Graphene and NTCDA adsorbed on Ag(111): Temperature-dependent binding distance and phonon coupling to the interface state

Flat lying physisorbates on surfaces can be expected to be lifted by heating. The details of this lattice expansion depend on the competition between two mechanisms. On the one hand, the interaction potential of each atom with the substrate is anharmonic and constitutes the reason for lifting. On the other hand, the internal bending stiffness of the planar adsorbate reduces its tendency to lift with temperature. Here we show that this competition can be investigated in a simplified molecular-dynamics simulation which considers only the vertical motion of the atoms of the adsorbate in respective model potentials. For two prototypical systems, i.e., graphene and 1,4,5,8-naphthalene-tetracarboxylic acid dianhydride (NTCDA) on Ag(111), we observe lifting by 0.04 Å (graphene) and 0.06 Å (NTCDA) between zero and room temperature. For the latter system, the impact of the temperature-dependent geometrical and vibrational properties on the occurring interface state (IS) is discussed. By introducing a local electron-phonon interaction model, we combine the indirect dependence of the IS on the average binding distance with direct coupling to the vibrational modes and obtain good agreement with the experimental results.

Figure 1

(a) Accoustical and optical branch of the out-of-plane phonon dispersion of graphene from our model potential in Eq. (3). (b) Energy increase $\mathrm{\Delta }E$ caused by a deflection $\mathrm{\Delta }z$ of one single carbon atom from a graphene layer. The squares are the energies extracted from DFT calculations and the solid line is a quadratic fit from our model $\mathrm{\Delta }E=6k\mathrm{\Delta }{z}^2$.

Figure 2

(a) Temperature-dependent average binding distance of the graphene layer from our simplified MD model for graphene. (b) Averaged histograms of the vertical coordinate of the single carbon atoms for several temperatures.

Figure 3

Energetics of NTCDA/Ag(111) with separate treatment of the 4 carboxyl-oxygen atoms and the mostly carbon based core (14 carbon and 2 anhydride oxygen atoms) of the molecule. The molecule is sketched in (h) in a top and side view, where the carboxyl-oxygen atoms are marked as filled red circles and the mostly carbon based core atoms of the molecule are indicated by the open blue circles. Additionally, the different atomic positions from Table 2 are defined. (a)–(c) show the DFT data for the external, internal and total binding potential for different average binding distances of the carboxyl and core atoms. A profile of the binding potential as indicated by the gray dashed line in (c) is printed in (g). (d) depicts the fit to the DFT data from (a) as defined in Eq. (9), (e) is the internal potential model based on Eq. (16) for the same configurations as in (b), and (f) is their sum in comparison to (c).

Figure 4

Kinetic and total energy, and internal and external potential over different timescales from our MD model calculation for NTCDA/Ag(111) at $369\mathrm{K}$. (a) and (b) show the behavior on two different timescales, after initial equilibration has taken place.

Figure 5

NTCDA/Ag(111) binding distances of the carboxyl-oxygen atoms and the mostly carbon based core as defined in Fig. 3. The bottom panel shows the average binding distance of the molecule for various temperatures. Additionally, the average binding distance of all atoms of the molecule is shown. The top panel displays the distribution of the binding distances of the core and carboxyl atoms for temperatures of 34, 80, 144, 211, 291, and $385\mathrm{K}$.

Figure 6

Charge density and surface charge of the clean Ag(111) surface Shockley state. (a) shows the line charge density from a calculation of a 120 layer Ag(111) slab along the $z$ axis for various ${\mathit{k}}_{\parallel }$ that has been integrated over the in-plane coordinates. The $z$ scale is indicated by the Ag layers that are printed below the data. The densities inside the crystal ${\rho }_{\mathrm{bulk}}$ and on the surface ${\rho }_{\mathrm{surf}}$ are colored in blue and red, respectively. The integral over the latter is printed in (b) with the black solid line for different parallel momenta including ${\mathit{q}}_{\max }$ that is defined in Fig. 8. For comparison, results of slab calculations with only 30 and 60 layers are plotted in blue and yellow. The black dashed line indicates our assumption for infinite number of layers and the gray area is the Gaussian approximation that we use to model the electron-phonon interaction.

Figure 7

Energy and dispersion of the IS from DFT calculations projected with a method from [6] for the room- and zero-temperature configurations as defined in the main text with vertical adsorption heights of 2.95 (gray) and $2.89\text{Å}$ (black). The energetic position of the curves is obtained from a sum of the calculated energy shift between interface and clean surface state and the measured energy onset of the surface state [57]. The onset energies at $\mathrm{\Gamma }$ ($\mathit{k}=0$) are compared to the modified IS model (see main text) from [18] in the inset of the figure. Additionally, a linear fit of the DFT data is drawn.

Figure 8

(a) Unit cell (top) and Brillouin zone (bottom) of the NTCDA/Ag(111) monolayer in the commensurate relaxed phase. The blue dashed lines denote the original unit cell or Brillouin zone with two molecules, one in on-top and one in bridge position. The solid light green line gives the unit cell with only one molecule without consideration of different adsorption sites. (b+c) IS onset energies calculated from the average binding distance of our simulation time steps. (b) shows the IS histograms for several temperatures where the open circles are the calculated data and the solid lines are Gaussian fit functions $f\left(E\right)=A{e}^{(\theta (E-E_{\max })m+\theta (E_{\max }-E)n){(E-E_{\max })}^2}$ which allow for different slopes at both flanks of the peaks. The recieved maxima $E_{\max }$ are plotted in (c) as blue-to-red triangles. Additionally, the resulting IS onset energies with the EPI correction from Eq. (52) are printed as gray triangles. For comparison, the green triangles denote the experimental IS energies from [17]. The dashed lines in the corresponding colors are linear fits to guide the eye.

Figure 9

FFT of a MD calculation with a harmonized form of the potential from Eq. (10) at $112\mathrm{K}$ with a run time of $5\mathrm{ns}$. The red lines are the FFTs of all atom coordinates printed on top of each other. The blue squares mark the 20 frequencies with largest oscillation strength which are listed in Table 3. The inset shows the normed elongation pattern for the first six eigenmodes $\nu$ of the NTCDA/Ag(111) system, i.e., with frequencies up to $2.1\times {10}^{13}{\mathrm{s}}^{-1}$.