Inelastic electron tunneling spectroscopy for probing strongly correlated many-body systems by scanning tunneling microscopy

We present an extension of the tunneling theory for scanning tunneling microscopy (STM) to include different types of electron-vibrational couplings responsible for inelastic contributions to the tunnel current in the strong-coupling limit. It allows for a better understanding of more complex scanning tunneling spectra of molecules on a metallic substrate in separating elastic and inelastic contributions. The starting point is the exact solution of the spectral functions for the electronically active local orbitals in the absence of the STM tip. This includes electron-phonon coupling in the coupled system comprising the molecule and the substrate to arbitrary order including the antiadiabatic strong-coupling regime as well as the Kondo effect on a free-electron spin of the molecule. The tunneling current is derived in second order of the tunneling matrix element which is expanded in powers of the relevant vibrational displacements. We use the results of an ab initio calculation for the single-particle electronic properties as an adapted material-specific input for a numerical renormalization group approach for accurately determining the electronic properties of a 1,4,5,8-naphthalene-tetracarboxylic acid dianhydride molecule on Ag(111) as a challenging sample system for our theory. Our analysis shows that the mismatch between the ab initio many-body calculation of the tunnel current in the absence of any electron-phonon coupling to the experimental scanning tunneling spectra can be resolved by including two mechanisms: (i) a strong unconventional Holstein term on the local substrate orbital leads to the reduction of the Kondo temperature and (ii) a further electron-vibrational coupling to the tunneling matrix element is responsible for inelastic steps in the $dI/dV$ curve at finite frequencies.

Figure 1

Cartoon of the setup: a system comprising a molecule and a substrate is coupled to an STM tip. The arrows indicate the two transmission paths for electrons tunneling from the tip to the system $S$. The interference of such multiorbital tunneling paths is responsible for Fano line shapes in the tunneling spectra [35].

Figure 2

Second-order Feynman diagram of the generating Luttinger-Ward functional in the system $S$. The full line represents the full local electron Green's function $G_d\left(z\right)$, the wiggled line the full phonon propagator. $i{\omega }_n=i\pi (2n+1)/\beta$ denote the fermionic Matsubara frequencies and $i{\omega }_n=i2\pi n/\beta$ the bosonic Matsubara frequencies [93].

Figure 3

Constant-current STM image of the relaxed phase of NTCDA on Ag(111) $\left(I=200\mathrm{pA},V=50\mathrm{mV}\right)$. Graphical representations of (gas-phase) NTCDA molecules have been overlaid over the bright and dark molecules. The white, gray, and red circles indicate hydrogen, carbon, and oxygen atoms of NTCDA, respectively. The length of the scale bar is 10 Å.

Figure 4

Constant-current STM image of the rippled phase of NTCDA on Ag(111) $\left(I=200\mathrm{pA},V=50\mathrm{mV}\right)$. The arrow indicates a line along which the character of the molecules changes gradually from bright to dark and vice versa. The length of the scale bar is 20 Å.

Figure 5

$dI/dV$ spectra of the bright and dark molecules acquired at the CH edge (left) and at the center (right) of the bright (red line) and dark (black line) molecules, respectively. The spectra are plotted on the bias-voltage axis as measured. A calibration of bias-voltage scale to symmetrize the inelastic features would require shifting the spectra by $2.25\mathrm{mV}$ to the right. Fano fits according to Eq. (39) are indicated by blue lines, the fit parameters are bright molecule, CH edge: $\delta =24.04,q=6.44$. Bright molecule, center: $\delta =32.50,q=1.74$. Dark molecule, CH edge: $\delta =25.42,q=6.95$. Dark molecule, center: $\delta =29.84,q=2.08$. Average fit parameters are listed in Table 1.

Figure 6

(a) Temperature evolution of the FWHM of the zero-bias peak of the bright molecule. (b) $dI/dV(V,z)$ maps for the bright (left) and dark (right) molecules, recorded after the formation of the tip-molecule bond. The meaning of the $z$ coordinate is shown schematically in the illustrations. (c) $dI/dV$ spectra at various $z$ in a magnetic field of $B=2.5\mathrm{T}$, measured at $T=4.3\mathrm{K}$.

