1. Introduction

In the last decade, solution-processed lead halide perovskites have shown great promise for high-efficiency solar cells with efficiencies exceeding 25% [1,2]. However, there remains toxicity and stability problems for these lead based perovskites. On the other hand, the recently emerged bismuth based double perovskites (DPs) with Bi³⁺ and a monovalent metal cation (e.g., Ag⁺) substituting two Pb²⁺ cations possess merits such as low-toxicity and high stability, yet their performance is far from satisfactory (structure shown in Fig. 1(a)). The ultimate performance of a semiconductor for solar cell application is related to some key optoelectronic properties such as the band gap energy (E_g), carrier mobility, carrier lifetime and light absorption coefficients, etc. Till now, these optoelectronic properties are still poorly understood, or under dispute for bismuth based DPs. For example, although Sb or Tl alloying reduces the band gap that is beneficial for solar light harvesting, the effects of alloying on other optoelectronic properties such as the phonon dynamics have never been investigated. Furthermore, for the most-frequently used DP: Cs₂AgBiBr₆, the measured band gap fluctuates in the range of 1.80-2.25 eV, [3–6] which is partially attributed to it possessing an indirect band gap with the band edge absorption characteristics difficult to measure. In this Letter, we present a comprehensive study on the coherent acoustic phonon (CAP) dynamics of the pure, Sb- and Tl-alloyed Cs₂AgBiBr₆ single crystals, not only to investigate the effects of alloying on phonon dynamics, but also to probe the band edge absorption characteristics of these DPs. We found that Sb or Tl alloying does not alter the CAP frequencies, but accelerates the CAP damping rates which is attributed to the increased weak absorption below the original indirect band gap of the pure DP. The increased sub-gap absorption may originate from the redshift of the indirect band gap and/or the increased density of band tail states by alloying. More importantly, we have developed a method based on the CAP damping mechanism to accurately extract the absorption spectrum on the absolute scale near the band edge for the DPs, which is too weak to be measured using conventional UV-vis spectrometer.

 

Fig. 1. (a) Crystal structure of Cs₂AgBiBr₆ DP. Typical images of as grown (b)Cs₂AgBiBr₆, (c) Cs₂AgBi_0.625Sb_0.375Br₆ and (d) Cs₂Ag_1-aBi_1-bTl_xBr₆ (x = a + b=0.075) single crystals.

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2. Experimental design

Following previous reports, [7,8] the Cs₂AgBiBr₆ single crystals were fabricated by dissolving CsBr (213 mg, 1.00 mmol), BiBr₃ (225 mg, 0.5 mmol), and AgBr (94 mg, 0.5 mmol) in 3 mL of 47% HBr and reacting at 120 °C for 24 h. Large single crystals showing red color were then precipitated after slow cooling the precursor solution to room temperature. The Sb-alloyed Cs₂AgBiBr₆ (Cs₂AgBi_0.625Sb_0.375Br₆) single crystals were prepared via the same procedures for the pure Cs₂AgBiBr₆, except that the 0.5 mmol BiBr₃ was replaced by 0.5 mmol mixture of BiBr₃ and SbBr₃ at a molar ratio of 3:2. For the growth of the Tl-alloyed Cs₂AgBiBr₆ (Cs₂Ag_1-aBi_1-bTl_xBr₆ (x = a + b=0.075)) single crystals, we followed the previous report by A. D. Slavney et al: [9] aqueous solution of TlBr in HBr (3.6 (1) mM), TlBr beads (∼50.0 mg, 0.176 mmol) were dissolved in 48 weight% HBr at 80° C for 2 h. The undissolved residues were removed by filtering through glass microfiber filter paper. The solution (2 mL) was then mixed with BiBr₃ (57 mg, 0.13 mmol), AgBr (24 mg, 0.13 mmol), and CsBr (53 mg, 0.25 mmol) and heated at 100 °C to yield a clear yellow solution. The solution was then cooled to room temperature over 3 days at a rate of 1 °C/hr and black crystals appear after the cooling and further storage. All the processes were done in a sealed vial to prevent any contact with the toxic Tl. Typical as-grown pure, Sb- and Tl-alloyed Cs₂AgBiBr₆ single crystals with sizes on the order of mm × mm × mm are shown in Fig. 1(b)-(d). For the calculations of the electron structure and mechanical properties of Cs₂AgBiBr₆, we employed the all-electron-like projector augmented wave (PAW) method [10] and the Perdew-Burke-Ernserhof revised for solids (PBEsol) exchange correlation potential [11] as implemented in the VASP code [12].

