10. Superconductivity

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					         10. Superconductivity
•   Experimental Survey
     –   Occurrence of Superconductivity
     –   Destruction of Superconductivity by Magnetic Fields
     –   Meisner Effect
     –   Heat Capacity
     –   Energy Gap
     –   Microwave and Infrared Properties
     –   Isotope Effect
•   Theoretical Survey
     –   Thermodynamics of the Superconducting Transition
     –   London Equation
     –   Coherence Length
     –   BCS Theory of Superconductivity
     –   BCS Ground State
     –   Flux Quantization in a Superconducting Ring
     –   Duration of Persistent Currents
     –   Type II Superconductors
     –   Vortex State
     –   Elimination of Hc1 and Hc2
     –   Single Particle Tunneling
     –   Josephson Superconductor Tunneling
     –   DC Josephson Effect
     –   AC Josephon Effect
     –   Macroscopic Quantum Interference
•   High-Temperature Superconductors
K.Onnes (1911) :
  ρ → 0 as T → TC
                    Experimental Survey
1. ρ → 0 for T < TC . Persistent current in ring lasts > 1 yr.
2. NMR: supercurrent decay time > 105 yrs.
3. Meissner effect: superconductor = perfect diamagnet.




             Normal state     SuperC state


4. BCS theory: Cooper pairs (k, –k ). See App. H & I.
            Occurrence of Superconductivity
         Occurrence:
         Metallic elements, alloys, intermetallic compounds,
         doped semiconductors, organic metals, …
Range of TC :
90K    for YBa2Cu3O7.                     Si: TC = 8.3K at P = 165 Kbar
.001K for Rh.
Destruction of Superconductivity by Magnetic Fields
Magnetic field destroys superconductivity.



                                                         H C  BaC       in CGS units


                                                         HC TC   0


                                                                           T 2 
                                                    H C T   H C  0   1    
                                                                           TC  
                                                                                  




            Magnetic impurities lower TC :
              10–4 Fe destroys superC of Mo (TC = 0.92K ).
              1% Gd lowers TC of La from 5.6K to 0.6K.
            Non-magnetic impurities do not affect TC .
                                       Meissner Effect

                                              B = 0 inside superC


Normal state            SuperC state


 For a long thin specimen with long axis // Ha,
 H is the same inside & outside the specimen (depolarizing field ~ 0)
       B  H a  4 M  0                      M      1
                                       →          
                                               Ha    4

 Caution: A perfect conductor (ρ = 0) may not exhibit Meissner effect.

    Ohm’s law          E j           →   E  0 if   0

                       1B                 B
               E         0         →      0            (B is frozen, not expelled.)
                       c t                t
   Also, a perfect conductor cannot maintain a permanent eddy current screen
   → B penetrates ~1 cm/hr.
                Alloys / Transition metals with high ρ.
Most elements
                   ρ = 0 but B  0 in vortex state.
                                                  HC2 ~ 41T for Nb3(Al0.7 Ge0.3).
                                                  HC2 ~ 54T for PbMo6S8.




Commercial superconducting magnets of ~1T are readily available.
                               Heat Capacity
                                            SS  S N
                                            → superC state is more ordered

                                        ΔS ~ 10–4 kB per atom
                                        → only 10–4 e’s participate in transition.




                                                                          Al




                SN
CN   T  T          → SN   T
                T
Ce S  e a / T   → energy gap
                                      Energy Gap
                                                                        Eg / 2 kB T                   Eg / kB T
Comparison with optical & tunneling measurements →          Ce S  e                   not Ce S  e




                                                                                           Eg  0 ~ 104  F




  For Ha = 0, n-s transition is 2nd order ( no latent heat, discontinuity in Ce , Eg →0 at TC ).
                 Microwave and Infrared Properties

EM waves are mostly reflected due to impedance mismatch at metal-vacuum boundary.
They can penetrate about ~20A into the metal.

Photons with ω < Eg are not absorbed
→ surface penetration is greater in superC than in normal state.

