Unraveling the Role of Aluminum in Boosting Lithium-Ionic Conductivity of LLZO

Abstract

The development of high-performance solid electrolytes is critical to advancing solid-state lithium-ion batteries (SSBs), with lithium lanthanum zirconium oxide (LLZO) emerging as a leading candidate due to its chemical stability and wide electrochemical window. In this study, we systematically investigated the effects of cation dopants, including aluminum (Al³⁺), tantalum (Ta⁵⁺), gallium (Ga³⁺), and rubidium (Rb⁺), on the structural, electronic, and ionic transport properties of LLZO using density functional theory (DFT) and ab initio molecular dynamics (AIMD) simulations. It appeared that, among all simulated results, Al-LLZO exhibits the highest ionic conductivity of 1.439 × 10⁻² S/cm with reduced activation energy of 0.138 eV, driven by enhanced lithium vacancy concentrations and preserved cubic-phase stability. Ta-LLZO follows, with a conductivity of 7.12 × 10⁻³ S/cm, while Ga-LLZO and Rb-LLZO provide moderate conductivity of 3.73 × 10⁻³ S/cm and 3.32 × 10⁻³ S/cm, respectively. Charge density analysis reveals that Al and Ta dopants facilitate smoother lithium-ion migration by minimizing electrostatic barriers. Furthermore, Al-LLZO demonstrates low electronic conductivity (1.72 × 10⁻⁸ S/cm) and favorable binding energy, mitigating dendrite formation risks. Comparative evaluations of radial distribution functions (RDFs) and XRD patterns confirm the structural integrity of doped systems. Overall, Al emerges as the most effective and economically viable dopant, optimizing LLZO for scalable, durable, and high-conductivity solid-state batteries.

1. Introduction

The improvement of high-performance solid electrolytes is critical for advancement of solid-state batteries (SSBs), offering enhanced safety, improved thermal stability, and higher energy densities compared to conventional liquid electrolyte-based systems [1]. Among the various candidates explored, lithium lanthanum zirconium oxide (LLZO) garnet-type electrolytes have emerged as particularly promising due to their favorable lithium ionic conductivity, wide electrochemical range of stability, and excellent chemical compatibility with lithium metal [2,3]. Structurally, LLZO exhibits a robust three-dimensional garnet framework composed of corner-sharing tetrahedral and octahedral polyhedra, with the general chemical formula Li₇La₃Zr₂O₁₂. Depending on synthesis conditions and composition, LLZO stabilizes in either a tetragonal (t-LLZO) or cubic (c-LLZO) phase. While t-LLZO demonstrates excellent structural stability at room temperature, its low ionic conductivity (~10⁻⁷ S/cm) limits practical applications. In contrast, c-LLZO achieves higher lithium-ion conductivities exceeding 10⁻⁴ S/cm at room temperature due to its favorable lithium sublattice and increased vacancy concentrations, making it the preferred phase for SSB applications [4,5,6]. Additionally, doped c-LLZO displays remarkable chemical stability against lithium metal, suppressing dendrite formation and mitigating electrolyte degradation, thereby reinforcing its potential as a next-generation solid electrolyte material [7,8].

Despite these advantages, intrinsic limitations of c-LLZO such as low grain boundary conductivity, poor densification, and phase instability hinder its widespread implementation in SSBs [9]. To overcome these challenges, doping strategies have been extensively explored to tailor LLZO’s structural and electrochemical properties. Cationic doping, in particular, has proven effective in generating lithium vacancies and interstitials, which facilitate lithium-ion migration and enhance overall conductivity [10,11,12]. Furthermore, dopants can induce local structural modifications, optimize lattice parameters, and stabilize the highly conductive cubic phase [13,14]. Improved electrochemical stability in both anodes and cathode materials have also been achieved through judicious doping, contributing to extended cycle life and enhanced safety in full-cell configurations [15,16,17].

To date, various doping strategies have been implemented to optimize LLZO for commercial applications. These include cationic doping—where lanthanum, zirconium, or lithium sites are partially substituted with dopants such as aluminum, gallium, or titanium [18,19,20], and anionic doping, which introduces non-oxygen anions like fluorine or sulfur to generate beneficial oxygen vacancies [21,22]. More recently, dual-doping approaches have also been proposed to synergistically combine the advantages of multiple dopants [2,20]. Among these, aluminum (Al³⁺), gallium (Ga³⁺), rubidium (Rb⁺), and tantalum (Ta⁵⁺) have demonstrated particularly promising results, with notable improvements in phase stability and lithium-ion conductivity.

Reported studies have shown that doping with Al³⁺ can raise ionic conductivity in LLZO to values as high as 6–10 × 10⁻⁴ S/cm, while Ta⁵⁺ doping can achieve conductivities of around 4–8 × 10⁻⁴ S/cm, depending on synthesis conditions and microstructure control [23]. However, achieving consistent, high-performance LLZO across different doping strategies remains a challenge, particularly when balancing conductivity, stability, and scalability for industrial applications. Furthermore, minimizing electronic conductivity while maximizing ionic transport is critical to suppress lithium dendrite growth and ensure long-term cycling stability in practical battery systems [24].

