Coupled Dirac Fermions and Neutrino-like Oscillations in Twisted

Letter pubs.acs.org/NanoLett

Coupled Dirac Fermions and Neutrino-like Oscillations in Twisted Bilayer Graphene Lede Xian,† Z. F. Wang,†,‡ and M. Y. Chou*,†,§ †

School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, United States Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112, United States § Institute of Atomic and Molecular Science, Academia Sinica, Taipei 10617, Taiwan ‡

S Supporting Information *

ABSTRACT: The low-energy quasiparticles in graphene can be described by a Dirac−Weyl Hamiltonian for massless fermions, hence graphene has been proposed to be an effective medium to study exotic phenomena originally predicted for relativistic particle physics, such as Klein tunneling and Zitterbewegung. In this work, we show that another important particle-physics phenomenon, the neutrino oscillation, can be studied and observed in a particular graphene system, namely, twisted bilayer graphene. It has been found that graphene layers grown epitaxially on SiC or by the chemical vapor deposition method on metal substrates display a stacking pattern with adjacent layers rotated by an angle with respect to each other. The quasiparticle states in two distinct graphene layers act as neutrinos with two flavors, and the interlayer interaction between them induces an appreciable coupling between these two “flavors” of massless fermions, leading to neutrino-like oscillations. In addition, our calculation shows that anisotropic transport properties manifest in a specific energy window, which is accessible experimentally in twisted bilayer graphene. Combining two graphene layers enables us to probe the rich physics involving multiple interacting Dirac fermions. KEYWORDS: Twisted bilayer graphene, interlayer interaction, wave packet, anisotropic transport, neutrino-like oscillation, tight-binding simulation ith the demonstration of field-effect transistors made of graphene layers in 2005,1 graphene has become the subject of great interest and widespread enthusiasm in the materials research field. Graphene consists of a single layer of carbon atoms forming a honeycomb lattice. It is a twodimensional zero-gap semiconductor with a linear band dispersion and a vanishing electron effective mass near the Fermi level.1−5 This results from the fact that there are two equivalent sublattices in graphene, which gives rise to the two degrees of freedom in the electronic structure. With this unique feature, the low-energy quasiparticles in graphene can be described by the Dirac-Weyl Hamiltonian6,7

W

H0 = vFσ ·p

expressed as a linear combination of several energy eigenstates.16 As it propagates through the space, the quantum mechanical phases of these energy eigenstates will advance at different rates. Therefore, the neutrino in one flavor can be turned into other flavors periodically. Studying neutrino oscillations is currently a major endeavor in particle physics17 with the goal to probe the fundamental physics associated with the existence of neutrino mass, charge-parity (CP) violation, and so forth.18 Given the similarities in the particle dynamics of neutrinos and the quasiparticles in graphene, it is expected that neutrino-like oscillations can also be found in graphene systems with a proper setting. In this work, we study the electronic structure and wave packet propagation in two layers of graphene that are rotated with respect to each other by an arbitrary angle θ between 0 and 60°.19−21 This type of twisted bilayer graphene (TBG) is found in samples grown either on the C face of the SiC substrate by thermal decomposition22,23 or on a metal surface by chemical vapor deposition.19,24−26 Distinct from graphite, TBG preserves many features associated with monolayer graphene, as seen in electron transport,23 infrared absorption

(1)

where vF ≈ 10 m/s is the Fermi velocity and σ is the Pauli matrix. Effectively, quasiparticles in graphene are massless Dirac particles with a “pseudospin” of 1/2 analogous to neutrinos.3,5 Such unique characteristics have profound implications from both fundamental and applied points of view. Relativistic phenomena, such as the Klein paradox8,9 and Zitterbewegung,10,11 have been investigated in the graphene systems. Intriguing physics arises if interaction exists between different “flavors” of massless Dirac particles. One particular example is the neutrino oscillation between electron, tau, and mu neutrinos12−15 resulting from the distinction between flavor and energy eigenstates. A neutrino created in one flavor can be 6

