Extended carrier lifetimes and diffusion in hybrid perovskites revealed by Hall effect and photoconductivity measurements

Impressive performance of hybrid perovskite solar cells reported in recent years still awaits a comprehensive understanding of its microscopic origins. In this work, the intrinsic Hall mobility and photocarrier recombination coefficient are directly measured in these materials in steady-state transport studies. The results show that electron-hole recombination and carrier trapping rates in hybrid perovskites are very low. The bimolecular recombination coefficient (10−11 to 10−10 cm3 s−1) is found to be on par with that in the best direct-band inorganic semiconductors, even though the intrinsic Hall mobility in hybrid perovskites is considerably lower (up to 60 cm2 V−1 s−1). Measured here, steady-state carrier lifetimes (of up to 3 ms) and diffusion lengths (as long as 650 μm) are significantly longer than those in high-purity crystalline inorganic semiconductors. We suggest that these experimental findings are consistent with the polaronic nature of charge carriers, resulting from an interaction of charges with methylammonium dipoles.

emf, as well as Schottky contact resistance, are also frequently overlooked by researchers in Hall effect measurements (for more discussion, see Ref. [1]).
Here, in order to address the problem of signal-to-noise ratio, we have employed an improved ac Hall measurement technique [ 4 ], in which a low frequency (f = 0.5 -5 Hz) ac magnetic field B (rms magnitude 0.23 T) is applied perpendicular to the sample's surface, while a dc current, I, is passed through the sample. An ac Hall voltage, V Hall , is detected across the channel at the same frequency, f, by a phase sensitive lock-in amplifier, which allows to drastically improve the signal-to-noise ratio. Using an ac Hall technique is found to be very important for reliable Hall effect measurements in hybrid perovskites. While dc Hall measurements do seem to reveal a Hall signal ( Fig. 3(d) of the main text), the poor signal-tonoise ratio makes it difficult to obtain quantitative and thus conclusive results. However, even with the much better signal-to-noise ratio in ac Hall measurements, one still needs to be extremely careful with certain spurious effects that often lead to data misinterpretation.
One of such artifacts is caused by the , making the task of discriminating these signal easier. However, in non-ideal systems, the phase angle between these signals can somewhat differ from /2, and the signals themselves can have both of their X and Y components non-zero, V Hall = (V Hall x , V Hall y ) and This situation is schematically depicted in Supplementary Figure 1. Thus, an additional effort is necessary to extract the true Hall voltage. In principle, the parasitic Faraday induction signal can be much reduced by arranging the sample's wiring in such a way as to minimize the area of the loop and by performing these measurements at a very low frequency (e.g., < 1 Hz two-step process: first, an ac voltage between the Hall leads is measured at I = 0, and then the measurement is repeated at I > 0, followed by the subtraction procedure and other data analysis Our starting point is the rate equation for carrier density n (Eq. 1 in the main text): .
For the definition of all the variables and parameters, please see the main text. In the regime dominated by bimolecular recombination, this equation is replaced with: ( Both the carrier density, n, and the photoexcitation density, G, in this equation are the threedimensional variables in the units of cm -3 and cm -3 s -1 , respectively. The Hall carrier density, n Hall , and the incident photon flux, F, experimentally determined in this work are on the contrary the two-dimensional parameters (a projected carrier density and a photon flux incident at the surface) in the units of cm -2 and cm -2 s -1 , respectively. Thus, these experimental parameters must be first converted to the three-dimensional variables in order for us to be able to use the above rate equations, Supplementary Equations (1, 2), for the analysis of the data shown in Fig. 4 (main text).
For our CH 3 NH 3 PbI 3 thin film samples, since the film thickness (100 -200 nm) is comparable to the light penetration length, α -1 ~ 80 nm (α is the optical absorption coefficient, not to be confused with the power exponent α in photoconductivity measurements), it is reasonable to use the light penetration length to define the photoexcitation density as: Similarly, one can define an effective three-dimensional carrier density as: n = n Hall /α -1 . Using thus obtained G and n, we extract the e-h recombination coefficient  by fitting the experimental data in Fig. 4 (b, c) of the main text with Supplementary Equation (2). In the regime where a bimolecular recombination dominates (that is, when a square-root power dependence, For the above analysis of the photocurrent excitation power dependence to be valid, the photoexcitation penetration length α -1 must be comparable to the thickness of the sample effectively occupied by photocarriers. This is because in defining the 3D variables, n and G, via the 2D n Hall and F, we have assumed that the volume occupied by photocarriers is comparable to the volume where photon absorption takes place. This assumption is justified in the case of thinfilm samples, such as above and in Fig. 4 (b, c) of the main text, whose thickness is ~ α -1 , and the carrier diffusion length l >> α -1 . In such a case, one can consider that the photoexcitation and charge carriers occupy the same volume of the sample and are both distributed approximately uniformly through the thickness of the sample, thus allowing to define the effective threedimensional carrier density as: n = n Hall /α -1 .
However, such an analysis becomes invalid in bulk CH 3  Equation (2) can then be rewritten in terms of the experimental (two-dimensional) parameters, n Hall and F, as: By fitting our experimental n Hall (F) dependence ( Fig. 4 (d) of the main text) with the above relationship, we experimentally find the slope   (d/) 1/2 . We then assume as usual that the photocarrier generation efficiency per photon is 100 % ( = 1), which leads to the estimate of the The much greater  and l extracted above in CH 3 NH 3 PbBr 3 single crystals appear to be mainly the result of a much thicker layer of the sample populated with photocarriers upon photoexcitation (afforded simply by the macroscopic thickness of these crystals), which leads to a significantly lower 3D carrier density than in thin films. Indeed, the lowest photoexcitation intensity in the single-crystal data in Fig. 4 (d) of the main text corresponds to n Hall = 310 11 cm -2 and n ~ n Hall /l = 4.610 12 cm -3 , which is almost two and a half orders of magnitude lower than the carrier density in our thin-film devices (~ 10 15 cm -3 ) at a similar photoexcitation intensity. In thin-film samples, all the photogenerated carriers are confined to the small physical thickness of the film, resulting in a high carrier density. In such a case, since the e-h recombination lifetime is inversely proportional to the carrier density  = (n) -1 , electrons and holes have much higher probability to recombine. Again, we emphasize that in the bimolecular recombination regime, both  and l depend on the carrier density n, and thus these parameters must always be mentioned with a reference to a particular carrier density.

