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Magnetoband structures of AB-stacked zigzag nanographite ribbons
Physics Letters A 306 (2002) 137–143 www.elsevier.com/locate/pla
Magnetoband structures of AB-stacked zigzag nanographite ribbons C.P. Chang a,b,∗ , C.W. Chiu b , F.L. Shyu c , R.B. Chen d , M.F. Lin b a Center for General Education, Tainan Woman’s College of Arts & Technology, 701 Tainan, Taiwan b Department of Physics, National Cheng Kung University, 701 Tainan, Taiwan c Department of Physics, Chinese Military Academy Kaohsiung, 830 Kaohsiung, Taiwan d Department of Electrical Engineering, Cheng Shiu Institute of Technology, 830 Kaohsiung, Taiwan
Received 11 October 2002; accepted 26 October 2002 Communicated by V.M. Agranovich
Abstract Magnetoband structures of AB-stacked zigzag nanographite ribbons are studied by the tight-binding model. The magnetic field changes band width, energy space, and energy dispersions (the produce of Landau subbands and Landau levels). It causes many zero energy points. Such points and corresponding localized states are studied in detail. There are certain important differences between localized states and edge states. Oscillation period of Landau subbands are determined by these points. The interribbon interactions also affect magnetoband structures, such as energy dispersions, band width, oscillation period of Landau subbands, and flux dependence of Hofstadter butterflies. 2002 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx; 71.15.Fv Keywords: Band structures; Nanographite ribbons; Hofstadter butterfly
Electronic properties of nanographite ribbons can be modified by changing width, edge shape, stacking sequence, and magnetic field. A one-dimensional nanographite monoribbon is obtained by cutting a twodimensional graphite sheet along the longitudinal direction (x). ˆ Band structure is dominated by geometric structures [1–10], which, thus, stimulates many studies, such as magnetic properties [11,12], optical
* Corresponding author.
E-mail address: [email protected] (C.P. Chang).
properties [13], electronic excitations [14], transport properties [15], and growth [16–18]. A zigzag ribbon, with zigzag structure along the x-axis, has very special band structure. The partial flat bands at the Fermi level result from the nonbonding edge states located at the outmost zigzag edges. They are expected to play a very important role on physical properties [11–15]. Wakabayashi et al. [11] had used the tight-binding model to study electronic and magnetic properties of nanographite monoribbon. The magnetic susceptibility is strongly dependent on edge shape and ribbon width. Especially, the Pauli paramagnetism originat-
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 1 5 7 8 - 5
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ing from the edge states of the zigzag ribbons can significantly contribute to the magnetic susceptibility. The relation between the edge states and the localized states induced by the magnetic flux has not been investigated in their study. On the other hand, Harigaya had studied the mechanism on the magnetism in the AA- and AB-stacked nanographite ribbons [12]. They found that the electronic spins localized in the open shell of the AB-stacked zigzag ribbons is the favorable condition for the magnetism. However, the effects of the interribbon interactions on the edge states are neglected. Energy spectra of Bloch electrons in the magnetic field have been widely studied in 2D square lattices [19–21], triangle lattices [21–23], hexagonal lattices [24–26], artificial structures [27], 3D cubic lattices [28,29], and 1D square lattices [30]. Such lattices only belong to hypertheoretical systems. Here, we study magnetoband structures of AB-stacked zigzag nanographite ribbons. This study shows that the magnetic field causes a lot of zero energy points. Such points are identified to be associated with localized states. The differences between localized states and edge states are examined. Magnetic field produces oscillational Landau subbands at low energy. Their oscillation periods are determined by zero energy points. The effects due to the interribbon interactions are also investigated. By incorporating the Peierl phase ei2πθij into the hopping integrals, we can couple a uniform magnetic field B, parallel to the z-axis, to electronic structures of nanographite ribbons in the tight-binding model. The Hamiltonian is given by γi,j ei2πθij ci+ cj + H.c. , H= (1) i,j
where the subindices i, j denote the summation over all sites and γi,j is the hopping integral. ci+ (cj ) is the creation (annihilation) operation, which generates (destroys) an electron at i (j ) site, and ei2πθij is the Peierl phase caused by the external field. θi,j is the line integral of the vector potential A from i to j in unit of j flux quantum Φ0 = ch/e, i.e., θi,j = (1/Φ0 ) i A dl. With the Landau gauge, the position-dependent vector potential is A = (−By, 0, 0). The geometric structure of the AB-stacked zigzag nanographite ribbons is shown in Fig. 1. Each ribbon consists of the benzene-ring carbon atoms. The
Fig. 1. The geometric structure of the Ny = 5 AB-stacked zigzag nanographite ribbons. γ0 and αi ’s indicate the atom–atom interactions.
nearest-neighbor C–C bond length is b = 1.42 √Å. The periodical length along the x-axis is Ix = 3 b for the zigzag chains, where B atoms are the nearestneighbors of A atoms. The width is defined by the number of the zigzag chains (Ny ) along the y-axis. The identical zigzag ribbons are periodically arranged in the AB sequence. A atoms are in a straight line along the z-axis, while B atoms form a zigzag texture in the y–z plane. That will make the local environment of A atoms different from that of B atoms. This difference, denoted as the chemical-shift α6 , will be taken into account in the band-structure calculations. The AB-stacked nanographite ribbons have two ribbons or 4Ny carbon atoms in a 2D primitive cell. The first Brillouin zone is a rectangle defined by −π/Ix kx π/Ix and −π/2Ic kz π/2Ic . Ic (= 3.35 Å) is the periodic distance between two neighboring ribbons. The atom–atom interactions are as follows. γ0 is the interaction between A atom and B atom on the same ribbon. α1 (α3 ) represents the interaction between two A (B) atoms from two neighboring ribbons. α5 (α2 ) is that for two A (B) atoms from two nextneighboring ribbons. α4 corresponds to the interaction between A atom and B atom, from two neighboring ribbons.
