# Global soil moisture from in-situ measurements using machine learning –
SoMo.ml

Sungmin O  Max Planck Institute for Biogeochemistry, Jena, D-07745, Germany
corresponding author(s): Sungmin O<EMAIL_ADDRESS> Rene Orth  Max
Planck Institute for Biogeochemistry, Jena, D-07745, Germany

###### Abstract

While soil moisture information is essential for a wide range of hydrologic
and climate applications, spatially-continuous soil moisture data is only
available from satellite observations or model simulations. Here we present a
global, long-term dataset of soil moisture generated from in-situ measurements
using machine learning, SoMo.ml. We train a Long Short-Term Memory (LSTM)
model to extrapolate daily soil moisture dynamics in space and in time, based
on in-situ data collected from more than 1,000 stations across the globe.
SoMo.ml provides multi-layer soil moisture data (0–10 cm, 10–30 cm, and 30–50
cm) at 0.25∘ spatial and daily temporal resolution over the period 2000–2019.
The performance of the resulting dataset is evaluated through cross validation
and inter-comparison with existing soil moisture datasets. SoMo.ml performs
especially well in terms of temporal dynamics, making it particularly useful
for applications requiring time-varying soil moisture, such as anomaly
detection and memory analyses. SoMo.ml complements the existing suite of
modelled and satellite-based datasets given its independent and novel
derivation, to support large-scale hydrological, meteorological, and
ecological analyses.

## Background & Summary

Soil moisture plays a key role in land-atmosphere interactions through its
control on water, energy, and carbon cycles [1, 2]. Weather and climate
variations are mediated by the soil moisture status [3, 4, 5, 6]. Therefore,
the spatiotemporal variations of soil moisture can influence the development
and the persistence of extreme weather events such as heat waves, droughts,
floods, and fires [7, 8, 9, 10, 11]. For these reasons, soil moisture
information is required to support a wide range of research and applications,
e.g. agricultural monitoring, flood and drought prediction, climate
projections, and carbon cycle modelling [12]. Consequently, soil moisture is
recognised as an Essential Climate Variable by the Global Climate Observing
System [13].

Despite its scientific and societal importance, large-scale long-term
observations of soil moisture are scarce. There is a significant number of in-
situ soil moisture measurement networks [14], but they are not uniformly
distributed. Satellite observations allow the derivation of global-scale soil
moisture estimates; however, they represent only the top few centimetres of
the soil. Moreover, satellite retrievals in areas with complex topography,
dense vegetation, and frozen or snow-covered soils are challenging, leading to
data gaps [15]. On the other hand, physically-based models can provide
seamless soil moisture data at the global scale, but large differences exist
across the models due to different and uncertain parameterisations of e.g. the
spatial heterogeneity of soils and vegetation, and the non-linear relationship
between soil moisture and evapotranspiration [16, 17]. In summary, each source
of soil moisture data has characteristic strengths and weaknesses.

Meanwhile, machine-learning (ML) presents an alternative opportunity to
produce seamless soil moisture data. The usefulness of ML algorithms for soil
moisture estimation or forecasting has been demonstrated in the previous
studies [18, 19, 20, 21]. ML algorithms are able to ‘learn’ the complex
relationship between soil moisture (target) and meteorological variables
(predictors) from training data. In this way, soil moisture information can be
inferred from readily observed predictor data in an empirical way without
explicit knowledge of the physical behaviour of the system (e.g. land surface
processes). The resulting soil moisture data is independent from, and can
complement existing satellite-based or model-derived datasets. Similar data-
driven approaches to derive gridded datasets using ML algorithms have been
successfully employed in the cases of land-atmosphere fluxes [22] and runoff
[23].

Here we present a novel global-scale gridded soil moisture dataset generated
through a data-driven approach (Fig. 1). Namely, we employ a Long Short-Term
Memory neural network (LSTM) [24] to build a soil moisture simulation model.
Daily meteorological time series and static features obtained from both
reanalysis and remote sensing datasets are used as predictor variables. As a
target variable, we use adjusted in-situ soil moisture measurements from
different depths obtained from the International Soil Moisture Network (ISMN)
[14] and the National Center for Monitoring and Early Warning of Natural
Disasters of Brazil (CEMADEN) [25]. The raw point-level data are scaled to
match respective means and variabilities obtained from European Centre for
Medium-Range Weather Forecasts (ECMWF) ERA5 gridded soil moisture to allow
seamless merging of measurements across different stations and time periods,
and to estimate soil moisture at a target grid-scale.

Our new global soil moisture dataset, SoMo.ml, provides soil moisture at three
different depths: 0-10 cm, 10-30 cm, and 30-50 cm, corresponding to Layer 1,
Layer 2, and Layer 3, respectively. The data has a spatiotemporal resolution
of 0.25∘ and daily, covering the period of 2000 to 2019. See Table 1 for more
details.

## Methods

### Target soil moisture data preparation

Target soil moisture data at 0.25∘ and daily resolution for model training is
constructed using the in-situ measurements. From the ISMN data only ‘good’
observations are selected, based on the quality flag [26]. CEMADEN provides
only useful-quality data [27]. Both datasets provide sub-daily data and daily
averages are computed for the days with at least six available sub-daily
estimates. Stations or sensors with less than 2 months of data are discarded.

In-situ measurements across the different sites are collected with various
sensor types, which have different calibrations. Therefore, the means and
variances of the obtained time series are not necessarily comparable, which
could introduce artifacts during the LSTM training. For this reason, we adjust
the mean and standard deviation of the daily in-situ time series to those of
the respective ERA5 grid-cell soil moisture within the overlapping period. As
ERA5 soil moisture is available at 0-7 cm, 7-28 cm, and 28-100 cm depths, it
is vertically interpolated into the target layer depths with a depth-weighted
averaging. If more than one in-situ measurement time series is available at
the same depth within the same grid cell (0.25∘), their average is taken. As a
result, the adjusted in-situ target data resembles ERA5 soil moisture in terms
of mean and standard deviation, while its daily temporal variations follow the
ground observations. Our approach is also based on the fact that temporal
variations from point-level data have a greater areal representation compared
to absolute soil moisture values [28, 29]. We can therefore assume that point-
level data contains sufficient information to infer soil moisture dynamics at
the grid scale.

For each soil layer, we preferentially select the adjusted in-situ measurement
taken at the mid-depth of the layer; i.e. 5 cm, 20 cm, and 40 cm,
respectively. If no data is available at the mid-depth, the measurement taken
closest to the mid-depth, and within the layer, is chosen, leading to a total
of 1114, 1064, and 683 grid pixels for the three layers, respectively. The
location of the grid cells with available target soil moisture is shown in
Fig. 2a. Selected depths and data lengths of target soil moisture data
employed for each layer are depicted in Fig. 2b. A considerable fraction of
the target data is obtained from North America across diverse hydro-climatic
regions (see Fig. 3). While training data from South America represents warm
and semiarid regions, those from Asia mostly cover relatively cold regions.

### Model training

LSTM is a special kind of recurrent neural networks that is capable of
learning long-term dependencies across time steps in sequential data [24]. It
has been widely used in land surface modelling such as runoff or soil moisture
simulations [20, 21, 30, 31]. An adapted version of the LSTM architecture,
Entity-Aware LSTM[30], that can ingest time-varying forcing and static inputs
separately is used in this study, thereby allowing the algorithm to explicitly
differentiate the two different types of information.

We model soil moisture using the Entity-Aware LSTM architecture (hereafter
referred to as ‘LSTM model’); the model consists of 1) 128 of hidden units, 2)
one LSTM layer with one dense layer, and 3) 0.5 of dropout rate. More details
about the model validation can be found in Supplementary Information (see
Figs. S1 and S2). The LSTM model is trained to learn the relationship between
the multiple predictor variables and the target soil moisture. The model is
trained separately for each soil layer. The predictor data used for the LSTM-
based soil moisture modelling is listed in Table 2. The meteorological inputs
during days t-364 to t are used to simulate soil moisture at day t; i.e. the
model can establish the relationship of present soil moisture with present and
past meteorological forcing over a full annual cycle. All input data are
normalised using their mean and standard deviation to enhance the training
efficiency [32]. We use the mean squared error divided by the standard
deviation of soil moisture at each individual grid cell as a loss function.
This scaling ensures comparative values of the loss function across wet and
dry regions with potentially different temporal variabilities [30].

Commonly used meteorological forcing variables [33, 34] are prepared from new
global atmospheric reanalysis ERA5 produced by ECMWF [35], as listed in Table
2. For the deeper layers, soil moisture simulated from the upper layer(s) is
further used as input data. ERA5 uses large amounts and diverse kinds of
observations such as synoptic station data, satellite radiances, and ground-
based radar precipitation information via the 4D-Var data assimilation. Its
enhanced quality as meteorological forcing, compared to its predecessor ERA-
Interim, has been demonstrated through an experiment with land surface models
[36]. ERA5 is available in near real-time, allowing corresponding future
updates of the SoMo.ml dataset.

For the static data, long-term mean precipitation and aridity over the period
of 2000-2019 is computed using the ERA5 data [35]. Aridity is defined as the
ratio of net radiation (converted into ${mm}$) divided by precipitation [37].
We characterise topography through mean and standard deviation of sub-grid
scale elevation, as obtained from the ETOPO1 digital elevation model [38]. In
addition, we use soil type and land cover information from the Global Land
Data Assimilation System (GLDAS) data archive [39]. GLDAS resampled soil
porosity and fractions of sand, silt, and clay from FAO datasets [40] into
0.25∘ spatial resolution. The land cover is based on MODIS-derived 20-category
vegetation data that uses a modified International Geosphere–Biosphere
Programme classification scheme [41]. We use GLDAS Dominant Vegetation Type
Data Version 2 which assigned the predominant vegetation type to each 0.25∘
grid cell.

### Global data generation

The LSTM model is trained using the entire training dataset which consists of
the available target soil moisture data and corresponding predictor data.
After establishing the internal relationships (‘learning’), the model is
applied using the predictor data over a quasi-global area of 90∘–60∘ at 0.25∘
spatial resolution. In order to account for the random initialisation of
LSTM’s trainable parameters, five simulations are performed and final soil
moisture values are computed as an average of the five simulations.

## Data Records

The SoMo.ml dataset can be accessed at https://figshare.com. Three compressed
files (.zip) contain data in NetCDF format for the three respective layers. An
example file name is ’SoMo.ml_v1_<LAYER>_<YYYY>.nc’, with LAYER and YYYY
standing for soil moisture layer depth and year, respectively.

## Technical Validation

### Comparison with independent in-situ measurements

Cross-validation (5-fold) is made through a direct grid-to-point comparison
between the SoMo.ml* and the in-situ measurements as done in many previous
studies [42, 43, 44, 45, 46]. The simulated soil moisture for the validation
is hereafter referred to as SoMo.ml*, as this simulation data differs somewhat
from the actual SoMo.ml because it is not based on training with all available
target data, but only with 80% of the data according to the 5-fold cross
validation approach. This validation also enables a comparative assessment of
modelled soil moisture from the LSTM with that of state-of-the-art global
gridded datasets such as ERA5, GLEAM[43], and the satellite-based ESA-CCI [15]
datasets. Established skill scores such as normalised root-mean-square error
(NRMSE), relative bias, and correlation coefficient are used to quantify the
agreement with the ground truth data.

