- Email: [email protected]
Spectral properties of multi-layered graphene in a magnetic field
Superlattices and Microstructures 86 (2015) 68–72
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Spectral properties of multi-layered graphene in a magnetic field E.N. Grishanov a, I.Yu. Popov b,⇑ a b
Department of Mathematics and IT, Ogarev Mordovia State University, Bolshevistskaya Str. 68, Saransk, Russia Department of Higher Mathematics, ITMO University, Kroverkskiy pr. 49, St. Petersburg 197101, Russia
a r t i c l e
i n f o
Article history: Received 9 July 2015 Accepted 10 July 2015 Available online 11 July 2015 Keywords: Graphene Hofstadter butterfly Spectrum
a b s t r a c t Solvable model of multi-layered graphene in a homogeneous magnetic field is constructed. The spectrum of the Hamiltonian has Hofstadter butterfly type. The comparison of the spectrum for single-layer graphene with that for two-layered and three-layered systems is made. Ó 2015 Published by Elsevier Ltd.
1. Introduction Starting from famous Hofstadter paper [1], fractal structure of the spectrum of nanosystems in a magnetic field attracts great attention both theoreticians and experimentalists. It is inspired by the peculiarities of the electron transport in such systems. Experimental observation of this effect in natural systems has remained elusive because the required experimental magnetic field is too high (approximately, 105 T) for a typical lattice constant of 0.1 nm. One way to circumvent this difficulty was to construct artificial superlattice to increase the lattice constant [2]. Particularly, one can find this type of the spectrum when dealing with periodic arrays of quantum dots [3–11]. One interesting nanostructure more is graphene. It demonstrates many intriguing properties [12]. We are interested in the spectral properties of electron in graphene in a magnetic field. It is related with the quantum Hall effect [13]. Graphene is two-dimensional (2D) Dirac-like electronic system. Due to this reason, its anomalous quantum Hall effect could be measured [14,15]. To calculate the spectrum for various magnetic field, one uses usually tight-binding approximation (see, e.g., [7]) or zero-range potentials model (see, e.g., [16,17]). The next interesting question is related with a comparison of the spectral results for single-layer graphene and multi-layered graphene structure. In bilayer graphene, the two atomic layers can be stacked together in many different ways, and even be twisted [8,18]. Of course, it has an influence on the spectrum [19,20]. An effect of number of layers is also important. In the present paper we describe the Hofstadter butterfly type structure of the spectrum for one, two and three graphene layers. We deal with layers stacked in the most natural way - the Bernal stacking configuration (see Fig. 1). 2. Model construction The starting point of the construction is the standard Hamiltonian H0 of free spinless charged particle (of mass m and charge e) in a homogeneous magnetic field B (Landau operator): ⇑ Corresponding author. E-mail address: [email protected] (I.Yu. Popov). http://dx.doi.org/10.1016/j.spmi.2015.07.025 0749-6036/Ó 2015 Published by Elsevier Ltd.
E.N. Grishanov, I.Yu. Popov / Superlattices and Microstructures 86 (2015) 68–72
69
Fig. 1. Geometric structure of bilayer graphene. The Bernal stacking configuration.
