of 10
PHYSICAL REVIEW B
94
, 075120 (2016)
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, California 91125, USA
2
Joint Center for Artificial Photosynthesis, California Institute of Technology, Pasadena, California 91125, USA
3
NG NEXT, 1 Space Park Drive, Redondo Beach, California 90278, USA
4
Materials and Process Simulation Center, California Institute of Technology, Pasadena, California 91125, USA
(Received 1 February 2016; revised manuscript received 26 July 2016; published 11 August 2016)
Ultrafast laser measurements probe the nonequilibrium dynamics of excited electrons in metals with increasing
temporal resolution. Electronic structure calculations can provide a detailed microscopic 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 phenomenological treatment of electron-phonon interactions. We
present
ab initio
predictions of the electron-temperature dependent heat capacities and electron-phonon coupling
coefficients of plasmonic metals. We find substantial differences from free-electron and semiempirical 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, facilitating complete theoretical predictions
of the time-resolved optical probe signatures in ultrafast laser experiments.
DOI:
10.1103/PhysRevB.94.075120
I. INTRODUCTION
Understanding the energy transfer mechanisms during ther-
mal nonequilibrium between electrons and the lattice is critical
for a wide array of applications. Nonequilibrium electron
properties on time scales of 10 s–100 s of femtoseconds are
most efficiently observed with pulsed laser measurement tech-
niques [
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 nonequilibrium
distribution has been the subject of intense research for two
decades. A majority of investigations so far employ various ap-
proximate models, typically based on free-electron models and
empirical electron-phonon interactions, to calculate the energy
absorption, electron-electron thermalization, and electron-
phonon relaxation [
8
19
]. However, a complete
ab initio
description of the time evolution and optical response of this
nonequilibrium electron gas from femtosecond to picosecond
time scales has remained elusive, especially because of the
empirical treatment of electron-phonon interactions [
20
].
The initial electron thermalization via electron-electron
scattering is qualitatively described within the Landau theory
of Fermi liquids [
21
24
]. The subsequent relaxation of the high
temperature electron gas with the lattice is widely described
by the two-temperature model (TTM) [
1
,
5
7
,
17
,
20
], 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
T
e
)
G
(
T
e
)
×
(
T
e
T
l
)
+
S
(
t
)
,
C
l
(
T
l
)
dT
l
dt
=∇·
(
κ
p
T
l
)
+
G
(
T
e
)
×
(
T
e
T
l
)
.
(1)
Here,
κ
e
and
κ
p
are the thermal conductivities of the electrons
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 describes energy deposition
by a laser pulse. In nanostructures, the temperatures become
homogeneous in space rapidly and the contributions of the
thermal conductivities drop out. A vast majority of studies,
both theoretical and experimental, treat the remaining material
parameters,
G
(
T
e
),
C
e
(
T
e
), and
C
l
(
T
l
), as phenomenological
temperature-independent constants [
25
32
].
Figure
1
schematically shows the time evolution of the
electron and lattice temperatures in a plasmonic metal like
gold, and the role of the
temperature-dependent
material
properties. The electronic density of states and the resultant
electronic heat capacity
C
e
(
T
e
) determine the peak electron
temperature
T
e
reached after electron-electron thermalization.
The electron-phonon matrix elements and the resulting cou-
pling strength
G
(
T
e
) determine the rate of energy transfer from
the electrons to the lattice, which along with
C
e
(
T
e
) determines
the rate of relaxation of
T
e
. Finally, the phonon density of states
and the resulting lattice heat capacity
C
l
(
T
l
) determine the rise
in lattice temperature
T
l
.
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 nanostructures to melt
the metal once the electrons and lattice have equilibrated
[
33
]. The highest electron temperature,
T
max
e
, accessible in
repeatable measurements is therefore limited only by the
equilibrated lattice temperature being less than the melting
temperature
T
m
of the metal [
34
], which yields the condition
T
max
e
T
m
dT
e
C
e
(
T
e
)
=
T
m
T
0
dT
l
C
l
(
T
l
). Starting at room tempera-
ture
T
0
=
300 K and using our 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 occupations of the
d
-bands
2 eV below the Fermi level. Consequently, it is
important to derive the temperature dependence of these
material parameters from electronic structure calculations
rather than free-electron like models [
20
].
2469-9950/2016/94(7)/075120(10)
075120-1
©2016 American Physical Society
ANA M. BROWN
et al.
PHYSICAL REVIEW B
94
, 075120 (2016)
FIG. 1. Schematic electron and lattice temperature evolution with
time following laser pulse illumination of a plasmonic metal like
gold, along with the relevant material properties that determine
this evolution. The vertical position of the gold atoms on the plot
corresponds to electron temperature, and the vibration marks around
the atoms schematically indicate lattice temperature. We show that
both the electron heat-capacity
C
e
(
T
e
) [from electronic density of
states (DOS)] that sets the peak electron temperature
T
e
, and the
electron-phonon coupling strength
G
(
T
e
) (from electron-phonon
matrix element
M
e-ph
) that affects the relaxation time of
T
e
, vary with
T
e
in a manner sensitive to details of
d
electrons in noble metals. Only
the lattice heat capacity
C
l
(
T
l
), that determines the lattice temperature
rise, does not vary substantially between the detailed phonon DOS
and simpler models.
Therefore to accurately predict the transient optical re-
sponse 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 response, carrier generation, and
electron transport in plasmonic metals in comparison with
experiment, with no empirical parameters [
35
]. In this article,
we calculate
C
e
(
T
e
),
G
(
T
e
) and the temperature and frequency-
dependent dielectric function,

