Hydrogenated graphene oxide (H-G-SiO₂) Janus structure: experimental and computational study of strong piezo-electricity response

Graphene was grown on a copper substrate in a horizontal quartz tube by chemical vapor deposition and then transferred onto a Si/SiO₂ substrate by a wet process as outlined in [10]. The grown SiO₂ has a thickness of 300 nm. The graphene is then hydrogenated using RF plasma in a hydrogen ambient. To perform the plasma hydrogenation, a parallel-plate reactive ion etching reactor operating at a frequency of 13.56 MHz was used. The samples are placed on the bottom electrodes with a diameter of 300 mm and to minimize the plasma reactivity and damages caused by energetic ions, hydrogen plasma treatment is performed at a low power of 13 W. During the process of plasma hydrogenation, the pressure of the chamber was 0.03 mbar and the gas flow was 200 SCCM (standard cubic per centimeter). To ensure about the saturation, nearly 30 min of hydrogen plasma treatment was required. When the graphene is incompletely hydrogenated, the material is called hydrogenated graphene rather than graphane. The hydrogenated graphene has reversible hydrogenation property, making it suitable for hydrogen based storage media [11]. To verify the reversibility of hydrogenation, we performed annealing of the samples in Ar atmosphere at 300 °C for 2 h. Confocal Raman microscopy is used to monitor the structural changes of graphene by hydrogenation and to study the induced strain to the hydrogenated graphene stack. Raman spectroscopy is carried out using a laser excitation with 532 nm wavelength. XPS was applied to analyze the chemical composition of the hydrogenated graphene oxide (HGO) stack and to examine the carbon oxygen and carbon hydrogen bonds. To extract data, a Gammadata-scienta ESC A200 hemispherical analyzer was used, equipped with an Al Kα (1486.6 eV) x-ray source. An Ntegra Prima AFM (NT-MDT) was used to perform the PFM measurements. The measurements were carried out by Pt-coated cantilevers obtained from NT-MDT: NSG10/Au (390 kHz resonance frequency, force constant 37.6 Nm⁻¹) and CSG10/Au (39 kHz resonance frequency, force constant 0.5 Nm⁻¹). In the contact mode, by CSG10/Au cantilevers, PFM and conductive AFM were performed. For the PFM measurements, practiced in a quasi-static regime, the frequencies were smaller than 100 kHz, and the a.c. excitation signal amplitude was ranging from 0 to 5 V. Also, The PFM amplitude was evaluated by the frequency-dependent measurements in the frequency range of 10–500 kHz.

Figure 1(a) illustrates the SEM image of the graphene sheet, transferred onto Si/SiO₂ substrate while figure 1(b) displays the Raman spectra of the graphene for three cases including: (i) before, (ii) after exposure to hydrogen plasma and (iii) after annealing. These Raman spectra are normalized to achieve similar intensity for the G peak. Before hydrogenation, the characteristic Raman peaks are D, G and 2D at wavenumbers of 1353 cm⁻¹, 1580 cm⁻¹ and 2670.5 cm⁻¹, respectively. In these spectra, D peak corresponds to the sp³ defects in the graphene lattice, G peak is attributed to the optical vibration of E_2g phonons being at the center of the Brillouin zone, while 2D is originated by two phonon scattering process, being the sum of two phonons having opposite momentums [19].The ratio of I_2D/I_G, indicates that the transferred graphene is a single layer. By hydrogenation, sharp D and D' peaks appear at 1347 cm⁻¹ and 1632 cm⁻¹ wavenumbers, respectively. The 2D peak at 2669 cm⁻¹ broadens slightly and its height decreases relative to that of the G peak (1583 cm⁻¹). Moreover, a new peak (D  +  D') appears at about 2955 cm⁻¹, which is the sum of two phonons having different momentums and dissimilar to the bands of the 2D and 2D', and its activation requires defect [19]. By annealing, all of the peaks (D, D' and D  +  D') related to the defect, were strongly suppressed and the peaks G and 2D (1581 cm⁻¹ and 2669 cm⁻¹), almost recovered to their original shape and position.

