Modeling of Structural Properties of Zr(Nb)-Doped c-LTO–Zr(Nb)

Materials With Spinel Structure for Li-Ion Batteries

MIRSALIM M. ASADOV1, 2,*, SOLMAZ N. MUSTAFAEVA3, SAIDA O. MAMMADOVA3,

ESMIRA S. KULI-ZADE1, VLADIMR F. LUKICHEV4

1Nagiyev Institute of Catalysis and Inorganic Chemistry,

Ministry of Science and Education of Azerbaijan,

Baku, AZ-1143,

AZERBAIJAN

2Scientific Research Institute of Geotechnological Problems of Oil, Gas and Chemistry,

Ministry of Science and Education of Azerbaijan,

Baku, AZ-1010,

AZERBAIJAN

3Institute of Physics, Ministry of Science and Education of Azerbaijan,

Baku, AZ-1143,

AZERBAIJAN

4Valiev Physics and Technology Institute,

Russian Academy of Sciences,

Moscow, 117218,

RUSSIA

*Corresponding Author

Abstract: - Using the density functional theory (DFT), we investigated the electronic, magnetic and energy

properties of Li4Ti5-xMxO12 (M = Zr, Nb; x = 0–0.01) (or c-LTO–Zr (Nb)) supercells, which are isostructural to

the high-temperature spinel (cubic; sp.gr. 

) modification of the Li4Ti5O12-based anode material (c-LTO)

for lithium-ion batteries (LIB). In DFT calculations with a generalized gradient approximation (functional

GGA), spin-orbit coupling (SOC) and the Hubbard correction were considered (DFT SGGA+ ). For the Ti 3d

and Zr(Nb) 4d states, pd and dd-model values of the U effective interaction energies were used. To account for

the effective Coulomb intra-atomic interaction (U) between particles, the DFT SGGA+U calculations were

performed using  = 4 eV, O 2p, Ti 3d, Zr (Nb) 4d. The band structure and density of states (DOS) of c-LTO

supercells calculated by DFT SGGA+ are consistent with both theoretical and experimental data for c-LTO.

The theoretical band gap () is smaller than the experimental value of c-LTO. Taking into account the

chemical potentials of the doping impurities, the formation energy of c-LTO–Zr(Nb) was calculated. The

decrease in the total energy due to the change in the partial densities of states (PDOS) of the filled spin-up and

spin-down subbands indicates that c-LTO–Zr(Nb) is thermodynamically stable. The density of spin-up d-states

exceeds the density of spin-down d-states in c-LTO–Zr(Nb), so the  values in these structures differ from

each other. Doping with Zr(Nb) narrows  and improves the electrical conductivity of c-LTO–Zr(Nb) due to

the contribution of Zr4+ 4d and Nb3+ 4d orbitals to the particle transport. The Fermi level of the band structure

of c-LTO–Zr(Nb) supercells shifts to the conduction band as a result of Zr(Nb) doping. The local magnetic

moments of Zr(Nb) impurity atoms in c-LTO–Zr(Nb) are calculated.

Key-Words: - New Battery Materials, Doped Lithium Titanate Oxide, c-LTO–Zr(Nb), Density Functional

Theory, DOS, Band Structure, Magnetic Moment.

Received: April 30, 2024. Revised: November 11, 2024. Accepted: December 1, 2024. Published: December 31, 2024.

WSEAS TRANSACTIONS on ELECTRONICS

DOI: 10.37394/232017.2024.15.19

Mirsalim M. Asadov, Solmaz N. Mustafaeva,

Saida O. Mammadova, Esmira S. Kuli-Zade,

Vladimr F. Lukichev

E-ISSN: 2415-1513

166

Volume 15, 2024

1 Introduction

Rechargeable lithium-ion batteries (LIBs) are

known to be alternative energy storage systems.

LIBs provide high energy density and long cycle

life, making them important for applications such as

portable electrical appliances, [1], [2], [3], [4], [5].

The power of LIBs depends on various electrode

materials and factors, including the rate of lithium-

ion and electron migration through the electrolyte

between the two electrodes. To produce high-power

and stable LIBs, it is necessary to develop new

materials and highly conductive nanostructures for

the electrodes. Graphite, graphene, and lithium

titanate oxide Li4Ti5O12 (с-LTO), in particular with

a nanocrystalline spinel structure (face-centered

cubic syngony; space group 

;   8.35 Å), are

used as anode materials. Such electrode

nanomaterials accelerate the transport of particles in

LIBs and ensure their stability.

In order to increase the service life of LIB, new

materials are being developed that effectively

reduce energy losses caused by battery polarization.

Materials such as с-LTO increase the service life of

LIB, compared to graphite materials, [6].

LTO can operate at high current densities

corresponding to the 60C mode, when charging and

discharging occur in 1 min. In this case, the decrease

in LIB capacity is  50% of the nominal capacity.

LTO has good cyclability due to the same specific

volume in the LIB charge/discharge process. The

oxidized and reduced compositions of c-LTO and

Li7Ti5O12 at the phase boundary of the c-LTO spinel

phase have an almost horizontal discharge curve.

This is due to the two-phase discharge mechanism

and low mutual solubility of c-LTO/Li7Ti5O12 at the

boundary of the spinel phase (111). One of the

disadvantages of с-LTO as a negative electrode

(anode during discharge) in LIB is its relatively low

theoretical specific capacity of 175 mAh/g (for

graphite this value is 372 mAh/g).

The reversible process of lithium implantation

in c-LTO at the phase boundary of the c-

LTO/Li7Ti5O12 spinel phase (111) reduces the

oxidation state of titanium from Ti+4 to Ti+3.4 in

batteries. This process in LIB takes place at a

potential of +1.5 V according to the reaction:

Li4Ti5O12 (spinel) + 3 Li+ + 3e ↔ Li7Ti5O12 (rock

salt).

(1)

The parameters of the crystal lattice of c-LTO,

as a negative electrode, do not change significantly

in this process. Titanium ions can also be reduced to

the oxidation state of Ti+3 by increasing the

concentration of implanted other ions, including

lithium ions in the c-LTO electrode. The theoretical

specific capacity of LIB in this process with lithium-

ion increases to 290 mAh/g. However, such

reactions with other metal ions can cause

irreversible structural changes in the c-LTO

electrode, and its capacity can decrease during LIB

cycling. Therefore, improving the characteristics of

the electrodes (increasing the electron conductivity

of the achieved discharge capacity, and reducing

material degradation during LIB cycling) is a

practically important task. For example, by doping

c-LTO, it is possible to introduce d-element ions

into the c-LTO lattice and reduce titanium ions to

Ti+3 without destroying the c-LTO lattice.

