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

Abstract

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.

ARTICLE

Received 1 Jun 2015 | Accepted 14 Jun 2016 | Published 1 Aug 2016

Y. Chen1,w, H.T. Yi1, X. Wu2, R. Haroldson3, Y.N. Gartstein3, Y.I. Rodionov4, K.S. Tikhonov5, A. Zakhidov3,4,X.-Y. Zhu2 & V. Podzorov1,6

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 coefcient 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 coefcient (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 mm) are signicantly longer than those in high-purity crystalline inorganic semiconductors. We suggest that these experimental ndings are consistent with the polaronic nature of charge carriers, resulting from an interaction of charges with methylammonium dipoles.

1 Department of Physics, Rutgers University, Piscataway, New Jersey 08854, USA. 2 Department of Chemistry, Columbia University, New York, New York 10027, USA. 3 Department of Physics and NanoTech Institute, University of Texas at Dallas, Richardson, Texas 75080, USA. 4 The Institute for Theoretical and Applied Electrodynamics, The National University of Science and Technology, MISIS, Moscow 119049, Russia. 5 Landau Institute for Theoretical Physics, Moscow 119334, Russia. 6 Institute for Adv. Mater. and Devices for Nanotech., Rutgers University, Piscataway, New Jersey 08854, USA. w Present address:

Department of Physics, South University of Science and Technology of China, Shenzhen, Guangdong, China. Correspondence and requests for materials should be addressed to V.P. (email: mailto:[email protected]

Web End [email protected] ).

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DOI: 10.1038/ncomms12253 OPEN

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

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12253

Hybrid (organicinorganic) perovskite solar cells represent the recent breakthrough in photovoltaic applications with reported power conversion efciencies reaching

20% (refs 13). In addition to the abundance of the applied studies on this topic, there is a great interest in understanding the fundamental transport and photophysical properties of these materials, the picture of which is not rmly established yet, thereby calling for more reliable experimental studies. For instance, a possibility of high charge-carrier mobility has been considered as one important factor that contributes to the excellent photovoltaic performance of hybrid lead-halide perovskites. Yet, an unambiguous determination of the intrinsic mobility in these materials is missing. Existing experimental values in similar materials range from 0.6 to 50 cm2 V 1 s 1 (refs 46). However, most of the important transport and photophysical parameters, including the carrier mobilities, lifetimes and recombination rates, were so far determined either in materials different from those relevant for high-performance solar cells (for instance, in metallic tin-halide instead of insulating lead-halide perovskites, or perovskites that adopt 2D layered rather than 3D cubic structure) or obtained indirectly, under the conditions less relevant to applications (for instance, in ultrafast spectroscopic experiments, rather than steady-state transport measurements).

Here, we report artefact-corrected Hall effect and steady-state photoconductivity measurements carried out in a range of thin lms and single crystals of exemplary hybrid perovskites, CH3NH3PbI3 and CH3NH3PbBr3, of current interest for photovoltaic applications. Hall effect allows us to directly and independently address the density of photogenerated carriers, nHall, and the intrinsic carrier mobility, mHall, without assumptions typical for other methods, such as in ultrafast spectroscopic techniques or space-charge-limited current measurements. We nd that, in a wide range of illumination intensities, the dynamics of photocarriers is governed by bimolecular electron-hole (e-h) recombination with a very small recombination coefcient g in the range of 10 11 to 10 10 cm3 s 1, which is comparable to the values observed in the best single-crystalline direct-band inorganic semiconductors, such as GaAs, even though the measured intrinsic Hall mobilities are moderate (mHall of up to 60 cm2 V 1 s 1 in perovskite single crystals) and smaller than m in typical inorganic semiconductors by 13 orders of magnitude. In addition, the carrier lifetime, t, and diffusion length, l, directly measured in our steady-state transport experiments are found to be remarkably long (t is up to 30 ms and l is up to 23 mm in polycrystalline lms, and up to 3 ms and 650 mm in single crystals, respectively). Our experiment thus provides a direct steady-state measurement quantitatively revealing a low-rate photocarrier recombination and negligible trapping, as well as extremely long carrier lifetimes and diffusion lengths in hybrid perovskites. While in agreement with some of the recent theoretical predictions710, these results accentuate important questions as of the physical origins of the found intrinsic carrier mobility, e-h recombination and trapping rates in these materials synthesized via inexpensive vapour- or solution-based routes at temperatures close to room temperature. We propose rationalization of some of our ndings based on the picture of re-organization of the methylammonium dipoles around the charge carriers. This interaction leads to carrier relaxation into polarons, whose properties differ from the bare band carriers1113.

Resultsa.c. Hall effect measurements of charge-carrier mobility. It should be emphasized that unambiguous determination of the

intrinsic (that is, not dominated by traps) charge-carrier mobility requires precise Hall effect measurements, which are quite challenging in highly resistive materials with relatively low m, such as organic semiconductors or the hybrid perovskites studied here (see, for example, ref. 14). The major challenges are associated with a very high resistivity of pure stoichiometric perovskites (greater than GO), related to the negligible (in the dark) density of charge carriers, and a poor signal-to-noise ratio in conventional d.c. Hall measurements of low-m materials.