Figure 4 shows the distribution of the NRMSE of SoMo.ml* across climate
regimes (left) and a comparison of these results with the respective
performances of the reference datasets (right). NRMSE is defined as the RMSE
divided by the means of ground truth. Although SoMo.ml* shows slightly higher
biases at some stations over warm and arid regions, there is no clear overall
climate dependency of the NRMSE. In Layer 1, while the median NRMSE of
SoMo.ml* is similar to that of ESA-CCI, which shows lowest NRMSE, a wider
spread of errors is observed. ERA5 and GLEAM tend to overestimate in-situ
measurements (see Fig. S3 in Supplementary Information for relative biases),
leading to slightly higher NRMSE values. In the deeper layers, where ESA-CCI
is not available, NRMSE values of SoMo.ml* are slightly lower but overall
similar to those of the ERA5 and GLEAM references. As a result, this
comparison highlights similar deviations of absolute soil moisture values from
in-situ measurements across the considered datasets.

Figure 5 shows results from a similar comparison, but focusing on the time-
variability of the soil moisture dataset as expressed by the correlation of
soil moisture anomalies with in-situ measurements. To exclude the impact of
the seasonal cycle, we consider short-term anomalies[47, 48]. For each soil
moisture at day d, a period P is defined as P= [d-17, d+17] (corresponding to
a 5-week window). If at least 10 data are available within the period, the
average soil moisture and corresponding anomaly are computed. Equations are
applied to each station and a grid pixel it lies on. No pronounced climate
dependency of the correlations is observed for SoMo.ml* (Fig. 5, left).
Comparing with the reference datasets, SoMo.ml* outperforms them for the top
layer. While overall anomaly correlations decrease in the deeper layers, also
for these layers SoMo.ml* shows closer agreement with the observations than
the reference datasets. The results underline the particular strength of
SoMo.ml*, and likely also the actual SoMo.ml, to represent the temporal
variability of soil moisture. This is somewhat expected; while this comparison
is done against independent in-situ measurements, the temporal dynamics of
SoMo.ml* are directly learned from (remaining) in-situ measurements. Similar
results are obtained when using the correlations of long-term absolute soil
moisture, and of anomalies derived by removing the mean daily averages (Figs.
S4 and S5, respectively).

Note that ESA-CCI has missing values in space and time and GLEAM is available
only until 2018, such that partly different spatiotemporal data are used among
datasets in the comparison. We repeat the analysis above using only data where
all datasets are available and find very similar results (not shown). In
summary, compared with state-of-the-art references, SoMo.ml* shows a
comparable performance in terms of biases, while outperforming the other
datasets in terms of temporal correlations.

### Global-scale comparison with existing gridded datasets

Next, we examine the spatial patterns of SoMo.ml at the global scale. Figure
6a presents the median soil moisture values over the entire period. Low values
in arid regions such as southwest North America, North Africa, central Asia,
and Australia and high values in more humid regions such as the northern
latitudes and Southeast Asia are well captured. A comparison of the
distributions of global median soil moisture between SoMo.ml and the
references can be found from Figs. S6 to S8. Figure 6b compares latitudinal
profile of SoMo.ml against that of the reference datasets (Fig.6b). Overall,
we find a satisfactory consistency between global patterns of SoMo.ml and the
reference datasets. For instance, the highest average soil moisture occurs
near the equator in the tropics, while driest soil moisture is found near
20∘N. These patterns are overall well reproduced in SoMo.ml. This is expected
to some extent because we rescale the target soil moisture using ERA5 means
and standard deviations, such that the LSTM algorithm will pick up these ERA5
characteristics in locations and at time steps with available in-situ
measurements. Nonetheless, SoMo.ml between 15∘N and 25∘N tends to be wetter
than the reference datasets (over the eastern part of the Sahara desert),
especially in the deeper layers. More generally, SoMo.ml might not properly
describe soil moisture in very-arid regions, which can be related to a lack of
training data from such regions (see Fig. 3). Different patterns found in ESA-
CCI are likely due to the missing data. Over very high latitudes over 60∘N, we
can observe relatively large differences across datasets, probably due to
different freezing and thawing patterns. Meanwhile, in-situ measurements (not
adjusted) do not show a meaningful pattern of latitudinal averages but large
variability across stations and sensors, whereby it is not clear to which
extent this is due to different sensor types and calibrations or due to actual
moisture differences caused by heterogeneous land surface characteristics.

## Usage Notes

We present a global, multi-layer, long-term soil moisture dataset generated
through a novel data-driven approach, and with comprehensive ground truth
data. For model training, we preprocess the in-situ measurements to obtain
more spatiotemporally consistent, grid-scale target soil moisture data by
adopting mean and standard deviation from ERA5 data while preserving the
observed temporal variations from the in-situ measurements. At the same time,
our newly generated soil moisture data outperforms other existing gridded
datasets, including ERA5, in terms of daily temporal dynamics as indicated by
highest temporal(anomaly) correlation with the ground observations.
Nonetheless, the data quality in conditions outside the spatiotemporal range
sampled within the observations is potentially uncertain. LSTM performance can
be significantly affected by the (lack of) hydro-climatic diversity in the
training data, even more than by the quantity of data [31]. As shown in Fig.
3, while the in-situ soil moisture measurements are obtained from networks
worldwide, the data does not cover all globally occurring hydro-climatic
conditions. Therefore, relatively high uncertainty outside the training
conditions such as at high latitudes and in arid regions is expected. However,
this lack of observations in particular conditions also presents a challenge
to other datasets/models [48, 49]. Therefore, for instance, using SoMo.ml
within an ensemble of independently derived datasets could be a promising
solution to obtain more reliable soil moisture information in these data-
sparse regions [50, 51]. As a result, our new independent soil moisture
dataset is a valuable addition to the existing suite of soil moisture
datasets, and can enhance future large-scale hydrologic and ecologic analyses,
and also benchmark studies to evaluate land surface models and remote sensing
data.

## Code availability

We implement the LSTM model by adopting python modules obtained from
https://github.com/kratzert/ealstm_regional_modeling. Python codes to generate
and/or analyse data are available from the corresponding author on reasonable
request.