Doping of c-LTO with lanthanides (Eu [3], La

[4], [5], [6], [7], [8], [9], [10], [11], [12], [13], Pr

[14], Nd [15]) can increase the discharge capacity

and cycling stability over a wide potential range.

However, the effect of doping with transition d-

elements on the electrochemical properties of c-

LTO has been little studied.

The results of the study of Zr doping of c-LTO

compound with a cubic structure at the Ti site show

that the electrochemical performance of the Li4Ti5-

xZrxO12 electrode is improved, [16], [17], [18]. The

reduction of titanium ions Ti+4→ Ti+3 in the c-LTO

electrode can also be achieved by Nb doping at the

Ti site of the c-LTO lattice [19], [20]. This can

increase the charge transfer efficiency of Li4Ti5-

xNbxO12, which in turn will improve the

performance of LIB.

Our work [17] presents the results of ab initio

modeling of the monoclinic modification of m-

LTOZr(Nb) (sp. gr. , No. 15; a = 8.35 Å, b =

8.32 Å, c = 13.17 Å). Zr(Nb) doping of the m-LTO

crystal in the lithium position enhances the spin-

orbit interaction in the electronic structure of m-

LTO–Zr(Nb). And this, in turn, can increase charge

transfer and improve LIB performance. However,

there is still no data on the electronic structure of c-

LTO–Zr(Nb) crystals with a cubic structure.

The aim of this work is ab initio modeling of the

structural properties of Zr(Nb)-doped (x = 0-0.01

mole fraction) c-LTO–Zr(Nb) materials with a cubic

structure (space group 

) for LIB. Changes in

the electronic band structure and energy of doped c-

LTO–Zr(Nb) nanocrystals with changes in

composition and structure were studied using ab

initio methods with the Hubbard correction.

Supercells were considered in which the interaction

of impurity Zr(Nb) atoms partially substituting for

WSEAS TRANSACTIONS on ELECTRONICS

DOI: 10.37394/232017.2024.15.19

Mirsalim M. Asadov, Solmaz N. Mustafaeva,

Saida O. Mammadova, Esmira S. Kuli-Zade,

Vladimr F. Lukichev

E-ISSN: 2415-1513

167

Volume 15, 2024

titanium atoms leads to changes in the properties of

c-LTO–Zr(Nb).

2 Calculation Methods

Below are the results of studying the change in the

electronic structure and energy of doped c-LTO–

Zr(Nb) nanocrystals under light doping conditions.

The electronic band structures of c-LTO-–Zr(Nb) of

the cubic modification were calculated using density

functional theory methods taking into account spin

polarization and the Hubbard correction (DFT

SGGA+). The c-LTO–Zr(Nb) supercells are

considered, in which titanium particles are partially

substituted by Zr(Nb) particles in the cell. For

Li4Ti5-xZrxO12 and Li4Ti5-xNbxO12, the impurity

concentration was in the range x = 0–0.01. The

structural parameters of the crystals were calculated

without taking into account van der Waals

interactions and the U correction. The interaction of

impurity Zr(Nb) particles with partial substitution of

titanium in the c-LTO cell does not qualitatively

change the lattice parameter of c-LTO–Zr(Nb)

crystals. The use of ab initio methods allows us to

compare and predict the physical and

physicochemical properties of c-LTO–Zr(Nb)

materials. Using the example of DFT GGA–PBE

[21], [22], [23] calculation of the electronic

properties of binary and ternary phases in [24], we

present a method for calculating the characteristics

of anode materials. In [25], the electronic-structural

and physical properties of partially doped SiC–Li

supercells were calculated for LIB.

The DFT method with the local density

approximation (LDA) functional is not able to

adequately predict the parameters of materials. The

DFT LDA calculation leads to an incorrect metallic

state, for example, for transition metal oxides. In

LDA, the exchange-correlation functional ()

depends only on the density 󰇛󰇜 in the coordinate

(󰇜 where it is calculated:



󰇛󰇜 󰇛󰇜󰇛󰇜 (2)

The c-LTO object is an example of a ternary

compound in which titanium ions contain partially

filled d-orbitals. In doped c-LTO, the DFT LDA

calculations show a metallic state of the electronic

structure of c-LTO. On the other hand, the LDA

functional in the calculations underestimates the

lattice constants of the crystal and overestimates the

binding energies. In other words, the preliminary

DFT LDA calculation reduced the accuracy of the

calculation. In addition, LDA functional is

ineffective in calculations for predicting equilibrium

configurations of surface and defect structures, in

particular, oxide materials.

Taking this into account, a modified LSDA

functional was used in the calculations, including

the spin-orbit interaction (SOC) for the atomic

electron:



󰇛 󰇜 󰇛 󰇜󰇛󰇜 (3)

In DFT calculations of the energy of c-LTO-

based structures, in addition to those indicated, the

effect of the Hubbard Coulomb repulsion energy

󰇛󰇜 on the total energy was also taken into account.

Total energy calculations were performed using

GGA–PBE software, [17], [24]. The use of the

LSDA+ functional improved the results of

calculations of the electronic structure of c-LTO–Zr

and c-LTO–Nb materials. These structures are

characterized by strong electronic interactions in

spatially localized states, in particular, d-orbitals, of

Zr(Nb) alloying components. Adding the correction

term  to the single-particle electronic potential

increases the accuracy of calculations.

DFT LSDA+ calculations of c-LTO–Zr(Nb)

supercells yield a smaller band gap  than the

experimental value. Increasing the size of c-LTO–

Zr(Nb) supercells does not improve the calculated

 value compared to the experimental value c-

LTO. DFT LSDA+ calculated  were still

smaller than the experimental  value.

Taking into account the above, the generalized

gradient approximation (GGA) functional was used

in the calculations. The GGA potential adds a term

to the DFT calculations that reflects the electron

density gradient at a given point. In DFT GGA

calculations, the non-uniform distribution of

electron density and non-local effects are partially

taken into account. DFT SGGA+ calculations of c-

LTO–Zr(Nb) parameters were closer to

experimental data compared to DFT LSDA+

calculations.

