Model: shreyaspullehf/superconductor-search-v7
Author: shreyaspullehf
Pipeline: sentence-similarity
Library: sentence-transformers
Tags: sentence-transformers, safetensors, bert, sentence-similarity, feature-extraction, dense, generated_from_trainer, dataset_size:6420, loss:MultipleNegativesRankingLoss, arxiv:2108.03655, arxiv:1908.10084, arxiv:1705.00652, base_model:sentence-transformers/all-MiniLM-L6-v2, base_model:finetune:sentence-transformers/all-MiniLM-L6-v2, text-embeddings-inference, endpoints_compatible, region:us
Base models (finetune): id: sentence-transformers/all-MiniLM-L6-v2
Card metadata: tags: sentence-transformers, sentence-similarity, feature-extraction, dense, generated_from_trainer, dataset_size:6420, loss:MultipleNegativesRankingLoss
base_model: sentence-transformers/all-MiniLM-L6-v2
widget: source_sentence: superconductor
sentences: A fault current limiter (FCL), also known as fault current controller (FCC), is a device which limits the prospective fault current when a fault occurs (e.g. in a power transmission network) without complete disconnection.  The term includes superconducting, solid-state and inductive devices.


== Applications ==
Electric power distribution systems include circuit breakers to disconnect power in case of a fault, but to maximize reliability, they wish to disconnect the smallest possible portion of the network.  This means that even the smallest circuit breakers, as well as all wiring to them, must be able to disconnect large fault currents.
A problem arises if the electricity supply is upgraded, by adding new generation capacity or by adding cross-connections.  Because these increase the amount of power that can be supplied, all of the branch circuits must have their bus bars and circuit breakers upgraded to handle the new higher fault current limit.
This poses a particular problem when distributed generation, such as wind farms and rooftop solar power, is added to an existing electric grid.  It is desirable to be able to add additional power sources without large system-wide upgrades.
A simple solution is to add electrical impedance to the circuit.  This limits the rate at which current can increase, which limits the level the fault current can rise to before the breaker is opened.  However, this also limits the ability of the circuit to satisfy rapidly changing demand, so the addition or removal of large loads causes unstable power.
A fault current limiter is a nonlinear element which has a low impedance at normal current levels, but presents a higher impedance at fault current levels.  Further, this change is extremely rapid, before a circuit breaker can trip a few milliseconds later.  (High-power circuit breakers are synchronized to the alternating current zero crossing to minimize arcing.)
While the power is unstable during the fault, it is not completely disconnected.  After the faulting branch is disconnected, the fault current limiter automatically returns to normal operation.


== Superconducting fault current limiter ==
Superconducting fault current limiters exploit the extremely rapid loss of superconductivity (called "quenching") above a critical combination of temperature, current density, and magnetic field.  In normal operation, current flows through the superconductor without resistance and negligible impedance.
If a fault develops, the superconductor quenches, its resistance rises sharply, and current is diverted to a parallel circuit with the desired higher impedance. (The structure is not usable as a circuit breaker, because the normally-conducting superconductive material does not have a high enough resistance.  It is only high enough to cause sufficient heating to melt the material.)
Superconducting fault current limiters are described as being in one of two major categories: resistive or inductive.
In a resistive FCL, the current passes directly through the superconductor.  When it quenches, the sharp rise in resistance reduces the fault current from what it would otherwise be (the prospective fault current). A resistive FCL can be either DC or AC. If it is AC, then there will be a steady power dissipation from AC losses (superconducting hysteresis losses) which must be removed by the cryogenic system. An AC FCL is usually made from wire wound non-inductively; otherwise the inductance of the device would create an extra constant power loss on the system.
Inductive FCLs come in many variants, but the basic concept is a transformer with a resistive FCL as the secondary. In un-faulted operation, there is no resistance in the secondary and so the inductance of the device is low. A fault current quenches the superconductor, the secondary becomes resistive and the inductance of the whole device rises. The advantage of this design is that there is no heat ingress through current leads into the superconductor, and so the cryogenic power load may be lower. However, the large amount of iron required means that inductive FCLs are much bigger and heavier than resistive FCLs.  The first successful field test of an HTS FCL of this type was by SC Power Systems, a division of Zenergy Power plc in 2009.
The quench process is a two-step process.  First, a small region quenches directly in response to a high current density.  This section rapidly heats by Joule heating, and the increase in temperature quenches adjacent regions.GridON Ltd has developed the first commercial inductive FCL for distribution & transmission networks. Using a unique and proprietary concept of magnetic-flux alteration - requiring no superconducting or cryogenic components - the self-triggered FCL instantaneously increases its impedance tenfold upon fault condition. It limits the fault current for its entire duration and recovers to its normal condition immediately thereafter. This inductive FCL is scalable to extra high voltage ratings.


== Solid state fault current limiter ==


== Inductive fault current limiter ==


== Development of the superconducting fault current limiters ==
FCLs are under active development. In 2007, there were at least six national and international projects using magnesium diboride wire or YBCO tape, and two using BSCCO-2212 rods. Countries active in FCL development are Germany, the UK, the US, Korea and China. In 2007, the US Department of Energy spent $29m on three FCL development projects.
High temperature superconductors are required for practical FCLs.  AC losses generate constant heat inside the superconductor, and the cost of cryogenic cooling at liquid helium temperatures required by low temperature superconductors makes the whole device uneconomic.
First applications for FCLs are likely to  be used to help control medium-voltage electricity distribution systems, followed by electric-drive ships: naval vessels, submarines and cruise ships. Larger FCLs may eventually be deployed in high-voltage transmission systems.


== See also ==
Current limiting
Power-system protection
Superconductivity
Magnesium diboride
YBCO


== References ==


== External links ==
Superconducting Fault Current Limiters
UK Government 2007 Report on FCLs
High-temperature superconductor fault current limiters: concepts, applications, and development status
Fault Current Limiter and Their Types
2012: YBCO-tape FCL enters service in German private grid, What is superconductivity? | Chemistry - Materials: How They Work | Suitable for teaching 11-16s. Suitable for teaching 14-16s. Superconductivity helps transport electricity more efficiently. Mark Miodownik describes the discovery.

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Create an account, subscribe to the channel and create playlists for different age groups, sets an, High-inductance superconducting qubits | Vladimir Manucharyan, Professor, University of Maryland (USA), Head of Superconducting Circuits laboratory of the Quantum Materials Center and the Joint Quantum Institute at the University of Maryland

The 1st international school
Superconducting Quantum Hardware

Superconducting technologies for quantum information processing
within the framework of the 6th International Conference on Quantum Technologies ICQT 2021
https://research.misis.ru/sqh_school_2021, source_sentence: superfluid vacuum theory
sentences: Superfluid vacuum theory | Wikipedia audio article | This is an audio version of the Wikipedia Article:
Superfluid vacuum theory


00:00:47 1 History
00:03:00 2 Relation to other concepts and theories
00:03:11 2.1 Lorentz and Galilean symmetries
00:05:04 2.2 Relativistic quantum field theory
00:06:42 2.3 Curved space-time
00:08:03 2.4 Cosmological constant
00:08:39 2.5 Gravitational waves and gravitons
00:10:13 2.6 Mass generation and Higgs boson
00:12:38 3 Logarithmic BEC vacuum theory
00:14:29 4 See also



Listening is a more natural way, In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called band gaps or forbidden bands).
Band theory derives these bands and band gaps by examining the allowed quantum mechanical wave functions for an electron in a large, periodic lattice of atoms or molecules. Band theory has been successfully used to explain many physical properties of solids, such as electrical resistivity and optical absorption, and forms the foundation of the understanding of all solid-state devices (transistors, solar cells, etc.).

Why bands and band gaps occur
The formation of electronic bands and band gaps can be illustrated with two complementary models for electrons in solids. The first one is the nearly free electron model, in which the electrons are assumed to move almost freely within the material. In this model, the electronic states rese, Superconducting radio frequency (SRF) science and technology involves the application of electrical superconductors to radio frequency devices.  The ultra-low electrical resistivity of a superconducting material allows an RF resonator to obtain an extremely high quality factor, Q.  For example, it is commonplace for a 1.3 GHz niobium SRF resonant cavity at 1.8 kelvins to obtain a quality factor of Q=5×1010.  Such a very high Q resonator stores energy with very low loss and narrow bandwidth. These properties can be exploited for a variety of applications, including the construction of high-performance particle accelerator structures.


== Introduction ==
The amount of loss in an SRF resonant cavity is so minute that it is often explained with the following comparison: Galileo Galilei (1564–1642) was one of the first investigators of pendulous motion, a simple form of mechanical resonance. Had Galileo experimented with a 1 Hz resonator with a quality factor Q typical of today's SRF cavities and left it swinging in an entombed lab since the early 17th century, that pendulum would still be swinging today with about half of its original amplitude.
The most common application of superconducting RF is in particle accelerators. Accelerators typically use resonant RF cavities formed from or coated with superconducting materials. Electromagnetic fields are excited in the cavity by coupling in an RF source with an antenna.  When the RF fed by the antenna is the same as that of a cavity mode, the resonant fields build to high amplitudes. Charged particles passing through apertures in the cavity are then accelerated by the electric fields and deflected by the magnetic fields.  The resonant frequency driven in SRF cavities typically ranges from 200 MHz to 3 GHz, depending on the particle species to be accelerated.
The most common fabrication technology for such SRF cavities is to form thin walled (1–3 mm) shell components from high purity niobium sheets by stamping. These shell components are then welded together to form cavities.
A simplified diagram of the key elements of an SRF cavity setup is shown below.  The cavity is immersed in a saturated liquid helium bath. Pumping removes helium vapor boil-off and controls the bath temperature.  The helium vessel is often pumped to a pressure below helium's superfluid lambda point to take advantage of the superfluid's thermal properties. Because superfluid has very high thermal conductivity, it makes an excellent coolant. In addition, superfluids boil only at free surfaces, preventing the formation of bubbles on the surface of the cavity, which would cause mechanical perturbations.  An antenna is needed in the setup to couple RF power to the cavity fields and, in turn, any passing particle beam.  The cold portions of the setup need to be extremely well insulated, which is best accomplished by a vacuum vessel surrounding the helium vessel and all ancillary cold components.  The full SRF cavity containment system, including the vacuum vessel and many details not discussed here, is a cryomodule.

Entry into superconducting RF technology can incur more complexity, expense, and time than normal-conducting RF cavity strategies.  SRF requires chemical facilities for harsh cavity treatments, a low-particulate cleanroom for high-pressure water rinsing and assembly of components, and complex engineering for the cryomodule vessel and cryogenics.  A vexing aspect of SRF is the as-yet elusive ability to consistently produce high Q cavities in high volume production, which would be required for a large linear collider.  Nevertheless, for many applications the capabilities of SRF cavities provide the only solution for a host of demanding performance requirements.
Several extensive treatments of SRF physics and technology are available, many of them free of charge and online.  There are the proceedings of CERN accelerator schools, a scientific paper giving a thorough presentation of the many aspects of an SRF cavity to be used in the International Linear Collider, bi-annual International Conferences on RF Superconductivity held at varying global locations in odd numbered years, and tutorials presented at the conferences.


== SRF cavity application in particle accelerators ==

A large variety of RF cavities are used in particle accelerators.  Historically most have been made of copper – a good electrical conductor – and operated near room temperature with exterior water cooling to remove the heat generated by the electrical loss in the cavity.  In the past two decades, however, accelerator facilities have increasingly found superconducting cavities to be more suitable (or necessary) for their accelerators than normal-conducting copper versions.  The motivation for using superconductors in RF cavities is not to achieve a net power saving, but rather to increase the quality of the particle beam being accelerated.  Though superconductors have little AC electrical resistance, the little power they do dissipate is radiated at very low temperatures, typically in a liquid helium bath at 1.6 K to 4.5 K, and maintaining such low temperatures takes a lot of energy.  The refrigeration power required to maintain the cryogenic bath at low temperature in the presence of heat from small RF power dissipation is dictated by the Carnot efficiency, and can easily be comparable to the normal-conductor power dissipation of a room-temperature copper cavity.  The principle motivations for using superconducting RF cavities, are:

High duty cycle or cw operation.  SRF cavities allow the excitation of high electromagnetic fields at high duty cycle, or even cw, in such regimes that a copper cavity's electrical loss could melt the copper, even with robust water cooling.
Low beam impedance.  The low electrical loss in an SRF cavity allows their geometry to have large beampipe apertures while still maintaining a high accelerating field along the beam axis.  Normal-conducting cavities need small beam apertures to concentrate the electric field as compensation for power losses in wall currents.  However, the small apertures can be deleterious to a particle beam due to their spawning of larger wakefields, which are quantified by the accelerator parameters termed "beam impedance" and "loss parameter".
Nearly all RF power goes to the beam.  The RF source driving the cavity need only provide the RF power that is absorbed by the particle beam being accelerated, since the RF power dissipated in the SRF cavity walls is negligible.  This is in contrast to normal-conducting cavities where the wall power loss can easily equal or exceed the beam power consumption.  The RF power budget is important since the RF source technologies, such as a Klystron, Inductive output tube (IOT), or solid state amplifier, have costs that increase dramatically with increasing power.
When future advances in superconducting material science allow higher superconducting critical temperatures Tc and consequently higher SRF bath temperatures, then the reduced thermocline between the cavity and the surrounding environment could yield a significant net power savings by SRF over the normal conducting approach to RF cavities.  Other issues will need to be considered with a higher bath temperature, though, such as the fact that superfluidity (which is presently exploited with liquid helium) would not be present with (for example) liquid nitrogen.  At present, none of the "high Tc" superconducting materials are suitable for RF applications.  Shortcomings of these materials arise due to their underlying physics as well as their bulk mechanical properties not being amenable to fabricating accelerator cavities.  However, depositing films of promising materials onto other mechanically amenable cavity materials may provide a viable option for exotic materials serving SRF applications.  At present, the de facto choice for SRF material is still pure niobium, which has a critical temperature of 9.3 K and functions as a superconductor nicely in a liquid helium bath of 4.2 K or lower, and has excellent mechanical properties.


== Physics of SRF cavities ==
The physics of Superconducting RF can be complex and lengthy.  A few simple approximations derived from the complex theories, though, can serve to provide some of the important parameters of SRF cavities.
By way of background, some of the pertinent parameters of RF cavities are itemized as follows.  A resonator's quality factor is defined by

  
    
      
        
          Q
          
            o
          
        
        =
        
          
            
              ω
              U
            
            
              P
              
                d
              
            
          
        
      
    
    {\displaystyle Q_{o}={\frac {\omega U}{P_{d}}}}
  
,
where:

ω is the resonant frequency in [rad/s],
U is the energy stored in [J], and
Pd is the power dissipated in [W] in the cavity to maintain the energy U.
The energy stored in the cavity is given by the integral of field energy density over its volume,

  
    
      
        U
        =
        
          
            
              μ
              
                0
              
            
            2
          
        
        ∫
        
          
            |
          
          
            
              H
              →
            
          
          
            
              |
            
            
              2
            
          
          d
          V
        
      
    
    {\displaystyle U={\frac {\mu _{0}}{2}}\int {|{\overrightarrow {H}}|^{2}dV}}
  
 ,
where:

H is the magnetic field in the cavity and
μ0 is the permeability of free space.
The power dissipated is given by the integral of resistive wall losses over its surface,

  
    
      
        
          P
          
            d
          
        
        =
        
          
            
              R
              
                s
              
            
            2
          
        
        ∫
        
          
            |
          
          
            
              H
              →
            
          
          
            
              |
            
            
              2
            
          
          d
          S
        
      
    
    {\displaystyle P_{d}={\frac {R_{s}}{2}}\int {|{\overrightarrow {H}}|^{2}dS}}
  
 ,
where:

Rs is the surface resistance which will be discussed below.
The integrals of the electromagnetic field in the above expressions are generally not solved analytically, since the cavity boundaries rarely lie along axes of common coordinate systems.  Instead, the calculations are performed by any of a variety of computer programs that solve for the fields for non-simple cavity shapes, and then numerically integrate the above expressions.
An RF cavity parameter known as the Geometry Factor ranks the cavity's effectiveness of providing accelerating electric field due to the influence of its shape alone, which excludes specific material wall loss.  The Geometry Factor is given by

  
    
      
        G
        =
        
          
            
              ω
              
                μ
                
                  0
                
              
              ∫
              
                
                  |
                
                
                  
                    H
                    →
                  
                
                
                  
                    |
                  
                  
                    2
                  
                
                d
                V
              
            
            
              ∫
              
                
                  |
                
                
                  
                    H
                    →
                  
                
                
                  
                    |
                  
                  
                    2
                  
                
                d
                S
              
            
          
        
      
    
    {\displaystyle G={\frac {\omega \mu _{0}\int {|{\overrightarrow {H}}|^{2}dV}}{\int {|{\overrightarrow {H}}|^{2}dS}}}}
  
 ,
and then

  
    
      
        
          Q
          
            o
          
        
        =
        
          
            G
            
              R
              
                s
              
            
          
        
        ⋅
      
    
    {\displaystyle Q_{o}={\frac {G}{R_{s}}}\cdot }
  

The geometry factor is quoted for cavity designs to allow comparison to other designs independent of wall loss, since wall loss for SRF cavities can vary substantially depending on material preparation, cryogenic bath temperature, electromagnetic field level, and other highly variable parameters.  The Geometry Factor is also independent of cavity size, it is constant as a cavity shape is scaled to change its frequency.
As an example of the above parameters, a typical 9-cell SRF cavity for the International Linear Collider (a.k.a. a TESLA cavity) would have G=270 Ω and Rs= 10 nΩ, giving Qo=2.7×1010.
The critical parameter for SRF cavities in the above equations is the surface resistance Rs, and is where the complex physics comes into play.  For normal-conducting copper cavities operating near room temperature, Rs is simply determined by the empirically measured bulk electrical conductivity σ by

  
    
      
        
          R
          
            s
             
            n
            o
            r
            m
            a
            l
          
        
        =
        
          
            
              
                ω
                
                  μ
                  
                    0
                  
                
              
              
                2
                σ
              
            
          
        
      