Supplementary Note 3. van Roosbroeck-Shockley theory of radiative recombination
and polaronic effects.
where and are the equilibrium (dark) recombination rate and concentration of electrons (equal to that of holes), respectively. Furthermore, is related thermodynamically to the optical properties of the system as [ 7 ]: Here c is the speed of light in vacuum, and the optical properties are described by the frequency ω-dependent refraction, , and extinction, , coefficients. An advantage of Supplementary Equation (5) is that it relates the rate to directly measurable experimental quantities and can be evaluated as such. For these calculations, we use here the accurate parameterized description [ 8 ] of optical parameters of CH 3 NH 3 PbI 3 perovskites derived from their ellipsometric measurements, which are displayed in Supplementary Figure 2 for reference.
In this parameterization, the onset of optical absorption ( ) takes place at eV.
On the other hand, there is no direct measurement of the equilibrium (dark) concentration .
We will explore below the influence of the interaction of conventional band carriers with the MA dipole reorganization ("dipolar phonons") on this quantity.

The electron-phonon coupling constant.
In the theory of polarons [ 9,10,11 ], it is common to define the dimensionless coupling constant between a band charge carrier (effective mass m) and the polar medium as: Here, and   are the static and high-frequency (optical) dielectric constants, respectively, k e the Coulomb's law constant, and ω 0 the frequency of the longitudinal dipolar vibrational (phonon) mode. To make numerical estimates, from a range of published dielectric constants, ≈ 18-36,   ~ 5-6 [ 8,12 ], we will specify here and (see Supplementary Figure 2 (b)), along with the effective band mass , where is the free electron mass [ 13 ].