For T << TC ,
          ρs = 0 for ω < Eg .
          ρs  ρn for ω > Eg . (sharp threshold at Eg)
For T  TC ,
          ρs  0 for all ω  0 ( screening of E incomplete due to finite inertia of e )
                                       Isotope Effect
Isotope effect:     M  TC  const

       → e-phonon interaction involved in superC.
                                  2                                TC  Debye  M 1/2        1
     Original BCS:      k BTC         D e1 / N    F   V
                                                                →                          
                                                                                               2
      Deviation from α = ½ can be caused by coulomb interaction between e’s.




            Absence of isotope effect due to band structure.
                  Theoretical Survey
   Thermodynamics of the Superconducting Transition
   London Equation
   Coherence Length
   BCS Theory of Superconductivity
   BCS Ground State
   Flux Quantization in a Superconducting Ring
   Duration of Persistent Currents
   Type II Superconductors
   Vortex State
   Estimation of Hc1 and Hc2
   Single Particle Tunneling
   Josephson Superconductor Tunneling
   DC Josephson Effect
   AC Josephon Effect
   Macroscopic Quantum Interference

              • Thermodynamics Considerations
              • Phenomenological Models
              • Quantum Theory
                 Thermodynamics of the Superconducting
Type I superC:                                                B  H  4 M
                                                                        Ba
                                                              W   M  d Ba
                                                                       0


                                                                  dF  M  d B a

                                                                            1
                                                                dFS          Ba d Ba
                                                                           4
                                                                                       2
                                                                                     Ba
                                                              FS  Ba   FS  0  
                                                                                     8

                                                              FN  BaC   FN  0

                                     FN  BaC   FS  BaC   F  0   BaC
                                                                                 2


                                                                         8
                                                                S




                                                   F  FN  0  FS  0 
                                                                             2
                                                                            BaC
                                                                            8

                                    dFN            dFS          → no latent heat
                                               
                                    dT    TC       dT    TC
                                                                ( 2nd order transition)
                                                                (continuous transition)
                                          London Equation
                                    c
London model:         j                 A          in London                           A  0         An surface with j0  0
                                  4 L
                                      2

                                                     gauge:
                                              c                        c
                →              j                 A                       B
                                          4 L   2
                                                                4 L
                                                                    2



              4                                     4           1          1
      B       j             →    B                j   2  A   2 B
               c                                      c          L         L

                                     B  2B

     →                1
              2B            B            London equation
                         2
                          L


                                                                                        λL = London penetration length

                                                                                             mc 2
                                          B//  x   B//  0 e
                                                                     x /  L
                                                                                     L              see flux quantization
                                                                                            4 nq 2


         • Meissner effect not complete in thin enough films.
         • HC of thin films in parallel fields can be very high.
                                       Coherence Length
      Coherence length ξ ~ distance over which nS remains relatively uniform.
              See Landau-Ginzburg theory, App I., for exact definition.
       Local properties = Average of non-local properties over regions ~ ξ .
       Minimum thickness of normal-superC interface ~ ξ .
        Spatial variation of ψ increases K.E.
        → High spatial variation of ψS can destroy superC.

Let           ei k x                 
                                              2
                                                
                                             1 i k  q  x i k x
                                                e          e                       * 
                                                                                                   1
                                                                                                   2
                                                                                                      2  ei q x  ei q x   1  cos qx
→           *  1                                                                           2     Lq
                                                            1  cos qx  dx  L                        1  e  i q x  dx
                                                     L /2
                                         
                                                                                                           L /2
                                                                                                sin
                                                      L /2                                   q      2     L /2
  p2                                  p2
                 k2  2                
                                                                                                           2 m  k  q  e 
                                                                                                              2
  2m                                                   1                   i  k  q  x                                    i k q x
                                      2m                        dx e                       e   i k x
                                                                                                                                           k 2ei k x 
                                                                                                                          2
                2m                              2                                                                                              

                                                        2 
                                                                dx  k  q  1  ei q x   k 2 1  ei q x 
                                                        1
                                                 2    2m  
                                               
