Graphene coherent states

DOI 10.1140/epjp/i2017-11794-y Regular Article Eur. Phys. J. Plus (2017) 132: 499 THE EUROPEAN PHYSICAL JOURNAL PLUS Graphene coherent states Erik D´ıaz-Bautistaaand David J. Fern´andezb Physics Department, Cinvestav, P.O. Box. 14-740, 07000 Mexico City, Mexico Received: 2 September 2017 / Revised: 21 October 2017 Published online: 29 November 2017 – c Societ`a Italiana di Fisica / Springer-Verlag 2017 Abstract. In this paper we will construct the coherent states for a Dirac electron in graphene placed in a constant homogeneous magnetic field which is orthogonal to the graphene surface. First of all, we will identify the appropriate annihilation and creation operators. Then, we will derive the coherent states as eigenstates of the annihilation operator, with complex eigenvalues. Several physical quantities, as the Heisenberg uncertainty product, probability density and mean energy value, will be as well explored. 1 Introduction In the quantum mechanical treatment of the harmonic oscillator, the first task typically is to determine the eigenstates and eigenvalues of the Hamiltonian, looking for them either by solving the corresponding second-order ordinary differential equation with appropriate boundary conditions or through algebraic techniques. It is just when analyzing the result of the calculation for the mean values of the position and momentum operators in these states, both becoming zero, that one realizes the impossibility of choosing directly the bound states as semiclassical models (see e.g., [1]). This is why Schr¨odinger looked for an alternative set of states, whose mean values would follow the oscillator classical trajectory in phase space [2]; nowadays they are called coherent states (CS), after Glauber coined the name in the sixties of the previous century, based on the coherence properties of these states [3,4]. The CS have proved very fruitful in several areas of physics, e.g., the coherent states are used often to analyze the border between classical and quantum mechanics (the so-called semiclassical regime), they have been employed as models of light in quantum optics, they supply alternatives to the coordinates or momentum representations, they have shed light onto the quantization procedures, etc. [5–8]. Retrospectively, one can observe that the coherent state approach represents an important second stage in the study of a quantum mechanical system. There are several coherent state definitions for systems different from the harmonic oscillator (see, e.g., [7]); thus, to proceed with a CS analysis first one has to fix the definition to be used: whenever is possible to identify an annihilation operator of the system, the coherent states which are eigenstates of such an operator seems the most natural available choice. On the other hand, it is well known that graphene is a single layer of carbon atoms arranged in a hexagonal honeycomb lattice, which is the basic structural element of other carbon allotropes. Since its conduction and valence bands meet at the so-called Dirac points on the edge of the Brillouin zone (see fig. 1), it is considered a zero-gap semiconductor [9–14]. At low energies, close to a Dirac point, the electrons can be described by an equation which is formally equivalent to the massless (or ultra-relativistic) Dirac equation: −ivFσ·∇Ψ(r)=EΨ(r),(1) where vF∼106m/s