The polaron energy spectrum.
As a result of the dressing by the phonon cloud [ 10 ], the standard energy-momentum relation for the band carriers, ⁄ and -⁄ At larger momenta, however, the modification of the polaronic spectrum is more significant than would be suggested just by mass renormalization (8) Non-parabolic dispersion curves in Supplementary Figure 3, of course, also exhibit polaron group velocities lower than would be suggested just by the renormalized mass (10). In fact, in Larsen theory, the group velocities vanish upon the approach of the momenta to their critical values (see discussions of this point in Refs. [10,17]). Lower group velocities would correspondingly result in lower carrier mobilities.

The application of vRS theory to the polaronic bands in the parabolic approximation.
It is also instructive to evaluate the radiative recombination rate within the confines of the standard theory of electron-hole recombination in semiconductors (see, e.g., where M fi is the transition (dipole) matrix element: and are the energies of the final and initial polaron, and of the photon states, respectively. The unit vector is n k = k/k. The parabolic dispersions for the electron and hole polarons read: The integral in Supplementary Equation (11) is taken over all possible final states of the irradiated photon and the possible states of the hole. The total equilibrium unit-volume recombination rate, R eq , reads: where the factor ( ) [( ( ))] corresponds to the averaging with thermal distribution functions over all possible initial states of the electron and hole polarons.
The calculations are particularly easy to perform for the equilibrium (dark) case. It is known (see also Supplementary Equation (4)) that the non-equilibrium (that is, under photoexcitation) recombination rate, R non-eq , is related to the equilibrium rate, R eq , as follows: , where n and p are the non-equilibrium (photoexcited) concentrations of electrons and holes (for which we assume, n  p), and n eq or p eq (n eq = p eq ) are the equilibrium (dark) concentrations of electrons and holes. The average effective recombination time is then defined as:

The computation of recombination rate R in hybrid perovskites.
The integral over the final states d 3 p f in Supplementary Equation (11) while the characteristic photon wave-vector is: Thus, for our calculation k ph k e , k h . Therefore, when dealing with energy δ-function, we may set . Rewriting the δ-function as ( ), we integrate over p i 2 dp i to obtain the so-called joint density of states factor √  . The matrix element |M fi | 2 is to be averaged over the direction of the optical emission and summed over possible photon polarizations: Taking into account the position of the Fermi level ( Supplementary Figure 4), we set . As a result, in the limit E g ≫ k B T, one gets the following expression for the recombination rate: The integral (20) can be taken analytically, given the simplification provided by the condition E g ≫ k B T. The dipole matrix element is expressed via the so-called Kane energy, E P : Kane energy E P in hybrid perovskites has been estimated in numerical simulations of the optical absorption of these materials [ 18 , 19 ]: Combining Eqs. (15), (16) and the identity for the equilibrium carrier concentration n eq : one arrives at the final expression for the carrier recombination time  as a function of photoexcited carrier density, p, and polaron effective mass, m p : 3.6. Estimates of the recombination time  and the bimolecular recombination coefficient .
As discussed at the setup, here our goal is not to calculate the polaron mass but to estimate it from the experimental data for mobility. Such an estimate for the polaron effective mass in perovskites can be obtained using the following qualitative arguments. The carrier mobility of the material can be expressed as follows: where l is the carrier momentum relaxation length, and v the thermal velocity. Within the polaron picture we discuss, it is clear that l should be at least of the order of one lattice constant: l > a. Otherwise, the transport would be of the hopping type that is appropriate for a very different type of so-called small-radius polarons. [ 10 ] With this limitation on l, , where for the carrier mobility we took the value in the middle of the range experimentally revealed in our Hall measurements, µ ~ 30 cm 2 V -1 s -1 (see main text). Hence, following Supplementary Equation (24) above, we extract the following e-h recombination time in our samples (at the lowest photoexcitation intensity used in our experiment):  > 1 ms.
In this computation, we used p = 4.6×10 12 cm -3 for the carrier concentration from the experiment (see main text), for the band mass m = 0.15m 0 , and for the band gap E g = 1.6 eV.
The bimolecular recombination coefficient  can also be estimated from Supplementary Equation (24):   (p) -1 < 210 -10 cm 3 s -1 (28) The resulting magnitudes (28) of the bimolecular recombination coefficient  (and recombination time  (27) for a specific carrier density) thus appear in a reasonably good agreement with the experiment.