(
ω,T
e
), from first principles.
These calculations implicitly include electronic-structure ef-
fects in the density of states and electron-phonon interaction
matrix elements, and implicitly account for processes such as
Umklapp scattering. We show substantial differences between
our predictions and those from simplified models due to the
energy dependence of the electron-phonon matrix elements,
especially at high electron 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, phonon coupling,
and temperature dependent dielectric function of plasmonic
materials (Sec.
II A
). In Sec.
II B
, we show calculations
of the electron heat capacity and its dependence on the
electron temperature due to the electronic density of states.
Analogously, Sec.
II C
presents the lattice-temperature de-
pendence of the lattice heat capacity due to the phonon
density of states. Next, in Sec.
II D
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, Sec.
II E
presents
the temperature and frequency dependence of the dielectric
function, including direct (interband), phonon-assisted, and
Drude intraband contributions. Section
III
summarizes our
results and discusses their application to plasmonic nanos-
tructures in various experimental regimes.
II. THEORY AND RESULTS
A. Computational details
We perform density-functional theory (DFT) calculations
of the electronic states, phonons, electron-phonon and op-
tical matrix elements, and several derived quantities based
on these properties, for four plasmonic metals, aluminum,
copper, silver, and gold. We use the open-source plane-wave
density-functional software named JDFTx [
36
] to perform
fully relativistic (spinorial) band structure calculations using
norm-conserving pseudopotentials at a kinetic energy cutoff
of 30 hartrees, and the PBEsol exchange-correlation func-
tional (Perdew-Burke-Ernzerhof functional reparametrized for
solids) [
37
] with a localized
+
U
correction [
38
]forthe
d
-bands in the noble metals. Reference [
39
] 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 matrix
elements using perturbations on a 4
×
4
×
4 supercell. In
our calculations, these matrix elements implicitly include
Umklapp-like processes. We then convert the electron and
phonon Hamiltonians to a maximally localized Wannier
function basis [
40
], with 12
3
k
-points in the Brillouin zone
for electrons. Specifically, we employ 24 Wannier centers
for aluminum and 46 spinorial centers for the noble met-
als which reproduces the density functional theory (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 Hamilto-
nians to arbitrary wave vectors and perform dense Monte
Carlo sampling for accurately evaluating the Brillouin 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. [
35
]for
further details on the calculation protocol and benchmarks of
the accuracy of the electron-phonon coupling (e.g., 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 important factor in
075120-2
Ab INITIO
PHONON COUPLING AND OPTICAL . . .
PHYSICAL REVIEW B
94
, 075120 (2016)
0.1
0.2
0.3
0.4
DOS [10
29
eV
-1
m
-3
]
(a) Al
PBEsol+U (this work)
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 for (a) Al,
(b) Ag, (c) Au, and (d) Cu from our relativistic PBEsol
+
U
calculations, previous semilocal PBE DFT calculations [
20
](less
accurate band structure), and a free electron model.
the electron-phonon coupling and dielectric response of hot
electrons. Above, the band index
n
implicitly counts spinorial
orbitals in our relativistic calculations, and hence we omit the
explicit spin degeneracy factor.
Figure
2
compares the DOS predicted by our relativistic
PBEsol
+
U
method with a previous nonrelativistic semilocal
estimate [
20
] using the PBE (Perdew-Burke-Ernzerhof) func-
tional [
41
], as well as a free electron model
ε
k
=

