In this paper, we model a heterogeneous dielectric medium exhibiting fractal geometry or disordered random structures by applying non-integer dimensions to determine its capacitance between two parallel plates. The capacitance depends on the fractional dimensions of the fractal or disordered dielectric slab, which may be obtained from the theoretical fractal dimension or box-counting method. The findings are verified by CST Studio Suite (Electromagnetic field simulation software), experimental measurements, and the equivalent capacitance method. Five common types of fractals (Cantor bars/plates, Sierpinski carpet, Sierpinski triangle, Haferman carpet, and Menger sponge) and random structures are tested with good agreement. There is also an effective gain of capacitance in using less amount of dielectric materials, which may be useful in material-savings of dielectrics. This research shows a useful tool in modeling the capacitance of heterogeneous materials, where fractals and disordered structures may be commonly encountered in organic materials and any dielectrics where precision and fabrication are not perfect.

## I. INTRODUCTION

Capacitance ($C$) of a capacitor describes its ability to store electric charges $Q$ at a given applied voltage $V$, which is defined as $C=Q/V$. The simplest capacitor is the parallel plate capacitor, which is composed of a dielectric slab of thickness $d$ and surface area $A$ being sandwiched by two electrodes with an electrical potential difference of $V$. If the size of $A$ is much larger than $d$, the capacitance is given by $C=\kappa \epsilon 0\xd7A/d$, where $\kappa $ is the dielectric constant of the slab. High capacitance is required for many applications such as high-k dielectric materials,^{1} super-capacitors,^{2} energy storage,^{2–4} breakdown of heterogeneous solid dielectrics,^{5} laser induced breakdown on dielectric,^{6} dielectric breakdown due to multipactor discharge,^{7–11} and others.

Ultra-thin dielectric materials^{12} and reliability of 2D dielectrics materials^{13} are important in various applications, such as electrical contacts of various 2D materials and junctions,^{14–20} van der Waals (vdW) heterostructures based electronics and optoelectronics,^{21–24} and interface exciton physics^{25,26} for which the roughness scale of the dielectric materials may not be ignored as compared to its thickness. One may then question if there is a simple way to determine the capacitance for an imperfect dielectric slab due to any geometrical variation. In this paper, we consider a dielectric slab is no longer a perfect three-dimensional (3D) solid, but a fractal or disordered object, which is described by a fractional dimension (less than 3) and we are interested to calculate its capacitance in a semi-analytical model.

Mandelbrot^{27} introduced the concept of fractal dimension to discriminate complex patterns from the normal objects with integer dimensions like 1D, 2D, and 3D. The fractal dimension is a fascinating concept used in many complex objects in diverse disciplines^{28} of physics, mathematics, engineering, and others. It has also been used for the study of continuum models by employing non-integer dimensional spaces.^{29–33} Fractional calculus has always been a diverse area to explore fractional characteristics in different regimes of applied mathematics fields.^{34–36} Some examples involving Maxwell equations in fractional dimensions are electromagnetic fields and wave phenomenon in fractional mediums,^{37} antennas radiation problems,^{38} scattering in perfect electromagnetic conducting medium using Kobayashi potential method,^{39} Green’s function,^{40} cloaking and magnification,^{41} planar interface of dielectric-chiral mediums,^{42} and many others.

Table I shows some common fractal objects, such as Cantor bars, Sierpinski triangle, Sierpinski carpet, Menger sponge, Haferman carpet, and their corresponding theoretical Hausdorff dimensions. For simplicity, we consider the capacitance of a fractal dielectric slab in a parallel plate setting, where the area has a length $L$ much larger than $d$, and the capacitance will have the following scaling:

where 1 $<\alpha \u2264$ 2 and $0<\beta \u22641$ are the fractional dimensions of the fractal slab for its area and depth, respectively. Please refer to the Appendix for the derivation of the scaling. For a perfect non-fractal object, we have $\alpha =2$ and $\beta =1$, so the total dimension is $\alpha +\beta =3$. The scaling law of Eq. (1) will be confirmed by CST simulation and experimental measurement.

