Investigations on the Carrier Mobility of Cs 2 NaFeCl 6 Double Perovskites

: Double perovskite materials have gradually become widely studied due to their potential applications in solar cells and other optoelectronic devices. We take Cs 2 NaFeCl 6 as an example to investigate the carrier mobility with respect to the acoustic phonon and the optical phonon scattering mechanisms. By considering the deformation potential, carrier effective mass, and bulk modulus, the longitudinal acoustic (LA) phonon-determined mobilities for electrons and holes in Cs 2 NaFeCl 6 are found to be µ e = 2886.08 cm 2 v − 1 s − 1 and µ h = 39.09 cm 2 v − 1 s − 1 , respectively. The optical scattering mechanism involves calculating the Fröhlich coupling constant, dielectric constant, and polaron mass to determine the multiple polar optical (PO) phonon-scattering-determined mobilities, resulting in µ e = 279.25 cm 2 v − 1 s − 1 and µ h = 21.29 cm 2 v − 1 s − 1 , respectively. By combining both interactions, the total electron mobility and hole mobility are determined to be 254.61 cm 2 v − 1 s − 1 and 13.78 cm 2 v − 1 s − 1 , respectively. The findings suggest that the polarization of both electrons and ions, small coupling constant, and bulk modulus in Cs 2 NaFeCl 6 ’s lattice make PO scattering a significant contribution to carrier mobility in this specific double perovskite, highlighting the importance of considering this in enhancing the optoelectronic properties of Cs 2 NaFeCl 6 and other double perovskites.


Introduction
Solar cells, such as silicon solar cells, organic solar cells, and perovskite solar cells (PSCs), are considered highly promising energy sources due to their renewable and clean properties [1][2][3][4][5].Research on halide perovskite materials has significantly advanced in recent years, with the power conversion efficiency (PCE) increasing from an initial value of 3.8% to a remarkable 26.41% in just 15 years [6,7].Furthermore, the theoretical PCE of PSCs can potentially reach 31%, surpassing the theoretical limit of 29% for silicon solar cells.These exceptional optoelectronic properties of halide perovskites stem from their adjustable band gap, strong absorption, and high carrier mobility [8][9][10][11].However, the investigation of organic-inorganic perovskites, such as MAPbBr 3 and MAPbI 3 , has revealed certain drawbacks like instability and Pb toxicity, which limit their practical applications.In response, extensive research efforts are being directed towards identifying new halide perovskites that exhibit improved stability and environmental compatibility, while retaining their exceptional optoelectronic properties.Consequently, interest in double perovskites is steadily growing among researchers.Double perovskites with the formula A 2 B ′ B ′′ X 6 are created by substituting the B-site (Pb 2+ ) cation with a lead-free combination of monovalent (B ′ ) and trivalent (B ′′ ) cation pairs.Researchers have demonstrated that the double perovskites Cs 2 AgBiX 6 (X = Cl, Br, I) have potential for optoelectronic applications through Crystals 2024, 14, 547 2 of 12 the precise control of surface defects [12].For instance, Cs 2 AgBiCl 6 has shown promise in high energy radiation detection [13,14] and as a white light emitter [15].Various strategies can be employed to adjust the physical properties of double perovskites.For instance, Cu doping has been shown to enhance the photoelectric performance of Cs 2 AgSbCl 6 [16].The band gap of double perovskites can also be modulated by changing A cations, as has been demonstrated theoretically [17].Additionally, studies have successfully synthesized (CH 3 NH 3 ) 2 AgBiI 6 with high stability in air, making it a viable option for tandem solar cells [18].
Despite advancements, the efficiency of double perovskite solar cells still lags behind that of traditional perovskite solar cells.Semiconductors' optoelectronic properties are influenced by various factors, including the absorption efficiency, quantum efficiency, and carrier mobility [19,20].Among these factors, the carrier mobility, which refers to the transport efficiency of light-generated electron/hole pairs, plays a crucial role in determining the power conversion efficiency (PCE) of solar cells.Several studies have indicated that CsPbBr 3 has an electron mobility of around 1000 cm 2 v −1 s −1 , which is comparable to that of Si [21,22].Additionally, the mobility of MAPbI 3 can reach up to 800 cm 2 v −1 s −1 , while MASnI 3 boasts a mobility exceeding 2000 cm 2 v −1 s −1 [23,24].These high mobilities play a crucial role in enhancing the optoelectronic performance of perovskite solar cells (PSCs).Conversely, Cs 2 AgBiBr 6 double perovskite demonstrates a relatively low electron/hole mobility of 16.7/14.2cm 2 v −1 s −1 [25], leading to subpar photovoltaic performance [26,27].As a result, extensive research efforts are being conducted to enhance the transport properties of double perovskites.For example, some studies have shown that impurity doping or adjusting the composition of Cs 2 AgInCl 6 can modulate the carrier mobility, resulting in the development of novel double perovskites [28].
Recently, a new double perovskite, Cs 2 NaFeCl 6 , was successfully synthesized, displaying environmental stability, thermal stability, and thermochromic cyclic stability [29,30].The exciton-binding energy of Cs 2 NaFeCl 6 is only 20 meV at 80 K, facilitating the separation of light-induced electron/hole pairs and leading to high optoelectronic efficiency [31].However, the experimentally measured carrier mobility of Cs 2 NaFeCl 6 at room temperature is initially low.To enhance the transport properties, various methods have been implemented.For example, Ag doping has been found to elevate the electron mobility from 1.06 cm 2 v −1 s −1 to 15.3 cm 2 v −1 s −1 .Additionally, the incorporation of Ag substantially boosts the carrier diffusion length from 3.7 nm to 311 nm.Notably, the measured hole carrier mobility is unexpectedly higher than the electron carrier mobility in Cs 2 NaFeCl 6 , which differs from other perovskites [32].In order to enhance the optoelectronic performance of Cs 2 NaFeCl 6 and other double perovskites for practical applications, a comprehensive understanding of their electronic properties is crucial.It is imperative to investigate how carriers move within the crystalline structure of Cs 2 NaFeCl 6 and identify the scattering mechanism that dictates the upper limit of mobility.Only by clarifying these aspects can we develop precise strategies to enhance the transport properties of Cs 2 NaFeCl 6 .To date, no research has addressed these critical factors.
Therefore, the acoustic and optical phonon scattering mechanisms in Cs 2 NaFeCl 6 were investigated using a semi-empirical model of a longitudinal acoustic (LA) phonon and polar optical (PO) phonon.It was observed that the mobilities of electrons/holes in Cs 2 NaFeCl 6 are primarily influenced by PO phonon scattering, attributed to its polarization of both electrons and ions, weak coupling, and small bulk modulus.These insights can potentially aid in enhancing the transport properties of Cs 2 NaFeCl 6 and other double perovskites.

