Electron-electron interactions in isolated and realistic quantum dots: A density functional theory study

We use Kohn-Sham spin-density-functional theory to study the statistics of ground-state spin and the spacing between conductance peaks in the Coulomb blockade regime for both two-dimensional isolated and realistic quantum dots. We make a systematic investigation of the effects of electron-electron interaction strength and electron number on both the peak spacing and spin distributions. A direct comparison between the distributions from isolated and realistic dots shows that, despite the difference in the boundary conditions and confining potential, the statistical properties are qualitatively the same. Strong even/odd pairing in the peak spacing distribution is observed only in the weak $e-e$ interaction regime and vanishes for moderate interactions. The probability of high spin ground states increases for stronger $e-e$ interaction and seems to saturate around $r_s\sim 4$. The saturated value is larger than previous theoretical predictions. Both spin and conductance peak spacing distributions show substantial variation as the electron number increases, not saturating until $N\sim 150$. To interpret our numerical results, we analyze the spin distribution in the even $N$ case using a simple two-level model.

Figure 1

(Color online) Upper panel: Contours of $V_{\text{ext}}$ in QOP system with parameters $a={10}^{-4}$, $b=\pi ∕4$, $\lambda =0.53$, and $\gamma =0.2$. Lower panel: Electron density at $N=200$ and $S=0$. It is obvious that the electron density in the isolated dot is far from uniform even within the classically allowed area; in the case shown here, the maximal density is $110\%$ larger than that in the center.

Figure 2

(Color online) Distributions of dimensionless peak spacing [Eq. (4)] for even (solid, red) and odd (dashed, blue) $N$ in isolated quantum dots at different interaction strengths. The inset of each figure shows the corresponding ground state spin distribution for $N$ even (solid, red) and odd (shaded, blue). A sliding window is used in estimating the probability density of peak spacing, yielding a smooth curve rather than a histogram; the width of the Gaussian window is taken as $\omega =\left({\xi }_{\max }-{\xi }_{\min }\right)∕\left(2\right(1+{\mathrm{Log}}_{2}n\left)\right)$ where $n$ is the number of data points. Note the striking evolution from clearly different even and odd distributions coupled with minimum ground state spin to very similar distributions with a large fraction of high spin ground states.

Figure 3

(Color online) Schematic illustration of a realistic quantum dot studied in this paper. The dot is formed based on a GaAs-AlGaAs heterostructure, which consists of (from the bottom) an undoped GaAs substrate, an undoped AlGaAs spacer layer, a $n$-doped AlGaAs layer, and finally a GaAs cap layer. The widths for the spacer, doped, and cap layers are denoted, as $s$, $d$, and $c$, respectively. In this paper, $s=15\text{nm}$, $d=10\text{nm}$, and $c=10\text{nm}$ are taken.

Figure 4

Illustration of the shape of top confining gate used in this study. A negative voltage is imposed on the shaded region, depleting the electrons underneath so that the motion of electrons is confined to a small region. To obtain enough statistics, 20 different irregular confining shapes are generated by taking different values of $x_1$, $x_2$, $y_1$, and $y_2$. The size of the confining gate, as well as the value of the imposed gate voltage, determines the electron density, and therefore the value of $r_s$. In the typical experimental regime, with $r_s\sim 1.0$ and $N\sim 100$, the size is about $L_x\sim L_y\sim 500\text{nm}$.

Figure 5

(Color online) Upper panel: Contours of a sample $V_{\text{ext}}^{\mathrm{R}2\mathrm{D}}$. With the shape of the confining gate shown in Fig. 4, the confining potential is very irregular; its classical dynamics is believed to be chaotic. Lower panel: Electron density at $N=200$ and $r_s\sim 1.0$ in a typical confining potential. It is much more uniform that that in the isolated QOP shown in Fig. 1.

Figure 6

(Color online) Comparison of QOP and R2D models: distributions of conductance peak spacing (solid for even $N$, dashed for odd $N$) and ground state spin (insets) in (a) QOP at $r_s\sim 1.0$; (b) R2D at $r_s\sim 1.0$; (c) QOP at $r_s\sim 1.5$; (d) R2D at $r_s\sim 1.5$. For the QOP results $N=80–160$, while for the R2D model the additional gate averaging allows a smaller range $N=110-130$. Results for the two types of dots are in qualitative agreement.

Figure 7

(Color online) Probabilities of different spin states as a function of $r_s$ for $N=11–40$ in realistic quantum dots (circles, $S=0$; squares, $S=1$; diamonds, $S=2$; up triangles, $S=1∕2$; down triangles, $S=3∕2$). The error bars shown in the plot are calculated from the standard bootstrap method. The ground state spin probabilities saturate at large $r_s$.

Figure 8

(Color online) Distributions of dimensionless peak spacing for even (solid, red) and odd (dashed, blue) $N$ in realistic quantum dots in different $N$ ranges at $r_s\sim 1.0$. There is a clear difference between even and odd distributions for small dots which, however, vanishes for large dots.

Figure 9

(Color online) Probabilities of different spin states for different $N$ ranges at $r_s\sim 1.0$ (circles, $S=0$; squares, $S=1$; up triangles, $S=1∕2$; down triangles, $S=3∕2$). The error bars shown in the plot are calculated from the standard bootstrap method. Note the substantial change from small to large dots.

Figure 10

(Color online) Two-level model analysis of spin distribution for $N=11–40$ as a function of interaction strength. (a) Probability of triplet ground state as a function of $r_s$ from TLM-ED (circles), TLM-HF (down triangles), and KS (up triangles). (b) Mean of diagonal interaction matrix elements, $J_{11}$, $J_{12}$, and $K_{12}$, as well as the mean of the off-diagonal matrix element $V_{1121}$, all in units of $\Delta$.

Figure 11

(Color online) Two-level model (TLM) analysis of spin distribution as a function of $N$. (a) Probability of triplet ground state as a function of $N$ from TLM-ED (circles), TLM-HF (down triangles), and KS (up triangles). (b) Means of diagonal interaction matrix elements, $J_{11}$, $J_{12}$, and $K_{12}$, as well as that of the off-diagonal matrix element $V_{1121}$, all in units of $\Delta$.