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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 dellAquila,
Italy
S. Massidda, C. Fran-
chini
Universit a degli Studi di Cagliari,
Italy
M. Luders
Daresbury Laboratory, Warring-ton WA4 4AD, UK
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Outline
1 DFT for superconductors
2 ResultsSimple MetalsMgB2Li and Al under pressure
3 Conclusions
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Outline
1 DFT for superconductors
2 ResultsSimple MetalsMgB2Li and Al under pressure
3 Conclusions
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DFT for superconductors
Goal
We want to describeConventional Superconductivity
Our goal isTo have a theory able to predict , fully ab-initio material specic
properties like T c and the gap 0
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f
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DFT for superconductors
Goal
We want to describeConventional Superconductivity
Our goal isTo have a theory able to predict , fully ab-initio material specic
properties like T c and the gap 0
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DFT f d
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DFT for superconductors
State of the Art
BCS Theory
The attractive interaction between the Cooper pairs is an empirical parameterBCS reproduces common features ( not material specic) of weak el-ph couplingsuperconductors (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 , usually tted to the
experimental T c
Not possible to perform a fully ab-initio calculation of superconductingproperties
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DFT for s percond ctors
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DFT for superconductors
State of the Art
BCS Theory
The attractive interaction between the Cooper pairs is an empirical parameterBCS reproduces common features ( not material specic) of weak el-ph couplingsuperconductors (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 , usually tted to the
experimental T c
Not possible to perform a fully ab-initio calculation of superconductingproperties
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DFT for superconductors
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DFT for superconductors
State of the Art
BCS Theory
The attractive interaction between the Cooper pairs is an empirical parameterBCS reproduces common features ( not material specic) of weak el-ph couplingsuperconductors (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 , usually tted to the
experimental T c
Not possible to perform a fully ab-initio calculation of superconductingproperties
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DFT for superconductors
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DFT for superconductors
SCDFT
DFT for Superconductors
Coulomb and el-ph interactions enter the theory on the same footing
No empirical parameter, like , is used
Allows to predict T c and 0 from rst principles
The order parameter of the singlet superconducting state
(r , r ) = (r ) (r )
is the most important ingredient of SCDFT, entering the theory as an extradensity
cond-mat/0408685, cond-mat/0408686(accepted in Phys. Rev. B)
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DFT for superconductors
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DFT for superconductors
SCDFT
DFT for Superconductors
Coulomb and el-ph interactions enter the theory on the same footing
No empirical parameter, like , is used
Allows to predict T c and 0 from rst principles
The order parameter of the singlet superconducting state
(r , r ) = (r ) (r )
is the most important ingredient of SCDFT, entering the theory as an extradensity
cond-mat/0408685, cond-mat/0408686(accepted in Phys. Rev. B)
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DFT for superconductors
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p
Hamiltonian
Our starting Hamiltonian is
H = H e + H n + H en ,with
H e = T e + W ee + Z d3 r n (r )v (r ) Z d3 r Z d3 r (r , r ) (r , r ) + H.c.H n =
T n +
W nn + Z d
3
N n (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 (r ) (r ) ; (r , r ) = (r ) (r )(R ) = (R 1 ) (R 2 ) (R 2 )(R 1 )
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p
Hohenberg-Kohn theorem
TheoremThere is the one-to-one correspondence
n (r ), (r , r ), (R ) v (r ), (r , r ), V (R )
As a consequence:
Theorem
All physical observables are functionals of {n (r ), (r , r ), (R )}
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Hohenberg-Kohn theorem
TheoremThere is the one-to-one correspondence
n (r ), (r , r ), (R ) v (r ), (r , r ), V (R )
As a consequence:
Theorem
All physical observables are functionals of {n (r ), (r , r ), (R )}
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Kohn-Sham scheme
Electronic KS equation
2
2 + v s(r ) u i (r ) + d3 r s(r , r )v i (r ) = E i u i (r )
2
2 + v s(r ) v i (r ) + d3 r s (r , r )u i (r ) = E i v i (r )
Nuclear KS equation
2
2M + V s(R ) n (R ) = E n n (R )
There exist functionals v s[n , ], s[n , ], and V s [n , ] such that the aboveequations reproduce the exact densities of the interacting system.
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Kohn-Sham potentials
The 3 KS potentials are dened as
v s(r ) = 0
|{z} v Z d3 R ZN (R )|r R |
| {z } v enH+ Z d3 r n (r )|r r |
| {z } v eeH+
F xcn (r )
| {z } v xc s(r , r ) = 0
|{z} + (r , r )|r r |
| {z } H+ F xc (r , r )
| {z } xcV s(R ) = 0
|{z} V
+ XZ Z
|R R |
| {z } W nn
X Z d3 r n (r )| r R |
| {z } V enH
+ F xc
(R )
| {z } V xc
Until here the theory is, in principle, exact: no approximation yet .
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Harmonic Approximation
In a solid, the atoms remain close to their equilibrium positions , so we canexpand 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
i
j V s R 0U i U
j
The linear term in U vanishes, as the atoms are in equilibrium, so we obtain
H n, KS =
q
q b q b q + 32
+ O(U 3 )
Similarly, we obtain a electron-phonon coupling term in H e, KS by expandingthe v enH + v xc terms.