2
k
2
2
m
e
for which
g
(
ε
)
=
ε
2
π
2
(
2
m
e

2
)
3
/
2
. The free electron model is a reasonable
approximation for aluminum and the PBE and PBEsol
+
U
density-functional calculations also agree reasonably well in
this case (
U
=
0 for aluminum). The regular 31
3
k
-point grid
used for Brillouin zone sampling introduces the sharp artifacts
in the DOS from Ref. [
20
], 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 semilocal PBE calculations
predict
d
-bands that are narrower and closer to the Fermi
level than the PBEsol
+
U
predictions. The
U
correction [
38
]
accounts for self-interaction errors in semilocal DFT and
positions the
d
-bands in agreement with ARPES measure-
ments (to within
0
.
1eV)[
39
]. Additionally, the DOS in the
nonrelativistic PBE calculations strongly peaks at the top of
the
d
-bands (closest to the Fermi level), whereas the DOS in
our relativistic calculations 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
inaccuracies in the DOS due to electronic structure methods
previously 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
2
4
6
8
C
e
[10
5
J/m
3
K]
(a) Al
Eq. 3 (this work)
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 function
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 Fig.
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 [
20
] over-
estimate it because their
d
-bands are too close to the Fermi level.
temperature (
T
e
), can be related to the DOS as
C
e
(
T
e
)
=
−∞
dε g
(
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 depends only
on electronic states within a few
k
B
T
e
of the Fermi level.
For the free electron model, Taylor expanding
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
=

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
,
23
],
which makes it possible for laser pulses to increase
T
e
by
10
3
–10
4
K, while
T
l
remains relatively constant [
6
,
42
,
43
].
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 temperatures (up to
2000 K) for all four
metals.
With increasing
T
e
,
∂f /∂ T
e
in (
3
) is nonzero increasingly
further away from the Fermi energy, so that deviations
from the free electron DOS eventually become important.
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 increase in
C
e
(
T
e
) once
T
e
is
high enough that
∂f /∂ T
e
becomes nonzero 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 of Ref. [
20
], which results in an overestimation of
075120-3
ANA M. BROWN
et al.
PHYSICAL REVIEW B
94
, 075120 (2016)
60
120
180
240
300
DOS [10
29
eV
-1
m
-3
]
(a) Al
Eq. 4 (this work)
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 DFT-calculated phonon density of states
and the Debye model for (a) Al, (b) Ag, (c) Au, and (d) Cu.
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