First Principle Calculations
First principle simulations were carried out using VASP 5.2 software to study the double perovskite material Cs 2 NaFeCl 6 , with the crystal structure (file 2001471) obtained from the CCDC database [33].Density functional theory was utilized [34] to analyze the crystal structure, bulk modulus, carrier effective mass, and dielectric constant of the Cs 2 NaFeCl 6 perovskite.The k-point mesh was set at 4 × 4 × 4 for lattice relaxation and 8 × 8 × 8 for electronic structure calculations.Spin-orbit coupling (SOC) was considered due to the heavy elements in Cs 2 NaFeCl 6 perovskite.The Projector-Augmented Wave (PAW) method with a cutoff energy of 400 eV was used to calculate the electronic wave functions.The convergence thresholds for the total energy and force in the iterative process are 10 −4 eV per atom and 0.01 eV per Å, respectively.

Mobility Calculations
As carriers traverse the vibrating lattice of solid materials, they undergo continuous collisions with lattice atoms or ionized impurities, leading to scattering that alters their trajectories and ultimately determines the upper limit of their carrier mobility.Various factors contribute to these scattering interactions, including acoustic phonon scattering and optical phonon scattering.At low temperatures, LA phonon scattering predominantly influences the acoustic phonon scattering.Specifically, the LA phonon induces changes in the band edges, resulting in potential disturbances and subsequent carrier scattering.The carrier mobility derived from the LA phonon model can be represented by Equation (1) [35].
where e, ℏ, and k B represent the unit charge, the reduced Planck constant, and the Boltzmann constant, respectively.T denotes the temperature, while C α stands for the bulk modulus, and E α is the volume deformation potential.
When the carriers are scattered by optical phonons, the polarized electric field accompanied by longitudinal optical vibration impacts a strong coupling interaction on electrons, known as Fröhlich action [36].Therefore, the mobility determined by polaron LO phonon scattering is referred to as the PO-determined carrier mobility in the subsequent sections.The degree of polarization is denoted by the electron-phonon coupling constant α [37], which can be calculated using the following formula: where ε ∞ is the high-frequency dielectric constant, ε s is the static dielectric constant, Ω is the optical branch frequency, m b is the effective carrier mass, and ε 0 is the vacuum dielectric constant.The PO-determined carrier mobility can be described using Equation (3) [38].
where G is the degeneracy factor, which usually varies between 0.8 and 2, and we set this value as 1 in this work [38].m p is the polaron mass, and ω LO is the optical branch frequency.

