1 Introduction

Carbon-based electromagnetic absorbing composites have become a research hot-spot in recent years because of their application potential in military and civil fields [1,2,3,4,5,6]. Magnetic modification has been widely studied as a reliable strategy to improve the impedance matching of microwave between the free space and absorbing composites. Accordingly, the magnetic loss mechanisms are widely discussed [7,8,9,10], but the dielectric loss mechanisms are frequently summarized by a variety of interfacial polarization relaxations or vague synergistic effects [11,12,13]. In fact, the dielectric constant of the less layer graphite-based or Mxene-based absorbing composites frequently demonstrated multi-peak oscillation with frequency [14, 15], and their Cole–Cole plots show complex curves with no obvious pattern, implying that the conventional analysis method based on ideal Debye relaxation theory is particularly idealized to provide sufficient information. To date, research on the physical mechanism of such oscillations is still limited in the literature, and uncertainty persists, Nonetheless, an extensive understanding of this mechanism may provide theoretical reference for the design and performance improvement of absorbing materials.

In this work, we present a strategy for studying the dielectric loss mechanism in quasi-2D nickel–iron layered double hydroxide (NiFe-LDH) and less-layer graphite (LLG) composites. The particle swarm optimization (PSO) algorithm was used to analyze the loss mechanism in the framework of Krammers–Kronig (KK) relation based on the experimental data of the absorbing properties in the NiFe-LDH/LLG samples. Further X-ray photoelectron spectroscopy (XPS) and Raman spectral characterization explored the origin of the loss mechanisms in the sample.

2 Experiment and discussion

The NiFe-LDH/LLG composites are synthesized using a method combining liquid phase shear and electrostatic self-assembly (the synthetic details are shown in the Supplementary information). In brief, the NiFe-LDH laminates prepared by hydrothermal method are positively charged, and the NiFe-LDH and expanded graphite (EG) are mixed into deionized water to form suspension.

The suspension is introduced into a home-made microcavity shear device. In the shear device, the suspension whirled at a high speed with a moving disk driven by a high-speed electric spindle. The liquid phase shear force generated by the velocity gradient realized the interlaminar peeling of NiFe-LDH and graphite, and the NiFe-LDH/LLG composite particles are formed by self-assembly.

Figure 1 demonstrates the morphologies and microstructures of NiFe-LDH and NiFe-LDH/LLG. Figure 1a shows a typical lamellar morphology of NiFe-LDH similar to the result of other literature [16]. Figure 1b, c shows the close bonding between the NiFe-LDH laminate and the carbon layer surface, The thin layers of hydrotalcite with the size of tens of nanometers are densely distributed on the surface of the graphite layer. The in-situ electron diffraction analysis result of Fig. 1d manifests a quasi-2D/2D structure between NiFe-LDH and LLG. The lattice fringes with an interplanar spacing of 0.34 nm correspond to the (0 0 2) plane of LLG, and the interplanar spacings of 0.39 nm and 0.26 nm correspond to the (0 0 6) and (0 1 2) planes of NiFe-LDH, respectively. Figure 1d illustrates the electron diffraction pattern of NiFe-LDH/LLG.

Fig. 1
figure 1

Morphologies and microstructures of NiFe-LDH and NiFe-LDH/LLG. SEM images of a NiFe-LDH and b NiFe-LDH/LLG; c TEM and d HRTEM images of NiFe-LDH/LLG

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The radii of the four diffraction rings from inside to outside are 0.26 nm, 0.21 nm, 0.15 nm and 0.11 nm, representing the (0 1 2), (0 1 5), (1 1 0) and (1 1 3) crystal planes of NiFe-LDH, respectively, which are consistent with the XRD results (Fig. S2).

The result of the electromagnetic wave reflection loss (RL) measured by coaxial method is shown in Fig. 2a. The image shows the characteristic of the broad frequency absorption common to carbon-based composite absorbing materials [17, 18]. Interestingly, the real and imaginary parts of the dielectric constant shows a multimodal oscillation with frequency similar to that of Mxene-based absorbing composites (Fig. 2b). Obviously, such dispersion relation deviates from the strict Debye relaxation theory. In the perfect Debye paradigm, the complex permittivity is expressed as follows:

$$\varepsilon_{r} = \varepsilon_{\infty } + (\varepsilon_{s} - \varepsilon_{\infty } )/(1 + j\omega \tau )$$
(1)
Fig. 2
figure 2

Reflection loss (RL) of the NiFe-LDH/LLG composites in different thicknesses (a); dispersion relation of NiFe-LDH/LLG between complex permittivity and frequency (b); standard Debye relaxation curve (c); Cole–Cole semicircle curve (d)

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When subjected to the KK relation, the real part \(\varepsilon^{\prime}\) and imaginary part \(\varepsilon^{\prime\prime}\) satisfy the famous Cole–Cole semicircle curve relationship [19]:

$$(\varepsilon^{\prime} - \frac{{\varepsilon_{s} + \varepsilon_{\infty } }}{2})^{2} + (\varepsilon^{\prime\prime})^{2} = (\frac{{\varepsilon_{s} - \varepsilon_{\infty } }}{2})^{2}$$
(2)

where \(\varepsilon_{s}\) and \(\varepsilon_{\infty }\) are the static permittivity and permittivity at infinite frequency, respectively; and \(\tau\) is the polarization relaxation time.

