3. Results and Discussion
The microscopic morphology of the cross-section of Cu/graphite (represented by Cu/Gr in
Figure 1) was initially examined. As shown in the enlarged region of the red box in
Figure 1a, the thickness of the Cu coatings is approximately 800 ± 50 nm.
Figure 1b indicates that the thickness of the Cu coating is approximately 1000 ± 50 nm. The thickness of the Cu coatings varied slightly in different positions, mainly due to the height fluctuations of the graphite substrate.
Figure 1c–e further highlights this undulation effect, which showcases the confocal 3D morphology of the pure Cu and Cu/graphite. The surface roughness values for the pure Cu, 800 nm Cu/graphite, and 1000 nm Cu/graphite were calculated as 60 nm, 432 nm, and 460 nm, respectively. These values were determined by the surface height differences measured using confocal microscopy.
Subsequently, both the 800 nm and 1000 nm Cu/graphite were examined regarding the trends in RF conductivity with frequency (represented by Cu/Gr in
Figure 2). A comparison with the trend observed for pure Cu is presented in
Figure 2a. The RF conductivity of the pure Cu decreases slightly with increasing frequency, ranging from 50 MS/m to 56 MS/m. In contrast, the RF conductivity of the Cu/graphite increases with rising frequency, as depicted in
Figure 2b, c. Notably, the RF conductivity of the Cu/graphite reaches the highest value at 40 GHz, with 2.4 MS/m of 800 nm Cu/graphite and 7.3 MS/m of 1000 nm Cu/graphite. To further investigate the RF conductivity trends with frequency for the Cu/graphite and pure Cu, an assessment was conducted to compare the RF conductivity differences between the pure Cu and Cu/graphite at 15 GHz, 21 GHz, and 30 GHz with those at 40 GHz, as shown in
Figure 2d. The RF conductivity of the pure Cu increases by more than 5% at 15–30 GHz, with increments of 4.0 MS/m, 3.8 MS/m, and 2.7 MS/m, respectively, compared to 40 GHz. Conversely, the RF conductivity of the 800 nm Cu/graphite decreases by 0.75 MS/m, 0.46 MS/m, and 0.28 MS/m, respectively, at 15–30 GHz compared to 40 GHz, resulting in more than a 10% decrease. Similarly, the RF conductivity of the 1000 nm Cu/graphite decreases by 3.3 MS/m, 2.3 MS/m, and 1.0 MS/m, respectively, at 15–30 GHz, exhibiting a decrease of more than 10% from the values observed at 40 GHz.
In both the 800 nm Cu/graphite and 1000 nm Cu/graphite, the thickness of the Cu coatings adequately exceeds the skin depth of Cu at 15–40 GHz (540 nm–330 nm), supporting the assumption that electromagnetic waves decay exclusively within Cu layer [
36]. Thus, the main difference between the Cu/graphite and pure Cu lies in their distinct surface roughness. According to Equations (8)–(13), the RF conductivity exhibits an inverse correlation with the microwave power loss of the material. Numerous studies have demonstrated that the additional absorption and reflection of microwaves can be non-negligible when the surface roughness exceeds a certain threshold value [
37,
38]. For the 800 nm Cu/graphite, the skin depth (approximately 550 nm) and surface roughness (432 nm) are comparable in magnitude at a test frequency of 15 GHz. Consequently, the influence of the surface roughness on the electromagnetic waves must be considered when calculating the conductor loss during waveguide transmission.
However, the exclusive influence of the surface roughness fails to account for the observed trend of rising RF conductivity in the Cu/graphite samples with frequency. For the pure Cu, it can be assumed that the power loss (
) of the electromagnetic wave transmission at four frequency points from 15 to 40 GHz is represented as
, with the corresponding RF conductivity denoted as
. The
primarily consists of two components, as shown in Equation (14): the power loss due to surface roughness
and the ohmic loss
, arising from the conductor’s inherent resistance [
39]:
With the increased frequency, the skin depth of the pure Cu () gradually decreases, while the fluctuation of the rough surface remains unchanged. Consequently, the ratio of the rough surface to the skin depth gradually increases, resulting in an increased power loss of the rough surface to the electromagnetic wave, i.e., . Because is solely related to the material itself, it can be concluded that based on Formula (14). Furthermore, based on the inverse relationship between RF conductivity and microwave power loss of conductor, it can be deduced that . Similarly, it can be concluded that .
