Novel Dielectric Nanogranular Materials with an Electrically Tunable Frequency Response

The electrical modulation of the functionality of matter is of significant interest in physics and multifunctional tunable electronic device applications. Here, new dielectric materials with nanogranular structures comprised of nano‐sized Co granular metals dispersed in a Mg‐fluoride‐based dielectric matrix are explored. The dielectric relaxation frequency (fr), which represents a sharp decrease in dielectric permittivity in dielectrics, can be tuned by a DC electric field (E). As E increases, the position of fr first shifts to the low‐frequency side and then to the high‐frequency side, achieving a tunable fr in a certain frequency range. The ability to electrically modulate the relaxation frequency may help construct novel tunable frequency filters. The dielectric properties are theoretically examined based on the asymmetric electron tunnelling model that considers the size difference of granular pairs, offering an insightful understanding of the structure‐property relationship in disordered granular solids.


Introduction
Frequency filters that alter the amplitude and phase of an electrical signal with respect to the frequency are extensively used in electronic applications such as telecommunication and signal processing systems, which can transmit the desired signal in a specific frequency range while rejecting or suppressing signals with the structural configuration. There are two mechanisms for AC conduction: electron tunnelling between small granules with small separations and intergranular capacitive transport for large ones with large separations. Thus, for a granular system with a wide size distribution ranging from a few to tens of nanometers, both cases may coexist. [25,27,28,29] In particular, the former is a quantum effect caused by electron tunnelling between adjacent granules separated by a sub-nanometer distance. [30] While dielectric divergence has been reported in the low-frequency range [31] or near the percolation threshold, [32] AC transport response to the DC electric field has not yet been discussed.
In this work, we demonstrate new dielectric nanogranular materials, where the dielectric relaxation frequency (f r ) of the AC transport response is tuned by a DC electric field. Using dielectric granular materials with electrically tunable frequency response, as shown in Figure 1a, device structures such as RF low pass filters and antennas may be simplified and miniaturized. The structure comprises nanometer-sized magnetic metals dispersed in an insulating matrix. Using different metallic Co fractions (x) of Co-MgF 2 films, f r can be tuned by the electric field, as shown in Figure 1b. Specifically, for x = 0.24, f r is controlled in the range from 1.5 MHz (OFF state) to 2.2 MHz (ON state) by increasing the electric field up to 14 kV cm −1 , as shown in Figure 1c. To explain this, we develop an asymmetric charge oscillation model by considering granular pairs with a size difference. Using this model, the results of tunable f r can be theoretically and numerically validated.

Structure and Granular Size Distribution of Nanogranular Films
In nanogranular films, metallic granules with diameters of several nanometers are randomly distributed in an insulating matrix with a sub-nanometer-scaled intergranular separation, as shown in Figure 2a. Charge oscillation (tunnel conductance) occurs between granules through the insulating matrix. Figure 2b shows a high-resolution transmission electron microscope image of a selected Co-MgF 2 film with a Co content of www.advelectronicmat.de x = 0.20, which comprises multiple dark circles dispersed in a host matrix. The circles are amorphous Co granules of several nanometers in diameter. The discontinuous bright area is an insulating matrix of MgF 2 with a visible lattice fringe that prevents nanometric granular cluster aggregation. A HAADF-STEM image with a Z-contrast shows an optimum complementary match between the bright Co granules with high atomic number (Z number) and the dark MgF 2 matrix (see Figure S1, Supporting Information), where the matrix's genuine composition deviates slightly from its stoichiometric composition (MgF 2 ) with a slight excess of F (see Figure S2, Supporting Information). The magnetic nature of Co granules allows quantitatively analyzing the distribution of granule size in such complex disordered solids using the Langevin function. [33] All films exhibit typical superparamagnetic behavior with no hysteresis (approximately zero coercivity), indicating that each granule is isolated from the others without any interactions or coupling between them (see Figure S3, Supporting Information). In Figure 2c, the granular size (d) distribution from Langevin fits to the room-temperature magnetization curves of the samples indicates that there is a slight increase in the statistical median d m from 1.8 nm (x = 0.16) and 2.5 nm (x = 0.20) to 2.7 nm (x = 0.24) and 3.0 nm (x = 0.27). Figure 3 shows the real parts of the dielectric permittivity (ε′) in the presence of an electric field (E). For E = 0, with x increased from 0.16 to 0.27, ε′ is enhanced over the frequency range despite there is a nonlinear dielectric change in the dielectric permittivity (ε′ 1k ) at 1 kHz as x varies. This granular content dependency of dielectric variation deviation has already been found in other nanogranular materials systems in either low [34] or high [15] granular content range. This is most likely owing to the lower electrical resistance for large x values greater than 0.2, which results in a decrease in tunnel conductance with aggregated granules. Each film has a typical relaxation process at a certain dielectric relaxation frequency (f r ) at which ε′ sharply decreases. The position of f r shifts toward the highfrequency end, ranging from kHz to MHz due to a decrease in the inter-granular spacing s 12 . The increasing granular pair density subsequently results in the position of f r shifting to the high-frequency side. For x = 0.16 with an increased E up to 12 kV cm −1 , as shown in Figure 3a, the value of ε′ decreases www.advelectronicmat.de over the frequency range, particularly at low frequencies. The films with other x values exhibit similar trends to E, as shown in Figure 3b-d. The solid lines represent the theoretical fits to the frequency response of ε′, as will be discussed later. The normalized ε′/ε′ 1k in Figure 3e indicates that the application of an electric field can effectively tune the dielectric response.

