The recently discovered tunnel magnetodielectric (TMD) effect—the magnetic field-induced increase in the dielectric permittivity (*ε*′) of nanogranular composites caused by the spin-dependent quantum mechanical charge tunneling—is of interest for both the scientific value that combines the fields of magnetoelectric and spintronics and multifunctional device applications. However, little is known about how large the maximum dielectric change Δ*ε*′/*ε*′ can achieve and why the Δ*ε*′/*ε*′ variations obey the dependence of square of normalized magnetization (*m*^{2}), which are critically important for searching and designing materials with higher Δ*ε*′/*ε*′. Here, we perform approximate theoretical derivation and reveal that the maximum Δ*ε*′/*ε*′ can be estimated using intrinsic tunneling spin polarization (*P*_{T}) and extrinsic normalized magnetization (*m*), that is, Δ*ε*′/*ε′* = 2*P*_{T}^{2}*m*^{2}. This formulation allows predicting over 200% of theoretical limit for *m* = 1 and accounts for the observed *m*^{2} dependence of Δ*ε*′/*ε*′ for a given *P*_{T}. We experimentally demonstrate that *x*-dependence of Δ*ε*′/*ε′* in (Co_{x}Fe_{100−x})–MgF_{2} films is phenomenologically consistent with this formulation. This work is pivotal to the design of ultra-highly tunable magnetoelectric applications of the TMD effect at room temperature.

The magnetoelectric (ME) effect has gained considerable research attention because of its ferroic properties and application in multifunctional devices, such as magnetic-field sensors, antennas, and storage devices.^{1–5} ME materials are typically single-phase compounds and composites that are either available in bulk or thin films. The progress in the theoretical model and fabrication techniques of ME materials have encouraged the development of various ME composites with numerous heterostructures, such as laminar,^{6–8} pillar,^{9–11} and granular nanostructures,^{12,13} with an intimate interface contact between the piezoelectric and magnetostrictive phases. These ME materials and structures increase the potential for multifunctional ME applications.

With an appropriate combination of magnetic nanogranules and insulating host matrix, nanogranular structures, a kind of disordered granular solids, can exhibit both magnetic and dielectric characteristics. We recently observed an appreciable increase in the dielectric permittivity (*ε′*) of this structure on the application of a magnetic field (*H*), which is a kind of magnetodielectric (MD) effect. Unlike the aforementioned mechanism, the MD effect is caused by the spin-dependent quantum–mechanical tunneling of charge carriers through interaction with the insulating matrix (tunnel barrier).^{14,15} This unique MD effect is a result of the collective behavior of numerous electric dipoles (granular pairs) as displayed in Fig. 1(a), and thus also termed as the tunnel magnetodielectric (TMD) effect. The magnetization (*M*) direction of a single granular pair tends to align with that of *H* in Fig. 1(b). The TMD effect features a considerable maximum dielectric change (Δ*ε*′/*ε′*) with frequency in Fig. 1(c). The resultant peak position is closely correlated with the average intergranular spacing (*s*)^{16,17} and granular distribution.^{18}

Since the discovery of the TMD effect, large experimental effort has been made to improve the magnitude of Δ*ε*′/*ε′*. For instance, given that the achieved Δ*ε*′/*ε′* of 0.1% at *H* = 1 kOe and 0.3% at *H* = 10 kOe of Fe_{9}Co_{8}–Mg_{26}F_{57} nanogranular film was not adequately high,^{14} a two-dimensional Co/AlF_{3} granular heterostructure^{19} that permits the introduction of a small fraction of ferromagnetic component was proven to substantially enhance the Δ*ε*′/*ε′* up to 0.8% at *H* = 1 kOe; this peak Δ*ε*′/*ε′* was subsequently increased to 1.45% as the thickness was reduced to the monolayer limit.^{20} Furthermore, engineering the refined granule–matrix interfaces further increased Δ*ε*′/*ε′* up to record highs of 2.1% and 8.5% at *H* = 1 and *H* = 10 kOe, respectively.^{21} Despite these experimental studies^{14,16,18,19} on the control or enhancement of the TMD effect that have greatly improved the understanding of the structure-property correlation, the prediction of the theoretical limit of Δ*ε*′/*ε′* has yet to be unraveled so far. On the other hand, although a prior study^{14} has illustrated the magnetic-field dependence of Δ*ε*′/*ε′*, the dependence on the square of the normalized magnetization (*m*^{2} = *M*^{2}/*M*_{s}^{2}) is yet to be understood despite this correlation (i.e., Δ*ε*′/*ε′* ∝ *m*^{2}) being centered around the spin-dependent quantum-mechanical charge tunneling mechanism, which has been observed in numerous nanogranular material systems.^{14–19} Therefore, understanding these behaviors from a theoretical perspective, specifically on how large Δ*ε*′/*ε′* can attain and how such a correlation can arise, is critical.

