Elastic modulus scaling in graphene-metal composite nanoribbons

Pd is used as the metal substrate for graphene-metal composite nanoribbons for the following reasons: (i) Pd is an excellent catalyst, has a high melting point and high solid carbon solubility which makes it suitable for segregation-driven graphene synthesis. These properties allow us to fabricate high-quality graphene via a rapid CVD process while avoiding solid-state dewetting which challenges the stability of thin films at high temperatures [17]. (ii) Graphene-Pd has large lattice mismatch (~3.3% theoretically-predicted lattice mismatch compressive strain on Pd (111) surface) [31] and large thermal expansion mismatch (Pd: $11.8\, \times {10^{ - 6}}\,{K^{ - 1}}$ and Gr: $\left( { - 8.0\, \pm 0.7} \right) \times {10^{ - 6}}\,{K^{ - 1}}$) [32]. Figure 1 illustrates the fabrication processes of freestanding PdGr composite nanoribbon for microbridge testing. A series of 4 µm-wide, 20 µm spaced thin Pd nanoribbon arrays with the thickness ranging from 36 nm to 300 nm are sputtered on a SiO₂ (300 nm)/Si substrate with deposition rate of 0.7 Å s⁻¹ at room temperature, as sketched in figure 1(a). The Pd nanoribbons are heated up to 550 °C and annealed for 3 h in helium environment. The annealing step leads to uniform columnar grain structures [16]. We use PMMA as a carrier layer to transfer the Pd nanoribbon arrays on a SiO₂/Si transmission electron microscopy (TEM) grid [33]. Notably, Pd films can readily delaminate from SiO₂ surface by dipping into 0.3 vol % hydrofluoric acid solution, due to the weak Pd-SiO₂ interfacial bonding. The TEM grid is made of SiO₂/Si and has a through slot with a 110 µm wide gap. The transferred Pd nanoribbon arrays are aligned and cross over the slot (figure 1(b)). After transfer, the PMMA carrier layer can be removed by Ar-O₂ reactive ion etching (RIE), leaving the suspended ultrathin Pd nanoribbons on the slotted TEM grid. The whole sample is then loaded in a high temperature furnace (1100 °C) for graphene synthesis using a rapid CVD process, which has been developed in our previous study (figure 1(c)) [17]. Scanning electron microscope (SEM) images in figures 1(d) and (e) present the uniform morphologies of as-grown PdGr nanoribbons suspended over the TEM grid. Table S1 in the supplementary information (SI) (stacks.iop.org/JphysD/53/185305/mmedia) lists the detailed dimensions of the nanoribbons. We confirm that graphene coating wraps around freestanding Pd nanoribbons using TEM and Raman spectroscopy, as shown in figures 1(f) and (g). TEM images show that we get continuous bilayer graphene coating the Pd, and they conformably wrap the Pd surface. In some localized regions we see trilayer graphene, but they are not continuous and their contribution to the overall behavior is negligible. At the same spot shown in figure 1(f), the characteristic Raman feature, the so-called G peak, lies at 1573 cm⁻¹ (405 nm laser excitation) confirming the sp² hybridized carbon on Pd nanoribbon. Notably, we also use 532 nm laser excitation for Raman spectroscopy analysis, whereas it has large spot size which includes PdGr nanoribbon edge effect (see figures S1(a) and (b)). G Raman frequency can be also observed on the backside of freestanding PdGr ribbon, as shown in figure S1(c). These suggest as-grown graphene effectively wraps the Pd nanoribbon surface. In addition, we also transfer as-grown graphene from Pd surface onto a clean SiO₂ substrate, and confirm the bilayer nature of graphene by Raman spectrum characterization (see figure S1(d) and S1(e) in SI).

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We developed and validated a model to precisely determine the elastic modulus of nanoribbons in a non-ideal microbridge geometry. Freestanding Pd nanoribbons shown in figure 1(b) usually have some slack on the TEM grid resulting during transfer. We measured the 3D profiles of freestanding Pd nanoribbons using a confocal laser profilometer (Keyence VK-X1000), as displayed in figure 2(a). Figure 2(b) shows a typical Z-profile of a slack bare Pd nanoribbon. A wedge indenter (200 ± 50 nm tip radius, 15 µm edge, 30° defining angle, Micro Star Tech.) applies line load in the middle of a freestanding nanoribbon for the microbridge testing. Notably, stresses produced during the CVD synthesis remove the slack and result in slightly taut nanoribbons after graphene growth, possibly due to thermal expansion mismatch between Pd and the SiO₂/Si substrate, as well as diffusion of Pd into the substrate at high temperature. Figures 2(c) and (d) compare the 3D and Z-profiles of as-grown PdGr nanoribbons with relatively taut morphology. Straining the slack Pd nanoribbons during nanoindentation can cause a displacement drift ($\Delta h$) in load-displacement ($P - h$) which can effectively soften the apparent stiffness during data analysis. This measurement uncertainty is nontrivial but frequently ignored in testing of freestanding thin films. In this study, we consider such displacement drift in microbridge nanoindentation model and develop a numerical analysis method to extract the elastic properties from indentation data properly.

