of 10
Ab initio
phonon coupling and optical response of hot electrons in plasmonic metals
Ana M. Brown,
1
Ravishankar Sundararaman,
2
Prineha
Narang,
1, 2, 3
William A. Goddard III,
2, 4
and Harry A. Atwater
1, 2
1
Thomas J. Watson Laboratories of Applied Physics, California Institute of Technology, Pasadena CA
2
Joint Center for Artificial Photosynthesis, California Institute of Technology, Pasadena CA
3
NG NEXT, 1 Space Park Drive, Redondo Beach CA
4
Materials and Process Simulation Center, California Institute of Technology, Pasadena CA
(Dated: February 2, 2016)
Ultrafast laser measurements probe the non-equilibrium dynamics of excited electrons in metals
with increasing temporal resolution. Electronic structure calculations can provide a detailed mi-
croscopic understanding of hot electron dynamics, but a parameter-free description of pump-probe
measurements has not yet been possible, despite intensive research, because of the phenomenolog-
ical treatment of electron-phonon interactions. We present
ab initio
predictions of the electron-
temperature dependent heat capacities and electron-phonon coupling coefficients of plasmonic met-
als. We find substantial differences from free-electron and semi-empirical estimates, especially in
noble metals above transient electron temperatures of 2000 K, because of the previously-neglected
strong dependence of electron-phonon matrix elements on electron energy. We also present first-
principles calculations of the electron-temperature dependent dielectric response of hot electrons in
plasmonic metals, including direct interband and phonon-assisted intraband transitions, facilitat-
ing complete
ab initio
predictions of the time-resolved optical probe signatures in ultrafast laser
experiments.
I. INTRODUCTION
Understanding the energy transfer mechanisms during
thermal non-equilibrium between electrons and the lat-
tice is critical for a wide array of applications. Non-
equilibrium electron properties on time scales of 10-100s
of femtoseconds are most efficiently observed with pulsed
laser measurement techniques.
1–7
Laser irradiation of a
metal film or nanostructure with an ultrashort laser pulse
pushes the electron gas out of equilibrium; describing the
evolution of this non-equilibrium distribution has been
the subject of intense research for two decades.
8–12
A
majority of investigations so far employ various approxi-
mate models, typically based on free-electron models and
empirical electron-phonon interactions, to calculate the
energy absorption, electron-electron thermalization and
electron-phonon relaxation.
13–18
However, a complete
ab
initio
description of the time evolution and optical re-
sponse of this non-equilibrium electron gas from fem-
tosecond to picosecond time scales has remained elusive,
especially because of the empirical treatment of electron-
phonon interactions.
19
The initial electron thermalization via electron-electron
scattering is qualitatively described within the Lan-
dau theory of Fermi liquids.
20–23
The subsequent relax-
ation of the high temperature electron gas with the lat-
tice is widely described by the two-temperature model
(TTM),
5–7,17,19
given by coupled differential equations for
the electron and lattice temperatures,
T
e
and
T
l
,
C
e
(
T
e
)
dT
e
dt
=
(
κ
e
5
T
e
)
G
(
T
e
)
×
(
T
e
T
l
) +
S
(
t
)
C
l
(
T
l
)
dT
l
dt
=
(
κ
p
5
T
p
) +
G
(
T
e
)
×
(
T
e
T
l
)
.
(1)
Here,
κ
e
and
κ
p
are the thermal conductivities of the elec-
trons and phonons,
G
(
T
e
) is the electron-phonon coupling
factor,
C
e
(
T
e
) and
C
l
(
T
l
) are the electronic and lattice
heat capacities, and
S
(
t
) is the source term which de-
scribes energy deposition by a laser pulse. In nanostruc-
tures, the temperatures become homogeneous in space
rapidly and the contributions of the thermal conductiv-
ities drop out. A vast majority of studies treat the re-
maining material parameters,
G
(
T
e
),
C
e
(
T
e
) and
C
l
(
T
l
),
as phenomenological temperature-independent constants.
A key challenge in the quantitative application of
TTM models is the determination of these
temperature-
dependent
material parameters. With pulsed lasers, it is
possible to absorb sufficient energy in plasmonic nanos-
tructures to melt the metal once the electrons and lat-
tice have equilibrated.
24
The highest electron temper-
ature,
T
max
e
accessible in repeatable measurements is
therefore limited only by the equilibrated lattice temper-
ature being less than the melting temperature
T
m
of the
metal,
25
which yields the condition
T
max
e
T
m
d
T
e
C
e
(
T
e
) =
T
m
T
0
d
T
l
C
l
(
T
l
). Starting at room temperature
T
0
= 300 K
and using our
ab initio
calculations of the electron and
lattice heat capacities,
C
e
(
T
e
) and
C
l
(
T
l
), we find
T
max
e
5700, 8300, 7500 and 6700 K respectively for aluminum,
silver, gold and copper. For gold and copper in particular,
these temperatures are sufficient to change the occupa-
tions of the
d
-bands
2 eV below the Fermi level. Conse-
quently, it is important to derive the temperature depen-
dence of these material parameters from electronic struc-
ture calculations rather than free-electron like models.
19
To accurately predict the transient optical response
of metal nanostructures, we account for the electron-
temperature dependence of the electronic heat capacity,
electron-phonon coupling factor and dielectric functions.
These properties, in turn, require accurate electron and
phonon band structures as well as electron-phonon and
optical matrix elements. We recently showed that
ab
initio
calculations can quantitatively predict optical re-
sponse, carrier generation and electron transport in plas-
arXiv:1602.00625v1 [physics.comp-ph] 1 Feb 2016
2
monic metals in comparison with experiment, with no em-
pirical parameters.
26
In this article, we calculate
C
e
(
T
e
),
G
(
T
e
) and the temperature and frequency-dependent di-
electric function,