To overcome these difculties, here, we have developed a specialized highly sensitive a.c. Hall measurement technique, corrected for the Faraday-induction artefacts, in which a low-frequency a.c. magnetic eld, B, is applied perpendicular to the samples surface, while a d.c. current, I, is passed through the sample, and an a.c. Hall voltage, VHall, is detected across the

channel by a phase-sensitive lock-in technique, which allows to markedly increase the signal-to-noise ratio15. A parasitic Faraday-induction electro-motive force, occurring in a.c. Hall measurements at the same frequency as VHall, is usually

comparable to the actual Hall voltage signal and can easily compromise these measurements. Therefore, Faraday-induction-corrected a.c. Hall measurements, as implemented here in perovskites, are absolutely necessary to obtain reliable data (see the Methods section, Supplementary Fig. 1 and Supplementary Note 1)15. Pure CH3NH3PbI3 samples are highly resistive in the dark (typical RZ100 GO), and thus we utilize a steady-state monochromatic photo-excitation (l 465 nm) to generate a

population of carriers and be able to measure Hall effect. In CH3NH3PbBr3 single crystals, we were able to measure a Hall effect in the dark as well, because these crystals are weakly conducting in the dark (at a level of 50 MO). We perform all our measurements in a 4-probe/Hall bar geometry to account for contact-resistance effects and ensure that channel conductivity, s, as well as the Hall mobility and carrier density, mHall and nHall, are

determined correctly.

Steady-state photoconductivity vs. light intensity. Three types of hybrid perovskite samples were used in our study (Fig. 1). (a) Semi amorphous, solution-grown thin lms, (b) polycrystalline, vapour-grown thin lms and (c) highly ordered, solution-grown single crystals. The microscopy and X-ray diffraction clearly show that the vapour-processed lms have a much better crystallinity than the solution-processed ones, and our single crystals have excellent quality (see Fig. 1 and see the Methods section for details).

We have found that all our high-quality (stoichiometric) samples exhibit a very low dark conductivity. Nevertheless, a signicant photoconductivity is observed in all of them. Figure 2 shows typical dependences of a steady-state photoconductivity, sPC, on photo-excitation density, G, which always follows a power law, sPCpGa, with the exponent a 1 or (linear or square-root

regimes). Photo-excitation density, G, is dened as the incident photon ux F (in cm 2 s 1) divided by the absorption length of the material.

The observed sPC(G) dependence can be understood in terms of charge-carrier monomolecular decay (trapping in the linear regime) or bimolecular decay (e-h recombination in the square-root regime). The photocarrier density, n (n neEnh), in

steady-state measurements is found from the following rate equation, with dn/dt 0 at dynamic equilibrium (see, for

example,16):

dndt kG

n ttr

gn2 0: 1

Here, kG is the rate of carrier generation via photon absorption resulting in production of free electrons and holes with

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12253 ARTICLE

a b c d

1 m 1 m 1 cm

e

Pbl2

Semi-amorphous CH3NH3Pbl3 film Polycrystalline CH3NH3Pbl3 film

CH3NH3PbBr3 single crystal

110 220

100

Intensity (a.u.)

200

110 210

10 20 30 40

2[afii9835] ()

Figure 1 | Morphological and structural characterization of hybrid perovskites used in this study. (a,b) Helium-ion microscope images of CH3NH3PbI3 thin lms grown by solution and vapour methods, respectively (see the Methods section). Vapour-grown lms clearly show much higher crystallinity.

(c) Optical photograph of solution-grown CH3NH3PbBr3 single crystals. (d) A reconstructed precession image of the hk0 level from single-crystal X-ray diffraction of these single crystals, showing perfect cubic crystalline structure with low density of defects. (e) Typical powder X-ray diffraction curves of all three types of samples shown in ac, semi amorphous lm (black), polycrystalline lm (green) and single crystals (red). The arrow indicates the peak corresponding to unreacted PbI2 precursor, and vertical dashed lines show the (110) and (220) peaks of the fully stoichiometric CH3NH3PbI3.

104

[afii9825] = 1/2

105

106

[afii9825] = 1 [afii9825] = 1/2

107

[afii9846] PC(S)

108

109

1010

CH3NH3PbBr3 single crystal Vapour-grown CH3NH3PbI3 thin film Solution-grown CH3NH3PbI3 thin film

Photoexcitation density, G (cm3 s1)

[afii9825] = 1

1011

1012

1016 1017 1018 1019 1020 1021 1022 1023

Figure 2 | Steady-state photoconductivity measured as a function of photo-excitation density in thin lms and single crystals of hybrid perovskites. Measurements are carried out by 4-probe technique under a cw photo-excitation with a blue light (l 465 nm) in CH3NH3PbBr3 single

crystals (red squares), vapour-grown polycrystalline CH3NH3PbI3 lms (green triangles) and solution-grown semi amorphous CH3NH3PbI3 lms (black circles). Dashed lines are the power law ts, sPCpGa, with the

exponents a 1 or (as indicated). It is clear that bimolecular eh

recombination (a ) dominates the behaviour at high photo-excitation

densities (note the double-log scale).