Geometric Structures
The relaxed Cs 2 NaFeCl 6 supercell is depicted in Figure 1a, with lattice parameters of 10.33 Å.The bond lengths of Cs-Cl, Na-Cl, and Fe-Cl are measured at 3.66 Å, 2.86 Å, and 2.31 Å, respectively.The structure of Cs 2 NaFeCl 6 adheres to the standard cubic double perovskite lattice, with the Na-Cl-Fe bond angles measuring 180.00 • .
2.31 Å, respectively.The structure of Cs2NaFeCl6 adheres to the standard cubic double perovskite lattice, with the Na-Cl-Fe bond angles measuring 180.00°.The stability of a perovskite can be evaluated using the tolerance factor t, which is defined in Equation ( 4).
were RA, RB, and RX represent the radii of the constituent ions in ABX3 perovskites.It is important to note that in double perovskite materials with the A2B′B″X6 structure, RB in Equation ( 4) can be calculated using the average atomic radii of the monovalent (B′) and trivalent (B″) cation pairs.To assess the stability of the Cs2NaFeCl6 lattice further, the tolerance factor t can be determined using the following equation: where <Cs-Cl> represents the interatomic distance of Cs and Cl atoms, namely, the Cs-Cl bond length.Meanwhile, <X-Cl>avg equals the average value of the Na-Cl and Fe-Cl bond lengths.A cubic structure is considered stable when the value of t falls between 0.9 and 1.0; otherwise, the structure may distort to enhance bonding interactions.The results for Cs2NaFeCl6 are presented in Table 1, revealing a tolerance factor of 1.00, indicating the stability of the cubic phase.This suggests that the Cs2NaFeCl6 lattice is isotropic with no lattice distortion, a finding supported by the identification of Raman active vibration modes.The polarization characteristics of these modes imply Fm-3m symmetry for the Cs2NaFeCl6 lattice, indicating an average ordered distribution of Fe and Na atoms throughout the lattice [39].

LA Phonon-Determined Carrier Mobility
As per Equation (1), detailed in Section 2.2, various factors must be determined prior to calculating the mobility of LA phonons.Of these factors, the effective mass of carriers holds paramount importance.Utilizing first principle calculations, we acquired the band structure of Cs2NaFeCl6, depicted in Figure 2b.To ensure precision, we accounted for the The stability of a perovskite can be evaluated using the tolerance factor t, which is defined in Equation ( 4).
were R A , R B , and R X represent the radii of the constituent ions in ABX 3 perovskites.It is important to note that in double perovskite materials with the A 2 B ′ B ′′ X 6 structure, R B in Equation ( 4) can be calculated using the average atomic radii of the monovalent (B ′ ) and trivalent (B ′′ ) cation pairs.To assess the stability of the Cs 2 NaFeCl 6 lattice further, the tolerance factor t can be determined using the following equation: where <Cs-Cl> represents the interatomic distance of Cs and Cl atoms, namely, the Cs-Cl bond length.Meanwhile, <X-Cl> avg equals the average value of the Na-Cl and Fe-Cl bond lengths.A cubic structure is considered stable when the value of t falls between 0.9 and 1.0; otherwise, the structure may distort to enhance bonding interactions.The results for Cs 2 NaFeCl 6 are presented in Table 1, revealing a tolerance factor of 1.00, indicating the stability of the cubic phase.This suggests that the Cs 2 NaFeCl 6 lattice is isotropic with no lattice distortion, a finding supported by the identification of Raman active vibration modes.The polarization characteristics of these modes imply Fm-3m symmetry for the Cs 2 NaFeCl 6 lattice, indicating an average ordered distribution of Fe and Na atoms throughout the lattice [39].