    
    {\displaystyle R_{s\ normal}={\sqrt {\frac {\omega \mu _{0}}{2\sigma }}}}
  
  .
For copper at 300 K, σ=5.8×107 (Ω·m)−1 and at 1.3 GHz, Rs copper= 9.4 mΩ.
For Type II superconductors in RF fields, Rs can be viewed as the sum of the superconducting BCS resistance and temperature-independent "residual resistances",

  
    
      
        
          R
          
            s
          
        
        =
        
          R
          
            B
            C
            S
          
        
        +
        
          R
          
            r
            e
            s
          
        
      
    
    {\displaystyle R_{s}=R_{BCS}+R_{res}}
  
  .
The BCS resistance derives from BCS theory.  One way to view the nature of the BCS RF resistance is that the superconducting Cooper pairs, which have zero resistance for DC current, have finite mass and momentum which has to alternate sinusoidally for the AC currents of RF fields, thus giving rise to a small energy loss.  The BCS resistance for niobium can be approximated when the temperature is less than half of niobium's superconducting critical temperature, T<Tc/2, by

  
    
      
        
          R
          
            B
            C
            S
          
        
        ≃
        2
        ×
        
          10
          
            −
            4
          
        
        
          
            (
            
              
                f
                
                  1.5
                  ×
                  
                    10
                    
                      9
                    
                  
                
              
            
            )
          
          
            2
          
        
        
          
            
              e
              
                −
                17.67
                
                  /
                
                T
              
            
            T
          
        
      
    
    {\displaystyle R_{BCS}\simeq 2\times 10^{-4}\left({\frac {f}{1.5\times 10^{9}}}\right)^{2}{\frac {e^{-17.67/T}}{T}}}
  
  [Ω],
where:

f is the frequency in [Hz],
T is the temperature in [K], and
Tc=9.3 K for niobium, so this approximation is valid for T<4.65 K.
Note that for superconductors, the BCS resistance increases quadratically with frequency, ~f 2, whereas for normal conductors the surface resistance increases as the root of frequency, ~√f.  For this reason, the majority of superconducting cavity applications favor lower frequencies, <3 GHz, and normal-conducting cavity applications favor higher frequencies, >0.5 GHz, there being some overlap depending on the application.
The superconductor's residual resistance arises from several sources, such as random material defects, hydrides that can form on the surface due to hot chemistry and slow cool-down, and others that are yet to be identified.  One of the quantifiable residual resistance contributions is due to an external magnetic field pinning magnetic fluxons in a Type II superconductor.  The pinned fluxon cores create small normal-conducting regions in the niobium that can be summed to estimate their net resistance.  For niobium, the magnetic field contribution to Rs can be approximated by

  
    
      
        
          R
          
            H
          
        
        =
        
          
            
              H
              
                e
                x
                t
              
            
            
              2
              
                H
                
                  c
                  2
                
              
            
          
        
        
          R
          
            n
          
        
        ≈
        9.49
        ×
        
          10
          
            −
            12
          
        
        
          H
          
            e
            x
            t
          
        
        
          
            f
          
        
      
    
    {\displaystyle R_{H}={\frac {H_{ext}}{2H_{c2}}}R_{n}\approx 9.49\times 10^{-12}H_{ext}{\sqrt {f}}}
  
  [Ω],
where:

Hext is any external magnetic field in [Oe],
Hc2 is the Type II superconductor magnetic quench field, which is 2400 Oe (190 kA/m) for niobium, and
Rn is the normal-conducting resistance of niobium in ohms.
The Earth's nominal magnetic flux of 0.5 gauss (50 μT) translates to a magnetic field of 0.5 Oe (40 A/m) and would produce a residual surface resistance in a superconductor that is orders of magnitude greater than the BCS resistance, rendering the superconductor too lossy for practical use.  For this reason, superconducting cavities are surrounded by magnetic shielding to reduce the field permeating the cavity to typically <10 mOe (0.8 A/m).
Using the above approximations for a niobium a SRF cavity at 1.8 K, 1.3 GHz, and assuming a magnetic field of 10 mOe (0.8 A/m), the surface resistance components would be

RBCS = 4.55 nΩ and
Rres = RH = 3.42 nΩ, giving a net surface resistance
Rs = 7.97 nΩ.  If for this cavity
G = 270 Ω then the ideal quality factor would be
Qo = 3.4×1010.
The Qo just described can be further improved by up to a factor of 2 by performing a mild vacuum bake of the cavity.  Empirically, the bake seems to reduce the BCS resistance by 50%, but increases the residual resistance by 30%.  The plot below shows the ideal Qo values for a range of residual magnetic field for a baked and unbaked cavity.

In general, much care and attention to detail must be exercised in the experimental setup of SRF cavities so that there is not Qo degradation due to RF losses in ancillary components, such as stainless steel vacuum flanges that are too close to the cavity's evanescent fields.  However, careful SRF cavity preparation and experimental configuration have achieved the ideal Qo not only for low field amplitudes, but up to cavity fields that are typically 75% of the magnetic field quench limit.  Few cavities make it to the magnetic field quench limit since residual losses and vanishingly small defects heat up localized spots, which eventually exceed the superconducting critical temperature and lead to a thermal quench.


== Q  vs E ==
When using superconducting RF cavities in particle accelerators, the field level in the cavity should generally be as high as possible to most efficiently accelerate the beam passing through it.  The Qo values described by the above calculations tend to degrade as the fields increase, which is plotted for a given cavity as a "Q vs E" curve, where "E" refers to the accelerating electric field of the TM01 mode.  Ideally, the cavity Qo would remain constant as the accelerating field is increased all the way up to the point of a magnetic quench field, as indicated by the "ideal" dashed line in the plot below.  In reality, though, even a well prepared niobium cavity will have a Q vs E curve that lies beneath the ideal, as shown by the "good" curve in the plot.
There are many phenomena that can occur in an SRF cavity to degrade its Q vs E performance, such as impurities in the niobium, hydrogen contamination due to excessive heat during chemistry, and a rough surface finish.  After a couple decades of development, a necessary prescription for successful SRF cavity production is emerging.  This includes:

Eddy-current scanning of the raw niobium sheet for impurities,
Good quality control of electron beam welding parameters,
Maintaining low cavity temperatures during acid chemistry to avoid hydrogen contamination,
Electropolishing of the cavity interior to achieve a very smooth surface,
High pressure rinsing (HPR) of the cavity interior in a clean room with filtered water to remove particulate contamination,
Careful assembly of the cavity and other vacuum apparatus in a clean room with clean practices,
A vacuum bake of the cavity at 120 °C for 48 hours; this typically improves Qo by a factor of 2.

There remains some uncertainty as to the root cause of why some of these steps lead to success, such as the electropolish and vacuum bake.  However, if this prescription is not followed, the Q vs E curve often shows an excessive degradation of Qo with increasing field, as shown by the "Q slope" curve in the plot below.  Finding the root causes of Q slope phenomena is the subject of ongoing fundamental SRF research.  The insight gained could lead to simpler cavity fabrication processes as well as benefit future material development efforts to find higher Tc alternatives to niobium.
In 2012, the Q(E) dependence on SRF cavities discovered for the first time in such a way that the Q-rise phenomenon was observed in Ti doped SRF cavity. The quality factor increases with increase in accelerating field and was explained by the presence of sharper peaks in the electronic density of states at the gap edges in doped cavities and such peaks being broadened by the rf current. Later the similar phenomenon was observed with nitrogen doping and which has been the current state-of-art cavity preparation for high performance.


== Wakefields and higher order modes (HOMs) ==
One of the main reasons for using SRF cavities in particle accelerators is that their large apertures result in low beam impedance and higher thresholds of deleterious beam instabilities.  As a charged particle beam passes through a cavity, its electromagnetic radiation field is perturbed by the sudden increase of the conducting wall diameter in the transition from the small-diameter beampipe to the large hollow RF cavity.  A portion of the particle's radiation field is then "clipped off" upon re-entrance into the beampipe and left behind as wakefields in the cavity.  The wakefields are simply superimposed upon the externally driven accelerating fields in the cavity.  The spawning of electromagnetic cavity modes as wakefields from the passing beam is analogous to a drumstick striking a drumhead and exciting many resonant mechanical modes.
The beam wakefields in an RF cavity excite a subset of the spectrum of the many electromagnetic modes, including the externally driven TM01 mode.  There are then a host of beam instabilities that can occur as the repetitive particle beam passes through the RF cavity, each time adding to the wakefield energy in a collection of modes.
For a particle bunch with charge q, a length much shorter than the wavelength of a given cavity mode, and traversing the cavity at time t=0, the amplitude of the wakefield voltage left behind in the cavity in a given mode is given by 

  
    
      
        
          V
          
            w
            a
            k
            e
          
        
        =
        
          
            
              q
              
                ω
                
                  o
                
              
              R
            
            
              2
              
                Q
                
                  o
                
              
            
          
        
         
        
          e
          
            j
            
              ω
              
                o
              
            
            t
          
        
         
        
          e
          
            −
            
              
                
                  
                    ω
                    
                      o
                    
                  
                  t
                
                
                  2
                  
                    Q
                    
                      L
                    
                  
                
              
            
          
        
        =
        k
        q
         
        
          e
          
            j
            
              ω
              
                o
              
            
            t
          
        
         
        
          e
          
            −
            
              
                
                  
                    ω
                    
                      o
                    
                  
                  t
                
                
                  2
                  
                    Q
                    
                      L
                    
                  
                
              
            
          
        
      
    
    {\displaystyle V_{wake}={\frac {q\omega _{o}R}{2Q_{o}}}\ e^{j\omega _{o}t}\ e^{-{\frac {\omega _{o}t}{2Q_{L}}}}=kq\ e^{j\omega _{o}t}\ e^{-{\frac {\omega _{o}t}{2Q_{L}}}}}
  
  ,
where:

R is the shunt impedance of the cavity mode defined by

  
    
      
        R
        =
        
          
            
              
                (
                
                  ∫
                  
                    
                      
                        E
                        →
                      
                    
                    ⋅
                    d
                    l
                  
                
                )
              
              
                2
              
            
            
              P
              
                d
              
            
          
        
        =
        
          
            
              V
              
                2
              
            
            
              P
              
                d
              
            
          
        
      
    
    {\displaystyle R={\frac {\left(\int {{\overrightarrow {E}}\cdot dl}\right)^{2}}{P_{d}}}={\frac {V^{2}}{P_{d}}}}
  
  ,
E is the electric field of the RF mode,
Pd is the power dissipated in the cavity to produce the electric field E,
QL is the "loaded Q" of the cavity, which takes into account energy leakage out of the coupling antenna,
ωo is the angular frequency of the mode,
the imaginary exponential is the mode's sinusoidal time variation,
the real exponential term quantifies the decay of the wakefield with time, and

  
    
      
        k
        =
        
          
            
              
                ω
                
                  o
                
              
              R
            
            
              2
              
                Q
                
                  o
                
              
            
          
        
      
    
    {\displaystyle k={\frac {\omega _{o}R}{2Q_{o}}}}
  
  is termed the loss parameter of the RF mode.
The shunt impedance R can be calculated from the solution of the electromagnetic fields of a mode, typically by a computer program that solves for the fields.  In the equation for Vwake, the ratio R/Qo serves as a good comparative measure of wakefield amplitude for various cavity shapes, since the other terms are typically dictated by the application and are fixed.  Mathematically,

  
    
      
        
          
            R
            
              Q
              
                o
              
            
          
        
        =
        
          
            
              V
              
                2
              
            
            
              
                ω
                
                  o
                
              
              U
            
          
        
        =
        
          
            
              2
              
                
                  (
                  
                    ∫
                    
                      
                        
                          E
                          →
                        
                      
                      ⋅
                      d
                      l
                    
                  
                  )
                
                
                  2
                
              
            
            
              
                ω
                
                  o
                
              
              
                μ
                
                  o
                
              
              ∫
              
                
                  |
                
                
                  
                    H
                    →
                  
                
                
                  
                    |
                  
                  
                    2
                  
                
                d
                V
              
            
          
        
        =
        
          
            
              2
              k
            
            
              ω
              
                o
              
            
          
        
      
    
    {\displaystyle {\frac {R}{Q_{o}}}={\frac {V^{2}}{\omega _{o}U}}={\frac {2\left(\int {{\overrightarrow {E}}\cdot dl}\right)^{2}}{\omega _{o}\mu _{o}\int {|{\overrightarrow {H}}|^{2}dV}}}={\frac {2k}{\omega _{o}}}}
  
  ,
where relations defined above have been used.  R/Qo is then a parameter that factors out cavity dissipation and is viewed as measure of the cavity geometry's effectiveness of producing accelerating voltage per stored energy in its volume.  The wakefield being proportional to R/Qo can be seen intuitively since a cavity with small beam apertures concentrates the electric field on axis and has high R/Qo, but also clips off more of the particle bunch's radiation field as deleterious wakefields.
The calculation of electromagnetic field buildup in a cavity due to wakefields can be complex and depends strongly on the specific accelerator mode of operation.  For the straightforward case of a storage ring with repetitive particle bunches spaced by time interval Tb and a bunch length much shorter than the wavelength of a given mode, the long term steady state wakefield voltage presented to the beam by the mode is given by

  
    
      
        
          V
          
            s
            s
             
            w
            a
            k
            e
          
        
        =
        
          V
          
            w
            a
            k
            e
          
        
        
          (
          
            
              
                1
                
                  1
                  −
                  
                    e
                    
                      −
                      τ
                    
                  
                  
                    e
                    
                      j
                      δ
                    
                  
                
              
            
            −
            
              
                1
                2
              
            
          
          )
        
      
    
    {\displaystyle V_{ss\ wake}=V_{wake}\left({\frac {1}{1-e^{-\tau }e^{j\delta }}}-{\frac {1}{2}}\right)}
  
  ,
where:

  
    
      
        τ
        =
        
          
            
              
                ω
                
                  o
                
              
              
                T
                
                  b
                
              
            
            
              2
              
                Q
                
                  L
                
              
            
          
        
      
    
    {\displaystyle \tau ={\frac {\omega _{o}T_{b}}{2Q_{L}}}}
  
 is the decay of the wakefield between bunches, and
δ is the phase shift of the wakefield mode between bunch passages through the cavity.
As an example calculation, let the phase shift δ=0, which would be close to the case for the TM01 mode by design and unfortunately likely to occur for a few HOM's.  Having δ=0 (or an integer multiple of an RF mode's period, δ=n2π) gives the worse-case wakefield build-up, where successive bunches are maximally decelerated by previous bunches' wakefields and give up even more energy than with only their "self wake".  Then, taking ωo = 2π 500 MHz, Tb=1 μs, and QL=106, the buildup of wakefields would be Vss wake=637×Vwake.  A pitfall for any accelerator cavity would be the presence of what is termed a "trapped mode".  This is an HOM that does not leak out of the cavity and consequently has a QL that can be orders of magnitude larger than used in this example.  In this case, the buildup of wakefields of the trapped mode would likely cause a beam instability.  The beam instability implications due to the Vss wake wakefields is thus addressed differently for the fundamental accelerating mode TM01 and all other RF modes, as described next.


=== Fundamental accelerating mode TM010 ===
The complex calculations treating wakefield-related beam stability for the TM010 mode in accelerators show that there are specific regions of phase between the beam bunches and the driven RF mode that allow stable operation at the highest possible beam currents.  At some point of increasing beam current, though, just about any accelerator configuration will become unstable.  As pointed out above, the beam wakefield amplitude is proportional to the cavity parameter R/Qo, so this is typically used as a comparative measure of the likelihood of TM01 related beam instabilities.  A comparison of R/Qo and R for a 500 MHz superconducting cavity and a 500 MHz normal-conducting cavity is shown below.  The accelerating voltage provided by both cavities is comparable for a given net power consumption when including refrigeration power for SRF.  The R/Qo for the SRF cavity is 15 times less than the normal-conducting version, and thus less beam-instability susceptible.  This one of the main reasons such SRF cavities are chosen for use in high-current storage rings.


=== Higher order modes (HOMs) ===

In addition to the fundamental accelerating TM010 mode of an RF cavity, numerous higher frequency modes and a few lower-frequency dipole modes are excited by charged particle beam wakefields, all generally denoted higher order modes (HOMs).  These modes serve no useful purpose for accelerator particle beam dynamics, only giving rise to beam instabilities, and are best heavily damped to have as low a QL as possible.  The damping is accomplished by preferentially allowing dipole and all HOMs to leak out of the SRF cavity, and then coupling them to resistive RF loads.  The leaking out of undesired RF modes occurs along the beampipe, and results from a careful design of the cavity aperture shapes.  The aperture shapes are tailored to keep the TM01 mode "trapped" with high Qo inside of the cavity and allow HOMs to propagate away.  The propagation of HOMs is sometimes facilitated by having a larger diameter beampipe on one side of the cavity, beyond the smaller diameter cavity iris, as seen in the SRF cavity CAD cross-section at the top of this wiki page.  The larger beampipe diameter allows the HOMs to easily propagate away from the cavity to an HOM antenna or beamline absorber.
The resistive load for HOMs can be implemented by having loop antennas located at apertures on the side of the beampipe, with coaxial lines routing the RF to outside of the cryostat to standard RF loads.  Another approach is to place the HOM loads directly on the beampipe as hollow cylinders with RF lossy material attached to the interior surface, as shown in the adjacent image.  This "beamline load" approach can be more technically challenging, since the load must absorb high RF power while preserving a high-vacuum beamline environment in close proximity to a contamination-sensitive SRF cavity.  Further, such loads must sometimes operate at cryogenic temperatures to avoid large thermal gradients along the beampipe from the cold SRF cavity.  The benefit of the beamline HOM load configuration, however, is a greater absorptive bandwidth and HOM attenuation as compared to antenna coupling.  This benefit can be the difference between a stable vs. an unstable particle beam for high current accelerators.