In view of the preceding analysis, an escalation in the surface roughness of the material contributes solely to a more evident inclination for the RF conductivity to decrease with increasing frequency. This finding has been supported by other studies as well [
40]. It is worth noting that the aforementioned analysis is predicated on the material being a pure metal. Regardless of whether the skin depth decreases or increases, the skin layer remains the metal itself. In other words,
is only associated with the material itself and does not increase with higher frequencies. In bilayer-structured materials, such as Cu/graphite in this study, when the frequency is below 15 GHz, the skin layer includes surface Cu coatings, interfaces, and substrates. Hence, it is essential to investigate whether the skin depth contains interfaces and substrates within the 15–40 GHz range. According to the skin effect theory, the propagation factor within a material is determined when a plane wave is incident on the smooth surface of a planar metal [
41]:
The depth at which the amplitude attenuates to
times its initial value is defined as the skin depth, denoted as
, provided by [
42]
It is essential to note that
represents only the region where 63.2% (
) of the current is concentrated. The region containing 95% of the current is concentrated within 3
, and the region containing 98.2% of the current is concentrated within 4
. Clearly, for pure metals, considering
or 4
does not alter the trend of decreasing RF conductivity with increasing frequency. However, for the Cu/graphite, focusing solely on
is not realistic due to the 36.8% current conduction in the interface and substrates. Considering the significant difference in intrinsic conductivity between the substrate graphite and Cu coatings, this scenario cannot be directly compared to the case of pure metal. Essentially, it is crucial for the Cu/graphite to consider the electromagnetic wave loss within 3
to assess the power loss of the material’s electromagnetic field with varying frequency at least. As a result, despite the thickness of the Cu coatings being higher than the
, it is not valid to assume that the electromagnetic wave only attenuates within the Cu coatings. The RF conductivity of the Cu/graphite shown in
Figure 2 actually corresponds to the effective conductivity, which encompasses the rough Cu coatings, the Cu/graphite interface, and the graphite substrate.
Based on the analysis above, the skin depth of Cu/graphite includes rough Cu film, Cu/graphite interface, and graphite substrate. To distinguish it from pure Cu, it can be assumed that the power loss (
) of electromagnetic wave transmission from 15 to 40 GHz is represented as
, and the corresponding RF conductivity is denoted as
. The
primarily consists of four aspects: the power loss due to surface roughness
, the ohmic loss
arising from the conductor’s inherent resistance, the power loss
due to Cu/graphite interface, and power loss
from graphite substrate. As the frequency increases, although the power loss of the rough surface gradually increases, the power loss caused by the graphite is significantly reduced due to the reduction in the thickness of the graphite substrate in the skin depth. Since the conductivity of graphite is three orders of magnitude lower than that of Cu, the reduction in power loss of graphite with the increased frequency is significantly lower than that increasing power loss by the rough surface. Therefore, it can be concluded that
. Similarly, it can be deduced that
, which is consistent with the trend shown in
Figure 2d.
According to the skin effect theory, the attenuation of electromagnetic waves in Cu/graphite should consider the relationship between the thickness of Cu coatings and
.
Figure 3 depicts the relationship between electromagnetic wave power loss and effective conductivity for the 800 nm Cu/graphite and 1000 nm Cu/graphite at frequencies ranging from 15 to 40 GHz, taking the
condition into account. When the skin depth of Cu (
) falls within 250 nm–650 nm and the thickness of Cu coatings (
) is 600–1200 nm, it can be segmented into two regions labeled as
≥
and
<
to elucidate the numerical relationship between
and
, highlighted by the light red and light blue areas in
Figure 3a. For a Cu thickness of 1000 nm, it falls within the
≥
region when the test frequency is 40 GHz. Moreover, all other testing conditions in this study fall within the
<
region.