ω
q
α
)
,
(4)
where

ω
q
α
are energies of phonons with polarization index
α
and wave vector
q
, directly determines the lattice heat capacity,
C
l
(
T
l
)
=
0
dε D
(
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 ap-
proximated 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
(

q
D
v
α
ε
)
/
(

v
α
)
3
, where
v
α
={
v
L
,v
T
,v
T
}
are
the speeds of sound for the one longitudinal and two degenerate
transverse phonon modes of the face-centered cubic metals
considered here [
34
].
Figure
4
compares the DFT-calculated 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, Fig.
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
/
at high
temperatures, which is insensitive to details in the phonon
DOS. In fact, the largest deviations of the Debye model are
below 100 K and less than 10% from the direct calculations for
all four metals. We therefore find that a simple model of the
phonons 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.
10
20
30
40
C
l
[10
5
J/m
3
K]
(a) Al
Eq. 5 (this work)
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 DFT and Debye model predictions of the
lattice heat capacity as a function of lattice temperature,
C
l
(
T
l
), for
(a) Al, (b) Ag, (c) Au, and (d) Cu. Despite large differences in the
density of states (Fig.
4
), the predicted lattice heat capacities of the
two models agree within 10%.
D. Electron-phonon coupling
In Sec.
II B
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 details in the electron-phonon matrix
elements calculated using DFT also play a significant role, in
addition to the electronic band structure, and compare previous
semiempirical estimates of the
T
e
-dependent phonon coupling
to our direct calculations.
The rate of energy transfer from electrons at temperature
T
e
to the lattice (phonons) at temperature
T
l
per unit volume
is given by Fermi’s golden rule as
dE
dt
G
(
T
e
)(
T
e
T
l
)
=
2
π

BZ
d
k
d
k
(2
π
)
6
nn
α
δ
(
b

ω
k
k
)
×

ω
k
k
g
k
k
k
n
,
k
n
2
S
T
e
,T
l
(
ε
k
n
k
n
,

ω
k
k
)
,
(6)
with
S
T
e
,T
l
(
ε,ε
,

ω
ph
)
f
(
ε,T
e
)
n
(

ω
ph
,T
l
)(1
f
(
ε
,T
e
))
(1
f
(
ε,T
e
))(1
+
n
(

ω
ph
,T
l
))
f
(
ε
,T
e
)
.
(7)
Here,
is the unit cell volume,

ω
q
α
is the energy of a phonon
with wave vector
q
=
k
k
and polarization index
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 oc-
cupation 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
(

ω,T
l
) being the Fermi and Bose distribution
075120-4
Ab INITIO
PHONON COUPLING AND OPTICAL . . .
PHYSICAL REVIEW B
94
, 075120 (2016)
function for the electrons and phonons, respectively. Using
the fact that
S
T
e
,T
e
=
0 for an energy-conserving process
ε
+

ω
ph
=
ε
by detailed balance, we can write the electron-
phonon coupling coefficient as
G
(
T
e
)
=
2
π

BZ
d
k
d
k
(2
π
)
6
nn
α
δ
(
ε
k
n
ε
k
n

ω
k
k
)
×

ω
k
k
g
k
k
k
n
,
k
n
2
(
f
(
ε
k
n
,T
e
)
f
(
ε
k
n
,T
e
))
×
n
(

ω
k
k
,T
e
)
n
(

ω
k
k
,T
l
)
T
e
T
l
.
(8)
This general form for
DFT-based
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.
[
44
]. Note that
unlike previous empirical models, here the coupling coefficient
depends on the lattice temperature
T
l
as well, but we omit the
T
l
label in
G
(
T
e
) to keep the notation consistent with previous
approaches [
20
], and present results below for
T
l
=
298 K
(ambient temperature).
The direct evaluation of
G
(
T
e
)using(
8
) requires 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 methods for these matrix elements
[
35
,
45
], and therefore our results are the first parameter-free
predictions of
G
(
T
e
), derived entirely from DFT.
Previous theoretical estimates of
G
(
T
e
) are semiempirical,
combining DFT electronic structure with empirical models
for the phonon coupling. For example, Wang
et al.
[
46
]
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 coupling coefficient as
G
(
T
e
)
πk
B

g
(
ε
F
)
λ
(

ω
)
2
−∞
dε g
2
(
ε
)
∂f
(
ε,T
e
)
∂ε
,
(9)
where
λ
is the electron-phonon mass enhancement parameter
and
(

