of 4
Supplemental Information for:
Ultrafast carrier experimental and
ab initio
dynamics in plasmonic nanoparticles
Ana M. Brown,
1
Ravishankar Sundararaman,
2
Prineha Narang,
1, 2, 3
Adam
M. Schwartzberg,
4
William A. Goddard III,
2, 5
and Harry A. Atwater
1, 2
1
Thomas J. Watson Laboratories of Applied Physics, California Institute of Technology
2
Joint Center for Artificial Photosynthesis, California Institute of Technology
3
NG NEXT, 1 Space Park Drive, Redondo Beach CA
4
The Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley CA
5
Materials and Process Simulation Center, California Institute of Technology,
1200 East California Blvd, Pasadena CA 91125 USA
(Dated: December 17, 2016)
SPATIAL DYNAMICS OF ELECTRONS
As experimental systems for ultrafast spectroscopy of
metals, both thin films and nanoparticles have distinct
advantages and disadvantages. Thin films are in gen-
eral a cleaner system, offering potentially much better
control over geometry, surface quality and grain-size /
crystallinity. However, modelling and interpreting ultra-
fast spectroscopy of these systems involves one additional
complication: the electron distributions vary in space as
well as in time. Within the two temperature model,
the spatial variation is usually handled via an electron
thermal conductivity term, but this description is only
valid over length scales much larger than the character-
istic mean free path on the order of tens of nanometers.
At intermediate dimensions, super-diffusive and ballistic
electron transport effects become important.
On the other extreme, in plasmonic nanoparticles with
dimensions on the order of these mean free paths and
smaller, the carrier distributions remain spatially homo-
geneous to an excellent approximation. We therefore pick
such plasmonic particles for a first joint experimental and
ab initio
study. Neglecting the spatial dependence al-
lows us to treat the time dynamics and spectral response
in much greater detail with electronic structure meth-
ods, than previously possible with empirical free-electron
models.
Extending such an analysis to the case with spatial
transport is the subject of future work. This requires
adding spatial degrees of freedom to the Boltzmann equa-
tion, and computing the
ab initio
collision integrals as
we do here, separately for different points in space. This
level of theory will naturally capture super-diffusive / bal-
listic transport as well as energy-dependence of the car-
rier mean free paths. Although the theoretical formula-
tion to include spatial dependence is straightforward, the
computational expense increases substantially, requiring
development of appropriate algorithms to make such cal-
culations practical.
ELECTRON-PHONON COLLISION INTEGRAL
Here we calculate the electron-phonon collision integral
for the interaction of an arbitrary hot electron distribu-
tion,
f
(
ε
), with a thermal phonon distribution
n
(
ω,T
l
),
given by the Bose distribution at lattice temperature
T
l
.
We start with the rate of energy transfer between the
electrons and lattice per unit volume, which is exactly
(6) and (7) from Ref. 1, except that we allow
f
(
ε
) to
be an arbitrary distribution (instead of restricting it to a
Fermi distribution at some temperature
T
e
),
dE
dt
e-ph
=
2
π
̄
h
BZ
d
k
d
k
(2
π
)
6
n
δ
(
ε
k
n
ε
k
n
̄
k
k
)
×
̄
k
k
g
k
k
k
n,
k
n
2
S
(
ε
k
n
k
n
k
k
) (1)
with
S
(
ε,ε
)
f
(
ε
)
n
(
ω
)(1
f
(
ε
))
(1
f
(
ε
))(1+
n
(
ω
))
f
(
ε
)
.
(2)
Here Ω is the unit cell volume,
ε
k
n
is the energy of elec-
tron with wave-vector
k
in band
n
, ̄
k
k
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
ab initio
electron-phonon ma-
trix element coupling this phonon to electronic states in-
dexed by
k
n
and
k
n
. The band index explicitly includes
spin as well in order to handle spinorial (relativistic) elec-
tronic states, and hence we do not include the conven-
tional spin degeneracy factor present in non-relativistic
expressions. (See Ref. 2 for more details.)
The above expressions involve double integrals over
the Brillouin zone of
ab initio
electron-phonon matrix
elements, and are expensive to evaluate even with the
Wannier-function-based formulation that we use,
2
espe-
cially if we need to calculate it repeatedly (once per time
step) for evaluating the collision integral in the Boltz-
mann equation. To arrive at a practical approximation
which retains electronic structure details, we note that
the phonon energy is negligible on the relevant electronic
scale ( ̄

ε,ε
). We can then Taylor expand the occu-
pation factors in the energy-conserving cases of
S
(
ε,ε
)
(which are the only ones that contribute in (1) above) as
S
(
ε,ε
+ ̄
hω,ω
)
≈−
f
(
ε
)(1
f
(
ε
))
∂f
∂ε
̄
[1 +
n
(
ω
)
f
(
ε
)] (3)
Further, making the high-temperature phonon occupa-
tion factor approximation (
n
(
ω
)
k
B
T
l
/
( ̄
)

1)
2
shown to be highly accurate for calculating the total
electron-phonon coupling strength in Ref. 3, we can sim-
plify the above expression to
S
(
ε,ε
+ ̄
hω,ω
)
≈−
f
(
ε
)(1
f
(
ε
))
∂f
∂ε
k
B
T
l
.
