Shift of Dirac points and strain induced pseudo-magnetic field in graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Shift of Dirac points and strain induced pseudo-magnetic field in graphene
Hua Tong Yang ∗ Center for Advanced Optoelectronic Functional Materials Research,Key Laboratory for UV-Emitting Materials and Technology of Ministry of Education,and School of Physics, Northeast Normal University, Changchun 130024, China
We propose that the strain induced effective pseudo-magnetic field in graphene can also be ex-plained by a curl movement of the Dirac points, if the Dirac points can be regarded as a slowlyvarying function of position. We also prove that the Dirac points must be confined within twotriangles, each one has 1/8 the area of the Brillouin zone.
PACS numbers: 73.22.Pr, 73.22.Dj, 73.22.Gk, 73.20.At
The discovery of graphene, a monolayer carbon atomsheet [1], and the development of experimental techniqueto manipulate this two-dimensional(2D) material have ig-nited intense interest in this system [2–5]. One of themost attractive characters of graphene is that its lowenergy excitation satisfies a massless 2D Dirac equation[6], and the chemical potential crosses its Dirac points(orFermi points) in neutral graphene. These special char-acters lead to many unusual properties and new phe-nomena [5, 7–9], such as the anomalous integer quantumHall effect(QHE) [8, 9]. Recently, experiments have con-firmed another remarkable effect that mechanical straincan induce a very strong effective pseudo-magnetic field,leading to a pseudo-QHE, which can be observed in zeromagnetic field [10, 11]. In this paper we propose that thestrain induced effective vector potential can also be ex-plained by shift δ K ( x ) of the Dirac points K ( x ), its effec-tive pseudo-magnetic field is in proportion to ∇ × K ( x ),only if the Dirac points K ( x ) can be regarded as a slowlyvarying function of position, and the Fermi velocity isgeneralized to a tensor[12]. We also prove that the Diracpoints can not be arbitrarily moved, they must be con-fined within two triangles, each one has 1/8 the area ofthe Brillouin zone(BZ).Firstly, consider a tight-binding Hamiltonian describ-ing a uniformly deformed honeycomb lattice with threedifferent nearest-neighbor hopping energies t , t , t [13–15]: ˆ H = − X < i a, j b> t i a, j b c † i a c j b + h.c., (1)where c j b ( c † i a ) are annihilation(creation) operators, i ( j )are position vectors of unit cells, a ( b ) denote two inequiv-alent atoms in a unit cell, t i a, j b is the electronic hoppingenergy from the j th unit cell b atom to i th unit cell a atom. Suppose that the deformed lattice remains invari-ant under spatial translation, i.e., t i a, j b only depends on i − j , but the three nearest-neighbor hopping energies t , , may be different owing to anisotropy of strains, asshown in Fig.1. The hopping parameters can be writtenas some 2 × t ( i − j ), whose elements are de-fined by [ t ( i − j )] a,b ≡ t i a, j b . For this nearest-neighbor a a t t t c c c xy a b FIG. 1: Unit cell and hopping parameters for deformedgraphene. tight-binding Hamiltonian, the non-vanishing hoppingmatrixes are t (0) = (cid:18) t t (cid:19) , t ( a ) = (cid:18) t (cid:19) , t ( a ) = (cid:18) t (cid:19) (2)and t ( − a ) = t † ( a ) , t ( − a ) = t † ( a ) . By Fourier trans-formation c j ,a ( b ) = 1 √ N X k c k ,a ( b ) exp( i k · j )with N a normalization