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Disorder effect on the integer quantum Hall effect in trilayer graphene
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 J. Phys.: Condens. Matter 25 495503 (http://iopscience.iop.org/0953-8984/25/49/495503) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 150.216.68.200 This content was downloaded on 26/10/2014 at 15:20
Please note that terms and conditions apply.
IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 25 (2013) 495503 (7pp)
doi:10.1088/0953-8984/25/49/495503
Disorder effect on the integer quantum Hall effect in trilayer graphene H Y Tian1,2 , R Ma3 , K S Chan4,5 and J Wang1 1
Department of Physics, Southeast University, Nanjing 210096, People’s Republic of China Department of Physics, Yancheng Institute of Technology, Jiangsu 224051, People’s Republic of China 3 School of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, People’s Republic of China 4 Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, People’s Republic of China 5 Shenzhen Research Institute, City University of Hong Kong, Shenzhen, People’s Republic of China 2
E-mail: [email protected]
Received 29 August 2013 Published 1 November 2013 Online at stacks.iop.org/JPhysCM/25/495503 Abstract We numerically investigate the disorder effect on the integer quantum Hall effect in a trilayer graphene (TLG) system by use of the Kubo formula. For a clean sample, both Bernal (ABA) and rhombohedral (ABC) stacked TLGs display the same quantum rule with abnormal quantized Hall plateaus σxy = νe2 /h (ν = ±6, ±10, ±14, . . .) in the band center and normal quantized Hall plateaus at the band edges. In the presence of disorder, the Hall plateaus become obscure and the higher plateaus disappear first with the increase of the disorder; however, the Hall plateaus of the ABA-stacked TLG are destroyed more readily in comparison with the ABC-stacked one. The longitudinal conductance minimums of the system corresponding to the Hall plateaus become narrower and thinner with disorder, and those of the ABC-stacked TLG are comparatively more stable than those of the ABA structure. The findings indicate that the l = 3 chiral quasiparticles with cubic energy dispersion in ABC-stacked TLG have comparatively stronger immunity to the disorder than the l = 1 and 2 chiral quasiparticles in the ABA counterpart. (Some figures may appear in colour only in the online journal)
1. Introduction
spatial inversion symmetry [9–11]. Different from monolayer graphene, the quasiparticles in the bilayer graphene have a Berry phase of 2π that leads to another type of IQHE [3], displayed as σxy = νe2 /h (ν = ±4, ±8, ±12, . . .), and the IQHE in bilayer graphene was found to be more stable than that in monolayer graphene when the disorder effect was taken into account [7, 8]. Recently, trilayer graphene (TLG) has also drawn much attention of both experimental and theoretic researchers [12–14, 4, 15–19]. TLG has two types of stacking order: Bernal (ABA) stacking and rhombohedral (ABC) stacking, obtained by rotating the top and bottom layers −120◦ or 120◦ with respect to the middle one, as shown in figure 1. Different stacking orders produce totally different low-energy electronic spectra [20–25]. For ABA-stacked TLG, the energy band structure in the vicinity of the
The unique chiral nature of the low-energy quasiparticles in graphene, characterized by a Berry phase π with linear dispersion, gives rise to an unusual integer quantum Hall effect (IQHE) [1–4], σxy = ±gs (n + 12 )e2 /h, where gs = 4 is the Landau-level (LL) degeneracy in a magnetic field due to the spin and valley degrees of freedom. For multilayer graphene, the interlayer coupling substantially influences the energy band structure as well as the low-energy electronic properties. For instance, in bilayer graphene in which the top layer is rotated with respect to the bottom one by 120◦ , the quasiparticle spectrum is modified from linear dispersion to quadratic dispersion [5, 6], due to the interlayer coupling. Bilayer graphene was also found to exhibit a tunable energy gap under a perpendicular electric field because of the broken 0953-8984/13/495503+07$33.00
1
c 2013 IOP Publishing Ltd Printed in the UK & the USA
J. Phys.: Condens. Matter 25 (2013) 495503
H Y Tian et al
wavefunctions of such degree-l chiral quasiparticles acquire a Berry phase of lπ on adiabatic propagation along a closed orbit. This paper is organized as follows. In section 2, we introduce the model Hamiltonian and the formulas to calculate the Hall conductivity as well as the longitudinal conductance. In section 3, detailed discussions of the disorder effect on the IQHE and longitudinal conductance are presented. A brief summary is drawn in section 4.
2. Model and formalism
Figure 1. Atomic structures of trilayer graphene with the ABA (Bernal) stacking order (a) and the ABC (rhombohedral) stacking order (b). The intralayer hopping integral t and interlayer coupling γ1 and γ3 are marked.
