We discuss our results in the following order: First, we discuss the parameters of the system for the energy spectrum and the electron density, including the spin ground-state configuration. Next, we calculate a complete phase diagram for the nonferromagnetic and Nagaoka ferromagnetic phases. This gives us insight on the configuration for which the Nagaoka ferromagnetism is the most robust. We then discuss a similar phase diagram but with a shallower potential. We investigate the case where we gradually change the configuration of dots to form a pseudo-1D chain of dots to understand the competition between the ferromagnetic and nonferromagnetic states and the effect of anisotropy on this competition. Lastly, we observe the effect on the Nagaoka phase of a changed potential of one of the quantum dots.
Nagaoka spin polarization
The anisotropy of the effective mass causes the x and y axes of the system to be inequivalent. Therefore, we vary both parameters, \(\mu _x\) and \(\mu _y\) to search for the high-spin configuration in the ground state. The potential and square root of the density of the ground state for different combinations of the parameters are plotted in Fig. 2. The potential depth considered for the following calculations is \(V_d=125\) meV. The densities are calculated using Eq. (5) for the ground state. For spacings \(\mu _x = 9\) nm, \(\mu _y = 7\) nm (Fig. 2a) the electrons are located near the center of four dots, the charge densities of the array are distinctly separated with little or no density overlap (Fig. 2b). Note that even though the dots are far away, the square root density in the center of dots is not circular, but is more spread in the x direction (Fig. 2a), which is a consequence of the mass anisotropy. For the rest of the plots with \(\mu _y\le 6\) nm the potential [see Fig. 2c,e,g] resembles two dumbbell-like shapes due to the electron tunneling in the zigzag direction, for which the assumed distance between the dots is smaller than in the armchair direction. The trace of electron tunneling between the dots can be observed in Fig. 2d for \(\mu _x = 8\) nm, \(\mu _y = 6\) nm. As we move the dots closer together in the y direction, we can observe in Fig. 2f that the square root densities of the top-bottom pair of dots now overlap considerably. Finally, for dots arranged in a rectangle with \(\mu _x = 6.8\) nm, \(\mu _y = 5.2\) nm, the square root electron density is highest along the edges of the rectangle.
The energy spectra for the same parameters are plotted in Fig. 3. For \(\mu _x = 9\) nm and \(\mu _y = 7\) nm, the ground state at \(B=0\) is a four-fold degenerate state with spin eigenvalue \(S=3/2\) but the spin doublet 1/2 is above with the energy difference of \(40\,\mu\)eV (Nagaoka gap \(\Delta E=-40\,\mu\)eV). The low-energy spectra are illustrated in Fig. 3a. Although the tunneling coupling between the dots is weak, it is already large enough to promote the spin-polarized quadruplet with the total spin eigenvalue of \(S=3/2\) and the z component eigenvalues \(\sigma _z=-3/2,-1/2,1/2,3/2\) in the ground state. An asymptotic case of large interdot distances corresponds to tunneling and interaction being negligible so that the quadruplet becomes degenerate with the spin \(S=1/2\) doublet.
Reducing the spacing between the dots to \(\mu _x = 8\) nm and \(\mu _y = 6\) nm, the energy gap between the lowest spin \(S=1/2\) and spin \(S=3/2\) states becomes \(100\,\mu\)eV with the high-spin ground state [see Fig. 3b].
The energy spectrum for \(\mu _x = 9\) nm and \(\mu _y = 5\) nm is plotted in Fig. 3c. In this case, the electrons interact weakly in the x-direction and strongly in the y-direction. We have effectively two extended quantum dots with an electron in one of the other two dots. Since the ground state of the two electrons for negligible magnetic field is a singlet state40, the top-bottom pair of dots will contain two electrons of opposite spins, resulting in the ground state being a spin \(S=1/2\) state due to the spin of the solitary electron. The ground state becomes spin-polarized by the Zeeman interaction only at a high magnetic field of about 1.9 T.