(
ω,T
e
) from first principles. These cal-
culations implicitly include electronic-structure effects in
the density of states and electron-phonon interaction ma-
trix elements, and implicitly account for processes such
as Umklapp scattering. We show substantial differences
between our fully
ab initio
predictions and those from
simplified models due to the energy dependence of the
electron-phonon matrix elements, especially at high elec-
tron temperatures.
The paper is organized as follows. We start with the
theoretical background and computational methods used
in the calculations of the electron heat capacity,
ab ini-
tio
phonon coupling and temperature dependent
ab initio
dielectric function of plasmonic materials (Section II A).
In Section IIB, we show calculations of the electron heat
capacity and its dependence on the electron temperature
due to the electronic density of states. Analogously, sec-
tion IIC presents the lattice-temperature dependence of
the lattice heat capacity due to the phonon density of
states. Next, in Section IID we show a key result of the
paper: temperature dependence of the electron-phonon
coupling strength accounting for energy dependence of
the electron-phonon matrix elements. Finally, section IIE
presents the temperature and frequency dependence of
the
ab initio
dielectric function, including direct (inter-
band), phonon-assisted and Drude intraband contribu-
tions. Section III summarizes our results and discusses
their application to plasmonic nanostructures in various
experimental regimes.
II. THEORY AND RESULTS
A. Computational details
We perform
ab initio
calculations of the electronic
states, phonons, electron-phonon and optical matrix el-
ements, and several derived quantities based on these
properties, for four plasmonic metals, aluminum, cop-
per, silver and gold. We use the open-source JDFTx
plane-wave density functional software
27
to perform fully
relativistic (spinorial) band structure calculations using
norm-conserving pseudopotentials at a kinetic energy cut-
off of 30 Hartrees, and the PBEsol exchange-correlation
functional
28
with a localized ‘+
U
’ correction
29
for the
d
-bands in the noble metals. Ref. 30 shows that this
method produces accurate electronic band structures in
agreement with angle-resolved photoemission (ARPES)
measurements within 0.1 eV.
We calculate phonon energies and electron-phonon ma-
trix elements using perturbations on a 4
×
4
×
4 supercell.
In
ab initio
calculations, these matrix elements implic-
itly include Umklapp-like processes. We then convert
the electron and phonon Hamiltonians to a maximally-
localized Wannier function basis,
31
with 12
3
k
-points in
the Brillouin zone for electrons. Specifically, we employ
24 Wannier centers for aluminum and 46 spinorial cen-
FIG. 1.
Figure
1 represents the evolution of the non-
equilibrium ‘hot’ electrons with time with different regimes
dominated by distinct material properties. The electronic den-
sity of states (DOS) determines the electronic heat capacity
C
e
(
T
e
) and the temperature to which the electrons equilibrate.
The electron-phonon coupling
G
(
T
e
) determines the dynam-
ics of energy transfer from the electrons to the phonons. The
phonon DOS determines the lattice heat capacity
C
l
and the
temperature to which the lattice equilibrates. All of these
properties are particularly sensitive to the electron tempera-
ture
T
e
, and are essential for a quantitative description of the
ultrafast response of plasmonic metals under laser excitation.
ters for the noble metals which reproduces the DFT band
structure exactly to at least 50 eV above the Fermi level.
Using this Wannier representation, we interpolate the
electron, phonon and electron-phonon interaction Hamil-
tonians to arbitrary wave-vectors and perform dense
Monte Carlo sampling for accurately evaluating the Bril-
louin zone integrals for each derived property below. This
dense Brillouin zone sampling is necessary because of
the large disparity in the energy scales of electrons and
phonons, and directly calculating DFT phonon properties
on dense
k
-point grids is computationally expensive and
impractical. See Ref. 26 for further details on the
ab ini-
tio
calculation protocol and benchmarks of the accuracy
