Anisotropy of Thermal Diffusivity in Lead Halide Perovskite Layers

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C: Energy Conversion and Storage; Energy and Charge Transport

Anisotropy of Thermal Diffusivity in Lead Halide Perovskite Layers Revealed by Thermal Grating Technique Patrik Scajev, Ramunas Aleksiejunas, Shinobu Terakawa, Chuanjiang Qin, Takashi Fujihara, Toshinori Matsushima, Chihaya Adachi, and Saulius Jursenas J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b02288 • Publication Date (Web): 28 May 2019 Downloaded from http://pubs.acs.org on May 28, 2019

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The Journal of Physical Chemistry

Anisotropy of Thermal Diffusivity in Lead Halide Perovskite Layers Revealed by Thermal Grating Technique P. Ščajev1, R. Aleksiejūnas1*, S. Terakawa2,3, C. Qin2,3, T. Fujihara4, T. Matsushima2,3,5, C. Adachi2,3,5, and S. Juršėnas1

1

Institute of Photonics and Nanotechnology, Vilnius University, Sauletekio Ave. 3, LT-10257,

Vilnius, Lithuania 2

Center for Organic Photonics and Electronics Research (OPERA), Kyushu University 744,

Motooka, Nishi, Fukuoka 819-0395, Japan 3

Japan Science and Technology Agency (JST), ERATO, Adachi Molecular Exciton Engineering

Project, 744 Motooka, Nishi, Fukuoka 819-0395, Japan 4

Innovative Organic Device Laboratory, Institute of Systems, Information Technologies and

Nanotechnologies (ISIT), Fukuoka Industry-Academia Symphonicity (FiaS) 2-110, 4-1 Kyudaishinmachi, Nishi, Fukuoka 819-0388, Japan 5

International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu University,

744 Motooka, Nishi, Fukuoka 819-0395, Japan

*[email protected]

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Abstract Heat management of optoelectronic devices is of critical significance in lead halide perovskites due to intrinsically-low thermal conductivity of this material. Despite its importance, the thermal conductivity remains understudied, particularly in polycrystalline perovskite layers with different halides. Here, we employ a novel method for investigation of thermal properties in perovskite layers, which is based on light-induced transient diffraction grating technique. We demonstrate the applicability of thermal grating technique by determining in all-optical way the thermo-optic coefficient, speed of sound, and thermal conductivity in vapor-deposited polycrystalline layers of MAPbX3 (X = Cl, Br, I), MAPbBr2I, and MAPbCl2Br perovskites. We reveal the spatial anisotropy of thermal conductivity, which is noticeably lower in the direction along the layer surface (0.2 – 0.5 W/Km) if compared to that across the layer (0.3 – 1.1 W/Km). Finally, we demonstrate that for both directions the thermal conductivity scales linearly with the average speed of sound in the perovskite layers.

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Introduction Lead halide perovskites (MAPbX3) earned their fame mostly as an efficient and potentially cheap materials for solar cells.1 Recent advances, however, encouraged to consider them for much wider range of applications, including light-emitting or even laser diodes.2–4 These photonic devices operate at high drive currents resulting in a considerable amount of heat that must be dissipated. Heat management turns out to be of a critical importance in perovskite devices due to their intrinsically-low thermal conductivity.5 Heating of a photonic device is detrimental in itself; for lead halide perovskites it is especially destructive due to their thermal instability,6 since heat promotes the evolution of volatile organic species and formation of lead halides.7 Therefore, there is a necessity in detailed understanding of heat transport in perovskites and methods to investigate them. Despite its practical importance, the heat transport in perovskites has been studied much less than, e.g. charge carrier transport, and is not fully understood yet. Two important questions are related to the role of compositional structure and phonon scattering at the grain boundaries in polycrystalline perovskite layers. Indeed, it is desirable to use low-cost and scalable deposition methods, but they are known to yield the polycrystalline perovskite layers.8 It is intuitive to expect then that thermal conductivity 𝜅 in these layers should decrease due to phonon scattering at grain boundaries; however, the experimental studies dealing with thermal conductivity in polycrystalline perovskites still are few, while the majority of published works are devoted to MAPbI3 single crystals. Even for lead iodide crystals, the reported 𝜅 values are somewhat scattered depending on measurement technique and individual sample: 𝜅 = 0.14 W/mK was obtained in crystal platelets micro-photoluminescence technique,9 𝜅 = 0.3 ― 0.42 W/mK was measured in single crystals by flash method,10,11 and 𝜅 = 0.5 W/mK – by using the 3