constant, the Hamiltonian (1)can be cast into the formˆ H = − X k (cid:2) c † k ,a , c † k ,b (cid:3)(cid:20) h aa ( k ) h ab ( k ) h ba ( k ) h bb ( k ) (cid:21)(cid:20) c k ,a c k ,b (cid:21) , (3)where h aa ( k ) = h bb ( k ) = 0 , h ab ( k ) = h ∗ ba ( k ) , and h ba ( k ) = t + t exp( i k · a ) + t exp( i k · a ) , (4)with a , a the lattice unit vectors. The energy bandsobtained by diagonalizing this Hamiltonian are[16] E ± ( k ) = ±| t + ˜ t ( k ) + ˜ t ( k ) | , (5)where ˜ t ( k ) = t e i k · a , ˜ t ( k ) = t e i k · a , the plus signcorresponds to the upper( π ) and minus to the lower( π ∗ )band respectively. From Eq.(5) we notice that if K is azero point of h ba ( K ), i.e., t + ˜ t ( K ) + ˜ t ( K ) = 0 , (6)then E + ( k ) and E − ( k ) will meet at K , i.e., E + ( K ) = E − ( K ) = 0, this K is known as the Dirac point. TheHamiltonian (3) can be expanded up to a linear order in p = k − K in a neighborhood of point K (cid:20) h aa ( k ) h ab ( k ) h ba ( k ) h bb ( k ) (cid:21) ≃ (cid:20) ~α ∗ · p ~α · p (cid:21) = v µν σ µ p ν , (7)where µ, ν = 1 , µ, ν is implied, ~α is a complex vector with Re( ~α ) = ( v , v ), Im( ~α ) =( v , v ), σ , are Pauli matrixes acting on the sublatticedegree of freedom, tensor v µν represents the anisotropyof the dispersion near the Dirac points, it only occursnoticeable departure from v F δ µν in a strongly deformedgraphene[12]. However, after this modification the straininduced effective vector potential will acquire a directphysical meaning. For a graphene under nonuniform butslowly varying strain, t i ( x ) and hence the Dirac point K ( x ) as well as v µν ( x ) can be regarded as some smoothfunctions of position x , the local linearized Hamiltonian v µν σ µ ( k ν − K ν ( x )) on the RHS of Eq.(7) can be cast into v µν ( x ) σ µ ( p ν − δK ν ( x )) , (8)where p − δ K ( x ) ≡ k − K ( x ), δ K ( x ) ≡ K ( x ) − K f with K f the corresponding Dirac point in strain-freegraphene. Unlike the usual explanation of the strain in-duced gauge field in graphene[5, 17], where the effectivevector potential is an auxiliary quantity and describesthe mixed effects of both anisotropy of v µν and the shiftof Dirac point, here the vector potential only representsthe relative translation of the Dirac points, ( e/c ) A ( x ) = δ K ( x ), its pseudo-magnetic field B ( x ) = ( c/e ) ∇ × K ( x ),and the physical effects are mainly determined by thepseudo-magnetic flux through a loop ( c/e ) H L K ( x ) · d x .In the following sections we shall discuss the propertiesof K ( x ), and illustrate how a curl field K ( x ) is inducedby a strain.From Eq.(6) we know that the vectors representing t ,˜ t ( K ), ˜ t ( K ) in the complex plane can form a directedtriangle for a Dirac point K , as illustrated in Fig.2a.According to the triangle inequality, we have the follow-ing necessary and sufficient conditions for the existenceof the Dirac points[13]: t + t ≥ t , t + t ≥ t , t + t ≥ t . (9)These conditions define a pyramidal domain in the( t , t , t ) space, shown in Fig.2b. If t , t , t satisfy in-equalities (9), then there exists two directed triangleswith the same edges t , t , t but different possible ori-entations, which determine two angles θ , θ satisfying t + t e iθ + t e iθ = 0, where θ , θ are given by the lawof cosine θ ± = ± (cid:2) π − arccos (cid:0) t + t − t t t (cid:1)(cid:3) ,θ ± = ± (cid:2) arccos (cid:0) t + t − t t t (cid:1) − π (cid:3) . (10) t t t t ˜ t ˜ t θ θ KM’’ M’ Ma b O’’O O’ FIG. 2: (color online). (a) Zero points of h ba ( k ) determinetwo directed triangles with edges t , , in the complex plane.For a given t and a fixed direction of ˜ t , the arguments of˜ t must satisfy conditions (15) to ensure t , ≥