of the electron-phonon coupling (eg. resistivity within 5%
for all four metals).
B. Electronic density of states and heat capacity
The electronic density of states (DOS) per unit volume
g
(
ε
) =
BZ
d
k
(2
π
)
3
n
δ
(
ε
ε
k
n
)
,
(2)
where
ε
k
n
are energies of quasiparticles with band index
n
and wave-vector
k
in the Brillouin zone BZ, directly
determines the electronic heat capacity and is an impor-
tant factor in the electron-phonon coupling and dielectric
response of hot electrons. Above, the band index
n
im-
plicitly counts spinorial orbitals in our relativistic calcu-
3
0.1
0.2
0.3
0.4
DOS [10
29
eV
-1
m
-3
]
a) Al
ab initio
Lin et al. 2008
free electron
0.5
1
1.5
2
2.5
b) Ag
0
0.5
1
1.5
-10
-5
0
5
10
DOS [10
29
eV
-1
m
-3
]
ε
-
ε
F
[eV]
c) Au
0
1
2
3
-10
-5
0
5
10
ε
-
ε
F
[eV]
d) Cu
FIG. 2.
Comparison of electronic density of states of for
(a) Al, (b) Ag, (c) Au and (d) Cu from our relativistic
PBEsol+
U
calculations (
ab initio
), previous semi-local PBE
DFT calculations
19
(less accurate band structure), and a free
electron model.
lations, and hence we omit the explicit spin degeneracy
factor.
Figure 2 compares the DOS predicted by our relativis-
tic PBEsol+
U
method with a previous non-relativistic
semi-local estimate
19
using the PBE functional,
32
as well
as a free electron model
ε
k
=
̄
h
2
k
2
2
m
e
for which
g
(
ε
) =
ε
2
π
2
(
2
m
e
̄
h
2
)
3
/
2
. The free electron model is a reasonable ap-
proximation for aluminum and the PBE and PBEsol+
U
density-functional calculations also agree reasonably well
in this case. The regular 31
3
k
-point grid used for Bril-
louin zone sampling introduces the sharp artifacts in the
DOS from Ref. 19, compared to the much denser Monte
Carlo sampling in our calculations with 640,000
k
-points
for Au, Ag, and Cu, and 1,280,000
k
-points for Al.
For the noble metals, the free electron model and the
density functional methods agree reasonably near the
Fermi level, but differ significantly
2 eV below the
Fermi level where
d
-bands contribute. The free electron
models ignore the
d
-bands entirely, whereas the semi-local
PBE calculations predict
d
-bands that are narrower and
closer to the Fermi level than the PBEsol+
U
predictions.
The
U
correction
29
accounts for self-interaction errors in
semi-local DFT and positions the
d
-bands in agreement
with ARPES measurements (to within
0
.
1 eV).
30
Addi-
tionally, the DOS in the non-relativistic PBE calculations
strongly peaks at the top of the
d
-bands (closest to the
Fermi level), whereas the DOS in our relativistic calcu-
lations is comparatively balanced between the top and
middle of the
d
-bands due to strong spin-orbit splitting,
particularly for gold. Below, we find that these inaccura-
cies in the DOS due to electronic structure methods pre-
viously employed for studying hot electrons propagates
to the predicted electronic heat capacity and electron-
phonon coupling.
The electronic heat capacity, defined as the derivative
of the electronic energy per unit volume with respect to
the electronic temperature (
T
e
), can be related to the
2
4
6
8
C
e
[10
5
J/m
3
K]
a) Al
ab-initio
Lin et al. 2008
Sommerfeld
4
8
12
b) Ag
0
4
8
12
0
2
4
6
8
C
e
[10
5
J/m
3
K]
T
e
[10
3
K]
c) Au
0
5
10
15
20
0
2
4
6
8
T
e
[10
3
K]
d) Cu
FIG. 3. Comparison of the electronic heat capacity as a func-
tion of electron temperature,
C
e
(
T
e
), for (a) Al, (b) Ag, (c)
Au and (d) Cu, corresponding to the three electronic density-
of-states predictions shown in Figure 2. The free electron
Sommerfeld model underestimates
C
e
for noble metals at high
T
e
because it neglects
d
-band contributions, whereas previous
DFT calculations
19
overestimate it because their
d
-bands are
too close to the Fermi level.
DOS as
C
e
(
T
e
) =
−∞
d
εg
(
ε
)
ε
∂f
(
ε,T
e
)
∂T
e
,
(3)
where
f
(
,T
e
) is the Fermi distribution function. The
term
∂f/∂T
e
is sharply peaked at the Fermi energy
ε
F
with a width
k
B
T
e
, and therefore the heat capacity de-
pends only on electronic states within a few
k
B
T
e
of the
Fermi level. For the free electron model, Taylor expand-
ing
g
(
ε
) around
ε
F
and analytically integrating (3) yields
the Sommerfeld model
C
e
(
T
e
) =
π
2
n
e
k
2
B
2
ε
F
T
e
, which is valid
for
T
e

