Quantifying the interplay between fine structure and geometry of an individual molecule on a surface

The pathway toward the tailored synthesis of materials starts with precise characterization of the conformational properties and dynamics of individual molecules. Electron spin resonance (ESR)-based scanning tunneling microscopy can potentially address molecular structure with unprecedented resolution. Here, we determine the fine structure and geometry of an individual titanium-hydride molecule, utilizing a combination of a newly developed millikelvin ESR scanning tunneling microscope in a vector magnetic field and ab initio approaches. We demonstrate a strikingly large anisotropy of the $g$ tensor, unusual for a spin doublet ground state, resulting from a nontrivial orbital angular momentum stemming from the molecular ground state. We quantify the relationship between the resultant fine structure, hindered rotational modes, and orbital excitations. Our model system provides avenues to determine the structure and dynamics of individual molecules.

Figure 1

Structure of the TiH molecule and influence of the surface. (a) Evolution of the ${}^2\mathrm{\Delta }$ and ${}^4\mathrm{\Phi }$ states of the TiH molecule as a function of distance from the surface of MgO, obtained from ab initio quantum chemistry (QC) calculations. The electronic ground state of the system changes from the ${}^4\mathrm{\Phi }$ to the ${}^2\mathrm{\Delta }$ state for the adsorbed TiH. The dashed line indicates the relaxed height obtained from DFT + $U$ and the dotted line the energy minimum of the states from QC calculations. (b) Illustration of the TiH molecule adsorbed on the oxygen site of an ionic ${\mathrm{Mg}}^{2+}{\mathrm{O}}^{2-}$ surface. The ${\tilde{V}}_{\mathrm{RF}}$ is applied on the magnetic Cr tip and is added to $V_{\mathrm{DC}}$ as schematically sketched. (c) Three-dimensional representation of a constant-current scanning tunneling microscopy (STM) image of a TiH molecule adsorbed next to two Fe atoms. Different apparent heights for Fe (151 ± 8 pm) and TiH (103 ± 8 pm) clearly distinguish both adsorbate types ($V_{\mathrm{DC}}=30\mathrm{mV}, I_t=10\mathrm{pA}$).

Figure 2

Millikelvin electron spin resonance (ESR) scanning tunneling microscopy (STM) of a TiH molecule in variable field orientations. (a) Sketch of the Zeeman diagram for the level splitting of the doublet state in different field orientations $B_{\parallel }$ (blue) and $B_{\perp }$ (red). Dashed arrows of the same lengths indicate the allowed $\mathrm{\Delta }\mathrm{\Omega }=\pm 1$ transitions for a specific $f$ (with $\mathrm{\Omega }=\mathrm{\Lambda }+\mathrm{\Sigma }$). (b) $B$-sweep mode ESR measurements with the same microtip for two TiH molecules with the magnetic field swept in $\pm B_{\parallel }$ (blue) or $\pm B_{\perp }$ (red) direction. Peak positions are extracted from Lorentzian fits and subsequently fitted with a linear model (dashed lines). For the same selected frequencies, they appear at very different magnetic fields for the two directions, revealing an anisotropic $g$ tensor with $g_{\parallel }=1.80\pm 0.02$ ([25.2 ± 0.2] GHz/T) and $g_{\perp }=0.63\pm 0.01$ ([8.8 ± 0.1] GHz/T) ($V_{\mathrm{DC}}=50\mathrm{mV}, I_t=2\mathrm{pA}, f_{\mathrm{chop}}=877\mathrm{Hz}, V_{\mathrm{RF}}=7.9\mathrm{mV}$). $f$-sweep mode ESR measurements in (c) $B_{\parallel }$ and (d) $B_{\perp }$ direction with the same microtip and on the same TiH molecule as in the $B_{\parallel }$-sweep in (b). Solid lines represent Lorentzian fits to the experimental data. Linear fits to the extracted peak positions [inset in (d)] reveal $g_{\parallel }=1.62\pm 0.06$ ([22.7 ± 0.9] GHz/T) and $g_{\perp }=0.66\pm 0.02$ ([9.3 ± 0.3] GHz/T) ($V_{\mathrm{DC}}=50\mathrm{mV}, I_t=2\mathrm{pA}, f_{\mathrm{chop}}=877\mathrm{Hz}, V_{\mathrm{RF}}=8.0\mathrm{mV}$).