== Cryogenics ==

A significant part of SRF technology is cryogenic engineering.  The SRF cavities tend to be thin-walled structures immersed in a bath of liquid helium having temperature 1.6 K to 4.5 K.  Careful engineering is then required to insulate the helium bath from the room-temperature external environment.  This is accomplished by:

A vacuum chamber surrounding the cold components to eliminate convective heat transfer by gases.
Multi-layer insulation wrapped around cold components.  This insulation is composed of dozens of alternating layers of aluminized mylar and thin fiberglass sheet, which reflects infrared radiation that shines through the vacuum insulation from the 300 K exterior walls.
Low thermal conductivity mechanical connections between the cold mass and the room temperature vacuum vessel.  These connections are required, for example, to support the mass of the helium vessel inside the vacuum vessel and to connect the apertures in the SRF cavity to the accelerator beamline.  Both types of connections transition from internal cryogenic temperatures to room temperature at the vacuum vessel boundary.  The thermal conductivity of these parts is minimized by having small cross sectional area and being composed of low thermal conductivity material, such as stainless steel for the vacuum beampipe and fiber reinforced epoxies (G10) for mechanical support.  The vacuum beampipe also requires good electrical conductivity on its interior surface to propagate the image currents of the beam, which is accomplished by about 100 μm of copper plating on the interior surface.
The major cryogenic engineering challenge is the refrigeration plant for the liquid helium.  The small power that is dissipated in an SRF cavity and the heat leak to the vacuum vessel are both heat loads at very low temperature.  The refrigerator must replenish this loss with an inherent poor efficiency, given by the product of the Carnot efficiency ηC and a "practical" efficiency ηp.  The Carnot efficiency derives from the second law of thermodynamics and can be quite low.  It is given by

  
    
      
        
          η
          
            C
          
        
        =
        
          
            {
            
              
                
                  
                    
                      
                        T
                        
                          c
                          o
                          l
                          d
                        
                      
                      
                        
                          T
                          
                            w
                            a
                            r
                            m
                          
                        
                        −
                        
                          T
                          
                            c
                            o
                            l
                            d
                          
                        
                      
                    
                  
                  ,
                
                
                  
                    
                      if 
                    
                  
                  
                    T
                    
                      c
                      o
                      l
                      d
                    
                  
                  <
                  
                    T
                    
                      w
                      a
                      r
                      m
                    
                  
                  −
                  
                    T
                    
                      c
                      o
                      l
                      d
                    
                  
                
              
              
                
                  1
                  ,
                
                
                  
                    
                      otherwise
                    
                  
                
              
            
            
          
        
      
    
    {\displaystyle \eta _{C}={\begin{cases}{\frac {T_{cold}}{T_{warm}-T_{cold}}},&{\mbox{if }}T_{cold}<T_{warm}-T_{cold}\\1,&{\mbox{otherwise}}\end{cases}}}
  

where

Tcold is the temperature of the cold load, which is the helium vessel in this case, and
Twarm is the temperature of the refrigeration heat sink, usually room temperature.
In most cases Twarm =300 K, so for Tcold ≥150 K the Carnot efficiency is unity.  The practical efficiency is a catch-all term that accounts for the many mechanical non-idealities that come into play in a refrigeration system aside from the fundamental physics of the Carnot efficiency.  For a large refrigeration installation there is some economy of scale, and it is possible to achieve ηp in the range of 0.2–0.3.  The wall-plug power consumed by the refrigerator is then

  
    
      
        
          P
          
            w
            a
            r
            m
          
        
        =
        
          
            
              P
              
                c
                o
                l
                d
              
            
            
              
                η
                
                  C
                
              
               
              
                η
                
                  p
                
              
            
          
        
      
    
    {\displaystyle P_{warm}={\frac {P_{cold}}{\eta _{C}\ \eta _{p}}}}
  
  ,
where

Pcold is the power dissipated at temperature Tcold .
As an example, if the refrigerator delivers 1.8 K helium to the cryomodule where the cavity and heat leak dissipate Pcold=10 W, then the refrigerator having Twarm=300 K and ηp=0.3 would have ηC=0.006 and a wall-plug power of Pwarm=5.5 kW.  Of course, most accelerator facilities have numerous SRF cavities, so the refrigeration plants can get to be very large installations.

The temperature of operation of an SRF cavity is typically selected as a minimization of wall-plug power for the entire SRF system.  The plot to the right then shows the pressure to which the helium vessel must be pumped to obtain the desired liquid helium temperature.  Atmospheric pressure is 760 Torr (101.325 kPa), corresponding to 4.2 K helium.  The superfluid λ point occurs at about 38 Torr (5.1 kPa), corresponding to 2.18 K helium.  Most SRF systems either operate at atmospheric pressure, 4.2 K, or below the λ point at a system efficiency optimum usually around 1.8 K, corresponding to about 12 Torr (1.6 kPa).


== See also ==
Cavity quantum electrodynamics
Circuit quantum electrodynamics


== References ==, source_sentence: superconductivity explained
sentences: A vortex (pl. vortices) is a spinning, often turbulent, flow of fluid. The motion of the fluid swirling rapidly around a center is called a vortex. The speed and rate of rotation of the fluid are greatest at the center and become smaller with distance from the center. Related pages Vorticity Wormhole Other websites Dust Devil Movie A short movie showing many spinning vortices of varying sizes Video of two water vortex rings colliding Archived 2005-12-09 at the Wayback Machine (MPEG) BubbleRings.com Web site on "bubble rings", which are underwater rings made of air formed from vortices. The site has some information on how these rings work. Chapter 3 Rotational Flows: Circulation and Turbulence Archived 2006-09-05 at the Wayback Machine Video of Vortex Formation and Annihilation in a Superconductor (By Joel Thorarinson Archived 2008-10-08 at the Wayback Machine), The ABSURDITY of Quantum Mechanics at LARGE SCALES! | Claim your SPECIAL OFFER for MagellanTV here: https://try.magellantv.com/arvinash  Start your free trial TODAY so you can watch “Virtual Universe” about computer simulations of the universe, and the rest of MagellanTV’s science collection: https://www.magellantv.com/series/cosmic-vistas

REFERENCES
Quantum tunneling: https://youtu.be/YstJxj30hzs
Superconductivity: https://youtu.be/vruYFOlM1-Q
Quantum entanglement: https://youtu.be/unb_yoj1Usk
Quantum mechanics explained: https://youtu.be/-hxljpx, INTRODUCTION
Question: What is the diﬀerence between a Fermi liquid and a non-Fermi liquid?
Answer: The diﬀerence is the same as the one between bananas and non-bananas.
Boris Altschuler (2001)
Unconventional or “nodal” superconductors appeared on the scene in 1979, when the heavy-fermion superconductor
CeCu2Si2 and the organic superconducting Bechgaard salts (TMTSF)2PF6 were discovered. Since then, many more
heavy-fermion superconductors [1] and organic superconductors [2] have been synthesized. This development was
followed by the epoch making discovery of high-Tc cuprate superconductor La2−xBaxCuO4 by Bednorz and M¨uller
[3] with the superconducting transition temperature Tc = 35K in 1986. Within a few years new classes of high-Tc
cuprates emerged, including YBa2Cu3O6+δ, La2−xSrxCuO4, Bi2Sr2Ca1−xYxCu2O8+δ (Bi2212); and HgBa2CaCu2O6
with Tc = 145K. The subsequent enthusiasm and confusion are well documented in an early review by Enz [4].
Confusion? Yes, initially it was thought that Landau’s Fermi liquid theory [5] and the BCS theory [6] were no longer
applicable.
Among many proposals one of the most inﬂuential were Anderson’s dogmas [7], which can be summarized as follows:
a. The action takes place in the CuO2 plane common to all high-Tc cuprate superconductors.
b. The undoped state is a Mott insulator with antiferromagnetic (AF) order. Upon doping superconductivity
appears. Therefore the simplest Hamiltonian is the two-dimensional (2D) one-band Hubbard model:
H = −t
X
⟨i,j⟩α
(c+
iαcjα + h.c) + U
X
i
nini,
(1)
where ⟨i, j⟩connects the nearest neighbors in the 2D square lattice. The c+
iα and ciα are creation and annihilation
operator for the holes at the site i with spin α and ni = c+
ici.
c. As a possible ground state of Eq. (1) Anderson proposed the resonating valence bond (RVB) state:
Ψ =
Y
i
(1 −di)|BCS⟩,
(2)
where |BCS⟩is the BCS state for s-wave superconductors [6], and Q
i(1 −di) with di = nini is called the
Gutzwiller operator. Q
i(1 −di) annihilates all doubly occupied states.
In spite of tremendous eﬀorts spent on both Eqs . (1) and (2) it has been diﬃcult to ﬁnd solutions in two dimensions.
On the other hand the 1D version of Eq. (1) is now completely understood [8,9]. In the meantime the perturbative
analyses based on Eq. (1) predict BCS d-wave superconductivity in high-Tc cuprates [10,11,12,13].
2
FIG. 1:
Local density of states around a single vortex for d-wave (left) and s-wave superconductivity.
High-quality single crystals of YBCO, LSCO and thin ﬁlms of Bi2212 became available around 1992. The d-wave
superconductivity in these high-Tc cuprates was established in 1994. Among many experiments, angle-resolved pho-
toemission spectroscopy (ARPES) [14] and Josephson interferometry [15,16] played a crucial role. Around this time
several theoretical groups started to analyze the physical properties of d-wave superconductivity within the BCS
framework [17,18,19,20]. In 1993 Patrick Lee [21] demonstrated the universal heat conduction in d-wave supercon-
ductors. Furthermore the thermal conductivity was shown to increase with increasing impurity scattering [19]. This
counterintuitive behavior was observed in the Zn-substituted YBCO in [22]. The electronic contribution to the ther-
mal conductivity in d-wave superconductors is proportional to T at low temperatures (i.e. T ≪∆where ∆is the
maximum value of the energy gap). Here we assume ∆(k) = ∆cos(2φ) and φ is the angle k in the a −b plane makes
from the a axis. Then in the limit of no impurity scattering (i.e. Γ →0 where Γ is the quasiparticle scattering rate
in the normal state) the thermal conductivity is given by
κ00
T
= k2
B
3ℏ
v
v2
n,
(3)
where κ00/T = limΓ→0 κ/T , and v/v2 = EF /∆and n is the quasiparticle density or the hole density. The velocities
v and v2 are deﬁned from the quasiparticle energy at the Dirac cone
Ek =
q
v2(k∥−kF )2 + v2
2k2
⊥,
(4)
where v is the Fermi velocity, k∥is the radial component of the wave vector and k⊥the component perpendicular to
the Fermi surface.
Later we shall derive Eq.(3) in Section IV. The thermal conductivity in single crystals of optimally doped YBCO
and Bi2212 below T = 1K was measured by May Chiao et al [23,24]. Making use of Eq. (3) they found ∆/EF = 1/10
and 1/14 for Bi2212 and YBCO respectively. These remarkable ratios imply:
a. High-Tc cuprates are described by the BCS theory of d-wave superconductivity. They are far away from the
Bose-Einstein condensation limit which requires ∆∼EF [25].
b. According to the Ginzburg criterion, ﬂuctuation eﬀects are of the order ∆/EF ∼(pF ξ)−1. In other words they
should be at most 10%. This appears to exclude large phase ﬂuctuations and stripe phases discussed in [26,27].
c. For ∆/EF = 1/10 there are hundreds of quasiparticle bound states around the core of a single vortex in d-wave
superconductors [28,29]. The radial (r) dependence of the local quasiparticle density of states is very similar
to the one obtained for s-wave superconductivity [30]. We show in Fig. 1 the local density of states around a
single vortex for a d-wave and s-wave superconductor. These are well-known bound states ﬁrst discovered by
Caroli, de-Gennes and Matricon [31,32] for s-wave superconductors.
In earlier works [33,34,35,36] it was asserted that there would be no bound state around a single vortex in d-wave
superconductors.
However in these works it was assumed ∆≃EF in order to facilitate the numerical analysis
3
0.05
0.15
0.25
100
500
0
x
T
Tc
T ∗
AF
d-wave SC
pseudogap phase
FIG. 2: The phase diagram of hole-doped high-Tc cuprate superconductors.
based on the lattice version of the Bogoliubov de Gennes equation.
This assumption (∆≃EF ) eliminates the
bound states in these numerical analyses.
If we limit ourselves to the experiments on single crystals of high-Tc
cuprate superconductors, we ﬁnd that most of the behaviors observed are consistent with the BCS theory of d-
wave superconductivity [37]. Also it is better to use the continuum version of the Bogoliubov de Gennes equation.
which is proposed in [38] and used in [28,29]. More recently a similar analysis is extended for a vortex in an f-wave
superconductor [39].
From a theoretical point of view the universality of the Landau Fermi liquid in 2D systems was demonstrated within
the renormalization group analysis [40,41,42]. The quasiparticles in the normal state of high-Tc cuprates appear to
be a Fermi liquid state. Furthermore the quasiparticles in d-wave superconductors are in a BCS-Fermi liquid state
with the quasiparticle energy
Ek =
q
v2(k∥−kF )2 + ∆2 cos2(2φ).
(5)
In the vicinity of the Dirac cone Eq. (5) reduces to Eq. (4).
Another consequence of the renormalization group analysis is that the instability of the normal Fermi liquid is
related to the infrared divergence in the particle-particle (and the hole-hole) channel or the particle-hole channel. The
former results in conventional or unconventional superconductivity and the latter in conventional or unconventional
density wave states. Therefore it is of great interest to look at the phase diagram of the high-Tc cuprates from this
point of view. We show a schematic phase diagram in Fig. 2.
Let us consider the hole-doped region. In the vicinity of x = 0 there is an antiferromagnetic (AF) insulating phase,
which is a Mott insulator (MI). As the hole-doping x is increased, the AF order is rapidly suppressed around x = 3%.
Then a superconducting region develops for 5% < x < 25%. This is sometimes called “the superconducting dome”.
Also in the underdoped region there is the pseudogap (PG) regime. The nature of the pseudogap is still hotly
debated. There is evidence that it is a d-wave density wave (dDW) [43,44,45,46] with an energy gap ∆(k) = ∆cos(2φ).
Earlier proposals [43,45] have only considered the commensurate case with Z2 symmetry something similar to a ﬂux
phase [47,48].
On the contrary, we consider the incommensurate d-wave density wave [44,46] case where the condensate has U(1)
symmetry as in conventional charge-density wave systems (CDW). Here the phase vortex [49] is the most common
topological defect. Moreover the phase diagram suggests a coexistence region of dDW and d-wave superconductivity
[43,45]. Then the phenomenological gap introduced by Tallon and Loram [50] should be the energy gap associated
with dDW.
Recently Laughlin [51] has pointed out that the wave function (2) is impractical, since the Gutzwiller operator has
no inverse. Instead, he proposed to analyze Eq. (2) with a modiﬁed Gutzwiller operator which has an inverse. For
example
Y
i
(1 −di)
→
Y
i
(1 −αdi),
(6)
4
FIG. 3: From top left: 2D f-wave in Sr2RuO4, dx2−y2-wave in CeCoIn5 and in κ-(ET)2Cu(NCS)2, (s + g)-wave in YNi2B2C,
(p + h)-wave in the A phase of PrOs4Sb12, (p + h)-wave in the B phase of PrOs4Sb12.
with α < 1. He called this “gossamer superconductivity”. “Gossamer” means ﬁlmy cobweb or something light, fragile
but strong. If we look at this new wave function with Eq. (6), we realize that this is an example of competing order
parameters [52,53]. We shall come back to this question at the end of our course.
In a seminal paper Volovik [54] has shown that the quasiparticle density of states in the vortex state of the d-wave
superconductors is calculable within a semiclassical approximation. The predicted
√
H dependence of the speciﬁc
heat has been conﬁrmed experimentally in single crystals of YBCO [55,56], LSCO [57] and Sr2RuO4 [58,59]. This
semiclassical approach has been extended in a variety of directions [60,61,62,63,64,65].
When high-quality single
crystals in the extremely clean limit (i.e. l ≫ξ where l is the quasiparticle mean free path and ξ the superconducting
coherence length) are available, the angle- dependent thermal conductivity in the vortex state provides unique access
to the gap symmetry ∆(k). In the last few years Izawa et al. have determined ∆(k) in Sr2RuO4 [66], CeCoIn5 [67],
κ-(ET)2Cu(NCS)2 [68], YNi2B2C [69] and PrOs4Sb12 [70,71]. These gap functions are shown in Fig. 3.
In the following, we ﬁrst focus on the quasiparticle spectrum in a variety of nodal superconductors. Then the eﬀect of
impurity scattering and the universal heat conduction is brieﬂy summarized. The quasiclassical approximation in the
vortex state in nodal superconductors is the central part of this course. Also the properties of nodal superconductivity
in YNi2B2C and PrOs4Sb12 are brieﬂy summarized. In the last chapter we discuss unconventional density wave and
gossamer superconductivity, which indicate new directions to follow in new materials.
5
II.
QUASIPARTICLE SPECTRUM IN NODAL SUPERCONDUCTORS
Following the BCS paper [5] we consider the eﬀective Hamiltonian
H =
X
k,α
ξ(k)c+
kαckα + 1
2
X
k,α,k′,α′
v(k, k′)c+
k′α′c+
−k′,−α′c−k,−αckα,
(7)
with
v(k, k′) = −⟨|f|2⟩−1V f(k)f(k′),
(8)
and
⟨|f|2⟩=
1
4π
Z
dΩ|f(k)|2
3D
(9)
=
1
(2π)2
Z
dχdφ|f(k)|2
2D
(10)
depending on whether the system is 3D or quasi-2D. In the following, we consider a group of quasi-2D superconductors,
whose quasiparticle density of states, thermodynamics etc.
are identical [64].
We deﬁne quasi 2D systems by a
cylindrical Fermi surface, as in high-Tc cuprates, Sr2RuO4, CeCoIn5, κ-(ET)2Cu(NCS)2 etc. Also we consider a
group of nodal superconductors with f(k) = cos(2φ), sin(2φ) (d-wave), e±iφ cos(χ) (f-wave), cos χ (d-wave), sin χ
(p-wave), e±iφ sin(χ) (d-wave), cos(2χ) (g-wave) etc. Here χ = ckz. Then it can be readily shown (Exercise 1) that
these superconductors have an identical quasiparticle density of states (DOS).
Within the mean-ﬁeld approximation, i.e. the BCS approximation, Eq. (7) is transformed as
H =
X
k,α
Ψ+
k,α(ξ(k)ρ3 + ∆(k)ρ1)Ψk,α −
X
k
|∆(k)|2
v
(11)
The corresponding Nambu-Gor’kov Green function [72,73] is given by
G−1(k, ω) = ω −ξ(k)ρ3 −∆(k)ρ1,
(12)
where the ρi’s are Pauli matrices operating on the Nambu spinor space. For simplicity we consider here only spin
singlet pairing and f as a real function.
Then the poles of the Green function Eq. (12) give the quasiparticle energy
ω = ±
p
ξ2(k) + ∆2(k) ≃±
q
v2(k∥−kF )2 + v2
2k2
⊥.
(13)
The last expression is an approximation near the Dirac cone.
From Eq. (13) the quasiparticle density of states is obtained as [17]
g(E) = Re
*
|E|
p
E2 −∆2|f|2
+
=



