Adiabatic and non-adiabatic phonon dispersion in a

awfulhihatUrban and Civil

Nov 15, 2013 (3 years and 7 months ago)

84 views

Matteo Calandra
,

Gianni
Profeta
,
Michele

Casula

and Francesco
Mauri

Adiabatic and non
-
adiabatic phonon dispersion in a
Wannier

functions approach:

applications to MgB
2

, CaC
6

and K
-
doped
Picene

M. Calandra
et al.
Phys.
Rev
. B
82
, 165111 (2010)

M.
Casula

et al.

to
appear

on Phys.
Rev
.
Lett
.

Motivation of the
work


Description of phonon dispersions in metals needs an
ultradense

sampling of the Fermi surface.


Impossible to sample a sharp

Fermi surface with a coarse grid.

300 K

An
ultradense

grid is needed.

Fermi Surface

Motivation of the
work


Description of phonon dispersions in metals needs an
ultradense

sampling of the Fermi surface.


A finite temperature
T
ph

is introduced

not sampling

sampling

N
k
(T
ph
) is the number of
k
-
points necessary to sample the Fermi surface a a given
T
ph
.

Typical values:
T
ph
=0.03
Ryd

≈4700 K
N
k
(T
ph
)=12
3


4000 K

Fermi Surface

Outline

Wannier

interpolation scheme to calculate adiabatic and non
-
adiabatic

phonon dispersions on
ultradense

electron and phonon momentum grids.

Phonon dispersion and electron
-
phonon coupling in MgB
2

Adiabatic and non adiabatic phonon dispersion in CaC
6

Theory:

Applications:

Electron
-
phonon coupling in K
-
doped
Picene

(78 atoms per cell)

Time
-
dependent force
-
constants matrix

cell

atom in the cell
at equilibrium

displacement from

equilibrium

Displacing atom I at time
t
’ induces a force
F
J
(t
) on atom J at time
t
.

The force
-
constants matrix is then

and its
ω

and Fourier transforms are defined as:

J

WARNING

Complex Quantity!

From

forces to phonon
frequencies

The force
-
constants matrix is complex, we define:

If

then the self
-
consistent equation

Gives the phonon frequencies.

Force
-
Constants

functional

We write the force constants in a functional form. Using linear response we look for a


with

and

is the time dependent charge density.

functional


such that:


The functional has the form:

where

is the
Hartree

and exchange correlation kernel

and T is the temperature.

is the number of
k
-
points necessary to converge the sum at a temperature T.

Baroni

et al.
,
Rev
.
Mod
. Phys.
73
, 515 (2001)

Gonze and Lee, PRB
55
, 10355 (1997)

M. Calandra
et al.

Phys.
Rev
. B
82
, 165111 (2010)


Force
-
Constants

functional

The first term contains the product of the screened potential matrix elements:

This term depends on
ω

explicitly in the denominator but also implicitly in
ρ

and
ρ
’.

Force
-
Constants

functional

(
details
)

+ ….

for

for

The solution of this equation requires self
-
consistency in
ω

!

Baroni

et al.
,
Rev
.
Mod
. Phys.
73
, 515 (2001)

Gonze and Lee, PRB
55
, 10355 (1997)

M. Calandra
et al.

Phys.
Rev
. B
82
, 165111 (2010)


The functional has the form:

Baroni

et al.
,
Rev
.
Mod
. Phys.
73
, 515 (2001)

Gonze and Lee, PRB
55
, 10355 (1997)

M. Calandra
et al.

arXiv:1007.2098

double
-
counting coulomb term

Force
-
Constants

functional

(
details
)

The
ω

dependence of

Difficulties

in
calculating

dynamical

force
-
constants

in
metals


In the force
-
constants definition, T=T
0
=300K=0.0019
Ryd

is the physical temperature

which, in metallic systems, requires an enormous number of
k
-
points to be evaluated.

should be calculated self
-
consistently and it is thus

very expensive.

Stationary

condition for
F
I J

The following condition holds:

A linear error in

affects the functional and the phonon frequencies at
second order!

