Role of the electric field in surface electron dynamics above the vacuum level

Scanning tunneling spectroscopy is used to study the dynamics of hot electrons trapped on a Cu(100) surface in field-emission resonances (FERs) above the vacuum level. Differential conductance maps show isotropic electron interference wave patterns around defects whenever their energy lies within a surface-projected band gap. Their Fourier analysis reveals a broad wave-vector distribution, interpreted as due to the lateral acceleration of hot electrons in the inhomogeneous tip-induced potential. A line-shape analysis of the characteristic constant-current conductance spectra permits establishing the relation between apparent width of peaks and intrinsic linewidth of FERs, as well as the identification of the different broadening mechanisms.

Figure 1

(Color online) (a) $dI∕dV$ map of the Cu(100) surface shown in (b) ($I=10\mathrm{nA}$; $V_S=4.9\mathrm{V}$). Standing waves are clearly seen around steps and surface and subsurface defects. (c) Constant-current $dI∕dV$ spectra around the first field-emission resonance. [(d)–(f)] $dI∕dV$ maps of a smaller region taken at the bias values indicated in the figure. [(g)–(i)] 2D Fourier transform of (d)–(f). The arrows indicate the inner and outside radii of the disk appearing in $K$ space (see the text).

Figure 2

(Color online) (a) Schematic representation of the nonconservation of parallel momentum due to the applied field by a tip of finite radius $R$. At a given value of electron energy $E=e{V}_s-\varphi$, the kinetic energy for parallel motion $E_{k_{\Vert }}$ increases from the tip position $r_{\Vert }=0$ to distant sites at the surface $r_{\Vert }=s$. (b), (c) (d) show the charge density of electron states confined in a large box with a potential $V\left(x\right)=V_0∕[1+{(x∕x_0)}^2]$ with energy below, slightly above and above, $V_0$, respectively [$V_0=0.03$ (arbitrary units) and $x_0=200$ (arbitrary units)]. (e), (f), and (g) show the corresponding FT. Curves (c), (d), (f), and (g) are shifted vertically for clarity. The arrows indicate the range of wave vectors with significant weight in the FT and, thus, confirm the measured behavior in Figs. 1g, 1h, 1i.

Figure 3

(Color online) $dI∕dV$ spectra taken for several current set points ranging from $0.6\text{to}60\mathrm{nA}$ $(V_{ac}=10\mathrm{mV}\mathrm{rms})$. The inset shows the width of the peaks from a Lorentzian fit, as well as values obtained from a phase coherence length analysis (Ref. 28). Shaded areas correspond to the energy location in $\overline{\Gamma }$ of copper bulk bands projected on the (100) surface (Ref. 29).

Figure 4

(Color online) (a) Comparison between a calculated $dI∕dV$ spectrum (black curve) and the transmission coefficient (red curve) at the tip Fermi level $T(E_F;V)$. The inset shows a comparison between a static calculation (black) of $T\left(E\right)$ at $Z\left(V\right)$ values of the first resonance (thus reflecting the intrinsic FER line shape), $T(E_F;V)$ (red), and $dI∕dV\left(V\right)$ (blue). (b) $dI∕dV$ spectra for two $V_{0i}$ values (see the text).