Abstract
When the motion of electrons is restricted to a plane under a perpendicular magnetic field, a variety of quantum phases emerge at low temperatures, the properties of which are dictated by the Coulomb interaction and its interplay with disorder. At very strong magnetic field, the sequence of fractional quantum Hall liquid phases^{1} terminates in an insulating phase, which is widely believed to be due to the solidification of electrons into domains possessing Wigner crystal^{2} order^{3,4,5,6,7,8,9,10,11}. The existence of such Wigner crystal domains is signalled by the emergence of microwave pinningmode resonances^{10,11}, which reflect the mechanical properties characteristic of a solid. However, the most direct manifestation of the broken translational symmetry accompanying the solidification—the spatial modulation of particles’ probability amplitudes—has not been observed yet. Here, we demonstrate that nuclear magnetic resonance provides a direct probe of the density topography of electron solids in the integer and fractional quantum Hall regimes. The data uncover quantum and thermal fluctuations of lattice electrons resolved on the nanometre scale. Our results pave the way to studies of other exotic phases with nontrivial spatial spin/charge order.
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Main
Wigner crystallization is a concept originally proposed to occur in a disorderfree dilute electron system at zero magnetic field, when the Coulomb energy dominates over the kinetic energy^{2}. Application of a strong perpendicular magnetic field (B) on a twodimensional electron system (2DES) also facilitates Wigner crystallization^{12,13,14,15}, as it quantizes the electron energy into a discrete spectrum known as Landau levels (LLs) and thereby quenches the kinetic energy. As the electrons are confined to orbits with spatial extent of the order of the magnetic length ℓ_{B} = (h/2πeB)^{1/2} (e: elementary charge, h: Planck’s constant), Wigner crystallization becomes governed not by the electron density n, but by the LL filling factor ν = nh/eB(= 2πnℓ_{B}^{2}). Evidence for the formation of Wigner crystal (WC) domains in the regime of integer^{16,17} and fractional^{10,11} ν is provided by the observation of resonances in the microwave conductivity. These resonances are interpreted as arising from shear modes of WC domains pinned by disorder and, consequently, as a manifestation of a finite shear modulus^{3,10,14}, a distinguishing feature of a solid. We use nuclear magnetic resonance (NMR) to demonstrate another defining feature of a solid related to its structural properties. The Knight shift measures the effective magnetic field that electron spins exert on the nuclei of the host material^{18}, making it a sensitive probe of the local electron density and its spatial modulation.
We studied a 2DES confined to a 27nmwide GaAs quantum well using a resistively detected NMR (RDNMR) technique^{19} in the millikelvin regime (Methods). Figure 1a shows resonance spectra of ^{75}As nuclei at B = 6.4 T, obtained at various filling factors around ν = 2, accessed using a gate voltage. At ν = 2, equal numbers of spinup and spindown electrons fill the lowest N = 0 LL, resulting in zero net spin polarization. The corresponding resonance spectrum thus represents the bare resonance frequency of the ^{75}As nuclei, which is unaffected by the electron spin. As we move away from ν = 2, the addition of spinup electrons to the first excited N = 1 LL or removal of spindown electrons from the filled N = 0 LL generates an effective filling factor ν^{∗} = ν − 2 with a finite net spin polarization (Fig. 1b), which shifts the spectra to lower frequencies (Knight shift).
In a simple picture, the net spin polarization is proportional to the number of electrons or holes added to the ν = 2 background, so that the Knight shift K_{s} should increase linearly with ν^{∗}. Notably, we observe an atypical nonlinear suppression of K_{s} at small ν^{∗}, accompanied by striking spectral anomalies. This becomes evident by comparing the experimentally observed spectra with simulations derived from a uniform 2DES. The simulations take into account the variation of the local electron (hole) density ρ(z) = n^{∗}ψ_{ν}(z)^{2} along the direction normal to the plane of the 2DES, with ψ_{ν}(z) being the subband wavefunction (Fig. 1c) and n^{∗}(≡ν^{∗}eB/h) the sheet density of the electrons or holes added to ν = 2. The simulation, which assumes the system to be uniform in the x–y plane, reproduces the spectra at ν = 2 − 1/3 and ν ≥ 2 + 1/5 (solid lines) well. These ‘normal’ lineshapes, characterized by sharp lowfrequency onsets and elongated highfrequency tails, can be understood as a spectroscopic map of ψ_{ν}(z)^{2} (refs 18, 19). However, the simulation fails to reproduce the spectra observed at 2 − 1/3 < ν < 2 and 2 < ν < 2 + 1/5 (dashed lines). K_{s}, measured at the peak of the spectra, is substantially below that expected from the simulation. Concurrently, we observe striking spectral anomalies, characterized by an elongated lowfrequency tail and a highfrequency cutoff, which are most pronounced at ν = 1.8 and 2.1. Intriguingly, around ν = 2.1, the lowfrequency tail exceeds the K_{s} value for a maximally polarized uniform 2DES.