© 2013 American Chemical Society

Received: July 3, 2013 Revised: September 27, 2013 Published: September 30, 2013 5159

dx.doi.org/10.1021/nl4024403 | Nano Lett. 2013, 13, 5159−5164

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spectroscopy,27 angle-resolved photoemission spectroscopy (ARPES),28 and scanning tunneling spectroscopy (STS).29 This arises from the weak interlayer coupling in the low energy region.20,30−32 However, the coupling can give rise to appreciable effects in the overall electronic structure, as found in our study. The original quasiparticles in each layer are analogous to the flavor states for neutrinos. Interlayer coupling creates a mixture of these layer states, leading to anisotropic quasiparticle propagation in a specific energy window that varies as the twist angle. This further yields a neutrino-like oscillation. The results demonstrate that the added layer in TBG provides another degree of freedom to manipulate the quasiparticle interaction, leading to a unique system with rich physics. In TBG, the Dirac cones associated with the top and bottom layers are also rotated by the same angle θ in momentum space. The two Dirac cones centered at K and Kθ are separated by ΔK ≈ 2|K| sin(θ/2), where |K| = 4π/(3a0) and a0 is the monolayer lattice constant. A schematic illustration of the interlayer coupling is given in Figure 1. On the plane cutting through the

interaction, we have two sets of hyperbolas as shown in Figure 1e, which will be deformed in the presence of interlayer coupling. A schematic plot of the final energy bands is shown in Figure 1f based on results from first-principles calculations. Two rather flat bands are found in the valence bands, which are expected to induce intriguing transport properties. The asymmetric flattening in Figure 1f arises from the differences in the interlayer coupling matrix elements for the electrons and for the holes. A twisted bilayer system exhibits a Moiré pattern in real space. In most cases, the system is no longer periodic. Commensurate structures can be found for rotational angles satisfying31,33 cos(θ) =

n2 + 4nm + m2 2(n2 + nm + m2)

(2)

where the integer pair (m,n) is used to label the structure. The lattice vectors of the supercell are t1 = na1 + ma2, and t2 = −ma1 + (n + m)a2, where a1 and a2 are the lattice vectors of the primitive unit cell in monolayer graphene (|a1| = |a2| = a0). Electronic properties can be calculated in detail for these commensurate structures. However, the size of the unit cell increases quickly as the twist angle decreases. Therefore, we first perform calculations using density functional theory (DFT) for a manageable set of commensurate structures; then tight-binding parameters are determined using the calculated DFT band dispersions as fitting references.36 A very good fit can be obtained, and the tight-binding model is able to provide a decent description of the band dispersion for the π-bands. In the following, we present the results for TBG with a twist angle of 7.34° and a supercell of 122 atoms in each layer. The conclusions will apply to cases with other twist angles larger than 1.5° and less than 58.5° (see Supporting Information). Complex electronic structure has been reported for extremely small values of twist angles37,38 (less than 1.5° or larger than 58.5°), which were not considered in our study. The calculated band dispersions using a tight-binding model36 are shown in Figure 2a, with the energy zero set at the Fermi level of a charge neutral bilayer. In the low energy region, the band structure displays the typical monolayer graphene feature, namely, linear dispersions around the Dirac points (K and K′). As explained previously, the interlayer coupling induces a gap near M, which is also the midpoint between the rotated Dirac cones associated with the top and bottom layers, respectively. For this particular twist angle, the gap at M is of the order of 0.2 eV (marked by dashed lines in Figure 2a) within which new features in the energy dispersions along other directions are of particular interest and will be the focus of the current work. To clearly see the symmetry of band dispersions, the constant-energy contours of the top two valence bands are shown in Figure 2b in the Brillouin zone of the supercell. In the low-energy region, the linear bands are isotropic, manifested by circular contours around the corners of the Brillouin zone similar to those in monolayer graphene. However, when the energy reaches the gap at M, the two coupled bands become anisotropic, as revealed by the two sets of intersecting triangular contours centered at the Γ point. The energy bands along the dashed line in Figure 2b are shown in Figure 2c, and we can clearly see the expected flat band around −0.6 eV (highlighted in red). Since the group velocity is perpendicular to the constant-energy contour, there are six preferred transport