Fractal capacitors . | Hausdorff dimension . |
---|---|

Cantor bars | 1.2619 |

Sierpinski triangle | 1.5850 |

Sierpinski carpet | 1.8926 |

Menger sponge | 2.7268 |

Haferman carpet | 2 |

Fractal capacitors . | Hausdorff dimension . |
---|---|

Cantor bars | 1.2619 |

Sierpinski triangle | 1.5850 |

Sierpinski carpet | 1.8926 |

Menger sponge | 2.7268 |

Haferman carpet | 2 |

Other than the fractals shown in Table I, we also create random disordered objects where the fractal dimensions can be determined by box-counting method.^{43,44} Note the Hausdorff dimension is for a perfect fractal object with infinite number $n$ of the pattern’s replication. For a realistic imperfect object, box-counting is better to capture its fractional dimension. Using the box-counting method and fractional modeling, useful and analytical models had been created to solve some practical objects, such as rough cathode of electron emission,^{45,46} porous solid of its charge transport,^{47} rough metal for its reflectivity,^{48} and non-3D materials for its binding energy,^{26} which all agree well with experimental measurements.

## II. FRACTIONAL DIMENSIONS OF *α* AND *β*

In Figs. 1(a)–1(d), we create four different fractal objects, namely, Cantor bars, Sierpinski carpet, Sierpinski triangle, Haferman carpet, and use CST simulation to calculate their capacitance at a fixed thickness of $d=1$ mm without any fractional dimension ($\beta =1$). For each fractal object, we have different regions either composed of dielectric material (color) at $\kappa =3.8$ (i.e., silicon dioxide) or free-space (white) at $\kappa =1$, respectively. To check the area scaling of $L\alpha $, we vary the length of the area from $L=10$ mm to 50 mm to obtain its capacitance from CST simulation (symbols) for a potential difference of 1 V. To verify the scaling law, we use two values of $\alpha $ based on the Hausdorff dimension (solid lines) from Table I and box-counting method (dashed lines). Here, we only have $n=3$ replication in creating the fractal pattern, so the box-counting based $\alpha $ is more accurate that the Hausdorff dimension to describe the real fractional dimensions. Thus, the box-counted $\alpha $ is in better agreement to the CST simulation. Our findings confirm that the area scaling of the capacitance for a fractal slab is indeed $C\u221dL\alpha $, where $\alpha $ is the fractal dimension of the area. With $\beta =1$ and 1 $<\alpha <2$, these fractal dielectric objects have a total fractal dimensional of $\alpha +\beta $ between 2 and 3.

In Fig. 2, we study the capacitance for a Menger sponge with different ratios of $L/d>1$. The total Hausdorff dimension for Menger sponge is $2.727=\alpha +\beta $, where $\alpha =1.818$ (two-third of 2.727) and $\beta =0.909$ (one-third of 2.727). By varying $L=5$ to 50 mm for two different $L/d=2$ and 8, we compare the simulated capacitance (symbols) with the proposed scaling laws of $C\u221dL\alpha /d\beta \u221dL1.818/L0.909\u221dL0.909$. The comparison confirms that the agreement at large $L/d=8$ is better due to the assumption of $L/d\u226b1$ for a parallel plate capacitor. Note here $d$ is varied accordingly when $L$ is increased from 10 to 50 mm for a fixed $L/d$.

## III. DISORDERED SURFACE AND SOLID

Other than the regular fractal objects, we also create disordered surfaces with randomly filled dielectric materials ($\kappa =3.8$) at different volumes (20%, 50%, and 80% of $f=0.2$, 0.5, and 0.8) as shown in Fig. 3. They are created by randomly assigned rods of $d$ = 1 mm ($\beta =1$) over a square surface of $L\xd7L$, where $L$ is varied from 10 to 50 mm. For each set of $f$ and $L$, we create three random structures and determine their $\alpha $ values by box-counting method and to calculate the capacitance using CST. The average values of the simulated capacitance are plotted in the figure to compare with the $L\alpha $ scaling based on their respective $\alpha =1.4564$, 1.7015, and 1.8137 for $f=0.2$, 0.5, and 0.8, respectively. Interestingly, the agreements are very good, which confirms the area scaling of $C\u221dL\alpha $ even for randomly disordered surfaces.

