Abstract

The quantum Hall effect (QHE) originates from discrete Landau levels forming in a two-dimensional electron system in a magnetic field1. In three dimensions, the QHE is forbidden because the third dimension spreads Landau levels into overlapping bands, destroying the quantization. Here we report the QHE in graphite crystals that are up to hundreds of atomic layers thick, a thickness at which graphite was believed to behave as a normal, bulk semimetal2. We attribute this observation to a dimensional reduction of electron dynamics in high magnetic fields, such that the electron spectrum remains continuous only in the field direction, and only the last two quasi-one-dimensional Landau bands cross the Fermi level3,4. Under these conditions, the formation of standing waves in sufficiently thin graphite films leads to a discrete spectrum allowing the QHE. Despite the large thickness, we observe differences between crystals with even and odd numbers of graphene layers. Films with odd layer numbers show reduced QHE gaps, as compared to films of similar thicknesses but with even numbers because the latter retain the inversion symmetry characteristic of bilayer graphene5,6. We also observe clear signatures of electron–electron interactions including the fractional QHE below 0.5 K.

Main

The Lorentz force imposed by a magnetic field, B, changes the straight ballistic motion of electrons into spiral trajectories aligned along B (Fig. 1a). This spiral motion gives rise to Landau bands that are characterized by plane waves propagating along the B direction but quantized in the directions perpendicular to the magnetic field7. The increase in B changes the distance between Landau bands, and the resulting crossings of the Fermi level, EF, with the Landau band edges lead to quantum oscillations. When only the lowest few Landau bands cross EF (ultra-quantum regime (UQR)), the magnetic field effectively makes the electron motion one-dimensional (1D) (with conductivity allowed only in the direction parallel to B). In normal metals such as copper, this dimensional reduction would require B > 10,000 T. In semimetals with their small Fermi surfaces (graphite2,4,8, for example), the dimensional reduction and the UQR can be reached in moderate B. Indeed, magnetotransport measurements in bulk crystals4,9,10,11 and films of graphite12,13 revealed quantum oscillations reaching the lowest Landau bands. However, no dissipationless QHE transport could be observed as expected for the 3D systems.

Fig. 1: QHE in 3D graphite.
Fig. 1

a, Ballistic electron trajectories in the absence of magnetic field (left schematic) undergo dimensional reduction into cyclotron spirals in high fields (middle). In thin crystals, the spiral trajectories can form standing waves (right). b, Transversal ρxy (black curve) and longitudinal ρxx (red) resistivity as a function of B, measured at 0.25 K in a 6-nm-thick graphite device. The density nb = −1.1 × 1012 cm−2 is induced by applying a back-gate voltage and the negative sign corresponds to holes. The inset shows a schematic of our hBN/graphite/hBN heterostructures. c, Hall conductivity σxy (black curve) and ρxx (red) as a function of back-gate voltage Vbg at B = 15.5 T. Grey curves: measurements at B from 13.5 to 17.5 T; T = 0.25 K; Vtg = 0 V; same device as in b. Inset: optical micrograph of the used Hall bar device (scale bar, 10 µm). d, ρxx in the Corbino geometry; T = 0.25 K; L = 10 nm. ρxx diverges at the integer filling factors ν (top x axis). Inset: false-colour scanning electron micrograph of the Corbino device. Scale bar, 5 µm.

The UQR in graphite can be reached when B > 7 T (ref. 4) where all electrons near the Fermi level belong to the two half-filled 1D bands emerging from the lowest (0 and 1) Landau levels of bilayer graphene that is the building block of Bernal-stacked graphite5. At the opposite edges of the 3D Brillouin zone (lines +KH and −KH, designated below as valleys) these 0 and 1 states reside on alternating atomic planes of carbon atoms, and in the bulk graphite crystal the inversion symmetry of its bilayer unit cell makes these spectra valley-degenerate. The 1D character of Landau bands in graphite in the UQR suggests the formation of charge-density-wave states14 at B > 25 T due to their half-filling and of an insulating phase15 at even higher B > 70 T, where the splitting between the 0 and 1 Landau bands exceeds their bandwidth. In our case of thin graphite crystals, these quasi-1D bands split into a set of standing waves separated by Δkz = π/L, where L is the crystal thickness (Fig. 1a). The standing waves lead to discretization of the electronic spectrum and, therefore, can in principle allow the QHE that is otherwise forbidden in three dimensions. Here we show that, in the UQR, high-quality graphite films of a submicrometre thickness indeed exhibit the fully developed QHE with quantized Hall and zero longitudinal resistances. Moreover, the standing waves retain the valley-layer correspondence of the 0 and 1 bulk Landau bands so that crystals with an even number of layers N retain the valley degeneracy in the standing-wave spectrum whereas the degeneracy is lifted for odd N because of the lack of inversion symmetry6. In a way, the described films exhibit the electronic properties of both 3D and 2D conductors and, hence, can be referred to as a 2.5D system.

