Relationship between changes in interface characteristics and external voltage under compressing force in metal–graphene–metal stacks

A key part of the tested samples was gold-supported monolayer graphene. For the supporting Au, a thin film was deposited by a dc magnetron sputtering method. The Au film thickness was about 100 nm and the root-mean-square (RMS) roughness less than 0.4 nm. The film was grown on a standard Si plate with an insulating SiO₂ layer with a thickness of about d_SiO2 = 275 nm. The samples were produced by wet transfer of a graphene layer on the surface of the Au film. A commercially available chemical vapor deposition (CVD) grown graphene monolayer (Graphenea) was used for this. The transfer method was described in detail in a previous study [24]. The substrate with the graphene was annealed in Ar atmosphere at 300 °C for 124 min before the investigation [22]. In this work, the complete samples consisted of the SiO₂/Si substrate, supporting Au film, a graphene layer and a conductive probe as the top electrode.

The outline of the sample arrangement is illustrated in figure 1(a). In this figure, a schematic is also shown of the electrical measurements combined with the mechanical force curve ramps. The supporting Au film was used as the bottom electrode (M2), whereas the probe of an SPM was the top electrode (M1) in these experiments. The probe also produced a compressing force perpendicular to the surface of the layered structure. Accepting that the graphene was fixed by the van der Waals forces on the Au film, a schematic of the deformations under the compressing force is shown in figure 1(b). In this schematic, the tip of the Pt probe is represented by a ball with the diameter equal to the tip diameter. The interlayer distances due to van der Waals forces are indicated by d_GM1 for the interlayer between the Pt probe (M1) and graphene (Gr) monolayer and by d_GM2 for the interlayer distance between the graphene (Gr) and Au film (M2).

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The force curves were measured using a Dimension 3100 SPM (Veeco Inc.). For this, the force was sequentially changed by pressing the probe into the surface in the approach and retract paths. A typical force curve is illustrated in figure 1(c) by the dependence of the tip–surface interaction force F on the height z equal to the length change of the piezoceramic driver. In the approach path, switching is typically detected for the probe when the non-contact state changes to the contact state for the tip of the probe. The switching corresponds to the height when the long-distance attraction forces exceed the deformation force of the cantilever. At this point, the snap-in force F_snap was measured at the apex of the triangle in the approach curve (figure 1(c)–A, right panel). When the probe was moved back from the surfaces, the tip was in contact with the surface until the deformation of the cantilever exceeded the adhesion forces. The adhesion forces F_adh were measured at the apex of the triangle in the path of the retract probe (figure 1(c)–R, left panel). The maximum compressing force was about F_max ∼ 190 nN at the maximal indentation in the surface. The SPM probe force was calibrated by measuring the force–distance curve on a hard clean surface (reference platinum grating sample (Bruker)) in contact mode. From a linear part of the curve, the cantilever deflection in nanometers was found, and using the nominal value of the cantilever spring constant k, the numerical force value F was calculated by Hooke's law [25].

The tunneling atomic force microscopy (TUNA, Veeco Inc.) module was connected to the SPM for the electrical measurements. An external constant dc voltage V_app was applied between the probe and the Au film by the TUNA during each force curve measurement. Between two sequential measurements, V_app was changed in a stepwise manner so that an individual pattern of the voltage sequences could be associated with a specific sequence of voltages collected from the interval −1.5 V ⩽ V_app ⩽ + 1.5 V. In these experiments, a pure Pt probe (Pt-rock, model RMN-12Pt400B, Bruker) was used as the base potential electrode. The geometry and dimensions of this probe-based electrode were described by a physical image of the probe end obtained from a special reconstruction experiment.

A commercial TipCheck (Budget Sensors) reference plate was used for probe tip reconstruction. The reconstruction of the tip was performed according to the proposed method [26]. The TipCheck plate surface covered by an intentionally produced rough layer was scanned by the probe pressed into the surface by a fixed force. The AFM topography images were analyzed by a tip characterization module in the commercial data processing software package scanning probe image processor (SPIP, Image metrology A/S). As a result, the probe tip was visualized by a 3D image acceptable for measurement of the dimensions. Typical characteristics of the probe tip are illustrated in figure 1(d) by the profile lines of the cross-sections obtained at four individual compressing forces F_load. In the inset, a flat representation of the tip image is visualized. The cross-section was obtained along the arrow in the inset in figure 1(d). According to the methodology [24], the accuracy of the tip visualization depends on F_load. For a comparably small F_load, the interaction between the tip and the sample surface is weak, and the accuracy of the topography measurement is low. Increasing the F_load increases the accuracy of the measurement. It is worth noting that the additional rock of the tip can be seen in figure 1(d). For the forces used in our experiments, this additional rock was 7–20 nm above the main rock of the tip and therefore did not have any influence on the experimental results. The accuracy of the visualization is illustrated by the dependences of the radius of the tip R_probe versus the compressing force F_load shown in figure 1(e). The radius was obtained from the cross-section in figure 1(d) by measuring the width of the tip at a fixed height H_prb-z from the apex of the tip. The dependences in figure 1(e) are presented for two heights H_prb-z equal to 0.3 nm and 1 nm from the apex. These results were used to model the long-range force interaction between the probe and the surface.