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Under the magnetic field, the AB-stacked zigzag ribbons still own two ribbons in a 2D primitive cell. The Hamiltonian of ribbons is a 4Ny × 4Ny Hermitian matrix. It can be arranged as 2 × 2 blocks. Each block matrix is a 2Ny × 2Ny matrix. The outmost zigzag chains of the two neighboring ribbons are out of line along the x-axis, so the center of each ribbon will not coincide with each other. When we choose the coordinate origin at the geometric center of one ribbon in calculating the Peierl phase θi,j , the diagonal matrix (Hij )1,1 differers from (Hij )2,2 . The two diagonal matrices are, respectively, α6 + β 2 α5 /2, if j = i; j = 2m − 1, β 2 α2 /2, if j = i; j = 2m, √
2γ0 cos 3 bkx /2 − π(m − [N])Φ , (Hij )1,1 = if j = i + 1; j = 2m, γ0 , if j = i + 1; j = 2m + 1, (2) 0, others, and (Hij )2,2 α + β 2 α5 /2, if j = i; j = 2m − 1, 6 2 2m, β α2 /2, √ if j = i; j =
2γ0 cos 3 bkx /2 − π m − [N] + 13 Φ , = if j = i + 1; j = 2m, γ0 , if j = i + 1; j = 2m + 1, (3) 0, others, where m labels the mth zigzag chain along the y-axis in each ribbon, and Φ in unit of Φ0 is the magnetic flux through the benzene ring. The difference between the centers of two ribbons gives rise to πΦ/3 in the cosine term in Eq. (3). [N] is defined as if Ny is even, Ny /2, [N] = (Ny + 1)/2, if Ny is odd. √ We can treat γ0 (γm = 2γ0 cos{ 3 bkx /2 − π(m − [N])Φ}) in Eq. (2) as the interchain (intrachain) interactions of zigzag chains on the same ribbon. γm is dependent on the wavevector kx , the ribbon width Ny , the position of zigzag chain m, and the magnetic flux Φ. The interaction between the atoms at the mth zigzag chain is switched off by the zero of γm and the interesting results thus emerge.
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The off-diagonal block matrix, which describes the interribbon interactions, is (Hij )1,2 βα1 , if j = i; j = 2m − 1, βα , if j = i; j = 2m, 3
√ 1 2βα4 cos 3 bkx /2 − π m − [N] + 3 Φ , if j = i + 1; j = 2m, √
2βα4 cos 3 bkx /2 − π(m − [N])Φ , = if j = i − 1; j = 2m − 1, βα 4 , if j = i + 1; j = 2m + 1, βα , if j = i + 3; j = 2m, 4
√ 1 2βα3 cos 3 bkx /2 − π m − [N] + 6 Φ , if j = i + 2; j = 2m, (4) 0, others, where β = 2 cos(kz Ic ). There is a simple relation between the two off-diagonal block matrices: (Hij )2,1 = (Hj i )1,2 . E c,v , energy dispersion, in unit of γ0 is obtained by solving the Hamiltonian matrix equations, where the superscript c (v) represents the unoccupied π ∗ band (the occupied π band). The values of γ0 and αi ’s are taken from those of graphite [31]: γ0 = 2.598 eV, α1 = 0.364 eV, α2 = −0.014 eV, α3 = 0.319 eV, α4 = 0.177 eV, α5 = 0.036 eV, and α6 = −0.026 eV. The Ny = 5 AB-stacked zigzag ribbons are chosen for a model study. Energy dispersions at Φ = 0 are shown in Fig. 2(a). The π -electronic structure along W Z is the same as that of a single ribbon because Eq. (4) vanishes at kz Ic = π/2 (β = 0). There are parabolic bands and low-energy partial flat bands. Energy dispersions of E v < 0 are symmetric to those of E c > 0 about EF = 0. Along ZΓ (β ranging from 0 to 2), the interribbon interactions cause band crossing, and change energy dispersions, energy spacing, and band width. The difference between energy dispersions along Γ X and those along ZW is mainly due to the interribbon interactions. From W to X (γm = 0 in Eq. (2) and β ranging from 0 to 2), the interribbon interactions split the energy dispersions near ±γ0 . The 1B1 atoms through the interchain interaction γ0 (the interribbon interaction βα4 ) interact with the 1A2 (2B1 ) atoms. Moreover, the 1A2 atoms interact with the 2A2 atoms by the interribbon interaction βα1 . The interribbon interactions 2α1 and 2α4 make the degen-
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Fig. 2. The electronic structures of the Ny = 5 AB-stacked ribbons at (a) Φ = 0 and (b) Φ = 0.25. The unit of Φ is Φ0 .
