Density Functional Theory for Superconductors

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15 Νοε 2013 (πριν από 4 χρόνια και 1 μήνα)

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Density Functional Theory for Superconductors
M.A.L.Marques
marques@tddft.org
IMPMC  Universit ´e Pierre et Marie Curie,Paris VI
GDR-DFT05,18-05-2005,Cap d'Agde
M.A.L.Marques (IMPMC)
DFT for superconductors
GDR-DFT 1/25
Co-workers
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
GDR-DFT 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
GDR-DFT 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
GDR-DFT 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
GDR-DFT 3/25
DFT for superconductors
Goal
We want to describe
Conventional Superconductivity
Our goal is
To have a theory able to
predict
,fully
ab-initio
material specic
properties like T
c
and the gap Δ
0
M.A.L.Marques (IMPMC)
DFT for superconductors
GDR-DFT 4/25
DFT for superconductors
Goal
We want to describe
Conventional Superconductivity
Our goal is
To have a theory able to
predict
,fully
ab-initio
material specic
properties like T
c
and the gap Δ
0
M.A.L.Marques (IMPMC)
DFT for superconductors
GDR-DFT 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 specic) of weak el-ph coupling
superconductors (e.g.the ratio 2Δ
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
tted 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
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 specic) of weak el-ph coupling
superconductors (e.g.the ratio 2Δ
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
tted 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
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 specic) of weak el-ph coupling
superconductors (e.g.the ratio 2Δ
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
tted 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
DFT for superconductors
SCDFT
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 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
cond-mat/0408685,cond-mat/0408686
(accepted in Phys.Rev.B)
M.A.L.Marques (IMPMC)
DFT for superconductors
GDR-DFT 6/25
DFT for superconductors
SCDFT
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 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
cond-mat/0408685,cond-mat/0408686
(accepted in Phys.Rev.B)
M.A.L.Marques (IMPMC)
DFT for superconductors
GDR-DFT 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
GDR-DFT 7/25
DFT for superconductors
Hohenberg-Kohn theorem
Theorem
There is the one-to-one 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
GDR-DFT 8/25
DFT for superconductors
Hohenberg-Kohn theorem
Theorem
There is the one-to-one 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
GDR-DFT 8/25
DFT for superconductors
Kohn-Shamscheme
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
GDR-DFT 9/25
DFT for superconductors
Kohn-Shampotentials
The 3 KS potentials are dened 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
GDR-DFT 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 electron-phonon coupling term in H
e,KS
by expanding
the v
en
H
+v
xc
terms.
M.A.L.Marques (IMPMC)
DFT for superconductors
GDR-DFT 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
BCS-like 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
GDR-DFT 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
BCS-like 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
GDR-DFT 12/25
DFT for superconductors
Construction of an approximate F
xc
We apply
G¨orling-Levy 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
GDR-DFT 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
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
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
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
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
GDR-DFT 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
TF-ME
TF-SK
TF-FE
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]
TF-ME
TF-SK
TF-FE
Al
Ta
Pb
Nb
M.A.L.Marques (IMPMC)
DFT for superconductors
GDR-DFT 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
TF-SK
TF-ME
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
TF-ME, T = 0 K
TF-SK, T = 0 K
TF-SK, T = 6 K
TF-SK, T = 7 K
Pb
M.A.L.Marques (IMPMC)
DFT for superconductors
GDR-DFT 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
GDR-DFT 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 Γ-A line (for π bands el-ph is roughly 3 times smaller)
Strong anisotropy,which leads to a k-dependent gap Δ = Δ(k)
Phys.Rev.Lett.94,037004 (2005)
M.A.L.Marques (IMPMC)
DFT for superconductors
GDR-DFT 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
GDR-DFT 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
GDR-DFT 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
GDR-DFT 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
Mc-Millan, 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
Mc-Millan, this work
Gubser [11]
Sundqvist [12]
fcc hR1 cI16
Li
Al
M.A.L.Marques (IMPMC)
DFT for superconductors
GDR-DFT 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 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 ) = w
el
xc
(i,j ) +w
ph
xc
(i,j )
M.A.L.Marques (IMPMC)
DFT for superconductors
GDR-DFT 24/25
Conclusions
Outlook
DFT of superconductivity offers,for the rst time,the possibility to perform
fully ab-initio
calculations of superconducting properties,like the transition
temperature,the gap,or the specic 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