Microscopic details of stripes and bubbles in the quantum Hall regime

We use a fully self-consistent laterally resolved Hartree-Fock approximation for numerically addressing the electron configurations at higher Landau levels in the quantum Hall regime for near-macroscopic sample sizes. Our results give microscopic details of stripe- and bubble-like charge density modulations and show how these emerge depending on the filling factor. We find that there exists a region at the boundaries of the stripes and bubbles with a density modulation that corresponds to a filling factor around half filling. The microscopic details of these boundary regions determine the geometrical boundary conditions for aligning the charge density modulation either as stripes or bubbles. Transport is modelled using a non-equilibrium network model giving a pronounced anisotropy in direction of the injected current in the stripe regime close to half filling. We obtain a stripe period of 2.9 cyclotron radii. Our results indicate the dominance of many particle physics in the integer quantum Hall regime and provide an intuitive understanding of its consequences in strong magnetic fields.

We use a fully self-consistent laterally resolved Hartree-Fock approximation for numerically addressing the electron configurations at higher Landau levels in the quantum Hall regime for nearmacroscopic sample sizes. Our results give microscopic details of stripe-and bubble-like charge density modulations and show how these emerge depending on the filling factor. We find that there exists a region at the boundaries of the stripes and bubbles with a density modulation that corresponds to a filling factor around half filling. The microscopic details of these boundary regions determine the geometrical boundary conditions for aligning the charge density modulation either as stripes or bubbles. Transport is modelled using a non-equilibrium network model giving a pronounced anisotropy in direction of the injected current in the stripe regime close to half filling. We obtain a stripe period of 2.9 cyclotron radii. Our results indicate the dominance of many particle physics in the integer quantum Hall regime and provide an intuitive understanding of its consequences in strong magnetic fields. In May 2019, the quantum Hall effect [1,2] was formally included among the select group of high-precision experiments to form the basis of a new SI system of units based on the "Planck constant h, the elementary charge e, the Boltzmann constant k, and the Avogadro constant N A " [3]. This had long been awaited and certainly represents a great achievement and fitting fulfillment of the vision for "natürliche Masseinheiten" [4] proposed by Max Planck [5]. In his essay to celebrate this achievement [3], von Klitzing also points out "that a microscopic picture of the quantum Hall effect for real devices with electrical contacts and finite current flow is still missing." Prominent examples of such microscopic details are so-called "bubble" and "stripe" phases [6]. They have been identified, e.g., by transport experiments in higher Landau levels (LLs) of ultra-high mobility samples [7][8][9] and are characterized by strong transport anisotropies (stripes) or reentrance effects (bubbles). It is believed that the phases correspond to density modulations with characteristic geometric non-uniformities due to the interplay of Coulomb interaction and the wave functions in higher Landau levels.
Early work in modelling density modulations in the quantum Hall regime started from the Chklovskii, Shklovskii and Glazman picture of uni-directional chargedensity waves (CDWs) [10][11][12] while a mean field treatment established the existence of various phases [13,14]. However, spatially resolved information does not yet exist in these phases. Experimentally this is due to the intrinsic challenge of using local scanning probes in low temperatures for such remotely doped systems [15][16][17]. Nevertheless, much indirect experimental evidence for the existence of bubble and stripe phases has now been accumulated [17][18][19][20][21][22][23][24][25][26][27]. Theoretical modelling has likewise concentrated on transport signatures of these phases [28][29][30][31] while spatially resolved models of bubbles and stripes are only available in clean systems [32]. The period of the stripe patterns has been predicted to follow d ∝ R c with R c = l B √ 2n + 1 the cyclotron radius in LL n at the Fermi energy [11,33] and l B = /eB the magnetic length. Experimentally, 1.5R c [17] and 3.6R c [34] have been reported while ∼ 2.7R c is predicted theoretically [33].