is the Fermi velocity, which replaces the velocity of light in Dirac theory, the components of the vector σare the Pauli matrices, Ψ(r) is the two-component wave function of the electrons and Eis its energy [15]. Consequently, the electrons and holes in graphene are called Dirac fermions [16–22], which emerge naturally from a tight-binding model for a generic hexagonal lattice in the low-energy regime [23]. In fact, graphene belongs to a class of systems in condensed matter for which the low-energy quasi-particles behave like massless or massive Dirac fermions. These systems are known as Dirac materials in the literature [24]. It is worth to notice that, when Dirac fermions are compared with ordinary electrons in magnetic fields, their behavior leads to new physical phenomena, such as the anomalous integer quantum Hall effect, the Zitterbewegung ae-mail: [email protected] be-mail: [email protected] Page 2 of 13 Eur. Phys. J. Plus (2017) 132: 499 Fig. 1. Left: Lattice structure of the graphene, where the sublattices are labeled by A and B. Right: Brillouin zone for the graphene. The Dirac cones appear at the Kand Kpoints. and the Klein paradox [16,25]. We shall see below that under particular physical conditions, a problem similar to that considered in [26] will arise naturally. Motivated by this, it seems clear to us the need to build up the coherent states for graphene in static magnetic fields, i.e., to address already the second stage in the quantum mechanical analysis of this problem. Let us note that, up to now, the only work we have detected in which the CS for this system have been studied, for a strong homogeneous magnetic field, was done by Wu et al. through group theoretical methods [27]. In order to start implementing our program, this paper has been organized as follows. In sect. 2 the DiracWeyl equation will be introduced and the physical problem to be considered will be briefly discussed. In sect. 3 the annihilation operator associated to our system will be defined, and the corresponding coherent states will be constructed as eigenstates of that operator. We will analyze as well several physical quantities for these states. Our conclusions will be presented in sect. 4. 2 Dirac-Weyl equation Let us suppose that the graphene is placed in a static magnetic field which is orthogonal to the material surface (the x−yplane) [15,28,29]. The interaction of a Dirac electron with such a field close to a Dirac point Kin the Brillouin zone is described by the Dirac-Weyl equation, which is obtained by replacing in eq. (1) the momentum operator p=−i∇by p+eA/c, leading to vFσ·p+eA cΨ(x, y)=EΨ(x, y),(2) where −eis the charge of the electron. Landau gauge is conveniently chosen, with the vector potential given by A=A(x)ˆeyand B=∇×A=B(x)ˆez. Note that there is a translational invariance along the y-direction, then the spinor Ψ(x, y) can be expressed as Ψ(x, y) = exp(iky)ψ+(x) iψ−(x),(3) with kbeing the wave number in the y-direction and ψ±(x) describing the electron amplitude on two adjacent sites in the unit cell of graphene. Thus, the solutions of the Dirac-Weyl equation have an internal degree of freedom that mimics the spin, called pseudospin. It admits the interpretation that each component is the projection of the particle wavefunction onto the sublattice A (spin up) or B (spin down). Substituting now eq. (3) in eq. (2), the Dirac-Weyl equation yields two coupled first-order linear differential equations, ±d dx+e cA(x)+kψ∓(x)= E vF ψ±(x),(4) which can be easily decoupled into two Schr¨odinger equations H±ψ±(x)=Eψ±(x), where [29] H±=−d2 dx2+V±,V ±=eA(x) c+k2 ±e c dA(x) dx,E=E2 2v2 F .