Predicting New BCS Superconductorsg p
Marvin L. CohenDepartment of Physics, University of
California and Materials Sciences DivisionCalifornia, and Materials Sciences Division,Lawrence Berkeley Laboratory
Berkeley, CA
CLASSES OFCLASSES OF SUPERCONDUCTORS-------------------------------
BCS : conventional metals, C60, some organics, doped semiconductors, MgB2,…
-------------------------“BCS” EXOTIC: copper oxides, heavyBCS EXOTIC: copper oxides, heavy
fermion metals, some organics,…
SEMIEMPIRICAL
GeTe SnTe SrTiO3GeTe, SnTe, SrTiO3
Standard ModelPlane Wave Pseudopotential Method
[PWPM][PWPM]
Plane WavePseudopotential MethodPseudopotential Method
(Standard Model of Solids)
For a broad class of solids, clusters, andmolecules, this method describesground-state and excited-stateproperties such as:
electronic structurecrystal structure and structural
transitionsstructural and mechanical propertiesvibrational propertiesp pelectron-lattice interactionssuperconductivityoptical propertiesoptical propertiesphotoemission properties
G T S T S TiO3SEMIEMPIRICAL
GeTe, SnTe, SrTiO3-------------------------------
AB INITIOhigh pressure:high pressure:
sh Si, hcp Si, Ge(?),GaAs,Nb3Nb
Calculational Methods
Also: An & Pickett, PRL 2001; Kortus, et al, PRL 2001; Liu, et al, PRL 2001; Kong, et al, PRB 2001; …
Superconductivity in the Eliashberg Formalism
BCS TheoryElectron pairingi h hvia phonon exchange
Main ingredient: momentum– and frequency-dependent Eliashberg function
where N(εF) = density of states per spin at Fermi level l t h t i l tgkk’ = electron-phonon matrix element
ωjq = frequency of phonon in jth branch with q=k-k’
Equivalently:q y
λ = < λ(k, k’, 0) >
Transition Temperature and Isotope EffectEffect
Anharmonicity --> small αΒ
H.J.CHOI et al
Distribution of Electron-Phonon Couplingsp g
σ−π
σ−σπ−π
Notes: 1) most metals: λ ~ 0.3 - 0.52) MgB2: <λ> = 0.61; specific heat data λ = 0.58, 0.62
Superconducting Gap at 4K
• Δ(k) on Fermi surface at T=4 K
• Large gap on cylindricalσ−sheets
• 2 dominant sets of gap values
Specific Heat of MgB2
R.M. Swift & D. White J. Am. Chem. Soc. 79, 3641 (1957)A layered material whose low temperature specific heat did
0.014not conform to the expectations of Debye theory.
0.01
0.012
K2)
MgB2
40K
0.006
0.008
/T (c
al/K 40K
9K
0 002
0.004
C/
0
0.002
0 1000 2000 3000 4000 5000T2 (K2)T2 (K2)
Thanks to Neil Ashcroft for sending me the paper and table
Summary
• MgB2 is a multi-gap, phonon-mediated superconductor
• Large electron-phonon coupling of the σ boron states responsible for high Tc
• Need to solve the k-dependent Eliashberg equations to obtain the correct Tc and other quantities
• First-principles results explained Tc, specific heat, isotope t h t i i t li d th d texponents, photoemission, tunneling, and other data
• Theory predicts at least 2 dominant superconducting gap values at low temperature and the two gap feature is robustvalues at low temperature, and the two-gap feature is robust against pressure, doping and impurity scattering.
RAISING Tc
WE TRIED TO USE THEORY TO SUGGEST HOW TO INCREASE THE TRANSITION
TEMPERATURE OF MAGNESIUM DIBORIDE SIGNIFICANTLY BUT FAILED!
THIS RESULT IS CONSISTENT WITH EXPERIMENTS UP TO NOWEXPERIMENTS UP TO NOW.
Phonon softening in superconductors
Phonon softening ↔ e ph coupling strengthTaC (Tc = 11K)HfC (Tc ~ 0K)
Phonon softening ↔ e-ph coupling strength
HfC (Tc ~ 0K)
H. G. Smith and W. Glaser, PRL 25, 1611 (1970).
L. Pinstchovius et al., PRL 54, 1260 (1985)L. Pinstchovius et al. PRB 28, 5866 (1983)L. Pintschovius, phys. stat. sol. (b) 242, 30 (2005)
E2g phonon in MgB2 and AlB22g p g 2 2
E2g phonon in MgB2 and AlB22g p g 2 2
E2g
AlB2MgB2
?
