Introduction

Owing to a weak van der Waals intelayer coupling, graphite exists in several polytypes, of which Bernal (AB) and rhombohedral (ABC) are the best studied in terms of their electronic properties3,6,8,9. One half of carbons in Bernal graphite are arranged into chains aligned with the vertical direction ("double-dimers”), whereas the other half appears to be in between empty centers of hexagons in graphene layers above and underneath. This determines the electronic properties of bulk graphite to be semimetallic with a Fermi surface that contains electron and hole pockets, as confirmed by Shubnikov-de Haas (SdH) oscillations3,10,11,11,12,13 and cyclotron resonance (CR) measurements1,3,14,15,16,17,18,19. In contrast, all carbons in rhombohedral graphite lattice are arranged into pair which appear on the top of each other in the consecutive layers (so-called “dimers”), which makes thin films of rhombohedral graphite insulating in their bulk, due to interlayer hybridization of carbon’s Pz orbitals. This films also feature surface states, which are topologically protected in the spirit of Su-Schrieffer-Heeger model20 and have already been observed experimentally9,21,22 demonstrating 2D semimetallic properties with a clearly manifested electron-hole asymmetry23.

Statistically, each of those bulk polytypes of graphite may contain sporadic stacking faults, modifying electronic properties of bulk material or a thin film. Here we present a model for electronic properties of an isolated stacking fault in Bernal graphite. Structurally, such a fault can be viewed as an ABC trilayer surrounded by Bernal bulk crystal, see Fig. 1a. While in Fig. 1a the fault looks like an offset of carbon chains in the upper and lower parts of semi-infinite Bernal crystals, however, as noted in ref. 24, such structural fault can not be generated by a mutual shift of two Bernal half-spaces. Moreover, due to the ABC-structure of the fault, it hosts localized two-dimensional (2D) states, inherited from the surface states of the rhombohedral trilayer, noted in several recent publications24,25,26,27,28,29. In this work we study the electronic properties of such a stacking fault in order to describe in detail the dispersion characteristics of the 2D bands that it hosts, in particular, to predict experimentally measurable characteristics such as the period of Shubnikov-de Haas oscillations and the cyclotron mass of 2D carriers at the Fermi level. In contrast to the earlier theories of fault-bound 2D bands, here, we account for the full set of Sloczweski-Weiss-McClure (SWMcC) parameters for graphite1,2,30 implemented in the hybrid k ⋅ p - tight binding theory, supplemented by the self-consistent Hartree computation of on-layer energies. Taking all SWMcC parameters into account (in particular, the next-neighbor/layer hoppings and coordination-dependent on-carbon potentials) is crucially important, as those lift an artificial degeneracy of band edges (similarly to what has been found in monolithic ABC films7,23) which haunt the minimal model with only the closest neighbor hopping24,25. The full set of SWMcC parameters in the model is also needed for correctly describing trigonal warping of the electronic band structure of bulk, surface and 2D fault states. As a result, for the fault-localized 2D bands, we identify a set of three distinct Fermi pockets characterized by SdH frequency νSdH ≈ 1 T (distinctively different from the values \({\nu }_{SdH}^{e}\approx 6.3\,{\rm{T}}\) and \({\nu }_{SdH}^{h}\approx 4.6\,{\rm{T}}\) reported for bulk Bernal graphite3,10,11,11,12,13) and cyclotron mass of mCR ≈ 0.04m0 (to be compared to mCR ≈ 0.06m0 related to CR of bulk electrons1,3,14,15,16,17,18,19, where m0 is the free electron mass). We also identify the differences between inter-Landau-level selection rules for cyclotron transitions in the 2D bands and in the bulk, and analyze the dependence of the fault characteristics on the values of fault-related tight-binding parameters.

Fig. 1: Stacking order of the embedded fault and corresponding band structure and Landau level spectrum.
figure 1

a Schematic of the ABC stacking fault embedded in Bernal graphite, with labeled hopping parameters. White-colored carbon orbitals correspond to high-energy dimer sites. Blue denotes non-dimer Bernal sites, and red indicates states localized on a stacking fault. b Low-energy bands of a 30+1+30 layer system with red and blue color indicating states localized on 3 ABC stacked layers of a stacking fault and the bulk graphite states respectively. Yellow contour is the Fermi level (E = 0) for an undoped system. The fault-bound 2D band is identified by the red colour and it features the Dirac points (DP) in three conic branches which merge together at energies below the Lifshitz transition (LT). c Corresponding Landau levels (black thin lines) superimposed on a density of states projected onto 3 ABC stacked layers of a stacking fault (in red). Horizontal arrows indicate the states in the band structure which develop into their corresponding LLs in finite B field.

