DFT Superconductors

8/13/2019 DFT Superconductors

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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

M. A. L. Marques (IMPMC) DFT for superconductors GDR-DFT 2 / 25


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Outline

1 DFT for superconductors

2 ResultsSimple MetalsMgB2Li and Al under pressure

3 Conclusions



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Outline



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


f


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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


DFT f d


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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


DFT for s percond ctors


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State of the Art

BCS Theory


Eliashberg Theory




experimental T c





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State of the Art

BCS Theory


Eliashberg Theory




experimental T c





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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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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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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



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


Results Simple Metals

O li


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Outline



3 Conclusions



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



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


Results MgB 2

Outline


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Outline



3 Conclusions


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)


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)


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


Results Li and Al under pressure

Outline


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Outline


2 Results

Simple MetalsMgB2Li and Al under pressure

3 Conclusions


Results Li and Al under pressure

Li and Al under pressure


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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


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 )


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


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.


DFT Superconductors

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