Recently we have re-investigated the integer quantum Hall (IQH) regime spatially resolved in low and high LLs by a fully self-consistent Hartree-Fock (HF) method [35,36] within a disordered system capable of modelling remotely doped systems [37]. Our results indicate the importance of many-body physics already in the IQH regime. We find essential modifications to the simple picture of a smoothly varying compressible bulk [10][11][12]: our results show how this bulk is microscopically formed by a mixture of clusters with full and empty LLs [36].
The key-mechanism is a Hund's rule behaviour for the occupation of the spin-split LLs. The resulting g-factor enhancement is then a local quantity depending on the local filling factor ν( r) = 2πl 2 B ρ( r) [35,36] with ρ( r) the local carrier density. This exchange-enhanced g-factor is a concept that allows to discuss the exchange interaction within the single electron picture [38,39]. By doing so, we find that the local variation of the enhancement of the Zeeman energy due to ν( r) has to be considered in ad- dition to the laterally varying Hartree potential, leading to a modified effective potential for the electrons that also strongly modifies the screening behaviour. In addition to the largely repulsive Hartree part of the selfconsistent Thomas-Fermi screening, the ν( r) dependence of enhanced Zeeman energy leads to a positive feedback loop in the self-consistent carrier redistribution and produces an instability of the electron density ρ, which may lead to jumps either to a locally full or locally empty LL [35,36], resulting in a clustering of the filling factor that is triggered by the disorder or edge potential. The boundaries of those clusters finally create narrow channels that align mainly along the edge or random potential fluctuations. In case of a very clean high mobility electron system such a trigger effect for the cluster formation by the random potential is missing and the electron system has to find such a cluster structure by self organization, resulting in the formation of stripes or bubbles.
The numerical simulations are performed as described in Refs. [35][36][37] via a variational minimization of the selfconsistent Hartree-Fock-Roothaan equation [40][41][42][43]. To simulate ultra-high mobility samples, the random potential strength is kept low. The filling factor and the lateral system size are made as large as possible with respect to the available computing power. Configurations up to 1 µm 2 are achievable as shown in Fig. 1 with spatial resolution of ∼ 4.4 nm well below l B for a magnetic field B varying from, e.g., 1 (l B ∼ 26 nm) to 6.5 Tesla (l B ∼ 10 nm). The random potential is generated by Gaussian impurity potentials of radius 40 nm, the number of impurities is N = 2000, their random placement results in a fluctuating potential of V max = 0.43 mV and V min = −0.50 mV. At a total filling factor of, e.g., ν = 6.54 (= ν ↓ + ν ↑ = 3.54 + 3.0) and B = 1.5 Tesla, this corresponds to more than 2000 electrons. In order to generate transport data, we employ a non-equilibrium network model (NNM) introduced previously [44]. A very large number of step by step calculations are required [45] in the NNM. Hence, for keeping within the available computing time, the transport simulations have been performed for smaller sample size such as 500 × 500 nm 2 .
In Fig. 1 we depict the variation in ν ↓ ( r) for three different densities at fixed magnetic field. Fig. 1(a) shows the situation far from any half-odd average ν ↓ . The required area of the filled (spin-down) clusters is only a minor part of the total area. As seen in the figure this can be achieved by a nearly evenly-spaced distribution of "bubble"-like shaped clusters. Since we are considering already the 4th partly-filled spin-split LL, the boundaries of the bubbles consist of three sub-stripes because of the 3 nodes of the Landau basis function for the 4th LL [16]. A boundary with such an internal structure takes up a substantial area as well and even tends to dominate the region of a single bubble as a whole. In other words, the area that one bubble needs is dominated by the width of the boundaries and not by the region of an assumed idealized full LL. When increasing the filling factor towards half filling, as shown in Fig. 1(b), it becomes clear that such a bubble-like geometry of the clusters is impossible to achieve with the given total ν, because round bubbles would leave too much unfilled space between the bubbles even if touching each other. The only way to remove the unfilled space between bubbles is a change of the geometry so that the boundaries of different clusters arrange almost in parallel, which means a transfer to a "stripe"like geometry. In order to demonstrate that the width of the boundaries of the stripes and bubbles depend on the LL index, Fig. 1(c) shows the half-filled LL at filling factor ν ≈ 4.5 instead of ν = 6.5 and one can see that there is one sub-stripe less at the boundary, allowing a more dense arrangement of stripes or bubbles than in the higher LLs. These results suggest that the higher the filling ν the higher the tendency for creating the stripe pattern near a half-filled LL. Overall, the figures show that our HF calculations can provide a fascinating insight in the spatial behaviour of bubble and stripe configurations.  2 shows the transport data obtained by applying the NNM at different carrier densities. Since the stripe alignment tends to be more horizontal, i.e. along the xdirection as shown in Fig. 1, the longitudinal resistance appears higher for vertical sample current and lower for horizontal current flow. This is consistent with experimental observations resulting in large R xx peaks for vertical current, while for a horizontal current the R xx peaks are hardly visible. When the difference between horizontal and vertical R xx is no longer prominent, we find that we have reached a bubble phase.