## References

  * [1] Daly, E. & Porporato, A.  A review of soil moisture dynamics: from rainfall infiltration to ecosystem response.  _Environ. Eng. Sci._ 22, 9–24, https://doi.org/10.1089/ees.2005.22.9 (2005).
  * [2] Seneviratne, S. I. _et al._ Investigating soil moisture–climate interactions in a changing climate: A review.  _Earth-Sci. Rev._ 99, 125–161, https://doi.org/10.1016/j.earscirev.2010.02.004 (2010).
  * [3] Koster, R. D. _et al._ Realistic initialization of land surface states: Impacts on subseasonal forecast skill.  _J. Hydrometeorol._ 5, 1049–1063, https://doi.org/10.1175/JHM-387.1 (2004).
  * [4] Orth, R. & Seneviratne, S. I.  Using soil moisture forecasts for sub-seasonal summer temperature predictions in Europe.  _Clim. Dyn._ 43, 3403–3418, https://doi.org/10.1007/s00382-014-2112-x (2014).
  * [5] Prodhomme, C., Doblas-Reyes, F., Bellprat, O. & Dutra, E.  Impact of land-surface initialization on sub-seasonal to seasonal forecasts over Europe.  _Clim. Dyn._ 47, 919–935, https://doi.org/10.1007/s00382-015-2879-4 (2016).
  * [6] Denissen, J. M., Teuling, A. J., Reichstein, M. & Orth, R.  Critical soil moisture derived from satellite observations over europe.  _J. Geophys. Res. Atmos._ 125, e2019JD031672, https://doi.org/10.1029/2019JD031672 (2020).
  * [7] Lorenz, R., Jaeger, E. B. & Seneviratne, S. I.  Persistence of heat waves and its link to soil moisture memory: persistence of heat waves.  _Geophys. Res. Lett._ 37, L09703, https://doi.org/10.1029/2010GL042764 (2010).
  * [8] Mueller, B. & Seneviratne, S. I.  Hot days induced by precipitation deficits at the global scale.  _PNAS_ 109, 12398–12403, https://doi.org/10.1073/pnas.1204330109 (2012).
  * [9] Whan, K. _et al._ Impact of soil moisture on extreme maximum temperatures in Europe.  _Weather. Clim. Extremes_ 9, 57–67, https://doi.org/10.1016/j.wace.2015.05.001 (2015).
  * [10] Sharma, A., Wasko, C. & Lettenmaier, D. P.  If precipitation extremes are increasing, why aren’t floods?  _Water Resour. Res._ 54, 8545–8551, https://doi.org/10.1029/2018WR023749 (2018).
  * [11] O, S., Hou, X. & Orth, R.  Observational evidence of wildfire-promoting soil moisture anomalies.  _Sci. Rep._ 10, 11008, https://doi.org/10.1038/s41598-020-67530-4 (2020).
  * [12] Brown, M. E. _et al._ NASA’s Soil Moisture Active Passive (SMAP) mission and opportunities for applications users.  _Bull. Amer. Meteor._ 94, 1125–1128, https://doi.org/10.1175/BAMS-D-11-00049.1 (2013).
  * [13] GCOS.  Systematic observation requirements for satellite-based products for climate, 2011 update.  _WMO GCOS Rep._ 154, 127pp (2011).  Available at https://library.wmo.int/doc_num.php?explnum_id=3710.
  * [14] Dorigo, W. A. _et al._ The International Soil Moisture Network: a data hosting facility for global in situ soil moisture measurements.  _Hydrol. Earth Syst. Sci._ 15, 1675–1698, https://doi.org/10.5194/hess-15-1675-2011 (2011).
  * [15] Dorigo, W. _et al._ ESA CCI Soil Moisture for improved Earth system understanding: State-of-the art and future directions.  _Remote Sens. Environ._ 203, 185–215, https://doi.org/10.1016/j.rse.2017.07.001 (2017).
  * [16] Dirmeyer, P. A. _et al._ GSWP-2: Multimodel analysis and implications for our perception of the land surface.  _Bull. Amer. Meteor._ 87, 1381–1398, https://doi.org/10.1175/BAMS-87-10-1381 (2006).
  * [17] Koster, R. D. _et al._ On the nature of soil moisture in land surface models.  _J. Clim._ 22, 4322–4335, https://doi.org/10.1175/2009JCLI2832.1 (2009).
  * [18] Gill, M. K., Asefa, T., Kemblowski, M. W. & McKee, M.  Soil moisture prediction using support vector machines.  _J. Am. Water Resour. Assoc._ 42, 1033–1046, https://doi.org/10.1111/j.1752-1688.2006.tb04512.x (2006).
  * [19] Ahmad, S., Kalra, A. & Stephen, H.  Estimating soil moisture using remote sensing data: A machine learning approach.  _Adv. Water Resour._ 33, 69–80, https://doi.org/10.1016/j.advwatres.2009.10.008 (2010).
  * [20] Adeyemi, O., Grove, I., Peets, S., Domun, Y. & Norton, T.  Dynamic neural network modelling of soil moisture content for predictive irrigation scheduling.  _Sensors_ 18, 3408, https://doi.org/10.3390/s18103408 (2018).
  * [21] Fang, K. & Shen, C.  Near-real-time forecast of satellite-based soil moisture using long short-term memory with an adaptive data integration kernel.  _J. Hydrometeorol._ 21, 399–413, https://doi.org/10.1175/JHM-D-19-0169.1 (2019).
  * [22] Jung, M. _et al._ The FLUXCOM ensemble of global land-atmosphere energy fluxes.  _Sci. Data_ 6, 74, https://doi.org/10.1038/s41597-019-0076-8 (2019).
  * [23] Ghiggi, G., Humphrey, V., Seneviratne, S. I. & Gudmundsson, L.  GRUN: an observation-based global gridded runoff dataset from 1902 to 2014.  _Earth Syst. Sci. Data_ 11, 1655–1674, https://doi.org/10.5194/essd-11-1655-2019 (2019).
  * [24] Hochreiter, S. & Schmidhuber, J.  Long short-term memory.  _Neural Computation_ 9, 1735–1780, https://doi.org/10.1162/neco.1997.9.8.1735 (1997).
  * [25] Zeri, M.  A soil moisture dataset over the Brazilian semiarid region, https://doi.org/10.17632/XRK5RFCPVG.2 (2020).
  * [26] Dorigo, W. _et al._ Global automated quality control of in situ soil moisture data from the international soil moisture network.  _Vadose Zone Journal_ 12, 21pp, https://doi.org/10.2136/vzj2012.0097 (2013).
  * [27] Zeri, M. _et al._ Tools for communicating agricultural drought over the brazilian semiarid using the soil moisture index.  _Water_ 10, 1421, https://doi.org/10.3390/w10101421 (2018).
  * [28] Mittelbach, H. & Seneviratne, S. I.  A new perspective on the spatio-temporal variability of soil moisture: temporal dynamics versus time-invariant contributions.  _Hydrol. Earth Syst. Sci._ 16, 2169–2179, https://doi.org/10.5194/hess-16-2169-2012 (2012).
  * [29] Mälicke, M., Hassler, S. K., Blume, T., Weiler, M. & Zehe, E.  Soil moisture: variable in space but redundant in time.  _Hydrol. Earth Syst. Sci._ 24, 2633–2653, https://doi.org/10.5194/hess-24-2633-2020 (2020).
  * [30] Kratzert, F. _et al._ Towards learning universal, regional, and local hydrological behaviors via machine learning applied to large-sample datasets.  _Hydrol. Earth Syst. Sci._ 23, 5089–5110, https://doi.org/10.5194/hess-23-5089-2019 (2019).
  * [31] O, S., Dutra, E. & Orth, R.  Robustness of process-based versus data-driven modelling in changing climatic conditions.  _J. Hydrometeor._ 1929–1944, https://doi.org/10.1175/JHM-D-20-0072.1 (2020).
  * [32] LeCun, Y. A., Bottou, L., Orr, G. B. & Müller, K.-R.  Efficient BackProp.  In _Neural networks: tricks of the trade, Second Edition_ , 9–48, https://doi.org/10.1007/978-3-642-35289-8_3 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012).
  * [33] Sheffield, J., Goteti, G. & Wood, E. F.  Development of a 50-year high-resolution global dataset of meteorological forcings for land surface modeling.  _J. Clim._ 19, 3088–3111, https://doi.org/10.1175/JCLI3790.1 (2006).
  * [34] Balsamo, G. _et al._ A revised hydrology for the ECMWF model: Verification from field site to terrestrial water storage and impact in the Integrated Forecast System.  _J. Hydrometeorol._ 10, 623–643, https://doi.org/10.1175/2008JHM1068.1 (2009).
  * [35] Hersbach, H. _et al._ The ERA5 global reanalysis.  _Q. J. R. Meteorol. Soc._ qj.3803, https://doi.org/10.1002/qj.3803 (2020).
  * [36] Albergel, C. _et al._ ERA-5 and ERA-Interim driven ISBA land surface model simulations: which one performs better?  _Hydrol. Earth Syst. Sci._ 22, 3515–3532, https://doi.org/10.5194/hess-22-3515-2018 (2018).
  * [37] Budyko, M.  Climate and life.  _Academic Press: New York, NY, USA_ (1974).
  * [38] Amante, C.  ETOPO1 1 Arc-Minute Global Relief Model: Procedures, Data Sources and Analysis, https://doi.org/10.7289/V5C8276M (2009).  Accessed at 1 July 2020.
  * [39] Rodell, M. _et al._ The Global Land Data Assimilation System.  _Bull. Amer. Meteor._ 85, 381–394, https://doi.org/10.1175/BAMS-85-3-381 (2004).
  * [40] Reynolds, C. A., Jackson, T. J. & Rawls, W. J.  Estimating soil water-holding capacities by linking the Food and Agriculture Organization Soil map of the world with global pedon databases and continuous pedotransfer functions.  _Water Resour. Res._ 36, 3653–3662, https://doi.org/10.1029/2000WR900130 (2000).
  * [41] Friedl, M. _et al._ Global land cover mapping from MODIS: algorithms and early results.  _Remote Sens. Environ._ 83, 287–302, https://doi.org/10.1016/S0034-4257(02)00078-0 (2002).
  * [42] Albergel, C., de Rosnay, P., Balsamo, G., Isaksen, L. & Muñoz-Sabater, J.  Soil moisture analyses at ECMWF: evaluation using global ground-based in situ observations.  _J. Hydrometeorol._ 13, 1442–1460, https://doi.org/10.1175/JHM-D-11-0107.1 (2012).
  * [43] Martens, B. _et al._ GLEAM v3: satellite-based land evaporation and root-zone soil moisture.  _Geosci. Model Dev._ 10, 1903–1925, https://doi.org/10.5194/gmd-10-1903-2017 (2017).
  * [44] Pablos, M., González-Zamora, A., Sánchez, N. & Martínez-Fernández, J.  Assessment of root zone soil moisture estimations from SMAP, SMOS and MODIS observations.  _Remote Sens._ 10, 981, https://doi.org/10.3390/rs10070981 (2018).
  * [45] Al-Yaari, A. _et al._ Validation of satellite microwave retrieved soil moisture with global ground-based measurements.  In _IGARSS 2018 - 2018 IEEE International Geoscience and Remote Sensing Symposium_ , 3743–3746, https://doi.org/10.1109/IGARSS.2018.8517557 (IEEE, Valencia, 2018).
  * [46] Li, M., Wu, P. & Ma, Z.  Comprehensive evaluation of soil moisture and soil temperature from third-generation atmospheric and land reanalysis datasets.  _Int. J. Climatol._ joc.6549, https://doi.org/10.1002/joc.6549 (2020).
  * [47] Albergel, C. _et al._ Skill and global trend analysis of soil moisture from reanalyses and microwave remote sensing.  _J. Hydrometeorol._ 14, 1259–1277, https://doi.org/10.1175/JHM-D-12-0161.1 (2013).
  * [48] Dorigo, W. _et al._ Evaluation of the ESA CCI soil moisture product using ground-based observations.  _Remote Sens. Environ._ 162, 380–395, https://doi.org/10.1016/j.rse.2014.07.023 (2015).
  * [49] Reichle, R. H. _et al._ Global assessment of the SMAP Level-4 surface and root-zone soil moisture product using assimilation diagnostics.  _J. Hydrometeorol._ 18, 3217–3237, https://doi.org/10.1175/JHM-D-17-0130.1 (2017).
  * [50] Guo, Z., Dirmeyer, P. A., Gao, X. & Zhao, M.  Improving the quality of simulated soil moisture with a multi-model ensemble approach.  _Q. J. R. Meteorol. Soc._ 133, 731–747, https://doi.org/10.1002/qj.48 (2007).
  * [51] Wang, A., Bohn, T. J., Mahanama, S. P., Koster, R. D. & Lettenmaier, D. P.  Multimodel ensemble reconstruction of drought over the continental United States.  _J. Clim._ 22, 2694–2712, https://doi.org/10.1175/2008JCLI2586.1 (2009).
  * [52] O, S. & Orth, R.  Global soil moisture from in situ measurements using machine learning - somo.ml, https://doi.org/10.17871/bgi_somo.ml_v1_2020 (2020).

## Acknowledgements

We would like to thank Ulrich Weber (Max Planck Institute for Biogeochemistry)
for preprocessing and providing the datasets. We also thank Dr. Sophia Walther
(Max Planck Institute for Biogeochemistry) for her valuable comments. This
study is supported by the German Research Foundation (Emmy Noether grant
391059971).

## Author contributions statement

SO and RO designed the study. SO performed the computations and data analysis.
All authors discussed the results and wrote the paper.

## Competing interests

The authors declare that they have no conflict of interests.

## Figures & Tables

Figure 1: Schematic of data-driven approach to generate global-scale gridded
soil moisture from in-situ measurements Figure 2: (a) Spatial distribution of
the target soil moisture data; 1114, 1064, and 683 grid cells are available
for the layers of 0–10 cm, 10–30 cm, and 30–50 cm, respectively. (b) Data
length and measurement depths of the target soil moisture over the period of
2000–2019. Figure 3: Distribution of target soil moisture across hydro-
climatic regimes for each layer. The total number of target data grid cells is
given for each continent. Global grid pixels are randomly sampled (5 %) from
all land pixels for brevity. Figure 4: Comparison of absolute soil moisture
between SoMo.ml* and in-situ data for each layer (top to bottom): (left) NRMSE
values of SoMo.ml* at each measurement station and (right) comparison with
other global gridded datasets. Box plot whiskers show the 0.2 to 0.8 quantiles
of the NRMSE across all measurement stations. The boxes are ranked according
to the median NRMSE so that the best performing data is positioned at the top.
Figure 5: Same as in Fig. 4, but for correlation coefficient of anomalies
where anomalies are determined by removing the mean of a surrounding 35-day
window for each value. Figure 6: (a) Global maps of 20-year long-term medians
of SoMo.ml. (b) Comparison of latitudinal median profiles among the considered
datasets. In the case of GLEAM, root-zone soil moisture is used for both Layer
2 and Layer 3.

Data type |  Gridded
---|---
Spatial Extent |  Quasi-global (90∘N–60∘S)
Temporal coverage |  2000 to 2019
Spatial Resolution |  0.25∘ x 0.25∘
Temporal Resolution |  daily
Variables |  Soil moisture at three layers (0–10 cm, 10–30 cm, and 30–50 cm)
Unit |  Volumetric soil moisture [${m{{}^{3}}m^{-3}}$]
File format |  NetCDF
Key strengths |  1) Global scale, long-term data. 2) Largely independent from available gridded soil moisture products. 3) Better agreement with in-situ measurements in terms of temporal soil moisture dynamics.
Limitations |  1) Performance depends on in-situ data availability, which is low in tropical regions including Africa. 2) Uncertainty and errors in measurements may affect the model performance. 3) ERA5-based scaling is necessary, making means and variabilities of SoMo.ml similar to ERA5 data.
Table 1: Specifications of SoMo.ml v1. |  Variable |  Source |  Description
---|---|---|---
Dynamic |  Air temperature |  ERA5[35] |  Daily meteorological forcing obtained from ECMWF reanalysis
|  Precipitation |  |
|  Specific humidity |  |
|  Net surface radiation |  |
|  Downward surface solar radiation |  |
|  Land surface temperature |  |
|  Soil moisture from upper layer(s) for second and third layers |  SoMo.ml[52] |  ML-based soil moisture produced in this study
Static |  Mean precipitation |  ERA5[35] |  Long-term mean precipitation
|  Aridity |  ERA5[35] |  Ratio of net radiation to precipitation
|  Topography |  ETOPO1[38] |  Mean and standard deviation of sub-grid scale elevation values at each grid cell
|  Vegetation type |  GLDAS[39] |  Predominant vegetation type (MODIS-derived) at each grid cell
|  Soil type |  GLDAS[39] |  Clay, sand and silt fractions based on FAO Soil Map of the World[40]
|  Soil porosity |  GLDAS[39] |  Soil porosity across layers, based on FAO Soil Map of the World[41]
Table 2: Predictor data used for the LSTM model.


