Phase transitions induced by a lateral superlattice potential in a two-dimensional electron gas

We study the phase transitions induced by a lateral superlattice potential (a metallic grid) placed on top of a two-dimensional electron gas (2DEG) formed in a semiconductor quantum well. In a quantizing magnetic field and at filling factor $\nu =1,$ the ground state of the 2DEG depends on the strength $V_g$ of the superlattice potential as well as on the number of flux quanta piercing the unit cell of the external potential. It was recently shown that in the case of a square lateral superlattice, the potential modulates both the electronic and spin density and in some range of $V_g$, the ground state is a two-sublattice spin meron crystal where adjacent merons have the global phase of their spin texture shifted by $\pi ,$ i.e., they are “antiferromagnetically” ordered. In this paper, we evaluate the importance of Landau-level mixing on the phase diagram obtained previously for the square lattice and derive the phase diagram of the 2DEG modulated by a triangular superlattice. When Landau-level mixing is considered, we find in this case that, in some range of $V_g,$ the ground state is a three-sublattice spin meron crystal where adjacent merons of the same vorticity have the global phase of their spin texture rotated by ${120}^{\circ }$ with respect to one another. This meron crystal is preceded in the phase diagram by another meron lattice phase with a very different spin texture that does not appear, at first glance, to resolve the spin frustration inherent to an antiferromagnetic ordering on a triangular lattice.

Figure 1

Behaviour of (a) the magnetic field $B$ and electronic density $n_e$ and (b) the cyclotron energy ratio $E_{\text{cyc}}/(e^2/\kappa \ell )$ as a function of the parameter $\mathrm{\Gamma }$ for the square and triangular lattices at filling factor $\nu =1$ and for a lattice constant of the external grid given by $a_0=50$ nm.

Figure 2

Phase diagram of the 2DEG under a square superlattice potential $V_g$. The number above each red symbol gives the magnetic field (in tesla) for that particular value of $\mathrm{\Gamma }.$ Note that the lines are only a guide to the eye. Only the symbols correspond to calculated values. The dashed line indicates the upper limit in $V_g$ of our numerical calculation.

Figure 3

Occupation (filling factor) of Landau levels $n=0,1$ with spin $\sigma =\pm 1$ as a function of the grid potential $V_g$ (in meV) for the square lattice at $\mathrm{\Gamma }=3/4.$ The vertical dashed lines indicate the phase boundaries.

Figure 4

Discontinuity in the $z$ component of the spin density as a function of $\mathrm{\Gamma }$ for the square and triangular grids. For $\mathrm{\Gamma }<1/2,$ the discontinuity is between the VORTEX1 and VORTEX2 phases for the square lattice and VORTEX3 and VORTEX4 for the triangular lattice, while for $\mathrm{\Gamma }>1/2$ it is between the VORTEX1 and CDW2 phases for the square lattice and between CDW1 and CDW2 for the triangular lattice.

Figure 5

Phase diagram of the 2DEG under a triangular superlattice potential $V_g$ (in meV). The number above each symbol gives the magnetic field (in tesla) for that particular value of $\mathrm{\Gamma }.$ Note that the lines connecting the symbols are only a guide to the eye. Only the symbols correspond to calculated values. The dashed line indicates the limit in $V_g$ of the numerical calculation.

Figure 6

Electronic density $n\left(\mathbf{r}\right)$ in units of $1/2\pi {\ell }^2$ and spin texture in the $x-y$ plane for the VORTEX3 state. Parameters: $\mathrm{\Gamma }=1/4,a_0=50$ nm and $\nu =1.$ The magnetic unit cell is indicated by the parallelogram.

Figure 7

Electronic density $n\left(\mathbf{r}\right)$ in units of $1/2\pi {\ell }^2$ and spin texture $\mathbf{S}\left(\mathbf{r}\right)$ in the $x-y$ plane for the VORTEX4 states. Parameters: $\mathrm{\Gamma }=1/4,a_0=50$ nm and $\nu =1.$ The magnetic unit cell is indicated by the parallelogram.

Figure 8

Filling factor ${\nu }_{n,\sigma }$ of the four Landau levels $n=0,1$ with spin $\sigma =\pm 1$ as a function of the superlattice potential $V_g/\left(e^2/\kappa \ell \right)$ for the triangular lattice at $\mathrm{\Gamma }=1/3.$ The vertical dashed lines indicate the boundaries of the different phases.

Figure 9

Behavior of the Hartree-Fock gap ${\mathrm{\Delta }}_{\text{eh}}$ as a function of the grid potential $V_g$ in the different phases of the triangular lattice at $\mathrm{\Gamma }=1/4.$ The gap is evaluated from the density of states, an example of which is given in the inset for $V_g=2.17$ meV, i.e., in the VORTEX3 phase. The blue line in the inset is the integrated density of states.

Figure 10

Electronic density $n\left(\mathbf{r}\right)$ and dipole field $\mathbf{d}\left(\mathbf{r}\right)$ in the $x-y$ plane for the VORTEX1 phase at $V_g/\left(e^2/\kappa \ell \right)=0.22$ for the square lattice with parameters $a_0=50$ nm, $\mathrm{\Gamma }=1/4$, and $\nu =1.$