Ultrasensitive detection of lateral atomic-scale interactions on graphite (0001) via bimodal dynamic force measurements

Enhanced sensitivity to lateral forces on the nominally flat and inert (0001) surface of graphite is demonstrated via room-temperature dynamic force microscopy using simultaneous excitation and FM detection of the lowest flexural and torsional cantilever resonance modes. The site-independent long-range tip-sample interaction causes no significant lateral force variations except near atomic steps but unprecedented sensitivity to short-range forces is achieved on flat terraces in the attractive range. Two-dimensional bimodal force vs distance maps confirm the stronger distance dependence of the torsional frequency shift compared to the flexural resonance shift. This agrees with model calculations based on theoretical expressions. The lateral force gradient is extracted from the measured torsional shift, and a lateral force variation in at most $\pm 20\text{pN}$ is obtained by integrating this gradient parallel to the surface. A further integration reveals a potential-energy variation in the attractive force range of only 3 meV.

Figure 1

(Color online) Schematic drawing of tip and sample in bimodal DFM with the first-flexural and torsional resonances modes. The lines and the arrows show the tip trajectories during oscillations toward and away from the sample, respectively.

Figure 2

(Color online) Distance dependence of (a) the long-range force $F_{\text{LR}}$, the vertical short-range force $F_{\text{SR}}$, and total interaction force $F_{\text{Total}}$ along ${\text{A-A}}^{\prime }$ in Fig. 1, and (b) vertical $F_Z^{\prime }$ and lateral force gradients $F_X^{\prime }$ all calculated with an assumed model tip-sample interaction potential (see text). (c) Frequency shift $\Delta f_{\text{TR}}$ of the torsional resonance computed with Eq. (1) (points) and Eq. (6) (curve). (d) Computed frequency shifts $\Delta f_{1\text{st}}$, $\Delta f_{2\text{nd}}$ of the first- and second-flexural modes, and of the first torsional resonance, $\Delta f_{\text{TR}}$, using Eqs. (3, 5, 6), respectively. Assumed parameters: $f_{1\text{st}}=150\text{kHz}$, $f_{2\text{nd}}=980\text{kHz}$, $f_{\text{TR}}=1480\text{kHz}$, $k_{1\text{st}}=25\text{N}/\text{m}$, $k_{2\text{nd}}=1450\text{N}/\text{m}$, $k_{\text{TR}}=2000\text{N}/\text{m}$, $A_{1\text{st}}=10\text{nm}$, $A_{\text{TR}}=10\text{pm}$, $R_{\text{tip}}=5\text{nm}$, and $\lambda =79\text{pm}$.

Figure 3

(Color online) (a) Topography and (b) $\Delta f_{\text{TR}}$ map of graphite (0001) in a scan area of $300\text{nm}\times 300\text{nm}$. (c) Topography and (d) $\Delta f_{\text{TR}}$ map inside the dashed square around the pit observed in (a); the scan area is $50\text{nm}\times 50\text{nm}$. Double-headed arrows schematically depict the directions of the tip oscillations used to record the data. The significance of the other arrows is discussed in the text. Imaging parameters; (a) and (b) $\Delta f_{1\text{st}}=-19.8\text{Hz}$, $A_{1\text{st}}=10\text{nm}$, $A_{\text{TR}}=38\text{pm}$, the sample bias $V_{\text{bias}}=0.244\text{V}$. (c) and (d) $\Delta f_{1\text{st}}=-20.0\text{Hz}$, $A_{1\text{st}}=15\text{nm}$, $A_{\text{TR}}=63\text{pm}$, $V_{\text{bias}}=0.464\text{V}$. $f_{1\text{st}}=152297\text{Hz}$, $f_{\text{TR}}=1493393\text{Hz}$, $Q_{1\text{st}}=34994$, and $Q_{\text{TR}}=135109$.

Figure 4

(Color online) (a) $\Delta f_{\text{TR}}$ map near the left side of the pit in Fig. 3d in a scan area of $15\text{nm}\times 15\text{nm}$ and (b) corresponding line profile along ${\text{A-A}}^{\prime }$. (c) $\Delta f_{\text{TR}}$ map on the flat part in a scan area of $810\text{pm}\times 970\text{pm}$ together with a superposed carbon hexagon properly placed with respect to the surrounding maxima (hollow sites), and (d) corresponding line profiles along ${\text{A-A}}^{\prime }$, ${\text{B-B}}^{\prime }$, and ${\text{C-C}}^{\prime }$. Imaging parameters; $\Delta f_{1\text{st}}=-248\text{Hz}$ in (a) and $\Delta f_{1\text{st}}=-249\text{Hz}$ in (c), $A_{1\text{st}}=15\text{nm}$, $A_{\text{TR}}=63\text{pm}$, and $V_{\text{bias}}=0.265\text{V}$.

Figure 5

(Color online) (a) Two-dimensional $\Delta f_{1\text{st}}$ and (b) $\Delta f_{\text{TR}}$ maps in the $X\text{−}Z$ plane over a flat region. The insets show the interaction force, extracted from the measured $\Delta f_{1\text{st}}$ in (a) and the magnified $\Delta f_{\text{TR}}$ in (b). (c) Extracted lateral force gradient from the measured $\Delta f_{\text{TR}}$, using Eq. (6). (d) Lateral force obtained by integrating the lateral force gradient. The cantilever used was the same as that in Fig. 4 but the tip apex condition had changed. Measurement parameters; $A_{1\text{st}}=15\text{pm}$, $A_{\text{TR}}=38\text{pm}$, and $V_{\text{bias}}=0.463\text{V}$.

Figure 6

(Color online) (a) Lateral interaction potential variation map $U_X$ obtained by integrating the lateral force shown in Fig. 5d in the $X$ direction.