Semiclassical quantization of electrons in magnetic fields: the generalized Peierls substitution
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Semiclassical quantization of electrons in magnetic fields: the generalized Peierlssubstitution
Pierre Gosselin , Hocine Boumrar , and Herv´e Mohrbach Institut Fourier, UMR 5582 CNRS-UJF, UFR de Math´ematiques,Universit´e Grenoble I, BP74, 38402 Saint Martin d’H`eres, Cedex, France Laboratoire de Physique Mol´eculaire et des Collisions, ICPMB-FR CNRS 2843,Universit´e Paul Verlaine-Metz, 57078 Metz Cedex 3, France and Laboratoire de Physique et Chimie Quantique, Universit´e Mouloud Mammeri -BP 17, Tizi Ouzou, Algerie
A generalized Peierls substitution which takes into account a Berry phase term must be consideredfor the semiclassical treatment of electrons in a magnetic field. This substitution turns out to be anessential element for the correct determination of the semiclassical equations of motion as well as forthe semiclassical Bohr-Sommerfeld quantization condition for energy levels. A general expressionfor the cross-sectional area is derived and used as an illustration for the calculation of energy levelsof Bloch and Dirac electrons.
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Semiclassical approaches are very important in manyarea of physics for the study of the short wave length be-havior of quantum systems, including Bloch electrons incrystals or Dirac particles in external fields. An essentialingredient of these approaches is the Bohr-Sommerfeldquantization condition, whose generalization from scalarto vector wave fields has revealed new gauge structuresrelated to Berry’s phases [1]. This paper presents a de-tailed study of the semiclassical quantization for a sin-gle quantum particle in a magnetic field, exemplified byelectrons in a crystal and by Dirac electrons. This uni-fied description of a particle in a magnetic field is basedon a method of semiclassical diagonalization for an arbi-trary matrix valued Hamiltonian developed previously [2](for a generalization to higher order in ~ see [3]). Thismethod results in an effective diagonal Hamiltonian interms of gauge-covariant but noncanonical, actually non-commutative, coordinates. It will be shown that a gen-eralized Berry’s phase dependent Peierls substitution isnecessary for the establishment of the full equations ofmotion including Berry’s phase terms. This substitutionturns out to be also an essential ingredient for the Bohr-Sommerfeld quantization condition of an electron in amagnetic field. Indeed, when reformulated in terms ofthe generalized Peierls substitution, this condition leadsto a modification of the semiclassical quantization rulesas well as to a generalization of the cross-sectional areaderived independently by Roth [4] and Fal’kovskii [5] inthe context of Bloch electrons. Semiclassical diagonalization.
Let us consider a systemof a quantum particle in an uniform external magneticfield B = ∇ × e A described by an arbitrary matrix val-ued Hamiltonian H ( Π , R ), where Π = P + e e A ( R ) is thecovariant momentum and e > H ( Π , R ) = H m ( Π ) + ϕ ( R ) where H m ( Π ) is the pure magnetic part and ϕ ( R ) is the ex-ternal electric potential. In this paper we will be mainly interested by the magnetic contribution. The exact di-agonalization of this matrix valued operator through anunitary matrix U ( Π ) is in general not known, and in thispaper we apply a recursive diagonalization procedure de-veloped previously by two of the authors. This procedureis based on a series expansion in the Planck constantof the required diagonal Hamiltonian [2][3]. By diago-nal Hamiltonian it is meant a matrix representation withblock-diagonal matrix elements associated with energyband subspaces . The method is based on the knowledgeof the zero-order diagonal representation ε = U H U +0 where U is the zero-order transformation matrix, H theHamiltonian H m , in which the components Π i are con-sidered formally as classical, and therefore commutingoperators. Quantum corrections are then re-introducedto yield the expression for the diagonal Hamiltonian H d = U ( Π ) HU + ( Π ) which, if we limit ourselves tothe semiclassical order (the semiclassical condition beingthat the radius of curvature of the orbit is large in com-parison with wavelength), has diagonal operator elementslabelled by the energy index n which reads :( H d ) nn = ε n ( π n ) + ϕ ( r n ) − e ~ M . B (1)Here ε n ( π ) is the zero-order matrix element of ε (it canitself be a