First-principles insight into the photoelectronic properties of Ge-based perovskites

Abstract

The crystal configuration, electronic structure, charge-carrier transport, and optical properties of Ge-based MAGeX₃ perovskites (MA = CH₃NH₃⁺; X = Cl⁻, Br⁻, and I⁻) and AGeI₃ (A = Cs⁺, MA, FA (HC(NH₂)₂⁺), MO (CH₃C(NH₂)₂⁺), and GA (C(NH₂)₃⁺)) were investigated using first-principles theory. The results showed that the increase in Ge–X bonds (from Cl⁻ to I⁻) in MAGeX₃ increased the volumes, weakened the covalent coupling of Ge–X, lowered the bandgaps, reduced the electron and hole effective masses, and red shifted the absorption spectra. Different A cations in the AGeI₃ systems altered the package of perovskite crystals and thus significantly influenced the electronic and optical properties of those perovskites. Electronic property analyses revealed that the valence band maxima (VBM) of AGeI₃ perovskites were mainly contributed by the I 5p and Ge 4s orbitals, whereas the conduction band minima (CBM) were dominated by Ge 4p orbitals. In AGeI₃ perovskites, the bandgap increased and the absorption spectrum blue shifted in the sequence of Cs⁺ → MA → FA → MO → GA. Our results highlighted the effects of A and X on the photoelectronic properties of Ge-based perovskites.

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1. Introduction

Hybrid organic–inorganic perovskites have opened a new horizon for solar energy utilization in the past several years owing to their cheap raw materials, simple fabrication, and high photoelectronic conversion efficiency.^1–5 In 2009, organic–inorganic perovskites were first observed to exhibit a power conversion efficiency of 3.8%, and an efficiency of over 20% was rapidly discovered afterwards.^6–8 Pb-Based halide perovskites involving MAPbI₃,^8,9 FAPbI₃,¹⁰ MAPbI_xCl_3−x (x = 0–3),¹¹ and MAPbI_xBr_3−x (x = 0–3),¹² have currently dominated mainstream research. However, challenges in Pb-based perovskite solar cells still exist, such as high toxicity and instability.¹³ Therefore, finding alternative materials for Pb is necessary. Ge, which belongs to the same subgroup (IVB) with Pb, exhibits photophysical properties similar to Pb-based materials to capture photon.¹⁴ Kanatzidis et al. experimentally found that the crystallographic characteristics of Ge-based perovskites strongly depended on cation sizes.¹⁵ Among these perovskites, CsGeI₃, MAGeI₃, and FAGeI₃ perovskites exhibited trigonal structure (R3m), MOGeI₃ and GAGeI₃ perovskites exhibited monoclinic structure (P2₁), TAGeI₃ (TA = (CH₃)₃NH⁺) perovskite exhibited hexagonal structure (P6₃), and PAGeI₃ (PA = (CH₃)₂C(H)NH₃⁺) perovskite exhibited tetragonal structure (I4̄2d). The bandgaps of these perovskites fluctuated from 1.6 eV (CsGeI₃) to 2.8 eV (TAGeI₃). Mhaisalkar et al. synthesized three AGeI₃ halide perovskites, namely, CsGeI₃, MAGeI₃, and FAGeI₃, with corresponding bandgaps of 1.63, 2.00, and 2.35 eV;¹⁶ the photocurrent values of CsGeI₃ and MAGeI₃ solar cells were 5.7 and 4.0 mA cm⁻², respectively. Although previous researches have confirmed that Ge-based halide perovskites could exhibit distinctive performance as nontoxic photoelectronic conversion materials, the bonding characteristics, electron density distribution, charge transfer, and the relationship between the crystal configurations and photoelectronic properties, among others, have yet to be convincingly elucidated.

In the present work, we investigated a series of nontoxic Ge-based halide perovskites by using first-principles theory. We used MAGeI₃ as initial model and replaced I⁻ with Cl⁻ and Br⁻ anions to build MAGeX₃ systems. In addition, we replaced MA cation in MAGeI₃ with a series of inorganic and organic cations to build AGeI₃ systems (A = Cs⁺, MA, FA, MO, and GA). The crystal configuration, electronic structure, charge-carrier transport, optical properties, and the relationship between structural optimization and photoelectronic properties were elucidated.

