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.

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=κε0×A/d, where κ 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 materials12 and reliability of 2D dielectrics materials13 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 physics25,26 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.

Mandelbrot27 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 disciplines28 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:

CLαdβ,
(1)

where 1 <α 2 and 0<β1 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 α=2 and β=1, so the total dimension is α+β=3. The scaling law of Eq. (1) will be confirmed by CST simulation and experimental measurement.

TABLE I.

Hausdorff dimension of some fractals.

Fractal capacitorsHausdorff dimension
Cantor bars 1.2619 
Sierpinski triangle 1.5850 
Sierpinski carpet 1.8926 
Menger sponge 2.7268 
Haferman carpet 
Fractal capacitorsHausdorff dimension
Cantor bars 1.2619 
Sierpinski triangle 1.5850 
Sierpinski carpet 1.8926 
Menger sponge 2.7268 
Haferman carpet 

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.

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 (β=1). For each fractal object, we have different regions either composed of dielectric material (color) at κ=3.8 (i.e., silicon dioxide) or free-space (white) at κ=1, respectively. To check the area scaling of Lα, 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 α 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 α is more accurate that the Hausdorff dimension to describe the real fractional dimensions. Thus, the box-counted α is in better agreement to the CST simulation. Our findings confirm that the area scaling of the capacitance for a fractal slab is indeed CLα, where α is the fractal dimension of the area. With β=1 and 1 <α<2, these fractal dielectric objects have a total fractal dimensional of α+β between 2 and 3.

FIG. 1.

Top views of the fractal slabs and their capacitance by CST simulation (symbols) as a function of L (length of the surface area). The thickness is d=1 mm with β=1. The comparison with the scaling of Lα is with α from Table I: Hausdorff dimension (solid lines) and from box-counting method (dashed lines).

FIG. 1.

Top views of the fractal slabs and their capacitance by CST simulation (symbols) as a function of L (length of the surface area). The thickness is d=1 mm with β=1. The comparison with the scaling of Lα is with α from Table I: Hausdorff dimension (solid lines) and from box-counting method (dashed lines).

Close modal

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=α+β, where α=1.818 (two-third of 2.727) and β=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 CLα/dβL1.818/L0.909L0.909. The comparison confirms that the agreement at large L/d=8 is better due to the assumption of L/d1 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.

FIG. 2.

(a) Top view of the Menger sponge and its simulated capacitance (symbols) as a function of L for (b) L/d=8 and (c) L/d=2. The scaling is CL0.909 (solid line) for Hausdorff dimension. In (b), the dashed line represents box-count scaling CL0.836 for a generated disordered slab with 20% of its volume composed of randomly placed dielectric materials. Note L/d=8 or 2 is fixed, and the value of d is varied accordingly for L=10–50 mm.

FIG. 2.

(a) Top view of the Menger sponge and its simulated capacitance (symbols) as a function of L for (b) L/d=8 and (c) L/d=2. The scaling is CL0.909 (solid line) for Hausdorff dimension. In (b), the dashed line represents box-count scaling CL0.836 for a generated disordered slab with 20% of its volume composed of randomly placed dielectric materials. Note L/d=8 or 2 is fixed, and the value of d is varied accordingly for L=10–50 mm.

Close modal

Other than the regular fractal objects, we also create disordered surfaces with randomly filled dielectric materials (κ=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 (β=1) over a square surface of L×L, where L is varied from 10 to 50 mm. For each set of f and L, we create three random structures and determine their α 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 Lα scaling based on their respective α=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 CLα even for randomly disordered surfaces.

FIG. 3.

Graphical representation of capacitance vs length L for randomly generated fractal disordered slab with only partially f filled dielectric materials with (a) f = 0.2 (20%) with α=1.4564, (b) f=0.5 (50%) with α=1.7015, and (c) f=0.8 (80%) with α=1.8137. α values are calculated by using the box-count method.

FIG. 3.