To prepare our devices, we cleaved natural graphite crystals along their basal planes using the micromechanical exfoliation technique16. Then we encapsulated the cleaved crystals in hexagonal boron nitride (hBN), expanding the technology of van der Waals heterostructures to 3D systems17. Due to a self-cleansing mechanism, this approach results in quality atomically flat interfaces with little electron scattering at the surfaces18,19. We focus on crystals with thicknesses above 10 graphene layers (L > 3.5 nm), which are commonly considered as 3D graphite electronically20. For the upper limit of L, we chose thicknesses of about 100 nm because an average distance between stacking faults in natural graphite is known to be 100–200 nm21,22 and those defects efficiently decouple electronic states in different parts of graphene stacks23. All of our devices showed a metallic temperature dependence of resistivity (see, for example, Supplementary Fig. 1). Above 30 K the mobility of charge carriers followed a 1/T dependence, in agreement with the earlier studies24,25, and saturated below 10 K to approximately 4 × 105 cm2 V−1 s−1.

As we turned the magnetic field on, our devices exhibited strong positive magnetoresistance such that the longitudinal resistivity ρxx notably increased with increasing B from 0 to 0.4 T (Supplementary Fig. 1), which is typical for compensated semimetals26. Already at fairly low B below 0.1 T, we observed clear Shubnikov–de Haas oscillations that developed into the QHE above approximately 7 T (Fig. 1b). Indeed, one can see that around B = nh/, the Hall resistivity ρxy plateaus out at unit fractions of h/e2, which is accompanied by vanishing ρxx. Here, n is the electrostatically controlled carrier density, h is the Planck constant, e is the elementary charge and ν is the Landau filling factor that relates n to the flux quantum, h/e. The QHE is also clearly seen if we fixed B and changed n (Fig. 1c). Note that current densities as low as 50 nA µm−1 were sufficient to suppress the QHE (Supplementary Fig. 2). Vanishing ρxx (and σxx) signifies dissipationless transport along the edges of the otherwise insulating films1. We confirmed these insulating states using the edgeless (Corbino) geometry: Fig. 1d shows the insulating behaviour of a 10-nm-thick Corbino device where, for integer ν, ρxx exceeds 0.1 GΩ.

In graphite, mobile charge carriers induced by gate voltages are mainly localized in the first couple of layers near the surface because of electrostatic screening. Since the 0 and 1 Landau bands cross EF at almost half-filling (Fig. 2c), their surface states coexist and mix with the bulk states, making the two graphite surfaces interconnected (correlated). However, for sufficiently high doping the situation is expected to change because higher Landau bands can also become occupied, which results in charge accumulation near the surfaces. The resulting surface states decay exponentially into the bulk, and this breaks down correlations between the two surfaces. To disentangle the contributions from the bulk and surface states to the observed QHE we carried out transport measurements in a double-gated Corbino device (Fig. 2) and electronic compressibility measurements in a double-gated capacitor device (Supplementary Fig. 3). With the two gates, we can induce charge carriers at the top (nt) and bottom (nb) surfaces independently. Longitudinal conductivity σxx(B) as a function of the electrostatically induced density ntot = nt + nb is shown as a colour map in Fig. 2a where metallic states (bright colours) fan out from the neutrality point, ntot = 0. The majority of the observed features can be attributed to the discussed 2.5D QHE states as in high fields they are linear in B, and their slopes B/ntot follow integer filling factors. However, there is a notable change at approximately 7 T, below which the QHE is obscured by a complicated criss-crossing behaviour (the existence of the critical field manifests itself even more clearly in thicker devices, see Supplementary Fig. 4). In addition, we also observe features that are not linear in B and do not follow the integer ν, examples of which are indicated by the dashed curves in Fig. 2a. We attribute those quantized levels to the 2D accumulation layers at the two graphite surfaces (for discussion, see Methods). The difference between the quantized features due to the surface states and those due to the 2.5D QHE is especially obvious from the maps of σxx (nt, nb) at a fixed B (Fig. 2b; also, Supplementary Fig. 3). The 2.5D QHE states formed by the standing waves in the bulk follow the diagonal lines on the map; that is, both top and bottom gates contribute equally to the filling of the Landau states that penetrate through the whole bulk. In contrast, the quantized states at the top and bottom surfaces follow horizontal and vertical lines, respectively, which indicates that the states are electronically decoupled and interactions between the two surfaces are screened out27. To better understand the origin of the surface states we performed self-consistent tight-binding/Hartree calculations (Methods), and the computed density of states and carrier density profiles are presented in Supplementary Fig. 3.

