Energy spectrum for charge carriers in graphene with folded deformations or uniaxial flexural modes: analogies to the quantum Hall effect under random pseudo-magnetic fields
EEnergy spectrum for charge carriers in graphene with folded deformations or withflexural modes with Gaussian and L´evy distributed random pseudo-magnetic fields
Abdiel E. Champo and Gerardo G. Naumis ∗ Departamento de Sistemas Complejos, Instituto de Fisica, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 20-364,01000,Ciudad de M´exico, M´exico. (Dated: February 16, 2021)The electronic behaviour in graphene under a flexural field with random height displacements,considered as pseudo-magnetic fields, is studied. General folded deformations (not necessarily ran-dom) were first studied, giving an expression for the zero energy modes. For Gaussian foldeddeformations, it is possible to use a Coulomb gauge norm for the fields allowing contact with pre-vious work on the quantum Hall effect with random fields, showing that the density of states has apower law behaviour and that the zero energy modes wavefunctions are multifractal. This hints ofan unusual electron velocity distribution. Also, an Aharonov-Bohm pseudo-effect is produced. Formore general non-folded general flexural strain, is not possible to use a Coulomb gauge. However, aRandom Phase Approximation (RPA) and the scheme of random matrix theory allows to tackle theproblem. For Gaussian distributed fields, the spectrum presents an average gap and for some cases,a breaking of the particle-hole symmetry. Finally, for the case of L´evy distributed fields, nearly flatbands are seen due to strong electron localization.
Keywords: graphene; random pseudo-magnetic fields; gaps; Aharonov-Bohm pseudo-effect; L´evy disorder.
I. INTRODUCTION
Recently, Dirac materials have attracted intense re-search interest following the celebrated discovery of atwo-dimensional (2D) hexagonal allotropic atomic car-bon, graphene [1], because of its peculiar band structureand its fascinating properties [2, 3] largely due to themassless Dirac fermion behavior of the charge carriers.Due to such excellent mechanical, magnetic and ther-mal properties of graphite monolayers, they can beused for the development of superconducting devices formicro-electromechanical and nano-electromechanical sys-tems, leading to the development of the next generationof nanoelectronics [4, 5]. As the use of graphene sheetsincreases, the understanding of the mechanical behaviouris necessary and important for the design and analysis ofgraphene nanostructures and nanosystems. This openeda new field of research known as straintronics, which aimsto refine the electronic and optical properties by apply-ing mechanical deformations [6]. Following this direc-tion, many theoretical works have been made studyingthe effect of mechanical strains on the electronic prop-erties [7, 8] using a tight-binding approach [9, 10] andeffective Hamiltonians for low energies in the vicinity ofDirac points [11–14]. These electronic degrees of free-dom are coupled to the structural lattice deformations,and this allows to modify its electronic properties in in-teresting ways [11, 15–18]. It has been shown that amodel to describe the coupling of the electrons to theout-of-plane deformation should be the Dirac equationin curved space [15, 19–21]. Such coupling is due to theappearance of pseudo-magnetic fields caused by the de-formations [6, 11–13, 22, 23]. ∗ naumis@fisica.unam.mx In recent years, experimental evidence has been foundthat for certain regimes, fluctuations in graphene mem-branes follow a Cauchy distribution that results in largemovements and sudden changes in curvature by means ofthe mirror buckling effect [24–26]. This mirror bucklingeffect was first related to the heating due to the scanningmicroscope. Later on, it was found that this mirror buck-ling is always presents and that the height of the flexu-ral vibrations follow a L´evy distribution with parameters α = 1 . , γ = 0 [25]. It was also found an unusual distribu-tion of electron velocities [25] and a theory has been pro-posed to explain it [27]. However, this last theory is basedon considering carbon atoms in the framework of theclassical kinetic theory of gases and the Fokker-Planck-Kolmogorov master equation, but this scheme does notexplicitly consider the contribution of out-of-plane acous-tic modes and that the membrane executes Brownian mo-tion with rare large height excursion indicative of L´evywalks. Thus, a more exhaustive study is needed concern-ing this point. Likewise, Mao et. al. [28] demonstratethat graphene monolayers placed on an atomically flatsubstrate can be forced to undergo a buckling transition,resulting in a periodically modulated pseudo-magneticfield, which in turn creates a ‘post-graphene’ materialwith flat