Ferromagnetism in quantum dot plaquettes

Following recent experimental progress concerning Nagaoka ferromagnetism in finite-size quantum dot plaquettes, a general theoretical analysis is warranted in order to ascertain in rather generic terms which arrangements of a small number of quantum dots can produce saturated ferromagnetic ground states and under which constraints on interaction and interdot tunneling in the plaquette. This is particularly necessary since Nagaoka ferromagnetism is fragile and arises only under rather special conditions. We test the robustness of ground state ferromagnetism in the presence of a long-range Coulomb interaction and long-range as well as short-range interdot hopping by modeling a wide range of different plaquette geometries accessible by arranging a few ($\sim 4$) quantum dots in a controlled manner. We find that ferromagnetism is robust to the presence of long-range Coulomb interactions, and we develop conditions constraining the tunneling strength such that the ground state is ferromagnetic. Additionally, we predict the presence of a partially spin-polarized ferromagnetic state for 4 electrons in a Y-shaped 4-quantum-dot plaquette. Finally, we consider 4 electrons in a ring of 5 dots. This does not satisfy the Nagaoka condition; however, we show that the ground state is spin 1 for strong, but not infinite, on-site interaction. Thus, even though Nagaoka's theorem does not apply, the ground state for the finite system with one hole in a ring of 5 dots is partially ferromagnetic. We provide detailed fully analytical results for the existence or not of ferromagnetic ground states in several quantum dot geometries which can be studied in currently available coupled quantum dot systems.

Figure 1

A depiction of the different 4-dot geometries studied in this work, numbered in bold according to the subsection numbers in which they are discussed in Secs. 2b and 3b. Solid lines depict nearest-neighbor hopping terms, and dashed lines next-nearest-neighbor hopping terms, which we consider in some cases. In all cases long-range Coulomb interactions are included.

Figure 2

$U_0^{\text{crit}}$ versus $V_a$ for three electrons in a four-dot square configuration. Here $V_d=V_a/\sqrt{2}$.

Figure 3

$E_{3/2}$ and $E_{1/2}$ versus $U_0$ for different values of $V_a$ for three electrons in a four-dot square configuration. The point where $E_{3/2}$ and $E_{1/2}$ cross is $U_0^{\text{crit}}$. Here $V_d=V_a/\sqrt{2}$.

Figure 4

Plot of $E_{3/2}$ and $E_{1/2}$ versus $t_d/t_a$ for three electrons in a four-dot square configuration with diagonal hopping in the infinite-$U$ limit. We see that ground state ferromagnetism is only possible for $t_d<t_a/4$.

Figure 5

Plot of $E_{3/2}$ and $E_{1/2}$ versus $t_b/t_a$ for three electrons in a four-dot rectangular configuration with no diagonal hopping in the infinite-$U$ limit.

Figure 6

Plot of $U_{\text{crit}}$ versus $t_b/t_a$ for three electrons in a four-dot rectangular configuration with $V=W=0$.

Figure 7

Plot of $E_{3/2}$, $E_{1/2}$, and the nearly degenerate excited state energy $E_{1/2}^{\text{ex}}$ versus $U$ for three electrons in a four-dot rectangular configuration with no diagonal hopping with $t_b/t_a=0.8$ (top) and $t_b/t_a=0.2$ (bottom). Here $V=W=0$.

Figure 8

Plot of $E_{3/2}$, $E_{1/2}$, and $E_{1/2}^{\text{ex}}$ versus $U$ for three electrons in a four-dot linear array for $V=0$ (top) and $V=5t_a$ (bottom). Here $W=V/4$.

Figure 9

A depiction of a ring of 5 dots. Solid lines depict nearest-neighbor hopping terms and Coulomb interactions, and dashed lines long-range Coulomb interactions.