LA Phonon-Determined Carrier Mobility
As per Equation (1), detailed in Section 2.2, various factors must be determined prior to calculating the mobility of LA phonons.Of these factors, the effective mass of carriers holds paramount importance.Utilizing first principle calculations, we acquired the band structure of Cs 2 NaFeCl 6 , depicted in Figure 2b.To ensure precision, we accounted for the SOC effect in our calculations.Figure 2b illustrates that both the valence band maximum (VBM) and the conduction band minimum (CBM) are situated at the Γ point, constituting a direct band gap with a value of 1.09 eV.This implies that the electron momentum can remain constant during the transition process, leading to enhanced luminous efficiency and making it well suited for the fabrication of photoelectric devices.The effective masses of the electron and hole are obtained by fitting the formula of m* = ℏ 2 /(∂ 2 E(k)/∂k 2 ) on the VBM and CBM, which are m h * = 2.22 m 0 and m e * = 0.42 m 0 , respectively.This shows that the hole's effective mass in Cs 2 NaFeCl 6 is about 5 times larger than the electron's effective mass, indicating that the electrons are more dominant than the hole in the carrier migration.In addition to effective masses, the bulk modulus and volume deformation potential play crucial roles in the LA phonon model.The bulk modulus characterizes the connection between the volumetric strain and average stress.The calculated bulk modulus of Cs 2 NaFeCl 6 is 20.32 GPa, a similar value to that of other halide perovskites [40].The bulk modulus is determined by analyzing the second derivative of the system's energy in relation to volume variations, thus showing sensitivity to the lattice phonon modes.In the case of halide perovskites, a smaller bulk modulus is linked to the presence of low-frequency phonon modes within the inorganic lattice [41].We believe that the small bulk modulus observed in Cs 2 NaFeCl 6 suggests that electrons mainly interact with these low-frequency lattice phonons, effectively reducing the charge reorganization.
Crystals 2024, 14, x FOR PEER REVIEW 5 of 12 SOC effect in our calculations.Figure 2b illustrates that both the valence band maximum (VBM) and the conduction band minimum (CBM) are situated at the Γ point, constituting a direct band gap with a value of 1.09 eV.This implies that the electron momentum can remain constant during the transition process, leading to enhanced luminous efficiency and making it well suited for the fabrication of photoelectric devices.The effective masses of the electron and hole are obtained by fitting the formula of m* = ℏ /(∂ 2 E(k)/∂k 2 ) on the VBM and CBM, which are mh* = 2.22 m0 and me* = 0.42 m0, respectively.This shows that the hole's effective mass in Cs2NaFeCl6 is about 5 times larger than the electron's effective mass, indicating that the electrons are more dominant than the hole in the carrier migration.In addition to effective masses, the bulk modulus and volume deformation potential play crucial roles in the LA phonon model.The bulk modulus characterizes the connection between the volumetric strain and average stress.The calculated bulk modulus of Cs2NaFeCl6 is 20.32 GPa, a similar value to that of other halide perovskites [40].The bulk modulus is determined by analyzing the second derivative of the system's energy in relation to volume variations, thus showing sensitivity to the lattice phonon modes.In the case of halide perovskites, a smaller bulk modulus is linked to the presence of low-frequency phonon modes within the inorganic lattice [41].We believe that the small bulk modulus observed in Cs2NaFeCl6 suggests that electrons mainly interact with these lowfrequency lattice phonons, effectively reducing the charge reorganization.With the introduction of strain and the subsequent vibration of the crystal lattice, the band edges also change accordingly, giving birth to the deformation potential.To simulate the lattice vibration under applied strain, we changed the lattice size of Cs2NaFeCl6 manually in the range of a [0.95, 1.05] strain, incrementing by 0.01 to create eleven strained configurations.Each configuration was optimized with fixed lattice parameters while allowing for relaxation of all interior atomic positions.Figure 3 displays some of the optimized structures.With the introduction of strain and the subsequent vibration of the crystal lattice, the band edges also change accordingly, giving birth to the deformation potential.To simulate the lattice vibration under applied strain, we changed the lattice size of Cs 2 NaFeCl 6 manually in the range of a [0.95, 1.05] strain, incrementing by 0.01 to create eleven strained configurations.Each configuration was optimized with fixed lattice parameters while allowing for relaxation of all interior atomic positions.Figure 3 displays some of the optimized structures.Figure 2a,c show the band structures of Cs2NaFeCl6 during stretching and compression, respectively.Both structures keep the direct band gap, and the band gaps of 1.05strained and 0.95-strained Cs2NaFeCl6 are 1.07 eV and 1.12 eV, respectively.The variation in band gaps is caused by changes in the band edges under strain.The lattice vibrations introduced by the LA phonon alter the positions of the band edges, resulting in deformation potentials of Cs2NaFeCl6.To obtain a more accurate position of the band edges, we conducted band alignment among these strained Cs2NaFeCl6 systems using the Cs-6s orbit as the base energy level.The results are depicted in Figure 2d.The volume deformation potentials are calculated using the formula E = ∂Eedge/∂lnV and are Ee = 1.94 eV and Eh = 2.08 eV for the electron and hole, respectively.The calculated deformation potentials are listed in Table 2.
Using these parameters obtained above, the carrier mobilities at room temperature (T = 300 K) were determined to be µe = 2886.08cm 2 v −1 s −1 and µh = 39.09cm 2 v −1 s −1 for LA phonon interactions.It was observed that in the LA phonon-dominant mode, the electron mobility significantly exceeds that of the hole, which is attributed to the smaller effective mass of electrons.