H0 ¼
2 h e p AðrÞ : c 2m
ð1Þ
Here p ¼ i hr is the momentum operator in R3 ; AðrÞ ¼ 12 B r is the vector potential of the magnetic field B (the symmetric gauge is chosen). It is assumed that the magnetic field is orthogonal to the graphene sheet plane. Below, the following notahxð‘ þ 1=2Þ; ‘ ¼ 0; 1; . . . are the Landau levels of the operator H0 tions are used: x ¼ jeBj=2c is the cyclotron frequency, e‘ ¼ (1), U0 ¼ 2p hc=jej is the magnetic flux quantum, n ¼ B=U0 is the density of the magnetic flux. It is well-known that operator H0 (1) has the continuous spectrum occupying the half-axis ½e0 ; 1Þ. More precisely, it consists of infinite number of branches: 1 [
½e‘ ; 1Þ
‘¼0
Let a be the interatomic distance in single-layer graphene. In real physical system one has a ¼ 0:142 nm (see, e.g., [8]). pffiffi pffiffiffi Vectors a1 ¼ ð 3a; 0; 0Þ and a2 ¼ 23 a; 32 a; 0 are the basic vectors of the Bravais lattice K for graphene with the basic cell
n o C K ¼ r 2 R3 : 0 6 r i 6 ai ði ¼ 1; 2Þ; r 3 2 R : The set of nodes (atoms) in the basic cell is marked as K 2 C K . The set K contains two elements for single-layer graphene, four elements for bilayer graphene, six elements for three-layered graphene, etc. Then, the set
C ¼ K þ K ¼ fj þ k : j 2 K; k 2 Kg is the set of all nodes of periodic system with the lattice K. As for real graphene samples, the model does not describe correctly the sample edge region. Fig. 1 shows the lattice structure of bilayer graphene. It corresponds to the most natural physical situation [21]. Let g ðg ¼ nða1 a2 ÞÞ be the number of the magnetic flux quanta through the basic cell of lattice K. The model Hamiltonian for spinless electron in our system is constructed as a perturbation of the operator (1) by zero-range potentials posed at the nodes of lattice K. Formally, the model operator can be written as
X ^ dðr cÞ; H ¼ H0 þ a
ð2Þ
c2C
^ is a real constant which characterizes the strength of the point-like potential. Correct mathematical description of where a the operator (2) is given in the framework of the theory of self-adjoint extensions of symmetric operators [17,22–26]. Let us describe the procedure. Let D be the following set
D ¼ ff 2 DðH0 Þ : f ðcÞ ¼ 0 8c 2 Cg: The operator S is the restriction of the operator H0 on D. It is a symmetric operator. The model Hamiltonian H is a self-adjoint extension of symmetric operator S. It can be described by the Krein resolvent formula (see, e.g., [27,28]). Let RðfÞ ¼ ðH fÞ1
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E.N. Grishanov, I.Yu. Popov / Superlattices and Microstructures 86 (2015) 68–72
and R0 ðfÞ ¼ ðH0 fÞ1 be the resolvents of the operators H and H0 , correspondingly. In accordance with the Krein formula, one has
RðfÞ ¼ R0 ðfÞ CðfÞ½QðfÞ A1 C ðfÞ;
ð3Þ
where CðfÞ; Q ðfÞ are C-field and Krein Q-function, correspondingly, A is a self-adjoint operator parameterizing the self-adjoint extensions of the operator S. The operator A acts in so-called space of boundary values (it is infinite-dimensional because the operator S has infinite deficiency indices). As the lattice of the system has identical nodes (there are identical atoms at all nodes), the operator A can be chosen in the following form: A ¼ aI, where I is the identical operator. The terms of formula (3) can be written explicitly. Let Gðr; r0 ; fÞ be the Green function of the operator H (i.e., the integral kernel of the resolvent RðfÞ) and G0 ðr; r0 ; fÞ be the Green function of the operator H0 . The explicit expression for G0 is well-known. It was obtained in [29]:
G0 ðr; r0 ; fÞ ¼ Uðr; r0 ÞF 1 ðr r0 ; fÞ ¼ Uðr; r0 ÞF 2 ðr r0 ; fÞ; rffiffiffiffi h i m n 2 Uðr; r0 Þ ¼ 2 exp pinðr r0 Þ pnðr? r0? Þ =2 ; p 2h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X exp 4pnð‘ þ 1=2Þ fjrjj j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L‘ ðpnr2? Þ; F 1 ðr; fÞ ¼ ‘ þ 1=2 f=4pn ‘¼0 h i Z 1 exp pn r2 =ðet 1Þ þ r2 =t ? jj 1 dt pffiffi ; F 2 ðr; fÞ ¼ pffiffiffiffi p 0 ð1 et Þ exp ½ð1=2 f=4pnÞt t
ð4Þ
ð5Þ
where rjj is the projection of r on the direction of the magnetic field B; r? ¼ r rjj , and L‘ ðxÞ is the ‘-th Laguerre polynomial. The Q-function can be represented as infinite matrix Q ðfÞ ¼ ðQ ðc; c0 ÞÞc;c0 2C with the following entries
8 < G0 ðc; c0 ; fÞ; c – c0 ; qffiffiffi Q ðc; c0 ; fÞ ¼ : m2 pn Z 12 ; 12 4pf n ; h 2
c ¼ c0 :
ð6Þ
Here Zðs; v Þ is the Hurwitz f-function [30]. The matrix Q ðfÞ A (f 2 nR) gives one invertible bounded linear operator in the Hilbert space ‘2 ðCÞ. Below, we will not distinguish the notations of the operator and its matrix and will mark the both by ½Q ðfÞ A1 . Using the Krein formula (3), one obtains the explicit expression for the Green function G of the operator H (analogously to [29,31]):
Gðr; r0 ; fÞ ¼ G0 ðr; r0 ; fÞ
X
0 0 ½Q ðfÞ A1 c;c0 G0 ðr; c; fÞG0 ðc ; r ; fÞ:
ð7Þ
c;c0 2C
Thus, the construction is complete.