We implemented pump-probe technique in reflection mode to characterize the CAP in the DP single crystals. In our measurements, a commercial white-light pump-probe setup (Helios, Ultrafast Systems LLC) was used. The 400 nm pump pulses were generated from a Ti-sapphire regenerative amplifier (Coherent Libra, 800 nm, 50 fs, 4 mJ, 1 kHz) that passed through a frequency-doubling barium borate (BBO) crystal. The probe pulses were generated by passing a small fraction of the 800 nm pulses from the regenerative amplifier through a 1 mm thick sapphire crystal. Both the pump and probe pulses have an incident angle < 10° and the incident direction is along the (111) orientation.

3. Results and discussion

Figure 2(a) shows the pseudo-color map of the transient reflectivity for a pure Cs₂AgBiBr₆ single crystal upon 400 nm photoexcitation (200 µJ cm^-2). Clearly it is found that for all the probe wavelengths larger than ∼480 nm, strong oscillations can be observed. Figure 2(b) depicts the transient reflectivity dynamics of Cs₂AgBiBr₆ bulk crystals at selected probe wavelengths (520 nm, 600 nm, 690 nm). After the fast initial decay within a few ps, the transient reflectivity is dominated by these strong oscillations. The fast initial decay is attributed to the electronic relaxation via optical and acoustic phonons. These strong oscillations with periods on the order of tens of ps, are attributed to CAPs, i.e., the photoexcitation induces lattice strain through mechanisms such as deformation potential, thermoelasticity, inverse piezoelectricity and electrostriction and the strain modifies the refractive index through the photoelastic effect [13]. The reflected probe beam by the surface of the sample and by the propagating CAP in the bulk then interfere with each other resulting in an oscillation.

 

Fig. 2. (a) Pseudo-color map of the transient reflectivity for Cs₂AgBiBr₆ single crystal. (b) Transient reflectivity of Cs₂AgBiBr₆ single crystal probed at 690 nm, 600 nm and 520 nm showing strong CAP oscillations. Black lines are the fitted curves using Eq. (3).

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According to Thomsen et al and Wu et al, in the opaque bulk crystals with thickness L >> 1/α that can be treated as a semi-infinite medium, the CAP oscillation in transient reflectivity is expressed by:[14,15]

We then fitted the transient reflectivity signals using the model given by:

