Generalized WKB theory for electron tunneling in gapped α-\mathcal{T}_3 lattices
Nicholas Weekes, Andrii Iurov, Liubov Zhemchuzhna, Godfrey Gumbs, Danhong Huang
GGeneralized WKB theory for electron tunneling in gapped α − T lattices Nicholas Weekes , Andrii Iurov ∗ , Liubov Zhemchuzhna , , Godfrey Gumbs , , and Danhong Huang , Department of Physics and Computer Science,Medgar Evers College of City University of New York,Brooklyn, NY 11225, USA Department of Physics and Astronomy,Hunter College of the City University of New York,695 Park Avenue, New York, New York 10065, USA Donostia International Physics Center (DIPC),P de Manuel Lardizabal, 4,20018 San Sebastian, Basque Country, Spain Space Vehicles Directorate,US Air Force Research Laboratory,Kirtland Air Force Base, New Mexico 87117, USA Center for High Technology Materials,University of New Mexico, 1313 Goddard SE,Albuquerque, New Mexico, 87106, USA (Dated: February 9, 2021)We generalize Wentzel-Kramers-Brillouin (WKB) semi-classical equations for pseudospin-1 α −T materials with arbitrary hopping parameter 0 < α <
1, which includes the dice lattice and grapheneas two limiting cases. In conjunction with a series-expansion method in powers of Planck constant (cid:126) ,we acquired and solved a system of recurrent differential equations for semi-classical electron wavefunctions in α − T . Making use of these obtained wave functions, we analyzed the physics-relatedmechanism and quantified the transmission of pseudospin-1 Dirac electrons across non-rectangularpotential barriers in α − T materials with both zero and finite band gaps. Our studies revealseveral unique features, including the way in which the electron transmission depends on the energygap, the slope of the potential barrier profile and the transverse momentum of incoming electrons.Specifically, we have found a strong dependence of the obtained transmission amplitude on thegeometry-phase φ = tan − α of α − T lattices. We believe our current findings can be applied toDirac cone-based tunneling transistors in ultrafast analog RF devices, as well as to tunneling-currentcontrol by a potential barrier through a one-dimensional array of scatters. I. INTRODUCTION
The quantum states and the motion of a charged carrier in a lattice are in many ways, not much different fromthose described by classical dynamics. Also, the semi-classical approximation is usually a useful and importanttool for providing a simplified description as well as further investigations of single-particle and collective propertiesof electronic states with high kinetic energies in various materials. Similar to standard quantum mechanics, theWentzel-Kramers-Brillouin (WKB) approximation for an α − T lattice can be made by expanding the electroneigenstate (or wave function) of the considered Hamiltonian as a power series of Planck’s constant (cid:126) . Such a methodis also employed for solving the second-order differential equations having coordinate-dependent coefficients, which aremathematically equivalent to the Schr¨odinger equation with a spatially varying potential. Although most physicalproblems studied by this method are one-dimensional, the lately-developed WKB theory has been generalized tomultiple dimensions for new two-dimensional materials.The α − T model represents the newest, and likely, the most technologically promising class of low-dimensionalmaterials with zero-mass Dirac fermions, and it has become one of the hot spots in condensed matter physicsafter the discovery of graphene and its gapless, linear and relativistic low-energy band structure. The