electronic bands. This buckling of 2D crys-tals offers a strategy for exploring interaction phenomenacharacteristic of flat bands.In addition, there is an growing interest in folded defor-mations due to transport properties of strained folds ingraphene exhibit a rich behavior ranging from Coulombblockade to Fabry-P´erot oscillations for different fold ori-entations. Those exhibiting strong confinement, behaveas electronic waveguides in the direction parallel to thefold axis, providing a new way to realize 1D conduct-ing channels in 2D graphene by strain engineering [29].In general, the mechanical displacements on graphene a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b causes strong changes in the vacuum-induced shifts ofthe transition frequency of some emitter and, because itslow mass and high Q factor, make it a particular attrac-tive candidate for a wide class of sensors [30].Although most of the work concerning this topic hasbeen focused in studying the electron mobility thoroughusing transport equations [31, 32].In this work we study the effects of the charge carri-ers energy spectrum in the presence of random pseudo-electromagnetic fields which models the case of the ver-tical fluctuations due to flexural modes, because arethe large phonon population of the low-energy flexuralmodes originating from the quadratic phonon dispersionis known to dominate the electron scattering [31] andthermal transport [33, 34]. In particular, we show thatfor certain kind of flexural fields, one can make close con-tact with previous works on Dirac fermions in randomelectromagnetic potentials, besides its close relationshipwith the phase transition between the plateaus in Hall’squantum states and the quasi-excitations in d -wave su-perconductors [35]. Then we show that for more generalfields, the Coulomb gauge condition can not be fulfilledand thus we use random matrix theory to obtain the spec-trum. As a result, we find that on average, an energy gapopens between the bands, as if the electron acquires aneffective mass during this process.It is important to remark that the methods pre-sented here can be extended to study other optoelec-tronic properties in 2D materials, such as phosphorene[36] or borophene [6], and these effects can also be stud-ied using the present methodology, as plane deformationsor flexural waves can be considered as random pseudo-electromagnetic waves; in addition, the present resultscan be extended for new Dirac materials [37, 38], as wellas for other probability distributions for the fields.The work is organized as follows. In Sec. II, we in-troduce the effective Hamiltonian for low energies andobtain the time-independent Schr¨odinger equation to besolved for the problem of the behavior of charge carri-ers under pseudo-electromagnetic fields in graphene. InSec. III, we analize specifically the electronic propertiesof graphene with Gaussian white-noise folded deforma-tions. In Sec. IV, we consider a general flexural strainand using the RPA method to obtain an expression forthe effective Hamiltonian in k space. Section V, we de-scribe the distribution of the effective hamiltonian eigen-values with the random matrix theory, which in the endis what will provide us with information about the spec-trum of allowed eigenenergies. A comparison is made be-tween results obtained theoretically with those obtainedby computer calculations. In Section VI we show someof the most important results for the L´evy’s distribution.And finally, we present the conclusions in Section VII. II. HAMILTONIAN MODEL
Out-of-plane acoustic modes are characteristic vibra-tions in graphene. These low frequency modes are easy toexcite and carry most of the vibrational energy [23, 39].They consist in a dynamic elongation, bending and tor-sion of the local bonds. The stretching or tension of thebonds is by far the most important for the electrons, sinceit causes a greater impact on the tunneling parameter [4].Some lattice deformations can be expressed by a gaugefield using a Hamiltonian at low energies [11, 23].The low-energy Hamiltonian for non-interacting elec-trons in deformed graphene is a Dirac-type Hamiltonianand is given by [14, 23, 40]: ˆ H η ( r ) = v F σ η · ( ˆ p − η A ( r , t )) + V ( r , t ) I × , (1)where r = ( x, y ) is the position vector, the subscript η = ± K , K (cid:48) respectively; v F isthe Fermi velocity ( v F /c ≈ /
300 with c is the vacuumspeed of light); ˆ p = (ˆ p x , ˆ p y ) is the moment operator forthe charge carriers, σ = ( ησ x , σ y ) is the Pauli matrixvector, and A and V are the pseudo vector and scalarpotentials respectively, given by [6, 13, 14, 23] V ( r , t ) = g ( ε xx + ε yy ) (2) A ( r , t ) = ( A x , A y ) = (cid:126) β a cc ( ε xx − ε yy , − ε xy ) (3)The parameter g takes values from 0 to 20 eV [11, 41], a cc = 1 .