ABA-stacked and ABC-stacked TLGs are composed of three coupled hexagonal lattice layers including pairs of inequivalent sublattices {Am , Bm }, where m = 1, 2, 3 is the layer index, as shown in figure 1. In ABAstacked TLG, both the top and bottom layers rotate +120◦ along the layer normal with respect to the middle layer, so the atom sites B1 , A2 , and B3 are arranged directly above or below each other [19, 21, 18]. The ABC stacking corresponds to the middle and top layers rotating 120◦ contiguously with regard to the bottom one in the same direction; as a result, site A3 is below site B2 , while site B3 lies below the center of the hexagons in the middle layer. A nearest-neighbor tight-binding model is employed to describe the TLG system within a perpendicular magnetic field, ! X iaij † H = −t e ciAm cjBm
Dirac point consists of superimposed linear and quadratic spectra [26, 27], while the cubic dispersion of the low-energy quasiparticles with l = 3 chirality appears in ABC-stacked TLG [20, 21]. In a perpendicular electric field, ABA-stacked TLG stays in the semimetallic [28, 18] state while an energy gap is opened at the Dirac point for ABC-stacked TLG [20–22, 25, 29]. When an external magnetic field B is √applied to a TLG, an ABA-stacked ordering exhibits both B-dependent monolayer-like LL and B-dependent bilayer-like LL [22, 5, 30], while,√for an ABC-stacked TLG, the LL is given by En ∝ B2/3 n(n − 1)(n − 2) with a Berry phase 3π [20–24]. Although the TLG energy spectra are different for these two stacking orders, the IQHE was predicted to be the same and quantized as σxy = νe2 /h (ν = ±6, ±10, ±14, . . .), which is independent of the stacking order [19, 23]. However, the prediction was based on the clean limit of the TLG sample and the disorder effect was not taken into account. Given the different immunity to disorder for linear and quadratic quasiparticles [7, 8], ABC-stacked and ABA-stacked TLGs may exhibit different disorder effects on the IQHE. Therefore, the study of the disorder effect on the IQHE is necessary to mark different chiral quasiparticles in these two types of TLG. In this work, we investigate numerically the disorder effect on the IQHE in ABA-stacked and ABC-stacked TLGs and focus on their different immunity to disorder. By use of the Kubo formula, we first recover the well-known Hall plateaus with the filling factor ν = ±6, ±10, ±14, . . . in a clean sample. In the presence of disorder, the Hall plateaus are gradually destroyed and the high filling-factor plateaus disappear first with the increase of disorder. The IQHE of ABC-stacked TLGs is comparatively more stable than that of ABA-stacked TLGs under disorder; that is, the l = 3 quasiparticles in the ABC structure have a stronger surviving capability in disorder in comparison to the l = 1 and 2 quasiparticles in ABA-stacked TLGs, which is also confirmed by the behavior of the longitudinal conductance in disorder. The conductance minimums (valleys) diminish gradually with disorder but the valleys of the ABA-stacked TLG disappear faster than those of the ABC-stacked counterpart. The index l comes from the effective Hamiltonian of nonlinear generalization of Dirac–Weyl quasiparticles, Hl ∼ vl |p|l , with v the electron velocity and p the momentum, so the
hmiji
−
X
(γ1 eiaij c†jB1 ciA2 + γ3 eiaij c†jB2 ciA1 )
h1,2iji
−
X
(γ1 eiaij c†jB3 (A3 ) ciA2 (B2 )
h2,3iji ia
+ γe ij c†jB2 (A2 ) ciA3 (B3 ) ) X + h.c. + wi (c†mi cmi ),
(1)
mi
where the first term describes the nearest-neighbor hopping in each hexagonal lattice layer, where c†iAm (cjBm ) is the creation (annihilation) operator at site Am (Bm ) in the mth layer (m = 1, 2, 3) of the TLG; the second term denotes the coupling between the top and middle graphene layer, and the third term represents the coupling between the lower two layers. t is the intralayer hopping integral of electrons between the Am and Bm sites, γ1 is the strong interlayer coupling between sites B1 –A2 –B3 in ABA stacking and B1 –A2 , B2 –A3 in ABC stacking, and γ3 is the weaker interlayer coupling between sites A1 –B2 –A3 in the ABA stacking and A1 –B2 , A2 –B3 in the ABC stacking. wi is the random disorder potential uniformly distributed in the interval wi ∈ [−W/2, W/2]t, with W the disorder strength. aij is the Peierls’ phase factor fromP the magnetic field, and the magnetic flux per hexagon φ = aij = 2π M is proportional to the applied magnetic field B with summation over a hexagon, where M is assumed to be any integer in principle and the lattice constant is taken to be unity. Note that the Zeeman 2
J. Phys.: Condens. Matter 25 (2013) 495503
H Y Tian et al
splitting from the magnetic field is neglected here, since it is much weaker than the LL energy, and the spin degeneracy is explicitly considered in the numerical results. We consider a finite-size system N = Lx × Ly × Lz with periodic conditions in the graphene plane to calculate the IQHE as well as the longitudinal conductance, where Lz = 3 denotes three graphene layers along the z direction (the layer–plane normal), Ly is the zigzag-chain number of a single monolayer of the TLG, and Lx is the atom-site number in each zigzag chain. The total flux through the sample is Nφ 4π , where N = Lx Ly /M is taken to be an integer, so that periodic boundary conditions for the single-particle magnetic translation operators can be used in both the x and y directions. When M is commensurate with Lx or Ly , the magnetic periodic boundary conditions are reduced to the ordinary periodic boundary conditions. The eigenvectors |αi and eigenenergies α of the ABA-stacked (ABC-stacked) TLG are obtained through exact diagonalization of the Hamiltonian equation (1). Therefore, the Hall conductivity σxy can be worked out by use of the Kubo formula, σxy =
ie2 h¯ X hα|Vx |βihβ|Vy |αi − h.c. , S α,β (α − β )2
the hopping integrals are taken as [12] t = 3.16 eV, γ1 = 0.39 eV, and γ3 = 0.315 eV, the temperature is taken as T = 0 K, and the disorder average is carried out over 800 sample configurations. Notice that three Slonczewski–Weiss–McClure parameters γ2 , γ4 , and γ5 are not taken into account here, which would destroy the electron–hole symmetry and valley degeneracy, and lead to the broken-symmetry quantum Hall states resulting from the splitting of Landau-level crossing points [12]. Since several measurements [13, 14] of the IQHE in ABC-stacked TLG did not show such a phenomenon, we ignore these parameters in the calculations, and compare the disorder effect on the IQHE in the ABC and ABA structures on the same footing. We first focus on the IQHE in a clean TLG with different stacking orders and plot the numerical results in figure 2, where σxy is shown as a function of the Fermi energy EF in the full band. It is seen that discrete LLs emerge in the system and constitute the nonzero density of states, which correspond to the connections of the two consecutive Hall plateaus. The central LL at EF = 0 is referred to as the n = 0 level, the one just above (below) as n = 1 (n = −1), and so on. According to the behaviors of σxy , the energy band is naturally divided into three different regimes. Around the band center, the Hall conductivity is quantized as σxy = νe2 /h, where ν = ±(k + 3/2)gs , with k an integer and