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
qν
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
qν
is
a
substantial
part of the
bandwith
?
K
3
Picene
–
Electron

phonon
coupling
DOS/spin
Fermi
functions
EP
matrix
element
Usually
ω
qν
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
qν
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
)
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