probability k per photon (the photocarrier-generation efciency). The second and third terms represent the two channels of carrier decay: the trapping and e-h recombination, where ttr is the trap-

limited carrier lifetime (an average time carriers diffuse before

being trapped), and g is the coefcient of e-h recombination. In the carrier density range probed here, we do not see any experimental evidence of the third-order (Auger) processes, which are therefore excluded from equation (1). At low photo-excitation intensities, when the concentration of electrons and holes is small, the dominant process limiting the carrier lifetime is trapping, and gn2 term in equation (1) can be neglected, leading to a linear regime in photoconductivity: sPC emn emkttrG,

where e is the elementary charge. With increasing excitation intensity, the bimolecular recombination eventually becomes dominant, resulting in a transition from the linear to a sublinear regime: sPC em (k/g)1/2 G1/2, obtained from equation (1) by

neglecting the n/ttr term. Figure 2 shows that these two regimes are indeed what is observed in hybrid perovskites. Highly crystalline samples exhibit bimolecular recombination regime (a 1/2) in a wider range of excitation intensities, which is

consistent with the higher sPC and mHall in the crystal samples (see below).

Hall measurements under photo-excitation and in the dark. To decouple the carrier density and mobility in sPC enm and obtain

the microscopic parameters describing the carrier dynamics, ttr

and g, from the rate equation for n (equation (1)) and experimental data, one needs to know the intrinsic charge-carrier mobility m. For this purpose, we have performed Hall effect measurements, as discussed above, using 4-probe/Hall-bar device structures (Fig. 3a) and a variant of an a.c. Hall measurement technique (Fig. 3b) specically adjusted for these materials (see the Methods section, Supplementary Fig. 1 and Supplementary Note 1). A typical measurement result is shown in Fig. 3c: a very clear and quiet a.c. Hall signal as detected in a solution-grown CH3NH3PbI3 thin lm that has a Hall mobility of only mHall

1.5 cm2 V 1 s 1. Reliable Hall measurements with such

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12253

a

b

Lock-in

ac B

I

S

D

1 cm

V

c

d

I = 4 nA

CH3NH3PbBr3 single crystal

0.5

0.0

0.5

100

50

0

ac V Hall(V)

CH3NH3Pbl3 thin film

dc V Hall(mV)

1.0

0.5

0.0

0.5

1.0

B(T)

I = 0

I = 200 nA

0 20 40

0 20 40 t (min)

t (min)

Figure 3 | Hall effect measurements in hybrid lead-halide perovskites. (a) A photo of typical solution-grown CH3NH3PbI3 thin lm on glass with Au contacts in a 4-probe/Hall bar geometry. (b) A diagram of the a.c. Hall effect measurement setup (see description in text and ref. 15). (c) A representative

Faraday-induction-corrected a.c. Hall measurement in a solution-grown CH3NH3PbI3 thin lm shown in a. When a d.c. excitation current of 4 nA is driven through the lm uniformly illuminated with a blue light (l 465 nm) and subjected to an a.c. B eld of r.m.s. magnitude 0.23 Tat 0.5 Hz, an a.c. Hall voltage

of 100 mV is detected above the zero-bias background by using a lock-in technique shown in b. The vertical and horizontal dashed lines show the moment d.c. excitation current is turned on (at t 28 min) and the two levels of the Hall voltage, at I 0 and 4 nA, respectively. Note that even though Hall mobility

of this sample is only mHall 1.5 cm2 V 1 s 1, the signal-to-noise ratio is excellent (the standard deviation in VHall among consecutive six

measurements is only about 0.5%). (d) A representative d.c. Hall measurement in CH3NH3PbBr3 single crystals. VHall is measured with an electrometer at a constant excitation current I 200 nA, while B eld is slowly swept between 0.5 T and 0.5 T. Despite the much higher mobility of this sample

(mHall 113 cm

2 V 1 s 1), the signal-to-noise ratio is much worse, with the large error imposed by the noise.

an excellent signal-to-noise ratio in highly resistive systems with carrier mobilities as low as 1 cm2 V 1 s 1 are unprecedented. For comparison, conventional d.c. Hall measurements performed in a perovskite single crystal with a much higher mobility, mHall 11 cm2 V 1 s 1, shown in Fig. 3d, evi

dently exhibit a much noisier signal. It is clear that Faraday-induction-corrected a.c. Hall measurements are by far superior to the conventional d.c. technique in terms of the signal-to-noise ratio, even though it uses a smaller magnetic eld (r.m.s. B 0.23 T).

As expected for pure undoped band insulators, our CH3NH3PbI3 samples are highly resistive in the dark (with a typical sample resistance 4100 GO). Thus, we used a cw photo-excitation to generate photocarriers and perform steady-state photo a.c. Hall measurements. In a system with photogenerated electrons and holes (neEnh n) and a negligible

concentration of dark carriers, Hall voltage is given by:

VHall

W

L B VL mh me; 2 where W and L are the channels width and length, respectively,

VL is the longitudinal voltage drop along the channel (corrected for contact effects by using the 4-probe technique), and mh, me are

the mobilities of holes and electrons, respectively. Equation (2) shows that photo Hall effect measurements yield the difference between the electron and hole mobilities, rather than their absolute values. In the context of the perovskites under study,

recent calculations showed that me and mh in these materials should differ from each other by an amount comparable to the mobilities themselves7,1720. Indeed, theoretical calculations have consistently indicated that while effective masses of electrons, me,

and holes, mh, have the same order of magnitude, there is also a noticeable difference between the two, ranging from 20 to 200%, depending on the specic computational method used18. A recent THz spectroscopy study also suggests a difference of a factor of 2 between the electron and hole mobilities21.