Fig. 2: Landau levels in thin graphite films.
Fig. 2

a, σxx as a function of B and ntot measured at nt = nb in a Corbino device with L = 11.6 nm; T = 0.25 K. The white dashed curves are guides to the eye for the features arising from the Landau quantization of electrostatically induced surface states. b, Map of σxx (nt, nb) at 18 T. The horizontal and vertical features at the edges of the map (shown by white contour lines) are due to the quantized states at the graphite surfaces. The diagonal features are ascribed to the kz-quantized lowest Landau bands that extend through the graphite bulk (2.5D QHE). Logarithmic colour scales for both a and b; navy to yellow is 50 nS to 50 µS. c, Dispersion relation for low Landau bands calculated using the tight-binding model at 15 T (Methods). Only the lowest two bands (labelled as 0 and 1, and split by δ10) cross the Fermi level.

The conductivity maps in Fig. 2 also show that the gaps along the integer filling factors can close and reopen as a function of both n and B, which is indicative of level crossings. To explain the observed patterns, we refer to the band structure of graphite in the Slonczewski–Weiss–McClure (SWMC) tight-binding model2,8,28,29 for Bernal-stacked graphite (see Methods). At zero field, the low-energy electronic structure is dominated by two bands in two valleys2,8,28 that are degenerate for kx,y = ±K and weakly disperse in kz (+KH and −KH valleys). In quantizing fields4,29, these are reduced to two nearly coinciding Landau bands 0 and 1 (Fig. 2c). They cross the Fermi level close to kF ≈ π/4c (c = 0.335 nm is the interlayer distance in graphite), exhibit the 1D Fermi velocity vF ≈ 15,000 m s−1 and are split by δ10 ≈ 0.37 meV T−1 (Methods).

In graphite crystals with a small L, electrons in the two Landau bands 0 and 1 form standing waves, which for different valleys reside on graphene layers of opposite parity: the states in the +KH valley live on the even layers, and the states in the −KH valley on the odd ones (see Methods). For even-N crystals, there is the same number of even and odd layers to support states in both +KH and −KH valleys, leading to the valley-degenerate ladder of states that are spaced in kz by the distance \(\frac{\rm{ \uppi }}{{c(N + 2)}}\). In odd-N crystals, the number of even and odd states differs by one, which results in a kz-shift of approximately \(\frac{\rm{ \uppi} }{{2c(N + 1)}}\) between the two ladders, around the Fermi level. This shift lifts the valley degeneracy and, therefore, reduces the energy spacing, δEodd, by nearly a factor of two as compared to the even-N films (see Fig. 3a). We refer to this difference between the standing-wave spectra in graphite with even and odd N as the layer-parity effect.

Fig. 3: Thickness and layer-parity dependence of the 2.5D QHE.
Fig. 3

a, E(kz) for the two lowest Landau bands in graphite (grey and red curves). The blue and orange dots are the standing-wave states caused by kz-quantization calculated for +KH and −KH valleys, respectively. The top and bottom panels are for even N = 34 and odd N = 35, respectively. b, σxx (B, ν) for 34- and 35-layer devices (top and bottom panels, respectively) measured at 1 K. Logarithmic greyscale: black to white is 0.1 to 5.6 µS. Note the double degeneracy (Δν = 2) for even-layered graphite and the fully lifted degeneracy (Δν = 1) for the odd-layered one. c, Calculated density of states as a function of B and ν for the graphite device in b. Logarithmic scale: black to white is 0.01 to 1.5 eV−1 nm−2 and 0.01 to 1 eV−1 nm−2 for the top and bottom panels, respectively. d, Energy gaps for the 2.5D QHE as a function of thickness. Shown are the maximal gaps found away from level crossings. Red symbols, graphite devices with even N; blue symbols, odd N; black symbols, unconfirmed parity. The red and blue lines are the gaps estimated from the free-particle model. Top inset: examples of Arrhenius plots for three devices. Bottom inset: relative strengths of δEeven/odd, δ10 and 2µBB (Zeeman splitting) gaps as a function of N and B.

To test this effect, we measured the 2.5D QHE in several pairs of Corbino devices, in which the thickness differed only by a single graphene layer. We fabricated these pairs from the same graphite crystals having monatomic steps on the surface (Methods). The transport measurements were performed in sufficiently high B to induce spin polarization; that is, 2µBB > kBT, where µB is the Bohr magneton and kB is the Boltzmann constant. Figure 3b shows that for the 34-layer graphite device the degeneracy of the QHE states was 2 whereas all of the unit steps in ν were observed for the device with N = 35. This indicates that all of the degeneracies were lifted in the latter case, including the valley degeneracy. Data for devices with N = 14 and 15 are presented in Supplementary Fig. 5.

The pronounced level crossings seen in the maps of Figs. 2 and 3 can also be related to specifics of the 2.5D QHE in graphite. This behaviour arises due to competition between the Landau band splitting, δ10, the spin splitting, 2µBB, and the B-independent standing wave spacing, δEeven/odd, which all lead to a complex hierarchy of energy gaps and numerous level crossings with changing B. The bottom inset in Fig. 3d summarizes the expected interplay between the gap sizes for different fields, thicknesses and parities (Methods and Supplementary Fig. 6). Figure 3 also shows the measured maximal QHE gaps, ΔE, for each of the studied samples with different L. The gaps were measured away from level crossings using the Arrhenius model for the dissipative conductivity in the QHE regime, \(\sigma _{xx} \propto {\rm e}^{ - \Delta E/2k_{\rm B}T}\) (top inset of Fig. 3d). The red and blue curves show the upper bounds for ΔE as found in our modelling for even and odd N, respectively.

