Coulomb and tunneling-coupled trilayer systems at zero magnetic field

The ground-state electronic configuration of three coupled bidimensional electron gases has been determined using a variational Hartree-Fock approach, at zero magnetic field. The layers are Coulomb coupled, and tunneling is present between neighboring layers. In the limit of small separation between layers, the tunneling becomes the dominant energy contribution, while for large distance between layers the physics is driven by the Hartree electrostatic energy. Transition from tunneling to Hartree dominated physics is shifted towards larger layer separation values as the total bidimensional density of the trilayers decreases. The interlayer exchange stabilizes a “balanced” configuration, where the three layers are approximately equally occupied; most of the experiments are performed in the vicinity of this balanced configuration. Several ground-state configurations are a consequence of a delicate interplay between tunneling and intersubband exchange.

Figure 1

Schematic view of the three layer system. Each layer of area $A$ has a two-dimensional electronic number density given by $n_1$, $n_2$, and $n_3$. Charge neutrality is provided by two positively charged layers located at $z=\pm h$ (not shown), with $h\gg d$. The layers are coupled by the hopping $t$ and the Coulomb interaction. ${\varepsilon }_1$, ${\varepsilon }_2$, and ${\varepsilon }_3$ are the site or layer energies.

Figure 2

Ground-state phase diagram, in the $r_s-{\varepsilon }_2^{*}$ plane, for $d^{*}=t^{*}=0$. The meaning of the different symbols is given in Table 1.

Figure 3

Ground-state phase diagram, in the $r_s-t^{*}$ plane, for $d^{*}={\varepsilon }_2^{*}=0$. The meaning of the different symbols is given in Table 2.

Figure 4

Ground-state phase diagram, in the $r_s-t^{*}$ plane, for $d^{*}=0$, and ${\varepsilon }_2^{*}=-0.5$. For $t^{*}\gtrsim 0.2$, all configurations are the same as in Fig. 3. For $0<t^{*}\lesssim 0.2$, the configurations A, B, and F are the same as in Fig. 2. The three remaining configurations C, D, and E are similar to the corresponding ones in Fig. 2 regarding the subband occupancies, but with $\theta \ne 0$.

Figure 5

Ground-state phase diagram, in the $r_s-d^{*}$ plane, for ${\varepsilon }_2^{*}=0$ and with the layer separation dependent hopping $t^{*}\left(d^{*}\right)$, as explained in the text. For small values of $d^{*}$, the ground-state configurations are the same as Fig. 3. For large values of $d^{*}$, the meaning of the different symbols is given in Table 3. The straight line corresponds to the approximation given in Eq. (13) for the boundary between $\alpha$ and $\beta$ configurations.

Figure 6

Ground-state configurations in the parameter space $r_s-d^{*}$, for ${\varepsilon }_2^{*}=-0.8$. The new configurations $P{3}^{\prime }$ and $P{3}^{\prime \prime }$ are as follows: $P{3}^{\prime }$, ${\eta }_{a\uparrow }={\eta }_{c\uparrow }=x$, ${\eta }_{b\uparrow }=1-2x,\theta =0$; $P{3}^{\prime \prime }$, ${\eta }_{a\uparrow }=x,{\eta }_{b\uparrow }=y,{\eta }_{c\uparrow }=1-x-y,\theta =\pi /2$. The two empty circles correspond to the experimental sample of Ref. [8] (see text).

Figure 7

Layer occupation factors for a trilayer with $d^{*}=2$, ${\varepsilon }_2^{*}=-0.8$, and $t^{*}(d^{*}=2)=0.05$. In the upper (lower) panel spin discriminated (undiscriminated) occupation factors are shown. At the tiny dashed vertical lines, there is a change in the ground-state configuration.