2. Computational methods

All of the first-principles calculations were performed by using the Vienna ab initio simulation package (VASP) based on density functional theory with all-electron projected augmented wave (PAW) method.^17,18 Semi-local generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) exchange correlation functional was employed. The valance wave functions were expanded in a plane-wave basis with cutoff energy of 500 eV. All atoms were allowed to relax in their geometry optimizations. We performed Brillouin-zone integrations using Monkhorst–Pack grids¹⁹ with (4 × 4 × 4) mesh for the structure optimization and more denser mesh (8 × 8 × 8) for the electronic and optical properties. All of the structures were relaxed with a conjugate-gradient algorithm until the energy on the atoms less than 1.0 × 10⁻⁵ eV. The van der Waals (VDW) interactions played a crucial role in the organic/inorganic hybrid materials, especially in their equilibrium geometries.²⁰ So, the VDW-DFT method was also involved in the structural optimizations for all of the perovskite systems. In addition, spin-orbital coupling (SOC) had a certain influence on the electronic structures of these perovskites, therefore, the band structures calculated with the PBE + SOC functional were considered (see Fig. S1 in ESI†). Since the results calculated by the PBE and PBE + SOC functionals were close, for clarity, only the results by the PBE functional were analyzed unless otherwise stated.

3. Results and discussion

3.1 Crystal configuration

Accurate structural characteristics are essential in predicting the electronic and optical properties of perovskites. Fig. 1 presents the optimized trigonal structures of MAGeX₃, in which the distorted GeX₆ octahedra interconnect through a corner-sharing pattern, and MA cations align along the c axis to balance the GeX₃ framework. Compared with the equivalent Pb–I bonds in tetragonal MAPbI₃ perovskite,²¹ the Ge–X bonds in trigonal MAGeX₃ are divided into two types of bond: one is short bond with a strong covalent interaction and the other is long bond with a weak interaction. Fig. 1a clearly shows the short Ge–I bonds, which exhibit an inverted pyramidal configuration with the Ge atom located at the base of the pyramid. As shown in Fig. 1b–d and Table 1, the bond distances are 2.44 and 3.09 Å for the Ge–Cl bonds, 2.60 and 3.16 Å for the Ge–Br bonds, and 2.80 and 3.30 Å for the Ge–I bonds, agreeing well with experimental data of 2.77 and 3.45 Å.¹⁵ These values indicate that the Ge–X bond distances increase when X goes from Cl⁻ to Br⁻ and I⁻ anions. The lattice parameters and volumes are calculated as follows: a = b = 7.28 Å, c = 10.61 Å, and V = 487.54 ų for MAGeCl₃; a = b = 7.64 Å, c = 10.91 Å, and V = 552.01 ų for MAGeBr₃; a = b = 8.20 Å, c = 11.34 Å, and V = 660.12 ų for MAGeI₃; these results demonstrate a gradually increasing trend for MAGeX₃ perovskites. The increased Ge–X bond distances and volumes would significantly influence the electronic structures and optical properties of perovskites.

Fig. 1 Optimized MAGeX₃ structures. (a) MAGeI₃ in ball and stick graph; (b) MAGeCl₃, (c) MAGeBr₃, and (d) MAGeI₃ in polyhedral graph.

Table 1 The calculated lattice parameters of the MAGeX₃ and AGeI₃ perovskites and the experimental results (in brackets)

Items	CsGeI3	MAGeCl3	MAGeBr3	MAGeI3	FAGeI3	MOGeI3	GAGeI3
Crystal system	Trigonal	Trigonal	Trigonal	Trigonal	Trigonal	Monoclinic	Monoclinic
Z	3	3	3	3	3	6	4
Ge–X bond distances (Å)	2.84	2.44	2.60	2.80	2.80–2.83	2.82–2.87	2.87–2.97
	3.15	3.09	3.16	3.30	3.64–3.91	3.60–3.92	3.38–3.55
Edge length (Å)	a = b = 8.40	a = b = 7.28	a = b = 7.64	a = b = 8.20	a = 8.71	a = 13.04	a = 9.44
	c = 10.44	c = 10.61	c = 10.91	c = 11.34	b = 8.67	b = 9.59	b = 7.58
	(a = b = 8.36, c = 10.61)			(a = b = 8.55, c = 11.16)	c = 12.50	c = 15.22	c = 18.48
					(a = b = 8.47, c = 11.73)	(a = 12.56, b = 9.34, c = 14.83)	(a = 9.23, b = 7.34, c = 17.97)
Angle (°)	α = β = 90.00	α = β = 90.00	α = β = 90.00	α = β = 90.00	α = 88.91	α = 90.00	α = 90.00
	γ = 120.00	γ = 120.00	γ = 120.00	γ = 120.00	β = 88.89	β = 113.10	β = 120.61
					γ = 121.11	γ = 90.00	γ = 90.00
V (Å^3)	637.61	487.54	552.01	660.12	806.58	1751.78	1137.50