In the DFT SGGA+ calculations, the basic

electron configurations of the components in the

following valence states were used: Li – 1s22s1, O –

1s22s22p4, Ti – [Ar] 3d24s2, Zr – [Kr]4d25s2, Nb –

[Kr] 4d45s1. The effect of d-orbitals of Zr(Nb)

doping of c-LTO was taken into account in the

calculations by the SOC contribution to the

electronic structure. In other words, partial

substitution of titanium in the c-LTO lattice by

doping Zr(Nb) metal atoms was taken into account

in the calculations by the SOC term.

WSEAS TRANSACTIONS on ELECTRONICS

DOI: 10.37394/232017.2024.15.19

Mirsalim M. Asadov, Solmaz N. Mustafaeva,

Saida O. Mammadova, Esmira S. Kuli-Zade,

Vladimr F. Lukichev

E-ISSN: 2415-1513

168

Volume 15, 2024

For the analysis of c-LTO–Zr(Nb) systems and

for the calculation of the total potential, we used the

effective Hubbard parameter as     ,

where the constant  = 0. In other words, for the

analysis of Ti 3d and Zr(Nb) 4d states taking into

account the occupation number of d-orbitals in the

calculation parameter , the constant was taken as

 = 0.

In the DFT SGGA+ calculations, the

contribution of the local Coulomb repulsion

between localized d-electrons was taken into

account by the effective Hubbard energy  = 4

eV. The kinetic energy of plane wave cutoff was

300 eV. In the supercell calculations, a k-grid

scheme constructed using the Monkhorst–Pack

method was used, and a flat grid with dimensions of

2 × 2 × 2 in the Brillouin zone was chosen.

Increasing the k-grid size in the irreducible wedge

of the first Brillouin zone of supercells using

nonequivalent k-points does not have a significant

effect on the experimental lattice parameters and

atomic positions in the calculations. The unit cell

parameters and atomic positions in the c-LTO–

Zr(Nb) lattice were optimized. The atomic positions

and cell parameters were determined by relaxing all

forces in the system to  0.01 eV/Å. Each alloying

element was considered as a substitutional defect in

the 32-atom c-LTO–Zr(Nb) supercell. The

formation energy was determined by the formula:

󰇛󰇜 󰇛󰇜 󰇛󰇜  󰇛

󰇜 

󰇛

󰇜 󰇛 

󰇜

(4)

where 󰇛󰇜 and 󰇛󰇜 are the total energy of the

supercell with c-LTO–Zr(Nb) and without the

alloying element Zr(Nb), , ,   are the

chemical potentials of the alloying element Zr or

Nb, Li, Ti and oxygen O, respectively. These values

were extracted from the total energy of each element

in its stable phase. For these calculations, the

Brillouin zones are selected using a k-point grid

with a density of at least 5000 points per inverse

atom.

3 Results and Discussion

3.1 Li–Ti–O System

When working with phase diagrams of ternary

systems, the following interrelated problems are

solved: a) the chemical composition of the ternary

alloy ABC (or chemical compound ABC) is

determined by the position of the figurative point on

the plane of the concentration triangle of the system

A–B–C; b) the figurative point on the plane of the

concentration triangle A–B–C is determined by the

specified composition of the ternary alloy ABC. In

our case, to analyze the equilibrium state of ternary

compounds (or phases) in Li–Ti–O alloys, it is

important to consider the quasi-binary section Li2O–

TiO2. The composition-property section diagram of

Li2O–TiO2 includes several ternary phases of

stoichiometric composition.

The phase diagram of the system contains four

ternary compounds, [17]. Figure 1 (Appendix)

shows our refined equilibrium phase diagram of the

Li–Ti–O system.

The LTO compound is one of the four ternary

intermediate phases in the Li–Ti–O system. The

LTO compound has several modifications (in

particular, monoclinic (m) and cubic (c) structure).

No noticeable deviation from the stoichiometry of

the LTO composition is observed. The temperature

of the phase transition from the monoclinic structure

to the high-temperature cubic structure m-LTO  c-

LTO is 930 С. As indicated above, we are

interested in the properties of с-LTO with a spinel

structure (cubic structure; space group 

󰇜.

3.2 Formation Energies

The formation energy of chemical bonds determines

the stability of the phase. Therefore, the formation

energy of the alloyed c-LTO–Zr(Nb) phases was

calculated. The outer level of Zr 4d25s2, Nb 4d45s1

contains four and three electrons, respectively,

located on the 4d and 5s sublevels. For Zr and Nb,

the oxidation state of +4 and +3, respectively, is

more typical, since a small amount of energy must

be spent to excite these atoms, i.e. to transfer

electrons from the 4d state to the 5s state. This is

completely covered by the formation energy of

chemical bonds of the alloyed c-LTO–Zr(Nb)

phases. The ionization potential of Zr and Nb (free

atom) is 6.84 and 6.88 eV (small), the values of

electron affinity (0.43 and 0.89 eV) and

electronegativity (1.33 and 1.6 on the Pauling scale)

are also small. Therefore, Zr and Nb, being active

metals, will exhibit reducing properties in the

alloying reaction of c-LTO. That is, Zr and Nb

atoms will give up electrons when interacting with

c-LTO.

Thus, it can be expected that doping will lead to

the formation of c-LTO–Zr(Nb) stable molecule.

That is, energy is required to break such a molecule.

The formation of a chemical bond reduces the

WSEAS TRANSACTIONS on ELECTRONICS

DOI: 10.37394/232017.2024.15.19

Mirsalim M. Asadov, Solmaz N. Mustafaeva,

Saida O. Mammadova, Esmira S. Kuli-Zade,

Vladimr F. Lukichev

E-ISSN: 2415-1513

169

Volume 15, 2024

energy of the system, since the bonding pair of

electrons is attracted simultaneously by both the

nucleus of the cation atoms (Li, Ti, Zr, Nb) and the

nucleus of the oxygen atom (O) in c-LTO–Zr(Nb).

In this case, the electrons will be distributed

unevenly. Since different cation and oxygen atoms

will attract electrons with different strengths. This is

also determined by the ionization energies of these

four different component atoms of c-LTO–Zr and c-

LTO–Nb.

In cases where the thermal and volume effects

of the reaction are large, i.e. when the energies of

formation of chemical bonds have values of the

order of 10 kJ/mol, the concentration of doping

atoms must be taken into account in the

calculations.