eracy of state energies E c,v = ±γ0 at W be destroyed at X. The interribbon interactions also lift off the degeneracy of the low-energy partial flat bands. The state energies associated with the 1A1 (1B5 ) atoms are close to the Fermi level because the effective atom–atom interactions of these atoms with its neighbors are very weak (α2 , α5 ; α6 ). The different interribbon interactions would split the state energies along XW . There exist two doubly degenerate energy bands, and the energy difference between them is maximum at X and minimum at W . The state energies at W are α6 and 0. If the chemical-shift α6 ≈ 0 is neglected, there are four degenerate states with zero energy. The corresponding eigenvectors are φ1A1 and φ2B1 (φ1B5 and φ2A5 ), where φAm (φBm ) is the π orbital at A (B) site in the mth zigzag chain. They are the π orbitals located at the cusp positions of the outmost zigzag chains. These four states are called “edge states”. In Fig. 2(b), we show energy dispersions at Φ = 1/4. They are strongly modified by the magnetic
flux. Along W Z (β = 0), electronic structures of the two independent ribbons are determined by Eqs. (2) and (3), respectively. The symmetry of band structure about EF remains unchanged at any Φ. There are two set of energy bands out of phase, mainly owing to the phase difference πΦ/3 in Eq. (3). The magnetic field leads to the subband crossing. The almost flat Landau levels are formed at high energy. They can also be called the partial flat bands. However, electrons are localized not only at the outmost zigzag positions. Such localized states are nonedge states. There are oscillational Landau subbands at low energy. Zero √ energy points appear at 3 bkx /2 = 0, π/4, and π/2, respectively. √ They are determined by the zeros of γm (or cos{ 3 bkx /2 − π(m − [N])Φm }√= 0 in Eq. (2)). The oscillation period is defined as 3 b(kx , where the wavevectors of two (kx is the difference between√ adjacent zero energy points. 3 b(kx /2 is equal to πΦ. The period is equal to π/2 for Φ = 1/4. Another ribbon exhibits zero energy points at π/12 and π/3. The localized states corresponding to the zero energy points deserve a closer investigation. At Z (kx = 0), γ1 = 0 and breaks the bonds between φ1A1 and its nearest neighbor φ1B1 The localized state is just the edge state φ1A1 . γ1 = 0 is equivalent to slit the ribbon along the first zigzag chain and thus produces new boundary. Another eigenvector corresponding to the boundary is Ψ = c1 φ1B1 + c2 φ1B2 + c3 φ1B3 + c4 φ1B4 + c5φ1B5 , where c1 = (γ2 · γ3 · γ4 · γ5 )/γ04 , c2 = −(γ3 · γ4 · γ5 )/γ03 , c3 = (γ4 · γ5 )/γ02 , c4 = −γ5 /γ0 , and√ c5 = 1. This is localized state is not a edge state. At 3 bkx /2 = π/4 (the middle point of W Z), the zero of γ2 closes the intrachain interaction in the second zigzag chain and thus cuts the ribbon along the second zigzag chain. The associated eigenvectors are Ψ = (γ0 /γ1 )·φ1A1 −φ1B1 and Ψ = c1 φ1B2 +c2 φ1B3 + c3 φ1B4 + c4 φ1B5 . They are not completely localized at the zigzag edges. Apparently, these localized states are not the same as the edge states related to φ1A1 and φ1B5 . At W , γ3 is equal to zero. The magnetic flux closes γ3 and divides one ribbon into two pieces along the third zigzag chain. The 10 × 10 Hamiltonian matrix is reduced to two identical 5 × 5 matrices. The localized states of the zero energy points are Ψ = c1 φ1A1 + c2 φ1A2 + c3 φ1A3 and Ψ = c1 φ1B5 + c2 φ1B4 + c3 φ1B3 , where c1 = γ02 /(γ1 · γ2 ), c2 = −γ0 /γ2 and c3 = 1. They do not belong to the edge states.
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Fig. 3. The Hofstadter butterflies for the Ny = 5 AB-stacked zigzag ribbons at Z, W , Γ , and X are shown in (a), (b), (c) and (d), respectively.
The energy dispersions are highly anisotropic. There are oscillational Landau subbands at low energy along Γ X, W Z, and Γ W , while Landau levels exist at low energy along ZΓ and XW . From W to X and at the neighborhood of W , the partial flat bands at EF are the localized states. The interribbon interactions split the partial flat bands from W to X. Along ZΓ , there also exist partial flat bands near to EF . They are localized states or edge states. Along Γ X (β = 2 in Eq. (4)), the interribbon interactions couple two nanographite ribbons. As a result, they could induce the differences in energy dispersions between W Z and Γ X, such as positions of zero energy points, band width, Landau levels, and Landau subbands near EF . The oscillation period of Landau subbands is absent. The magnetic-flux-dependent Hofstadter butterflies are useful in understanding magneto-electronic states. We first see those at Z (kx = 0 and β = 0 in Eqs. (2)– (4)). The interribbon interactions vanish, i.e., there are two independent ribbons. Both band width and energy spacing vary with magnetic flux. One Hofstadter
