Small polaron formation is known to limit ground-state mobilities in metal oxide photocatalysts. However, the role of small polaron formation in the photoexcited state and how this affects the photoconversion efficiency has yet to be determined. Here, transient femtosecond extreme-ultraviolet measurements suggest that small polaron localization is responsible for the ultrafast trapping of photoexcited carriers in haematite (α-Fe_2O_3). Small polaron formation is evidenced by a sub-100 fs splitting of the Fe 3p core orbitals in the Fe M_(2,3) edge. The small polaron formation kinetics reproduces the triple-exponential relaxation frequently attributed to trap states. However, the measured spectral signature resembles only the spectral predictions of a small polaron and not the pre-edge features expected for mid-gap trap states. The small polaron formation probability, hopping radius and lifetime varies with excitation wavelength, decreasing with increasing energy in the t_(2g) conduction band. The excitation-wavelength-dependent localization of carriers by small polaron formation is potentially a limiting factor in haematite’s photoconversion efficiency

Alivisatos, A. Paul

Carneiro, Lucas M.

Cushing, Scott K.

Leone, Stephen R.

Liu, Chong

Su, Yude

Yang, Peidong

Caltech Authors - Main

In the format provided by the authors and unedited.
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DOI: 10.1038/NMAT4936
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Excitation Wavelength Dependent Small Polaron Trapping of 
Photoexcited Carriers in -Fe2O3 
Lucas M. Carneiro,1,4,† Scott K. Cushing,1,4,† Chong Liu,1, ‡ Yude Su1, Peidong Yang,1,2,5,6  
A. Paul Alivisatos,1,2,5,6 and Stephen R. Leone*,1,3,4 
 
1Department of Chemistry, 2Department of Materials Science and Engineering, and 3Department 
of Physics, University of California, Berkeley, California 94720, United States 
ͶChemical Sciences Division and 5Material Sciences Division, Lawrence Berkeley National 
Laboratory, Berkeley, California 94720, United States 
6Kavli Energy NanoScience Institute, Berkeley, California 94720, USA 
‡Present Address: Department of Chemistry and Chemical Biology, Harvard University, 
Cambridge, MA 02138, USA 
†Equal Contribution 
Corresponding Author 
*E-mail: srl@berkeley.edu. 
  
Excitation-wavelength-dependent small polaron 
trapping of p toexcited carriers in α-Fe2O3
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Charge Transfer Multiplet and Polaron Model of Differential Absorption 
a) Charge Transfer State The XUV absorption for the initial optically excited state is 
predicted using a charge transfer multiplet calculation by CTM4XAS (Reference 29, main paper) 
with a crystal field splitting of 10Dq = 1.45 eV following Reference 26 from the main paper. The 
optically induced charge transfer can be included by designating the final state to be Fe2+, or by 
setting the difference between the core hole and Hubbard potential to 5 eV and the hopping 
parameter for the t2g and eg band to 2 eV. The latter introduces a ligand-hole on the oxygen site, 
introducing the optical charge transfer into the CTM4XAS calculation. In order to account for 
shorter Auger lifetimes at higher energy, the broadening of the x-ray transitions is represented by 
a Lorentzian with a linearly increasing width from 0.5 eV at 54.5 eV to 2 eV at 56 eV and a Fano 
asymmetry parameter of 3.5.  Instrument resolution is included by a 0.5 eV width Gaussian. The 
differential absorption is calculated using the difference between the excited state, charge transfer 
model and the experimentally measured core-hole modified ground state in Figure 1d. For the 
differential spectrum, a 0.5 eV Lorentzian broadening of the core-hole modified ground state 
absorption was also found to improve agreement, consistent with an electronic-excited state 
broadening.  
b) Polaron State The polaron state was predicted by convolving the ground state plus core 
hole XUV absorption with a three-fold energy splitting of the 3p core level using the values and 
weighting predicted for Fe3+ in FePO4 in Reference 30 of the main paper. The experimental ground 
state absorption was convolved with a three-fold delta function at energies of 0 eV, 1 eV, and 2.5 
eV with weighting of 1/3, 1/2, and 1/6, respectively, which best-fit the data. The differential 
absorption was calculated by taking the difference between the polaron state absorption and the 
experimental ground state absorption. 
 
