Phonon-mediated tunneling into a two-dimensional electron gas on the Be(0001) surface

The electronic properties of the Be(0001) surface are investigated on the atomic scale at low temperature. Utilizing laterally resolved scanning tunneling spectroscopy (STS) a pronounced surface state is identified, additionally manifested by the observation of bias-dependent standing-wave patterns. Fourier analysis reveals a parabolic dispersive behavior, being characteristic for a two-dimensional free-electron gas confined to the Be(0001) surface. Moreover, signatures of the opening of inelastic phonon-mediated tunnel channels for electrons near the Fermi level are observed by STS.

Figure 1

Constant current STM images of the Be(0001) surface. (a) Atomically flat terraces are separated by an atomic step edge. High-symmetry directions of the crystal are indicated with arrows ($T=4.2\mathrm{K}, I=100\mathrm{pA}, U=1.4\mathrm{V}$). (b) High-resolution STM image and its Fourier spectrum (inset). (c) The transformation of the spot features reveals the atomic lattice, whereas (d) the ring feature of the Fourier spectrum corresponds to standing waves ($T=60\mathrm{K}, I=10\mathrm{nA}, U=10\mathrm{mV}$).

Figure 2

Scanning tunneling spectroscopy on Be(0001). (a) Typical tunneling spectroscopy curve. A shoulder indicates the onset of the surface state (arrow). Bulk contributions are separated from the surface state by fitting with a polynomial (red line) ($T=4.2\mathrm{K}$). (b) Surface-state contribution extracted from the spectrum in (a). (c) Top: Higher resolution spectroscopy around the surface state onset (red). A symmetric conductance step is sketched in gray. Bottom: numerical derivative $d^{2}I/d{U}^2$. The peak position determines the onset energy $E_{\mathrm{s}}$, and fitting the rising flank with a Lorentzian peak (blue) results in the lifetime ${\mathrm{\Gamma }}_{\mathrm{s}}$.

Figure 3

Laterally resolved scanning tunneling spectroscopy on Be(0001). (a) Bias-dependent $dI/dU$ maps, acquired at values of $U$ as indicated. The periodic length clearly changes with $U$ ($T=4.2\mathrm{K}, I=8$, 5, $3\mathrm{nA}$). (b) Constant current STM image of standing waves emanating from a step edge that runs along a high-symmetry direction ($T=4.2\mathrm{K}, I=1\mathrm{nA}, U=-100\mathrm{mV}$). The propagation direction of standing electron waves is along $\left[10\overline{1}0\right]$. (c) Color plot of tunneling spectra taken as a function of distance $x$ from an atomic step edge. Bias ranges: $[-3.0;-1.6]\mathrm{V}$; $[-1.5;-0.3]\mathrm{V}$; $[-0.3;0.7]\mathrm{V}$. Stabilization conditions: ($-3.0\mathrm{V}, 2\mathrm{nA}$); ($-1.5\mathrm{V}, 2\mathrm{nA}$); ($-0.3\mathrm{V}, 1\mathrm{nA}$). Each horizontal row is normalized to its mean value. (d) Normalized Fourier components along the $\overline{\mathrm{\Gamma }}\text{−}\overline{M}$ direction calculated from (c). A parabolic dispersion of the surface state is observed (solid line).

Figure 4

(a) STS of Be(0001), measured with a W tip ($T=4.2\mathrm{K}, U_{\mathrm{stab}}=-200\mathrm{mV}, I_{\mathrm{stab}}=1.83\mathrm{nA}, U_{\mathrm{mod}}=2\mathrm{mV}$ RMS) and (b) its numerical differentiation $d^{2}I/d{U}^2$. The insets schematically depict signatures of inelastic excitations. (c) Schematic representation of the elastic (green) and inelastic (red) tunneling processes involving surface-state electrons (orange) originating from Be(0001). Filled (empty) bulk bands are indicated in dark (light) gray. (d) Averaged $d^{2}I/d{U}^2$ signal over the positive and negative bias absolute values in (b). Arrows mark the peaks at 41 and 67 mV, respectively. The region of sizable inelastic excitations is marked in gray.

Figure 5

(a) Constant-current STM images of the standing-wave pattern acquired at different bias voltages as indicated ($I=1\mathrm{nA}$; $T=40\mathrm{K}$). Cross sections taken along the lines in the images indicate respective typical wave amplitudes, ranging between 1 and 4 pm. (b) Detailed view of the FFT-STS obtained around $E_{\mathrm{F}}$. A pronounced spectral weight of the standing wave is observed within the energy window of maximum phonon energies (arrow). (c) Thermovoltage $U_{\mathrm{th}}$, as derived from the laterally resolved spectroscopy data and projected to a situation where the sample and the tip are at a temperature of 150 and $300\mathrm{K}$, respectively. A significant variation of $U_{\mathrm{th}}$ is found on the standing-wave pattern.