Predicting New BCS Superconductors. Marvin L. Cohen Department of Physics, University of. Lawrence Berkeley Laboratory Berkeley, CA

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1 Predicting New BCS Superconductors Marvin L. Cohen Department of Physics, University of California, and Materials Sciences Division, Lawrence Berkeley Laboratory Berkeley, CA

2 CLASSES OF SUPERCONDUCTORS BCS : conventional metals, C60, some organics, doped semiconductors, MgB2, BCS EXOTIC: copper oxides, heavy fermion metals, some organics,

3 SEMIEMPIRICAL GeTe, SnTe, SrTiO3

4 Standard Model Plane Wave Pseudopotential Method [PWPM]

5 Plane Wave Pseudopotential Method (Standard Model of Solids) For a broad class of solids, clusters, and molecules, this method describes ground-state and excited-state properties such as: electronic structure crystal structure and structural transitions structural and mechanical properties vibrational properties p electron-lattice interactions superconductivity optical properties photoemission properties

6 SEMIEMPIRICAL GeTe, SnTe, SrTiO AB INITIO high pressure: sh Si, hcp Si, Ge(?),GaAs, Nb3Nb

8 Calculational Methods

9 Also: An & Pickett, PRL 2001; Kortus, et al, PRL 2001; Liu, et al, PRL 2001; Kong, et al, PRB 2001;

10 Superconductivity in the Eliashberg Formalism BCS Theory Electron pairing via phonon exchange Main ingredient: momentum and frequency-dependent Eliashberg function where N(ε F ) = density of states per spin at Fermi level g kk = electron-phonon matrix element ω jq = frequency of phonon in jth branch with q=k-k Equivalently: λ = < λ(k, k, 0) >

11 Transition Temperature and Isotope Effect Anharmonicity --> small α Β H.J.CHOI et al

12 Distribution of Electron-Phonon Couplings σ π π π σ σ Notes: 1) most metals: λ ~ ) MgB 2 : <λ> = 0.61; specific heat data λ = 0.58, 0.62

13 Superconducting Gap at 4K Δ(k) on Fermi surface at T=4 K Large gap on cylindrical σ sheets 2 dominant sets of gap values

14 Specific Heat of MgB2

15 R.M. Swift & D. White J. Am. Chem. Soc. 79, 3641 (1957) A layered material whose low temperature specific heat did not conform to the expectations of Debye theory MgB 2 K 2) C/ /T (cal/k K 40K T 2 (K 2 ) Thanks to Neil Ashcroft for sending me the paper and table

16 Summary MgB 2 is a multi-gap, phonon-mediated superconductor Large electron-phonon coupling of the σ boron states responsible for high T c Need to solve the k-dependent Eliashberg equations to obtain the correct T c and other quantities First-principles results explained T c, specific heat, isotope exponents, photoemission, i tunneling, and other data Theory predicts at least 2 dominant superconducting gap values at low temperature, and the two-gap feature is robust against pressure, doping and impurity scattering.

17 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 NOW.

18 Phonon softening in superconductors Phonon softening e-ph coupling strength TaC (Tc = 11K) 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)

19 E2g pphonon in MgB g 2 and AlB2

20 E 2g phonon in MgB 2 and AlB 2 E 2g? AlB 2 MgB 2 E 2g K. P. Bohnen, R. Heid, and B. Renker, PRL 86, 5771 (2001).

21 Doping dependence of phonon renormalization in MgB 2 Both electron and hole dopings result in reduced phonon renormalization. hole-doping electron-doping. Zhang, Louie, Cohen, PRL 94 (2005)

24 FOR C60, ROUGHLY THE ELECTRON -PHONON PARAMETER IN BCS λ~nv SO TO GET HIGHER Tc s: ] 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,?

27 BN/C60 Peapods

28 T c formulas T c 1 = 1.14ω ph exp BCS, 1957 N(0) V TD 1.04(1 + λ) Tc = exp * McMillan, λ μ ( λ ) ω log 1.04(1 + λ) T = exp for λ < 1.5 c * 1.20 λ μ ( λ) Allen and Dynes, T = λ ω for λ > 10, μ * =0 c λ max 2 ω 2 1 α F ( ω ) ω 0 2 dω + anisotropic electrons, anharmonic phonons, etc

29 ELECTRON-PHONON COUPLING λ ω 2 η i = i M i SO λ CAN BE VIEWED AS THE RATIO OF AN ELECTRONIC SPRING CONSTANT η AND A LATTICE SPRING CONSTANT

30 Numerical results C, diamond C, graphite BN, zincblende Si, diamond

31 Values of η η (ev/å 2 2 ) (K) EXP λ ω C (diamond)* ~ 10 C (graphite)* ? BN* ? Si* ~10 *at peak of η(e) J. MOUSSA et al

32 Strong coupling limit T c λ ω λ ~ Ω 2 ph ω ω 2 ph 2 ph electron-phonon coupling strength Or the electronic spring constant /ionic mass η / M Tc α Ω ω αω 2 2 stability of ph ph ph bare lattice

34 Diamond

35 Graphene Electronic Structure E Energy EE E F k x ' unoccupied k y ' occupied r E = hv F k k y k x E 2 = p 2 c 2 2D massless Dirac fermion system

36 2-D graphene as physical realization of (2+1)D QED Single particle energy dispersion ARPES, S. Y. Zhou et al, Nature Phys.2, 595 (2006) Massless Dirac equation with c * ~c/300~10 6 m/s Quantum Hall effect in graphene obseved Electric field induced half-metallic states in graphene nanoribbons

37 Electron and Phonon Self-energy

38 Bloch to Wannier Representation Bloch Wannier F. GIUSTINO et al

39 Wannier Representation Bloch Wannier

40 Wannier Representation

FIG. 1: (color online) Left: phonon dispersions

with the experimental data of Ref. 23 (circles).

dispersions of TaC after rescaling the Ta mass

The arrows indicate the wavevectors exhibiting

41 FIG. 1: (color online) Left: phonon dispersions of (a) TaC and (b) HfC (solid lines), together with the experimental data of Ref. 23 (circles). The dashed lines in (b) correspond to the dispersions of TaC after rescaling the Ta mass to the Hf value. The arrows indicate the wavevectors exhibiting Kohn anomalies. Right: Fermi surfaces of TaC (c) and HfC (d). J. NOFFSINGER et al

42 Electron Self-energy YIELDS A MASS ENHANCEMENT AND ASSOCIATED KINK AT THE FERMI SURFACE. KINKS HAVE BEEN OBSERVED IN ARPES DATA AND INTERPRETETED AS SIGNATURES OF STRONG ELECTRON-PHONON COUPLING.

43 Electron-Phonon Interaction in the Photoemission Spectrum of La 2-x Sr x CuO4 from First Principles 2 x x p Kink (for example, Lanzara et al, Nature) 2001 By measuring the change in slope, the electron-phonon coupling is estimated k k

46 CONCLUSION BASED ON THE WANNIER FORMALISM FOR CALCULATING ELECTRON-PHONON SELF- ENERGIES, THE COUPLING IS 1/7 OF WHAT IS NEEDED TO REPRODUCE THE OBSERVED ARPES KINKS F.GIUSTINO et al--on the web

47 THE END

arxiv:cond-mat/ v1 9 Nov 2001

arxiv:cond-mat/ v1 9 Nov 2001 Superconducting energy gaps, low temperature specific heat, and quasiparticle spectra of MgB 2 Hyoung Joon Choi, David Roundy, Hong Sun, Marvin L. Cohen, & Steven G. Louie Department of Physics, University