Motivation for Rate Equation Model 
Generally, hot electron-phonon equilibration is described in a two-temperature model, where 
the average occupation statistics of each system are described using their excited state temperature. 
The temperature is then related to the total energy of the system through the heat capacity, allowing 
a thermodynamic balance model to be used to describe the thermalization rates. In a two-
temperature model, optical excitation first creates a hot electron population. Next, electron-phonon 
scattering thermalizes the hot electrons by creating a non-thermal optical phonon population. The 
transfer rate between the two systems is then determined by the electron-phonon scattering time, 
which must be fit to the data, as well as the relative temperature of each system.   
Here, in addition to electron-phonon scattering, an optical phonon and electron can recombine 
to localize the electron in a small polaron, requiring an additional scattering time to be fit. The 
exact relationship between the temperature of hot electrons and the amplitude of the charge transfer 
hybridized state is unknown. Additionally, the optical phonon temperature cannot be directly 
measured. It is, however, expected that the charge transfer hybridized state amplitude will decrease 
as the hot electron distribution cools since optical phonon scattering changes the local electronic 
environment. To simultaneously describe hot carrier thermalization and polaron formation, the 
two-temperature model can instead be converted into a rate equation with average populations for 
the hot electrons and optical phonons.  
Using the change in the average population instead of a temperature allows a bimolecular 
recombination term between an electron and optical-phonon to be included to describe small 
polaron formation. The feedback between hot electron and hot phonon populations, the cooling 
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rate of hot electrons by scattering with optical phonons, and the possibility of a hot electron 
combining with an excited phonon to create a polaron can then be described simultaneously. 
Inclusion of electron and optical phonon scattering with acoustic phonons in the rate equation 
model does not alter the dynamics because of the low temperatures (<350 K) and short time scales 
(<3 ps) modeled. Only the polar optical phonons take part in polaron formation, so acoustic 
phonons would not be expected to directly change the polaron formation rate, only indirectly on 
longer time scales by influencing the electron or optical phonon population. In this model, small 
polaron formation is treated statistically, ignoring the spatial nature of the electron and phonon 
that creates the polaron after excitation. 
 
Derivation of Rate Equation Model for Polaron Formation 
The hot electron occupation of a given band k, considering all optical phonon absorption and 
emission processes but ignoring state filling effects, is given by the differential equation: 
𝑛𝑛?̇?𝑘 =
1
𝜏𝜏𝑂𝑂𝑂𝑂
[−(?̅?𝑛 + 1)𝑛𝑛𝑘𝑘 + ?̅?𝑛𝑛𝑛𝑘𝑘−𝑞𝑞 + (?̅?𝑛+ 1)𝑛𝑛𝑘𝑘+𝑞𝑞 − ?̅?𝑛𝑛𝑛𝑘𝑘]   (1) 
where 𝜏𝜏𝑂𝑂𝑂𝑂 is the electron-phonon scattering time, ?̅?𝑛 is the average occupation of the phonon mode 
(Maxwell-Boltzmann distribution), and 𝑛𝑛𝑖𝑖 is the electron occupation at the given momentum. The 
density of states and momentum dependence of 𝜏𝜏𝑂𝑂𝑂𝑂 is neglected. The total phonon population for 
momentum q would then go as 
𝑛𝑛?̇?𝑞 = − ∑ 𝑛𝑛𝑘𝑘,𝑞𝑞̇𝑘𝑘      (2) 
since a phonon is created or destroyed for each emission or absorption process. The transient XUV 
measurement only gives information about the ensemble-averaged occupation over all energy and 
momentum, as the transient amplitudes only show to what extent the charge transfer state and 
polaron state are occupied or not, independent of energy and momentum. To simulate the 
experimental observables, the sum over all the k and k±q connected states can be performed. 
Assuming the excited electron populations are described by a Boltzmann distribution, this allows 
Equation 1 to be rewritten as the difference in population between any two states separated by 
phonon momentum q as 
∑Δ𝑛𝑛𝑘𝑘̇ = ∑
1
𝜏𝜏𝑂𝑂𝑂𝑂
[−(?̅?𝑛 + 1)Δ𝑛𝑛𝑘𝑘 + ?̅?𝑛Δ𝑛𝑛𝑘𝑘] = −∑
Δ𝑛𝑛𝑘𝑘
𝜏𝜏𝑂𝑂𝑂𝑂 
    (3) 
with Δ𝑛𝑛𝑘𝑘 = 𝑛𝑛𝑘𝑘+𝑞𝑞 − 𝑛𝑛𝑘𝑘 
The sum gives an ensemble averaged change between any two states separated by a phonon of 
momentum q with an average scattering time of 𝜏𝜏𝑂𝑂𝑂𝑂. To turn Equation 3 into a thermodynamic 
balance model, the sum can be done over the energy of each state k, giving the ensemble average 
of the change in energy of the system. This leads to the Two Temperature Model as 
𝑑𝑑𝑈𝑈𝑒𝑒
𝑑𝑑𝑑𝑑
= −
𝐶𝐶𝑉𝑉,𝑒𝑒(𝑇𝑇𝑒𝑒)⋅(𝑇𝑇𝑒𝑒−𝑇𝑇𝑂𝑂𝑂𝑂)
𝜏𝜏𝑂𝑂𝑂𝑂
   (4) 
𝑑𝑑𝑈𝑈𝑂𝑂𝑂𝑂
𝑑𝑑𝑑𝑑
=
𝐶𝐶𝑉𝑉,𝑒𝑒(𝑇𝑇𝑒𝑒)⋅(𝑇𝑇𝑒𝑒−𝑇𝑇𝑂𝑂𝑂𝑂)
𝜏𝜏𝑂𝑂𝑂𝑂
   (5) 
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where 𝑈𝑈 = 𝐶𝐶𝑉𝑉(𝑇𝑇) ⋅ 𝑇𝑇 is the energy for each system at constant-volume through the specific heat 
𝐶𝐶𝑉𝑉(𝑇𝑇) and temperature T. In equation (4) and (5) the kinetics are summarized in the time dependent 
change of the average system temperature and dependence on temperature of the specific heat. 
Here, the relationship between the temperature of the electron, phonon, and polaron and the 
measured amplitude is unknown. The heat capacity of the polaron is also unknown. Therefore, the 
sum in equation 3 can instead be taken only over the population, giving the average change in 
population between any two k states separated by phonon momentum q as 
?̇?𝑛𝑒𝑒 = −
ne−𝑛𝑛𝑂𝑂𝑂𝑂
𝜏𝜏𝑂𝑂𝑂𝑂 
    (6) 
?̇?𝑛𝑂𝑂𝑂𝑂 =
ne−𝑛𝑛𝑂𝑂𝑂𝑂
𝜏𝜏𝑂𝑂𝑂𝑂 
    (7) 
This allows the polaron to be added in as a bimolecular change in both the electron and hole 
population. The polaron formation time then describes taking an excited electron at some k and a 
phonon at some q and removing that population from the ensemble excited state.  
?̇?𝑛𝑒𝑒 = −
ne−𝑛𝑛𝑂𝑂𝑂𝑂
𝜏𝜏𝑂𝑂𝑂𝑂 
−
ne⋅𝑛𝑛𝑂𝑂𝑂𝑂
𝜏𝜏𝑂𝑂𝑃𝑃𝑃𝑃 
    (8) 
?̇?𝑛𝑂𝑂𝑂𝑂 =
ne−𝑛𝑛𝑂𝑂𝑂𝑂
𝜏𝜏𝑂𝑂𝑂𝑂 
−
ne⋅𝑛𝑛𝑂𝑂𝑂𝑂
𝜏𝜏𝑂𝑂𝑃𝑃𝑃𝑃 
    (9) 
?̇?𝑛𝑂𝑂𝑃𝑃𝑃𝑃 =
ne⋅𝑛𝑛𝑂𝑂𝑂𝑂
𝜏𝜏𝑂𝑂𝑃𝑃𝑃𝑃 
      (10) 
Equations 8-10 are fit to the experimental data to obtain the electron-phonon scattering time, the 
polaron formation time, and two amplitude fit coefficients to relate the predicted population to the 
change in the experimental differential absorption for the polaron and charge transfer hybridized 
states. The fitting was done using a robust, multi-start fit method to find the global minimum.  
  