Figure 3(a) shows the CAP oscillation frequency as a function of the probe energy for the pure, Sb- and Tl-alloyed Cs₂AgBiBr₆ single crystals. It is found that the CAP frequency increases linearly with the probe energy for E < E_g. However, when E > E_g, the linear relation does not hold. From Eq. (1), we know that the CAP frequency is related to the refractive index. The nonlinear relation above the band gap is therefore attributed to the significant contribution by interband transitions on the dielectric constants. The solid line is the linear fitting using Eq. (1) for the data below the band gap of Cs₂AgBiBr₆, which yields a fitted nv_s = 6660 m/s. J. Steele et al calculated the high-frequency optical dielectric constant ɛ_o=5.4 [17]. We used it to calculate the refractive index far below the band gap and obtained a sound velocity v_s = 2.87 ×10³ m/s. The extracted sound velocity is then compared with the theoretical prediction. With the aid of first-principle calculations, we obtained the stiffness coefficients of Cs₂AgBiBr₆: c₁₁=52.06 GPa, c₁₂=18.08 GPa, and c₄₄=7.57 GPa. The elastic constant along the (111) direction is expressed by: C₍₁₁₁₎=1/3(c₁₁+2 c₁₂+4 c₄₄) = 39.5 GPa. We then obtained the theoretical sound velocity of 2.83 ×10³ m/s along the (111) direction using ${v_{\rm{s}}} = \sqrt {C/\rho } {\rm{ }}$, where ρ is the material mass density, which is in excellent consistency with the experimental values. We found Sb or Tl alloying have negligible effect on the CAP oscillation frequency. This means the refractive indices are less affected by the alloying in the probe wavelength range. In linear response theory, the dielectric function of a semiconductor is contributed by different types of oscillators with different resonance frequencies and strength, e.g., excitons, phonons, etc [18]. Near the band gap, the dielectric constant is strongly affected by absorption via interband or sub-gap defect transitions. For the pure, Sb- and Tl-alloyed Cs₂AgBiBr₆, it was shown that they all possess indirect band gaps, with the direct absorption bands a few hundred meV above the indirect band gaps. Alloying Sb or Tl reduces the indirect band gap of the pure Cs₂AgBiBr₆ as observed in diffuse reflectance and predicted by the density function theory calculations. [6,9] Furthermore, alloying may also introduce lattice disorder which increases the band tail absorpion below the band gap as discussed later. We note that both the indirect absorption and the band tail absorption are much weaker than the direct absorption. Therefore, we speculate that the alloying only alters the weak indirect or band tail transition energies/strength, while those direct transitions with strong oscillator strength above the indirect band gap are little affected by Sb or Tl alloying. As a result, the real part of the dielectric constant does not change much as inferred from the Kramers-Kronig relation due to the fact that only the weak transitions are modified by the alloying. Our speculation is consistent with the results that when we fabricated thin films using the black Tl-alloyed single crystals, the absorption spectrum is almost the same with the films prepared by the pure single crystals (Supplement 1, Fig. S1). In a thin film with the thickness of ∼100 nm, the indirect absorption is too weak to be measured. Therefore, it proves that the direct transitions above the indirect band gap are less affected by the alloying. The difference in color in single crystals (red versus black) for the pure and Tl-alloyed DPs arises from the change of absorption by weak transitions which become apparent when light passes through these thick crystals.

 

Fig. 3. (a) The CAP frequency and (b) damping rate as a function of the probe energy for the pure and alloyed Cs₂AgBiBr₆ DPs.

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Figure 3(b) shows the CAP damping rate (k_d=1/τ_d) as a function of the probe energy for the pure and alloyed DP single crystals. At the low energy side far below the band gap, the damping rates for all the crystals have little dependence on the probe energy and approaches the order of 10⁻¹ ns^-1. According to Eq. (2), this corresponds to an intrinsic acoustic phonon lifetime on the order of 10 ns, regardless of the alloying. The similarly ultra-long acoustic phonon lifetime implies that in all the three types of single crystals the interaction between defects and the low frequency acoustic phonons is weak. As the increase of the probe energy, the damping rates for all the crystals rapidly soar by orders and reaches 15 ns^-1 when E is around 2.5 eV. We cannot further extract the damping rate for E > 2.6 eV due to the dominance of the electronic signal and the diminishing CAP oscillations. The increase of the damping rates corresponds to a decreased penetration depth of the probe light according to Eq. (2). Compared to the pure DP, The CAP damping of Sb- and Tl-alloyed DPs shows similar contribution from the intrinsic phonon as observed at the low energy side. However, as the increase of the probe energy, it is found the damping rate is: k_pure<k_Sb<k_Tl, until the energy reaches ∼2.5 eV. Since the intrinsic phonon lifetime is little affected by the alloying, the increased damping rate is most probably due to different absorption above the indirect band gap by Sb or Tl alloying. Therefore, the rate of CAP damping may provide us important information on the absorption characteristics near the band edge for these indirect band gap semiconductors.