Graphical representation of capacitance vs length L for randomly generated fractal disordered slab with only partially f filled dielectric materials with (a) f = 0.2 (20%) with α=1.4564, (b) f=0.5 (50%) with α=1.7015, and (c) f=0.8 (80%) with α=1.8137. α values are calculated by using the box-count method.

Close modal

Similarly, four disordered solids are constructed by randomly assigning an array of 128×128×16 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=α+β, and α=1.673 (two-third of 2.5098) and β=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 CLα/dβL1.673/L0.836L0.836 (dashed line) as shown in Fig. 2(b), which confirms again the scaling is also valid for disordered solid.

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 α=1.8926 (solid line) and box-counting α=1.7696 (dash lines) have shown pretty good agreement with the area scaling law of CLα.

FIG. 4.

Experimentally measured capacitance (symbols) for a Sierpinski carpet of L up to 120 mm. The CLα scaling for α of Hausdorff dimension (solid line) and box-counting (dashed line).

FIG. 4.

Experimentally measured capacitance (symbols) for a Sierpinski carpet of L up to 120 mm. The CLα scaling for α of Hausdorff dimension (solid line) and box-counting (dashed line).

Close modal

To verify the thickness scaling of Cdβ, 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 β=0.6309, 0.7565, and 0.8181, which are calculated by using Hausdorff’s formula of β=log(2)/log[(11/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 Cdβ in Fig. 5.

FIG. 5.

Experimentally measured capacitance (symbols) for Cantor plates as a function of d between the electrodes in comparison with scaling law of Cdβ (lines) at various β.

FIG. 5.

Experimentally measured capacitance (symbols) for Cantor plates as a function of d between the electrodes in comparison with scaling law of Cdβ (lines) at various β.

Close modal

For the fractal structures studied above with fixed d at β=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 κ (=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=(1f)×Co/κ, and Cs=f×Co. Here, Co is the capacitance of a fully filled dielectric slab: Co=κε0×A/d. In the normalized form of C¯=C/Co, we have

C¯=1κ+f(11κ).
(2)

For f goes to 0 and 1, we have C¯=1/κ and C¯=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¯ is also shown in Table III, which agrees very well (less than 3% in error) with Eq. (2).

TABLE II.

Fraction of dielectric material used, f, normalized to a non-fractal slab for different fractals at its n replication, where n = 0 is the perfect non-fractal slab. Here, we use L = 50 mm and d = 1 mm for Menger sponge, we have L/d = 8.

Level (n)f
Cantor barsSierpinski carpetSierpinski triangleHaferman carpetMenger sponge
0.4444 0.8888 0.7500 0.4444 0.7407 
0.1975 0.7901 0.5625 0.7531 0.5487 
0.0878 0.7023 0.4218 0.5816 0.4064 
0.0390 0.6243 0.3164 0.6768 0.3011 
Level (n)f
Cantor barsSierpinski carpetSierpinski triangleHaferman carpetMenger sponge
0.4444 0.8888 0.7500 0.4444 0.7407 
0.1975 0.7901 0.5625 0.7531 0.5487 
0.0878 0.7023 0.4218 0.5816 0.4064 
0.0390 0.6243 0.3164 0.6768 0.3011 
TABLE III.

Normalized capacitance C¯=C/Co, where Co is the capacitance of non-fractal structure at n = 0.

Level (n)C¯
Cantor barsSierpinski carpetSierpinski triangleHaferman carpetMenger sponge
0.5995 0.9196 0.8210 0.5995 0.7235 
0.4207 0.8478 0.6876 0.8224 0.5657 
0.3369 0.7832 0.5850 0.6935 0.4713 
0.3003 0.7248 0.5059 0.7603 0.4048 
Level (n)C¯
Cantor barsSierpinski carpetSierpinski triangleHaferman carpetMenger sponge
0.5995 0.9196 0.8210 0.5995 0.7235 
0.4207 0.8478 0.6876 0.8224 0.5657 
0.3369 0.7832 0.5850 0.6935 0.4713 
0.3003 0.7248 0.5059 0.7603 0.4048 

For more complicated structures β1 in Fig. 2(a), we may consider a disordered slab (with square surface area) is composed of an array of N×N×M 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×N capacitors are connected in parallel, the normalized equivalent capacitance is given by

C¯=1N2i=1Nj=1N1κ(1fij)+fij,
(3)

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×M capacitors is considered to be connected in parallel and the resulting N equivalent capacitors are connected in series, we have the normalized equivalent capacitance,

1C¯=κMi=1M1(1fi)+κfi,
(4)

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¯ 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¯ 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¯ 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¯/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.