The strict Debye relaxation theory of permittivity as a function of frequency presents the curves shown in Fig. 2c, and the corresponding Cole–Cole plot appears as a semicircle pattern specified in Eq. (2). However, in NiFe-LDH/LLG composites, the Cole–Cole diagram demonstrated in Fig. 2d consists some irregular semicircles and some complex curves, and a linear upward trend exists in the larger portion of \(\varepsilon^{\prime}\), which is a common conductivity loss feature in carbon-based absorbing materials [20].

The comparison of Fig. 2b, c indicated that \(\varepsilon^{\prime}\) exhibits the characteristics of oscillation based on the monotonic drop similar to the Debye relaxation in the frequency range of 2–12 GHz, and \(\varepsilon^{\prime\prime}\) behaves as approximate Debye behavior. In a higher frequency range, both \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) completely deviate from the standard Debye relaxation mode and keep some oscillations. Considering the possibility that Debye relaxation cannot follow the change of the microwave field at a high frequency, we speculated that more than one dielectric loss mechanism existed in the sample of NiFe-LDH/LLG composites, i.e., Debye relaxation may be responsible for the downtrend of \(\varepsilon^{\prime}\) with frequency, while the oscillation characteristics may be related to the Drude–Lorentz resonance [21]. Accordingly, the Debye empirical formula of the multi-relaxation process is introduced to simulate the experimental data, and the resonance absorption data are extracted by subtracting the simulation data of Debye theory from the experimental data. The empirical formula of Debye theory can be expressed as Eq. (3), and the specific real and imaginary parts of the dielectric constant are shown in Eqs. (4) and (5) [22].

$$\varepsilon_{r} = \varepsilon_{\infty } + (\varepsilon_{s} - \varepsilon_{\infty } )/1 + (j\omega \tau )^{1 - \alpha }$$
(3)
$$\varepsilon^{\prime} = \varepsilon_{\infty } + \frac{{(\varepsilon_{s} - \varepsilon_{\infty } )[1 + \sin (\frac{\pi \alpha }{2})(\omega \tau )^{1 - \alpha } ]}}{{1 + 2\sin (\frac{\pi \alpha }{2})(\omega \tau )^{1 - \alpha } + (\omega \tau )^{2(1 - \alpha )} }}$$
(4)
$$\varepsilon^{\prime\prime} = \frac{{(\varepsilon_{s} - \varepsilon_{\infty } )\cos (\frac{\pi \alpha }{2})(\omega \tau )^{1 - \alpha } }}{{1 + 2\sin (\frac{\pi \alpha }{2})(\omega \tau )^{1 - \alpha } + (\omega \tau )^{2(1 - \alpha )} }} + \frac{\sigma }{{\omega \varepsilon_{0} }}$$
(5)

Parameter \(\alpha\) is used to indicate the departure of standard Debye theory, and its value is set to \(0 < \alpha < 1\). The resonance absorption data are simulated using particle swarm optimization algorithm (PSO), and the calculation formula of Drude–Lorentz oscillator is shown in Eq. (6) [23, 24]:

$$\varepsilon_{r} = \sum\limits_{i = 1}^{N} {\varepsilon_{i\infty } + \frac{{\Delta_{i\varepsilon } \omega_{ip}^{2} }}{{ - \omega^{2} + j\gamma_{i} \omega + \omega_{i0}^{2} }} - \frac{{\omega_{ip}^{2} }}{{\omega (\omega + j\gamma_{i} )}}}$$
(6)

where \(\Delta_{i\varepsilon } = \varepsilon_{is} - \varepsilon_{i\infty }\), \(\omega_{ip}\) is the plasma angular frequency, \(\omega_{i0}\) is the resonance frequency, and \(\gamma_{i}\) is the damping term.

Only the appropriate parameters can ensure the validity of the oscillation data extracted from the experimental data, and the simulation of the data by PSO can be achieved. We compared different parameter combinations used in the Debye relaxation simulation. The appropriate parameters are \(\varepsilon_{s} = 20\), \(\varepsilon_{\infty } = 9.39\), \(\tau = 3.2 \times 10^{ - 11} s\) and \(\alpha = 0.01\). The simulation results of the dielectric real part and the dielectric imaginary part are shown in Fig. 3a, b, respectively. The oscillation data are extracted and presented in Fig. 3c, d as relevant results and named as the Data except Debye. The simulated results of the PSO are plotted in Fig. 3c, d, and named the Drude–Lorentz model.

