Electronic and Optical Properties of CsGeX3 (X= Cl, Br, and I) Compounds.

We used first-principles calculations to investigate the electrical and optical properties of CsGeX3 (X = Cl, Br, and I) compounds. These materials present rich and unique physical and chemical phenomena, such as the optimal geometric structure, the electronic band structure, the charge density distribution, and the special van Hove singularities in the electronic density of states. The optical properties cover a slight red shift of the optical gap, corresponding to weak electron-hole interactions, strong absorption coefficients, and weak reflectance spectra. The presented theoretical framework will provide a full understanding of the various phenomena and promising applications for solar cells and other electro-optic materials.

Introduction

Nowadays, solar cell devices have become one of the outstanding systems in engineering applications and basic science research studies[1−4] because they can provide eco-friendly and renewable energy and are efficient by using a special route to convert photon energy into electricity.[5−13] The principal architectures of the perovskite solar cell include the absorber layer sandwich between the electron-transporting and hole-transporting layers, in which the first component is very important since it absorbs the electromagnetic wave and generates the electron–hole pairs. Apparently, the geometric, electronic, and optical features of these components are regarded as very important characteristics. These factors should be critical in determining the efficiency of light-generated currents and finding the best combination of the three kinds of core components.

Up to now, inorganic perovskite materials,[14−16] such as CsPbX3 (X = Cl, Br, and I),[17,18] have attracted more and more research attention for energy harvesting applications due to their excellent photoelectronic properties. However, there are still many disadvantages,[17] for example, the limit of large-scale applications owing to the toxicity of the lead element.[19] Currently, the lead-free CsGeX3 (X = Cl, Br, and I) compounds are other candidates to replace CsPbX3 perovskites for solar cell and other applications.[19,20] In fact, such compounds cover many interesting properties/novel features and have been investigated by both previous theoretical[21−28] and experimental methods.[29−32]

On the experimental side, Li et al. have successfully synthesized the CsGeI3 compound and demonstrated that this material processes a high photocurrent (6 mA cm2), which means that it is suitable for photovoltaic applications.[29] To enhance the photocurrent generation of WS2 nanoflakes, WS2/CsGeBr3 has been fabricated, with an external quantum yield and a responsivity of about ∼151% and 6.4 (A/W), respectively.[30] The absorption edges of CsGeCl3 and CsGeBr3 are very sensitive and significantly red-shift under applied external hydrostatic pressure. This phenomenon was verified by the experiment of Schwarz and his co-workers.[31] Besides, photoluminescence and absorption measurements of CsGeCl3, CsGeBr3, and CsGeI3 have been reported.[32]

On the theoretical aspects, the fundamental electrical and optical properties of the CsGeCl3, CsGeBr3, and CsGeI3 materials have been investigated with various approaches, such as first-principles calculations,[26,33] the effective mass approximation,[33] and the tight-binding model;[23,34] among them, the first-principles calculations are the most effective approach to identify the optimal geometric, electronic, and optical properties of these compounds. According to current theoretical predictions, CsGeX3 belongs to the direct gap semiconductor materials,[21] with their gap values ranging from 0.82 to 7.91 eV, depending on the approximations.[22−27] For example, using the Perdew–Burke–Ernzerhof (PBE)-sol functional, Jong et al.[33] provided the energy band gaps of 1.19, 1.46, and 2.13 eV[28] for CsGeI3, CsGeBr3, and CsGeCl3, respectively. The accuracy of the energy gap could be improved with the typical values of 1.64, 2.34, and 3.24 eV for CsGeI3, CsGeBr3, and CsGeCl3, respectively, when adopting the hybrid functional (HSE) approximation.

Generally, the geometric and electronic properties of CsGeX3 have been studied; however, the critical factors, such as the orbital hybridizations[35] in the chemical bonds, have not been discussed in detail. Only a few studies have been performed on the optical properties so far;[20,26,33,36] most theoretical investigations are based on the density functional theory (DFT), and thus, the theoretical predictions on optical properties of CsGeX3 compounds do not accommodate the experiment results.[32] Note that the operation of a solar cell is highly dependent on the formation of excitons since the charge separations are hugely influenced by these electron–hole bound states. Although the excitonic effects in the broad energy range of materials,[33,37] including CsGeX3,[33] have been qualitatively estimated by effective mass models, the quantitative description of these bound states is rather limited. There are few works that relied on ab initio Bethe–Salpeter equation (BSE) calculations;[20,38−40] however, the formation as well as the nature of the excitonic states has not been elucidated yet.