42 ˚A is the interatomic distance for undeformedgraphene lattice, the dimensionless coefficient β ≈ . g refers to flexural changes inthe membrane while the term (cid:126) β/ a cc refers to changesin bond length, as we know it requires more energy tomake bond length changes to a rearrangement in the po-sitions of the atoms on the membrane [41]. Thereforedepending on how “strong” the deformations are we canvary the value of g within a range of energies, while thefactor (cid:126) β/ a cc remains approximately constant (withinthe validity range of our model).In general, we can consider a displacement outside theplane h = h ( r , t ), and a displacement inside the plane u = u ( r , t ). The stress tensor ε µν is given by ε µν = 12 ( ∂ µ h∂ ν h ) + 12 ( ∂ µ u ν + ∂ ν u µ ) , µ, ν = x, y. (4)We shall consider the simplest case, in which the defor-mation is only perpendicular to the plane, i.e., u = 0, sofrom Eq. (4) ε xx = 12 ( ∂ x h ) ε yy = 12 ( ∂ y h ) ε xy = 12 ( ∂ x h ) ( ∂ y h ) (5)We introduce new variables defined as l ( r , t ) ≡ ( ∂ x h ) − ( ∂ y h ) l ( r , t ) ≡ ∂ x h ) ( ∂ y h ) (6)which will give us information about how “strong” thevertical displacements are. On the other hand, by makinguse of the Eqs. (2), (3),(5) and (6), we can rewrite thescalar and pseudo-vector potentials, V ( r , t ) = g (cid:113) l ( r , t ) + l ( r , t ) A ( r , t ) = (cid:126) β a cc [ l ( r , t )ˆ x − l ( r , t )ˆ y ] . (7)From Eq. (1) and (7), the Hamiltonian is ˆ H η ( r ) = ˆ H ( r ) + W ( r , t ) + g | l ( r , t ) | (8)where the hat is used to denote differential operators asfollows, ˆ H ( r ) = v F (cid:18) η ˆ p x − i ˆ p y ) η ˆ p x + i ˆ p y (cid:19) W ( r , t ) = (cid:18) − η ˜ βl ( r , t ) − η ˜ βl ∗ ( r , t ) 0 (cid:19) (9)where l ( r , t ) ≡ ηl ( r , t ) + il ( r , t ) and we defined theparameter ¯ β as,¯ β = v F (cid:126) β a cc ≈ . η ( r , t ) follows atime-dependent Schr¨odinger type equation i (cid:126) ∂∂t Ψ η ( r , t ) = ˆ H η ( r )Ψ η ( r , t ) (11)where Ψ η ( r , t ) = (cid:18) ψ ηA ( r , t ) ψ ηB ( r , t ) (cid:19) (12)It is straightforward to prove that the Schr¨odinger typeequation (11) can be rewritten as i (cid:126) ∂ψ ηA ∂t = g | l | ψ ηA + (cid:2) v F ( η ˆ p x − i ˆ p y ) − η ¯ βl (cid:3) ψ ηB ,i (cid:126) ∂ψ ηB ∂t = g | l | ψ ηB + (cid:2) v F ( η ˆ p x + i ˆ p y ) − η ¯ βl ∗ (cid:3) ψ ηA . (13)Notice how the magnitude of the disorder enters in theDirac equation through the parameters ¯ β and g . While g plays the role of a random local chemical potential, ˜ β is a random local magnetic field.Eq. (13) is a complex stochastic equation. Insteadof solving the time-dependent problem, we consider thatthe deformation process is adiabatic in the time scaleof the electron dynamics. In such a case, we can sup-pose that the disorder is quenched and thus l and l are time-independent. In such a case, Eq. (9) becomesa time-independent Hamiltonian with a spatial randompotential l ( r , t ) = l ( r ).Returning to Eq. (11) becomes the time-independentSchr¨odinger equation ˆ H η ( r )Ψ η ( r ) = E Ψ η ( r ) and we areinterested in finding the distribution of the Hamiltonianeigenvalues of ˆ H η ( r ). III. FOLDED DEFORMATIONS
To understand the changes induced by random flexuraldeformations, we study in this section folded deforma-tions. Such kind of fields have been observed experimen-tally in deformed graphene [42–44] and there are somestudies for particular deformations [29, 45, 46]. In a gen-eral folded deformation, the field does not vary in onedirection. Therefore, it can be written as, h ( y ) = k c (cid:88) k = − k c a k exp( iky ) (14)with a − k = a ∗ k as h ( y ) is a real, and the coefficients a k can be deterministic or random variables. k c is a cut-off parameter and in what follows all sums are under-stood to use it. As explained in the next section, forthermally activated fields, k c can be estimated from theBose-Einstein distribution and the frequency dispersionof flexural modes.From Eqs. (3) and (4) , the vectorial potential hasonly one component different from zero, A x ( y ) = (cid:126) β a cc (cid:34)(cid:88) k a k k exp( iky ) (cid:35) (15)The advantage of this particular deformation is that A ( r ) is in the Coulomb gauge ∇ · A ( r ) = 0, thereforecan be obtained as the derivative of a scalar field, A i = (cid:15) ij ∂ j Φ( r ) (16)where (cid:15) ij is the 2D Levi-Civita tensor with i = x, y and j = x, y . For this particular case, we express Φ( y ) interms of the following Fourier decomposition,Φ( y ) = Φ ( y ) + (cid:88) k (cid:54) =0 e iky ˜Φ( k ) (17)with, Φ ( y ) = (cid:126) β a cc (cid:32)(cid:88) k k | a k | (cid:33) y (18)and, ˜Φ( k ) = − i (cid:126) β a cc k (cid:34)(cid:88) k (cid:48) a k a ∗ k (cid:48) − k k (cid:48) ( k (cid:48) − k ) (cid:35) (19)The associated pseudomagnetic field is B = ∇ Φ( r ).It is worthwhile noticing that although