Therefore, even though precise values of me and mh cannot be obtained from photo Hall measurements, the difference,

Dm mhmeBm, extracted from equation (2) would yield a

faithful representation. In turn, the carrier density obtained from the photo Hall effect measurements is: n ne nh sPC/(e m),

which is also a good approximation for the actual density of photogenerated charges. The data presented below are analysed using this association. The dark Hall effect measurements possible in weakly conducting single-crystal CH3NH3PbBr3 samples give further credence to the approach.

Figure 4 presents Hall effect data for a variety of hybrid perovskite samples. Fig. 4a shows that Hall mobility, mHall

80.4 cm2 V 1 s 1, measured in a polycrystalline CH3NH3PbI3 lm remains almost constant over the range of nearly three orders of magnitude in light intensity (the error is dened by the uctuations in Fig. 4a). The 4-probe photoconductivity, sPC, and the density of photogenerated charge carriers, nHall, determined from the simultaneous longitudinal

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a b

9

7 1019 1020 1021 1022

1019 1020 1021 1022 1014 1015 1016 1017

106

Polycrystalline CH3NH3Pbl3 film

Polycrystalline CH3NH3Pbl3 film

CH3NH3PbBr3 single crystal

2 V1 s1 )

3 )

8

1017

1016

1015

1013

1012

photo [afii9846] 4 probe (S)

photo

107

105

[afii9839] Hall(cm

108

1019

1020 1021 1022

Absorbed photon density, G (cm3 s1) Absorbed photon density, G (cm3 s1)

Absorbed photon density, G (cm3 s1) Incident photon flux, F (cm2 s1)

2 ) n Hall (cm

c d

104

[afii9839]Hall in (cm2 V1 s1):

3 )

1016

1015

0.5, Solution1.6, Solution2.5, Vapour

n Hall(cm

[afii9846] 4 probe(S)

n Hall(cm

3.5, Vapour 8, Vapour 10, Vapour

[afii9839]Hall = 60 cm2 V1 s1

Figure 4 | Steady-state photoconductivity and Hall effect measurements in perovskite lms and crystals. (a) mHall in a vapour-grown 100 nm-thick polycrystalline CH3NH3PbI3 lm (similar to that shown in Fig. 1b), measured at different photo-excitation densities, is nearly a constant (the dashed line indicates an average value ofE8 cm2 V 1 s 1). (b) Photoconductivity and Hall carrier density measured in this lm as a function of absorbed photon density. (c) nHall(G) in six thin-lm samples with very different morphologies and mHall values, showing that bimolecular recombination (nHallpG1/2) governs all these samples. Hall mobility and the method of fabrication are indicated for each lm in the legend. (d) Photoconductivity and Hall carrier density measured in a macroscopic, bulky CH3NH3PbBr3 single crystal (as the ones shown in Fig. 1c) with a (dark) Hall mobility mHallE60 cm2 V 1 s 1 (for holes). Dashed lines in bd are the power law ts with exponent a : n or s constant G1/2.

sPC and Hall effect measurements in these polycrystalline lms are plotted in Fig. 4b. Within the entire measurement range, the carrier density dependence on the illumination intensity exhibits a power law: nHallpG1/2. We emphasize that independent

measurements of mHall as a function of photo-excitation density are essential for obtaining nHall(G) dependence and therefore for determination of the microscopic transport parameters, ttr and g, by using equation (1) to t the data. With the density of photogenerated carriers determined experimentally, we can now interpret their dynamics with the help of equation (1). As pointed out above, under a steady-state photo-excitation, equation (1) gives n (k/g)1/2 G1/2 in the regime governed by a bimolecular

recombination, that is, when the carrier trapping time ttr well

exceeds the time of e-h recombination tr, ttr44tr (gn) 1.

Fitting the experimental Hall carrier density in Fig. 4b with this n(G) relationship yields the upper bound for the bimolecular e-h recombination coefcient g in polycrystalline perovskite lms, gr3 10 11 cm3 s 1 (for the photocarrier-generation

efciency kr100%).

The measurements shown in Fig. 4 can also be used to directly estimate the steady-state charge-carrier lifetime, either ttr or tr,

and diffusion length, l. Indeed, since bimolecular recombination (a ) apparently dominates in the entire range of photon

densities in Fig. 4b, we must have the condition n/ttroogn2 (or,

ttr44tr (gn) 1) fullled for all the incident photon intensities,

including the lowest one, at which nHall 9 1014 cm 3

(Fig. 4b). Therefore, the effective trap-limited carrier lifetime ttr

in the disordered polycrystalline lm in Fig. 4b must be longer than the lifetime limited by bimolecular e-h recombination, ttr4trE30 ms. This estimate shows that even in disordered CH3NH3PbI3 thin lms, charge carriers have an extremely long lifetime, as far as trapping is concerned, and thus the effective density of the corresponding deep traps must be very low, or the

traps must be electronically passivated. The corresponding lower bound of trap-limited carrier diffusion lengths in these polycrystalline perovskite lms is: l (Dttr)1/2B23 mm. Here,

we needed the Hall mobility again to calculate the diffusion coefcient, DBmHallkBT/e, where kB is the Boltzmann constant

and T 300 K. Note that negligible trapping and a very long

diffusion length, greater than the grain size in our polycrystalline lms (Fig. 1b), are consistent with the recent theoretical studies predicting the presence of only shallow traps and benign grain boundaries that do not trap carriers in perovskites9,10. Of course, given the dominant e-h recombination, the actual carrier lifetime in the a regime is a decreasing function of photo-excitation

density G: tr (gn) 1 (see also16).