FIG. 6.

(a) Electrostatic potential across the parallel plates with Menger sponge at level 2 sandwiched in between. (b) Schematic of the partition scheme for Menger sponge at level 2. Capacitance for region 1 is calculated using Eq. (3) with red boxes, indicating that the capacitors of columns are connected in parallel. Capacitance for region 2 is calculated using Eq (4) with red boxes indicating that the capacitors of layers are connected in series. The resulting capacitors of the two regions are connected in series.

FIG. 6.

(a) Electrostatic potential across the parallel plates with Menger sponge at level 2 sandwiched in between. (b) Schematic of the partition scheme for Menger sponge at level 2. Capacitance for region 1 is calculated using Eq. (3) with red boxes, indicating that the capacitors of columns are connected in parallel. Capacitance for region 2 is calculated using Eq (4) with red boxes indicating that the capacitors of layers are connected in series. The resulting capacitors of the two regions are connected in series.

Close modal
TABLE IV.

Normalized capacitance and relative error (in parentheses) of Menger sponge plotted in Fig. 2(b)—dots.

Level (n)C¯
fEq. (2)Eq. (3)Eq. (4)Partitioning by Eqs. (3) and (4)
0.7047 0.8090 (11.81%) 0.7036 (2.75%) 0.7749 (7.11%) 0.7273 (0.53%) 
0.5487 0.6675 (17.99%) 0.5344 (5.53%) 0.6252 (10.52%) 0.5724 (1.18%) 
0.4064 0.5626 (19.38%) 0.4350 (7.71%) 0.5235 (11.08%) 0.4741 (0.60%) 
0.3011 0.4850 (19.81%) 0.3748 (7.41%) 0.4529 (11.88%) 0.4099 (1.26%) 
Level (n)C¯
fEq. (2)Eq. (3)Eq. (4)Partitioning by Eqs. (3) and (4)
0.7047 0.8090 (11.81%) 0.7036 (2.75%) 0.7749 (7.11%) 0.7273 (0.53%) 
0.5487 0.6675 (17.99%) 0.5344 (5.53%) 0.6252 (10.52%) 0.5724 (1.18%) 
0.4064 0.5626 (19.38%) 0.4350 (7.71%) 0.5235 (11.08%) 0.4741 (0.60%) 
0.3011 0.4850 (19.81%) 0.3748 (7.41%) 0.4529 (11.88%) 0.4099 (1.26%) 
TABLE V.

Capacitance gain per unit of dielectric materials, C¯/f.

Level (n)C¯/f
Cantor barsSierpinski carpetSierpinski triangleHaferman carpetMenger sponge
1.3490 1.0346 1.0946 1.3490 0.9767 
2.1301 1.0730 1.2224 1.0920 1.0309 
3.8371 1.1152 1.3869 1.1924 1.1597 
7.7000 1.1609 1.5989 1.1234 1.3444 
Level (n)C¯/f
Cantor barsSierpinski carpetSierpinski triangleHaferman carpetMenger sponge
1.3490 1.0346 1.0946 1.3490 0.9767 
2.1301 1.0730 1.2224 1.0920 1.0309 
3.8371 1.1152 1.3869 1.1924 1.1597 
7.7000 1.1609 1.5989 1.1234 1.3444 

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 β=1 cases studied in Figs. 1 and 3. It is not a trivial matter to construct a right equivalent circuit model for β1 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 α and β 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 (α and β) 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

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

The authors have no conflicts to disclose.