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As can be seen from figure 1(b(iii)), after annealing at a temperture of 300 °C, the ID/IG ratio significantly decreases. Previous studies show that the frequencies corresponding to G and 2D band peaks in graphene, shift with the applied uniaxial [20–22] and biaxial strain [23, 24]. Also ${{\omega }_{{\rm G}}}$ and ${{\omega }_{2{\rm D}}}$ bands depend strongly on dopings, both electrically [25, 26] or chemically [27, 28]. Hence, to correctly estimate the strain, it is appropriate to separate the effects of the mechanical strain and the charge doping from the shift in the Raman peaks. To be able to do so, we use the correlation analysis between the two modes of graphene (${{\omega }_{{\rm G}}}$, ${{\omega }_{2{\rm D}}}$) [29]. Typically, the compressive and tensile strains of the graphene lattice result into a phonon hardening (frequency upshift) and phonon softening, respectively. The shifts in the position of the peak caused by the exerted strain is the material intrinsic property and is quantified by Grüneisen parameters, describing the effect of volume modulation of a crystal lattice on the vibrational properties [30]. Since covalent bonds form in both X and Y directions, the Grüneisen parameter for biaxial strain $\frac{\partial {{\omega }_{{\rm G}}}}{\partial \varepsilon }=-69.1\,{\rm c}{{{\rm m}}^{-1}}\,{{\%}^{-1}}$ is utilized in this work [29, 31, 32]. It is worth mentioning that for the phonons below 20 THz, Grüneisen parameter for graphene and hydrogenated-graphene are very similar [33, 34]. The magnitude of strain in a single layer graphene is given by $\varepsilon =\frac{\left( \omega _{{\rm G}}^{{\rm s}}-\omega _{{\rm G}}^{0} \right)}{{}^{\partial {{\omega }_{{\rm G}}}}\diagup{}_{\partial \varepsilon } }$, where $\omega _{{\rm G}}^{0}$ is the Raman frequency in the absence of strain and doping and $\omega _{{\rm G}}^{{\rm s}}$ is the Raman frequency of the G band due to the strain (without the doping effect), $\Delta \omega =\omega _{{\rm G}}^{{\rm s}}-\omega _{{\rm G}}^{0}$ is the strain based shift of the Raman frequency, and ε is the applied strain on the sample. $\omega _{{\rm G}}^{0}$ of the G-band, where the strain and doping is assumed to be zero, is approximately 1582 cm⁻¹ [29]. By using the correlation analysis used in [29], the values of the $\omega _{G}^{s}$ of the G band of the graphene/SiO₂, hydrogenated graphene/SiO₂ and annealed sample are equal to 1578 cm⁻¹, 1575.7 cm⁻¹ and 1576.5 cm⁻¹, respectively.

By using strain equation $\varepsilon =\frac{\left( \omega _{{\rm G}}^{{\rm s}}-\omega _{{\rm G}}^{0} \right)}{{}^{\partial {{\omega }_{{\rm G}}}}\diagup{}_{\partial \varepsilon } }$, an in-plane strain of 0.09% is obtained after hydrogenation of graphene on Si/SiO₂ substrate. It is also worth noting that the values of strain before hydrogenation and after annealing are 0.06% and 0.08%, respectively. The topography measurements of the hydrogenated-graphene structure deposited on SiO₂ substrate were implemented using a conventional tapping mode AFM (figure 1(c)) and its height profile is represented in figure 1(d).

To investigate the chemical modification of the CVD monolayer graphene, transferred on Si/SiO₂ substrate and treated by hydrogen (H₂) plasma, XPS is used. XPS is very appropriate tool to identify the chemical composition in carbonaceous samples [35–37]. The lorentzian-gaussian curve components were extracted by the deconvolution of XPS spectra at the different regions. Figure 2(a), shows the low-resolution XPS scan of the as-grown graphene film grown on Cu substrate, the graphene film transferred on Si/SiO₂ substrate and the hydrogenated-graphene layer on the Si/SiO₂ substrate.