After subtracting the intrinsic phonon contribution and using the sound velocity derived above, we are able to provide absorption spectra on the absolute scale for the pure and alloyed Cs₂AgBiBr₆ single crystals near the band edge (1.7-2.5 eV) as shown in Fig. 4. We found the absorption coefficients of the pure DP show excellent consistency with those measured using UV-vis spectrometer for a polycrystalline thin film (∼ 100 nm thick) for E >2.3 eV. The spectrum in this region is mainly attributed to the direct exciton absorption located at X and Γ points of the band structure [19,20]. Below E∼2.3 eV, the steady-state absorption spectroscopy cannot provide accurate measurement for weak absorption in the thin film, and the signal is immersed in noise. On the other hand, the method used in this work can still yield good results for absorption coefficient on the order of 10² cm^-1. The indirect band gap determined by Tauc plot is around 2.25 eV (Fig. 4 inset), consistent with reports using diffused reflectance spectroscopy, photothermal deflection spectroscopy etc [20,21]. Below the indirect band gap, there is still a long tail of weak absorption extending to ∼1.6 eV, which is attributed to the band tail states by thermal fluctuation or impurities/defects. The light absorption coefficients for the Sb- and Tl- alloyed single crystals are increased in the energy region of 1.7-2.5 eV. This is consistent with the color change and the reduced indirect band gap for the alloyed single crystals. Above E > 2.5 eV, the absorption shows little difference (Fig. 3(b) & Fig. 4), which is congruent with our speculation above that the direct transitions are little affected by the alloying. In previous reports, using diffuse reflectance spectroscopy, it is determined the indirect band gap for Cs₂AgBi_0.625Sb_0.375Br₆ is around 1.86 eV, and for Cs₂Ag_1-aBi_1-bTl_xBr₆ (x = a + b=0.075) is around 1.4 eV [6,9]. For the latter, we could not determine the band gap in our measurements due to the spectral limitation of our spectrometer (λ < 800 nm). However, Tauc plot also does not show band gap like characteristics at around 1.86 eV for the former. We propose that it possibly arises from an overlapping of the absorption by the original band tail states and the reduced indirect band gap by Sb alloying. On one hand, the alloying tunes the lattice parameters and shrinks the band gap; on the other hand, the incorporation of foreign atoms may increase the lattice disorder and the density of sub-gap band tail states. The absorption arising from the former has a square-like dependence on the photon energy between the original and new band gaps, while the absorption from the latter shows an exponential decay with the decrease of photon energy (Supplement 1, Fig. S2). Both of them are comparably weak which leads to the complex absorption characteristics when adding together. Therefore, inferred from our measurements, those band gaps determined by the diffuse reflectance spectroscopy in previous reports may beg the question as the band tail absorption may also contribute. This may also explain why the band gaps predicted by theoretical calculations and determined by different experimental techniques show large discrepancies.

 

Fig. 4. Near-band gap absorption spectrum measured using the CAP damping method. Inset: The indirect band gap determined using the CAP damping method for the pure Cs₂AgBiBr₆ single crystal.

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4. Conclusion

In summary, we have performed a comprehensive study on the dynamics of CAPs in pure, Sb- and Tl-alloyed Cs₂AgBiBr₆ DP single crystals. We found Sb or Tl alloying increases the weak absorption below the original band gap of the pure DP by red-shifting the band gap and/or increasing the band tail states, which has negligible effect on the CAP frequency due to these weak transitions contributing insignificantly to the real part of the dielectric constant. However, the increased weak absorption can drastically accelerate the CAP damping rate which is highly sensitive to the light penetration depth. Based on the mechanism of CAP damping, we developed a method to measure the weak absorption near the indirect band gap of the pure and alloyed Cs₂AgBiBr₆ DP single crystals with coefficients on the order of 10² cm^-1, which can be hardly measured using existing techniques. The method may be further applied to many other materials showing CAPs such as Si, III-V compounds, superlattice, etc, which is a promising complement to the existing optical absorption measurement techniques.