                                                                             2
                                                                                                             

                                                1 2                   2
                                                     k  q  k   k 2  k q
                                                                  2
                                                              2
                                                                                                                         for q << k
                                                2 2m 
                           2
                                                                      2m
          K .E.            kq
                       2m
                   2                                                                             2
     K .E.         kq   →   Critical modulation for destroying superC is                         k q0  Eg
                  2m                                                                            2m
                                            2
                                             kF     vF
  Intrinsic coherence length:       0                                      see Table 5
                                           2mEg   2 Eg

  ξ in impure material is smaller than ξ0 . (built-in modulation)

                                                       1        1        1
   ξ & λ depend on normal state mfp .                                               see Tinkham, p.7 & 113.
                                                               0       l

                                                  Pure sample:
                                                         0                   L
                                                                                                           L
                                                                                            →            
                                                                                                           0
                                                  Dirty sample:
                                                                                       0                        L
                                                         0 l                 L            →         
                                                                                       l                         l


                                                           1
ξ0 = 10 λL                                                                 Type I
                                                            2
                                                           1
                                                                           Type II
                                                            2
                BCS Theory of Superconductivity
BCS = Bardeen, Cooper, Schrieffer
BCS wavefunction = Cooper pairs of electrons k and –k (s-wave pairing)

Features & accomplishments of BCS theory :
• Attractive e-e interaction –U → Eg between ground & excited states.
• Eg dictates HC , thermal & EM properties.
• –U is due to effective e-ph-e interaction.
• λ , ξ , London eq. (for slowly varying B ), Meissner effect, …
• Quantization of magnetic flux involves unit of charge 2e.


                                           1       
   U D(εF) << 1 :     TC 1.14  exp                 θ = Debye temperature
                                        U D  F  

                       Eg  0  3.528 kBTC

        Higher ρ → Higher TC (worse conductor → better superC)
                               BCS Ground State


                                 T=0

                                                                                   Cooper pair:
                                                                                   1-e occupancy
                                                                                   with Teff = TC



 Normal state                                   Super state:
                                                Cooper pair mixes e’s from below & above εF




 K.E.normal   K.E.super   but     Enormal  Esuper    due to –U.


Cooper pair: ( k , –k ) → spin = 0 (boson)                    see App. H
          Flux Quantization in a Superconducting Ring
                                                                        1                   1
                                                                          E r               B r          n r  
                                                                                   2                    2
Energy intensity for large number of photons:                  I                      
                                                                       4                  4

    →         E r      4      n  r  ei   r           E * r       4        n  r  ei  r 


Let ψ(r) be the super state wave function.                        Particle density n = ψ*ψ

 n = constant →         n ei   r              *  n ei  r 
                                1    q  1     q 
 Velocity operator:        v     p  A     A
                                m    c  m i   c 

                                                q                          n      q 
                                   n
                           * v   e  i     A  ei                        A 
  Particle flux:                                                             m      c 
                                   m        i   c 

                                                          nq     q 
   Electric current density:         j  q * v              A 
                                                          m      c 
                   nq     q        n q2                                                               mc 2
             j         A        B                     London eq. with  L 
                   m      c        mc                                                                4 nq 2
     nq     q 
j        A 
     m      c 

Meissner effect: B = j = 0 inside superC →                                 q
                                                                          A
                                                                           c

                                                      q
                                    C
                                           d l 
                                                      c      C
                                                                  A dl

                                                q                q          q
                                                c S
                                                    A  dσ   B  dσ  
                                                                 c S        c

                               ψ measurable → ψ single-valued                      → Δ=2πs              sZ

                 hc
                 s       Flux quantization
                 q

                         hc
  q = –2e →       0        2.0678 107 gauss cm2                      = fluxoid or fluxon
                         2e

 Flux through ring :        ext   sc  s  0
                                                                                    see Tinkham, p.121, for a
                                                                                    derivation via Sommerfeld
         Φext not quantized → Φsc must adjust                                       quantization rule
                            Duration of Persistent Currents
 Thermal fluctuation : superC → normal : fluxoid escapes from ring

  Transition rate W = ( attempt freq ) ( Boltzmann factor for activation barrier )

   Boltzmann factor for activation barrier = exp( −β ΔF )

Free energy of barrier = ΔF = (minimum volume) (excess free energy density of normal state)

    minimum volume  R ξ2 .       R = wire thickness                                           R  2 HC
                                                                                                      2

    excess free energy density of normal state = HC2 / 8 π.                               F 
                                                                                                 8
R = 10−4 cm, ξ = 10−4 cm,                 HC = 103 G, gives ΔF  10−7 erg.