thermocouples attached to the investigated layers.5 As to the role of compositional structure, two recent studies addressed the dependence of 𝜅 on halide X in MAPbX3 single crystals: ultralow values within the range 𝜅 = 0.3 ― 0.73 W/mK were reported independently on halide, with general tendency of 𝜅 increasing from -I3 to -Br3 to -Cl3.11,12 Both latter studies concluded that low thermal conductivity in lead halide perovskites has an intrinsic nature determined by low acoustic speed in soft material and strong phonon-phonon scattering due to lattice vibration anharmonicity. This conclusion was further supported by calculations using the equilibrium molecular dynamics.13 Resonant scattering was argued to be of lesser importance, at least at room temperatue,12,14,15 in contrast to what had been proposed before.5 The impact of layer polycrystallinity remains even less clear. In several studies, 𝜅 was measured in polycrystalline MAPbX3 samples with X = I, Br, and Cl, but the obtained values strikingly resembled those in single crystals with corresponding halide.5,14,16 All these measurements, however, were carried out on samples prepared by hot pressing procedure, which resulted in laterally large grains of dimensions on micron scale. G. Elbaz et al. concluded that ~70% of phonons determining the thermal conductivity have mean free path below 100 nm in organic-inorganic perovskites,12 suggesting that thermal conductivity should not be affected in the layers with large grains as in the mentioned studies. A possible impact of grain boundary scattering was proposed by Z. Guo et al, who demonstrated that the thermal conductivity is larger at the layer surface than that deeper in the sample (0.5 vs 0.3 W/mK), probably due to the increasing lateral dimensions of grains when going further away from the substrate.17 In this paper, we present a new method of measuring the thermal properties of perovskite layers by employing the light-induced transient grating (LITG) technique. We upgraded the setup that was used to study the carrier transport in perovskite layers18,19 by expanding the probe 4


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delay time up to several microseconds, which allowed the observation of thermal diffraction grating dynamics. We used thermal grating technique then to obtain in all-optical manner the thermo-optic coefficient, average speed of sound, and thermal diffusivity in vapor-deposited polycrystalline MAPbX3 layers with different halides and much smaller crystallite size. We reveal the anisotropy of thermal diffusivity, which noticeably differs in directions parallel and perpendicular to layer surface. We discuss the possible reasons of this anisotropy in terms of phonon scattering at grain boundaries of non-spherical and spatially-oriented crystallites. We demonstrate that in both directions the thermal conductivity scales linearly with the average speed of sound in the layer. Experimental Methods Materials MAPbI3, MAPbBr3, MAPbCl3, and two mixed halide perovskite MAPbBr2I, MAPbCl2Br polycrystalline layers of smooth surfaces and different bandgaps were vapor-deposited on fused silica substrates. For the detailed description of sample growth procedures, please refer to our earlier work.19 The parameters of the layers that are used in current study are provided in Table 1. The thickness and average grain size of perovskite layers were determined from scanning electron microscopy (SEM) images shown in the Supporting Information (Figure S1). The layer thickness in the main set varied from 460 to 600 nm; we also included two thicker layers of MAPbI3 and MAPbBr3 (1100 and 1300 nm) for comparison. The average grain size in the samples was 130 – 160 nm, only in MAPbI3 layer the grains were smaller of ~100 nm size. The values of absorption coefficient at the pump wavelength 𝛼𝑝𝑢𝑚𝑝, band gap 𝐸𝐺, and the carrier lifetime (determined from photoluminescence decay) 𝜏0𝑃𝐿 are used as determined in the previous work.19

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Table 1. Parameters of the Perovskite Layers.* Sample

Thickness, nm

Grain size, nm

Pump wavelength, nm

pump, 105 cm-1

Eg, eV

MAPbI3

600 (1100)

100 ± 50

527

1.2

1.65

0.5

MAPbBr2I

480 ± 50

150 ± 50

527

0.77

1.96

8

MAPbBr3

500 (1300)

160 ± 50

527

0.66

2.36

0.6

MAPbCl2Br

460 ± 50

140 ± 50

351

1.2

2.42

0.11

MAPbCl3

500 ± 50

130 ± 50

351

1.2

3.11

0.15

PL, ns

* - numbers in parenthesis correspond to thicker layers.