Fig. 2. Hofstadter butterfly (energy-flux diagram) of a single layer graphene. The energy is measured in eV, the unit of flux is the magnetic flux quantum.
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E.N. Grishanov, I.Yu. Popov / Superlattices and Microstructures 86 (2015) 68–72
3. Results and discussion It is known that the point f 2 R belongs to the spectrum of the operator H is the resolvent RðfÞ does not exist or exists but is not bounded. In the case of rational values of the magnetic flux number g ¼ N=M (N 2 Z; M 2 N) through the basic cell of the lattice K, the spectrum of the operator H can be found by expansion of the representation of the magnetic translation (by vectors of lattice K) group in respect to the spectrum of the irreducible subrepresentations [32]. These subrepresentations e ðp; fÞ are parameterized by points of two-dimensional torus T2 ¼ ½0; 1=MÞ ½0; 1Þ. For each point p; p 2 T2 , we mark as Q g
g
the matrix of size jKjM jKjM with the following entries
e ðp; m; j; m0 ; j0 ; fÞ ¼ exp½piðm m0 Þnðj a2 Þ Q exp
n
1 X
Q ðk1 a1 þ ðk2 M þ m m0 Þa2 þ j; j0 ; fÞ
k1 ;k2 ¼1
o g pin½j ðk1 a1 þ k2 Ma2 Þ 2pi kp þ k1 ðk2 M þ m þ m0 Þ ; 2
ð8Þ
where j; j0 2 K m; m0 ¼ 0; . . . ; M 1.
Fig. 3. Hofstadter butterfly (energy-flux diagram) of a bilayer graphene. Each layer is splitted into two sublayers. The energy is measured in eV, the unit of flux is the magnetic flux quantum.
Fig. 4. Hofstadter butterfly (energy-flux diagram) of a three-layer graphene. Each layer is splitted into three sublayers. The energy is measured in eV, the unit of flux is the magnetic flux quantum.
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The spectrum of the operator H consists of the spectrum of the operator H0 and roots of the dispersion equation
h i e ðp; fÞ a ¼ 0: det Q
ð9Þ
The spectral structure dependence on the magnetic field is shown in Figs. 2–4 (we does not show the unperturbed levels (the spectrum of the operator H0 ) in the figures). Three cases are considered: single-layer graphene, bilayer graphene, three-layer graphene. Of course, in all cases one observe the Hofstadter-type spectral structure. It can be noted that increasing of the magnetic field gives one some upward shift (as a trend) of the spectrum. Transition from single-layer system to multi-layered systems leads to additional splitting. One see that for n-layered structure an additional splitting of each level into n sublevels takes place. Our results are in good correlation with [18]. As for the results of [8], there is some difference. Our ‘‘energy – magnetic flux’’ diagrams do not contain intersections of subbands trajectories (if these subbands are induced by transition from single-layer to multi-layered structure). Such intersections exist in diagrams from [8] and do not exist in diagrams from [18]. Acknowledgements This work was partially financially supported by the Government of the Russian Federation (grant 074-U01), by the Ministry of Education and Science of the Russian Federation (GOSZADANIE 2014/190, Projects No. 14.Z50.31.0031 and No. 1.754.2014/K), by grant MK-5001.2015.1 of the President of the Russian Federation. References [1] D.R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14 (1976) 2239–2249. [2] C. Albrecht, J.H. Smet, K. von Klitzing, D. Weiss, V. Umansky, H. Schweizer, Evidence of Hofstadter’s fractal energy spectrum in the quantized Hall conductance, Phys. Rev. Lett. 86 (2001) 147–150. [3] V.A. Geyler, I.Yu. Popov, A.V. Popov, A.A. Ovechkina, Fractal spectrum of periodic quantum systems in a magnetic field, Chaos, Solitons Fract. 