Similarly, four disordered solids are constructed by randomly assigning an array of $128\xd7128\xd716$ cubes of the dielectric material with a volume of 20% ($f=0.2$) of the slab. The fractional dimension of this disordered solid is obtained by the box-counting method, which gives $2.5098=\alpha +\beta $, and $\alpha =1.673$ (two-third of 2.5098) and $\beta =0.836$ (one-third of 2.5098). By varying $L$ from 5 to 50 mm at fixed $L/d=8$, the simulated capacitance (symbols) is calculated using CST, which shows excellent agreement with the proposed scaling law (dashed line) of $C\u221dL\alpha /d\beta \u221dL1.673/L0.836\u221dL0.836$ (dashed line) as shown in Fig. 2(b), which confirms again the scaling is also valid for disordered solid.

## IV. EXPERIMENTAL VERIFICATION

For experimental verification, we consider Sierpinski carpets of different replications $n=2$, 3, and 4, which are fabricated by a laser machine on an acrylic board with a thickness of $d=5$ mm as shown in Fig. 4. The capacitance was measured by using an ISO-TECH LCR-1701 meter at 10 kHz. The experimental results (symbols) in comparison with the Hausdorff dimension of $\alpha =1.8926$ (solid line) and box-counting $\alpha =1.7696$ (dash lines) have shown pretty good agreement with the area scaling law of $C\u221dL\alpha $.

To verify the thickness scaling of $C\u221dd\u2212\beta $, the fractal dimension of $d$ is represented by Cantor plates with different removal factors ($r=3$, 5, and 7) and their corresponding fractal dimension are $\beta =0.6309$, 0.7565, and 0.8181, which are calculated by using Hausdorff’s formula of $\beta =\u2212log\u2061(2)/log\u2061[(1\u22121/r)/2]$. These Cantor plates are fabricated by 3D printing using poly-lactic acid and the capacitance is measured by using an LCR meter at 10 kHz testing frequency. We confirm that the measurements agree pretty well with the scaling of $C\u221dd\u2212\beta $ in Fig. 5.

## V. COMPARISON WITH EQUIVALENT CIRCUIT MODEL

For the fractal structures studied above with fixed $d$ at $\beta =1$, we can use the equivalent capacitance method to calculate the capacitance if we know the fraction ($f<1$) of the slab, which is composed of the dielectric material of $\kappa $ (=3.8 as studied here). The total capacitance is the sum of capacitance calculated from the material, $Cs$ and the free space, $Cv$, i.e., $C=Cv+Cs$, where $Cv=(1\u2212f)\xd7Co/\kappa $, and $Cs=f\xd7Co$. Here, $Co$ is the capacitance of a fully filled dielectric slab: $Co=\kappa \epsilon 0\xd7A/d$. In the normalized form of $C\xaf=C/Co$, we have

For $f$ goes to 0 and 1, we have $C\xaf=1/\kappa $ and $C\xaf=1$, respectively. Note that the values of $f$ will depend on the types of fractal objects and their replication levels $n$ as shown in Table II. The CST obtained normalized capacitance $C\xaf$ is also shown in Table III, which agrees very well (less than 3% in error) with Eq. (2).

Level (n) . | f
. | ||||
---|---|---|---|---|---|

Cantor bars . | Sierpinski carpet . | Sierpinski triangle . | Haferman carpet . | Menger sponge . | |

0 | 1 | 1 | 1 | 1 | 1 |

1 | 0.4444 | 0.8888 | 0.7500 | 0.4444 | 0.7407 |

2 | 0.1975 | 0.7901 | 0.5625 | 0.7531 | 0.5487 |

3 | 0.0878 | 0.7023 | 0.4218 | 0.5816 | 0.4064 |

4 | 0.0390 | 0.6243 | 0.3164 | 0.6768 | 0.3011 |

Level (n) . | f
. | ||||
---|---|---|---|---|---|

Cantor bars . | Sierpinski carpet . | Sierpinski triangle . | Haferman carpet . | Menger sponge . | |