(5) Eur. Phys. J. Plus (2017) 132: 499 Page 3 of 13 For a constant magnetic field, orthogonal to the graphene surface and pointing in the positive z-direction (B=B0ˆez with B0>0), the vector potential is selected as A=B0xˆey. Introducing now the constant ωas B0=c 2eω→ω=2eB0 c, whose dimensions are (lenght)−2, the potentials in eq. (5) become two shifted oscillators of the form V±=ω2 4x+2k ω2 ±1 2ω. (6) Thus, the eigenvalues E± nfor the Hamiltonians H±are related as follows: E− 0=0,E− n=E+ n−1=nω, n =1,2,..., (7) and the associated eigenfunctions are those of the standard harmonic oscillator, ψ± n(x)=1 2nn!ω 2π1/2Hnω 2x+2k ωexp −ω 4x+2k ω2,(8) where Hn[·] denotes the Hermite polynomial of degree n∈N. We conclude that the complete solution of the corresponding Dirac-Weyl equation in a constant magnetic field consists of the eigenvalues E± n=±vF√nω, n =0,1,..., (9) where the plus (minus) sign refers to the enegy electrons (holes), and the normalized eigenvectors Ψ0(x, y) = exp(iky)0 iψ− 0(x),Ψ n(x, y)=exp(iky) √2ψ+ n−1(x) iψ− n(x),n=1,2,... . (10) 3 Annihilation operator Since the eigenstates of the previous Dirac-Weyl equation are expressed in terms of the eigenfunctions of the standard harmonic oscillator, it seems natural to look for an annihilation operator for the Hamiltonian in eq. (2). In fact, let ˆ A−be the operator defined by ˆ A−=f1(ˆ N)ˆ ϑ−0 0f(ˆ N+ˆ 1)ˆ ϑ−,(11) where ˆ ϑ±,ˆ Nare given by ˆ ϑ−=1 √2(z+∂z),ˆ ϑ+=1 √2(z−∂z),ˆ N=ˆ ϑ+ˆ ϑ−, with z=ω/2(x+2k/ω), and f,f1are two real adjustable functions which will be used to guarantee that ˆ A−Ψn= cnΨn−1. Then, ˆ A−Ψn= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0forn=0 f(1) √2Ψ0for n=1 1 √2exp(iky)√n−1f1(n−2)ψn−2 √nf(n)iψn−1for n=2,3,... . (12) In order to ensure that ˆ A−Ψn=cnΨn−1,(13) it must happen that √n−1f1(n−2) = √nf(n),n=2,3,... . (14) Page 4 of 13 Eur. Phys. J. Plus (2017) 132: 499 In such a case it is obtained that cn=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0forn=0, f(1) √2for n=1, √nf(n)forn=2,3,..., (15) and the explicit expression for the annihilation operator ˆ A−turns out to be ˆ A−=⎛ ⎜ ⎝ˆ N+ˆ 2 ˆ N+ˆ 1 f(ˆ N+ˆ 2)ˆ ϑ−0 0f(ˆ N+ˆ 1)ˆ ϑ− ⎞ ⎟ ⎠.(16) Equation (15) indicates that the explicit form of the function f(n) is required to determine the complete action of the annihilation operator ˆ A−onto the eigenstates Ψn. It is also quite important for the properties of the graphene coherent states, as it will be immediately seen. 3.1 Coherent states as eigenvectors of ˆ A− Let us define the coherent states Ψαas eigenstates of the annihilation operator ˆ A−with complex eigenvalue α ˆ A−Ψα=αΨα,α∈C.(17) Expressing Ψαas a linear combination of the states Ψn,wehave Ψα(x, y)= ∞  n=0 anΨn(x, y)=a00 iψ− 0(x)+∞  n=1 an √2ψ+ n−1(x) iψ− n(x)exp(iky).(18) Using eq. (17) we find a recurrence relation for the coefficients anwhich depends on the value taken by f(1). We can identify two different cases. 3.1.1 Case with f(1) =0 First of all suppose that f(n)=0∀n=1,2,... . Thus we get a1=√2αa0/f(1) and an+1 =αna1 (n+ 1)!f(n+1)···f(2) =√2αn+1 a0 (n+ 1)! [f(n+ 1)]! ,(19) where [f(n)]! ≡1forn=0, f(1) ···f(n)forn=1,2,... . The free constant a0is used to normalize Ψα; we obtain: Ψα(x, y)=1+ ∞  n=1 2|α|2n n!