The spin singlet state is removed from the ground state when the tunneling in the x direction is enhanced for a reduced distance in x direction to \(\mu _x=6.8\) nm with \(\mu _y=5.2\) nm we obtain the ground state, which is again spin polarized with a large energy gap of \(\Delta E=-230\,\mu\)eV in Fig. 3d. This is the maximal gap that we obtain for the applied single-dot potential depth.
The energy gap \(\Delta E\) for various geometry of the quantum dot array is given in a phase diagram that displays the nonferromagnetic and Nagaoka ferromagnetic phases in Fig. 4. The diagram is calculated for a small magnetic field of 1 mT. For the case with \(\mu _y\ge 7\) nm, the four dots are located far away from each other, rendering tunneling negligible, and hence the lowest spin \(S=1/2\) and spin \(S=3/2\) states are nearly degenerate, as seen on the right side of the figure. In the case where the dots are located much closer to each other in y direction (left side of the diagram), the ground state tends to be the spin-1/2 state as in Fig. 3c. This is a quantitative result for the strong tunnel coupling forming two double-dot subsystems. The region in between corresponds to the extended ferromagnetic system with a spin-polarized ground state. In this region, the tunneling between neighbouring dots is enough for the ’hole’ to move around the four dots. The largest gap is \(\Delta E=-230\,\mu\)eV and is located at \(\mu _x=6.8\) nm, \(\mu _y=5.2\) nm.
To understand in detail the effects of the interactions depending on the location of the dots, we study the cross sections of the phase diagram to see the energy gap as a function of the parameter \(\mu _y\) for various \((\mu _x-\mu _y)\) in Fig. 5a. The top-most (bluest) line shows the change in the energy gap \(\Delta E\) with \(\mu _y\) for \(\mu _x-\mu _y=1\) nm. It barely goes below the zero line, indicating a very weak Nagaoka ferromagnetic phase near \(\mu _y=6\) nm. As the difference \(\mu _x-\mu _y\) increases, the next four lines from the top slowly start to go much lower than the zero line, indicating that the system is in a stronger ferromagnetic phase and that it would take more energy to invert one of the spins and break the phase. The lowest line is the one that goes the lowest under the zero line, which is for the difference \(\mu _x-\mu _y=1.6\) nm. The second line from the bottom (greenest) is for the difference of 1.9 nm and is now higher than the difference of 1.6 nm. At this point, the electron density begins to take the shape of two dumbbells with decreasing x-direction overlap.
Fig. 5b shows the energy gap as a function of \(\mu _x\) for different y spacings. The plot has few interesting features that give insight on the system. First, every plot has a peak at low values of \(\mu _x\) slightly below the fixed \(\mu _y\) distance. The initial growth of the energy gap at the left side of the plot is due to an extension of the region accessible for electrons that has a larger influence on the spin-unpolarized state than on the spin-polarized state for which the electrons cannot occupy the same location anyway. When the system is separated to single-electron islands maxima the lines dip to a certain value before tending to a constant value as \(\mu _x\) increases. This constant itself tends to zero for the lines as the parameter \(\mu _y\) becomes large, indicating a vanishing tunnel coupling. In order from topmost (red) line in Fig. 5b, for \(\mu _y=4.5\) nm the energy gap is always positive and for this y spacing, the system never attains the ferromagnetic ordering. For other parameters, a ferromagnetic ground state is found for a range of \(\mu _x\).
The maximal Nagaoka gap is expected to shift on the \(\mu _x,\mu _y\) plane for a varied potential depth that affects the strength of the tunnel coupling. The phase diagram for the potential depth of \(V_d=60\) meVis plotted in Fig. 6. The results are qualitatively similar to the case of \(V_d=125\) meV presented above. The largest Nagaoka gap \(\Delta E = -211\,\mu\)eV occurs for the parameters \(\mu _x=7.52\) nm and \(\mu _y=5.35\) nm. Because of the shallower potential, the electrons are less localized within the separate dots. This results in a larger tunnel coupling and the Nagaoka phase is achieved for larger inter-dot distances compared to the case of \(V_d=125\) meV. Note that the parameter \(\mu _x\) changed from 6.8 nm to 7.5 nm, while the parameter \(\mu _y\) changed only about 0.15 nm. This can be attributed to the heavier electron effective mass in y direction and the tunneling energy changing more strongly with the interdot distance.
