Theoretical Study of the Impact of Al, Ga and In Doping on Magnetization, Polarization, and Band Gap Energy of CuFeO₂

Abstract

We have conducted a first-time investigation into the multiferroic properties and band gap behavior of ${\mathrm{CuFeO}}_2$ doped with Al, Ga, and In ions at the Fe site, employing a microscopic model and Green’s function formalism. The tunability of the band gap across a broad energy spectrum highlights the potential of perovskite materials for advanced applications, including photovoltaics, photodetectors, lasers, light-emitting diodes, and high-energy particle sensors. The disparity in ionic radii between the dopant and host ions introduces local lattice distortions, leading to modifications in the exchange interaction parameters. As a result, the influence of ion doping on various properties of ${\mathrm{CuFeO}}_2$ has been elucidated at microscopic level. Our findings indicate that Al doping enhances magnetization and reduces the band gap energy. In contrast, doping with Ga or In results in a decrease in magnetization and an increase in band gap energy. Additionally, it is demonstrated that ferroelectric polarization can be induced either via external magnetic fields or by Al substitution at the Fe site. The theoretical results show good qualitative agreement with experimental data, confirming the validity of the proposed model and method.

1. Introduction

${\mathrm{CuFeO}}_2$ (CFO) belongs to the delafossite family of minerals, generally represented by the formula ${\mathrm{ABO}}_2$. This structural class is distinguished by layers of linearly coordinated A-site cations interleaved with edge-sharing ${\mathrm{BO}}_6$ octahedral networks [1,2]. CFO finds utility in a range of energy-related applications, such as solar cells, photocathodes, photoelectrochemical water splitting, and as a cathode material for lithium-ion batteries. It is recognized as a prototypical type-II multiferroic compound, in which electric polarization arises from complex non-collinear spin arrangements. In CFO, ferroelectric polarization can be induced either through the application of an external magnetic field or by substituting the Fe site with dopant ions such as Al, Ga, Rh, Ni, Mn, or Co [3,4,5,6,7]. The effects of Ga and Sc doping on the dielectric polarization of CFO have been systematically investigated by Shi et al. [8] and Elkhouni et al. [9], respectively. Their results indicated that the polarization increases with higher Ga content, while Sc doping suppresses the polarization. This doping sensitivity is thought to arise from the modulation of superexchange pathways and local structural distortions, which influence magnetic interactions and polar order. Recent experimental and computational studies suggest that dopant-induced modifications in bond geometry, especially Fe-O-Fe angles, play a pivotal role in tuning the magnetic frustration and spin–lattice coupling. Consequently, precise control over chemical substitution provides a viable strategy to tailor CFO’s multiferroic response, offering new avenues for the development of advanced multifunctional devices in electronics, spintronics, and energy conversion technologies.

In contrast to CFO, compounds such as ${\mathrm{CuCrO}}_2$ and ${\mathrm{AgCrO}}_2$ exhibit spin-induced ferroelectricity below their Neel temperature $T_N$ without the need for an external magnetic field [10]. Investigations of the dielectric constant measured both parallel and perpendicular to the crystallographic c-axis in CFO reveal that magnetic phase transitions have a pronounced influence on its dielectric behavior [11]. Terada et al. [12,13] measured the temperature-dependent dielectric constant ${ϵ}_{\left[110\right]}$ in single crystals of Ga- and Al-doped CFO, observing significant anomalies associated with spin-lattice interactions. Complementary studies by Tamatsukuri et al. [14,15] examined the dielectric response of CFO under uniaxial pressure, providing further evidence of strong spin–lattice coupling. The impact of Al doping on the stability of the ground-state spin configuration was investigated by Ajiro et al. [16] using pulsed magnetic fields, with the results interpreted through a magnetic domain framework. These collective findings illustrate the intricate interplay between magnetic ordering, lattice distortions, and impurity effects in determining the multiferroic properties of CFO. Notably, pressure and chemical doping can shift the energetic balance among competing magnetic ground states, thereby enabling external control over ferroelectric polarization. Such understanding is essential for advancing the development of tunable multiferroic systems that respond predictably to external fields or mechanical strain, a key requirement for their integration into next-generation adaptive electronic and spintronic platforms.