gs = 4 for each LL due to the valley and spin degeneracies. As each additional LL is occupied, the total Hall conductivity is increased by gs e2 /h. At the particle–hole symmetry point EF = 0, the energy level has twelvefold degeneracy, and there is no σxy = 0 plateau unless there is an energy gap at the Dirac point, which may come from the external electric field [32]. At the band edges, the conventional IQHE emerges together with several unconventional Hall plateaus. Similar to bilayer graphene [8], there exist two critical energy regions which do not have quantized Hall conductivity. These crossover regions also correspond to a transport regime, where the Hall resistance changes sign and the longitudinal conductivity exhibits metallic behavior. The Van Hove singularity in the electron density of states may be responsible for the singular behavior of the Hall conductivity in the crossover regions. The inset in figure 2 shows the Hall conductivity with different magnetic flux in the band center. It is shown that, on reducing the magnetic flux from φ = 2π/12 to 2π/48, more quantized Hall plateaus emerge at the band center following the same quantization rule. Although ABA-stacked and ABC-stacked TLGs display the same IQHE in the clean limit, they may have different immunity to disorder due to the different chirality of the quasiparticles. In figures 3(a) and (b), several Hall plateaus around the band center are shown for the ABA and ABC structures, respectively. At a relatively weak disorder W = 0.45, the Hall plateaus ν = ±6, ±18, ±30 remain well quantized, whereas the plateaus ν = ±10, ±14, ±22, ±26 become unclear, because the widths of these plateaus are small or the neighboring Landau-level spans are small. For ν = ±34, ±38, the Hall plateaus in the ABA-stacked TLG case disappear, whereas those in the ABC structure can still survive, although they are not very clear. With the further
(2)
where S is the area of the sample and Vx and Vy are the velocity operators. Although the above Kubo formula can also be used to calculate the longitudinal conductivity σxx , we choose to employ the nonequilibrium Green’s function technique [31] to compute the longitudinal conductance G = Ly σxx , since it is very efficient and can save much computational time, especially for calculation of the disorder effect. The conductance formula is given by G=
e2 Tr[0L Gr 0R Ga ], h
(3)
† where 0L(R) = i(6L(R) − 6L(R) ) is the line width function with 6L(R) the self-energy function of the left and right implicit TLG leads, which does not have any disorder or interaction that influences the electron energy. In fact, 0L(R) also denotes the escaping rate of electrons and is proportional to the velocity V. 6L(R) can be obtained from † 6L(R) = Hs,L(R) gL(R) HL(R),s , where Hs,L(R) is the coupling between the scattering region and the left (right) electrode and gL(R) is the surface Green’s function of the two ideal semi-infinite leads. The retarded Green’s function Gr = (Ga )† = [EI − H − 6L − 6R ]−1 is calculated by a recursive method, where H is the Hamiltonian matrix of equation (1) and I is a unit matrix. It is noted that we do not need the periodic condition along the x-axis for calculating the longitudinal conductance, while, along the y-axis, the periodic condition remains unchanged.
3. Calculations and discussions In this section, we shall calculate the IQHE σxy and the conductance G according to equations (2) and (3), and focus on the disorder effect. In the calculations, 3
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H Y Tian et al
Figure 2. Quantum Hall conductivity σxy and electron density of states in the full energy band for the ABA-stacked TLG in (a) and ABC-stacked TLG in (b). The sample size is N = 96 × 24 × 3, the magnetic flux is φ = 2π/48, and the disorder strength is W = 0. In the inset, the Hall conductivity at the band center with φ = 2π/12, 2π/24, 2π/48 is plotted.
at W = 0, as shown in figure 4(a), since the Landau-level span is proportional to the magnetic field regardless of whether l = 1, 2, or 3 chiral quasiparticles are considered. Meanwhile, it is seen that, when the magnetic flux decreases, the Hall plateaus can be destroyed more easily by the disorder, as shown in figure 4(b). For a weak magnetic flux, for instance, φ = 2π/96, the larger ν Hall plateau is also destroyed easily with the increase of the disorder strength, as can be seen from figure 4(c). The ABC-stacked TLG obeys the same disorder rule except that the Hall plateaus are more stable than the ABA-stacked TLG at the same W and φ. In figure 4(d), it is shown that, at W = 0.5, the ABC-stacked TLG exhibits more quantized Hall plateaus than the ABA structure, in which higher plateaus are floated up severely. In order to further explore the disorder effect on the IQHE or the transport of different chiral quasiparticles in the two stacking-order TLGs, we calculate the longitudinal conductance G for different disorder strengths. The conductances are calculated according to equation (3) by using the nonequilibrium Green’s function method, which is much more efficient than the Kubo formula in equation (2). The conductance results are presented in figures 5(a) and (c) for the ABA-stacked TLG and figures 5(b) and (d) for the ABC-stacked TLG, respectively. It is seen that zero conductance corresponds to the Hall plateau, while the conductance peaks coincide with the plateau connections, where the Fermi energy overlaps exactly with one LL. When the disorder increases, the LL broadening (energy uncertainty due to disorder scattering) becomes larger, so the Hall plateau becomes smaller. Finally, the conductance minimum will increase from zero to a finite value when the Hall plateau entirely disappears; this can be seen from figures 5(c) and (d). As the disorder W changes from 0.5 to 0.7, the width of the conductance valley becomes narrower, while the higher valley
Figure 3. Calculated Hall conductivity σxy of ABA-stacked (a) and ABC-stacked (b) TLG as a function of the Fermi energy EF . Parameters are the magnetic flux φ = 2π/48 and the system size N = 96 × 24 × 3.
increase of the disorder W, the high plateaus with a large filling factor ν are destroyed first, and the ABA Hall plateaus are destroyed faster than the ABC ones. For instance, when W = 0.8 and 1.25, the plateaus ν = 30 and 18 still have a definite width for the ABC system, while they entirely disappear for the ABA structure. Therefore, the IQHE in ABC-stacked TLG seems to have a comparatively stronger capability to survive in disorder than that of ABA-stacked TLG, especially for the large ν case. For the ν = ±6 plateaus at the band center, both the ABA and ABC structures display the same immunities to disorder, and the plateaus can even survive for disorder strength W > 2.0. In figure 4, we present the IQHE under weak magnetic fields φ = 2π/96, 2π/192, and 2π/288, with and without disorder. It is shown that more quantized Hall plateaus emerge 4
J. Phys.: Condens. Matter 25 (2013) 495503
H Y Tian et al
Figure 4. Calculated Hall conductivity σxy with different magnetic fluxes and disorder strengths for the ABA-stacked TLG. A comparison between ABA-stacked and ABC-stacked TLGs at W = 0.5 and φ = 2π/96 is presented in (d).
