Hofstadter's Butterfly in Moire Superlattices: a Fractal Quantum Hall Effect
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Hofstadter's butterfly in moir´esuperlattices: A fractal quantum Hall effect C. R. Dean1;2, L. Wang2, P. Maher3, C. Forsythe3, F. Ghahari3, Y. Gao2, J. Katoch4, M. Ishigami4, P. Moon5, M. Koshino5, T. Taniguchi6, K. Watanabe6, K. L. Shepard1, J. Hone2, and P. Kim3 1Department of Electrical Engineering, Columbia University, New York, NY 2Department of Mechanical Engineering, Columbia University, New York, NY 3Department of Physics, Columbia University, New York, NY 4Department of Physics and Nanoscience Technology Center, University of Central Florida, Orlando, FL 5 Department of Physics, Tohoku University, Sendai, Japan and 6National Institute for Materials Science, 1-1 Namiki, Tsukuba, Japan PACS numbers: xx Electrons moving through a spatially periodic of electrons subjected simultaneously to both a periodic lattice potential develop a quantized energy spec- electric field and a magnetic field can be simply param- trum consisting of discrete Bloch bands. In two eterized by the dimensionless ratio φ/φ0 where φ = BA dimensions, electrons moving through a magnetic is magnetic flux per unit cell. The general solution of field also develop a quantized energy spectrum, this problem, however, exhibits a rich complexity due consisting of highly degenerate Landau energy to the incommensurate periodicities between the Bloch levels. In 1976 Douglas Hofstadter theoretically and Landau states21. In his seminal work1, Hofstadter considered the intersection of these two prob- showed that for commensurate fields, corresponding to lems and discovered that 2D electrons subjected rational values of φ/φ0 = p=q, where p and q are co- to both a magnetic field and a periodic electro- prime integers, the single-particle Bloch band splits into static potential exhibit a self-similar recursive en- q subbands (beginning with the Landau level descrip- ergy spectrum1. Known as Hofstadter's butter- tion it is equivalently shown2 that at these same rational fly, this complex spectrum results from a delicate values each Landau level splits into p subbands). This interplay between the characteristic lengths as- results in a quasi-continuous distribution of incommen- sociated with the two quantizing fields1{11, and surate quantum states that exhibits self-similar recursive represents one of the first quantum fractals dis- structure, yielding the butterfly-like fractal energy dia- covered in physics. In the decades since, exper- gram (see SI). imental attempts to study this effect have been Important insight into this problem was subsequently limited by difficulties in reconciling the two length provided by Wannier2, who considered the density of scales. Typical crystalline systems (< 1 nm peri- charge carriers, n, required to fill each fractal subband. odicity) require impossibly large magnetic fields Replotting the Hofstadter energy spectrum as integrated to reach the commensurability condition, while density versus field, Wannier realized that all spectral in artificially engineered structures (& 100 nm), gaps are constrained to linear trajectories in the density- the corresponding fields are too small to com- field diagram. This can be described by a simple Dio- pletely overcome disorder12{20. Here we demon- phantine relation strate that moir´esuperlattices arising in bilayer graphene coupled to hexagonal boron nitride pro- vide a nearly ideal-sized periodic modulation, en- (n=no) = t(φ/φo) + s (1) abling unprecedented experimental access to the fractal spectrum. We confirm that quantum Hall where n=no and φ/φo are the normalized carrier density effect features associated with the fractal gaps and magnetic flux, respectively, and s and t are both are described by two integer topological quantum integer valued. Here n=no represents the Bloch band numbers, and report evidence of their recursive filling fraction, which is distinct from the usual Landau structure. Observation of Hofstadter's spectrum level filling fraction, ν = nφ0=B (the two are related by in graphene provides the further opportunity to the normalized flux, i.e. n=no = νφ/φo). The physi- arXiv:1212.4783v1 [cond-mat.mes-hall] 19 Dec 2012 investigate emergent behaviour within a fractal cal significance of the s and t quantum numbers became energy landscape in a system with tunable inter- fully apparent with the discovery of the integer quantum nal degrees of freedom. Hall effect22 in 1980, after which it was shown, simulta- neously by both Stˇrreda4 and Thouless et al5, that the The total number of electron states per area of a com- Hall conductivity associated with each mini-gap in the 2 pletely filled Bloch band is exactly n0 = 1=A, where A fractal spectrum is quantized according to σxy = te =h. is the area of the unit cell of the periodic potential. In The second quantum number, s, physically corresponds a magnetic field, the number of states per area of each to the Bloch band filling index in the fractal spectrum6. filled Landau level (LL) is given by B/φ0 where φ0 = h=e This