We show more details of the level crossing behaviour in Fig. 4. Figure 4a compares the measured and simulated σxx (B, ν) for the 11.6 nm device in Fig. 2. In both cases there are two periods, Δν, at which crossings occur: Δν = 4 and 8. For N > 20, we estimate (see Methods) that δEeven/odd < δ10 for the experimental range of B and, therefore, the level crossings are dominated by fast changes in the Landau band splitting. Figure 4b highlights the complex hierarchy of gaps for different B. For example, at about 13 T, 4 valley-degenerate levels approach each other, which effectively results in relatively large gaps between crossings and, therefore, in robust quantization such that steps with Δν = 8 persist to kBT ≈ δEeven ≈ 94N−1 meV (this dependence is also shown by the red curve in Fig. 3d). On the other hand, at around 10 T, valley-degenerate levels cross at some distance from each other, resulting in approximately twice smaller gaps and steps with Δν = 4, as observed in both experiment and simulations. Similarly, for films with odd N, in which the valley degeneracy is lifted, the upper limit for the 2.5D QHE gaps is set by δEodd ≈ 47N−1 meV (blue curve in Fig. 3d). Note that the observed dependences δE(N) imply that, at millikelvin temperatures, the 2.5D QHE should be observable even in micrometre-thick graphite, if no stacking faults are present that would probably suppress the quantization.

Fig. 4: Hierarchy of QHE gaps and level crossings in the UQR.
Fig. 4

a, Left panel: map of σxx (B,ν) for the device in Fig. 2; T = 3 K. Logarithmic grey scale: black to white is 0.6 to 5.6 µS. Right panel: calculated density of states as a function of B and ν. Colour scale: black to white is 0.1 to 2 eV−1 nm−2. b, Calculated 0 (black) and 1 (red) Landau bands of 34-layer graphite. The solid (dashed) lines are \(\uparrow\) \(\left( \downarrow \right)\) spin states. The Landau level crossings result in QHE-gap closures in a. c, Left panel: the same as in the left panel of a but the measurements were performed at 1.2 K. Logarithmic scale: black to white is 0.1 to 7.4 µS. Right panel: zoomed-in region of b. Gap closures in the left panel of c and the level crossings in the right panel of c are labelled by the corresponding spin states \(\uparrow\) and \(\downarrow\). The coloured background specifies ν = −8 (pink), −6 (yellow), −4 (violet), −2 (orange) and 0 (blue).

The free-particle (non-interacting) model describes all of the main characteristics of the observed QHE confirming that the phenomenon arises due to the standing waves and is controlled mostly by graphite’s thickness and layer parity. However, at our highest B and lowest T, the devices also exhibit features indicating that electron–electron interactions contribute to the electronic properties. For example, Fig. 4c and Supplementary Fig. 7a show that crossings between the same-spin states are much broader than those between the opposite-spin states, which suggests some magnetic ordering30. Besides, even-layer graphite devices exhibit spontaneous valley polarization at T < 0.5 K (Supplementary Fig. 7f–g). For relatively thin devices (L = 3–4 nm) we also observed the onset of the fractional QHE (Supplementary Fig. 8). The general expectation is that electron–electron interactions can trigger a variety of phase transitions in the form of density waves, excitonic condensation or Wigner crystals15,31,32, and the high-quality graphite films introduced in this work offer a venue (especially, at millikelvin temperatures) to study such transitions with many competing order parameters.

Methods

Device fabrication

Our hBN/graphite/hBN heterostructures were assembled using the dry transfer technology17,33. Graphite and hBN flakes were first mechanically exfoliated onto oxidized Si wafers. Then the top hBN and graphite crystals were sequentially picked up by a polymer bilayer of polydimethylsiloxane and polymethylmethacrylate (PMMA). The hBN/graphite stack was released onto the bottom hBN crystal, which completed the heterostructure. It is known that accurate crystallographic alignment between graphite and hBN crystal lattices can result in superlattice effects and, therefore, we intentionally misaligned the graphite with respect to both hBN crystals by angles larger than 5°. This makes the moiré period small and shifts the band structure reconstruction to energies inaccessible in transport measurements. Raman spectroscopy was also employed to confirm that the chosen graphite crystals had the Bernal (ABA) stacking34 (Supplementary Fig. 9).

