Visualizing the multifractal wave functions of a disordered two-dimensional electron gas

The wave functions of a disordered two-dimensional electron gas at the quantum-critical Anderson transition are predicted to exhibit multifractal scaling in their real space amplitude. We experimentally investigate the appearance of these characteristics in the spatially resolved local density of states of the two-dimensional mixed surface alloy ${\mathrm{Bi}}_{\mathrm{x}}{\mathrm{Pb}}_{(1-\mathrm{x})}/\mathrm{Ag}\left(111\right)$, by combining high-resolution scanning tunneling microscopy with spin- and angle-resolved inverse-photoemission experiments. Our detailed knowledge of the surface alloy's electronic band structure, the exact lattice structure, and the atomically resolved local density of states enables us to construct a realistic Anderson tight binding model, and to directly compare the measured local density of states characteristics with those from our model calculations. The statistical analyses of these two-dimensional local density of states maps reveal their log-normal distributions and multifractal scaling characteristics of the underlying wave functions with a finite anomalous scaling exponent. Finally, our experimental results confirm theoretical predictions of an exact scaling symmetry for Anderson quantum phase transitions in the Wigner-Dyson classes.

Figure 1

Multifractal wave-function scaling in two dimensions. (a) Calculated LDOS ($|\mathrm{\Psi }\left(\vec{r}\right){|}^2$) map (approx. $50\times 50$ lattice sites) of a two-dimensional electron gas exhibiting multifractal scaling behavior with power-law decay characteristics. The LDOS scales with different exponents $\alpha$ along different topographic lines (black solid lines). (b) Schematic plot of the singularity spectrum $f\left(\alpha \right)$ for a $d=2$ dimensional system with multifractal (purple) and metallic (black) scaling behavior.

Figure 2

Electronic properties and atomic lattice structure of the mixed surface alloy. (a) STM topography of the ${\text{Bi}}_{0.79}{\text{Pb}}_{0.21}\text{/Ag(111)}$ mixed surface alloy (setpoint $V=3\mathrm{mV}$, $I=7\mathrm{nA}$). Scale bar 5 nm. (b) Schematic sketch of the mixed surface alloy's atomic lattice structure. The alloy atoms substitute every third Ag atom of the surface layer, forming a $\sqrt{3}\times \sqrt{3}\text{R}{30}^{\circ }$ lattice structure. $\mathrm{\Delta }\mathrm{z}$ denotes the apparent height difference between the Bi and Pb atoms as measured with the STM; X denotes an alloy lattice vacancy. (c) Histogram analysis of the topography in (a) displaying the occurrence of the relative STM tip apparent height. (d) Band structure of the Bi/Ag(111) surface alloy along $\overline{\mathrm{\Gamma }}\overline{K}$ measured by spin-resolved IPE. The data points indicate the peak positions with the solid lines as guides to the eye. Purple (blue) colors refer to the spin-quantization axis forming a right (left) handed coordinate system with the $\overline{\mathrm{\Gamma }}\overline{K}$ line, and the sample normal. (e) $dI/dV$ tunneling spectra (setpoint $V=-1\mathrm{V}$, $I=1\mathrm{nA}$, $V_{\mathrm{mod}}=25\mathrm{mV}$) measured with the STM tip located at different surface locations, which are indicated in (a). The spectral features are labeled according to their band character.

Figure 3

(a) STM topography of the ${\text{Bi}}_{0.79}{\text{Pb}}_{0.21}\text{/Ag(111)}$ mixed surface alloy (setpoint $V=3\mathrm{mV}$, $I=7\mathrm{nA}$). Scale bar 10 nm. The purple box ($20\times 20{\mathrm{nm}}^2$) highlights the area in which the normalized $dI/dV$ maps have been measured: (b) at the $s{p}_{\mathrm{z}}$-band edge ($+0.08\mathrm{eV}$), (c) within the $s{p}_{\mathrm{z}}$ band ($-0.4\mathrm{eV}$) (setpoint $I=1\mathrm{nA}$, $V_{\mathrm{mod}}=25\mathrm{mV}$). (d) Reconstructed lattice model of the mixed surface alloy, where the topography in (a) serves as structural input for the Anderson tight-binding model calculation. Panel (e) and (f) show the calculated normalized LDOS maps at the $s{p}_{\mathrm{z}}$-band edge and inside the $s{p}_{\mathrm{z}}$ band, respectively (please refer to Table 1 for numerical parameters).

Figure 4

Multifractal analysis of the measured and calculated LDOS maps. (a) Left panel: Statistical distribution of the $dI/dV$ amplitude for the map measured at $V=80$ mV near the $s{p}_{\mathrm{z}}$-band edge in Fig. 3. Right panel: The same analysis applied to the map measured at $V=-400\mathrm{mV}$ inside the $s{p}_{\mathrm{z}}$ band in Fig. 3. The black lines represent the corresponding calculated distribution functions. (b) Left panel: Statistical distribution of the LDOS amplitude for the map calculated at the $s{p}_{\mathrm{z}}$-band edge in Fig. 3. Right panel: The same analysis applied to the calculated map inside the $s{p}_{\mathrm{z}}$-band in Fig. 3. (c) $f\left(\alpha \right)$ of the $dI/dV$ maps in Fig. 3 (purple circles) and Fig. 3 (blue diamonds). (d) $f\left(\alpha \right)$ of the calculated LDOS maps in Fig. 3 (purple circles) and Fig. 3 (blue diamonds). (e) Bottom panel: Magnification of the $f\left(\alpha \right)$ spectra shown in (c). The black solid lines are the quadratic fits to the data. Top panel: Magnified view around the $f\left(\alpha \right)$ maxima.

Figure 5

Calculated band structure of the $s{p}_{\mathrm{z}}$ band. The corresponding tight-binding parameter are given in Table 1. The energy scale is given in units of $E/t$.

Figure 6

NPR as a function of the energy, $E$, from the band onset, $E_0=0$, for each calculated eigenstate in the $s{p}_{\mathrm{z}}$ band. The shaded Gaussian shows the weighted integration interval for the calculation of the LDOS map in Fig. 3. The insets show the spatially resolved eigenstate densities, $|\mathrm{\Psi }(\vec{r},E){|}^2$, at the respective energies (purple/white: high/low values, edge length 20 nm).

Figure 7

(a) Normalized $dI/dV$ map [area: purple box in STM topography of Fig. 3] measured around the band minimum of the $m_{\mathrm{j}}=3/2$-band (setpoint $V=1\mathrm{V}$, $I=1\mathrm{nA}$, $V_{\mathrm{mod}}=25\mathrm{mV}$, scale bar 5 nm). (b) Statistical distribution of the corresponding $dI/dV$ amplitude.