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Results

Tight-binding model for a fault in Bernal graphite

The structure of an ABC stacking fault is sketched in Fig. 1a, where we also label all the involved SWMcC tight-binding model parameters. In this sketch we use the following color-coding: The white-colored sites correspond to carbon dimers hybridized by the strongest interlayer coupling γ1. Non-dimer sites of Bernal graphite are identified using blue color. The sites marked by red are those which host the localized states at the fault with energies close to the Fermi level of graphite.

The primary choice of the values of SWMcC parameters is based on the values determined for bulk graphite in ref. 19: γ0 = 3.16 eV, γ1 = 375 meV, γ2 = − 20 meV, γ3 = 315 meV, \({\tilde{\gamma }}_{3}=315\,{\rm{meV}}\), γ4 = 44 meV, \({\tilde{\gamma }}_{4}=315\,{\rm{meV}}\), γ5 = 38 meV6,19,31,32 (with graphene lattice constant a = 0.246nm). This choice was made because this parameter set describes well CR and SdH experiments (for details see Fig. S9). While in this primary set we do not distinguish between bulk hopping parameters and those within the fault layers, however, we later account for possible differences between “non-dimer" to “non-dimer" hopping γ3vs “non-dimer" to “half-dimer" hopping \({\tilde{\gamma }}_{3}\). These parameters are implemented in the k ⋅ p-theory – tight-binding Hamiltonian Eq.(S1) described in Supplementary Information. We also use the inset in Fig.1a to show the top view of the set of red-painted cites of the ABC trilayer at the fault to illustrate trigonal symmetry of the structure which implies that \({\tilde{\gamma }}_{3}\) parameters are the same for all “non-dimer" to “half-dimer" hoppings. In addition, we allow for a variation of fault-related hopping \({\tilde{\gamma }}_{2,4}\) from the bulk values (γ2,4), due to the difference of their nearest-neighbour environment. In addition we account for the onsite energy shift, \(\Delta^{\prime} =25\,{\rm{meV}}\) produced for each carbon Pz orbital by another carbon above or below: this will produce an energy shift \(2\Delta^{\prime}\) for all dimer sites (white circles) and \(\Delta^{\prime}\) for all sites in the middle plane of the fault (light red).

As localization of electron states at the fault is associated with charge redistribution between nearby layers and the bulk, it is necessary to account for the resulting variation of on-layer energies self-consistently. Here we implement it using Hartree approximation. A representative example of self-consistently calculated density and on-layer potential distribution is shown in Fig. S2 in the supplementary material for the primary choice of tight-binding model parameters.

To realize numerical analysis of electronic states of a single fault we numerically compute the spectra of graphitic film with 61 layers where the fault resides exactly on the middle layer of the film. These numerical calculations produce a combination of bulk bands, surface states6,33 and fault-localized bands. We also repeat the calculations for the same system in magnetic field and analyze its Landau level (LL) spectrum. In particular, projecting the LL states onto three layers that constitute the fault structure. For graphic representation of the data, we applied broadening Γ = 0.2 meV of all LLs which enables us to plot the density of states (DoS) in this system, in particular, local DoS on the fault (projected onto the 3 middle layers of the film). In parallel, we compute the 3D band structure (both at B = 0 and finite magnetic field) of a 3D “crystal" constructed of a periodic sequence of faults and anti-faults separated from each other by 23 layers of Bernal graphite. In this case, for each in-plane momentum (kx, ky), the extended bulk states produce broad “1D" bands of kz dispersion (with negligible gaps), whereas the states localized on the fault can be identified as exponentially narrow (non-dispersive in kz) bands. The latter manifest themselves as peaks in the overall density of states, so that they give E(kx, ky) dispersion of fault bound states even without the need to project their wave functions onto the fault layers. The agreeable comparison of the results of these two methods (with the same broadening of all states applied in both cases) gives us a confidence in the identified 2D dispersion and LL spectrum of the fault-related states.