Electrical contacts to graphite were prepared by electron-beam lithography, which created openings in a PMMA resist and allowed us to remove the top hBN layer using reactive ion etching. Electron-beam evaporation was then used to deposit metal contacts Cr/Au (3 nm/80 nm) into the etched areas. Another round of electron-beam lithography and electron-beam evaporation was implemented to define the top gate, which also served as an etch mask to define the Hall bar geometry. In the case of Corbino devices, crosslinked PMMA made by overexposure of the electron-beam resist was used as an insulating bridge between the inner and outer contacts.

To investigate the layer-parity effect, pairs of Corbino devices were fabricated on the same graphite flakes that were chosen to have isolated monatomic steps on the surface. The selection of graphite was based on the optical contrast but the step height was subsequently checked by atomic force microscopy. We then determined the layer parity of the graphite at each side of the step based on the degeneracy period of the fan diagram, which, in combination with measured thickness, allowed us to unambiguously identify the number of layers.

In total, we studied approximately 20 devices in both Hall bar and Corbino geometries with thicknesses ranging from 3.5 to 93 nm. In addition, to distinguish between surface and bulk effects, we fabricated several devices on quartz substrates to carry out capacitance measurements. Furthermore, to check whether stacking faults in graphite destroy the observed 2.5D QHE, we prepared special hBN/graphite-graphite/hBN heterostructures in which the graphite film consisted of two crystals that were intentionally slightly misaligned to form an artificial stacking fault. Such devices exhibited no insulating states essential for observation of the QHE (Supplementary Fig. 9a,b). We also note that hBN encapsulation is crucial for the observation of the 2.5D QHE as witnessed in experiments using similar devices but with graphite placed in direct contact with the rough quartz substrate (no bottom hBN). Its absence ruined the electronic quality of the bottom graphite surfaces, which was sufficient to smear the 2.5D QHE gaps completely (Supplementary Fig. 9d,e). On the other hand, for the properly encapsulated graphite the 2.5D QHE was quite robust and observed in all of our devices, even for lateral sizes up to tens of micrometres (limited only by the maximum area we could find that was free from contamination bubbles) (Supplementary Fig. 9f).

Transport and capacitance measurements

The longitudinal and Hall voltages were recorded using an SR830 lock-in amplifier and applying a.c. currents typically of about 10 nA. Such unusually small currents were found necessary to avoid the suppression of the observed 2.5D QHE (Supplementary Fig. 2). For measurements on the Corbino devices, a small bias (typically 0.1 mV) was applied between the inner and outer contacts, and the current was recorded using again an SR830 amplifier. The capacitance was measured using an on-chip cold bridge based on a high-electron mobility transistor35. The typical excitation voltages were in the range from 1 to 10 mV, depending on the thickness of the hBN dielectric layer.

Away from the QHE regime (high temperatures and low magnetic fields), σxx and σxy in semimetallic graphite can be described by the expressions \(\sigma _{xx}\left( {n_i,\mu _i,B} \right) = \mathop {\sum}\nolimits_i {\frac{{n_ie\mu _i}}{{1 + \mu _{_i}^2B^2}}}\) and \(\sigma _{xy}\left( {n_i,\mu _i,B} \right) = \mathop {\sum}\nolimits_i {\frac{{n_ie\mu _i^2B}}{{1 + \mu _i^2B^2}}}\), where ni and µi are the carrier densities and mobilities, respectively, and the index i runs through different types of charge carrier. Using the above expressions, we fitted the measured B and T dependences for σxx and σxy and found that the model with just two types of carrier provides good agreement with the experiment. This also allowed us to estimate the charge densities and mobilities in the absence of electrostatic doping (Supplementary Fig. 1).

Tight-binding description of Landau bands in the UQR

The SWMC model2,8,29 has been widely employed to describe the band structure of graphite. While values of most of the hopping amplitudes (γ0, γ1, γ3 and γ4) have been well established in the earlier studies, there is no clear consensus about the values of γ2, γ5 and a dimer site energy shift Δ; hence, we adjusted them to get the best fitting to the experimental results obtained in the present work. Namely, we chose γ2, γ5 and Δ to match the slope of 0, 1 Landau bands dispersion near the Fermi level (which is given by γ2 with an extra correction coming from γ3) and the distance δ10 between the 0 and 1 Landau bands. Thus, the values of the SWMC parameters we used are γ0= 3.16 eV, γ1 = 0.39 eV, γ2 = −17 meV, γ3 = 0.315 eV, γ4 = 70 meV, γ5 = 38 meV and Δ = −5 meV.