Variation of Dielectric Permittivity of Nanogranular Films in a DC Electric Field
With increased E to 6 kV cm −1 , ε′/ε′ 1k normalized by ε′ at 1 kHz first shifts to the low-frequency side, and then, upon further increasing E, it moves to the high-frequency side. This tunable response can be also observed for films with other x values (see Figure S4, Supporting Information). Further, the dielectric variations in the electric field, (ε′ 0 −ε′ E )/ε′ 0 , are maximum at relatively low E up to 6 kV cm −1 , accompanied by a slight shift to the low-frequency side from 340 to 146 kHz (see Figure  S5, Supporting Information). Further, this peak position at a fixed E is related to the variation x (see Figure S6, Supporting Information).

Theoretical Fits to the Electric Field-Induced Frequency Variations
The change in the dielectric permittivity is intimately related to the nanometric structure, including the granular size distribution and inter-granular spacing. Since the Maxwell-Wagner model [35] is established based on the charge build-up between different component dielectrics with different conductivities, it is unable to explain the dielectric relaxation process and the variation of f r . Hence, the Debye-Fröhlich model [36,37] is used to simulate the electric polarization of dielectrics in the form of granular pairs, which allows describing the variation and distribution of f r . The dielectric permittivity is written as follows: where β is a parameter representing the distribution of f r , [38] which follows an ideal Debye relaxation for β = 1 and may have a distribution for 0 < β < 1. A larger β value generally indicates a narrower distribution of f r . Figure 4 summarizes the derived f r by quantitatively reproducing the experimental results of dielectric response to E in To explain how f r varies in an electric field, it is critical to understand the polarization mechanism in these granular systems. For the Debye-Fröhlich model, electric polarization is caused by the symmetric charge oscillation between granular pairs with the same granular size, [15] as shown in Figure 5a  www.advelectronicmat.de cannot match this model. Thus, we develop a generalized model by introducing the distribution of granular size-a natural consequence of granular systems. Then, the charge tunnelling phenomenon becomes an asymmetric oscillation between granular pairs with different diameters (d 1 ≠ d 2 ). Based on this model, the variation of f r may vary depending on the applied E (see Section 4 for the details on the relaxation frequency f r controlled by E), which is expressed as where f r0 is the pristine relaxation frequency obtained by performing theoretical fits to the frequency response of ε′ in the absence of a DC electric field, E c and ΔE c are the average and difference of the charging energies, respectively, E ci ≈ e 2 /(ε 0 d i ) (i = 1, 2), d 12 is the distance between the centers of the two nearest neighboring granules, k B is Boltzmann constant, and T is the temperature. By assuming a statistical median d m value in Figure Table S1, Supporting Information). The theoretical fits indicate that the electric polarization upon applying E can be viewed as the collective behavior of the numerous granular pairs with various combinations of granular sizes.

Asymmetric Charge Oscillation Model in a DC Electric Field
To probe the origin of the electric-field induced tunable f r in this work, we focus on the Debye-Fröhlich model. [36,37] As described earlier, charge oscillation is typically considered based on the granular pairs of the same size, forming a double potential well in the AC electric field E ω (t), as shown in Figure 5a. This model explains the distribution of f r by introducing a distribution function. Nonetheless, the granular size difference and resulting charge energy difference in each granular pair are ignored. In the absence of charging energy difference (ΔE c = 0) in Equation (2), the variation of f r is monotonically increased as

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E is gradually applied, as shown in Figure 5b, which is unable to reproduce and explain the calculation result in Figure 4b-e. By introducing the charging energy difference (ΔE c ≠ 0), which is a natural consequence of granular systems, we propose a generalized asymmetric model, as shown in Figure 5c. For this model, there exists a difference in granular size (d 1 > d 2 ), which causes the potential shift to be asymmetric in the electric field is the AC electric field and E is the DC electric field. Based on this model, the calculation results in Figure 5d indicate that f r first decreases in small E, and then increases upon further increasing E, which is consistent with the results in Figure 4b-e. Further, the theoretical calculations reveal that the granular size difference (d 1 -d 2 ) is proportional to the decreased magnitude of f r in small E. In other words, as the size difference increases, f r decreases sharply. For a specific combination of d 1 and d 2 (e.g., d 1 = 3.0 nm, d 2 = 2.0 nm), an increasingly large intergranular separation (s 12 ) may result in a sharp increase while maintaining a negligibly small decrease in f r with increased E in the inset of Figure 5d. This model allows establishing the structure-property relationship in disordered granular solids and quantitively analyzing the effect of structural parameters on the tunability of f r .