matrix as for a Dirac Hamiltonian, in whichcase a block-diagonalization is considered) in which clas-sical variables are now replaced by the quantum opera-tors π n = Π + ~ A π and r n = R + ~ A r , where we havedefined the Berry connection as being the projection onthe n th energy band A r/π = P n (cid:2) A r/π (cid:3) of the matrix A r/π = ± i (cid:2) U ∇ π/r U + (cid:3) . It turns out that in the mag-netic case the Berry connection in momentum has thefollowing expression A π = − e A r × B . Looking at Eq. (1)we have proven that instead of the Peierls substitution[6], which amounts to replace the canonical momentum P by the covariant one Π in the energy band ε n , one hasto consider a generalization of the Peierls substitution viaTypeset by REVTEXthe noncanonical covariant momentum π n = Π − e ~ A × B (2)(where we now use the notation A ≡ A r ) . The lastterm in Eq. (1) is the coupling between the uniformmagnetic field and the magnetic moment defined as M ( π ) = i P n ([ ε, A ] × A ) = ~ P n (cid:18) · A × A (cid:19) . Note thatit is common (especially in solid state physics [7]) towrite the matrix elements of the components of M as M i = i ~ ε ijk P m = n ( · A j ) nm ( · A k ) mn ε n − ε m (where we used · A nm = i ~ ( ε n − ε m ) A nm ), which thus depend on the band-to-band matrix element of A .The appearance of the Berry connection allows usto define naturally non-Abelian (in general) Berry cur-vatures Θ ij ( π ) = ∂ r i A j − ∂ r j A i + (cid:2) A i , A j (cid:3) where forsimplicity we omit now band indices. Position oper-ators then satisfy an unusual non-commutative alge-bra (cid:2) r i , r j (cid:3) = i ~ Θ ij . The generalized covariant mo-mentum satisfy an algebra (cid:2) π i , π j (cid:3) = − ie ~ ε ijk B k + ie ~ ε ipk ε jql Θ pq B k B l slightly corrected with respect tothe usual one (cid:2) Π i , Π j (cid:3) = − ie ~ ε ijk B k by a term of order ~ B which can in general be neglected. The Heisenbergrelations between the coordinate and the momentum (cid:2) r i , π j (cid:3) = i ~ δ ij + ie ~ ε jlk Θ il B k is also slightly changedbut by a term of order ~ B . This contribution which is adirect consequence of introducing the generalized covari-ant momentum was overlooked in previous works, withthe exception of Bliokh’s work on the specific case of theDirac equation [8]. It turns out that this term is essentialfor the determination of the genuine semiclassical equa-tions of motion which are · r = ∂ E /∂ π − ~ ˙ π × Θ( π )˙ π = − e E − e ˙ r × B (3)where we defined E ≡ ε − e ~ M . B . As consequence ofthe non-commutative algebra, the velocity equation iscorrected by an anomalous velocity term ˙ π × Θ, wherethe vector Θ defined as Θ i = ε ijk Θ jk / n th band,associated to the electron motion in the n th energy band.These equations of motion, where first derived in solidstate physics context in [9] (see also [10]) by consideringthe evolution of the wave packet of a Bloch electron inan electromagnetic field. In this picture, it is the meanover wave packets of the operator r corresponding thusto the wave-packet center r c and the mean of π givingthe mean wave vector π c that are the variables in Eq (3).The operatorial approach reveals first that the operator π is in fact a generalized covariant momentum operatorwhich replaces the Peierls substitution, and second, thatthe operatorial equations of motion are not restricted toBloch electrons in a magnetic field but are valid for anyphysical system described by an arbitrary matrix valued Hamiltonian of the kind H ( Π , R ) = H m ( Π ) + ϕ ( R ) . Inparticular they are also valid for Dirac particles movingin an electromagnetic field.Note that in solids, for crystals with simultaneoustime-reversal and spatial inversion symmetry, the Berrycurvature and the magnetic moment vanish identicallythroughout the Brillouin zone [9]. This is the case formost applications in solid state physics, but there are sit-uations where these symmetries are not simultaneouslypresent as in GaAs, where inversion symmetry is broken,or in ferromagnets, which break time reversal symme-tries. In the same way, the presence of a strong mag-netic field, the magnetic Bloch bands corresponding tothe unperturbated system breaks the time inversion sym-metries. In all these cases the dynamical and transportproperties must be described by the full equations of mo-tion given by Eq. (3). In the case of Dirac particles, boththe Berry curvature and the magnetic moment are nonzero and the full equations of motion have to be consid-ered [2][8].