Figure 5. Longitudinal conductance G as a function of the Fermi energy EF for ABA-stacked TLG in (a), (c) and ABC-stacked TLG in (b), (d), with different disorder strengths. Parameters are φ = 2π/48 and N = 96 × 24 × 3.
It is noted that the width of the ν = 34 conductance valley or Hall plateau for ABA-stacked TLG is wider than that of ABC-stacked TLG at W = 0.1 in figures 5(a) and (b), whereas it diminishes faster with the disorder as shown figures 5(c) and (d), i.e., even the original Hall plateau width or the Landau-level span is wider at some filling factors ν in
disappears first. As can be seen in figure 5(c), when W = 0.5, the ν = 34 conductance dip is very clear, while at W = 0.6 this valley becomes much shallower and then disappears entirely at W = 0.7. For the ABC case, the conductance valley ν = 34 experiences the same process, while it is clearly present even at W = 0.7 in figure 5(d). 5
J. Phys.: Condens. Matter 25 (2013) 495503
H Y Tian et al
Figure 6. Longitudinal conductance G as a function of the Fermi energy EF for ABA-stacked and ABC-stacked TLGs with different disorder strengths. Parameters are the same as those in figure 3.
4. Conclusion
the ABA structure, but it may still disappear faster than that of the ABC structure. Therefore, the l = 3 chiral quasiparticles in ABC-stacked TLG have a comparatively strong capability to survive in disorder, while, for the ABA stacking order, the quasiparticles have both l = 1 and 2 chirality but no l = 3 chirality, since the system can be regarded as a superposition of monolayer and bilayer graphene. This confirms the findings of a previous study [7, 8], namely that the l = 2 chiral quasiparticles have a stronger surviving capability in disorder than the l = 1 chiral quasiparticles of monolayer graphene. In figure 6, we proceed to calculate the conductance G with four different disorder strengths that are the same as those in figure 3. It is seen that, for the wider conductance valleys ν = ±18, ±30, the ABC-stacked valleys are more stable; For example, they almost have the same width at W = 0.45 in figure 6(a), whereas at W = 0.8 the ν = 30 valley of the ABA structure is more severely destroyed than that of the ABC stacking, as shown in figure 6(b). And so is the case for the conductance valley ν = ±18, which at W = 1.25 is much deeper for the ABC structure, as shown in figure 6(c). It is also seen that the ν = 34 conductance valley in the ABA structure disappear entirely when the disorder W is enhanced from 0.45 to 0.8, whereas it still has a clear dip for the ABC counterpart, as shown in figure 6(b). This is consistent with the results in figure 3, where at W = 0.8 the Hall plateaus v = ±34 have a fluctuation in ABC-stacked TLG, while they decay quickly in ABA-stacked TLG. Meanwhile, we note that the most stable Hall plateaus ν = ±6 in figure 3 coincide with the most stable conductance valleys, as shown in figure 6, which nearly have the same width under the four different disorder strengths.
To conclude, we have investigated the disorder effect on the IQHE in TLGs and compared the relative stability of the different chiral quasiparticles in the ABA and ABC stacking orders. By use of the Kubo formula, we first computed the well-known IQHE σxy = νe2 /h (ν = ±6, ±10, ±14, . . .) in the clean limit, which is the same for both stacking orders. When the disorder is turned on, the Hall plateaus are destroyed through the float-up of extended levels toward the band center, and high filling-factor plateaus always disappear first. The IQHE in ABC-stacked TLG is relatively more stable in the presence of disorder than that in ABA-stacked TLG. The longitudinal conductance calculations further confirm that the l = 3 chiral quasiparticles of ABC-stacked TLG have a stronger surviving capability in disorder than the l = 1 and 2 chiral quasiparticles in ABA-stacked TLG.
Acknowledgments We are very grateful to Professor Sheng for stimulating discussions. The work was supported by the NSFC under grant Nos 11274260, 11274059, 11074032, and 11104146.