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To explore the effect of A cations on the photoelectronic characteristics of perovskites, we replace MA with Cs⁺, FA, MO, and GA on the basis of the high-performing MAGeI₃ perovskite, as shown in Fig. 2. From the inorganic cation Cs⁺ to organic cations MA, FA, MO, and GA, their sizes increase gradually, and thus increasing the Ge–I bond distances and the volumes of perovskites, as shown in Table 1. The calculated lattice constants of AGeI₃ perovskites are in good agreement with the experimental results¹⁵ (see Table 1), justifying the reliability of our model and levels of method. For the same reason, MOGeI₃ and GAGeI₃ perovskites prefer the monoclinic crystal rather than the trigonal crystal similar to that of CsGeI₃, MAGeI₃, and FAGeI₃ perovskites. To quantitatively characterize the crystal configuration and evaluate the effect of different A cations on crystal stability, the tolerance factor (t) was brought up using the following formula,^22,23

where R_A, R_B, and R_I represent the ionic radii of A, B, and I ions, respectively. The tolerance factor t, as well as the octahedral factor u (ratio of R_B/R_X), could give a accurate description of perovskites' stability with the limit of u > 0.41 and t ≤ 1.06. For the AGeI₃ perovskites, with the increase size of A cations, the crystals could adopt highly distorted coordination due to a stereoactive lone pair. Thus the hybrid Ge-based perovskites are exceptive with the octahedral factor significantly smaller than 0.41 and tolerance factor larger than 1.06,²⁴ such as, the tolerance factor of 1.17 for MOGeI₃.

Fig. 2 Optimized A cations ((a) caesium (Cs⁺), (b) formadinium (FA), (c) acetamidinium (MO), and (d) gunidinium (GA), from top to bottom) and corresponding AGeI₃ perovskites structures in each row. The first three panels are presented in ball and stick graph; the right panel is presented in polyhedral graph.

Our results exhibit that the perovskites containing Cs⁺, MA, and FA cations are 3D structures. With the increased sizes of the organic cations, the 3D framework is distorted (quasi-3D structure) for MOGeI₃ and completely collapses to form a 1D chain structure for GAGeI₃ perovskite (Fig. 2c and d). These cations that pad the innerspace of GeI₃ framework affect the package of these crystals. For the trigonal crystals, the GeI₆ octahedra interconnect through corner-sharing pattern, whereas the GeI₆ octahedra interconnect through face-sharing pattern for the monoclinic crystals. The MA and FA cations align along the c axis, and GA cations are tilted by ∼30° from the c axis. The padding of the MO cations are more disordered than that of the GA cations because of the asymmetry of the former.

3.2 Electronic structure

As the light harvesters in solar cells, the electronic structures of perovskites are crucial factors for the efficient spectral absorption.²⁵ Fig. 3 and 4 show the band structures and density of states (DOS) with the Fermi level set at 0 eV. As shown in Fig. 3a–c, the band structures of MAGeX₃ (X = Cl⁻, Br⁻, and I⁻) are similar to each other, in which the VBM and CBM are located at the same Z (0, 0, 0.5) position in the Brillouin zone. This indicates that the MAGeX₃ perovskites are direct-bandgap semiconductors. The calculated bandgap of MAGeX₃ from Cl⁻, Br⁻, to I⁻ decreases from 1.91, 1.60, to 1.20 eV. As shown in Fig. 3d and e, the trigonal Ge-based perovskites of CsGeI₃ and FAGeI₃ are also direct-bandgap semiconductors showing the direct bandgap nature located at Z (0, 0, 0.5) symmetry point. However, the monoclinic Ge-based perovskites of MOGeI₃ and GAGeI₃ are indirect-bandgap semiconductors with the VBM and CBM located at different points. Both the top valence band and the bottom conduction band of monoclinic crystals do not fluctuate as strongly as those of trigonal crystals, indicating that the states of monoclinic crystals are greatly delocalized. This phenomenon increases the bandgaps of monoclinic crystals, and thus hindering their efficient excitations and transportation. The increased sizes of A cations, as we mentioned above, could affect their electronic structures of the perovskites. When A is changed in the sequence of Cs⁺ → MA → FA → MO → GA, the bandgaps of AGeI₃ perovskites exhibit an increasing trend, as shown in Fig. 3. In addition, the band structures of all perovskites are also calculated using the PBE + SOC functional, as shown in Fig. 3h. The bandgaps show similar alteration trend but smaller than those obtained by using the PBE functional (within 0.30 eV), except for MAGeCl₃ perovskite. When compared with the experimental values of 1.60, 1.90, and 2.20 eV for CsGeI₃, MAGeI₃, and FAGeI₃,¹⁵ results from both DFT methods unavoidably underestimate their bandgaps but exhibit perfectly consistent changing trend.

