Density Functional Theory for Superconductors

Transcription

1 Density Functional Theory for Superconductors M. A. L. Marques IMPMC Université Pierre et Marie Curie, Paris VI GDR-DFT05, , Cap d Agde M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 1 / 25

2 Co-workers L. Fast, N. Lathiotakis, A. Floris, E. K. U. Gross Institut für Theoretische Physik, FU Berlin, Germany G. Profeta, A. Continenza Università degli studi dell Aquila, Italy S. Massidda, C. Franchini Università degli Studi di Cagliari, Italy M. Lüders Daresbury Laboratory, Warrington WA4 4AD, UK M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 2 / 25

3 Outline 1 DFT for superconductors 2 Results Simple Metals MgB 2 Li and Al under pressure 3 Conclusions M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 3 / 25

4 Outline 1 DFT for superconductors 2 Results Simple Metals MgB 2 Li and Al under pressure 3 Conclusions M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 3 / 25

5 Outline 1 DFT for superconductors 2 Results Simple Metals MgB 2 Li and Al under pressure 3 Conclusions M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 3 / 25

6 Goal DFT for superconductors We want to describe Conventional Superconductivity Our goal is To have a theory able to predict, fully ab-initio material specific properties like T c and the gap 0 M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 4 / 25

7 Goal DFT for superconductors We want to describe Conventional Superconductivity Our goal is To have a theory able to predict, fully ab-initio material specific properties like T c and the gap 0 M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 4 / 25

8 DFT for superconductors State of the Art BCS Theory The attractive interaction between the Cooper pairs is an empirical parameter BCS reproduces common features (not material specific) of weak el-ph coupling superconductors (e.g. the ratio 2 0 /k B T c) Eliashberg Theory Strong coupling theory But el-ph and Coulomb interactions are not treated on the same footing Coulomb repulsion is normally included through the parameter µ, usually fitted to the experimental T c Not possible to perform a fully ab-initio calculation of superconducting properties M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 5 / 25

9 DFT for superconductors State of the Art BCS Theory The attractive interaction between the Cooper pairs is an empirical parameter BCS reproduces common features (not material specific) of weak el-ph coupling superconductors (e.g. the ratio 2 0 /k B T c) Eliashberg Theory Strong coupling theory But el-ph and Coulomb interactions are not treated on the same footing Coulomb repulsion is normally included through the parameter µ, usually fitted to the experimental T c Not possible to perform a fully ab-initio calculation of superconducting properties M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 5 / 25

10 DFT for superconductors State of the Art BCS Theory The attractive interaction between the Cooper pairs is an empirical parameter BCS reproduces common features (not material specific) of weak el-ph coupling superconductors (e.g. the ratio 2 0 /k B T c) Eliashberg Theory Strong coupling theory But el-ph and Coulomb interactions are not treated on the same footing Coulomb repulsion is normally included through the parameter µ, usually fitted to the experimental T c Not possible to perform a fully ab-initio calculation of superconducting properties M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 5 / 25

11 SCDFT DFT for superconductors DFT for Superconductors Coulomb and el-ph interactions enter the theory on the same footing No empirical parameter, like µ, is used Allows to predict T c and 0 from first principles The order parameter of the singlet superconducting state χ(r, r ) = ˆψ (r) ˆψ (r ) is the most important ingredient of SCDFT, entering the theory as an extra density cond-mat/ , cond-mat/ (accepted in Phys. Rev. B) M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 6 / 25

12 SCDFT DFT for superconductors DFT for Superconductors Coulomb and el-ph interactions enter the theory on the same footing No empirical parameter, like µ, is used Allows to predict T c and 0 from first principles The order parameter of the singlet superconducting state χ(r, r ) = ˆψ (r) ˆψ (r ) is the most important ingredient of SCDFT, entering the theory as an extra density cond-mat/ , cond-mat/ (accepted in Phys. Rev. B) M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 6 / 25

13 Hamiltonian DFT for superconductors Our starting Hamiltonian is Ĥ = Ĥe + Ĥn + Ĥen, with Z Z Z Ĥ e = ˆT e + Ŵee + d 3 r ˆn(r)v(r) d 3 r d 3 r ˆˆχ(r, r ) (r, r ) + H.c. Z Ĥ n = ˆT n + Ŵnn + d 3 N n ˆΓ(R)V (R), v(r) = external potential acting on the electrons (e.g. applied voltage) (r, r ) = external pairing potential (e.g. proximity induced) V (R) = external potential acting on the nuclei n(r) = X σ ˆψ σ (r) ˆψ σ(r) ; χ(r, r ) = ˆψ (r) ˆψ (r ) Γ(R) = ˆφ (R 1 ) ˆφ (R 2 ) ˆφ(R 2 ) ˆφ(R 1 ) M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 7 / 25