2
π|x|K(x) for |x| < 1,
2
πK(1/x) for |x| > 1,
(14)
where x = |E|/∆and K(k) is the complete elliptic integral of the ﬁrst kind. We show in Fig. 4 the quasiparticle
density of states versus x, compared with the one for s-wave superconductor with a full energy gap.
For small energies, the density of states can be expanded as
g(E) ≃|E|/∆
(15)
for |E|/∆≪1.
6
0
0.5
1
1.5
2
E/∆
0
1
2
3
4
N(E)/N0
FIG. 4: The quasiparticle density of states for nodal superconductors (solid line) and for s-wave superconductors (dashed line).
The mean-ﬁeld approximation also gives the gap equation
λ−1 = 2πT ⟨|f|2⟩−1 X′
n
*
f 2
p
ω2n + f 2
+
= ⟨|f|2⟩−1
Z E0
0
dE Re
*
|E|
p
E2 −∆2|f|2
+
tanh( E
2T ),
(16)
which we have written in 2 alternative forms.
Here λ = νN0 is the dimensionless coupling constant, ωn is the
Matsubara frequency and E0 is the cut-oﬀenergy. Also the ωn sum in the ﬁrst equation has to be cut oﬀat ωn ≃E0.
Then within the weak coupling limit we ﬁnd [74]
∆(0)/Tc = 2.14,
(17)
∆(t)/∆(0) ≃
p
1 −t3,
(18)
where t = T/Tc. In Fig. 5 we show ∆(t)/∆(0) versus t together with the approximate expression Eq.(18).
As to the thermodynamics, it is convenient to start with the entropy S given by
S = −4N0
Z ∞
0
dE g(E)[f ln f + (1 −f) ln(1 −f)]
= 4N0
Z ∞
0
dE g(E)[ln(1 + e−βE) + βE(1 + eβE)].
(19)
Here N0 is the quasiparticle density of states at the Fermi surface in the normal state and f = (1 + eβE)−1 is the
Fermi function.
From S, the speciﬁc heat and the thermodynamic critical ﬁeld are obtained by
Cs/T = ∂S
∂T ,
(20)
and
1
8π H2
c (T ) = Fn(T ) −Fs(T ) =
Z Tc
T
dT ′ (Sn(T ′) −Ss(T ′)).
(21)
7
0
0.2
0.4
0.6
0.8
1
t
0
0.2
0.4
0.6
0.8
1
∆(t)/∆(0)
FIG. 5: The temperature dependence of the order parameter (dashed - approximate, solid - exact
Using the density of states g(E), we obtain
Cs
γsT = 27ζ(3)
2π2
T
∆
for T ≪∆,
(22)
and
Hc(0) =
p
2πN0∆(0).
(23)
Similarly the superﬂuid density within the a −b plane is given by
ρs(T )/ρs(0) = 2πT
X
n

∆2f 2
(ω2n + ∆2f 2)3/2

.
(24)
In the limit T →0 this reduces to
ρs(T )/ρs(0) = 1 −2 ln 2
T
∆(0).
(25)
In other words one can obtain ∆(0) from the T linear slope of the superﬂuid density.
There is still a controversy as to the correct expression of the c axis superﬂuid density. The simplest assumptions
[75] give
ρs,c(T )/ρs,c(0) = π
2
∆(t)
∆(0)⟨f tanh(∆f/(2T ))⟩
≃1 −π2
6

T
∆(0)
2
.
(26)
Finally, the spin susceptibility and the nuclear spin lattice relaxation rate are given by
χs/χn = 1 −ρs(T )/ρs(0),
(27)
8
and
T −1
1
/T −1
1n =
Z ∞
0
dE
2T g2(E)sech2( E
2T )
≃π2
3
 T
∆
2
.
(28)
We stress again that these expressions are not only valid for d-wave superconductors in the weak-coupling limit
as in high-Tc cuprates, but for all nodal superconductors with f(k) given above. Therefore in order to explore the
individual gap symmetry, we have to look at other properties.
It is somewhat surprising that all the energy gaps of the identiﬁed nodal superconductors in the quasi-2D systems
belong to the above class of f’s.
Exercises 1.
1.1 Within the weak-coupling theory calculate the jump in the speciﬁc heat at T = Tc
a. for s-wave superconductivity. (Answer: ∆C/Cn = 12/(2ζ(3)) = 1.43)
b. for nodal superconductivity with ∆(k) = ∆f(k). (Answer: ∆C/Cn = 12/(7ζ(3))⟨|f|4⟩/⟨|f|2⟩2)
1.2 Evaluate the ratio ∆(0)/Tc within the weak-coupling theory
a. for s-wave superconductivity. (Answer: ∆(0)/Tc = π/γ = 1.76)
b. for nodal superconductivity. (Answer: ∆(0)/Tc = (π/γ) exp(−⟨f 2⟩−1⟨f 2 ln f⟩))
1.3 Show that the quasiparticle density of states for a group of f’s discussed is the same as given by Eq. (14).
1.4 Express Re⟨f 2/
p
x2 −f 2⟩in terms of complete elliptic integrals. Answer:
Re
*
f 2
p
x2 −f 2
+
=



2
π(K(x) −E(x))
for x < 1,
2
πx(K(1/x) −E(1/x)) for x > 1.
(29)
III.
EFFECT OF IMPURITY SCATTERING
In metals the presence of impurities or foreign atoms is unavoidable. Also, they provide the simplest agents of
quasiparticle relaxation. Therefore the study of impurity scattering is crucial to understand quasiparticle transport
such as electric conductivity and thermal conductivity.
In the early sixties the eﬀect of impurity scattering in s-wave superconductivity was systematically studied in
[76,77,78].
As is well known, nonmagnetic impurity scattering has little eﬀect in s-wave superconductors.
The
superconducting transition temperature and the thermodynamics are almost unaﬀected. The most dominant eﬀect
is a reduction of the quasiparticle mean-free path, as in the normal state, and its consequence on the magnetic
penetration depth. On the other hand, magnetic impurities have a profound eﬀect on s-wave superconductivity. The
superconducting transition temperature is sharply reduced. Also in some cases gapless superconductivity is induced
[78,79].
In contrast, the nonmagnetic impurities have profound eﬀects on nodal superconductors as pointed out in [80,81,82].
In particular, resonant impurity scattering appears to be prevalent. The extreme limit is the unitary limit where a
resonance occurs at E = 0. Such a model has been discussed for d-wave superconductivity in high-Tc cuprates in
[83,84]. The ﬁrst self-consistent studies of impurity scattering in d-wave superconductors were performed in [18,19,20].
For simplicity we shall limit ourselves to the unitary limit. Also we assume that the impurity is point like, i.e. it has
only s-wave scattering amplitude. Then the eﬀect of impurity scattering can be incorporated by ω →˜ω in Eq. (12)
where ˜ω is the renormalized frequency given by [18]
˜ω = ω −π
2 Γ
√
1 −˜x2
˜x

K

1
√
1 −˜x2
−1
,
(30)
9
FIG. 6:
The quasiparticle density of states in the presence of impurities, for several impurity concentrations.
and ˜x = ˜ω/∆and Γ = ni(πN0)−1 is the quasi-particle scattering rate in the normal state. Then the quasiparticle
density of states in the presence of impurities is given by
g(E, Γ) = 2
π Re

˜x
√
1 −˜x2 K

1
√
1 −˜x2

.
(31)
The quasiparticle density of states in the presence of impurities is shown in Fig. 6. We note a rapid appearance of
disorder-induced spectral weight at E = 0. Indeed g(0, Γ) is given by
g(0, Γ) =
C0
p
1 + C2
0
K
 
1
p
1 + C2
0
!
,
(32)
where C0 is obtained from
C2
0
p
1 + C2
0
= π
2
Γ
∆K−1
 
1
p
1 + C2
0
!
.
(33)
In the limit Γ/∆→0, both Eq. (32) and Eq. (33) reduce to
g(0, Γ) = C0 ln( 4
C0
) ≃
 
πΓ
2∆ln−1
 
4
r
2∆
πΓ
!!1/2
,
(34)
and
C2
0 ln(4/C0) ≃πΓ
2∆.
(35)
10
FIG. 7: ∆(0, Γ)/∆00 (dashed line), Tc/Tc0 (solid line) and g(0, Γ)/N0 (dashed-dotted line) are shown as a function of Γ/Γc in
the unitary limit.
The gap equation in the presence of impurity scattering is given by
λ−1 = 2πT ⟨f 2⟩−1 X′
n
*
f 2
p
˜ω2n + ∆2f 2
+
= 8T
∆
X′
n
p
1 + ˜x2n
 
E
 
1
p
1 + ˜x2n
!
−
˜x2
n
1 + ˜x2n
K
 
1
p
1 + ˜x2n
!!
,
(36)
where ˜xn = ˜ωn/∆and ˜ωn is the renormalized Matsubara frequency.
Then in the limit ∆→0, we obtain the
Abrikosov-Gor’kov formula:
−ln
 Tc
Tc0

= Ψ
1
2 +
Γ
2πTc

−Ψ
1
2

,
(37)
where Tc (Tc0) is the superconducting transition temperature in the presence (absence) of impurities. Here Ψ(z) is
the digamma function.
Note that Eq. (37) is the same as for s-wave superconductors in the presence of magnetic impurities [78]. In nodal
superconductors Γ is due to nonmagnetic impurities, whereas in s-wave superconductors Γ is associated with magnetic
scattering which involves spin ﬂipping.
At T = 0K Eq. (36) reduces to
−ln
∆(Γ, 0)
∆(0, 0)

= 2

f 2 ln(C0 +
q
C2
0 + f 2)

−2Γ
∆
Z ∞
C0
dx x2(1 −E/K)[(1 + x2)E −K],
(38)
where E = E(1/
√
1 + x2) and K = K(1/
√
1 + x2) and C0 has already been deﬁned in Eq. (33). We show in Fig. 7
Tc/Tc0, ∆(0, Γ)/∆(0, 0, ) and g(0) versus Γ/Γc where Γc = πTc0/(2γ) ≃0.8819Tc0.
Surprisingly, ∆(0, Γ)/∆(0, 0) follows very closely Tc/Tc0. Also g(0, Γ) increases very rapidly with Γ. This rapid
increase in the DOS has been measured by the low temperature speciﬁc heat in doped LSCO [85]. Note Cs/(γsT ) =
g(0). As seen from FIG. 8 the experimental data agree very well with the theoretical prediction. Especially the
agreement is almost perfect in the vicinity of the optimally doped LSCO. This suggests strongly that the weak-coupling
BCS theory for d-wave superconductivity is adequate for LSCO. We have already mentioned that ∆(0)/Tc = 2.14 in
11
the weak-coupling limit. We can deduce this ratio for optimally doped LSCO, YBCO and Bi2212. These are 2.15,
2.64 and 5.3 respectively. Indeed LSCO appears to be in the weak-coupling limit whereas YBCO may be in the
intermediate regime. Since the large ratio 5.3 for Bi2212 is rather strange, we wonder if another order parameter is
FIG. 8:
Experimental values for Tc(Γ) for doped LSCO
hidden to make this large energy gap possible (see Section VIII).
The planar superﬂuid density in the presence of impurities is given by
ρs(T, Γ)
ρs(0, 0) = 2πT
∞
X
n=0

∆2f 2
(˜ω2n + ∆2f 2)3/2

.
(39)
For T = 0 this reduces to
ρs(T, Γ)
ρs(0, 0) = 1 −
Γ
∆C0
+ Γ
∆
Z ∞
C0
dx
x2

1 −E
K
2
.
(40)
This sharp decrease in superﬂuid density in Zn-substituted YBCO has been observed by the µ-SR experiments [86].
In Fig. 9 we show ρs(T, Γ)/ρs(0, 0) versus T/Tc0.
Exercises 2.
2.1. Derive Eq. (30).
2.2. Derive Eq. (34) and Eq. (35).
IV.
UNIVERSAL HEAT CONDUCTION
In 1993, Patrick Lee [21] pointed out that the thermal conductivity in d-wave superconductors is linear in T for
T ≪∆(0), and that it takes the universal value κ00/T = k2
Bvn/(3ℏv2) in the limit Γ →0. A further study indicates
that κ/T increases very rapidly with Γ, the quasiparticle scattering due to impurities [19]. This result was used by
May Chiao et al [23,24] to extract ∆(0)/EF of optimally doped Bi2212 and YBCO. By transforming the expression
of the thermal conductivity given by Ambegaokar and Griﬃn [87], the thermal conductivity for T ≪∆(0) is given by
12
[19]
κxx/κn = κyy/κn = Γ
∆

C2
0
(C2
0 + f 2)3/2

= 2Γ
π∆
1
p
1 + C2
0
E
 
1
p
1 + C2
0
!
,
(41)
where C0 and ∆= ∆(0, Γ) have been given in Eq. (33) and Eq. (38) respectively and E(k) is the complete elliptic
integral of the second kind. For later purposes, it is more convenient to normalize κxx by κc
n, the normal state thermal
conductivity, by Γ = Γc = 0.882Tc0 the critical scattering where superconductivity disappears.
κc
n = π2T n
6Γcm.
(42)
Then we can rewrite Eq. (41) as
κxx/κc
n = 2Γc
π∆
1
p
1 + C2
0
E
 
1
p
1 + C2
0
!
= I1(Γ/Γc).
(43)
I1(Γ/Γc) versus Γ/Γc is shown in Fig. 10.
We note that κxx increases monotonically with Γ. In other words, the thermal conductivity increases with the
impurity scattering. This counterintuitive behavior is understood, if one realizes that the impurity scattering produces
quasiparticles due to the pair-breaking eﬀect [19]. Indeed, the predicted Γ dependence of κxx was veriﬁed in Zn-doped
YBCO [22] and more recently in Sr2RuO4 [88]. If we substitute in Eq. (41) f = sin(2φ), cos χ, e±iφ cos χ, sin χ,
e±iφ sin χ or cos(2χ), we will obtain the same result. The planar thermal conductivity can thus not discriminate
between diﬀerent nodal superconductors [89]. The result for the out-of-plane thermal conductivity is of more interest.
We obtain
κzz/κc
n = Γc
∆

(1 −cos(2χ))
C2
0
(C2
0 + |f|2)3/2

= I1(Γ/Γc)
(44)
FIG. 9:
The superﬂuid density versus temperature for various impurity concentrations.
13
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
κxx/κn,    κzz/κn 
Γ/Γ0
I1(Γ/Γ0)
I2(Γ/Γ0)
I3(Γ/Γ0)
FIG. 10:
The functions I1(Γ/Γc), I2(Γ/Γc), I3(Γ/Γc)
for f = cos(2φ), sin(2φ) and cos(2χ). But
κzz/κc
n = 4Γc
π∆
1
p
1 + C2
0
 
(1 + C2
0)E
 
1
p
1 + C2
0
!
−K
 
1
p
1 + C2
0
!!
= I2(Γ/Γc)
(45)
for f = cos χ, e±iφ cos χ and
κzz/κc
n = 4Γc
π∆
C2
0
p
1 + C2
0
 