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Figure 2(b) depicts the high resolution XPS scan of the C1s for the graphene film grown on the Cu substrate. The peaks at 284.6 eV and 285.3 eV energies can be attributed to sp²-hybridized and sp³-hybridized carbon, respectively. The other four components of C–OH, C–O, C=O and COOH bnods are related to the peaks of 285.9 eV, 286.6 eV, 287.5 eV and 288.4 eV, respectively [36–38]. The presence of sp³ component in the deconvoluted spectra of C1s may be originated from the CH₄ trapping or absorption, during the process of the CVD growth. Since the D band in the Raman spectrum of the figure 1(b(i)) is not strong, it is improbable that the C–H bonds are to be formed on as-grown CVD graphene [39]. The ratio of tetragonal to trigonal carbon bonds in graphene structure is given by ϴ/(1  +  ϴ), in which ϴ is the ratio of sp³ to sp²-hybridized carbon component intensity [39].

The value of this ratio in the pristine CVD graphene is 0.09, so that the C–H/C–C bond ratio will be about 0.09/1.09  =  8.2%. About 13.2% of the total area of the C1s peak corresponds to the oxygen-related components resulting from decomposing the peak in the curve components, which might be the result of insufficient vacuum during the growth [37].

For the graphene, transferred onto the Si/SiO₂ substrate (figure 2(c)), the C1s high-resolution XPS measurement depicts the peaks at the same energies of figure 2(b), that is at C sp² with binding energy of (284.4 eV), C sp³ (285.1 eV), C–OH (285.8 eV), C–O (286.5 eV), C=O (287.0 eV), and COOH (288.5 eV), respectively. Upon transfer, the oxygen related components bonded with carbon are increased, from the original 13.2% to 22.3%. The enhanced sp³ components of the C1s peak (in comparison with the sp³ components of the C1s peak in figure 2(b)), corroborates the chemical configuration modification of the transferred layer due to possible bonding of carbon atoms with underlying oxygen atom from SiO₂ substrate. The value of ϴ/(1  +  ϴ) in figure 2(c), is about 18.9%. By deducing this quantity from its value obtained in figure 2(b), the oxygen coverage of graphene (C–O bond) is obtained to be about 10%. Using XPS, the hydrogen coverage on graphene has been evaluated. Figure 2(d) shows the C1s spectra of graphene transferred on Si/SiO₂ substrate after hydrogenation (13 W, 0.03 mbar). The C1s spectra were fitted again with six components as same as the ones in figures 2(b) and (c) but it is worth mentioning that the ratio of the hybridized C components of the sp³/sp² increases and the value of ϴ/(1  +  ϴ) became 31.8%, which is an indication of the formation of C–H bonds. Using this value and comparing it with the previous values of this quantity, one can conclude that the hydrogen coverage of graphene is about 12.9%.

Figure 2(e) depicts Si2p peak corresponding to the bare Si/SiO₂ substrate. As it can be seen, there is only one peak at 103.3 eV corresponding to the silicon oxygen bonding and it is consistent with literature [40, 41]. After transferring of graphene on Si/SiO₂ substrate (figure 2(f)), a week shoulder toward the lower binding energies of the main Si2p peak is observed, causing a new peak after deconvolution. The intensity of this peak is weaker and we speculate this to be due to a Si–O–C bonding which is another evidence of C–O bonding [42]. The intensity ratio of Si–O–C bond to Si2p peak is about 10% which was extracted from the graphene C1s peak.

To detect tiny vibrations of graphene surface between the tip and the substrate, an a.c. voltage of 5 V at a frequency of 90 kHz is applied. Measurements are performed in the contact mode. The piezo-response has negative sign and the induced amplitude of vibrations is about 0.04–0.05 nm (figure 3(a)). To detect PFM signals, a voltage spectroscopy measurement is performed in the step mode on the graphene, GO and on hydrogenated graphene oxide stack, HGO. For this purpose, between the tip and the substrate, an a.c. voltage of about 1 V at a frequency of 20 kHz is applied. By sweeping the d.c. bias from  −10 V to 10 V, the piezoresponse on HGO stack and graphene oxide (GO) stack (before hydrogenation) was studied (figure 3(b)). The strain along the out-of-plane direction is modeled by considering the initial thickness of the layer L as: $S=\frac{\Delta l}{L}$, where $\Delta l$ is displacement of the atomic position extracted experimentally by PFM along the out-of-plane direction (at quasi static regime with a frequency of 20 kHz and a displacement amplitude of about 0.02 nm (see figure 3(b)).

