Josephson  junc,on  in  the  magne,c  field
Spin-­‐filter  Josephson  junc/ons,  Kar/k  Senapa/,
Mark  G.  Blamire  &  Zoe  H.  Barber,    Nature
Materials  10,  849–852  (2011)
NbN–GdN–NbN
In-­‐plane  magne/c  field  dependence  of  junc/on  cri/cal  current  for  a  5±2   nm  GdN   barrier,  showing  Fraunhofer-­‐like  oscilla/ons  at  4.2  and  8.2
K.  The  solid  line  shows  a   fit  to  the  8.2
K  data,  where  the  effec/ve  field  seen  by  the  junc/on  has  an  addi/onal
10   Oe  origina/ng  from  the  barrier  moment,  which  reverses  smoothly  at  low  field.
The  calculated  Fraunhofer  paZerns  at  4.2
K  (using  a  constant  field-­‐shi[)  are  shown   as  doZed  lines,  to  emphasize  the  suppression  of  side-­‐lobe  cri/cal  current  Ic2.

ac  Josephson  effect:
Voltage  oscilla/ons  for   I>I c
O.  Monnoye,  Congrès  de  métrologie,  Toulon,
(2003)

Pairing  mechanism    in  conven,onal  superconductors

Proof  of  the  phonon  pairing  mechanism?

Isotope  effect:
Isotope  Effect  in  the  Superconduc/vity  of  Mercury
Phys.  Rev.  78,  477  –  Published  15  May  1950
Emanuel  Maxwell;
Superconduc/vity  of  Isotopes  of  Mercury
Phys.  Rev.  78,  487  –  Published  15  May  1950
C.  A.  Reynolds,  B.  Serin,  W.  H.  Wright,  and  L.  B.  NesbiZ
Deforma/on  poten/al  model:     λ g
F
BCS: T c
T
D
=  ω
D
=
1.13
T
D exp
(
−
2 / g
F
λ )
=  s / a
0
; s
= v
F m / M
~
T
D
∝
1 / M
Ξ 2
( k
F a
0
)
3
: independent of M
Ms 2 E
F

1.    In  heavy  elements,  rela/ve  changes  of  masses    are  small:  the  case  for  power-­‐law  scaling
is  not    very  convincing.
2.    In  compounds,  one  can    replace  only  one  one  element.
3.    The  coupling  constant  is  independent  of   M  only  in  a  very  simple  model.
4.   The  exponent  varies  drama,cally  even  within  “low  Tc”  superconductors  and  some,mes
is  even  nega,ve  (“inverse  isotope  effect”).

T c
∝
M
− α
phonon model:
α
=1/2
CaC
6
( T c
≈
11 K) :
α
B
≈
0.5
(Ba,K)BiO
MgB
2
( T c
3
( T c
≈
30 K):
≈
40 K) :
α
B
≈
α
O
0.3
≈
0.4
A.  Bill,  V.Z.  Kresin,  and  S.A.  Wolf
in:   Many  Fermion  Systems ,  edited  by  V.Z.  Kresin,
(Plenum  Press,  New  York,  1998)  p.  25;  arXiv:9801222

Iron-­‐based  superconductors:  inverse  isotope  effect
T c
∝
M
− α
Fe phonon model:
α
=1/2

Another  confirma/on  of  e-­‐ph  pairing  :
Strong-­‐coupling  (Eliashberg)  theory
λ = g
F
λ
/ 2
= ⎨
1.6, Hg
1.4, Pb
Treats  electron-­‐phonon  interac/on  explicitly
(rather  than  as  a  model  aZrac/on  in  the  BCS  model)
⎩⎪ 2.1, Pb
0.65
Bi
0.35
McMillan-­‐Dynes  formula
T c
=
 ω
D
1.12
⎜⎜
⎝
⎛ exp
−
1.04(1
+
λ − µ c
(
λ
)
1
+
0.62
λ
) ⎟⎟
⎞
⎠
;
µ c
=
"Coulomb pseudopotential"
λ =
2
∫ d
ω
ω
α 2
( ) matrix element g ph
( ) phonon density of states
First-principle calculations of
α 2
( )
, g ph
( ) ⇒
T c
Sa,sfactory    explana,ons  of  the  experiment  ( T c
,   tunneling  spectra)  but  not     a  single  (correct)  predic,on  for  a  new  superconductor
Strong  cri/que  of  the  e-­‐ph  coupling  model  (and  BCS  in  general):
J.  E.  Hirsch,  Phys.  Scripta   80   (2009)  035702