Fault-related electronic dispersion and Landau levels

In Fig. 1b we display the computed dispersions, E(kx, ky), of all 2D sub-bands (representative of size-quantization of electron states in the film along the K-H line in the 3D Brillouin zone of graphite) in a film with 61 layers and the stacking fault exactly in its middle. In this plot, we use color-coding to distinguish between bulk (blue) and fault-located (red) sub-bands, which were identified by projecting their wave functions onto the three middle layers of the film. The fault-related states feature three cone-like dispersions (anisotropic due to trigonal warping and with a pronounced electron-hole asymmetry), each ending with a Dirac-like singularity (DP) at the energy E − EF ≈ 6 meV above the self-consistently calculated Fermi level (the latter is dominated by the bulk states). Such a feature is reminiscent of bands characteristic for an ABC trilayer7 (illustrated in supplementary Fig. S8). These Dirac-like features are characterized by Berry phases π, which we confirmed by calculating \(\phi =-\oint {\rm{Im}} \left\langle \Psi \right\vert {\nabla }_{{\boldsymbol{p}}}\left\vert \Psi \right\rangle \cdot d{\boldsymbol{p}}\) along each of the three Fermi-lines using the wave-function computed for those 2D bands. The apparent electron-hole asymmetry of the 2D spectrum is the result of the upper branches of the ABC trilayer spectrum blending into the bulk states at the energies above 6 meV. The result in Fig. 1b demonstrates that the hole-type 2D bands are well isolated from the bulk bands across the energy range ~ ± 5 meV near the Fermi level, with the fault based bands traceable down to much lower energies where the individual Dirac cones merge into one single dispersion branch, upon experiencing a Lifshits transition (LT) at E − EF ≈ − 14 meV. This is inherited from the ABC-trilayer spectrum too, together with Berry phase 3π that we computed for the constant-energy cut through the spectrum below the LT. Such a behavior was reproduced by the band analysis in a 3D “crystal" constructed of a periodic sequence of faults and anti-faults separated from each other by 23 layers of Bernal graphite. To mention, in the opposite corner of graphite’s Brillouin zone (line K’-H’) this anisotropy of the dispersion will be inverted as prescribed by time inversion symmetry.

This band structure is mirrored by the magnetic field dependence of the local DoS on the stacking fault shown in Fig. 1c with red color, where we observe a characteristic \(-\sqrt{nB}\) fan of triply degenerate LLs originating from E − EF ≈ 6 meV corresponding to the three Dirac points. In addition to the layer projection, we applied broadening Γ = 0.2 meV to all computed LLs. The zoomed-in panel in Fig. 2 shows that, at small magnetic fields, we can trace the convergence of these graphene-like 2D LLs towards Dirac points identified in the B = 0 spectrum in Fig. 1. We also note that the triple degeneracy of graphene-like LLs is lifted around the Lifshits transition energy E − EF ~ − 12 meV (due to magnetic breakdown between different cones), so that at lower energies one can trace the fan plot of non-degenerate LLs related to a single connected dispersion surface, still well separated from the bulk states (visible in supplementary Fig. S3). At positive energies above the Dirac point we see little to no fault-localized LL formation, which happens because the stacking fault band continuously transforms into the bulk states. All these features are present in both the film calculation and in the analysis of kz bands in the “3D crystal", as compared in Fig. 2 (more detailed analysis of the spectra from the “3D crystal" calculation is presented in Fig. S7, including the kz band projections onto the fault layers).

Fig. 2: Fault bound carrier Landau level spectrum.
figure 2

(left) The total density of states for an infinite periodic array of stacking faults, each separated by 23 Bernal-stacked layers in a finite magnetic field. (Right) Density of states projected onto the three ABC-stacked layers, calculated for a finite 30+1+30 film with a single stacking fault. Lower image is a zoom at low magnetic fields.

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SdH oscillation frequency and cyclotron mass of the fault-related 2D band

Having identified a distinct fault-related band, below, we analyse whether its spectroscopic and transport characteristics would be distinguishable from the bulk states in Bernal graphite. As Shubnikov-de Haas (SdH) and de Haas-van Alphen oscillations have been well studied in graphite3,10,12,13,34, we compare their established frequencies (\(6.0 < {\nu }_{SdH}^{e} < 6.6\,{\rm{T}}\) and \(4.5 < {\nu }_{SdH}^{h} < 4.8\,{\rm{T}}\) for the electron and hole pockets of graphite’s Fermi surface respectively) with the expected SdH oscillation frequency for the 2D band. Also, cyclotron mass in graphite (mCR = 0.058 ± 0.001m0) has been identified with the electron branch of the bulk Fermi surface1,3,11,14,15,16,17,18,35, so that we will use the 2D band cyclotron mass as another characteristic to compare 2D vs bulk states. To mention, the SWMcC parameters used in our calculations where chosen to reproduce the above-mentioned bulk graphite characteristics (for details see supplementary section S4).