Thermal Diffraction Grating Measurements We use the LITG technique for determination of the thermal diffusion coefficient, thermooptic coefficient, and the average speed of sound in the lead halide perovskite layers of different composition. This versatile pump-probe type method is based on measuring the diffraction efficiency of a probe beam on a periodically modulated refractive index 𝑛, which is created by exposing the sample to the interference field of light.20 In this work, we exploit the fact that there are several physical mechanisms causing the modulation of refractive index and they appear on distinctly different timescales. Initially, the refractive index is altered proportionally to the density of photogenerated excess carriers; this regime becomes increasingly popular for determination of carrier diffusion coefficient in perovskites.18,19,21,22 The upper limit for the decay time of free carrier grating is set by the excess carrier lifetime, which in highly excited perovskite layers is of the order of up to tens of nanoseconds.18,19 We note that there are multiple reports of very long carrier lifetimes, up to the microsecond scale;23,24 however, such a long lifetime is characteristic for low carrier densities when carrier transport is strongly influenced by carrier localization processes. 6


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The second stage of LITG decay takes place on considerably longer timescale, where the refractive index is modulated due to thermo-optic effect by periodical temperature variation; we refer to this stage as thermal (diffraction) grating.20 The heating of the sample occurs in the illuminated areas due to non-radiative carrier recombination. The thermal grating decays with characteristic time 𝜏𝐺, due to (i) thermal diffusion along the grating vector (parallel to sample’s surface) and (ii) thermal diffusion towards the substrate with characteristic times 𝜏𝐷 and 𝜏𝑇ℎ, respectively. The grating decay time 𝜏𝐺 can be expressed as:20 1

1

1

1

𝜏𝐺 = 𝜏𝑇ℎ + 𝜏𝐷 = 𝜏𝑇ℎ +

4𝜋2𝐷𝑇ℎ Λ2

(1)

,

here Λ is the period of thermal grating and 𝐷𝑇ℎ is the thermal diffusion coefficient. 𝐷𝑇ℎ is related to thermal conductivity 𝜅 as 𝐷𝑇ℎ = 𝜅/𝜌𝐶𝑃, where 𝜌 and 𝐶𝑝 are the material density and heat capacity, respectively. By measuring a set of decay transients for several different Λ values, one can determine 𝐷𝑇ℎ, according to Eq. (1). This method has been shown as a suitable tool for thermal diffusion investigation in both bulk materials and layers on a substrate as well;25 we used it previously to determine the thermo-optic coefficient in SiC.26 Finally, the heating of the lattice results in the additional mechanism of refractive index modulation due to the local changes in material density. As a result, the spatially-periodic thermal grating gives rise to the travelling acoustic wave that can be seen in LITG kinetics as periodic decoration of otherwise continuous decay. From the period of these oscillations, the speed of sound can be determined in various materials,20 as it was demonstrated explicitly for 4H-SiC.26

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In this work, we used 10 ps duration pulses from Nd-YLF laser at 527 nm or 351 nm wavelength to record the transient in a sample. The pump wavelength for a given layer was chosen so that the quanta energy would exceed the bandgap of studied perovskite material. The diameter of pump beams was 750  20 m, while the pump energy fluence was 1.1 mJ/cm2 for 351 nm and 1.9 – 2.5 mJ/cm2 for 527 nm excitations, respectively. The transient grating was monitored by diffraction of electronically delayed probe pulse at 1064 nm that was delivered by 10 ns duration Nd-YAG laser (probe beam diameter was 200  30 m). The period Λ of the transient grating was varied between 1.9, 2.3, 2.9, 3.8 and 7.8 m, by using the holographic beam splitters of different period and a telescope that projected the image of the splitters onto the sample surface. Results and Discussion Thermo-Optic Coefficient We begin with the determination of the thermo-optic coefficient 𝑑𝑛/𝑑𝑇 in the perovskite layers with different halides. As it is shown in Ref.26, the diffraction efficiency 𝜂(𝑡), which is measured experimentally as the ratio between the intensities of diffracted and transmitted parts of the probe, for a thermal grating can be expressed in the following form: 𝜋