(4)
Now, we can substitute 4 and insert the identity
dεδ
(
ε
ε
k
n
) into (1), and rearrange it to collect contri-
butions with same initial electron-energy
dE
dt
e-ph
=
dεH
(
ε
)
[
f
(
ε
)(1
f
(
ε
)) +
∂f
∂ε
k
B
T
l
]
,
(5)
with the definition
H
(
ε
) =
2
π
̄
h
BZ
d
k
d
k
(2
π
)
6
n
δ
(
ε
ε
k
n
)
×
δ
(
ε
k
n
ε
̄
k
k
) ̄
k
k
g
k
k
k
n,
k
n
2
.
(6)
Finally, to calculate the electron-phonon contribution
to the collision integral Γ
e-ph
[
f
(
ε,t
)
,T
l
] =
df
(
ε
)
dt
e-ph
, we
note that the contribution to
dE/dt
from electrons with
energy
ε
corresponds to energy exchange between the lat-
tice and electrons of energy
ε
+ ̄
, where ̄
is negligi-
ble on the energy scale of the electrons. Therefore we
can equate the energy flow from the electrons to the lat-
tice (the integrand in (5) above) to an energy flow from
electrons with energy
ε
to electrons with energy
ε
+
,
resulting in the differential equation
H
(
ε
)
[
f
(
ε
)(1
f
(
ε
)) +
∂f
∂ε
k
B
T
l
]
=
∂ε
[
g
(
ε
)
df
(
ε
)
dt
︷︷
Γ
e-ph
]
,
(7)
where
g
(
ε
) is the electronic density of states. Integrating
by parts over
ε
then yields the desired collision integral
Γ
e-ph
[
f
(
ε
)
,T
l
] =
1
g
(
ε
)
∂ε
[
H
(
ε
)
(
f
(
ε
)(1
f
(
ε
)) +
∂f
∂ε
k
B
T
l
)]
,
(8)
which is the same as (4) in the main text.
In this approximate form,
H
(
ε
) includes the detailed
electronic structure, including energy dependence of the
DFT-calculated density of states and electron-phonon
matrix elements, but it only needs to be computed once
for a material using the computationally-expensive (6).
Subsequently, the collision integral given by (8) only in-
volves a single integral over the electron energy which is
computationally feasible for efficient solution of the Boltz-
mann equation. Appendix A presents a numerical tabu-
lation of
H
(
ε
) for the commonly used plasmonic metals,
the noble metals and aluminum, which will be useful for
implementing this efficient strategy in other analyses of
pump probe spectroscopy of plasmonic metals.
Figure 1 plots
H
(
ε
) for the noble metals and aluminum.
Note that it varies by over two orders of magnitude with
10
8
6
4
2
0
2
4
ε
ε
F
[eV]
10
0
10
1
10
2
10
3
H
(
ε
)
[ps
1
nm
3
]
Au
Ag
Cu
Al
FIG. 1.
Ab initio
calculations of the energy-resolved electron-
phonon coupling strength
H
(
ε
) as a function of energy for the
noble metals and aluminum, which allows retaining electronic-
structure effects in the electron-phonon relaxation at low com-
putational expense. See Appendix A for a numerical tabula-
tion of these functions.
large increases below the Fermi level for noble metals due
to
d
bands, while aluminum exhibits only small varia-
tions. The shape of
H
(
ε
) resembles the density of states
g
(
ε
), but it is not strictly proportional to it because the
involved electron-phonon matrix elements also vary with
energy. We discuss this point at length in Ref. 1, where
we plot the quantity
h
(
ε
)
H
(
ε
)
/g
(
ε
).
1
A. M. Brown, R. Sundararaman, P. Narang, W. A. God-
dard III, and H. A. Atwater, Phys. Rev. B
94
, 075120
(2016).
2
A. Brown, R. Sundararaman, P. Narang, W. A. Goddard III,
and H. A. Atwater, ACS Nano
10
, 957 (2016).