The zero-field spectrum of graphite, determined by the SWMC model, contains four bands. Two high-energy bands, \({\it{\epsilon }}_ \pm ^{{\mathrm{split}}}\left( {k_z,p} \right)\) and

$$\begin{array}{*{20}{l}}{\it{\epsilon }}_ \pm ^{{\mathrm{split}}}\left( {k_z,p} \right) & = & \frac{1}{2}\left( {E_ \pm + E_3} \right)\\ && \pm \sqrt {\frac{1}{4}\left( {E_ \pm - E_3} \right)^2 + \left( {1 \mp \frac{{\gamma _4{\it{\Gamma }}}}{{\gamma _0}}} \right)^2\left( {vp} \right)^2} \end{array}$$

form a minority hole Fermi surface at the H point (kz = π/2c) whereas the other two degenerate low-energy bands, \({\it{\epsilon }}_ \pm ^{{\mathrm{low}}}\left( {k_z,p} \right)\), which is reminiscent of the bilayer graphene low-energy bands5, and

$$\begin{array}{*{20}{l}}{\it{\epsilon }}_ \pm ^{{\mathrm{low}}}\left( {k_z,p} \right) & = & \frac{1}{2}\left( {E_ \mp + E_3} \right) \\ && \pm \sqrt {\frac{1}{4}\left( {E_ \mp - E_3} \right)^2 + \left( {1 \pm \frac{{\gamma _4{\it{\Gamma }}}}{{\gamma _0}}} \right)^2\left( {vp} \right)^2} \end{array}$$

form majority hole and electron Fermi surfaces. Here, \(v = \frac{{\sqrt 3 \gamma _0a}}{{2\hbar }}\), \(p = \sqrt {p_x^2 + p_y^2}\), \(E_3 = \frac{1}{2}\gamma _2{\it{\Gamma }}^2\), \(E_ \pm = {\it{\Delta }} \pm \gamma _1{\it{\Gamma }} + \frac{1}{2}\gamma _5{\it{\Gamma }}^2\), \({\mathrm{\Gamma }} = 2\cos ck_z\), a = 0.246 nm, c = 0.335 nm and ħ = h/2π.

In a magnetic field oriented along the z axis, the in-plane momentum p has to be extended using Kohn–Luttinger substitution, \({\bf{p}} \to - i\hbar {\bf {\nabla}} + e{\bf{A}}\), and this leads to the Landau quantization of the lateral motion of electrons and formation of Landau bands26,34,35. For γ3 → 0 there is an analytic expression2,8,29,36,37 for both valley-degenerate (the symbol ± below is not affected by the choice of the valley) high-energy (‘split’) bands

$$\begin{array}{l}{\it{\epsilon }}_ \pm ^{{\mathrm{split}}}\left( {k_z,m} \right) \approx E_ \pm \left( {k_z} \right) + eB\hbar v^2\left( {2m - 1} \right)\omega _ \pm ,\\(m = 2,3, \ldots ),\omega _ \pm = \frac{{\left( {1 \mp \frac{{\gamma _4{\it{\Gamma }}}}{{\gamma _0}}} \right)^2}}{{E_ \pm \left( {k_z} \right) - E_3\left( {k_z} \right)}}\end{array}$$

and low-energy bands (m = 2, 3, …)

$$\begin{array}{*{20}{l}}{\it{\epsilon }}_ \pm ^{{\mathrm{low}}}\left( {k_z,m} \right) & \approx & E_3\left( {k_z} \right) - eB\hbar v^2 \\ && \left[ {\left( {m - \frac{1}{2}} \right)\left( {\omega _ + + \omega _ - } \right) \mp \sqrt {\left( {m - \frac{1}{2}} \right)^2\left( {\omega _ + - \omega _ - } \right)^2 + \omega _ + \omega _ - } } \right]\end{array}$$

Additionally, the two lowest energy bands (m = 0, 1) are described by

$${\it{\epsilon }}_0\left( {k_z} \right) = 2\gamma _2\cos ^2ck_z$$

and

$${\it{\epsilon }}_1\left( {k_z} \right) = 2\gamma _2\cos ^2ck_z - eB\hbar v^2\left( {\omega _ + + \omega _ - } \right)$$

for which the wavefunctions are localized at even (odd) layers in +KH (−KH) valleys. In magnetic fields B > 7 T, the large distance to the higher Landau bands drives graphite into the UQR, leaving only \({\it{\epsilon }}_{0,1}\left( {k_z} \right)\) bands to cross the Fermi level at \(k_{\rm F} \approx {\rm{\pi}} /4c\). This yields the 1D Fermi velocity along the z axis, \(\hbar v_{{\rm F}z} \approx - 2\gamma _2c\) for the electron motion across the film, and the splitting energy between 0 and 1 Landau bands

$${\rm \delta} _{10} = \varepsilon _1 - \varepsilon _0 \approx eB\hbar v^2\left[ {\frac{{\gamma _5 + {\it{\Delta}} }}{{\gamma _1^2}} + \frac{{4\gamma _4}}{{\gamma _0\gamma _1}}} \right] \approx 6.15\mu _{\rm B}B$$

where \(\mu _{\rm B} = \frac{{e\hbar }}{{2m_e}}\) is the Bohr magneton. For the parameters listed above, we estimate δ10 ≈ 0.37 meV T−1, which is about 3 times larger than the single-particle Zeeman energy for electrons in graphite. The Fermi velocity vFz and the splitting δ10 are the parameters that we have matched to the measured gaps and level crossings to fix the above-quoted values of the tight-binding model parameters γ2, γ5 and \({\mathit{\Delta}}\). In Supplementary Fig. 6, we compare the values of δ10 to the Zeeman splitting and the standing-wave level spacing in graphite of different thicknesses and layer parities.