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The amplitude of piezo-response increases for the applied negative d.c. voltages and it decreases for positive d.c. voltages. By applying a negative electric field to the tip, C–O and C–H dipoles, become closer to the vertical direction and hence the net polarization increases. By applying a positive electric field to the tip, dipoles will get away from vertical direction so that the net polarization decreases [8]. No hysteresis is observed, because the orientation of the fixed dipoles cannot be altered by any external electric field. To study the linearity and the piezo-electric nature of the response, the signal dependency to the frequency under various a.c. voltages is investigated. Figure 3(c) shows the amplitude of PFM signal versus different a.c. voltages at the cantilever fundamental resonance. The cantilever on graphene (HGO) resonates at a frequency of about 112 kHz. The inset of figure 3(c) shows the linear behaviour of amplitude versus a.c. voltage up to 2 V, which demonstrates the piezoelectricity of HGO layer (since no hysteresis is observed).

The governing physics interpreting the piezoelectric response of the hydrogenated graphene oxide is that by transferring the graphene layer onto SiO₂/Si substrates, carbon atoms of graphene interacts chemically with the terminated oxygen atoms of SiO₂, and C–O bonds are formed. Because of the electronegativity of oxygen, C–O bonds are polar and a net dipole moment in the system forms which in turn induces an anisotropic strain. By functionalizing the surface of graphene with hydrogen atoms, C–H bonds are formed with free carbon atoms of graphene that did not interact with oxygen atoms.

According to [6], if graphene is doped with adatoms like hydrogen, exerting an equi-biaxial in-plane strain to graphene induces a change in the polarization and increases the polarization of C–H bonds and the net polarization increases (in this case, hydrogenated graphene oxide, HGO). By applying an electric field with the aids of the AFM tip, the C–O and C–H dipoles re-orient to become closer to the normal direction and a PFM signal is detected.

Based on the mirror charge method calculations to obtain the electric field 'E', in the carbon oxide layer, the average uniform electric field in the hydrogenated graphene oxide layer below the tip is (supplementary information of [8]):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E\approx \frac{{{\varepsilon }_{{\rm Si}{{{\rm O}}_{2}}}}}{{{\varepsilon }_{{\rm HGO}}}}\frac{{{V}_{{\rm tip}}}}{{{R}_{0}}}\nonumber \end{align} \tag{ 1 }$$

where ${{\varepsilon }_{{\rm Si}{{{\rm O}}_{2}}}}=3.9$ is the SiO₂ dielectric constant [43] and according to our simulations (refer to supplementary information (stacks.iop.org/JPhysD/53/175303/mmedia)), the dielectric constant of hydrogenated graphene oxide stack is ${{\varepsilon }_{{\rm HGO}}}=4.7$. In equation (1), the applied voltage on the tip is ${{V}_{{\rm tip}}}=1\,{\rm V}$ and ${{R}_{0}}=30\,{\rm nm}$ is the radius of the AFM tip. The higher value of the dielectric constant in HGO structure compared to that of GO could be explained by extra bond between C and H in HGO that permits electrons to polarize along the bond more freely than in GO. This is in agreement with the numerical results as will be shown later, an increase of the polarizability is expected.

To obtain the piezoelectric coefficient d₃₃, by using the vertical strain induced by the intrinsic (${{\mathcal{P}}_{{\rm s}}}$) and the inductive (${{\mathcal{P}}_{{\rm i}}}$) polarizations in the presence of direct and alternating external electric fields, we derive the displacement amplitude normal to the surface detected by the PFM and calculate the result using the measured data. Induced strain can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\mathcal{S}}_{33}}={{G}_{33}}{{\mathcal{P}}^{2}}\nonumber \end{align} \tag{ 2 }$$

where $\mathcal{P}={{\mathcal{P}}_{{\rm s}}}+{{\mathcal{P}}_{{\rm i}}}={{\mathcal{P}}_{{\rm s}}}+\left( \varepsilon -1 \right){{\varepsilon }_{0}}E$. By casting this relation in equation (2) and excluding the electrostatic term (${{G}_{33}}\mathcal{P}_{{\rm s}}^{2}$) and by using ${{d}_{33}}=2{{G}_{33}}{{\mathcal{P}}_{{\rm s}}}\left( \varepsilon -1 \right){{\varepsilon }_{0}}$:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\mathcal{S}}_{33}}={{d}_{33}}E+{{G}_{33}}{{\left( \varepsilon -1 \right)}^{2}}\varepsilon _{0}^{2}{{E}^{2}}.\nonumber \end{align} \tag{ 3 }$$