The delafossite compound CFO functions as a p-type oxide semiconductor, featuring a direct band gap of approximately 1.5 eV [17], which positions it as a strong candidate for solar energy harvesting, photocatalysis, and next-generation oxide-based electronic devices. In a comparative study, Shah et al. [18] reported that ${\mathrm{CuAlO}}_2$ exhibits a relatively narrower band gap than CFO, highlighting the tunability of optical and electronic properties within the delafossite family. Further experimental research conducted by Nie et al. [19] examined bipolar doping behavior and band gap anomalies in ${\mathrm{CuM}}^{III}$${\mathrm{O}}_2$ (M = Al, Ga, In) transparent conductive oxides, revealing a counterintuitive trend of increasing band gap with heavier group III cations. This trend-observed in ${\mathrm{CuAlO}}_2$, ${\mathrm{CuGaO}}_2$, and ${\mathrm{CuInO}}_2$-contrasts sharply with the behavior of traditional semiconductors, where increasing atomic number typically results in a decrease in band gap. Such anomalous electronic characteristics are believed to arise from unique interactions between Cu 3d and O 2p orbitals and the specific nature of the ${\mathrm{M}}^{3+}$ cation, which influence both the valence band structure and hybridization effects. Understanding these unconventional trends is essential for the rational design of delafossite-based optoelectronic materials with targeted band gap and carrier transport properties.

The magnetic characteristics of CFO have been the subject of extensive investigation through diverse theoretical and computational methodologies aimed at capturing its complex spin dynamics and multiferroic behavior. Petrenko et al. [20] demonstrated that a Heisenberg spin model with weak magnetic anisotropy offers a more precise description of CFO’s magnetic interactions compared to the traditionally used two-dimensional Ising model, which inadequately accounts for the system’s non-collinear magnetic ground states. To model the rich and intricate magnetic phase diagrams observed in delafossite multiferroic antiferromagnets, Ribeiro et al. [21] employed a Landau phenomenological approach, incorporating higher-order spin interactions and symmetry-breaking terms. In parallel, Zhong et al. [22] utilized first-principles density functional theory (DFT) to investigate the microscopic origins of ferroelectricity in CFO, examining both collinear and noncollinear spin arrangements and highlighting the importance of spin–lattice coupling in stabilizing ferroelectric polarization. More recently, Zhu et al. [7] developed a magnetic cluster expansion framework that systematically includes all symmetry–allowed spin interactions in CFO, successfully reproducing the multiferroic properties of both pristine and Al-substituted systems through a generalized spin-current model. However, inelastic neutron scattering studies performed by Haraldsen et al. [23] on CFO doped with 3.5% Ga unveiled a distorted screw-type magnetic ground state, indicating the presence of a magnetoelectric coupling mechanism distinct from the conventional spin-current model, possibly involving more complex exchange striction or lattice-mediated effects. These observations emphasize the nuanced and multifaceted nature of spin-lattice entanglement in doped CFO systems. Furthermore, they point to the necessity of integrating multiple theoretical frameworks—ranging from atomistic models to continuum-level approaches—to fully capture the interplay of magnetic, electric, and structural degrees of freedom. The development of such comprehensive models not only advances fundamental understanding but also provides predictive capabilities essential for the tailored design of CFO-based multiferroic materials. Ultimately, elucidating these mechanisms opens the door to engineering materials with finely tuned magnetoelectric responses for integration into multifunctional spintronic and energy-conversion devices.

In our prior work [24], we investigated the effects of ion substitution on the magnetic and electronic properties of CFO nanoparticles, specifically examining Mn, Sc, and Mg doping at the Fe site, as well as Li and Ca doping at the Cu site. In the present study, we focus on the impact of Al, Ga, and In doping at the Fe site on the multiferroic behavior and band gap characteristics of bulk CFO, employing the s-d exchange model in conjunction with Green’s function theory. This approach allows for a microscopic understanding of how different dopant ions influence the spin–charge coupling and electronic structure in the delafossite lattice.