This property can be used to efficiently calculate adiabatic

and non
-
adiabatic phonon dispersion in a

NON SELF
-
CONSISTENT WAY


and a symmetric one on
ρ(
r
’).

Approximated

force constants
functional

We then define an approximate force constant functional:

Where:

is not anymore evaluated self
-
consistently at the physical temperature T=T
0


The
ω

dependence of

is neglected and the static limit

is considered.
SELF
-
CONSISTENCY at finite
ω

and at T=T
0

is not needed.

but at a much hotter one T=T
ph
≈0.03
Ryd

at which the phonon calculation is carried out.

Converging at
T
ph

requires much less
k
-
points,

N
k
(T
ph
) <<N
k
(T
0
)

The error in the phonon frequencies and on the functional is of order

2

From

theory

to a
practical

calculation

scheme
.

Passing in Fourier space, summing and subtracting the standard adiabatic force constants

calculated from first principles at a temperature
T
ph
, namely

-

where:

And the deformation potential matrix element (electron
-
phonon coupling) is:

with

From

theory

to a
practical

calculation

scheme
.

The DYNAMICAL force constants on an ULTRADENSE
k
-
point grid N
k
(T
0
)

at very low temperature T
0


are obtained from


the calculation of the STATIC force constants on a COARSE grid
N
k
(T
ph
) and a hot
temperature
T
ph


If a fast calculation of the deformation potential in throughout the BZ is available.

To interpolate the deformation potential matrix element we use Maximally localized
Wannier

functions

implementing the method proposed in

Giustino
et al.

PRB
76
, 165108 (2007)

N.
Marzari

and D. Vanderbilt, PRB
56
, 12847 (1997)

I.
Souza

et al.,
PRB
65
, 035109 (2002)

Mostofi

et al.

Comput. Phys.
Comm
.
178
,685 (2008)

APPLICATIONS

MgB
2

Adiabatic

phonon dispersion in MgB
2

Substantial enhancement of the in
-
plane

E
2g

Kohn anomaly related to inter
-
cylinders

nesting.

Kortus

et al.

PRL
86
, 4656 (2001)

M. Calandra
et al.

Phys.
Rev
. B
82
, 165111 (2010)

A.
Shukla

et al.
, PRL
90
, 095506 (2003)

Adiabatic

phonon dispersion in MgB
2

Substantial enhancement of the in
-
plane

E
2g

Kohn anomaly related to inter
-
cylinders

nesting.

A Kohn
-
anomaly appears on E
2g


and B
1g

branches along ΓA

The
ultradense

k
-
point sampling leads to phonon frequencies

in better agreement with experiments

Kortus

et al.

PRL
86
, 4656 (2001)

M. Calandra
et al.

Phys.
Rev
. B
82
, 165111 (2010)

A.
Shukla

et al.
, PRL
90
, 095506 (2003)

Accuracy

of
Wannier

interpolation

Linear response

Wannier

interpolation

Effect

on EP
coupling

λ
=0.74 (This Work)

In agreement with other calculations on

“sufficiently large grids”
-
>
λ
=0.73
-
0.77

Electron
-
phonon coupling almost converged (with time) ?

Does denser
k
-
point sampling in the calculation of
phonon frequencies have any effect on EP coupling ?

Ahn

and
Pickett
, PRL
86
, 4366 (2001)

Kong
et al.,
PRB
64
, 020501 (2001)

Bohnen

et al,

PRL
86
, 5771 (2001)

Liu,
Mazin

Kortus
, PRL
87
, 087005 (2001)

Choi
et al.

, Nature
418
, 758 (2002)

Eiguren

and C.
Ambrosch
-
Draxl
, PRB
78
, 045124 (2008)

Effect

on
Eliashberg

function

Significant discrepancy in the main peak position

of the
Eliashberg

function (E
2g

mode)

Effect

on
Eliashberg

function

Reduction of the Energy position of the E
2g

mode

respect to previous works with improved

sampling on phonon frequencies.