To analyse SdH oscillations, we make cuts of the local DoS (computed with broadened LLs) at several energies in the range ± 4 meV near the Fermi level and plot those against 1/B, see Fig. 3. Then, we perform a Fourier transform in the range of 1/B corresponding to 0.2 < B < 5T for each dataset, which gives us the frequency of SdH oscillations, νSdH(E − EF), plotted in the main panel of Fig.3 with the red line. The obtained frequencies agree with the areas, \({\mathcal{A}}(E)\) encircled by the individual Fermi contour in each of the three 2D dispersion pockets in Fig. 1b and recalculated into \({\nu }_{SdH}=\frac{{\mathcal{A}}}{he}\) (solid black line). As illustrated in the insets in Fig. 3, one side of the constant-energy contour of the fault approaches bulk bands at lower energies, so that at E < EF, it eventually touches the bulk bands, so that the encircled area can only be estimated by extrapolation (dashed black line). Despite that, the frequency of SdH oscillations is clearly identifiable based on the finite-B DoS data and appears to be close to the extrapolated values. Moreover, based on the Berry phase π calculated for Dirac-like dispersion branches in Section B, we expect a π-shift of the SdH oscillations phase specific to the fault-bound bands, as compared to bulk graphite11.

Fig. 3: Frequency and amplitude of SdH oscillations as a function of energy (E = 0 corresponds to the expected Fermi level for undoped system), results were obtained by a Fourier transform in the interval 0.2 < B < 5 T for a 30+1+30 system.
figure 3

The black line corresponds to the area of the red Fermi contour when it is closed. Insets show the Fermi contours at energies − 2, 0, 2 meV respectively. We observe that a SdH signal survives even in the regime when the states localized on the stacking fault do not form a closed Fermi contour: scattering to the bulk hole bands and back helps to close the path.

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For the effective mass, mCR, of 2D holes in the fault band, we use its relation36,

$${m}_{CR}=2\pi {\hslash }^{2}\,\text{DoS}=\frac{{\hslash }^{2}}{2\pi }\int_{\text{pocket}\,}\delta (E({\boldsymbol{p}})-{E}_{F}){d}^{2}{\boldsymbol{p}},$$
(1)

to the density of states (DoS) determined for a single 2D dispersion pocket at the Fermi level. The calculated mass agrees well with the computed inter-LL separation, ℏωc = ℏeB/mCR, (for pairs of LLs closest to the Fermi level) across the magnetic field interval 0.1 T < B < 1 T. For the spectrum shown in Fig. 2 this yields mCR = 0.042m0, which is about 2/3 of the bulk cyclotron mass in graphite.

As mentioned in Tight-binding model for a fault in Bernal graphite, the interplane couplings γ3 and γ2 could be different for the layers adjacent to the fault as compared to SWMcC parameters in the bulk. For this reason we explore the sensitivity of the values νSdH and mCR to the choice of \({\tilde{\gamma }}_{3}\) and \({\tilde{\gamma }}_{2}\) parameters. In Fig. 4 we illustrate those dependences, which indicate that within a reasonably wide range of \({\tilde{\gamma }}_{2,3}\) the sizes of the analysed fault observables remain distinctively different from the values of the corresponding characteristics of bulk graphite.

Fig. 4: Dependence of SdH oscillations frequency and cyclotron mass of stacking fault states on the poorly-known parameters \({\tilde{\gamma }}_{3}\) (top panel) and \({\tilde{\gamma }}_{2}\) (bottom panel).
figure 4

Blue lines are for νSdH and red lines are for mCR. For comparison, the bulk graphite values are νSdH ≈ 4.5 and 6.1T10,11 for holes and electrons, respectively, and mCR ≈ 0.058m019 (for electrons that dominate in CR). To mention, self-consistent on-layer potentials were calculated separately for each set of tight-binding parameters.