# Recent developments in plasmon-assisted photocatalysis - a personal
perspective

Yonatan Sivan  School of Electrical and Computer Engineering, Ben-Gurion
University, Beer-Sheva, 8410501, Israel<EMAIL_ADDRESS>Yonatan Dubi
Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva,
8410501, Israel

###### Abstract

It has been known for many years that metallic nanoparticles can catalyse
various chemical reactions, both in the dark and under illumination, through
different mechanisms. In the last decade or so, many claims of plasmon-
assisted “hot” electron driven catalysis of bond-dissociation reactions have
been put forward. These claims were challenged in a recent series of papers,
where both the underlying theory of “hot” electron generation and the use of
specific experimental setups to discover them in chemical reactions were
examined in detail. The conclusion that arose from these works is that as long
as temperature gradients exist inside the system (as for typical experimental
setups) a quantification of non-thermal effects is close to impossible.
Instead, a standard thermal theory was shown to be capable of explaining the
experimental findings quite accurately. Here, we review the central lines of
thought that led to these conclusions, from a personal perspective. We lay out
the key aspects of the theory, and point to the specific caveats one must be
aware of in performing photo-catalysis experiments. Finally, we provide some
future directions of study.

††preprint: AIP/123-QED

Metal nanoparticles (NPs) are known to exhibit unusual optical properties, in
particular, for having absorption and scattering cross-sections that exceed
their physical size. This was predicted to enable a long list of potential
applications in diverse fields Giannini_chemrev . Unfortunately, however, the
strong absorption in the metal severely limits many of these applications
Khurgin_Boltasseva_MRS ; Khurgin_loss_Nat_nanotech . Since the energy absorbed
in the metal gets converted to heat almost in its entirety Dubi-Sivan ; Dubi-
Sivan-Faraday , in more recent years, attention was diverted into applications
that exploit the absorption, like high temperature chemistry, thermal
treatment of cancer, water purification etc. thermo-plasmonics-review ;
Baffou_solvothermal ; refractory_plasmonics ; solar_steam_apps ;
solar_steam_apps_2 .

Adding to these, one of the most promising such applications was predicted to
be the speeding up of chemical reactions, usually referred to as plasmon-
assisted photocatalysis. In particular, it was envisioned that the absorption
of photons promotes the generation of a non-equilibrium (also known as non-
thermal, or “hot”) carrier distribution such that electrons in the high-energy
tail of this distribution can tunnel out of the metal into high-energy
orbitals of the surrounding molecules, and then catalyse the chemical
reaction. This was usually referred to as the “hot carrier mechanism” (or
“indirect” plasmon-assisted photocatalysis wei2018quantum ;
zhang2016fundamental ; seemala2019plasmon ). We note that the term “hot
carriers” is somewhat unfortunate, as it implies that the carriers are in
thermal equilibrium (i.e. their distribution is Fermi-like) albeit with a
temperature higher than the ambient (such that the system in general is out of
equilibrium). Yet, the common practice in the field refers to “hot electrons”
as having a distribution which is different than a thermal distribution with
the ambient temperature (see, e.g., the title of Ref. [Halas_Science_2018, ]).
Hereafter, we use the terms “non-thermal” and “hot” intermittently, as is
accustomed in the literature.

The “hot” electron mechanism was introduced to explain two main phenomena.
First, it explained the ability of a high energy metal electron to tunnel
through the Schottky barrier to an adjacent semiconductor and then be used for
photodetection at frequencies lower than the gap energy of the
semiconductorrussell2003room ; sandhu2000near ;
Moskovits_hot_es_water_splitting ; Moskovits_photosensitization ;
Uriel_Schottky ; Uriel_Schottky2 ; Valentine_hot_e_review . Elaborate models
of this effect provided a very good quantitative agreement with experimental
findings (see, e.g., [Giulia_Nat_Comm_2018, ]). Second, motivated by this
first effect, the “hot carrier mechanism” was also employed to explain why
noble metals, which are not known as particularly good catalysts, did provide
significant catalysis when subjected to optical illumination. The effect was
primarily explained as enabling low activation energy pathways. In some
studies (e.g. [Halas_dissociation_H2_TiO2, ; Halas_H2_dissociation_SiO2, ;
christopher2011visible, ; plasmonic_photocatalysis_Linic, ;
Halas_Science_2018, ]), it was claimed that the mere presence of “hot”
carriers reduces the activation energy of the favourable reaction pathways, as
a function of their number density (hence, as a function of the incoming light
intensity). This approach was predicted to surpass the efficiency of
traditional catalysis approaches liu2018metal ; chem_rev_photochemistry_2006
and to circumvent the well-known limitations associated with catalysis using
high temperatures. The latter includes high energy-consumption, shortened
catalyst lifetimes thermal_shortening_catalyst_lifetime  through sintering
deterioration, and especially non-selectivity which enables undesired
reactions to take place, hence, to loss of yield and efficiency, see e.g.,
discussion in [hot_e_review_Purdue, ]. This conclusion led to a rapid growth
of interest in plasmon-assisted photocatalysis, mostly as a viable pathway
towards cheap and efficient way to produce “green” fuels
hirakawa2004photoinduced ; plasmonic-chemistry-Baffou ;
plasmonic_photocatalysis_Clavero ; hot_es_review_2015_Moskovits ;
Valentine_hot_e_review ; hou2013review ; wu2017direct ; brooks2018toward ;
zhang2017surface .

Potentially because of the complex multi-disciplinary nature of this problem,
the exact manner in which “hot” electrons assist the reaction rate was not
supported by a quantitative, first-principle type theory, but instead, has
remained at the phenomenological level. Moreover, not only was it unclear how
the “hot” electrons mechanism related to the so-called “direct” mechanism
Wolf-Ertl ; Petek_Nat_Phot_2017  traditionally associated with “standard”
metal catalysts such as Pd, Pt, Ru, Rh etc. (see discussions in [Y2-eppur-si-
riscalda, ; Naldoni-tutorial-2020, ]), this explanation also ignored the role
of mere near-field enhancement Petek_Nat_Phot_2017 , as well as the build up
of temperature that follows the decay (thermalization) of the “hot” electrons
Aeschliman_e_photoemission_review . The relative importance of these effects
is particularly interesting in the context of standard catalysts (see e.g.,
[Liu-Everitt-Nano-Letters-2019, ]), and even in the so-called antenna-reactor
systems (see, e.g., [JACS_Xiong, ; Swearer_antenna_reactor, ;
Boltasseva_LPR_2020, ]) which combine a plasmonic metal (as a light harvester)
and a catalytic metal (as a reaction site).

From a more general perspective, the relative importance of thermal and non-
thermal effects in illuminated plasmonic NPs remained an issue under debate
culver1996temperature ; leenheer2014solar ; Baldi-ACS-Nano-2018 ;
sarhan2019importance . In this context, while in some works the importance of
thermal effects was acknowledged and harnessed for useful applications thermo-
plasmonics-review ; Govorov_thermoplasmonics ; refractory_plasmonics , many
early (experimental as well as theoretical) studies overlooked thermal effects
or concluded that they were small. In some cases, this may have originated
from the conceptually difficult distinction between thermal and non-thermal
effects, or from the common incorrect conception that at low illumination
intensity, non-thermal effects dominate over thermal effects Dubi-Sivan ;
Dubi-Sivan-Faraday . Other studies estimated thermal effects crudely and/or
employed too simplistic control experiments, most likely because of the
limitations of existing reliable thermometry techniques, especially in the
early stages of this line of research. The fact that quantum mechanical
effects on the nanoscale are more attractive than “macroscale” thermal effects
may have also offered some incentive for the downplaying of the latter.

In a series of recent works, we have shown how to separate thermal and non-
thermal effects using a simple addition to standard theoretical approaches
Dubi-Sivan ; Dubi-Sivan-Faraday . We have also shown that standard modelling
and careful temperature measurements can provide a purely thermal quantitative
explanation to many (although not all) reports of faster chemical reactions in
the presence of illuminated metal nanoparticles anti-Halas-Science-paper ;
Y2-eppur-si-riscalda ; R2R ; Un-Sivan-sensitivity .

Below, we review this line of work, aiming at the non-expert audience.
Notably, we do not aim to review all the vast literature on the topic
(especially not all the experimental work), noting that since our first
publications in the field Dubi-Sivan-Faraday ; anti-Halas-Science-paper ,
several review papers and viewpoints have already been published
thm_hot_e_faraday_discuss_2019 ; dyn_hot_e_faraday_discuss_2019 ; Baffou-
Quidant-Baldi ; Khurgin-Faraday-hot-es ; Khurgin-nanophotonics-2020 ;
Jain_viewpoint ; Naldoni-tutorial-2020 . Rather, we wish to take the reader
through the reasoning that guided us in our studies and how it collides with
many of the conclusions drawn previously in the literature. More generally,
our aim in this perspective is to promote the implementation of simple,
intuitive and quantitative physical arguments before employing exciting yet
speculative and/or highly sophisticated ones.

Evidence for thermally-driven photo-catalysis from theory and analysis of
experimental data. The starting point of our work was a simple question - what
happens to a small piece of metal when it is continuously illuminated? This
simple looking question turns out to be hard to answer. Specifically, there
seems to be a clash between the näive intuitive answer “it heats up” and the
strict physical statement that since the metal is out of equilibrium,
temperature is no longer well-defined and one cannot talk about heating at
all. This may have been the reason that several theoretical papers tried to
answer the aformentioned question by considering only how light would affect
the electron distribution inside the NP (see e.g., Refs.
[Manjavacas_Nordlander, ; Govorov_1, ]) or considering heating by taking the
temperature of the electrons as a fixed parameter (and guessing it rather than
calculating it), and even worse, assuming that phonons do not heat up at all
Govorov_ACS_phot_2017 . It is, however, not hard to appreciate that while
ignoring heat generation and heat leakage to the environment may be valid at
the early stages of an ultrashort excitation, these effects cannot be ignored
when studying the steady-state case.

To reconcile the apparent paradox described above, in Ref. [Dubi-Sivan, ] we
took a simple route of using the Boltzmann equation, which takes into account
all possible energy transfer pathways in the system, and augmenting it with
nothing more than the law of energy conservation in the whole system, i.e.,
including not only the photons and electrons, but also the lattice and the
environment. More specifically, we demanded that at the steady-state, all the
power that is pumped into the system via photon absorption must leave it via
heat current to the surrounding host. Further simplification was obtained by
naturally separating the electron distribution into a “thermal” and ‘non-
thermal” parts, where the thermal part is a Fermi distribution, with a
temperature which is different than the ambient temperature. These two points
allowed us to calculate correctly both the electron temperature and the
steady-state electron non-equilibrium distribution (including the “hot”
electrons) under continuous wave illumination. Thus, unlike previous studies
of the electron non-equilibrium, this approach allowed us to evaluate the full
non-equilibrium electron distribution, while naturally accounting for the
effect of nanoparticle size and shape, as well as the thermal properties of
the host on the rate of heat transfer to the environment.

