Density Functional Theory for Superconductors
M.A.L.Marques
marques@tddft.org
IMPMC Universit ´e Pierre et Marie Curie,Paris VI
GDRDFT05,18052005,Cap d'Agde
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 1/25
Coworkers
L.Fast,N.Lathiotakis,
A.Floris,E.K.U.Gross
Institut f ¨ur Theoretische Physik,
FU Berlin,Germany
G.Profeta,A.Conti
nenza
Universit a degli studi dell'Aquila,
Italy
S.Massidda,C.Fran
chini
Universit a degli Studi di Cagliari,
Italy
M.L¨uders
Daresbury Laboratory,Warring
ton WA4 4AD,UK
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 2/25
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
GDRDFT 3/25
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
GDRDFT 3/25
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
GDRDFT 3/25
DFT for superconductors
Goal
We want to describe
Conventional Superconductivity
Our goal is
To have a theory able to
predict
,fully
abinitio
material specic
properties like T
c
and the gap Δ
0
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 4/25
DFT for superconductors
Goal
We want to describe
Conventional Superconductivity
Our goal is
To have a theory able to
predict
,fully
abinitio
material specic
properties like T
c
and the gap Δ
0
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 4/25
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 specic) of weak elph coupling
superconductors (e.g.the ratio 2Δ
0
/k
B
T
c
)
Eliashberg Theory
Strong coupling theory
But elph and Coulomb interactions are
not
treated on the same footing
Coulomb repulsion is normally included through the parameter
µ
∗
,usually
tted to the
experimental T
c
Not possible to perform a fully abinitio calculation of superconducting
properties
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 5/25
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 specic) of weak elph coupling
superconductors (e.g.the ratio 2Δ
0
/k
B
T
c
)
Eliashberg Theory
Strong coupling theory
But elph and Coulomb interactions are
not
treated on the same footing
Coulomb repulsion is normally included through the parameter
µ
∗
,usually
tted to the
experimental T
c
Not possible to perform a fully abinitio calculation of superconducting
properties
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 5/25
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 specic) of weak elph coupling
superconductors (e.g.the ratio 2Δ
0
/k
B
T
c
)
Eliashberg Theory
Strong coupling theory
But elph and Coulomb interactions are
not
treated on the same footing
Coulomb repulsion is normally included through the parameter
µ
∗
,usually
tted to the
experimental T
c
Not possible to perform a fully abinitio calculation of superconducting
properties
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 5/25
DFT for superconductors
SCDFT
DFT for Superconductors
Coulomb and elph interactions enter the theory on the same footing
No empirical parameter,like µ
∗
,is used
Allows to
predict
T
c
and Δ
0
from
rst principles
The order parameter of the singlet superconducting state
χ(r,r
) =
ˆ
ψ
↑
(r)
ˆ
ψ
↓
(r
)
is the most important ingredient of SCDFT,entering the theory as an extra
density
condmat/0408685,condmat/0408686
(accepted in Phys.Rev.B)
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 6/25
DFT for superconductors
SCDFT
DFT for Superconductors
Coulomb and elph interactions enter the theory on the same footing
No empirical parameter,like µ
∗
,is used
Allows to
predict
T
c
and Δ
0
from
rst principles
The order parameter of the singlet superconducting state
χ(r,r
) =
ˆ
ψ
↑
(r)
ˆ
ψ
↓
(r
)
is the most important ingredient of SCDFT,entering the theory as an extra
density
condmat/0408685,condmat/0408686
(accepted in Phys.Rev.B)
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 6/25
DFT for superconductors
Hamiltonian
Our starting Hamiltonian is
ˆ
H =
ˆ
H
e
+
ˆ
H
n
+
ˆ
H
en
,
with
ˆ
H
e
=
ˆ
T
e
+
ˆ
W
ee
+
Z
d
3
r
ˆ
n(r)
v(r)
−
Z
d
3
r
Z
d
3
r
ˆ
ˆχ(r,r
)
Δ
∗
(r,r
)
+H.c.
˜
ˆ
H
n
=
ˆ
T
n
+
ˆ
W
nn
+
Z
d
3
N
n
ˆ
Γ(R
)
V(R
)
,
v(r)
= external potential acting on the electrons (e.g.applied voltage)
Δ
∗
(r,r
)
= external pairing potential (e.g.proximity induced)
V(R
)
= external potential acting on the nuclei
n(r)
=
X
σ
ˆ
ψ
†
σ
(r)
ˆ
ψ
σ
(r);
χ(r,r
)
=
ˆ
ψ
↑
(r)
ˆ
ψ
↓
(r
)
Γ(R
)
=
ˆ
φ
†
(R
1
)
ˆ
φ
†
(R
2
) ∙ ∙ ∙
ˆ
φ(R
2
)
ˆ
φ(R
1
)
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 7/25
DFT for superconductors
HohenbergKohn theorem
Theorem
There is the onetoone correspondence
n(r)
,
χ(r,r
)
,
Γ(R
)
←→
v(r)
,
Δ
∗
(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
GDRDFT 8/25
DFT for superconductors
HohenbergKohn theorem
Theorem
There is the onetoone correspondence
n(r)
,
χ(r,r
)
,
Γ(R
)
←→
v(r)
,
Δ
∗
(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
GDRDFT 8/25
DFT for superconductors
KohnShamscheme
Electronic KS equation
−
2
2
+
v
s
(r)
−µ
u
i
(r) +
d
3
r
Δ
s
(r,r
)
v
i
(r
) = E
i
u
i
(r)
−
−
2
2
+
v
s
(r)
−µ
v
i
(r) +
d
3
r
Δ
∗
s
(r,r
)
u
i
(r
) = E
i
v
i
(r)
Nuclear KS equation
α
−
2
α
2M
+
V
s
(R
)
Φ
n
(R
) = E
n
Φ
n
(R
)
There exist functionals
v
s
[n,χΓ]
,
Δ
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
GDRDFT 9/25
DFT for superconductors
KohnShampotentials
The 3 KS potentials are dened as
v
s
(r)
= 0
{z}
v
−
Z
d
3
R
ZN(R)
r −R