𝑑𝑛

𝜂(𝑡) =± 𝑑𝑇𝐶 𝜌𝜆 𝑝

𝑝𝑟𝑜𝑏𝑒

𝑡

𝑝𝑢𝑚𝑝𝑑

𝐼0(1 ― 𝑒 ― 𝛼

)𝑒

―𝜏

,

𝐺

(2)

here 𝜆𝑝𝑟𝑜𝑏𝑒 is the probe wavelength, 𝐼0 is the energy fluence of the pump beam, and 𝑑 is the layer thickness. The 𝐶𝑝values were taken from Ref. 27 and are shown in Table 2. Since we didn’t find any data on 𝐶𝑝 in mixed halide perovskites, it was obtained by interpolation between the corresponding single halide perovskite values according to the molar halide composition. 8


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Figure 1. Absolute values of thermo-optic coefficient as a function of the perovskite band gap 𝐸𝐺. The solid line shows the fit according to Eq. (3) with the free fitting parameters A = (4.2 ± 0.3)10-5 K-1, B = (2.4 ± 0.2)10-5 eVK-1, and  = 0.0134 eV.

The thermo-optic coefficient values calculated from the measured 𝜂(𝑡) using Eq. (2) are shown in Figure 1 as a function of the band gap of the perovskite. As it can be seen, 𝑑𝑛/𝑑𝑇 decreases with 𝐸𝐺 from 0.92 to 0.5 K-1 within the investigated range. This dependence fits well to the model proposed by X. Ziang et al.,28 which we modify assuming a single resonance at 𝐸𝐺: 𝑑𝑛 𝑑𝑇

=𝐴+(

𝐵(𝐸𝐺 ― 𝐸)

(3)

,

𝐸𝐺 ― 𝐸)2 + Γ2

here E = 1.1654 eV is the photon energy of the probe beam, while A, B, and 𝛤 are the free fitting parameters, with A representing the thermo-optic coefficient value far below the band gap, and Γ – the damping constant. The solid line in Figure 1 shows the fit to Eq. (3) with the fitting coefficient values of A = (4.2 ± 0.3)10-5 K-1, B = (2.4 ± 0.2)10-5 eVK-1, and  = 0.0134 eV. We note that qualitatively similar dependencies of n on temperature and wavelength are assumed for all perovskites, which is feasible since the induced temperature modulation is small and the difference 𝐸𝐺 ―𝐸 is large for all investigated perovskites. The calculated dependence of thermooptic coefficient on band gap is shown by the solid line in Figure 1. To compare the obtained

9



values to published data, we estimated the thermo-optic coefficient value dn/dT = -(0.8 ± 0.3)10-4 K-1 at E = 1.1654 eV using the ellipsometry results obtained at several temperatures in MAPbI3.29 This is in fair agreement with our value of -(0.9 ± 0.1)10-4 K-1. Please note that the thermo-optic coefficient is negative in perovskites since their band gap decreases with temperature, contrary to conventional semiconductors. In addition, we applied the thermal grating technique to measure the thermo-optic coefficient in CdTe where the reference data is more readily available. Our value of 1.810-4 K-1 was in good agreement with the published one of 2.010-4 K-1;30 we used 𝐶𝑝𝐶𝑑𝑇𝑒= 0.209 J/gK and 𝜌𝐶𝑑𝑇𝑒 = 5.86 g/cm3.31 Speed of Sound To determine the speed of sound in the perovskite layers, we used an intense excitation at 351 nm wavelength, which ensured rapid local heating of the lattice and resulted in the appearance of acoustic waves and oscillations in the initial parts of thermal grating transients (Figure 2).