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Figure 2(e) presents the typical $P - h$ curves from nanoindentation measurements on a Pd nanoribbon and a PdGr nanoribbon. Deformation of the slack geometry causes a noticeable hysteresis in the first one or two cycles of loading and unloading curves for Pd, however, this is negligible for PdGr. This hysteresis can add uncertainty in displacement measurement and affect the fitting results. To minimize such measurement uncertainty, we applied cyclic loadings on the same Pd ribbon, as shown in figure 2(f). After the first loading cycle (the black line), the subsequent loading curves almost coincide which indicates that the slack nanoribbon adjusts itself perpendicularly to the indenter edge and the nanoribbon deflection becomes normal.

The elastic properties of a freestanding pre-strained nanoribbon can be extracted from the $P - h$ behaviors using the reported indentation models [25]. However, in our case, the softening effect caused by specimen slack needs to be taken into consideration. We analyze the deformation of a freestanding slack Pd nanoribbon, as sketched in figure 3(a). The dark black line represents the Z-profiles of a Pd nanoribbon before CVD synthesis. The slack of the bare Pd nanoribbon is ${h0} = 0.8\:\:\mathrm{\mu m}$. After cyclic loading, Pd nanoribbon adjusts itself to the load applied by a wedge indenter and the slack geometry becomes straightened. Depending on the initial slack geometry, a Pd nanoribbon is subjected to flexural and tension deformation during straightening. The light gray line ${l_1}$ in figure 3(a) illustrates how the nanoribbon becomes taut with a deflection ${h_1}$, and from this point onward, stretching along the ribbon length direction dominates the nanoribbon load-deformation. As indentation proceeds, the nanoribbon deflects to ${h_2}$, as shown in figure 3(a) line ${l_2}$. Previous studies have shown that, except for a very small range of initial deformation, the bending deformation does not affect the calculation of the elastic properties in microbridge testing and can be ignored for ultrathin film materials [23, 25]. In this study, we focus on the nanoribbon's deflection from ${h_1}$ to ${h_2}$ with the following assumptions: the bending moment can be ignored, nanoribbon is stretched and elastic during indentation, nanoribbon is effectively clamped on a rigid substrate without sliding. This can be confirmed from the negligible difference among the cyclic loading curves in figure 2(f). A new analysis is carried out to accurately extract the modulus from the nanoindentation load displacement of nanoribbon having slack, and more details are presented in the SI. In brief, the load-displacement behavior of a slack Pd nanoribbon can be expressed as:

$$\begin{equation}{P_2} = \left( {\frac{{8AE}}{{l_0^3}} - \frac{{8A{\sigma _1}}}{{l_0^3}}} \right)h_2^3 + \left( { - \frac{{4AE}}{{{l_0}}} + \frac{{4AE}}{{{l_1}}} + \frac{{4A{\sigma _1}}}{{{l_0}}}} \right){h_2}; \end{equation} \tag{ 1 }$$

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Notably, due to the slack, ${h_2}$ is greater than the real indentation-induced displacement of the Pd nanoribbon as shown in figure 3(a). In addition, there is a preload in nanoindentation for zero-deflection point detection. In this study, the preload is around 2 µN. ${h_2}$ can be expressed as the sum of the measured deflection data and slack height ${h_0}$ as well as the pre-deflection ${h_{pre}}$.

$$\begin{equation}{h_2} = h + {h_0} + {h_{pre}} = h + {h_1}\!;\end{equation} \tag{ 2 }$$

$$\begin{equation}{P_2} = P + {P_{pre}}\!;\end{equation} \tag{ 3 }$$

Based on this analysis, instead of the cubic-linear relation in the membrane model reported in previous studies [23, 25]. here we use a full 3rd order polynomial function (equation (4)) to analyze the $P - h$ data:

$$\begin{equation}P = {f_1}{h^3} + {f_2}{h^2} + {f_3}h + {f_4};\end{equation} \tag{ 4 }$$

which gives precise and repeatable results of the elastic moduli taking into consideration the non-ideal initial slack geometry.