{z
}
v
en
H
+
Z
d
3
r
n(r
)
r −r


{z
}
v
ee
H
+
δF
xc
δn(r)

{z
}
v
xc
Δ
s
(r,r
)
= 0
{z}
Δ
+
χ(r,r
)
r −r


{z
}
Δ
H
+
δF
xc
δχ(r,r
)

{z
}
Δ
xc
V
s
(R
)
= 0
{z}
V
+
X
αβ
Z
α
Z
β
R
α
−R
β


{z
}
W
nn
−
X
α
Z
d
3
r
n(r)
r −R
α


{z
}
V
en
H
+
δF
xc
δΓ(R
)

{z
}
V
xc
Until here the theory is,in principle,exact:no approximation yet.
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 10/25
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

R
0
∙ U
+
1
2
3
ij
µν
∂
µ
i
∂
ν
j
V
s
R
0
U
µ
i
U
ν
j
The linear term in U
vanishes,as the atoms are in equilibrium,so we obtain
ˆ
H
n,KS
=
q
Ω
q
ˆ
b
†
q
ˆ
b
q
+
3
2
+O(U
3
)
Similarly,we obtain a electronphonon coupling term in H
e,KS
by expanding
the v
en
H
+v
xc
terms.
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 11/25
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 Δ
s
can be cast into a
BCSlike gap
equation
.
Δ
s
(j ) = −
1
2
X
j
w
eff
(i,j )
tanh
“
β
2
ξ
j
”
ξ
j
Δ
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
GDRDFT 12/25
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 Δ
s
can be cast into a
BCSlike gap
equation
.
Δ
s
(j ) = −
1
2
X
j
w
eff
(i,j )
tanh
“
β
2
ξ
j
”
ξ
j
Δ
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
GDRDFT 12/25
DFT for superconductors
Construction of an approximate F
xc
We apply
G¨orlingLevy perturbation theory
ˆ
H =
ˆ
H
KS
+
ˆ
H
1
In rst order we have 4 contributions to F
xc
F
xc
=
+
+
+
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 13/25
DFT for superconductors
The gap equation
Δ(n,k) = −Z
ph
(n,k)Δ(n,k) −
n
k
K
ph
+K
el
Δ(n
,k
)
2E
n
,k
tanh
βE
n
,k
2
where E
n,k
=
q
(
n,k
−µ)
2
+Δ(n,k)
2
Features
BCS form but
parameter free
effective interaction K = K
ph
+K
el
is
calculated
abinitio
static
(frequency independent) but with retardation effects included in the
Z and K functionals
kspace formalism allows to calculate the (possibly)
anisotropic
nature of
the gap
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 14/25
DFT for superconductors
The gap equation
Δ(n,k) = −Z
ph
(n,k)Δ(n,k) −
n
k
K
ph
+K
el
Δ(n
,k
)
2E
n
,k
tanh
βE
n
,k
2
where E
n,k
=
q
(
n,k
−µ)
2
+Δ(n,k)
2
Features
BCS form but
parameter free
effective interaction K = K
ph
+K
el
is
calculated
abinitio
static
(frequency independent) but with retardation effects included in the
Z and K functionals
kspace formalism allows to calculate the (possibly)
anisotropic
nature of
the gap
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 14/25
Results
Simple Metals
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
GDRDFT 15/25
Results
Simple Metals
Simple Metals
0 2 4 6 8 10
Experimental T
c
[K]
0
2
4
6
8
10