Figure 4c shows nHall and mHall in six different solution

and vapour-grown polycrystalline thin lms. In general, vapour-grown samples have an appreciably higher mHall,

consistent with their better crystallinity. In sharp contrast, nHall(G) does not seem to correlate much with the preparation

method and mobility. Indeed, while mHall differs among the six samples by as much as a factor of 20, nHall varies only within a factor of two. Correspondingly, g values are also very similar for all these samples, gB(15) 10 11 cm3 s 1. This suggests that

while lm morphology has a clear effect on charge transport, it has little effect on photocarrier generation and recombination. This is consistent with the notion that bimolecular recombination in this system is not governed by carrier diffusion, therefore we do not see a strong correlation between the recombination dynamics and charge-carrier mobility.

Finally, we have performed Hall measurements in CH3NH3PbBr3 single crystals (Fig. 4d). One important difference in this case is that these crystals are weakly conducting in the dark, which allows us to perform Hall effect measurements in the dark and obtain the mobility of holes (mHall 605 cm2 V 1 s 1),

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utilizing the frequency o-dependent refraction, nr(o), and

extinction, k(o), coefcients. An assessment of equation (3) can be made using model considerations (as illustrated in Supplementary Note 3). Even more attractively, one can use the actual experimental optical data to evaluate Req in equation (4), for which here we use the optical parameters (see Supplementary Fig. 2 and Supplementary Note 3) of CH3NH3PbI3 perovskite extracted from accurate ellipsometric measurements in ref. 28. On the other hand, the dark carrier concentration at the thermal equilibrium neq is not measured directly. If one were to use the standard textbook expression for a non-degenerate semiconductor with energy gap Eg separating two parabolic bands27:

n2eq

without having an ambiguity of charge compensation as in photo Hall measurements. Photoconductivity in these crystals is much higher than that in thin lms and also exhibits a well-dened a behaviour (Fig. 4d). A similar analysis of

nHall(G) dependence shows that the e-h recombination coefcient in these crystals is gB8 10 11 cm3 s 1 (for more details on

the extraction procedure see Supplementary Note 2). By dening the effective recombination-limited carrier lifetime again as tr (gn) 1 and using the experimental carrier density from

Hall effect measurements, we can determine the carrier lifetime and diffusion length in these single crystals. At the lowest incident photo-excitation ux in Fig. 4d, corresponding to the measured projected carrier density nHall 3 1011 cm 2 and the effective

bulk carrier density nB4.6 1012 cm 3 (Supplementary

Note 2), we nd: trB2.7 ms, and l (Dtr)1/2B650 mm, which are

remarkably long for a solution-grown semiconductor. We emphasize that these values represent the lower limit for the trap-limited carrier lifetime, ttr, and diffusion length, ltr, since the condition ttr44tr must be satised in the entire regime dominated by a bimolecular recombination, thus indicating again that trapping is strongly suppressed in these materials. We must add that sPC(G) in

analogous lead iodide (CH3NH3PbI3) single crystals (not shown here) are qualitatively similar, except that these crystals are highly insulating in the dark, and thus only photo Hall measurements were possible, yielding an estimate for mBfew cm2 V 1 s 1.

Theoretical estimates of e-h recombination coefcients. E-h recombination in semiconductors is a fundamentally important process, and various approaches have been developed for assessing the corresponding kinetic coefcient g. In one approach, for instance, g is associated with the product sv of the Coulomb capture cross section s and carrier thermal velocity v (ref. 16). In the case of disordered organic and inorganic semiconductors, recombination of charge carriers is often described by the Langevin model22,23, which leads to eme mhe0er for the recombination coefcient, where e0 and er are the dielectric permittivities of vacuum and the material, respectively. Evaluating these expressions for our systems would yield estimates of g on the order of 10 6 cm3 s 1 or higher, that is 45 orders of magnitudes greater than what we nd experimentally. One, of course, realizes that the above models, evidently not applicable to our case, refer to the e-h collision events rather than to the radiative recombination per se. A more appropriate approach that was successfully applied to the actual radiative recombination in inorganic semiconductors is based on the van RoosbroeckShockleys theory rooted in the principle of detailed balance (for reviews, see, for instance, refs 24,25). This theory, in particular, explains well the values of gB10 10 cm3 s 1 exhibited by high-purity direct-band inorganic semiconductors, such as, for instance, GaAs26,27 (see also http://www.ioffe.ru/SVA/NSM/Semicond/index.html

Web End =http://www.ioffe.ru/SVA/NSM/Semicond/index.html ), which, remarkably, are comparable to the values we extract from our observations in hybrid perovskites. In the van Roosbroeck Shockleys approach, the radiative recombination coefcient g is established by the systems properties in the thermal equilibrium (that is, in the dark):

g Req=n2eq; 3

where, Req and neq are the equilibrium (dark) recombination rate and concentration of electrons (equal to that of holes), respectively. Furthermore, Req is related thermodynamically to the optical properties of the system as25:

Req

2 p2c3

memhkBT 3

2p3 6 e Eg=kBT ; 5

equation (3) then yields the recombination coefcient:

g 1:4 10 10 eEg Eopt=kBT cm3s 1: 6

Here, the prefactor was calculated with T 300 K and

meCmh m 0.2m0 (m0 being the free electron mass).