We can explain both the atypical spectral lineshapes and the strong suppression of K_{s} coherently if we assume that the electrons have solidified into a periodic lattice. We note that the formation of an electron solid around integer filling factor is nontrivial^{16}, most of previous WC observations being made in the fractional regime. Figure 2a depicts ρ(x, y), the calculated spatial variation of the local electron density in the x–y plane for a Wigner solid. The calculation for ν < 2 assumes the Maki–Zotos ansatz (MZA) wavefunction^{14}, in which a hole in the filled N = 0 LL, described by a singleparticle harmonicoscillator wavefunction, is localized at each lattice site. For ν > 2 we extended the MZA by replacing the singleparticle wavefunction with that for the N = 1 LL, which gives rise to the characteristic ringshaped structure in Fig. 2a. The MZA is a meanfield Hartree–Fock solution, which describes an uncorrelated WC at T = 0 K. We therefore introduced a tunable parameter σ, which controls the Gaussian blurring that is added to the inplane density variation via the function to model correlation and finitetemperature effects. The spatially varying probability density of a WC induces a spatially varying Knight shift which deforms the NMR spectrum. For example, nuclei located in the lowdensity regions that form between lattice points give rise to spectral weight at high frequencies with much reduced Knight shifts. By taking into account both the inplane and outofplane density variation given by ρ(x, y) and ψ_{ν}(z)^{2}, respectively, we are able to simulate the NMR spectrum for a Wigner solid (Supplementary Information). Figure 2b demonstrates that simulations using σ as a fitting parameter faithfully reproduce the anomalous NMR spectra observed around ν = 1.8 and 2.1. The values for σ in the N = 1 LL are typically larger than those in the N = 0 LL, owing to the difference in the shape and spatial extent of the LL wavefunctions (Fig. 2c).
The spectra provide farreaching insight into the internal architecture of the solid, such as the quantum properties of its constituent particles. In the lower panels of Fig. 2d, the measured K_{s} is plotted against ν^{∗} and compared with calculations using σ = 0 and ℓ_{B} (thin dashed and solid lines, respectively). In contrast to the conduction onset in the N = 0 and N = 1 LLs situated at similar ν^{∗} values, as manifested by a rise in the longitudinal resistance (Fig. 2d, upper panels), the transition from a solid to a liquid phase occurs at much smaller ν^{∗} (∼0.18) in the N = 1 LL than in the N = 0 LL (ν^{∗}∼ 0.28), reflecting the larger spatial extent of the N = 1 LL wavefunction. Furthermore, the lowfrequency shoulder, peculiar to N = 1 LL spectra around ν = 2.1, is a hallmark of the nodal structure of the second LL wavefunction. These characteristics, determined by the Landau orbital index, highlight the quantum constitution of the WCs and their liquid–solid phase competition. Our simulations show that the lowfrequency shoulder at ν = 2.1 is sensitive to the σ value (Supplementary Information), which in turn allows us to quantify the deviation from the T = 0 Hartree–Fock WC.
Figure 3a illustrates the spectral evolution at elevated temperatures. Although the spectra shift to lower frequencies, with the peak approaching the position expected for a uniform electron liquid, the data clearly show that the WCs have not yet melted even at 350 mK. The evolution of the spectra can be reproduced by exploiting σ as a temperaturedependent parameter that describes the particles’ displacements from their equilibrium positions, induced by thermally excited phonon modes. Physically, σ^{2} represents the extra mean square displacement that is added to the zeropoint fluctuation at T = 0, which is given by λ^{2} = 2ℓ_{B}^{2} and 4ℓ_{B}^{2} for the uncorrelated WC in the N = 0 and N = 1 LLs, respectively (Supplementary Information). In Fig. 3b, we plot (σ/a_{ν})^{2} versus temperature for ν = 1.8, 1.9 and 2.1, obtained from the fits, where is the lattice constant. These values can be translated into the Lindemann parameter 〈u^{2}〉/a_{ν}^{2} ≡ (λ^{2} + σ^{2})/a_{ν}^{2}, as indicated on the right axis. Consistent with theory^{13}, (σ/a_{ν})^{2} increases almost linearly with T. However, the data reveal the temperature coefficients to be greater by a factor of 3–5 than expected from the simple model^{13}, assuming a realistic size (L ≈ 0.5–2 μm) (refs 10, 20) of the WC domains. Such deviation can be attributed to the finite thickness of the 2DES, which weakens the Coulomb interaction and thus reduces the magnetophonon frequency through the shear modulus^{21}. For ν = 2.1, a linear extrapolation finds a finite σ of ∼ 0.8ℓ_{B} at T = 0 K, which suggests correlation effects that require a comparison with more elaborate theories^{15,22,23,24}.