On the basis of the above analyses, the relationship between electromagnetic wave power loss and effective conductivity for the 800 nm Cu/graphite and 1000 nm Cu/graphite at frequency points from 15 to 40 GHz is classified into two cases, illustrated in
Figure 3b,c. When
<
, electromagnetic waves are incident vertically on the Cu surface from air and undergo reflection and transmission phenomena at the interface, as elucidated by electromagnetic wave transmission theory. This superposition of incident and reflected waves within Cu coatings results in a specific level of attenuation. The electromagnetic waves then propagate through graphite until complete attenuation, as illustrated in
Figure 3b. The attenuation process of electromagnetic waves in Cu/graphite can be characterized by the reflection coefficient [
43], denoted as
. According to Ye and Bai’s study, the total reflection coefficient of the bilayer structure can be calculated as follows:
According to the boundary conditions at the interface of “air/Cu coatings”,
Similarly, at the interface of the “Cu/graphite substrate”,
where
is propagation constant in the thin film,
here,
is the skin depth in Cu coatings
is imaginary unit, and
is the permeability of the vacuum. Similarly, the propagation constant in substrate is
where
is permittivity of vacuum,
is relatively permittivity of substrate. Total reflection coefficient of the bilayer structure can be obtained by combining Equations (17) and (18):
To obtain Equation (22), magnetic field is related to electric field by intrinsic impedance as
. Here,
, intrinsic impedance for air, and
, intrinsic impedance for thin film, can be calculated by
Therefore,
in the Equation (22) can be expressed as follows:
And
is the intrinsic impedance for substrate.
Equations (22)–(24) indicate that the total reflection coefficient
depends on the electrical properties of the Cu coatings
and substrate
/
, as well as the thickness of the Cu film
and the operating frequency
[
29,
44].
The energy of the attenuated electromagnetic waves primarily dissipates in the form of power loss, which is a function of the effective conductivity of Cu/graphite. Therefore, the relationship between electromagnetic wave attenuation and power loss can be established through the effective conductivity of Cu/graphite. Similarly, when
≥
, the electromagnetic waves are completely attenuated in the Cu coatings, and the total reflection coefficient is only related to
,
, and
. On the basis of
Figure 3, we can infer that, although the direct determination of the influence of rough surfaces, interfaces, and substrates on RF conductivity is challenging, a quantitative analysis can be achieved. The impact of rough surfaces, interfaces, and substrates on RF conductivity can be quantified by the correlation between effective conductivity and electromagnetic wave power loss. This entails comparing the power loss of a conductor with rough surfaces, interfaces, and substrates to that of a smooth conductor. Therefore, it is imperative to quantitatively calculate the electromagnetic wave power loss caused by rough surfaces, interfaces, and substrates.
Figure 4 illustrates the process of determining the ratio of power loss between the rough and dual layered conductors and smooth conductors. To simplify the problem, we initially consider the scenario where
, as depicted in
Figure 4a. In this case, the effective conductivity of the rough conductor is solely affected by the presence of rough surfaces on the power loss. This effect can be determined by calculating the ratio of power loss between conductors with rough surfaces and smooth conductors. The known parameters include
,
,
,
, and the characteristic values of the rough surface, such as
and
, which can be obtained through confocal microscopy. The loaded quality factor of the system
can be obtained through separate resonant cavity testing. The effective conductivity
, incorporating the rough surface, can be derived using Formulas (9)–(13). The losses in the system encompass not only those of the sample but also the losses of the cavity wall itself, denoted by
and
. Therefore, it is essential to independently quantify the additional power loss attributed to the rough surface (
). Subsequently,
can be calculated with Formulas (9)–(13). Combined with the inverse relationship between RF conductivity and power loss,
, denoted by the introduced parameter
, can be obtained. Furthermore, when
, as indicated in
Figure 4b, the
of the rough surface can be determined. Additionally,
and
can be obtained through separate resonant cavity testing. Using the theoretical formula for
, the conductivity of the Cu part contained in
, denoted as
, and the interface conductivity
can be determined [
44]. Based on the theoretical relationship between conductivity and power loss, the power loss ratio of Cu/graphite and smooth pure Cu can be calculated. Similarly,
can be obtained, represented by the introduced parameter
, as illustrated in
Figure 4b.
Following the theoretical calculation process depicted in
Figure 4, our initial step is modeling surface roughness to calculate the additional
. This modeling approach enables us to quantitatively evaluate the impact of surface roughness on the RF conductivity of Cu/graphite. In contrast to many numerical models for roughness [
40], this study employs fractal theory for modeling rough surfaces. Originally formulated by Mandelbrot, the fractal theory has evolved over the years and gained popularity among researchers due to its ability to more accurately describe surfaces that are neither periodic nor entirely random [
45,
46].