More details on the methodology of the investigation of the mechanical and electrical properties in an M–G–M vertical structure were presented in a previous work [22].

Accepting d_GM1 ≪ R_tip in the model, three mechanisms of the interaction forces between the components were included. First, in the classical approach, the electrostatic force between an SPM probe and the conductive surface is typically described as the force between the electrodes in a flat capacitor F_C. Second, at very short distances (<1 nm) a redistribution of electrons cannot be neglected in a metal–graphene system. The extra doping of the graphene results in a short-range force F_tQ in the interaction between the probe and the surface. Third, microcapillary and van der Waals-originated forces F_surf are typically detected due to surface molecular water and technology built-in material features in the SPM experiments. Assuming acceptable the sum of the independent components, the probe–surface interaction can be described by the total force

$$\begin{equation}{F_{{\text{adh}}}} = {F_{\text{C}}}\left( V \right) + {F_{{\text{tQ}}}}\left( V \right) + {F_{{\text{surf}}}}.\end{equation} \tag{ 1 }$$

Due to the mechanisms of the interaction, the first two terms F_C and F_tQ in (1) are dependent on the applied external voltage V.

The force F_C between the electrodes in the plane parallel capacitor can be described by the simple relationship

$$\begin{equation}{F_{\text{C}}}\left( V \right) = \frac{1}{2}\frac{{\partial C}}{{\partial {L_{\text{C}}}}}{V^2} = \frac{{ - 1}}{2}{\varepsilon _{{\text{gap}}}}{\varepsilon _0}\pi {\left( {\frac{{{R_{{\text{tip}}}}}}{{{L_{\text{C}}}}}} \right)^2}{V^2},\end{equation} \tag{ 2 }$$

where C is the capacitance of the capacitor, R_tip is the Pt probe radius, L_C is the distance between the M1 and M2 electrodes and _gap is the dielectric permittivity of the substance in the capacitor. In equilibrium, it can be written

$$\begin{equation}{L_{\text{C}}} = {d_{{\text{MGM}}}} = {d_{{\text{GM}}1}} + {d_{{\text{GM}}2}},\end{equation} \tag{ 3 }$$

where d_MGM is the total thickness of the M–G–M system, and d_GM1 and d_GM2 are the interface distances between the graphene sheet and the top (Pt) and bottom (Au) electrodes, respectively.

The force F_tQ between the Pt probe and the electric charge Q originated from the extra charge in the graphene due to the electrons being transferred between the graphene and the probe. The charge was quantitatively determined by the relative density of the transferred electrons N_gQ for the unit cell of graphene. The force F_tQ can be described by a model based on an electric charge Q in a homogeneous electric field E produced between the electrodes by an external dc voltage. It can be obtained by a simple description:

$$\begin{equation}{F_{{\text{tQ}}}}\left( V \right) = QE = \frac{1}{2}\left( {e\frac{{\pi{\!R}_{{\text{tip}}}^2}}{{{\sigma _{{\text{cell}}}}}}{N_{{\text{gQ}}}}} \right) \cdot \left( {\frac{V}{{{d_{{\text{GM}}1}}}}} \right),\end{equation} \tag{ 4 }$$

where the graphene unit cell volume σ_cell = 5.18 Ų, and e is the electron charge.

The third force term in equation (1), F_surf, is accepted as a constant force component producing a shift of the base signal independent of the applied voltage in an individual force curve measurement cycle. As the F_surf term, by our definition, contains van der Waals-originated forces, all voltage-dependent force components are included in the first and second terms of equation (1), while the components in the third term are voltage-independent.