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).

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

We consider a parallel plate capacitor with a slab of spatially disordered dielectric material of dielectric constant, ε. 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

d2V¯dz¯2=0.
(A1)

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¯ being the potential normalized by R02. The slab is considered to be a fractal continuum, which can be characterized by a fractional dimension D (3) = α1+α2+β, with 0<α1,α2,β1. Here, α1, α2, and β 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

β2V¯=1c2(β,z¯)(d2V¯dz¯2β1z¯dV¯dz¯)=0,
(A2)

where

c(β,z¯)=πβ/2Γ(β/2)z¯β1,

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

V¯=Az¯β+B.
(A3)

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

V¯=V0¯d¯βz¯β.
(A4)

The normalized electric field is

E¯=βV¯=βΓ(β/2)πβ/2V0¯d¯βz^.
(A5)

At the capacitor plate, the normalized surface charge density, ρs¯=εE¯z^. Thus, the normalized charge is given by

Q¯=sρs¯dSα=ρs¯dx¯α1dy¯α2,
(A6)

where dx¯α1=πα1/2Γ(α1/2)|x|α11dx¯ and dy¯α2=πα2/2Γ(α2/2)|y|α21dy¯. Note that 0<α=α1+α22 is the fractional dimension of the area. For simplicity, we consider α1=α2=0.5×α and the side length of the parallel square plates to be L. Thus, the normalized charge is given by

Q¯=ρs¯πα/2Γ(α/2)4α2L¯α/2L¯α/2,
(A7)

and the normalized capacitance is

C¯=|Q¯|V0¯=εV0¯βV0¯4α2Γ(β/2)Γ(α/2)πα/2πβ/2L¯αd¯β.
(A8)

Equation (A8) suggests the following scaling relation:

CLαdβ,
(A9)

with 0<α2 and 0<β1. For comparison to measured or simulated capacitance, the following unnormalized form of Eq. (A8) is recovered:

C=R0×G×ε(L/R0)α(d/R0)β,
(A10)

where

G=β4α2Γ(β/2)Γ(α/2)πα/2πβ/2.

At α=2 and β=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.

