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Home Search Collections Journals About Contact us My IOPscience Theory of Green functions of free Dirac fermions in graphene This content has been downloaded from IOPscience Please scroll down to see the full text 2016 Adv Nat Sci: Nanosci Nanotechnol 015013 (http://iopscience.iop.org/2043-6262/7/1/015013) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 186.56.39.178 This content was downloaded on 24/06/2016 at 16:57 Please note that terms and conditions apply | Vietnam Academy of Science and Technology Advances in Natural Sciences: Nanoscience and Nanotechnology Adv Nat Sci.: Nanosci Nanotechnol (2016) 015013 (11pp) doi:10.1088/2043-6262/7/1/015013 Theory of Green functions of free Dirac fermions in graphene Van Hieu Nguyen1,2, Bich Ha Nguyen1,2 and Ngoc Dung Dinh1 Advanced Center of Physics and Institute of Materials Science, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay District, Hanoi, Vietnam University of Engineering and Technology, Vietnam National University, 144 Xuan Thuy, Cau Giay District, Hanoi, Vietnam E-mail: nvhieu@iop.vast.ac.vn Received 10 November 2015 Accepted for publication December 2015 Published 12 February 2016 Abstract This work is the beginning of our research on graphene quantum electrodynamics (GQED), based on the application of the methods of traditional quantum field theory to the study of the interacting system of quantized electromagnetic field and Dirac fermions in single-layer graphene After a brief review of the known results concerning the lattice and electronic structures of single-layer graphene we perform the construction of the quantum fields of free Dirac fermions and the establishment of the corresponding Heisenberg quantum equations of these fields We then elaborate the theory of Green functions of Dirac fermions in a free Dirac fermion gas at vanishing absolute temperature T=0, the theory of Matsubara temperature Green functions and the Keldysh theory of non-equilibrium Green functions Keywords: Dirac fermions, Heisenberg quantum equation of motions, Green functions Classification numbers: 2.01, 3.00, 5.15 electromagnetic interaction processes This work is the first step in the establishment of the basics of graphene quantum electrodynamics: the construction of the theory of Green functions of free Dirac fermions in graphene Since throughout the present work we often use knowledge of the lattice structure of graphene as well as expressions of the wave functions of Dirac fermions with the wave vectors near the corners of the Brillouin zones of the graphene lattice, first we present a brief review of this knowledge in section In the subsequent section 3, the explicit expressions of the quantum field of free Dirac fermions in graphene and the corresponding Heisenberg quantum equations of motion are established Section is devoted to the study of Green functions of Dirac fermions in a free Dirac fermion gas at vanishing absolute temperature T=0 The theory of Matsubara temperature Green functions of free Dirac fermions is presented in section 5, and the content of section is the Keldysh theory of nonequilibrium Green functions The conclusion and discussions are presented in section The unit system with c = = will be used Introduction In the comprehensive review [1] on the rise of graphene as the emergence of a new bright star ‘on the horizon of materials science and condensed matter physics’, Geim and Novoselov have remarked exactly that, as a strictly two-dimensional (2D) material, graphene ‘has already revealed a cornucopia of new physics’ It is the physics of graphene and graphene-based nanosystems, including graphene quantum electrodynamics (GQED) In the language of another work by Novoselov et al [2], GQED (‘resulting from the merger’ of the traditional quantum field theory with the dynamics of Dirac fermions in graphene) would ‘provide a clear understanding’ and a powerful theoretical tool for the investigation of a huge class of physical processes and phenomena talking place in the rich world of graphene-based