2. Model and Method

The compound CFO crystallizes in the R$\overline{3}$m space group, characterized by a layered structure composed of alternating non-magnetic ${\mathrm{Cu}}^{+}$-${\mathrm{O}}^{2-}$-${\mathrm{Cu}}^{+}$ planes and magnetic ${\mathrm{Fe}}^{3+}$ layers aligned along the crystallographic c-axis. In this arrangement, the ${\mathrm{Fe}}^{3+}$ ions occupy a two-dimensional triangular lattice, giving rise to geometrically frustrated antiferromagnetic interactions. In addition to its magnetic behavior, CFO also exhibits semiconducting characteristics, which can be effectively modeled using the s-d exchange interaction:

$$H=H_{sp}+H_{el}+H_{sp-el}.$$

(1)

$H_{sp}$ is the Heisenberg Hamiltonian:

$$\begin{array}{ccc}\hfill H_{sp}& =& -\sum _{ij}(1-x)J_{1ij}{\mathbf{S}}_i·{\mathbf{S}}_j-\sum _{ij}x{J}_{dij}{\mathbf{S}}_i·{\mathbf{S}}_j-\sum _{il}{J}_{2il}{\mathbf{S}}_i·{\mathbf{S}}_l\hfill \\ & -& D\sum _i{\left(S_i^z\right)}^2-g{\mu }_B\mathbf{h}\sum _i{\mathbf{S}}_i.\hfill \end{array}$$

(2)

In this model, ${\mathbf{S}}_i$ and its z component $S_i^z$ represent the spin operators associated with the localized magnetic moments at lattice site i. The exchange interaction between nearest-neighbor spins is characterized by the parameter $J_{ij}$, which includes both intra-layer ($J_1<0$) and inter-layer ($J_2<0$) antiferromagnetic couplings. Next-nearest-neighbor interactions within the hexagonal planes are neglected due to their relatively weak influence. The parameter $J_d$ accounts for the exchange interaction introduced by dopant ions, and x denotes the concentration of dopants at the Fe sites. The single-ion anisotropy is described by a positive parameter $D>0$, favoring spin alignment along the easy axis, while h represents an external magnetic field applied to the system. The magnetic properties of the system originate from the ${\mathrm{Fe}}^{3+}$ ions, which possess a spin quantum number S = 5/2. Incorporating dopant effects through $J_d$ and x enables a detailed analysis of how chemical substitution influences the magnetic phase stability and spin dynamics in the compound CFO. The term $H_{el}$ denotes the Hamiltonian for the conduction electrons expressed in the Wannier representation and is given as follows:

$$H_{el}=\sum _{ij\sigma }{t}_{ij}{c}_{i\sigma }^{+}{c}_{j\sigma },$$

(3)

where $t_{ij}$ represents the hopping integral, quantifying the probability amplitude for an electron to transfer from site i to site j. The operators $c_{i\sigma }$ and $c_{i\sigma }^{+}$ correspond to the fermionic annihilation and creation operators, respectively, for an electron with spin $\sigma$ at lattice site i. This Hamiltonian captures the kinetic energy contribution of the conduction electrons within the crystal lattice. Including the hopping terms allows for the analysis of electron mobility and its coupling with the localized magnetic moments in the system.