Figure 1. Schematic illustrations of interlayer coupling in twisted bilayer graphene. (a) A representative plane cutting through the two original Dirac points K and Kθ associated with the top and bottom layer, respectively; (b) corresponding energy bands on this cross section without interlayer coupling; (c) modified energy bands after taking into account interlayer coupling. (d) A representative cross section without cutting through the original Dirac points; (e) corresponding energy bands on this cross section without interlayer coupling; (f) modified energy bands after taking into account interlayer coupling.

two original Dirac points as shown in Figure 1a, one can see two sets of linear bands in Figure 1b if there were no interlayer interaction. With the interlayer coupling included, the two sets of bands are expected to interact, and two gaps will be opened above and below the charge neutrality point and near the kpoint midway between K and Kθ, as shown in Figure 1c. The two linear bands can still be preserved near K and Kθ for most θ values. However, the following two new features will emerge: (1) The effective Fermi velocity will be reduced due to the interaction; a significant decrease of the Fermi velocity can be found for small angles. This has been discussed in the literature based on first-principles and tight-binding calculations.33 (2) Near the gap the density of states will exhibit van Hove singularities for a 2D system. This has been verified by STS measurements.34,35 Moving away from the plane cutting through the two original Dirac points, the cross section shown in Figure 1d exhibits even more interesting features that have not been explored previously. Without the interlayer 5160


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Figure 2. Anisotropic band structures of twisted bilayer graphene. (a) Energy dispersions of twisted bilayer graphene with a twist angle of 7.34°. The energy gap at M is marked by dashed lines. (b) Constant-energy contours for the top two valence bands in the first Brillouin zone with a contour interval of 0.084 eV. One representative pair of intersecting triangular contours are highlighted by thick solid and dashed lines. (c) Band structure along the dashed line in (b) with the interaction-derived pair of bands highlighted. (d) Atomic arrangements of the twisted bilayer system in real space showing the Moiré pattern. The arrows show the directions of the group velocity associated with the triangular contours in (b).

directions for the quasiparticles in real space, as indicated by the arrows in Figure 2d. The energy window for the anisotropic transport coincides with the gap at M, making it possible to observe this unique phenomenon experimentally. This interesting transport property can be modeled by wave packet simulations, which were performed previously for monolayer graphene.39−42 We have implemented a scheme to carry out these calculations using a tight-binding Hamiltonian. The initial wave packet |ϕ0⟩ is constructed by imposing a realspace Gaussian function on an energy eigenstate |ψ1⟩ with wave vector k1 and energy E1 associated with band 1 in Figure 1f ⎡ (x − x )2 + (y − y )2 ⎤ 0 0 ⎥ |ϕ0⟩ = exp⎢ − |ψ ⟩ ⎢⎣ ⎥⎦ 1 2σ 2

(3)

where (x0, y0) is the initial center position for the wave packet. The time evolution of the wave packet ⎛ Ĥ ⎞ |ϕ(τ )⟩ = exp⎜ −i τ ⎟|ϕ0⟩ ⎝ ℏ ⎠

(4)

is calculated by solving the time-dependent Schrödinger equation with a Chebyshev scheme.43,44 By using a system size of 500 Å × 1300 Å and a Gaussian function with a spatial extent of 2σ = 34 Å in all directions, we find that the wave packet in this TBG system can maintain its integrity over a simulation length of a few hundreds of angstroms. The time-integrated probability density of the wave packet in TBG with θ = 7.34°, k1 = (−0.062 Å−1, −0.0025 Å−1), and E1 = −0.64 eV is shown in Figure 3a,b. Clearly, the electron beam follows one of the preferred propagation directions, x-direction in this case, with little spreading along the y-direction, which is consistent with the anisotropic band structure discussed above.

Figure 3. Time-integrated probability density of electron wave packets. (a,b) Wave packet transport in twisted bilayer graphene with a twist angle θ = 7.34°. The initial wave packets is constructed with a Gaussian envelope of 2σx = 34 Å imposed on the eigenstate with E1 = −0.64 eV. (c) Wave packet transport on monolayer graphene with the same energy. (d,e) Probability density profiles sampled along the lines in panels a,c, respectively. (f,g) Wave packet propagation in twisted bilayer graphene with an initial wave packet on only the top layer.