T
F
(
10
5
K). Above,
n
e
= 3
π
2
k
3
F
,
ε
F
=
̄
h
2
k
2
F
2
m
e
and
k
F
are respectively the number density, Fermi energy
and Fermi wave-vector of the free electron model.
At temperatures
T
e

T
F
, the electronic heat
capacities are much smaller than the lattice heat
capacities,
5,10,22
which makes it possible for laser pulses
to increase
T
e
by 10
3
10
4
Kelvin, while
T
l
remains rela-
tively constant.
6,33,34
Figure 3 compares
C
e
(
T
e
) from the
free-electron Sommerfeld model with predictions of (3)
using DOS from PBE and PBEsol+
U
calculations. The
free-electron Sommerfeld model is accurate at low tem-
peratures (up to
2000 K) for all four metals.
With increasing
T
e
,
∂f/∂T
e
in (3) is non-zero increas-
ingly further away from the Fermi energy, so that devia-
tions from the free electron DOS eventually become im-
portant. For aluminum, the DOS remains free-electron-
like over a wide energy range and the Sommerfeld model
remains valid throughout. For the noble metals, the
increase in DOS due to
d
-bands causes a dramatic in-
crease in
C
e
(
T
e
) once
T
e
is high enough that
∂f/∂T
e
be-
comes non-zero in that energy range. Copper and gold
have shallower
d
-bands and deviate at lower temperatures
compared to silver. Additionally, the
d
-bands are too
close to the Fermi level in the semilocal PBE calculations
4
60
120
180
240
300
DOS [10
29
eV
-1
m
-3
]
a) Al
ab initio
Debye
60
120
180
240
300
b) Ag
0
60
120
180
240
0
0.02
0.04
0.06
DOS [10
29
eV
-1
m
-3
]
ε
[eV]
c) Au
0
60
120
180
240
0
0.02
0.04
0.06
ε
[eV]
d) Cu
FIG. 4. Comparison of
ab initio
phonon density of states and
the Debye model for (a) Al, (b) Ag, (c) Au and (d) Cu.
of Ref. 19 which results in an overestimation of
C
e
(
T
e
)
compared to our predictions based on the more accurate
relativistic PBEsol+
U
method.
C. Phonon density of states and lattice heat
capacity
Similarly, the phonon DOS per unit volume
D
(
ε
) =
BZ
d
q
(2
π
)
3
α
δ
(
ε
̄
q
α
)
,
(4)
where ̄
q
α
are energies of phonons with polarization in-
dex
α
and wave-vector
q
, directly determines the lattice
heat capacity,
C
l
(
T
l
) =
0
d
εD
(
ε
)
ε
∂n
(
ε,T
l
)
∂T
l
,
(5)
where
n
(
ε,T
l
) is the Bose occupation factor.
Within the Debye model, the phonon energies are
approximated by an isotropic linear dispersion relation
ω
q
α
=
v
α
q
up to a maximum Debye wave vector
q
D
chosen to conserve the number of phonon modes per
unit volume. This model yields the analytical phonon
DOS,
D
(
ε
) =
ε
2
(2
π
2
)
α
θ
( ̄
hq
D
v
α
ε
)
/
( ̄
hv
α
)
3
, where
v
α
=
{
v
L
,v
T
,v
T
}
are the speeds of sound for the one longitu-
dinal and two degenerate transverse phonon modes of the
face-centered cubic metals considered here.
25
Figure 4 compares the
ab initio
phonon DOS with
the Debye model predictions, and shows that the Debye
model is a good approximation for the DOS only up to
0.01 eV. However, Figure 5 shows that the corresponding
predictions for the lattice heat capacities are very similar,
rapidly approaching the equipartition theorem prediction
of
C
l
= 3
k
B
/
Ω at high temperatures, which is insensitive
to details in the phonon DOS. In fact, the largest devia-
tions of the Debye model are below 100 K and less than
10 % from the
ab initio
predictions for all four metals.
We therefore find that a simple model of the phonons
10
20
30
40
C
l
[10
5
J/m
3
K]
a) Al
ab initio
Debye
10
20
30
40
b) Ag
0
10
20
30
0
0.5
1
1.5
2
C
l
[10
5
J/m
3
K]
T
l
[10
3
K]
c) Au
0
10
20
30
0
0.5
1
1.5
2
T
l
[10
3
K]
d) Cu
FIG. 5. Comparison of
ab initio
and Debye model predictions
of the lattice heat capacity as a function of lattice tempera-
ture,
C
l
(
T
l
), for (a) Al, (b) Ag, (c) Au and (d) Cu. Despite
large differences in the density of states (Figure 4), the pre-
dicted lattice heat capacities of the two models agree within
10%.
is adequate for predicting the lattice heat capacity, in