In this study, we combine the Debye–Fröhlich model with the spin-dependent charge tunneling effect to theoretically derive the limit of Δ*ε*′/*ε′* and show that the TMD ratio can be expressed as Δ*ε*′/*ε′* = 2*P*_{T}^{2}*m*^{2}, where *P*_{T} is the tunneling spin polarization (*P*_{T}) of the magnetic nanogranules and *m* is the normalized magnetization (*m* = *M*/*M*_{s}). According to this equation, for a sufficiently large magnetic field (*m* = 1), Δ*ε*′/*ε′* becomes 2*P*_{T}^{2}, whereby it allows predicting a theoretical limit of Δ*ε*′/*ε′* exceeding 200%, and for a small *P*_{T}, Δ*ε*′/*ε′* is proportional to *m*^{2}. We finally investigate the *x*-dependence of Δ*ε*′/*ε′* in (Co_{x}Fe_{100−x})–MgF_{2} films that can be consistently explained by the formulation.

For a nanogranular structure with numerous nanogranules, the distribution of intergranular spacing (*s*) and the various resultant relaxation times (*τ*_{0}, *τ*_{1}, *τ*_{2}, *τ*_{3},…,) are displayed in Fig. 1(a). The distribution of *τ* can be quantitively depicted using the following Cole–Cole equation:^{22}

where *β* (0 < *β* ≤ 1) represents a measure of *τ* distribution that remains centered around *τ*_{r}. When *β* = 1, the structure is an ideal Debye relaxation state and possesses a distribution for 0 < *β* < 1. The magnetization (*M*) direction is random in the absence of the magnetic field (*H* = 0); it aligns with that of *H* for considerably high *H* and approaches the saturated magnetization (*M*_{s}) state in Fig. 1(b), that is, *M*/*M*_{s} = 1. The *H*-induced variation in *τ*_{0} conforms to the following rule:

where *m* is the relative or normalized magnetization *M*/*M*_{s}, *P*_{T} is the tunneling spin polarization and refers to the difference between the number of spin-up and spin-down electrons divided by their sum, i.e., (D↑− D↓)/(D↑ + D↓), where *D _{σ}* is the density of states at the Fermi energy

*E*

_{F}for electrons with up (

*σ*= ↑) and down (

*σ*= ↓) spin.

^{23}So, the

*P*

_{T}should be restricted in the range of 0–1 and may vary depending on the magnetic materials and tunnel barrier materials used.

^{24}In particular, the half-metallic ferromagnets can ideally have a complete spin polarization, i.e.,

*P*

_{T}= 1.

^{25}

^{,}

*τ*

_{0}

^{H}and

*τ*

_{0}are the relaxation times with and without applied

*H*, respectively, during which times the charge carrier tunnels back and forth between a pair of two granules shown in the right side of Fig. 1(b). The tunneling rate

*γ*

_{12}of the charge carrier relies on the charging energy

*E*

_{12}, the intergranular spacing (

*s*), tunnel barrier height, and the relative orientation of

*M*between two granules,

^{23,26}

where *κ* is the decay rate of the electron wave-functions of the insulator matrix^{27} and *k*_{B} is the Boltzmann constant. Upon *H*, a gradually increasing *M* increases *γ*_{12} in Eq. (3), which induced in a decrease in *τ* in Eq. (2). As a result, dielectric enhancement over the frequency range was observed in Fig. 1(c). The dielectric permittivity *ε*′(*ω*) can be illustrated using the Debye–Fröhlich model that is able to illustrate the dielectric polarization of such disordered granular solid, as follows:

where Δ*ε*′ is the dielectric strength (*ε*′_{1k} − *ε*′_{∞}), and it is approximately determined by the number of density of granular pairs, *ω* = 2*πf*. The Δ*ε*′/*ε* can approach its maximum value near the dielectric relaxation process in Fig. 1(c), that is, *f* = 1/*τ*_{0}.