The capability of NMR to probe the internal structure of a 2DES also allows us to identify the origin of the insulating phase at low ν. At ν < 1/3, where transport measurements indicate that the system enters an insulating phase (Supplementary Information), NMR shows spectral anomalies (Fig. 4a) and strong suppression of K_{s} (Fig. 4b), which are found to be very similar to those observed for 5/3 < ν < 2 if particle–hole symmetry ν ↔ 2 − ν is invoked. Thus, we can say unambiguously that the insulating behaviour at ν < 1/3 is due to electron solidification, and not disorderinduced singleparticle localization of an Andersontype insulator^{25}. (See Supplementary Information for the roles of disorder on RDNMR spectra.) Moreover, the behaviour of K_{s} versus ν clearly shows that, under our experimental conditions, the localized particles in the WC are electrons and not fractionally charged quasiparticles which would appear at higher fields and in cleaner samples^{26}. The existence of a solid phase at ν = 1/5 and in the vicinity of ν = 1/3 is not trivial. Theories demonstrate that, in the thermodynamic limit, the Laughlin liquid state, being stabilized by longrange quantum fluctuations, is the ground state at these ν values^{15,22,23,24}. In real systems, however, disorder may introduce finitesize effects into the 2DES, where a correlated crystalline phase can have a lower energy at ν = 1/5 (ref. 27). Figure 4b reveals that as ν increases above 1/5, the system starts to deviate rapidly from the uncorrelated WC, which is indicated by the increase in σ (Fig. 4b inset). This continues until the system gives way to the liquid phase at ν = 1/3. Our data thus constitute spectroscopic evidence for the evolution of quantum electron solids driven by the interplay between disorder and quantum fluctuations.
NMR spectroscopy of 2DESs has mostly been used to probe the electron spin polarization via the Knight shift. Our NMR experiments have demonstrated that it is a far more powerful tool, capable of revealing nanoscale details of the spatial variation of spin/charge density that emerges when the translational symmetry is broken. This work provides the framework for future experiments that probe other exotic phases with nontrivial spatial spin/charge order, including stripe/bubble phases^{28}, reentrant insulating phases^{29}, and quantum Hall nematic phases^{30}.
Methods
The sample is a 100μmwide Hall bar with the 2DES confined to a 27nmwide gallium arsenide (GaAs) quantum well, grown on a n^{+}substrate that serves as back gate. The sample is inside the mixing chamber of a dilution refrigerator and surrounded by a threeturn coil which is connected to a frequency generator. Resonance spectra were obtained, following ref. 19, by monitoring the changes in the longitudinal resistance R_{xx} near ν_{read} = 0.59, induced by radiofrequency excitations (−17 dBm) at the filling factors of interest (that is, ν = 2 ± ν^{∗} and ν < 1/3).
References
Tsui, D. C., Stormer, H. L. & Gossard, A. C. Twodimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).
Wigner, E. P. On the interaction of electrons in metals. Phys. Rev. 46, 1002–1011 (1934).
Andrei, E. Y. et al. Observation of a magnetically induced Wigner solid. Phys. Rev. Lett. 60, 2765–2768 (1988).
Willett, R. L. et al. Termination of the series of fractional quantum Hall states at small filling factors. Phys. Rev. B 38, 7881–7884 (1988).
Jiang, H. W. et al. Quantum liquid versus electron solid around ν = 1/5 Landaulevel filling. Phys. Rev. Lett. 65, 633–636 (1990).
Goldman, V. J., Santos, M., Shayegan, M. & Cunningham, J. E. Evidence for twodimensional quantum Wigner crystal. Phys. Rev. Lett. 65, 2189–2192 (1990).
Williams, F. I. B. et al. Conduction threshold and pinning frequency of magnetically induced Wigner solid. Phys. Rev. Lett. 66, 3285–3288 (1991).
Buhmann, H. et al. Novel magnetooptical behavior in the Wignersolid regime. Phys. Rev. Lett. 66, 926–929 (1991).
Kukushkin, I. V. et al. Evidence of the triangular lattice of crystallized electrons from time resolved luminescence. Phys. Rev. Lett. 72, 3594–3597 (1994).