3
Z. Lin and L. V. Zhigilei, Physical Review B , 075133 (2008).
3
Appendix A: Numerical tabulation of
H
(
ε
)
ε
ε
F
[eV]
H
(
ε
) [ps
1
nm
3
]
Au
Ag
Cu
Al
-10.00
0.597480
0.000000
0.816735
6.198524
-9.90
0.670881
0.000000
0.856266
6.689361
-9.80
0.774425
0.000000
0.902880
7.187018
-9.70
0.898215
0.000000
0.961690
7.703302
-9.60
1.032793
0.000000
1.046594
8.235448
-9.50
1.184179
0.000000
1.189638
8.765580
-9.40
1.363770
0.000000
1.410654
9.294696
-9.30
1.577994
0.000000
1.660172
9.808935
-9.20
1.826637
0.000000
1.890825 10.321572
-9.10
2.113965
0.000000
2.112152 10.842666
-9.00
2.457445
0.000000
2.343315 11.405445
-8.90
2.868770
0.000000
2.598360 12.008095
-8.80
3.386541
1.475556
2.862264 12.595983
-8.70
4.107117
1.618258
3.126024 13.145285
-8.60
5.192705
1.786228
3.413060 13.684712
-8.50
6.908772
1.986710
3.723084 14.266588
-8.40
9.319275
2.230077
4.030691 14.860899
-8.30
12.043378 2.531880
4.331233 15.458478
-8.20
14.865087 2.916826
4.685048 16.055364
-8.10
18.082123 3.427609
5.044037 16.612520
-8.00
22.197636 4.148272
5.371962 17.192309
-7.90
27.323185 5.268445
5.762644 17.790330
-7.80
33.069582 7.184325
6.232967 18.385237
-7.70
38.250519 10.484126
6.730229 18.943394
-7.60
40.531103 15.844491
7.255323 19.462155
-7.50
40.306331 23.601246
7.801365 19.988145
-7.40
39.763118 33.675677
8.411933 20.522898
-7.30
39.405506 46.299891
9.071933 21.033789
-7.20
38.417504 61.757664
9.768299 21.546958
-7.10
36.329505 77.907890 10.523777 22.063230
-7.00
33.787195 88.939596 11.379665 22.524911
-6.90
32.813062 93.446226 12.408457 22.967377
-6.80
35.252292 92.037536 13.606140 23.426765
-6.70
39.545822 86.882304 14.819928 23.907181
-6.60
42.771785 81.543107 16.025767 24.383931
-6.50
43.428595 80.743230 17.459897 24.839701
-6.40
41.676827 81.827254 19.290432 25.273430
-6.30
37.914873 83.165421 21.489945 25.677661
-6.20
33.068636 80.826450 24.210219 26.062922
-6.10
30.885413 76.448005 27.830557 26.463662
-6.00
35.033425 72.838389 32.547644 26.861327
-5.90
42.512403 71.883561 38.280315 27.198252
-5.80
44.670602 76.628940 45.358527 27.488676
-5.70
41.661348 90.777077 52.954444 27.797102
-5.60
38.624055 103.900060 59.872873 28.141901
-5.50
36.820101 110.604850 66.502617 28.477667
-5.40
35.857701 105.218450 73.623761 28.807147
-5.30
35.530336 90.396281 82.379442 29.146016
-5.20
36.110198 86.604494 94.233431 29.483757
-5.10
35.968756 91.477162 108.827669 29.828703
-5.00
31.962004 96.512997 123.718213 30.138029
-4.90
28.101706 98.116865 135.581203 30.396355
-4.80
29.842037 101.840320 141.566473 30.646848
-4.70
36.189737 103.235373 143.726323 30.976934
-4.60
42.932975 101.059671 147.998533 31.302648
-4.50
47.643976 93.836750 153.294295 31.499090
-4.40
49.697666 86.398932 159.238665 31.547582
-4.30
49.433814 81.130231 170.388386 31.643203
-4.20
49.803150 78.124840 188.216701 31.864007
-4.10
52.936707 77.546116 209.783632 32.114375
ε
ε
F
[eV]
H
(
ε
) [ps
1
nm
3
]
Au
Ag
Cu
Al
-4.00
56.637420 78.815377 231.675668 32.376151
-3.90
57.186996 79.820041 253.170326 32.665279
-3.80
53.446850 74.878951 274.375328 33.000097
-3.70
48.705341 60.174976 292.369870 33.438294
-3.60
45.743259 43.903692 299.672044 33.973505
-3.50
43.877397 29.405280 281.384359 34.497851
-3.40
42.289515 19.209973 265.960167 35.001015
-3.30
40.984378 13.209101 262.028505 35.595797
-3.20
40.247768 9.827595 256.885351 36.285612
-3.10
40.237327 7.799188 254.971351 37.029560
-3.00
40.701507 6.471954 266.348686 37.832214
-2.90
41.660842 5.542253 283.624598 38.587687