The characteristics of the microscopic structure of the stripes consist on one hand in the periodicity of the stripe pattern and on the other hand in the microscopic details of the boundaries of the stripes. Together with the geometric shape of the stripes (and, indeed, the bubles), these determine the relation between the areas of full and empty LLs. Focusing first on the periodicity, Fig. 3 presents the 2D-Fourier transformation of the lateral carrier density at various magnetic fields from B = 1 Tesla to B = 6.5 Tesla at a fixed filling factor of ν = 4.5. While Fig. 3 (a) shows only a selection of a few spectra, Fig.  3 (b) displays the trend of all evaluated spectra. As can be seen in x-direction there is a smooth and somewhat smeared out distribution that starts from zero, indicating that there is no clear periodicity in x-direction. In contrast, in y-direction, there is a well pronounced maximum which matches the reciprocal stripe period in y-direction as also seen in Figs. 1 (b+c). For B = 1.5 Tesla the reciprocal period (wave number) appears to be close to 6 [46] and for B = 6 Tesla it matches 12, which is consistent with a √ B dependence. Furthermore, we extract the mean period of the corresponding stripe patterns, similar to those shown in Fig. 1, to be approximately 175 nm for B = 1.5 Tesla and 81 nm for B = 6 Tesla. Extending the analysis to other B values, we indeed find a clear √ B behaviour as shown in Fig. 3 (b). Previous theoretical results on the period d of the stripes have led to the expectation d = αR c = α (2n + 1) B/e [11,33]. Experimentally, α = 1.5 [17] and 3.6 [34] have been estimated. For n = 3 and B = 1.5 Tesla, these give 88 nm and 210 nm, respectively. Obviously, only the latter one is compatible with our result at 1.5 Tesla. Conversely, from Fig. 3 (b), we extract a value α = 2.9 ± 0.1. This agrees very well with previous straight-line CDW-based predictions of 2.7 [33] and 2.8 [12].
Friess et al. [17] investigated the Knight shift in nuclear magnetic resonance (NMR) spectra to demonstrate the co-existence of regions with different spin polarization due to the periodic variation of the filling factor in stripe and bubble phases. Their method provides direct information about the area fractions, although there is no direct information about geometry and periodicity. In order to extract also microscopic details of the structural information, they use a semiclassical model [17,47] based on superpositions of the single electron densities obtained from the Landau basis functions. In our case, we can similarly model the NMR intensity as Gaussian G describing the absorption spectrum of individual nuclei [47], f the NMR frequency, K max the maximal Knight shift for the fully spin-polarized LL at odd ν and f 0 the frequency of the non-shifted NMR line for the non spin-polarized situation at even filling factor ν − 1.
