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Pedagogical Overview
Unconventional Superconductivity (The Phenomenon) and Unconventional Superconductors (The Materials)
• Conventional Superconductivity • Unconventional Superconductivity and Superconductors • Some Personal Favorites
The Essence of Superconductivity -- With the Benefit of Hindsight
! pi= "int #$( )% cm r$( ) = "int #$( )% cm r$( ) e&i
Internal structure of a pair
• q = 2 • Pairing symmetry, pair size • Single par6cle excita6ons • Density of states • Material parameters of GL theory • The mechanism
Microscopic Specific Proper-es
ri
e
e !"
!BCS•
CM wave func5on of pair
Macroscopic Emergent Proper-es
• GL Order parameter/GL theory • R = 0, B = 0, • Type II superconduc6vity • Josephson effect
! = !0 = hc / q
ith pair
Superconductivity arises when phases lock !1 =!2 =!3 = ......!"..... =!
Some History
Adapted from a DoE Report
Pr(O-‐F)FeAs
BCS Theory
Conven6onal Superconduc6vity GL Theory, BCS, Gorkov and Eliashberg
Conven6onal Era Unconven6onal Era
Key Results of BCS Theory
• Superconductivity arises when electrons bind into pairs and these pairs lock their mutual phases to form a macroscopic quantum state. • The single-particle excitations of a superconductor are comprised of a phase- coherent linear combinations of electrons and holes. • , which implies that if no other phases intervene, all normal metals will become superconductors as T -> 0. Here λ is the attractive interaction parameter (not necessarily the electron-phonon interaction). • Size of Cooper pair
Tc = !oe"1/#
(k !,"k #)
! ks = uk! k
e + vkei"! k
h
!
•k !
• ! k "
• !k "
•! "k #
(k !,"k #)$ ( %k !," %k #)FS
In general, time reversed pairs (Anderson)
! " !vF / kTc
The Triumvirate Defining Conventional Superconductivity A Complete Theory (BCS + Mechanism + Equations for the Material Parameters)
BCS
Elisashberg Ginzburg-‐Landau
Tc =!0e"1
#"µ*
µ*= µ
1+ µ !n !F"D
#$%
&'(
F(r) = d 3! r " # 2 + 1
2$ # 4 !2
2m*%i&% e*
!cA'
()*+,#
2
+ B2
8-./0
10
230
40
Conventional Superconductivity
Unconventional Superconductivity
Even More Unconventional
Superconductivity Weak coupling λ << 1
Strong coupling λ > 1 Maximum in Tc
Very strong coupling λ >> 1 Phase fluctua5ons – Tc à 0
Mean field theory Quantum fluctua5ons when ξ is small
Bose metal of pairs?
S-‐wave singlet Time reversed pairs
Higher angular momentum pairing
p, d, …… pairing
S-‐wave triplet odd-‐ frequency pairing Broken TRS pairs – s + is, d + is, etc
Retarded el-‐ph interac5on Any other mechanism: Electronic interac5ons -‐-‐
spin or charge
Non-‐Fermi liquid in normal state
Quantum cri5cal points Compe5ng ordered phases
Anderson’s theorem Tc independent of disorder
Enhanced coulomb repulsion due to disorder
Disorder driven Superconductor/Insulator
Transi5on
Tc =!0e"1!