14 DFT for superconductors Hohenberg-Kohn theorem Theorem There is the one-to-one correspondence { n(r), χ(r, r ), Γ(R) } { v(r), (r, r ), V (R) } As a consequence: Theorem All physical observables are functionals of {n(r), χ(r, r ), Γ(R)} M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 8 / 25

15 DFT for superconductors Hohenberg-Kohn theorem Theorem There is the one-to-one correspondence { n(r), χ(r, r ), Γ(R) } { v(r), (r, r ), V (R) } As a consequence: Theorem All physical observables are functionals of {n(r), χ(r, r ), Γ(R)} M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 8 / 25

16 DFT for superconductors Kohn-Sham scheme Electronic KS equation ] [ v s(r) µ ] [ v s(r) µ α u i (r) + d 3 r s (r, r )v i (r ) = E i u i (r) v i (r) + d 3 r s (r, r )u i (r ) = E i v i (r) Nuclear KS equation [ ] 2 α 2M + V s(r) Φ n (R) = E n Φ n (R) There exist functionals v s [n, χγ], s [n, χγ], and V s [n, χγ] such that the above equations reproduce the exact densities of the interacting system. M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 9 / 25

17 DFT for superconductors Kohn-Sham potentials The 3 KS potentials are defined as v s(r) = 0 {z} v s(r, r ) = 0 {z} V s(r) = 0 {z} V Z d 3 R ZN(R) r R {z } v H en + χ(r, r ) r r {z } H + X αβ + Z αz β R α R β {z } W nn + δf xc δχ(r, r ) {z } xc X α Z d 3 r n(r ) r r {z } v H ee Z d 3 r Until here the theory is, in principle, exact: no approximation yet. n(r) r R α {z } V H en + δfxc δn(r) {z } v xc + δf xc δγ(r) {z } V xc M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 10 / 25

18 DFT for superconductors Harmonic Approximation In a solid, the atoms remain close to their equilibrium positions, so we can expand all quantities around these values. For example V s (R) = V s (R 0 + U) = V s (R 0 ) + V s R0 U µ i j ν V R0 s U µ i Uj ν ij µν The linear term in U vanishes, as the atoms are in equilibrium, so we obtain Ĥ n, KS = ( Ω q ˆb ˆb q q + 3 ) + O(U 3 ) 2 q Similarly, we obtain a electron-phonon coupling term in H e, KS by expanding the vh en + v xc terms. M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 11 / 25

19 DFT for superconductors Decoupling approximation The KS equations for the electrons involve two very different energy scales, the Fermi energy, and the gap energy. It is possible to decouple them with the help of the decoupling approximation. We write u i (r) u i ϕ i (r) ; v i (r) v i ϕ i (r) where the ϕ i are solutions of the normal state KS equation Near the transition temperature, χ 0, the equation for s can be cast into a BCS-like gap equation. s(j) = 1 2 X j tanh β 2 ξ j w eff (i, j) ξ j s(j) where the matrix elements of the effective interaction w eff (r, r, x, x ), and w eff (r, r, x, x ) = δ 2 F xc[n, χ] δχ (r, r )δχ(x, x ) χ=0 M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 12 / 25

20 DFT for superconductors Decoupling approximation The KS equations for the electrons involve two very different energy scales, the Fermi energy, and the gap energy. It is possible to decouple them with the help of the decoupling approximation. We write u i (r) u i ϕ i (r) ; v i (r) v i ϕ i (r) where the ϕ i are solutions of the normal state KS equation Near the transition temperature, χ 0, the equation for s can be cast into a BCS-like gap equation. s(j) = 1 2 X j tanh β 2 ξ j w eff (i, j) ξ j s(j) where the matrix elements of the effective interaction w eff (r, r, x, x ), and w eff (r, r, x, x ) = δ 2 F xc[n, χ] δχ (r, r )δχ(x, x ) χ=0 M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 12 / 25

21 DFT for superconductors Construction of an approximate F xc We apply Görling-Levy perturbation theory Ĥ = ĤKS + Ĥ1 In first order we have 4 contributions to F xc F xc = M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 13 / 25