The Hamiltonian $H_{sp-el}$ describes, within the framework of the single-band approximation, the exchange interaction between conduction electrons and localized magnetic moments, and is expressed as follows:

$$H_{sp-el}=-\sum _i{I}_i{\mathbf{S}}_i{\mathbf{s}}_i,$$

(4)

where I denotes the strength of the short-range exchange interaction between the localized spin operator ${\mathbf{S}}_i$ at site i and the conduction electron spin ${\mathbf{s}}_i$ at the same site. Due to its localized nature, this interaction is limited to on-site coupling, implying that a conduction electron interacts only with the magnetic ion at the same lattice position. The components of the conduction electron spin ${\mathbf{s}}_i$ can be expressed in terms of the fermionic creation and annihilation operators as follows: $s_i^{+}=c_{i+}^{+}{c}_{i-}$, $s_i^z=(c_{i+}^{+}{c}_{i+}-c_{i-}^{+}{c}_{i-})/2$. This Hamiltonian plays a central role in the s-d model, containing the spin-dependent scattering processes that influence both the magnetic and electronic transport properties of the material.

The magnetization is observed from

$$M=\langle S^z\rangle ={\displaystyle \frac{1}{N}}\sum _{ij}\left[(S+0.5)coth\left[(S+0.5)\beta E_{ij}\right)]-0.5coth\left(0.5\beta E_{ij}\right)\right].$$

(5)

Here, S denotes the spin quantum number, $\beta =1/k_{B}T$. $E_{ij}$ represents the spin excitations obtained from the spin Green’s function $G_{ij}=\ll S_i^{+};S_j^{-}\gg$, calculated using the method of Tserkovnikov [25].

The following is a brief description. After a formal integration of the equation of motion for the retarding two-time GF

$$G_{ij}\left(t\right)=\langle \langle S_i^{+}\left(t\right);S_j^{-}\rangle \rangle$$

(6)

one obtains

$$G_{ij}\left(t\right)=-i\theta \left(t\right)\langle [S_i^{+};S_j^{-}]\rangle exp(-i{E}_{ij}\left(t\right)t),$$

(7)

with $\theta \left(x\right)$ = 1 for $x>0$, $\theta \left(x\right)$ = 0 for $x<0$,

$$\begin{array}{ccc}\hfill E_{ij}\left(t\right)=E_{ij}& -& {\displaystyle \frac{i}{t}}{\int }_0^{t}d{t}^{\prime }{t}^{\prime }\left({\displaystyle \frac{\langle [j_i\left(t\right);j_j^{+}\left(t^{\prime }\right)]\rangle }{\langle [S_i^{+}\left(t\right);S_j^{-}\left(t^{\prime }\right)]\rangle }}\right.\hfill \\ & -& \left.{\displaystyle \frac{\langle [j_i\left(t\right);S_j^{-}\left(t^{\prime }\right)]\rangle \langle [S_i^{+}\left(t\right);j_j^{+}\left(t^{\prime }\right)]\rangle }{{\langle [S_i^{+}\left(t\right);S_j^{-}\left(t^{\prime }\right)]\rangle }^2}}\right)\hfill \end{array}$$

(8)

and $j_i\left(t\right)=\langle [S_i^{+}\left(t\right),H_{interaction}]\rangle$. The time-independent term

$$E_{ij}={\displaystyle \frac{\langle [[S_i^{+},H];S_j^{-}]\rangle }{\langle [S_i^{+};S_j^{-}]\rangle }}$$

(9)

is the excitation energy in the generalized Hartree–Fock approximation. The time-dependent term in Equation (8) includes damping effects.

In CFO with easy-axis magnetic anisotropy, the observed ferroelectric coupling cannot be accounted for by the inverse Dzyaloshinskii–Moriya mechanism [26] or by the standard spin-current model [27,28]. Arima [29] proposed that in crystals exhibiting triclinic, monoclinic, or trigonal symmetry, ferroelectricity may arise from proper screw-type magnetic ordering via modulation of metal–ligand hybridization influenced by spin–orbit coupling. In the case of CFO, ${\mathrm{Fe}}^{3+}$ ions possess three 3d electrons occupying a narrow, fully filled 3d ${\mathrm{t}}_{2g}$ band in a high-spin configuration. According to Arima [29], the d-p hybridization-altered by spin–orbit interactions in partially filled ${\mathrm{t}}_{2g}$ orbitals can generate a microscopic electric polarization aligned along the axis of the magnetic helicoid. This form of local polarization can be expressed as follows:

$${\mathbf{P}}_{ij}=const[({\mathbf{S}}_i·{\mathbf{e}}_{ij}){\mathbf{S}}_i-({\mathbf{S}}_j·{\mathbf{e}}_{ij}){\mathbf{S}}_j].$$