Despite extensive experimental and theoretical efforts, comparative insights into how these dopants influence the fundamental structural, electronic, and ionic transport mechanism of LLZO under consistent conditions remain limited [25,26,27]. Therefore, in this study, we employ density functional theory (DFT)-based first-principles calculations and ab initio molecular dynamics (AIMD) simulations to systematically evaluate the effects of Al³⁺, Ga³⁺, Rb⁺, and Ta⁵⁺ doping on LLZO. We investigate the resulting changes in ionic conductivity, activation energy, phase stability, diffusion mechanisms, and electronic properties to identify the optimal dopant for enhancing LLZO’s performance in next-generation solid-state battery technologies, with an emphasis on balancing performance and scalability for real-world deployment.

2. Methodology

2.1. Electronic Structure

Kohn–Sham equations have been employed to self-consistent DFT functions to obtain better approximation. The main equation is a fictitious one-electron Kohn–Sham equation [28].

$$\left(-{\displaystyle \raisebox{1ex}{${\mathrm{\hslash }}^2$}\!\left/ \!\raisebox{-1ex}{$2m$}\right.}{\nabla }^2+v\left[\left\{{\overrightarrow{R}}_I\right\},\overrightarrow{r,}\rho \left(\overrightarrow{r}\right)\right]\right){\phi }_i^{KS}\left(\left\{{\overrightarrow{R}}_I\right\},\overrightarrow{r}\right)={\epsilon }_i{\left(\left\{{\overrightarrow{R}}_I\right\}\right)\phi }_i^{KS}\left(\left\{{\overrightarrow{R}}_I\right\},\overrightarrow{r}\right)$$

(1)

where the first term indicates kinetic energy with the Laplacian operator on the Kohn–Sham orbital.

In Equation (1), there is a set of one-electron orbitals ${\phi }_i^{KS}\left(\left\{{\overrightarrow{R}}_I\right\},\overrightarrow{r}\right)$ and corresponding energies ${\epsilon }_i$. The orbitals and their energies parametrically depend on nuclear position $\left\{{\overrightarrow{R}}_I\right\}$. The orbitals are combined with orbital phi, which is the occupation function to construct the total density of electrons $\rho \left(\overrightarrow{r}\right)$.

$$\rho \left(\overrightarrow{r}\right)={\sum }_i{f_i\phi }_i^{K{S}^{*}}\left(\overrightarrow{r}\right){\phi }_i^{KS}\left(\overrightarrow{r}\right)$$

(2)

Density consists of pairs of orbitals with coinciding indices. Total electron density determines the external potential acting on the electrons, $v\left[\overrightarrow{r,}\rho \right]$, which is calculated as the derivative of the difference in the total energy $E^{tot}\left[\rho \right]$ and the kinetic energy of interacting electron system $T\left[\rho \right]$ with respect to the electron density, as shown in Equation (3).

$$v\left[\overrightarrow{r,}\rho \right]=\delta /\delta \rho (E^{tot}\left[\rho \right]-T\left[\rho \right])$$

(3)

Equations (1)–(3) were then solved by an iterative approach, self-consistent method through using the Vienna Ab initio Software Package (VASP 5.4.4) according to the Perdew–Burke–Ernzerhof technique (PBE-functional), although other functionals can be used [29,30].

The trajectory was computed in two phases using first-principles-based molecular dynamics: first, heating and second, molecular dynamics.

2.2. Heating

The thermal control algorithm adjusts the system’s temperature by reheating or cooling it, based on whether the average atomic momentum exceeds or falls below the target temperature. Following preliminary computations performed with DFT using VASP, the system is subsequently raised to a designated temperature, thereby enhancing its kinetic energy. The atomic system is coupled to a thermostat at a specified temperature, where the particle momenta ${\overrightarrow{P}}_I$ are rescaled accordingly at every time step $\sum _I^{N_I}{\displaystyle \frac{{\overrightarrow{P}}_I^2}{2M_I}}={\displaystyle \frac{3}{2}}{N}_I{k}_{B}T$.

$${\sum }_{I=1}^N{\displaystyle \frac{M_I{\left(\frac{d{\overrightarrow{R}}_I}{dt}{|}_{t=0}\right)}^2}{2}}={\sum }_{I=1}^N{\displaystyle \frac{{\left({\overrightarrow{P}}_I\right)}^2}{2M_I}}={\displaystyle \frac{3}{2}}{Nk}_{B}T$$

(4)

where $N$ represents the total number of ions, $M$ is the mass of an ion, and $k_B$ is the Boltzmann constant in eV.

The system is subsequently permitted to evolve for an infinitesimal duration, allowing redistribution between the potential and kinetic energy domains. This process is iterated multiple times until the kinetic energy converges to the desired value. The heating phase establishes the initial conditions for the subsequent molecular dynamics simulation. The atomic positions $\left\{{\overrightarrow{R}}_I\left(t\right)\right\}$ and momenta $\left\{{\overrightarrow{P}}_I\left(t\right)\right\}$ obtained at the final step of the heating phase serve as the initial condition parameters for the molecular dynamics simulation.