nanosystems and their Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI 2043-6262/16/015013+11$33.00 © 2016 Vietnam Academy of Science & Technology Adv Nat Sci.: Nanosci Nanotechnol (2016) 015013 V H Nguyen et al Figure Lattice structure (a) and the first Brillouin zone (b) of graphene Definitions and notations According to the review [3] on the electronic properties of graphene, each graphene single layer is a 2D lattice of carbon atoms with the hexagonal structure presented in figure 1(a) It consists of two interpenetrating triangular sublattices with the lattice vectors a a l1 = (3, ) , l2 = (3, - ) (1 ) 2 where a is the distance between the two nearest carbon atoms a≈1.42 The reciprocal lattice has the following lattice vectors k1 = 2p (1, 3a ) , k2 = 2p (1, - ) 3a (2 ) Vectors li and ki satisfy the condition ki l j = 2pdij Figure Hexagonal lattice of the corners of all BZs in the reciprocal lattice (3 ) The first Brillouin zone (BZ) is presented in figure 1(b) Two inequivalent corners K and K′ with the coordinate vectors K= 3p ⎛ ⎞ 3p ⎛ ⎞ ⎜1, ⎟ , K¢ = ⎜1, ⎟ 2a ⎝ 2a ⎝ 3⎠ 3⎠ (4 ) are called the Dirac points Each of them is the common vertex of two consecutive cone-like energy bands of Dirac fermions The corners of all BZs in the reciprocal lattice form a new hexagonal lattice of the points equivalent to the Dirac points K and K′ in the first BZ (figure 2) This new hexagonal lattice also consists of two interpenetrating triangular sublattices with the lattice vectors l1¢ = 3p 3p ⎛ 3⎞ (1, 0) , l ¢2 = ⎟⋅ ⎜ , a a ⎝2 ⎠ (5 ) As an example let us consider the sublattice of all points equivalent to the corner K They form a triangular lattice with the natural parallelogram elementary cell drawn in the left part of figure For avoiding the presence of four equivalent corners in each natural parallelogram elementary cell, in the sequel we shall use the symmetric Wigner–Seitz elementary cell drawn in the right part of figure instead of the parallelogram one The wave vector k is called to be near the corner K if it is contained inside the symmetric Wigner–Seitz Figure Natural parallelogram elementary cell (left part) and symmetric Wigner–Seitz elementary cell (right part) in the triangular lattice of the points equivalent to the Dirac point K in the reciprocal lattice elementary cell around this corner With respect to the sublattice of all points equivalent to the corner K´ we also have a similar result We chose the length unit such that the area of elementary cell is equal to Adv Nat Sci.: Nanosci Nanotechnol (2016) 015013 V H Nguyen et al Quantum field of free Dirac fermions space with the Cartesian coordinate system Being the spinors with respect to the rotations in some fictive 3D Euclidean space, they are similar to the isospinor called nucleon N with proton p and neutron n as its two components In order to establish explicit expressions of the quantum field of free Dirac fermions it is necessary to have formulae of the K,K ¢ wave functions of these quasiparticles Denote Fk, E (r) the wave function of the state with the wave vector k near the Dirac points K or K´ and the energy E It was known that K ⎧ iKr j K (r) , ⎪ F k, E (r ) = e k, E ⎨ ¢ ¢ K i K r K¢ ⎪F ⎩ k, E (r ) = e j k, E (r ) , N= in nuclear physics [4] and elementary particle physics [5–8] In order to distinguish the spinors (11) and (12) from the usual Pauli spinors let us call them Dirac spinors, quasispinors or pseudo-spinors It is worth investigating the symmetry with respect to the rotations in the abovementioned fictive 3D Euclidean space The Hamiltonian of the quantum field of free Dirac fermions is (6 ) K,K ¢ where j k, are the solutions of the 2D Dirac equations E (r) v F ( - it) j kK, E (r) = Ej kK, E (r) , (7 ) v F ( - it⁎) j kK,¢E (r) = Ej kK,¢E (r) , (8 ) where two components τ1 and τ2 of vector matrix τ are two matrices H0 = v F ( ) E(k ) = v F k , (9 ) i and two eigenfunctions j kK,,EK¢(k ) (r) = eikruK , K ¢ (k) , uK = uK ¢ ⎛ e-iq (k) ⎞ ⎜ iq (k) ⎟ h , ⎝e ⎠ ⎛ eiq (k) ⎞ = ⎜ ⎟ h¢, ⎝ e-iq (k) ⎠ (10) ⎛k ⎞ q (k) = arctg ⎜ ⎟ , k2 (11) i ảY K  (r , t ) = v F ( - it⁎) Y K  (r , t ) , ảt (18) K,K ¢ (19) Consider now the free Dirac fermion gas at vanishing absolute temperature T=0 In this case it is convenient to work in the electron hole formalism Denote EF the Fermi level and | Gñ the state vector of the ground state of the Dirac fermion gas in which all levels with energies larger than EF are empty and all those with energies less than EF are fully occupied The ground state | Gñ is expressed in terms of the Dirac fermion creation operators and the state vector | 0ñ of the vacuum (13) Y (r , t ) = eiKrY K (r , t ) + eiK ¢ r Y K  (r , t ) ảY K , K ¢ (r , t ) = - [H0, Y K , K  (r , t )] ảt (12) η and η′ are two arbitrary phase factors | h | = | h ¢ | = The quantum field of free Dirac fermions in the hexagonal graphene lattice has the expression Y K , K ¢ (r , t ) = (17) can be rewritten in the form of the Heisenberg quantum equation of motion where with the following expansion of Y ¶Y K (r , t ) = v F ( - it) Y K (r , t ) , ¶t i k12 + k 22 , (16) From the expansion formula (15) and the canonical anticommutation relations between destruction and creation operators akKn, K ¢ and (akKn, K ¢ )+ it follows that Dirac equations Equations (7) and (8) both have two solutions corresponding to two eigenvalues k = | k| = ò dr {YK (r, t )+(-it) YK (r, t ) + Y K ¢ (r , t )+( - it⁎) Y K ¢ (r , t )} t1 = , t2 = -i ⋅ i ( ) ( np) | Gñ = (14) å å En (k) EF En (k ¢) EF (r, t ): å å ei [kr- En (k ) t] unK ,K ¢ (k) akKn,K ¢, (15) Nc k n = (akKn )+ (akK¢ n¢ ¢)+ | 0đ (20) With respect to the ground state | Gñ the destruction/ creation operator akKn, K ¢ (akKn, K ¢ )+ of the Dirac fermion with energy less than EF becomes the creation/destruction operator of the Dirac hole in the corresponding state with the momentum and energy which will be specified in each separate case Since the reasonings for the states with wave vectors k near K and K′ are the same, until the end of this section we shall omit the indices K and K′ in the notations of field operators, destruction and creation operators as well as of the wave functions for simplifying the formulae where akKn, K ¢ is the destruction operator of the Dirac fermion with the wave function being the plane wave whose wave vector k satisfies the periodic boundary condition for a very large square graphene lattice containing Nc elementary cells Note that the role of the electron spin was omitted and electrons are considered as the spinless fermions Twocomponent wave functions (11) and (12) are not the usual spinors (Pauli spinors) in the three-dimensional (3D) physical Adv Nat Sci.: Nanosci Nanotechnol (2016) 015013 V H Nguyen et al Figure Energy bands when (a) EF=0, (b) EF>0 and (c) EF0 (figure 4(b)) All states with energies E+(k ) > EF are empty and for them we set E-(k ) = EF - Eh (k ) , ak - = b-+k , u-(k) = v ( - k) E+(k ) = EF + Ee (k ) , ak + = ak , u+(k) = u (k) In this case we obtain All states with energies E+(k ) < EF are occupied and for them we set (2 ) {ei [kr- Ee (k ) t ] u(2) (k) ak(2) å Nc k eiEF t Y (r , t ) = + q [E-(k ) - EF ] ei [kr- Ee (1 ) E+(k ) = EF - Eh(1) (k ) , ak + = b-(1k) + , + q [EF - u+(k) = v (1) ( - k) (k ) t ] u(1) (k) a (1) k E-(k )] e-i [kr- Eh (k ) t ] v (k) b k+}⋅ (23) All states with energies E-(k ) are occupied and for them we set E-(k ) = ak - = b-(2k) + , Instead of the quantum fields Y (r, t ) we use the new ones EF - Eh(2) (k ) , u-(k) = v (2) ( - k) ˆ (r , t ) = eiEF t Y (r , t )⋅ Y From formulae (20)–(23) it follows that the new fields (24) satisfy the new Heisenberg quantum equation of motion In this case we obtain eiEF t Y (r , t ) = å {q [E+(k ) - EF ] ei [kr- Ee (k ) t] u (k) ak Nc k (k ) t ] v (1) (k) b (1) + k (2 ) e-i [kr- Eh (k ) t ] v (2) (k) b k(2) +}⋅ i ˆ (r , t ) ¶Y ˆ (r , t )] = - [H0Â, Y ảt (25) {Ee (k) ak+ak + Eh (k) b k+b k}, (26) where + q [EF - E+(k )] e-i [kr- Eh (1 ) + (24) H0¢ = (22) k Adv Nat Sci.: Nanosci Nanotechnol (2016) 015013 V H Nguyen et al in the case with EF=0, H0¢ = å {q [E+(k ) - Introduce the Fourier transformation of Green functions (29) and (30) EF ] Ee (k ) ak+ak