0 | 1 | 1 | 1 | 1 | 1 |

1 | 0.4444 | 0.8888 | 0.7500 | 0.4444 | 0.7407 |

2 | 0.1975 | 0.7901 | 0.5625 | 0.7531 | 0.5487 |

3 | 0.0878 | 0.7023 | 0.4218 | 0.5816 | 0.4064 |

4 | 0.0390 | 0.6243 | 0.3164 | 0.6768 | 0.3011 |

Level (n)
. | $C\xaf$ . | ||||
---|---|---|---|---|---|

Cantor bars . | Sierpinski carpet . | Sierpinski triangle . | Haferman carpet . | Menger sponge . | |

0 | 1 | 1 | 1 | 1 | 1 |

1 | 0.5995 | 0.9196 | 0.8210 | 0.5995 | 0.7235 |

2 | 0.4207 | 0.8478 | 0.6876 | 0.8224 | 0.5657 |

3 | 0.3369 | 0.7832 | 0.5850 | 0.6935 | 0.4713 |

4 | 0.3003 | 0.7248 | 0.5059 | 0.7603 | 0.4048 |

Level (n)
. | $C\xaf$ . | ||||
---|---|---|---|---|---|

Cantor bars . | Sierpinski carpet . | Sierpinski triangle . | Haferman carpet . | Menger sponge . | |

0 | 1 | 1 | 1 | 1 | 1 |

1 | 0.5995 | 0.9196 | 0.8210 | 0.5995 | 0.7235 |

2 | 0.4207 | 0.8478 | 0.6876 | 0.8224 | 0.5657 |

3 | 0.3369 | 0.7832 | 0.5850 | 0.6935 | 0.4713 |

4 | 0.3003 | 0.7248 | 0.5059 | 0.7603 | 0.4048 |

For more complicated structures $\beta \u22601$ in Fig. 2(a), we may consider a disordered slab (with square surface area) is composed of an array of $N\xd7N\xd7M$ cubes with each cube filled with or without the dielectric material. If each column of $M$ cubes are considered to be capacitors connected in series and the resulting $N\xd7N$ capacitors are connected in parallel, the normalized equivalent capacitance is given by

where $N$ is the number of cubes along $x$ and $y$ (area) direction and $fij$ is the fraction of column $ij$, which is composed of the dielectric material. If each layer of $M\xd7M$ capacitors is considered to be connected in parallel and the resulting $N$ equivalent capacitors are connected in series, we have the normalized equivalent capacitance,

where $M$ is the number of cubes along $z$ (thickness) direction and $fi$ is the fraction of layer $i$, which is composed of the dielectric material. Equation (4) provides an estimated $C\xaf$ of < 5% relatively error (as compared to CST simulated results) for the four generated disordered solids [studied in Fig. 2(b)—asterisks].

However, Eqs. (2)–(4) provide estimated $C\xaf$ of error up to 20% (as summarized in Table IV) for the regular Menger sponge at all four levels [Fig. 2(b)—dots]. Having investigated the distribution of the electrostatic potential across the parallel plates, it is observed that the variations near the bottom plate are uniform, which leads to similar potential differences throughout the layers and, thus, allows us to apply Eq. (4) in the bottom region. On the other hand, the variations near the top plate are relatively more drastic as shown in Fig. 6(a) for Menger sponge (at level 2 as an example). Thus, we may apply Eq. (3) in the top region. For simplicity, we simply divide the Menger sponge [as illustrated in Fig. 6(b)] into two roughly equal regions and apply Eqs. (3) and (4) to the top and bottom regions, respectively. As shown in Table IV, this arbitrary partitioning technique in using Eqs. (3) and (4) has improved the accuracy with an error of less than 1.3% to the $C\xaf$ for the Menger sponge at all four levels. While good agreement is obtained, this partitioning is not trivial, which requires information on the field distribution of the complex structure. Thus, it may be simpler in using the proposed fractional model as suggested in this paper. For completeness, we also show the ratio of $C\xaf/f$, which measures the effective gain of capacitance per amount of materials used in Table V. This implies we can use fewer materials to obtain certain values of capacitance.