It follows from equations (2) and (4) that the force terms F_C and F_tQ are related to the parameters d_MG1, d_MG2 and N_gQ that are the characteristics of the interface properties in the M–G–M system. These parameters can be calculated for the M–G–M system in the equilibrium state by the model proposed initially in earlier work and successively adapted to M–G–M systems in a recent work [16, 22]. The phenomenological version of the model was used to perform a quantitative comparison between the model and the experiments. It must be noted here that the acceptability of the model was limited to the weak interaction between graphene and metal layers that was valid for Au and Pt in this work. There was also another model limitation on the energy interval defined by ±1 eV from the Dirac point, within which the Fermi level shift can be digitally analyzed assuming linear dependence of the density of the electronic states. This restriction was acceptable here in the force model analysis because the compression force produced a lower shift in the Fermi level as it was proved for the flat construction of the M–G–M system [22]. The Fermi level shift ΔE_F-eq-MGM important for the present study is illustrated in figure 2(a) by a dependence on the total thickness d_MGM of the M–G–M system in equilibrium. The numerical results of the model analysis were obtained assuming a balance between the components of the M–G–M system defined by the same Fermi level across a stack of flat layers. Therefore, the results in figure 2(a) can be understood as a visualized set of the related parameters (ΔE_F-eq-MGM, d_MGM) defining the equilibrium state in the M–G–M flat layer stack. It follows from these results that the shift was within the limitation |ΔE_F-eq-MGM| < 1 eV if the M–G–M stack was thicker than about d_MGM > 5.1 Å. In addition, the equilibrium state limited splitting into the related interface thicknesses d_MGM= d_GM1 + d_GM2 is graphically presented in figure 2(a) as the dependences of the split components d_GM1 and d_GM2 versus d_MGM. The changes in the M–G–M system due to the applied dc voltage V_app were obtained by modifying the model from previous work and including V_app in the relationship:

$$\begin{equation}{W_{{\text{GM}}}}\left( {{d_{{\text{GM}}}}} \right) = {W_{\text{M}}} - {W_{\text{G}}} - \Delta {V_{\text{c}}} - {V_{{\text{app}}}},\end{equation} \tag{ 5 }$$

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where W_M and W_G are the work functions of the metal and graphene, respectively, and ΔV_C is the barrier height in the metal–graphene contact [16]. Accepting equation (5), the model characteristics were calculated in the interval of the voltages −1.5 V ⩽ V_app ⩽ + 1.5 V. The results of the numerical analysis of the M–G–M system are graphically illustrated by typical dependences in figure 2(b). In the graphics, the amount of electrons N_gQ transferred to (N_gQ < 0) and from (N_gQ > 0) graphene are plotted as a function of V_app for the both graphene–metal contacts under the equilibrium condition of an identical Fermi level in the entire M–G–M stack. The dependences of N_gQ on V_app were sensitive to the changes in the equilibrium distance d_MGM and, consequently, these changes were reliably detected in the interaction force F_adh obtained from the numerical simulation of the model described by equations (1)–(4). The sensitivity of F_adh on the graphene–metal distance d_GM2 is illustrated in figure 3(a).

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In figure 3(a) the dependence of the adhesion force F_adh on the applied voltage V_app is graphically plotted for diverse fixed distances from the interval 2.98 Å ⩽ d_GM2 ⩽ 3.34 Å. It is helpful to note here that a reference M–G–M system was introduced for the detection of the changes in F_adh vs. V_app produced by a change in the selected parameters. The reference system was described by a set of numerical values of the parameters: d_GM2= 3.14 Å, R_tip= 150.0 Å, _gap= 6.5, F_surf= 47.0 nN, V_S= −45.0 mV (table 1). The parameter V_S was interpreted as an additional voltage shift (V= V_app + V_S) originating from uncontrolled extra electrical charges related to doping, irreversible changes and similar non-predictable effects in the system. It is useful to mention that this reference set of parameters led to model forces quantitatively close to the measured forces, with similar essential features of the dependence F_adh vs. V_app. Although the details of fitting the model to the experiment are presented below in this text, here the reference set was used to compare the influence of the adjustable parameters on the results of the model calculations presented in figure 3.

Table 1. Parameters used for fitting the model calculations to the experiment. Graphene–Pt distance d_GM1, graphene–Au distance d_GM2, effective tip radius R_tip, dielectric permittivity of the interface gap _gap, extra force due to the surrounding influence on the tip F_surf, and internal extra potential difference V_S.

Identity	dGM1 (Å)	dGM2 (Å)	Rtip (Å)	gap (a.u.)	VS (mV)	Fsurf (nN)
Figure (3) Reference	2.88	3.14	150.0	6.5	−45.0	34.0
Figure 4(a)-1	3.00	3.34	150.0	6.5	300.0	38.0
Figure 4(a)-2(ref)	2.88	3.14	150.0	6.5	−45.0	38.0
Figure 4(a)-3	2.91	3.19	200.0	6.5	35.0	40.7
Figure 4(b)-1	2.91	3.19	200.0	6.5	35.0	53.0
Figure 4(b)-2(ref)	2.88	3.14	150.0	6.5	−45.0	49.0
Figure 4(d)-1	2.91	3.19	55.0	1.0	0.0	−1.15
Figure 4(d)-2	2.91	3.19	60.0	1.0	0.0	−1.45
Figure 4(d)-3	2.91	3.19	25.0	1.0	0.0	−1.0
Figure 4(d)-4	2.91	3.19	200.0	1.0	0.0	−1.0

It can be understood from the comparison of the model calculations for diverse d_GM2 with that for the reference system in figure 3(a) that a wave-like variation in F_adh vs. V_app is highly sensitive to the changes in the interface distances. This wave-like variation was practically suppressed by a higher than ±6% deviation from the reference system. In addition, an asymmetric form of the dependence was obtained with respect to V= 0. Unexpectedly, the voltage shift V_S produced completely the opposite effect on the dependence F_adh vs. V_app compared to that in figure 3(a). The V_S effect is illustrated in figure 3(b) by a set of calculated dependences F_adh vs. V_app for the model with individual voltage shift from the interval −300 mV ⩽ V_S ⩽ 300 mV.