contrast to the remaining quantities we consider below
which are highly sensitive to details of the phonons and
their coupling to the electrons.
D. Electron-phonon coupling
In Section IIC we have shown that the electronic heat
capacity, which determines the initial temperature that
the hot electrons equilibrate to, is sensitive to electronic
structure especially in noble metals at high
T
e
where
d
-
bands contribute. Now we analyze the electron-phonon
coupling which determines the subsequent thermalization
of the hot electrons with the lattice. We show that de-
tails in the
ab initio
electron-phonon matrix elements also
play a significant role, in addition to the electronic band
structure, and compare previous semi-empirical estimates
of the
T
e
-dependent phonon coupling to our direct
ab ini-
tio
calculations.
The rate of energy transfer from electrons at temper-
ature
T
e
to the lattice (phonons) at temperature
T
l
per
unit volume is given by Fermi’s golden rule as
d
E
d
t
G
(
T
e
)(
T
e
T
l
)
(6)
=
2
π
̄
h
BZ
Ωd
k
d
k
(2
π
)
6
nn
α
δ
(
b
̄
k
k
)
×
̄
k
k
g
k
k
k
n
,
k
n
2
S
T
e
,T
l
(
ε
k
n
k
n
,
̄
k
k
)
with
S
T
e
,T
l
(
ε,ε
,
̄
ph
)
f
(
ε,T
e
)
n
( ̄
ph
,T
l
)(1
f
(
ε
,T
e
))
(1
f
(
ε,T
e
))(1 +
n
( ̄
ph
,T
l
))
f
(
ε
,T
e
)
.
(7)
Here, Ω is the unit cell volume, ̄
q
α
is the energy of a
phonon with wave-vector
q
=
k
k
and polarization in-
5
dex
α
, and
g
k
k
k
n
,
k
n
is the electron-phonon matrix element
coupling this phonon to electronic states indexed by
k
n
and
k
n
.
Above,
S
is the difference between the product of occu-
pation factors for the forward and reverse directions of the
electron-phonon scattering process
k
n
+
q
α
k
n
, with
f
(
ε,T
e
) and
n
( ̄
hω,T
l
) being the Fermi and Bose distribu-
tion function for the electrons and phonons respectively.
Using the fact that
S
T
e
,T
e
= 0 for an energy-conserving
process
ε
+ ̄
ph
=
ε
by detailed balance, we can write
the electron-phonon coupling coefficient as
G
(
T
e
) =
2
π
̄
h
BZ
Ωd
k
d
k
(2
π
)
6
nn
α
δ
(
ε
k
n
ε
k
n
̄
k
k
)
×
̄
k
k
g
k
k
k
n
,
k
n
2
(
f
(
ε
k
n
,T
e
)
f
(
ε
k
n
,T
e
))
×
n
( ̄
k
k
,T
e
)
n
( ̄
k
k
,T
l
)
T
e
T
l
(8)
This general form for
ab initio
electronic and phononic
states is analogous to previous single-band / free electron
theories of the electron-phonon coupling coefficient, see
for example the derivation by Allen et al.
35
The direct
ab initio
evaluation of
G
(
T
e
) using (8) re-
quires a six-dimensional integral over electron-phonon
matrix elements from DFT with very fine
k
-point grids
that can resolve both electronic and phononic energy
scales. This is impractical without the recently-developed
Wannier interpolation and Monte Carlo sampling meth-
ods for these matrix elements,
26,36
and therefore our re-
sults are the first fully
ab initio
predictions of
G
(
T
e
).
Previous theoretical estimates of
G
(
T
e
) are semi-
empirical, combining DFT electronic structure with em-
pirical models for the phonon coupling. For example,
Wang et al.
37
assume that the electron-phonon matrix
elements averaged over scattering angles is independent
of energy and that the phonon energies are smaller than
k
B
T
e
, and then approximate the electron-phonon cou-
pling coefficient as
G
(
T
e
)
πk
B
̄
hg
(
ε
F
)
λ
( ̄
)
2
−∞
d
εg
2
(
ε
)
∂f
(
ε,T
e
)
∂ε
,
(9)
where
λ
is the electron-phonon mass enhancement pa-
rameter and
( ̄
)
2
is the second moment of the phonon
spectrum.
8,19,38
Lin et al.
19
treat
λ
( ̄
)
2
as an empir-
ical parameter calibrated to experimental
G
(
T
e
) at low
T
e
obtained from thermoreflectance measurements, and
extrapolate it to higher
T
e
using (9). See Refs. 37 and 19
for more details.
For clarity, we motivate here a simpler derivation of an
expression of the form of (9) from the general form (8).
First, making the approximation ̄
q
α

