Orbital effect, subband depopulation, and conductance fluctuations in ballistic quantum dots under a tilted magnetic field

Using two-dimensional electron gases (2DEGs) confined to wide and narrow quantum wells, we study the magnetoconductance of ballistic quantum dots as a function of the well width and the tilt angle of the magnetic field $B$ with respect to the 2DEG. Both the wide and narrow quantum well dots feature magnetoconductance fluctuations (MCFs) at intermediate tilt angles, due to the finite thickness of the electron layer and the field-induced orbital effect. As $B$ approaches a strictly parallel configuration, a saturation of the MCFs’ spectral distribution is observed, combined with the persistence of a limited number of frequency components in the case of the narrow quantum well dot. It is found that the onset of saturation strongly depends on the width of the confining well. Using the results of self-consistent Poisson-Schrödinger simulations, the magnetoconductance is rescaled as a function of the Fermi level in the 2DEG. We perform a power spectrum analysis of the parallel field-induced MCFs in the energy space and find a good agreement with theoretical predictions.

Figure 1

Schematic of the experimental setup with in situ tilting of the sample. The tilt angle $\theta$ is determined by monitoring the 2DEG transverse magnetoresistance in the Hall bar located next to the dot. Upper inset: Schematic diagram of the dot. Dark regions represent etched areas of the 2DEG. The dot is connected to 2D reservoirs through two quantum point contacts.

Figure 2

Fast Fourier transform (FFT) of Shubnikov–de Haas oscillations for the NQW and WQW. The positions of the FFT peaks correspond to the subband densities.

Figure 3

(a) Magnetoconductance $g$ of the NQW dot at $T=0.3\mathrm{K}$ under a strictly perpendicular $B$. (b) Magnetoconductance of the NQW dot for tilt angles close to 90° ($\theta =89°$, 89.3°, 89.6°, 89.9°, and 90° from the upper to the lower curve). The traces are shifted for clarity in steps of $0.2e^2∕h$ with respect to the $\theta =90°$ curve. Inset: Transverse resistance of the parent 2DEG at indicated tilt angles, measured on the adjacent Hall bar (see Fig. 1).

Figure 4

(a) $T$ dependence of the NQW dot magnetoconductance under a strictly perpendicular $B$. The temperatures are 2.1, 1.1, and $0.3\mathrm{K}$ from top to bottom. (b) The temperature dependence of the NQW dot magnetoconductance under a strictly parallel $B$. The temperatures are 2.8, 1.5, 0.8, and $0.3\mathrm{K}$ from top to bottom, respectively. The gray area indicates the $B$ range for which the MCFs’ peak-to-peak amplitude is determined (see the text). The traces in panels (a) and (b) are shifted in steps of $0.1e^2∕h$ with respect to the $T=0.3\mathrm{K}$ curves.

Figure 5

(a) MCFs amplitude vs $T$ at $\theta =0°$ (standard deviation, ◇) and $\theta =90°$ (peak to peak, ∎) in the NQW. The dashed curves are two-parameter power law fits to data points. (b) $T$ dependence of the dephasing time ${\tau }_{\phi }$ extracted from Eq. (1) for the MCFs at $\theta =0°$ in the NQW.

Figure 6

(a) Low-temperature magnetoconductance of the WQW dot under a strictly perpendicular $B$. (b) Magnetoconductance of the WQW dot at indicated tilt angles. The traces are shifted in steps of $0.5e^2∕h$ with respect to the $\theta =90°$ magnetoconductance curve. Inset: Transverse magnetoresistance of the 2DEG measured on the adjacent Hall bar (see Fig. 1). Note the parabolic-like variation of $\Delta R$ already present for the NQW dot on the transverse magnetoresistance of Fig. 3c and the step around $4.2\mathrm{T}$.

Figure 7

(a) Temperature dependence of the WQW dot magnetoconductance under a strictly perpendicular $B$. The temperatures are 2.5, 1.5, and $0.3\mathrm{K}$ from top to bottom. (b) The magnetoconductance curve of the WQW dot under a strictly parallel $B$ at various $T$’s. From the upper to the lower curve, the temperatures are 0.3, 0.6, 1.5, and $2.5\mathrm{K}$, respectively. The gray area indicates the $B$ range for which the fluctuation’s amplitude is determined (see the text). The traces in panels (a) and (b) are shifted in steps of $0.4e^2∕h$ with respect to the $T=0.3\mathrm{K}$ magnetoconductance curves.