According to the results of the model simulation in figure 3(c), the sensitivity of the force detection can be significantly increased with an increase in the tip radius from 50 Å to 200 Å. In contrast to that, the force originating from the electrostatic interaction mechanism was noticeably reduced with the increase in dielectric permittivity _gap of the substance between the metal electrodes, as demonstrated by the model calculations graphically presented in figure 3(d). The plotted dependences F_adh vs. V_app were obtained, accepting individual permittivity from the interval 1.0 ⩽ _gap ⩽ 7.0 for the M–G–M stack in the model calculations.

The retrace part of the force curve is directly dependent on the properties of the hard contact between the tip and the surface. Therefore, the pull-off triangle in the SPM force curve (figure 1(c)) was experimentally studied for quantitative analysis of the interaction between the components in the M–G–M system. The force at the triangular apex F_adh was obtained from the measurements with voltage V_app applied between the probe and the graphene-supporting Au film. Typical experimental dependences of F_adh on V_app are graphically presented by the points in figures 4(a) and (b). In the figures, each individual point was detected under constant V_app in an individual force curve cycle of 15 s duration. In order to produce the dependences on the voltage, V_app was changed stepwise between the cycles according to the special algorithms. Two algorithms are graphically presented in figure 4(c). First, the magnitude V_app was increased linearly for positive (0 → + |V_max|) and then negative (0 → − |V_max|) voltages (1 in figure 4(c)). Second, V_app was increased like a zigzag path by switching the polarity from positive to negative for the same magnitude and back to positive in the next V_app rise step (2 in figure 4(c)). It was set to |V_max| = 1.5 V in all the experiments. For both algorithms of the voltage increase, the general shape of the experimental dependences F_adh vs. V_app was similar to a wave-like form variation of F_adh with V_app as in the model dependences in figure 3. The model dependences fitted to the experiment are also graphically presented by the corresponding lines with label 1 in figures 4(a) and (b), respectively. The numerical values of the tuned parameters are listed in table 1. For comparison, the reference model dependences are also presented by lines labeled with the number 2 in figures 4(a) and (b). Line 3 in figure 4(a) shows the similarity between the results corresponding to the individual algorithm of the V_app increase in figure 4(c). It must be noted here that the parameter F_surf was freely used to shift the dependences vertically without noticeable distortion of the key features. The numerical values of all the parameters can be compared for the plotted model dependences in table 1.

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The part of the force curve obtained during the approach of the probe to the surface was also analyzed using the same model as in equation (1). In this part, there was no contact between the tip and the surface, and only the long length forces were important. Consequently, the term F_tQ can be excluded from equation (1). The results of the experiments and the model simulation are illustrated in figure 4(d). The experimental results (points in figure 4(d)) are presented by the force F_snap detected at the apex of the snap-in triangle of the force curve. The dependences of F_snap on V_app were obtained using the same algorithms for the V_app increase sequences as in figure 4(c). The lines in figure 4(d) were calculated from the model in equation (1) without F_tQ. A parabola-like curve was used to describe the dependences of F_snap on V_app in figure 4(d). The model was fitted with the experiment for the set of parameters listed in table 1. It was quite complicated to identify the differences between the experiments with individual algorithms of the applied voltage by fitting the model with the experiment within the measurement errors. A comparison of the two sets of parameters in table 1 quantitatively describes the sensitivity of the model to the intentional changes in the tuned parameters. It is practically impossible to distinguish between the model dependences calculated for 50 Å ⩽ R_tip ⩽ 60 Å (lines 1 and 2 in figure 4(d)). The lowest part of the tip R_tip = 25 Å was clearly too small to result in the snap-in forces at higher applied voltages (line 3 in figure 4(d)). On the other hand, the snap-in forces were obviously too low for the probe R_tip = 200 Å (line 4 in figure 4(d)) that, on the contrary, fitted the model calculations of the pull-off forces in figures 4(a) and (b) well. It is worth noting here that some discrepancies between the experimental results corresponding to the individual algorithms were detectable on the basis of the model comparison with the experiment. However, the influence of the measurement history on the interaction forces was much less detectable in the M–G–M stack in figure 4(d) than in figures 4(a) and (b).