Bohr-Sommerfeld quantization.
Having shown the ne-cessity of the generalized Peierls substitution for the de-termination of the semiclassical equations of motion, wenow investigate the relevance of this new concept at thelevel of the semiclassical quantization of the energy levelsfor an electron motion in an external uniform magneticfield only ( ϕ = 0), so that Eq. (3) becomes ˙r = D (cid:0) ∂ E ∂ π (cid:1) and · π = − eD (cid:0) ∂ E ∂ π × B (cid:1) with D − = 1 − e ~ BΘ . For con-venience, B is chosen to point in the z -direction B = B k . Consequently the orbits satisfies the conditions E =constand π z =const. The semiclassical quantization of energylevels can be done according to the Bohr-Sommerfeldquantization rule I P ⊥ dR ⊥ = 2 π ~ ( n + 1 /
2) (4)where P ⊥ is the canonical momentum in the plane per-pendicular to the axis π z = cte. The integration is takenover a period of the motion and n is a large integer.Now, it turns out to be convenient to choose the gauge e A y = BX, e A x = e A z = 0 . In this gauge, one has π z = P z = cte , and the usual covariant momentumΠ y = P y + eBX. As BX = B ( x − ~ A x ) the general-ized covariant momentum defined as π y = Π y + e ~ B A x becomes π y = P y + eBx (5)which is formally the same relation as the one betweenthe canonical variables, but now relating the new co-variant generalized dynamical operators. This relationwith the help of the equations of motion gives · P y = · π y − eB · x = 0 thus P y is a constant of motion so that H P y dY = P y H dY = 0 and Eq.(4) becomes simply H P x dX = 2 π ~ ( n + 1 / . Now using the definition ofthe generalized momentum P x = π x + e ~ A y B and thedifferential of the canonical position dX = dx − ~ d A x = dπ y eB − ~ d A x , the Bohr-Sommerfeld condition Eq. (4) be-comes I π x dπ y = 2 π ~ eB (cid:18) n + 12 − π I A ⊥ d π ⊥ (cid:19) (6)where the integral is now taken along a closed trajec-tory Γ in the π space and π H A ⊥ d π ⊥ = φ B is theBerry phase for the orbit Γ. It is interesting to notethat in terms of the usual covariant momentum (Peierlssubstitution) we have instead of Eq. (6) the condition H Π x d Π y = 2 π ~ eB ( n + 1 / . The integration in Eq.(6) defines the cross-sectional area S ( ε, π z ) of the or-bit Γ which is the intersection of the constant energysurface ε ( π ) =const and the plane π z =const. There-fore the condition Eq. (6) implicitly determines the en-ergy levels ε n ( π z ) . Computing now the cross-sectionalarea S ( E , π z ) = S ( ε − e ~ M z B, π z ) ≈ S ( ε, π z ) + dS, with dS = H dκd π ⊥ the area of the annulus betweenthe energy surface ε =const and the surface ε + dε with dε = − e ~ M z B, and where dκ = q dπ x + dπ y is an ele-mentary length of the π orbit. Then, as dS can be written dS = H dεdκ | ∂ε/∂ π ⊥ | = − e ~ B H M z dκ | ∂ε/∂ π ⊥ | where the integral istaken over the orbit Γ, one has finally S ( E , π z ) = 2 π ~ eB (cid:18) n + 12 − φ B − π I M z ( π ) dκ | ∂ε/∂ π ⊥ | (cid:19) (7)It is common to write S ( E , π z ) = 2 π ~ eB ( n + γ ) definingthus the coefficient γ − = − φ B − π H M z dκ | ∂ε/∂ π ⊥ | . Thiscoefficient can also be written in a different form γ −