Φ ( y ) does notproduce a pseudomagnetic field, it produces an likeAharonov-Bohm effect as it leads to a constant A ( r ).An interesting consequence of having a field derivedfrom the potential is that for any flexural field, beingdeterministic or random, the zero-mode can always beconstructed. From the Schr¨odinger and Eq. (16), weobtain that for E = 0 and g = 0 the wave function is, ψ ± ( r ) = ( const. )(1 ± σ z ) (cid:18) e Φ( y ) e − Φ( y ) (cid:19) (20)where σ z is the Pauli z matrix. Similar functions werestudied years ago in the context of the integer quantumHall transition [47]. It can be proved that for a randommagnetic field in which the vector potential satisfies aGaussian white-noise distribution with mean zero andvariance ∆ A such that the average coefficients in Eq. (17)are, (cid:104) ˜Φ( k ) ˜Φ( k (cid:48) ) (cid:105) = (2 π ) δ ( k − k (cid:48) ) ∆ A k (21)the resulting wave-function is multifractal [47]. In a sam-ple of size L × L , the moments of the participation ratio P q ( L ) that measures a multifractal localization [48], P q ( L ) = (cid:104)| ψ ( r ) | q (cid:105) (22)are given by [47], P q ( L ) ≈ L τ ( q ) (23)with, τ ( q ) = 2( q −
1) + ∆ A π q (1 − q ) (24)where q need not be integer. For big samples, the multi-fractal spectrum is dominated by its maximal value, fromwhere the typical participation is [47], P typical ( L ) = e (cid:104) ln | Ψ | (cid:105) ≈ L A /π (25)Around these states and near the Fermi energy, the den-sity of states (DOS) is [47], ρ ( E ) = E − zz (26)with z = 2 + ∆ A /π . Such wavefunction multifractalityand DOS means that an unusual electron velocity distri-bution will appear even in the simplest case of a Gaussianrandom flexural field, without restoring to Levy distribu-tions of membrane jumps in graphene. In any case, theLevy jumps will induce an even more unusual distribu-tion. We end up the case of a folded flexural case by consid-ering the particular contribution of the Aharonov-Bohmterm, which for some geometries produces interesting ef-fects in graphene [49]. First we write the Fourier coef-ficients a k as the sum of an average plus a fluctuationpart, a k = (cid:104) a k (cid:105) + δa k . If a k is Gaussian distributed withzero mean we have,Φ ( y ) = (cid:126) β a cc (cid:88) k ( δa k ) k y ≈ π (cid:126) βa cc ∆ A k c y (27)and thus the phase difference between particles, with thesame start and end points, but travelling along two dif-ferent paths is,∆ φ = (cid:18) d Φ ( y ) dy A (cid:19) e (cid:126) = π βa cc (∆ A k c A ) e (28)where A is the area bounded by the two paths. For ther-mally activated fields, k c is determined from the temper-ature ( T ) population given by the Bose-Einstein distri-bution (see next section). As ∆ A ∼ k B T , Eq. (28) im-plies a very strong temperature dependent phase shift.This result is in agreement with recent first-principlescalculations based on density functional theory and theBoltzmann equation[50]. In the following sections, wewill explore more general fields. IV. GENERAL FLEXURAL FIELDS
In this section, we will consider a more general kindof disorder due to the perpendicular deformations in theHamiltonian ˜ H η ( r ). The displacement outside the planecan be written as, h ( r ) = kc (cid:88) k = − k c a k exp( i kr ) (29)It is important to remark that for graphene, the roomtemperature is far below the Debye temperature, which isabout 1000–2300 K [33, 34], and therefore (cid:126) ω ( k c ) ≤ k B T ,and as the purely harmonic flexural dispersion goes as[31] ω ( k ) = α | k | with α = 4 . m /s , it follows that, | k c | ≈ (cid:114) k B Tα (cid:126) (30)In what follows, all sums are understood to have a cut-off.Notice that k c is a strain dependent quantity [31, 51]. Infact, for free-standing graphene at thermal equilibrium, (cid:10) a k (cid:11) = (cid:126) (1 + 2 n B ( ω ( k )))2 M C ω ( k ) ≈ k B TM C ω ( k ) (31)where n B ( ω ( k )) is the thermal population of mode k , M C is the carbon mass and the second equality holds when (cid:126) ω ( k ) (cid:28) k B T . For purely harmonic flexural modes thefluctuation (cid:104) a k (cid:105) diverges as | k | − for small k . In real sam-ples, however, it is known that the singularity gets renor-malized due to lattice imperfections (i.e., by anharmoniceffects). The resulting dispersion can be parametrized as ω ( k ) = α (cid:112) k + k − τ k τc for τ >
0, from where it followsthat the quadratic mean displacement of each field mode,in the long wavelenght, is given by [52], (cid:104)| a k | (cid:105) ∝ k B Tk − τ k τc (32)where τ depend on the physical mechanism of renormal-ization. The physical scenarios are [52] : a) substratepinning that opens a gap in the phonon spectrum, corre-sponding to τ = 4; b) strain which makes the dispersionlinear at long wavelengths, τ = 2, c) anharmonic effectswhich yield τ = 0 .