The applied electric field consists of direct ${{E}_{{\rm d}}}$ and alternating ${{E}_{{\rm a}}}$ components. By insertion of these components into equation (3) and discarding static terms (${{d}_{33}}{{E}_{{\rm d }~ }},{{G}_{33}}{{\left( \varepsilon -1 \right)}^{2}}\varepsilon _{0}^{2}E_{{\rm d}}^{2}$) and leaving only first order components depending on the electric field ${{E}_{{\rm a}}}$, the equation (3) will be:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\mathcal{S}}_{33}}=\left( {{d}_{33}}+2{{G}_{33}}~{{\left( \varepsilon -1 \right)}^{2}}\varepsilon _{0}^{2}{{E}_{{\rm d}}} \right){{E}_{{\rm a}}}.\nonumber \end{align} \tag{ 4 }$$

Considering the alternating electric field in the form of ${{E}_{{\rm a}}}={{E}_{0}}{\rm Cos}\left( \omega t+{{\varphi }_{0}} \right)$, vertical to the surface displacement being detected by PFM, is expressed in the form of $\mathcal{Z}_{3}^{{\rm a}}={{\mathcal{Z}}_{1\omega }}{\rm Cos}(\omega t+{{\varphi }^{\prime }})$, and it is related to the vertical strain as $\mathcal{Z}_{3}^{{\rm a}}\approx h{{\mathcal{S}}_{33}}$ which $h$ is the thickness of hydrogenated graphene oxide layer (HGO). So the first harmonic amplitude can be obtained as;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\mathcal{Z}}_{1\omega }}\approx h\left( {{d}_{33}}+2{{G}_{33}}{{\left( \varepsilon -1 \right)}^{2}}\varepsilon _{0}^{2}{{E}_{{\rm d}}} \right){{E}_{0}}.\nonumber \end{align} \tag{ 5 }$$

Assuming ${{E}_{{\rm d}}}=0$, it leads to ${{d}_{33}}=\frac{{{\mathcal{Z}}_{1\omega }}}{h{{E}_{0}}}$, and by entering the electric field from equation (1), which is now only an alternating field, the longitudinal piezoelectric coefficient takes the form,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{d}_{33}}\approx \frac{{{\varepsilon }_{{\rm HGO}}}}{{{\varepsilon }_{{\rm Si}{{{\rm O}}_{2}}}}}\frac{{{R}_{0}}}{h}\frac{{{\mathcal{Z}}_{1\omega }}}{{{V}_{{\rm tip}}}}\nonumber \end{align} \tag{ 6 }$$

where ${{\mathcal{Z}}_{1\omega }}$ is the amplitude of the first harmonic and the magnitude of a.c. electric component (E_a.c.) is E₀. The thickness of hydrogenated graphene oxide layer is $h=3.2\,{{\rm }\mathring{\rm A} }$ (the bonding length of carbon–hydrogen bond is about 1.09 Å [44] and for carbon-oxygen bond is about 1.2–1.5 Å [45]). At low frequencies (~20 KHz), the values ${{\mathcal{Z}}_{1\omega }}\cong 0.019\,{\rm nm}\,{{{\rm V}}^{-1}}\,{\rm and}\,0.015\,{\rm nm}\,{{{\rm V}}^{-1}}$ are obtained from figure 3(c), at zero d.c. bias for HGO and GO stacks, respectively. By substituting these values in equation (6), the piezo-electric coefficients are obtained as ${{d}_{33}}=2.14\,{\rm nm}\,{{{\rm V}}^{-1}}\,{\rm and}\,1.65\,{\rm nm}\,{{{\rm V}}^{-1}}$, respectively, which are about two times higher than the value obtained for the graphene transferred onto the Si/SiO₂ substrate (GO stack) and about  ×  1.5 the ${{d}_{33}}$ value reported for graphene on SiO₂ grating substrate. This value are significantly larger than piezo-electric ceramics of the lead zirconate titanate family.