PO Phonon-Determined Carrier Mobility
The interaction between carriers and the macroscopic electric field generated by the LO phonons in polar semiconductors is referred to as the Fröhlich interaction.Equation (2) allows us to determine the electron-phonon coupling constant α by considering the high-frequency dielectric constant, the static dielectric constant, the effective mass of the carrier, and the characteristic phonon angular frequency.
The dielectric constant is the main parameter representing the dielectric or polarization properties of a material under the static electric field.From the first principle calculations, we obtained the contributions from electrons and ions to the dielectric function of Cs2NaFeCl6 separately, which are shown in Figure 4. Using the high-frequency dielectric constant relationship ε∞ = εe|ω=0 and the static dielectric constant relationship εs = εe|ω=0 + εi|ω=0, the high-frequency dielectric constant and the static dielectric constant were derived Figure 2a,c show the band structures of Cs 2 NaFeCl 6 during stretching and compression, respectively.Both structures keep the direct band gap, and the band gaps of 1.05-strained and 0.95-strained Cs 2 NaFeCl 6 are 1.07 eV and 1.12 eV, respectively.The variation in band gaps is caused by changes in the band edges under strain.The lattice vibrations introduced by the LA phonon alter the positions of the band edges, resulting in deformation potentials of Cs 2 NaFeCl 6 .To obtain a more accurate position of the band edges, we conducted band alignment among these strained Cs 2 NaFeCl 6 systems using the Cs-6s orbit as the base energy level.The results are depicted in Figure 2d.The volume deformation potentials are calculated using the formula E = ∂E edge /∂lnV and are E e = 1.94 eV and E h = 2.08 eV for the electron and hole, respectively.The calculated deformation potentials are listed in Table 2. Using these parameters obtained above, the carrier mobilities at room temperature (T = 300 K) were determined to be µ e = 2886.08cm 2 v −1 s −1 and µ h = 39.09cm 2 v −1 s −1 for LA phonon interactions.It was observed that in the LA phonon-dominant mode, the electron mobility significantly exceeds that of the hole, which is attributed to the smaller effective mass of electrons.