In general, the formation energy  (or enthalpy

of formation ) of the c-LTO–Zr(Nb) compound,

taking into account equation (1) and individual

components (i), is determined by the formula:

    󰇛󰇜 (5)

   󰇛󰇜



  󰇛 

 󰇛󰇜󰇜

  󰇛  󰇜

 

 ,

(5a)

where  is the chemical potential of element-

component ,  is the amount of element i in the c-

LTO–Zr(Nb) compound. The methodical part above

indicated that as a standard it is accepted that the

chemical potential of each species is in the ground

state of the element in the DFT calculation of the

total energy. The formation energy calculated with

this choice is valid for a temperature of 0 K.

The DFT+U calculation included Coulomb

interactions in localized d-orbitals with a U

correction. Then, taking into account the GGA+U

functional, the total energy can be determined from

the known expression:



  

 



󰇟







󰆓



󰆓󰆓

󰇠,

(6)

where 

 and  are the spherically averaged matrix

elements of the on-site Coulomb interactions of the

3d-orbital, n is the filling matrix of the 3d-orbital at

the on-site, obtained by projecting the wave function

onto the 3d atomic states. Here  or 󰆒= −2,

−1,0,1,2 are different d-orbitals,  =1 or −1 is the

electron spin. The site occupancy matrix was

expressed in explicit spin and orbital representation.

The calculated total energies were insensitive to .

To take into account the Hubbard correction 

to the GGA functional in DFT calculations in the

Coulomb repulsion effect, we neglected the

secondary effects of the Coulomb and magnetic

interactions. In other words, the parameter 

describing these effects was taken equal to zero,

     . In this case, the 



 correction

to the energy functional, significantly simplifies the

DFT SGGA+ calculation equations.

The value of  of ternary compounds of the

quasi-binary system  was calculated by

DFT LSDA method, [17]. To calculate the total

energy of the compounds, the reference values of



 of the corresponding binary oxides of the

Li–Ti–O system were used. For binary oxides of

titanium and lithium, the following values of



 (kJ/mol) were adopted: -1518 󰇛󰇜, -943

󰇛󰇜, - 68 󰇛󰇜, -598 󰇛󰇜 DFT   

calculations of the  showed a value that is 0.1

eV higher than the value obtained from DFT LSDA

calculations  = -3.21 eV, [17].

Fig. 2: Formation energy of ternary oxides as a

function of the composition of the   

system in DFT SGGA+ ( = 4 eV) calculations.

1 – , 2 – , 3 – , 4 –



The DFT SGGA+ ( = 4 eV, O 2p-, Ti 3d-,

Zr 4d- and Nb 4d-orbitals) formation energy of the

ternary compounds  (1),  (2),

 (3) and  (4) is shown in Figure

These negative  values indicate that the

ternary compounds are stable in the solid state.

Calculated DFT SGGA+ ( = 4 eV) formation

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energy for Li4Ti4.99Zr0.01O12 and Li4Ti4.99Nb0.01O12 is

–3.11 eV.

The difference in  can be related to the used

LSDA [17] or SGGA+ functional and the change

in the distribution of Ti particles and Zr(Nb)

alloying particles in the octahedral and tetrahedral

voids of the c-LTO–Zr(Nb) cell.

3.3 Atomic Structure of c-LTO

In the spinel structure of с-LTO, oxygen ions (32e)

form a cubic close-packed (ccp) array [4]. And the

gaps in the crystal lattice are partially occupied by

Li and Ti cations with tetrahedral (8a, 8b, 48c) and

octahedral (16c, 16d) coordination (Table 1). The

atomic structure of the c-LTO cell is shown in

Appendix in Figure 3(a) and Figure 3(b).

Table 1. Structural parameters of c-LTO

Atom Position Atom coordinates Occupancy

Li1 8a 0 1.000

Li2 16d 0.625 0.503

Ti 16d 0.625 0.497

O 32e 0.389 1.000

The calculated structural parameters of the

relaxed lattice of c-LTO are consistent with the

known data. For example, the DFT SGGA

calculated lattice parameter a = 8.3523 Å is in good

agreement with the data 8.3525 Å, [26].

3.4 Electronic Structure of c-LTO

The theoretical atomic parameters of a chemical

compound and the contributions to the chemical

bond and electron structure calculated on their basis

depend on the choice of the theoretical difference of

the exchange integrals for s-, p- and d-electrons. In

the considered c-LTO compound, this can be s-p-d-

local exchange interactions (s-p-d-model).

In the band structure calculations of the c-LTO

system, a difference in energy was found between

the spin "up ↑" and spin "down ↓" states. Recall that

each atom consists of protons and neutrons, as well

as orbiting electrons in a chemical compound.

Therefore, if we assume that these atoms have a spin

quantum number of ½, then the protons can also

exist in two spin states: spin "up" and spin "down".

The protons of the atoms have a positive charge and

can generate magnetic dipole moments. When a

magnetic field is applied to the dipole of a proton,

the dipole will either align itself "parallel" or

"antiparallel" to the direction of the magnetic field

depending on its spin state.

The "parallel" (low energy) and "antiparallel"

(high energy) states have a difference in energy

(ΔE) which is known to be proportional to the

magnetic field strength (B0), the gyromagnetic ratio

(γ) and Planck's constant (h):

      󰇛 

󰇜. (7)

The ratio of the number of nuclei in states with

high () and low () energy is determined as

a function of the difference in energy levels (ΔE),

the temperature (T) of the system and the Boltzmann

constant (kB):

/  󰇟 

󰇠. (8)

The results of DFT SGGA calculations of the

electronic structure of c-LTO are analyzed by

comparing the band structure and density of states

(DOS and partial PDOS) of c-LTO with the known

calculated data ( = 2 eV [26]). The band gap of c-

LTO, found experimentally by extrapolating the

linear part of the UV-visible spectra to the energy

axis, is 3.55 eV, [27].

The results of DFT SGGA calculations of the

electronic structure of c-LTO are analyzed by

comparing the band structure and density of states

(DOS and partial PDOS) of c-LTO with the known

calculated data ( = 2 eV [26]). The band gap of c-

LTO, found experimentally by extrapolating the

linear part of the UV-visible spectra to the energy

axis, is 3.55 eV, [27].

As can be seen, the theoretical band gap

obtained by us with the single-electron approach

using DFT GGA was significantly smaller than the

experimental value. To overestimate the calculated

band gap of c-LTO, large computational resources

are required. More precisely, for such systems

(insulators and semiconductors), a significant

increase in the unit cell is necessary, so they require

a corresponding increase in computational

resources. The theoretical band structure of c-LTO

calculated with two potentials is shown in Figure 4

(Appendix).