(10)

${\mathbf{r}}_{12}$ denotes the unit vector connecting adjacent ions i and j. This framework effectively accounts for the emergence of ferroelectricity in systems such as ${\mathrm{CuFe}}_{1-x}$${\mathrm{Al}}_x$${\mathrm{O}}_2$ (CFAO), which exhibits a proper screw-type magnetic order, and is likewise applicable to CFO. According to Arima’s mechanism, a reversal in the helicity of the magnetic screw structure leads to a corresponding reversal in the electric polarization—a phenomenon that has been experimentally verified by CFAO [30]. Consequently, the ferroelectric behavior observed in the triangular-lattice antiferromagnet CFO holds promise for applications involving the detection and manipulation of spin chirality.

Upon rotating the system about the c-axis by ${\mathrm{60}}^0$, such that the a-axis aligns with the [110] crystallographic direction, the unit vector ${\mathbf{e}}_{ij}$ as well as the spin vectors ${\mathbf{S}}_i$ and ${\mathbf{S}}_j$, adopt the following coordinate representations:

$${\mathbf{e}}_{ij}(1,0,0),{\mathbf{S}}_i(S_i^x,0,S_i^z),{\mathbf{S}}_j(S_j^x,0,S_j^z).$$

(11)

In the new coordinate system, the spins associated with the helical magnetic structure now reside within the $xz$ plane. This configuration facilitates a more straightforward evaluation of the polarization using Equation (10). Inserting these expressions in Equation (10) and using $S_i^x=0.5(S_i^{+}+S_i^{-})$ we obtain after long calculations

$$P_{ij\left[110\right]}={\displaystyle \frac{1}{2}}[(S_i^z-S_j^z)+(S_i^{-}{S}_i^{+}-S_j^{-}{S}_j^{+})].$$

(12)

Finally, we can write the expression for the polarization $P_{\left[110\right]}$ in ${\mathrm{CuFeO}}_2$ as

$$P_{\left[110\right]}={\displaystyle \frac{1}{N^2}}\sum _{ij}\langle P_{ij\left[110\right]}\rangle$$

(13)

where N represents the number of lattice sites.

Let us emphasize, that in certain compounds, such as CFO, the induced polarization is parallel to the vector of the spiral magnetic structure. This immediately rules out the possibility of the appearance of P via the “inverse” Dzyaloshinskii–Moriya mechanism or the magnetostrictive mechanism. Only with an appropriate combination of spin structure and crystal symmetry (magnetic materials with triangular unit cells and non-centrosymmetric lattices) can a macroscopic electric polarization be observed, as described by the Arima’s model. At high magnetic fields, the noncollinear magnetic structure becomes destabilized, and the polarization vanishes. The layered structure implies that local dipole moments induced by exchange striction will be oriented in opposite directions and will mutually cancel each other. At high impurity concentrations, secondary phases and clusters are formed, which alter the structure. The model we have constructed does not consider cluster formation. The fact that there is experimental confirmation of our numerical results supports the conclusion that the model adequately describes the system under investigation.

The band gap energy $E_g$ was calculated using the following relation:

$$E_g={ϵ}^{+}(\mathbf{k}=0)-{ϵ}^{-}(\mathbf{k}={\mathbf{k}}_{\sigma }),$$

(14)

where ${ϵ}^{+}(\mathbf{k}=0)$ and ${ϵ}^{-}(\mathbf{k}={\mathbf{k}}_{\sigma })$ denote the energies at the top of the valence band and the bottom of the conduction band, respectively. The electronic energy levels are given by the expression:

$${ϵ}_{ij}^{\pm }={ϵ}_{ij}-{\displaystyle \frac{\sigma }{2}}IM,$$

(15)

where ${ϵ}_{ij}$ represents the band electron energies in the paramagnetic state, $\sigma$ = ±1 corresponds to the spin index, and M is the magnetization. These energies are determined from the poles of the electronic Green’s function $g_{ij\sigma }=\ll c_{i\sigma };c_{j\sigma }^{+}\gg$. This approach allows for a self-consistent evaluation of the band structure modifications induced by the magnetic ordering in the system.