PO Phonon-Determined Carrier Mobility
The interaction between carriers and the macroscopic electric field generated by the LO phonons in polar semiconductors is referred to as the Fröhlich interaction.Equation (2) allows us to determine the electron-phonon coupling constant α by considering the highfrequency dielectric constant, the static dielectric constant, the effective mass of the carrier, and the characteristic phonon angular frequency.
The dielectric constant is the main parameter representing the dielectric or polarization properties of a material under the static electric field.From the first principle calculations, we obtained the contributions from electrons and ions to the dielectric function of Cs 2 NaFeCl 6 separately, which are shown in Figure 4. Using the high-frequency dielectric constant relationship ε ∞ = ε e | ω=0 and the static dielectric constant relationship ε s = ε e | ω=0 + ε i | ω=0 , the high-frequency dielectric constant and the static dielectric constant were derived as ε ∞ = 16.42 and ε s = 24.51,respectively.These obtained permittivities are rather large, indicating that the electrostatic shielding effect inside the Cs 2 NaFeCl 6 lattice is strong.Some studies proposed that the ∆ε = ε s − ε ∞ is determined by the polarization of electrons and ions.The value of ∆ε for Cs 2 NaFeCl 6 in this work is 8.09, which is significantly higher than typical non-polar semiconductor materials.For example, the theoretical/experimental ∆ε values of GaAs, AlAs, and GaP are 1.6/2.31.9/1.9 and 1.8/2.1,respectively [35].That is, this large value of ∆ε indicates strong polarization of both electrons and ions for the Cs 2 NaFeCl 6 double perovskites.Therefore, the coupling constant α, calculated using these aforementioned parameters, is relatively small, indicating that the electron-phonon coupling generated by the LO phonon in Cs 2 NaFeCl 6 is a weak coupling effect.
Crystals 2024, 14, x FOR PEER REVIEW 7 of 12 as ε∞ = 16.42 and εs = 24.51,respectively.These obtained permittivities are rather large, indicating that the electrostatic shielding effect inside the Cs2NaFeCl6 lattice is strong.Some studies proposed that the Δε = εs − ε∞ is determined by the polarization of electrons and ions.The value of Δε for Cs2NaFeCl6 in this work is 8.09, which is significantly higher than typical non-polar semiconductor materials.For example, the theoretical/experimental Δε values of GaAs, AlAs, and GaP are 1.6/2.31.9/1.9 and 1.8/2.1,respectively [35].That is, this large value of Δε indicates strong polarization of both electrons and ions for the Cs2NaFeCl6 double perovskites.Therefore, the coupling constant α, calculated using these aforementioned parameters, is relatively small, indicating that the electron-phonon coupling generated by the LO phonon in Cs2NaFeCl6 is a weak coupling effect.Previous reports have indicated the presence of multiple optical phonon branches in the Raman spectra of Cs2NaFeCl6, suggesting that there is not a single characteristic angular frequency for the optical phonon.In order to gain a comprehensive understanding of the mobility of Cs2NaFeCl6, we calculated the coupling constants for each optical phonon branch with multiple frequencies, rather than simply average these frequencies for each optical branch.At room temperature, the wave numbers of these multiple optical phonons are approximately 55.80 cm −1 , 163.80 cm −1 , and 292.00 cm −1 , corresponding to 1.67 THz, 4.91 THz, and 8.75 THz, respectively [39].In polar and ionic crystals, the combination of electrons in the conduction band and the resulting lattice distortions form what is known as a polaron.The movement of charge carriers can be simplified as the movement of these polarons.Consequently, the masses of the polarons play a vital role in understanding their transport properties, which can be determined using the following formula: The polaron masses can be determined through the Fröhlich interaction with the corresponding charge carriers [42].By utilizing Equations ( 2) and ( 6), we were able to derive the distinct coupling constants and corresponding polaron masses for different optical phonon branches, which are presented in Table 3.Previous reports have indicated the presence of multiple optical phonon branches in the Raman spectra of Cs 2 NaFeCl 6 , suggesting that there is not a single characteristic angular frequency for the optical phonon.In order to gain a comprehensive understanding of the mobility of Cs 2 NaFeCl 6 , we calculated the coupling constants for each optical phonon branch with multiple frequencies, rather than simply average these frequencies for each optical branch.At room temperature, the wave numbers of these multiple optical phonons are approximately 55.80 cm −1 , 163.80 cm −1 , and 292.00 cm −1 , corresponding to 1.67 THz, 4.91 THz, and 8.75 THz, respectively [39].In polar and ionic crystals, the combination of electrons in the conduction band and the resulting lattice distortions form what is known as a polaron.The movement of charge carriers can be simplified as the movement of these polarons.Consequently, the masses of the polarons play a vital role in understanding their transport properties, which can be determined using the following formula: The polaron masses can be determined through the Fröhlich interaction with the corresponding charge carriers [42].By utilizing Equations ( 2) and ( 6), we were able to derive the distinct coupling constants and corresponding polaron masses for different optical phonon branches, which are presented in Table 3.
It can be seen from Table 3 that both the coupling constants of the electron and hole have low values, which is ascribed to the fact that the effective masses of the electron and hole are relatively small, and the values of the static dielectric constant and high-frequency dielectric constant are relatively large, leading to an insignificant Fröhlich interaction in Cs 2 NaFeCl 6 .Notably, the coupling constant of the hole is significantly larger than that of the electron.By employing Equation ( 7), we calculated the total polaronic phonon-determined mobility by considering multiple optical phonon branches, where the reciprocity of the total mobility is equal to the sum of the reciprocities of the mobilities for each branch.
Using Equations ( 3) and ( 7), we were able to calculate the electron and hole mobilities for each optical branch, as well as the total mobility within the polar phonon model, which are also listed in Table 3.As can be seen, the total mobilities for the electrons and holes are µ e = 279.25 cm 2 v −1 s −1 and µ h = 21.29 cm 2 v −1 s −1 , respectively.Similar to the result discussed above, the mobility of the electrons is still larger than that of the holes in the PO phonon-dominated modes.These analyses reveal that the deformation potential, bulk modulus, and dielectric constant have similar effects on the mobility of both electrons and holes.However, the effective mass of a hole is five times that of an electron.Based on Equations ( 1)-( 3) discussed above, we can conclude that the mobility of an electron is much larger than that of a hole.Our results are consistent with the experimental and theoretical observations on the mobility of most perovskites [24,25,28].Moreover, among the mobilities obtained from the LA scattering model and PO scattering model, we found that the mobility that was dominated by PO phonon scattering aligns more closely with the experimentally measured mobility of Cs 2 NaFeCl 6 double perovskites.Therefore, it can be inferred that the carrier mobility in Cs 2 NaFeCl 6 is predominantly determined by the PO phonon model.