Taking into account the  correction in the

calculations of the c-LTO band structure does not

significantly increase the value of . In the band

structure, the forbidden band is located between the

occupied p-states of oxygen and the half-empty d-

states of Ti.

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(a)

(b)

Fig. 3: Atomic structure of c-LTO: primitive unit

cell (a), crystal structure (b)

DFT SGGA calculations of  values were

performed for both spin-up ↑= 2.34 eV and spin-

down ↓=2.19 eV electron states.

The energy gap is mainly determined by the

occupied O 2p states and Ti 3d states in the c-LTO

lattice. DFT SGGA+ ( = 4 eV, O 2p, Ti 3d)

calculations yield ↑= 3.56 eV, ↓ = 2.4 eV for c-

LTO.

The theoretical densities of electron states of the

c-LTO supercell calculated with the SGGA+

potential are shown in Figure 5 (Appendix). From

the DOS analysis it follows that the valence band of

c-LTO consists mainly of p-orbitals of O and d-

orbitals of Ti with a small admixture of ss-orbitals

of the components. In the conduction band of the

DOS, the s- and p-orbitals approximately coincide.

In Figure 5 (Appendix), along with the spin-

resolved with full DOS, the partial densities of the

state of Ti atoms of the c-LTO compound. The

PDOS shows sharp peaks at 2 and 4.2 eV, which

arise from the ground spin states of Ti. These Ti 3d2

states are highly localized since their energy width

is small. The minority spin s-, and p-states of Ti are

insignificant and are limited to the energy range

from −16 to 16 eV.

The energy characteristics indicate that in c-

LTO, the highest occupied molecular orbital is

mainly composed of oxygen (p-orbital). And the

lowest unoccupied molecular orbital is composed of

titanium (d-orbital). This makes the electronic

structure of c-LTO similar to insulators.

Thus, the valence band of c-LTO mainly

consists of O 2p states, while the conduction band is

dominated by the Ti 3d orbital states. The overlap of

Ti 3d and O 2p orbital states in DOS indicates a

strong O-Ti interaction in the TiO6 octahedra of the

c-LTO lattice.

3.5 Electronic Structure of c-LTO–Zr

There are several grades of cubic c-LTO powders,

differing in their properties. The differences in

properties are due to the structure of c-LTO grains

grown under different thermodynamic and kinetic

conditions. The formation of properties is associated

with the crystal structure and electronic interaction

of the particles of the substance.

Control over the type, and concentration of

point defects and impurities, as well as the

dislocation structure, enables the creation of

crystalline materials with specified properties.

However, the known technologies for the synthesis

of c-LTO do not fully allow the influence of

impurities on the properties to be controlled during

crystal growth. There are few works in the literature

devoted to the study of the electronic structural

properties of doped c-LTO. The parameters of the

crystal structure and electrochemical properties of

materials based on c-LTO are analyzed.

It can be assumed that the 4d-element impurities

Zr(Nb) do not have a significant effect on the

stoichiometry of c-LTO and the parameters of the

crystal structure when substituting the 3d-element

titanium. The sizes of Zr (atomic radius 1.60 Å;

ionic.

Radius 0.79 Å) and Nb (atomic radius 1.46 Å;

ionic radius 0.69 Å) are close to the sizes of the

substituted titanium (atomic radius 1.47 Å; ionic

radius 0.94 Å). In other words, Zr(Nb) impurities

should form substitution solutions in c-LTO. An

increase in the impurity concentration can distort the

cubic lattice of c-LTO and reduce its symmetry. A

characteristic difference between the substituted c-

LTO–Zr(Nb) material and pure c-LTO is the small

contribution of Zr(Nb) 4d-orbitals to the zone

structure.

In doped c-LTO–Zr(Nb) materials, in addition

to the electron charge, we used its spin to control

currents. The addition of 4d elements to c-LTO

supercells was carried out using the known

technique, [17]. The parameters of the crystal

structures of the alloying Zr(Nb) atoms and their

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atomic (ionic) radii differ from the parameters of c-

LTO. Therefore, the solubility of Zr(Nb) atoms in c-

LTO was taken to be several percent. That is, it was

assumed that c-LTO–Zr(Nb) forms dilute solid

solutions, where Zr(Nb) substitutes Ti. Determining

the substitution coefficient of Ti for Zr(Nb) in c-

LTO is a separate problem. Its solution will allow us

to establish where Zr(Nb) is located in the c-LTO

lattice sites when Ti is substituted.

The composition, properties, and electronic

structure of c-LTO are known, but there is still no

consensus on the type of exchange interaction of

particles upon doping c-LTO. Approaches to

calculating the electronic properties of doped c-LTO

include various modifications of the electron density

functional method.

The band structure models of c-LTO–Zr(Nb)

can be related to the relative energy positions of the

Fermi level (), the valence band, and the impurity

band, i.e. the band formed by the 4d electrons of the

Zr(Nb) impurity. In the double exchange model, the

d level of the transition metal Zr(Nb) should be in

the band gap, and in the p-d exchange model, the d

level should lie below the edge of the valence p

band. In such materials, the Fermi level can be in

the impurity band, and by changing its position, the

properties of the material can be altered. However,

the 4d electrons of Zr(Nb) can also hybridize with

the valence band at low binding energies (less than

4 eV).

Thus, taking into account also the known

experimental data (resonance photoemission,

photoelectron spectroscopy, infrared optical data,

etc.), we can conclude the following. The 4d states

of Zr(Nb) impurities can be located both in the

valence band of the initial c-LTO and correspond to

the s-d, p-d, and s-p-d of local exchange interaction

models. In this case, the maximum valence band of

c-LTO can be near  and hybridized with the

impurity 4d states of Zr(Nb). That is, the Fermi

level can be in the impurity band of Zr(Nb), and it is

possible that it overlaps with the valence band. In

semiconductors, for example, the impurity band and

the band of the initial semiconductor are mixed and

overlap. Therefore, the valence band and impurity

band of the c-LTO–Zr(Nb) structures can be taken

as conditional for analysis.