Shift from the rectangular geometry
In order to investigate the robustness of the spin-polarized ground state we deform the rectangular arrangement of dots so that electrons are trapped in a finite pseudo-1D chain. Starting from the initial state with the largest energy gap, i.e. \(\mu _x=6.8\) nm, \(\mu _y=5.2\) nm, we shift the location of the top-right dot in both x and y directions separately. The energy gap as a function of the shift \(\Delta \mu\) is plotted in Fig. 7. The anisotropy in the effective mass makes the nature of Nagaoka transition different in armchair (x) and zigzag (y) direction. For a shift in armchair direction, the spin-1/2 states have much lower energy than the high-spin state and the electrons from a 1D pseudo chain structure as seen in inset (f) of Fig. 7. The electron occupancy of the lower right dot is low for the shift of \(\Delta \mu _y=3.5\) nm in the zigzag direction. One of the electrons get fixed in the shifted dot, which is far away from the rest, which minimizes the Coulomb interaction energy. The two remaining electrons occupy the deeper left dumbbell instead of the shallower lower right dot [see inset (a) and (b) of Fig. 7]. No tunneling is possible between the rightmost dots and only a trace tunneling in the topmost dots. A similar effect is seen in the x-direction shift, but the geometry of the dots is such that the electrons form a chain with no tunneling between the two upper dots. Ref.24 showed that when the dot arrays are deformed to form a quantum dot chain, the spin polarization in the ground state is excluded by the Lieb-Mattis theorem40, which restricts the ground state solutions of such a 1D chain to low spin values. However, the anisotropy of the masses leads to an asymmetry in the tunneling of electrons between the dots. In the case of a shift in the zigzag direction, the system can be divided into two parts: the left double-dot subsystem and two single dots. As in the case of Fig. 3c, the double dot holds the singlet state of two electrons, and the third electron lowers the Coulomb repulsion by occupying the shifted dot. Tunnel coupling in the y direction decreases rapidly due to the higher effective mass, resulting in a transition occurring at a shift of about \(\sim 1.5\) nm. In contrast, lifting the Nagaoka ground-state polarization requires a shift of approximately 1.85 nm in the x-direction.
Potential detuning
In addition to moving the dots, it is possible to assess the tolerance of the ferromagnetic state to disorder by changing the potential depth of one of the dots. We vary the potential depth \(V_d\) of just the top right dot by an amount dV. Fig. 8 shows the Nagaoka gap as a function of the change dV in the range \(-20\) meV to 20 meV. The changes in the electron density as the potential changes are also shown in the insets of Fig. 8.
The most evident feature of the plot in Fig. 8 is the asymmetry in the energy gap variation for negative and positive potential changes. More precisely, for the positive change dV the Nagaoka ferromagnetic transitions to the low-spin state at \(dV \approx 7.0\) meV, while the same transition occurs at a bit smaller change of \(dV \approx -5.4\) meV for the negative change. The transition to the low-spin ground state for positive detuning is due to the localization of an electron in the detuned dot and the transition for negative detuning is due to the delocalization of an electron from the detuned dot which is excluded from the array by the energy mismatch leaving the three dots with an exact half-filling. When the top right dot is made deeper [insets (c) and (d)] the electron occupancy of this dot becomes larger than 1. When the dot is made shallower [insets (a) and (b)] the dot is emptied. In both cases the conditions for observation of the itinerant ferromagnetism are lifted, and the ground state acquires the low spin.