Density functional theory of phonondriven superconductivity
A. Linscheid, A. Sanna, E. K. U. Gross The success of density functional theory (DFT) for electronic structure calculations is at the
basis of modern theoretical condensed matter physics. The original theorem of Hohenberg
and Kohn (HK) and the reproducibility of the exact electronic density in a noninteracting
Kohn Sham (KS) system both extend to, in principle, any electronic phase, including
magnetism and superconductivity.
However the practical applicability of KSDFT depends on the availability of density
functionals for the relevant observables of the system. As a matter of fact to derive
density functionals able to describe the features of symmetry broken phases,
in particular the order parameter (OP) of that phase, turns out to be a task of outstanding
complexity.
A scheme to circumvent this problem is to generalize the HK theorem to include the
OP as an additional density. The corresponding KS system then reproduces both
the electronic and the additional density.In the case of superconductivity the original
formulation of a DFT scheme (SCDFT) is due to Oliveira, Gross and Kohn
^{[1]}
where the additional density is the order parameter of superconductivity
χ(r,r') = 〈Ψ↑(r)Ψ↓(r')⟩.
With a further development of DFT to include the nuclear degrees of freedom^{
[2]},
in recent years an approximate exchange correlation potential
F_{xc} for the KS BogoliubovdeGennes
system has been derived which features the electronphonon (eph) and the
electronelectron (ee) interaction on the same footing^{
[3]}.
This leads to a selfconsistent equation for the superconducting OP that depends on the phononic features and on the normalstate electronic structure.
In the space of single particle KS orbitals it is formally equivalent to a BCS gap equation
where n and k , respectively, are the electronic band index and the wave
vector inside the Brillouin zone. β is the inverse temperature and
are the excitation energies of the KS system, defined in terms of the gap
function Δ_{nk} and the KS eigenvalues
ξ_{n}k measured with respect to the
Fermi energy.
The kernel, κ , consists of two contributions
κ = κ ^{
eph}+κ^{
ee}, representing the effects of the eph and of the ee
interactions, respectively. The gap function is related to the the OP in the KS basis by
Compared to manybody perturbation theory, SCDFT features two major achievements:
1) It is completely free of adjustable parameters.
Coulomb and phonon mediated interactions are included without the need of introducing a
phenomenological μ^{∗}.
2) all the frequency summations are performed analytically in the construction of
F_{xc}.
Retardation effects can be exactly included but at the same time the gap equation has
still the form of a static BCS equation.
The formal simplicity of Eq. 1 then allows to account for the anisotropy
of real
systems at a low computational cost.
Fig. 1 Left: Superconducting gap of holedoped graphane (hydrogenated graphene), as a function of the energy distance from the Fermi level. Right: the Superconducting gap of CaC_{6}, nkresolved on the Fermi Surface (the colorscale gives the SCDFT gap in meV).
At the Fermi energy (ξ_{n}k = 0)
the form of Δ_{nk}
is determined mostly by the attractive phononic term
K^{eph}.
Beyond the phononic energy range the interaction becomes repulsive due the direct
Coulomb interaction between electrons in
K^{ee}.
The system then maximizes its condensation energy by including a sign change in
Δ_{nk}.
In accordance with Eq. 1, when both the interaction and
Δ_{nk}
change sign, then the overall contribution becomes once again attractive.
This mechanism takes the name Coulomb renormalization.
The typical behavior of Δ_{nk}
versus ξ_{n}_{k} is plotted
in Fig. 1
for graphane (hydrogenated graphene C_{2}H_{2}).
We use a logarithmic scale to enhance the behavior at the Fermi energy.
This system shows a characteristic two gap structure, i.e.
Δ_{nk}
at the Fermi energy shows two distinctly different values corresponding to the presence of
two Fermi surfaces.
A similar behaviour, but with three distinct gap values at the Fermi energy,
is found for hydrogen under pressure. We predict that this material has a critical
temperature of 242 K at 450 GPa
^{[4]}.
The more anisotropic the Fermi surface and the electronphonon coupling are the more
structured becomes the gap function at the Fermi energy. An example is CaC_{6}
shown in
Fig. 1
where the superconducting gap closely reflects the phononic
anisotropy [5].
To go beyond this reciprocal space description, we have recently implemented a
transformation of the superconducting OP
χ_{n}_{k} back into real space.
Fig.2 χ(R,0) of CaC_{6} (left) and C_{2}H_{2} (right)
This means to multiply the KS basis {φ_{nk}(r)}
of the initial expansion:
where R=(r+r')/2 and s=(rr') are
respectively the center of mass and the relative coordinate of the cooper pair.
We are thereby able to connect the chemical bonding properties with superconducting
features in a very graphic and compact way. As an example we
show χ(R,0) of CaC_{6} and
C_{2}H_{c} in Fig. 2. The electronic bonds giving the
largest contribution to superconductivity are clearly visible.
In graphane the large positive values come from the sp^{2} carbon bonds.
In CaC_{6} the dominant contribution arises from the πstates
as well as d_{z2} Ca orbitals and interlayer states.
The Coulomb renormalization contributions is provided by the CH bonds in graphane
and by the sp^{2} states of CaC_{6} (blue regions).
Although this real space representation is formally equivalent to the reciprocal
space one it results more natural when describing systems with large unit cells and
complex geometries like surfaces. We have recently studied a lead monolayer
deposited on the Si(111) surface. As shown in Fig. 3.
From the real space description one can clearly appreciate the localization of the
SC order parameter on the Pb layer, making this system a prototype for two
dimensional superconductivity.
Fig. 3 Crystal structure and SC order parameter for a Pb monolayer on the Si(111) surface. The lack of hybridization between lead and substrate makes the superconducting condensation extremely localized in real space.
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. Lueders, M.A.L. Marques, N.N. Lathiotakis, A. Floris, G. Profeta,
L. Fast, A. Continenza, S. Massidda, E.K.U. Gross , Phys. Rev. B 72, 024545 (2005),
M.A.L. Marques et al., Phys. Rev. B 72, 024546 (2005)
 [4]

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)
 [5]

A. Sanna, G. Profeta, A. Floris, A. Marini, E.K.U.
Gross and S. Massidda, Phys. Rev. B 75, 020511 (2007)
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