Fig. 3 Band structures of perovskites using the PBE functional: (a) MAGeCl₃, (b) MAGeBr₃, (c) MAGeI₃, (d) CsGeI₃, (e) FAGeI₃, (f) MOGeI₃, and (g) GAGeI₃. (h) Comparison of perovskites' bandgaps using the PBE and PBE + SOC functionals.

Fig. 4 (a) DOS and PDOS of CsGeI₃ perovskite with the inserted contour plots of wave functions of the VBM (under) and CBM (upper), (b) PDOS of Ge p orbitals in AGeI₃ perovskites, and (c) DOS of MAGeX₃ perovskites.

The DOS and partial DOS (PDOS) of CsGeI₃ perovskite are chosen to elucidate the frontier orbital compositions of AGeI₃ given that these perovskites possess similar electronic properties, as depicted in Fig. 4a. Cs⁺ does not contribute to the band edge states because the states of Cs⁺ do not appear around the Fermi level. In addition, no obvious DOS overlap occurs between Cs⁺ and I⁻, indicating that Cs⁺ weakly interacts with the halide anion. The VBM of CsGeI₃ is mainly contributed by the I 5p and Ge 4s orbitals, whereas the CBM is dominated by Ge 4p orbitals. The contour plots of GeI₆ wave functions, as inserted in the DOS, present the possible electron transition pathways from I 5p (Ge 4s) orbitals to Ge 4p orbitals during photoexcitation. Moreover, electron transition possibly occurs between the Ge 4s and Ge 4p orbitals. This “intra-atomic” bandgap structure usually leads to unfavorable bandgap behavior, where the increase in lattice constant gives rise to further energy splitting between these orbitals and the corresponding increase in bandgap.¹⁶ This phenomenon accounts for the increase in bandgaps of AGeI₃ perovskites resulting from the increase in the radii of A cations. Furthermore, the replacement of Cs⁺ with larger cations could trigger the stereochemically active 4s² lone pair electrons,¹⁵ causing the bandgaps of AGeI₃ perovskites to widen. The bandgaps of AGeI₃ perovskites calculated from the DOS (see Fig. S2 in ESI†) show an increasing trend in the sequence of Cs⁺ → MA → FA → MO → GA, consistent with the results of band structure analyses. The organic cations show no obvious states around Fermi level, indicating that A cations do not directly participate in electron transition. The CBM of trigonal CsGeI₃, MAGeI₃, and FAGeI₃ perovskites are dominated mainly by Ge p_z orbital, whereas the CBM of monoclinic MOGeI₃ and GAGeI₃ perovskites, are dominated mainly by Ge p_y orbital, as shown in Fig. 4b. This phenomenon depends on the microstructure of the crystals. The GeI₃ pyramidal configuration, composed of three short Ge–I bonds, lies along the [001]-direction for the trigonal crystals and induces the excited electrons to jump preferentially into the p_z orbital in CBM. In MOGeI₃ perovskite, the MO cations twist the GeI₃ pyramids and result in rearrangement towards the [010]-direction. In GAGeI₃ perovskite, a 1D chain is formed along the [010]-direction. Fig. 4c shows the TDOS of the MAGeX₃ perovskites with the Fermi level set at the VBM of MAGeI₃. In contrast to that of AGeI₃, the bandgap of MAGeX₃ perovskites decreases as X ionic radius increases from Cl⁻ to I⁻. The VMB is dominated mainly by the antibonding Ge s–X p orbitals. The energy level of I 5p orbitals is higher than that of Cl 3p and Br 4p orbitals, so the VBM up-shifts, reducing bandgaps of MAGeX₃ perovskites as Cl⁻ is replaced by Br⁻ and I⁻.