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We also note that inter-Landau-level selection rules for cyclotron transitions in the 2D bands differ from those of bulk graphite (where trigonal warping of dispersion permits super-harmonics ωCR/ωc = ± 1 + 3N, N = . . . , − 2, − 1, 0, 1, 2, . . . ). To analyse the optical oscillator strength, βn, of various cyclotron harmonics, we employ the kinetic equation approach developed in Ref. 17, which relates coupling of 2D carriers with light characterised with a polarization vector l as,

$$\begin{array}{r}{\beta }_{n}\propto {\left|\mathop{\int}\nolimits_{0}^{2\pi }{\boldsymbol{l}}\cdot {{\boldsymbol{v}}}_{{\boldsymbol{p(s)}}}{e}^{-ins}\frac{ds}{2\pi }\right|}^{2};\end{array}$$
(2)
$$\begin{array}{r}ds=\frac{d{p}_{\parallel }}{{m}_{CR}| {{\boldsymbol{v}}}_{{\boldsymbol{p(s)}}}| }\,\,\,{{\boldsymbol{v}}}_{{\boldsymbol{p}}}\equiv {{\boldsymbol{\nabla }}}_{p}E.\end{array}$$
(3)

Here s/ωc is the time of flight along the cyclotron orbit in real space37 (the latter has the shape of the Fermi contour rescaled by 1/(eB) and rotated by 90°), which we use to parameterize the time-dependence of the instant momentum of the charge carrier at the Fermi level. The resulting strengths of couplings with circularly-polarized light at frequencies ωCR = nωc are shown in Fig. 5 in comparison with the relative strengths of various cyclotron harmonics detected in bulk graphite18,19. This comparison shows that, independently of the fault SWMcC parameters, the CR of the holes in the 2D band of the fault is dominated by the principal cyclotron lines ωCR = ± ωc with weak satellites only at ± 2ωc.

Fig. 5: Relative strength of CR harmonics for a stacking fault band in circularly-polarized light.
figure 5

Negative harmonics correspond to the opposite directions of cyclotron motion vs circular polarization of light. For comparison, a plot for graphite, extracted from experimental data in19 is also presented.

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Effects of Finite Film Thickness

Up to this point, we have focused on ABC stacking faults embedded deep within thick multilayers. Here, we briefly explore the influence of finite thickness on these systems. Figure 6 displays the electronic dispersions for an 11-layer Bernal-stacked system with an ABC stacking fault positioned at different layers. For each fault location, we self-consistently calculate the on-layer potentials under the constraint of overall charge neutrality. In a symmetric 5+1+5 configuration, inversion symmetry preserves three gapless Dirac points associated with bands localized at the stacking fault. However, as the fault shifts toward the surface, a small gap begins to open at these Dirac points, reaching approximately 10 meV when the fault lies at the surface. While the areas of the hole pockets shrink slightly, our primary conclusions remain robust. Notably, when the fault approaches the surface, its electronic properties become more readily tunable via electrostatic gating.

Fig. 6: Effect of finite thickness and asymmetry on electronic dispersions.
figure 6

Shown are the band structures for undoped 11-layer systems with an ABC stacking fault positioned at varying depths, calculated with self-consistent on-layer potentials. The energy scale spans 40 meV from top to bottom. A gap emerges at the Dirac points due to the loss of inversion symmetry.

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Discussion

Overall, we used the tight-binding model of graphite, based on the full set of the Slonczewski-Weiss-McClure parameters, to show that a stacking fault in Bernal graphite hosts a 2D band separated from the continuum spectrum of bulk graphite, in both valleys. In particular, we found that this 2D band determines three distinct Fermi lines for the holes, which lie outside the Fermi surface of Bernal graphite. Our analysis predicts the frequency of Shubnikov-de Haas oscillations, νSdH ≈ 1 T, which substantially differs from the bulk values for graphite, \({\nu }_{SdH}^{e}\approx 6.3\,{\rm{T}}\) and \({\nu }_{SdH}^{h}\approx 4.6\,{\rm{T}}\), which are reproduced by the same tight-binding model. We also predict the cyclotron mass of fault-bound 2D electrons to be about 2/3 of the cyclotron mass for bulk carriers. In addition, the spectral analysis extended over a broad range of magnetic fields enables us to demonstrate the Dirac-like character of these three separate dispersion branches at energies above the Fermi level as well as the Lifshitz transition into a single connected Fermi line at energies ~ 15 meV below EF. These Dirac-like spectral features of the 2D band comply with Berry phases π computed for each of the three Fermi lines and the overall Berry phase 3π calculated for a constant energy contour below the Lifshitz transition energy, which is inherited from the Berry phase properties of ABC trilayer as a determining feature of the fault structure.

