Resistively detected NMR spectra of the crystal states of the two-dimensional electron gas in a quantizing magnetic field

Transport experiments on the two-dimensional electron gas (2DEG) confined into a semiconductor quantum well and subjected to a quantizing magnetic field have uncovered a rich variety of uniform and nonuniform phases such as the Laughlin liquids, the Wigner, bubble, and Skyrme crystals, and the quantum Hall stripe state. Optically pumped nuclear magnetic resonance (OP-NMR) has also been extremely useful in studying the magnetization and dynamics of electron solids with exotic spin textures such as the Skyrme crystal. Recently, it has been demonstrated that a related technique, resistively-detected nuclear magnetic resonance (RD-NMR), could be a good tool to study the topography of the electron solids in the fractional and integer quantum Hall regimes. In this work, we compute theoretically the RD-NMR line shapes of various crystal phases of the 2DEG and study the relation between their spin density and texture and their NMR spectra. This allows us to evaluate the ability of the RD-NMR to discriminate between the various types of crystal states.

Figure 1

(a) RD-NMR spectra for the Wigner crystal phase (BC1) at different filling factors $\nu$ in Landau level $N=0$ with $\sigma =0$ and for $\nu =0.3$ with $\sigma =1.4$. (b) Probability density $P\left(m_z\right)$ in the Wigner crystal phase for $\nu =0.2,0.3$, and $\sigma =0.$

Figure 2

(a) RD-NMR spectra for the Wigner crystal phase (BC2) at different filling factors $\nu$ in Landau level $N=0$ with $\sigma =0;$ (b) Probability density $P\left(m_z\right)$ in the Wigner crystal phase for $\nu =0.7\text{and}0.8$ with $\sigma =0.$

Figure 3

RD-NMR spectra in Landau level $N=0$ for (a) the Skyrme crystal phases (SK1) at different filling factors $\nu$ and for a Zeeman coupling $\gamma =0.015$ (unless indicated otherwise) with $\sigma =0;$ (b) the triangular (TLF) and square (SLA) Skyrme crystals for $\gamma =0.015\text{and}\nu =0.9$; (c) Knight shift (measured at the peak position) and spatially averaged polarization $〈m_z〉$ as a function of the filling factor.

Figure 4

RD-NMR spectra for (a) the BC1 phase with $M=1$ (Wigner crystal) at filling factor $\nu =0.1$ in Landau levels $N=0,1,2,\text{and}3$ and for the uniform phase at $\nu =0.1;$ (b) the BC1, SC1, and BC2 phases in Landau level $N=1$ at different filling factors; (c) probability density $P\left(m_z\right)$ for the same phases as in (a). The function $P\left(m_z\right)$ for $N=0$ extends to $m_z=1$ (not shown on the graph).

Figure 5

Traces of the spin density $m_z\left(\mathbf{r}\right)$ along $x$ for $y=0$ and at different filling factors in $N=2$ for the : (a) $M=1$ bubble crystals; (b) $M=2$ bubble crystals; (c) $M=3$ bubble crystals (bottom $x$ axis) and stripe phase (top $x$ axis). The numbers identify a region that contributes to a peak in the function $P\left(m_z\right)$ shown in Fig. 6. Here, $a$ is the lattice spacing and $\ell$ is the magnetic length. The dashed curve in (b) shows the magnetization trace for $N=3\text{and}M=2$ at $\nu =0.25.$

Figure 6

(a) Probability density $P\left(m_z\right)$ for some BC1 and SC1 phases in Landau level $N=2$ at different filling factors $\nu <0.5.$ Each curve is offset by 8 from the one below for clarity. This offset must be deducted from the value of $P\left(m_z\right)$; (b) corresponding RD-NMR spectra for the crystal phases in (a) (full black curve) and for the uniform phases (dashed blue curves). Each curve is offset by 1 from the one below for clarity. The intensity is in arbitrary units, but the same for all curves. The dashed-dotted (red)curves are the spectra of the crystal phases with a blurring factor $\sigma =1.4.$

Figure 7

RD-NMR spectra for the BC2 crystal phase (full black curve) and uniform phase (blue dashed curves) in Landau level $N=2$ for filling factor $0.5<\nu <1.$ The curves are offset by $0.5$ for clarity. The intensity is in arbitrary units, the same for all curves.

Figure 8

$N=2$ RD-NMR spectra (full curves) for the $M=1$ and 2 bubble crystals at $\nu =0.15$ and $0.3$ and for the stripe phase at $\nu =0.45$ calculated with a linewidth $\mathrm{\Gamma }=0.5$ kHz. The dashed curves are the spectra of the corresponding uniform phases.