Spin-dependent localization of electrons in quadruple quantum dots

An all electrical manipulation with singlet and triplet states of an electron pair in a quadruple quantum dot (QD) has been studied by a configuration interaction method. We have investigated two nanodevices that consist of four weakly coupled lateral QDs, in which the electron localization can be changed by tuning the external voltages: the gate voltage in the first nanodevice and the source-drain voltage in the second. The electrons that—in the initial state—singly occupy the two identical input QDs are transferred to the two asymmetric output QDs by the sufficiently high electric field. We have shown that—for the QDs with suitably chosen parameters—the localization of the electrons in the output QDs depends on the spin states of the electron pair and the final spin state can be uniquely determined by a single-shot measurement of the electric charge gathered in only one of the output QDs. This leads to a possibility of performing the controlled operations on the electron-spin qubits in the quadruple QDs. We have found that the classical exclusive OR (XOR) logic gate can be performed with the electric field-induced transitions and the output read out by the spin-to-charge conversion.

Figure 1

(Color online) Potential energy $U$ of the electron in the quadruple QD as a function of $x$ and $y$ coordinates in the final state of devices (a) A and (b) B described in the text. Ovals schematically show the sizes of the QDs labeled by $l1$, $l2$, $r1$, and $r2$.

Figure 2

(Color online) Potential energy $U$ of the electron in the quadruple QD as a function of $x$ for $y$ fixed at the values corresponding to the straight lines joining the centers of pairs of QDs $l1\text{−}r1$ and $l2\text{−}r2$ for device (a) A and (b) B. Dashed (blue) [solid (red)] curves show the initial $\left(i\right)$ [final $\left(f\right)$] potential-energy profiles.

Figure 3

(Color online) (a) Energy levels $E_0$, $E_1$, and $E_2$ of one-electron states ${\phi }_0$, ${\phi }_1$, and ${\phi }_2$ as functions of energy $W$ of the potential-well bottom of left QDs for device A. (b) Energy of singlet [$E_S$, solid (red) curve] and triplet [$E_T$, dashed (blue) curve)] states and exchange energy $J=E_T-E_S$ [(green) curve with square symbols] as functions of $W$ for device A. Vertical dashed lines correspond to the selected characteristic values of $W$.

Figure 4

(Color online) (a) Energy levels $E_0$, $E_1$, and $E_2$ of one-electron states ${\phi }_0$, ${\phi }_1$, and ${\phi }_2$ as functions of electric field $F$ for device B. (b) Energy of singlet [$E_S$, solid (red) curve] and triplet [$E_T$, dashed (blue) curve)] states and exchange energy $J=E_T-E_S$ [(green) curve with square symbols] as functions of electric field $F$ for device B. Vertical dashed lines correspond to the selected characteristic values of $F$.

Figure 5

(Color online) Contours of one-electron density distribution on $x\text{−}y$ plane for singlet (left panels) and triplet (right panels) states in device A. Panels (a)–(h) display the results for $W=W_0,W_1,W_2,W_3$ marked in Fig. 3. For these nanodevice parameters the localization of electrons in the final states [panels (g) and (h)] is different for both the spin states and the measurement of the electric charge in one of the right QDs allows us to distinguish the singlet and triplet states. QPC1 and QPC2 show two alternative positions of the quantum point contact.

Figure 6

(Color online) Contours of one-electron density distribution on $x\text{−}y$ plane for singlet (left panels) and triplet (right panels) states in device B. Panels (a)–(h) display the results for electric field $F=F_0,F_1,F_2,F_3$ marked in Fig. 4. For these nanodevice parameters the localization of electrons in the final states [panels (g) and (h)] is different for both the spin states and the measurement of the electric charge in one of the right QDs allows us to distinguish the singlet and triplet states. QPC1 and QPC2 show two alternative positions of the quantum point contact.

Figure 7

(Color online) Contours of one-electron probability density on $x\text{−}y$ plane in the singlet [panels (a) and (b)] and triplet [panels (c) and (d)] final states for device A. The occupancy of the right QDs by the electrons is the same in each spin state since the size of $\text{QD}\left(r2\right)$ is [(a) and (c)] too small and [(b) and (d)] too large.

Figure 8

(Color online) Contours of one-electron probability density on $x\text{−}y$ plane in the singlet [panels (a) and (b)] and triplet [panels (c) and (d)] final states for device B. The occupancy of the right QDs by the electrons is the same in each spin state since the size of $\text{QD}\left(r2\right)$ is [(a) and (c)] too small and [(b) and (d)] too large.

Figure 9

(Color online) Regimes I–V (separated by solid curves) of parameters $R_{r1}$ and $U_{r1}^0$ characterizing the $\text{QD}\left(r1\right)$. The final localization of the electrons in the right QDs is schematically shown in the right upper inset. In nanodevices A and B with the parameters from regimes II and IV (gray), the final localization of the electron pair is uniquely determined by its spin. The left upper inset is the zoom of the rectangular region shown on the main panel: solid (red) [dashed (green)] curves display the boundaries of regimes I–V obtained for device A (B).

Figure 10

(Color online) (a) Truth table of the classical XOR logic gate. In columns $X$ and $Y$ $\left(Z\right)$ the input (output) logical values are listed, numbers (1)–(4) in the first column identify the four XOR gate operations. (b) Outline of the nanodevice with the localization of electrons in the QDs (circles) schematically shown for operations (1)–(4). The input logical value 0 (1) is encoded as the spin-up (-down) state of the electron localized in one of the left QDs. Quantum point contacts QPC1 and QPC2 measure the electric charge of $\text{QD}\left(r1\right)$ and $\text{QD}\left(r2\right)$, respectively. The output logical values are defined as charge states $q\left(r1\right)$ $\left[q\left(r2\right)\right]$ of $\text{QD}\left(r1\right)$ $\left[\text{QD}\left(r2\right)\right]$ as follows: $q\left(r1\right)=0$ $\left[q\left(r2\right)=-2e\right]$ corresponds to logical value 1 and $q\left(r1\right)=-e$ $\left[q\left(r2\right)=-e\right]$ corresponds to logical value 0. The one-electron-spin states are depicted by arrows: up (down) arrow corresponds to the spin up (down). The double (blue) [solid (red)] arrows schematically show the localization of electrons in the input [output] states.