Note that the graphite spectrum is not particle–hole symmetric because of the non-zero value of the hopping parameter γ2. This is seen in Fig. 2a and Supplementary Fig. 3a, where the experimental maps do not exhibit electron–hole symmetry. For instance, the localized surface states of electron–hole asymmetric Landau bands 2+ and 2 (see Fig. 2c), which are responsible for the surface accumulation layers and screening of the gate potential, are reached by the Fermi level and get filled in different B for the same level of n- and p-doping.

Standing waves in the UQR and the layer-parity effect

For our thin graphite devices, it has been found that measurable QHE gaps start to appear in B as low as 0.5 T (Supplementary Fig. 4). However, for crystals with hundreds of atomic layers, the gaps become sufficiently large, clearly visible and regularly structured only for B > 7 T (in the UQR), where only the 0 and 1 Landau bands, \({\it{\epsilon }}_{0,1}\), are left to cross the Fermi level. The spectrum of such films can be described using kz standing waves within the quasi-1D bands \({\it{\epsilon }}_{0,1}\) and the allowed energies are determined by size quantization of kz. Electrons in the two Landau bands 0 and 1 form standing waves with the maxima on even graphene layers for the valley +KH and on odd layers for the valley −KH (swapping valleys for B → −B). For a crystal with an even number of layers, \(N = 2{\cal{N}}\), there is the same number of even and odd layers to support states in +KH and −KH valleys, which determines the valley-degenerate standing waves with the amplitude \(\sin (\frac{{{\rm \uppi} nj}}{{{\cal N} + 1}})\) on the jth graphene layer among even or odd sequences. This corresponds to wavenumbers \(k_n = \frac{{{\rm \uppi} n}}{{2c\left( {{\cal N} + 1} \right)}}\), where \(n = 1 \ldots {\cal{N}}\), and Landau band energies ε0(kn) and ε1(kn). In graphite with an odd number of layers, \(N = 2{\cal{N}}-1\), there are \(\cal {N}\) odd and \({\cal{N}} - 1\) even layers and the standing waves are such that for the valley –KH (on odd layers) \(k_n = \frac{{{\rm \uppi} n}}{{2c\left( {{\cal N} + 1} \right)}}\) whereas for the valley +KH (on even layers) \(k_n^\prime = \frac{{{\rm \uppi} n}}{{2c{\cal N}}}\). The difference between standing-wave spectra with even and odd N is illustrated in Fig. 3a and referred to as the layer-parity effect.

Near the Fermi level (\(n_{\rm F} \approx \cal{N}/2\)) quantized states in each valley form a staircase of levels separated by \(\frac{{{\rm \uppi} v_{\rm F}}}{{2{\cal N}c}} \approx \frac{{{\rm \uppi} \left| {\gamma _2} \right|}}{{\cal N}}\). For even N, these spectra are valley-degenerate and their spacing is given by \({\rm \updelta} E_{\rm{even}} \approx \frac{{{\rm \uppi} \left| {\gamma _2} \right|}}{{\cal N}} \approx 94N^{ - 1}\) meV whereas for odd N, the energy states in the valley +KH are shifted by \(\frac{{{\rm \uppi} v_{\rm F}}}{{4{\cal N}c}}\) with respect to those in the valley –‍KH. This lifts the valley degeneracy and, also, reduces the level spacing to \(\updelta E_{\rm{odd}} \approx \frac{{{\rm \uppi} \left| {\gamma _2} \right|}}{{2{\cal N}}} \approx 47N^{ - 1} \,{\rm{meV}}\). These energy spacings should be compared to the Landau band splitting given by δ10 ≈ 0.37 meV T−1, and the Zeeman splitting, 2µBB, because altogether they determine the gaps in the quantized spectrum and the hierarchy of 2.5D QHE plateaux.

In the case of even N, valley degeneracy may be broken by an electric field applied in the z direction (Supplementary Fig. 7). However, this effect should disappear in symmetrically gated devices (nt = nb) due to the recovery of inversion symmetry. For odd-N devices, the symmetric gating would still lead to corrections in the valley splitting, because states in Landau bands 0 and 1 are located on the first and last layer in one valley (and, therefore, are more strongly influenced by gating) whereas states in the other valley are located on the second and penultimate layers and, therefore, are more weakly affected by the same gating. Such corrections can explain a noticeable difference between the measured and simulated fan diagrams for the 35-layer graphite device (Fig. 3b,c) as well as the deviations of the measured gap energies for odd-layer graphite in Fig. 3d from the theoretical curve. In the latter case, for the two blue points that fall onto the red line, the maximal gaps were found at large filling factors. As, for odd N, gate-induced energy shifts of nearby states in different valleys are different, the simple free-particle model is expected to work well only for the first few filling factors where the gate-induced electric field is small. Therefore, notable deviations from the model predictions for these two devices are not surprising.

