When a two-dimensional electron system (2DES) is cooled to low temperatures and exposed to a strong magnetic field the energy spectrum consists of discrete and highly degenerate energy levels (Landau levels). Within these Landau levels all electrons have the same kinetic energy and the system is dominated by electron-electron interactions. Facilitated by this fortunate situation exotic, correlated phases emerge, such as the fractional quantum Hall effect and so-called density modulated phases. In the latter case the homogeneous electron system breaks up into clusters of electrons which arrange either in a triangular lattice ("bubble" phase) or form periodically ordered stripes ("stripe" phase). Standard electron transport experiments reveal their existence by a reappearance of the integer quantum Hall effect in the case of the bubble phase or by strong transport anisotropy for the stripe phase. Beside their behavior in electron transport experiments, up-to-date little is known about the microscopic structure of these density modulated phases.
Fig. 1: (a) Schematic of the sample configuration in Van-der-Pauw geometry. An in-situ grown backgate provides fast tunability of the electron density. To perform NMR experiments, a coil was wound around the sample. (b) Measurement scheme for resistively detected NMR experiments. (c) Longitudinal resistance Rxx as a function of perpendicular magnetic field B^ for different tilt angles (offset). Measurements were done at base temperature (T≈20mK). Shown in blue is the alternating current flow IAC along the in-plane magnetic field component B∥, whereas the perpendicular case is represented by the red curve. The current direction is indicated by red arrows.[less]
Fig. 1: (a) Schematic of the sample configuration in Van-der-Pauw geometry. An in-situ grown backgate provides fast tunability of the electron density. To perform NMR experiments, a coil was wound around the sample. (b) Measurement scheme for resistively detected NMR experiments. (c) Longitudinal resistance Rxx as a function of perpendicular magnetic field B^ for different tilt angles (offset). Measurements were done at base temperature (T≈20mK). Shown in blue is the alternating current flow IAC along the in-plane magnetic field component B∥, whereas the perpendicular case is represented by the red curve. The current direction is indicated by red arrows.
We have used nuclear spins as local detectors to probe the electronic density distribution of a stripe phase in the quantum Hall regime. The technique relies on a characteristic shift in the nuclear magnetic resonance (NMR) frequency of a nucleus being in contact to a spin polarized electron system - an effect also known as Knight shift. Each individual nucleus experiences a change of its Zeeman energy depending on the electronic spin polarization it is surrounded by. This effect arises from the hyperfine coupling between electronic and nuclear spins. In the case of a non-homogeneous electron spin density the Knight shift varies locally due to the changing electron spin polarization. This presence of different Knight shifts throughout the sample can be observed in the integrated NMR lineshape measured in experiment. Because of this, NMR is well suited to study the spatial electron distribution. The detection of the nuclear resonance condition is done conveniently by measuring the sample resistance while scanning the frequency of a radio frequency (RF) signal applied to the sample. For this purpose a coil is wound around the sample as shown in Fig. 1(a). If the radio frequency is resonant with the Zeeman splitting of the nuclei, the nuclear spin polarization decreases and, as a consequence, the electronic Zeeman energy changes. As in the case of the Knight shift, this effect is mediated by the hyperfine interaction. The influence of the nuclear spin polarization on the electron Zeeman energy allows detecting the NMR by changes in the sample resistance. However, the sensitivity of this resistive detection method depends strongly on the filling factor. In order to optimize for highest sensitivity we separate the detection of the nuclear spin polarization from their depolarization. Details of the measurement sequence are depicted in Fig. 1(b). In the first step a radio frequency close to the expected resonance frequency of the nuclei is applied to the coil while the electron system is at filling factor νprobe. It is the electron spin distribution of this filling factor which is reflected in the nuclear resonance frequency. After a waiting period the RF-induced change in the nuclear spin polarization is detected by measuring the sample resistance at the detection filling factor νdetect. Both steps are repeated multiple times, while for each cycle a slightly lower RF frequency is applied at νprobe in order to scan across the nuclear resonance. The measurement sequence requires that the nuclear spin polarization of the first step is preserved until its detection in the second step. Therefore, fast control of the filling factor is necessary. This is achieved by tuning the electron density electrostatically with the help of a backgate (Fig. 1(a)). In this way the filling factor can be changed quickly with no need for time consuming changes of the magnetic field.