82. Finally, when the flexural field ismade from a local mechanical distortion, the componentsare given by the Fourier transform of the deformationprofile.Using Eqs. (3) and (5) l ( r ) = (cid:88) k , k (cid:48) a k a ∗ k (cid:48) ( k x k (cid:48) x − k y k (cid:48) y ) e i ( k − k (cid:48) ) · r l ( r ) = (cid:88) k , k (cid:48) a k a ∗ k (cid:48) ( k x k (cid:48) y + k (cid:48) x k y ) e i ( k − k (cid:48) ) · r (33)Although some works assume the Coulomb norm forgeneral pseudo-electromagnetic fields [53], let us showthat in general such deformation can not be written asthe derivative of a scalar field. This can be proved asfollows, if we consider A i ( r ) = (cid:15) ij ∂ j Φ( r ) with Φ( r ) = (cid:80) k b k exp( i k · r ) it holds that b k = − iv F ˜ βk y (cid:88) k (cid:48) a k a ∗ k (cid:48) ( k x k (cid:48) x − k y k (cid:48) y ) e − i k (cid:48) · r and b k = − iv F ˜ βk x (cid:88) k (cid:48) a k a ∗ k (cid:48) ( k x k (cid:48) y + k y k (cid:48) x ) e − i k (cid:48) · r , (34)in general, the system of equations in Eq. (34) has nosolutions for b k except for few particular cases, as thefolded potential.For a random field, the set of coefficients a k are ran-dom variables given by a certain distribution. Once arealization of the field is given, we need to find the spec-trum and then average over an ensemble of realizations.However, for distributions having a finite first moment,let us write each a k in Eq. (33) as an average plus afluctuation part a k = (cid:104) a k (cid:105) + δa k . Therefore, for examplewe have, l ( r ) = (cid:88) k , k (cid:48) (cid:2) (cid:104) a k (cid:105)(cid:104) a ∗ k (cid:48) (cid:105) + δa k δa ∗− k (cid:48) (cid:3) × ( k x k (cid:48) x − k y k (cid:48) y ) e i ( k − k (cid:48) ) · r (35)For an independently distributed a k , only the terms with k = k (cid:48) and k = − k (cid:48) will survive as on average the sum over fluctuations is zero. Also, if (cid:104) a k (cid:105) = 0, we end uphaving, l ( r ) = (cid:88) k | a k | (cid:0) k x − k y (cid:1) (cid:104) − e i (2 k · r + φ k ) (cid:105) (36)For distributions without a finite first moment, the pre-vious analysis is not completely valid, but still is possibleto use a random phase approximation (RPA) in whichwe assume that the incoherent part of Eq. (33) is zero.Only terms with k = ± k (cid:48) survive. In any case, l ( r ) ≈ λ (0) + (cid:88) k λ ( k ) e i k · r (37)where the Fourier expansion coefficients are, λ (0) = (cid:88) k | a k | | k | cos 2 θ ≈ k (cid:54) = 0 λ ( k ) = −| a k/ | | k / | cos 2 θe iφ k / (39)where θ is the polar angle of k and φ k / the phase of thecoefficient a k / . A similar expansion holds for l ( r ), suchthat λ (0) ≈ k (cid:54) = 0, λ ( k ) ≈ −| a k / | | k / | sin 2 θe iφ k / (40)The Fourier components of the field can also be writtenas the complex number Λ( k ) ≡ λ ( k ) + iλ ( k ) or,Λ( k ) = −| a k / | | k / | e i ( φ k / +2 θ ) (41)As | a k | and φ k are random independent variables, weend up having a complex random field Λ( k ), in whichits components λ ( k ) and λ ( k ) can be taken as inde-pendent random variables. This field is then used in theoriginal Hamiltonian to obtain a Hamiltonian in momen-tum space, i.e., ˜ H η, k ≡ H η, k + g | Λ( k ) | (42)with, H η, k = ˜ H − ¯ β ( λ ( k ) , λ ( k )) · σ (43)and ˜ H = i (˜ k x , ˜ k y ) · σ where ˜ k x,y ≡ (cid:126) v F ηk x,y . Finally,it is convenient to define a shifted energy ε such that ε = E − g (cid:104)| l ( r ) |(cid:105) /
2. In the following section, we discussthe corresponding spectrum.
V. GAUSSIAN DISTRIBUTED FLEXURALFIELDS: RANDOM MATRIX THEORY
A matrix like ˜ H η, k , whose inputs are random variables,is called a random matrix . Of particular interest are thoserealizations that satisfy the condition of stability: theprobability distribution is “stable” if its shape is invariantunder convolution, a rigorous definition is given in Ref.[54]. Eigenvalues and eigenvectors are also random andthe main objective is to understand their distribution.We remark that the λ ( k ) and λ ( k ) distribution dependson | a k / | | k / | , as seen in Eqs. (39), (40). To simplify,in what follows we will work in the case of flexural strainin which τ = 2 from where | a k | ∝ /k . This results ina k independent distribution and thus we can remove thelabel k having only two random independent variables λ and λ . Notice that other cases are obtained by a simplerenormalization of our results by taking a k dependent ¯ β with the replacement ¯ β → ¯ β/k − τ .Also, consider first the case g = 0 in order to under-stand the pure stochastic matrix H η, k . Any ensemble ofrandom matrices is