0. (b) Diracpoints exist if t i satisfy inequalities (9), which describe a pyra-midal domain in ( t , t , t ) space, if ( t , t , t ) goes beyond thisdomain, an energy gap will be opened. Thus the Dirac points K can be determined by lettingexp( i K · a ) = exp( iθ ) , exp( i K · a ) = exp( iθ ) , (11)so we have K = 12 π (cid:0) θ b + θ b (cid:1) + K , (12)with b , b the reciprocal lattice vectors defined by a i · b j = 2 πδ ij , and K = n b + m b with n, m arearbitrary integers. Notice that if t + ˜ t + ˜ t = 0, then t + ˜ t ∗ + ˜ t ∗ = 0, this implies that there exists two Diracpoints K ( x ) and − K ( x ). However, if ( t , t , t ) exactlylocates on the boundary surface of the pyramid, e.g., t = t + t , then the two triangles will mutually coincideand ˜ t = ˜ t ∗ , ˜ t = ˜ t ∗ (see Fig.2a), hence K ( x ) and − K ( x )become equivalent, and ~α = i ( t a + t a ) becomes a pureimaginary vector, so the Fermi velocity in the directionsperpendicular to ~α vanishes(Fig.3b and 3d)[14, 15, 18]. If( t , t , t ) goes beyond the domain defined by Eq.(9), e.g., t > t + t , Eq.(6) will have no any root, an energy gapwith magnitude E g = 2( t − t − t ) will occur at the cor-responding points K ± = ± / b + b )(see Fig.3c)[19],and the effective Hamiltonian (8) must be further modi-fied by adding a mass term and some second order terms.Another important property is the range of K ( x ). Weshall prove that the Dirac points must be confined withinsome special regions of the BZ. To this end, notice thatif a Dirac point K = (1 / π )( θ , θ ) is given, then itsassociated t , , can also be determined up to an arbitraryfactor, except six special cases of θ , θ = 0 , ± π (see Fig.2a). If θ , θ = 0 , ± π . According to the law of sines wehave t t = sin θ sin( θ − θ ) , t t = sin θ sin( θ − θ ) , (13) FIG. 3: (color online) (a) Energy band when two Dirac pointsare very close, where t = 2 . , t , = 1 . , (b), (d) t =2 . , t , = 1 . , two Dirac points are equivalent(superposed),(c) t = 2 . , t , = 1 . , an energy gap occurs. or ( t , t , t ) ∝ (sin( θ − θ ) , − sin θ , sin θ ) . (14)For the six special cases we have: if ( θ , θ ) = ± ( π, π ) ,t = t + t ; if ( θ , θ ) = ± ( π, , t = t + t ; if ( θ , θ ) = ± (0 , π ) , t = t + t . From Eq.(14) we can find that the θ , θ must satisfy some constrain conditions to guarantee t , , ≥
0, as illustrated in Fig.2a. For an arbitrary t anda fixed θ (direction of ˜ t ), ˜ t must point in a directionbetween the directions of − ˜ t and negative real axis, i.e.,argument θ , θ must satisfy θ + π < θ < π, θ ∈ ( − π, , − π < θ < θ − π, θ ∈ (0 , π ) . (15)These two inequalities respectively determine the rangeof K ( x ) and − K ( x ). They describe two open trian-gles △ M M ′ M ′′ and △ M ′ M M ′′ in reciprocal space, asshown in Fig.4, each one has 1/8 the area of a unit cell ofthe reciprocal space(the parallelogram M ′′ M ′′ M ′′ M ′′ ),and each Dirac point is confined within a triangle, so,the Dirac points K and − K can meet(become equivalent)only at the vertexes of △ M M ′ M ′′ and △ M ′ M M ′′ . Theremaining hexagon(blue in Fig.4) is a forbidden regionfor the Dirac