The material system selected for this study has strong advantages for mechanical load transfer. It is known that Pd strongly interacts with graphene as also manifested by the small separation between the Pd and C atoms (0.23–0.25 nm on Pd (111) surface), a value smaller than the interlayer separation in graphite and smaller than the separation between graphene and other transition metals [34, 35]. The interfacial mechanics is governed by in-plane strain mismatch between graphene and Pd (111) for instance is ~3.3% in compression [31]. The sputtered Pd thin films after annealing are predominantly (111), see electron backscattering diffraction (EBSD) analyses in figure S9. Moreover, due to mismatch between the thermal expansion coefficients of graphene and the seed metal, graphene is usually in strain after synthesis.

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The interfacial strain in as-grown PdGr nanoribbons can be characterized using Raman spectroscopy because it affects the orbital hybridization of the carbon atoms hence it shifts the G and 2D peak frequencies. Figure 4(a) shows the Raman spectroscopy of PdGr nanoribbons clamped on a TEM grid and used for indentation. We observe two types of Raman spectra. Firstly, on the clamped region where the PdGr is supported on SiO₂/Si TEM grid, Raman spectroscopy shows single sharp G peak at ~1606.3 cm⁻¹ and 2D peak at ~2695.4 cm⁻¹. Whereas, there is an obvious red shift in G mode frequency to ~1551.6 cm⁻¹ for the freestanding region on the same Pd nanoribbon. It is known that this Raman shift is due to not only the strain in graphene but also the doping effect from metallic substrates [36, 37]. We can extract the strain effect in graphene by isolating the Pd substrate effect from Raman frequencies correlation, as shown in figure 4(b). There is an offset in the correlation of Raman ${\omega _G} - {\omega _{2D}}$ frequencies between the pristine graphene (with no strain no doping) and as-grown graphene on the Pd substrate. It has been investigated that graphene supported on strongly interacting substrates, e.g. Ni and Pd [34, 35], can be doped and show blue shifts in Raman frequencies compared to the pristine graphene [37]. The purple line in figure 4(b) represents the ${\omega _G} - {\omega _{2D}}$ correlation of graphene with different doping levels but in the same strain state. As a result, graphene grown on the supported Pd area is slightly compressed. This can be attributed to the lattice mismatch strain (${\varepsilon _{mis}}$) between graphene and Pd, as well as the polycrystalline substrate texture [30]. Notably, there is obvious red shift in ${\omega _G} - {\omega _{2D}}$ correlation of graphene grown on the freestanding region of a PdGr nanoribbon, indicating that graphene is stretched in these region [36]. The apparent slope in figure 4(b) is the strain slope, where the data points at the bottom left corner are under the largest tensile mismatch strains. This agrees with the straightening of the nanoribbons and the associated decrease in slack observed in the 3D profile measurements. The exact absolute value of the strain cannot be precisely determined from these measurements since it requires calibration to the Raman spectrum of graphene sitting epitaxially on Pd but without any mismatch strain. We use the results of figure 4 to qualitatively demonstrate the tensile strains in the graphene in the suspended nanoribbons.

Using these insights, we analyze the contributions of the graphene and the interfacial mismatch strains on the measured moduli of PdGr. We adopt a new 3D/2D rule of mixture approach that treats the PdC as a 3D 'bulk' material, while treating the graphene as a 2D 'surface'. The advantage of this approach is it allows us to use the reported properties of 2D modulus of graphene and other nanomaterials without assumptions on the graphene thickness. The number of layers of graphene can simply be incorporated in the model. Further, the critical role playes by the nonlinear modulus component of the graphene (softening) is considered especially given the existence of strain mismatch between the graphene and the nanoribbon. The tension per unit width in the deflected nanoribbons is

$$\begin{align}N & = {E_{PdC}}{t_{PdC}}\left( {\varepsilon - {\varepsilon _{mis}}} \right) + 2nE_{Gr}^{2D}\left( {{\varepsilon _{mis}} + \varepsilon } \right)\nonumber\\[2pt] & + 2nD_{Gr}^{2D}{\left( {{\varepsilon _{mis}} + \varepsilon } \right)^2} + {\sigma _r};\end{align} \tag{ 5 }$$