Using this technique we have studied the stripe phase emerging at filling factor 5/2 when the sample is tilted with respect to the external magnetic field. Fig. 1(c) shows the sample resistance in two orthogonal current directions for different tilt angles. The measurements were done at the base temperature of a dilution refrigerator (≈20mK). In the case of a perpendicular magnetic field both current directions exhibit similar behavior. Both traces reveal well developed 7/3, 8/3 and 5/2 fractional quantum Hall states. Upon tilting of the sample, the 5/2 state becomes weaker and eventually vanishes while at the same time a strong transport anisotropy develops. The hard axis is oriented along the in-plane magnetic field component B∥. This transport anisotropy indicates an electron density modulation in a stripe-like fashion. To test this prediction we measured nuclear resonance spectra at various filling factors between ν=2 und 3 at a fixed tilt angle of 60 degrees and constant total magnetic field of 6.9T. All the resonance spectra were taken for 75As nuclei as they yielded the strongest signal due to their high abundance. The recorded NMR spectra are displayed in Fig. 2(b). The corresponding transport behavior under these conditions is shown in Fig. 2(a). It differs from the measurements in Fig. 1(c) because of the heating induced by the RF radiation. Despite the higher temperature (≈70mK) the transport anisotropy remains strong.
Fig. 2: (a) Longitudinal resistance between ν=2 and 3 for fixed rotation angle of 60° and constant total magnetic field Btot=6.9T. The measurement was done under the influence of off-resonant RF radiation, resulting in a higher electron temperature (T≈70mK). (b) Resistively detected NMR spectra of 75As nuclei taken at various filling factors as indicated by colored bars in (a). NMR induced resistance changes ΔR were inverted, normalized and off-set for clarity. (c) Color plot of simulated NMR response. The calculations are based on the two-dimensional electron distribution of a stripe pattern with stripe period 2.6 magnetic lengths.[less]
Fig. 2: (a) Longitudinal resistance between ν=2 and 3 for fixed rotation angle of 60° and constant total magnetic field Btot=6.9T. The measurement was done under the influence of off-resonant RF radiation, resulting in a higher electron temperature (T≈70mK). (b) Resistively detected NMR spectra of 75As nuclei taken at various filling factors as indicated by colored bars in (a). NMR induced resistance changes ΔR were inverted, normalized and off-set for clarity. (c) Color plot of simulated NMR response. The calculations are based on the two-dimensional electron distribution of a stripe pattern with stripe period 2.6 magnetic lengths.
The evolution of the spectral lineshape as a function of filling factor in Fig. 2(b) starts with a single narrow resonance at ν=2. By gradually increasing νprobe a second resonance appears. This double resonance structure persists over a broad filling factor range coincident with the region displaying large transport anisotropy. Both peaks shift to lower frequencies with increasing filling factor. Ultimately, the two resonances merge back into a single, broad resonance at ν=3. The steady trend to lower resonance frequencies while approaching ν=3 reflects the increasing degree of electron spin polarization. Starting from the unpolarized quantum Hall state at ν=2, the population of the spin-up branch of the second Landau level rises with increasing filling factor. This change of polarization results in a down-shift of the nuclear resonance frequency. At ν=3 the electron spin polarization reaches its highest value. The substantial broadening of the resonance lineshape with increasing spin polarization can be understood as a consequence of the finite extent of the electronic wavefunction in the direction perpendicular to the quantum well.
Of main interest here is the appearance of a second resonance peak in the region of large transport anisotropy. It is attributed to a modulation of the electron density in the plane of the 2DES. The clear two-fold resonance shown in Fig. 2(b) suggests the spatial electron distribution to be dominated by two different densities. The region with higher density has a larger spin polarization and shifts the nuclear resonance to lower frequencies. To corroborate this interpretation we modeled the observed resonance behavior based on alternating, parallel stripes with filling factor 2 and 3. The relative stripe width sets the overall (average) filling factor and the stripe period is treated as a free parameter. The result is shown in Fig. 2(c). It matches the measured behavior in Fig. 2(b) qualitatively well showing the existence of two resonance peaks over a broad filling factor range as well as the evolution back to a single peak at ν=3. Apparent discrepancies between experiment and simulation may largely be attributed to the fact that our model does not take into account the electron distribution in the z-direction. As mentioned before, this would lead to a broadening of the resonance lineshape at higher filling factors.
From the best fit of our model we can extract details of the microscopic stripe phase structure. The stripe period is determined as 2.6 magnetic lengths and the electron density modulation, which is necessary to produce the splitting of the resonance line, is more than 20%. These results confirm the assumption of a stripe-like modulation of the electron density in the anisotropic phases.