K
 
1
p
1 + C2
0
!
−E
 
1
p
1 + C2
0
!!
= I3(Γ/Γc)
(46)
for f = sin χ and e±iφ sin χ. We show I1(Γ/Γc), I2(Γ/Γc) and I3(Γ/Γc) versus Γ/Γc in FIG. 10. It is clear that
κzz →0 for f = sin φ and e±iφ sin φ in the limit Γ →0. There is no universal heat conduction. Very recently
thermal conductivity data for κxx and κzz at T = 0.4K and H ∥ˆz in UPd2Al3 was reported [89,90]. This is shown
in Fig. 11. Although the ﬁeld dependence of κii is not the same as its Γ dependence, we conclude ∆(k) in UPd2Al3
is more consistent with ∆(k) ∼cos(2χ) rather than ∆(k) ∼cos χ. The latter has been proposed in [91] based on
antiparamagnon exchange with Q = (0, 0, π/c) [92].
Exercises 3.
3.1. Derive Eq. (41) for a variety of f’s.
3.2. Calculate κzz for these f’s.
14
FIG. 11: Thermal conductivity data for UPd2Al3
V.
QUASICLASSICAL APPROXIMATION
A.
Vortex state
In the presence of a magnetic ﬁeld, all known nodal superconductors enter into the vortex state when the magnetic
ﬁeld exceeds Hc1(T ), the lower critical ﬁeld.
In this vortex state the quantized vortex lines form a regular two
dimensional lattice in the equilibrium conﬁguration. Also each vortex line carries a unit ﬂux φ0 = hc/(2e) = 2.06 ·
10−7Gcm2. The marvelous theory of type II superconductivity was created by Abrikosov [93] in 1957, based on
the phenomenological Ginzburg-Landau theory [94].
The microscopic foundation of GL theory was provided by
Gor’kov [95] after the appearance of the BCS theory [6]. In s-wave superconductors the vortex lattice is usually
hexagonal. But in d-wave superconductors the square vortex lattice is more stable for T/Tc ≤0.9 [96,97]. Indeed
the square vortex lattice was recently observed in LSCO by neutron scattering experiments [98].
Also unlike in
s-wave superconductors, the quasiparticles dominate the low-temperature transport properties in the whole vortex
state in nodal superconductors. The quasiclassical approximation then provides the most practical way to handle the
quasiparticle transport in the vortex state for T ≪∆.
B.
Quasiparticle spectrum in the vortex state
In 1993 Volovik [54] showed how to calculate the quasiparticle DOS in the vortex state of d-wave superconduc-
tors,when T ≪∆(0). Here we shall follow this procedure. First we note that the quasiparticle energy Ek is shifted
to Ek −v · q = Ek −k · vs in the presence of a superﬂow [99]. Here v, 2q and vs are the quasiparticle velocity, the
pair momentum and the superﬂuid velocity respectively. Also for a class of nodal superconductors we considered in
Section II, we obtain g(E) = |E|/∆for |E| ≪∆.
In the presence of a superﬂow this is generalized as
g(E, H) = ∆−1⟨|E −v · q|⟩,
(47)
or
g(0, H) ≡g(H) = ∆−1⟨|v · q|⟩,
(48)
where ⟨. . . ⟩denotes the averages over the Fermi surface and over the vortex lattice. Here v · q is called the Doppler
shift.
15
FIG. 12:
Quasi-two-dimensional cylindrical Fermi surface
Let us consider d-wave superconductors in a magnetic ﬁeld H ∥ˆc. Then Eq. (48) becomes
g(H) =
4v
π∆d2
Z d
0
r dr
2π
Z π/2
0
dα cos α
Z 2π
0
dφ
2π δ(cos(2φ))
= 2
π2
v
√
eH
∆
.
(49)
Here d = 1/
√
eH and we assumed for simplicity a square vortex lattice with lattice constant d. Also, in the present
calculation we took q = ˆφ/(2r), where r is the distance from the center of a vortex and α is the angle between v and
q. We treated the average over the vortex lattice `a la Wigner-Seitz. Finally in the earlier treatments [60,61,63,64] the
factor π−1 coming from (2π)−1 R 2π
0
dφ δ(cos(2φ)) was missing. From Eq. (49) the speciﬁc heat and other observables
[100] are obtained as
Cs/γnT = g(H),
κs/κn = g(H),
ρs(H)
ρs(0) = 1 −g(H).
(50)
We have already mentioned that the
√
H dependence of the speciﬁc heat in the nodal superconductors was observed
in YBCO [55,56], LSCO [57] and Sr4RuO4 [58,59]. However, in the earlier analysis [55,56] a factor ∼0.3 was missing.
We believe the π−1 we ﬁnd here accounts for this missing factor. In other words, the semiclassical result is not only
qualitatively but also quantitatively accurate. It is very easy to work out ⟨|v · q|⟩for other classes of ∆(k) as discussed
in Section II. Then it is easily seen that the conﬁguration H ∥ˆc cannot discriminate d-wave superconductors from
other nodal superconductors.
C.
Extension to the arbitrary ﬁeld orientation
In order to get a handle on the gap symmetry of ∆(k), it is necessary to consider the case of arbitrary ﬁeld
orientations. For simplicity we limit ourselves to the case where H is in the a −b plane with an angle φ from the
a axis. Precisely this conﬁguration is considered in [62]. Unfortunately, however, a rather unrealistic Fermi surface
was considered. Instead we consider the quasi-2D Fermi surface shown in Fig. 12 [101]. In the present conﬁguration
the vortex loses circular symmetry around the vortex axis. Therefore we ﬁnd it is very useful to introduce a scale
16
transformation as in [102] to make a circular vortex. For f = cos(2φ) we ﬁnd [63]
g(H) = 2
π ˜v
√
eH
q
sin2 χ + sin2(φ −φ′)

≃2
π ˜v
√
eH(0.955 + 0.0285 cos(4φ)),
(51)
where ˜v = √vcv and vc is the Fermi velocity parallel to the c axis and φ is the angle H makes from the a axis. We
thus ﬁnd a 3% fourfold term (∼cos(4φ)) with a ≤.3% angular variation of g(H) The φ dependence of the speciﬁc
heat in YBCO was studied by Wang et al [103]. They could not ﬁnd this small fourfold term which is within their
experimental error.
When ∆(k) has a horizontal node as in Sr2RuO4 [66] g(H) does not exhibit the φ dependence. In particular for
f = e±iφ cos χ we obtain
⟨|vq|⟩= 2
π2 ˜v
√
eH
√
2E(1/
√
2) = 1.216 2
π2 ˜v
√
eH
(52)
Therefore the speciﬁc heat of Sr2RuO4 by Deguchi et al [104] appears somewhat puzzling. Their data for T > 0.5K is
consistent with 2D f-wave superconductor. However, something diﬀerent appears to happen below T = 0.3K. From
experience with YNi2B2C we know sharp cusps in g(H) imply point nodes and not line nodes [105]. Therefore the
speciﬁc heat for T < 0.3K suggests the appearance of point like minigaps for k ∥(1, 0, 0), (0, 1, 0) etc. These may be
reproduced with the secondary energy gap
∆2(k) ∼e±iφ cos χ(1 −a cos(4φ) cos2 χ)
with a ≲1. Then the secondary gap has the same symmetry as a 2D f-wave superconductor.
Exercises 4.
4.1. Calculate g(H) when H is rotated within the z −x plane. Consider here f = cos χ, sin χ and cos(2χ).
4.2. Calculate g(H) when H is rotated within the x −y plane. Consider the same set of f’s as above.
VI.
MAGNETOTHERMAL CONDUCTIVITY
Experiments on the angular dependent thermal conductivity in YBCO [106,107,108,109] indicate that the thermal
conductivity is sensitive to the nodal directions, although theoretical interpretation of this data has been given only
recently [65]. In order to calculate the thermal conductivity it is crucial to incorporate the eﬀects of impurity scattering
and of the Doppler shift on an equal footing. Then there arise natually two limiting cases: a) the superclean limit
(Γ∆)1/2 ≪< |v · q| > and b) the clean limit (Γ∆)1/2 ≫< |v · q| >≫Γ.
First it is necessary to determine the quasiparticle lifetime in the presence of impurity scattering and the Doppler
shift. This is given by [63]
C0 = πΓ
2∆[< C0 ln(
2
p
C2
0 + x2 ) + x tan−1(x/C0) >]−1
(53)
where ∆C0 = Im(˜ω) at ω = 0 is the quasi-particle relaxation rate on the Fermi surface and x = |v · q|/∆. This
formula applies to the class of f’s introduced in II, which include d-wave superconductivity with f = cos(2φ). Also
in the later analysis we assume that H lies in the ab-plane.
A.
Superclean limit
In the superclean limit we can assume that x ≫C0. Then Eq.(52) is solved as
C0 =
Γ
∆< x >−1 −2
π ( Γ
∆)2 < x >−3 [ln(2/x) −1]
(54)
17
On the other hand, in the clean limit we obtain
C2
0 ln(2/C0) = πΓ
2∆−1
2 < x2 > + . . .
(55)
Also when H is perpendicular to the a-b plane we obtain
< x2 > =
1
4π
v2eH
∆2
ln
∆
v
√
eH
(56)
The thermal conductivity within the a-b plane and in the limit T ≪∆is given by [63]
κ∥/κn(≡κxx/κn = κyy/κn) = πΓ
4∆
*
1 +
C2
0+x2−f 2
|(C0+ix)2+f 2|
Re
p
(C0 + ix)2 + f 2
+
(57)
where κn = π2T n
6Γm is the thermal conductivity in the normal state. Then in the superclean limit Eq.(56) reduces to
κ∥/κn = πΓ
2∆
θ(x2 −f 2)
xC0
(x2 −f 2)1/2

(58)
=
πΓ
4∆C0
< x >= π
4 < x >2
(59)
=
1
π3
v2(eH)
∆2
(60)
In the last step we assumed H ∥c. The H-linear thermal conductivity was ﬁrst observed in Sr2RuO4 at T = 0.3K [67],
which indicates nodal superconductivity with f = e±iφ cos(χ) in Sr2RuO4 and that the system is in the superclean
limit. More recently the H-linear thermal conductivity was observed in PrOs4Sb12 [71] for T ≤0.3K.
Now in the presence of a magnetic ﬁeld in the a-b plane, we must ﬁrst generalize Eq. 57 as [63]
κxx/κn = π
4 < x >< (1 + cos(2φ
′))x >
(61)
κyy/κn = π
4 < x >< (1 −cos(2φ
′))x >
(62)
and
κxy/κn = π
4 < x >< sin(2φ
′)x >
(63)
where the angle φ
′ refers to the direction of the quasiparticle wave vector within the a-b plane. Then for f = cos(2φ)
Eq.(58) gives
κ∥/κn(≡κxx/κn = κyy/κn) = ˜v2eH
π3∆2 (0.955 + 0.0285 cos(4φ))2
(64)
and
κxy/κn = −˜v2eH
π3∆2 (0.265 sin(2φ)(0.955 + +0.0285 cos(4φ))
(65)
Here ˜v = √vvc. Therefore κ∥should exhibit the fourfold term with a magnitude variation of the order of ∼6%.
In early experiments [106,107,109] a fourfold term of comparative magnitude was found in YBCO but of opposite
sign. Similarly Oca˜na and Esquinazi [108] found the Hall thermal conductivity ∼sin(2φ), but of opposite sign. These
problems were clariﬁed in [65]. All experiments on YBCO were performed at T ≫< |v · q| >, whereas the present
theory applies only for T ≪< |v · q| >.
Also, thermal conductivity measurements in CeCoIn5 [68] and κ-(ET)2Cu(NCS)2 [69] have seen evidence for a
fourfold term. If we assume that these experiments are done in the region T ≪< |v · q| >, we have to interpret
the data as dxy-wave superconductivity and dx2−y2 symmetry for CeCoIn5 and κ-(ET)2Cu(NCS)2 respectively. The
former identiﬁcation appears to be consistent with the magnetospeciﬁc heat data of CeCoIn5 reported in [110]. For
f = e±iφ cos χ, we obtain [64]
κxx/κn = ˜v2(eH)
∆2
× 1.479(1 −0.0416 cos(2φ))
(66)
The magnitude of the twofold term seen in [66] is consistent with Eq.(66).
18
B.
Clean limit
In the clean limit Eq.(57) is transformed as
κ∥/κ0 = 1 + < x2 >
3c2
0
(67)
≃1 +
1
6π2 ln
 
2
r
2∆
πΓ
!
v2(eH)
Γ∆
ln(
∆
v
√
eH
)
(68)
where κ0 = κ∥(H →0) and we have assumed H ∥c. In the clean limit the ﬁeld dependence is given by κ∥∼
H ln(H0/H). In a magnetic ﬁeld within the a-b plane we obtain
κxx/κ0 = 1 +
1
6π2 ln
 
2
r
2∆
πΓ
!
˜v2(eH)
Γ∆
[ln(
∆
˜v
√
eH
) −0.072 + 0.041 cos(4φ)]
(69)
and
κxy/κ0 =
1
3c2
0
< sin(2φ)x2 >
(70)
= −(˜v)2eH
3π2Γ∆ln(2
r
2∆
πΓ ) sin(2φ)[ln(
∆
˜v
√
eH
) −0.42]
(71)
where we have assumed f = cos(2φ).
The b-axis thermal conductivity of κ-(ET)2Cu(NCS)2 determined by Izawa et al [68] exhibits not only the four-fold
term predicted in Eq. 65, but also a two-fold term that was not predicted. We can interpret the two-fold term
as being due to the admixture of an s-wave component. The experimental data is most naturally interpreted as
∆(k) ∼cos(2φ) −0.067 [111]. Finally for T ≫< |v · q| > we obtain [65]
κxx/κn = 7π2
10 ( T
∆)2(1 + ( 2
π )2 ln2( 2∆
1.76T )) −
1
(2π)2 ln( 2∆
1.76T ) ˜v2(eH)
∆2

ln(
4∆
˜v
√
eH
−1
16(1 −cos(4φ))