1.
G. D.
Wilk
,
R. M.
Wallace
, and
J. M.
Anthony
, “
High-κ gate dielectrics: Current status and materials properties considerations
,”
J. Appl. Phys.
89
,
5243
5275
(
2001
).
2.
A.
Afif
,
S. M.
Rahman
,
A. T.
Azad
,
J.
Zaini
,
M. A.
Islan
, and
A. K.
Azad
, “
Advanced materials and technologies for hybrid supercapacitors for energy storage—A review
,”
J. Energy Storage
25
,
100852
(
2019
).
3.
S.
Fleischmann
,
Y.
Zhang
,
X.
Wang
,
P. T.
Cummings
,
J.
Wu
,
P.
Simon
,
Y.
Gogotsi
,
V.
Presser
, and
V.
Augustyn
, “
Continuous transition from double-layer to faradaic charge storage in confined electrolytes
,”
Nat. Energy
7
,
222
(
2022
).
4.
Y. V.
Lim
,
S.
Huang
,
Q.
Wu
,
D.
Kong
,
Y.
Wang
,
Y.
Zhu
,
Y.
Wang
,
Y.-X.
Wang
,
H.-K.
Liu
,
S.-X.
Dou
,
L. K.
Ang
, and
H. Y.
Yang
, “
Super kinetically pseudocapacitive MnCo2S4 nanourchins toward high-rate and highly stable sodium-ion storage
,”
Adv. Funct. Mater.
30
,
1909702
(
2020
).
5.
I. V.
Lisitsyn
,
H.
Inoue
,
I.
Nishizawa
,
S.
Katsuki
, and
H.
Akiyama
, “
Breakdown and destruction of heterogeneous solid dielectrics by high voltage pulses
,”
J. Appl. Phys.
84
,
6262
6267
(
1998
).
6.
D.
Du
,
X.
Liu
,
G.
Korn
,
J.
Squier
, and
G.
Mourou
, “
Laser-induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs
,”
Appl. Phys. Lett.
64
,
3071
3073
(
1994
).
7.
D.-Q.
Wen
,
P.
Zhang
,
Y.
Fu
,
J.
Krek
, and
J. P.
Verboncoeur
, “
Temporal single-surface multipactor dynamics under obliquely incident linearly polarized electric field
,”
Phys. Plasmas
26
,
123509
(
2019
).
8.
P.
Zhang
,
Y. Y.
Lau
,
M.
Franzi
, and
R. M.
Gilgenbach
, “
Multipactor susceptibility on a dielectric with a bias dc electric field and a background gas
,”
Phys. Plasmas
18
,
053508
(
2011
).
9.
L.
Wu
and
L. K.
Ang
, “
Multipactor discharge in a dielectric-loaded accelerating structure
,”
Phys. Plasmas
14
,
013105
(
2007
).
10.
L.-K.
Ang
,
Y. Y.
Lau
,
R. A.
Kishek
, and
R. M.
Gilgenbach
, “
Power deposited on a dielectric by multipactor
,”
IEEE Trans. Plasma Sci.
26
,
290
(
1998
).
11.
R. A.
Kishek
,
Y. Y.
Lau
,
L. K.
Ang
,
A.
Valfells
, and
R. M.
Gilgenbach
, “
Multipactor discharge on metals and dielectrics: Historical review and recent theories
,”
Phys. Plasmas
5
,
2120
2126
(
1998
).
12.
T.
Li
, “
A native oxide high-gate dielectric for two-dimensional electronics
,”
Nat. Electron.
3
,
473
(
2020
).
13.
Y.
Ji
,
C.
Pan
,
M.
Zhang
,
S.
Long
,
X.
Lian
,
F.
Miao
,
F.
Hui
,
Y.
Shi
,
L.
Larcher
,
E.
Wu
, and
M.
Lanza
, “
Boron nitride as two dimensional dielectric: Reliability and dielectric breakdown
,”
Appl. Phys. Lett.
108
,
012905
(
2016
).
14.
M. J. Y.
Y. Illarionov
and
T.
Knobloch
, “
Insulators for 2D nanoelectronics: The gap to bridge
,”
Nature Commun.
11
,
3385
(
2020
).
15.
Q.
Wang
,
L.
Cao
, and
S. J.
Liang
, “
Efficient Ohmic contacts and built-in atomic sublayer protection in MoSi2N4 and WSi2N4 monolayers
,”
npj 2D Mater Appl.
5
,
71
(
2021
).
16.
L.
Cao
,
G.
Zhou
,
Q.
Wang
,
L. K.
Ang
, and
Y. S.
Ang
, “
Two-dimensional van der Waals electrical contact to monolayer MoSi2N4
,”
Appl. Phys. Lett.
118
,
013106
(
2021
).
17.
L.
Cao
,
G.
