Probing the Nodal Structure of Landau Level Wave Functions in Real Space

The inversion layer of $p\text{−}\mathrm{InSb}(110)$ obtained by Cs adsorption of 1.8% of a monolayer is used to probe the Landau level wave functions within smooth potential valleys by scanning tunneling spectroscopy at 14 T. The nodal structure becomes apparent as a double peak structure of each spin polarized first Landau level, while the zeroth Landau level exhibits a single peak per spin level only. The real space data show single rings of the valley-confined drift states for the zeroth Landau level and double rings for the first Landau level. The result is reproduced by a recursive Green function algorithm using the potential landscape obtained experimentally. We show that the result is generic by comparing the local density of states from the Green function algorithm with results from a well-controlled analytic model based on the guiding center approach.

Figure 1

(a) Disorder potential of 2DES as determined by the average energy of the two spin components of ${\mathrm{LL}}_0$ [14]. (b)–(g) LDOS within the spin level $\uparrow$ of ${\mathrm{LL}}_0$ and ${\mathrm{LL}}_1$, calculated by recursive Green function algorithm [15] using $E_{\mathrm{pot}}(\mathbf{r})$ of (a), $m^{*}=0.03m_e$, $g=-21$, ${\alpha }_R=1\mathrm{eV}\AA$, $B=14\mathrm{T}$, and smooth potential boundary condition ($>4l_B$) outside the displayed area; energies with respect to the LL center, determined by spatially averaging the LDOS, are marked on top; red arrows mark potential valley where distinct doubling of lines is observed. (h) Single $dI/dV$ curve recorded within a potential valley. ${\mathrm{LL}}_n$, $\hslash {\omega }_c$, and opposite spin levels are highlighted, $B=14\mathrm{T}$, $V_{\mathrm{stab}}=50\mathrm{mV}$, $I_{\mathrm{stab}}=150\mathrm{pA}$, $V_{\text{mod}}=0.75{\mathrm{mV}}_{\mathrm{rms}}$. Differences between the two $\hslash {\omega }_c$’s relate to the known nonparabolicity of the InSb conduction band [32]. Note the two double peaks in ${\mathrm{LL}}_1$.

Figure 2

(a) Cross section through the potential minimum of (b) (black) and energy cuts shifted by LL energy and Zeeman energy as indicated by ${\mathrm{LL}}_n$ and spin arrows. (b) $E_{\mathrm{pot}}(\mathbf{r})$ of potential minimum deduced from average ${\mathrm{LL}}_0$ peak energy [14] and subsequently shifted down by $\hslash {\omega }_c/2$. (c) Left: $dI/dV$ spectrum recorded in the minimum of the potential valley in (b), $B=14\mathrm{T}$, $V_{\mathrm{stab}}=50\mathrm{mV}$, $I_{\mathrm{stab}}=150\mathrm{pA}$, $V_{\text{mod}}=0.75{\mathrm{mV}}_{\mathrm{rms}}$, where dashed lines highlight voltages of $dI/dV$ images in (d)–(k); right: $\mathrm{LDOS}(E)$ in the potential minimum of (b) calculated by recursive Green function approach, $m^{*}=0.028m_e$, $g=-21$, ${\alpha }_R=0.5\mathrm{eV}$, $B=14\mathrm{T}$. The LDOS peaks are broadened by a Lorentzian function representing a lifetime broadening of $0.05\hslash {\omega }_c$ in order to mimic the smallest peak width found within the experiment. Dashed lines highlight the energies of the LDOS images in (l)–(s), pink dots labeled ${\overline{E}}_{\mathrm{LL}n,\uparrow }$ mark the average LL energy obtained in the area of Fig. 1. (d)–(k) $dI/dV$ images recorded at the voltages marked in (c), same parameters as (c). (l)–(s) Calculated LDOS images at the energies marked in (c), same parameters as (c). Image sizes of (d)–(k) and image sizes of (l)–(s) are identical.

Figure 3

(a)–(c) LDOS images at the energies marked on top as calculated for $E_{\mathrm{pot}}(\mathbf{r})$ of Fig. 2 by recursive Green function approach; parameters as in Figs. 1. (d)–(f) LDOS images using the same potential and calculated by the analytic guiding center approach including ${\alpha }_R$ [16]; same parameters as in (a)–(c). (g)–(i) Same as (d)–(f) but neglecting ${\alpha }_R$.

Figure 4

(a) Sketch of the squared wave functions $|\mathrm{\Psi }(x,y){|}^2$ of ${\mathrm{LL}}_0$ and ${\mathrm{LL}}_1$ (gray, blue, red, and violet full lines) within the $E_{\mathrm{pot}}(\mathbf{r})$ (blue-green and black dashed line energetically shifted by $\hslash {\omega }_c$); resulting energies of the LL wave functions are indicated by the equally colored, horizontal, dotted lines cutting the energy axis of the middle plot; the tip (triangle of blue circles) tunnels exclusively into the $|{\mathrm{\Psi }}_{{\mathrm{LL}}_n}(x,y){|}^2$ at the position marked by a vertical, black dotted line. This results in the spectrum sketched in the middle plot. Additional Zeeman splitting leads to doubling of the spectral features as sketched in the right-hand plot. (b),(c) $dI/dV$ spectra recorded at the positions marked by the equally colored crosses in (d)–(g), $B=14\mathrm{T}$, $V_{\mathrm{stab}}=50\mathrm{mV}$, $I_{\mathrm{stab}}=150\mathrm{pA}$, $V_{\text{mod}}=0.75{\mathrm{mV}}_{\mathrm{rms}}$; dashed lines mark the voltages of the $dI/dV$ images in (d)–(g). (d)–(g) $dI/dV$ images at the voltages marked in (c). Note the nearly identical pattern and energy splitting of (d) and (f), respectively, (e) and (g).