References [1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V and Firsov A A 2005 Nature 438 197 [2] Zhang Y, Tan Y W, Stormer H L and Kim P 2005 Nature 438 201 6
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[15] Sena S H R, Pereira J M, Peeters F M and Farias G A 2011 Phys. Rev. B 84 205448 [16] Henriksen E A, Nandi D and Eisenstein J P 2012 Phys. Rev. X 2 011004 [17] Koshino M and McCann E 2010 Phys. Rev. B 81 115315 [18] Koshino M and McCann E 2009 Phys. Rev. B 79 125443 [19] Yuan S, Rold’an R and Katsnelson M I 2011 Phys. Rev. B 84 125455 [20] Koshino M and McCann E 2009 Phys. Rev. B 80 165409 [21] Koshino M 2010 Phys. Rev. B 81 125304 [22] Guinea F, Castro Neto A H and Peres N M R 2006 Phys. Rev. B 73 245426 [23] Min H and MacDonald A H 2008 Phys. Rev. B 77 155416 [24] Min H and MacDonald A H 2008 Prog. Theor. Phys. Suppl. 176 227 [25] Avetisyan A A, Partoens B and Peeters F M 2010 Phys. Rev. B 81 115432 [26] Guinea F, Castro Neto A H and Peres N M R 2007 Solid State Commun. 143 116 [27] Koshino M and Ando T 2007 Phys. Rev. B 76 085425 [28] Craciun M F, Russo S, Yamamoto M, Oostinga J B, Morpurgo A F and Tarucha S 2009 Nature Nanotechnol. 4 383 [29] Bao W et al 2011 Nature Phys. 7 948 [30] Ezawa M 2007 J. Phys. Soc. Japan 76 094701 [31] Keldysh L V 1965 Sov. Phys.—JETP 20 1018 [32] Ma R 2013 Eur. Phys. J. B 86 6
[3] Novoselov K S, McCann E, Morozov S V, Fal’ko V I, Katsnelson M I, Zeitler U, Jiang D, Schedin F and Geim A K 2006 Nature Phys. 2 177 [4] Jhang S H et al 2011 Phys. Rev. B 84 161408 [5] McCann E and Fal’ko V I 2006 Phys. Rev. Lett. 96 086805 [6] Nilsson J, Castro Neto A H, Peres N M R and Guinea F 2006 Phys. Rev. B 73 214418 [7] Sheng D N, Sheng L and Weng Z Y 2006 Phys. Rev. B 73 233406 [8] Ma R, Sheng L, Shen R, Liu M and Sheng D N 2009 Phys. Rev. B 80 205101 [9] Castro E V, Novoselov K S, Morozov S V, Peres N M R, Lopes dos Santos J M B, Nilsson J, Guinea F, Geim A K and CastroNeto A H 2007 Phys. Rev. Lett. 99 216802 [10] Oostinga J B, Heersche H B, Liu X, Morpurgo A F and Vandersypen L M K 2008 Nature Mater. 7 151 [11] Zhang Y, Tang T T, Girit C, Hao Z, Martin M C, Zettl A, Crommie M F, Shen Y R and Wang F 2009 Nature 459 820 [12] Taychatanapat T, Watanabe K, Taniguchi T and Herrero P J 2011 Nature Phys. 7 621 [13] Zhang L Y, Zhang Y, Camacho J, Khodas M and Zaliznyak I 2011 Nature Phys. 7 953 [14] Kumar A, Escoffier W, Poumirol J M, Faugeras C, Arovas D P, Fogler M M, Guinea F, Roche S, Goiran M and Raquet B 2011 Phys. Rev. Lett. 107 126806
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Disorder effect on the integer quantum Hall effect in trilayer graphene
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 J. Phys.: Condens. Matter 25 495503 (http://iopscience.iop.org/0953-8984/25/49/495503) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 150.216.68.200 This content was downloaded on 26/10/2014 at 15:20
Please note that terms and conditions apply.
IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 25 (2013) 495503 (7pp)
doi:10.1088/0953-8984/25/49/495503
Disorder effect on the integer quantum Hall effect in trilayer graphene H Y Tian1,2 , R Ma3 , K S Chan4,5 and J Wang1 1
Department of Physics, Southeast University, Nanjing 210096, People’s Republic of China Department of Physics, Yancheng Institute of Technology, Jiangsu 224051, People’s Republic of China 3 School of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, People’s Republic of China 4 Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, People’s Republic of China 5 Shenzhen Research Institute, City University of Hong Kong, Shenzhen, People’s Republic of China 2
E-mail: [email protected]
Received 29 August 2013 Published 1 November 2013 Online at stacks.iop.org/JPhysCM/25/495503 Abstract We numerically investigate the disorder effect on the integer quantum Hall effect in a trilayer graphene (TLG) system by use of the Kubo formula. For a clean sample, both Bernal (ABA) and rhombohedral (ABC) stacked TLGs display the same quantum rule with abnormal quantized Hall plateaus σxy = νe2 /h (ν = ±6, ±10, ±14, . . .) in the band center and normal quantized Hall plateaus at the band edges. In the presence of disorder, the Hall plateaus become obscure and the higher plateaus disappear first with the increase of the disorder; however, the Hall plateaus of the ABA-stacked TLG are destroyed more readily in comparison with the ABC-stacked one. The longitudinal conductance minimums of the system corresponding to the Hall plateaus become narrower and thinner with disorder, and those of the ABC-stacked TLG are comparatively more stable than those of the ABA structure. The findings indicate that the l = 3 chiral quasiparticles with cubic energy dispersion in ABC-stacked TLG have comparatively stronger immunity to the disorder than the l = 1 and 2 chiral quasiparticles in the ABA counterpart. (Some figures may appear in colour only in the online journal)
1. Introduction
spatial inversion symmetry [9–11]. Different from monolayer graphene, the quasiparticles in the bilayer graphene have a Berry phase of 2π that leads to another type of IQHE [3], displayed as σxy = νe2 /h (ν = ±4, ±8, ±12, . . .), and the IQHE in bilayer graphene was found to be more stable than that in monolayer graphene when the disorder effect was taken into account [7, 8]. Recently, trilayer graphene (TLG) has also drawn much attention of both experimental and theoretic researchers [12–14, 4, 15–19]. TLG has two types of stacking order: Bernal (ABA) stacking and rhombohedral (ABC) stacking, obtained by rotating the top and bottom layers −120◦ or 120◦ with respect to the middle one, as shown in figure 1. Different stacking orders produce totally different low-energy electronic spectra [20–25]. For ABA-stacked TLG, the energy band structure in the vicinity of the
The unique chiral nature of the low-energy quasiparticles in graphene, characterized by a Berry phase π with linear dispersion, gives rise to an unusual integer quantum Hall effect (IQHE) [1–4], σxy = ±gs (n + 12 )e2 /h, where gs = 4 is the Landau-level (LL) degeneracy in a magnetic field due to the spin and valley degrees of freedom. For multilayer graphene, the interlayer coupling substantially influences the energy band structure as well as the low-energy electronic properties. For instance, in bilayer graphene in which the top layer is rotated with respect to the bottom one by 120◦ , the quasiparticle spectrum is modified from linear dispersion to quadratic dispersion [5, 6], due to the interlayer coupling. Bilayer graphene was also found to exhibit a tunable energy gap under a perpendicular electric field because of the broken 0953-8984/13/495503+07$33.00
1
c 2013 IOP Publishing Ltd Printed in the UK & the USA
J. Phys.: Condens. Matter 25 (2013) 495503
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wavefunctions of such degree-l chiral quasiparticles acquire a Berry phase of lπ on adiabatic propagation along a closed orbit. This paper is organized as follows. In section 2, we introduce the model Hamiltonian and the formulas to calculate the Hall conductivity as well as the longitudinal conductance. In section 3, detailed discussions of the disorder effect on the IQHE and longitudinal conductance are presented. A brief summary is drawn in section 4.