In this work, the first-principles calculations were utilized to investigate the geometric and electronic properties and effects of the electron–hole interactions on the optical properties of the CsGeX3 (X = Cl, Br, and I) compounds. The theoretical framework is based on the examination of the geometric structure, the accurate quasi-particle energy band structure, the special van Hove singularities on the electronic density of states (DOS), the band decomposed charge densities, the dielectric functions in the case of presence/absence of excitonic effects, the exciton wave functions, and other related optical properties. As a result, the influences of excitonic effects on the optical absorption spectrum and its nature have been clarified. The CsGeCl3, CsGeBr3, and CsGeI3 compounds are excellent candidates for solar cell and opto-electronic applications. The presented theoretical studies in this work are useful for solar cells and other materials.

Computational Details

We investigated the DFT based on the VASP Package[41] to present the optimized structure of the CsGeCl3, CsGeBr3, and CsGeI3 compounds. The related problems to exchange–correlation functions were solved by adopting the PBE form of the generalized gradient approximation (GGA).[42] To treat the core issues, the projector augmented wave (PAW) pseudopotential was used.[43] The cutoff energy of 500 eV was used for the expansion of the plane wave. The Brillouin zone was integrated with a sufficient k-point mesh of 20 × 20 × 20 in the Γ-centered sampling technique for optimizing the geometric structure. Between two successive simulation steps, the ground state’s convergence condition was set to 10–8 eV. During the geometric optimization, all atoms and the cell could fully relax until the Hellmann–Feynman force acting on each atom was smaller than 0.01 eV.[28,44]

In the current research, the electronic properties of CsGeCl3, CsGeBr3, and CsGeI3[45,46] were studied by using the GW method (G stands for Green’s function and W represents the screened Coulomb potential), in which the quasi-particle energies were obtained within the G3W0 approximation (three self-consistent updates for the quasi-particle Green’s function) for the self-energy,[47] the response function’s cutoff energy was set to 200 eV, and 12 × 12 × 12-centered k-points sampling was used to represent reciprocal space. The quasi-particle band structure was plotted by using the Wannier 90 code.[48,49] Simultaneously, the DFT electronic band structures in the case of with/without spin–orbit coupling (SOC) are also calculated for comparison. However, the SOC effects have been ignored in our GW calculations due to the extremely expensive many-body effects. Based on the electronic wave functions, the single-particle excitations were described by Fermi’s golden rule[50]in which the intensity of the excitation peaks is described by the matrix element, |e⟨v||c⟩|2, while the available transition channels are defined by the joined DOS, .

In addition to the independent particle excitations, the presence of exciton states may have a significant impact on the optical responses. The wave functions related to these bound states of electrons and holes could be expressed by using the following expressionin which the amplitude A is determined by solving the standard BSE[51]where EQP and EQP, respectively, are the quasi-particle energies of the valence and the conduction states as obtained with the GW method. Keh is the kernel describing the correlated electron–hole pairs, and Ωis the energy of the excited states. The imaginary part of the dielectric function ϵ2(ω) is calculated from the excitonic states as

In this part, the Tamm–Dancoff approximation[52] was used; moreover, the energy cutoff and k-point sampling are set resemblances as in the GW calculations. Lorentzian with 100 meV broadening was used to replace the delta function. Since we are dealing with the low-frequency part of the absorption spectra, the four lowest conduction bands (CBs) and the seven highest valence bands (VBs) in the Bethe–Salpeter kernel are sufficient to describe the excitonic effects. All parameters in this works have carefully tested for the convergence of the calculations (Figure ).

Figure 1

Convergence of the electronic band gap of CsGeCl3 with respect to (a) self-consistent updates, (b) k-grid, and (c) cutoff frequency.