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Fig. S1 Differential Absorption Compared to Polaron Models. The differential absorption at 
Time = 3 ps is compared to two polaron models predicted using the splitting of the Fe 3p core level 
in either the ground state or the excited, charge transfer hybridized state. The values of the core-
level splitting and the assumed amplitude of the polaron state are taken to be the same for both 
models. Only the polaron model using the ground state can be related to the observed differential 
absorption. The model using the photoexcited charge transfer hybridized state, consisting of the 
photoexcited electron and hole, cannot describe the differential absorption. The discrepancy occurs 
because the charge transfer hybridized state is already red-shifted from the ground state. When the 
polaron-induced core-level splitting is applied, the spectrum is further redshifted. The change in 
absorbance below 57 eV is then overestimated compared to polaron splitting of the core-levels in 
the ground state.  
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Fig. S2 Possible Recombination Pathways After Optical Excitation. a, The experimental data 
are compared to the possible recombination pathways after optical excitation. The feature that 
characterizes the formation of the polaron state is the blue-shift of the zero-crossing around 57 eV, 
circled for clarity in each plot. b, Predicted differential absorption of the charge transfer state and 
polaron formation. c, Charge transfer state and recombination. d, Charge transfer state and 
renormalization representing carrier-carrier, optical phonon, or acoustic phonon based energy 
shifts. e, Thermal lattice expansions. and f, Charge transfer state followed by transfer to a defect 
state. The key regions of absorption are circled for clarity. Only for the polaron formation in b 
does the simulation reproduce the blue-shift of the zero crossing around 57 eV. The charge transfer 
and polaron state in b are simulated as described in the main text. Recombination in c is 
approximated by a three exponential decay of the charge transfer state amplitude with time 
constants of 100 fs, 1 ps, and 10 ps, matching those commonly inferred from visible light transient 
absorption literature. Renormalization in d is approximated by a 200 fs, 500 meV red-shift of the 
charge transfer, representing the upper limit of the usually sub 100 meV renormalization caused 
by carrier-carrier, optical phonon, or acoustic phonon interactions as discussed in the main text. 
Thermal effects in e are approximated by a constant 500 meV red-shift of the ground state, 
representing lattice expansion. Trap state absorption in f is approximated by a 5 percent Gaussian 
decrease in absorption centered at 53 eV with a width of 500 meV and a 1 ps transfer time from 
the charge transfer state. A trap state absorption or bleach could occur at the pre-edge of either 
multiplet peak, however, in neither case is the experimental data in part (a) replicated as well as 
the polaron model of (b). 
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Fig. S3 Polaron Formation Model. a, Following excitation, the optically excited electrons in the 
charge transfer state thermalize by scattering off of and exciting optical phonons. The optical 
phonon and electron can then combine to create a polaron. b, The rate equation model representing 
the polaron formation process. Here, G(t) represents the excitation source. ni and τi are the 
population and scattering time for i=OP the optical phonons, i=e the electrons, and i=Pol the 
polarons. The rate equation model is described in detail at the beginning of the Supplementary 
Information. The fit is performed using a robust, multi-start algorithm. 
 
