Wigner solids of domain wall skyrmions

Abstract

Detection and characterization of a different type of topological excitations, namely the domain wall (DW) skyrmion, has received increasing attention because the DW is ubiquitous from condensed matter to particle physics and cosmology. Here we present experimental evidence for the DW skyrmion as the ground state stabilized by long-range Coulomb interactions in a quantum Hall ferromagnet. We develop an alternative approach using nonlocal resistance measurements together with a local NMR probe to measure the effect of low current-induced dynamic nuclear polarization and thus to characterize the DW under equilibrium conditions. The dependence of nuclear spin relaxation in the DW on temperature, filling factor, quasiparticle localization, and effective magnetic fields allows us to interpret this ground state and its possible phase transitions in terms of Wigner solids of the DW skyrmion. These results demonstrate the importance of studying the intrinsic properties of quantum states that has been largely overlooked.

Fig. 1. Nonlocal resistance measurements of the Ising quantum Hall ferromagnet (IQHF) at filling factor ν = 2.

a Schematics of a two-dimensional electron gas (2DEG, dashed line) Hall bar with six terminals (1–4, S, and D) surrounded by a coil that produces a continuous-wave radiofrequency (RF) field. The tilt angle θ is defined between the magnetic field B and its perpendicular component B_perp. b Schematics of the Landau level (LL) splitting as a function of θ. The ν = 2 IQHF occurs when the two LLs with orbital index n = 0 and n =1 intersects, where the Zeeman splitting $E_z~B_{\mathrm{perp}}/\cos \theta$ and the cyclotron gap $E_{\mathrm{c}}~B_{\mathrm{perp}}$ are made equal by adjusting θ. An upward (downward) arrow is for spin-up (spin-down). c Four-terminal resistance $R_{kl,mn}$ versus B at T = 1 K, $I_{kl}$ = 31.6 nA, and θ = 64°. The longitudinal resistance R_SD,12 is given by the ratio of a voltage drop V₁₂ between terminals 1 and 2 to current flow I_SD between terminals S and D, and the nonlocal resistance R_23,14 (R_14,23) is calculated by V₁₄/I₂₃ (V₂₃/I₁₄). Note the different scales for longitudinal and nonlocal resistances. The shaded area represents the ν = 2 (B ~ 13 T, dashed line) IQHF region. d R_23,14 versus B at different temperatures. The arrow indicates increasing temperature. e Localization length ξ versus B obtained from a fit (Supplementary Fig. 4) to the B dependence of R_23,14 (red solid dots), R_14,23 (blue solid dots), and R_SD,12 (open dots) at different temperatures, respectively. f Current (I₂₃) dependence of R_23,14 versus B at T = 1 K.

Fig. 2. Temperature and magnetic field dependence of T₁ and T₂ in the NRDNMR measurement of the Ising quantum Hall ferromagnet (IQHF) at ν = 2.

a $R_{\mathrm{14,23}}^{+B}$ (right y axis) as a function of B (or ν) swept upwards (solid line) and downwards (dash-dotted line) with a sweep rate of 1.7 mT s⁻¹ at I₁₄ = 3.16 μA, θ = 64°, and T = 1 K. The B oriented parallel (opposite) to the normal direction of the 2DEG is written as $+\left(-\right)$B. $△R_{\mathrm{hys}}$ measures the magnitude of the resistance hysteresis at a fixed B. The NRDNMR signal occurs around ν = 2 where the B (or ν) dependence of T₁ (left y axis) is obtained. The dashed line marks the phase boundary (see main text). Note that error bars for all data in this study are the standard deviation of the mean. b 1/T₁ versus T at different B. The solid line is a fit to the data at 12.82 T (13.18 T) using the Arrhenius law with an activation energy ${△}_{\mathrm{sw}}$ of 1.36 (0.42) meV plus a T-independent constant of 0.02 s⁻¹. The dashed line is a linear fit, and its extrapolation to the x axis defines T_c at which 1/T₁ is zero. Inset shows the ν dependence of T_c. c 1/T₁ versus $R_{\mathrm{14,23}}^{+B}$ obtained by comparing the T dependence of $R_{\mathrm{14,23}}^{+B}$ at B = 13 T and I₁₄ = 3.16 μA (inset) and that of 1/T₁ (data at B = 13 T, b) point by point. The solid line is a guide to the eye. d ${\omega }_{\mathrm{R}}^2{T}_2$ versus B (or ν) at different temperatures. The dashed line marks the phase boundary that is the same as that in a. The Rabi frequency ${\omega }_{\mathrm{R}}=\gamma B_{\mathrm{RF}}/2$ (where $\gamma$ = 9.36 MHz/T is the gyromagnetic ratio of ¹¹⁵In) is about 10 Hz (see “Methods” section). Inset shows ${\omega }_{\mathrm{R}}^2{T}_2$ as a function of B_RF tuned by the RF output power. A good fit ($\propto B_{\mathrm{RF}}^2$, solid line) to the data indicates a linear dependence of ${\omega }_{\mathrm{R}}$ on B_RF.

Fig. 3. T₁ and T₂ results of the quantum Hall ferromagnet at ν = 2 with zero effective field b^*.

a Contour plot of B and $\theta$ dependence of $△R_{\mathrm{hys}}/R_{\mathrm{14,23}}^{+B}$ at a field-sweep rate of 1.7 mT s⁻¹, I₁₄ = 3.16 μA, and T = 1 K. Red dots indicate the position of maximum $△R_{\mathrm{hys}}/R_{\mathrm{14,23}}^{+B}$, 1/T₁, and $△R_{14,23}/R_{14.23}^{\mathrm{sat}}$. The solid line is a guide to the eye and the dashed line is calculated for the B and $\theta$ dependence of ν = 2. Temperature dependence of 1/T₁ (b) and ${\omega }_{\mathrm{R}}^2{T}_2$ (c) versus ν where the data (symbols) are in one-to-one correspondence with the red dots in a by $\nu =n_{\mathrm{s}}h/eB\cos \theta$ (where h is Planck’s constant and e the electron charge). It is clear that the data encircled by an oval in a exhibit a strong temperature dependence of T₁ and T₂.