Figure 7

$dI/dV$ spectra of the different NTCDA molecules in the rippled phase, recorded directly after the formation of the bond of the tip to the molecules. The data were acquired at $T=4.3\mathrm{K}$. Spectra are vertically offset by $5\mu \mathrm{S}$ for clarity.

Figure 8

Local density of states (LDOS) of the LUMO of NTCDA calculated $4\AA$ above the gas-phase molecule (left panel). A graphical representation of the gas-phase NTCDA molecule has been overlaid for clarity. The right panel shows the top view of the LUMO of the gas-phase NTCDA molecule. The different colors indicate the positive $\left[\mathrm{\Psi }\right(r)>0]$ and negative $\left[\mathrm{\Psi }\right(r)<0]$ contributions of the wave function.

Figure 9

The panel on the left shows a STM topography image measured at constant height above the molecules of the rippled phase $\left(V=47\mathrm{mV}\right)$. The panel on the right shows the corresponding $d^{2}I/d{V}^2$ image. The length of the scale bar is 10 Å.

Figure 10

(Left) $dI/dV$ spectra of the different NTCDA molecules in the rippled phase at $T=4.3\mathrm{K}$. The spectra have been recorded at the CH edge of each molecule. Different colors correspond to the different molecules as indicated by the colored frames around the molecules. The color coding is the same as in Fig. 7. (Right) $dI/dV$ spectra of the different NTCDA molecules in the rippled phase at $T=4.3\mathrm{K}$, measured at the center of each molecule. Colors as in (b). Spectra are vertically offset by $1\mathrm{nS}$ for clarity.

Figure 11

Plot of the step height $\mathrm{\Delta }{\sigma }_{\mathrm{center}}$ measured at the center of the molecules in the rippled phase as a function of the height ${\sigma }_{\mathrm{CH}}\left(0\right)$ of the Kondo resonance at the CH edge. The values are obtained from the spectra displayed in Fig. 10. Color coding as in Fig. 10. Data points have been fitted by a linear function of the form $\mathrm{\Delta }{\sigma }_{\mathrm{center}}=c\times {\sigma }_{\mathrm{CH}}\left(0\right)+b$, fit parameters are $c=0.13,b=0.30\mathrm{nS}$.

Figure 12

PDOS of the LUMO of NTCDA/Ag(111) as calculated by a combination of DFT and MBPT. The latter includes quasiparticle corrections. (a) On-top molecule, (b) bridge molecule.

Figure 13

Comparison between the experimental $dI/dV$ spectra recorded on NTCDA/Ag(111) (black lines) and the results of NRG calculations for the SIAM with the PDOS of Fig. 12 as input (blue lines). The NRG spectra have been adjusted with a constant offset $[{\rho }_{\text{offset}}=0.2\mathrm{nS}$ in (a), ${\rho }_{\text{offset}}=1.1\mathrm{nS}$ in (b)] to account for an experimental background signal such that the zero-bias peak heights and the high-frequency tails are matched to the experimental $dI/dV$ curve.

Figure 14

Ratio $T_K\left({\lambda }_c\right)/T_K({\lambda }_c=0)$ calculated for the symmetric single-impurity Anderson model plus unusual Holstein coupling ${\lambda }_c$ for a single vibrational mode ${\omega }_0$. The main panel shows the Kondo temperature as function of the coupling ${\lambda }_c$ for $U/{\mathrm{\Gamma }}_0=10$ and different values for the vibrational frequency ${\omega }_0$, normalized at ${\lambda }_c=0$. The inset depicts the same quantity but plotted against the polaronic energy shift $E_p={\lambda }_c^2/{\omega }_0$.