Two central findings came out of this approach. The first is that, due to the
fact that the electron-electron relaxation time is incredibly short in metals,
almost all of the power coming from the illumination goes into heating the
electrons and the phonons (which maintain almost the same steady-state
temperature), and only a tiny fraction goes into skewing the electron
distribution and generating “hot” electrons. Importantly, our calculations
have shown clearly that the dominance of thermal effects over non-thermal
(“hot” carrier) effects become more significant as the illumination intensity
becomes lower. This result invalidates a common claim Govorov_ACS_phot_2017
that since the temperature rise associated with low illumination intensity is
small with respect to the ambient temperature, then, “hot” carrier effects are
dominant for low intensities. The second finding is that in spite of the first
point above, the number of “hot” electrons, i.e., the number of electrons with
an energy excess of what they would have within a thermal-only distribution,
increases by many orders of magnitude.

These two findings seem to be at odds with many claims of photocatalysis being
accelerated by illuminated plasmonic nanoparticles, most specifically with
some of the seminal papers in the field Halas_dissociation_H2_TiO2 ;
Halas_H2_dissociation_SiO2 ; christopher2011visible ;
plasmonic_photocatalysis_Linic ; Halas_Science_2018 . Specifically, simple
estimates of heating suggest much higher heating than observed in these
studies (the reasons for that will be discussed below). In addition, only a
1-2 orders of magnitude rise in reaction rates was observed with respect to
the reaction rate in the dark, although the $8-10$ orders of magnitude
increase of the number of “hot” electrons implies that a similar $8-10$ orders
of magnitude enhancement should be observed in the reaction rate Dubi-Sivan-
Faraday .

Following these apparent discrepancies, the question arises: can experimental
reports of faster reaction rates in the presence of metal NPs be explained
solely based on thermal effects, assuming that the only thing that is
happening in experiments is that the sample heats up? The answer to this
question turns out to be yes.

It requires, however, recognizing that the measured temperature, $T_{M}$, may
be lower than the actual temperature $T$ of the catalyst (this assumption will
be proven correct in the following sections, as was discussed in Refs. [anti-
Halas-Science-paper, ; Y2-eppur-si-riscalda, ]). Why is this important?
because, according to the standard Arrhenius theory of chemical reactions, the
reaction rates obey

$R\propto\exp\left(-\frac{\mathcal{E}_{a}}{k_{B}T}\right),$ (1)

where $\mathcal{E}_{a}$ is the activation energy, $k_{B}$ is the Boltzmann
constant. Thus, if the measured temperature, $T_{M}$, is smaller than the
actual temperature $T$ of the catalyst (as shown in Y2-eppur-si-riscalda
111Indeed, in Y2-eppur-si-riscalda  we have shown that $T_{M}$ was several
tens of degrees lower than the actual temperature, and sometimes
(Halas_Science_2018 ; R2R ) even several hundreds of degrees lower. ), it
would appear that the reaction rates overshoot the Arrhenius law - a result
which thus was interpreted as enhanced reaction rate coming from “hot”
electrons.

To demonstrate that this is a plausible claim, we rely on the simple
connection between the catalyst temperature, the measured temperature and the
incident illumination intensity $I_{inc}$,

$T=T_{dark}+aI_{inc}=T_{M}+\tilde{a}I_{inc}.$ (2)

where $T_{dark}$ is the temperature of the reactor when no illumination is
present. The photothermal conversion coefficient $a$ depends on a number of
system-specific parameters (NP size and shape, material, density and number,
illumination wavelength, thermal properties of the host etc.) thermo-
plasmonics-basics ; thermo-plasmonics-review ;
Baffou_pulsed_heat_eq_with_Kapitza ; thermo-plasmonics-multi_NP . As explained
in detail in [R2R, ], Eq. (2) can be easily extended to account for the slower
rate of heating occurring at relatively high illumination levels, e.g., by
adding a nonlinear contribution of $I_{inc}$ to the temperature. However, this
is necessary only in very few experiments, see [Y2-eppur-si-riscalda, ] and
[thm_hot_e_faraday_discuss_2019, , p. 270-271].

Eqs. (1) and (2) have only two unknowns for any given photocatalytic reaction,
namely the activation energy $\mathcal{E}_{a}$ and the photo-thermal
conversion coefficient $\tilde{a}$. In order to obtain these from published
data, all one needs is the reaction rate in the dark (which provides
$\mathcal{E}_{a}$) and a single point of reaction rate under illumination at a
given illumination intensity (which provides $\tilde{a}$). This strikingly
simple procedure is enough to reproduce essentially all the data of Refs.
[Halas_dissociation_H2_TiO2, ; Halas_H2_dissociation_SiO2, ;
christopher2011visible, ; plasmonic_photocatalysis_Linic, ;
Halas_Science_2018, ] with remarkable accuracyanti-Halas-Science-paper ;
Y2-eppur-si-riscalda ; R2R . In Fig. 1 we show a collage of experimental data
from Refs. [Halas_Science_2018, ](a), [Halas_dissociation_H2_TiO2, ](b),
[plasmonic_photocatalysis_Linic, ](c) and [christopher2011visible, ](d), along
with a fit to an Arrhenius curve (solid line), demonstrating the agreement
between the data and the theory described above.

Nevertheless, as noted in Ref. Y2-eppur-si-riscalda, , from an Arrhenius fit
alone, one cannot determine with absolute certainty if the illumination only
changes the temperature without changing the activation energy, or contributes
to both effects (a point which was later raised also by Jain
jain2019phenomenological ; response_2_Jain ). Thus, in order to gain further
support to the pure thermal interpretation, the fits shown in Fig. 1 were then
validated with independent thermal calculations which accounted for the
details of the metal NPs used (shape, size, density, ..), the host material
and illumination via an effective medium theory. The excellent agreement
between the values of the calculated and fitted photothermal conversion
coefficient $a$ (e.g., Ref. anti-Halas-Science-paper, ) indicates that one
does not need to invoke any mechanism of change in the activation energy in
order to explain the experimental results.

Figure 1: Experimental data (symbols) and a fit (lines) using the classical
Arrhenius theory of Eq. (1), showing remarkable agreement between theory and
experiment. Data are collected from Refs. [Halas_Science_2018, ](a),
[Halas_dissociation_H2_TiO2, ](b), [plasmonic_photocatalysis_Linic, ](c) and
[christopher2011visible, ](d). Figures are reproduced with permission from
Refs. [anti-Halas-Science-paper, ](a) and [Y2-eppur-si-riscalda, ](b-d).

On the importance of understanding temperature gradients.  The remarkable
reconstruction of the experimental data with Arrhenius theory confirms that
there is a substantial difference between the temperature of the catalysts
($T$) and the measured temperature ($T_{M}$) in the studies analyzed in Fig.
1. The reconstruction is also based on the assumption that one can
characterize the catalysts using a single temperature. But what is this
temperature and how does it relate to the reality of the experiments? These
questions led us to a comprehensive inquiry into the experimental procedures
of the papers we analyzed Halas_Science_2018 ; Halas_dissociation_H2_TiO2 ;
christopher2011visible ; plasmonic_photocatalysis_Linic ; Halas_Science_2018 .
We discovered a series of flaws in the experimental setups, that may have led
the authors of these papers to exaggerate the importance of “hot” electrons
compared to thermal effects. A detailed account of proper experimental
practices required to separate thermal from “hot” electron contributions has
been given by us Y2-eppur-si-riscalda  and by others Liu-Everitt-Nano-
Letters-2019 ; Liu-Everitt-Nano-research-2019 ; Jain_viewpoint ; Baffou-
Quidant-Baldi .

Here, we wish to stress what we believe is a key issue in all experiments
attempting to separate thermal from non-thermal contributions, namely, the
correct account for temperature gradients. This topic has been discussed at
length in [Liu-Everitt-Nano-Letters-2019, ; R2R, ], so we wish to demonstrate
its importance via an example. Consider a (rather standard) photo-catalysis
experiment, where a $\sim 1$mm sample is illuminated from above and heated
from below. How would one go about and isolate the non-thermal contribution to
the reaction rate? A naïve approach would be to measure the reaction rate
under illumination, record the surface temperature, measure the reaction rate
at the recorded temperature in the dark, and subtract the latter reaction rate
from the former. Such a protocol has indeed been employed on various works to
extract the “hot” electron contribution to the reaction rate
Halas_Science_2018 ; Halas-Nature-Catalysis-2020 .

However, there is an essential flaw in this protocol, as it completely
neglects temperature gradients Liu-Everitt-Nano-Letters-2019 , and here is
why. When the reaction rate is measured under illumination, the surface of the
sample is much hotter than its bulk, because the penetration of light is
rather small (typically tens of microns). Thus, effectively, the bulk of the
sample will contribute to the catalysis at a reaction rate which is defined by
Eq. (1) with a temperature which is much lower than the surface temperature.

On the other hand, heating the sample using an external heater (i.e., in the
dark), typically leads to a more uniform temperature distribution (in fact,
there might be temperature gradients in the reverse direction to the
illuminated case, see [Liu-Everitt-Nano-Letters-2019, ] in the presence of a
bottom heater; this possibility is ignored for the sake of simplicity). In
these cases, when heating to the surface temperature, the entire bulk is at
the surface temperature recorded in the photocatalysis experiment. Therefore,
subtracting from the illuminated reaction rate the reaction rate in the dark
at the surface temperature is meaningless. Similarly, the non-negligible
transverse gradients (see [Un-Sivan-sensitivity, ]) are likely to cause
additional differences between the experiment and its control; higher-order
moments of the temperature distribution may be playing a role, yet to be
understood Liu-Everitt-Nano-Letters-2019 , 222Note that the transverse
gradients were not accounted for in that work..

The only way around this problem, at least for thick samples, is to have a
thermal control experiment in the dark that fully reproduces the temperature
gradient under illumination. While attempts in this direction have been made
Liu-Everitt-Nano-research-2019 , in reality this is probably an impossible
task, essentially because of the exponential sensitivity of the reaction rate
on the temperature. The remaining alternative is to perform a thermocatalysis
control in which the sample is hotter than in the photocatalysis experiment,
see, e.g., [yu2019plasmonic, ]; this approach may highlight the contribution
of the ‘hot” electrons, but is incapable of properly quantifying it.

More systematic evaluation of the temperature distribution in plasmon-assisted
photocatalysis samples.  In [Un-Sivan-sensitivity, ], we generalized the
discussion of the temperature calculations for plasmon-assisted photocatalysis
samples. First, we validated the effective medium approach presented in
[Y2-eppur-si-riscalda, ], and provided a simple analytic formula for the
temperature rise in characteristic samples. This formula enables an immediate
evaluation of the importance of the thermal effects; it also opens the way to
inclusion of fluid dynamics effects, see below. Then, we studied the
dependence of the temperature distribution on the various parameters of the
system. We showed that since typically these samples are designed to absorb
all the incoming light, the steady-state temperature distribution shows
overall insensitivity to the illumination spectra, temporal pattern, particle
density, size and shape; this result contradicts a wide range of popular
claims that can be found in prior literature and verbally-presented
conceptions, which usually do not follow from a proper thermal analysis. In
that sense, the temperature distribution is determined by macroscopic
considerations, i.e., the balance of total heat generated in the sample and
the heat transfer to the surroundings (rather than by the microscopic
details). This also shows that while the use of metal nanoparticles for
heating was initially motivated by the localized nature of heat generation at
the nano-proximity of the particles, in practice, the (steady-state)
temperature distribution generated by an ensemble of NPs is not so different
from that created by an external heat source.