2. Model and formalism
Figure 1. Atomic structures of trilayer graphene with the ABA (Bernal) stacking order (a) and the ABC (rhombohedral) stacking order (b). The intralayer hopping integral t and interlayer coupling γ1 and γ3 are marked.
ABA-stacked and ABC-stacked TLGs are composed of three coupled hexagonal lattice layers including pairs of inequivalent sublattices {Am , Bm }, where m = 1, 2, 3 is the layer index, as shown in figure 1. In ABAstacked TLG, both the top and bottom layers rotate +120◦ along the layer normal with respect to the middle layer, so the atom sites B1 , A2 , and B3 are arranged directly above or below each other [19, 21, 18]. The ABC stacking corresponds to the middle and top layers rotating 120◦ contiguously with regard to the bottom one in the same direction; as a result, site A3 is below site B2 , while site B3 lies below the center of the hexagons in the middle layer. A nearest-neighbor tight-binding model is employed to describe the TLG system within a perpendicular magnetic field, ! X iaij † H = −t e ciAm cjBm
Dirac point consists of superimposed linear and quadratic spectra [26, 27], while the cubic dispersion of the low-energy quasiparticles with l = 3 chirality appears in ABC-stacked TLG [20, 21]. In a perpendicular electric field, ABA-stacked TLG stays in the semimetallic [28, 18] state while an energy gap is opened at the Dirac point for ABC-stacked TLG [20–22, 25, 29]. When an external magnetic field B is √applied to a TLG, an ABA-stacked ordering exhibits both B-dependent monolayer-like LL and B-dependent bilayer-like LL [22, 5, 30], while,√for an ABC-stacked TLG, the LL is given by En ∝ B2/3 n(n − 1)(n − 2) with a Berry phase 3π [20–24]. Although the TLG energy spectra are different for these two stacking orders, the IQHE was predicted to be the same and quantized as σxy = νe2 /h (ν = ±6, ±10, ±14, . . .), which is independent of the stacking order [19, 23]. However, the prediction was based on the clean limit of the TLG sample and the disorder effect was not taken into account. Given the different immunity to disorder for linear and quadratic quasiparticles [7, 8], ABC-stacked and ABA-stacked TLGs may exhibit different disorder effects on the IQHE. Therefore, the study of the disorder effect on the IQHE is necessary to mark different chiral quasiparticles in these two types of TLG. In this work, we investigate numerically the disorder effect on the IQHE in ABA-stacked and ABC-stacked TLGs and focus on their different immunity to disorder. By use of the Kubo formula, we first recover the well-known Hall plateaus with the filling factor ν = ±6, ±10, ±14, . . . in a clean sample. In the presence of disorder, the Hall plateaus are gradually destroyed and the high filling-factor plateaus disappear first with the increase of disorder. The IQHE of ABC-stacked TLGs is comparatively more stable than that of ABA-stacked TLGs under disorder; that is, the l = 3 quasiparticles in the ABC structure have a stronger surviving capability in disorder in comparison to the l = 1 and 2 quasiparticles in ABA-stacked TLGs, which is also confirmed by the behavior of the longitudinal conductance in disorder. The conductance minimums (valleys) diminish gradually with disorder but the valleys of the ABA-stacked TLG disappear faster than those of the ABC-stacked counterpart. The index l comes from the effective Hamiltonian of nonlinear generalization of Dirac–Weyl quasiparticles, Hl ∼ vl |p|l , with v the electron velocity and p the momentum, so the
hmiji
−
X
(γ1 eiaij c†jB1 ciA2 + γ3 eiaij c†jB2 ciA1 )
h1,2iji
−
X
(γ1 eiaij c†jB3 (A3 ) ciA2 (B2 )
h2,3iji ia
+ γe ij c†jB2 (A2 ) ciA3 (B3 ) ) X + h.c. + wi (c†mi cmi ),
(1)
mi
where the first term describes the nearest-neighbor hopping in each hexagonal lattice layer, where c†iAm (cjBm ) is the creation (annihilation) operator at site Am (Bm ) in the mth layer (m = 1, 2, 3) of the TLG; the second term denotes the coupling between the top and middle graphene layer, and the third term represents the coupling between the lower two layers. t is the intralayer hopping integral of electrons between the Am and Bm sites, γ1 is the strong interlayer coupling between sites B1 –A2 –B3 in ABA stacking and B1 –A2 , B2 –A3 in ABC stacking, and γ3 is the weaker interlayer coupling between sites A1 –B2 –A3 in the ABA stacking and A1 –B2 , A2 –B3 in the ABC stacking. wi is the random disorder potential uniformly distributed in the interval wi ∈ [−W/2, W/2]t, with W the disorder strength. aij is the Peierls’ phase factor fromP the magnetic field, and the magnetic flux per hexagon φ = aij = 2π M is proportional to the applied magnetic field B with summation over a hexagon, where M is assumed to be any integer in principle and the lattice constant is taken to be unity. Note that the Zeeman 2
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splitting from the magnetic field is neglected here, since it is much weaker than the LL energy, and the spin degeneracy is explicitly considered in the numerical results. We consider a finite-size system N = Lx × Ly × Lz with periodic conditions in the graphene plane to calculate the IQHE as well as the longitudinal conductance, where Lz = 3 denotes three graphene layers along the z direction (the layer–plane normal), Ly is the zigzag-chain number of a single monolayer of the TLG, and Lx is the atom-site number in each zigzag chain. The total flux through the sample is Nφ 4π , where N = Lx Ly /M is taken to be an integer, so that periodic boundary conditions for the single-particle magnetic translation operators can be used in both the x and y directions. When M is commensurate with Lx or Ly , the magnetic periodic boundary conditions are reduced to the ordinary periodic boundary conditions. The eigenvectors |αi and eigenenergies α of the ABA-stacked (ABC-stacked) TLG are obtained through exact diagonalization of the Hamiltonian equation (1). Therefore, the Hall conductivity σxy can be worked out by use of the Kubo formula, σxy =