Based on the electronic structure of c-LTO, it

can be assumed that after doping, the d-states of Ti

should be partially filled. Such filling of the d-state

of Ti can also take place, for example, after Li-

intercalation of c-LTO. Then, taking into account

the model of the two-phase coexistence of the c-

LTO/Li7Ti5O12 phases (Figure 6, Appendix), it can

be assumed that near the spinel phase (111) the

structure will have a metallic character.

The energy of this phase boundary of с-

LTO/Li7Ti5O12 calculated by the DFT GGA+

method is small since even nanosized Li-

intercalated Li4+xTi5O12 nanocrystals can contain the

(111) phase boundary.

The atomic structure of Zr-doped c-LTO is

shown in Figure 7 (Appendix). Here, the Zr atom

substitutes Ti in the c-LTO lattice.

The electronic properties and band structure of

c-LTO–Zr along high symmetry points were

calculated using the DFT SGGA+ method ( =

4 eV, O 2p, Ti 3d, Zr 4d) (Figure 8, Appendix).

Taking into account the spin states of DFT

SGGA+ ( = 4 eV, O 2p, Ti 3d, Zr 4d), the

energy of the forbidden band calculated by us in the

band structure of lightly doped c-LTO–Zr (c-

Li4Ti4.99Zr0.01O12) had the following value: ↑=

2.68 eV and ↓ = 1.95 eV. For the band structure

of the lightly doped c-LTO–Zr supercell, the

valence band maximum (VBM) is located at the Γ

point, while heavy doping can shift the conduction

band minimum (CBM) towards the X point.

Heavy doping of a c-LTO–Zr can lead to the

fact that the transition of an electron between the

conduction band and the valence band (Γ–X) can be

accompanied by a change in momentum, i.e. an

indirect transition can take place. With such an

indirect transition, an indirect form of the forbidden

zone is obtained in the quasi-momentum space

between the valence band and the conduction band

of the lattice of the substance. The transition of an

electron between zones in a c-LTO–Zr structure can

be accompanied by a change in its momentum,

carried away by an additional particle, for example,

a phonon.

Thus, the DFT SGGA+ calculated band gap

energy of lightly doped c-LTO–Zr is on average

2.32 eV. This value is consistent with calculations

[23] and lower than the experimental value of 3.55

eV, [24]. The calculated total DOS and partial

densities of states PDOS for c-LTO–Zr are shown in

Figure 9 (Appendix), a, b. DFT SGGA+ 

calculated the partial magnetic moment of

zirconium in Li4Ti4.99Zr0.01O12 is 1.002 .

3.6 Electronic Structure of c-LTO–Nb

The atomic structure of c-LTO–Nb is shown in

Figure 10. Here, the Nb atom substitutes Ti in the c-

LTO lattice.

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The results of DFT SGGA+U calculations of the

electronic properties of c-LTO–Nb along the high

symmetry points are shown in Figure 11.

In the band structure of c-LTO–Nb, the valence

band maximum (VBM) is located at the Γ point, as

in c-LTO–Zr, and the conduction band minimum

(CBM) is slightly shifted towards the X point due to

light doping. The electron transition between the

conduction and valence bands (Γ–X) under such

doping is a direct transition.

Fig. 10: Atomic structure of doped c-LTO–Nb. The

Nb atom substitutes Ti in the lattice c-Li4Ti5-

xNbxO12

Fig. 11. Calculated DFT SGGA+ ( = 4 eV, O

2p, Ti 3d, Nb 4d) band structure of c-LTO–Nb,

where the Nb atom substitutes the Ti atom in the c-

Li4Ti4.99Nb0.01O12 lattice

The calculated SGGA+ band gap energy of c-

LTO–Nb was  = 2.09 eV. The total DOS and

partial densities of states PDOS for c-LTO–Nb are

shown in Appendix in Figure 12(a), and Figure

12(b).

The observed sharp peaks in the DOS spectra

arise from the ground spin states of Ti 3d and the

impurity Zr(Nb) 4d. The spin states of Ti and

Zr(Nb) are strongly localized in c-LTO–Zr(Nb)

because their energy width is small. The difference

in energy between the spin states of s, p, and d

electrons arises from the contribution of the

Hubbard parameter ( = 4 eV) to the

calculations. The value of  = 4 eV used for Ti 3d

and Zr(Nb) 4d in the DFT SGGA+ calculations

shifts the minority spin states of d electrons near the

Fermi energy. This is probably why the non-ground

spin states of d-electrons have low energy, and

electrons can be excited in them.

Therefore, these minor spin states can be filled

with electrons and affect the physical properties of

c-LTO–Zr(Nb). DFT SGGA+ calculated the local

partial magnetic moment of niobium is 0 with

Nb(Ti) substitution in c-Li4Ti4.99Nb0.01O12 lattice.

3.7 Conductivity of Doped c-LTO–Zr

Li4Ti5-xZrxO12 (x = 0.05, 0.1, 0.2) electrodes have

better electronic and ionic conductivity than the

undoped LTO electrode. This is indicated by the

results of the electrochemical study of Li4Ti5-

xZrxO12, [16]. For example, for the optimal Li4Ti5-

xZrxO12 sample with a composition of 0.1 Zr, the

charge transfer resistance and the solution resistance

are 2.0 and 99.6 Ohm, respectively.

The Li4Ti5-xZrxO12 electrode containing a doping

impurity of 0.1 Zr possesses better electronic

conductivity and ionic conductivity. With heavy

doping, part of the Zr impurity cannot substitute Ti

in the c-LTO lattice. This prevents the transport of

Li ions and electrons in c-LTO–Zr. Thus, the

lithium-ion conductivity and electronic conductivity

of c-LTO-Zr, compared to c-LTO (Figure 13) is

reduced. Electrodes made of c-LTO–Zr

compositions have similar redox characteristics.

This indicates that light Zr doping has little effect on

the parameters of the electrochemical reaction

involving the c-LTO–Zr anode. Doped Li4Ti5-

xZrxO12 (x = 0, 0.05, 0.1, 0.2) samples also have high

discharge capacity and cyclic stability at the rate of

0.5C and 1.0C compared to pure c-LTO. With an

increase in the rate from 0.5C, 1C, 3C, 5C, and 10C

to 20C, the capacity of c-LTO–Zr decreased rapidly,

[16], [26].