where $N$ is the tensile force per unit width during indentation, ${t_{PdC}}$ is the nanoribbon thickness, $n$ is the number of graphene layers (for example $n$ for bilayer graphene), here we ignore the interlayer sliding in few-layer graphene and assume each layer effectively interacts with the underlying Pd substrate. This is a suitable assumption due to the small strains applied. $\varepsilon$ is the uniaxial tensile strain in PdGr composite nanoribbon during nanoindentation, ${\sigma _r}$ is the residual stress in the nanoribbon due to stretching over the trench during growth, and the factor 2 in equation (5) accounts for the top and bottom surfaces of the nanoribbon. Here, we ignore the thickness of graphene layer and introduce the 2D elastic constants $E_{Gr}^{2D}\, = \,340\:\:{\rm{N}}\,{{\rm{m}}^{ - 1}}$. Since molecular dynamic simulations and experiments suggest that graphene shows quadratic dependence on strain [13, 38], we consider the nonlinear elastic behavior of as-grown graphene layer on Pd nanoribbons with a third-order elastic constant $D_{Gr}^{2D}\, = \, - 690\:\:{\rm{N}}\,{{\rm{m}}^{ - 1}}$. The nonlinear elastic constitutive behavior of graphene can be expressed as ${\sigma _{Gr}} = E_{Gr}^{2D}\varepsilon + D_{Gr}^{2D}{\varepsilon ^2}$ [13]. We define the volume fraction $\nu \, \equiv \,\frac{{{t_{PdC}}}}{t}$. Straining of the Pd surface by graphene synthesis could introduce the surface stress, which can contribute to the measured modulus when the nanoribbon is strained due to the quadratic term.

In this study, we use very small nanoindentation strains so that $\varepsilon \, \ll \,{\varepsilon _{mis}}$. As a result, we can expand equation (5) and ignore the resulting ${\varepsilon ^2}$ term, giving:

$$\begin{align}N & = {\sigma _r} + 2nE_{Gr}^{2D}{\varepsilon _{mis}} + 2nD_{Gr}^{2D}\varepsilon _{mis}^2 - {E_{PdC}}{t_{PdC}}{\varepsilon _{mis}} \nonumber\\ & \,+ \left( {{E_{PdC}}\nu + \frac{{2nE_{Gr}^{2D} + 4nD_{Gr}^{2D}{\varepsilon _{mis}}}}{t}} \right)t\varepsilon; \end{align} \tag{ 6 }$$

where the interfacial strain due to growth cancels out $2nE_{Gr}^{2D}{\varepsilon _{mis}} + 2nD_{Gr}^{2D}\varepsilon _{mis}^2 - {E_{PdC}}{t_{PdC}}{\varepsilon _{mis}} = 0$. The term in the bracket of equation (6) ${E_{PdC}}\nu + \frac{{2nE_{Gr}^{2D} + 4nD_{Gr}^{2D}{\varepsilon _{mis}}}}{t}$ is then equivalent to the measured elastic modulus of PdGr composite nanoribbon as it depends on the applied indentation strain $\varepsilon$. Therefore,

$$\begin{equation*}{E_{PdC}}\nu + \frac{{2nE_{Gr}^{2D} + 4nD_{Gr}^{2D}{\varepsilon _{mis}}}}{t} = {E_{PdGr}};\end{equation*}$$

$$\begin{equation} \Rightarrow {E_{PdGr}} - \nu {E_{PdC}} = \frac{{2nE_{Gr}^{2D} + 4nD_{Gr}^{2D}{\varepsilon _{mis}}}}{t}.\end{equation} \tag{ 7 }$$

A couple of insightful conclusions can be drawn from this simple analysis: (i) the quadratic modulus of graphene leads to the dependence of the measured modulus on the mismatch strain ${\varepsilon _{mis}}$; and (ii) the term ${E_{PdGr}} - \nu {E_{PdC}}$ can be a directly measured and used to analyze the relative contributions of the graphene modulus $2nE_{Gr}^{2D}$ and the mismatch strain $4nD_{Gr}^{2D}{\varepsilon _{mis}}$ on the nanoribbon elastic response. Figure 5(a) shows the measured moduli of Pd, PdGr and PdC with different film thicknesses. Indeed, experimentally, the elastic modulus of bare Pd does not vary within the thickness range from 36 to 300 nm in this study. On the other hand, as discussed before, the elastic modulus of PdGr thin films significantly increases after the synthesis of graphene, and this increase is scale dependent. More specifically, the PdGr modulus increases notably as film thickness reduces. This not only stems from the contribution of high in-plane stiffness of graphene in PdGr composite, but is also affected by the graphene nonlinear elastic term $D_{Gr}^{2D}$and the large Pd-graphene interfacial stress. Notably, after etching the graphene layer away, we also release the interfacial elastic energy installed between graphene and Pd. This confirms the nontrivial contribution of the Pd-graphene interfacial stress on the elastic property in PdGr composites.