22 DFT for superconductors The gap equation (n, k) = Z ph (n, k) (n, k) n k [ Kph + K el ] (n, k ) 2E n,k q where E n,k = (ɛ n,k µ) 2 + (n, k) 2 ( ) βen tanh,k 2 Features BCS form but parameter free effective interaction K = K ph + K el is calculated ab-initio static (frequency independent) but with retardation effects included in the Z and K functionals k-space formalism allows to calculate the (possibly) anisotropic nature of the gap M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 14 / 25

23 DFT for superconductors The gap equation (n, k) = Z ph (n, k) (n, k) n k [ Kph + K el ] (n, k ) 2E n,k q where E n,k = (ɛ n,k µ) 2 + (n, k) 2 ( ) βen tanh,k 2 Features BCS form but parameter free effective interaction K = K ph + K el is calculated ab-initio static (frequency independent) but with retardation effects included in the Z and K functionals k-space formalism allows to calculate the (possibly) anisotropic nature of the gap M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 14 / 25

24 Outline Results Simple Metals 1 DFT for superconductors 2 Results Simple Metals MgB 2 Li and Al under pressure 3 Conclusions M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 15 / 25

25 Simple Metals Results Simple Metals Calculated T c [K] Mo TF-ME TF-SK TF-FE Ta Pb Nb Al Experimental T c [K] Calculated 0 [mev] Al TF-ME TF-SK TF-FE Ta Pb Nb Experimental 0 [mev] M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 16 / 25

26 Gap of Pb Results Simple Metals 0 [mev] Pb Experiment TF-SK TF-ME T [K] [mev] Pb Experiment TF-ME, T = 0 K TF-SK, T = 0 K TF-SK, T = 6 K TF-SK, T = 7 K ξ [ev] M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 17 / 25

27 Outline Results MgB 2 1 DFT for superconductors 2 Results Simple Metals MgB 2 Li and Al under pressure 3 Conclusions M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 18 / 25

28 MgB 2 Results MgB 2 Why such a high T c (39.5 K)? Strong coupling of σ bands with the optical E2g phonon mode for q along the Γ-A line (for π bands el-ph is roughly 3 times smaller) Strong anisotropy, which leads to a k-dependent gap = (k) Phys. Rev. Lett. 94, (2005) M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 19 / 25

29 MgB 2 - Results Results MgB 2 [mev] Iavarone et al. Szabo et al. Schmidt et al. Gonnelli et al. present work (a) C el (T)/C el,n (T) T [K] Bouquet et al. Putti et al. Yang et al. present work T/T c (b) M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 20 / 25

30 Results MgB2 Gap of MgB2 8 σ σ Average π π Average Experiments [mev] M. A. L. Marques (IMPMC) ε µ [ev] DFT for superconductors 1 10 GDR-DFT 21 / 25

31 Outline Results Li and Al under pressure 1 DFT for superconductors 2 Results Simple Metals MgB 2 Li and Al under pressure 3 Conclusions M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 22 / 25

32 Results Li and Al under pressure Li and Al under pressure T c (K) SCDFT, this work Mc-Millan, this work Lin [5] Shimizu [6] Struzhkin [7] Deemyad [8] Li fcc hr1 ci Pressure (GPa) SCDFT, this work Mc-Millan, this work Gubser [11] Sundqvist [12] Al Pressure (GPa) M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 23 / 25

33 Conclusions What can we learn? Suppose we had a very good approximation for the functional F xc [n, χ]. What could we learn about the mechanism leading to superconductivity in the high-t c materials? Remember: The functional F xc [n, χ] is universal, i.e., the same functional for all materials. By solving the KS equation for the particular material we can understand the mechanism in retrospect by studying the effective interaction w eff (i, j) = wxc(i, el j) + wxc ph (i, j) M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 24 / 25

34 Outlook Conclusions DFT of superconductivity offers, for the first time, the possibility to perform fully ab-initio calculations of superconducting properties, like the transition temperature, the gap, or the specific heat. Until now, we obtained very promising results for simple metals MgB 2 Li and Al under pressure However, further work is necessary More applications to benchmark the theory: doped fullerenes, nanotubes, high-t c s, etc. Replace the Thomas-Fermi interaction by a RPA. Development of new (better) functionals for the electron-phonon interaction. M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 25 / 25