To determine the maximum limit of Δ*ε*′/*ε′*, we first derive the real parts of the dielectric permittivity *ε*′ as follows:

Here, *β* is a critical parameter that can influence the width of the frequency response of Δ*ε*′/*ε′* and its magnitude.^{16} A larger *β* value typically suggests a higher Δ*ε*′/*ε′*. The details of the effect of *β* on the magnitude and width of Δ*ε*′/*ε′* are presented in the supplementary material. If *β* was not equal to 1, this disordered granular structure may display a certain distribution of inter-granular spacing(*s*) with a variance (Δ*s*), consequently enabling a positive TMD ratio distributed over a wide frequency range. In other words, the granular pairs with large *s* may contribute more to the low-frequency side of the TMD effect, whereas granular pairs with small *s* are responsible for the high-frequency region in Fig. 1(c). Therefore, the Δ*s* is more likely undesired because it may inevitably result in a reduction in peak magnitude.^{18} To accurately estimate the maximum Δ*ε*′/*ε′*, *β* can be reasonably assumed to be 1, so that the intergranular distance (*s*) for each granular pair was assumed to be almost the same as that of a single *τ*_{0}. This ideal structural configuration resembles that of CoFeB/MgO/CoFeB magnetic tunnel junctions with an MgO spacer layer between CoFeB electrodes.^{28} Assuming *β* = 1 also simplifies Eq. (5), and the dielectric permittivity without (*ε*′) and with (*ε*′_{H}) applied *H* can be expressed as follows:

This expression is known as the Debye model, which allows us to disregard the distribution state of *τ*_{0}. According to Eqs. (6a) and (6b), the dielectric change (i.e., TMD ratio) that is defined as the change of *ε*′ with and without *H* can be derived as

The maximum Δ*ε*′/*ε′* is known to appear near *f* = 1/*τ*_{0}, as presented in Fig. 1(c). We, thus, reasonably assume *ω* = 2*πf* = 2*π*/*τ*_{0}. By substituting Eq. (2) in Eq. (7), the TMD ratio can be expressed, based on the aforementioned assumption, as follows:

This formulation allows estimating accurate Δ*ε*′/*ε′* that is determined by the *P*_{T} and *m*. We notice that both *P*_{T} and *m* should be restricted between 0 and 1; hence, *P*_{T}^{2}*m*^{2} should be inevitably in the range 0–1, which results in the (1+*P*_{T}^{2}*m*^{2})^{2} value becoming considerably smaller than 4*π*^{2}. Therefore, the following expression can be simplified as follows:

Considering that the *P*_{T}^{4}*m*^{4} value is negligibly smaller than 2*P*_{T}^{2}*m*^{2}, the aforementioned formulation can be further approximately simplified to enable a better understanding of the correlations between Δ*ε*′/*ε′* and the parameters (*P*_{T} and *m*),

To predict the maximum Δ*ε*′/*ε′*, an infinitely high *H* is reasonably assumed, that is, *m* = *M*/*M*_{s} = 1. The Δ*ε*′/*ε′* can, thus, be denoted as Δ*ε*′/*ε′* = 2*P*_{T}^{2}. We then investigated the difference between Eqs. (8) and (10) without and with approximation for *m* = 1 and plotted the results in Fig. 2. At a large *P*_{T} > 0.5, a large deviation was observed, and Δ*ε*′/*ε′* from Eq. (8) could theoretically reach 270% for *P*_{T} =1, which is higher than the predicted 200% from the approximately derived Eq. (10) in Fig. 2(a). Nevertheless, at small *P*_{T} ≤ 0.5, the results from the two formulas are consistent from the inset of Fig. 2(a), suggesting that Δ*ε*′/*ε′* from Eq. (8) can be estimated by 2*P*_{T}^{2} using Eq. (10) at small *P*_{T} range.