The following two fine features can be identified: (i) The intensity of the electron beams are almost the same on the top 5161


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(Figure 3a) and bottom (Figure 3b) layers, since the initial wave packet is constructed out of the eigenstate |ψ1⟩ that has equal contributions from both layers. (ii) The intensity of the electron beams is not uniformly distributed along the propagation direction; the high-density regions coincide with the AA stacking regions. This is consistent with the finding in previous calculations that electrons in TBG tend to have a higher concentration in the AA regions of the Moiré pattern.33 In comparison, the wave packet propagation in monolayer graphene will be dramatically different because of its isotropic energy bands. The time-integrated probability density of the wave packet in monolayer graphene with the same E1 = −0.64 eV is shown in Figure 3c. The density distribution spreads out with an angle depending on the size of the Gaussian envelope and the k1 vector.41,42 To illustrate the differences, in Figure 3d,e we plot the probability density along the vertical lines (186 Å from x0) shown in Figure 3a,c, respectively. The density profiles for TBG and monolayer graphene in Figure 3d,e can be fitted with a Gaussian function of 2σ = 37 and 198 Å, respectively. The spatial extension at x0 + 186 Å in TBG is comparable to its initial value of 2σ = 34 Å at x0, indicating a good collimation behavior. In contrast, the wave packet spreads out considerably in monolayer graphene. A more realistic experimental setup is to introduce the initial wave packet in only one of the layers. The wave packet can be found in another layer as it propagates because of the interlayer coupling. The time-integrated probability density of such a wave packet in the top and bottom layers of TBG is shown in Figure 3f,g, respectively, which exhibits the expected collimation behavior in both layers. More interestingly, an oscillation exists as the peak intensity in the top layer corresponds to a minimum in the bottom layer and vice versa, as shown by the dashed lines in Figure 3f,g. This interesting oscillation behavior is clearly illustrated in Figure 4a showing the probability density of the propagating wave packet at serval time steps in one period. In this process, the quasiparticle wave packet in the top layer gradually transfers to the bottom layer and back to the top layer. We plot the layer-integrated probability density versus time in Figure 4b. The probability density oscillates between the top and bottom layers, while the sum remains constant. The oscillation period is estimated to be 7.2 fs for this specific twist angle. This quasiparticle oscillation behavior in TBG is closely related to the neutrino oscillation in particle physics. Each graphene layer in TBG has its own low-energy massless quasiparticles just as neutrinos have flavors. As shown schematically in Figure 1, the interlayer coupling creates new energy eigenstates that are linear combinations of original single-layer states. Specifically, the energy eigenstates (|ψ1⟩ and |ψ2⟩) can be written as ⎛ |ψ ⟩ ⎞ ⎛ |ψ ⟩ ⎞ ⎛ ⎜ 1 ⎟ = ⎜ cos α −sin α ⎞⎟⎜ t ⎟ ⎜|ψ ⟩⎟ ⎝ sin α cos α ⎠⎜|ψ ⟩⎟ ⎝ 2 ⎠ ⎝ b⎠

Figure 4. Quasiparticle oscillations in twisted bilayer graphene. (a) Probability density of an electron wave packet on top and bottom layers at different time steps. (b) Time-dependent layer-integrated probability density of an electron wave packet. Initial wave packet setup is the same as the one in Figure 3f,g.

⎛ Ĥ ⎞ ⎛ E ⎞ |ϕ(τ )⟩ = exp⎜ −i τ ⎟|ψt ⟩ = cos α exp⎜ −i 1 τ ⎟|ψ1⟩ ⎝ ℏ ⎠ ⎝ ℏ ⎠ ⎛ E ⎞ + sin α exp⎜ −i 2 τ ⎟|ψ2⟩ ⎝ ℏ ⎠

(6)

where E1 and E2 are the energy of the eigenstates |ψ1⟩ and |ψ2⟩, respectively. The probability of finding the particle in the bottom layer will oscillate with time in the same way as a neutrino changes its flavor ⎛ ΔEτ ⎞ ⎟ Pb = |⟨ψb|ϕ(τ )⟩|2 = sin 2(2α)sin 2⎜ ⎝ 2ℏ ⎠