(72)
κxy/κn =
1
2π2 sin(2φ) ˜v2(eH)
∆2
ln( 2∆
1.76T ) ln(
4∆
˜v
√
eH
)
(73)
It appears that both Eq.(67) and (68) describe consistently the angle-dependent magnetothermal conductivity observed
in YBCO [108,109].
Exercises 5.
5.1 Indicate how to derive Eq.(58) from Eq.(57).
5.2 Consider κzz for f’s with horizontal nodes (i.e. f = sin χ, cos χ, and cos 2χ) when the magnetic ﬁeld is rotated
within the z-x plane.
VII.
SUPERCONDUCTIVITY WITH POINT NODES
A.
Borocarbides YNi2B2C and LuNi2B2C
This class of superconductivity was discovered in 1994 [112]. Their relatively high transition temperatures (Tc = 15.5
K and 16.5 K for YNi2B2C and LuNi2B2C, respectively) and their interplay between magnetism and superconductivity
are of great interest. In the following we will focus on the two superconducting borocarbides YNi2B2C and LuNi2B2C,
which have no indication of magnetism.
The presence of a substantial s-wave component in the superconducting order parameter ∆(k) in YNi2B2C was
established by substituting Ni by a small amount of Pt. An opening of the energy gap was observed by speciﬁc heat
measurements [113]. On the other hand, this superconductivity exhibits a number of peculiarities unexpected for
s-wave superconductors. For example the
√
H dependence of the speciﬁc heat in the vortex state [114,115] indicates
a nodal superconductor. Furthermore the presence of de-Haas-van-Alphen (dHvA) oscillation in the vortex state in
19
LuNi2B2C down to H = 0.2Hc2 suggests again nodal superconductivity.[116,117] Also T −1
1
in NMR exhibits a T 3
power law consistent with nodal superconductivity [118]. In addition ∆(k) exhibits a fourfold symmetry as seen
from the angular dependence of the upper critical ﬁeld Hc2 when the magnetic ﬁeld is rotated within the a-b plane
[119,120]. Also, unlike the superconductors discussed in previous sections, here we are dealing with a superconductor
with a 3D Fermi surface.
Then we postulate [105]
∆(k) = (∆/2)(1 −sin4(θ) cos(4φ))
(74)
where θ and φ are polar coordinates describing k, the quasiparticle wave vector. This order parameter belongs to
the hybrid representation of s+g-wave superconductivity. The precise matching of the s and g-wave components is
necessary in order to ﬁnd the nodal excitations as observed experimentally. Within a model Hamiltonian the stability
of such a precise matching is considered in [121]. The corresponding ∆(k) is shown in Fig. 13. The quasiparticle
density of states is given by
FIG. 13: Order parameter for the borocarbides YNi2B2C and LuNi2B2C
G(E) = |x|
Z 2
0
dy F(y)Re
1
(x2 −y2)1/2
(75)
where x = E/∆and
F(y) = 1
2
Z u0
0
dz
((1 −z2)4 −(1 −u2
0)4)
(76)
with u0 =
p
(1 −(|1 −y|)1/2). We note that F(2 −y) = F(y). The DOS is shown in Fig. 14. For |E|/∆= x ≪1
the quasiparticle density of states can be approximated by
0
0.5
1
1.5
2
0
0.5
1
1.5
N(E)/N0
E/∆
Γ/∆=0.00
Γ/∆=0.02
Γ/∆=0.20
Γ/∆=1.00
0
0.1
0
0.05
0.1
N(E)/N0
E/∆
FIG. 14:
Quasi-particle density of states for the borocarbides YNi2B2C and LuNi2B2C
20
g(E) = π|E|
8∆(1 + (9/(8π))(|E|/∆) + . . .)
(77)
which gives
Cs/γNT ≃27ζ(3)
8π
(T/∆)
(78)
Unlike the point nodes discussed in Ref. [1], the point nodes here are quadratic which gives N(E) ∼|E| and Cs ∼T 2.
In the presence of a magnetic ﬁeld the quasiparticle density of states is given by
g(0, H) = π
8∆< |v · q| >= ˜v(eH)1/2I(θ, φ)
2∆
(79)
where ˜v = √vavc and
I(θ, φ) = 1
2((1 −sin2 θ sin2 φ)1/2 + (1 −sin2 θ cos2 φ)1/2)
(80)
Here θ and φ are the direction of the magnetic ﬁeld where the polar axis is taken parallel to the c axis. We show in
Fig. 15 I(θ, φ). In particular for θ = π/2 the DOS exhibits cusps at φ = 0, π/2, π, 3π/2. These cusps appear also in
-45
0
45
90
135
Φ(deg.)
0.6
0.8
1
I(θ,Φ)/I(θ,45)
90
o
60
o
θ=45
o
s+g-wave
[100]
[010]
FIG. 15: The angular function I(θ, φ)
the speciﬁc heat, since
Cs/γNT = g(0, H)
(81)
for T ≪∆. Indeed, such cusps have been seen by Park et al [122.] Unfortunately, however, their data are limited to
θ = π/2 and cannot explore the whole extent of I(θ, φ).
Due to the presence of an s-wave component, the eﬀect of impurity scattering is very diﬀerent from the usual nodal
superconductors discussed in III and IV.[123] First of all there is no resonant scattering. So the Born approximation
suﬃces. Also, unlike in other nodal superconductors the energy gap opens up immediately in the presence of impurities.
The energy gap ωg is given in a good approximation by ωg = Γ(1+ Γ
∆)−1. Therefore the speciﬁc heat and the thermal
conductivity decrease as T →0 like Cs/T ∼κ/T ∼(ωg/T )2e−ωg/T . Hence, there is no universal heat conduction.
The gap equation is solved in the presence of impurities. We ﬁnd
−ln(Tc/Tc0) = 0.203
1.203[Ψ(1
2 +
Γ
2πTc
) −Ψ(1
2)]
(82)
which is compared with Eq.(37) for usual nodal superconductors. We show in Fig. 16 Tc/Tc0, ∆(0, Γ)/∆00 versus
Γ/∆00. Both Tc/Tc0 and, ∆(0, Γ)/∆00 decrease much more slowly as Γ/∆00 increases.
21
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Tc/Tc0 and ∆(Γ,T=0)/∆00
Γ/∆00
s+g-wave: Tc/Tc0
∆(Γ,T=0)/∆00
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Tc/Tc0 and ∆(Γ,T=0)/∆00
Γ/∆00
d-wave: Tc/Tc0
∆(Γ,T=0)/∆00
FIG. 16: Tc and ∆(0, Γ) for borocarbides with impurities
θ
φ
a
b
c // q
H
0.095
0.090
κzz (W/Km)
135
90
45
0
-45
φ  (degree)
θ = 90º 
 60º 
 45º 
T =0.43 K
µ0H  = 1 T 
FIG. 17: Thermal conductivity in vortex state in YNi2B2
Finally the thermal conductivity in the vortex state is given by [124]
κzz/κn =
x
2 ln(2/x)[(1 −y2)3/2 −3
2y(arccosy −y(1 −y2)1/2)]θ(1 −y)
(83)
≃
x
2 ln(2/x)
(84)
where x = ˜v(eH)1/2I(θ,φ)
∆
, y =
Γ
∆x. We show in Fig. 17 data by Izawa at al [69], where the magnetic ﬁeld is rotated
conically around the c-axis. The clear cusps for θ = π/2 disappear as θ is decreased. This is the clear sign of point
nodes at k = (100), (010), (−100) and (0 −10). The eﬀect of Pt-substitution of Ni in Y(Ni1−xPtx)2B2C with x=0.05
is studied by Kamata et al [125]. As seen from Fig. 19 the angular dependence in κzz disappeared completely for 5%
substitution of Pt. From Eq.(78) and Tc = 13.1K of the 5 percent Pt substituted system, we can deduce Γ = 23.8K.
22
FIG. 18: Eﬀect of impurities on thermal conductivity - theoretical
FIG. 19: Eﬀect of impurities on thermal conductivity - experimental
This Γ is much larger than Tc0. Clearly parallel experiments with Pt-substitution less than 1% are highly desirable.
The present ∆(k) describes very well the ultrasonic attenuation data of YNi2B2C [126,127].
B.
Skutterudite PrOs4Sb12
The skutterudite PrOs4Sb12 is a heavy-fermion superconductor with transition temperature Tc ∼1.8K [128,129,
130]. The angle-dependent magnetothermal conductivity data of this material inidcates an interesting multiphase
structure characterized by the gap functions ∆A(k) and ∆B(k) with point nodes.[70,71,131] More recently, there has
been mounting evidence for triplet superconductivity in this compound. First, from µ-SR measurement Aoki et al
[132] discovered a remnant magnetization in the B phase of this compound indicating triplet pairing. Also more
recent thermal conductivity data by Izawa et al is not consistent with the singlet model [131], but is consistent with
p+h-wave superconductors [71]. Finally, Tou et al [133] reported NMR data of PrOs4Sb12 of which the Knight shift
suggests the triplet pairing.
23
The triplet pairing together with the position of point nodes give almost uniquely [71]
∆A(k) = 3
2d∆e±iφi(1 −k1
4 −k2
4 −k3
4)
(85)
∆B(k) = d∆e±iφ3(1 −k3
4)
(86)
where e±iφ1 = (k2 ± ik3)/
p
(k2
2 + k2
3), etc. Here φi in Eq.(85) is one of φ1, φ2 and φ3. Also we have chosen the
nodal direction of ∆B parallel to (0 0 1) in Eq.(86). Also |∆A(k)| has the cubic symmetry consistent with the
crystal symmetry of PrOs4Sb12, while |∆B(k| has the axial symmetry. We show in Fig. 20 |∆A(k| and |∆B(k)|.
Very recently the magnetic penetration depth of the B-phase in PrOs4Sb12 has been reported [134]. They applied
FIG. 20: Proposed A-phase (left) and B-phase order parameters for PrOs4Sb12
a magnetic ﬁeld parallel to each of the 3 crystal axes and determined the magnetic penetration depth and found
isotropic superﬂuid density. At ﬁrst sight this is clearly in contradiction to our ∆B(k) in Eq. (86). However, this is
understood, if we assume that the stationary magnetic ﬁeld controls the symmetry axis of ∆B(k) [135]. Indeed the
ground state energy is favorable when the nodes are aligned parallel to H. Also the T 2 dependence of the superﬂuid
density for T ≪∆follows from this assumption. As in all the triplet superconductors, our superconducting order
parameter breaks the chiral symmetry. So both order parameters are six-fold degenerate.
The quasiparticle DOS of these states is given by
g(E) = |x|Re
*
1
p
x2 −|f|2
+
(87)
for x = |E|
∆. This is evaluated numerically and shown in Fig. 21. For |x| ≪1 we obtain
g(E) ≃π|x|/4 A −phase
(88)
≃π|x|/8 B −phase
(89)
In the B-phase there is a logarithmic singularity at E = ∆as in d-wave superconductors, whereas in the A-phase the
singularity is split into 2 cusps at E/∆= 3/4 and 1. In the vortex state the quasiparticle DOS is given by
gA(H) = 1
2
v
√
eH
∆
IA(θ, φ) A −phase
(90)
= 1
4
v
√
eH
∆
IB(θ, φ) B −phase
(91)
where
IA(θ, φ) = sin θ + (1 −cos2 θ sin2 φ)1/2 + (1 −cos2 θ cos2 φ)1/2
(92)
IB(θ, φ) = sin θ.
(93)
24
0
0.5
1
1.5
2
E/∆
0
2
4
N(E)/N0
B-phase
A-phase
FIG. 21: Quasiparticle density of states for p+h-wave superconductor
In the B-phase we have assumed that the nodes are parallel to (0 0 1). Then the low-temperature speciﬁc heat, etc.
are given by
CS/γNT = g(H),
χS/χN = g(H)
(94)
ρS(H)/ρS(0) = 1 −g(H) (A −phase)
(95)
ρS∥(H)/ρS∥(0) = 1 −3g(H) (B −phase)
(96)
The last expression means the superﬂuid density measured parallel to the nodal directions.
In p+h-wave superconductors the impurity scattering is handled similarly to other nodal superconductors. For
example, the superconductivity in the A-phase exhibits the universal heat conduction with κ00/T = π2nEF /12∆(0).
On the other hand in the B-phase the heat current has to be parallel to the nodal direction. Then we will have
κ00/T = π2nEF /8∆(0). When the heat current is perpendicular to the nodal direction in the B-phase, the thermal
conductivity vanishes like T 2.
Now let us consider κzz in the vortex state when H is rotated in the z-x plane with θ the angle H makes from the
z axis. Then in the superclean limit ((Γ∆)1/2 ≪v
√
eH) we obtain
κzz
κn
= v2eH
8∆2 sin2 θ A −phase
(97)
= 3v2eH
64∆2 sin2 θ B −phase
(98)
where in the B-phase we assumed that the nodes are parallel to (0 0 1). Similarly in the clean limit ((Γ∆)1/2 ≫v
√
eH)
we obtain
κzz/κ0 = 1 + 3v2eH
40Γ∆ln(
r
2∆
Γ ) sin2 θ ln(
∆
v
√
eH sin θ
) (A −phase)
(99)
= 1 + v2eH
12Γ∆ln(
r
2∆
Γ ) sin2 θ ln(
∆
v
√
eH sin θ
) (B −phase)
(100)
where κ0 = κzz(H = 0). In Fig. 22 we show the experimental data at T = 0.35 K together with the theoretical
ﬁt from Eq.(90) and (91). As is readily seen the ﬁts are excellent with ∆/v
√
eH = 5 and 3 for data at H = 0.5 T
(B phase) and H= 1.2 T (A-phase) respectively. Then making use of the weak-coupling theory gaps ∆A(0) = 4.2K
and ∆B(0) = 3.5K for the A-phase and B-phase respectively, we can deduce v = 0.96 × 107 cm/sec and Γ ≃0.1K.
These values are very reasonable. From the de Haas-van Alphen measurement v is estimated to be 0.7 × 107 cm/sec
25
0.0
0.2
0.4
0.6
0.8
1.0
θ/π
0.070
0.075
0.080
0.085
0.090
κzz(θ)
H = 1.2, φ = 0
H = 1.2, φ = π/4
H = 0.5, φ = 0
H = 0.5, φ = π/2
FIG. 22: Thermal conductivity of PrOs4Sb12 in the vortex state
(α band), 0.66 × 107 cm/sec (β band), and 0.23 ×107 cm/sec (γ band) [136]. Perhaps the triplet superconductivity
in PrOs4Sb12 is not so surprising. The superconductivity in Sr2RuO4, UPt3, UBe13, URu2Si2 and UNi2Al3 appears
to be triplet [137]. More surprising is the high degeneracy of the ground state due to the chiral symmetry-breaking.
Both the A-phase and the B-phase have six-fold degeneracy. This means that a variety of topological defects in both
the A-phase and the B-phase are likely, which deserve further study. The multi-phase structure is rare but at least
we have seen it in superﬂuid 3He and in UPt3.
On the other hand the exact location of the A-B phase boundary is still controversial. A recent thermodynamical
study [138] indicates that the A-B phase boundary is rather parallel to the upper critical ﬁeld Hc2 of the A phase.
Somewhat surprisingly the less symmetric B phase is realized for the low-ﬁeld region. Is this related to the antiferro-
magnetic quadrupolar (AFQ) state which appears above H ≃4T [139]? It looks like both the superconductivity in
YNi2B2C and in PrOs4Sb12 have great futures.
C.
Summary on ∆(k)
a). Quasi-2D systems
1. d-wave superconductors (f = cos(2φ); hole doped high-Tc cuprates (YBCO, LSCO, Bi2212, Tl2201), electron-
doped cuprates (NCCO, PCCO), CeCoIn5 and κ −(ET )2Cu(NCS)2. As to the gap symmetry of CeCoIn5 there is
controversy between dx2−y2-wave versus dxy-wave.
2. Sr2RuO4. Early experiments established triplet pairing and chiral symmetry breaking [140,141,142]. When the
high quality single crystals became available, both the speciﬁc heat data [58] and the magnetic penetration data [143]
exhibited characteristics of nodal superconductors. Therefore the p-wave superconductivity proposed in [145] is clearly
out. In order to save this situation a multigap model was proposed by Zhitomirsky and Rice [146]. However, our
analysis of the optical conductivity [147] indicates that there is little room for p-wave superconductivity. Therefore
the important question is: where are the line nodes in ∆(k).
Clearly the magnetothermal conductivity data [66] and the ultrasonic attenuation data [148] support horizontal
nodes as in f = e±iφ cos χ. Therefore, except for the speciﬁc heat data by Deguchi et al [104] we mentioned earlier all
available data are consistent with the 2D f-wave model. Also we have proposed that the optical conductivity [147],
the Raman spectra [149], and the supercurrent experiment [150,151] below T =0.1 K will provide the deﬁnitive test
of ∆(k) in Sr2RuO4. The review paper by Mackenzie and Maeno [152] is excellent, but does not cite the signiﬁcant
evidence for nodal superconductivity in Sr2RuO4, if we concentrate on the data from high quality single crystals. On
the other hand, we agree that deﬁnitive experiments below T = 100 mK are highly desirable.
26
3. UPd2Al3 As brieﬂy described in IV, we have proposed f = cos(2χ).[89]
4. We are awaiting the magnetothermal conductivity data of UNi2Al3, URu2Si2 and CeCu2Si2.
5) 3D systems
a). YNi2B2C, LuNi2B2C. ∆(k) = (∆/2)(1 −sin4 θ cos(4φ))
b). PrOs4Sb12, p+h-wave superconductivity.
VIII.
D-WAVE DENSITY WAVE
In the preceding lectures we have reviewed a variety of nodal superconductors which have appeared since 1979.
For some of them we have succeeded in identifying the gap symmetry. All these analyses are based on the BCS
theory of nodal superconductors. Also the semiclassical approximation played an important part in elucidating the
quasiparticle properties in the vortex state.
Therefore it is very natural to contemplate a parallel development in charge density wave (CDW) and spin density
wave [2,153] systems The unconventional density wave (UDW) was speculated as a possible electronic ground state
in excitonic insulators in 1968 [154]. However, the recent surge of interest in UDW is in part due to the proposal
that the pseudogap phase in high-Tc cuprate superconductors is a d-wave density wave (d-DW).[43,44,45,46] It is
important to point out that two kinds of d-DW have been considered. The ﬁrst one is commensurate and individual
square lattice enclosing a circulating current [43,45] analogous to the ﬂux phase [47]. Therefore the ground state has
Z2 symmetry and “visons” as topological defects. [48] On the other hand we consider a d-DW, which is in general
incommensurate. [46] Therefore as in a superconductor the ground state has U(1) symmetry associated with the axial
gauge transformation
c†
k →eiφ/2c†
k,
c†
k+Q →e−iφ/2c†
k+Q
(101)
The d-DW can have “phase vortices” similar to the ones in conventional CDW [49].
Here are a few characteristics of unconventional density waves (UDW) [155]. First of all, the transition from the
normal state to UDW is a metal-to-metal transition. Though the quasiparticle density decreases due to the opening
of a partial energy gap, UDW are conductors down to T = 0 K. In high-Tc cuprates the resistivity ∼T 2 in the
normal state changes to the resistivity ∼T for example. Also, we shall see that the quasiparticles in UDW are a
standard Fermi liquid. Therefore this anomalous resistivity behavior does not imply the existence of a “non-Fermi
liquid”. Since ⟨∆(k)⟩=0, there will be no x-ray or neutron signal for charge density or spin density. Therefore UDW
is sometimes called a condensate with a hidden order parameter [45].
The quasiparticle Green function in UDW is given by
G−1(k, ω) = ω −η(k) −ξ
′(k)ρ3 −∆(k)ρ1
(102)
where ξ
′(k) = 1
2(ξ(k) −ξ(k −Q)), η(k) = 1
2(ξ(k) + ξ(k + q)) and Q is the nesting vector. Also the ρ′s are the Pauli
matrices operating on the spinor space made of |c†
k⟩and |c†
k−Q⟩. Here we consider only UCDW for simplicity and
spin indices are dropped. In the following we drop the prime for ξ
′(k), since there will be no confusion. Then the
quasi-particle energy is given by
ω = η(k) ±
p
ξ2(k) + ∆2(k)
(103)
which is identical to that for nodal superconductors except for η(k), the imperfect nesting term.
Here are two urgent questions:
Where can we ﬁnd UDW? If they exist, how can we identify them? As to the second question, we believe that two
hallmarks of UDW are the angle-dependent magnetoresistance and the giant Nernst eﬀect [155]. Both of these are
consequences of the Landau quantization of the quasiparticle spectrum in UDW or Nersesyan’s eﬀect.[156,157]
A.
The Nersesyan eﬀect
In 1989 Nersesyan et al pointed out that the quasiparticle motion in UDW is quantized in the presence of a magnetic
ﬁeld perpendicular to the conducting plane. Then the QP spectrum becomes
E±
n = η ±
p
2nvv2e|B cos θ|
(104)
27
where n=0,1,2,3... and the n ̸= 0 states are doubly degenerate and v2/v = ∆/EF [158]. Here ∆is the maximal energy
gap in UDW and we assumed η (the chemical potential) independent of k. First the magnetoresistance is given by
R(B, θ)−1 = 2σo + 4σ1
 e−x1 + cosh(ξ0)
cosh x1 + cosh ξ0