ω
)
2
is the second moment of the phonon spectrum
[
8
,
20
,
47
]. Lin
et al.
[
20
] treat
λ
(

ω
)
2
as an empirical
parameter calibrated to experimental
G
(
T
e
)atlow
T
e
obtained
from thermoreflectance measurements, and extrapolate it to
higher
T
e
using (
9
). See Refs. [
46
] and [
20
] 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

ω
q
α

T
e
(which is reasonably
valid for
T
e
above room temperature) allows us to approximate
the difference between the electron occupation factors in the
second line of (
8
)by

ω
q
α
∂f /∂ ε
(using energy conservation).
Additionally, for
T
e
T
l
, the third line of (
8
) simplifies
to
k
B
/
(

ω
k
k
). With no other approximations, we can
then rearrange (
8
) to collect contributions by initial electron
energy,
G
(
T
e
)
πk
B

g
(
ε
F
)
−∞
dε h
(
ε
)
g
2
(
ε
)
∂f
(
ε,T
e
)
∂ε
,
(10)
0
200
400
600
800
-10
-5
0
5
h(
ε
) [meV
2
]
ε
-
ε
F
[eV]
(a) Al
Eq. 11 (this work)
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, and (d) Cu. For the noble
metals,
h
(

F
) is substantially larger than its value in the
d
-bands,
which causes previous semiempirical estimates [
20
] using a constant
h
(
ε
) to overestimate the electron-phonon coupling [
G
(
T
e
)] at
T
e

3000 K, as shown in Fig.
7
.
with
h
(
ε
)
2
g
(
ε
F
)
g
2
(
ε
)
BZ
d
k
d
k
(2
π
)
6
nn
α
δ
(
ε
ε
k
n
)
×
δ
(
ε
k
n
ε
k
n

ω
k
k
)

ω
k
k
g
k
k
k
n
,
k
n
2
.
(11)
Therefore, the primary approximation in previous semiem-
pirical estimates [
20
,
46
] is the replacement of
h
(
ε
)byan
energy-independent constant
λ
(

ω
)
2
,usedasanempirical
parameter.
Figure
6
compares our calculations of this energy-resolved
electron-phonon coupling strength,
h
(
ε
), with previous empir-
ical estimates of
λ
(

ω
)
2
, and Fig.
7
compares the resulting
temperature dependence of the electron-phonon coupling,
G
(
T
e
), from (
8
) and semiempirical 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
)tobeanempirical
constant [
17
,
20
] results in a significant overestimate of
G
(
T
e
)
at high
T
e
, compared to the direct calculations. Additionally,
the shallowness of the
d
-bands in the semilocal PBE band
structure used in Ref. [
20
] lowers the onset temperature of
the increase in
G
(
T
e
), and results in further overestimation
compared to our predictions.
Our predictions agree very well with the experimental
measurements of
G
(
T
e
) available at lower temperatures for
noble metals [
3
,
14
,
15
,
32
,
48
]. In fact, the semiempirical calcu-
lation based on
λ
(