2.3. Molecular Dynamics

Following the thermalization phase, the molecular dynamics simulation commences. Once thermal equilibrium is achieved, the atomic momenta serve as input for computing the trajectory of adiabatic ground-state molecular dynamics using Newton’s laws of motion:

$${\displaystyle \frac{d^2}{d{t}^2}}{\overrightarrow{R}}_I={\overrightarrow{F}}_I(\left[\rho (\overrightarrow{r})\right])/M_I$$

(5)

where the path ${\overrightarrow{R}}_I\left(t\right)$ of each ion is determined by solving Newton’s equations of motion, starting from initial conditions set during the heating phase. In AIMD, at every time step, the force ${\overrightarrow{F}}_I\left(t\right)$ is recalculated as an observable from the electronic structure information utilizing the Hellmann–Feynman theorem.

The diffusion characteristics were determined by analyzing the trajectories of lithium ions R_I(t) achieved from AIMD simulations. The shift ΔR_I of ion I between times $t_1$ and $t_2$ can be determined as:

$$\mathsf{\Delta }{R}_I\left(\mathsf{\Delta }t\right)=\left|\overrightarrow{R_I}\left(t_2\right)-\overrightarrow{R_I}\left(t_1\right)\right|$$

$$=\sqrt{{{[x}_I\left(t_2\right)-x_I\left(t_1\right)]}^2+{[y_I\left(t_2\right)-y_I(t_1\left)\right]}^2+{[z_I\left(t_2\right)-z_I(t_1\left)\right]}^2}$$

(6)

where Δt = $t_2$ − $t_1$ represents the duration of the time span.

As the ions are bound to their sites in solid electrolytes, they are not in continuous movement; for certain temperature and at certain times when an ion gains sufficient kinetic energy, it makes a jump from one site to another neighboring site, and this jump is termed rise time τ. Hence, the rate X of the Li-ion movement is calculated by taking the inverse of the rise time:

$$X={\displaystyle \frac{1}{\tau }}$$

$$\Rightarrow \mathit{log}X=\mathit{log}{\displaystyle \frac{1}{\tau }}$$

(7)

Rates are calculated at different temperatures for a simulation period of 50 ps, and the whole MD trajectory is observed, hypothesizing that “rise time” is shorter at higher temperature, i.e., rates are higher at higher temperature. The rate can also be represented using an Arrhenius-type expression, as shown below:

$$X\left(T\right)=A\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{-E_a}{k_{B}T}}\right)$$

(8)

where $A$ is a constant pre-factor, also known as pre-exponential parameter; $T$ represents absolute temperature; and $E_a$ refers to activation energy.

In case the numerical values of rates X₁(T₁) and X₂(T₂) are available, the energy activation E_a is calculated using the following formula:

$$E_a=-k_B{\displaystyle \frac{ln{X}_2\left(T_2\right)-ln{X}_1\left(T_1\right)}{\frac{1}{T_2}-\frac{1}{T_1}}}$$

(9)

The overall squared displacement of N migrating ions during the time duration of Δt is computed as $\sum _{i=1}^N\left(\right|\Delta R_I(\mathsf{\Delta }t){|}^2)$. This represents the motion of all N migrating ions over a time duration Δt. The total mean squared displacement (TMSD) of all diffusing ions is computed as:

$$\mathrm{T}\mathrm{M}\mathrm{S}\mathrm{D}(\mathsf{\Delta }\mathrm{t})={\sum }_{i=1}^N\langle |R_I(\mathsf{\Delta }t)-R_I\left(0\right){|}^2\rangle$$

(10)

The mean square displacement (MSD) is calculated as the TMSD per migrating ion, given by:

$$\mathrm{M}\mathrm{S}\mathrm{D}(\mathsf{\Delta }\mathrm{t})={\displaystyle \frac{1}{N}}\mathrm{T}\mathrm{M}\mathrm{S}\mathrm{D}(\mathsf{\Delta }\mathrm{t})$$

(11)

The self-diffusion coefficient D of Li ions can be evaluated using the Einstein relation by taking the average over the diffusion behavior:

$$D= {\displaystyle \frac{MSD\left(\Delta t\right)}{2d\left(\Delta t\right)}}$$

(12)

where d = 3 denotes the dimensionality of the lattice in which diffusion occurs, and $〈{\left[R\left(\Delta t\right)\right]}^2〉$ represents the squared displacement of lithium ions.

From the diffusivity D, ionic conductivity can be computed according to the Nernst–Einstein relationship [31,32]:

$$\sigma = {\displaystyle \frac{N{q}^2}{V{k}_{B}T}}D$$

(13)

where V represents the overall system volume, and q denotes the charge of the ionic species.

2.4. Formation Energy and Electronic Conductivity

Using DFT calculations, the formation energy of any dopant can be calculated by the following equation:

$$E_{FE}=E_{doped-LLZO}^{tot}-E_{undoped-LLZO}^{tot}\pm \sum n_i{u}_i$$

(14)

where $E_{undoped-LLZO}^{tot}$ is the total energy of undoped LLZO, $n_i$ represents the number of atoms of species i that are introduced or eliminated, and $u_i$ is the chemical potential of species $i$.