butterfly, as shown in Fig. 3(a) by the bold dots, is symmetric to Φ = 1/2. The zero energy points occur at magnetic flux Φm = ±1/4, ±1/2 and ±3/4. Φm ceases the intrachain interaction at the mth zigzag chain (γm = 0) and thus determines these zero energy points. At Φ1 (Φ5 ) = ±1/4 and ±3/4, γ1 (γ5 ) at the first (fifth) zigzag chain is vanishing. The zero of γ1 (γ5 ) isolates the φ1A1 (φ1B5 ) atom from its nearest neighboring atom φ1B1 (φ1A5 ). This is to say, the localized states are just the edge states. For Φ2 (Φ4 ) = 1/2, both γ2 and γ4 are equal to zero. γ2 (γ4 ) = 0 closes the intrachain interaction in the second (fourth) zigzag chain, which is equivalent to cut the ribbon along the second and the fourth zigzag chains at the same time. Therefore, the 10 × 10 Hamiltonian matrix is decomposed into two 3 × 3 matrices and one 4 × 4 matrix. √ degenerate eigenvalues √ The five doubly are 0, ± 5 γ0 and (1 ± 2 )γ0 . The localized states corresponding to zero energy points are√Ψ = (φ1A1 + √ 2φ1A2 )/ 5 and Ψ = (φ1B5 + 2φ1B4 )/ 5. Evidently, they are not the same as the edge states. Another Hofstadter butterfly is described by Eq. (3). They are
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asymmetric to Φ = 1/2, mainly due to the phase shift πΦ/3 in two different ribbons. cos{π(m − 3 + 1/3)Φm } = 0 in Eq. (3) determines zero energy points. Φ1 = 3/10(9/10), Φ4 = 3/8, and Φ5 = 3/14 make intrachain interactions γ1 , γ4 , and γ5 vanish at m = 1, 4, and 5 zigzag chains, respectively. The localized states associated with zero energy points are not the edge states. For example, at Φ4 = 3/8, the eigenvectors are Ψ = (γ5 /γ0 )φ2B4 − φ2B5 and Ψ = c1 φ2A1 + c2 φ2A2 + c3 φ2A3 + c4 φ2A4 , where c1 = −γ03 /(γ1 γ2 γ3 ), c2 = γ02 /(γ2 γ3 ), c3 = −γ0 /γ3 , and c4 = 1. √ At W ( 3 bkx /2 = π/2) and (β = 0), two ribbons are effectively decoupled. One exhibits the nondispersive Hofstadter butterfly at EF , as shown in Fig. 3(b) by the bold dots. There are five doubly degenerate energy levels, which are symmetric to Φ = 1/2. Band width and energy spacing vary with magnetic flux. The intrachain interaction γ3 is always equal to zero at any Φ. The 10 × 10 Hamiltonian matrix is changed into two identical √ 5 × 5 matrices. The five eigenvalues are 0, ±(a ± b/2)1/2 , a = (γ12 + γ22 )/2 + γ02 and b = (γ12 − γ22 )2 + 4γ22 γ02 , where γ1 (γ2 ) is the intrachain interaction at the first (second) zigzag chain. The eigenvalues are strongly dependent on the magnetic flux except E c,v = 0. The localized states of the zero energy point are Ψ = c1 φ1A1 + c2 φ1A2 + c3 φ1A3 and Ψ = c1 φ1B5 + c2 φ1B4 + c3 φ1B3 , where c1 = 1, c2 = −γ1 /γ0 and c3 = (γ1 γ2 )/γ02 . Also noticed that only at integral flux Φ = 0, ±1, . . . , the localized states are just the edge states. That the intrachain interactions γ1 and γ2 are equal to zero is the main reason. Another ribbon can exhibit the oscillational Hofstadter butterfly at low energy. The zero energy point is determined by the condition sin{π(m − 3 + 1/3)Φm } = 0 (Eq. (3)). The magnetic flux Φm = ±1/(m − 3 + 1/3), ±2/(m − 3 + 1/3), . . . , i.e., Φ1 = 3/5, Φ3 = 0, Φ4 = 3/4, and Φ5 = 3/7 and 6/7. The associated localized states are nonedge states except at Φ = 0. For example, at Φ4 = 3/4, the corresponding eigenvectors are√Φ = c1 φ2A1 + c2√ φ2A2 + c3 φ2A3 + c4 φ2A4 and Ψ = ( 2 φ2B4 + φ2B5 )/ 3. In short, localized states are nonedge states except that the intrachain interaction vanishes at the outmost zigzag chains. Finally, we study the effect of interribbon interactions on the Hofstadter butterfly. At Γ , the interribbon interactions are maximum. Comparing Fig. 3(c)
with Fig. 3(a), they destroy the state degeneracy at Φ < 0.15 and make the asymmetry of electronic states about Φ = 1/2 more serious. The band width decreases in the increasing of Φ, since the interribbon interactions are suppressed by it. However, the interribbon interactions hardly affect Φm and deeper occupied states (E v < −2γ0). The Hofstadter butterfly at X (Fig. 3(d)) is also compared with that at W (Fig. 3(b)). The symmetry of the bold Hofstadter butterfly in Fig. 3(b) is destroyed. Except at low magnetic field, the nondispersive Hofstadter butterfly is absent in the presence of the interribbon interactions. It is replaced by the oscillational Landau subbands. The zero energy points are at Φ = 0, 3/7, 0.5, 3/5, 3/4, and 6/7. Except Φ = 0.5, they are the same as those of the light Hofstadter butterfly in Fig. 3(b). In summary, we employ the tight-binding model to calculate magnetoband structures of AB-stacked zigzag nanographite ribbons. The magnetic field can induce Landau subbands and Landau levels, and change energy spacing and band width. The magnetoband structures are strongly anisotropic. At low energy, there are Landau levels along ZΓ and XW , while oscillational Landau subbands occur along Γ X, W Z, and Γ W . Magnetic field causes many zero energy points. In the case of the AB-stacked zigzag nanographite ribbons along W Z, zero energy points and associated localized states can be analytically obtained. It is very helpful for calculating the oscillation period of Landau subbands at low energy and clarifying the relation between localized states and edge states. Localized states differ from edge states except that the intrachain interaction is turned off at the outmost zigzag chain. The interribbon interactions significantly affect energy dispersions and band width. They also destroy the symmetry of the Hofstadter butterfly and the nondispersive Hofstadter butterfly.
Acknowledgements
The authors gratefully acknowledge the support of the Taiwan National Science Council under the contract numbers NSC 89-2112-M-268-002 and NCS 90-2112-M-006-020.