The results of this analysis reflects some rather intuitive conclusions.
First, thermal effects accumulate with the number of NPs, whereas “hot”
carrier effects do not - for a given level of illumination, they are
independent of the number of NPs. This implies that further to the local
considerations presented above (see [Dubi-Sivan-Faraday, ]), reactions in
dense systems of NPs are more likely to be catalyzed by heating rather than by
non-thermal effects.

Second, while the initial intention was to speed up chemical processes with
high activation energy (e.g., [Halas_dissociation_H2_TiO2, ]), these are
particularly sensitive to increasing temperature Un-Sivan-sensitivity  (via
the Arrhenius Law (1)). Conversely, low activation energy reactions (i.e.,
those which are typically efficient at room temperature) are those that may
benefit much more from “hot” electrons; this was indeed demonstrated recently
in [Boltasseva_LPR_2020, ].

All the above implies, somewhat ironically, that “hot” electrons are
(typically) useful only under conditions which are the exact opposite to those
in which they were originally envisioned to be efficient! Whether there is a
practical advantage for speeding up chemical reactions occurring across a
large volume using such dilute suspensions of metal NPs remains to be proven.

Outlook.  Our re-interpretation anti-Halas-Science-paper ; Y2-eppur-si-
riscalda , as well as similar work by others Baffou-Quidant-Baldi ; Khurgin-
Faraday-hot-es ; Khurgin-nanophotonics-2020 ; Khurgin-Levy-ACS-Photonics-2020
; Jain_viewpoint  shows that thermal effects are frequently important and even
dominant factors in plasmon-assisted photocatalysis, such that their
quantification is essential for the understanding of the mechanisms underlying
every specific chemical reaction. However, while the simplistic thermal theory
(1) was sufficient to explain several “high-profile” studies, there are quite
a few studies it cannot explain. When such a discrepancy is observed, the
catalytic enhancement is usually ascribed to non-thermal carriers, see, for
example, [Baldi-ACS-Nano-2018, ; Frontiera-2018, ; Liu-Everitt-Nano-
research-2019, ; yu2019plasmonic, ; Giulia_2019, ; Boltasseva_LPR_2020, ], to
name just a few. However, the exponential sensitivity mentioned above implies
that in order to rectify such conclusions, one has to improve the model’s
validity and to reach a quantitative match of the theory to the experiments.

First, the quantitative thermal theory has to be complemented with a
quantitative model of chemical interactions, including electron transfer
between the NPs and the chemical moieties as well as the reaction dynamics.
This is essential for the isolation of thermal and non-thermal contributions
when they co-exist. For instance, in [plasmonic_photocatalysis_Linic, ],
employing a Kinetic Isotope (KIE) measurement showed that the strong ($\sim
100\%$) thermal contribution is accompanied by a small ($\sim 1-3\%$)
contribution which was interpreted as non-thermal action (although a thermal
origin for this effect has been suggested, see discussion in [Y2-eppur-si-
riscalda, ]). Another example is low activation energy reactions in which the
thermal channel is simultaneously relatively efficient and insensitive to
further illumination Boltasseva_LPR_2020 . A third example is reactions which
exhibit both temperature-dependent reaction rates and photo-selectivity, which
cannot be simply explained with Arrhenius theory Yugang_Sun ; Halas-Nature-
Energy-2020 ; Yugang_Sun ; yu2018plasmonic ; yu2019plasmonic ; JACS_Xiong ;
Giulia_2019 . This extension of the theory will also allow one to take into
consideration the type of chemical reaction; specifically, as so far we have
re-analyzed studies involving only bond dissociation, it would be interesting
to examine also reduction-oxidation reactions in which charge transfer is an
integral part of the reaction.

Second, while our work discussed at length photo-catalysis experiments in
hydrostatic and dry conditions, many of the experiments in the field involve
fluid flow and electro-chemical charge transfer (in solution) Tatsuma_2004 ;
Willets_electrochemistry_2018 ; Caleb_Hill_Heating_Electrochemical_JPCC_2019 ;
Cortes_Nano_lett_2019 ; Lange_2020 ; Lopez_electrochemistry_2020 ;
Lu_electrochemistry_2020 . Understanding these experiments requires inclusion
of various experimental elements such as heat convection and stirring
yu2019plasmonic , reactant pressure and distribution within the catalyst bed
Liu-Everitt-Nano-Letters-2019 , the thermal properties of the electro-chemical
cell, charge transfer properties (including the effects of voltage bias) etc..
First steps towards achieving this were accomplished in [Un-Sivan-sensitivity,
] (where the effective medium thermal calculations of [Y2-eppur-si-riscalda, ]
were rectified vs. exact simulations, as a first step towards combining heat
and fluid dynamics) and in [Caleb_Hill_Heating_Electrochemical_JPCC_2019, ]
(where a formulation combining heat and charge transfer was described).

Third, one has to account for the effect of high illumination intensities and
high temperatures on the optical properties of the system. Indeed, in many
cases, the ambient temperature is increased significantly, to increase the
(dark) reaction yield, so that the material permittivities change
significantly (see e.g., Shalaev_ellipsometry_gold ;
Shalaev_ellipsometry_silver ; PT_Shen_ellipsometry_gold  for some recent high
temperature ellipsometry studies of metals). Furthermore, in some experiments
(e.g., [Halas_Science_2018, ]), the light-induced heating reached several and
even many hundreds of degrees Kelvin, so that the reaction rate ceased to grow
linearly (see detailed discussion in [thm_hot_e_faraday_discuss_2019, , p.
271], as well as in [Y2-eppur-si-riscalda, ] and [R2R, ]). Since the thermal
nonlinearity of the metal NPs (and even of the host dielectric) may be
relatively strong (see [Sivan-Chu-high-T-nl-plasmonics, ; Gurwich-Sivan-CW-
nlty-metal_NP, ; IWU-Sivan-CW-nlty-metal_NP, ]), it it likely to be stronger
than non-thermal effects.

Finally, the theory should include, when necessary, also considerations
related with the non-locality of the electron response, as well as quantum
mechanical considerations associated with electron state discretization
Govorov_1 ; hot_es_Atwater ; GdA_hot_es ; Khurgin_Landau_damping ; Khurgin-
Faraday-hot-es ; Khurgin-Levy-ACS-Photonics-2020 ; Lischner_hot_es , electron
tunneling out of the metal NP Khurgin-nanophotonics-2020 ;
Giulia_Nat_Comm_2018  etc. all of which are essential for the quantification
of the reaction process as a whole. One of the immediate applications of such
theory would be a quantification of the efficiency of promising pathways such
as the elongation of tunnelled carrier lifetimes - via hole scavengers
yu2019plasmonic , dielectric cores (in the spirit of photodetection
experiments) Boltasseva_LPR_2020  or core-shell antenna-reactor structures
JACS_Xiong .

Many of the above elements were already developed. However, a complete theory
that includes all these components is yet to be compiled and implemented for a
quantitative comparison to experimental data. This will allow us, as a
community, to embark upon a proper re-evaluation of past results, on one hand,
and to design meaningful experiments and interpret them thoroughly, on the
other.

In parallel to the extension of the theoretical framework, the experimental
characterization of the temperature profile has to be improved. Indeed,
measuring the temperature at only two points (as e.g., in [Liu-Everitt-Nano-
Letters-2019, ; Liu-Everitt-Nano-research-2019, ; Halas-Nature-Energy-2020, ])
gives no information on unavoidable Un-Sivan-sensitivity  transverse
gradients. Such gradients may strongly affect reaction rates, yet are many
times un-accounted for. High resolution thermometry techniques have been
developed Cahill_T_measure ; Baumberg_SERS_T_measure ;
Pramod_Reddy_T_measure_2015 ; Lukin_T_measure ; NV_centers_T_measurement_NTU ;
Orrit-Caldarola_T_measure ; Orrit-Caldarola_T_measure_transient  and
implemented in the context of photocatalysis Cortes_Nano_lett_2019 , however,
they are far from being sufficiently robust Orrit-
Caldarola_T_measure_transient . Indeed, similar difficulties to measure the
temperature distribution were observed also in other contemporary problems in
nanophotonics, in particular, in perovskite films, see
[perovskite_solar_cells, ; perovskite_solar_cells_comment, ] and the ongoing
exchange.

While the above elements are necessary for the characterization of the
macroscopic properties, many recent experiments addressed the challenging
problem of measuring the reaction rate on the single NP level
Vadai_Dionne_hydrogenation ; Gross_Nature_2017 ;
Giessen_single_NP_hydrogenation ; Cortes_Nano_lett_2019 ; Peng_Chen . Beyond
the obvious difficulty to obtain a measurable reaction rate from such small
entities, this context poses several additional challenges such as the need to
quantify distributions and currents created by gradients of local
electromagnetic fields, temperature and charge, heat transfer on the particle
level (including the Kapitza resistance) etc.. Furthermore, reactions in the
presence of single nanoparticles are sensitive to specific shape, size and
composition, which affects the catalytic properties of the nanoparticle; e.g.,
sharp edges and corners are known to contribute to the catalytic enhancement -
with and without “hot” electrons El_Sayed_review .

Final Thoughts.  The comprehensive critical studies published over the last
year or so (see [dyn_hot_e_faraday_discuss_2019, ; Khurgin-Faraday-hot-es, ;
thm_hot_e_faraday_discuss_2019, ; anti-Halas-Science-paper, ; Y2-eppur-si-
riscalda, ; R2R, ; Jain_viewpoint, ; Baffou-Quidant-Baldi, ; Khurgin-
nanophotonics-2020, ]) and the vibrant ongoing discussion in the field seem to
have re-directed the study of plasmon-assisted photocatalysis into a more
reliable, quantitative scientific routine. Detailed thermal calculations are
becoming more common and accessible (e.g., [Baffou-Quidant-Baldi, ;
Boltasseva_LPR_2020, ; yu2019plasmonic, ; Y2-eppur-si-riscalda, ; Un-Sivan-
sensitivity, ]) and the awareness to the characteristics of the temperature
distribution Y2-eppur-si-riscalda ; Un-Sivan-sensitivity ; Boltasseva_LPR_2020
and to proper temperature measurements has grown Cahill_T_measure ;
metal_luminescence_Cahill_PNAS ; Cortes_Nat_Comm_2017 ; Baldi-ACS-Nano-2018 ;
Orrit-Caldarola_T_measure ; Cortes_Nano_lett_2019 ; Liu-Everitt-Nano-
research-2019 ; Orrit-Caldarola_T_measure_transient . Advanced techniques are
being developed and the use of fluorescence and Raman techniques is becoming
more abundant in the context of plasmon-assisted photocatalysis Majima_1 ;
Majima_1 ; Baumberg_SERS_T_measure ; Baumberg-Faraday ; Peng_Chen ;
Frontiera-2018 ; Orrit-Caldarola_T_measure_transient . Indeed, several
convincing demonstrations of “hot” electron-driven reactions with proper
thermal control have been reported Baldi-ACS-Nano-2018 ; yu2018plasmonic ;
yu2019plasmonic ; Boltasseva_LPR_2020 .