Charge carrier concentration for a specific dopant at a given temperature is given by:

$$n_e=N_c\exp \left(-{\displaystyle \frac{{E_c-E}_F}{k_{B}T}}\right)$$

(15)

$$n_h=N_v\exp \left(-{\displaystyle \frac{E_F-E_V}{k_{B}T}}\right)$$

(16)

where $E_F$ is the Fermi-level energy, determined by solving the charge neutrality condition numerically at the temperature T, considering all relevant charged defects and carriers. $N_c=2{\left({\displaystyle \frac{2\pi m_e^{*}{k}_{B}T}{h^2}}\right)}^{3/2}$ and $N_v=2{\left({\displaystyle \frac{2\pi m_h^{*}{k}_{B}T}{h^2}}\right)}^{3/2}$ are the effective density of states for the conduction band and valence band, respectively, where $m_e^{*}$ and $m_h^{*}$ are the effective mass of the electron and hole, respectively. $E_c$ and $E_v$ are the energies of the conduction and valence band, respectively.

Using the Drude model and Boltzmann transport equation, the mobility of the carrier can be found using the following equation:

$$\mu ={\displaystyle \frac{e\tau }{m^{*}}}$$

(17)

where e is the charge of an electron, and τ is the relaxation time, which represents the average time between two consecutive events of scattering for any charge carrier with a typical value of 10⁻¹⁴ s, and m* is the effective mass of the charge carriers.

Using the mobility and concentration of the carriers, the electronic conductivity of doped LLZO was calculated as:

$$\sigma =n_{e}e\mu$$

(18)

2.5. Binding Energy

Binding energies between atoms can be calculated using Equation (19), where the total energy of the individual molecules (Li, La, Zr, O, and dopant) are subtracted from the total energy of the whole molecule.

$$E_{BE}=E_{Li}^{tot}+E_{La}^{tot}+E_{Zr}^{tot}+E_O^{tot}+E_{dopant}^{tot}-E_{doped-LLZO}^{tot}$$

(19)

where $E_{doped-LLZO}^{tot}$ is the total energy of the doped LLZO, and $E_{Li}^{tot}, E_{La}^{tot}, E_{Zr}^{tot}, E_O^{tot}, {and E}_{dopant}^{tot}$ represent total energies of Li, La, Zr, O, and dopants, respectively.

2.6. Radial Distribution Function

The radial distribution function (RDF), also known as the pair correlation function, describes how the number of ions or density of ions varies in a system as a function of distance from a reference ion. It simply indicates the probability of finding an ion in a molecular system at a specified distance from a certain reference ion.

At every time step along with the molecular dynamics (MD) trajectory, the atomic positions {R_I(t)} are recorded, enabling single-point electronic structure calculations. In a few cases, structural reconfigurations in the model may be triggered by thermal equilibration, promoting processes such as desorption, adsorption, or the formation of chemical bonds. RDF calculation for specific interatomic distances is found along the MD trajectory:

$$RDF\left(t,r\right)={\displaystyle \frac{1}{4\pi r^2}}{\sum }_{IJ}\delta \left(r-\left|{\overrightarrow{R}}_I\left(t\right)-{\overrightarrow{R}}_J\left(t\right)\right|\right)$$

(20)

where ${\overrightarrow{R}}_I\left(t\right)$ and ${\overrightarrow{R}}_J\left(t\right)$ are the positions of the I^th and J^th ion, respectively.

3. Computational Methodology

In this study, DFT calculations were accomplished using VASP software, and the plane wave basis set was used [33,34]. The projector-augmented wave (PAW) correlation functional was used for treating the interaction between core states [35]. A plane-polarized wave cutoff energy of 520 eV with the corresponding pseudopotentials was used. Total energy calculations were executed using the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA) [29,30]. The Monkhorst–Pack scheme was used for generation of the K-space matrix, and a 1 × 1 × 1 k-point mesh was automatically generated for Brillouin zone sampling using the Monkhorst–Pack scheme within VASP. Appropriate mesh dimensions were automatically determined by the software based on the supercell size, ensuring adequate sampling for geometry optimizations and static energy calculations. For well visualization, a c-LLZO (COD-7206766) and t-LLZO (mp-942733) unit cell model of 8 per formula unit (pfu) (Li₅₆La₂₄Zr₁₆O₉₆) comprising 192 atoms is shown in Figure 1. To model the cubic phase, the fractional occupancies of Li ions on the 24d tetrahedral and 96h octahedral sites were represented using an ordered configuration within the supercell, consistent with the overall Li stoichiometry, where oxidation states of the constituent atoms were Li⁺, La³⁺, Zr⁴⁺, and O²⁻ respectively; in other words, Li⁺ was coordinated to 4 or 6 oxygen atoms, and La³⁺ and Zr⁴⁺ were coordinated to 2 and 8 oxygen atoms, respectively [36,37]. Although these geometries have void vacancies, they did manage to charge the balance stoichiometry, where La, Zr, and O formed stable scaffolds while Li cations filled the voids. In addition, a site occupancy factor (SOF) of g = 0.42 for the tetrahedral 24d site and g = 0.48 for the octahedral 96h site was used in c-LLZO model, but g = 1 was used for all sites (tetrahedral 8a, 16f octahedral, and 32g octahedral) in t-LLZO.