Zhou
,
Q.
Wu
,
S. A.
Yang
,
H. Y.
Yang
,
Y. S.
Ang
, and
L. K.
Ang
, “
Electrical contact between an ultrathin topological Dirac semimetal and a two-dimensional material
,”
Phys. Rev. Appl.
13
,
054030
(
2020
).
18.
S.
Banerjee
,
L.
Cao
,
Y. S.
Ang
,
L. K.
Ang
, and
P.
Zhang
, “
Reducing contact resistance in two-dimensional-material-based electrical contacts by roughness engineering
,”
Phys. Rev. Appl.
13
,
064021
(
2020
).
19.
Q.
Wu
and
L. K.
Ang
, “
Giant tunneling magnetoresistance in atomically thin VSi2N4/MoSi2N4/VSi2N4 magnetic tunnel junction
,”
Appl. Phys. Lett.
120
,
022401
(
2022
).
20.
Q.
Wu
,
Y. S.
Ang
,
L.
Cao
, and
L. K.
Ang
, “
Design of metal contacts for monolayer Fe3GeTe2 based devices
,”
Appl. Phys. Lett.
115
,
083105
(
2019
).
21.
S.
Wang
,
X.
Pan
,
L.
Lyu
,
C.-Y.
Wang
,
P.
Wang
,
C.
Pan
,
Y.
Yang
,
C.
Wang
,
J.
Shi
,
B.
Cheng
,
W.
Yu
,
S.-J.
Liang
, and
F.
Miao
, “
Nonvolatile van der Waals heterostructure phototransistor for encrypted optoelectronic logic circuit
,”
ACS Nano
16
,
4528
4535
(
2022
).
22.
S.-J.
Liang
,
B.
Cheng
,
X.
Cui
, and
F.
Miao
, “
Van der Waals heterostructures for high-performance device applications: Challenges and opportunities
,”
Adv. Mater.
32
,
1903800
(
2020
), available at https://onlinelibrary.wiley.com/doi/10.1002/adma.201903800.
23.
M.
Wang
,
S.
Cai
,
C.
Pan
,
C.
Wang
,
X.
Lian
,
Y.
Zhuo
,
K.
Xu
,
T.
Cao
,
X.
Pan
,
B.
Wang
,
S.-J.
Liang
,
J. J.
Yang
,
P.
Wang
, and
F.
Miao
, “
Robust memristors based on layered two-dimensional materials
,”
Nat. Electron.
1
,
130
(
2018
).
24.
E.
Linardy
,
M.
Trushin
,
K.
Watanabe
,
T.
Taniguchi
, and
G.
Eda
, “
Electro-optic upconversion in van der Waals heterostructures via nonequilibrium photocarrier tunneling
,”
Adv. Mater.
32
,
2001543
(
2020
).
25.
J.
Wang
,
Y. J.
Zhou
,
D.
Xiang
,
S. J.
Ng
,
K.
Watanabe
,
T.
Taniguchi
, and
G.
Eda
, “
Polarized light-emitting diodes based on anisotropic excitons in few-layer ReS2
,”
Adv. Mater.
32
,
2001890
(
2020
).
26.
S.
Ahmad
,
M.
Zubair
,
O.
Jalil
,
M. Q.
Mehmood
,
U.
Younis
,
X.
Liu
,
K. W.
Ang
, and
L. K.
Ang
, “
Generalized scaling law for exciton binding energy in two-dimensional materials
,”
Phys. Rev. Appl.
13
,
064062
(
2020
).
27.
B. B.
Mandelbrot
,
The Fractal Geometry of Nature
(
W.H. Freeman
,
New York
,
1982
), Vol. 1.
28.
K.
Falconer
,
Fractal Geometry: Mathematical Foundations and Applications
(
John Wiley & Sons
,
2004
).
29.
V. E.
Tarasov
, “
Fractional hydrodynamic equations for fractal media
,”
Ann. Phys.
318
,
286
307
(
2005
).
30.
V. E.
Tarasov
, “
Continuous medium model for fractal media
,”
Phys. Lett. A
336
,
167
174
(
2005
).
31.
F. H.
Stillinger
, “
Axiomatic basis for spaces with noninteger dimension
,”
J. Math. Phys.
18
,
1224
1234
(
1977
).
32.
C.
Palmer
and
P. N.
Stavrinou
, “
Equations of motion in a non-integer-dimensional space
,”
J. Phys. A: Math. Gen.
37
,
6987
(
2004
).
33.
V. E.
Tarasov
, “
Acoustic waves in fractal media: Non-integer dimensional spaces approach
,”
Wave Motion
63
,
18
22
(
2016
).
34.
V. E.
Tarasov
, “
Fractional vector calculus and fractional Maxwell’s equations