Ye, P. D. et al. Correlation lengths of the Wignercrystal order in a twodimensional electron system at high magnetic fields. Phys. Rev. Lett. 89, 176802 (2002).
Chen, Y. P. et al. Melting of a 2D quantum electron solid in high magnetic field. Nature Phys. 2, 452–455 (2006).
Lozovik, Yu. E. & Yudson, V. I. Crystallization of a twodimensional electron gas in a magnetic field. JETP Lett. 22, 11–12 (1975).
Ulinich, F. P. & Usov, N. A. Phase diagram of a twodimensional Wigner crystal in a magnetic field. Zh. Eksp. Teor. Fiz. 76, 288–294 (1979).
Maki, K. & Zotos, X. Static and dynamic properties of a twodimensional Wigner crystal in a strong magnetic field. Phys. Rev. B 28, 4349–4356 (1983).
Lam, P. K. & Girvin, S. M. Liquid–solid transition and the fractional quantumHall effect. Phys. Rev. B 30, 473–475 (1984).
Chen, Y. et al. Microwave resonance of the 2D Wigner crystal around integer Landau fillings. Phys. Rev. Lett. 91, 016801 (2003).
Zhu, H. et al. Pinningmode resonance of a Skyrme crystal near Landaulevel filling factor ν = 1. Phys. Rev. Lett. 104, 226801 (2010).
Kuzma, N. N., Khandelwal, P., Barrett, S. E., Pfeiffer, L. N. & West, K. W. Ultraslow electron spin dynamics in GaAs quantum wells probed by optically pumped NMR. Science 281, 686–690 (1998).
Tiemann, L., Gamez, G., Kumada, N. & Muraki, K. Unraveling the spin polarization of the ν = 5/2 fractional quantum Hall state. Science 335, 828–831 (2012).
Martin, J. et al. Localization of fractionally charged quasiparticles. Science 305, 980–983 (2004).
Ettouhami, A. M., Klironomos, F. D. & Dorsey, A. T. Static and dynamic properties of crystalline phases of twodimensional electrons in a strong magnetic field. Phys. Rev. B 73, 165324 (2006).
Yi, H. & Fertig, H. A. Laughlin–Jastrowcorrelated Wigner crystal in a strong magnetic field. Phys. Rev. B 58, 4019–4027 (1998).
Shibata, N. & Yoshioka, D. Ground state phase diagram of 2D electrons in high magnetic field. J. Phys. Soc. Jpn 72, 664–672 (2003).
Chang, CC., Jeon, G. S. & Jain, J. K. Microscopic verification of topological electronvortex binding in the lowest Landaulevel crystal state. Phys. Rev. Lett. 94, 016809 (2005).
Kivelson, S., Lee, DH. & Zhang, SC. Global phase diagram in the quantum Hall effect. Phys. Rev. B 46, 2223–2238 (1992).
Zhu, H. et al. Observation of a pinning mode in a Wigner solid with ν = 1/3 fractional quantum Hall excitations. Phys. Rev. Lett. 105, 126803 (2010).
Chang, CC., Töke, C., Jeon, G. S. & Jain, J. K. Competition between compositefermioncrystal and liquid orders at ν = 1/5. Phys. Rev. B 73, 155323 (2006).
Lilly, M. P., Cooper, K. B., Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Evidence for an anisotropic state of twodimensional electrons in high Landau levels. Phys. Rev. Lett. 82, 394–397 (1999).
Eisenstein, J. P., Cooper, K. B., Pfeiffer, L. N. & West, K. W. Insulating and fractional quantum Hall states in the first excited Landau level. Phys. Rev. Lett. 88, 076801 (2002).
Xia, J., Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Evidence for a fractionally quantized Hall state with anisotropic longitudinal transport. Nature Phys. 7, 845–848 (2011).
Acknowledgements
The authors would like to acknowledge R. Morf for insightful discussions and valuable comments. Appreciation also goes to N. Kumada for experimental support and T. Higashi for sharing the results of his theoretical calculations.
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L.T. and T.D.R. performed measurements. L.T. wrote the measurement program. T.D.R. and K.M. performed simulations of the NMR spectra. N.S. provided density matrix renormalization group calculations for comparison and provided theoretical support. K.M. grew heterostructures and fabricated the sample. L.T., T.D.R. and K.M. analysed the data and wrote the paper with input from N.S.
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Tiemann, L., Rhone, T., Shibata, N. et al. NMR profiling of quantum electron solids in high magnetic fields. Nature Phys 10, 648–652 (2014). https://doi.org/10.1038/nphys3031
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DOI: https://doi.org/10.1038/nphys3031
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