss>>sn we can show that the surface resistance, Rsurf, is This leads to three key facts about SRF Rs is proportional to sn which is proportional to the mean free path, this is odd and is the opposite of intuition. Rs is proportional to lL3 hence we need a large concentration of charge carriers Rs is proportional to w2 hence SRF is not efficient at high frequency (above ~4 GHz)

RF superconductivity However the London penetration depth also varies with temperature as the number of cooper pairs decreases as we approach Tc. The exact dependence is of Rs is given by BCS theory which predicts the properties of SC materials.

BCS Theory Expressions for the surface resistance have been developed by Mattis and Bardeen based on BCS theory. These equations are difficult to solve but computer programmes have been used to numerically solve them. A good fit to the results for Niobium (for T=Tc/2 and frequencies below 1 THz) is given by

Residual Resistance An SRF cavity will decrease its resistance with temperature in theory. However in practice there is often a minimum resistance due to the effects of normal conducting impurities in the niobium. One of the main effects is flux pinning where magnetic fields are frozen into normal conducting impurities inside the superconductor. This can be avoided by shielding the cavity from magnetic fields during cooldown.

Residual Resistance Ratio (RRR) As we decrease the temperature of a normal conducting metal the DC conductivity increases. However at some temperature the increase saturates to a level determined by impurities. The ratio of the resistance at 300 K to the resistance at low temperature (normal) is known as the residual resistance ratio (RRR). The theoretical maximum RRR for Niobium is around 35,000 due to electron-phonon scattering.

Thermal Conductivity At low temperature the thermal conductivity is dominated by electronic scattering as the phonon contribution scales as T3. However in a superconductor the number of electrons decrease as they condense in copper pairs. However for large grain Nb a peak in the phonon contribution can occur at 2K (very material dependant). In fine grain materials this phonon peak is suppressed due to scattering at the grain boundary. Thermal conductivity and DC electrical conductivity (normal) are both dominated by electronic scattering at low temperature. At 4.2 K the thermal conductivity is about 25% of the RRR. RRR=4ks Thermal conductivity hence increases with the purity of the material.

Thermal Breakdown When a superconductor reaches Tc it becomes a normal conductor (known as a thermal breakdown of superconductivity). One of the major causes of this is when a normal conducting impurity near the surface is heated by the time varying magnetic field causing the superconductor around it to also heat up. When the temperature reaches Tc the breakdown quickly spreads throughout the cavity as the normal conducting region grows.

Field limits due to Thermal Breakdown If we have a normal conducting impurity, of radius a in a sheet of Nb of thickness d, the power dissipation/heat flow into the impurity can be given as If we assume a<<d and that the bath temperature at d is Tb and the maximum temperature is Tc, we can solve and rearrange For example at a bath temperature of 2K, a 50 micron impurity ar Rn=10mW in RRR=300 Nb, will breakdown at Bmax=82 mT Thermal conductivity Defect radius um

Field Emission Another cause of thermal breakdown is when electrons that are emmitted from the surface due to field emission are accelerated in the RF fields and hit the cavity surface. When this happens the energy is deposited as heat which can cause a thermal breakdown or simply heat up the surface causing a higher surface resistance in the superconducting material. The biggest cause of field emission is field enhancement at sharp edges (usually foreign objects that have contaminated the surface).

Reducing thermal breakdown Approaches to avoiding thermal breakdown are To use high RRR material (however remember the surface resistance also increases with mean free path so not too large a RRR) To carefully etch and clean the cavity to remove impurities. Typically we use RRR~250 for SRF cavities

End of Lecture 1

Thermodynamic Critical Field When electrons condence into cooper pairs the superconducting state becomes more ordered than the normal conducting state and hence the free energy is lower. F = Uint-TS When an excternal magnetic field is applied to the superconductor, supercurrents flow to cancel the field and hence the free energy increases. When the external field rises to such a level, Hc, that the superconducting state and normal conducting state have an equal amount of free energy the states are in equilibrium. The free energy of a superconductor of volume, Vs, is given by

Thermodynamic critical field When the external field reaches Hc all the flux enters the superconductor. This is known as the thermodynamic critical field. This is also related to the critical current density. BCS theory leads to an expression for the zero temperature critical field Where g is the electronic specific heat. Hc is around 200 mT for Nb at T= 0 K. As the temperature is increased the thermodynamic critical field is reduced as

Type I/II superconductors Consider a magnetic field decaying in a thick sheet of superconductor such that the thickness, d, is larger than the penetration depth. Now consider a thin sheet where d<lL. It can be seen that the magnetic field does not reach zero. As a result less work is done excluding the field and the free energy is lower.