Figure 2. Initial parts of LITG transients under strong 351 nm excitation and Λ = 0.93 𝜇m grating period in the perovskite layers. The dots show the experimental data, the lines – the calculated decay transients according to Eq. (4). The oscillations mark the contribution of photo-acoustic effect.

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The oscillating parts of LITG transients were fitted to the following relation:26

(

𝜂(𝑡) = 𝐴 𝑒

𝑡

―𝜏

𝐺

+ 𝐴1sin

2

( ) + 𝐴 sin ( )) , 2𝜋𝑡 𝑇1

2𝜋𝑡

2

(4)

𝑇2

here the first term describes the exponential decay of thermal grating according to Eq. (1), while the second and third terms account for oscillatory component caused by transverse and longitudinal phonons, respectively; A stands for the corresponding amplitudes and T – periods of these components. A and T served as free parameters to fit the data in Figure 2 to Eq. (4). From the obtained oscillation periods T, we calculated the sound speed for transverse 𝑣𝑇 = Λ/𝑇1 and longitudinal 𝑣𝐿 = Λ/𝑇2 sound waves. The average sound speed 𝑣𝑎𝑣 was calculated as: 𝑣𝑎𝑣 =

𝐴1𝑣𝑇 + 𝐴2𝑣𝐿 𝐴1 + 𝐴2

.

(5)

In Table 2, we list the measured transverse, longitudinal, and average sound velocities in our samples. While we could not find any data for mixed perovskites, the speed of sound obtained by LITG in MAPbX3 (X = Cl, Br, I) is in reasonable agreement with that obtained using the elastic modulus values measured by nanoindentation experiments.12 Table 2. The Speed of Sound Measured by LITG Method in Perovskite Layers. Sample

v T, m/s

vL, m/s

vav,* m/s

Vav,** m/s

MAPbI3

1292

1603

1404

1390

MAPbBr2I

1603

2163

1790

-

MAPbBr3

1676

2480

1944

1717

MAPbCl2Br

1788

2385

1987

-

MAPbCl3

2104

3020

2397

2194

11



* - The estimated experimental error of speed of sound does not exceed 50 m/s. ** - The average values of sound speed reported by G. Elbaz et al.12

Thermal Diffusivity Now we apply the thermal diffraction gratings to investigate heat dynamics in the perovskite layers. To determine the thermal diffusivity and thermal conductivity in the samples, we recorded the transient gratings for various grating periods Λ. Figure 3 (a) shows the

Figure 3. (a) Diffraction efficiency transients recorded in MAPbBr3 at free different grating periods Λ. The solid lines indicate the decay traces modelled using Eq. (7). (b) Diffraction efficiency transients recorded in MAPbBr3 at one fixed grating period and different excitation energy fluencies. The numbers near the transients indicate the excitation energy fluencies and time constants of grating decay. The inset: the dependence of peak diffraction efficiency value vs excitation energy fluence. The number shows the slope of the dependence in log-log representation. (c) Grating decay rate 1/𝜏𝐺 as a function of 1/Λ2. The solid lines show the modelled D. Open points show data for two thicker layers.

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diffraction efficiency transients for different grating period values in MAPbBr3; the decay kinetics in other samples look similarly. The time constant of grating decay 𝜏𝐺 decreases for smaller periods indicating the contribution of heat diffusion. We note that in this study we investigate the processes much slower than free carrier recombination. As it was already mentioned, the lifetime of free carriers reaches up to ~10 ns in these layers, as determined from the time resolved PL measurements (see Table 1). To illustrate the rate of free carrier recombination, we show in Figure 3 (b) the transient of differential transmission (DT), which signal amplitude is proportional to free carrier density. To additionally confirm that the investigated LITG dynamics is not affected by free carrier recombination, we measured the decay kinetics for different excitation energy fluencies. Figure 3 (b) shows that while the LITG signal amplitude scales with excitation squared (see the inset), the grating decay time remains independent on excitation. This fact excludes the impact of free carriers recombination, where the recombination rate at given carrier densities should increase with excitation due to band-toband and Auger recombination.32 The decay time constants 𝜏𝐺 were used to build the plots of 1/𝜏𝐺 against 1/Λ2 (Figure 3 (c)) and to determine the thermal diffusion coefficient DTh and the characteristic time Th according to Eq. (1); the obtained values are listed in Table 3. We attribute the periodindependent term 1 𝜏𝑇ℎ to the rate of heat transport towards the substrate serving as a heat sink. This assumption is supported by the values of Th, which are larger for thicker layers (see Table 3, numbers in parenthesis for MAPbI3 and MAPbBr3), since the phonon travel distance is longer.