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Magnetoband structures of AB-stacked zigzag nanographite ribbons C.P. Chang a,b,∗ , C.W. Chiu b , F.L. Shyu c , R.B. Chen d , M.F. Lin b a Center for General Education, Tainan Woman’s College of Arts & Technology, 701 Tainan, Taiwan b Department of Physics, National Cheng Kung University, 701 Tainan, Taiwan c Department of Physics, Chinese Military Academy Kaohsiung, 830 Kaohsiung, Taiwan d Department of Electrical Engineering, Cheng Shiu Institute of Technology, 830 Kaohsiung, Taiwan
Received 11 October 2002; accepted 26 October 2002 Communicated by V.M. Agranovich
Abstract Magnetoband structures of AB-stacked zigzag nanographite ribbons are studied by the tight-binding model. The magnetic field changes band width, energy space, and energy dispersions (the produce of Landau subbands and Landau levels). It causes many zero energy points. Such points and corresponding localized states are studied in detail. There are certain important differences between localized states and edge states. Oscillation period of Landau subbands are determined by these points. The interribbon interactions also affect magnetoband structures, such as energy dispersions, band width, oscillation period of Landau subbands, and flux dependence of Hofstadter butterflies. 2002 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx; 71.15.Fv Keywords: Band structures; Nanographite ribbons; Hofstadter butterfly
Electronic properties of nanographite ribbons can be modified by changing width, edge shape, stacking sequence, and magnetic field. A one-dimensional nanographite monoribbon is obtained by cutting a twodimensional graphite sheet along the longitudinal direction (x). ˆ Band structure is dominated by geometric structures [1–10], which, thus, stimulates many studies, such as magnetic properties [11,12], optical
* Corresponding author.
E-mail address: [email protected] (C.P. Chang).
properties [13], electronic excitations [14], transport properties [15], and growth [16–18]. A zigzag ribbon, with zigzag structure along the x-axis, has very special band structure. The partial flat bands at the Fermi level result from the nonbonding edge states located at the outmost zigzag edges. They are expected to play a very important role on physical properties [11–15]. Wakabayashi et al. [11] had used the tight-binding model to study electronic and magnetic properties of nanographite monoribbon. The magnetic susceptibility is strongly dependent on edge shape and ribbon width. Especially, the Pauli paramagnetism originat-
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 1 5 7 8 - 5
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ing from the edge states of the zigzag ribbons can significantly contribute to the magnetic susceptibility. The relation between the edge states and the localized states induced by the magnetic flux has not been investigated in their study. On the other hand, Harigaya had studied the mechanism on the magnetism in the AA- and AB-stacked nanographite ribbons [12]. They found that the electronic spins localized in the open shell of the AB-stacked zigzag ribbons is the favorable condition for the magnetism. However, the effects of the interribbon interactions on the edge states are neglected. Energy spectra of Bloch electrons in the magnetic field have been widely studied in 2D square lattices [19–21], triangle lattices [21–23], hexagonal lattices [24–26], artificial structures [27], 3D cubic lattices [28,29], and 1D square lattices [30]. Such lattices only belong to hypertheoretical systems. Here, we study magnetoband structures of AB-stacked zigzag nanographite ribbons. This study shows that the magnetic field causes a lot of zero energy points. Such points are identified to be associated with localized states. The differences between localized states and edge states are examined. Magnetic field produces oscillational Landau subbands at low energy. Their oscillation periods are determined by zero energy points. The effects due to the interribbon interactions are also investigated. By incorporating the Peierl phase ei2πθij into the hopping integrals, we can couple a uniform magnetic field B, parallel to the z-axis, to electronic structures of nanographite ribbons in the tight-binding model. The Hamiltonian is given by γi,j ei2πθij ci+ cj + H.c. , H= (1) i,j
where the subindices i, j denote the summation over all sites and γi,j is the hopping integral. ci+ (cj ) is the creation (annihilation) operation, which generates (destroys) an electron at i (j ) site, and ei2πθij is the Peierl phase caused by the external field. θi,j is the line integral of the vector potential A from i to j in unit of j flux quantum Φ0 = ch/e, i.e., θi,j = (1/Φ0 ) i A dl. With the Landau gauge, the position-dependent vector potential is A = (−By, 0, 0). The geometric structure of the AB-stacked zigzag nanographite ribbons is shown in Fig. 1. Each ribbon consists of the benzene-ring carbon atoms. The
Fig. 1. The geometric structure of the Ny = 5 AB-stacked zigzag nanographite ribbons. γ0 and αi ’s indicate the atom–atom interactions.
nearest-neighbor C–C bond length is b = 1.42 √Å. The periodical length along the x-axis is Ix = 3 b for the zigzag chains, where B atoms are the nearestneighbors of A atoms. The width is defined by the number of the zigzag chains (Ny ) along the y-axis. The identical zigzag ribbons are periodically arranged in the AB sequence. A atoms are in a straight line along the z-axis, while B atoms form a zigzag texture in the y–z plane. That will make the local environment of A atoms different from that of B atoms. This difference, denoted as the chemical-shift α6 , will be taken into account in the band-structure calculations. The AB-stacked nanographite ribbons have two ribbons or 4Ny carbon atoms in a 2D primitive cell. The first Brillouin zone is a rectangle defined by −π/Ix kx π/Ix and −π/2Ic kz π/2Ic . Ic (= 3.35 Å) is the periodic distance between two neighboring ribbons. The atom–atom interactions are as follows. γ0 is the interaction between A atom and B atom on the same ribbon. α1 (α3 ) represents the interaction between two A (B) atoms from two neighboring ribbons. α5 (α2 ) is that for two A (B) atoms from two nextneighboring ribbons. α4 corresponds to the interaction between A atom and B atom, from two neighboring ribbons.