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Figure 5(b) plots the equation (7) with the measured ${E_{PdGr}}$ and ${E_{PdC}}$. By the least square fitting, $2nE_{Gr}^{2D} + 4nD_{Gr}^{2D}{\varepsilon _{mis}} \approx 1251.1\,J\,{m^{ - 2}}$. This suggests that graphene-induced surface strain from high temperature synthesis can significantly modify thin film materials' elastic modulus. Using the values from [13, 38] and $n$ for bilayer graphene, we find a mismatch strain ${\varepsilon _{mis}}$ (tensile). More generally, it shows that graphene is an effective reinforcement in Pd up to volume fraction of ~0.3%.

Next, we try to study the origin of increase of the PdC modulus (${E_{PdC}}$) by 14.8% compared to that of Pd (${E_{Pd}}$) even when the graphene is removed by etching. Besides the expected interstitial carbon reinforcement effect, we also investigated the formation of palladium carbide-like phase close to the Pd surface after graphene growth. Figure 6(a) shows a cross sectional TEM image of a PdGr thin film which shows the graphene layers as well as the fringes associated with the atomic spacings of the Pd. We measured the Pd lattice fringes near the Pd-graphene interface and noted that they are expanded to about 7.2% in comparison with Pd bulk values (figure S10). This expansion can be related to carbon incorporation into the Pd lattice especially in the sub-surface sites [39]. Importantly, carbon atoms in this carbon-rich layer close to the Pd surface have preferred interstitial sites and can form a carbide like Pd-C phase [40–42]. The formation process of this Pd-C phase is unclear, nevertheless, there is a number of theoretically reported stable Pd-C phases exhibiting very high stiffness and hardness [43]. In light of these considerations, it is reasonable to propose a model with Pd/Pd-C/Gr laminated structure for the grown PdGr composite thin film, as shown in figure 6(b). In order to confirm the existence of Pd-C layer in Pd subsurface, x-ray photoelectron spectroscopy (XPS) is used to track the core level shift in Pd peaks. XPS measurements are made using a Kratos Axis Ultra x-ray photoelectron spectrometer using monochromatic Al Kα radiation (1486.6 eV). The x-ray detection depth is about 10 nm. The binding energies are referenced to the graphitic C 1s signal at 284.4 eV. We tilt the PdGr surface so that more subsurface signal could be collected. Figures 6(c) and (d) present the Pd 3d and C 1s XP spectra of as-grown PdGr thin film. As expected, the carbonaceous species shift the binding energy (BE) of surface Pd atoms to the high BE side of the bulk Pd signal (core level of Pd 3d_5/2~335.1 eV). The BE for this carbon-rich species is found at about 335.7 eV, which is 0.6–0.8 eV shifted from Pd 3d core level. This agrees well with the experimental and calculated results for the sub-Pd surface Pd-C phase [40, 41]. In figure 6(d), the C 1s peak at BE = 284.4 eV is assigned to graphene related peaks. The peak asymmetry toward lower BE with a low intensity peak at ~283.7 eV is a signature of the presence of carbide, which can be related to Pd-C bonds.

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With the presence of the carbide phase near the Pd-graphene interface, we can separate the 3D elastic modulus of PdC into the carbide boundary layer modulus ${E_S}$ and the pure Pd modulus ${E_{Pd}}$ with the rule of mixture as ${E_{PdC}} = {E_S}{\upsilon _S} + {E_{Pd}}\left( {1 - {\upsilon _S}} \right)$, where the surface layer thickness is ${t_S}$ and the volume fraction ${\upsilon _S} \equiv \frac{{{t_S}}}{{{t_{PdC}}}}$. The thickness of ${t_S}$ can be estimated from the fringe spacings in the TEM of figures S6 and S7. For example, for PdGr-4, ${t_{PS}} \approx 3.7\,\rm{nm}$ and ${t_{PdC}} = 189\:\:\rm{nm}$, with the measured ${E_{PdC}} = 116.8 \pm 2.5\:\:\rm{GPa}$. Using the measured value of ${E_{Pd}} = 108.0 \pm 2.6\:\:\rm{GPa}$, we can estimate the modulus for carbide phase about ${E_S} = 332.8\:\:\rm{GPa}$, which is in the range of the reported theoretical results [43]. It is possible that multiple types of stable carbide phases coexist in PdGr thin film. However, this simple analysis leverages the imaging and XPS to rationalize the measured increase in modulus of the nanoribbons even when the graphene layer is etched.