Methods

Zero magnetic field Hamiltonian

We illustrate the ABC stacking fault in Bernal graphite again in Fig. S1 (in the supplementary material): there are m and n AB-stacked layers separated by a single layer where the string of vertical ABAB alignment of atoms is terminated and shifted. In the terminology of Ref. 38, this is an (m, 1, n) stack. In the basis of sublattice amplitudes for electron states in A and B sublattices of an N = m + 1 + n film, \({\Psi }^{\dagger }={\left(\begin{array}{cccc}{\psi }_{{A}_{1}},&{\psi }_{{B}_{1}},&\cdots ,&{\psi }_{{A}_{N}},{\psi }_{{B}_{N}}\end{array}\right)}^{\dagger }\), the k ⋅ p Hamiltonian is written as

$$\begin{array}{ll}{\mathcal{H}}\,=\,\left(\begin{array}{ccccccccc}{H}_{+}^{s}&{V}^{\dagger }&W&0&0&0&0&0&0\\ V&{H}_{-}&V&W&0&0&0&0&0\\ {W}^{\dagger }&{V}^{\dagger }&{H}_{+}&{V}^{\dagger }&W&0&0&0&0\\ 0&{W}^{\dagger }&V&{H}_{-}&{V}_{2}&\tilde{W}&0&0&0\\ 0&0&{W}^{\dagger }&{V}_{2}^{\dagger }&{H}_{{\rm{g}}}&{V}_{1}&W&0&0\\ 0&0&0&{\tilde{W}}^{\dagger }&{V}_{1}^{\dagger }&{H}_{+}&{V}^{\dagger }&W&0\\ 0&0&0&0&{W}^{\dagger }&V&{H}_{-}&V&W\\ 0&0&0&0&0&{W}^{\dagger }&{V}^{\dagger }&{H}_{+}&{V}^{\dagger }\\ 0&0&0&0&0&0&{W}^{\dagger }&V&{H}_{-}^{s}\end{array}\right),\\ {H}_{+}\,=\,{H}_{{\rm{g}}}+\left(\begin{array}{ll}2\Delta {\prime} &0\\ 0&0\end{array}\right),\,{H}_{-}={H}_{{\rm{g}}}+\left(\begin{array}{cc}0&0\\ 0&2\Delta {\prime} \end{array}\right)\\ {H}_{+}^{s}\,=\,{H}_{{\rm{g}}}+\left(\begin{array}{ll}\Delta {\prime} &0\\ 0&0\end{array}\right),\,{H}_{-}^{s}={H}_{{\rm{g}}}+\left(\begin{array}{ll}0&0\\ 0&\Delta {\prime} \end{array}\right),\\ {H}_{g}\,=\,{v}_{0}\left(\begin{array}{ll}0&{\pi }_{\xi }\\ {\pi }_{\xi }^{* }&0\end{array}\right),\quad V=\left(\begin{array}{ll}-{v}_{4}{\pi }_{\xi }^{* }&{\gamma }_{1}\\ -{v}_{3}{\pi }_{\xi }&-{v}_{4}{\pi }_{\xi }^{* }\end{array}\right),\\ {V}_{1}\,=\,\left(\begin{array}{ll}-{\tilde{v}}_{3}{\pi }_{\xi }^{* }&{\gamma }_{1}\\ -{\tilde{v}}_{3}{\pi }_{\xi }&-{\tilde{v}}_{4}{\pi }_{\xi }^{* }\end{array}\right),{V}_{2}=\left(\begin{array}{ll}-{\tilde{v}}_{4}{\pi }_{\xi }^{* }&{\gamma }_{1}\\ -{\tilde{v}}_{3}{\pi }_{\xi }&-{\tilde{v}}_{3}{\pi }_{\xi }^{* }\end{array}\right),\\ \tilde{W}\,=\,\left(\begin{array}{cc}0&0\\ {\gamma }_{2}/2&0\end{array}\right),\quad W=\left(\begin{array}{ll}{\gamma }_{5}/2&0\\ 0&{\gamma }_{2}/2\end{array}\right),\\ {v}_{i}\,=\,\frac{\sqrt{3}a}{2\hslash }{\gamma }_{i}.\end{array}$$
(4)