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Under the magnetic field, the AB-stacked zigzag ribbons still own two ribbons in a 2D primitive cell. The Hamiltonian of ribbons is a 4Ny × 4Ny Hermitian matrix. It can be arranged as 2 × 2 blocks. Each block matrix is a 2Ny × 2Ny matrix. The outmost zigzag chains of the two neighboring ribbons are out of line along the x-axis, so the center of each ribbon will not coincide with each other. When we choose the coordinate origin at the geometric center of one ribbon in calculating the Peierl phase θi,j , the diagonal matrix (Hij )1,1 differers from (Hij )2,2 . The two diagonal matrices are, respectively, α6 + β 2 α5 /2, if j = i; j = 2m − 1, β 2 α2 /2, if j = i; j = 2m, √
2γ0 cos 3 bkx /2 − π(m − [N])Φ , (Hij )1,1 = if j = i + 1; j = 2m, γ0 , if j = i + 1; j = 2m + 1, (2) 0, others, and (Hij )2,2 α + β 2 α5 /2, if j = i; j = 2m − 1, 6 2 2m, β α2 /2, √ if j = i; j =
2γ0 cos 3 bkx /2 − π m − [N] + 13 Φ , = if j = i + 1; j = 2m, γ0 , if j = i + 1; j = 2m + 1, (3) 0, others, where m labels the mth zigzag chain along the y-axis in each ribbon, and Φ in unit of Φ0 is the magnetic flux through the benzene ring. The difference between the centers of two ribbons gives rise to πΦ/3 in the cosine term in Eq. (3). [N] is defined as if Ny is even, Ny /2, [N] = (Ny + 1)/2, if Ny is odd. √ We can treat γ0 (γm = 2γ0 cos{ 3 bkx /2 − π(m − [N])Φ}) in Eq. (2) as the interchain (intrachain) interactions of zigzag chains on the same ribbon. γm is dependent on the wavevector kx , the ribbon width Ny , the position of zigzag chain m, and the magnetic flux Φ. The interaction between the atoms at the mth zigzag chain is switched off by the zero of γm and the interesting results thus emerge.
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The off-diagonal block matrix, which describes the interribbon interactions, is (Hij )1,2 βα1 , if j = i; j = 2m − 1, βα , if j = i; j = 2m, 3
√ 1 2βα4 cos 3 bkx /2 − π m − [N] + 3 Φ , if j = i + 1; j = 2m, √
2βα4 cos 3 bkx /2 − π(m − [N])Φ , = if j = i − 1; j = 2m − 1, βα 4 , if j = i + 1; j = 2m + 1, βα , if j = i + 3; j = 2m, 4
√ 1 2βα3 cos 3 bkx /2 − π m − [N] + 6 Φ , if j = i + 2; j = 2m, (4) 0, others, where β = 2 cos(kz Ic ). There is a simple relation between the two off-diagonal block matrices: (Hij )2,1 = (Hj i )1,2 . E c,v , energy dispersion, in unit of γ0 is obtained by solving the Hamiltonian matrix equations, where the superscript c (v) represents the unoccupied π ∗ band (the occupied π band). The values of γ0 and αi ’s are taken from those of graphite [31]: γ0 = 2.598 eV, α1 = 0.364 eV, α2 = −0.014 eV, α3 = 0.319 eV, α4 = 0.177 eV, α5 = 0.036 eV, and α6 = −0.026 eV. The Ny = 5 AB-stacked zigzag ribbons are chosen for a model study. Energy dispersions at Φ = 0 are shown in Fig. 2(a). The π -electronic structure along W Z is the same as that of a single ribbon because Eq. (4) vanishes at kz Ic = π/2 (β = 0). There are parabolic bands and low-energy partial flat bands. Energy dispersions of E v < 0 are symmetric to those of E c > 0 about EF = 0. Along ZΓ (β ranging from 0 to 2), the interribbon interactions cause band crossing, and change energy dispersions, energy spacing, and band width. The difference between energy dispersions along Γ X and those along ZW is mainly due to the interribbon interactions. From W to X (γm = 0 in Eq. (2) and β ranging from 0 to 2), the interribbon interactions split the energy dispersions near ±γ0 . The 1B1 atoms through the interchain interaction γ0 (the interribbon interaction βα4 ) interact with the 1A2 (2B1 ) atoms. Moreover, the 1A2 atoms interact with the 2A2 atoms by the interribbon interaction βα1 . The interribbon interactions 2α1 and 2α4 make the degen-
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Fig. 2. The electronic structures of the Ny = 5 AB-stacked ribbons at (a) Φ = 0 and (b) Φ = 0.25. The unit of Φ is Φ0 .