The derived results from Eq. (10) also show that for a specific *P*_{T}, Δ*ε*′/*ε′* increases in proportion to the square of the normalized magnetization (*m*^{2}), that is, Δ*ε*′/*ε′* ∝ *m*^{2}, if each nanogranule behaves like a superparamagnet, and the magnetic interactions among them were negligibly small. For a given *m* = 1 and *P*_{T} = 1, Δ*ε*′/*ε′* approached the theoretical limit in Fig. 2(b). Notably, the *m*^{2} dependence of Δ*ε*′/*ε′* in the inset of Fig. 2(b) has been used to explain the observed data in various nanogranular materials systems^{14,16,18} based on the Inoue–Maekawa model^{23} in which the average angle (cos *θ*) of the magnetization direction over a large number of *θ* values can be regarded as *m*^{2}, that is, cos *θ* = *m*^{2}. Therefore, the derived Δ*ε*′/*ε′* ∝ *m*^{2} offers direct evidence to explain the consistence of the relationship between Δ*ε*′/*ε′* and *m*^{2}.

To prove the validity of the derived Eq. (10), we designed quasi-binary alloys of Co_{x}Fe_{100−}_{x} nanogranules with various Co/(Co+Fe) ratios (*x*, in percentage) to control the *P*_{T} value. We experimentally deposited (Co_{x}Fe_{100−}_{x})–MgF_{2} nanogranular films (thickness ∼600 nm), which required many sequential preparations^{29–31} in Fig. 3(a), with MgF_{2} and Fe targets placed in different chambers. Adjustable chip number of Co (5 × 5 mm^{2}) placed on the Fe target (3 in.) allowed the deposition of Co_{x}Fe_{100−}_{x} granules with various *x* ranging from 0 to 100. The film composition was denoted as (Co_{x}Fe_{100−}_{x})–(MgF_{2}). The x-ray photoelectron spectroscopy (XPS) was used to examine the compositions of each film. The standard samples of metallic Co and Fe as well as stoichiometric Co_{50}Fe_{50} alloy and MgF_{2} ceramic were used as reference materials for corrections to accurately check the composition variation of Co and Fe contents in the films. Details of the small variations in the CoFe content are summarized in the supplementary material. The film for *x* = 46 exhibited a uniform nanogranular structure in Fig. 3(b). The crystalline matrix was confirmed from the diffraction pattern in the inset of Fig. 3(b). The dark dots likely correspond to the amorphous granules (∼2 nm) embedded in a host matrix, and the bright contrast likely correspond to the MgF_{2} matrix in Fig. 3(c). The corresponding elemental maps of Co and Fe in Figs. 3(d) and 3(e) indicate that the distribution of Co matched well with that of Fe, clearly suggesting that the granules consisted of CoFe alloys. The variations in the granular size and its distribution are negligible because the variation of granular content is limited to within a small range of 15%–17%. The clear phase separation between the CoFe and MgF_{2} was probably attributed to the relatively large enthalpy of the formation of the MgF_{2} matrix and the CoFe alloy compared to pure Co or Fe.^{32} The XRD peaks around *θ* = 27° correspond to the (110) peaks of MgF_{2} with a tetragonal structure (JCPDS No. 72-1150) in Fig. 3(f). The magnetization curve with *x* = 46 shows an unsaturated state under large *H* up to 20 kOe in Fig. 3(g). The film lacks hysteresis under small *H* = 500 Oe in the inset of Fig. 3(g), indicating that each CoFe granule behaved like a superparamagnet.