Yet, some problematic practices still persist, for instance, the use of
different samples for the photocatalysis experiment and the thermocatalysis
control Halas-Nature-Energy-2020 , incorrect normalization by catalyst mass
Halas-Nature-Energy-2020  (see detailed discussion in [R2R, ]) and wrong
extraction of “hot” electron contribution to the reaction rates Halas-Nature-
Catalysis-2020 . Additional common misconceptions (such as the incorrect claim
on the dominance of non-thermal effects over thermal effects at low
illumination intensity Govorov_ACS_phot_2017 , the absence of transverse
temperature uniformities, the dependence of the number of “hot” electrons on
particle size and shape (see correction of Govorov_ACS_phot_2017  in Dubi-
Sivan-Faraday ) or the ability of gas flow to eliminate temperature gradients)
still need to be corrected.

We wish to end this perspective with a more general note. We believe that the
ongoing debate reviewed above proves that there is room within the current
publishing climate for more critical (self-)reflective research of the type
that we described in our recent manuscripts; this may even be considered as a
sort of necessary gadfly. Thus, addressing the scientific criticism in one’s
own papers, encouraging debate and publishing controversial (and sometimes
editorially inconvenient) papers, will catalyse uprooting of incorrect claims,
unfounded beliefs and wrong practices.

Erratum. Following lengthy discussions with L. Besteiro, we realized that one
aspect of the criticism described in Ref. [Dubi-Sivan, ; Dubi-Sivan-APL-
Perspective, ] over his previous work Govorov_ACS_phot_2017  is unjustified.

In particular, the formulation developed in Ref. [Govorov_ACS_phot_2017, ]
relies on a unique parameter defined as $T_{eff}$. While this parameter has no
analog in similar formulations (see e.g., Refs. [hot_es_Atwater, ; GdA_hot_es,
; Dubi-Sivan, )], and its origin is not clearly explained, its description can
be easily lead one to understand that it represents the electron temperature.
However, Besteiro clarified that this parameter is extracted from the
numerical evaluation of the electron excitation/relaxation rate (Fig. 7(b)-(c)
in Ref. [Govorov_ACS_phot_2017, ]), such that despite its suggestive notation,
it does not correspond to the electron temperature.

In that sense, our comparison of the choices of $T_{eff}$ to the predictions
of classical heat equations (in Refs. [Dubi-Sivan, ; Dubi-Sivan-APL-
Perspective, ]) is irrelevant. We apologize for the confusion caused.

This issue, however, has only a small impact on the overall criticism
expressed on the formulation of Ref. [Govorov_ACS_phot_2017, ] in Refs. [Dubi-
Sivan, ; Dubi-Sivan-Faraday, ; Dubi-Sivan-APL-Perspective, ]. In a nutshell,
that formulation assumes that the thermal and non-thermal distributions can be
decoupled, and treats the latter only. As shown in Ref. [Dubi-Sivan, ; Dubi-
Sivan-Faraday, ], this decoupling is not valid when going beyond the
relaxation time approximation for the electron-phonon interaction (as adopted
in Ref. [Govorov_ACS_phot_2017, ]) such that global energy conservation in the
whole system (consisting of the photons, electrons, phonons and the
environment) is considered. A potential exception is the case of very weak
illumination. This subtle yet fundamental issue will be clarified in a future
publication in detail.

Data Availability Statement Data sharing is not applicable to this article as
no new data were created or analyzed in this study.