-2.80
43.653436 4.848347 290.475994 39.005172
-2.70
46.340845 4.331875 283.355087 39.013894
-2.60
46.431320 3.962417 273.305008 38.849016
-2.50
41.809658 3.657115 270.749946 38.551685
-2.40
34.813331 3.412770 279.261668 38.130836
-2.30
28.241631 3.227020 296.272016 37.666112
-2.20
22.429107 3.075512 308.724473 37.184839
-2.10
17.106065 2.935300 283.311241 36.680719
-2.00
12.882255 2.810574 216.920580 36.238897
-1.90
9.923748
2.715421 139.326820 35.893158
-1.80
7.845475
2.638158
83.873314 35.701858
-1.70
6.309495
2.551878
54.253277 35.779615
-1.60
5.119061
2.480718
39.054058 36.231247
-1.50
4.263923
2.439441
30.371743 37.060418
-1.40
3.717504
2.413600
24.933018 37.955130
-1.30
3.380527
2.397134
21.328588 38.833731
-1.20
3.151459
2.404588
18.811727 39.745956
-1.10
2.989413
2.428734
17.076024 40.598065
-1.00
2.863580
2.436585
15.837471 41.359983
-0.90
2.778031
2.437883
14.817745 42.005387
-0.80
2.720992
2.447703
14.017466 42.249750
-0.70
2.648223
2.470743
13.302102 42.020193
-0.60
2.574284
2.493939
12.686451 41.680184
-0.50
2.525533
2.528236
12.257100 41.536025
-0.40
2.523077
2.557014
12.105565 41.832011
-0.30
2.542511
2.585195
12.021962 42.414980
-0.20
2.522894
2.587301
11.740343 43.004344
-0.10
2.470244
2.581560
11.451441 43.693398
0.00
2.435814
2.583180
11.200920 44.425385
0.10
2.426847
2.583427
11.034614 45.085237
0.20
2.436780
2.588506
10.984070 45.737887
0.30
2.464336
2.576051
10.923256 46.300993
0.40
2.463778
2.540278
10.881696 46.642588
0.50
2.464721
2.555209
10.751503 46.838757
0.60
2.495020
2.575880
10.415702 47.001483
0.70
2.559410
2.578451
10.146843 47.058244
0.80
2.614610
2.559763
10.053222 47.040557
0.90
2.623679
2.563904
10.052537 47.077239
1.00
2.630884
2.591037
9.973968 47.203988
1.10
2.642963
2.579932
9.841346 47.458961
1.20
2.634060
2.535060
9.741046 47.693959
1.30
2.620393
2.500295
9.654080 47.965374
1.40
2.620866
2.473010
9.540269 48.386402
1.50
2.604098
2.463454
9.465990 48.708489
1.60
2.577070
2.483351
9.371522 48.851114
1.70
2.543569
2.513114
9.260463 49.012515
1.80
2.496389
2.493051
9.079118 49.208001
1.90
2.450117
2.448111
8.805104 49.394081
2.00
2.416159
2.401856
8.480309 49.578514
2.10
2.396226
2.352809
8.237696 49.799448
2.20
2.383551
2.297894
8.031179 50.062242
4
ε
ε
F
[eV]
H
(
ε
) [ps
1
nm
3
]
Au
Ag
Cu
Al
2.30
2.348383
2.255700
7.898258 50.334948
2.40
2.304990
2.232271
7.810425 50.620405
2.50
2.276492
2.189032
7.704899 50.869306
2.60
2.271009
2.133390
7.566762 51.151426
2.70
2.303656
2.075882
7.378927 51.444578
2.80
2.403414
2.019139
7.153443 51.774267
2.90
2.631845
1.992539
6.967334 52.105309
3.00
3.095318
1.991042
6.871189 52.426393
3.10
3.862063
1.965651
6.784856 52.837046
3.20
4.511806
1.926301
6.650920 53.332478
3.30
4.628620
1.917617
6.459822 53.879606
3.40
4.404569
1.957138
6.255090 54.475118
3.50
4.069953
2.078739
6.106165 55.069544
3.60
3.649925
2.360859
6.008816 55.644777
3.70
3.212520
2.885108
6.009063 56.202618
3.80
2.850649
3.573692
6.112833 56.758981
3.90
2.583920
4.075775
6.350068 57.387389
4.00
2.385781
4.336034
6.838766 58.061439
4.10
2.220298
4.482171
7.942753 58.727943
4.20
2.064424
4.548858
9.876789 59.331686
4.30
1.927370
4.589624
12.049057 59.980713
4.40
1.815514
4.593069
13.493986 60.693005
4.50
1.737646
4.518367
14.244065 61.418616
4.60
1.694126
4.373779
14.762899 62.177836
4.70
1.676210
4.202975
15.175684 62.867071
4.80
1.681104
4.069866
15.490545 63.497082
4.90
1.703042
3.982175
15.635429 64.230583
5.00
1.745026
3.900388
15.587448 64.964196