Results and Discussion

The primitive cell of inorganic CsGeX3 (X = Cl, Br, and I) perovskite materials is depicted in Figure a. These structures are part of the R3m space group. A unit cell consists of one Cs atom, one Ge atom, and three Cl/Br/I atoms. The optimized lattice parameters have been calculated and are shown in Table after achieving full lattice and atomic relaxations, the minimum of total energy. The calculated lattice constants of the these compounds agree with the available experimental and theoretical results.[26,28] The basic architecture of CsGeX3 includes the Cs atom with a preferred 12 Cl/Br/I atom coordinate number, and the Ge atom is centered in the slightly distorted octahedron with three different Ge–Cl/Ge–Br/Ge–I chemical bonds (Figure b). This type of octahedral distortion causes electric polarization which occurs spontaneously, which can improve the separation of charge carriers and allow the photovoltage to exceed the band gap.[53,54] Furthermore, the slightly non-uniform environment demonstrates the rather complex hybridizations of orbitals in the chemical bonds, which are responsible for the essential electronic and optical properties.

Figure 2

(a) Geometric structure of CsGeX3; the unit cell is dashed black. (b) Two kinds of chemical bondings: [CsX12] and [GeX6] (X = Cl, Br, and I).

Table 1

Number of Bonds, the Bond Length of Chemical Bonding of Cs–Cl/Cs–Br/Cs–I and Ge–Cl/Ge–Br/Ge–I Bonds, and the Lattice Constant of the CsGeX3 (X = Cl, Br and I) Compounds

				lattice constant
space group (R3m)	atom–atom	number of bonds	bond length (Å)	a (Å)	α (deg)
CsGeCl3	Cs–Cl	12	3.90,3.91, 3.97	5.53	89.66
				5.33a	89.88a
	Ge–Cl	6	2.41,3.14	5.53b	88.98b
				5.44c	89.63c
CsGeBr3	Cs–Br	12	4.03,4.09, 4.17	5.77	88.61
				5.56a	89.15a
	Ge–Br	6	2.58,3.21	5.78b	88.35b
				5.63c	88.74c
CsGeI3	Cs–I	12	4.30,4.40	6.10	88.61
				5.94a	88.68a
	Ge–I	6	2.79,3.30	6.15b	87.78b
				5.98c	88.61c

The lattice parameters were determined from the PBE functional from ref (33)

The lattice parameters were determined from the PBE functional from ref (20)

These values are from experiments.[60]

The CsGeX3 perovskite compounds sketch unique electronic properties which are entirely reflected in the energy band structure along the high symmetry points in the first Brillouin zone, named Γ–––Γ. The effective energy which is related to the main electronic features of these compounds lies mostly in the energy range of −8.0–6.0 eV (left panel in Figure ). Obviously, a lot of sub-bands are formed by specific atoms with numerous orbitals in the unit cell. The energy dispersions of these states reveal the parabolic, saddle, and camelback shapes (−5 to −2 eV). Many maximum and minimum points in the wave-vector space generate van Hove singularities in the DOS. The CsGeX3 compounds belong to direct band gap semiconductors, in which the lowest unoccupied states and the highest occupied ones locate on the same Z point. The effects of SOC make a minor reduction of the gap; these effects are significantly smaller than that of CsSnX3[34] or CsPbX3[34,55] due to the lighter Ge atom. In addition, the relativity effects also cannot create the higher spin degeneracy breaking in the electronic state vicinity of the Fermi level, and thus, the dark excitons due to spin-forbidden transitions are totally absent. The gap values of the DFT calculations in the absence and presence of SOC are summarized in Table and Figure S1 (Supporting Information), and these results are in accordance with previous works.[16−18] After the GW approximations are adopted, the band gap values are increased to 4.01, 3.05, and 1.88 eV, respectively, for CsGeCl3, CsGeBr3, and CsGeI3 compounds. The slightly over-estimated band gap values originate from the absence of SOC.

Figure 3

Left panels show the GW quasi-particle band structures of the CsGeCl3, CsGeBr3, and CsGeI3 compounds (solid black). The green arrows at the Z point indicate the direct gaps. In addition, DFT electronic band structures are also added for comparisons (dash red). Right panels indicate the van Hove singularities in the DOS of CsGeCl3, CsGeBr3, and CsGeI3 compounds.