Numerically, I ν (f ) is calculated by evaluating our interacting ν( r) at each r and summing over all points of a typically 229 × 229 grid. In Fig. 4, we show that I ν (f ) exhibits features that red-shift to lower frequencies when increasing ν, e.g., from the non-spin polarized ν = 4 to a fully spin polarized ν = 5 (see supplement [48] for ν = 2 → 3). At intermediate filling factors the spectral response splits, mainly into double/triple peak structures. This indicates the co-existence of regions with different filling factor as expected from a stripe or bubble like electron distribution (see supplement [48]). Around ν ∼ 4.4 one can recognise three peaks with changing weights in the superposition while varying the total filling factor. This is in good agreement with the experimental results [17], but naturally less so when compared to a single-particle modelling. Intuitively, the driving force for the formation of bub-bles and stripes can be understood as follows: the Hund's rule behaviour causes a g-factor enhancement that maintains the tendency to fill up just one spin-level as much as possible before starting to fill up the next spin-level. When entering a stripe/bubble from outside (e.g. ν = 4.0, not spin polarized) to inside (e.g. ν = 5.0, fully spin polarized) the g-factor enhancement also drastically increases. Inside the stripe/bubble the spin-levels are strongly pushed apart, corresponding to a significantly lower energy of the occupied lower spin state, while outside there is minimal g-factor enhancement at even filling factor ν = 4.0. In terms of an effective single particle picture the electrons in the stripes/bubbles encounter a potential well established by the local variation of the g-factor enhancement and which dominates over the repulsive Hartree interaction. From a qualitative point of view, the stripes can be understood as leaky non-coupled electron wave guides that consist of self assembled onedimensional potential wells and the leakage shows up as the boundary region as discussed before.
In conclusion, we have computed the microscopic picture of stripe and bubble phase for the IQH regime at high LLs and in weak disorder. Our results rely on spatially-resolved, self-consistent HF calculations of nearly macroscopic sizes of O(1 µm 2 ) and, for transport calculations, are coupled to device contacts with finite currents. The existence of microscopic stripes and bubbles at weak disorder is thus confirmed. Their spatial features show intriguing extended oscillations along the stripes and surrounding the bubbles. These are clearly due to the structure of the underlying Landau states. We find that the stripe period scales with B −1/2 as expected and agrees in detail with previous experimental measurements. Overall, together with results from HF calculations in the strong disorder regime [35][36][37], this shows that the IQH regime can now be described spatially resolved with high accuracy from microscopic to nearmacroscopic length scales. Our results shine new light on the understanding of the microscopic picture of the IQH effect and demonstrate the permanent dominance of many particle physics for quantum Hall physics. The demonstrated Hund's rule behavior in context with the g-factor enhancement allows to incorporate the exchange interaction into an intuitive understanding of the major effects driving weakly and strongly disordered quantum Hall systems.

Supplemental Material
In Fig. S1 we show the variation in ν( r) for three different densities at fixed magnetic field B. These results are identical to Fig. 1 of the main text for ν ↓ ( r) but also show ν ↑ ( r) in the left column. The choice of colors is as in Refs. [1,2].
In order to show that the presence of the remnants of the LL wave functions around each stripe is significant, we perform the calculations of I ν (f ) for three test pat-terns for ν( r). The results are given in Fig. S2. We find that only the variation given by ν( r) as calculated in HF can reproduce essential global features of the experimental NMR results presented in Ref. [3].
In Fig. S3 we show the behavior of I ν (f ) for stripe-and bubble-like charge density waves. As in Fig. S2, the HF results for ν( r) lead to good qualitative agreement with the non-interacting model used in Ref. [3]. Nevertheless, the details around, e.g., ν = 2.5 are rather different, highlighting the importance of interactions. Lateral carrier density distribution mapped on the filling factor scale ν( r) for different total filling factors (a) ν = 6.20, (b) ν = 6.54 and (c) ν = 4.54. The left and right columns contains results for ν ↑ and ν ↓ , respectively. The color shades represent the filling factor range, where green indicates the second LL for ν ↓,↑ = 1 → 2, red the third LL, light blue the fourth and yellow denoting LL 5. These colours are as in Refs. 1 and 2 and the value indicated for ρ, B denote the electron density in units of 10 11 cm −2 and 1 Tesla, respectively. .