According to fractal theory, the function expression for the roughness model is provided as follows [
47,
48,
49]:
here,
represents the height of the random surface profile, and
signifies the position coordinates of the profile.
is the fractal dimension, and
is the characteristic scale parameter reflecting the amplitude magnitude and determining the specific size of
.
represents the spatial frequency of the profile, where
= 1.5 is applicable to the randomness of high-frequency spectral density and phase for profiles that follow a normal distribution. Since the roughness profile is a non-stationary stochastic process, the relationship between the lowest frequency of the profile structure and the length of the roughness sample is provided by
, the initial term of the fractal function, which is an integer, and
, which is the sampling length of the roughness sample.
The derived power spectral density function is as follows:
is the radial function, defined as
The continuous power spectrum can be approximated as
It is evident that the continuous power spectrum
enables a power law, which is fundamental to the fractal characterization of surface micro-topography. The fractal dimension
is linked to the extent of variation in the amplitude of the surface topography. A higher
value suggests the existence of more high-frequency components and richer details on the surface. The relationship between the parameters in the fractal method and the roughness parameters of the surface is as follows:
the relationship between
and the amplitude coefficient
is
.
Generally, a smaller
corresponds to a larger fractal parameter
. Based on the established fractal model, the power loss of a semi-infinite planar waveguide is calculated. The variation range of the surface contour lines is taken from
to
on the three-dimensional model of the waveguide. The conductor is defined as a smooth plane when
. Assuming that the varying magnetic field
propagates along the surface
in the Z direction, the eddy currents propagate in the
plane. The small changes in the magnetic field caused by finite conductors are negligible. Due to the small surface roughness induced by height variation, it is assumed that the magnetic field
remains unchanged due to variations in the
direction. The calculation of the eddy currents assumes no variation in
along the Z direction. Within the given volume
in the conductor, the eddy current losses are [
40,
50,
51,
52]
For conductors,
. Therefore, the displacement current can be ignored. For the alternating magnetic field in the conductor, the equation is satisfied as follows:
where
is the conduction current density vector.
Since
only has a Z component, we have
, where
is the unit vector in the Z direction. Therefore, we obtain
Therefore,
where
is the unit vector in the
direction, we obtain
Based on Green’s formula, if there is any continuous differentiable function
,
, we obtain
where
is the closed surface surrounding
and
is the outer normal direction of
.
Order
,
, we obtain
Order
,
, we obtain
Adding Equations (38) and (39) together provides
From
, the following can be obtained:
Therefore,
We can obtain
Taking a differential element
of the conductor, with width z in the
direction and unit length
in the Z direction and extending infinitely deep in the
direction. Since
, the surface integral at the Z terminal is 0; on the surfaces at
and
, the integral results cancel each other out; as
approaches
, the field tends to 0, so only a portion of the upper surface
remains in the integral over the closed surface
. At this point,
, so we have
Using Green’s formula again, let
,
, and
,
, respectively. Based on the calculated results and Equations (41) and (42),
Substituting Equation (45) into Equation (44),
Substituting
into Equation (46),
Because
is a plural number, it can be separated from integration in Equation (47) as follows:
Based on the fundamental properties of complex numbers, namely
Therefore,
Because the volume integral is performed in three directions,
,
, and
, we can transform the triple integral in
volume interval (Equation (50)) into a double integral in the
plane.
Therefore,
where
is the power dissipated in the infinitesimal conductor element
.
If the conductor has a rough surface, according to the previously established fractal surface roughness model for determining the surface profile, i.e.,
, the power dissipated in the infinitesimal conductor element is provided by
If the conductor has a smooth surface, i.e.,
, then
is the following:
Therefore,
It should be noted that, when deriving the formula for the power loss in a semi-infinite conducting plane, it is assumed that the varying magnetic field is denoted as
and for the sake of simplification. It is considered that
does not vary with the
and
coordinates. However, in actual waveguide structures, the magnetic field on the conductor surface has three components:
,
, and
, which do vary with the
,
and
coordinates. For the cylindrical resonant cavity in this work, the power loss of the resonant cavity has already been provided by Equations (9)–(11). Therefore, the difference in power loss between rough surface and smooth surface only lies in the loss
on the
surface of the test metal plate. For the rough surface with a contour
, the expression for
is
Therefore, for the cylindrical resonant cavity studied in this research, Equation (32) can be modified by defining
as the ratio of power loss between rough and smooth surfaces as follows:
Based on Equations (6)–(13) and (33)–(35), it is possible to determine the specific values of at different frequencies.