The doped LLZO models used in this study were constructed by substituting specific cation sites in the parent cubic LLZO (Li₅₆La₂₄Zr₁₆O₉₆, 8 pfu model) structure. The doping site was determined by (1) comparing the defect formation energy (described in Section 2.4), while the site that resulted in the lowest formation energy was selected to ensure thermodynamic favorability, and (2) considering ionic size compatibility, which led to a stable structural framework. Specifically, Al³⁺ and Ga³⁺, whose radii are small and close to that of Li⁺, were doped at the Li⁺ site (24d/96h) by removing 12 Li atoms and adding 4 Al atoms or 4 Ga atoms, leading to Li₄₄Al₄La₂₄Zr₁₆O₉₆ and Li₄₄Ga₄La₂₄Zr₁₆O₉₆. Rb⁺ has a large ionic radius, which helps preserve the structural framework, and was introduced at the La³⁺ site by replacing 1 La atom with 1 Rb atom and adding 2 Li atoms (to maintain electrical neutrality), leading to Li₅₈La₂₃RbZr₁₆O₉₆. Ta⁵⁺ was doped to the Zr⁴⁺ site by removing 8 Zr atoms and 8 Li atoms and adding 8 Ta atoms, leading to Li₄₈La₂₄Zr₈Ta₈O₉₆.

AIMD simulations were executed in the NVT ensemble (constant number of particles, volume, and temperature) using a Nosé–Hoover thermostat [38,39] to maintain the target temperature(s) of 750 K, 1000 K, 1250 K, 1500 K, and 1750 K. Temperature in a range of up to 1750 K was chosen to ensure enhanced atomic mobility while preserving the solid-state framework necessary for studying realistic diffusion mechanisms. A time interval of 1 femtosecond was used for integrating the equations of motion. The systems were initially equilibrated during the heating stage (as described in Section 2.2), followed by production runs of 50 ps for trajectory analysis (Section 2.3).

4. Results and Discussion

With the DFT method, we calculated the ionic conductivity, electronic conductivity, activation energy, and formation energy of pure LLZO and different doped LLZO. In calculating the ionic conductivity, we ran molecular dynamics for 50 ps to ensure data stability. We also ensured a proper linear fit to the MSD–time relationship.

4.1. Ionic Conductivity and Activation Energy

The highest ionic conductivity for different cation (Al, Ga, Rb, and Ta)-doped LLZO are plotted in Figure 2a, and their corresponding the lowest activation energies are plotted in Figure 2b. The highest ionic conductivities of other doped LLZO are given in Table S1, which is available in the Supplementary Materials. As one can see from

Table 1. Ionic conductivity and activation energy.

	c-LLZO	t-LLZO	Al-LLZO	Ga-LLZO	Rb-LLZO	Ta-LLZO
Ionic conductivity (S/cm)	2.38 × 10−3	3.03 × 10−7	1.439 × 10−2	3.73 × 10−3	3.32 × 10−3	7.12 × 10−3
Activation energy (eV)	0.227	0.496	0.138	0.197	0.212	0.191
Experimental ionic conductivity (S/cm) and activation energy (eV)	2 × 10−4 (0.25) [40]	1.3 × 10−6 (--) * [41]	4.96 × 10−4 (0.28) [42]	1.46 × 10−3 (0.25) [43]	--*	--*

* Indicates not reported for this chemical composition.

, among these doped LLZO, the Al-LLZO exhibits very high ionic conductivity (1.439 × 10⁻² S/cm), which is one order higher than others, and Ta-LLZO shows an ionic conductivity of 7.12 × 10⁻³ S/cm. These calculated ionic conductivities are a little bit higher than the experimentally reported ones because grain boundary resistance, interstitial defects, and analytical error from simulation are not considered, and the error comes from measurement techniques [23]; moreover, ionic conductivity varies depending on the synthesis technique [2].

It is evident that proper elemental doping increases ionic conductivity, as one can see from Figure 2a, in which doped LLZO exhibits greater ionic conductivity compared to the undoped cubic LLZO [44,45,46]. There is an inverse relationship between ionic conductivity and activation energy; tetragonal-phase LLZO has the highest activation energy, which in turns shows the lowest ionic conductivity.

4.2. Analysis Based on Electronic Properties

The density of charge is contingent upon how electric charge is distributed, and it can manifest as either positive or negative. This density is quantified as the amount of electric charge per unit area of a surface, or per unit volume of an object or field. It elucidates the extent to which electric charge accumulates within a specific field. The charge density of molecules significantly influences chemical and separation processes. The distribution of charge density affects the connectivity of ionic pathways. Well-distributed negative charge densities can create continuous and less tortuous pathways for lithium ions, enhancing ionic conductivity. High-charge-density regions can create strong electrostatic interactions, increasing the energy barrier for ion movement. Conversely, regions with optimized charge density can lower the activation energy, facilitate easier ion migration, and improve conductivity. A higher positive charge correlates with increased instability. To put it simply, lower charge density results in greater stability of positive charge. By comparing the charge densities of the LLZO variants depicted in Figure 3, it becomes evident that Al-LLZO and Ta-LLZO possess well-distributed lower charge density comparable to that of c-LLZO. This low charge density makes the Li ions in Al-LLZO and Ta-LLZO migrate easily from one site to another by breaking bonds because of the lower electrostatic potential.