same α − T model bears all crucial electronic properties of graphene, yet it is still remarkably distinguished by the presence of anadditional flat band at the Dirac point of its energy dispersion. This dispersionless energy band remains stable andpersists in the presence of charged disorder states, external electric, magnetic and optical fields, or a time-dependentmodulation potential. Consequently, the observed energy spectrum of α − T appears as metallic, i.e., all these threebands intersect at the corners of Brillouin zone. ∗ E-mail contact: [email protected], [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b From the perspective of the atomic structure, the difference between an α − T and graphene honeycomb latticesappears through an additional fermionic atom situated at the center of each hexagon, which is referred to as a hubC atom. The hopping integral between the hub and one of the A and B rim atoms is different from that betweennearest-neighbor rim atoms of hexagon, and the ratio of these hopping parameters is quantified by a variable α . Themaximum value for α is 1 corresponding to the dice lattice, while its minimum value is 0 for graphene for a completelydecoupled set of hub atoms. From this point of view, the α − T model can be essentially viewed as an interpolationbetween graphene and a dice lattice as α increases continuously from 0 to 1. Recently, models for stronger interactionswith α > α − T , and especially the dice lattices, has recently beenfound in a number of existing and experimentally synthesized materials. These include three-layer arrangement ofSrTiO /SrIrO /SrTiO lattices , Lieb and the Kagome optical lattices and waveguides , Josephsonarrays, , Hg − x Cd x quantum well . A comprehensive review of all dice-like systems with a flat band can be foundin Ref. [25]. Very recently, a band structure involving flat bands was realized in In . Ga . As/InP semiconductorquantum wells along with lateral geometry. The α − T model reveals a number of promising electronic, collective, magnetic, optical, and transport features, such as topological Dirac semimetals and tilted Dirac cone materials. All α − T materials demonstrate unique symmetry and topological properties,especially in the presence of a dressing irradiation. For example, a photo-induced topological phase transitionis observed under a non-resonant optical field with a specific polarization , and the lattice is turned from asemimetal to Haldane-like Chern insulator in this case.The Klein paradox, defined as complete electron tunneling independent of barrier height and width, is one of thelandmark properties of all Dirac materials including graphene. This paradoxical behavior was predicted for α − T for all possible parameters α and becomes asymmetric (i.e., observed at a finite electron incidence angle) underlinearly-polarized irradiation. In graphene, unimpeded transmission also exists for a trapezoidal (not square)potential barrier facilitated by a finite electric field in the barrier region.
The question whether such an effectcould also be observed for α − T with α > (cid:126) . Making use of thisexpansion, we obtain closed-form analytic expressions for the wave function including the phase factor in additionto its spatially dependent amplitude. In Sec. III, we apply our theory for calculating the transmission of electrons invarious cases with non-square potential barriers, such as a trapezoidal barrier imposed by a linear-potential profile.Meanwhile, we also consider gapped α − T materials and compute their energy dispersion, semi-classical action,classically forbidden regions, and amplitude of transmission. Finally, some remarks are present in Sec. IV regardingderivation of the set of WKB equations and their application to electron transmission and other numerical results. II. SEMI-CLASSICAL SOLUTION OF α − T MODEL