ω
)
2
underestimates the room temperature
electron-phonon coupling for these metals; the significant
overestimation of
G
(
T
e
) seen in Fig.
7
is despite this partial
cancellation of error. This shows the importance of detailed
DFT electron-phonon matrix elements in calculating the
coupling between hot electrons and the lattice.
Experimental measurements of the electron-phonon cou-
pling in noble metals are reliable because of the reasonably
075120-5
ANA M. BROWN
et al.
PHYSICAL REVIEW B
94
, 075120 (2016)
2
4
6
8
G [10
17
W/m
3
K]
(a) Al
Eq. 8 (this work)
Lin et al. 2008
Hostetler et al. 1999
0.2
0.4
0.6
0.8
(b) Ag
Eq. 8 (this work)
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
Eq. 8 (this work)
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
Eq. 8 (this work)
Lin et al. 2008
Elsayed-Ali et al. 1987
Hohlfeld et al. 2000
FIG. 7. Comparison of predictions of the electron-phonon cou-
pling strength as a function of electron temperature,
G
(
T
e
), for (a) Al,
(b) Ag, (c) Au, and (d) Cu, with experimental measurements where
available [
3
,
14
,
15
,
32
,
48
]. The DFT-based semiempirical predictions
of Lin
et al.
[
20
] overestimate the coupling for noble metals
at high temperatures because they assume an energy-independent
electron-phonon coupling strength (Fig.
6
) and neglect the weaker
phonon coupling of
d
-bands compared to the conduction band.
The experimental results (and hence the semiempirical predictions)
for aluminum underestimate electron-phonon coupling because they
include the effect of competing electron-electron thermalization
which happens on the same time scale.
clear separation between a fast electron-electron thermal-
ization rise followed by a slower electron-phonon decay in
the thermoreflectance signal. In aluminum, these time scales
significantly overlap resulting in strong nonequilibrium effects
and making experimental determination of the equilibrium
electron-phonon coupling
G
(
T
e
) difficult. Consequently, the
value of
G
(
T
e
) for Al is not well agreed upon [
32
,
49
].
Using a simplified single-band free-electron-like model of
the electrons, Ref. [
50
] estimates
G
2
.
9
×
10
7
W
/
m
3
Kfor
thermalized electrons at 2000 K, which is 1
.
5
×
larger than
G
1
.
9
×
10
7
W
/
m
3
K for nonthermalized electrons with
the same amount of energy. In Fig.
7(a)
, our predictions
using (
8
) which assumes equilibrium are 2
×
larger than
the experimental estimates [
32
] which implicitly include the
nonequilibrium effects. On the other hand, the semiempirical
model of Ref. [
20
] assumes thermalized electrons, but fits
to experimental data that includes nonthermal effects (and
matches experiment by construction). The single-band-model
nonequilibrium predictions do not match experiment because
it assumes a simple model for electron-phonon matrix elements
that ignores Umklapp processes [
50
]. Ultimately, quantitative
agreement with experiments for aluminum (for the right
reasons) therefore requires an extension of our nonempirical
DFT approach (
8
) to include nonequilibrium effects, a subject
of current work in our group.
E. Dielectric function
The final ingredient for a complete theoretical descrip-
tion of ultrafast transient absorption measurements is the
temperature-dependent dielectric function of the material. We
previously showed [
35
] that we could predict the imaginary
part of the dielectric function Im

(
ω
) of plasmonic metals in
quantitative agreement with ellipsometric measurements for a
wide range of frequencies by accounting 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 contributions
and focus on their electron temperature dependence below;
see Ref. [
35
] 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 relaxation
time
τ
calculated using the linearized Boltzmann equation with
collision integrals based on DFT [
35
]. 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

ω
)
ˆ
λ
·
p
k
n
n
2
,
(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

ω

ω
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

ω
+
+
p
k
n
n
1
g
k
k
k
n
1
,
k
n
ε
k
n
1
ε
k
n

ω
k
k
+
)
2
,
(14)
capture the contributions due to direct interband excitations
and phonon-assisted intraband excitations, respectively. Here
p
k
n
n
are matrix elements of the momentum operator,
ˆ
λ
is the electric field direction (results are isotropic for crys-
tals with cubic symmetry), and all remaining 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 quasipar-
ticles.
The dielectric function calculated using (
12
)–(
14
) depends
on the electron temperature
T
e
in two ways. First, the electron
occupations
f
k
n
directly depend on
T
e
. Second, the phase
space for electron-electron scattering increases with electron
temperature, which increases the Lorentzian broadening in the
energy conserving
δ
functions in (
13
) and (
14
).
075120-6