The formation energy of defects significantly influences the electronic conductivity of LLZO. Lower formation energy leads to higher defect concentrations, enhancing the availability of charge carriers and potentially increasing electronic conductivity. However, this must be balanced with the mobility of carriers, as excessive defect concentrations can lead to increased scattering and reduced mobility. Understanding and optimizing defect formation energies and their resultant carrier concentrations are essential for improving the electronic properties of solid electrolytes. On the other hand, the formation energy of dopants in LLZO plays a critical role in determining their effectiveness in modifying the material’s properties. Low-formation-energy dopants are more easily incorporated, enhancing solubility, stability, and defect formation in beneficial ways. They can increase lithium vacancy concentration, thereby improving ionic conductivity. Therefore, the number of Li ions present and the vacancy concentration must be balanced to create a more disordered Li pattern, between 44 and 48 for an 8 pfu model, as shown in Figure S1 (available in the Supplementary Materials) [47]. High-formation-energy dopants, on the other hand, may lead to phase separation and limited solubility, potentially destabilizing the material. Understanding and optimizing dopant formation energies is essential for developing high-performance LLZO-based solid electrolytes, as shown in

Table 2. Formation energy and electronic conductivity of Al-LLZO, Ga-LLZO, Rb-LLZO, and Ta-LLZO.

	Al-LLZO	Ga-LLZO	Rb-LLZO	Ta-LLZO
Formation energy (eV)	−0.56	−0.47	−0.43	−0.51
Electronic conductivity (S/cm)	1.72 × 10−8	7.86 × 10−8	2.23 × 10−9	4.33 × 10−8

. Rb-LLZO shows the lowest electronic conductivity, which makes it more reliable in terms of dendrite formation, followed by Al-LLZO; Ga-LLZO has the highest electronic conductivity, making it less reliable because of the ease of carrier transfer between electrolyte and electrode in the process of dendrite formation [24].

The electronic conductivity of an insulator depends on its bandgap. When the bandgap decreases, the jumping tendency of available electrons in the valence band increases with the increase in temperature. The calculated bandgap for our bulk c-LLZO and t-LLZO model is found to be 4.0 eV and 4.15 eV, respectively, as shown in the total density of states (TDOSs) in Figure 4, which is in agreement with the published results [48,49]. The smaller DFT calculated bandgap compared to the experimental bandgap in LLZO is primarily due to the limitations of standard exchange-correlation functionals (GGA), the neglect of many-body effects, and the idealized nature of the calculations. The self-interaction error in the GGA functional leads to an underestimation of the repulsion between electrons. This results in narrower bandgaps, as the energy levels are not correctly separated. DFT calculations are usually considered an ideal, defect-free crystal structure. Impurities, vacancies, and other defects present in experimental samples can affect the bandgap. These imperfections can introduce states in the bandgap, effectively reducing the measured bandgap in some cases, but usually the experimental bandgap is larger due to the intrinsic electronic structure being different in real materials.

Materials possessing a wide bandgap exhibit a restricted array of energy levels accessible to electrons, resulting in diminished electrical conductivity. Among all of them, the highest bandgap is observed in t-LLZO, followed by Rb-LLZO. The lowest bandgap is found in Ga-LLZO; this is clearly due to doping, except Rb doping electronic conductivity increases because of the shrinkage in bandgap. When electronic conductivity increases, the reliability decreases due to the formation of dendrites, which reduces the life span of the electrolyte material.

Figure S2 (available in the Supplementary Materials) shows the vibrational frequency spectrum of different cation-doped LLZO. The attempt frequency can be thought of as the frequency with which an ion vibrates in its potential well before it attempts to jump to a neighboring site. The calculated attempt frequencies at 1000 K for c-LLZO, t-LLZO, Al-LLZO, Ga-LLZO, Rb-LLZO, and Ta-LLZO are 13.35 THz, 15.75 THz, 12.13 THz, 12.27 THz, 12.35 THz, and 12.20 THz, respectively. We found that the material with a high attempt frequency however brought about a lower ionic conductivity. It can be explained by high activation energy in the case of high attempt frequency, as revealed by the Meyer–Neldel rule [50,51,52], where attempt frequency $v\propto e^{{\displaystyle \frac{E_a}{E_{mn}}}}$, indicating that the attempt frequency is exponentially proportional to the activation energy ($E_a$) and inverse to the Meyer–Neldel energy $E_{mn}$ (with a typical value varying from 0.05 to 0.2 eV). This is in line with the activation energy result depicted in Figure 2b, where t-LLZO, which presents the highest attempt frequency, possesses the highest activation energy, whereas Al-LLZO, which presents the lowest attempt frequency, possesses the lowest activation energy.

4.3. Analysis Based on Structural Properties

Reduced ionic radius of the dopant and doping site is correlated with increased ionic conductivity of any complex. Ionic bonding strength is contingent upon ionic radius, as smaller ions permit tighter packing within the ionic lattice, which reduces the distance between two sites and decreases the activation energy. Consequently, closer proximity between oppositely charged ions intensifies electrostatic forces, thereby reinforcing the bond. The ionic radii of these elements are given below in

Table 3. Ionic radius of Li, La, Zr, and dopants.

