Systematic investigation of pseudogaps in In, Al, and Pb islands

The novel pseudogap state has been systematically studied on flat-top In, Al, and Pb islands/films grown on $\mathrm{Si}\left(111\right)$ surfaces with scanning tunneling microscopy. Our observations and analysis suggest that the interplay between electron-phonon interaction and quantum confinement, the dynamic Coulomb blockade, and the orthodox Coulomb blockade are, respectively, the dominant mechanism of the formation of such pseudogap states in the flat-top metal islands at different island size scales. Our results can help to settle the dispute on the origins of the pseudogap state in such systems.

Figure 1

Pb islands grown at different temperatures with a Pb coverage of about 3.2 ML. The lateral size of the Pb islands decreases from $\sim 10000\mathrm{n}{\mathrm{m}}^2$ with substrate temperature of 270 K (a) to $\sim 10\mathrm{n}{\mathrm{m}}^2$ with substrate temperature of 200 K (c). The volcano shaped islands grown at 270 K could change into flat-top ones (b) after annealing at RT for 30 min. The image size is $300\times 300\mathrm{n}{\mathrm{m}}^2$ and the imaging conditions are $V_{\mathrm{sample}}=-2\mathrm{V}$ and $I_t=0.3\mathrm{nA}$.

Figure 2

An In island grown on Pb wetting layer at RT. Image size is $300\times 300\mathrm{n}{\mathrm{m}}^2$ and the imaging conditions are $V_{\mathrm{sample}}=-2\mathrm{V}$ and $I_t=0.3\mathrm{nA}$.

Figure 3

(a) Al islands grown at RT on an Al wetting layer prepared at 150 K. The inset shows the small islands of (001) orientation among the large islands. The image size is $300\times 300\mathrm{n}{\mathrm{m}}^2$ and the imaging conditions are $V_{\mathrm{sample}}=-2\mathrm{V}$ and $I_t=0.3\mathrm{nA}$. (b) Al film with a thickness of $\sim 7\mathrm{ML}$ grown at 150 K.

Figure 4

$dI/dV$ curves demonstrating the QWSs of Pb, In, and Al islands with various thickness.

Figure 5

STM images of a large Pb island (a) and a large In island (d) of similar size. Inset: QWSs of 9 ML Pb island and 9 ML In island measured at the positions marked “×” in the figures. At the same positions, a set of selected differential conductance $dI/dV$ spectra at various temperatures measured with setpoint $V_{\mathrm{sample}}=-20\mathrm{mV}$ and $I_t=1\mathrm{nA}$ are shown in (b) and (e), respectively, for Pb and In islands. The deduced ZBC data versus temperature for Pb island and In island are plotted in (c) and (f).

Figure 6

Pseudogap state in Al island. (a) STM image of connected Al islands. (b) QWSs of a 9 ML Al island, measured at the setpoint $V_{\mathrm{sample}}=-2\mathrm{V}$ and $I_t=1\mathrm{nA}$. (c) $dI/dV$ spectra of the Al island around the Fermi level at various temperatures. The setpoint for these $dI/dV$ spectra is $V_{\mathrm{sample}}=-100\mathrm{mV}$ and $I_t=1\mathrm{nA}$.

Figure 7

Pseudopeak state in 8 ML thick Al island. (a) $dI/dV$ spectrum of the 8 ML Al island around the Fermi level. The setpoint for this $dI/dV$ spectra is $V_{\mathrm{sample}}=-1\mathrm{V}$ and $I_t=0.3\mathrm{nA}$. (b) Simulated $dI/dV$ spectrum with electron-phonon scattering model which shows the pseudopeak feature (solid line). Without the electron-phonon scattering effect, the $dI/dV$ peak is shown in the dashed line.

Figure 8

Pseudopeak states in 18 ML thick Pb island. (a) and (b) $dI/dV$ spectra around the Fermi level measured at different positions on an 18 ML Pb island. The setpoint for these $dI/dV$ spectra is $V_{\mathrm{sample}}=-100\mathrm{mV}$ and $I_t=1\mathrm{nA}$. (c) and (d) Simulated $dI/dV$ spectra with electron-phonon scattering model. (e) $dI/dV$ spectra at various temperatures showing the evolution of superconductivity and pseudopeak for this Pb island. The superconductivity feature has been broadened by the instrument.

Figure 9

Transition from pseudopeak states to pseudogap states. (a) QWS near the Fermi level measured at different positions on an 8 ML Pb island at 8 K. The setpoint $V_{\mathrm{sample}}=-2\mathrm{V}$ and $I_t=1\mathrm{nA}$. (b) Pseudogap or pseudopeak state on this 8 ML Pb island with slightly different QWS positions. The setpoint $V_{\mathrm{sample}}=-20\mathrm{mV}$ and $I_t=1\mathrm{nA}$. The dashed line is the $dI/dV$ curve measured on a 9 ML Pb island for reference. (c) The simulated QWS at different energy positions induced by changes in the phase shift. (d) The simulated pseudogap and pseudopeak features.

Figure 10

(a) STM images of a number of Pb islands of different sizes. (b) $dI/dV$ spectra of 9 ML Pb islands with different lateral sizes measured at 8 K. All the spectra are measured at the setpoint of $V_{\mathrm{sample}}=-20\mathrm{mV}$ and $I_t=1\mathrm{nA}$. (c) Calculated differential conductance $dI/dV$ spectra of 9 ML Pb islands with DCB model.

Figure 11

(a)–(c) $dI/dV$ spectra of Al islands with different lateral sizes measured at 4.5 K together with simulated results with different models which are presented in the text. First two spectra are measured at the setpoint of $V_{\mathrm{sample}}=-20\mathrm{mV}$ and $I_t=1\mathrm{nA}$, the third spectrum is taken at the setpoint of $V_{\mathrm{sample}}=2\mathrm{V}$ and $I_t=1\mathrm{nA}$. Values of parameters used in the calculation are as follows: (a) reflectivity of interface $R=0.6$, phase shift $\mathrm{\Phi }=0.55\pi$; (b) $R=0.11R_K, C=15aF$, where $R_K$, the resistance quantum, is given by $R_K=h/e^2=25.8K\mathrm{\Omega }$; (c) $R_T=2G\mathrm{\Omega }, R=25M\mathrm{\Omega },C_T=0.617aF,C=4.9aF,Q_0=-0.380e$.

Figure 12

Comparison of the width (a) and the depth (b) of pseudogap of islands formed by different materials at 8 K. The Debye energies are taken from Ref. [37] and $\lambda =1.55$ for Pb, $\lambda =0.95$ for In, and $\lambda =0.44$ for Al are taken from Refs. [36, 38, 39] for the respective bulk materials.

Figure 13

Comparison between experimental pseudogap and simulated pseudogap by Eqs. (1, 2, 3). (a) For Pb, with simulations employed different phonon spectra. (b) For In, with simulations employed different phonon spectra. (c) For Al, with simulations employed different phonon spectra.