(105)
where x1 = β
p
2evv2|B cos θ|, ξ0 = βη and β = 1/kBT . Here we considered only the 2 lowest Landau levels. A
slight modiﬁcation of this simple formula can describe the low-temperature phase of α-(ET)2KHg(SCN)4 [159] and
the metallic phase in the Bechgaard salts [160,161]. See Fig. 23. Second, in the presence of an electric ﬁeld the QP
FIG. 23: Phase diagrams for low-temperature phase of α-(ET)2KHg(SCN)4 [159](left) and the metallic phase in the Bechgaard
salts(right)
orbits drift with vD = (E×B)/|B|2. This gives rise to a large negative Nernst eﬀect [158]. We obtain
αxy = −SR
B = 2eR(ln 2 + 2 ln(2 cosh(x1/2)) −x1 tanh(x1/2))
(106)
where S is the entropy associated with the QPs.
Indeed a large negative Nernst eﬀect has been observed in α-
(ET)2KHg(SCN)4 [162], in the pseudogap region of high-Tc cuprates [163,164,165], in the CDW region of NbSe2 [166]
and more recently in CeCoIn5[167,168]. Recently we have analyzed the Nernst coeﬃcient observed in CeCoIn5 [168]
in terms of Eq. 105. This is shown in Fig 24 a) and b). We obtain an excellent ﬁt as is seen readily. On the other
hand, in order to describe the temperature dependence we have to assume
∆(T )v(T ) = a + bT 4
(107)
whose origin is unclear. However, it is possible that the above temperature dependence indicates the presence of a
quantum critical point (QCP) in CeCoIn5.
In addition, we have shown recently the giant Nernst eﬀect observed in the underdoped region of YBCO, LSCO and
Bi-2212 [163,165] are described in terms of dDW [169]. Very recently a similar but positive giant Nernst eﬀect has
been reported in the AF phase in URu2Si2 [170]. Indeed there has been a suggestion that CDW in NbSe2 is UCDW
[171] and the AF phase in URu2Si2 is USDW [172,173]. So far we have identiﬁed 7 candidates for UDW which include
the 3 high-Tc cuprate superconductors YBCO, LSCO and Bi-2212 in the underdoped region.
Coming back to high-Tc cuprates, 1) ∆(k) = ∆cos(2φ) in the pseudogap region is determined by ARPES [174].
2) The similar phase diagram as in Fig. 2 is obtained when T is replaced by the low-temperature energy gap which
is measured by STM [175,176], ARPES [14,177] and the universal heat conduction [178]. In particular, if one puts
∆= 2.14T ∗these two phase diagrams match almost perfectly. Needless to say, ∆/Tc = 2.14 is also valid for d-wave
density wave in the weak-coupling limit when |η| ≪∆[17,74].
IX.
GOSSAMER SUPERCONDUCTIVITY
In the ﬁrst lecture we discussed Laughlin’s critique [51] on Anderson’s RVB state (Eq.(2)). The new wave function
suggested by Laughlin can be interpreted as d-wave superconductor in the presence of another order parameter [52,53].
From the phase diagram for high-Tc cuprates, the most relevent state is a d-wave superconductor in the presence
28
FIG. 24: The magnetic ﬁeld dependence of the Nernst coeﬃcient is plotted for (left) T = 1.3 K, 1.65 K, 2.5 K, 3.5 K and 4.8
K from top to bottom, and for (right) T = 7.3 K, 10.5 K and 15 K from bottom to top. The circles denote the experimental
data, while the solid line is our ﬁt.
of d-wave density wave. Also this is the prevailing feature in heavy-fermion systems like URu2Si2, CeCu2Si2 [179],
CeCoIn5, CeRhIn5, UPd2Al3, and organic conductors like κ −(ET )2 salts [180,181,182].
Coming back to high-Tc cuprate superconductors, ∆(k) for d-DW can be readily identiﬁed as the order parameter
phenomenologically introduced by Tallon and Loram [50].
For example, the superﬂuid density in the gossamer
superconductor at T=0 K is given by
ρs(0, η) =
∆2
1(0)
∆2
2(0) + ∆2
1(0)
(108)
while the c-axis superﬂuid density is
ρs,c(0, η) =
∆2
1(0)
p
∆2
2(0) + ∆2
1(0)
(109)
where ∆1(0) and ∆2(0) are the maximal gaps of d-wave superconductor and d-wave density wave, respectively, at T=
0 K.
It is of great interest to explore a variety of transport properties in the gossamer superconductivity. For example,
if η, the chemical potential is negligible (|η| ≪T ), we obtain again the universal heat conduction (Eq.(3)) where now
∆=
p
∆2
1 + ∆2
2. Also, ARPES would see the energy gap [174]
∆(k) =
q
(∆2 cos(2φ) −η)2 + ∆2
1 cos2(2φ)
(110)
≃|∆cos(2φ) −η|
(111)
If you look carefully at the phase diagram of Bi-2212 (see for example [177]), unlike LSCO and YBCO the supercon-
ducting dome in Bi-2212 is completely covered by the pseudogap phase. This suggests that the superconductivity in
Bi-2212 is gossamer for the whole doping range. This may be one of the reasons why ∆(0)/Tc ∼5 in Bi-2212 is so
large. In this sense the systematic study of the optical conductivity, the Raman spectra and the thermal conductivity
(with special attention to the doping dependence of these quantities) is of great interest.
In summary, in exploring unconventional or nodal superconductors, we have encountered a vast forest inhabited by
many unconventional density wave (UDW) and gossamer superconductors where these two order parameters coexist.
Surprisingly, all of these ground states have been expected from the infrared instability of the 2D and 3D Fermi liquid.
Therefore we can restore the legacy of Landau’s Fermi liquid theory and BCS theory of superconductivity in a proper
perspective. Then armed with the Green function methodology as in Abrikosov, Gor’kov and Dzyaloshinski (AGD)
[183] we will be fully prepared to explore the plethora of new ground states.
Acknowledgments
We have beneﬁtted from close collaboration with Thomas Dahm, Balazs Dora, Koichi Izawa, Konomi Kamata,
Masaru Kato, Hae-Young Kee, Yong Baek Kim, Bojana Korin-Hamzi´c, Yuji Matsuda, Peter Thalmeier, Silvia Tomi´c
29
and Attila Virosztek. Also KM and HW gratefully acknowledge the hospitality and the support of the Max-Planck
Institute for the Physics of Complex Systems at Dresden for many summers since 1995.
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32
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New York 1975)., source_sentence: high-tc superconductivity
sentences: Seven distinct families of superconductors with critical temperatures at
ambient pressure that equal or surpass the historic 23 K limit for Nb3Ge have
been discovered in the last 25 years. Each family is reviewed briefly and their
common chemical features are discussed. High temperature superconductors are
distinguished by having a high (\geq 50%) content of nonmetallic elements and
fall into two broad classes. 'Metal-nonmetal' superconductors require a
specific combination of elements such as Cu-O and Fe-As which give rise to the
highest known Tc's, probably through a magnetic pairing mechanism.
'Nonmetal-bonded' materials contain covalently-bonded nonmetal anion networks
and are BCS-like superconductors. Fitting an extreme value function to the
distribution of Tc values for the known high-Tc families suggests that the
probability of a newly discovered superconductor family having maximum Tc > 100
K is ~0.1-1%, decreasing to ~0.02-0.2% for room temperature superconductivity., Levitating Superconductor Experiment | Watch Dr Agustin Schiffrin demonstrate the levitating superconductor experiment.

Discover more: https://www.monash.edu/science/schools/physics, superconductor | one of my child genius ideas at around 7 years of age (without any prior knowledge of any of this) was of a perpetually propelled train by the use of magnets of consecutive opposite polarities on the track and train. this borders with the (currently) pseudoscience concept of perpetuum mobile but through means of electrically interchanging the polarity of each consecutive magnet it might not be as unrealistic to use for the gradual propelling of the train. as we see here in a snippet from The B1M, source_sentence: what is abrikosov vortex
sentences: Applications of Superconductors I B. Tech. I B. Sc. I AKTU Physics | This lecture deals with the Applications of Superconductors.
Link of the New Syllabus
https://drive.google.com/drive/folders/17wvsgFDoIhOjrHcIYyNcLxEoBOXF3Fqb?usp=share_link
Link for Old Syllabus, Assignments, and Notes
https://drive.google.com/drive/folders/1Frt1JEqvX4fLkach4220bKkwU_mFB37i?usp=share_link 
Link of Notes and Assignment for the session 2022-2023 (NEW)
https://drive.google.com/drive/folders/1RWrFDthqVIhUxFrG5R3l1_b03FOye6EE?usp=share_link, In superconductivity, a fluxon (also called an Abrikosov vortex or quantum vortex) is a vortex of supercurrent in a type-II superconductor, used by Soviet physicist Alexei Abrikosov to explain magnetic behavior of type-II superconductors. Abrikosov vortices occur generically in the Ginzburg–Landau theory of superconductivity.


== Overview ==
The solution is a combination of fluxon solution by Fritz London, combined with a concept of core of quantum  vortex by Lars Onsager.
In the quantum vortex, supercurrent circulates around the normal (i.e. non-superconducting) core of the vortex. The core has a size 
  
    
      
        ∼
        ξ
      
    
    {\displaystyle \sim \xi }
  
 — the superconducting coherence length (parameter of a Ginzburg–Landau theory). The supercurrents decay on the distance about 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 (London penetration depth) from the core. Note that in type-II superconductors 
  
    
      
        λ
        >
        ξ
        
          /
        
        
          
            2
          
        
      
    
    {\displaystyle \lambda >\xi /{\sqrt {2}}}
  
. The circulating supercurrents induce magnetic fields with the total flux equal to a single flux quantum 
  
    
      
        
          Φ
          
            0
          
        
      
    
    {\displaystyle \Phi _{0}}
  
. Therefore, an Abrikosov vortex is often called a fluxon.
The magnetic field distribution of a single vortex far from its core can be described by the same equation as in the London's fluxoid  

where 
  
    
      
        
          K
          
            0
          
        
        (
        z
        )
      
    
    {\displaystyle K_{0}(z)}
  
 is a zeroth-order Bessel function. Note that, according to the above formula, at 
  
    
      
        r
        →
        0
      
    
    {\displaystyle r\to 0}
  
 the magnetic field 
  
    
      
        B
        (
        r
        )
        ∝
        ln
        ⁡
        (
        λ
        
          /
        
        r
        )
      
    
    {\displaystyle B(r)\propto \ln(\lambda /r)}
  
, i.e. logarithmically diverges. In reality, for 
  
    
      
        r
        ≲
        ξ
      
    
    {\displaystyle r\lesssim \xi }
  
 the field is simply given by

where κ = λ/ξ is known as the Ginzburg–Landau parameter, which must be 
  
    
      
        κ
        >
        1
        
          /
        
        
          
            2
          
        
      
    
    {\displaystyle \kappa >1/{\sqrt {2}}}
  
 in type-II superconductors.
Abrikosov vortices can be trapped in a type-II superconductor by chance, on defects, etc. Even if initially type-II superconductor contains no vortices, and one applies a magnetic field 
  
    
      
        H
      
    
    {\displaystyle H}
  
 larger than the lower critical field 
  
    
      
        
          H
          
            c
            1
          
        
      
    
    {\displaystyle H_{c1}}
  
 (but smaller than the upper critical field 
  
    
      
        
          H
          
            c
            2
          
        
      
    
    {\displaystyle H_{c2}}
  
), the field penetrates into superconductor in terms of Abrikosov vortices. Each vortex obeys London's magnetic flux quantization and carries one quantum of magnetic flux 
  
    
      
        
          Φ
          
            0
          
        
      
    
    {\displaystyle \Phi _{0}}
  
. Abrikosov vortices form a lattice, usually triangular, with the average vortex density (flux density) approximately equal to the externally applied magnetic field. As with other lattices, defects may form as dislocations.


== See also ==


== References ==, Superconductivity is a set of physical properties observed in superconductors: materials where electrical resistance vanishes and magnetic fields are expelled from the material. Unlike an ordinary metallic conductor, whose resistance decreases gradually as its temperature is lowered, even down to near absolute zero, a superconductor has a characteristic critical temperature below which the resistance drops abruptly to zero. An electric current through a loop of superconducting wire can persist indefinitely with no power source.
The superconductivity phenomenon was discovered in 1911 by Dutch physicist Heike Kamerlingh Onnes. Like ferromagnetism and atomic spectral lines, superconductivity is a phenomenon which can only be explained by quantum mechanics. It is characterized by the Meissner effect, the complete cancellation of the magnetic field in the interior of the superconductor during its transitions into the superconducting state. The occurrence of the Meissner effect indicates that superconductivity cannot be understood simply as the idealization of perfect conductivity in classical physics.
In 1986, it was discovered that some cuprate-perovskite ceramic materials have a critical temperature above 35 K (−238 °C). It was shortly found (by Ching-Wu Chu) that replacing the lanthanum with yttrium, i.e. making YBCO, raised the critical temperature to 92 K (−181 °C), which was important because liquid nitrogen could then be used as a refrigerant. Such a high transition temperature is theoretically impossible for a conventional superconductor, leading the materials to be termed high-temperature superconductors. The cheaply available coolant liquid nitrogen boils at 77 K (−196 °C) and thus the existence of superconductivity at higher temperatures than this facilitates many experiments and applications that are less practical at lower temperatures.


== History ==

Superconductivity was discovered on April 8, 1911, by Heike Kamerlingh Onnes, who was studying the resistance of solid mercury at cryogenic temperatures using the recently produced liquid helium as a refrigerant. At the temperature of 4.2 K, he observed that the resistance abruptly disappeared. In the same experiment, he also observed the superfluid transition of helium at 2.2 K, without recognizing its significance. The precise date and circumstances of the discovery were only reconstructed a century later, when Onnes's notebook was found. In subsequent decades, superconductivity was observed in several other materials. In 1913, lead was found to superconduct at 7 K, and in 1941 niobium nitride was found to superconduct at 16 K.
Great efforts have been devoted to finding out how and why superconductivity works; the important step occurred in 1933, when Meissner and Ochsenfeld discovered that superconductors expelled applied magnetic fields, a phenomenon which has come to be known as the Meissner effect. In 1935, Fritz and Heinz London showed that the Meissner effect was a consequence of the minimization of the electromagnetic free energy carried by superconducting current.


=== London constitutive equations ===
The theoretical model that was first conceived for superconductivity was completely classical: it is summarized by London constitutive equations. It was put forward by the brothers Fritz and Heinz London in 1935, shortly after the discovery that magnetic fields are expelled from superconductors. A major triumph of the equations of this theory is their ability to explain the Meissner effect, wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold. By using the London equation, one can obtain the dependence of the magnetic field inside the superconductor on the distance to the surface.
The two constitutive equations for a superconductor by London are:

  
    
      
        
          
            
              ∂
              
                j
              
            
            
              ∂
              t
            
          
        
        =
        
          
            
              n
              
                e
                
                  2
                
              
            
            m
          
        
        
          E
        
        ,
        
        
          ∇
        
        ×
        
          j
        
        =
        −
        
          
            
              n
              
                e
                
                  2
                
              
            
            m
          
        
        
          B
        
        .
      
    
    {\displaystyle {\frac {\partial \mathbf {j} }{\partial t}}={\frac {ne^{2}}{m}}\mathbf {E} ,\qquad \mathbf {\nabla } \times \mathbf {j} =-{\frac {ne^{2}}{m}}\mathbf {B} .}
  

The first equation follows from Newton's second law for superconducting electrons.


=== Conventional theories (1950s) ===
During the 1950s, theoretical condensed matter physicists arrived at an understanding of "conventional" superconductivity, through a pair of remarkable and important theories: the phenomenological Ginzburg–Landau theory (1950) and the microscopic BCS theory (1957).
In 1950, the phenomenological Ginzburg–Landau theory of superconductivity was devised by Landau and Ginzburg. This theory, which combined Landau's theory of second-order phase transitions with a Schrödinger-like wave equation, had great success in explaining the macroscopic properties of superconductors. In particular, Abrikosov showed that Ginzburg–Landau theory predicts the division of superconductors into the two categories now referred to as Type I and Type II. Abrikosov and Ginzburg were awarded the 2003 Nobel Prize for their work (Landau had received the 1962 Nobel Prize for other work, and died in 1968). The four-dimensional extension of the Ginzburg–Landau theory, the Coleman-Weinberg model, is important in quantum field theory and cosmology.
Also in 1950, Maxwell and Reynolds et al. found that the critical temperature of a superconductor depends on the isotopic mass of the constituent element. This important discovery pointed to the electron–phonon interaction as the microscopic mechanism responsible for superconductivity.
The complete microscopic theory of superconductivity was finally proposed in 1957 by Bardeen, Cooper and Schrieffer. This BCS theory explained the superconducting current as a superfluid of Cooper pairs, pairs of electrons interacting through the exchange of phonons. For this work, the authors were awarded the Nobel Prize in 1972.
The BCS theory was set on a firmer footing in 1958, when N. N. Bogolyubov showed that the BCS wavefunction, which had originally been derived from a variational argument, could be obtained using a canonical transformation of the electronic Hamiltonian. In 1959, Lev Gor'kov showed that the BCS theory reduced to the Ginzburg–Landau theory close to the critical temperature.
Generalizations of BCS theory for conventional superconductors form the basis for the understanding of the phenomenon of superfluidity, because they fall into the lambda transition universality class. The extent to which such generalizations can be applied to unconventional superconductors is still controversial.


=== Niobium ===
The first practical application of superconductivity was developed in 1954 with Dudley Allen Buck's invention of the cryotron. Two superconductors with greatly different values of the critical magnetic field are combined to produce a fast, simple switch for computer elements.
Soon after discovering superconductivity in 1911, Kamerlingh Onnes attempted to make an electromagnet with superconducting windings but found that relatively low magnetic fields destroyed superconductivity in the materials he investigated. Much later, in 1955, G. B. Yntema succeeded in constructing a small 0.7-tesla iron-core electromagnet with superconducting niobium wire windings. Then, in 1961, J. E. Kunzler, E. Buehler, F. S. L. Hsu, and J. H. Wernick made the startling discovery that, at 4.2 kelvin, niobium–tin, a compound consisting of three parts niobium and one part tin, was capable of supporting a current density of more than 100,000 amperes per square centimeter in a magnetic field of 8.8 tesla. The alloy was brittle and difficult to fabricate, but niobium–tin proved useful for generating magnetic fields as high as 20 tesla.
In 1962, T. G. Berlincourt and R. R. Hake discovered that more ductile alloys of niobium and titanium are suitable for applications up to 10 tesla. Commercial production of niobium–titanium supermagnet wire immediately commenced at Westinghouse Electric Corporation and at Wah Chang Corporation. Although niobium–titanium boasts less-impressive superconducting properties than those of niobium–tin, niobium–titanium became the most widely used "workhorse" supermagnet material, in large measure a consequence of its very high ductility and ease of fabrication. However, both niobium–tin and niobium–titanium found wide application in MRI medical imagers, bending and focusing magnets for enormous high-energy-particle accelerators, and other applications. Conectus, a European superconductivity consortium, estimated that in 2014, global economic activity for which superconductivity was indispensable amounted to about five billion euros, with MRI systems accounting for about 80% of that total.


=== Josephson effect ===
In 1962, Josephson made the important theoretical prediction that a supercurrent can flow between two pieces of superconductor separated by a thin layer of insulator. This phenomenon, now called the Josephson effect, is exploited by superconducting devices such as SQUIDs. It is used in the most accurate available measurements of the magnetic flux quantum Φ0 = h/(2e), where h is the Planck constant. Coupled with the quantum Hall resistivity, this leads to a precise measurement of the Planck constant. Josephson was awarded the Nobel Prize for this work in 1973.
In 2008, it was proposed that the same mechanism that produces superconductivity could produce a superinsulator state in some materials, with almost infinite electrical resistance. The first development and study of superconducting Bose–Einstein condensate (BEC) in 2020 suggested a "smooth transition between" BEC and Bardeen-Cooper-Shrieffer regimes.