(7)

where ΔE = E1 − E2. As expected, the frequency of the oscillation depends on the energy splitting due to the interlayer coupling. A splitting of 0.54 eV found in the band structure in Figure 2c at k1 gives an oscillation period of 7.6 fs, which is comparable to the value of 7.2 fs obtained from our numerical results in Figure 4b. The small difference between these two values may be due to the higher-order mixing of the wave functions in the full-scale band structure. The small time scale of the oscillations makes it difficult to detect them directly. However, the time-integrated electron beam will propagate along six preferred directions and exhibit bright-dark patterns in real space with a characteristic period of the order of 50 Å in Figure 3f,g. It should be detectable using the technique of scanning probe microscopy (SPM).45−47 In these experiments, one scans a negatively charged atomic force microscopy (AFM) tip over the sample and simultaneously measures the conductance between two ohmic contacts as a function of the tip position. The electron density profile is locally perturbed by the AFM tip mostly in the top layer. Therefore, the resulting conductance changes provide a real-space image of the twodimensional electron flow in the top layer.

(5)

where |ψt⟩ and |ψb⟩ are monolayer eigenstates for the top and bottom layers, respectively. The mixing matrix is a unitary one that is analogous to the Pontecorvo−Maki−Nakagawa−Sakata (PMNS)12−14 matrix for neutrino oscillations. Therefore, injecting charge carriers into one of the layers is similar to creating neutrinos of one flavor. For this particular example, we have tan α = 0.98. If an initial wave packet is created in the top layer, its evolution in the bilayer system will be 5162


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DMR-08−20382). Z.F.W. acknowledges additional support from ARL (Cooperative Agreement No. W911NF-12−2− 0023). This research used computational resources at the National Energy Research Scientific Computing Center (supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02−05CH11231).

The energy window for the collimated propagation depends on the twist angle of TBG. It lies in the gap near the M point of the supercell Brillouin zone. The center of this energy window can be estimated by the following equation Ec = ℏvF

ΔK 2

■

(8)

As the rotational angle changes, this energy value can easily vary in the range of 0.1−0.8 eV. External doping48−50 or gating of the graphene system can tune the Fermi level to fall within this energy window in order to reveal the interesting quasiparticle behavior discussed in this work. Alternatively, high-energy electrons can be injected into the system with small point contacts to study such phenomenon together with scanning tunneling microscopy (STM) tips. Outside the window toward the original Dirac points, the linear bands dominate and the electron propagation would be similar to the monolayer case shown in Figure 3c. In conclusion, we have shown that twisted bilayer graphene is an intriguing system exhibiting surprising properties that have not been explored previously. There exists an energy window in which the band dispersion is highly anisotropic leading to the possibility of generating collimated electron beams in certain directions. In addition, an interesting quasiparticle oscillation can take place in this energy window similar to the neutrino oscillation in particle physics with the layer degree of freedom playing the role of the flavor in neutrinos. The energy eigenstates contain a mixture of states from different layers due to the interlayer coupling. Experimental measurements of neutrino oscillations give insight into the properties of and the interaction among neutrinos, while the quasiparticle oscillation in twisted bilayer graphene reflects the intrinsic coupling between the layers that can be tuned by changing the rotational angles. This allows for further tests of concepts in neutrino physics in the multilayer graphene system. Moreover, the fact that the neutrino-like quasiparticle oscillation and anisotropic electron transport only take place in a specific energy window makes it feasible to control such phenomena by varying the doping level or the energy of injected electrons with the possibility of developing various device applications.

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ASSOCIATED CONTENT

S Supporting Information *

More details on the wave packet simulation, additional data from the calculations, and further analysis of the results are provided. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Drs. Markus Kindermann, Jia-An Yan, WenYing Ruan, and Salvador Barraza-Lopez for insightful discussions. We acknowledge the support by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DEFG02−97ER45632 and by the Georgia Tech MRSEC (funded by the National Science Foundation under Grants 5163


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