Level (n)
. | . | $C\xaf$ . | |||
---|---|---|---|---|---|

f
. | Eq. (2) . | Eq. (3) . | Eq. (4) . | Partitioning by Eqs. (3) and (4) . | |

1 | 0.7047 | 0.8090 (11.81%) | 0.7036 (2.75%) | 0.7749 (7.11%) | 0.7273 (0.53%) |

2 | 0.5487 | 0.6675 (17.99%) | 0.5344 (5.53%) | 0.6252 (10.52%) | 0.5724 (1.18%) |

3 | 0.4064 | 0.5626 (19.38%) | 0.4350 (7.71%) | 0.5235 (11.08%) | 0.4741 (0.60%) |

4 | 0.3011 | 0.4850 (19.81%) | 0.3748 (7.41%) | 0.4529 (11.88%) | 0.4099 (1.26%) |

Level (n)
. | . | $C\xaf$ . | |||
---|---|---|---|---|---|

f
. | Eq. (2) . | Eq. (3) . | Eq. (4) . | Partitioning by Eqs. (3) and (4) . | |

1 | 0.7047 | 0.8090 (11.81%) | 0.7036 (2.75%) | 0.7749 (7.11%) | 0.7273 (0.53%) |

2 | 0.5487 | 0.6675 (17.99%) | 0.5344 (5.53%) | 0.6252 (10.52%) | 0.5724 (1.18%) |

3 | 0.4064 | 0.5626 (19.38%) | 0.4350 (7.71%) | 0.5235 (11.08%) | 0.4741 (0.60%) |

4 | 0.3011 | 0.4850 (19.81%) | 0.3748 (7.41%) | 0.4529 (11.88%) | 0.4099 (1.26%) |

Level (n)
. | $C\xaf/f$ . | ||||
---|---|---|---|---|---|

Cantor bars . | Sierpinski carpet . | Sierpinski triangle . | Haferman carpet . | Menger sponge . | |

0 | 1 | 1 | 1 | 1 | 1 |

1 | 1.3490 | 1.0346 | 1.0946 | 1.3490 | 0.9767 |

2 | 2.1301 | 1.0730 | 1.2224 | 1.0920 | 1.0309 |

3 | 3.8371 | 1.1152 | 1.3869 | 1.1924 | 1.1597 |

4 | 7.7000 | 1.1609 | 1.5989 | 1.1234 | 1.3444 |

Level (n)
. | $C\xaf/f$ . | ||||
---|---|---|---|---|---|

Cantor bars . | Sierpinski carpet . | Sierpinski triangle . | Haferman carpet . | Menger sponge . | |

0 | 1 | 1 | 1 | 1 | 1 |

1 | 1.3490 | 1.0346 | 1.0946 | 1.3490 | 0.9767 |

2 | 2.1301 | 1.0730 | 1.2224 | 1.0920 | 1.0309 |

3 | 3.8371 | 1.1152 | 1.3869 | 1.1924 | 1.1597 |

4 | 7.7000 | 1.1609 | 1.5989 | 1.1234 | 1.3444 |

## VI. CONCLUSION

In conclusion, we report that the electrostatic capacitance for a fractal-like disordered dielectric slab in a parallel plate configuration is following a general scaling law, which depends only on the fractal dimensions of the fractal dielectric slabs. These fractal dimensions can be obtained either from the Hausdorff dimension or the box-counting method. Using CST simulation, we have confirmed the scaling for five common fractal objects and randomly disordered structures. Selected fractals like Sierpinski carpet and Cantor bars have also been fabricated to measure their capacitance, which confirms the scaling laws. If the ratio of the filled dielectric materials to the total volume of the slab is known, equivalent capacitance models are constructed to provide good estimation with less than 5% in error. However, the equivalent circuit model may be limited to only less complicated structures like $\beta =1$ cases studied in Figs. 1 and 3. It is not a trivial matter to construct a right equivalent circuit model for $\beta \u22601$ cases like Menger sponge as shown in Table IV. Thus, the semi-analytical fractional model and its scaling suggested here provide a useful approach that once the scaling law is bench-marked at a given $L$ and $d$, it can be used to calculate the capacitance for other $L$ and $d$ once the fractional dimensions of $\alpha $ and $\beta $ are determined from the box-counting method. The model can also be used to characterize the degree of disorder for a given imperfect dielectric slab by determining its fractional dimensions ($\alpha $ and $\beta $) in performing an optimal fitting of its measured capacitance to our model. Our findings also suggest an effective gain of capacitance in using less amount of electric materials to obtain a fixed capacitance. This paper shows a useful tool (semi-analytical model) for calculating the capacitance of heterogeneous materials, where fractals and disordered structures may be commonly encountered in organic materials and any dielectrics where precision and fabrication are not perfect. In future, it may be interesting to explore if such a fractional modeling approach can be extended to study the quantum transport in fractal geometry and networks.^{49,50}