The Dominant Factor for Mobility
Based on those analyses of the acoustic phonon and the optical phonon scattering for carrier mobility, the total carrier mobility of Cs 2 NaFeCl 6 can be calculated via the following equation: 1 Table 4 presents the electron mobility calculation for Cs 2 NaFeCl 6 , yielding a value of 254.61 cm 2 v −1 s −1 , which is dominated by PO phonon scattering.The reason for the dominant contribution from the PO scattering is that the relatively small electron effective mass of Cs 2 NaFeCl 6 leads to a more profound Fröhlich coupling interaction in the polarization field generated by optical lattice waves.In contrast, the hole mobility in Cs 2 NaFeCl 6 is calculated to be 13.78 cm 2 v −1 s −1 , with predominant contributions from PO phonon scattering and noticeable effects from LA phonon scattering.This indicates that the low coupling constant also affects the dominant role of optical phonon scattering in the hole mobility of Cs 2 NaFeCl 6 .The relatively larger effective mass of the hole is more susceptible to the low frequency phonon and is more easily affected by the bulk modulus, thus showing a mobility that is dominated by PO phonon scattering, with a considerable effect from LA phonon scattering.Our theoretical results are basically in agreement with the experimentally observed data and can provide elaborate explanations for their intrinsic mechanism.Furthermore, we applied the same methodology to analyze the variation in carrier mobility of Cs 2 NaFeCl 6 with temperature, which is important for the real applications of this material.Figure 5 illustrates that within the temperature range of 100-300 K, the mobilities of both electrons and holes decrease with the increase in temperature.Moreover, the mobilities of electrons and holes dominated by LA phonon scattering are always higher than that determined by PO phonon scattering.This suggests that PO phonon scattering continues to dominate the carrier mobility of Cs 2 NaFeCl 6 in the 100-300 K range.Our theoretical results are basically in agreement with the experimentally observed data and can provide elaborate explanations for their intrinsic mechanism.Furthermore, we applied the same methodology to analyze the variation in carrier mobility of Cs2NaFeCl6 with temperature, which is important for the real applications of this material.Figure 5 illustrates that within the temperature range of 100-300 K, the mobilities of both electrons and holes decrease with the increase in temperature.Moreover, the mobilities of electrons and holes dominated by LA phonon scattering are always higher than that determined by PO phonon scattering.This suggests that PO phonon scattering continues to dominate the carrier mobility of Cs2NaFeCl6 in the 100-300 K range.Previous research has shown that an acoustic phonon limits the mobility of non-polar semiconductors, while an optical phonon is the primary source of carrier scattering in polar semiconductors [43,44].The high Δε value suggests that Cs2NaFeCl6 is highly polar and is prone to PO phonon scattering.As a result, applying an electric field to Cs2NaFeCl6 causes distortion in the electron cloud and shifts in ion positions, ultimately resulting in carrier scattering.Recent studies have analyzed the polarity of organic-inorganic hybrid perovskites through theoretical and experimental approaches, suggesting that their polarity is influenced by factors such as composition, processing, and environment.This dynamic nature of polarity can impact the carrier mobility in various manners, highlighting the need for additional research on Cs2NaFeCl6 and other double perovskites.

Conclusions
In summary, this study focused on the theoretical investigation of the carrier mobility of Cs2NaFeCl6.By analyzing the electronic, dielectric, and elastic properties of the cubic double perovskite structure of Cs2NaFeCl6, we aimed to identify the key factors that influence the carrier mobility in this unique material.The carrier mobility of Cs2NaFeCl6 was calculated considering both acoustic phonon and optical phonon scattering mechanisms.The deformation potential was determined by applying strain to the original Cs2NaFeCl6 supercell, resulting in values of 2.08 eV for the VBM and 1.94 eV for the CBM.Using the calculated carrier effective mass and bulk modulus, we finally obtained the LA phonon-determined mobilities of Cs2NaFeCl6 for electrons and holes with values of µe = 2886.08cm 2 v −1 s −1 and µh = 39.09cm 2 v −1 s −1 , respectively.In a model of the optical scattering Previous research has shown that an acoustic phonon limits the mobility of non-polar semiconductors, while an optical phonon is the primary source of carrier scattering in polar semiconductors [43,44].The high ∆ε value suggests that Cs 2 NaFeCl 6 is highly polar and is prone to PO phonon scattering.As a result, applying an electric field to Cs 2 NaFeCl 6 causes distortion in the electron cloud and shifts in ion positions, ultimately resulting in carrier scattering.Recent studies have analyzed the polarity of organic-inorganic hybrid perovskites through theoretical and experimental approaches, suggesting that their polarity is influenced by factors such as composition, processing, and environment.This dynamic nature of polarity can impact the carrier mobility in various manners, highlighting the need for additional research on Cs 2 NaFeCl 6 and other double perovskites.