According to the theoretical calculation process on the right side of
Figure 4, it is first necessary to obtain
and
at different frequencies to calculate the specific values at different frequencies. The value of
at different frequencies can be directly measured through the separated resonant cavity.
has been obtained by numerous researchers through testing [
18]. According to the work of Takashi Shimizu, the effective interfacial conductivity of the composite material can be provided by the following equation [
53]:
where
,
, and
are represented as follows:
Whereas, due to the perturbations of
,
, and
, the variations in each resonance frequency
,
, and
are calculated based on rigorous analysis using the mode matching method [
53].
According to the derivation process shown in
Figure 4, combined with Equations (36)–(39), the quantitative calculation is performed to evaluate the effects of surface roughness, interface, and substrate on the effective conductivity of composite materials under the condition of
. The ratio of power loss between the defined rough surface composite material and the smooth surface pure metal is defined as
. The value of
is provided by
Based on Equations (56) and (61),
and
are calculated at different frequencies, and the fitting curves of their variations with skin depth are plotted. The comparison with the measured values
is shown in
Figure 5.
On the one hand,
Figure 5 demonstrates the trend of the RF conductivity test values of the Cu/graphite samples with respect to the skin depth of Cu. The blue triangles represent the RF conductivity test values for the 1000 nm Cu/graphite and the red circles represent the RF conductivity test values for the 800 nm Cu/graphite. It can be found that the RF conductivity of the Cu/graphite gradually decreases as the Cu skin depth increases. This is mainly because the proportion of the graphite in the skin layer of the Cu/graphite gradually increases with increasing skin depth, resulting in a significant increase in the microwave power loss.
On the other hand,
Figure 5 also displays the RF conductivity values of the Cu/graphite calculated by only considering the additional power loss of the surface roughness of the Cu film (denoted by
), and the trends of
with the Cu skin depth are shown by the blue and red dashed lines, representing the cases of the 1000 nm Cu and 800 nm Cu, respectively. Correspondingly, the calculated value of the RF conductivity of the Cu/graphite when simultaneously considering the power loss from the surface roughness of the Cu film, the Cu/graphite interface, and the graphite substrate is denoted by
. Similarly, the trend of
with the Cu skin depth is denoted by the blue and red solid lines, representing the cases of 1000 nm Cu and 800 nm Cu, respectively. For the 1000 nm Cu, both the
and
values differed little from the test at a Cu skin depth of about 330 nm. However, as the skin depth increases, the difference between the value of
and the tested value gradually becomes larger. However, the value of
is always less different from the tested value. This is due to the fact that the coating thickness of the 1000 nm Cu is greater than
, which makes the electromagnetic wave completely attenuated in the Cu coatings when the skin depth of the Cu film is 330 nm. It can be seen from
Figure 3c that the surface roughness of the Cu coatings is the main factor affecting the Cu/graphite RF conductivity, leading to the values of
and
at this point being less different from the test values of the RF conductivity.
When the skin depth of Cu is greater than 330 nm, the electromagnetic waves in both the 1000 nm Cu/graphite and 800 nm Cu/graphite penetrate the Cu coatings and Cu/graphite interface, and finally are completely lost in the graphite substrate. Therefore, the loss of the electromagnetic wave is composed of the loss caused by the rough surface of the Cu coatings, the Cu/graphite interface, and the graphite substrate, as shown in
Figure 3b. With the increase in the skin depth, the proportion of the graphite gradually increases, resulting in the proportion of the graphite loss gradually increasing. Therefore, the error of only considering the loss of the rough Cu coatings’ surface gradually increases. With the increase in skin depth, the difference between the value of
and the test value gradually increases. Correspondingly, the difference between
and the test value is always small. It indicates that, when the thickness of the Cu coatings is less than
, it is more accurate to evaluate the RF conductivity of the Cu coatings by considering the losses caused by the rough surface of the Cu coatings, the Cu/graphite interface, and the graphite substrate.
Table 1 presents the differences between the fitting values of
and
and the measured value
for the Cu skin depths corresponding to four frequency points between 15 and 40 GHz. For the 800 nm Cu/graphite, the difference between
and
(represented by
in
Table 1) exceeds 30% and increases gradually with the increase in the skin depth, even exceeding 60%. For the 1000 nm Cu/graphite, the difference between
and
(represented by
in
Table 1) is only 3.29% at
, and it increases gradually with the increase in the skin depth, reaching a maximum of over 40%. The difference between the fitting value of
and the measured value
is less than 7% (represented by
and
in
Table 1), and it fluctuates only slightly with the variation in the skin depth.