determined through the probabilitydistribution function (PDF) p ( H η ) that depends on thematrix inputs (˜ h ) ij , 1 ≤ i, j ≤ n .In our first case we will consider λ , as Gaussian dis-tributed random variables with mean zero and standarddeviation s , i.e., p ( λ , ) ∼ N (0 , s ) = 1 √ πs exp ( − λ / s ); (44)As λ and λ are Gaussian distributed components of acomplex number Λ, this complex number and its normare distributed according to [55–58], | Λ | ∼ R (0 , s ) = λs exp ( − λ / s ) , Λ ∼ ˜ N (0 , s ) , (45)where R is the Rayleigh distribution (see Appendix A)and ˜ N is the normal distribution over C . Therefore therandom matrix H η, k is distributed as ˜ N ( ˜ H , ⊗ Σ ) withmean (cid:104) H η, k (cid:105) = ˜ H and covariance cov( H η, k ) = ⊗ Σ and Σ = diag( s , s ). We further observe that H † η, k H η, k is a complex non-central Wishart matrix. The eigenvaluedensities of complex central Wishart matrices are studiedin Ref. [59, 60] and the non-central Wishart matrices arestudied in Ref. [61]. Using the results in Ref. [61], we findthat the PDF for the amplitude of eigenvalues { ξ i } i =1 ofthe operator H η, k is p Rice ( | ξ i | ) = | ξ i | s exp (cid:32) − (cid:0) | ξ i | + ξ (cid:1) s (cid:33) × I (cid:18) | ξ i | ξ s (cid:19) Θ( | ξ i | ) , (46)where I is the modified Bessel function of the first kindof order zero, Θ is the Heaviside theta function and ξ = (cid:126) v F | k | , s = ˜ βs . Because ξ = − ξ ≥
0, then p Rice ( ξ ) = ξ s exp (cid:32) − (cid:0) ξ + ξ (cid:1) s (cid:33) × I (cid:18) ξ ξ s (cid:19) Θ( ξ ) , (47a) p Rice ( ξ ) = − ξ s exp (cid:32) − (cid:0) ξ + ξ (cid:1) s (cid:33) × I (cid:18) − ξ ξ s (cid:19) Θ( − ξ ) , (47b)This distribution is called the Rice distribution and itsexpected values (cid:104) ξ i (cid:105) and (cid:104) ξ i (cid:105) for i = 1 , (cid:104)| ξ i |(cid:105) = (cid:114) π s L / ( − ξ s ) , (48a) (cid:104)| ξ i | (cid:105) = 2 s + ξ , (48b)where L q ( x ) denotes a Laguerre polynomial. In the fol-lowing subsection, we will plot such distributions andcompare with some numerical calculations.Let us know return to the more general problem ofincluding a g (cid:54) = 0. If χ = ( g/ | Λ | then χ ∼ R (0 , s ) (a)(b) Figure 1. a) Velocity Fermi for flexural strain disorderedgraphene ( τ = 2) as function of ˜ k x , ˜ k y in units of Fermi veloc-ity of non-deformed graphene, as obtained from Eq. (50). b)Density plot of Fermi velocity, we observe that it is isotropic. a)b)Figure 2. Distribution the eigenvalues E and E of theHamiltonian Eq. (42) under for Gaussian disorder in the case g = 0 and ¯ β ≈ . ˜ k points in ˜ k space. In a), ˜ k = ( − . ,
0) and b) ˜ k = (3 . , g = 0, the energy distribution is symmetric withrespect to the Fermi energy, E F = 0. where s = ( g/ s . From Eq. (42), the eigenvalues of ˜ H η, k are E i = χ + ξ i , i = 1 , (cid:104) E i (cid:105) = (cid:104) χ (cid:105) + sgn( ξ i ) (cid:104)| ξ i |(cid:105) = (cid:114) π (cid:18) s + sgn( ξ i ) s L / (cid:18) − ξ s (cid:19)(cid:19) (49) The Fermi velocity for deformed graphene is v DF,i = 1 (cid:126) ∇ k (cid:104) E i (cid:105) = sgn( ξ i ) (cid:126) (cid:114) π s L / ( − ξ s ) − L − / ( − ξ s ) ξ × ∇ k ξ ( k )= (cid:114) π s L / ( − ξ s ) − L − / ( − ξ s ) ξ v F,i (50)where v F,i is the Fermi velocity for undeformed graphene.In the Fig. 1 we present the magnitude of the resultingFermi velocity in deformed graphene, normalized by theFermi velocity of non-deformed graphene. From Fig. 1,we can make two observations, the isotropic behavior andthe velocity reduction as the momentum gets closer to theDirac point. For L´evy’s general distributions, similar ex-pressions have not been found, but there are some studiesin this direction [62, 63].
A. Comparison with numerical results at somespecific k points.
To obtain the eigenvalues distribution of the hamilto-nian Eq. (9), a code written in Python was used. Someimportant limiting cases are discussed below:
1. Case g = 0 (no scalar field). In this case, the distribution of probabilities for eachof the eigenvalues E , E , given in Eqs. (47) are similar.The only change is that each of the these distributions iscentred at values close to ± ξ . As E and E are the en-ergies allowed for the conduction and valence bands, thisimplies that these bands will be symmetric with respectto the Fermi energy (in the undeformed case) E F = 0.Fig. 2 presents the distributions ˜ p ( E ) , ˜ p ( E ). Asstated by the theoretical approach, the distributions aresymmetric. In Fig. 2 we also compare these analytic re-sults with the numerical prediction. An excellent agree-ment is seen between the theory and the simulation.