11 (2000) 281–288. [4] Y. Hasegawa, M. Kohmoto, Quantum Hall effect and the topological number in graphene, Phys. Rev. B 74 (2006) 155415. [5] J.G. Analytis, S.J. Blundell, A. Ardavan, Landau levels, molecular orbitals, and the Hofstadter butterfly in finite systems, Am. J. Phys. 72 (2004) 613–618. [6] R. Rammal, Landau level spectrum of Bloch electrons in a honeycomb lattice, J. Phys. 46 (1985) 1345–1354. [7] N. Nemec, G. Cuniberti, Hofstadter butterflies of carbon nanotubes: pseudofractality of the magnetoelectronic spectrum, Phys. Rev. B 74 (2006) 165411. [8] N. Nemec, G. Cuniberti, Hofstadter butterflies of bilayer graphene, Phys. Rev. B (Rapid Comm.) 75 (2007) 201404 (R). [9] D. Weiss, K. von Klitzing, K. Ploog, G. Weimann, Magnetoresistance oscillations in a twodimensional electron gas induced by a submicrometer periodic potential, Europhys. Lett. 8 (1989) 179–184. [10] R.R. Gerhardts, D. Weiss, U. Wulf, Magnetoresistance oscillations in a grid potential: Indication of a Hofstadter-type energy spectrum, Phys. Rev. B 43 (1991) 5192–5195. [11] V. Gudmundsson, R.R. Gerhardts, Manifestation of the Hofstadter butter in far-infrared absorption, Phys. Rev. B 54 (1996) 5223–5227. [12] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature 438 (2005) 197–200. [13] V.P. Gusynin, S.G. Sharapov, Unconventional integer quantum Hall effect in graphene, Phys. Rev. Lett. 95 (2005) 146801. [14] Y. Zhang, Y.-W. Tan, H.L. Stormer, P. Kim, Experimental observation of the quantum Hall effect and Berry’s phase in graphene, Nature 438 (2005) 201– 204. [15] E. McCann, V.I. Fal’ko, Landau-level degeneracy and quantum Hall effect in a graphite bilayer, Phys. Rev. Lett. 96 (2006) 086805. [16] J. Brüning, V.V. Demidov, V.A. Geyler, Hofstadter-type spectral diagrams for the Bloch electron in three dimensions, Phys. Rev. B 69 (2003) 033202. [17] V.A. Geyler, B.S. Pavlov, I.Yu. Popov, One-particle spectral problem for superlattice with a constant magnetic field, Atti. Sem. Mat. Fis. Univ. Modena 46 (1998) 79–124. [18] Z.F. Wang, F. Liu, M.Y. Chou, Fractal Landau-level spectra in twisted bilayer graphene, Nano Lett. 12 (2012) 3833–3838, http://dx.doi.org/10.1021/ nl301794t. [19] H. Min, A.H. MacDonald, Electronic structure of multilayer graphene, Prog. Theor. Phys. Suppl. 176 (2008) 227–252, http://dx.doi.org/10.1143/ PTPS.176.227. [20] A.S. Mayorov, D.C. Elias, M. Mucha-Kruczynski, R.V. Gorbachev, T. Tudorovskiy, A. Zhukov, S.V. Morozov, M.I. Katsnelson, V.I. Fal’ko, A.K. Geim, K.S. Novoselov, Interaction-driven spectrum reconstruction in bilayer graphene, Science 333 (2011) 860–863, http://dx.doi.org/10.1126/science.1208683. [21] J. Hass, W.A. de Heer, E.H. Conrad, The growth and morphology of epitaxial multilayer graphene, J. Phys.: Condens. Matter 20 (2008) 323202 (27 pp). [22] V.A. Geyler, I.Yu. Popov, The spectrum of a magneto-Bloch electron in a periodic array of quantum dots: explicitly solvable model, Z. Phys. B 93 (1994) 437–439. [23] V.A. Geyler, I.Yu. Popov, Periodic array of quantum dots in a magnetic field: irrational; honeycomb lattice, Z. Phys. B 98 (1995) 473–477. [24] S. Albeverio, V.A. Geyler, E.N. Grishanov, D.A. Ivanov, Point perturbations in constant curvature spaces, Int. J. Theor. Phys. 49 (2010) 728–758. [25] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, in: P. Exner (Ed.), Solvable Models in Quantum Mechanics, second ed., AMS Chelsea Publishing, Providence, R.I., 2005. with an appendix by. [26] D.A. Eremin, E.N. Grishanov, D.A. Ivanov, A.A. Lazutkina, E.S. Minkin, I. Yu. Popov, An explicitly solvable model for tunneling through a quantum dots array in a magnetic field, Chin. J. Phys. 52 (2014). 1119–112. [27] S. Albeverio, P. Kurasov, Singular perturbations of differential operators. Solvable Schrödinger type operators, London Mathematical Society Lecture Notes, vol. 27, Cambridge Univ. Press, Cambridge, 2000. [28] B.S. Pavlov, The theory of extensions and explicity-solvable models, Russ. Math. Surv. 42 (6) (1997) 127–168. [29] V.A. Geyler, V.A. Margulis, Anderson localization in the nondiscrete Maryland model, Theor. Math. Phys. 70 (1987) 133–140. [30] H. Bateman, A. Erdélyi, Higher Transcendental Functions, vol. I, McGraw-Hill, New York, 1953. [31] V.A. Geyler, V.V. Demidov, Spectrum of three-dimensional Landau operator perturbed by a periodic point potential, Theor. Math. Phys. 103 (1995) 561–569. [32] V.A. Geyler, The two-dimensional Scrödinger operator with a uniform magnetic field, and its perturbation by periodic zero-range potentials, St. Petersburg Math. J. 3 (1992) 489–532.