Fig. 3
figure 3

Simulation results of the dielectric constant with Debye theory, a for real part and b for imaginary part; simulation results of the oscillation data by PSO, c and d for real and imaginary parts, respectively

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In the 2–18 GHz frequency range, the simulation results are fairly consistent with the oscillation data. The specific fitting parameters are shown in Table 1. Therefore, one can evidently conclude that the Drude–Lorentz resonance loss mechanism exists in this composites. Notably, the imaginary part of the dielectric constant has a positive contribution to microwave absorption. However, the imaginary part in the Debye mode is small and almost unchanged in the high frequency band. Moreover, the Snoek limit restricts the absorbing effect of the magnetic loss, Consequently, the common microwave absorbing materials do not have ideal absorbing performance. In Fig. 3d, the dielectric imaginary part in NiFe-LDH/LLG oscillates and rises to a certain extent, which can improve the high frequency absorbing properties. The NiFe-LDH/LLG sample has a performance of more than 5 dB attenuation in the range of 14–18 GHz, even if the thickness is only 1.5 mm, as shown in Fig. 2a.

Table1 Fitting parameters of NiFe-LDH/LLG
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As previously mentioned, the Drude–Lorentz resonance is beneficial to microwave high-frequency absorption, but what the physical root is remains unclear. We suspect that the excitation of plasmons at the 2D interface plays an important role in the Drude–Lorentz resonance absorption. Theoretically, the plasmons of bulk materials have bandgaps and are generally difficult to be excited by external fields, but the plasmon dispersion relation of two-dimensional electron gas is expressed as \(\omega_{p}^{2D} = \left( {\frac{{2\pi ne^{2} q}}{\varepsilon m * }} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}\) In the formula, \(\omega_{p}^{2D}\) is the angular frequency of electron collective oscillation, \(m *\) is the effective mass, \(\varepsilon\) is the dielectric constant, \(n\) is the electron density, \(q\) is the wave number, and the dispersion relation without the bandgap determines its easy excitation characteristics, which has been widely confirmed in the semiconductor MOS inversion layer. From the perspective of dispersion relationship, a considerable range of frequency ranges can be obtained by adjusting the electron density and effective mass. Resonant absorption occurs when the incident electromagnetic wave and the plasmon frequency match. This resonance absorption mechanism is extended to absorbing materials, which is beneficial to the broadening of the effective bandwidth [21]. Multiple oscillators may exist in a given system [25, 26], specifically in the case of the NiFe-LDH/LLG composite in this paper, XPS and Raman spectroscopy confirm the presence of electron migration at the NiFe-LDH and LLG interfaces, SEM and TEM characterization shows that the composite material is a quasi-two-dimensional system, so after the migration of electrons must be accompanied by two-dimensional collective oscillation phenomenon. As shown in Table 1, the broader resonance absorption peak in the text is the result of the superposition of multiple single oscillators. The spin–orbit peaks of Fe 2p3/2 and Fe 2p1/2 slightly move from 710.43 eV and 724.17 eV in NiFe-LDH to 711.43 eV and 724.08 eV compared with those in NiFe-LDH/LLG, as shown in Fig. 4b, c. Similar slight movements also occur in Ni 2p3/2 and Ni 2p1/2, from 855.33 and 872.88 eV to 855.58 and 873.58 eV. The deviation of the binding energy indicates that some electron migration occurs at the interface between NiFe-LDH and LLG. Further Raman characterization is shown in Fig. 4d, the D band of carbon in NiFe-LDH/LLG becomes higher and broader, suggesting a higher degree of disorder in LLG during the functionalization process. Furthermore, the G band of carbon up shifts by 12 cm−1 from 1580 cm−1 in EG to 1592 cm−1 with an asymmetrical shoulder peak in NiFe-LDH/LLG. The G-band in the Raman spectrum of the carbonaceous material corresponds to the sp2 carbon stretching modes [27]. Such Raman shifts of the G band for NiFe-LDH/LLG provide evidence for the charge transfer between the NiFe-LDH laminates and LLG sheets, suggesting a strong interaction between the NiFe-LDH and the LLG sheet. In addition, a large number of laminates are dispersed on the sheets of the LLG shown in Fig. 1c. Thus, it is reasonable to believe that a quasi-2D interface system is formed between the NiFe-LDH and the LLG sheets, and it promotes Drude–Lorentz resonance absorption.

Fig. 4
figure 4

a XPS analysis of NiFe-LDH and NiFe-LDH/LLG; high-resolution b Fe 2p, c Ni 2p; d Raman spectra of the LLG and NiFe-LDH/LLG composites

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3 Conclusion

In this work, we present a strategy to promote the Drude–Lorentz resonance absorption in quasi-2D graphite composites. We first synthesize the NiFe hydrotalcite/less layer graphite composites by utilizing the method combining liquid phase shear and electrostatic self-assembly. The experimental absorbing data of the composites are used to investigate the loss mechanism in the KK relation framework using the PSO algorithm. We clarify that the excitation of plasmons at the 2D interface plays an important role in the Drude–Lorentz resonance absorption, which is beneficial to high-frequency microwave absorption. Finally, the XPS and Raman spectral characterization are shown to further confirm the loss mechanisms in these samples. We believe that this concept can give rise to improvements in the design and preparation of high-performance absorbing materials.