2. Case g/ (cid:54) = ¯ β . Since χ and ξ i are dependent random variables, it isnecessary to find the joint probability distribution andthus calculate the distribution of P ( E ); however, sincethese two are distributed with Rice distributions, we haveproposed the following Rice-type distribution (see Ap- a)b)Figure 3. Energy distribution for Gaussian strain flexuraldisorder in the Hamiltonian Eq. (42) for the case g ≈ ¯ β ,using g = 7 eV and ¯ β ≈ . ˜ k = (0 , σ , ν , µ , R ) = (7 . , . × − , . × − , . σ , ν , µ , R ) = (0 . , . × − , . × − , . R is the coefficient of de-termination. The real values of parameters must be ν , =0 , µ , = 0 and σ = 0 . σ = 6 . pendix B) for EP ( E ) = ( E − µ ) σ exp (cid:18) − ( E − µ ) + | ν | σ (cid:19) × I (cid:18) ( E − µ ) | ν | σ (cid:19) Θ( E − µ ) (51)where σ, ν, µ are parameters to be found using the realmoments of E , those computed using the joint probabil-ity distribution. The main advantage of this approximat-ing distribution is to be able to obtain the approximatemoments of E from Eq. (B3).We obtain the parameters using a curve fitting andin Fig. 3 we compare the obtained parameters with the a)b)Figure 4. Energy distribution for Gaussian disorder in theHamiltonian Eq. (42) for the case g > ¯ β , using g = 15eV and ¯ β ≈ . ˜ k points: a) ˜ k = ( − . ,
0) and b) ˜ k = (3 . , σ , ν , µ , R ) = (10 . , − . , . , . σ , ν , µ , R ) = (3 . , − . , − . , . σ , ν , µ , R ) = (10 . , − . , . , . σ , ν , µ , R ) = (5 . , . , − . , . E and E are shifted toward positive val-ues of E when compared with the case g = 0. We observethat that the distribution for E is narrower than for E . Ingeneral, this is the expected type of distributions when wedeform graphene because g involves the flexural effects of themembrane, while the ¯ β coefficient is almost constant as onlyinvolves bond length changes. real parameters in the case ξ = 0 and g/ ≈ ˜ β . Insuch situation the real distribution for ξ i are reduced to aRayleigh type distributions, then E i ∼ R (0 , (( g/ ± β ) ),i.e. from Eq. (51), µ = 0 , ν = 0 and σ = g ± ˜ β .In this case σ ≈ , σ (cid:29) s , and using Eq. (51) wehave that the PDF for E will be very narrow comparedto that of E . On the other hand, the distribution for E ,given that σ (cid:29) σ , is expected to be wider thus givingrise to the average valence band. In other words, the E average, and the values for the conduction band will beshifted to more positive values, so we expect a gap to beopen.As in previous case, we consider the case g/ > ˜ β usingthe parameters from Eq. (51). This behavior is seen inFig. 4, where we present the distributions for E and E at some selected values of k x , i.e., in panel a) for ( − . , . , E when comparedwith Fig. 2. In addition, there is an overlap betweenboth distributions. Deformations of this type can exciteelectrons of the bands of valence to bands of conductionalthough the average values are different. This is simpleto understand as g plays the role of an effective strongchemical potential in Eq. (8). B. Comparison of the energy dispersion withnumerical results.
In this subsection, we studied the changes in the energydispersion due to disorder, but take notice that all ofour calculations are for strain flexural disorder, and thuschange for other physical cases with different exponent τ . In Fig. 5 shows the expected energy (cid:104) E ( k ) (cid:105) and (cid:104) E ( k ) (cid:105) obtained from the numerical and theoretical cal-culations and we compare with the undeformed casewhere the dispersion is linear around the Dirac point k = (0 , k = (0 ,
0) and from Eq. (49), the average gap sizeis ∆ E = √ πs L / (cid:18) − ξ ( k = )2 s (cid:19) ≈ . s = ˜ β. (52)This shows that the principal factor for the gap size is ˜ β due to its relation with the hopping parameter.Also, in Fig. 5 we plot the energy dispersion sepa-rated from the average by ± s i , i = 1 ,
2, where s i refersto the standard deviation of the data, for different limitsof disorder a) g = 0 (no scalar field), b) g/ ≈ ˜ β and c) g/ > ˜ β (big scalar field).We observe that g (cid:54) = 0 breaks the particle-hole sym-metry and its effect by changing s i . In addition, as g is aparameter related to the purely mechanical deformationand it can vary in a wide range of values, then g/ > ˜ β is the case most likely to occur for an experimental real-ization. (a) g = 0(b) g ≈ β (c) g > β Figure 5. Energy spectrum for Gaussian distributed disorder, N (0 , k =( k x ,
0) for the case, a) g = 0 and ¯ β ≈ . g/ ≈ ¯ β ,using g = 7 eV, c) g/ > ¯ β , using g = 15 eV. The dottedlines indicate the average energies for each band, (cid:104) E i (cid:105) , i =1(green) , ± s i of the averageenergies, i.e. (cid:104) E i (cid:105)± s i , i = 1(blue) , s i refers to the standard deviation of the data. The symbolsare the numerical computations. VI. ENERGY SPECTRUM FOR L´EVY’SDISORDER.