ie2 h¯ X hα|Vx |βihβ|Vy |αi − h.c. , S α,β (α − β )2
the hopping integrals are taken as [12] t = 3.16 eV, γ1 = 0.39 eV, and γ3 = 0.315 eV, the temperature is taken as T = 0 K, and the disorder average is carried out over 800 sample configurations. Notice that three Slonczewski–Weiss–McClure parameters γ2 , γ4 , and γ5 are not taken into account here, which would destroy the electron–hole symmetry and valley degeneracy, and lead to the broken-symmetry quantum Hall states resulting from the splitting of Landau-level crossing points [12]. Since several measurements [13, 14] of the IQHE in ABC-stacked TLG did not show such a phenomenon, we ignore these parameters in the calculations, and compare the disorder effect on the IQHE in the ABC and ABA structures on the same footing. We first focus on the IQHE in a clean TLG with different stacking orders and plot the numerical results in figure 2, where σxy is shown as a function of the Fermi energy EF in the full band. It is seen that discrete LLs emerge in the system and constitute the nonzero density of states, which correspond to the connections of the two consecutive Hall plateaus. The central LL at EF = 0 is referred to as the n = 0 level, the one just above (below) as n = 1 (n = −1), and so on. According to the behaviors of σxy , the energy band is naturally divided into three different regimes. Around the band center, the Hall conductivity is quantized as σxy = νe2 /h, where ν = ±(k + 3/2)gs , with k an integer and gs = 4 for each LL due to the valley and spin degeneracies. As each additional LL is occupied, the total Hall conductivity is increased by gs e2 /h. At the particle–hole symmetry point EF = 0, the energy level has twelvefold degeneracy, and there is no σxy = 0 plateau unless there is an energy gap at the Dirac point, which may come from the external electric field [32]. At the band edges, the conventional IQHE emerges together with several unconventional Hall plateaus. Similar to bilayer graphene [8], there exist two critical energy regions which do not have quantized Hall conductivity. These crossover regions also correspond to a transport regime, where the Hall resistance changes sign and the longitudinal conductivity exhibits metallic behavior. The Van Hove singularity in the electron density of states may be responsible for the singular behavior of the Hall conductivity in the crossover regions. The inset in figure 2 shows the Hall conductivity with different magnetic flux in the band center. It is shown that, on reducing the magnetic flux from φ = 2π/12 to 2π/48, more quantized Hall plateaus emerge at the band center following the same quantization rule. Although ABA-stacked and ABC-stacked TLGs display the same IQHE in the clean limit, they may have different immunity to disorder due to the different chirality of the quasiparticles. In figures 3(a) and (b), several Hall plateaus around the band center are shown for the ABA and ABC structures, respectively. At a relatively weak disorder W = 0.45, the Hall plateaus ν = ±6, ±18, ±30 remain well quantized, whereas the plateaus ν = ±10, ±14, ±22, ±26 become unclear, because the widths of these plateaus are small or the neighboring Landau-level spans are small. For ν = ±34, ±38, the Hall plateaus in the ABA-stacked TLG case disappear, whereas those in the ABC structure can still survive, although they are not very clear. With the further
(2)
where S is the area of the sample and Vx and Vy are the velocity operators. Although the above Kubo formula can also be used to calculate the longitudinal conductivity σxx , we choose to employ the nonequilibrium Green’s function technique [31] to compute the longitudinal conductance G = Ly σxx , since it is very efficient and can save much computational time, especially for calculation of the disorder effect. The conductance formula is given by G=
e2 Tr[0L Gr 0R Ga ], h
(3)
† where 0L(R) = i(6L(R) − 6L(R) ) is the line width function with 6L(R) the self-energy function of the left and right implicit TLG leads, which does not have any disorder or interaction that influences the electron energy. In fact, 0L(R) also denotes the escaping rate of electrons and is proportional to the velocity V. 6L(R) can be obtained from † 6L(R) = Hs,L(R) gL(R) HL(R),s , where Hs,L(R) is the coupling between the scattering region and the left (right) electrode and gL(R) is the surface Green’s function of the two ideal semi-infinite leads. The retarded Green’s function Gr = (Ga )† = [EI − H − 6L − 6R ]−1 is calculated by a recursive method, where H is the Hamiltonian matrix of equation (1) and I is a unit matrix. It is noted that we do not need the periodic condition along the x-axis for calculating the longitudinal conductance, while, along the y-axis, the periodic condition remains unchanged.
3. Calculations and discussions In this section, we shall calculate the IQHE σxy and the conductance G according to equations (2) and (3), and focus on the disorder effect. In the calculations, 3
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Figure 2. Quantum Hall conductivity σxy and electron density of states in the full energy band for the ABA-stacked TLG in (a) and ABC-stacked TLG in (b). The sample size is N = 96 × 24 × 3, the magnetic flux is φ = 2π/48, and the disorder strength is W = 0. In the inset, the Hall conductivity at the band center with φ = 2π/12, 2π/24, 2π/48 is plotted.
