Nuclear magnetic resonance line shapes of Wigner crystals in ${}^{13}\mathrm{C}$-enriched graphene

Assuming that the nuclear magnetic resonance (NMR) signal from a ${}^{13}\mathrm{C}$-isotope-enriched layer of graphene can be made sufficiently intense to be measured, we compute the NMR line shape of the different crystals' ground states that are expected to occur in graphene in a strong magnetic field. We first show that in nonuniform states there is, in addition to the frequency shift due to the spin hyperfine interaction, a second contribution of equal importance from the coupling between the orbital motion of the electrons and the nuclei. We then show that if the linewidth of the bare signal can be made sufficiently small, the Wigner and bubble crystals have line shapes that differ qualitatively from that of the uniform state at the same density while crystal states that have spin or valley pseudospin textures do not. Finally, we find that a relatively small value of the bare linewidth is sufficient to wash out the distinctive signature of the crystal states in the NMR line shape.

Figure 1

NMR line shapes of Wigner crystals (full lines) and corresponding uniform states (dashed lines) at Zeeman coupling ${\mathrm{\Delta }}_Z=0.03$ for: (a) the electron Wigner crystal at filling factor ${\nu }_n=0.2$ in Landau levels $n=0,1,2;$ (b) the Wigner crystal at ${\nu }_2=0.2$ and the bubble crystal with two electrons at ${\nu }_2=0.4$ in level $n=2;$ (c) the electron Wigner crystal at ${\nu }_2=0.2$ and the hole Wigner crystal at ${\nu }_2=3.8$, both for $n=2.$ The linewidth $\mathrm{\Gamma }=1.0\mathrm{kHz}$ in all cases.

Figure 2

NMR line shape calculated with only the orbital shift, only the spin shift, or with both shifts (“total” line) for the Wigner crystal in Landau level $n=1$ at Zeeman coupling ${\mathrm{\Delta }}_Z=0.03$ and with linewidth $\mathrm{\Gamma }=1\mathrm{kHz}$.

Figure 3

NMR line shapes of : (a) the meron crystal with valley pseudospin texture in Landau level $n=1$ at filling factor ${\nu }_1=0.8$ and Zeeman coupling ${\mathrm{\Delta }}_Z=0.03$; (b) the crystal in (a) calculated with different values $\mathrm{\Gamma }$ of the linewidth as indicated. The dashed lines give the line shapes for the corresponding uniform state in each case.

Figure 4

(a) Contour plot of the electronic density $n\left(\mathbf{r}\right)$ and spin texture in the spin Skyrmion crystal in Landau level $n=1$ at filling factor ${\nu }_1=2.2$ for Zeeman coupling ${\mathrm{\Delta }}_Z=0.01;$ (b) contour plot of valley pseudospin component $P_z\left(\mathbf{r}\right)$ and valley pseudospin texture for the meron crystal in Landau level $n=0$ at filling factor ${\nu }_0=0.8.$

Figure 5

Evolution of the NMR line shape with Zeeman coupling in Landau level $n=1$ at filling factor ${\nu }_1=2.2:$ triangular Wigner crystal at ${\mathrm{\Delta }}_Z=0.03;$ spin Skyrmion and triangular Wigner crystals at ${\mathrm{\Delta }}_Z=0.02$ (full red line is the spin Skyrmion crystal and the dashed black line is the triangular Wigner crystal); spin Skyrmion crystal at ${\mathrm{\Delta }}_Z=0.01.$ The linewidth $\mathrm{\Gamma }=1\mathrm{kHz}$ in all cases. The dashed lines give the NMR signal from the corresponding uniform states.

Figure 6

NMR line shapes calculated by keeping only the spin shift of Wigner crystals (full lines) and corresponding uniform states (dashed lines) at Zeeman coupling ${\mathrm{\Delta }}_Z=0.03$ for the electron Wigner crystal at filling factor ${\nu }_n=0.2$ in Landau levels $n=0,1,2.$ The linewidth $\mathrm{\Gamma }=1\mathrm{kHz}$.