Table 2

Electronic Properties of CsGeX3 Compounds: Fundamental Electronic Band Gap from DFT, DFT + SOC, HSE, HSE + SOC, and GW Approximationsa

	band gap (eV)
systems	DFT	DFT + SOC	HSE	HSE + SOC	GW	EX
CsGeCl3	2.12	2.05	2.81	2.28b	4.01	3.40d
	2.13b	2.08b	3.24b		4.37c
	2.07c	2.01c
	1.97c
CsGeBr3	1.40	1.35	1.98	1.77b	3.05	2.39d
	1.46b	1.42b	2.34b		2.69c
	1.44c	1.39c
	1.09d
CsGeI3	1.05	0.88	1.44	1.52b	1.88	1.63e
	1.19b	1.06b	1.64b		1.69c
	1.14c	0.99c
	0.80d

The experimental (EX) results are also listed for comparison.

The electronic band gap estimated from ref (33)

The electronic band gap estimated from ref (20)

These values are from experiments.[20]

Reference.[61]

The effective mass of the hole/electron, being inversely proportional to the second-order derivative of energy, is able to reveal the carrier mobility. Since the carrier mobility in most cases is not strongly dependent on the band gap and the GW corrections insignificantly modify the band curvatures, for simplicity, we evaluate the effective mass at the DFT level of theory. According to delicate analysis, we can evidence that the hole states in the top of the VB are faster than the electron ones in the bottom of the CB for all CsGeX3 compounds. Additionally, we can also see that the effective mass/kinetic energy decreases/increases as the halogen changes from Cl and Br to I (Table ). This behavior is related to the interactions between the nuclear charge and valence charge of the halogen atoms. In fact, the interactions of valence charge in the halogen atoms with its nuclear charge decrease in the following order: Cl > Br > I atom. The weaker interaction of the latter mostly comes from the larger atomic radius, and thus, its carriers are less tightly bound to the nuclei/higher mobility than the formers. The electronic band structure, the carrier mobility, and projected DOS together with the band decomposed charge density can be used to fully understand the origination and behavior of excitons in the CsGeX3 compounds.

Table 3

Effective Mass of Electrons, Holes, and Reduced Mass Calculated with the PBE Functionala

				ε1(0)
systems	me*	mh*	μ	DFT	GW	GW + BSE	Eb(eV)
CsGeCl3	0.272	0.284	0.139	3.36	1.70	2.45	0.28
	0.450b	0.300b	0.140c	3.24b
	0.270c	0.280c
CsGeBr3	0.176	0.181	0.089	4.29	1.97	2.85	0.25
	0.191b	0.197b	0.09c	3.88b
	0.180c	0.190c
CsGeI3	0.137	0.143	0.070	5.86	2.53	4.19	0.14
	0.176b	0.186b	0.080c	4.83b
	0.150c	0.160c

The static dielectric constants of with and without excitonic effects and exciton binding energy calculated from the GW + BSE method.

The effective mass of electrons and holes was calculated with the DFT method.[61]

Referene.[33]

In addition to quasi-particle energy band structures, the DOS could provide complete information about the electronic properties as well as the orbital hybridizations in the chemical bonds. As clearly indicated in the right panels in Figure , the DOS of three-dimensional CsGeX3 compounds exhibits a lot of symmetric/asymmetric and shoulder structures arising from the parabolic, saddle, and dispersionless energy sub-bands in the band structure. The van Hove singularities of different orbitals mixed with each other are evidence of the complicated orbital hybridizations in the chemical bonds. Around the Fermi level, the DOS disappears and creates a sizable energy band gap of 4.01, 3.05, and 1.88 eV for CsGeCl3, CsGeBr3, and CsGeI3, respectively. According to the orbital-projected DOS, the occupied states around the Fermi level can be divided into three paths, for example, for CsGeCl3, (i) −8 eV < Ev < −5 eV is due to the strong orbital hybridizations of Ge-4p and Cl-3p states and (ii) −5 eV < Ev < −2 eV is co-dominated by the Ge-4s and Cl-3p orbitals. Apparently, due to the very high ionization energy, the contributions of the Cl-3s orbital lie very deep and thus do not hybridize with the others. As for the CsGeBr3 and CsGeI3 cases, these states move farther up, reflecting the higher atomic energy levels in the latter elements of the periodic table. For the unoccupied states, (iii) 2 eV < Ec < 6 eV is mainly dominated by Cl-3p and Ge-4p orbitals and only has slight contributions to the Cs-6s states. Similar phenomena are also observed for the CsGeBr3 and CsGeI3 compounds. The relationship between the energy band structures, the projected DOS, and the band decomposed charge densities (discussed later) can identify the orbital hybridizations in all Cs–Cl/Cs–Br/Cs–I and Ge–Cl/Ge–Br/Ge–I bonds of the 3D CsGeCl3, CsGeBr3, and CsGeI3 compounds, respectively.