In this section, the results of our computational modelling for the piezoelectric behaviour of the H-C-SiO₂ structure is discussed. The piezoelectric effect enhancement, resulting from the co-adsorption of oxygen atoms on a monolayer graphene is first investigated. In the hydrogen/graphene/SiO₂ structure, hydrogen atoms are assumed to be adsorbed on the surface of the graphene sheet, while oxygen atoms of SiO₂ surface are adsorbed on the other side to form a H-Graphene-O Janus structure. The side and top views of the atomistic configuration of H-C-SiO₂ are shown in figure 4, where x, y , and z are denoted as the zigzag, armchair and out-of-plane directions of the graphene layer, respectively. The optimum H-C-SiO₂ structure with the least total energy is determined by choosing a supercell as can be seen in figures 4(a) and (d) with the lattice constants of a  =  4.53 Å and b  =  5.12 Å along the X and Y directions. To model the structure as a quasi-2D martial, the out-of-plane lattice constant is equal to c  =  49.43 Å, being more than twice the thickness of the structure which is 22.24 Å.

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Now, to evaluate the piezoelectric behavior of the structure, the piezoelectric coefficients are calculated. The piezoelectric coefficients are generally described by the full third-rank piezoelectric tensors as ${{e}_{ijk}}$ and ${{d}_{ijk}}$. Due to the crystal symmetry, in the Voigt notation, the mentioned third-rank tensors are reduced to ${{e}_{il}}$ and ${{d}_{il}}$ tensors, with l  =  1..., 6. Accordingly, ${{e}_{33}}$ and ${{d}_{33}}$ as vertical tensors are defined as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{e}_{33}}=\frac{\partial {{P}_{3}}}{\partial {{\varepsilon }_{33}}}=\frac{\partial {{\sigma }_{33}}}{\partial {{E}_{3}}}\nonumber \end{align} \tag{ 7 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{d}_{33}}=\frac{\partial {{P}_{3}}}{\partial {{\sigma }_{33}}}=\frac{\partial {{\varepsilon }_{33}}}{\partial {{E}_{3}}}\nonumber \end{align} \tag{ 8 }$$

where, ${{P}_{3}}$, ${{\varepsilon }_{33}}$, ${{\sigma }_{33}}$ and ${{E}_{3}}$ denote surface polarization, strain, stress and macroscopic electric fields along the out-of-plane direction, respectively. To calculate the piezoelectric coefficients ${{e}_{33}}$ and ${{d}_{33}}$, it is necessary to compare the developing dipole moments originated by the exertion of the strain along the out-of-plane uniaxial direction from the initial polarization moment of the undistorted structure (the unstrained relaxed structure with the minimum total energy). Using Berry phase expressions implemented in VASP, the total contribution of the ionic and electronic dipole moments change due to the strain are calculated [16–18]

Then the obtained dipole moment through a volume consisting of the unit cell area and a specific effective thickness (d_eff) as shown in figure 4(a), is turned into the three dimensional (3D) polarization in which the distortion of ions effectively gives rise to polarization modulation.

The compressive and tensile strains are modeled by lowering and heightening the upper hydrogen atoms to a fixed out of plane atomic coordinate respectively, while; the atomic positions of the structure are allowed to relax in all directions. Appling an array of out-of-plane strains from  −4% (compressive strain) to  +4% (tensile strain) on the H/C/SiO₂ structure shown in figure 4(a), reveals that lower oxygen atoms closest to the graphene layer are the only ions of SiO₂ substrate being distorted by the initial strain (next group of oxygen atoms stayed undistorted), thereby the polarization change is mostly originated from the distortion of the relaxed ions along the distance between the upper hydrogen and lowermost oxygen atoms closest to the graphene layer. This distance is denoted as the effective distance ${{d}_{{\rm ef\,\!f}}}$ being equal to 3.214 Å for the undistorted condition. Therefore, the effective strain as a post relaxation computation is obtained by $\frac{\Delta {{d}_{{\rm ef\,\!f}}}}{{{d}_{{\rm ef\,\!f}}}}$, in which $\Delta {{d}_{{\rm ef\,\!f}}}$ is the amount of effective thickness modulation and the ${{d}_{{\rm ef\,\!f}}}$ is the undistorted effective thickness.