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Fig. 13: Cyclic volt-ampere characteristics of

Li4Ti5-xZrxO12 samples (x = 0, 0.05, 0.1, 0.2). Scan

rate: 0.2 mV/s, [16]

3.8 Conductivity of c-LTO–Nb

The c-LTO–Nb electrode has a higher specific

capacity and better cycling performance than the

LTO electrode. The c-LTO–Nb electrode exhibits a

high charge rate with a reversible capacity of 135

mAh/g at 1C, 127 mAh/g at 20C, and 80 mAh/g at

40C.

Fig. 14: Cycling characteristics of c-LTO and

Li4Ti4.95Nb0.05O12 at (a) 1C and (b) 10C, [27]

The conductivity of doped samples is also

improved (Figure 14). The conductivity of

Li4Ti4.95Nb0.05O12 and c-LTO is 1.127×10−9 and

6.615×10−10 S/cm, respectively, [27]. The

conductivity of Li4Ti4.95Nb0.05O12 also has a higher

electrical conductivity than that of LTO, [28]. At

room temperature, the conductivity of

Li4Ti4.95Nb0.05O12 is higher (2×10–5 S/cm) than that

of c-LTO (1.87 × 10–6 S/cm).

4 Conclusions

The compound of the composition Li4Ti5O12 (c-

LTO; cubic structure, space group – 

; spinel

structure) is used as an anode (along with other

high-voltage materials) in lithium-ion batteries

(LIB). However, c-LTO crystals as an anode have

disadvantages in their electronic, energy, and cyclic

characteristics. Our calculations allow us to

conclude that doping with d-elements improves the

structural and electrical properties of c-LTO

crystals.

This paper presents the results of DFT

SGGA+ 2  2  2 supercell calculations of cubic

spinel of partially substituted particles (atoms

and/or ions) of titanium on zirconium (niobium)

particles c-LTO–Zr(Nb). Calculations of lightly

doped nanocrystals c-LTO; Li4Ti4.99Zr0.01O12 and

Li4Ti4.99Nb0.01O12 (x = 0–0.01) showed the

following. Using the SGGA+

functional (the effective Hubbard parameter

was = 4 eV) and the spin polarization effect, it

is possible to adequately describe the structures

with localized states of Ti 3d and Zr(Nb) 4d

orbitals in c-LTO–Zr(Nb).

The geometry of the crystal structure with the

optimized lattice parameter of c-LTO (8.352 Å)

and c-LTO–Zr(Nb) (8.362 Å) includes an

interconnected network of Ti, Zr(Nb), and O layers

in the form of [TiO6] and partially substituted

octahedra [Zr(Nb)O6]. The proximity of the particle

sizes of titanium and 4d elements with partial

substitution of titanium ions by zirconium (or

niobium) particles in the c-LTO spinel structure

does not significantly distort the c-LTO–Zr(Nb)

structure.

The formation energy  = -3.11 eV for

Li4Ti4.99Zr0.01O12 and Li4Ti4.99Nb0.01O12 calculated

using DFT SGGA+ ( = 4 eV, O 2p-, Ti 3d-,

Zr 4d- and Nb 4d-orbital) differs little from the

value  = –3.21 eV for the monoclinic

modification of m-LTO calculated using the DFT

LSDA method. The difference in  is associated

both with the functional used and can be associated

with a change in the distribution of Ti cations and

Zr(Nb) impurity particles in the octahedral and

tetrahedral voids of the c-LTO–Zr(Nb) unit cell.

The energy spectra of the position of the

valence and conduction band edges relative to the

vacuum level and the density of states DOS of

electrons in c-LTO–Zr(Nb) indicate the following.

DOS of c-LTO–Zr(Nb) crystals contains sharp

peaks near the Fermi energy . For the

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compositions Li4Ti4.99Zr0.01O12 and

Li4Ti4.99Nb0.01O12, this energy in DOS is about –1.7

eV.

The energy near  also includes the spin-orbit

interaction (SOC) related to the spatial asymmetry

of the states. The SOC arises from the ground spin

states of the Ti 3d2, Zr 4d2 and Nb 4d4 orbitals. The

SOC energy includes the exchange energy and the

energy of the direct Coulomb interaction. The SOC

energy depends on the energy of different spin

configurations of the d-elements and the degree of

partial substitution of Zr(Nb) particles for titanium

particles. The ground spin states of the Ti 3d and

Zr(Nb) 4d orbitals are strongly localized in c-LTO–

Zr(Nb) because their energy width is small. At

weak SOC, the transitions between singlet and

triplet states of the polarized levels in c-LTO–

Zr(Nb) can be preserved. The minority spin states

of Zr(Nb) are limited by the energy range and are

negligible in DOS. Increasing the degree of partial

substitution of titanium atoms by 4d Zr(Nb) atoms

can lead to the formation of an indirect forbidden

transition in c-LTO–Zr(Nb).

DFT SGGA+ ( = 4 eV, O 2p-, Ti 3d-, Zr

4d- and Nb 4d-orbital) calculated band gap  of

doped c-LTO–Zr(Nb) structures with partial

substitution of titanium by zirconium (or niobium)

[Zr(Nb) Ti] in the spin-up (↑) and spin-down (↓)

states differ from each other. Calculations of c-

LTO–Zr(Nb) give the following average values of

, eV: с-LTO 2.98 (х = 0), Li4Ti4.99Zr0.01O12 2.32,

Li4Ti4.99Nb0.01O12 2.09.

The difference in the calculated energy

between the two spin states (spin up ↑ and spin

down ↓) is due to the parameter  used for Ti 3d,

Zr 4d and Nb 4d orbitals. Using = 4 eV affects

the shift of the minority spin states near the Fermi

energy. Since the minority spin states, for example

in Ti 3d orbitals in c-LTO have relatively high

energy, electrons cannot be potentially excited in

them. However, the 4d orbitals of the dopant

Zr(Nb) can fill the empty states of the Ti 3d

orbitals, which will affect the physical properties of

c-LTO–Zr(Nb).

Magnetic properties of a substance are usually

determined by the magnetic properties of electrons

and atoms (orbital motion of electrons; magnetic

moment of an electron; magnetic moment of an

atomic nucleus). The magnetic field caused by the

magnetic moment of a nucleus is much smaller

than the magnetic field created by the orbital

motion of electrons and the spin of electrons.