3. Numerical Results and Discussion

The numerical calculations were conducted using the JAVA software platform, employing a self-consistent iterative method. The initial parameters for the iterations correspond to the model values specified below at T = 0 K. In each subsequent step, the output from the preceding iteration serves as the input for the next. The iterative process proceeds until the difference between successive iterations falls below a predefined convergence threshold. Convergence was achieved when successive iterations yielded stable values for the magnetization, polarization, and band gap, ensuring the reliability of the computed results. The parameters utilized in the simulations are: $J_1$ = −0.456 meV [31], $J_2$ = −0.132 meV [31], D = 0.068 meV [31], spin quantum number S = 5/2, and exchange interaction strength I = 0.2 eV [32].

3.1. Al Doping Dependence of the Magnetization in CFO

The magnetization of CFO can be altered by substituting various ions at the Fe sites. The introduction of dopant ions in place of Fe leads to modifications in the microstructure, causing lattice distortions. The magnetic properties of CFO are mainly governed by the 3d orbitals of Fe, especially through Fe-Fe exchange interactions. Therefore, substituting Fe sites to introduce different magnetic interactions has been extensively utilized to investigate the magnetism in CFO. In this study, we focus at first on the effect of ${\mathrm{Al}}^{3+}$ doping on the magnetic behavior of CFO. The ionic radius of the host ${\mathrm{Fe}}^{3+}$ ion is 0.69 $\dot{A}$, whereas that of the substituted ${\mathrm{Al}}^{3+}$ ion is slightly smaller, namely 0.675 $\dot{A}$. This size difference generates compressive strain, resulting in a reduction in the lattice parameters. Within our theoretical framework, this implies that the exchange interaction constant in the doped state, denoted by $J_d$, which is inversely related to the lattice parameters, becomes larger than that in the undoped material ($J_d>J$). Consequently, the magnetization is enhanced in Al-doped CFO. The calculated magnetization as a function of doping concentration x (ranging from 0 to 0.02) is presented in Figure 1, showing good agreement with experimental observations reported by Seki et al. [3].

Let us note that within the framework of our developed model, we operate in such concentration ranges of the dopants that the solubility limit (referring to solid solutions) is not exceeded, i.e., no clusters or secondary phases of the dopant ions are formed. Beyond this limit, synthesis processes become highly influenced, difficult to control, and unsuitable for the fabrication of components in device engineering, as the resulting experimental outcomes are non-reproducible. We assume thermodynamic stability to be present, as we are working with materials for which experimental measurements exist and the results are reproducible, which implies such thermal stability.

The Neel temperature $T_{N1}$ = 14 K remains essentially unchanged upon doping. Notably, at low Al doping levels, the system exhibits predominantly antiferromagnetic order, followed by a regime where antiferromagnetism coexists with ferromagnetic components.

These results underscore the sensitivity of the magnetic ground state to subtle lattice distortions induced by ionic substitution. Further investigations are required to elucidate the interplay between doping concentration, strain effects, and magnetic phase transitions in CFO.

