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Density functional theory of phonon-driven superconductivityA. Linscheid, A. Sanna, and E. K. U. Gross

The success of density functional theory(DFT) for electronic structure calculations isat the basis of modern theoretical condensedmatter physics. The original theorem of Ho-henberg and Kohn (HK) and the reproducibil-ity of the exact electronic density in a non-interacting Kohn-Sham (KS) system both ex-tend to, in principle, any electronic phase,including magnetism and superconductivity.However the practical applicability of KS-DFTdepends on the availability of density function-als for the relevant observables of the system.As a matter of fact to derive density function-als able to describe the features of symmetrybroken phases, in particular the order parame-ter (OP) of that phase, turns out to be a task ofoutstanding complexity.

A scheme to circumvent this problem is togeneralize the HK theorem to include the OPas an additional density. The correspondingKS system then reproduces both the electronicand the additional density. In the case of super-conductivity the original formulation of a DFTscheme (SCDFT) is due to Oliveira, Gross andKohn [1] where the additional density is the or-der parameter of superconductivity χ(r, r′) =〈ψ↑(r)ψ↓(r′)〉. With a further development ofDFT to include the nuclear degrees of free-dom [2], in recent years an approximate ex-change correlation functionial Fxc for the KSBogoliubov-de-Gennes system has been de-rived which features the electron-phonon (e-ph) and the electron-electron (e-e) interactionon the same footing [3]. An important prereq-uisite for constructing such functional approxi-mations is the knowledge of exact properties atfinite temperature [4]. Ultimately the formal-ism leads to a BCS-type gap equation

∆nk=Znk∆nk −∑

n′k′K nk

n′k′

tanh(

βEn′k′

2

)

2En′k′∆n′k′ , (1)

where n and k, respectively, are the electronicband index and the wave vector inside the Bril-

louin zone. β is the inverse temperature and

Enk =

√

ξ2nk + |∆nk|

2 are the excitation ener-gies of the KS system, defined in terms ofthe gap function ∆nk and the KS eigenvaluesξnk measured with respect to the Fermi energy.The kernel, K , consists of two contributionsK = K e−ph

+ K e−e, representing the effectsof the e-ph and of the e-e interactions, respec-tively. The gap function is related to the OP inthe KS basis by χnk =

∆nk2Enk

tanh(

β

2Enk

)

.Compared to many-body perturbation the-

ory, SCDFT features two major achievements:1) It is completely free of adjustable param-eters. Coulomb and phonon mediated inter-actions are included without the need of in-troducing a phenomenological µ∗. 2) All thefrequency summations are performed analyti-cally in the construction of Fxc. Retardationeffects can be exactly included but at the sametime the gap equation has still the form of astatic BCS equation. The formal simplicity ofeq. 1 then allows to account for the anisotropyof real systems at a low computational cost.

Fig. 1: Superconducting gap of hole-dopedgraphane (hydrogenated graphene).

At the Fermi energy (ξnk = 0) the formof ∆nk is determined mostly by the attractivephononic term K e−ph. Beyond the phononicenergy range the interaction becomes repulsivedue the direct Coulomb interaction betweenelectrons in K e−e. The system then maximizesits condensation energy by including a signchange in ∆nk. In accordance with Eq. 1, whenboth the interaction and ∆nk change sign, then

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Selected Results

the overall contribution becomes once againattractive. This mechanism takes the nameCoulomb renormalization.

The typical behavior of ∆nk versus ξnk isplotted in Fig. 1 for graphane (hydrogenatedgraphene C2H2). We use a logarithmic scale toenhance the behavior at the Fermi energy. Thissystem shows a characteristic two gap struc-ture, i.e. ∆nk at the Fermi energy shows twodistinctly different values corresponding to thepresence of two Fermi surfaces. A similar be-haviour, but with three distinct gap values atthe Fermi energy, is found for hydrogen underpressure. We predict that this material has acritical temperature of 242 K at 450 GPa [5].

The more anisotropic the Fermi surface andthe electron-phonon coupling are the morestructured becomes the gap function at theFermi energy. An example is CaC6 shown inFig. 2 where the superconducting gap closelyreflects the phononic anisotropy.

Fig. 2: Fermi surface and nk-resolved supercon-ducting gap in CaC6. The color scale indicates theSCDFT gap (meV).

To go beyond this reciprocal space descrip-tion, we have recently implemented a trans-formation of the superconducting OP χnk backinto real space.

This means to multiply the KS basis {ϕnk(r)}of the initial expansion:

χ(R, s) =∑

nk

χnkϕnk(r)ϕ∗nk(r′) (2)

where R = (r + r′)/2 and s = (r − r′) are re-spectively the center of mass and the relativecoordinate of the Cooper pair. We are therebyable to connect the chemical bonding proper-ties with superconducting features in a verygraphic and compact way. As an example weshow χ(R, 0) of CaC6 and C2H2 in Fig. 3. The

Fig. 3: χ(R, 0) of CaC6 (top) and C2H2 (bottom)

electronic bonds giving the largest contribu-tion to superconductivity are clearly visible. Ingraphane the large positive values come fromthe sp2 carbon bonds. In CaC6 the dominantcontribution arises from the π-states as well asfrom dz2 Ca orbitals and interlayer states.

References

[1] L. N. Oliveira, E. K. U. Gross, and W. Kohn,Phys. Rev. Lett. 60, 2430 (1988)

[2] T. Kreibich and E. K. U. Gross, Phys. Rev. Lett.86, 2984 (2001)

[3] M. Luders et al., Phys. Rev. B 72, 024545 (2005);M. A. L. Marques et al., Phys. Rev. B 72, 024546(2005)

[4] S. Pittalis, C. Proetto, A. Floris, A. Sanna,C. Bersier, K. Burke, and E. K. U. Gross, Phys.Rev. Lett (2011), accepted, arXiv: 1008.0586

[5] P. Cudazzo, G. Profeta, A. Sanna, A. Floris,A. Continenza, S. Massidda, and E. K. U. Gross,Phys. Rev. Lett. 100, 257001 (2008); Phys. Rev.B 81, 134505 (2010); Phys. Rev. B 81, 134506(2010)

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