Local magnetism in c-LTO–Zr(Nb) is formed

by the contribution of spin energy (exchange

energy and direct Coulomb interaction energy) to

the total moment of an electron. When calculating

the magnetic moment in c-LTO–Zr(Nb), the

contribution of the spin magnetic moments of

electrons forming the electron shell of Zr(Nb)

atoms was taken into account. Thus, the local

magnetic moment is the result of the spatial

distribution of electrons and atoms of the c-LTO–

Zr(Nb) cell, including impurity d-orbitals of

Zr(Nb). The local magnetic moment of the partial

4d-orbitals of Zr(Nb) in the Li4Ti4.99Zr0.01O12 and

Li4Ti4.99Nb0.01O12 compounds calculated by the

   ( = 4 eV) had the following

values: 1.002  with [ZrTi] substitution and 0

 with [NbTi] substitution. An increase in the

degree of substitution of titanium particles by

impurity Zr(Nb) particles affects the calculated

magnetic moment of the partial components in c-

LTO–Zr(Nb).

The known experimental results show that the

alloyed c-LTO–Zr(Nb) materials possess improved

electronic conductivity and electrochemical

performance in LIB. c-LTO–Zr(Nb) anodes have

higher specific capacity and better cyclic

performance in LIB. For Li4Ti4.95Nb0.05O12, the

capacity is 169.1 mAh/g at 1C and 115.7 mAh/g at

10C after 100 cycles, which is significantly higher

than the capacity of the c-LTO anode. The

maximum electronic conductivity corresponds to

the Li4Ti4.95Zr0.05O12 and Li4Ti4.95Nb0.05O12

compositions. These compositions also have a

higher lithium-ion diffusion coefficient compared

to pure c-LTO.

One of the possible options for using the

obtained DFT results of calculation of the structural

properties of doped materials c-LTO–Zr(Nb) may

be their application in difficult-to-obtain structures

and areas where it is not possible to repeatedly

synthesize a nanostructure with a given

composition, to interfere with the formation of the

desired polytype of the structure and physical

properties of materials.

DFT results can be used to predict whether a

given c-LTO–Zr(Nb) cubic structure will transform

to another modification or polytype during LIB

operation. Phase transitions in c-LTO–Zr(Nb)

compounds can be associated with instability of the

crystal lattice in the highly symmetric cubic phase.

In this class of compounds, both homogeneous

non-polar distortions of the crystal lattice and

distortions associated with a change in the volume

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of the unit cell of the crystal are observed in low-

temperature phases. Using theoretical and DFT

data, important physical properties of c-LTO–

Zr(Nb) such as the density of electron states, bulk

modulus, lattice constants, permittivity, electrical

conductivity, and others can be further determined.

The practical value of the results is that, given

a method for determining the morphology and

structural parameters of batteries, the DFT

calculation data can be used to warn the user before

the physical parameters, including the crystal

lattice parameters of the LTO electrode, are

violated.

Such a forecast can save time and money for

the manufacturer and the consumer to prevent

failures in the operation of the electrochemical

system containing c-LTO. Data on the patterns of

change in physical and chemical properties during

doping will enable the preparation for electrode

damage, poor reproducibility, and failure situations

in advance and ensure the smooth operation of с-

LIB.

The obtained data can be used to evaluate the

influence of other alloying elements on the

structure, properties, design, and performance

improvement problems of LIB. Using the modeling

method with the fitting of physical and structural

properties in the prediction process, it is possible to

optimize the prediction performance and the

operation of LIB.

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Vladimr F. Lukichev

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

- Mirsalim Asadov, Solmaz Mustafaeva, Saida

Mammadova equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

- Esmira Kuli-Zade and Vladimr Lukichev was

responsible for the Statistics.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

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APPENDIX

Fig. 1: Refined phase diagram of the system   . 1 –  ; 2 –    (liquid); 3 –  

; 4 –    󰇛󰇜, where ss – solid solutions; 5 –    󰇛󰇜; 6 –

    󰇛󰇜; 7 –   󰇛󰇜; 8 –   󰇛󰇜 +    9 –   󰇛󰇜; 10

–   󰇛󰇜  󰇛  󰇜 11 –   󰇛󰇜 +    12 –   󰇛󰇜

+ ; 13 – 󰇛󰇜; 14 –   󰇛󰇜   ; 15 –   󰇛󰇜 ; 16 –

  󰇛󰇜 ; 17 –   ; 18 –   󰇛󰇜 ; 19 –   󰇛󰇜 

a) (a) b)(b)

Fig. 4: Calculated band structure in a 2 × 2 × 2 supercell of c-LTO spinel. a)(a) DFT LSDA calculation, b)(b)

DFT SGGA+ calculation ( = 4 eV, for 2p electrons of O, 3d electrons of Ti). In the calculations, only

local exchange integrals for 2p- and 3d-electrons (p-d and d-d model) were taken into account. Zero of the

energy scales corresponds to , i.e., the Fermi level is 0

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Fig. 5: Density of states of atoms in the 2 × 2 × 2 spinel supercell of c-LTO. a)(a) DOS of the c-LTO cell

calculated by the DFT SGGA+ method ( = 4 eV, O 2p, Ti 3d). (b) DOS of c-LTO, [28]. (c) PDOS of Ti in

c-LTO [26]. Zero of the energy scales corresponds to , i.e., the Fermi level is 0

Fig. 6: Structure of the two-phase model c-LTO/Li7Ti5O12 with (111) interface planes as phase boundaries

(dashed lines), [28]. In the Ti–O framework, the Li filling sites vary from 8a in c-LTO to 16c in Li7Ti5O12. The

TiO6 octahedra have a stacking sequence ABAB. . . along the <111> spinel directions in the cell

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Fig. 7: Atomic structure of Zr-doped c-LTO–Zr. The Zr atom substitutes Ti in the lattice c-Li4Ti5-xZrxO12

Fig. 8: DFT SGGA+ ( = 4 eV, O 2p, Ti 3d, Zr 4d) calculated band structure of the 2 × 2 × 2 supercell of

c-LTO–Zr spinel, where the Zr atom substitutes the Ti atom in the lattice c- Li4Ti4.99Zr0.01O12. Zero of the

energy scales corresponds to , i.e., the Fermi level is 0

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(a) (b)

Fig. 9: Spin-allowed, total DOS (a) and partial density of states (PDOS) of d-states in c-Li4Ti4.99Zr0.01O12 (b)

(a) (b)

Fig. 12: Calculated by DFT SGGA+ ( = 4 eV, O 2p, Ti 3d, Nb 4d) total density of states (a) and atomic-

partial density of d-states (PDOS) in c-Li4Ti4.99Nb0.01O12 (b) for the energy range from −16 to 16 eV

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