Finally, the stress corresponding to each exerted strain is computed. So, under an out-of-plane strain, in addition to the ionic relaxation, the volume and shape of the unit cell are first relaxed. For a precise evaluation, the lattice vector along the z-direction is fixed, whereas the length of lattice vectors along the x and y  directions are altered iteratively, so that the in-plane stress becomes smaller than 10–4 Gpa. Then the sum of the forces on the hydrogen atoms is extracted as the uniaxial out-of-plane force exerted on the unit cell corresponding to each strain.

To compute the piezoelectric coefficients e₃₃ in equation (7), it is required to evaluate 3D polarization modulations $\partial {{P}_{3}}$ under an array of exerting strains. Figure 5, depicts the extracted 3D polarization modulation versus the applied compressive and tensile strains computed by the presented DFT procedure. As can be seen in figure 5, the H-C-SiO₂ structure performs two different piezoelectric responses for small and large values of strain. Also it can be seen that under smaller strains from  −0.025 of compressive to 0.015 of tensile strains, the 3D polarization modulation is less than its piezoelectric response for larger strains. It can be indicated that the piezoelectric behavior of the structure largely depends on the distortion of the symmetric configuration of the structure. Due to the rigid chemical bonding between graphene layer and the hydrogen and oxygen atoms, by small out-of-plane strains, the symmetric structural configuration is retained more than the large exerted strains. Consequently, the modulation of ionic dipole and electronic moments resulting from the small displacement of the ions would not be as much as large strains.

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Furthermore, the abrupt rise of polarization modulation at the 0.015 tensile strains compared to the  −0.025 compressive strains can be seen in figures 5 and 6 which can be attributed to higher sensitivity of the H-C-SiO₂ material to tensile strains. On the other side, as can be seen in figure 5, further increase of tensile and compressive strains more than 0.015 and  −0.025, decreases the polarization modulation. This contrary behavior is correlated to the decrease of ionic and electronic dipole moment modulations resulting from the electronic transition of the H-C-SiO₂ material under an out-of-plane strain. As both the compressive and tensile strains are increased, the bandgap of the graphene layer narrows more and finally, a semiconductor to metal phase transition occurs. The bandgap of a buckled 2D material decreases with the exertion of an out-of-plane compressive uniaxial strain. Due to the co-adsorption of hydrogen and oxygen at the surface of graphene, the layer is buckled enough, being subject to semiconductor to metal phase transition under the mentioned strain.

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For the evaluation of piezoelectric coefficients d₃₃ by the equation (8), 3D polarization modulation $\partial {{P}_{3}}$ under the out-of-plane stress must be extracted. As explained before, the stress corresponding to each strain can be computed. Figures 6(a) and (b) show the 3D polarization changes versus stress for both small and larger values of stress. Now by using 3D polarization modulations under the exerted strains and their corresponding stresses, one can calculate the ${{e}_{33}}$ and ${{d}_{33}}$ as shown in figures 5(a), (b) and 6(a), (b). The slopes of 3D polarization modulation plots versus strain and versus stress in figures 5 and 6(a), (b) are denoted as e₃₃ and ${{d}_{33}}$, respectively. Due to the different piezoelectric response of the structure for different strain values, two different ${{e}_{33}}$ and ${{d}_{33}}$ coefficients as 643 C m⁻² and 1.831 C m⁻² and 1.7 nm V⁻¹ and 5.1 pm V⁻¹ were computed for small and large strain regimes, respectively.

To compare the results of the theoretical modeling with the experimental physical findings, the ${{e}_{33}}$ and ${{d}_{33}}$ coefficients were computed by differentiating the distorted (strained) polarization with the undistorted (unstrained) condition for each applied strain. The computed results for some of the small and large strains are collected in table 1.

Table 1. Piezoelectric coefficients d₃₃ and e₃₃.