Figure 8

$T$ dependence of the dephasing time ${\tau }_{\phi }$ extracted from Eq. (1) for the $\theta =0°$ (▵) and $\theta =90°$ (●) MCFs in the WQW. The dashed lines are two-parameter power law fits to the data points.

Figure 9

MCFs of the NQW (upper row) and WQW (lower row) under a strictly perpendicular (left column) and parallel (right column) $B$. To isolate the MCFs, a slowly varying background corresponding to the high-$T$ magnetoconductances (Figs. 4, 7) was removed from the raw curves of Figs. 3, 6, at the corresponding tilt angles.

Figure 10

The effective tilt angle ${\theta }_{B_c}$ calculated from the correlation field $B_c$ (see the text) is shown as a function of the angle $\theta$ extracted from Hall measurements. Here ${\theta }_{B_c}$ is calculated for $\theta ⩾84°$ for both the WQW (◻) and NQW (▴) dots. The plain and dotted curves are a guide to the eye. Upper inset: Zoom into the $\theta ⩾89.5°$ range for the NQW dot. Lower inset: Power spectra of the WQW dot MCFs for $\theta =0°$, 89°, and 90°. The dashed curve is a fit to the $\theta =0°$ data (see the text).

Figure 11

(Color online) Dispersion relations of the unpatterned 2DEG confined in the NQW sample at indicated values of the parallel $B$. The 2DEG has a sheet density $n_s=2.2\times {10}^{11}{\mathrm{cm}}^{-2}$. The conduction band edge $E_C$ is calculated with respect to the Fermi energy $E_F=0$.

Figure 12

(Color online) Dispersion relations of the unpatterned 2DEG confined in the WQW sample at indicated values of parallel $B$. The conduction band edge $E_C$ is calculated with respect to the Fermi energy $E_F=0$. The 2DEG has a sheet density $n_s=3.2\times {10}^{11}{\mathrm{cm}}^{-2}$ with two occupied subbands at $B=0$. The lower (upper) subband $E_C\left(k_x\right)$ relations are represented as plain (dashed) curves.

Figure 13

Self-consistent Fermi lines at indicated values of the magnetic field $B$ and for (a) the ground state of the NQW, (b) the upper, and (c) lower occupied subbands of the WQW sample. The right panel in (a) is a closeup of the Fermi lines in the small window indicated on the left panel. In (b), the range of investigated $B$’s is limited to $4\mathrm{T}$, as the subband is fully depleted for $B>4.2\mathrm{T}$.

Figure 14

(a) Self-consistent calculation of the variation of $E_F$ with $B$ for the ground state of the NQW and for the two occupied subbands of the WQW. Here $E_F$ is calculated with respect to the corresponding conduction band edges. Inset: The calculated $E_F\left(B\right)$ in the NQW. (b) Calculated sheet densities $n_s$ of the lower and upper occupied subbands in the WQW with increasing magnetic field $B$. The total density is $3.2\times {10}^{11}{\mathrm{cm}}^{-2}$. The gray regions indicate the $B$ range where only one subband is occupied in the WQW sample.

Figure 15

(a) Variation with magnetic field $B$ of the self-consistent effective Fermi energy $\delta E_F$ in the narrow (plain curve) and wide (dotted curve) quantum wells. For the WQW, the two $E_F\left(B\right)$ relations for the upper and lower occupied subbands shown in Fig. 14a were merged into a single $\delta E_F\left(B\right)$ using Eq. (15). MCFs of the narrow and wide quantum wells as a function of $\delta E_F$. The gray region indicates the energy range where only one subband is occupied in the WQW sample.

Figure 16

(Color online) Frequency analysis as a function of energy of the WQW dot MCFs. Here $\delta g\left(\delta E_F\right)$ is Fourier transformed on the selected energy ranges (see the text) to obtain the average power spectra $S_E$ shown in the figure as dashed red and blue curves. The plain red and blue curves are one-parameter fits (see the text) to these $S_E\left(f_E\right)$ spectra. Insets: Dispersion relations of the wide quantum well at $B=0$ [$\delta g\left(E_F\right)=0$, two occupied subbands, lower panel] and $B=6\mathrm{T}$ [$\delta g\left(E_F\right)=3\mathrm{meV}$, one occupied subband, upper panel]. The position of the Fermi level is indicated as a dotted line.