,”
Ann. Phys.
323
,
2756
2778
(
2008
).
35.
V. E.
Tarasov
,
Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media
(
Springer Science & Business Media
,
2011
).
36.
K.
Diethelm
,
V.
Kiryakova
,
Y.
Luchko
,
J.
Machado
, and
V. E.
Tarasov
, “
Trends, directions for further research, and some open problems of fractional calculus
,”
Nonlinear Dyn.
107
,
3245
3270
(
2022
), available at https://link.springer.com/article/10.1007/s11071-021-07158-9.
37.
M.
Zubair
,
M. J.
Mughal
, and
Q. A.
Naqvi
,
Electromagnetic Fields and Waves in Fractional Dimensional Space
(
Springer Science & Business Media
,
2012
).
38.
M. J.
Mughal
and
M.
Zubair
, “Fractional space solutions of antenna radiation problems: An application to Hertzian dipole,” in 2011 IEEE 19th Signal Processing and Communications Applications Conference (SIU) (IEEE, 2011), pp. 62–65.
39.
Q. A.
Naqvi
, “
Scattering from a perfect electromagnetic conducting (PEMC) strip buried in non-integer dimensional dielectric half-space using Kobayashi potential method
,”
Optik
149
,
132
143
(
2017
).
40.
M.
Abbas
,
A. A.
Rizvi
, and
Q. A.
Naqvi
, “
Two dimensional Green’s function for non-integer dimensional dielectric half space geometry
,”
Optik
127
,
8530
8535
(
2016
).
41.
M.
Nisar
and
Q. A.
Naqvi
, “
Cloaking and magnifying using radial anisotropy in non-integer dimensional space
,”
Phys. Lett. A
382
,
2055
2060
(
2018
).
42.
S.
Kanwal
and
Q. A.
Naqvi
, “
Electromagnetic behavior of a planar interface of non-integer dimensional dielectric-chiral mediums
,”
Optik
181
,
105
114
(
2019
).
43.
E.
Davis
,
Y.
Liu
,
L.
Jiang
,
Y.
Lu
, and
S.
Ndao
, “
Wetting characteristics of 3-dimensional nanostructured fractal surfaces
,”
Appl. Surf. Sci.
392
,
929
935
(
2017
).
44.
S.-Y.
Chen
,
G.-Z.
Ma
,
H.-D.
Wang
,
P.-F.
He
,
M.
Liu
,
H.-J.
Wang
, and
B.-S.
Xu
, “
Comparison of solidity and fractal dimension of plasma sprayed splat with different spreading morphologies
,”
Appl. Surf. Sci.
409
,
277
284
(
2017
).
45.
M.
Zubair
and
L. K.
Ang
, “
Fractional-dimensional child-Langmuir law for a rough cathode
,”
Phys. Plasmas
23
,
072118
(
2016
).
46.
M.
Zubair
,
Y. S.
Ang
, and
L. K.
Ang
, “
Fractional Fowler–Nordheim law for field emission from rough surface with nonparabolic energy dispersion
,”
IEEE Trans. Electron Devices
65
,
2089
2095
(
2018
).
47.
M.
Zubair
,
Y. S.
Ang
, and
L. K.
Ang
, “
Thickness dependence of space-charge-limited current in spatially disordered organic semiconductors
,”
IEEE Trans. Electron Devices
65
,
3421
(
2018
).
48.
M.
Zubair
,
Y. S.
Ang
,
K. J. A.
Ooi
, and
L. K.
Ang
, “
Fractional Fresnel coefficients for optical absorption in femtosecond laser-induced rough metal surfaces
,”
J. Appl. Phys.
124
,
163101
(
2018
).
49.
S. N.
Kempkes
,
R. N.
Slotand
,
S. E.
Freeney
,
S. J. M.
Zevenhuizen
,
D.
Vanmaekelbergh
,
I.
Swart
, and
C. M.
Smith
, “
Design and characterization of electrons in a fractal geometry
,”
Nat. Phys.
15
,
127
131
(
2019
).
50.
X.-Y.
Xu
,
X.-W.
Wang
,
D.-Y.
Chen
,
C. M.
Smith
, and
X.-M.
Jin
, “
Quantum transport in fractal networks
,”
Nat. Photonics
15
,
703
710
(
2021
).