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Table 3. Thermal Properties of Perovskite Layers.* Sample

MAPbI3

, g/cm3

Cp,

dn/dT,

DTh,

D||,

D┴,



┴

J/gK

10-4 K-1

10-3 cm2/s

10-3 cm2/s

10-3 cm2/s

W/mK

W/mK

4.16

0.306

0.93

1.6

2470

1.5

2.2

0.19

0.28

(1.6)

(3950)

(1.5)

(2.1)

(0.19)

(0.27)

Th, ns

MAPbBr2I

3.932

0.337

0.76

3.0

920

2.6

4.0

0.34

0.52

MAPbBr3

3.818

0.355

0.65

3.2

1440

2.9

3.5

0.39

0.47

(2.6)

(9000)

(2.5)

(3.7)

(0.34)

(0.5)

MAPbCl2Br

3.387

0.436

0.59

3.4

910

3.0

4.8

0.44

0.71

MAPbCl3

3.171

0.492

0.53

3.9

703

3.5

6.0

0.5

1.1

* - numbers in parenthesis correspond to the thicker layers

To better describe the heat transport under our experimental conditions, we adopted the model of heat diffusion supplemented with periodic heat generation term:20 𝑑Δ𝑇(𝑥,𝑦,𝑧) 𝑑𝑡

= ∇ ∙ (𝐷(𝑥,𝑦,𝑧)∇Δ𝑇(𝑥,𝑦,𝑧)) +𝐺(𝑥,𝑦,𝑧,𝑡),

(6)

here ∇ is the differential operator, 𝐺(𝑥,𝑦,𝑧,𝑡) = 𝑃(𝑥,𝑦,𝑧,𝑡)𝛼𝑝𝑢𝑚𝑝 𝜌𝐶𝑝 is the temperature generation function, 𝑃(𝑥,𝑦,𝑧,𝑡) is the photoexcitation power, related to the excitation energy fluence as ∫𝑃(𝑥,𝑦,𝑧,𝑡)𝑑𝑡 = 𝐼(𝑥,𝑦,𝑧). In a thin perovskite layer deposited on silica substrate and photoexcited by a periodic interference pattern, it is reasonable to assume that temperature gradients are formed along two perpendicular directions, one along the grating vector (in our case – along the layer surface) and the other – perpendicular to the surface, or along the direction towards the substrate; we denote these two directions as x and z, respectively. We denote the diffusion coefficient components as 𝐷𝑥 = 𝐷𝑦 = 𝐷 ∥ (𝑧) (diffusion coefficient along the surface) and 𝐷𝑧 = 𝐷 ⊥ (𝑧) (diffusion coefficient across the surface). We assume 𝐷 ∥ (𝑧) being constant in

14


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the (x, y) plane, but both diffusivity components changes towards the substrate. Under these assumptions, Eq. (6) is transfromed into the following form: ∂Δ𝑇(𝑥,𝑧,𝑡) ∂𝑡

= 𝐷 ∥ (𝑧)

∂2Δ𝑇(𝑥,𝑧,𝑡) ∂𝑥

2

+ 𝐷 ⊥ (𝑧)

∂2Δ𝑇(𝑥,𝑧,𝑡) ∂𝑧

2

+

∂𝐷 ⊥ (𝑧)∂Δ𝑇(𝑥,𝑧,𝑡) ∂𝑧

∂𝑧

+𝐺(𝑥,𝑧,𝑡),

(7)

With initial spatial distribution Δ𝑇(𝑥,𝑧):

(

Δ𝑇(𝑥,𝑧) = Δ𝑇0 1 + cos

2𝜋𝑥

𝐼0𝛼𝑝𝑢𝑚𝑝

𝛬

𝜌𝐶𝑝

( ))exp ( ―𝛼𝑧), 𝑧 < 𝑑, Δ𝑇0 =

.

