“Artificial Atoms” in Magnetic Fields: Wave-Function Shaping and Phase-Sensitive Tunneling

We demonstrate the possibility to influence the shape of the wave functions in semiconductor quantum dots by the application of an external magnetic field $B_z$. The states of the so-called $p$ shell, which show distinct orientations along the crystal axes for $B_z=0$, can be modified to become more and more circularly symmetric with an increasing field. Their changing probability density can be monitored using magnetotunneling wave function mapping. Calculations of the magnetotunneling signals are in good agreement with the experimental data and explain the different tunneling maps of the $p^{+}$ and $p^{-}$ states as a consequence of the different sign of their respective phase factors.

Figure 1

Top: Schematic sample structure. Bottom: Capacitance-voltage traces, showing the subsequent filling of the InAs islands with 1–6 electrons. The first two electrons form the so-called “$s$ shell,” the 3rd to 6th electron the “$p$ shell.” With increasing magnetic field $B_z$, the $p$ shell exhibits Zeeman splitting with two states decreasing in energy ($p^{-}$) and two increasing ($p^{+}$).

Figure 2

Experimental maps of the tunneling current (capacitance amplitude) as a function of the in-plane magnetic field ($B_x,B_y$) for different constant perpendicular fields $B_z$. With increasing $B_z$, the maps develop from a $x\mathrm{\text{−}}y$ symmetry towards circular symmetry.

Figure 3

Calculated momentum space representation of the lowest quantum dot states ($s$, $p^{+}$, and $p^{-}$, leftmost column) and three degenerate emitter states of the lowest Landau level (angular momentum $l=0,-1,-2$, top row). The center panels show the overlap integrals of these states in matrix form. The number in each panel indicates the maximum of the color scale. The rightmost column depicts the sum of the overlap integrals, which corresponds to the calculated magnetotunneling signal. The plots scan a momentum range of $\pm 8\times {10}^8{\mathrm{m}}^{-1}$, which corresponds, according to Eq. (1), to a field of $\pm 13\mathrm{T}$.