([f(n)]!)2−1/2Ψ0(x, y)+ ∞  n=1 √2αn √n![f(n)]!Ψn(x, y).(20) 3.1.2 Case with f(1) = 0 If f(1) = 0 we obtain that a0= 0 and the following recurrence relationship: an+1√n+1f(n+1)=αa n,n=1,2,... . (21) Now, depending on the value of f(2), two possibilities appear once again. Eur. Phys. J. Plus (2017) 132: 499 Page 5 of 13 A. Case with f(2) =0. If we suppose that f(n)=0∀n=2,3,... and define g(n)≡f(n+ 1), eq. (21) leads to an+1 =αn (n+ 1)![g(n)]!a1.(22) Substituting this expression in eq. (18) and then normalizing we obtain Ψα(x, y)=∞  n=0 |α|2n (n+ 1)! ([g(n)]!)2−1/2∞  n=0 αn (n+ 1)! [g(n)]!Ψn+1(x, y).(23) B. Case with f(2) = 0. On the other hand, if f(2) = 0 and f(n)=0∀n=3,4,..., the normalized coherent states turn out to be now Ψα(x, y)=∞  n=0 |α|2n (n+ 2)! ([h(n)]!)2−1/2∞  n=0 αn (n+ 2)! [h(n)]!Ψn+2(x, y),(24) where h(n)≡f(n+ 2). Let us notice that the graphene coherent states of eqs. (20), (23), (24) look similar to the so-called vector coherent states. For information concerning the last states, the reader can refer to, e.g., [30–32] and references therein. 3.2 Mean values and the Heisenberg uncertainty relation Let the dimensionless position and momentum operators be given by ˆz=1 √2(ˆ ϑ++ˆ ϑ−),ˆp=i √2(ˆ ϑ+−ˆ ϑ−).(25) In units of , the Heisenberg uncertainty relation is expressed by σ2 zσ2 p≥1 4,(26) where σ2 S≡ˆ S2−ˆ S2for an arbitrary observable ˆ S. We will calculate next these quantities for some examples of the coherent states, which will stress the important role played by the function f(n) in our treatment. 3.2.1 The case with f(1) =0 Let us consider first the particular choice f(ˆ N)=ˆ 1. Thus, eq. (20) leads to Ψα(x, y)= 1 2exp(r2)−1Ψ0(x, y)+ ∞  n=1 √2αn √n!Ψn(x, y),(27) where r=|α|. Using these coherent states, the mean values for the operators ˆz,ˆpof eq. (25) as well as their squares become ˆzα=√2Re(α) 2exp(r2)−1exp(r2)+ ∞  n=1 r2n Γ(n)Γ(n+2),(28a) ˆz2α=1 4exp(r2)−21+4r2exp(r2) + 2[[Re(α)]2−[Im(α)]2]exp(r2)+ ∞  n=1 √n+1r2n Γ(n)Γ(n+3),(28b) ˆpα=√2Im(α) 2exp(r2)−1exp(r2)+ ∞  n=1 r2n Γ(n)Γ(n+2),(28c) ˆp2α=1 4exp(r2)−21+4r2exp(r2)−2[[Re(α)]2−[Im(α)]2]exp(r2)+ ∞  n=1 √n+1r2n Γ(n)Γ(n+3).(28d) Through them it is straightforward to calculate (σz)2 α(σp)2 α(see fig. 2). Note that in the limit α→0wehave (σz)2 α(σp)2 α→1/4. Page 6 of 13 Eur. Phys. J. Plus (2017) 132: 499 Fig. 2. Heisenberg uncertainty relation (σz)2 α(σp)2 αas function of αfor f(n)=1. 3.2.2 The case with f(1) = 0 As we saw in sect. 3.1.2, when f(1) = 0 two options appear, which depend on the value taken by f(2). A. The case with f(2) =0. Let us choose now f(ˆ N+ˆ 1) = g(ˆ N)= √ˆ N √ˆ N+ˆ 1,sothatf(n)=0∀n=2,3,... . From eq. (23), the explicit form for the normalized coherent states becomes Ψα(x, y) = exp(−r2/2) ∞  n=0 αn √n!Ψn+1(x, y).(29) The mean values for the operators of eq. (25) and their squares become ˆzα=Re(α) √21 + exp(−r2)∞  n=0 √n+2r2n Γ(n+1)Γ(n+2),(30a) ˆz2α=exp(−r2)∞  n=0 (n+1)r2n Γ(n+1) +[[Re(α)]2−[Im(α)]2] 21+exp(−r2)∞  n=0 √n+3r2n Γ(n+1)Γ(n+2),(30b) ˆpα=Im(α) √21 + exp(−r2)∞  n=0 √n+2r2n Γ(n+1)Γ(n+2),(30c) ˆp2α=exp(−r2)∞  n=0 (n+1)r2n Γ(n+1) −[[Re(α)]2−[Im(α)]2] 21+exp(−r2)∞  n=0 √n+3r2n Γ(n+1)Γ(n+2).(30d) In the limit α→0 it turns out that (σz)2 α(σp)2 α→1 (see fig. 3). B. The case with f(2) = 0. Let us consider finally that f(ˆ N+ˆ 2) = h(ˆ N)= ˆ N√ˆ N+ˆ 1 √ˆ N+ˆ 2. The explicit expression for the normalized coherent states arises from eq. (24): Ψα(x, y)= 1 0F2(1,2; r2) ∞  n=0 αn n!