=== 2D materials ===
Multiple types of superconductivity are reported in devices made of single-layer materials. Some of these materials can switch between conducting, insulating, and other behaviors.
Twisting materials imbues them with a "moiré" pattern involving tiled hexagonal cells that act like atoms and host electrons. In this environment, the electrons move slowly enough for their collective interactions to guide their behavior. When each cell has a single electron, the electrons take on an antiferromagnetic arrangement; each electron can have a preferred location and magnetic orientation. Their intrinsic magnetic fields tend to alternate between pointing up and down. Adding electrons allows superconductivity by causing Cooper pairs to form. Fu and Schrade argued that electron-on-electron action was allowing both antiferromagnetic and superconducting states.
The first success with 2D materials involved a twisted bilayer graphene sheet (2018, Tc ~1.7 K, 1.1° twist). A twisted three-layer graphene device was later shown to superconduct (2021, Tc ~2.8 K). Then an untwisted trilayer graphene device was reported to superconduct (2022, Tc 1-2 K). The latter was later shown to be tunable, easily reproducing behavior found in millions of other configurations. Directly observing what happens when electrons are added to a material or slightly weakening its electric field enables quick testing of an unprecedented number of recipes to see which lead to superconductivity.
In four and five layer rhombohedral graphene, a form of superconductivity with spontaneously broken time reversal symmetry known as "chiral superconductivity" was recently observed. These systems were not observed to have any superlattice effects, and they can flip between two possible magnetic states without exiting the superconducting phase. This is in strong contrast to other observations of superconductivity and magnetic fields. 
These devices have applications in quantum computing.
2D materials other than graphene have also been made to superconduct. Transition metal dichalcogenide (TMD) sheets twisted at 5 degrees intermittently achieved superconduction by creating a Josephson junction. The device used used thin layers of palladium to connect to the sides of a tungsten telluride layer surrounded and protected by boron nitride. Another group demonstrated superconduction in molybdenum telluride (MoTe₂) in 2D van der Waals materials using ferroelectric domain walls. The Tc was implied to be higher than typical TMDs (~5–10 K).
A Cornell group added a 3.5-degree twist to an insulator that allowed electrons to slow down and interact strongly, leaving one electron per cell, exhibiting superconduction. Existing theories do not explain this behavior.
Fu and collaborators proposed that electrons arrange to form a repeating crystal that allows the electron grid to float independently of the background atomic nuclei and the electron grid to relax. Its ripples pair electrons the way phonons do, although this is unconfirmed.


== Classification ==

Superconductors are classified according to many criteria. The most common are:


=== Response to a magnetic field ===
A superconductor can be Type I, meaning it has a single critical field, above which superconductivity is lost and below which the magnetic field is completely expelled from the superconductor; or Type II, meaning it has two critical fields, between which it allows partial penetration of the magnetic field through isolated points called vortices. Furthermore, in multicomponent superconductors it is possible to combine the two behaviours. In that case the superconductor is of Type-1.5.


=== Theory of operation ===
A superconductor is conventional if it is driven by electron–phonon interaction and explained by the BCS theory or its extension, the Eliashberg theory. Otherwise, it is unconventional. Alternatively, a superconductor is called unconventional if the superconducting order parameter transforms according to a non-trivial irreducible representation of the system's point group or space group.


=== Critical temperature ===
A superconductor is generally considered high-temperature if it reaches a superconducting state above a temperature of 30 K (−243.15 °C); as in the initial discovery by Georg Bednorz and K. Alex Müller. It may also reference materials that transition to superconductivity when cooled using liquid nitrogen – that is, at only Tc > 77 K, although this is generally used only to emphasize that liquid nitrogen coolant is sufficient. Low temperature superconductors refer to materials with a critical temperature below 30 K, and are cooled mainly by liquid helium (Tc > 4.2 K). One exception to this rule is the iron pnictide group of superconductors that display behaviour and properties typical of high-temperature superconductors, yet some of the group have critical temperatures below 30 K.


=== Material ===

Superconductor material classes include chemical elements (e.g. mercury or lead), alloys (such as niobium–titanium, germanium–niobium, and niobium nitride), ceramics (YBCO and magnesium diboride), superconducting pnictides (like fluorine-doped LaOFeAs), single-layer materials such as graphene and transition metal dichalcogenides, or organic superconductors (fullerenes and carbon nanotubes; though perhaps these examples should be included among the chemical elements, as they are composed entirely of carbon).


== Elementary properties ==

Several physical properties of superconductors vary from material to material, such as the critical temperature, the value of the superconducting gap, the critical magnetic field, and the critical current density at which superconductivity is destroyed. On the other hand, there is a class of properties that are independent of the underlying material. The Meissner effect, the quantization of the magnetic flux or permanent currents, i.e. the state of zero resistance are the most important examples. The existence of these "universal" properties is rooted in the nature of the broken symmetry of the superconductor and the emergence of off-diagonal long range order. Superconductivity is a thermodynamic phase, and thus possesses certain distinguishing properties which are largely independent of microscopic details. Off diagonal long range order is closely connected to the formation of Cooper pairs.


=== Zero electrical DC resistance ===

The simplest method to measure the electrical resistance of a sample of some material is to place it in an electrical circuit in series with a current source I and measure the resulting voltage V across the sample. The resistance of the sample is given by Ohm's law as R = V / I. If the voltage is zero, this means that the resistance is zero.
Superconductors are also able to maintain a current with no applied voltage whatsoever, a property exploited in superconducting electromagnets such as those found in MRI machines. Experiments have demonstrated that currents in superconducting coils can persist for years without any measurable degradation. Experimental evidence points to a lifetime of at least 100,000 years. Theoretical estimates for the lifetime of a persistent current can exceed the estimated lifetime of the universe, depending on the wire geometry and the temperature. In practice, currents injected in superconducting coils persisted for 28 years, 7 months, 27 days in a superconducting gravimeter in Belgium, from August 4, 1995 until March 31, 2024. In such instruments, the measurement is based on the monitoring of the levitation of a superconducting niobium sphere with a mass of four grams.
In a normal conductor, an electric current may be visualized as a fluid of electrons moving across a heavy ionic lattice. The electrons are constantly colliding with the ions in the lattice, and during each collision some of the energy carried by the current is absorbed by the lattice and converted into heat, which is essentially the vibrational kinetic energy of the lattice ions. As a result, the energy carried by the current is constantly being dissipated. This is the phenomenon of electrical resistance and Joule heating.
The situation is different in a superconductor. In a conventional superconductor, the electronic fluid cannot be resolved into individual electrons. Instead, it consists of bound pairs of electrons known as Cooper pairs. This pairing is caused by an attractive force between electrons from the exchange of phonons. This pairing is very weak, and small thermal vibrations can fracture the bond. Due to quantum mechanics, the energy spectrum of this Cooper pair fluid possesses an energy gap, meaning there is a minimum amount of energy ΔE that must be supplied in order to excite the fluid. Therefore, if ΔE is larger than the thermal energy of the lattice, given by kT, where k is the Boltzmann constant and T is the temperature, the fluid will not be scattered by the lattice. The Cooper pair fluid is thus a superfluid, meaning it can flow without energy dissipation.
In the class of superconductors known as type II superconductors, including all known high-temperature superconductors, an extremely low but non-zero resistivity appears at temperatures not too far below the nominal superconducting transition when an electric current is applied in conjunction with a strong magnetic field, which may be caused by the electric current. This is due to the motion of magnetic vortices in the electronic superfluid, which dissipates some of the energy carried by the current. If the current is sufficiently small, the vortices are stationary, and the resistivity vanishes. The resistance due to this effect is minuscule compared with that of non-superconducting materials, but must be taken into account in sensitive experiments. However, as the temperature decreases far enough below the nominal superconducting transition, these vortices can become frozen into a disordered but stationary phase known as a "vortex glass". Below this vortex glass transition temperature, the resistance of the material becomes truly zero.


=== Phase transition ===

In superconducting materials, the characteristics of superconductivity appear when the temperature T is lowered below a critical temperature Tc. The value of this critical temperature varies from material to material. Conventional superconductors usually have critical temperatures ranging from around 20 K to less than 1 K. Solid mercury, for example, has a critical temperature of 4.2 K. As of 2015, the highest critical temperature found for a conventional superconductor is 203 K for H2S, although high pressures of approximately 90 gigapascals were required. Cuprate superconductors can have much higher critical temperatures: YBa2Cu3O7, one of the first cuprate superconductors to be discovered, has a critical temperature above 90 K, and mercury-based cuprates have been found with critical temperatures in excess of 130 K. The basic physical mechanism responsible for the high critical temperature is not yet clear. However, it is clear that a two-electron pairing is involved, although the nature of the pairing (
  
    
      
        s
      
    
    {\displaystyle s}
  
 wave vs. 
  
    
      
        d
      
    
    {\displaystyle d}
  
 wave) remains controversial.
Similarly, at a fixed temperature below the critical temperature, superconducting materials cease to superconduct when an external magnetic field is applied which is greater than the critical magnetic field. This is because the Gibbs free energy of the superconducting phase increases quadratically with the magnetic field while the free energy of the normal phase is roughly independent of the magnetic field. If the material superconducts in the absence of a field, then the superconducting phase free energy is lower than that of the normal phase and so for some finite value of the magnetic field (proportional to the square root of the difference of the free energies at zero magnetic field) the two free energies will be equal and a phase transition to the normal phase will occur. More generally, a higher temperature and a stronger magnetic field lead to a smaller fraction of electrons that are superconducting and consequently to a longer London penetration depth of external magnetic fields and currents. The penetration depth becomes infinite at the phase transition.
The onset of superconductivity is accompanied by abrupt changes in various physical properties, which is the hallmark of a phase transition. For example, the electronic heat capacity is proportional to the temperature in the normal (non-superconducting) regime. At the superconducting transition, it suffers a discontinuous jump and thereafter ceases to be linear. At low temperatures, it varies instead as e−α/T for some constant, α. This exponential behavior is one of the pieces of evidence for the existence of the energy gap.
The order of the superconducting phase transition was long a matter of debate. Experiments indicate that the transition is second-order, meaning there is no latent heat. However, in the presence of an external magnetic field there is latent heat, because the superconducting phase has a lower entropy below the critical temperature than the normal phase. It has been experimentally demonstrated that, as a consequence, when the magnetic field is increased beyond the critical field, the resulting phase transition leads to a decrease in the temperature of the superconducting material.
Calculations in the 1970s suggested that it may actually be weakly first-order due to the effect of long-range fluctuations in the electromagnetic field. In the 1980s it was shown theoretically with the help of a disorder field theory, in which the vortex lines of the superconductor play a major role, that the transition is of second order within the type II regime and of first order (i.e., latent heat) within the type I regime, and that the two regions are separated by a tricritical point. The results were strongly supported by Monte Carlo computer simulations.


=== Meissner effect ===

When a superconductor is placed in a weak external magnetic field H, and cooled below its transition temperature, the magnetic field is ejected. The Meissner effect does not cause the field to be completely ejected but instead, the field penetrates the superconductor but only to a very small distance, characterized by a parameter λ, called the London penetration depth, decaying exponentially to zero within the bulk of the material. The Meissner effect is a defining characteristic of superconductivity. For most superconductors, the London penetration depth is on the order of 100 nm.
The Meissner effect is sometimes confused with the kind of diamagnetism one would expect in a perfect electrical conductor: according to Lenz's law, when a changing magnetic field is applied to a conductor, it will induce an electric current in the conductor that creates an opposing magnetic field. In a perfect conductor, an arbitrarily large current can be induced, and the resulting magnetic field exactly cancels the applied field.
The Meissner effect is distinct from this – it is the spontaneous expulsion that occurs during transition to superconductivity. Suppose we have a material in its normal state, containing a constant internal magnetic field. When the material is cooled below the critical temperature, we would observe the abrupt expulsion of the internal magnetic field, which we would not expect based on Lenz's law.
The Meissner effect was given a phenomenological explanation by the brothers Fritz and Heinz London, who showed that the electromagnetic free energy in a superconductor is minimized provided 
  
    
      
        
          ∇
          
            2
          
        
        
          H
        
        =
        
          λ
          
            −
            2
          
        
        
          H
        
        
      
    
    {\displaystyle \nabla ^{2}\mathbf {H} =\lambda ^{-2}\mathbf {H} \,}
  
 where H is the magnetic field and λ is the London penetration depth.
This equation, which is known as the London equation, predicts that the magnetic field in a superconductor decays exponentially from whatever value it possesses at the surface.
A superconductor with little or no magnetic field within it is said to be in the Meissner state. The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value Hc. Depending on the geometry of the sample, one may obtain an intermediate state consisting of a baroque pattern of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value Hc1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength Hc2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.


=== London moment ===

Conversely, a spinning superconductor generates a magnetic field, precisely aligned with the spin axis. The effect, the London moment, was put to good use in Gravity Probe B. This experiment measured the magnetic fields of four superconducting gyroscopes to determine their spin axes. This was critical to the experiment since it is one of the few ways to accurately determine the spin axis of an otherwise featureless sphere.


== High-temperature superconductivity ==


== Applications ==

Superconductors are promising candidate materials for devising fundamental circuit elements of electronic, spintronic, and quantum technologies. One such example is a superconducting diode, in which supercurrent flows along one direction only, that promise dissipationless superconducting and semiconducting-superconducting hybrid technologies.
Superconducting magnets are some of the most powerful electromagnets known. They are used in MRI/NMR machines, mass spectrometers, the beam-steering magnets used in particle accelerators and plasma confining magnets in some tokamaks. They can also be used for magnetic separation, where weakly magnetic particles are extracted from a background of less or non-magnetic particles, as in the pigment industries. They can also be used in large wind turbines to overcome the restrictions imposed by high electrical currents, with an industrial grade 3.6 megawatt superconducting windmill generator having been tested successfully in Denmark.
In the 1950s and 1960s, superconductors were used to build experimental digital computers using cryotron switches. More recently, superconductors have been used to make digital circuits based on rapid single flux quantum technology and RF and microwave filters for mobile phone base stations.
Superconductors are used to build Josephson junctions which are the building blocks of SQUIDs (superconducting quantum interference devices), the most sensitive magnetometers known. SQUIDs are used in scanning SQUID microscopes and magnetoencephalography. Series of Josephson devices are used to realize the SI volt. Superconducting photon detectors can be realised in a variety of device configurations. Depending on the particular mode of operation, a superconductor–insulator–superconductor Josephson junction can be used as a photon detector or as a mixer. The large resistance change at the transition from the normal to the superconducting state is used to build thermometers in cryogenic micro-calorimeter photon detectors. The same effect is used in ultrasensitive bolometers made from superconducting materials. Superconducting nanowire single-photon detectors offer high speed, low noise single-photon detection and have been employed widely in advanced photon-counting applications.
Other early markets are arising where the relative efficiency, size and weight advantages of devices based on high-temperature superconductivity outweigh the additional costs involved. For example, in wind turbines the lower weight and volume of superconducting generators could lead to savings in construction and tower costs, offsetting the higher costs for the generator and lowering the total levelized cost of electricity (LCOE).
Promising future applications include high-performance smart grid, electric power transmission, transformers, power storage devices, compact fusion power devices, electric motors (e.g. for vehicle propulsion, as in vactrains or maglev trains), magnetic levitation devices, fault current limiters, enhancing spintronic devices with superconducting materials, and superconducting magnetic refrigeration. However, superconductivity is sensitive to moving magnetic fields, so applications that use alternating current (e.g. transformers) will be more difficult to develop than those that rely upon direct current. Compared to traditional power lines, superconducting transmission lines are more efficient and require only a fraction of the space, which would not only lead to a better environmental performance but could also improve public acceptance for expansion of the electric grid. Another attractive industrial aspect is the ability for high power transmission at lower voltages. Advancements in the efficiency of cooling systems and use of cheap coolants such as liquid nitrogen have also significantly decreased cooling costs needed for superconductivity.


== Nobel Prizes ==
As of 2022, there have been five Nobel Prizes in Physics for superconductivity related subjects:

Heike Kamerlingh Onnes (1913), "for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium".
John Bardeen, Leon N. Cooper, and J. Robert Schrieffer (1972), "for their jointly developed theory of superconductivity, usually called the BCS-theory".
Leo Esaki, Ivar Giaever, and Brian D. Josephson (1973), "for their experimental discoveries regarding tunneling phenomena in semiconductors and superconductors, respectively" and "for his theoretical predictions of the properties of a supercurrent through a tunnel barrier, in particular those phenomena which are generally known as the Josephson effects".
Georg Bednorz and K. Alex Müller (1987), "for their important break-through in the discovery of superconductivity in ceramic materials".
Alexei A. Abrikosov, Vitaly L. Ginzburg, and Anthony J. Leggett (2003), "for pioneering contributions to the theory of superconductors and superfluids".


== See also ==
Hydrogen cryomagnetics – Use of cryogenic liquid hydrogen to cool an electromagnet
MXenes – Class of two-dimensional inorganic compounds
Potential applications of graphene
Superconducting magnetic energy storage – Energy storage technique


== References ==


== Further reading ==
IEC standard 60050-815:2000, International Electrotechnical Vocabulary (IEV) – Part 815: Superconductivity Archived 2020-03-07 at the Wayback Machine.
Kleinert, Hagen (1989). "Superflow and Vortex Lines". Gauge Fields in Condensed Matter. Vol. 1. World Scientific. ISBN 978-9971-5-0210-2.
Larkin, Anatoly; Varlamov, Andrei (2005). Theory of Fluctuations in Superconductors. Oxford University Press. ISBN 978-0-19-852815-9.
Lebed, A. G. (2008). The Physics of Organic Superconductors and Conductors. Vol. 110 (1st ed.). Springer. ISBN 978-3-540-76667-4.
Matricon, Jean; Waysand, Georges; Glashausser, Charles (2003). The Cold Wars: A History of Superconductivity. Rutgers University Press. ISBN 978-0-8135-3295-0.
"Physicist Discovers Exotic Superconductivity". ScienceDaily. 17 August 2006.
Tinkham, Michael (2004). Introduction to Superconductivity (2nd ed.). Dover Books. ISBN 978-0-486-43503-9.
Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 978-0-7167-4345-3.
O'Mahony, Shane M.; University of Oxford (2022). "On the electron pairing mechanism of copper-oxide high temperature superconductivity". Proceedings of the National Academy of Sciences. 119 (37) e2207449119. arXiv:2108.03655. Bibcode:2022PNAS..11907449O. doi:10.1073/pnas.2207449119. PMC 9477408. PMID 36067325.
Charlie Wood, Quanta Magazine (2022). "High-Temperature Superconductivity Understood at Last".


== External links ==

Video about Type I Superconductors: R=0/transition temperatures/ B is a state variable/ Meissner effect/ Energy gap(Giaever)/ BCS model
Lectures on Superconductivity (series of videos, including interviews with leading experts)
YouTube Video Levitating magnet
DoITPoMS Teaching and Learning Package – "Superconductivity"
The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity – The Feynman Lectures on Physics.
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