Elements	Al3+	Ga3+	Li+	Rb+	La3+	Ta5+	Zr4+
Ionic Radius (Å)	0.54	0.62	0.76	1.52	1.06	0.79	0.72

Green: Li-site doping; yellow: La-site doping; blue: Zr-site doping.

[53]. A significant mismatch in ionic radius between the dopant and the host ion can introduce lattice strain, which can affect structural stability. For example, Rb⁺ (1.52 Å) replacing La⁺ (1.06 Å) shows a huge radius mismatch that is significantly larger than that of the host ion. This can cause local distortions and stress within the lattice. Moreover, excessive lattice strain due to a size mismatch can lead to phase instability, potentially resulting in phase separation or the formation of secondary phases that are detrimental to the electrolyte’s performance. When Ta⁵⁺ (0.79 Å) replaces Zr⁴⁺ (0.72 Å), one lithium interstitial is created for every Ta⁵⁺ dopant. This enhances lithium-ion conductivity, but the larger ionic size of Ta⁵⁺ compared to Zr⁴⁺ might cause significant lattice distortions. Mathematically, $E_{binding}\propto {\displaystyle \frac{z_1{z}_2{e}^2}{{4\pi \epsilon }_{0}r}}$, where, $z_1$ and $z_2$ represent atomic charges, and r represents interatomic distance, which identifies smaller ionic radii that can create a tighter pack, resulting in shorter migration distance and ultimately reducing the activation energy to increase the mobility. Therefore, when Al³⁺ (0.54 Å) replaces the host Li⁺ (0.76 Å), it shows close packing with minimal distortion and enhanced mobility due to the increased number of mobile carriers and vacancies.

The bottleneck size of the triangular window as shown in Figure 5 plays a very important role in the diffusion of Li; the smaller the triangle is, the denser the structure is, which in turn increases the chance of Li migration and thus increases ionic conductivity. The approximate distance covered by a Li ion for different pathways is given in

Table 4. Approximate distance of pathways between sites.

Migration Pathway	Pathway 1	Pathway 2	Pathway 3
Distance (Å)	≈2.2	≈3.7	≈0.9

, and the approximate length of the arms of the bottleneck triangle are given in

Table 5. Size of hands of bottleneck triangle in different doped LLZO.

Triangle Arm Length (Å)	c-LLZO	t-LLZO	Al-LLZO	Ga-LLZO	Rb-LLZO	Ta-LLZO
Side A	3.11	3.12	3.08	3.20	3.12	3.10
Side B	3.046	3.09	3.10	3.15	3.20	2.96
Side C	3.17	3.26	2.88	2.96	2.96	3.10
Average	3.11	3.16	3.02	3.10	3.09	3.06

. Most of the migrating jump of Li occurs using pathway 3 because it is easier to achieve, whereas it is very highly unlikely to observe the jump using pathway 2, as shown in Figure S3 available in the Supplementary Materials [49].

Doping not only helps increase the ionic conductivity of the structure but also helps improve the stability, which is evident from

Table 6. Binding energy of the optimized structures of pristine LLZO and doped LLZO.

	c-LLZO	t-LLZO	Al-LLZO	Ga-LLZO	Rb-LLZO	Ta-LLZO
Binding energy (eV)	−1107.46	−1101.5746	−1121.39	−1110.29	−1108.62	−1113.64

. Binding energy refers to the energy required to separate a particle from a system or to disassemble a system of particles into its individual components. In the context of ionic conductivity, binding energy often pertains to the energy holding an ion in its lattice site or the energy required to separate an ion pair into an electrolyte. Higher binding energy generally implies stronger interactions between ions and the lattice or between ion pairs. When ions are more tightly bound, the activation energy for their movement increases because more energy is required to overcome these strong interactions. Conversely, lower binding energy means weaker interactions, resulting in lower activation energy and easier ion movement. High binding energy leads to high activation energy, making it difficult for ions to move. This typically results in lower ionic conductivity because fewer ions can successfully jump between sites at a given temperature. Table 6 shows that Al-LLZO exhibits the lowest binding, whereas t-LLZO exhibits the highest binding energy, and it is in perfect agreement with the results shown in Figure 2.

Figure 6 shows the visual correlation of how the ionic radius of dopants influences both ionic conductivity and binding energy in LLZO-based solid electrolytes. The trends suggest that smaller ionic radii (e.g., Al³⁺) are associated with higher conductivity and more negative binding energy, implying enhanced mobility and structural stability. Larger ionic radii (e.g., Rb⁺) tend to decrease conductivity and reduce binding energy, likely due to lattice strain and phase instability.

4.4. Cubic-Phase Preservation and Model Validation

RDF serves to elucidate the arrangement of atoms around each other within a given system. Figure 7 describes the arrangement of Li atoms with respect to the first Li atom. Figure 7a indicates the clear distinction between two phases of LLZO: The blue curve contains distinct peaks, which are an indication of uniform arrangement of Li atoms in t-LLZO, whereas the red curve reveals Li atoms in c-LLZO, which are not in uniform arrangement. In addition, the interatomic distance between Li atoms is less in c-LLZO than in t-LLZO, resulting in a closely packed or dense structure for c-LLZO.