Type I/II superconductors Is it therefore energetically favourable for the flux to enter a thick superconductor through a number of thin normal conducting layers? However there is a surface energy associated with the boundary between normal and superconducting regions. For some superconductors this surface energy is negative hence the flux starts to enter the material through normal conducting regions at a magnetic field below Hc. These are Type II superconductors

Type I/II superconductors When the density of cooper pairs is suppressed over the coherence length the free energy per unit area is increased by In an external magnetic field, He, the free energy per unit area is lowered by Hence the net boundary energy per unit area is given by As the coherence length and the penetration depth vary from material to material the boundary energy can be positive or negative.

Type I/II superconductors The ratio of the penetration depth to the coherence length is known as the Ginzberg-Landau parameter and its value tells us if flux will enter a superconductor below Hc (Type II) or at Hc (Type I). This can also be explained by the fact that if the coherence length is large the transition between normal and superconducting regions is not sharp. Lead is a Type I superconductor with KGL~0.45 Niobium is a weak type II superconductor with KGL~0.8-1 Most high T superconductors are strong Type II as coherence length is inversely proportional to Tc. Type I Type II

Type II superconductors For Type II superconductors the flux begins to enter the bulk at a field, Hc1, and the bulk breaks up into a periodic lattice of normal conducting zones each with a flux vortice. All the flux enters the superconductor at a higher magnetic field, Hc2, when the normal conduction regions grow enough to overlap. Ginzburg-Landau theory predicts the field at which all the flux enters as There is no formula for Hc1 but a good approximation is

RF critical magnetic field The above staatements are for the DC case. However just because the flux enters a superconductor in DC, doesn’t mean that there cannot exist a “superheated” superconducting state for RF magnetic fields. In fact superconductivity can persist metastabily up to a RF critical magnetic field, Hsh. For Type I superconductors the superconducting state exists up until the boundary energy is zero ie. For Type II Ginzburg-Landau theory predicts . kGL<<1 kGL~1 kGL>>1

Niobium The properties of high purity polycrystalline Niobium are hence [Bahte 1997] Tc =9.2 K London penetration depth = 30 nm Coherence length = 39 nm GL parameter = 0.8 Bsh= 190 mT @4.2 K, 240 mT @0 K Bc1= 130 mT @4.2 K, 164 mT @0 K Bc= 158 mT @4.2 K, 200 mT @0 K

Impurities Oxygen on the surface trapping flux Hydrogen Q-disease Lossy metal contamination (covered in L1) Field emitters (covered in L1) Bulk contamination decreasing purity and mean free path

Mean Free Path The coherence length varies with mean free path, l, as And the london penertaion depth becomes In the dirty limit (l<<z) the BCS surface resistance increases as the London penetration depth decreases In the clean limit (l>z) scattering is dominated by the the applied magnetic field hence the length scale is roughly l not l.

Mean Free Path As a result the minimum BCS resistance occurs at a mean free path of around 200 nm for Nb. High RRR Nb typically has a larger mean free path and is in the clean limit. Baked cavities have a shorter mean free path (due to oxygen diffusion from the surface into the penetration depth) and are in the dirty limit.

Residual Resistance and Impurities Residual resistance is normally dominated by Loss Nb-H formation at the RF surface (Q disease) Flux pinning due to oxygen on the cavity surface. Hydride formation occurs when hydrogen in the bulk nucleates at the surface. This occurs in cavities with high hydrogen contamination levels (normally from a BCP gone wrong) and cavities that have been cooled down too slowly. Hydride phases start at about 150 K but hydrogen mobility reduces with temperature, so the time to cool from 150-100 K is critical. When the cavity is cooled down under a dc magnetic field flux gets trapped in the oxide layer on the surface, causing a normal conducting zone around the flux. Hence the cavity is normally shielded from the earths magnetic field.