3.3 Charge-carrier transport

Photogenerated electrons and holes in Ge-based perovskites thermally relax to the CBM and VBM, respectively. The effective masses (m*) of carriers existing around the CBM and VBM are estimated to evaluate the charge transport property by using the PBE functional according to the following equation,²⁶where ε(k) is the band edge eigenvalue, and k is the wavevector. The calculated effective masses of electron and hole ( and , relative to the electron static mass m_o) along the Z (0, 0, 0.5) → Γ (0, 0, 0) direction in CsGeI₃, MAGeI₃, and FAGeI₃ are 0.22/0.23, 0.27/0.29, and 0.66/0.76m_o, respectively. This result indicates that CsGeI₃ perovskite offers the great advantage for the carrier transport owing to its small effective masses. For the MAGeX₃ perovskites, their effective masses of carriers decrease from Cl⁻ to I⁻ (see Table S1 in ESI†). The effective masses of electron and hole of MAGeCl₃ perovskite are 0.77 and 0.39m_o, and the ratio of 0.77/0.39 is considerably larger than 1, indicating that MAGeCl₃ is a hole-transporting semiconductor.²⁷ The MAGeBr₃ and MAGeI₃ perovskites demonstrate the ability for bipolar carrier transport with effective electron and hole masses of 0.33/0.33m_o and 0.27/0.29m_o, respectively. The SOC interaction has little influence on the effective masses of these Ge-based perovskites (see Table S1 in ESI†).

Table 2 provides the Bader charges to estimate the amount of charge transfer of perovskites. The Bader charges of A, Ge, and X ions fluctuate from 0.76 to 0.86 e, 0.67 to 1.08 e, and −0.49 to −0.64 e, respectively. The Bader charges analyses indicate that electrons are transferred from the A and Ge cations to X halide anions. The charges of Ge and X atoms exhibit large deviations from pure ionic interactions (i.e., Ge²⁺ and I⁻), suggesting that the combined covalent and ionic interactions lie between Ge and X atoms.²⁸ The Ge–I covalent bonding characteristic can be supported by the hybridization of the Ge 4s and 4p states with I 5p states at the top valence bands shown in Fig. 4a. The Bader charges of A cations decrease slightly along Cs⁺ → MA → FA → MO → GA in AGeI₃ perovskites, indicating that their charge contributions decrease gradually. As mentioned above, given that A cations do not participate in direct electronic transition, they act as charge donors to supply GeX₃ framework with ∼0.8 e. The Bader charges of MA and Ge in MAGeX₃ perovskites decrease as X goes from Cl⁻ to I⁻, and this phenomenon is attributed to the decreasing abilities of these ions to obtain charges.

Table 2 Bader charges of perovskites

Bader charges	CsGeI3	MAGeCl3	MAGeBr3	MAGeI3	FAGeI3	MOGeI3	GAGeI3
A (e)	0.86	0.85	0.82	0.79	0.78	0.76	0.76
Ge (e)	0.67	1.08	0.91	0.67	0.71	0.72	0.74
X (e)	−0.51	−0.64	−0.58	−0.49	−0.50	−0.50	−0.50

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Fig. 5 shows the electron density of MAGeX₃ perovskites to further illustrate the bonding characteristics of Ge–X. A strong coupling exists between Ge and Cl atoms, as shown in Fig. 5a. However, the long Ge–Cl bond (encircled with the red dotted line) shows no obvious electron density plot, indicating that the covalent coupling interaction of the short Ge–Cl bond is stronger than that of the long Ge–Cl bond. When X goes from Cl⁻ to I⁻, the coupling of Ge–X decreases continuously, as depicted in Fig. 5a–c. Moreover, the interaction intensity can be reflected by the bonding energy of MAGeX₃. A strong coupling corresponds to a large absolute value of bonding energies of MAGeX₃ (X = Cl⁻, Br⁻, and I⁻) are −168.97, −163.73, and −158.00 eV, exhibiting a decreasing trend. MAGeCl₃ is the most stable crystal, whereas MAGeI₃ is the most unstable crystal. This means an additional 10.97 eV energy is therefore required to form MAGeCl₃ than MAGeI₃.

Fig. 5 Electron density at the 0.04a₀⁻³ isodensity surface (a₀: Bohr radius) in (a) MAGeCl₃, (b) MAGeBr₃, and (c) MAGeI₃. The long and short Ge–Cl bonds are encircled with red and blue dotted lines, respectively.