eracy of state energies E c,v = ±γ0 at W be destroyed at X. The interribbon interactions also lift off the degeneracy of the low-energy partial flat bands. The state energies associated with the 1A1 (1B5 ) atoms are close to the Fermi level because the effective atom–atom interactions of these atoms with its neighbors are very weak (α2 , α5 ; α6 ). The different interribbon interactions would split the state energies along XW . There exist two doubly degenerate energy bands, and the energy difference between them is maximum at X and minimum at W . The state energies at W are α6 and 0. If the chemical-shift α6 ≈ 0 is neglected, there are four degenerate states with zero energy. The corresponding eigenvectors are φ1A1 and φ2B1 (φ1B5 and φ2A5 ), where φAm (φBm ) is the π orbital at A (B) site in the mth zigzag chain. They are the π orbitals located at the cusp positions of the outmost zigzag chains. These four states are called “edge states”. In Fig. 2(b), we show energy dispersions at Φ = 1/4. They are strongly modified by the magnetic
flux. Along W Z (β = 0), electronic structures of the two independent ribbons are determined by Eqs. (2) and (3), respectively. The symmetry of band structure about EF remains unchanged at any Φ. There are two set of energy bands out of phase, mainly owing to the phase difference πΦ/3 in Eq. (3). The magnetic field leads to the subband crossing. The almost flat Landau levels are formed at high energy. They can also be called the partial flat bands. However, electrons are localized not only at the outmost zigzag positions. Such localized states are nonedge states. There are oscillational Landau subbands at low energy. Zero √ energy points appear at 3 bkx /2 = 0, π/4, and π/2, respectively. √ They are determined by the zeros of γm (or cos{ 3 bkx /2 − π(m − [N])Φm }√= 0 in Eq. (2)). The oscillation period is defined as 3 b(kx , where the wavevectors of two (kx is the difference between√ adjacent zero energy points. 3 b(kx /2 is equal to πΦ. The period is equal to π/2 for Φ = 1/4. Another ribbon exhibits zero energy points at π/12 and π/3. The localized states corresponding to the zero energy points deserve a closer investigation. At Z (kx = 0), γ1 = 0 and breaks the bonds between φ1A1 and its nearest neighbor φ1B1 The localized state is just the edge state φ1A1 . γ1 = 0 is equivalent to slit the ribbon along the first zigzag chain and thus produces new boundary. Another eigenvector corresponding to the boundary is Ψ = c1 φ1B1 + c2 φ1B2 + c3 φ1B3 + c4 φ1B4 + c5φ1B5 , where c1 = (γ2 · γ3 · γ4 · γ5 )/γ04 , c2 = −(γ3 · γ4 · γ5 )/γ03 , c3 = (γ4 · γ5 )/γ02 , c4 = −γ5 /γ0 , and√ c5 = 1. This is localized state is not a edge state. At 3 bkx /2 = π/4 (the middle point of W Z), the zero of γ2 closes the intrachain interaction in the second zigzag chain and thus cuts the ribbon along the second zigzag chain. The associated eigenvectors are Ψ = (γ0 /γ1 )·φ1A1 −φ1B1 and Ψ = c1 φ1B2 +c2 φ1B3 + c3 φ1B4 + c4 φ1B5 . They are not completely localized at the zigzag edges. Apparently, these localized states are not the same as the edge states related to φ1A1 and φ1B5 . At W , γ3 is equal to zero. The magnetic flux closes γ3 and divides one ribbon into two pieces along the third zigzag chain. The 10 × 10 Hamiltonian matrix is reduced to two identical 5 × 5 matrices. The localized states of the zero energy points are Ψ = c1 φ1A1 + c2 φ1A2 + c3 φ1A3 and Ψ = c1 φ1B5 + c2 φ1B4 + c3 φ1B3 , where c1 = γ02 /(γ1 · γ2 ), c2 = −γ0 /γ2 and c3 = 1. They do not belong to the edge states.
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Fig. 3. The Hofstadter butterflies for the Ny = 5 AB-stacked zigzag ribbons at Z, W , Γ , and X are shown in (a), (b), (c) and (d), respectively.
The energy dispersions are highly anisotropic. There are oscillational Landau subbands at low energy along Γ X, W Z, and Γ W , while Landau levels exist at low energy along ZΓ and XW . From W to X and at the neighborhood of W , the partial flat bands at EF are the localized states. The interribbon interactions split the partial flat bands from W to X. Along ZΓ , there also exist partial flat bands near to EF . They are localized states or edge states. Along Γ X (β = 2 in Eq. (4)), the interribbon interactions couple two nanographite ribbons. As a result, they could induce the differences in energy dispersions between W Z and Γ X, such as positions of zero energy points, band width, Landau levels, and Landau subbands near EF . The oscillation period of Landau subbands is absent. The magnetic-flux-dependent Hofstadter butterflies are useful in understanding magneto-electronic states. We first see those at Z (kx = 0 and β = 0 in Eqs. (2)– (4)). The interribbon interactions vanish, i.e., there are two independent ribbons. Both band width and energy spacing vary with magnetic flux. One Hofstadter