E2g
K. P. Bohnen, R. Heid, and B. Renker, PRL 86, 5771 (2001).
Doping dependence of phonon renormalization in MgB2
Both electron and holedopings result in reduced
h li tiphonon renormalization.
hole-doping electron-dopinghole doping electron doping
. Zhang, Louie, Cohen, PRL 94 (2005)
FOR C60, ROUGHLY THE ELECTRON -PHONON PARAMETER IN BCS λ~NV
SO TO GET HIGHER Tc’s: ------------------------------------------
1] INCREASE “N” WITH CARRIERS BY DOPING OR BAND STRUCTURE EFFECTS--LIKE d-BANDS
2] CAN INCREASE “V” BY INCREASING THE “CURVATURE”
GRAPHENE, GRAPHITE, NANOTUBE, C60, C36,?
BN/C60 Peapods
Tc formulas
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
VNT phc )0(
1exp14.1 ω BCS, 1957
⎟⎟⎠
⎞⎜⎜⎝
⎛+−+
−=)6201(
)1(04.1exp451 * λμλ
λDc
TT McMillan, 1968
⎠⎝ )(
⎟⎟⎠
⎞⎜⎜⎝
⎛++
−=)6201(
)1(04.1exp201 *
log
λμλλω
cT for λ < 1.5
⎠⎝ + )62.01(45.1 λμλ
⎟⎠
⎜⎝ +− )62.01(20.1 λμλ
2183.0 ωλ=cT for λ > 10, μ*=0Allen and Dynes, 1975
∫ −max 12 )(2ω
λ dF∫≡0
12 )(2 ωωωαλ dF+ anisotropic electrons, anharmonic phonons, etc…
ELECTRON-PHONON COUPLING
∑ iηωλ 2
ELECTRON PHONON COUPLING
∑=i i
i
Mηωλ
SO λ CAN BE VIEWED AS THE RATIO OF ANELECTRONIC SPRING CONSTANT η AND A LATTICESPRING CONSTANT
Numerical resultsC, diamond
C, graphite
BN, zincblendeSi, diamond
Values of η
η (eV/Å2) (K) EXPC (diamond)* 54 290 ~ 10
2183.0 ωλ
C (diamond) 54 290 10
C (graphite)* 48 270 ?
BN* 36 240 ?BN 36 240 ?
Si* 10 82 ~10
*at peak of η(E)
J. MOUSSA et al
Strong coupling limit
21830 ωλ≤T electron-phonon183.0 ωλ≤cT
Or the electronic spring
electron-phononcoupling strength
2
22
~ phph
ωω
λ−Ω
constant /ionic mass
η / Mphω η / M
phphphcT Ω≤−Ω≤ αωα 22 stability ofbare latticebare lattice
DiamondDiamond
Graphene Electronic StructureEn
ergy
E
unoccupiedr EE
kx' ky'occupied
E =hvF ′ k EF
kx
ky
E2 = p2c2
2D massless Dirac fermion system
2-D graphene as physical realization of (2+1)D QED
Single particle energy dispersion ARPES, S. Y. Zhou et al,
M l Di ti ith * /300 106 /
Nature Phys.2, 595 (2006)
Massless Dirac equation with c ~c/300~106m/s
Quantum Hall effect in graphene obseved
Electric field induced half-metallic states in graphene nanoribbons
Electron and Phonon Self-energygy
Bloch to Wannier Representationp
Bloch Wannier
F. GIUSTINO et al
Wannier RepresentationWannier RepresentationBloch Wannier
Wannier Representation
FIG. 1: (color online) Left: phonon dispersions of (a) TaCand (b) HfC (solid lines), together with the experimental dataof Ref. 23 (circles). The dashed lines in (b) correspond tothe dispersions of TaC after rescaling the Ta mass to the Hfthe dispersions of TaC after rescaling the Ta mass to the Hfvalue. The arrows indicate the wavevectors exhibiting Kohnanomalies. Right: Fermi surfaces of TaC (c) and HfC (d).
J. NOFFSINGER et al
Electron Self-energygy
YIELDS A MASS ENHANCEMENT AND ASSOCIATED “KINK” AT THE FERMI SURFACE.”KINKS” HAVE BEENKINK AT THE FERMI SURFACE. KINKS HAVE BEEN OBSERVED IN ARPES DATA AND INTERPRETETED AS SIGNATURES OF STRONG ELECTRON-PHONON COUPLINGCOUPLING.
Electron-Phonon Interaction in the Photoemission Spectrumof La2-xSrxCuO4 from First Principles
“Kink” (for example, Lanzara et al, Nature) 2001
2 x x p
By measuring the change in slope, the electron-phonon coupling is estimated
0 1 0 10 1 0 1k k
CONCLUSION
BASED ON THE WANNIER FORMALISM FOR CALCULATING
ELECTRON-PHONON SELF-ENERGIES, THE COUPLING IS 1/7
OF WHAT IS NEEDED TOOF WHAT IS NEEDED TO REPRODUCE THE OBSERVED
ARPES “KINKS”ARPES “KINKS” F.GIUSTINO et al--on the web
THE ENDTHE END