Thin Films Since thermal breakdown is such a major problem, many have sought to grow a thin superconducting film on a copper cavity substrate. This film has the SRF properties of the film but the thermal properties of the bulk hence avoids thermal breakdown. However many impurities and dislocations are introduced when grown and there are compressive stresses from the substrate and this alters the properties of the film compared to the bulk SRF magterial. For example Nb films have a large Ginzburg-Laundau parameter (3.5-12) and a higher Tc (up to 9.6 K). In addition there seems to be a high drop in Q with increasing field that cannot be fully explained, this limits their operation to low fields. On the plus side their residual resistance is less sensitive to external magnetic fields than bulk Nb.

Sputtering Nb on Cu There are two main methods of sputtering Nb onto copper, bias sputtering and magnetron sputtering. In bias sputtering a DC potential is placed between a Niobium cathode and a cavity anode and a small grid with a bias potenitial is placed between them. The cavity is back filled with some inert gas (usually argon) which is ionised. Argon ions strike the Nb cathode releasing Nb atoms which then impinge on the copper substrate.

Sputtering Nb on Cu In magnetron sputtering a magnetic field is used to cause the Nb ions to impinge on the substrate with a transverse momentum and there is no bias. New techniques are looking at high power impulse magnetron sputtering to use high power for short times as energetic deposition gives better film quality. Magnetron sputtering has given the best results to date for elliptical cavities but is difficult to get right on complex structures hence bias sputtering is usually better on QWR’s.

Nb sputtered results Unfortuntely sputtered cavity show a low field Q drop (the Q drops very fast with increasing field) This limits the maximum gradient in these cavities. They are however less sensitive to external magnetic fields.

High Tc superconductors There have been many superconductors found to have transition temperatures above the 23 K prediction for metallic BCS superconductors. The three most common types are iron based, fullerites and copper oxide based. These materials have very high transition temperatures up to 133 K for copper oxides (HgBa2Ca2Cu3Ox). This could allow accelerators with LN2.

RF properties of High Tc SC High Tc means high Hc High Tc means low coherence length High Tc means large penetration depth due to low numbers of charge carriers New non-metallic SC means high surface resistance

RF properties of High Tc SC Obviously a high Tc is really good as this in turn leads to a large accelerating voltage possibly hundreds of MV/m. This means shorter accelerators. However the higher surface resistance and shorter coherence length can be a major problem. The grain boundaries tend to be on the order of a few nm so if the coherence length is less than this then RF losses grow substantially as the grains are weakly coupled and a supercurrent flows between grains by tunneling. Above a threshold surface field (related to the critical current) the grains are decoupled leading to additional losses. This is the current theory but might prove to be incorrect. In addition most high Tc superconductors would have to be grown as a thin film on the cavity surface as they are not machinable or formable.

A15’s The highest useful Tc for an SRF cavity is about 20 K where the coherence length is about 5 nm (for Nb3Sn). There are a number of alloys that are superconducting at Tc~20 K, these are known as the A15’s and they all have the form A3B. The most successful so far has been Nb3Sn.

Nb3Sn Tc = 18.2 K Hsh = 400 mT Coherence length = 6 nm London penetration depth = 60 nm Kappa GL = 20 Nb3Sn is a promising material as its higher Tc and Hsh than Nb would allow a high gradient (100 MV/m) linear collider at 4.2 K (as opposed to 1.8K) The only fabrication technique explored has been vapour diffusion of tin into niobium at 900-1200 degC (this could allow upgrading of existing cavities). The difficulty is not to create other non-SC phases. Results have found a higher low field Q value than niobium (1011) at 4.2 K and the highest gradients have been 16 MV/m. There is also a high increase in resistance at higher fields believed to be due to intergrain losses.

The End