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Spectral properties of multi-layered graphene in a magnetic field E.N. Grishanov a, I.Yu. Popov b,⇑ a b
Department of Mathematics and IT, Ogarev Mordovia State University, Bolshevistskaya Str. 68, Saransk, Russia Department of Higher Mathematics, ITMO University, Kroverkskiy pr. 49, St. Petersburg 197101, Russia
a r t i c l e
i n f o
Article history: Received 9 July 2015 Accepted 10 July 2015 Available online 11 July 2015 Keywords: Graphene Hofstadter butterfly Spectrum
a b s t r a c t Solvable model of multi-layered graphene in a homogeneous magnetic field is constructed. The spectrum of the Hamiltonian has Hofstadter butterfly type. The comparison of the spectrum for single-layer graphene with that for two-layered and three-layered systems is made. Ó 2015 Published by Elsevier Ltd.
1. Introduction Starting from famous Hofstadter paper [1], fractal structure of the spectrum of nanosystems in a magnetic field attracts great attention both theoreticians and experimentalists. It is inspired by the peculiarities of the electron transport in such systems. Experimental observation of this effect in natural systems has remained elusive because the required experimental magnetic field is too high (approximately, 105 T) for a typical lattice constant of 0.1 nm. One way to circumvent this difficulty was to construct artificial superlattice to increase the lattice constant [2]. Particularly, one can find this type of the spectrum when dealing with periodic arrays of quantum dots [3–11]. One interesting nanostructure more is graphene. It demonstrates many intriguing properties [12]. We are interested in the spectral properties of electron in graphene in a magnetic field. It is related with the quantum Hall effect [13]. Graphene is two-dimensional (2D) Dirac-like electronic system. Due to this reason, its anomalous quantum Hall effect could be measured [14,15]. To calculate the spectrum for various magnetic field, one uses usually tight-binding approximation (see, e.g., [7]) or zero-range potentials model (see, e.g., [16,17]). The next interesting question is related with a comparison of the spectral results for single-layer graphene and multi-layered graphene structure. In bilayer graphene, the two atomic layers can be stacked together in many different ways, and even be twisted [8,18]. Of course, it has an influence on the spectrum [19,20]. An effect of number of layers is also important. In the present paper we describe the Hofstadter butterfly type structure of the spectrum for one, two and three graphene layers. We deal with layers stacked in the most natural way - the Bernal stacking configuration (see Fig. 1). 2. Model construction The starting point of the construction is the standard Hamiltonian H0 of free spinless charged particle (of mass m and charge e) in a homogeneous magnetic field B (Landau operator): ⇑ Corresponding author. E-mail address: [email protected] (I.Yu. Popov). http://dx.doi.org/10.1016/j.spmi.2015.07.025 0749-6036/Ó 2015 Published by Elsevier Ltd.
E.N. Grishanov, I.Yu. Popov / Superlattices and Microstructures 86 (2015) 68–72
69
Fig. 1. Geometric structure of bilayer graphene. The Bernal stacking configuration.