(8)

𝐷 ⊥ (𝑧) is expected to be inhomogeneous near the substrate, continously transforming from 𝐷 ⊥ 𝑝𝑒𝑟𝑜𝑣𝑠𝑘𝑖𝑡𝑒 to 𝐷 ⊥ 𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒, which is taken into account in Eq. (7) by the derivative ∂𝐷 ⊥ (𝑧)/∂𝑧. We assume this derivative to have a sharp Gaussian shape: ∂𝐷 ⊥ (𝑧) ∂𝑧

=

(𝐷 ⊥ 𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 ― 𝐷 ⊥ 𝑝𝑒𝑟𝑜𝑣𝑠𝑘𝑖𝑡𝑒) 𝜋𝑧0

(

exp ―

).

(𝑧 ― 𝑑)2 𝑧20

(8)

The integration constant in (8) was chosen such that ∫

(

∂𝐷(𝑧) ∂𝑧

)𝑑𝑧 = 𝐷 ⊥ 𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 ― 𝐷 ⊥ 𝑝𝑒𝑟𝑜𝑣𝑠𝑘𝑖𝑡𝑒 ,

(9)

and the value z0 ≈ 30 nm was used. Fused silica (FS) parameters used in the modelling are the following: 𝐶𝑝 𝐹𝑆 = 0.74 J/gK, 𝜌𝐹𝑆 = 2.2 g/cm3, 𝐷𝑇ℎ 𝐹𝑆 = 8.2  10-3 cm2/s, and 𝑑𝑛/𝑑𝑇(𝐹𝑆) = 9  10-6 K-1.33,34 The described model provided a good agreement between the calculated and measured dynamics of thermal grating (the calculated transients for MAPbBr3 are shown by solid lines in Figure 3 (a)) and allowed for estimation of heat diffusion coefficient in directions parallel and perpendicular to the sample‘s surface. As it can be seen from the Table 3, 𝐷𝑇ℎ and 𝐷 ∥ values are

15



rather close. This fact leads to the conclusion that the standard procedure of diffusion coefficient determination from Eq. (1) can be used for estimation of heat diffusivity and thermal conductivity of a perovskite layer with a good presission. More thorough analysis, however, allows for some further insigths into the role of crystallite grains in thermal conductivity of perovskite layers.

Figure 4. Dependence of thermal conductivity on the average speed of sound for the directions parallel ( 𝜅 ∥ navy symbols) and perpendicular (𝜅 ⊥ olive symbols) to the layer surface. The solid lines show the linear fits.

The values of 𝐷 ∥ , 𝐷 ⊥ , 𝜅 ∥ , and 𝜅 ⊥ obtained from modelling are listed in Table 3, while Figure 4 shows the calculated 𝜅 ∥ and 𝜅 ⊥ values as functions of the average sound speed (Table 2). Several conclusions can be drawn by analysing the data in Figure 4. First of all, the thermal conductivity in the studied polycrystalline layers is small (0.2 – 1.1 W/mK) and fits reasonably well within the limits reported by other authors. However, a clear spatial anysotropy of thermal conductivity can be seen, which is 1.5 – 2.2 times higher in the direction perpendicular to the surface than that along the surface. Finally, the measured thermal conductivity increases linearly with the speed of sound in the layers for both directions. The latter result agrees well with that reported by Elbaz et al., where it was proposed that the mean free path of acoustic phonons is