Equation (6) features the absorption onset parameter denoted Eopt, which is equal to 1.553 eV in the parameterization of ref. 28.

The excitonic (e-h attraction) effects11,25 are known to reduce the onset of optical absorption in comparison with the semiconductor band-gap in perovskites (see, for example, the discussion of excitons in ref. 29). One can approximately assess the resulting exponential factor in equation (6) from the corresponding exciton binding energy EX. For m 0.2m0 and

the relative permittivity equal to 5, for instance, EXC54 meV,

indicating that the exponential factor is of the order of 10, which would make result (6) for g larger than our experimental values by about one order of magnitude. The discrepancy here might result from the underestimate of neq by equation (5) for the conventional band carriers, and the larger values of neq would lead to a better agreement with the experiment. We suggest that larger concentrations neq might be actually realized in perovskites due to the interaction of band charge carriers with methylammonium dipoles, as illustrated in Fig. 5.

Accounting for polaronic effects. Theoretical calculations by Frost et al.7 demonstrate that the methylammonium dipoles in hybrid perovskites carry a signicant dipole moment, P 2.29 D,

+

Figure 5 | Dipolar polarons in hybrid perovskite lattice. Schematically, charge carriers can induce rotational re-organization of the surrounding CH3NH3 dipoles (green arrows), leading to the formation of dipolar polarons.

Such polarons could account for the relatively low intrinsic charge-carrier mobilities and reduced bimolecular recombination coefcients experimentally revealed in this work. The structure of hybrid perovskite unit cell, with PbI6 or

PbBr6 octahedra and a polar methylammonium molecular cation at the centre, is shown in the upper left corner. The green arrows in the main panel represent these polar methylammonium cations.

Z

1

0

n2rko3do exp o=kBT

1

; 4

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neq. The data in Fig. 6 show that this effect can be substantial: in this illustration, the volume of phase space available for polarons with energies Ep(k) Ep(0)r:o0 increases approximately by a

factor of 1.543C3.7 relative to the bare band carriers (compare the solid red and dashed black lines). Thereby the denominator in equation (3) could additionally increase by about an order of magnitude.

DiscussionWhile currently there is no analytical framework that would afford quantitatively reliable calculations of the interplay of excitonic and polaronic effects in the relevant parameter range (all energy parameters: kBT, :o0, polaronic and excitonic bindings, are of the same order of magnitude), the above estimates and analysis demonstrate that the polaronic effects might provide an explanation that brings the experimental optical data and our measurements of the radiative recombination coefcient g in a reasonably good agreement with each other.

Indeed, the experimental estimates for g derived above from the Hall effect measurements span the range of (18) 10 11

cm3 s 1. On the other hand, the polaronic effects we discussed evidently lead to a substantial decrease of the value in equation (6) evaluated without such effects and likely reducing this estimate below 10 10 cm3 s 1. Given the fact that photo

Hall measurements could precisely address only the difference of the hole and electron mobilities, and some uncertainty in theoretical estimates, the consistency of our results appears quite satisfactory. In addition, we note that the non-parabolic polaron dispersion displayed in Fig. 6 clearly features carrier group velocities vgr(k) qEp(k)/:qk considerably lower than

those, :k/m, of the bare band carriers. This, of course, leads to lower carrier mobilities as indeed observed in our experiments.

Another important experimental observation is that we have not found the monomolecular decay regime (trapping) down to rather low carrier densities, nE1015 cm 3 in thin lms, and 5 1012 cm 3 in single crystals (Fig. 4). If present,

such a regime would manifest itself as a linear sPC(G)

dependence (a 1). The absence of trapping is exactly the

reason why bimolecular e-h recombination dominates down to such a low carrier density, leading to a remarkably long carrier lifetime and diffusion length at diluted photo-excitation densities, consistent with prior observations33,34. In direct-band inorganic semiconductors, even though similarly small values of g 10 1110 10 cm3 s 1 are observed in high-purity

crystalline samples, achieving a millisecond carrier lifetime and nearly a millimetre-long diffusion length is unheard of. Ordinarily, other recombination mechanisms such as trapping on recombination centres (or even Auger processes) would start to dominate at a higher crossover carrier density, given the rather small eh recombination probability. In fact, in well-optimized high-purity single crystals of GaAs, InP or InAs, t and l are a few ms and a few tens of mm at best26,27 (see also http://www.ioffe.ru/SVA/NSM/Semicond/index.html