formalism suggests several unique and unambiguous is the magnetic flux quanta. The quantum description experimental signatures associated with the Hofstadter 2 energy spectrum that are distinct from the conventional a b quantum Hall effect: i) the Hall conductance can vary graphene non-monotonically and can even fluctuate in sign, ii) Hall BN conductance plateaus together with vanishing longitudi- θ nal resistance can appear at non-integer LL filling frac- tions, iii) the Hall conductance plateau remains quan- SiO2 2 Si tized to integral multiples of e =h, however, the quanti- 1 μm 30 nm zation integer is not directly associated with the usual LL c 14 d 35 6 80 filling fraction. Instead, quantization is equal to the slope 200K 150K B=1T 12 60 100K T = 300 mK 30 4 50K /h) of the gap trajectory in the n=no versus φ/φo Wannier 2 1.7K RXX (e 40 10 σ 25 R XY 2 diagram, in accordance with the Diophantine equation. 20 8 ) 20 ) ) 0 Ω Ω Ω -40 -20 0 20 40 0 Mini-gaps within the fractal energy spectrum become (k (k Vgate (Volts) 6 XX 15 XY R (k R p R significant only once the magnetic length (lB = =eB), -2 ~ 4 which characterizes the cyclotron motion, is of the same 10 -4 order as the wavelength of the periodic potential, which 2 5 0 0 -6 characterizes the Bloch waves. For usual crystal lattices, -40 -20 0 20 40 -40 -20 0 20 40 Vgate (Volts) Vgate (Volts) where the inter-atomic spacing is a few ˚angstroms,the necessary magnetic field is impossibly large, in excess of 10,000 T. The main experimental effort therefore has FIG. 1. Moir´e superlattice (a) Cartoon schematic of 12{20 been to lithographically define artificial superlattices graphene on hBN showing the emergence of a moir´epat- with unit cell dimension of order tens of nanometers so tern. The moir´ewavelength varies with the mismatch angle, that the critical magnetic field remains small enough to θ. (b) Left shows an AFM image of a multi-terminal Hall be achievable in the lab, yet still large enough so that bar. Right shows a high resolution image in a magnified re- the quantum Hall effect is fully resolved without being gion. The moir´epattern is evident as a triangular lattice (up- smeared out by disorder. Fabricating the optimally-sized per inset shows a further magnified region). FFT of the scan periodic lattice, while maintaining coherent registry over area (lower inset) confirms a triangular lattice symmetry with the full device and without introducing substantial dis- period 15:5 ± 0:9 nm. (c) Resistance measured versus gate voltage at zero magnetic field. Inset shows the corresponding order has proven a formidable technical challenge. Pat- conductivity versus temperature, indicating that the satellite terned GaAs/AlGaAs heterostructures with ∼ 100 nm features disappear above ∼100 K. (d) Longitudinal resistance periodic gates provided the first experimental support (left axis) and Hall resistance (right axis) versus gate volt- for Hofstadter's predictions17{19. However, limited abil- age at B = 1 T. The Hall resistance inverts sign and passes ity to tune the carrier density or reach the fully devel- through zero at the same gate voltage as the satellite peaks. oped quantum Hall effect (QHE) regime in these sam- ples has made it difficult to map out the complete spec- trum. While similar concepts have also been pursued in non-solid-state model systems23,24, the rich physics of the with triangular symmetry. Fast Fourier transform (FFT) Hofstadter spectrum remains largely unexplored. analysis of the image, shown in the inset, indicates that Heterostructures consisting of atomically thin materi- the moir´ewavelength is 15:5 ± 0:9 nm. This is compa- als in a multi-layer stack provide a new route towards rable to the maximal wavelength of ∼ 14 nm expected realizing a two-dimensional system with laterally mod- for graphene on hBN25{27 (set by the 1.8% lattice mis- ulated periodic structure. In particular, coupling be- match between the two crystals), suggesting that in this tween graphene and hexagonal boron nitride (hBN), device the BLG lattice is oriented with near zero angle whose crystal lattices are isomorphic, results in a periodic mismatch to the underlying hBN lattice. moir´epattern. The moir´ewavelength is directly related Fig. 1c shows transport data measured from the same to the angular rotation between the two lattices25{27, device. In addition to the usual resistance peak at the and is tunable through the desired length scales with- charge neutrality point (CNP), occurring at gate voltage 9,10 out the need for lithographic techniques . Moreover Vg ∼ 2 V, two additional satellite resistance peaks ap- hBN provides an ideal substrate for achieving high mo- pear, symmetrically located at Vsatl ∼ ±30 V away from bility graphene devices, crucial for high resolution quan- the CNP. These satellite features are consistent with a tum Hall measurements28,29, while field effect gating in depression in the density of states (DOS) at the super- graphene allows the Fermi energy to be continuously var- lattice Brillouin zone band edge, analogous to previous ied through the entire moir´eBloch band.