## ACKNOWLEDGMENTS

This work was supported by the USA ONRG (Grant No. N62909-19-1-2047). S.K. acknowledges support from a MOE PhD scholarship.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Samra Kanwal:** Data curation (lead); Investigation (equal); Software (equal); Validation (equal); Writing – original draft (lead); Writing – review and editing (equal). **Chun Yun Kee:** Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – review and editing (equal). **Samuel Y. W. Low:** Data curation (equal); Investigation (equal); Visualization (equal); Writing – review and editing (equal). **Muhammad Zubair:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – review and editing (equal). **L. K. Ang:** Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review and editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: DERIVATION OF FRACTIONAL MODEL FOR PARALLEL PLATE CAPACITOR

We consider a parallel plate capacitor with a slab of spatially disordered dielectric material of dielectric constant, $\epsilon $. Its plates are placed at $z=0$ and $z=d$ with lower plate grounded and upper plate at potential $V0$. The electrostatic potential between the plates is governed by the Laplace equation

Here, in normalized form, the overhead bar will denote the quantity normalized to $R0$, a characteristic length of the system, with the exception of $V\xaf$ being the potential normalized by $R02$. The slab is considered to be a fractal continuum, which can be characterized by a fractional dimension $D$ ($\u22643$) = $\alpha 1+\alpha 2+\beta $, with $0<\alpha 1,\alpha 2,\beta \u22641$. Here, $\alpha 1$, $\alpha 2$, and $\beta $ denote the fractional dimension along directions $x$, $y$, and $z$, respectively.

Following the typical approach,^{45} the differential operator in Eq. (A1) is replaced by the fractional differential operator

where

and $\Gamma $ is the Gamma function. The solution to Eq. (A2) is in the following form:

With the boundary conditions of $V\xaf(0)=0$ and $V\xaf(d\xaf)=V0\xaf$, the solution to Eq. (A2) is

The normalized electric field is

At the capacitor plate, the normalized surface charge density, $\rho s\xaf=\epsilon E\xaf\u22c5z^.$ Thus, the normalized charge is given by

where $dx\xaf\alpha 1=\pi \alpha 1/2\Gamma (\alpha 1/2)|x|\alpha 1\u22121dx\xaf$ and $dy\xaf\alpha 2=\pi \alpha 2/2\Gamma (\alpha 2/2)|y|\alpha 2\u22121dy\xaf$. Note that $0<\alpha =\alpha 1+\alpha 2\u22642$ is the fractional dimension of the area. For simplicity, we consider $\alpha 1=\alpha 2=0.5\xd7\alpha $ and the side length of the parallel square plates to be $L$. Thus, the normalized charge is given by

and the normalized capacitance is

Equation (A8) suggests the following scaling relation:

with $0<\alpha \u22642$ and $0<\beta \u22641$. For comparison to measured or simulated capacitance, the following unnormalized form of Eq. (A8) is recovered:

where

At $\alpha =2$ and $\beta =1$, we have $G=1$ and recover the classical results. Note that $R0$ can be determined by fitting with simulation or measurement. Once it is fixed, the equation is valid for any $L$ and $d$.

## REFERENCES

*2011 IEEE 19th Signal Processing and Communications Applications Conference (SIU)*(IEEE, 2011), pp. 62–65.