points. This confinement also limits theorder of magnitude of ∇ × K ( x ), i.e., the strain inducedpseudo-magnetic field. In order to show the underlyingregularity, here we have ignored the variations of b , b with the deformation of lattice, and simply sketch all K = ( k , k ) in the same affine frame. After translatingto the first BZ of graphene, △ M M ′ M ′′ and △ M ′ M M ′′ are equivalent to a ringlike region consists of six triangles △ M KM ′ , △ M K M ′′ , △ M ′′ K M ′ , etc. M ′ ′ O O ′ M KK ′ M ′ K ′ M ′ ′ M ′ ′ K K M O ′ O K ′ M ′ ′ b b O ′ ′ O ′ ′ M ′ xy FIG. 4: (color inline) Rang of the Dirac points consists of sixtriangles in first the BZ, or △ MM ′ M ′′ and △ M ′ M M ′′ . In order to illustrate how a non-vanishing ∇ × K is in-duced by strain, we only need to analyze three ideal cases,in which only one t i is slightly changed, t i → t + δt i ,while the other two t j,k remain constant, t j = t k = t ,which can also be roughly regarded as that the bond c i is elongated(or compressed) while the other two bonds c j , c k and their directions remain fixed(see Fig.5b). No-tice that the Dirac points only depend on the relativeproportions of t , t , t , so, as an equivalent case, wecan always assume that t remains constant and only t , t are variables. Moreover, in these equivalent casesthe ˜ t and ˜ t can be determined by the end of the vec-tor t + ˜ t = − ˜ t , denoted by P in Fig.5a. So, wecan represent the variation of the Dirac points by theshift of the point P. To this end, we have to determinethe corresponding P of the three classes of characteris-tic points in the range of the Dirac points: (1) K (or K ′ ) etc.(see Fig.4), according to Eq.(13), Dirac pointslocate at these two points only if t = t = t , theircorresponding P is located at K (or K ′ ) in Fig.5a; (2)critical points M ( M ) = ( ± / , M ′ ( M ′ ) = (0 , ± / M ′′ ( M ′′ ) = ( ± / , ± / , in these cases there ex-ist only one Dirac point since the points K and − K areequivalent, their corresponding ( t , t , t ) are located onthe boundary of the pyramidal domain, while their cor-responding P are located at the real axis in Fig.5a; (3) O , O ′ , O ′′ etc., their corresponding P are the centers oftwo circles and the infinite limit points of the straightline KK ′ in Fig.5a, which respectively correspond tothe limits of t → t → t → t = t → ∞ ). Now we analyze the shifts of the Diracpoints in the three ideal situations. (1) t , t remain con-stant and t = t , only t is variable, the trajectory of thecorresponding P is a circle with radius t and centered M O M ′ O ′ M ′ ′ θ θ PKK ′ a Im Re xy δ K δ K δ K b t t t c c c FIG. 5: (a) The trajectories of P in three ideal cases. (b) Aschematic diagram of the shift of Dirac point δ K and its curl, δ K perpendicular to c i , if bond c i is slightly elongated. at the point ( t ,
0) in Fig.5a, so the arguments ( θ , θ )satisfy θ − θ − π = 0 , θ ∈ ( − π, ,θ − θ + 2 π = 0 , θ ∈ (0 , π ) . (16)They describe line segments M ′ O and M ′ O in Fig.4; (2) t (= t ) remain constant while t is variable, the trajec-tory of corresponding P is another circle with radius t centered at the origin, its associated ( θ , θ ) satisfy θ − θ + 2 π = 0 , θ ∈ ( − π, ,θ − θ − π = 0 , θ ∈ (0 , π ) , (17)which describe M O ′ and M O ′ in Fig.4; (3) t remainsconstant while t , t are variable but t = t (or vice versa, t is variable, t (= t ) remain constant), the trajectoryof P is straight line KK ′ , ( θ , θ ) satisfy θ + θ = 0 , π < | θ | < π, (18)which describe M ′′ O ′′ and M ′′ O ′′ in Fig.4. Summariz-ing Eqs.