Accurate
k
-
point sampling on
λ

only is not sufficient,

phonon frequencies need to be accurately converged!

Much lower value of
ω
log

APPLICATIONS

CaC
6

Adiabatic

phonon dispersion in CaC
6


Many Kohn anomalies occur at all

energy scales in the phonon spectrum.

(see black arrows)

The low energy anomaly on
Ca
xy

phonon modes

is not at X (as it was inferred on the basis of

Fourier interpolated branches) but nearby.

M. Calandra and F.
Mauri
, PRB
74
, 094507 (2006)

M. Calandra and F.
Mauri
, PRL
95
, 237002 (2005)

J. S. Kim
et al.
, PRB
74
, 214513 (2006)

Adiabatic

phonon dispersion in CaC
6


Many Kohn anomalies occur at all

energy scales in the phonon spectrum.

(see black arrows)

The low energy anomaly on
Ca
xy

phonon modes

is not at X (as it was inferred on the basis of

Fourier interpolated branches) but nearby.

The anomaly is present at all energy scales

(nesting)

Non
adiabatic

(NA) phonon dispersion in CaC
6


Giant NA effects predicted at zone center

seen in Raman scattering.

Saitta

et al.,
PRL
100
, 226401 (2008)

Dean et al., PRB
81
, 045405 (2010)

It is unclear to what extent NA effects extend

from zone center.

Can NA effects be relevant for superconductivity ?

Non
adiabatic

(NA) phonon dispersion in CaC
6


Giant NA effects predicted at zone center

seen in Raman scattering.

Saitta

et al.,
PRL
100
, 226401 (2008)

Dean et al., PRB
81
, 045405 (2010)

It is unclear to what extent NA effects extend

from zone center.

Can NA effects be relevant for superconductivity ?

NA effects are not localized at zone center but extend

throughout the full
Brillouin

zone!

Raman

C
22
H
14

Picene

Phenantrene

C
14
H
10

Coronnene

C
24
H
12

SUPERCONDUCTING HYDROCARBONS

T
c
= 7K or 18K

T
c
= 5K

T
c
= 3
-
18 K ???

two

phases ??

Insulating

molecular

crystal

becoming

superconducting

upon

K intercalation.

Mitsuhashi

et al.
, Nature
464

76 (2010)

Wang
et al.
, arXiv:1102.4075

Kubozono

et al.
,
unpublished

M.
Casula
, M. Calandra, G.
Profeta

and F.
Mauri
, To
appear

on PRL

C
22
H
14

K
3

Picene

-

Structure

Molecule

Crystal
with

K
3

intercalation

K
-
intercalation

changes the angle
between

the
molecules

from

61
°

(
Picene
) to 114
°

(K
3

Picene
).

Two

molecules/cell

= 78
atom/cell

T.
Kosugi

et al.
, J. Phys. Soc.
Japan

78
, 11 (2009)

Rigid

doping of
undoped

Picene

completely

unjustified
!

T.
Kosugi

et al.
, J. Phys. Soc.
Japan

78
, 11 (2009)

T.
Kosugi

et al.

arXiv:1109.2059

H. Okazaki, PRB
82
, 195114 (2010)

K
3

Picene



Electronic

structure

Very

narrow

Bandwith

≈0.3 eV

Substantial

mixing

of the LUMO+1

with

other

electronic

states.

Substantial

variation of

the DOS on the phonon

frequency

energy

scale
!

Ultradense

k
-
point

sampling

needed

(
we

need

a
smearing

smaller

than

0.1 eV
at

least).

Difficulties

with

the
electron
-
phonon

calculation
:

The full K
3

Picene

crystal

structure
needs

to
be

taken

into

account

(78
atoms/cell
)

A CHALLENGE FOR FIRST PRINCIPLES CALCULATIONS

K
3

Picene



Electron
-
phonon

coupling

Bandwith

≈0.3 eV

Strong

variation of the
electron
-
phonon

coupling

detectable

for
σ

<
Bandwidth


N
k
=120
3

78
atoms/cell

All
curves

converged

K
3

Picene

is

not a
molecular

crystal

(for
what

concerns

the EP
coupling
).