When we discuss the 2.5D QHE we use the analogy between the Landau level states in bilayer graphene5 and the states in the lowest two Landau bands in the UQR graphite. That is, we attribute an e2/h contribution towards σxy from each filled standing wave (each quantized kz) of 0 and 1 Landau bands in a given spin and valley state, counting n-type- and p-type-doped filling from the neutrality condition, ntot = 0. As a result, σxy at integer filling factors is exactly proportional to the charge density induced electrostatically in graphite: σxy = νe2/h = ntote/B, in agreement with the case of the conventional QHE.

Surface states

To understand the additional features in the observed Landau fan diagrams (Fig. 2a and Supplementary Fig. 3a), related (as indicated by Fig. 2b and Supplementary Fig. 3b) to the formation of surface states, we appended the SWMC model by the self-consistent analysis of gate-voltage-induced surface charge accumulation, implemented in the Hartree approximation (see ref. 38). In Supplementary Fig. 3c,d we show examples of numerically stable self-consistent solutions for potential and charge distributions on few surface layers for several values of gate-induced charge carrier densities nt. For large nt, one can see in Supplementary Fig. 3c the regions of enhanced compressibility (higher density of states) that correspond to filling localized surface states that split-off from the higher (m = 2± and 3±) Landau bands of bulk graphite with electrons trapped in a surface accumulation layer by the self-consistent potential U(z) (Supplementary Fig. 3d). In particular, the 2+ Landau band forms a surface state for electrons that spawns from the neutrality point at B ≈ 7.5 T in experimental data (Supplementary Fig. 3a) and at B ≈ 6 T in theoretical modelling (Supplementary Fig. 3c). Similarly, the localized states chipping off the higher-energy Landau band 3+ on the n-doping side, and also the Landau band 2 on the p-doping side, can be identified at smaller B in Supplementary Fig. 3.

Data availability

All relevant data are available from the corresponding authors on reasonable request.

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Acknowledgements

This work was supported by the EU Graphene Flagship Program, the European Research Council, the Royal Society and the Engineering and Physical Sciences Research Council. J.Y. and A.M. acknowledges the support of EPSRC Early Career Fellowship EP/N007131/1.

Author information

Affiliations

  1. National Graphene Institute, University of Manchester, Manchester, UK

    • Jun Yin
    • , Sergey Slizovskiy
    • , Yang Cao
    • , Sheng Hu
    • , Yaping Yang
    • , Seok-Kyun Son
    • , Servet Ozdemir
    • , Kostya S. Novoselov
    • , A. K. Geim
    • , Vladimir Fal’ko
    •  & Artem Mishchenko
  2. School of Physics and Astronomy, University of Manchester, Manchester, UK

    • Jun Yin
    • , Sergey Slizovskiy
    • , Yaping Yang
    • , Seok-Kyun Son
    • , Servet Ozdemir
    • , Kostya S. Novoselov
    • , Francisco Guinea
    • , A. K. Geim
    • , Vladimir Fal’ko
    •  & Artem Mishchenko
  3. College of Chemistry and Chemical Engineering, Collaborative Innovation Center of Chemistry for Energy Materials, Xiamen University, Xiamen, China

    • Yang Cao
    •  & Sheng Hu
  4. State Key Laboratory of Physical Chemistry of Solid Surface, Xiamen University, Xiamen, China

    • Yang Cao
  5. Laboratoire National des Champs Magnétiques Intenses, LNCMI-CNRS-UGA-UPS-INSA-EMFL, Grenoble, France

    • Inna Lobanova
    •  & Benjamin A. Piot
  6. National Institute for Materials Science, Tsukuba, Japan

    • Takashi Taniguchi
    •  & Kenji Watanabe
  7. IMDEA Nanoscience, Madrid, Spain

    • Francisco Guinea
  8. Henry Royce Institute for Advanced Materials, Manchester, UK

    • Vladimir Fal’ko

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Contributions

A.M., A.K.G. and J.Y. conceived the experiments. J.Y., I.L., S.O. and B.P. conducted the transport measurements. J.Y., Y.C., S.H., Y.Y. and S.-K.S. prepared the samples. A.M. and J.Y. performed data analysis. S.S. and V.F. developed theory and S.S., V.F. and F.G. interpreted the data and performed the tight-binding simulations. T.T. and K.W. provided hBN crystals. A.M. wrote the manuscript with input from V.F., A.K.G., J.Y., S.S., K.S.N and F.G.

Competing interests

The authors declare no competing interests.

Corresponding authors

Correspondence to A. K. Geim or Vladimir Fal’ko or Artem Mishchenko.

Supplementary information

  1. Supplementary Information

    Supplementary Figures 1–9.

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https://doi.org/10.1038/s41567-019-0427-6