H0 ¼
2 h e p AðrÞ : c 2m
ð1Þ
Here p ¼ i hr is the momentum operator in R3 ; AðrÞ ¼ 12 B r is the vector potential of the magnetic field B (the symmetric gauge is chosen). It is assumed that the magnetic field is orthogonal to the graphene sheet plane. Below, the following notahxð‘ þ 1=2Þ; ‘ ¼ 0; 1; . . . are the Landau levels of the operator H0 tions are used: x ¼ jeBj=2c is the cyclotron frequency, e‘ ¼ (1), U0 ¼ 2p hc=jej is the magnetic flux quantum, n ¼ B=U0 is the density of the magnetic flux. It is well-known that operator H0 (1) has the continuous spectrum occupying the half-axis ½e0 ; 1Þ. More precisely, it consists of infinite number of branches: 1 [
½e‘ ; 1Þ
‘¼0
Let a be the interatomic distance in single-layer graphene. In real physical system one has a ¼ 0:142 nm (see, e.g., [8]). pffiffi pffiffiffi Vectors a1 ¼ ð 3a; 0; 0Þ and a2 ¼ 23 a; 32 a; 0 are the basic vectors of the Bravais lattice K for graphene with the basic cell
n o C K ¼ r 2 R3 : 0 6 r i 6 ai ði ¼ 1; 2Þ; r 3 2 R : The set of nodes (atoms) in the basic cell is marked as K 2 C K . The set K contains two elements for single-layer graphene, four elements for bilayer graphene, six elements for three-layered graphene, etc. Then, the set
C ¼ K þ K ¼ fj þ k : j 2 K; k 2 Kg is the set of all nodes of periodic system with the lattice K. As for real graphene samples, the model does not describe correctly the sample edge region. Fig. 1 shows the lattice structure of bilayer graphene. It corresponds to the most natural physical situation [21]. Let g ðg ¼ nða1 a2 ÞÞ be the number of the magnetic flux quanta through the basic cell of lattice K. The model Hamiltonian for spinless electron in our system is constructed as a perturbation of the operator (1) by zero-range potentials posed at the nodes of lattice K. Formally, the model operator can be written as
X ^ dðr cÞ; H ¼ H0 þ a
ð2Þ
c2C
^ is a real constant which characterizes the strength of the point-like potential. Correct mathematical description of where a the operator (2) is given in the framework of the theory of self-adjoint extensions of symmetric operators [17,22–26]. Let us describe the procedure. Let D be the following set
D ¼ ff 2 DðH0 Þ : f ðcÞ ¼ 0 8c 2 Cg: The operator S is the restriction of the operator H0 on D. It is a symmetric operator. The model Hamiltonian H is a self-adjoint extension of symmetric operator S. It can be described by the Krein resolvent formula (see, e.g., [27,28]). Let RðfÞ ¼ ðH fÞ1
70
E.N. Grishanov, I.Yu. Popov / Superlattices and Microstructures 86 (2015) 68–72
and R0 ðfÞ ¼ ðH0 fÞ1 be the resolvents of the operators H and H0 , correspondingly. In accordance with the Krein formula, one has
RðfÞ ¼ R0 ðfÞ CðfÞ½QðfÞ A1 C ðfÞ;
ð3Þ
where CðfÞ; Q ðfÞ are C-field and Krein Q-function, correspondingly, A is a self-adjoint operator parameterizing the self-adjoint extensions of the operator S. The operator A acts in so-called space of boundary values (it is infinite-dimensional because the operator S has infinite deficiency indices). As the lattice of the system has identical nodes (there are identical atoms at all nodes), the operator A can be chosen in the following form: A ¼ aI, where I is the identical operator. The terms of formula (3) can be written explicitly. Let Gðr; r0 ; fÞ be the Green function of the operator H (i.e., the integral kernel of the resolvent RðfÞ) and G0 ðr; r0 ; fÞ be the Green function of the operator H0 . The explicit expression for G0 is well-known. It was obtained in [29]:
G0 ðr; r0 ; fÞ ¼ Uðr; r0 ÞF 1 ðr r0 ; fÞ ¼ Uðr; r0 ÞF 2 ðr r0 ; fÞ; rffiffiffiffi h i m n 2 Uðr; r0 Þ ¼ 2 exp pinðr r0 Þ pnðr? r0? Þ =2 ; p 2h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X exp 4pnð‘ þ 1=2Þ fjrjj j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L‘ ðpnr2? Þ; F 1 ðr; fÞ ¼ ‘ þ 1=2 f=4pn ‘¼0 h i Z 1 exp pn r2 =ðet 1Þ þ r2 =t ? jj 1 dt pffiffi ; F 2 ðr; fÞ ¼ pffiffiffiffi p 0 ð1 et Þ exp ½ð1=2 f=4pnÞt t