  
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Fig. S4 Experimental Differential Absorption Plots for Several Pump Wavelengths. At each 
pump wavelength indicated on the figure, the zero crossing point near 56-57 eV between the 
positive and negative features is seen to blue-shift within ~500 fs. The blue-shift consistently stops 
within ~2-3 ps.  
  
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Fig. S5 Differential Absorption Lineouts for Several Pump Wavelengths. Each time point has 
been averaged over the surrounding 6 time points on a logarithmic scale, isolating the key time 
periods during relaxation. The zero-crossing between the positive and negative features 
representing the formation of the polaron is at 56-57 eV. For each pump wavelength, the zero-
crossing blue-shifts rapidly within ~500 fs and continues to blue-shift until ~2-3 ps.  
  
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Fig. S6 Polaron Formation Model fit to several Pump Wavelengths. The multivariate 
regression amplitudes for the charge transfer hybridized and polaron state (circles and squares, 
respectively) are shown for each pump wavelength indicated on the figure, along with the fit (solid 
lines) using the rate equation model described in the beginning of the Supporting Information and 
Figure S2. The scaled phonon population that results from the rate equation model fit is shown for 
comparison as a dashed grey line. The phonon population is not directly measured in the 
experiment, so here it is scaled the same as the electron population, which would represent one 
electron scattering to create one phonon, which then creates one polaron. The deviations from this 
picture caused by multi-phonon effects are discussed in the text. For each pump wavelength the 
fitted electron-phonon scattering constant is 30-60 fs, with the insensitivity to electron-phonon 
scattering time coming from the 50 fs instrument response function of the pump pulse. The fitted 
polaron formation times are 97±4 fs for 400 nm excitation, 87±5 fs for 480 nm excitation, 88±3 fs 
for 520 nm excitation, and 100±5 fs for 560 nm excitation. 
 
 
 
 
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Fig. S7 Experimental Differential Absorption Plots for Long Time Scales at Several Pump 
Wavelengths. The lifetime of the polaron signal (blue, positive change in absorption) decreases 
with increasing excitation wavelength. At 560 nm excitation, the polaron signal has decayed to 
within noise level by ~250 ps.  
 
  
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Fig. S8 Stretched-Exponential Fit to Long Time Decay. Squares represent the experimental 
data, calculated by summing over the polaron peak in Figure S7. Solid lines represent the best fit 
using the stretched exponential function described in the main text. The excited-state lifetime and 
the hopping radius increase with increasing excitation energy. The best-fit parameters are 
tabulated in Table S1. The curves have been offset for clarity.  
  
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Table S1 Best fit coefficients for the stretched exponential function, 𝐴𝐴 ⋅ exp (− (𝑡𝑡
𝜏𝜏
)
𝛽𝛽
) . Error 
bars represent the standard error of the fit. 
Wavelength (nm) Amplitude, A Lifetime, 𝝉𝝉 (ps) Stretching Factor, 𝜷𝜷 
400 nm 0.98±0.04 370±50 0.45±0.06 
480 nm 0.94±0.03 345±35 0.78±0.1 
520 nm 0.96±0.03 344±30 0.76±0.1 
560 nm 1.01±0.04 189±15 1±0.15 
 


Excitation-wavelength-dependent small polaron trapping of photoexcited carriers in α-Fe_2O_3

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Excitation-wavelength-dependent small polaron trapping of photoexcited carriers in α-Fe_2O_3

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