at W = 0, as shown in figure 4(a), since the Landau-level span is proportional to the magnetic field regardless of whether l = 1, 2, or 3 chiral quasiparticles are considered. Meanwhile, it is seen that, when the magnetic flux decreases, the Hall plateaus can be destroyed more easily by the disorder, as shown in figure 4(b). For a weak magnetic flux, for instance, φ = 2π/96, the larger ν Hall plateau is also destroyed easily with the increase of the disorder strength, as can be seen from figure 4(c). The ABC-stacked TLG obeys the same disorder rule except that the Hall plateaus are more stable than the ABA-stacked TLG at the same W and φ. In figure 4(d), it is shown that, at W = 0.5, the ABC-stacked TLG exhibits more quantized Hall plateaus than the ABA structure, in which higher plateaus are floated up severely. In order to further explore the disorder effect on the IQHE or the transport of different chiral quasiparticles in the two stacking-order TLGs, we calculate the longitudinal conductance G for different disorder strengths. The conductances are calculated according to equation (3) by using the nonequilibrium Green’s function method, which is much more efficient than the Kubo formula in equation (2). The conductance results are presented in figures 5(a) and (c) for the ABA-stacked TLG and figures 5(b) and (d) for the ABC-stacked TLG, respectively. It is seen that zero conductance corresponds to the Hall plateau, while the conductance peaks coincide with the plateau connections, where the Fermi energy overlaps exactly with one LL. When the disorder increases, the LL broadening (energy uncertainty due to disorder scattering) becomes larger, so the Hall plateau becomes smaller. Finally, the conductance minimum will increase from zero to a finite value when the Hall plateau entirely disappears; this can be seen from figures 5(c) and (d). As the disorder W changes from 0.5 to 0.7, the width of the conductance valley becomes narrower, while the higher valley
Figure 3. Calculated Hall conductivity σxy of ABA-stacked (a) and ABC-stacked (b) TLG as a function of the Fermi energy EF . Parameters are the magnetic flux φ = 2π/48 and the system size N = 96 × 24 × 3.
increase of the disorder W, the high plateaus with a large filling factor ν are destroyed first, and the ABA Hall plateaus are destroyed faster than the ABC ones. For instance, when W = 0.8 and 1.25, the plateaus ν = 30 and 18 still have a definite width for the ABC system, while they entirely disappear for the ABA structure. Therefore, the IQHE in ABC-stacked TLG seems to have a comparatively stronger capability to survive in disorder than that of ABA-stacked TLG, especially for the large ν case. For the ν = ±6 plateaus at the band center, both the ABA and ABC structures display the same immunities to disorder, and the plateaus can even survive for disorder strength W > 2.0. In figure 4, we present the IQHE under weak magnetic fields φ = 2π/96, 2π/192, and 2π/288, with and without disorder. It is shown that more quantized Hall plateaus emerge 4
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Figure 4. Calculated Hall conductivity σxy with different magnetic fluxes and disorder strengths for the ABA-stacked TLG. A comparison between ABA-stacked and ABC-stacked TLGs at W = 0.5 and φ = 2π/96 is presented in (d).
Figure 5. Longitudinal conductance G as a function of the Fermi energy EF for ABA-stacked TLG in (a), (c) and ABC-stacked TLG in (b), (d), with different disorder strengths. Parameters are φ = 2π/48 and N = 96 × 24 × 3.
It is noted that the width of the ν = 34 conductance valley or Hall plateau for ABA-stacked TLG is wider than that of ABC-stacked TLG at W = 0.1 in figures 5(a) and (b), whereas it diminishes faster with the disorder as shown figures 5(c) and (d), i.e., even the original Hall plateau width or the Landau-level span is wider at some filling factors ν in
disappears first. As can be seen in figure 5(c), when W = 0.5, the ν = 34 conductance dip is very clear, while at W = 0.6 this valley becomes much shallower and then disappears entirely at W = 0.7. For the ABC case, the conductance valley ν = 34 experiences the same process, while it is clearly present even at W = 0.7 in figure 5(d). 5
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Figure 6. Longitudinal conductance G as a function of the Fermi energy EF for ABA-stacked and ABC-stacked TLGs with different disorder strengths. Parameters are the same as those in figure 3.
4. Conclusion
the ABA structure, but it may still disappear faster than that of the ABC structure. Therefore, the l = 3 chiral quasiparticles in ABC-stacked TLG have a comparatively strong capability to survive in disorder, while, for the ABA stacking order, the quasiparticles have both l = 1 and 2 chirality but no l = 3 chirality, since the system can be regarded as a superposition of monolayer and bilayer graphene. This confirms the findings of a previous study [7, 8], namely that the l = 2 chiral quasiparticles have a stronger surviving capability in disorder than the l = 1 chiral quasiparticles of monolayer graphene. In figure 6, we proceed to calculate the conductance G with four different disorder strengths that are the same as those in figure 3. It is seen that, for the wider conductance valleys ν = ±18, ±30, the ABC-stacked valleys are more stable; For example, they almost have the same width at W = 0.45 in figure 6(a), whereas at W = 0.8 the ν = 30 valley of the ABA structure is more severely destroyed than that of the ABC stacking, as shown in figure 6(b). And so is the case for the conductance valley ν = ±18, which at W = 1.25 is much deeper for the ABC structure, as shown in figure 6(c). It is also seen that the ν = 34 conductance valley in the ABA structure disappear entirely when the disorder W is enhanced from 0.45 to 0.8, whereas it still has a clear dip for the ABC counterpart, as shown in figure 6(b). This is consistent with the results in figure 3, where at W = 0.8 the Hall plateaus v = ±34 have a fluctuation in ABC-stacked TLG, while they decay quickly in ABA-stacked TLG. Meanwhile, we note that the most stable Hall plateaus ν = ±6 in figure 3 coincide with the most stable conductance valleys, as shown in figure 6, which nearly have the same width under the four different disorder strengths.
To conclude, we have investigated the disorder effect on the IQHE in TLGs and compared the relative stability of the different chiral quasiparticles in the ABA and ABC stacking orders. By use of the Kubo formula, we first computed the well-known IQHE σxy = νe2 /h (ν = ±6, ±10, ±14, . . .) in the clean limit, which is the same for both stacking orders. When the disorder is turned on, the Hall plateaus are destroyed through the float-up of extended levels toward the band center, and high filling-factor plateaus always disappear first. The IQHE in ABC-stacked TLG is relatively more stable in the presence of disorder than that in ABA-stacked TLG. The longitudinal conductance calculations further confirm that the l = 3 chiral quasiparticles of ABC-stacked TLG have a stronger surviving capability in disorder than the l = 1 and 2 chiral quasiparticles in ABA-stacked TLG.
Acknowledgments We are very grateful to Professor Sheng for stimulating discussions. The work was supported by the NSFC under grant Nos 11274260, 11274059, 11074032, and 11104146.
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