where we show an explicit example for N = 3 + 1 + 3 system shown in Fig. S1, while the generalization to thicker Bernal stacks is done by straightforward repetition of V†, V and H−, H+ patterns along the diagonals. Here, πξ ≡ ξpx + ipy, with p = (px, py) being the valley momentum measured from \(\hslash {{\boldsymbol{K}}}_{\xi }=\hslash \xi \frac{4\pi }{3a}(1,0)\). Matrices V and W describe the nearest and next-nearest layer couplings, and they are assumed to be independent of the distance to the surface layers. Below we use the following values of parameters implemented in Eq. (4), with graphene lattice constant a = 0.246 nm and hopping energies γ0 = 3.16 eV, γ3 = 315 meV, γ4 = 44 meV, \({\tilde{\gamma }}_{4}=44\,{\rm{meV}}\), \({\tilde{\gamma }}_{3}=315\,{\rm{meV}}\), γ1 = 375 meV, \(\Delta {\prime} =25\,{\rm{meV}}\), γ2 = − 20 meV, γ5 = 38 meV6,19,31,32; the hopping parameters on the stacking fault are expected to be similar to ABC stacked graphene multilayers and have not been precisely fixed in the literature; we choose the same values as for Bernal graphite for our main figures and study the dependence of CR and SdH oscillations on \({\tilde{\gamma }}_{2,3}\) in Fig. 4. The parameter \(\Delta {\prime}\) accounts for energy difference of dimer and non-dimer sites (pale blue/red vs filled circles in Fig. S1 and we assume that this energy difference doubles if there are Carbon atoms both directly above and below the given one (white circles in Fig. S1), as is the case in Bernal stacking. In addition to SWMcC Hamiltonian, we account for the mean-field electrostatic interactions between the layers by including self-consistently calculated on-layer potentials Ui, i = 1, . . n + m + 1 at the diagonal, accounting for the vertical electric field between the layers similarly to39. The results for self-consistent potential and charge density induced on graphene layers is shown in Fig. S2. There, the potential saturates to ≈ 4 meV away from the defect and surfaces, corresponding to graphite Fermi level − 4 meV (or, − 4 + γ2 = − 24 meV in the notations typical for graphite literature). The stacking fault is polarized with hole-doping on the middle layer and electrons on the adjacent layers.

Finite magnetic field Hamiltonian

The effect of an out-of-plane magnetic field B is included by implementing the Luttinger substitution40. We express all momentum terms in Eq. (4) by using the raising (\({\hat{a}}^{\dagger }\)) and lowering (\(\hat{a}\)) operators of the quantum harmonic oscillator,

$${\pi }_{+}=\hslash \frac{\sqrt{2}}{{l}_{B}}\hat{a},\quad \quad {\pi }_{+}^{\dagger }=\hslash \frac{\sqrt{2}}{{l}_{B}}{\hat{a}}^{\dagger },$$
$${\pi }_{-}=-\hslash \frac{\sqrt{2}}{{l}_{B}}{\hat{a}}^{\dagger },\quad \quad {\pi }_{-}^{\dagger }=-\hslash \frac{\sqrt{2}}{{l}_{B}}\hat{a},$$

where \({l}_{B}=\sqrt{\hslash /eB}\) is the magnetic length. To compute Landau levels, we use the basis \(\{{\left\vert n\right\rangle }_{l}^{{\rm{A}}},{\left\vert n\right\rangle }_{l}^{{\rm{B}}}\}\), where l is the layer index and \(\left\vert n\right\rangle\) is the eigenstate of order n of the harmonic oscillator41, such that \({\hat{a}}^{\dagger }\left\vert n\right\rangle =\sqrt{n+1}\left\vert n+1\right\rangle\) and \(\hat{a}\left\vert n\right\rangle =\sqrt{n}\left\vert n-1\right\rangle\). The series is truncated at a level Nc sufficiently large to ensure spectral convergence. As noted in Ref. 42, the maximum value of Nc also depends on the sublattice index and the layer index, which avoids the appearance of spurious states at low energies.