T
e
(which is
reasonably valid for
T
e
above room temperature) allows
us to approximate the difference between the electron oc-
cupation factors in the second line of (8) by ̄
q
α
∂f/∂ε
(using energy conservation). Additionally, for
T
e

T
l
,
the third line of (8) simplifies to
k
B
/
( ̄
k
k
). With
no other approximations, we can then rearrange (8) to
0
200
400
600
800
-10
-5
0
5
h(
ε
) [meV
2
]
ε
-
ε
F
[eV]
a) Al
ab initio
Lin et al. 2008
0
50
100
150
-6
-3
0
3
ε
-
ε
F
[eV]
b) Ag
0
15
30
45
60
-8
-4
0
h(
ε
) [meV
2
]
ε
-
ε
F
[eV]
c) Au
0
50
100
150
200
250
300
-8
-4
0
4
ε
-
ε
F
[eV]
d) Cu
FIG. 6.
Energy-resolved electron-phonon coupling strength
h
(
ε
), defined by (11), for (a) Al, (b) Ag, (c) Au, (d) Cu.
For the noble metals,
h
(

F
) is substantially larger than its
value in the
d
-bands, which causes previous semi-empirical
estimates
19
using a constant
h
(
ε
) to overestimate the electron-
phonon coupling (
G
(
T
e
)) at
T
e
>
3000 K, as shown in Fig. 7.
collect contributions by initial electron energy,
G
(
T
e
)
πk
B
̄
hg
(
ε
F
)
−∞
d
εh
(
ε
)
g
2
(
ε
)
∂f
(
ε,T
e
)
∂ε
(10)
with
h
(
ε
)
2
g
(
ε
F
)
g
2
(
ε
)
BZ
Ωd
k
d
k
(2
π
)
6
nn
α
δ
(
ε
ε
k
n
)
×
δ
(
ε
k
n
ε
k
n
̄
k
k
) ̄
k
k
g
k
k
k
n
,
k
n
2
.
(11)
Therefore, the primary approximation in previous semi-
empirical estimates
19,37
is the replacement of
h
(
ε
) by an
energy-independent constant
λ
( ̄
)
2
, used as an empir-
ical parameter.
Fig. 6 compares
ab initio
calculations of this energy-
resolved electron-phonon coupling strength,
h
(
ε
), with
previous empirical estimates of
λ
( ̄
)
2
, and Fig. 7
compares the resulting temperature dependence of the
electron-phonon coupling,
G
(
T
e
), from the
ab initio
(8)
and semi-empirical methods(9). For noble metals,
G
(
T
e
)
increases sharply beyond
T
e
3000 K because of the
large density of states in the
d
-bands. However,
h
(
ε
) is
smaller by a factor of 2
3 in the
d
-bands compared
to near the Fermi level. Therefore, assuming
h
(
ε
) to be
an empirical constant
17,19
results in a significant overes-
timate of
G
(
T
e
) at high
T
e
, compared to the direct
ab
initio
calculations. Additionally, the shallowness of the
d
-bands in the semi-local PBE band structure used in
Ref. 19 lowers the onset temperature of the increase in
G
(
T
e
), and results in further overestimation compared to
our
ab initio
predictions.
The
ab initio
predictions agree very well with the ex-
perimental measurements of
G
(
T
e
) available at lower tem-
peratures for noble metals.
3,14,15,39,40
In fact, the semi-
empirical calculation based on
λ
( ̄
)
2
underestimates
6
2
4
6
8
G [10
17
W/m
3
K]
a) Al
ab-initio
Lin et al. 2008
Hostetler et al. 1999
0.2
0.4
0.6
0.8
b) Ag
ab-initio
Lin et al. 2008
Groeneveld et al. 1990
Groeneveld et al. 1995
0
0.4
0.8
1.2
0
2
4
6
8
G [10
17
W/m
3
K]
T
e
[10
3
K]
c) Au
ab-initio
Lin et al. 2008
Hostetler et al. 1999
Hohlfeld et al. 2000
0
2
4
0
2
4
6
8
T
e
[10
3
K]
d) Cu
ab-initio
Lin et al. 2008
Elsayed-Ali et al. 1987
Hohlfeld et al. 2000
FIG. 7.
Comparison of predictions of the electron-phonon
coupling strength as a function of electron temperature,
G
(
T
e
), for (a) Al, (b) Ag, (c) Au and (d) Cu, with exper-
imental measurements where available.
3,14,15,39,40
The DFT-
based semi-empirical predictions of Lin et al.
19
overestimate
the coupling for noble metals at high temperatures because
they assume an energy-independent electron-phonon coupling
strength (Figure 6) and neglect the weaker phonon coupling
of
d
-bands compared to the conduction band. The experimen-
tal results (and hence the semi-empirical predictions) for alu-
minum underestimate electron-phonon coupling because they
include the effect of competing electron-electron thermaliza-
tion which happens on the same time scale.
the room temperature electron-phonon coupling for these
metals; the significant overestimation of
G
(
T
e
) seen in
Fig. 7 is in despite this partial cancellation of error. This
shows the importance of
ab initio
electron-phonon matrix
elements in calculating the coupling between hot electrons
and the lattice.
Experimental measurements of the electron-phonon
coupling in noble metals are reliable because of the rea-
sonably clear separation between a fast electron-electron
thermalization rise followed by a slower electron-phonon
decay in the thermoreflectance signal. In aluminum, these
time scales significantly overlap making an unambiguous
experimental determination of
G
difficult. Consequently,
the value of
G
for Al is not well agreed upon.
41
For ex-
ample, in Ref. 39, there is no fast transient free-electron
spike and
G
is extracted from the lattice temperature
variation instead. However, the measured rate for the
lattice temperature rise includes competing contributions
from electron-electron and electron-phonon thermaliza-
tion; attributing the entire rate to electron-phonon cou-
pling only provides a lower bound for
G
. Indeed, fig-
ure 7(a) shows that this experimental estimate
39
and its
phenomenological extension to higher
T
e
19
significantly
underestimate our
ab initio
predictions by almost a fac-
tor of two. Note that density-functional theory is highly
reliable for the mostly free-electron-like band structure
of aluminum, and the
ab initio
electron-phonon matrix
elements are accurate to within 5 %.
26
We therefore con-
clude that electron-electron thermalization is only about
two times faster then electron-phonon thermalization in
aluminum, causing the significant discrepancy in experi-
mental measurements. This further underscores the im-
portance of
ab initio
calculations over phenomenological
models of electron-phonon coupling.
E. Dielectric Function
The final ingredient for a complete
ab initio
descrip-
tion of ultrafast transient absorption measurements is the
temperature-dependent dielectric function of the mate-
rial. We previously showed
26
that we could predict the
imaginary part of the dielectric function Im