Calculated Tc [K]
Al
TFME
TFSK
TFFE
Ta
Pb
Nb
Mo
0 0.5 1 1.5 2
Experimental Δ
0
[meV]
0
0.5
1
1.5
2
Calculated Δ0 [meV]
TFME
TFSK
TFFE
Al
Ta
Pb
Nb
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 16/25
Results
Simple Metals
Gap of Pb
0 2 4 6 8
T [K]
0.0
0.4
0.8
1.2
Δ
0 [meV]
Experiment
TFSK
TFME
Pb
0.0001 0.001 0.01 0.1 1 10
ξ [eV]
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Δ [meV]
Experiment
TFME, T = 0 K
TFSK, T = 0 K
TFSK, T = 6 K
TFSK, T = 7 K
Pb
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 17/25
Results
MgB
2
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
GDRDFT 18/25
Results
MgB
2
MgB
2
Why such a high T
c
(39.5K)?
Strong coupling of σ bands with the optical E2g phonon mode for
q along the ΓA line (for π bands elph is roughly 3 times smaller)
Strong anisotropy,which leads to a kdependent gap Δ = Δ(k)
Phys.Rev.Lett.94,037004 (2005)
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 19/25
Results
MgB
2
MgB
2
 Results
0 10 20 30 40
T [K]
0
2
4
6
8
Δ [meV]
Iavarone et al.
Szabo et al.
Schmidt et al.
Gonnelli et al.
present work
0 0.5 1
T/T
c
0
0.5
1
1.5
2
2.5
Cel(T)/Cel,N(T)
Bouquet et al.
Putti et al.
Yang et al.
present work
(a)
(b)
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 20/25
Results
MgB
2
Gap of MgB
2
0.001 0.01 0.1 1 10
ε − µ [eV]
2
0
2
4
6
8
Δ [meV]
σ
σ Average
π
π Average
Experiments
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 21/25
Results
Li and Al under pressure
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
GDRDFT 22/25
Results
Li and Al under pressure
Li and Al under pressure
0 10 20 30 40 50 60
Pressure (GPa)
0
4
8
12
16
20
Tc (K)
SCDFT, this work
McMillan, this work
Lin [5]
Shimizu [6]
Struzhkin [7]
Deemyad [8]
0 2 4 6 8
Pressure (GPa)
0
0.2
0.4
0.6
0.8
1
1.2
SCDFT, this work
McMillan, this work
Gubser [11]
Sundqvist [12]
fcc hR1 cI16
Li
Al
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 23/25
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 highT
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 ) = w
el
xc
(i,j ) +w
ph
xc
(i,j )
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 24/25
Conclusions
Outlook
DFT of superconductivity offers,for the rst time,the possibility to perform
fully abinitio
calculations of superconducting properties,like the transition
temperature,the gap,or the specic 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,highT
c
s,etc.
Replace the ThomasFermi interaction by a RPA.
Development of new (better) functionals for the electronphonon
interaction.
M.A.L.Marques (IMPMC)
DFT for superconductors
GDRDFT 25/25
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