k Dab (r , t ) = + q [EF - E+(k )] Eh(1) (k ) b k(1) +b k(1) + Eh(2) (k ) b k(2) +b k(2)} k E-(k )] Eh (k ) b k+b k} (28) in the case with EF0 we have ˜ ab (k, w ) = q [E+(k ) - EF ] ua (k) ub (k)* D w - Ee (k ) + i0 Green functions of Dirac fermions in the free Dirac fermion gas at T=0 + ˆ aK (r , t ) Y ˆ bK (r¢ , t ¢)+]| G đ , - r¢ , t - t ¢) = - i G | T [Y (29) w + Eh(1) (k ) - i0 va(2) ( - k) vb(2) ( - k)* w + Eh(2) (k ) - i0 , (35) while in the third case with EF0, and + q [EF - 1 åeikr 2p Nc k ¶ K¢ K Dab (r - r¢ , t - t ¢) - v F ( - it ⁎)ag D gb (r - r , t - t Â) ảt = dab d (t - t ¢) d (r - r¢) K Dab (r - r¢ , t - t ¢) (32) = -i Explicit expressions of Green functions (29) and (30) depend on the position of the Fermi level EF For simplifying formulae let us omit again the indices K and K′ until the end of this section Depending on the value of EF there exist three different cases In the first case with EF=0 the operator Yˆ a (r, t ) is expressed in terms of the components ua (k) and va (k) by means of formula (21), in the second case with EF>0 it is expressed in terms of the components ua (k), va(1) (k) and va(2) (k) by means of formula (22), while in the third case with EF 0 we have (48) Dab (r - r¢ , t )M = K = Tr {e ¢ Tr {e-b T H0 } åeik (r- r¢) Nc k ⎧ q (t ) e-tEe (k ) - q ( - t ) e-(b T + t ) Ee (k ) ´ ⎨q [E+(k ) - EF ] ⎩ + e-b T Ee (k ) ⁎ ´ ua (k) ub (k) K ¯ b (r¢ , t ¢)M ] đ Dab (r - r¢ , t - t ¢)M = Tt [Y aK (r , t )M Y ¯ bK (r¢ , t ¢)M ]} t )M Y (52) (47) The Matsubara temperature Green functions of Dirac fermions are defined by the following formula Tt [Y aK (r , åeik (r- r¢) Nc k Dab (r - r¢ , t )M = ¯ aK ¢ (r , t )M ¶Y ¯ gK ¢ (r , t )M = - [v F ( - it)ag - EF dag ] Y ảt -b T H0 (51) Now let us derive the explicit expressions of the Green K¢ K functions Dab (r - r¢ , t )M Since the (r - r¢ , t )M and Dab reasonings and calculations not depend on the presence of the indices K and K´, we shall omit both these indices until the end of this section There are three different cases depending on the position of the Fermi level EF By means of standard calculations we obtain following result in the case with EF=0: ¶Y aK (r , t )M = - [v F ( - it)ag - EF dag ] Y gK (r , t )M , (45) ¶t ¯ aK (r , t )M ¶Y = -[v F ( - it⁎)ag - EF dag ] ¶t ¯ gK (r , t )M , ´Y - r¢ , t - t ¢)M = dab d (r - r¢) d (t - t ¢) From this common form it is easy to derive concrete forms of the differential equations for different fields K¢ K YaK (r, t )M , Y aK ¢ (r, t )M and Y¯ a (r, t )M , Y¯ a (r, t )M We obtain ¶Y aK ¢ (r , t )M = -[v F ( - it⁎)ag - EF dag ] ¶t ´ Y gK ¢ (r , t )M , (50) + [v F ( - it)ag - EF dag ] ảt K ´ D gb (r - r¢ , t - t ¢)M (43) They obey the Heisenberg quantum equation of motion ¶Y aK , K ¢ (r , t )M = [H0¢, Y aK , K ¢ (r , t )M ] , ảt aK , K  (r , t )M ảY aK , K  (r , t )M ] = [H0Â, Y ảt K b (r¢ , t ¢)M - q (t ¢ - t ) Y ¯b = q (t - t ¢) Y aK (r , t )M Y q (t ) e(t - b T ) Eh (1 ) + q [EF - E+(k )] , ´ (49) + - q ( - t ) etEh (1 ) (k ) (1 ) e-b T Eh (k ) va(1) ( - k) vb(1) ( - k)⁎ q (t ) e(t - b T ) Eh (2 ) and a similar one obtained from this formula after the replacement K K ¢ , where Tτ denotes the operation of ordering the product of operators along the decreasing 1+ (k ) 1+ (k ) - q ( - t ) etEh (2 ) e-b T Eh (k ) (2 ) (k ) ⎫ va(2) ( - k) vb(2) ( - k)⁎⎬⋅ ⎭ (53) Adv Nat Sci.: Nanosci Nanotechnol (2016) 015013 V H Nguyen et al Similarly, in the case with EF0, and ua(2) (k) ub(2) (k)⁎ ien + Ee(2) (k ) + q [E-(k ) - (63) (58) ien - Eh(2) (k ) ˜ ab (k, en)M = D ¯ b (r¢ , z¢)C ]} Tr {e-b T H ¢0 TC [Y aK ¢ (r , z)C Y (56) in the case with EF=0, + (60) K¢ ¯ bK ¢ (r¢ , z¢)C ] đ Dab (r , r¢ ; z - z¢)C = - i TC [Y aK ¢ (r , z)C Y Inverting the expansion formulae of the functions (52)– (54), we obtain + q [EF - E+(k )] Tr {e-b T H ¢0 } and 1 åeik (r- r¢) b Nc k T bT ¯ b (r¢ , z¢)C ]} Tr {e-b T H ¢0 TC [Y aK (r , z)C Y K = -i i ua(1) (k) ub(1) (k)⁎ EF ] ien + Ee(1) (k ) + q [EF - E-(k )] ( - k)⁎ va ( - k) vb ien - Eh (k ) i (59) (64) ảY aK  (r , z)C = [v F ( - it*)ag - EF dag ] Y gK ¢ (r , z)C , (65) ¶z i in the case with EF