(
ω
) of plas-
monic metals in quantitative agreement with ellipsomet-
ric measurements for a wide range of frequencies by ac-
counting for the three dominant contributions,
Im

(
ω
) =
4
πσ
0
ω
(1 +
ω
2
τ
2
)
+ Im

direct
(
ω
) + Im

phonon
(
ω
)
.
(12)
We briefly summarize the calculation of these contribu-
tions and focus on their electron temperature dependence
below; see Ref. 26 for a detailed description.
The first term of (12) accounts for the Drude response
of the metal due to free carriers near the Fermi level, with
the zero-frequency conductivity
σ
0
and momentum relax-
ation time
τ
calculated using the linearized Boltzmann
equation with
ab initio
collision integrals. The second
and third terms of (12),
Im

direct
(
ω
) =
4
π
2
e
2
m
2
e
ω
2
BZ
d
k
(2
π
)
3
n
n
(
f
k
n
f
k
n
)
δ
(
ε
k
n
ε
k
n
̄
)
ˆ
λ
·〈
p
k
n
n
2
,
and
(13)
Im

phonon
(
ω
) =
4
π
2
e
2
m
2
e
ω
2
BZ
d
k
d
k
(2
π
)
6
n
±
(
f
k
n
f
k
n
)
(
n
k
k
+
1
2
1
2
)
δ
(
ε
k
n
ε
k
n
̄
̄
k
k
)
×
ˆ
λ
·
n
1
g
k
k
k
n
,
k
n
1
p
k
n
1
n
ε
k
n
1
ε
k
n
̄
+
+
p
k
n
n
1
g
k
k
k
n
1
,
k
n
ε
k
n
1
ε
k
n
̄
k
k
+
2
,
(14)
capture the contributions due to direct interband excita-
tions and phonon-assisted intraband excitations respec-
7
tively. Here
p
k
n
n
are matrix elements of the momen-
tum operator,
ˆ
λ
is the electric field direction (results are
isotropic for crystals with cubic symmetry), and all re-
maining electron and phonon properties are exactly as
described previously. The energy-conserving
δ
-functions
are replaced by a Lorentzian of width equal to the sum of
initial and final electron linewidths, because of the finite
lifetime of the quasiparticles.
The dielectric function calculated using (12-14) de-
pends on the electron temperature
T
e
in two ways. First,
the electron occupations
f
k
n
directly depend on
T
e
. Sec-
ond, the phase-space for electron-electron scattering in-
creases with electron temperature, which increases the
momentum relaxation rate (
τ
1
) in the first Drude term
of (12) and the Lorentzian broadening in the energy con-
serving
δ
-function in (13) and (14).
We calculate
ab initio
electron linewidths using
Fermi golden rule calculations for electron-electron and
electron-phonon scattering at room temperature, as de-
tailed in Ref. 26. These calculations are computation-
ally expensive and difficult to repeat for several electron
temperatures; we use the
ab initio
linewidths at room
temperature with an analytical correction for the
T
e
de-
pendence. The electron-phonon scattering rate depends
on the lattice temperature, but is approximately indepen-
dent of
T
e
because the phase space for scattering is de-
termined primarily by the electronic density-of-states and
electron-phonon matrix elements, which depend strongly
on the electron energies but not on the occupation factors
or
T
e
. The phase space for electron-electron scattering,
on the other hand, depends on the occupation factors and
T
e
because an electron at an energy far from the Fermi
level can scatter with electrons close to the Fermi level.
The variation of this phase-space with temperature is pri-
marily due to the change in occupation of states near the
Fermi level, and we can therefore estimate this effect in
plasmonic metals using a free electron model.
Within a free electron model, the phase-space for
electron-electron scattering grows quadratically with en-
ergy relative to the Fermi level, resulting in scattering
rates
(
ε
ε
F
)
2
at zero electron temperatures, as is
well-known.
2,42
We can extend these derivations to fi-
nite electron temperature to show that the energy and
temperature-dependent electron-electron scattering rate
τ
1
ee
(
ε,T
e
)
D
e
̄
h
[(
ε
ε
F
)
2
+ (
πk
B
T
e
)
2
]
(15)
for
|
ε
ε
F
| 
ε
F
and
T
e