3.2. Al Doping Dependence of the Band Gap Energy in CFO

The band gap is a crucial parameter governing the optoelectronic characteristics of materials, making its controllability highly important. The capability to tune the band gap across a broad energy spectrum has established perovskites as promising materials for diverse applications, including photovoltaics, lasers, light-emitting devices, photodetectors, and high-energy particle detectors. In particular, semiconductors with narrower band gaps are essential for improving absorption within the visible solar spectrum. Delafossite CFO is a p-type oxide semiconductor with an approximate band gap of 1.5 eV [17], attracting considerable attention for solar energy conversion and oxide electronic devices. The band gap in CFO mainly arises from charge-transfer transitions between O-2p and Fe-3d electronic states. In this study, we focus on the effects of ${\mathrm{Al}}^{3+}$ doping at the Fe sites in CFO. As previously noted, the substitution induces compressive strain, leading to an increase in the exchange interaction constant ($J_d>J$). Using Equations (11) and (12), we derive the doping dependence of the band gap $E_g$ in Al-doped CFO, observing that the magnetization M increases with the doping concentration x. The results, depicted in Figure 2, reveal a reduction in the band gap energy $E_g$ with increasing Al content, resulting in enhanced transparency within the visible spectral region. This finding aligns well with experimental observations reported by Shah et al. [18], validating the effectiveness of the proposed theoretical model in capturing the electronic and magnetic behavior of Al-doped CFO.

Nevertheless, comprehensive experimental investigations and advanced theoretical analyses remain necessary to fully elucidate the underlying mechanisms by which Al doping influences the structural, magnetic, and electronic properties of CFO. Such studies will contribute to optimizing CFO-based materials for future optoelectronic and spintronic applications.

3.3. Al Doping Dependence of the Polarization in CFO

The compound CFO exhibits multiferroic behavior, wherein electric polarization P can be induced either through the application of an external magnetic field h or by chemical substitution at the Fe site with various dopant ions such as Al, Ga, Rh, Ni, Mn, or Co [3,4,5,6,7]. The dependence of the polarization component $P_{\left[110\right]}$ on the Al concentration under zero magnetic field (h = 0) and constant temperature (T = const) has been analyzed. As shown in Figure 3, the polarization $P_{\left[110\right]}$ increases monotonically with doping concentration x, remaining zero for $x\le 0.01$. This behavior indicates a threshold concentration necessary to induce spontaneous polarization in the absence of an external field. The calculated trend $P\left(x\right)$ is in good agreement with experimental results reported by Seki et al. [3], Nakajima et al. [30], and Shi et al. [8].

To further substantiate the magnetoelectric coupling, the field dependence of the polarization $P_{\left[110\right]}$ was investigated. In the ferroelectric incommensurate phase, a finite polarization along the [110] direction emerges under the influence of a magnetic field applied parallel to the c-axis ($h_c$). This phenomenon has been examined through various experimental techniques, including bulk magnetoelectric measurements [3], synchrotron X-ray diffraction [15], neutron scattering [5], electron spin resonance [11], terahertz spectroscopy [10], and Moessbauer spectroscopy [33]. At zero doping (x = 0), a polarization $P_{\left[110\right]}$ is still induced when the magnetic field exceeds a threshold value, as depicted in Figure 4. Below approximately $h_c$ = 60 kOe, $P_{\left[110\right]}$ remains negligible, after which it increases with the field strength. However, at higher magnetic fields, polarization is gradually suppressed, indicating a magnetic-field-induced transition from an incommensurate polar phase to a nonpolar spin configuration. This field-induced suppression of polarization is characteristic of a spin reorientation transition, confirming the delicate balance between magnetic and electric orders in CFO. The calculated polarization response under magnetic field agrees well with experimental data by Seki et al. [3] for Al-doped CFO. Comparable magnetoelectric responses under ion doping and magnetic fields have been observed for Rh- and Ga-substituted CFO, as reported by Pachoud [34] and Haraldsen et al. [23], respectively.

These findings underscore the tunability of multiferroic properties in CFO through external magnetic field and chemical perturbations, making it a promising platform for magnetoelectric device applications. Further studies on field orientation and temperature dependence could provide deeper insights into the phase boundaries and coupling mechanisms.