3.4 Optical property

The optical properties can be obtained from dielectric functions, which are usually used to describe the linear response of the crystal system to electro-magnetic radiation. The imaginary part ε₂(ω) of dielectric function is derived from the appropriate momentum matrix elements between the occupied and the unoccupied wave functions within the selection rules over the Brillouin zone,²⁹ and the real part ε₁(ω) of dielectric function follows the Kramer–Kronig relationship. The absorption coefficient I(ω) is related directly to the band structure of the materials and describes their absorptive behavior. Thus, the absorption coefficient I(ω) is determined according to the explicit expression,³⁰

Fig. 6 shows the calculated wavelength dependences of I(ω) of MAGeX₃ and AGeI₃ perovskites. When X goes from Cl⁻ to I⁻, the absorption spectrum of MAGeX₃ is red shifted because of the reduced bandgap. MAGeI₃ shows a strong absorption peak at ∼500 nm, which corresponds to the electron transition from the occupied state I 5p (Ge 4s) to the unoccupied state Ge 4p. To compare with the prototype MAPbI₃, we included the absorption coefficient of MAPbI₃ in Fig. 6a. In the region of 300–450 nm, MAPbI₃ shows a superior absorption intensity than MAGeI₃. In the long wavelength region of 450–800 nm, the absorption coefficient of MAGeI₃ is higher than that of MAPbI₃. The Ge–I bond, rather than Ge–Cl (Br) bond, makes MAGeI₃ the best candidate among the MAGeX₃ perovskites with the superior dielectric property owning to its favorable band structure and effective masses of electron and hole. In AGeI₃ systems, the absorption spectrum is blue shifted and absorption intensity in the visible region decreases in the sequence of Cs⁺ → MA → FA → MO → GA. CsGeI₃ exhibits a better performance compared with the other AGeI₃ perovskites. The Cs⁺ cation, which possesses the smallest ionic radius, not only shortens Ge–I bond to enhance the coupling but also provides more charges to GeI₃ framework, facilitating electron excitation and transport. Taking stability, electronic structure, charge-carrier transport, and optical properties into consideration, CsGeI₃ and MAGeI₃ offer a competitive potential use in perovskite solar cells.

Fig. 6 The absorption coefficient I(ω) of (a) MAGeX₃ and MAPbI₃; (b) AGeI₃ perovskites.

4. Conclusions

First-principles calculations were employed to investigate the structural, electronic, and optical properties of Ge-based MAGeX₃ (X = Cl⁻, Br⁻, and I⁻) and AGeI₃ (A = Cs⁺, MA, FA, MO, and GA) perovskites. In MAGeX₃ systems, MA cations do not directly participate in electronic transition, but act as charge donors to supply GeX₃ framework with ∼0.8 e. The altered Ge–X bonds from Cl⁻ to Br⁻ and I⁻ can increase the volumes, weaken the covalent coupling of Ge–X, lower the bandgaps, reduce the electron and hole effective masses, and cause the red shifting of the absorption spectrum. In AGeI₃ systems, the CsGeI₃, MAGeI₃, and FAGeI₃ perovskites exhibit trigonal structures showing direct-bandgap nature, whereas the MOGeI₃ and GAGeI₃ perovskites exhibit monoclinic structures showing indirect-bandgap nature. Weak interactions exist between A cations and GeI₃ framework, whereas both covalent and ionic interactions exist between Ge cations and I anions. The photo-excited electrons transit from the I 5p–Ge 4s antibonding states to Ge 4p empty states, and the absorption spectrum is blue shifted as A is changed in the sequence of Cs⁺ → MA → FA → MO → GA. Our results not only highlight the effects of A and X on photoelectronic properties of Ge-based perovskites, but also provide theoretical guidance for the design and screening of environmentally friendly materials for perovskite solar cells.

Acknowledgements

This work was supported by NSFC (21303266), the Fundamental Research Funds for the Central Universities (15CX05050A, 15CX08010A, and 14CX02214A), and the Postgraduate's Innovation Project (YCXJ2016084).

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Footnotes

† Electronic supplementary information (ESI) available: The band structures by the PBE + SOC functional, density of states, and electron and hole effective masses of the trigonal perovskites are presented. See DOI: 10.1039/c6ra18534g

‡ These authors have made an equal contribution to this work.

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