Conclusions
In summary, this study focused on the theoretical investigation of the carrier mobility of Cs 2 NaFeCl 6 .By analyzing the electronic, dielectric, and elastic properties of the cubic double perovskite structure of Cs 2 NaFeCl 6 , we aimed to identify the key factors that influence the carrier mobility in this unique material.The carrier mobility of Cs 2 NaFeCl 6 was calculated considering both acoustic phonon and optical phonon scattering mechanisms.The deformation potential was determined by applying strain to the original Cs 2 NaFeCl 6 supercell, resulting in values of 2.08 eV for the VBM and 1.94 eV for the CBM.Using the calculated carrier effective mass and bulk modulus, we finally obtained the LA phonon-determined mobilities of Cs 2 NaFeCl 6 for electrons and holes with values of µ e = 2886.08cm 2 v −1 s −1 and µ h = 39.09cm 2 v −1 s −1 , respectively.In a model of the optical scattering mechanism, the Fröhlich coupling constant ascribed for the electron-phonon coupling, the dielectric constant, and the polaron mass were calculated, and then, the multiple PO phonon-scattering-determined mobilities of Cs 2 NaFeCl 6 were obtained accordingly for electrons and holes, with values of µ e = 279.25 cm 2 v −1 s −1 and µ h = 21.29 cm 2 v −1 s −1 , respectively.Finally, by combining the acoustic phonon and optical phonon interactions, we obtained the electron mobility and the hole mobility of 254.61 cm 2 v −1 s −1 and 13.78 cm 2 v −1 s −1 , respectively.According to our calculated results, we revealed that due to the polarization of both electrons and ions in Cs 2 NaFeCl 6 , the PO scattering makes significant contributions to the carrier mobility of this double perovskite.More importantly, this gives indications for us that the double perovskites show complicated scattering mechanisms for the electrons and holes, which should be considered to further promote the optoelectronic properties of Cs 2 NaFeCl 6 and other double perovskites.

Figure 1 .
Figure 1.(a) The geometric structures and (b) metal octahedron frameworks of Cs2NaFeCl6.The dark purple, light purple, light green, and dark blue balls represent Cs, Na, Cl, and Fe atoms, respectively.The numbers in (a) denote the bond lengths and the bond angles.

Figure 1 .
Figure 1.(a) The geometric structures and (b) metal octahedron frameworks of Cs 2 NaFeCl 6 .The dark purple, light purple, light green, and dark blue balls represent Cs, Na, Cl, and Fe atoms, respectively.The numbers in (a) denote the bond lengths and the bond angles.

Figure 3 .
Figure 3. Optimized geometric structures of (a) 0.95-strained and (b) 1.05-strained Cs2NaFeCl6 supercells.Numbers in all figures represent the bond lengths and the bong angles, respectively.The dark purple, light purple, light green, and dark blue balls represent Cs, Na, Cl, and Fe atoms, respectively.

Figure 3 .
Figure 3. Optimized geometric structures of (a) 0.95-strained and (b) 1.05-strained Cs 2 NaFeCl 6 supercells.Numbers in all figures represent the bond lengths and the bong angles, respectively.The dark purple, light purple, light green, and dark blue balls represent Cs, Na, Cl, and Fe atoms, respectively.

Figure 4 .
Figure 4. (a) The electronic and (b) ionic contributions to the dielectric function of Cs2NaFeCl6, respectively.

Figure 4 .
Figure 4. (a) The electronic and (b) ionic contributions to the dielectric function of Cs 2 NaFeCl 6 , respectively.

Figure 5 .
Figure 5.The mobilities of Cs2NaFeCl6 dominated by LA and PO scattering for (a) µe and (b) µh in the temperature range of 100 K-300 K, respectively.

Figure 5 .
Figure 5.The mobilities of Cs 2 NaFeCl 6 dominated by LA and PO scattering for (a) µ e and (b) µ h in the temperature range of 100 K-300 K, respectively.

Table 1 .
The lattice parameter, the bond lengths, and the tolerance factor t of Cs2NaFeCl6.

Table 1 .
The lattice parameter, the bond lengths, and the tolerance factor t of Cs 2 NaFeCl 6 .

Table 2 .
The calculated effective mass, bulk modulus, volume deformation potentials, and LA-dominated carrier mobility of Cs2NaFeCl6 under room temperature.
m* means the effective carrier mass.

Table 2 .
The calculated effective mass, bulk modulus, volume deformation potentials, and LAdominated carrier mobility of Cs 2 NaFeCl 6 under room temperature.
m* means the effective carrier mass.

Table 3 .
The calculated electron-phonon coupling constants α, the polaron masses, and the carrier mobilities, determined by the PO phonons of Cs2NaFeCl6 with multiple optical vibrations.

Table 3 .
The calculated electron-phonon coupling constants α, the polaron masses, and the carrier mobilities, determined by the PO phonons of Cs 2 NaFeCl 6 with multiple optical vibrations.

Table 4 .
The total carrier mobility (µTotle) of Cs2NaFeCl6 under room temperature.