To further comprehend the orbital hybridizations of the Cs–Cl/Cs–Br/Cs–I and Ge–Cl/Ge–Br/Ge–I chemical bonds, we have constructed the band decomposed charge densities (Figure a–c) for three parts of the valence and conduction states for the CsGeCl3, CsGeBr3, and CsGeI3 compounds. Since the plots for the CsGeCl3 compound look so similar to those of CsGeBr3 and CsGeI3, we only present the figures for CsGeCl3 as an example. As seen in Figure a, the lower portion of the occupied states (−8 eV < Ev < −5 eV) clearly presents the π bonding character of the Ge-4p and Cl-3p orbitals, in which the Cl-3p charge density exhibits an important contribution. The charge density on the top of the VBs (−5 eV < Ev < −2 eV) (Figure b) is dominated by σ bonding states formed from the comparable contribution of the Cl-3p and Ge-4s states. On the other hand, the lower portion of the unoccupied states (2 eV < Ec < 6 eV) (Figure c) clearly shows the σ bonding behavior originating from Cl-3p and Ge-4p states. The Ge states in this portion have a somewhat higher density than those on the top of the VBs because of the large contribution of Ge-4p orbitals. For the entire energy spectrum, the charge density associated with the Cs element is insignificant since it is related only to the 6s orbital. The band decomposed charge density, together with the energy band structure and the partial charge DOS, is useful for comprehending the orbital characters of the exciton states in the optical properties.

Figure 4

Band decomposed charge densities of CsGeCl3: (a) lower portion of VBs (−8 eV < Ev < −5 eV), (b) upper portion of the VBs (−5 eV < Ev < −2 eV), and (c) bottom portion of the CBs (2 eV < Ec < 6 eV).

To understand the optical properties, the real part [ε1(ω)] and imaginary part [ε2(ω)] of the dielectric functions in the absence/presence of the excitonic effects of the CsGeX3 systems were investigated (Figure ). In the absence of hole and electron couplings, the real part of the dielectric function ε1(ω) is weakly dependent on the energy in the inactive region, and the dielectric constant at zero energy ε1(0) is about 1.7, 2.0, and 2.7 eV, respectively, for the CsGeCl3, CsGeBr3, and CsGeI3 compounds. These values increase after the excitonic effects have been adopted (details in Table ). The ordering of the dielectric constant of CsGeX3 is satisfied with its band gap values and strongly affects the exciton binding (discussed later). Moreover, the dispersionless feature of the real part at the low frequency is also an important factor to determine the vanishing range of the absorption coefficients α(ω) and the reflectance coefficient R(ω) in the low energy region.

Figure 5

(a–c) Real part ε1(ω) and (d–f) the imaginary part ε2(ω) of the dielectric functions with/without excitonic effects for the CsGeCl3, CsGeBr3, and CsGeI3 compounds, respectively. (g–i) Enlarged figures for the imaginary part ε2(ω) of the dielectric functions for the CsGeCl3, CsGeBr3, and CsGeI3 compounds, respectively. The dash-blue line represents E1 and E2 exciton peaks, and the green line indicates the fundamental gap.

Concerning the imaginary part, ε2(ω), the optical gap (green color) is about 4.00, 3.03, and 1.89 eV, respectively, for the CsGeCl3, CsGeBr3, and CsGeI3 compounds in the absence of the excitonic effects. These values are equal to the fundamental direct gap Egi in Figure owing to the conservation of the momentum. After taking into account the Coulomb interactions, the excitation strength is enhanced due to the enhancement of the electron–hole wave function overlap. Moreover, the presence of the excitonic effects also creates a slightly red-shifted threshold energy. The exciton binding is defined as the difference between the fundamental band gap and the optical gap after taking the Coulomb interaction into account; furthermore, the binding energy is also strongly proportional/inversely proportional to the effective mass of the carriers/dielectric screening constants. According to the delicate analysis, the typical exciton binding energy of CsGeX3 lies in the following order: Cl > Br > I atom (Table and Figure g–i); this trend agrees well with the effective mass model.[33,56] It is very important to note that the electron–hole interactions in CsGeI3 are relatively weak and comparable with those in the CsPbI3 solar cell;[55] the meta-stable exciton states easily break down to create free electron and hole carriers and thus efficiently generate the electrical currents.