Strain [%]	Stress [GPa]	${{e}_{33}}$ (C m−2)	${{d}_{33}}$ (nm V−1)
−0.04	−15.5	530	1.36
−0.035	−13.5	608	1.57
−0.025	−8.85	854	2.4
−0.02	−7.43	1.7	4.7  ×  10−3
0.005	1.82	3.82	1  ×  10−3
0.01	3.28	2.19	6.7  ×  10−3
0.015	5.15	1381	4.02
0.025	8.46	822	2.43
0.035	13.4	583	1.51

Our procedure to relax the H-O-SiO₂ (assuming crystalline SiO₂) as can be seen in the figure 4, reveals the presence of C–O and C–H bonds. This is consistent with the XPS experimental measurements as shown in figure 2. The theoretical and experimental results extracted in this work are compared with some other works in table 2. As seen in table 2, the extracted theoretical result for ${{d}_{33}}$ is close to that of the experimental one in this work. Both show the same order of magnitudes in the range of a few nm V⁻¹ with about 0.7 nm V⁻¹ deviation. This difference might be attributed to some usual non-idealities in the experimental materials, procedure and the environmental conditions. For instance, for the DFT calculations we assumed a crystalline SiO₂ and the graphene layer with no defects or impurity. Also to model a quasi-2D structure a vacuum space was assumed on the upper area of the 2D structure, while in the experimental work, such condition is not met and the measurement is performed in an air ambient.

Table 2. Piezoelectric coefficients of some reported 2D structures.

Structures	${{e}_{33}}$ (C m−2) (theoretical)	${{d}_{33}}$ (pm V−1) (theoretical)	${{d}_{33}}$ (pm V−1) (experimental)
This work (H-Si-SiO2)	643, 1.831	1703, 5.1	2146
Graphene on SiO2 [8]	—	—	1400
PZN-8%PT [47]	—	—	2500
PbTiO3 single crystal [48]	—	—	143
MoSSe, MoSeTe, MoSTe, WSSe, WSeTe, WSTe [43]	0.352, 0.428, 0.835, 0.294, 0.374, 0.752	5.248, 6.217, 10.575, 5.319, 6.710, 9.279

According to table 2, both the extracted theoretical and experimental results reveal that the studied structure performs a significant piezoelectric behavior comparable to best bulk piezo-layers such as polycrystalline materials and single crystals of relaxor perovskite.

Table 2, also shows the calculated out-of-plane piezoelectric coefficients of mono and few layers MXY (M  =  Mo or W, X/Y  =  S, Se, or Te) as an uttermost recent theoretical study [46]. The structure studied in this work possesses piezoelectric coefficients that are almost three orders of magnitude larger than that of MXY structures under large values of strain. Even under small values of strain, the studied H-C-SiO₂ structure exhibits piezoelectric coefficients comparable to that of MXY structures.

Such a high piezoelectric response of our structure compared to MXY in [46] is attributed to the higher efficacy of different types of oxygen and hydrogen ions for the increment of dipole moments. It is similar to the enhancement of the piezoelectric response which has been reported in the [8]. In [8] the graphene layer on the SiO₂ substrate has the ${{d}_{33}}$ equal to 1.4 nm V⁻¹. On the other hand in [17] it is shown, that the pristine graphene and the circular porous graphene nanoribbon does not exhibit any piezoelectric behavior, while graphene sheets containing triangular holes have a piezoelectric coefficient of about 0.124 C m⁻². Compared to our structure, the studies depict that the co-adsorption of different ionic atoms on the graphene sheet is more effective and an easier approach than an arbitrary intentional alteration of the structure to bring sufficient dipole moment modulations. For example, the work presented in [49] depicts that clamped-ion and relaxed-ion of 2D transition metal dichalcogenides (TMDCs) benefit from the 2D in-plane d11 piezoelectric coefficient from 0.60 pm V⁻¹ to 4.60 pm V⁻¹. Although compared to [49], our structure may benefit from the higher piezoelectric coefficients. More investigation on the out-of-plane piezoelectric coefficients of clamped-ion and relaxed-ion of 2D TMDCs is required to fairly compare the structures.