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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 writeu i (r ) u i i (r ) ; v i (r ) v i i (r )
where the i are solutions of the normal state KS equation
Near the transition temperature, 0, the equation for s can be cast into a BCS-like gapequation .
s( j ) = 12 X j w eff (i , j )
tanh 2 j j
s( j )
where the matrix elements of the effective interaction w eff (r , r , x , x ) , and
w eff (r , r , x , x ) = 2 F xc [n , ]
(r , r ) (x , x ) = 0
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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 writeu i (r ) u i i (r ) ; v i (r ) v i i (r )
where the i are solutions of the normal state KS equation
Near the transition temperature, 0, the equation for s can be cast into a BCS-like gapequation .
s( j ) = 12 X j w eff (i , j )
tanh 2 j j
s( j )
where the matrix elements of the effective interaction w eff (r , r , x , x ) , and
w eff (r , r , x , x ) = 2 F xc [n , ]
(r , r ) (x , x ) = 0
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Construction of an approximate F xc
We apply Gorling-Levy perturbation theory
H = H KS + H 1
In rst order we have 4 contributions to F xc
F xc = +
+ +
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The gap equation
( n , k ) = Z ph
(n , k )( n , k ) n k
K ph
+ K el
( n , k )
2E n ,k tanh
E n ,k
2
where E n , k = q ( n , k )2 + |( 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 theZ and K functionals
k-space formalism allows to calculate the (possibly) anisotropic nature ofthe gap
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DFT for superconductors
Th i
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The gap equation
( n , k ) = Z ph
(n , k )( n , k ) n k
K ph
+ K el
( n , k )
2E n ,k tanh
E n ,k
2
where E n , k = q ( n , k )2 + |( 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 theZ and K functionals
k-space formalism allows to calculate the (possibly) anisotropic nature ofthe gap
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Results Simple Metals
O li
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Outline
1 DFT for superconductors
2 ResultsSimple MetalsMgB2Li and Al under pressure
3 Conclusions
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Results Simple Metals
Si l M t l
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Simple Metals
0 2 4 6 8 10Experimental T c [K]0
2
4
6
8
10
C a l c u
l a t e d
T c
[ K ]
Al
TF-METF-SKTF-FE
Ta
Pb Nb
Mo
0 0.5 1 1.5 2Experimental 0 [meV]0
0.5
1
1.5
2
C a l c u
l a t e d
0
[ m e V
]
TF-METF-SKTF-FE
Al
TaPb
Nb
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Results Simple Metals
G f Pb
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Gap of Pb
0 2 4 6 8T [K]
0.0
0.4
0.8
1.2
0
[ m e V
]
ExperimentTF-SKTF-ME
Pb
0.0001 0.001 0.01 0.1 1 10 [eV]
-0.4
-0.2
0.0
0.20.4
0.6
0.8
1.0
1.2
1.4
[ m
e V ]
ExperimentTF-ME, T = 0 KTF-SK, T = 0 KTF-SK, T = 6 KTF-SK, T = 7 K
Pb
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Results MgB 2
Outline
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Outline
1 DFT for superconductors
2 ResultsSimple MetalsMgB2Li and Al under pressure
3 Conclusions
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Results MgB 2
MgB
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MgB2
Why such a high T c (39.5 K)?
Strong coupling of bands with the optical E2g phonon mode forq along the -A line (for bands el-ph is roughly 3 times smaller)Strong anisotropy, which leads to a k-dependent gap = ( k )
Phys. Rev. Lett. 94, 037004 (2005)
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Results MgB 2
MgB Results
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MgB2 - Results
0 10 20 30 40T [K]
0
2
4
6
8
[ m e V
] Iavarone et al.Szabo et al.Schmidt et al.Gonnelli et al.present work
0 0.5 1T/T c
0
0.5
1
1.5
2
2.5
C e l
( T ) / C
e l , N
( T ) Bouquet et al.
Putti et al.Yang et al.
present work
(a)
(b)
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Results MgB 2
Gap of MgB 2
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Gap of MgB 2
0.001 0.01 0.1 1 10 [eV]
-2
0
2
4
6
8
[ m e V
]
Average Average
Experiments
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Results Li and Al under pressure
Outline
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Outline
1 DFT for superconductors
2 Results
Simple MetalsMgB2Li and Al under pressure
3 Conclusions
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Results Li and Al under pressure
Li and Al under pressure
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Li and Al under pressure
0 10 20 30 40 50 60Pressure (GPa)
0
4
8
12
16
20
T c
( K )
SCDFT, this work
Mc-Millan, this work Lin [5]Shimizu [6]Struzhkin [7]Deemyad [8]
0 2 4 6 8Pressure (GPa)
00.2
0.4
0.6
0.8
1
1.2
SCDFT, this work Mc-Millan, this work Gubser [11]Sundqvist [12]
fcc hR1 cI16
LiAl
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Conclusions
What can we learn?
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What can we learn?
Suppose we had a very good approximation for the functional F xc [n , ].What could we learn about the mechanism leading tosuperconductivity in the high- T c materials?
Remember: The functional F xc [n , ] is universal , i.e., the samefunctional for all materials.
By solving the KS equation for the particular material we canunderstand the mechanism in retrospect by studying the effective
interaction w eff (i , j ) = w elxc (i , j ) + w phxc (i , j )
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Conclusions
Outlook
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Outlook
DFT of superconductivity offers, for the rst time, the possibility to performfully ab-initio calculations of superconducting properties, like the transitiontemperature, the gap, or the specic heat. Until now, we obtained verypromising results for
simple metals
MgB2
Li and Al under pressure
However, further work is necessary
More applications to benchmark the theory: doped fullerenes,nanotubes, high-
T cs, etc.
Replace the Thomas-Fermi interaction by a RPA.
Development of new (better) functionals for the electron-phononinteraction.
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