 Note: estimate is good for T = 0 to 0.8 TC while ΔF → 0 as T → TC−

   β   10−15   erg at T = 10K → e  F  e10
                                                           8
                                                                        
                                                                     4.3  107   
                                                                  10

   Attempt freq  Eg /  10−15 / 10−27  1012 s−1
                    4.3  107     10 4.3 10  s1
                                                  7
                                                               Age of universe ~ 1018 s
     W  10 10
            12



  Exceptions: Near TC or in Type II materials.
Type II Superconductors

                Ha < HC1         B=0
                HC1 < Ha < HC2 B  0
                                 fluxoid penetration
                HC2 < Ha         M=0


                 Type I    Type II
                                             0 l
                ξ > λ2    ξ < λ2




                                      Electronic structure
                                      not much affected
                                                                              Ref: W.Buckel,
            Normal-Super Conductor Interface                                  “Superconductivity”




                                                          H2
Lowering of energy due to field penetration      B   A
                                                          8
                                                                    2
Increase of energy due to destruction of Cooper pairs:  C   A  HC
                                                                  8
                                                                 2
Normal:  B   C  0             Bulk superC:     B   C 
                                                               HC
                                                                   V
                                                               8
                                                     HC2
                                                            0      Type I
 Interface energy at H = HC   C   B      A           for
                                                     8     0     Type II
   HC2 for Nb3Sn ~ 100kG.


              Type I                  Type II
      surface energy > 0       surface energy < 0




                                                           Thin films with H
                                                           normal to surface



Type I: Intermediate state   Type II: Vortex state




  Fluxoid penetration reduces increase of energy due to flux repulsion.
                                   Vortex State




Meissner effect starts breaking down when a normal core can be substained.
   Normal core radius is always  ξ ; otherwise it’ll be bridged by surrounding ψS .
Fluxoids well separated:                0
                               H C1           = Field for nucleation of single fluxoid
fluxoid radius  λ                       2

 Closed-packed fluxoids:                 0
                               HC2 
 fluxoid radius  ξ                      2

   Type II: κ > 1 → λ > ξ       → HC1 < HC2            Vortex state allowed.
                                                       SuperC destroyed before fluxoid allowable
   Type I: κ < 1 → λ < ξ        → HC1 > HC2            → no vortex state.
                                               Flux lattice in NbSe2 at 1000 G & 0.2K.
                                               STM showing DOS at εF .


                                                EN – ES = Stabilization energy
                                                                        1 2
                                                     →      f core       HC   2
                                                                       8

                                                Decrease of E for allowing H
                                                penetration:          1 2
                                                           f mag      Ba   2
                                                                     8
                                                 Total core energy wrt super-state

                                                                          8
                                                                             
                                                     f  f core  f mag  1 H 2  2  B 2 2
                                                                             C         a        
                                                                                 
  Threshold for stable fluxoid: f = 0 at Ba = HC1.        → H C  H C1              H C1 
                                                                                 
         0
H C1 
          2                                                             
                →   H C 2   2 H C1   HC    HC1 HC 2           HC        H C 2   1H C 2
          0                                                              
HC2 
          2
Single Particle Tunneling

                            2 metals I and II
                            separated by insulator
                            C.




                                                        Al     Sn
                                                 TC   1.14K   3.72K




 Glass +      + Al strip       + Al2O3
 In contact   1mm wide         20-30 A: S.P.T.        + Sn strip
              2000A thick      10A: J.T.