ð4Þ
ð5Þ
where rjj is the projection of r on the direction of the magnetic field B; r? ¼ r rjj , and L‘ ðxÞ is the ‘-th Laguerre polynomial. The Q-function can be represented as infinite matrix Q ðfÞ ¼ ðQ ðc; c0 ÞÞc;c0 2C with the following entries
8 < G0 ðc; c0 ; fÞ; c – c0 ; qffiffiffi Q ðc; c0 ; fÞ ¼ : m2 pn Z 12 ; 12 4pf n ; h 2
c ¼ c0 :
ð6Þ
Here Zðs; v Þ is the Hurwitz f-function [30]. The matrix Q ðfÞ A (f 2 nR) gives one invertible bounded linear operator in the Hilbert space ‘2 ðCÞ. Below, we will not distinguish the notations of the operator and its matrix and will mark the both by ½Q ðfÞ A1 . Using the Krein formula (3), one obtains the explicit expression for the Green function G of the operator H (analogously to [29,31]):
Gðr; r0 ; fÞ ¼ G0 ðr; r0 ; fÞ
X
0 0 ½Q ðfÞ A1 c;c0 G0 ðr; c; fÞG0 ðc ; r ; fÞ:
ð7Þ
c;c0 2C
Thus, the construction is complete.
Fig. 2. Hofstadter butterfly (energy-flux diagram) of a single layer graphene. The energy is measured in eV, the unit of flux is the magnetic flux quantum.
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E.N. Grishanov, I.Yu. Popov / Superlattices and Microstructures 86 (2015) 68–72
3. Results and discussion It is known that the point f 2 R belongs to the spectrum of the operator H is the resolvent RðfÞ does not exist or exists but is not bounded. In the case of rational values of the magnetic flux number g ¼ N=M (N 2 Z; M 2 N) through the basic cell of the lattice K, the spectrum of the operator H can be found by expansion of the representation of the magnetic translation (by vectors of lattice K) group in respect to the spectrum of the irreducible subrepresentations [32]. These subrepresentations e ðp; fÞ are parameterized by points of two-dimensional torus T2 ¼ ½0; 1=MÞ ½0; 1Þ. For each point p; p 2 T2 , we mark as Q g
g
the matrix of size jKjM jKjM with the following entries
e ðp; m; j; m0 ; j0 ; fÞ ¼ exp½piðm m0 Þnðj a2 Þ Q exp
n
1 X
Q ðk1 a1 þ ðk2 M þ m m0 Þa2 þ j; j0 ; fÞ
k1 ;k2 ¼1
o g pin½j ðk1 a1 þ k2 Ma2 Þ 2pi kp þ k1 ðk2 M þ m þ m0 Þ ; 2
ð8Þ
where j; j0 2 K m; m0 ¼ 0; . . . ; M 1.
Fig. 3. Hofstadter butterfly (energy-flux diagram) of a bilayer graphene. Each layer is splitted into two sublayers. The energy is measured in eV, the unit of flux is the magnetic flux quantum.
Fig. 4. Hofstadter butterfly (energy-flux diagram) of a three-layer graphene. Each layer is splitted into three sublayers. The energy is measured in eV, the unit of flux is the magnetic flux quantum.
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The spectrum of the operator H consists of the spectrum of the operator H0 and roots of the dispersion equation
h i e ðp; fÞ a ¼ 0: det Q
ð9Þ
The spectral structure dependence on the magnetic field is shown in Figs. 2–4 (we does not show the unperturbed levels (the spectrum of the operator H0 ) in the figures). Three cases are considered: single-layer graphene, bilayer graphene, three-layer graphene. Of course, in all cases one observe the Hofstadter-type spectral structure. It can be noted that increasing of the magnetic field gives one some upward shift (as a trend) of the spectrum. Transition from single-layer system to multi-layered systems leads to additional splitting. One see that for n-layered structure an additional splitting of each level into n sublevels takes place. Our results are in good correlation with [18]. As for the results of [8], there is some difference. Our ‘‘energy – magnetic flux’’ diagrams do not contain intersections of subbands trajectories (if these subbands are induced by transition from single-layer to multi-layered structure). 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