(16)(17)(18) and comparing with Fig.4, we ob-serve that if a band, e.g., c is slightly elongated (orcompressed) along its direction, c → (1 + δ ) c , whilethe other two bonds c , c remain fixed, then t willbe slightly changed while t (= t ) remain constant, theDirac point K will be slightly moved in the directionperpendicular to c , i.e., δK y = 0(see Fig.4, K movestowards O ′′ if t decreases, towards M ′′ if t increases), K ′ (= − K ) is moved in the opposite direction. Thus, ifthe elongation of c is slowly varying in the x-direction,i.e, ∂t /∂x = 0, then ∂K y /∂x = 0, the other two casesare similar. So, a nonuniform strain as schematicallyshown in Fig.5b can induce a curl field K ( x ), ∇ × K = 0. We thank Yugui Yao, Chengshi Liu, Yichun Liu fortheir helpful discussions. This work was supportedby the National Science Foundation of China(GrantNos.10974027, 50725205, 50832001). ∗ Electronic address: [email protected][1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y.Zhang, S. V. Dubonos, I. V. Gregorieva, and A. A. Firsov,Science , 666(2004); K. S. Novoselov, D. Jiang, F.Schedin, T. J. Booth, V. V. Khotkevich, S. M. Morozov,A. K. Geim, Proc. Natl. Acad. Sci. , 10451(2005).[2] M. A. H. Vozmediano, M. P. Lopez-Sancho, and F.Guinea, Phys. Rev. Lett. , 166401(2002).[3] J.C. Meyer, K. Geim, M.I. Katsnelson, K.S. Novoselov,T.J. Booth, and S. Roth, Nature , 60(2007).[4] A.K. Geim, and K.S. Novoselov, Nature Materials ,183(2007); C.W.J. Beenakker, Rev. Mod. Phys. ,1337(2008).[5] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K.S. Novoselov, and A. K. Geim, Rev. Mod. Phys. ,109(2009).[6] G. W. Semenoff, Phys. Rev. Lett. , 2449(1984).[7] V. P. Gusynin, and S. G. Sharapov, Phys. Rev. Lett. ,146801(2005); V. P. Gusynin, V. A. Miransky, and S. G.Sharapov, Phys. Rev. B , 195429(2006).[8] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,M. I. Katsnelson, I. V. Gregorieva, S. V. Dubonos, andA. A. Firsov, Nature , 197(2005).[9] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim,Nature , 201(2005).[10] F. Guinea, M. I. Katsnelson, and A. K. Geim, NuaturPhysics , 30(2010).[11] N. Levy, S.A.Burke, K.L.Meaker, M.Panlassigui, A.Zettl,F.Guinea, and A.H. Crommie, Science , 544 (2010).[12] S. L. Zhu, B. Wang, and L.-M. Duan, Phys. Rev. Lett. , 260402(2007); O. Bahat-Treidel, O. Peleg, M. Grob-man, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys.Rev. Lett. , 063901(2010).[13] Y. Hasegawa, R. Konno, H. Nakano, and M. Kohmoto,Phys. Rev B , 033413(2006).[14] P. Dietl, F. Pi´echon, and G. Montambaux, Phys. Rev.Lett. , 236405(2008).[15] G. Montambaux, F. Pichon, J.-N. Fuchs, and M. O. Go-erbig, Phys. Rec. B 80, 153412(2009); G. Montambaux,F. Pichon, J.-N. Fuchs and M. O. Goerbig, Eur. Phys. J.B , 509(2009).[16] P. R. Wallace, Phys. Rev. , 622(1947).[17] C.L.Kane and E.J.Mele Phys. Rev. Lett. , 1932(1997);H. Suzuura and T. Ando, Phys. Rev. B , 235412(2002);J.L.Ma˜ne, Phys. Rev. B , 045430(2007).[18] V.M. Pereira, A.H. Castro Neto, and N.M.R. Peres,Phys. Rev. B , 045401(2009).[19] Other mechanisms of gap opening see e.g., S.Y. Zhou,G.H.Gweon, A.V. Fedorov, P.N.First, W.A.De Heer,D.H.Lee, F.Guinea, A.H.Castro Neto, and A.Lanzara,Nature Materials , 770(2007); G. Giovannetti, P. A.Khomyakov, G. Brocks, P.J. Kelly, and J. van denBrink, Phys. Rev. B76