Web End =http://www.ioffe.ru/ http://www.ioffe.ru/SVA/NSM/Semicond/index.html

Web End =SVA/NSM/Semicond/index.html ). This accentuates the question of why charge carriers in the (disordered) hybrid perovskites are less affected by trapping. The electronic aspects of the unusual defect physics in CH3NH3PbI3 perovskites have already been discussed10. Here, we are wondering if the interaction of defects with methylammonium dipoles could also contribute to the suppression of trapping. Indeed, typical medium-energy traps in semiconductors have energies dUtr 0.10.3 eV relative to

the band edge. Typical physical size of the traps is of the order of a lattice constant, drtrE5 . The local rearrangement of methylammonium dipoles by the effective forces in the vicinity of a defect could then occur, if the potential barrier for the methylammonium dipole rotation, UrotE10 meV, is smaller

and create a rough potential landscape at the nanoscale, but can be easily rotated or locally aligned by overcoming a small rotational energy barrier, UrotE1 kJ mol 1 (1.6 10 21 J

or 10 meV per dipole). Here, we propose that a charge carrier moving through the perovskite lattice may itself induce a local orientational rearrangement of the surrounding methylammonium dipoles tending to align them along its electric eld (Fig. 5), thus resulting in a type of a dipolar polaron, conceptually similar to polarons known in ionic crystals and polar semiconductors1113. The estimates outlined in Supplementary Note 3 show that such polarons in perovskites should be characterized as intermediate-coupling polarons12, with the dimensionless electronphonon coupling constant ae phC2.5 being a good representative value for the interaction

of the band carriers with the longitudinal dipolar vibrational modes of energy :o0B10 meV. The dressing of a band carrier by a phonon cloud is known to change its properties1113: the standard band carrier dispersion E(k) :2k2/2m would be

modied to the polaronic energy-momentum relation Ep(k).

Two aspects of this modication are important here.

First, polaron formation is energetically favourable, resulting in the polaronic energy shift Ep(0)C ae

ph:o0. The effective band-gap for the equilibrium concentration of electron- and hole-polarons is thus reduced: Eg ! Eg Eep0 Ehp0. With

the estimates above, this reduction largely negates the effect of the exciton binding EX and substantially decreases the exponential factor in equation (6). (One could also say that the thermal dissociation of an exciton into a polaron pair is more efcient than into a pair of band carriers.)

Second, polarons are heavier than the band carriers. While this is qualitatively clear already at the level of the effective mass renormalization12: m-mpCm(1 ae

ph/6), the actual changes are even more signicant, as the polaron dispersion becomes non-parabolic (see Fig. 6 with the accompanying caption, Supplementary Figs 3,4 and Supplementary Note 3)3032. This results in the increased density of the polaronic states and corresponding increase in the equilibrium carrier concentration

2.0

1.5

(E p(k) E p(0)) / h[afii9853] 0

1.0

0.5

0

0

0.5

1.0

1.5

2.0

k / kp

Figure 6 | Polaron energy-momentum relation. This plot compares the shape of the energy-momentum curves for different models calculated with the electronphonon coupling constant ae phC2.5. The energy is

measured in units of the vibrational energy :o0 and the wave number in units of kp (2mo0/:)1/2. The dashed lines show the parabolic dispersion,

for the bare band carrier (black) and for the polaronic carrier with renormalized mass mp (blue). The solid lines display the non-parabolic dispersion obtained with variational calculations, according to LeeLow Pines theory11,30 (green curve) and according to Larsen theory13,31 (red curve). The latter is shown in the limited range of its applicability, but it is this type of the dispersion that was actually conrmed in the state-of-theart diagrammatic Monte-Carlo calculations32.

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12253

than the energy gain associated with a dipole-trap interaction, for instance, if estimated as: P 1e dUtrdrtrB1075 meV (for

P 2.29 D). The re-arranged dipoles would thus reduce the

defects trapping cross section. Such a defect decoration would be reminiscent of the recently observed trap healing effect at two-dimensional semiconductor/polymer interfaces35, with the important distinction that in hybrid perovskites the functional (rotationally responsive) dipoles are naturally available throughout the entire bulk of the sample. The question arises if the dipole rearrangement effect could be greater than that afforded in the standard continuous-dielectric electrostatics. Detailed microscopic computational studies are needed to clarify this issue.

To conclude, we have measured Hall effect in hybrid (organoinorganic) perovskites lms and single crystals and found Hall mobilities ranging from 0.5 to 60 cm2 V 1 s 1, depending on the sample composition and crystallinity. Concurrent measurements of a steady-state photoconductivity and Hall carrier density have allowed to directly determine bimolecular recombination coefcients (as low as 10 10 to 10

11 cm3 s 1), carrier lifetimes (up to 30 ms and 2.7 ms in polycrystalline lms and single crystals, respectively) and diffusion lengths (up to 23 and 650 mm in lms and crystals, respectively).

These measurements provide a direct and conclusive evidence of low eh recombination rates and remarkably weak charge trapping in hybrid perovskites. They show that photophysical properties of these materials are quite different from those of conventional organic or inorganic semiconductors. We emphasize that our study determines these important transport parameters directly, from steady-state transport measurements, relevant to practical applications. The dipolar polaron model has been found helpful to explain the observed relatively low intrinsic carrier mobilities and radiative recombination rates.