We also compared the RDF arrangement between Li-Li in Al-LLZO and Ta-LLZO with c-LLZO. We observed that although they show similar RDF trends, Li-Li distance decreases due to the doping effect, as shown in the green and cyan curves in Figure 7b. The smeared peaks of the green and cyan curves reflect the randomness of the position of the Li ions, which is responsible for the higher ionic conductivity of doped LLZO.

The phase transition or structural integrity of a crystal can also be determined using the RDF. As shown in Figure 8, the interatomic distance-versus-RDF trend is similar, which indicates the intact phase preservation of Al- and Ta-doped LLZO.

RDF reveals atomic arrangements and helps assess both crystallinity and phase identity in support of ionic mobility within crystal. In Figure 7a, the sharp peaks for t-LLZO indicate an ordered structure, while the broader peaks and shorter Li–Li distances in c-LLZO suggest a more densely packed, disordered arrangement that supports higher ionic conductivity. Figure 7b shows that Al and Ta doping further reduces Li–Li distances and broadens RDF peaks, indicating increased Li-ion disorder, which enhances ionic mobility. Additionally, Figure 8 confirms that the doped systems maintain the original LLZO phase, as the RDF trends remain consistent, demonstrating successful phase preservation despite doping.

Moreover, by comparing the experimental XRD patterns of Al-LLZO and Ta-LLZO with those of the optimized structure, we found similar patterns, which indicate the phase preservation of our model that was used for this study, as shown in Figure 9 and Figure 10, where the trends of the peaks are in good agreement. Similar phase preservation of doped LLZO has also been reported by Feng et al. [54] and Nguyen et al. [27]. Distinct and prominent diffraction peaks associated with the cubic garnet structure suggest excellent crystallinity and structural uniformity. The lack of additional phase signal verifies that both Al and Ta dopants effectively maintain stable cubic phase, which is essential for improved lithium-ion transport.

From the simulation results, we can see that Al-LLZO, with a conductivity of 1.4369 × 10⁻² S/cm, is apparently superior to other doped LLZO, while Ta-LLZO exhibits a moderate conductivity of 7.12 × 10⁻³ S/cm, which is approximately double than of the conductivities of 3.73 × 10⁻³ S/cm and 3.32 × 10⁻³ S/cm for Ga-LLZO and Rb-LLZO, respectively. Higher ionic conductivity means more efficient lithium-ion transport, which is crucial for battery performance. Al-LLZO demonstrated excellent chemical stability in contact with lithium metal and various cathode materials [57]. This stability is crucial for maintaining the integrity and longevity of the battery. While Ta-LLZO is also stable, Al-doped LLZO’s stability is often reported as more consistent across a wider range of conditions [23]. Al is more abundant and less expensive compared to Ta. The lower cost and greater availability of aluminum make Al-doped LLZO a more economically viable option for large-scale battery production, and the process of doping LLZO with aluminum is generally more straightforward and less costly compared to doping with tantalum [58]. This ease of synthesis can lead to more uniform material properties and easier scalability in manufacturing [55]. On the other hand, Rb-LLZO exhibits good stability, but it has the drawback of having poor ionic conductivity and long-term stability compared to Al-LLZO and is not cost-effective [59].

5. Conclusions

Through systematic DFT and AIMD simulations, we evaluated the impacts of Al³⁺, Ga³⁺, Rb⁺, and Ta⁵⁺ cation doping on the properties of LLZO solid electrolytes. The results reveal that Al³⁺ and Ta⁵⁺ are relatively more effective dopants for simultaneously enhancing ionic conductivity and stabilizing the desirable cubic phase. However, while Ta⁵⁺ offers significant performance improvements, its high cost presents a barrier. Ga³⁺ doping and Rb⁺ doping provide moderate improvement, positioning it as a potential compromise. Computational simulations proved essential for this comparative analysis, enabling the isolation of intrinsic dopant effects on the LLZO lattice, electronic structure, and ion transport dynamics, decoupled from experimental variables like synthesis and microstructure.

Based on this comprehensive computational investigation, aluminum emerges as the most promising dopant overall. It achieves the highest ionic conductivity (1.439 × 10⁻² S/cm) coupled with the lowest activation energy (0.138 eV), attributed to favorable structural factors, including a comparable ionic size to host cations and the creation of the smallest Li-ion migration bottleneck window. Furthermore, Al-doping yields an exceptionally low electronic conductivity (1.72 × 10⁻⁸ S/cm), supported by the maintenance of a large electronic bandgap, which is critical for mitigating dendrite formation risks. Thermodynamically, Al-LLZO demonstrates high stability, evidenced by favorable defect formation energy and the most favorable overall binding energy among the systems studied. Moreover, RDF and XRD analyses confirm that Al doping effectively preserves the cubic garnet structure essential for high ionic mobility. The synergistic combination of superior ionic transport, low electronic conductivity, high stability, structural integrity, abundance, and cost-effectiveness makes Al³⁺ a suitable dopant for LLZO for practical application in solid-state lithium-ion batteries.