butterfly, as shown in Fig. 3(a) by the bold dots, is symmetric to Φ = 1/2. The zero energy points occur at magnetic flux Φm = ±1/4, ±1/2 and ±3/4. Φm ceases the intrachain interaction at the mth zigzag chain (γm = 0) and thus determines these zero energy points. At Φ1 (Φ5 ) = ±1/4 and ±3/4, γ1 (γ5 ) at the first (fifth) zigzag chain is vanishing. The zero of γ1 (γ5 ) isolates the φ1A1 (φ1B5 ) atom from its nearest neighboring atom φ1B1 (φ1A5 ). This is to say, the localized states are just the edge states. For Φ2 (Φ4 ) = 1/2, both γ2 and γ4 are equal to zero. γ2 (γ4 ) = 0 closes the intrachain interaction in the second (fourth) zigzag chain, which is equivalent to cut the ribbon along the second and the fourth zigzag chains at the same time. Therefore, the 10 × 10 Hamiltonian matrix is decomposed into two 3 × 3 matrices and one 4 × 4 matrix. √ degenerate eigenvalues √ The five doubly are 0, ± 5 γ0 and (1 ± 2 )γ0 . The localized states corresponding to zero energy points are√Ψ = (φ1A1 + √ 2φ1A2 )/ 5 and Ψ = (φ1B5 + 2φ1B4 )/ 5. Evidently, they are not the same as the edge states. Another Hofstadter butterfly is described by Eq. (3). They are
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asymmetric to Φ = 1/2, mainly due to the phase shift πΦ/3 in two different ribbons. cos{π(m − 3 + 1/3)Φm } = 0 in Eq. (3) determines zero energy points. Φ1 = 3/10(9/10), Φ4 = 3/8, and Φ5 = 3/14 make intrachain interactions γ1 , γ4 , and γ5 vanish at m = 1, 4, and 5 zigzag chains, respectively. The localized states associated with zero energy points are not the edge states. For example, at Φ4 = 3/8, the eigenvectors are Ψ = (γ5 /γ0 )φ2B4 − φ2B5 and Ψ = c1 φ2A1 + c2 φ2A2 + c3 φ2A3 + c4 φ2A4 , where c1 = −γ03 /(γ1 γ2 γ3 ), c2 = γ02 /(γ2 γ3 ), c3 = −γ0 /γ3 , and c4 = 1. √ At W ( 3 bkx /2 = π/2) and (β = 0), two ribbons are effectively decoupled. One exhibits the nondispersive Hofstadter butterfly at EF , as shown in Fig. 3(b) by the bold dots. There are five doubly degenerate energy levels, which are symmetric to Φ = 1/2. Band width and energy spacing vary with magnetic flux. The intrachain interaction γ3 is always equal to zero at any Φ. The 10 × 10 Hamiltonian matrix is changed into two identical √ 5 × 5 matrices. The five eigenvalues are 0, ±(a ± b/2)1/2 , a = (γ12 + γ22 )/2 + γ02 and b = (γ12 − γ22 )2 + 4γ22 γ02 , where γ1 (γ2 ) is the intrachain interaction at the first (second) zigzag chain. The eigenvalues are strongly dependent on the magnetic flux except E c,v = 0. The localized states of the zero energy point are Ψ = c1 φ1A1 + c2 φ1A2 + c3 φ1A3 and Ψ = c1 φ1B5 + c2 φ1B4 + c3 φ1B3 , where c1 = 1, c2 = −γ1 /γ0 and c3 = (γ1 γ2 )/γ02 . Also noticed that only at integral flux Φ = 0, ±1, . . . , the localized states are just the edge states. That the intrachain interactions γ1 and γ2 are equal to zero is the main reason. Another ribbon can exhibit the oscillational Hofstadter butterfly at low energy. The zero energy point is determined by the condition sin{π(m − 3 + 1/3)Φm } = 0 (Eq. (3)). The magnetic flux Φm = ±1/(m − 3 + 1/3), ±2/(m − 3 + 1/3), . . . , i.e., Φ1 = 3/5, Φ3 = 0, Φ4 = 3/4, and Φ5 = 3/7 and 6/7. The associated localized states are nonedge states except at Φ = 0. For example, at Φ4 = 3/4, the corresponding eigenvectors are√Φ = c1 φ2A1 + c2√ φ2A2 + c3 φ2A3 + c4 φ2A4 and Ψ = ( 2 φ2B4 + φ2B5 )/ 3. In short, localized states are nonedge states except that the intrachain interaction vanishes at the outmost zigzag chains. Finally, we study the effect of interribbon interactions on the Hofstadter butterfly. At Γ , the interribbon interactions are maximum. Comparing Fig. 3(c)
with Fig. 3(a), they destroy the state degeneracy at Φ < 0.15 and make the asymmetry of electronic states about Φ = 1/2 more serious. The band width decreases in the increasing of Φ, since the interribbon interactions are suppressed by it. However, the interribbon interactions hardly affect Φm and deeper occupied states (E v < −2γ0). The Hofstadter butterfly at X (Fig. 3(d)) is also compared with that at W (Fig. 3(b)). The symmetry of the bold Hofstadter butterfly in Fig. 3(b) is destroyed. Except at low magnetic field, the nondispersive Hofstadter butterfly is absent in the presence of the interribbon interactions. It is replaced by the oscillational Landau subbands. The zero energy points are at Φ = 0, 3/7, 0.5, 3/5, 3/4, and 6/7. Except Φ = 0.5, they are the same as those of the light Hofstadter butterfly in Fig. 3(b). In summary, we employ the tight-binding model to calculate magnetoband structures of AB-stacked zigzag nanographite ribbons. The magnetic field can induce Landau subbands and Landau levels, and change energy spacing and band width. The magnetoband structures are strongly anisotropic. At low energy, there are Landau levels along ZΓ and XW , while oscillational Landau subbands occur along Γ X, W Z, and Γ W . Magnetic field causes many zero energy points. In the case of the AB-stacked zigzag nanographite ribbons along W Z, zero energy points and associated localized states can be analytically obtained. It is very helpful for calculating the oscillation period of Landau subbands at low energy and clarifying the relation between localized states and edge states. Localized states differ from edge states except that the intrachain interaction is turned off at the outmost zigzag chain. The interribbon interactions significantly affect energy dispersions and band width. They also destroy the symmetry of the Hofstadter butterfly and the nondispersive Hofstadter butterfly.
Acknowledgements
The authors gratefully acknowledge the support of the Taiwan National Science Council under the contract numbers NSC 89-2112-M-268-002 and NCS 90-2112-M-006-020.
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