## References

  * (1) V. Giannini, A. I. Fernández-Domínguez, S. C. Heck, and S. A. Maier,  Chem. Rev. 111, 3888 (2011).
  * (2) J. B. Khurgin and A. Boltasseva,  MRS Bulletin 37, 768 (2012).
  * (3) J. B. Khurgin,  Nature Nanotechnology 10, 2 (2015).
  * (4) Y. Dubi and Y. Sivan,  Light: Science and Applications: Nature 8, 89 (2019).
  * (5) Y. Sivan, I. W. Un, and Y. Dubi,  Faraday Discussions 214, 215 (2019).
  * (6) G. Baffou and R. Quidant,  Laser Photon. Rev. 7, 171 (2013).
  * (7) H. M. L. Robert, F. Kundrat, E. B.-U. na, H. Rigneault, S. Monneret, R. Quidant, J. Polleux, and G. Baffou,  ACS Omega 1, 2 (2016).
  * (8) U. Guler, A. Boltasseva, and V. M. Shalaev,  Science 334, 263 (2014).
  * (9) Y. Lin, H. Xu, X. Shan, Y. Di, A. Zhao, Y. Hua, and Z. Gan,  J. Mater. Chem. A 7, 19203 (2019).
  * (10) M. Gao, L. Zhu, C. K. Peh, and G. W. Ho,  Energy Environ. Sci. 12, 841 (2019).
  * (11) Q. Wei, S. Wu, and Y. Sun,  Advanced Materials 30, 1802082 (2018).
  * (12) Y. Zhang, C. Yam, and G. C. Schatz,  The Journal of Physical Chemistry Letters 7, 1852 (2016).
  * (13) B. Seemala, A. J. Therrien, M. Lou, K. Li, J. P. Finzel, J. Qi, P. Nordlander, and P. Christopher,  ACS Energy Letters 4, 1803 (2019).
  * (14) L. Zhou, D. F. Swearer, C. Zhang, H. Robatjazi, H. Zhao, L. Henderson, L. Dong, P. Christopher, E. A. Carter, P. Nordlander, and N. J. Halas,  Science 362, 69 (2018).
  * (15) K. Russell, I. Appelbaum, H. Temkin, C. Perry, V. Narayanamurti, M. Hanson, and A. Gossard,  Applied physics letters 82, 2960 (2003).
  * (16) J. Sandhu, A. Heberle, B. Alphenaar, and J. Cleaver,  Applied Physics Letters 76, 1507 (2000).
  * (17) J. Lee, S. Mubeen, X. Ji, G. Stucky, and M. Moskovits,  Nano Lett. 12, 5014 (2012).
  * (18) S. Mubeen, G. Hernandez-Sosa, D. Moses, J. Lee, and M. Moskovits,  Nano Lett. 11, 5548 (2011).
  * (19) I. Goykhman, B. Desiatov, J. Khurgin, J. Shappir, and U. Levy,  Nano Lett. 11, 2219 (2011).
  * (20) I. Goykhman, B. Desiatov, J. Khurgin, J. Shappir, and U. Levy,  Opt. Exp. 20, 28594 (2012).
  * (21) W. Li and J. Valentine,  Nanophotonics 6, 177 (2016).
  * (22) G. Tagliabue, A. S. Jermyn, R. Sundararaman, A. J. Welch, J. S. DuChene, R. Pala, A. R. Davoyan, P. Narang, and H. A. Atwater,  Nature Communications 9, 3394 (2018).
  * (23) S. Mukherjee, F. Libisch, N. Large, O. Neumann, L. V. Brown, J. Cheng, J. B. Lassiter, E. A. Carter, P. Nordlander, and N. J. Halas,  Nano Lett. 13, 240 (2013).
  * (24) S. Mukherjee, L. Zhou, A. Goodman, N. Large, C. Ayala-Orozco, Y. Zhang, P. Nordlander, and N. J. Halas,  J. Am. Chem. Soc. 136, 64 (2014).
  * (25) P. Christopher, H. Xin, and S. Linic,  Nature Chemistry 3, 467 (2011).
  * (26) P. Christopher, H. Xin, A. Marimuthu, and S. Linic,  Nat. Materials 11, 1044 (2012).
  * (27) L. Liu and A. Corma,  Chem. Rev. 118, 4981 (2018).
  * (28) K. Watanabe, D. Menzel, N. Nilius, and H.-J. Freund,  Chem. Rev. 106, 4301 (2006).
  * (29) C. T. Campbell, S. C. Parker, and D. E. Starr,  Science 298, 811–814 (2002).
  * (30) A. Naldoni, F. Riboni, U. Guler, A. Boltasseva, V. M. Shalaev, and A. V. Kildishev,  Nanophotonics 5, 112 (2016).
  * (31) T. Hirakawa and P. V. Kamat,  Langmuir 20, 5645 (2004).
  * (32) G. Baffou and R. Quidant,  Chem. Soc. Rev. 43, 3898 (2014).
  * (33) C. Clavero,  Nat. Photon. 8, 95 (2014).
  * (34) M. Moskovits,  Nature Nanotech. 10, 6 (2015).
  * (35) W. Hou and S. B. Cronin,  Advanced Functional Materials 23, 1612 (2013).
  * (36) X. Wu, E. Jaatinen, S. Sarina, and H. Y. Zhu,  Journal of Physics D 50, 283001 (2017).
  * (37) J. L. Brooks, C. L. Warkentin, D. Saha, E. L. Keller, and R. R. Frontiera,  Nanophotonics 7, 1697 (2018).
  * (38) Y. Zhang, S. He, W. Guo, Y. Hu, J. Huang, J. R. Mulcahy, and W. D. Wei,  Chemical reviews 118, 2927 (2017).
  * (39) M. Bonn, S. Funk, C. Hess, D. N. Denzler, C. Stampfl, M. Scheffler, M. Wolf, and G. Ertl,  Science 285, 1042 (1999).
  * (40) S. Tan, A. Argondizzo, J. Ren, L. Liu, J. Zhao, and H. Petek,  Nature Photonics 11, 806–812 (2017).
  * (41) Y. Sivan, I. W. Un, and Y. Dubi,  Chemical Sciences 11, 5017 (2020).
  * (42) L. Mascaretti and A. Naldoni,  J. Appl. Phys. 128, 041101 (2020).
  * (43) M. Bauer, A. Marienfeld, and M. Aeschlimann,  Progress in Surface Science 90, 319–376 (2015).
  * (44) X. Li, X. Zhang, H. O. Everitt, and J. Liu,  Nano Letters 19, 1706 (2019).
  * (45) H. Huang, L. Zhang, Z. Lu, R. Long, C. Zhang, Y. Lin, K. Wei, C. Wang, L. Chen, Z.-Y. Li, Q. Zhang, Y. Luo, and Y. Xiong,  J. Am. Chem. Soc. 138, 6822 (2016).
  * (46) F. Swearer, H. Zhao, L. Zhou, C. Zhang, H. Robatjazi, J. M. P. Martirez, C. M. Krauter, S. Yazdi, M. J. McClain, E. Ringe, E. A. Carter, P. Nordlander, and N. J. Halas,  Proc. Natl. Acad. Sci. U.S.A 113, 8916 (2016).
  * (47) X. Xu, A. Dutta, J. Khurgin, A. Wei, V. M. Shalaev, and A. Boltasseva,  Laser and Photonics Reviews 14, 1900376 (2020).
  * (48) J. Culver, M. Li, Z.-J. Sun, R. Hochstrasser, and A. Yodh,  Chemical physics 205, 159 (1996).
  * (49) A. J. Leenheer, P. Narang, N. S. Lewis, and H. A. Atwater,  Journal of Applied Physics 115, 134301 (2014).
  * (50) R. Kamarudheen, G. W. Castellanos, L. P. J. Kamp, H. J. H. Clercx, and A. Baldi,  ACS Nano 12, 8447 (2018).
  * (51) R. M. Sarhan, W. Koopman, R. Schuetz, T. Schmid, F. Liebig, J. Koetz, and M. Bargheer,  Scientific reports 9, 3060 (2019).
  * (52) A. O. Govorov and H. H. Richardson,  Nano Today 2, 30 (2007).
  * (53) Y. Sivan, J. Baraban, I. W. Un, and Y. Dubi,  Science 364, eaaw9367 (2019).
  * (54) Y. Sivan, J. Baraban, and Y. Dubi,  OSA Continuum 3, 483 (2020).
  * (55) I. W. Un and Y. Sivan,  Nanoscale, accepted; https://arxiv.org/abs/2007.03421 (2020).
  * (56) J. Aizpurua, F. Baletto, J. Baumberg, P. Christopher, B. d. Nijs, P. Deshpande, Y. Diaz Fernandez, L. Fabris, S. Freakley, S. Gawinkowski, A. Govorov, N. Halas, R. Hernandez, B. Jankiewicz, J. Khurgin, M. Kuisma, P. V. Kumar, J. Lischner, J. Liu, A. Marini, R. J. Maurer, N. S. Mueller, M. Parente, J. Y. Park, S. Reich, Y. Sivan, G. Tagliabue, L. Torrente-Murciano, M. Thangamuthu, X. Xiao, and A. Zayats,  Faraday Discussions 214, 245 (2019).
  * (57) J. Aizpurua, M. Ashfold, F. Baletto, J. Baumberg, P. Christopher, E. Cortes, B. de Nijs, Y. Diaz Fernandez, J. Gargiulo, S. Gawinkowski, N. Halas, R. Hamans, B. Jankiewicz, J. Khurgin, P. V. Kumar, J. Liu, S. Maier, R. J. Maurer, A. Mount, N. S. Mueller, R. Oulton, M. Parente, J. Y. Park, J. Polanyi, J. Quiroz, S. Rejman, S. Schlucker, Z. Schultz, Y. Sivan, G. Tagliabue, M. Thangamuthu, L. Torrente-Murciano, X. Xiao, A. Zayats, and C. Zhan,  Faraday Discussions 214, 123 (2019).
  * (58) G. Baffou, I. Bordacchini, A. Baldi, and R. Quidant,  Light: Science and Applications 9, 108 (2020).
  * (59) J. Khurgin,  Faraday Discussions 214, 35 (2019).
  * (60) J. B. Khurgin,  Nanophotonics 9, 453 (2020).
  * (61) P. K. Jain,  J. Phys. Chem. 123, 24347 (2019).
  * (62) A. Manjavacas, J. G. Liu, V. Kulkarni, and P. Nordlander,  ACS Nano 8, 7630 (2014).
  * (63) A. O. Govorov, H. Zhang, and Y. K. Gun’ko,  J. Phys. Chem. C 117, 16616 (2013).
  * (64) L. V. Besteiro, X.-T. Kong, Z. Wang, G. Hartland, and A. O. Govorov,  ACS Photonics 4, 2759 (2017).
  * (65) Indeed, in Y2-eppur-si-riscalda  we have shown that $T_{M}$ was several tens of degrees lower than the actual temperature, and sometimes (Halas_Science_2018 ; R2R ) even several hundreds of degrees lower.
  * (66) G. Baffou, R. Quidant, and F. J. G. de Abajo,  ACS Nano 4, 709 (2010).
  * (67) G. Baffou and H. Rigneault,  Phys. Rev. B 84, 035415 (2011).
  * (68) G. Baffou, P. Berto, E. B. Urena, R. Quidant, S. Monneret, J. Polleux, and H. Rigneault,  ACS Nano 7, 6478 (2013).
  * (69) P. K. Jain,  Chem. Sci (2020).
  * (70) Y. Dubi, I. W. Un, and Y. Sivan,  Chem. Sci (2020).
  * (71) X. Li, H. O. Everitt, and J. Liu,  Nano Research 19, 1706 (2019).
  * (72) H. Robatjazi, J. L. Bao, M. Zhang, L. Zhou, P. Christopher, E. A. Carter, P. Nordlander, and N. J. Halas,  Nature Catalysis 3, 564 (2020).
  * (73) Note that the transverse gradients were not accounted for in that work.
  * (74) S. Yu and P. K. Jain,  Nature Communications 10, 2022 (2019).
  * (75) J. B. Khurgin and U. Levy,  ACS Photonics 7, 547 (2020).
  * (76) E. L. Keller and R. R. Frontiera,  ACS Nano 12, 5848 (2018).
  * (77) A. J. Welch, J. S. DuChene, G. Tagliabue, A. Davoyan, W.-H. Cheng, and H. A. Atwater,  ACS Appl. Energy Mater. 2, 164 (2019).
  * (78) X. Dai, Q. Wei, T. Duong, and Y. Sun,  Chem. Nano. Mat. 5, 1000 (2019).
  * (79) L. Zhou, J. M. P. Martirez, J. Finzel, C. Zhang, D. F. Swearer, S. Tian, H. Robatjazi, M. Lou, L. Dong, L. Henderson, P. Christopher, E. A. Carter, P. Nordlander, and N. J. Halas,  Nature Energy 5, 61 (2020).
  * (80) S. Yu, A. J. Wilson, J. Heo, and P. K. Jain,  Nano letters 18, 2189 (2018).
  * (81) Y. Tian and T. Tatsuma,  Chem. Commun. , 1810 (2004).
  * (82) Y. Yu, V. Sundaresan, and K. A. Willets,  J. Phys. Chem. C 122, 5040 (2018).
  * (83) M. Maley, J. W. Hill, P. Saha, J. D. Walmsley, and C. M. Hill,  The Journal of Physical Chemistry C 123, 12390 (2019).
  * (84) E. Pensa, J. Gargiulo, A. Lauri, S. Schlücker, E. Cortés, and S. A. Maier,  Nano Lett. 19, 1867 (2019).
  * (85) M. Rodio, M. Graf, F. Schulz, N. S. Mueller, M. Eich, and H. Lange,  ACS Catalysis 10, 2345 (2020).
  * (86) N. B. Schorr, M. J. Counihan, R. Bhargava, and J. Rodríguez-Loṕez,  Anal. Chem. 92, 3666 (2020).
  * (87) W. Ou, B. Zhou, J. Shen, T. W. Lo, D. Lei, S. Li, J. Zhong, Y. Y. Li, and J. Lu,  Angew. Chem. Int. Ed. 59, 6790 (2020).
  * (88) H. Reddy, U. Guler, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev,  Optical Materials Express 6, 2776 (2016).
  * (89) H. Reddy, U. Guler, K. Chaudhuri, A. Dutta, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva,  ACS Photonics 4, 1083 (2017).
  * (90) P.-T. Shen, Y. Sivan, C.-W. Lin, H.-L. Liu, C.-W. Chang, and S.-W. Chu,  Opt. Exp. 24, 19254 (2016).
  * (91) Y. Sivan and S.-W. Chu,  Nanophotonics 6, 317 (2017).
  * (92) I. Gurwich and Y. Sivan,  Phys. Rev. E 96, 012212 (2017).
  * (93) I. W. Un and Y. Sivan,  https://arxiv.org/abs/2006.11884 .
  * (94) A. M. Brown, R. Sundararaman, P. Narang, W. A. Goddard, and H. A. Atwater,  ACS Nano 10, 957 (2016).
  * (95) J. R. M. Saavedra, A. Asenjo-Garcia, and F. J. G. de Abajo,  ACS Photonics 3, 1637 (2016).
  * (96) J. Khurgin, W.-Y. Tsai, D. P. Tsai, and G. Sun,  ACS Photonics 4, 2871 (2017).
  * (97) S. D. Forno, L. Ranno, and J. Lischner,  J. Phys. Chem. C. 122, 8517 (2018).
  * (98) X. Xie and D. G. Cahill,  Appl. Phys. Lett. 109, 183104 (2016).
  * (99) J. T. Hugall and J. J. Baumberg,  Nano Lett. 15, 2600 (2015).
  * (100) K. Kim, B. Song, V. Fernández-Hurtado, W. Lee, W. Jeong, L. Cui, D. Thompson, J. Feist, M. T. H. Reid, F. J. García-Vidal, J. C. Cuevas, E. Meyhofer, and P. Reddy,  Nature 528, 387 (2015).
  * (101) G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin,  Nature 500, 54 (2013).
  * (102) Y.-K. Tzeng, P.-C. Tsai, H.-Y. Liu, O. Chen, H. Hsu, F.-G. Yee, M.-S. Chang, and H.-C. Chang,  Nano Lett. 15, 3945 (2015).
  * (103) A. Carattino, M. Caldarola, and M. Orrit,  Nano Lett. 18, 874 (2017).
  * (104) T. Jollans, M. Caldarola, Y. Sivan, and M. Orrit,  Journal of Physical Chemistry A, accepted; https://arxiv.org/abs/2003.08790 (2020).
  * (105) H. Tsai, R. Asadpour, J.-C. Blancon, C. C. Stoumpos, O. Durand, J. W. Strzalka, B. Chen, R. Verduzco, P. M. Ajayan, S. Tretiak, J. Even, M. A. Alam, M. G. Kanatzidis, W. Nie, and A. D. Mohite,  Science 360, 67 (2018).
  * (106) N. Rolston, R. Bennett-Kennett, L. T. Schelhas, J. M. Luther, J. A. Christians, J. J. Berry, and R. H. Dauskardt,  Science 368, eaay8691 (2020).
  * (107) M. Vadai, D. Angell, F. Hayee, K. Sywtu, and J. Dionne,  Nature Communications 9, 4658 (2018).
  * (108) C.-Y. Wu, W. J. Wolf, Y. Levartovsky, H. A. Bechtel, M. C. Martin, F. D. Toste, and E. Gross,  Nature 541, 511 (2017).
  * (109) J. Karst, F. Sterl, H. Linnenbank, T. Weiss, M. Hentschel, and H. Giessen,  Science Advances 6, eaaz0566 (2020).
  * (110) N. Zou, G. Chen, X. Mao, H. Shen, E. Choudhary, X. Zhou, and P. Chen,  Nano Lett. 12, 5570 (2018).
  * (111) M. B. Mohamed, R. Narayanan, and M. A. El-Sayed,  Acc. Chem. Res. 46, 1795 (2013).
  * (112) J. Huang, W. Wang, C. J. Murphy, and D. G. Cahill,  Proc. Nat. Acad. Sci. U.S.A 111, 906 (2014).
  * (113) E. Cortés, W. Xie, J. Cambiasso, A. S. Jermyn, R. Sundararaman, P. Narang, S. Schlücker, and S. A. Maier1,  Nature Communications 8, 14880 (2017).
  * (114) Z. Zheng, T. Tachikawa, and T. Majima,  J. Am. Chem. Soc. 136, 6870 (2014).
  * (115) J. Huang, B. de Nijs, S. Cormier, K. Sokolowski, D.-B. Grys, C. A. Readman, S. J. Barrow, O. A. Schermanb, and J. J. Baumberg,  Faraday Discussions 214, 455 (2019).
  * (116) Y. Sivan and Y. Dubi,  Appl. Phys. Letters 117, 130501 (2020).