As displayed in Fig. 4(a), the peak Δ*ε*′/*ε′* of the (Co_{x}Fe_{100−}_{x})–MgF_{2} film at *f* = 150–300 kHz increased gradually with increase in *x* and reached the maximum values in the range *x* = 0.3–0.7. A further increase in *x* decreased Δ*ε*′/*ε′*. Based on a prior theoretical investigation^{33} on the calculated *P*_{T} vs *x*, the theoretical fitting (solid line) using Eq. (10) is consistent with the experimental Δ*ε*′/*ε′*. The marginal discrepancy possibly occurred because of the composition variation of the CoFe granular content, which resulted in a small change in its magnitude and peak position near 150–300 kHz, as indicated from the frequency dependence of the Δ*ε*′/*ε′* with selected *x* in the inset of Fig. 4(a). Dielectric properties of CoFe–MgF_{2} films were provided in the supplementary material. It is worthy noted that the peak Δ*ε*′/*ε′* is still far from the maximum theoretical limit. The origin of the large difference between experiment and theory mainly arises from two reasons: (1) the *P _{T}* and

*m*used in the theoretical model are far from reaching the saturated values and, furthermore, this difference may become even larger due to the square dependence of

*P*and

_{T}*m*on Δ

*ε*′/

*ε*′; (2) the theoretical derivation is established based on the assumption that the intergranular spacing (

*s*) of the structure remains single value with no variance (Δ

*s*= 0), but the genuine disordered structure inevitably displays a certain distribution of

*s*. Whereas the former case, indeed, points out that the theoretical limit of Δ

*ε*′/

*ε′*is estimated at infinitely high

*H*and

*P*

_{T}that is difficult to achieve, the latter case suggests a strategy for increasing the TMD ratio from a structural standpoint, namely, artificially adjusting the distribution of

*s*in a tailorable way. The dependence of Δ

*ε*′/

*ε*′ on

*H*at

*f*= 150 kHz for

*x*= 46 in Fig. 4(b) resembles that derived from Eq. (10) shown in Fig. 2(b). The film showed a large Δ

*ε*′/

*ε*′ of 4.3% under

*H*= 10 kOe for

*x*= 46. As suggested by Eq. (10) and the Inoue–Maekawa model,

^{23}the increased trend of Δ

*ε*′/

*ε*′ should obey a linear relationship with the square of normalized magnetization (

*m*

^{2}), i.e., Δ

*ε*′/

*ε′*∝

*m*

^{2}. Herein, we confirmed that, for

*x*= 46, Δ

*ε*′/

*ε*′ is in good agreement with (

*M*/

*M*

_{20k})

^{2}.

The constraints of the derived formula, Δ*ε*′/*ε′* = 2*P*_{T}^{2}*m*^{2}, are discussed from the viewpoint of structural configuration and materials selection. First, the theoretical model holds true based on the randomly disordered granular structure in that the magnetic interactions among granules can be negligibly small; the heterostructure such as the proposed two-dimensional granular structure^{19} that deviates marginally from this formula due to the presence of magnetic couplings/interactions is otherwise not suitable to this model. Second, the *P*_{T} of the magnetic materials should be restricted in the range of 0–1 as previously depicted in Eq. (9). In other words, materials with *P*_{T} value beyond this range are not applicable to this theoretical model.

In summary, we performed a theoretical study of the TMD effect and found that the dielectric variations Δ*ε*′/*ε*′ can be estimated by both intrinsic *P*_{T} and extrinsic *m*, that is, Δ*ε*′/*ε′* = 2*P*_{T}^{2}*m*^{2}. We reached two key conclusions: (1) the theoretical limit of Δ*ε*′/*ε′* exceeding 200% can be predicted through Δ*ε*′/*ε′* = 2*P*_{T}^{2} for *m* = 1 and (2) the observed *m*^{2} dependence of the Δ*ε*′/*ε′* can be explained well for a given *P*_{T}. We further presented an experimental verification on the *x*-dependence of Δ*ε*′/*ε′* of (Co_{x}Fe_{100−x})–MgF_{2} films with the tunable *x*, which can be fitted by the formula, Δ*ε*′/*ε′*∝*P*_{T}^{2}. This study offers a design protocol for exploring various nanogranular composites with much focus on tailoring granular distribution, which will raise the possibility of high-performance magnetoelectric applications of the TMD effect.

See the supplementary material for theoretical results on the effect of the fitting parameter on tunnel magnetodielectric response and experimental magnetic and dielectric properties of the prepared film samples.

This study was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (Grants-in-Aid Nos. 21K18810 and 20H02447). Dr. T. Miyazaki and Dr. M. Nagasako were acknowledged for their help with the TEM and HAADF-STEM observations, respectively.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts of interest to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.