s-­‐wave  superconductors:
Con,nuum:  Δ(k)=const
La^ce:  Δ(k)>0  for  all  k
Non-­‐s-­‐wave:  nodes  Δ(k n
)=0
Non-­‐s-­‐wave  symmetry  of  the  order  parameter:  p-­‐wave  (l=1),  d-­‐wave  (l=2),
f-­‐wave  (l=3)…
Spin  singlet  even  orbital  momentum  (s,d…)
Spin  triplet  odd  orbital  momentum  (p,f…)
Triplet  superconduc,vity:  He 3  (superfluid),  Sr
2
RuO
4
,  UPt
3
,  UGe
2
(?)
Spin-­‐singlet,  d-­‐wave:  cuprates
Iron-­‐based  superconductors:  spin-­‐singlet.  Orbital  symmetry:  the  jury  is  s,ll  out.
One  possibility:  s±
NB:  unconven,onal≠HTC  (Sr
2
RuO
4
,  UPt
3
,  UGe
2
:   T c
~1  K)

3
J.  Wheatley,  RMP   47 ,  415  (1975)

B-­‐phase:  L=1,S=1,  superposi/on  of   the    S z
=-­‐1,0,1  states
A-­‐phase:  L=1,  S=1,  S z
=+1/-­‐1

UPt
3
Singlet  state:     spin  suscep/bility=0
at  T=0
B-­‐phase:  χ=(1/3)χ
A-­‐phase:  :  χ=χ
N
N
Gannon  et  al.
PRL   86 ,  104510  (2012)
Wheatley,  1975
Mackenzie  &  Maeno,  RMP   75 ,  657

Cuprates:  spin  singlet  (NMR)  s  or   d.
How  do  we  know  that  this  is  a  d-­‐wave?
Normal  quasipar,cles  reside  near  nodal  lines  even  at  T=0:
Thermodynamics  of  a  nodal  superconductor   is  very  much  different  from  s-­‐wave.   s-wave:
Λ ∝
1 / n s n
− n s
∝ exp(
−
2
Δ
/ T )
Λ = Λ ( ) + c exp
( −
2
Δ
/ T
) d -wave: n
− n s
Λ
=
Λ
∝
T
( ) + cT
Hardy  et  al.  PRL   70 ,  3999  (1993)

Possible symmetries of the order parameter on a square lattice s:
Δ ( ) = const s d d
− x
2 xy
:
− y
2
:
:
Δ
Δ
Δ
( )
( )
( )
=
=
=
Δ
Δ
Δ
⎡⎣
⎡⎣ cos cos sin k x k k x x a
+ cos k a
− cos k a sin k y a y y a a
⎤⎦
⎤⎦

Urbana  experiment:  Wollman  et  al.  PRL   71 ,  2134  (1993)
φ a
− φ b
=
2
π Φ
0 s-wave:
δ d-wave:
δ ab ab
=
=
0
π
+ δ ab
Edge  junc/on:  both  junc/ons  on  the  same  side
NB:  R  is  90  shi[ed  compared  to  I

CaZerjee  et  al.  Nature  Phys.   6 ,  99  (2009)
Observa/on  of  a   d -­‐wave  nodal  liquid  in  highly  underdoped  Bi
2
Sr
2
CaCu
2
O
8+ δ