12 = − π I [ e v × A + M ] z dκ | ∂ε/∂ π ⊥ | (8)with e v ≡ ∂ε/∂ π . Eq. (8) is a generalization of a previousresult found by Roth [4] and Fal’kovskii [5], in the spe-cific context of Bloch electrons in a magnetic field. Theconnection with Berry’s phase was seen later by Mik-itik and Sharlai [11]. In both [4] and [11], the term[ e v × A + M ] was written as P n (cid:2)(cid:0) Π m + v (cid:1) × A (cid:3) where v = Π m + ~ · A is the velocity operator before projectionon a band, and Π = m · R , a relation valid only for aHamiltonian whose kinetic energy is Π / m. ThereforeEq. (7) is more general and has a broader field of appli-cation, as it is a general result which applies for any kindof single quantum particle system in a magnetic field,including Bloch and Dirac electrons. Importantly thederivation provided here is new, and it turns out to bethe result of the generalized Peierls substitution in theBohr-Sommerfeld condition.
Bloch electron.
In a crystal, the Berry gauge A ( k ) is Abelian (a scalar operator), written in termsof the periodic part of the Bloch wave | u n ( k ) i as A ( k ) = i h u n ( k ) | ∂ k | u n ( k ) i , where k is the generalized co-variant pseudo momentum ( k = π / ~ ). Application ofEq. (7) for electron trajectories in a crystal with timereversal and spatial inversion symmetry, where it is ex-pected that, both Θ and M vanish in the Brillouin zone,has been studied by Mikitik and Sharlai [11]. But theseauthors also pointed out the fact that the Berry’s phaseis non zero when the electron orbit surrounds the band-contact line of a metal, actually φ B = ± /
2. Conse-quently, γ = 0 in this case, instead of the previously sup-posed constant value γ = 1 / γ can allows the detectionof band contact lines.As a simple application of Eq. (7) consider a crys-tal with time reversal and spatial inversion symmetry,and where the Fermi surface is an ellipsoid of revolutioncharacterized by two effective masses, a transverse m ⊥ and a longitudinal m l one. The energy levels can eas-ily be deduced. Indeed E = ~ (cid:16) k ⊥ m ⊥ + K z m l (cid:17) and thecross-sectional area S ( E , K z ) is a disc of radius square k ⊥ = 2 m ⊥ (cid:16) E / ~ − K z m l (cid:17) so that the energy levels are E n = eB ~ m ⊥ (cid:0) n + (cid:1) + ~ K z m l which actually coincide withthe exact ones because the energy levels of an harmonicoscillator keep their form at large n . Dirac electron.