3.4. Ga and In Doping Dependence of the Magnetization and Band Gap Energy in CFO

To further substantiate the capability of the proposed model to accurately describe the properties of doped CFO, we extend our analysis to include the band gap energies of ${\mathrm{CuAlO}}_2$, ${\mathrm{CuGaO}}_2$, and ${\mathrm{CuInO}}_2$, as experimentally investigated by Nie et al. [19]. The ionic radius of ${\mathrm{Al}}^{3+}$ (0.675 $\dot{A}$) is smaller than that of ${\mathrm{Ga}}^{3+}$ (0.76 $\dot{A}$), which in turn is smaller than that of ${\mathrm{In}}^{3+}$ (0.94 $\dot{A}$), whereas the ionic radius of ${\mathrm{Fe}}^{3+}$ is 0.69 $\dot{A}$-positioned between those of Al and Ga. Let us consider, firstly, the case of Ga doping of CFO. Ga has a larger radius compared to that of Fe, i.e., there appears a tensile strain. We have the relation $J_d<J$ which leads to a decrease in M and from Equations (11) and (12) an increase in the band gap energy $E_g$ (see Figure 1 and Figure 2, curve 2). In analogy, the magnetization of In-doped CFO is smaller whereas the band gap energy is larger than that of undoped CFO (see Figure 1 and Figure 2, curve 3). It must be noted that because the difference between the radii of In and Fe is larger than that between Ga and Fe, we have to choose a smaller $J_d$ value compared to the previous case. The observed results are in very good agreement with those reported by Nie et al. [19], Yanagi et al. [35,36], and Ueda et al. [37] for ${\mathrm{CuAlO}}_2$, ${\mathrm{CuGaO}}_2$ and ${\mathrm{CuInO}}_2$. It can be seen from Figure 2 that for x = 1 we would observe that the band gap $E_g$ increases from ${\mathrm{CuAlO}}_2$ to ${\mathrm{CuGaO}}_2$, and further to ${\mathrm{CuInO}}_2$, as reported by Nie et al. [19].

In principle, we would obtain the same result, an increasing band gap energy $E_g$ from Al to In for x = 1, also by substituting ${\mathrm{CuAlO}}_2$ with Ga, as well as ${\mathrm{CuGaO}}_2$ with In, where tensile stress and an increase in the band gap are also observed. Further quantitative studies incorporating strain-tuned exchange parameters could deepen our understanding of structure-property correlations in these systems.

Let us note that this increasing band gap from M = Al, Ga, to In, is an interesting result not seen in conventional semiconductors. Contrary to our and to the experimental results, the direct band gap of ${\mathrm{CuAlS}}_2$ (3.49 eV), ${\mathrm{CuGaS}}_2$ (2.43 eV), and ${\mathrm{CuInS}}_2$ (1.53 eV) decreases when the atomic number of the group-III elements increases [19]. This case will be studied in a next paper.

4. Conclusions

Using, for the first time, the s-d exchange model in conjunction with Green’s function theory, we have investigated the dependence of magnetization and band gap energy on the concentration of various doping ions, such as Al, Ga, and In in CFO comparing the results between them and with available experimental data. A microscopic mechanism has been proposed to elucidate how ion doping alters these physical properties. Specifically, the exchange interaction constants-being inversely related to the lattice parameters-are modifiable via ion substitution, which introduces lattice strain due to the size mismatch between dopant and host ions. In the case of Al doping, the resulting compressive strain enhances the exchange coupling, leading to an increase in the magnetization M and a concurrent decrease in the band gap energy $E_g$. Furthermore, we demonstrated that electric polarization can be induced either through external magnetic field application or via substitution of ${\mathrm{Fe}}^{3+}$ with ${\mathrm{Al}}^{3+}$ ions. Conversely, doping with larger ions such as ${\mathrm{Ga}}^{3+}$ and ${\mathrm{In}}^{3+}$ introduces tensile strain, resulting in reduced magnetization and increased band gap energy. The theoretical predictions obtained show good qualitative agreement with available experimental observations. These findings highlight the critical role of ionic radii and lattice distortions in tuning the multifunctional properties of delafossite materials for potential device applications.