Methods

Growth of hybrid perovskite thin lms and single crystals. For fabrication of thin-lm perovskite samples, we followed the published procedures36,37. Solution of PbI2 in dimethylformamide was spin-coated onto cleaned glass substrates, resulting in a homogeneous PbI2 lms. For solution-grown samples, PbI2 lms were immersed in a solution of methylammonium iodide in isopropanol. For vapour-grown perovskite lms, PbI2 was annealed in a saturated methylammonium iodide vapour at 190 C. The resulting CH3NH3PbI3 thin lms were uniform, large-area (centimetre-scale) lms on glass. CH3NH3PbBr3 single crystals were grown through anti-vapour diffusion process38. In brief, 1:1 ratio of methylammonium bromide (CH3NH3Br, synthesized following ref. 39) and lead bromide (PbBr2, Sigma Aldrich, Z98%) were dissolved in N,N-dimethylformamide. This solution was ltered into an inner container. The inner container was then put into a bigger container with dichloromethane. The outer container was sealed and kept at room temperature. Square bulky CH3NH3PbBr3 single crystals (35 mm on a side) grew at the bottom of the inner container within a few days.

Device fabrication. Devices in Hall bar geometry were fabricated by depositing Au or Ti through a shadow mask on freshly grown lms or crystals. After wiring, the devices were capped under vacuum (10 5 Torr) with a protective PFPE (peruoropolyether) oil, which is a chemically and electrically inert peruorinated polymer. We nd that PFPE can effectively protect samples from degradation due to moisture and other environmental factors. Control tests showed that PFPE does not cause any qualitative changes in the electrical properties of hybrid perovskite samples.

Magneto-transport and photoconductivity measurements. All measurements in this work were carried out at room temperature. a.c. Hall measurements were performed in an a.c. magnetic eld of B 0.23 T (r.m.s.), referenced to a Stanford

Research lock-in amplier that measures a.c. VHall. To reduce the parasitic Faraday induction, frequencies below 1 Hz were typically used. More importantly, VHall generated across the sample at a non-zero d.c. excitation (Ia0) was always compared with that generated at zero current (I 0), which yields a pure,

Faraday-induction-corrected, Hall voltage15. We have veried that VHall in all our samples was independent of the frequency in the range 0.3 to 5 Hz, which conrms that undesirable Faraday-induction signal was eliminated. Keithley 6221 current source was used to drive a d.c. excitation current through the sample. Calibration of our a.c. Hall setup has been done by carrying out measurements of a control Si

sample with known carrier density and mobility. Longitudinal (photo)conductivity was measured by 4-probe technique, which ensured that ambiguities associated with contact-resistance were eliminated. It is important to emphasize that although 2-probe sPC also exhibits a sublinear power dependence (sPCpGa, with ao1), the power exponent a in 2-probe measurements may vary in a wider range from sample to sample due to contact effects. In contrast, 4-probe photoconductivity systematically shows a at high illumination intensities, which signies a regime

governed by bimolecular recombination. Photo-excitation was achieved by illumination with a calibrated blue LED (max. power: 20 W, l 465 nm) driven by

a Keithley 2400 source meter. The highest photon ux used was close to that of one sun (integrating over the part of the spectrum absorbed by the perovskites). Thus, measurements in this work were performed within the range of light intensities relevant for solar cell applications. The errors in Hall mobility values obtained for polycrystalline and single-crystal samples (mHall 80.4 and 605 cm

2 V 1 s 1,

respectively) are dened by standard deviation in VHall measurements.

Data availability. The data that support the ndings of this study are available from the corresponding author on request.

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Acknowledgements

We thank Hang-Dong Lee and Torgny Gustafsson for their help with helium-ion microscopy, Pavel Irkhin for his help with the calibration of light sources, Szu-Ying Wang for her help with thin-lm sample fabrication. Y.C., H.T.Y. and V.P. thankthe National Science Foundation for the nancial support of this work under the grant

DMR-1506609 and the Institute for Advanced Materials and Devices for Nanotechnology (IAMDN) of Rutgers University for providing necessary facilities.X.-Y.Z. acknowledges support by the US Department of Energy, Ofce of ScienceBasic Energy Sciences, Grant ER46980, A.Z. thanks the Increase Competitiveness Program of NUST )MISIS* (No. K2-2015-014 ) for partial support and appreciates the Welch

Foundation for their partial support under grant AT-1617, Y.N.G. is grateful for support from the Department of Energy, Ofce of Basic Energy Science (DOE/OBES) grant DE-SC0010697.

Author contributions

V.P. designed the research project and supervised the experiment. Y.C. and H.T.Y. performed device fabrication and measurements. Y.C., X.W., R.H., X.-Y.Z and A.Z. grew perovskite thin lms and single crystals, Y.N.G., Y.I.R. and K.S.T. performed theoretical calculations, Y.C., Y.N.G. and V.P. wrote the paper. All authors discussed the results.

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How to cite this article: Chen, Y. et al. Extended carrier lifetimes and diffusion in hybrid perovskites revealed by Hall effect and photoconductivity measurements. Nat. Commun. 7:12253 doi: 10.1038/ncomms12253 (2016).

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