In this section, we will explore the consequences of aL´evy distributed disorder due to flexural strain in which τ = 2 from where | a k | ∝ /k . The one-dimensionalL´evy’s distributions is given by [55, 62, 64] L ( x ; α, γ, µ, c ) = 12 π (cid:90) ∞−∞ dt ϕ ( t ; α, γ, µ, c ) e − ixt , (53)where ϕ ( t ; α, γ, µ, c ) is the characteristic function. It canbe written as ϕ ( t ; α, γ, µ, c ) = exp ( itµ − | ct | α (1 + iγ sgn( t )Φ)) , (54)where sgn( t ) is the sign of t andΦ = (cid:26) tan( πα ) , α (cid:54) = 1 − π log( | t | ) , α = 1 , (55) µ ∈ R is a shift parameter; γ ∈ [ − , skewness parameter , and is a measure of asymmetry.The c is a scale factor; and for α < λ , distributedwith ∼ L ( x ; α, γ, µ, c ), where α = 1 . , γ = 0 , µ = 0 , c = 1.We observe that the distributions follow the asymptoticbehaviour ρ ( E ) ∝ E − (1+ α ) , characteristic of L´evy’s dis-tributions.In figure 7 we show the energy spectrum for a pathalong ( k x ,
0) . We note that for all cases g = 0 , ¯ β ≈ . g/ ≈ ¯ β , c) g/ > ¯ β the energy differ-ence |(cid:104) E (cid:105) − (cid:104) E (cid:105)| ≈
20 meV. Moreover, flat bands areformed for average energies. This ensures the existence oflocalized states, in agreement with the results reported byPereira et. al. [32], where the electronic transport prop-erties of graphene in the presence of a L´evy-type disorderwere analyzed. They showed that a system with Diraccarriers presents an anomalous localization[32], whichmeans that average transmission decays as, (cid:104) T (cid:105) ∝ L − α (56)with the system length L and 0 < α ≤
2. Meanwhile, for α >
VII. CONCLUSIONS.
We studied the effects in the electronic propertiesof graphene of random flexural deformations, whichare equivalent to random pseudo electromagnetic fields.First, we studied general folded deformations, giving anexpression for the zero-modes. For random Gaussiandistributed folded deformations, we made contact withworks on the quantum Hall effect with random fields, a)b)c)Figure 6. Energy distribution for L´evy disorder in theHamiltonian Eq. (9) for the case g/ > ¯ β , using g = 15 eVand ¯ β ≈ . k points a) ( − . , . , , ρ ( E ) ∝ E − (1+ α ) , characteristic of L´evy’s distributions.The solid lines are given as 10 η E − δ and its determinationcoefficient R and for each case we provide these values inthe form ( η, δ, R ),then a) (1 . , − . , . . , − . , . . , − . , . . , − . , . . , − . , . . , − . , . δ ≈ α with α = 1 . showing that the density of states has a power law be-1 (a) g = 0(b) g ≈ β (c) g > β Figure 7. Energy spectrum for L´evy disorder ( α = 1 . , γ = 0)in the Hamiltonian Eq. (9) along the path k = ( k x ,
0) forthe case, a) g = 0 and ¯ β ≈ . g/ ≈ ¯ β , using g = 7 eV, c) g/ > ¯ β , using g = 15 eV. We can observe that,in general, flat bands are formed for the average energies andthe size of the average gap does not depend on g . havior. Also, we showed a remarkable Aharonov-Bohmpseudo-effect and the wavefunction multifractality, whichmeans that an unusual electron velocity distribution will appear. For non-folded general flexural strain, we firstshowed that it is not possible to use a Coulomb normin order to obtain a magnetic potential. However, weused an RPA and the scheme of random matrix theoryfinding an average gap opening. This is mainly due tothe change of bond lengths, ˜ β , while the parameter g produces a shift in the values and a overlap between theoriginal conduction and valence bands, ˜ β induces a metal-insulator transition. For Gaussian disorder, such infor-mation is captured by the variance of the distribution,therefore are important to understand the electronic be-havior. An alternative way to explain this phenomenais that the electron is immersed in a region of pseudo-electromagnetic fields and therefore acquires an effectivemass.Finally, we studied a L´evy type of disorder, withinRPA and flexural strain. We observed that the aver-age bands are flat and this is evidence that buckling of2D graphene offers a strategy for exploring interactionphenomena characteristic of flat bands, similar to thatfound in [28].
ACKNOWLEDGEMENTS
We are grateful to Alejandro P´erez Riascos for theirsupport and feedback in carrying out this work. Wethank UNAM-DGAPA PAPIIT project IN102620 andCONACYT project 1564464. A.E. thanks CONACYTfor providing a schoolarship.
Appendix A
Here are some of the distributions used in this work, p R ( x ) = xs e − x s Θ( x ) , s > p Rice ( x ) = xs e − x ν s I (cid:16) xνs (cid:17) Θ( x ); s > , ν ≥ Appendix B
Since both χ and ξ are distributed, in general, with p Rice ( x ) distributions, then we propose that E i is dis-tributed following an approximated distribution given by P ( E ) = ( E − µ ) σ exp (cid:18) − ( E − µ ) + | ν | σ (cid:19) × I (cid:18) ( E − µ ) | ν | σ (cid:19) Θ( E − µ ) (B1)where σ, ν, µ are parameters to be found using the realmoments of E ; for example, using the Eq. (B1), the first2moment is (cid:104) E (cid:105) = (cid:114) π σL / (cid:18) − ν σ (cid:19) + µ (B2)and from Eq. (49) we obtain the first equation for σ, ν, µ .The main advantage of this approximated distribution is being able to obtain the approximate moments of E from (cid:104) ( E − µ ) k (cid:105) = σ k k Γ(1 + k/ L k/ (cid:18) − ν σ (cid:19) (B3) [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A.Firsov. Electric field effect in atomically thin carbonfilms. Science , 306(5696):666–669, 2004.[2] Alessandro Cresti, Norbert Nemec, Blanca Biel, GabrielNiebler, Fran¸cois Triozon, Gianaurelio Cuniberti, andStephan Roche. Charge transport in disorderedgraphene-based low dimensional materials.
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