Hot-carrier cooling and photoinduced refractive index changes in organic–inorganic lead halide perovskites

Metal-halide perovskites are at the frontier of optoelectronic research due to solution processability and excellent semiconductor properties. Here we use transient absorption spectroscopy to study hot-carrier distributions in CH3NH3PbI3 and quantify key semiconductor parameters. Above bandgap, non-resonant excitation creates quasi-thermalized carrier distributions within 100 fs. During carrier cooling, a sub-bandgap transient absorption signal arises at ∼1.6 eV, which is explained by the interplay of bandgap renormalization and hot-carrier distributions. At higher excitation densities, a ‘phonon bottleneck' substantially slows carrier cooling. This effect indicates a low contribution from inelastic carrier-impurity or phonon–impurity scattering in these polycrystalline materials, which supports high charge-carrier mobilities. Photoinduced reflectivity changes distort the shape of transient absorption spectra and must be included to extract physical constants. Using a simple band-filling model that accounts for these changes, we determine a small effective mass of mr=0.14 mo, which agrees with band structure calculations and high photovoltaic performance.


Models of carrier induced change in absorption
Supplementary Figure 13a shows the steady state absorption of CH 3 NH 3 PbI 3 , taken by a combination of reflection corrected UV-Vis and photothermal deflection spectroscopy. By carefully accounting for reflection and minimising light scatter we attain a spectrum better suited for extraction of optical parameters.
We fit the spectrum using Elliot's theory of Wannier excitons in bulk inorganic semiconductors 2 , 3 . We fit our broadening parameters by closely analysing the tail of the absorption for three orders of magnitude (Supplementary Figure 13b). By obtaining such a clean absorption spectrum with minimal scatter and reflection we are able to avoid an overestimation of our broadening parameters (and hence overestimation of exciton binding energy). A pseudo-Voigt profile was used initially to account for both homogeneous and inhomogeneous broadening. However, as can be seen in Supplementary Figure 10b, the Lorentzian portion of this profile was found to be small (Lorentzian weighting factor of <0.0001) and so the absorption near the band-edge must be dominated by inhomogeneous broadening and was convoluted with a simple Gaussian. We find a best fit for a small exciton binding energy, B = 17 meV, with a Gaussian broadening parameter Γ = 45 meV, and band-gap of 1.645 eV.
We can extend this theory in a basic fashion to help explain carrier-induced changes in absorption coefficient, which are modelled in SupplementaryFigure 14.
Our models are based primarily on simple theories of state-filling. Our hot-carrier analysis has shown good agreement with a simple state-filling model for states near the re-normalized bandedge. Others 1,4 have also shown that, for energies away from the band-edge and for large carrier densities, the transient absorption and photoluminescence spectra are consistent with a simple band-filling model. We thus assume a carrier dependent absorption coefficient given by 5 , Supplementary Figure 14 shows a modelled change in absorption spectra based on band-filling and an initial absorption coefficient, α 0 (E), calculated from Elliot's theory, as described above.
The ∆α curves of Supplementary Figure 14 are examples of representative spectra for a chosen set of parameters. The red, blue and green curves illustrate how the ∆α spectrum can be composed of contributions from both discrete and continuum states, where their bleach is due to a reduction in oscillator strength that is modelled here as a simple occupation effect but may also be due to screening of Coulombic enhancement 8though we note that our attempts to model this screening by including even the smallest reduction in exciton binding energy result in simulated spectra that do not match experiment. The black curve is a model of ∆α with the same reduced effective mass as the red curve, but for a model that does not include any Coulomb enhancement term or discrete excitonic states. We see that (for energies above ~1.7 eV), ∆α converges for the two theories, as Elliot's absorption theory converges towards the simple ( ) = 1 √ − g parabolic band-to-band absorption coefficient.

Photo-induced changes in refractive index
We account for the change in refractive index in our model by employing the Kramers-Kronig relations to obtain ∆n from the principal integral of ∆α 9 , Taking the total derivative, (1 + ) 2 ( g + ) 2 (1 + ⅇ −2 (1 − ) 2 ( g − ) 2 (1 + ) 2 ( g + ) 2 − A global fit of this expression for continuous states including Coulombic enhancement is shown in Supplementary Figure 16. The reduced effective mass from this analysis, m r = 0.14 m o agrees with the value obtained from fits of the simpler model (described in Figure 4) that does not include the possible contributions from Coulombic or interference effects.
To model the angular dependence, we must take the general case.
where 0, 1 , 2 are the angular and polarization dependent Fresnel equations' electric field reflection coefficients (from the first three reflections) , 1 , 3 the transmission coefficients (from the first and third transmitted beams), is the angle of the probe inside the absorbing perovskite film, and is the phase shift difference between each interfering beam given by, The transmitted intensity is similarly given by, where 0 , 2 are the angles of the beams before and after they have entered the perovskite film.
These angles are all calculated from Snell's law. We then obtain expressions for the curves shown in Supplementary Figure 7 by taking the partial derivatives with respect to α and n, and dividing by R+T to obtain where e is the charge of an electron, 0 the vacuum permittivity, and γ is the scattering rate from carrier-carrier and carrier-phonon interactions 10 . An estimate of this scattering rate can be obtained by comparing our hot carrier cooling data to a basic expression for the carrier loss rate, where LO is the average energy loss rate, ℏ LO the LO phonon energy, L is the lattice temperature and c the effective carrier temperature. For an LO phonon energy of 13 meVthe same as lead iodide, the data fits the expression for an average energy loss constant of LO = 20 fs, which we use as an estimate of γ, the scattering rate. This is in broad agreement with a theoretical calculation 11 of LO = 8 fs, based on values of static and optical dielectric constants from 12 . For this value of γ, the free carrier absorption in the visible region is negligible. Even if the scattering rate were two orders of magnitude greater, the free carrier term still has little effect on the TA spectra.
Many treatments have given expressions for the expected band gap shift due to the presence and interaction of free carriers 13,14,15,16 . This band-gap renormalization is always calculated to show a weak dependence on carrier density, i.e. ΔE scales with N q where q < ½ . Supplementary   Figure 9d shows the effect of varying the band-gap shift at a given carrier density for a carrier distribution at 300 K. For carriers at 300 K the spectral shape from our model is not significantly changed for small to moderate values (ΔE <15 meV at N = 6.4x10 18 cm -3 ). We find good fits to experimental data with a model of the density dependence of the band-gap renormalization shift with a phenomenological form based on III-V semiconductors 17 , ∆ BGR = ( − ) 1/3 , with N c = 1x10 16 cm -3 and B = 1.2x10 4 eV m 3/2 .