Isotope effect in acetylene ${\mathrm{C}}_2{\mathrm{H}}_2$ and ${\mathrm{C}}_2{\mathrm{D}}_2$ rotations on Cu(001)

A comprehensive analysis of the elementary processes behind the scanning tunneling microscope controlled rotation of ${\mathrm{C}}_2{\mathrm{H}}_2$ and ${\mathrm{C}}_2{\mathrm{D}}_2$, isotopologues of a single acetylene molecule adsorbed on the Cu(001) surface, is given, with a focus on the isotope effects. With the help of density-functional theory we calculate the vibrational modes of ${\mathrm{C}}_2{\mathrm{H}}_2$ and ${\mathrm{C}}_2{\mathrm{D}}_2$ on Cu(001) and estimate the anharmonic couplings between them, using a simple strings-on-rods model. The probability of the elementary processes, nonlinear and combination band, is estimated using the Keldysh diagram technique. This allows us to clarify the main peculiarities and the isotope effects of the ${\mathrm{C}}_2{\mathrm{H}}_2$ and ${\mathrm{C}}_2{\mathrm{D}}_2$ on Cu(001) rotation, discovered in the pioneering work [B. C. Stipe et al., Phys. Rev. Lett. 81, 1263 (1998)], which have not been previously understood.

Figure 1

Calculated equilibrium configuration of acetylene molecule (${\mathrm{C}}_2{\mathrm{H}}_2$) on Cu(001). The C–C and C–H bond lengths and C–C–H bond angle are given in the text.

Figure 2

The calculated anharmonic coefficients $\mathcal{K}{}_{\nu ,{\nu }^{\prime },{\nu }^{\prime \prime }}$ [(a) and (b)] of the ${\mathrm{C}}_2{\mathrm{H}}_2$/Cu(001), $\nu =1$ and 2, respectively; (c), (d) of the deuterated acetylene ${\mathrm{C}}_2{\mathrm{D}}_2$/Cu(001), $\nu =1$ and 2, respectively.

Figure 3

Sketch of the linear and nonlinear excitations of the RC mode. Three cases are considered: (a) low RC barrier ${\Omega }_r<{\varepsilon }_B<2{\Omega }_r$; (b) intermediate RC barrier $2{\Omega }_r<{\varepsilon }_B<3{\Omega }_r$; (c) high RC barrier $3{\Omega }_r<{\varepsilon }_B<4{\Omega }_r$.

Figure 4

Comparison of the rotation yield per electron $Y$ as a function of bias voltage for two isotopologues of acetylene ${\mathrm{C}}_2{\mathrm{H}}_2$ and ${\mathrm{C}}_2{\mathrm{D}}_2$ on Cu(001) surface. Symbols are the experimental data from Ref. [5]. Partial processes $R_A\left(V\right)$, $R_B\left(V\right)$, and $R_C\left(V\right)$ of the rotation rate of the ${\mathrm{C}}_2{\mathrm{H}}_2$ molecule are shown with short dashed, dashed-dotted, and dashed lines, respectively. Solid lines correspond to the sum of all partial processes.

Figure 5

(a), (b) Rotation yield per electron $Y$ as a function of tunneling current for two isotopologues of acetylene ${\mathrm{C}}_2{\mathrm{H}}_2$ and ${\mathrm{C}}_2{\mathrm{D}}_2$ on the Cu(001) surface. Lines are calculated results for different bias voltages: $V=400$, 449, 500, 600 mV. Thin dashed-dotted lines are the linear and nonlinear contributions to the rotational yield for $V=600$ mV. Dots in panel (a) correspond to the experimental data from Ref. [5]. (c) The dependence of the molecule-tip hybridization parameter ${\Delta }_t$ on the tunneling current for different bias voltages.

Figure 6

Dependence of the rotation rate per electron of the ${\mathrm{C}}_2{\mathrm{D}}_2$ molecule on bias voltage. Squares are the experimental data from Ref. [5]. The fitted rates $R_A\left(V\right)$, $R_B\left(V\right)$, and $R_C\left(V\right)$ are shown as short dashed, dashed-dotted, and dashed lines, respectively. The solid black line is the sum of all partial processes.