Apparently, the first two peaks (E1 and E2) in Figure are due to the excitonic effects since the excitation spectra without Coulomb attraction interactions nearly disappear in the ranges of 0 to 4 eV, 0 to 3 eV, and 0 to 2 eV for the CsGeCl3, CsGeBr3, and CsGeI3 compounds, respectively. These sharp peaks originated from the coherent superposition of exciting electrons and holes. According to the exciton wave functions, as clearly depicted, the exciton states are located in the reciprocal space. Going from chlorine to iodine, the effective mass of carriers and the band gap decreases, and as a consequence, the extent of the exciton in k-space decreases. The first and second exciton states, respectively, arise from the vertical transitions from the last valence states to the first and second conduction states of the band extreme in the vicinity of the Z points (the blue and red circles in Figure ), in which the excited holes and the excited electrons are characteristic for the Ge-4s, Cl-3p/Br-4p/I-5p, and Ge-4p orbitals, respectively. Obviously, owing to the high excitation strength around the visible region and beyond, the unique optical excitations of the CsGeCl3, and CsGeBr3 compounds can be used for electro-optics applications; specially, the CsGeI3 compound can be exploited for photovoltaic applications.

Figure 6

Exciton wave functions of the CsGeCl3/CsGeBr3/CsGeI3 compound are predicted as fat-band pictures. The red and blue circles show two E1 and E2 exciton peaks in the imaginary part of the dielectric functions, respectively.

The absorption coefficient α(ω),[57] the imaginary part of the refractive index k(ω), the real part of the refractive index[58]n(ω), and the reflectivity R(ω)[59] have been calculated and are shown in Figure a–c to better elucidate the optical properties of the CsGeCl3, CsGeBr3, and CsGeI3 compounds. Apparently, these compounds reveal strong absorption and weak reflectance coefficients. The α(ω) is equal to 0, and the R(ω) is almost independent of the energy in the visible region as they lack electronic excitation contributions. The fast increase of the absorption coefficient is contributed by the various inter-band transitions. The inverse values of the absorption coefficient are equal to 400, 200, and 100 Å for CsGeCl3, CsGeBr3, and CsGeI3, respectively. These findings imply that photons propagating in the medium are easily absorbed by the rich electronic excitations. As for the reflectance coefficient, the static reflective index is = 0.05, 0.08, and 0.13 eV for the CsGeCl3, CsGeBr3, and CsGeI3 compounds, respectively. A significant enhancement and a huge fluctuation are presented at a higher frequency. Obviously, these materials reveal low reflectivity and strong absorption, making them excellent candidates for solar cell applications.

Figure 7

Absorption coefficient α(ω), the imaginary part of the refractive index k(ω), the real part of the refractive index n(ω), and the reflectivity R(ω) for the CsGeCl3 (a), CsGeBr3, (b) and CsGeI3 (c) compounds with and without excitonic effects.

Conclusion and Remarks

To summarize, accurate first-principles calculations were performed to calculate the rich and unique electronic and optical properties of CsGeX3 (X = Cl, Br, and I) compounds. Consequently, the essential properties and potential applications for solar cell materials can be fully comprehended. The featured characteristics of these materials include an unusual geometric structure, unique electronic properties with a direct electronic band gap nature, the complicated orbital hybridizations comprehended from the electronic band structure, the band decomposed charge density, and the special van Hove singularities in the DOS. In addition, the exciton states and their nature have been qualitatively and quantitatively evaluated through delicate analysis on the mobility of carriers, the dielectric screening, orbital hybridizations in the initial and final states, and the exciton wave functions; other optical properties such as reflectance spectra and absorption coefficient are also achieved. Most importantly, the exciton binding of these materials is reliable to allow for the production of an electrical current upon sunlight adsorption. As a result, CsGeX3 (X = Cl, Br, and I) compounds emerge as promising candidates for solar cell and electro-optics applications.