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