Let us consider the Dirac Hamiltonian H = α . Π + βm in the presence of an uniform magneticfield, with α and β the usual (4 ×
4) Dirac matrices. Thesemiclassical block-diagonalization followed by a projec-tion on, say, the positive energy subspace, leads to the(2 ×
2) matrix valued energy operator E = ε − e ~ M . B where ε = √ π + m ( c = 1) and the magnetic momentis given by M = σ ε − L ε , with L = π × A representingthe intrinsic orbital angular momentum [8][2]. It turnsout that for Dirac, the magnetic moment can also be ex-pressed as M = ε Θ , with the curvature vector given bythe matrix [8][2] Θ ( π ) = − ε (cid:20) m σ + ( σ . π ) π ε + m (cid:21) with σ the Pauli matrices. Berry’s connection is definedas A= i h + , π | ∂ π | + , π i where | + , π i is two componentsspinor of the positive energy subspace. Consider B point-ing in the z -direction so that π z = P z =const, with thegoal to compute the Landau energy levels (LEL) as an ap-plication of Eq. (7) . As the cross-sectional area S ( ε, P z )is a disc of radius square π ⊥ = ε − m − P z , the applica-tion of Eq. (7) consists in replacing ε by E in π ⊥ so thatwe have S ( E , P z ) = π (cid:0) E n − m − P z (cid:1) , which yields thesemiclassical quantized LEL through the relation E n − m − P z = 2 ~ eB (cid:18) n + 12 − φ B − π I M z dκ | ∂ε/∂ π ⊥ | (cid:19) Now from the Berry connection A = π × σ ε ( ε + m ) we de-duce the Berry’s phase φ B = − τ + τ (cid:16) m ε + P z ε ( ε + m ) (cid:17) where τ = ± σ z . Berry’s phase is the sum of a topological part − τ and a non-topological τ (cid:16) m ε + P z ε ( ε + m ) (cid:17) one. The contri-bution from the magnetic moment yields π H M z dκ | ∂ε/∂ π ⊥ | = − τ (cid:16) m ε + P z ε ( ε + m ) (cid:17) a term which exactly cancels the non-topological contribution of φ B , so that finally E n = s m + 2 ~ Be (cid:18) n + 12 + τ (cid:19) + P z It turns out in this example that the semiclassical energyquantization coincides also with the exact result. It isusually expected that for a massless Dirac particle theBerry’s phase takes the topological value φ B = ± /
2, asa consequence of the band degeneracy at zero momentum[13]. This is not the case here because the magnetic fieldlifts this degeneracy as P z is not zero in, general. Butit turns out that the magnetic moment contribution ex-actly compensates for the non-topological Berry’s phasecontribution. This cancellation can be easily understoodfrom the expression Eq. (8) for the coefficient γ. Indeedfrom the equality [ e v × A + M ] z = (cid:2) π × A ε (cid:3) z + τ ε − L z ε = τ ε we deduce the expected result γ = + τ = 0 or 1.For a two-dimensional Dirac system it is therefore ex-pected that the magnetic moment for massless parti-cles exactly vanishes, and that the Berry’s phase takesthe topological value φ B = ± / . The electron motionin graphene is an interesting physical situation whichillustrates this assertion. Indeed, graphene is a two-dimensional carbon crystalline honeycomb structure withinversion symmetry so that M =0. The hexagonal Bril-louin zone has two distinct and degenerate Dirac pointsor valleys (labelled by τ ±
1) where the conduction andvalence bands meet and the electronic excitations behavelike massless relativistic fermions, so that φ B = ± / E n = ± q ~ eB (cid:0) n + + τ (cid:1) [14]. Thereforethe ground state is not degenerate as there is only onepossibility to realize it n = 0. This result explains thepeculiar quantum Hall effect of graphene [15]. Summary . We have shown that a generalized Peierlssubstitution including a Berry phase term must be con-sidered for a correct semiclassical treatment of electronsin a magnetic field. This substitution is essential for thedetermination of the full semiclassical equations of mo-tion, as well as for the semiclassical Bohr-Sommerfeld quantization condition for energy levels. Indeed, the sub-stitution in the Bohr-Sommerfeld condition leads to anexpression for the cross-sectional area which in some sortgeneralizes the formula found by Roth and Fal’kovskii inthe context of Bloch electrons in a crystal. Application ofthis formula to Dirac electrons shows the subtle cancella-tion mechanism between the magnetic moment and thenon-topological part of the Berry’s phase, which yieldsthe Landau energy levels.
Acknowledgement . The authors acknowledge fruitfuldiscussions with F. Pi´echon and J.N. Fuchs. We alsothank L. A. Fal’kovskii for having drawn our attentionto his own work on this subject. [1] W. G. Flynn and R. G. Littlejohn, Phys. Rev. Lett 66(1991) 2839; R. G. Littlejohn and W. G. Flynn, Phys.Rev. A 44 (1991) 5239.[2] P. Gosselin, A. B´erard and H. Mohrbach, Eur. Phys. J.B 58 (2007) 137;[3] P. Gosselin, J. Hanssen and H. Mohrbach, Phys. Rev. D77 (2008) 085008, P. Gosselin and H. Mohrbach,
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