Conclusion
In summary, we demonstrate how the relaxation frequency of dielectric nanogranular materials is controlled by a DC electric field (E). When E is small, the relaxation frequency, shifts to the low-frequency side, and then to the high-frequency side when E is increased further. To explain this, we establish a generalized model based on asymmetric charge oscillation, a natural consequence of complex granular materials, which allows reproducing the results of tunable relaxation frequency. Our results provide insights into the electric polarization of disordered granular solids to electric fields associated with nanometric structures, which is of importance in the field of dielectric and spintronics physics and may have application in compact filters and antennas.

Experimental Section
Film Sample Preparation: The Co-MgF 2 nanogranular thin films were sputter-deposited on Si/SiO 2 /Ti/Au and quartz substrates by an RF-magnetron co-sputtering apparatus under an Ar gas pressure of 0.5 Pa, using 2-in. Co and MgF 2 disk targets. To achieve a homogeneous and random distribution of granules, the substrate was rotated at a constant speed of 10 rpm. The composition ratio (x) of Co (granule) and Mg + F (matrix) in Co x -(MgF 2 ) 1−x films was adjusted by changing the sputtering power of each target. Film samples with different compositions were denoted as Co 16 Mg 24 F 60 (x = 0.16), Co 20 Mg 21 F 59 (x = 0.20), Co 24 Mg 20 F 56 (x = 0.24), and Co 27 Mg 18 F 55 (x = 0.27). The film thickness was regulated in the range of 800-1000 nm. A small portion of the substrate was covered prior to deposition to allow access to the bottom electrode for subsequent electrical and dielectric measurements. All measurements were performed at room temperature.
Composition and Structural Characterization: The chemical compositions of the films were determined by an X-ray fluorescence spectrometer. Structural observations were performed using a field-emission transmission electron microscope (FETEM, JEOL JEM-2100F) and Cs-corrected 200 kV high-angle annular dark-field (HAADF)-type scanning transmission electron microscope (STEM) (HAADF-STEM, JEM-ARM200F). The magnetic properties were measured using a vibrating sample magnetometer (TOEI PV-M20-5S). The DC electrical resistivity of the as-deposited film samples was measured using the fourprobe method. All film samples had electrical resistivities > 10 9 µΩ m.
Measurements of the Capacitance and its Variation in a DC Bias: For the capacitance measurement, the indium slat was cut into 1-mm-long pieces and pressed onto the film surface as the top electrode. Since the nonuniformity of the electric field may impact the variation of the relaxation frequency via altering the distribution state of electric charges and the resulting polarization. Thus, to make sure that a uniform distribution of electric field is obtained, a few indium slats as top electrodes were pasted to the film surface to reproduce the measured values. The capacitance (C p ) of the film samples was measured by an impedance analyzer (Keysight, E4990A) in the frequency range of 1 kHz-10 MHz at a constant oscillation voltage of 0.5 V. Before measurement, the equipment was adjusted with standard "short" and "open" corrections. A tunable DC bias (V b ) in the range 0−1.4 V was applied normal to the film surface to detect the capacitance variation. The applied DC bias and the resulting electric field (E) follow the relation, Calculation of Dielectric Permittivity of Nanogranular Films: Considering the film-type structure, the dielectric permittivity (ε′) was calculated using the relation, ε ε where C p is the measured capacitance, d is the film thickness, ε 0 ( = 8.85 × 10 12 F m −1 ) is the vacuum permittivity, and S is the area of the top electrode.

Derivation of Relaxation Frequency Variation in the Electric Field Based on the Asymmetric Charge Oscillation Model:
Electronic transport in a granular system relies on a significant number of nanometer-sized metallic granules randomly dispersed in an insulating matrix, largely dominated by thermally activated charge carriers that can tunnel from one granule to the nearest neighboring one by interacting with an insulating barrier. A charge carrier activated in one granule may tunnel to another depending on the separation and tunnel barrier height. A model for a pair of granules 1 and 2 and the corresponding double potential well is schematically shown in Figure 5a. In the absence of the AC electric field, the transition rate γ ij of the charge carrier from granule i to granule j, where i and j are either 1 or 2, is determined by the charging energies and the tunnelling process: [23,39] where γ 0 is the pristine transition frequency, s ij is the distance along the center-to-center axis between the granular surfaces, κ is the decay rate of the electron wave functions in the insulating region of the matrix related to the crystallinity of the matrix, k B is the Boltzmann constant, T is the temperature, and ΔE ij is the difference of the charging energy involved in the tunnelling process: [39] ( )

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The Debye-Fröhlich model [36,37] allowed describing the rate equations for the occupation probabilities (P 1 and P 2 ) of a thermally activated electron in granules 1 and 2, and is expressed as = − + where Δµ i (t) = eφ 1 + eφ iω (t) (i = 1, 2) is the chemical potential with φ 1 = +d 12 E/2, φ 2 = −d 12 E/2, φ 1ω (t) = +ed 12 E ω (t)/2, φ 2ω (t) = −ed 12 E ω (t)/2, The charging state is activated in either granule 1 or 2, for which P 1 + P 2 is time-independent and may be given by For a granular system, the broad distribution of granular size is a natural consequence. Thus, the difference in charging energies in every single granule should be introduced as shown in Figure 5c, by which the charging energy terms can be expressed as