Figure 15

Symmetric single-impurity Anderson model plus unusual Holstein coupling ${\lambda }_c$ for a single vibrational mode ${\omega }_0$. Renormalized hybridization ${\mathrm{\Gamma }}_0\to {\mathrm{\Gamma }}_{\mathrm{eff}}\left({\lambda }_c\right)$ as function of ${\lambda }_c$ for two different values for ${\omega }_0$ at $U/{\mathrm{\Gamma }}_0=10$. The inset depicts the same quantity for ${\omega }_0/{\mathrm{\Gamma }}_0=0.2,{\lambda }_c/{\mathrm{\Gamma }}_0=0.35$ as function of the Coulomb interaction.

Figure 16

Symmetric single-impurity Anderson model plus unusual Holstein coupling ${\lambda }_c$ for a single vibrational mode ${\omega }_0$. $T_K$ plotted against the inverse of the renormalized hybridization. Different Coulomb interactions are indicated by different colors and different vibrational frequencies ${\omega }_0$ by different points. For comparison, we added the textbook expression for $T_K$ of the SIAM as the dashed line.

Figure 17

Contributions to the elastic spectrum due to tunneling into the molecular orbital $d_{0\sigma }$ and the local surface orbital $c_{0\sigma }$, leading to the constituents (a) ${\rho }_{d_{0\sigma },d_{0\sigma }^{†}}\left(\omega \right)$, (b) ${\rho }_{c_{0\sigma },c_{0\sigma }^{†}}\left(\omega \right)$ and (c) ${\rho }_{d_{0\sigma },c_{0\sigma }^{†}}\left(\omega \right)={\rho }_{c_{0\sigma },d_{0\sigma }^{†}}\left(\omega \right)$. The spectral functions have been calculated with NRG. We set $U/{\mathrm{\Gamma }}_0=10,{\omega }_0/{\mathrm{\Gamma }}_0=0.1$ and different colors indicate various unconventional Holstein couplings ${\lambda }_c$.

Figure 18

Contributions to the inelastic spectrum due to tunneling into the molecular orbital $d_{0\sigma }$, the local surface orbital $c_{0\sigma }$, and a coupling ${\lambda }_{\mu \nu }^{\text{tip}}$ of the vibrational mode ${\omega }_0$ to the STM tip, leading to the constituents (a) ${\rho }_{{\widehat{X}}_0{d}_{0\sigma },{\widehat{X}}_0{d}_{0\sigma }^{†}}\left(\omega \right)$, (b) ${\rho }_{{\widehat{X}}_0{d}_{0\sigma },d_{0\sigma }^{†}}\left(\omega \right)={\rho }_{d_{0\sigma },{\widehat{X}}_0{d}_{0\sigma }^{†}}\left(\omega \right)$, and (c) ${\rho }_{{\widehat{X}}_0{d}_{0\sigma },c_{0\sigma }^{†}}\left(\omega \right)={\rho }_{c_{0\sigma },{\widehat{X}}_0{d}_{0\sigma }^{†}}\left(\omega \right)$. The spectral functions have been calculated with NRG. We set $U/{\mathrm{\Gamma }}_0=10,{\omega }_0/{\mathrm{\Gamma }}_0=0.1$ and different colors indicate various Holstein couplings ${\lambda }_c$.

Figure 19

NRG results for the Kondo temperature $T_K$ of the bright/on-top and dark/bridge molecules as a function of the unconventional Holstein coupling ${\lambda }_c$ for two different vibrational energies ${\omega }_0=41.6\text{meV},50.4\text{meV}$.

Figure 20

Elongation pattern of $B_{3g}$ vibrational mode No. $3\left({\omega }_0=50.4\text{meV}\right)$ from Table 2.

Figure 21

Individual contributions to the (a) elastic and (b) inelastic spectrum for the bright (on-top) molecule that are combined in Fig. 22 to fit the experimental $dI/dV$ spectra. Note that at the CH edge we only include the spectra displayed with solid lines, whereas the Fano effect in the center of the molecule leads to additional contributions which are plotted as dashed lines in Fig. 21. Results for the dark (bridge) molecule are qualitatively the same. Parameters as in Fig. 22.