                        Direct measurement of J.T. requires double junctions.
                        See W.Buckel, “Superconductivity”, p.85.
I, II both normal: line 1
                               
      N S    Nn  0 
                              2  2




                            I normal, II super:
                             T = 0, line 2
T0                         T  0, line 3
I , II both super: T  0
      Josephson Superconductor Tunneling


• DC Josephson effect:
       DC current when E = B = 0
• AC Josephson effect:
       rf oscillation for DC V.
• Macroscopic long-range quantum interference:
       B across 2 junctions → interference effects on IS
                                            DC Josephson Effect
                     1                                2
              i           T 2                    i         T 1                     T = transfer frequency
                     t                                 t

j         nj e
                  ij
                                                                                                                        2  1

             1  1  n1                                               1  n1                       n2 i 
 →                        i 1   1  i T  2                                  i 1  i T 2  i T      e
             t  2 n1  t    t                                         2 n1  t    t       1          n1

             2  1  n 2                                              1 n2                        n1
                          i 2   2  i T  1                                   i 2  i T 1  i T          ei 
             t  2 n2  t    t                                         2 n2  t    t       2          n2

                         1  n1         n2                        n1                                       n1           n2
Real parts:                      T        sin                            2T    n1 n2 sin                     
                        2 n1  t        n1                       t                                        t             t
                                                         →
                         1 n2            n1                     n2
                                  T        sin                          2T     n1 n2 sin 
                        2 n2  t          n2                         t

                           1          n2
 Imaginary                        T      cos                1 1 
                           t                                                  n1n2 cos 
                                                              t  n1 n2 
 parts:                                 n1
                                                                        
                           2          n1
                                  T      cos                 n1  n2     0
                          t            n2
      n1
J                 → J  J 0 sin              n1  n2 → DC current up to iC while V = 0.
       t
                                         AC Josephson Effect
                                          1                                          2
V across junction:                  i          T  2  eV 1                   i           T  1  eV 2                      q  2e
                                          t                                           t

    1  1  n1                                                            1  n1                      n2 i   eV
                 i 1   1  i T  2  i
                                             eV
                                                1                                     i 1  i T             e i
    t  2 n1  t    t                                                      2 n1  t    t                 n1

    2  1  n 2                                                           1 n2                       n1                 eV
                 i 2   2  i T   i e V                                         i 2  i T                 ei   i
    t  2 n2  t    t              1          2                            2 n2  t    t                 n2

                   1  n1           n2                           n1                                                    n1          n2
Real                       T          sin                             2T         n1 n2 sin                                 
parts:            2 n1  t          n1                          t                                                      t           t
                                                     →
                   1 n2                n1                      n2
                            T            sin                         2T          n1 n2 sin 
                  2 n2  t              n2                       t

                   1                                                          1 1                             2 eV
                                n2         eV                                                  n1n2 cos  
Imaginary                 T      cos                                       t  n1 n2 
parts:             t           n1                                                       

                   2                                                                                   2 eV
                                n1         eV                                   n1  n2     0              t
                          T      cos  
                  t            n2
                                                                                          2 eV                                       Precision
                                        2 eV            AC current with  
         J  J 0 sin   J 0 sin   0         t                                                                                   measure
                                                                                                                                   of e/
                                                                                         483.6 Mhz for V = 1 μV
                     Macroscopic Quantum Interference
                                                                   q      2e
Around closed loop enclosing flux Φ:              2 s  2             
                                                                   hc      c

                                         For B = 0,       a   2 a  1a   b   2b  1b

                                                                   2e
                                          For B  0,      b   a    
                                                                     c
                                                                   e                             e
                                               or        b  0                a  0          
                                                                    c                             c

                                   e                 e                         e
J tot  J b  J a  J 0 sin   0       sin   0        2 J 0 sin  0 cos
                                    c                 c                         c

                                                                                                zero offset due to
                                                               periodicity = 39.5 mG            background B
                                                               Imax = 1 mA


                                                                Junction area = 3 mm  0.5 mm

                                                                 periodicity = 16 mG
                                                                 Imax = 0.5 mA                        Prob 6
High-Temperature Superconductors

TC ceiling for intermetallic compounds = 23K.

				
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