16


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similar in all MAPbX3 perovskites and the differences in heat conductivity result mainly from the variations in the speed of sound.12 Thus, we show that the linear dependence applies for the grained layers as well. We attribute the spatial anisotropy of 𝜅 to the influence of crystallite grains to heat conductivity, which we believe can be two-fold. Firstly, the difference in 𝜅 values may originate from the non-equal scattering rates of acoustic phonons at grain boundaries, due to non-spherical shape of the grains. SEM images (see Supporting Information, Figure S1) show the grainy character of the layers, with the mean grain size within 100 – 160 nm range (see Table 1). A closer inspection reveals that grains at least in some layers are not spherical but rather prolongated towards the substrate. As a result, the mean distance between the grain boundaries across the sample is generally larger than that along the surface and considerably exceeds 100 nm. It was predicted that only the grains of dimensions around 100 nm or less effectively participate in phonon scattering because the majority of phonons have the mean free path below 100 nm.12 Therefore, the phonons travelling towards the substrate might experience less scattering, which leads to higher heat conductivity. We also note that phonons travelling along the surface can experience additional scattering due to surface roughness, which might be reason behind the lowest 𝜅 value in iodide films. Secondly, XRD spectra (Supporting Information, Figure S2) suggest a high level of orientational ordering of crystallites in some layers (-PbCl3, PbBr2I). Therefore, the anisotropy of 𝜅 in part can be related to differences of sound speed along the different crystallographic directions, as it was shown by numerical simulations for single MAPbX3 crystals.11 Conclusions

17



In this work, we applied a novel method of thermal gratings to study the heat transport properties in polycrystalline vapor-deposited MAPbX3 and mixed perovskite layers. This method can be attractive as it allows determining in all-optical way the material parameters like thermooptic coefficient, speed of sound, and thermal conductivity. We reveal the anisotropy of thermal conductivity, which is 1.5 – 2.2 times higher across the layer than that along the surface. We speculate that this difference can be related to phonon scattering at grain boundaries, which is different in mentioned directions due to non-spherical shape of crystallites in the layers; also, orientation of crystallites within the layer can play role due to different sound speed along different crystallographic directions. Finally, we show by independent measurements that thermal conductivity scales linearly with the speed of sound, which varies in perovskites with different halides, confirming that the speed of sound is the reason behind different thermal conductivity in organic-inorganic lead halide perovskite layers. The results on anisotropy of thermal transport can be especially important for designing the active regions of grained perovskite high power light-emitting and laser diodes. Supporting Information Cross-sectional SEM images of the investigated layers, XRD patterns of the investigated layers. This information is available free of charge via the Internet at http://pubs.acs.org Acknowledgements Vilnius University team acknowledges the financial support provided by Research Council of Lithuania under the project No. S-MIP-17-71. Additionally, this work was supported by the Japan Science and Technology Agency (JST), ERATO, Adachi Molecular Exciton Engineering Project (grant number JPMJER1305), JSPS KAKENHI (grant numbers JP15K14149 and JP16H04192), and The Canon Foundation. 18


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TOC Graphic

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Figure 1. Absolute values of thermo-optic coefficient as a function of the perovskite band gap EG. The solid line shows the fit according to Eq. (3). 70x58mm (300 x 300 DPI)



Figure 2. Initial parts of LITG transients under strong 351 nm excitation and Λ = 0.93 μm grating period in the perovskite layers. The dots show the experimental data, the lines – the calculated decay transients according to Eq. (4). The oscillations mark the contribution of photo-acoustic effect. 67x50mm (300 x 300 DPI)


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(a) Diffraction efficiency transients recorded in MAPbBr3 at free different grating periods Λ. The solid lines indicate the decay traces modelled using Eq. (7). 67x56mm (300 x 300 DPI)



(b) Diffraction efficiency transients recorded in MAPbBr3 at one fixed grating period and different excitation energy fluencies. The numbers near the transients indicate the excitation energy fluencies and time constants of grating decay. The inset: the dependence of peak diffraction efficiency value vs excitation energy fluence. The number shows the slope of the dependence in log-log representation. 65x55mm (300 x 300 DPI)


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Grating decay rate 1/τG as a function of 1/Λ2. The solid lines show the modelled DII. Open points show data for two thicker layers. 69x56mm (300 x 300 DPI)



Dependence of thermal conductivity on the average speed of sound for the directions parallel (κII navy symbols) and perpendicular (κ⊥olive symbols) to the layer surface. The solid lines show the linear fits. 70x56mm (300 x 300 DPI)


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Anisotropy of Thermal Diffusivity in Lead Halide Perovskite Layers

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