(n+ 1)!Ψn+2(x, y),(31) where pFqis a generalized hypergeometric function defined by pFq(a1,...,a p,b 1,...,b q;x)= Γ(b1)...Γ(bq) Γ(a1)...Γ(ap) ∞  n=0 Γ(a1+n)...Γ(ap+n) Γ(b1+n)...Γ(bq+n) xn n!. Eur. Phys. J. Plus (2017) 132: 499 Page 7 of 13 Fig. 3. Heisenberg uncertainty relation (σz)2 α(σp)2 αas function of αfor f(n)=√n−1/√n. The mean values for the operators in eq. (25) and their squares are now ˆzα=Re(α) √20F2(1,2; r2)0F2(2,2; r2)+ ∞  n=0 √n+3r2n n![Γ(n+ 2)]3Γ(n+3),(32a) ˆz2α=1 20F2(1,2; r2)2∞  n=0 (n+2)r2n Γ(n+ 2)[Γ(n+ 1)]2+[[Re(α)]2−[Im(α)]2] ×0F2(2,3; r2) 2+∞  n=0 √n+3r2n Γ(n+1) Γ(n+ 2)[Γ(n+ 3)]3,(32b) ˆpα=Im(α) √20F2(1,2; r2)0F2(2,2; r2)+ ∞  n=0 √n+3r2n n![Γ(n+ 2)]3Γ(n+3),(32c) ˆp2α=1 20F2(1,2; r2)2∞  n=0 (n+2)r2n Γ(n+ 2)[Γ(n+ 1)]2−[[Re(α)]2−[Im(α)]2] ×0F2(2,3; r2) 2+∞  n=0 √n+3r2n Γ(n+1) Γ(n+ 2)[Γ(n+ 3)]3.(32d) In the limit α→0 we get (σz)2 α(σp)2 α→4 (see fig. 4). As we can see, the Heisenberg uncertainty relation depends strongly on the coherent states under consideration. Thus, for the states in eq. (27) it takes its minimum at α= 0, while for those in eqs. (29) and (31) their maxima are reached at the same point; this is so since the lowest-energy eigenstate involved in the corresponding linear combination is different if different families of coherent states are taken into account (see also [33–35]). 3.3 Magnetic field and probability density The probability density ρ=Ψ† αΨαwill be used to analyze the properties of the graphene coherent states. It will depend on the following matrix elements [28]: ρn,m(x):=ψ+ n−1(x)ψ+ m−1(x)+ψ− n(x)ψ− m(x)=ρm,n(x).(33) Note also that, according to eq. (8), it depends on the magnetic field intensity B0through the parameter ω. Page 8 of 13 Eur. Phys. J. Plus (2017) 132: 499 Fig. 4. Heisenberg uncertainty relation (σz)2 α(σp)2 αas function of αfor f(n)=(n−2)√n−1/√n. Fig. 5. Probability density ρ(x, r, θ)forf(n)=1withB0=1/8 (above), B0= 2 (below), θ=0,π/4, π/2 (left to right, respectively) and k= 1. The blue, red and brown lines correspond to r=1,4,5, respectively. 3.3.1 Probability density for f(1) =0 A straightforward calculation using eq. (27) leads to ρ(x, r, θ)= 1 2exp(r2)−1∞  m=1 ∞  n=1 rm+ncos (n−m)θ √m!n!ρn,m(x)+2 ∞  n=1 rncos(nθ) √n!ψ− n(x)ψ− 0(x)+(ψ− 0(x))2,(34) where θ=Arg[α], r=|α|. Plots of this probability density for two magnetic field intensities and different θ’s and r’s are shown in fig. 5. Eur. Phys. J. Plus (2017) 132: 499 Page 9 of 13 Fig. 6. Probability density ρ(x, r, θ)forf(n)=√n−1/√nwith B0=1/8 (above), B0= 2 (below), θ=0,π/4, π/2(leftto right, respectively) and k= 1. The blue, red and brown lines correspond to r=1,3,5, respectively. Fig. 7. Probability density ρ(x, r, θ)forf(n)=(n−2)√n−1/√nwith B0=1/8 (above), B0= 2 (below), θ=0,π/4, π/2 (left to right, respectively), and k= 1. The blue, red and brown lines correspond to r=1,50,100, respectively. 3.3.2 Probability density for f(1) = 0 A. Case with f(2) =0. On the other hand, for the states of eq. (29) we get (see fig. 6) ρ(x, r, θ)=exp(−r2) 2 ∞  m=0 ∞  n=0 rm+ncos (n−m)θ Γ(m+1)Γ(n+1)ρn+1,m+1(x).(35) B. Case with f(2) = 0. Finally, by employing the states of eq. (31) we arrive at (see fig. 7) ρ(x, r, θ)= 1 20F2(1,2; r2) ∞  m=0 ∞  n=0 rm+ncos (n−m)θ m!n!Γ(m+2)Γ(n+2)ρn+2,m+2(x).(36)