Figure 3

Giant $g$-tensor anisotropy of TiH. (a) Extracted electron spin resonance (ESR) peak positions from 21 datasets on different molecules (indicated by different symbols) measured in two $B$-field directions $\perp$ (red) or $\parallel$ (blue). Filled or open symbols correspond to $B$- or $f$-sweep mode, respectively. Measurement parameters throughout all experiments were the same ($V_{\mathrm{DC}}=50\mathrm{mV}, I_t=2\mathrm{pA}, V_{\mathrm{RF}}=8\mathrm{mV}$), except for the inset [$V_{\mathrm{DC}}=50\mathrm{mV}, I_t=10\mathrm{pA}, {\tilde{V}}_{\mathrm{RF}}=4.466\mathrm{V}$ (uncalibrated $V_{\mathrm{RF}}$)]. (b) Experimental $g$ factors obtained from linear fits to the data in (a) as well as additionally included datasets with varied experimental parameters and tips (see Fig. S7 in the Supplemental Material [30] for a plot of all data). From 30 datasets in total, we obtain $g_{\parallel }=1.67\pm 0.16$ ([23.4 ± 2.29] GHz/T) and $g_{\perp }=0.61\pm 0.09$ ([8.49 ± 1.19] GHz/T). (c) Plots of the full width at half maximum (FWHM) (top) and ESR peak intensity (bottom) from $V_{\mathrm{RF}}$ power-dependent measurements of a TiH molecule extracted from fitting Fano lineshapes to the experimental data. Dashed lines indicate simultaneous fits within the 1 and 2 pA datasets, respectively, and reveal an asymptotic trend for the FWHM with $V_{\mathrm{RF}}$.

Figure 4

Modeling the $g$ tensor, the molecular geometry, and the hindered rotations. (a) Illustrations indicating the structural degrees of freedom of the TiH molecule: (i) bonding distance between Ti and O ($d$) or Ti and H ($R$), (ii) zero-point motion of H, and (iii) excited rotational mode. (b) Calculated in-plane (blue) and out-of-plane (red) $g$ factors of the TiH without (solid lines) or with (dashed lines) rotational dynamics. The $g$ tensor is highly sensitive to the adsorption height of the molecule. At $d=2.42\AA$, the minimum found in quantum chemistry (QC) calculations [Fig. 1], the experimentally observed anisotropic $g$ tensor can very well be reproduced. The inset shows the small variations in $g$ for changes of $R$. (c) Calculated potential wells (solid lines) for both ${}^2\mathrm{\Delta }$ orbital states and densities ρ (dashed lines) of the wave functions for the corresponding equidistant energy levels as a function of inclination angle θ. Both are reminiscent of an anharmonic two-dimensional quantum oscillator, where an amplitude of $\approx {15}^{\circ }$ can be deduced for the zero-point motion of the hydrogen.

Figure 5

Relation between $g_{\parallel }$ and $g_{\perp }$ for the ${}^2\mathrm{\Delta }$ state. The curve represents the one-dimensional space in the theoretical model, parametrized by the ratio between the energy splitting and spin-orbit coupling. Results from the electrostatic model and embedded cluster calculations are represented by dots, together with experimental results given in gray with error bars. We note that the black and red dots nearly overlap.

Figure 6

Potential energy curves of the ${}^2\mathrm{\Delta }$ state as a function of the height above the surface in various embedded cluster calculations.

Figure 7

Energy splitting and spin-orbit coupling constant of the ${}^2\mathrm{\Delta }$ state as a function of the height above the surface in various embedded cluster calculations.

Figure 8

$g$ factors of the ${}^2\mathrm{\Delta }$ state as a function of the height above the surface in the electrostatic model and using various embedded cluster calculations. Dotted vertical lines denote the optimal height above the surface in that method.