Molecular


Crystal

Not a
molecular


crystal

K
3

Picene



Electron
-
phonon

coupling

DOS/spin

Fermi
functions

EP
matrix

element

Usually

ω


is

neglected

in the Dirac
function

and the Fermi
functions

differences

is

replaced

with

its

derivative
,
leading

to

Is
this

justified

in K
3

Picene

where

ω


is

a
substantial

part of the
bandwith

?

K
3

Picene



Electron
-
phonon

coupling

DOS/spin

Fermi
functions

EP
matrix

element

Usually

ω


is

neglected

in the Dirac
function

and the Fermi
functions

differences

is

replaced

with

its

derivative
,
leading

to

Is
this

justified

in K
3

Picene

where

ω


is

a
substantial

part of the
bandwith

?

NO (17%
reduction

of the
electron
-
phonon

coupling
) !!!!

A.
Subedi

et al.

PRB
84
, 020508 (2011)

Rigid

band doping of
undoped

picene

λ
AD
=0.78,
ω
log
=126
meV

ω
log
=18
meV

K
3

Picene



Electron
-
phonon

coupling

40% of the
electron
-
phonon

coupling

comes

from

K and
intermolecular

modes


(~80% if
we

restrict

to
intramolecular

electronic

states!).

Conclusion

We develop a method to calculate adiabatic and non
-
adiabatic (clean limit) phonon
dispersion in metals.


The dispersions are calculated in a non self
-
consistent way using a
Wannier
-
based

interpolation scheme that allows integration over ultra dense phonon and electron

momentum grids.

MgB
2

: occurrence of new Kohn anomalies. An accurate determination of phonon
frequencies is necessary to have well converged
Eliashberg

functions and
ω
log

.

CaC
6

:

non adiabatic effects are not localized at zone center but extend throughout
the full BZ on
C
xy

vibrations.

Linear response: Quantum
-
Espresso code

Wannierization

: Wannier90

K
3

Picene

:
Intercalant

and intermolecular phonon
-
modes contribute substantially
(40%) to the EP coupling. 17% reduction of
λby

inclusion of
ω
.

ω
log
=18
meV

M.
Casula,
et

al.

To
appear

on PRL

See

also

M.
Casula

poster on K
3
Picene

Electronic

structure
undoped

versus
doped

Picene

K
3

Picene

*
Picene

E
f

PICENE

K
3

PICENE

Totally

different

electronic

structure and Fermi surface.

T.
Kosugi

et al.
, J. Phys. Soc.
Japan

78
, 11 (2009)

K
-
induced

FS

K
3

Picene



Role

of K

Energy

(eV)

K
3

Picene

*
Picene

(
compensating

background and K
3

picene

crystal

structure)

Additional

Fermi

Surface
when


adding

K

Eliashberg

function
:
Casula

v.s
.
Subedi

α
2
F(ω)/N(0)

M.
Casula,
et

al.

To
appear

on PRL

A.
Subedi

et al.

PRB
84
, 020508 (2011)

Low

energy

phonon
spectrum

Wannierized

band structures

MgB
2

CaC
6

Adiabatic

CaC
6

phonon dispersion

Eliashberg

function

in CaC
6

From

forces to phonon
frequencies

The force
-
constants matrix is complex, we define

If

then the relation

Gives Allen formula.

Baroni

et al.
,
Rev
.
Mod
. Phys.
73
, 515 (2001)

Gonze and Lee, PRB
55
, 10355 (1997)

M. Calandra
et al.

arXiv:1007.2098

double
-
counting coulomb term

In standard time
-
independent linear
-
response theory the same object is written as:

Product of matrix elements involving the derivative of the external potential and the




screened potential is present.

No double
-
counting term is present.

At convergence of the self
-
consistent process they must lead to the
same

result!

Force
-
Constants

functional

(
details
)

We define the following functional:

What is the advantage of introducing this functional formulation ?

Force
-
Constants

functional

(
details
)