ε
F
/k
B
.
Within the
free electron model, the constant of proportionality
D
e
=
m
e
e
4
4
π
̄
h
2
(

0
b
)
2
ε
3
/
2
S
ε
F
(
4
ε
F
ε
S
4
ε
F
+
ε
S
+ tan
1
4
ε
F
ε
S
)
, where
the background dielectric constant

0
b
and the Thomas-
Fermi screening energy scale
ε
S
are typically treated as
empirical parameters.
2
Here, we extract
D
e
by fitting (15)
to the
ab initio
electron-electron scattering rates at room
temperature
T
0
.
26
The resulting fit parameters are shown
in Table I. We then estimate the total scattering rates at
other temperatures by adding (
D
e
/
̄
h
)(
πk
B
)
2
(
T
2
e
T
2
0
)
to the total
ab initio
results (including electron-phonon
scattering) at
T
0
.
TABLE I. Parameters to describe the change in dielectric
function with electron temperature using (16), extracted from
fits to
ab initio
calculations. The energies and effective masses
for the parabolic band approximation for the
d
s
transition
in noble metals are indicated Figure 12(a).
Al
Ag
Au
Cu
Physical constants:
ω
p
[eV/ ̄
h
]
15.8
8.98
9.01
10.8
τ
1
[eV/ ̄
h
]
0.0911
0.0175
0.0240
0.0268
Fits to
ab initio
calculations:
D
e
[eV
-1
]
0.017
0.021
0.016
0.020
A
0
[eV
3
/
2
]
-
70
22
90
ε
c
[eV]
-
0.31
0.96
0.98
ε
0
[eV]
-
3.36
1.25
1.05
m
v
/m
c
-
5.4
3.4
16.1
-1
-0.5
0
0.5
1
(
ω
/
ω
p
)
2
ε
(
ω
)
a) Al
Im(
ε
) ab initio
Im(
ε
) experiment
Re(
ε
) ab initio
Re(
ε
) experiment
-0.5
0
0.5
1
1.5
2
b) Ag
-1
-0.5
0
0.5
1
1.5
1
3
5
(
ω
/
ω
p
)
2
ε
(
ω
)
Frequency [eV]
c) Au
-1
-0.5
0
0.5
1
1.5
1
3
5
Frequency [eV]
d) Cu
FIG. 8.
Predicted complex dielectric functions for (a) Al,
(b) Ag, (c) Au, (d) Cu at room temperature (300 K) com-
pared with ellipsometry measurements.
43
The
y
-axis is scaled
by
ω
2
2
p
in order to represent features at different frequen-
cies such as the Drude pole and the interband response on the
same scale.
Finally, we use the Kramers-Kronig relations to cal-
culate Re(

(
ω,T
e
)) from Im(

(
ω,T
e
)). Figure 8 com-
pares the predicted dielectric functions with ellipsometry
measurements
43
for a range of frequencies spanning from
near-infrared to ultraviolet. Note that we scale the
y
-axis
by (
ω/ω
p
)
2
, where
ω
p
=
4
πe
2
n
e
/m
e
is the free-electron
plasma frequency, in order to display features at all fre-
quencies on the same scale. We find excellent agreement
for aluminum within 10 % of experiment over the entire
frequency range, including the peak around 1.6 eV due to
an interband transition. The agreement is reasonable for
noble metals with a typical error within 20 %, but with
a larger error
50 % for certain features in the inter-
band
d
s
transitions due to inaccuracies in the
d
-band
positions predicted by DFT (especially for silver).
Figures 9, 10 and 11 show the change of the com-
plex dielectric function upon increasing the electron tem-
perature
T
e
from room temperature to 400 K, 1000 K
and 5000 K respectively, while the lattice remains at