To  second  order  in  a  short-­‐range  repulsive  interac/on,  the   effec6ve  interac/on  in  the
Cooper  channel    is   ATTRACTIVE   at  least  for  odd   L   and  for   L>>1.
W.  Kohn,  and  J.  M.  Luvnger,   New  Mechanism   for  Superconduc6vity ,  Physical  Review  LeZers,
Vol.  15,  No.  12,  pp.  524–526  (1965)
Recent  review:  S.  Mai/  (UF)  and
A.  V.  Chubukov,   arXiv:1305.4609
U>0
Original KL result: pairing amplitude in L -channel:
Γ L c
=
U
δ
L ,0
+ ( )
L
U 2 c
L
L 4
, L  1

Myth  and  truth  about  the  KL  mechanism
Myth:  the  mechanism  produces  only  ridiculously  low   T c
Origin  of  the  myth:  KL  obtained  the  result  valid  only  for   L>>1   and  applied  it  to   L=2.
( ) =
4
π a
δ 3
( ) m a
= scattering length.
For He 3 : k
F a
≈
2
Truth:  Values  of   T c
are  reasonable
For  L~1.
Lay  and  Layzer,  PRL   20 ,  187  (1968)
L
=
1 : T c
/ E
F
≈
10
−
3
Superfluid He 3 (Lee, Oscheroff, Richardson 1972) :
T c
/ E
F
≈
10
−
3 !

Nodes  are  a  consequence  of  repulsive  interac/on
Suppose that the pairing interaction is anisotropic. BCS equation
Δ ( ) =
(
−
V
If V k , k '
∑ k , k '
)
Δ ( )
(
1
−
2 n k '
) k '
2 E k '
>
0,
Δ ( )
must change sign for the eq-n to have a solution .
Another  consequence  of  repulsion:  enhanced  magne/sm   p p ,
Fermi liquid interaction function
F
α
αγ
,
βδ
( ) =
F s
( ) δ
αγ
δ
βδ
+
F a p ,
γ
( ) σ
 p ,
αγ
α i σ

βδ
θ k k ,
δ
F
αγ
,
βδ
( ) = k ,
β
F
αγ
,
βδ
( ) = δ
αγ
δ
βδ
⎣⎢
⎡
U
( ) −
−
1
2
U
⎝⎜
⎛
2 p
F k ,
δ sin
θ
2
⎠⎟
⎞ ⎤
⎦⎥
− k ,
β
1
2
U
⎝⎜
⎛
2 p
F sin
θ
2
⎠⎟
⎞
σ

αγ i σ

βδ p ,
γ

Singlet:  the  orbital  part  of  the  wave  func/on  is  even
Triplet:  the  orbital  part  of  the  wavefunc/on  is  odd
Repulsive  interac/on  favors  larger  distances  between  par/cles:  triplet  pairing  is  preferred
Strong  ferromagne/c  fluctua/ons  favor  triplet  pairing  (He 3 ,  Sr
2
RuO
4
…)

2
Superconduc/vity:  A  different  class?
Gilbert  G.  Lonzarich
Nature  Physics  3,  453  -­‐  454  (2007)

an6
cuprates
Fe-­‐based  superconductors
(pnic,des)

Doped  an/ferromagne/c  insulator   Staggered  spin  suscep/bility   q
π
= ( ) χ ( ) = ∑ r
<

S r i
(
−

S r
+ a
)
> e i q i r
Effec/ve  interac/on
~ ( q
− q
π
1
)
2 + ξ −
2
H
⇒ int
=
∑
∑ c †
α
, k
+ q
σ
 k , q ,
α
,
β c †
α
, k
+ q
σ
 k , q ,
α
,
β
αβ c
αβ
β
, k c
β
, k i S
− q i c †
γ
, p
− q
σ

γδ
+ ∑ q c p ,
δ
S q i S
− q
χ ( )
χ ( )
⇒
The  interac,on  is  repulsive  and  strongest  along  the  diagonals:  the   d x
2
-­‐y
2   state  wins!