Tc = e!1
!!µ*
Tc =!0e"1
#"µ*"$µ*
Unconventional Superconductivity
!(k) = dk 3" V (k,k ')!(k ')
! " !vF / kTpTc = min T! ,Tp{ }
Conventional Classic BCS and
Extensions
Unconventional Superconductivity
More Unconventional Superconductivity
Single band superconductivity
Weakly coupled bands/orbitals
Josephson effects in k space
Q = 0 pairing Finite Q pairing
Pair density waves
Quasi-particles Phase coherent
superposition of e’s and h’s
SN proximity effect In the presence of a
Dirac point
Majorana Fermions
Earthly Other worldly
Unconventional Superconductivity
Adapted from a DoE Report
Pr(O-‐F)FeAs BC
S Theory
Conven6onal Era Unconven6onal Era
Unconven5onal Superconduc5vity Along the El-‐Ph Line
BaPb1-‐xBixO3
12K
BaPb1-‐xBixO3
12K
SEMICONDU
CTING
SUPE
R-‐C
30K
Ba1-‐xKxBiO3
Ba
Nega5
ve-‐U CDW
Phase Diagrams of the Superconducting Bismuthates
Nega5
ve-‐U CDW
Note similarity to the phase diagrams of the cuprate superconductors but where the ordered insulating state is in the charge sector
BaBiO3
Pb Doping Ba
BiO3
K Doping
Charge-Disproportionated (Negative U) Superconductors (e.g., BaBiO3) A Failure of LDA Theory à A new class of correlated insulator
Bi4+ Bi4+ Bi4+ Bi4+ Bi4+
Bi3+ Bi3+ Bi3+ Bi5+ Bi5+
Metal
Oxygen
2Bi4+ (4s1)! Bi3+ (4s2 ) + Bi5+ (4s0 )and
Oxygen atoms move to screen charge (Breathing mode)
Nega6ve U -‐-‐ Involves both charge and laZce
Insulator
e e e e e
ee ee ee U
Superconductivity arises upon doping (Ba1-xKxBO3 and BaPb1-xBixO3)
Tc0 =!0e"1
#"µ*
Tcd =!0e"1
!"µ*""µ*
Disorder in the High Tc Bismuthates
BaPb1-xBixO3
Neg
ativ
e-U
CD
W
Orth
orho
mbi
c
Orth
orho
mbi
c
Mon
oclin
ic
BaPb1-xBixO3
Tetra
gona
l/O
rthor
hom
bic
λ = el-‐ph interac6on parameter μ* = retarded coulomb repulsion
• Clean limit (BCS/Eliashberg):
Conven5onal theory for Tc
• Dirty limit (BCS/Eliashburg/Fukuyama):
!µ * = increase in μ* due to disorder-‐induced localiza6on
Measured Tc = Tcd and independently es6mated permits es6mate of Tc0 !µ *
Disorder ma^ers in region of highest Tc
Heavy Fermions and Magnetic Mechanisms
Adapted from a DoE Report
Pr(O-‐F)FeAs BC
S Theory
Conven6onal Era Unconven6onal Era
Cuprates and Strong Correlation – A Mixed Blessing
Adapted from a DoE Report
Pr(O-‐F)FeAs BC
S Theory
Conven6onal Era Unconven6onal Era
In Nature the Transition Temperature Can Be “Astronomically” High
Alfred and Schmitt RMP 80 (2008)
Quark-‐Gluon Plasma
Color-‐Flavor Locked Quark Ma^er
With Some Experimental Evidence
Local pairing,small unit cell and 3D (to minimize phase fluctuations), strong interactions, mean-field breakdown, not a single band.
Any Room Temperature Superconductor Will be Extremely Unconventional Σιζε οφ
Χοοπερ παιρσ
Room Temperature
Superconductor Room Temperature
Pairing Temperature TP (K)
Siz
e of
Coo
per P
air ξ
(Å)
Room Temperature
BCS Theory (vF const.) ! ! !vF / kTc
If you want to find something unconventional, don’t look under the street light.
Empirical Guidance on Specific Interactions Material Archetype
Tc Interac6on Guidance
Bismuthates (i.e., doped BaBiO3)
30K Charge + Lattice
Doped Negative U Insulator
Cs3C60
40K Lattice + Correlation (charge)
El-Ph Covalent Bonds
MgB2
40K Lattice El-Ph Covalent Bonds
Prediction
Fe-Based 50K Spin Antiferromagnetism Multiple orbitals
Cuprates 130K Spin
Doped Antiferromagnetic
Positive U Mott Insulator
Trace High Tc Anomalies
> Room Temperature
? Shouldn’t Ignore
Electronic (charge and spin) interactions look good
To Determine Tc There are Two Characteristic Temperatures to Consider
Superconductivity arises when electrons (or holes) form pairs and the quantum phases of these pairs order (lock) to form a coherent macroscopic quantum state with a single phase.
e e e e
e e e e
e e e e
e e e e
e e e e
! p = ! p ei"
! p!Pairs form
Pairing interaction
SC arises
Phase locking
! sc!
! pi
!i ! =!1 =!2 =!3 i i i
Each process has its own characteristic temperature
Phase locking
Macroscopic quantum state
Theoretical Guidance An always controversial subject
• How to op5mize the el-‐ph mechanism – Modern electronic structure calcula5ons (It is clear that theory could have predicted the high Tc superconduc5vity of Mg B2)
• New mechanisms • Electronic structure calcula5ons as a tool in the search • Strong interac5ons, but honor the “Goldilocks Principle”
E.g., from weak coupled BCS (delocalized) limit to strong coupled (local) limit. Typically also for doping.
H = !t cm† cm" !U np#np$" ! t% cm
† cp"Berg, Orgad and Kivelson PRB 78, 094509 (2008)
High Tc and High Pair Density Superconductor Using a Normal Metal/Negative-U Insulator Proximity Effect:
Normal Metal
Nega6ve-‐U Insulator
Tc
t!
…. And Maybe We Can Have Our Cake and Eat It Too