Figure 22

Comparison between theoretical (blue) and experimental (black) $dI/dV$ spectra for both types of molecules (bright and dark), each measured at the CH edge and at the center of a molecule. Two vibrational energies have been used: ${\omega }_0=50.4\text{meV}$ and ${\omega }_1=41.6\text{meV}$. Parameters are (a) bright/on-top molecule, CH edge: ${\lambda }_c=200\mathrm{meV}$ for ${\omega }_0,{\lambda }_c=0$ for ${\omega }_1,{\lambda }_{00}^{\mathrm{tip}}=0.25,{\lambda }_{01}^{\mathrm{tip}}=0.35,t_c/t_d=0,{\rho }_{\mathrm{offset}}=0$; (b) bright/on-top molecule, center: ${\lambda }_c=200\mathrm{meV}$ for ${\omega }_0,{\lambda }_c=0$ for ${\omega }_1,{\lambda }_{00}^{\mathrm{tip}}=0.45,{\lambda }_{01}^{\mathrm{tip}}=1.0,t_c/t_d=0.5,{\rho }_{\mathrm{offset}}=1.85\mathrm{nS}$; (c) dark/bridge molecule, CH edge: ${\lambda }_c=220\mathrm{meV}$ for ${\omega }_0,{\lambda }_c=0$ for ${\omega }_1,{\lambda }_{00}^{\mathrm{tip}}=0.28,{\lambda }_{01}^{\mathrm{tip}}=0.40,t_c/t_d=0,{\rho }_{\mathrm{offset}}=0.52\mathrm{nS}$; (d) dark/bridge molecule, center: ${\lambda }_c=220\mathrm{meV}$ for ${\omega }_0,{\lambda }_c=0$ for ${\omega }_1,{\lambda }_{00}^{\mathrm{tip}}=0.35,{\lambda }_{01}^{\mathrm{tip}}=0.75,t_c/t_d=0.5,{\rho }_{\mathrm{offset}}=2.18\mathrm{nS}$.

Figure 23

(a) Spectral function ${\rho }_{d_{\sigma },d_{\sigma }^{†}}\left(\omega \right)$ of the molecular orbital for the particle-hole-symmetric antiadiabatic regime of the conventional Holstein model. (b) Spectral evolution of ${\rho }_{d_{\sigma },d_{\sigma }^{†}}\left(\omega \right)$ for increasing ${\lambda }_d>{\lambda }_d^c$. The case with ${\lambda }_d=0$ is added to (a) and (b) for comparison. (c) The spectral data of (b) plotted on a larger energy interval. Parameters: $\rho =\text{const},D/{\mathrm{\Gamma }}_0=10,U/{\mathrm{\Gamma }}_0=-2{\varepsilon }_d/{\mathrm{\Gamma }}_0=10$, and ${\omega }_0={\mathrm{\Gamma }}_0$.

Figure 24

Spectral functions of the molecular orbital in the antiadiabatic regime of the conventional Holstein model for local particle-hole asymmetry. (a) $\mathrm{\Delta }\varepsilon /{\mathrm{\Gamma }}_0=0.004$ and (b) $\mathrm{\Delta }\varepsilon /{\mathrm{\Gamma }}_0=0.01$. NRG parameters as in Fig. 23.

Figure 25

All contributions to the STS spectra in the antiadiabatic regime of the Holstein model, for three ${\lambda }_d/{\mathrm{\Gamma }}_0$ ratios as stated in (a). Particle-hole symmetry is assumed. (a) Spectral function taken from Fig. 23. (b) ${\overline{\rho }}^{\left(2\right)}\left(\omega \right)$. (c) ${\overline{\rho }}^{\left(1\right)}\left(\omega \right)$. NRG parameters as in Fig. 23.

Figure 26

Sum of all contributions to the STS spectra that are displayed in Fig. 25 (antiadiabatic, particle-hole-symmetric regime of the Holstein model), for ${\lambda }_d/{\mathrm{\Gamma }}_0=2.27$ and two values of ${\lambda }^{\mathrm{tip}}$. NRG parameters as in Fig. 23.