In this section, we derive the semi-classical wave functions of a gapless α − T lattice up to the first order of theseries expansion in powers of (cid:126) . An important step of this derivation involves calculating the semi-classical action ofwhich the spatial derivative in Eq. (23) corresponds to a position-dependent longitudinal electron momentum givenin Eq. (24). The mere knowledge of the action S ( ξ ), or momentum component π x ( ξ ), in terms of the scaled length ξ defined in Eq. (13), only provides a possibility to evaluate the electron transmission through a non-square barrierunder a finite longitudinal electric field. However, an analytical expression for wave function either does not exist oris too complicated to obtain as we deal with gapped α − T materials.We begin our study with the pseudospin-1 low-energy Hamiltonian for the α − T model, i.e.,ˆ H α ( k | τ, φ ) = (cid:126) v F k τ − cos φ k τ + cos φ k τ − sin φ k τ + sin φ , (1)where k τ ± = τ k x ± ik y , and valley index τ = ± K and K (cid:48) valleys.Phase φ (0 ≤ φ ≤ π/
4) introduced in Eq. (1) relates to α parameter by α = tan φ , and therefore, the limiting casesfor graphene and dice lattice correspond to φ = 0 and φ = π/
4, respectively.The Hamiltonian in Eq. (1) could be constructed using the following two φ -dependent 3 × S ( φ ) = (cid:110) ˆ S x ( φ ) , ˆ S y ( φ ) (cid:111) , where ˆ S x ( φ ) = φ φ φ φ , (2)ˆ S y ( φ ) = i − cos φ φ − sin φ φ . (3)As a result, we have ˆ H α ( k | τ, φ ) = v F ˆ S ( φ ) · {− i (cid:126) ∇ τ } + V ( x ) ˆΣ (3)0 , (4)where ˆΣ (3)0 is a 3 × V ( x ) represents a position-dependent electrostatic potential, and ∇ τ = { τ ∂/∂x, ∂/∂y } . In fact, the introduced matrices presented in Eqs. (2) and (3) are a φ -dependent generalizationof 3 × (3) x = 1 √ , (5)ˆΣ (3) y = i √ − −
10 1 0 . (6)where φ = π/ φ →
0, matrices in Eqs. (2) and (3) reduce to 2 × / (3) z = − , (7)so as to introduce an energy gap to a pseudospin-1 Hamiltonian. Three eigen-energies associated with the Hamiltonian in Eq. (1) are easily found to be ε γ = ± τ, φ ( k ) = γ (cid:126) v F k (8)with γ = − γ = +1) for the valance (conduction) band, and ε γ =0 τ, φ ( k ) = 0 . (9)for the remaining flat (or dispersionless) band. Here, all three bands in Eqs. (8) and (9) do not show any dependenceon phase φ (or parameter α ). Furthermore, two wave functions corresponding to the valence and conduction bandsin Eq. (8) take the form Ψ γ = ± τ, φ ( k ) = 1 √ τ cos φ e − iτθ k γτ sin φ e + iτθ k , (10)where θ k = arctan( k y /k x ) is the angle of wave vector k = { k x , k y } made with the x -axis. The other wave functionfor the flat band is Ψ γ =0 τ, φ ( k ) = sin φ e − iτθ k − cos φ e + iτθ k . (11)Here, we would like to indicate that the energy bands in Eqs. (8) and (9), as well as the wave functions in Eqs. (10)and (11), are obtained for a spatially-uniform potential independent of position coordinates x and y .As a generalization, we now consider an x -dependent potential V ( x ) so that the translational symmetry is keptonly along the y direction, and the wave function changes to Ψ( x, y ) (cid:118) ψ ( x ) e ik y y . Correspondingly, the previousHamiltonian in Eq. (4) is modified intoˆ H ( x, k y | τ, φ ) = (cid:126) v F V ( x ) cos φ ( − i (cid:126) τ ∂/∂x − ip y ) 0cos φ ( − i (cid:126) τ ∂/∂x + ip y ) V ( x ) sin φ ( − i (cid:126) τ ∂/∂x − ip y )0 sin φ ( − i (cid:126) τ ∂/∂x − ip y ) V ( x ) , (12)where p y = (cid:126) k y is conserved in the tunneling process. Following the approach and notations adopted in Ref. [4], werewrite our Hamiltonian in Eq. (12) and the corresponding eigenvalue equation through the following dimensionlessvariables, i.e., x → ξ , E → ε , p x,y → π x,y and ξ = xW B , ε = EV and ν ( x ) = V ( x ) V , π y = v y p y V . (13)Finally, using the fact that ∂/∂x → ∂/ ( W B ∂ξ ), we replace Planck constant (cid:126) by a dimensionless one (cid:126) , yielding (cid:126) ↔ (cid:18) v F W B V (cid:19) (cid:126) , (14)where V is the height of the square barrier in the absence of a longitudinal electric field. In Eq. (13), we avoidedusing an energy scale E F = (cid:126) v F k F (cid:118) (cid:126) v F /L with a unit length L for electron Fermi energy since we do not wishto introduce additional (cid:126) -related terms in the eigenvalue equation. This implies that the energy scale for incomingparticles can be large classically and not limited by values in units of E F .By using the dimensionless variables defined in Eq. (13), the eigenvalue equation becomesˆ H α ( ξ, π y | τ, φ ) Ψ γ ( ξ, π y | φ, τ ) = ε Ψ γ ( ξ, π y | φ, τ ) , (15)Ψ γ ( ξ, π y | φ, τ ) = Ψ γx ( ξ | φ, τ ) exp (cid:18) i π y η (cid:126) (cid:19) = φ A ( ξ | φ, τ ) φ H ( ξ ) φ B ( ξ | φ, τ ) exp (cid:18) i π y η (cid:126) (cid:19) , (16)and the Hamiltonian in Eq. (15) is nowˆ H α ( ξ, π y | τ, φ ) = ˆΣ (3)0 ν ( x ) + ˆΣ (3) x ( φ ) (cid:18) − i (cid:126) τ ∂∂ξ (cid:19) + ˆΣ (3) y ( φ ) π y = ν ( x ) cos φ ( − i (cid:126) τ ∂/∂ξ − iπ y ) 0cos φ ( − i (cid:126) τ ∂/∂ξ + iπ y ) ν ( x ) sin φ ( − i (cid:126) τ ∂/∂ξ − iπ y )0 sin φ ( − i (cid:126) τ ∂/∂ξ + iπ y ) ν ( x ) , (17)where ˆΣ (3)0 represents a 3 × (cid:126) (cid:118) (cid:126) ,namely,Ψ( ξ, π y | φ, τ ) = exp (cid:26) i (cid:126) S ( x ) (cid:27) ∞ (cid:88) λ =0 ( − i (cid:126) ) λ Ψ λ ( x ) = exp (cid:26) i (cid:126) S ( x ) (cid:27) (cid:2) Ψ ( x ) − i (cid:126) Ψ ( x ) − (cid:126) Ψ ( x ) + · · · (cid:3) , (18)where S ( x ) represents the semi-classical action in the WKB approximation, and our goal is obtaining a differentialequation with respect to x , which connects consecutive terms in the expansion in Eq. (18). From Eq. (17), however,we find that only a term involving (cid:126) can serve for this purpose. Explicitly, we write down such an equation asˆΣ (3) x ( φ ) (cid:26) ∂∂ξ Ψ λ ( ξ, π y | φ, τ ) (cid:27) − √ O T ( ξ, π y | φ τ ) Ψ λ +1 ( ξ, π y | φ, τ ) = 0 , (19)where λ = 0 , , , , · · · , and Ψ λ = − ( ξ, π y | φ, τ ) ≡
0. Here, the transport operator ˆ O T ( ξ, π y | φ, τ ), connectingconsequent terms of expansion in Eq. (18), is easily found to beˆ O T ( ξ, π y | φ, τ ) = κ ( ξ ) cos φ ( τ ∂ S ( ξ ) /∂ξ − iπ y ) 0cos φ ( τ ∂ S ( ξ ) /∂ξ + iπ y ) κ ( ξ ) sin φ ( τ ∂ S ( ξ ) /∂ξ − iπ y )0 sin φ ( τ ∂ S ( ξ ) /∂ξ + iπ y ) κ ( ξ ) , (20)where κ ( ξ ) = ν ( ξ ) − ε . Specifically, by setting λ = −
1, Eq. (19) gives rise toˆ O T ( ξ, π y | φ, τ ) Ψ ( ξ, π y | φ, τ ) = 0 . (21)For a linear homogeneous Eq. (21), a non-trivial solution exists only if its determinant is zero, i.e.,( ε − ν ( ξ )) (cid:34) (cid:18) ∂ S ( ξ ) ∂ξ (cid:19) + π y − ( ε − ν ( ξ )) (cid:35) = 0 , (22)which is independent of φ . Generally speaking, we know ν ( ξ ) (cid:54) = ε , and then Eq. (22) leads us to S ( ξ ) − S ( ξ ) = ξ (cid:90) ξ π x ( η ) dη , (23)where π x ( ξ ) = ± (cid:113) [ ε − ν ( ξ )] − π y (24)represents the position-dependent longitudinal momentum of electrons, while the transverse momentum π y remainsas a constant in the tunneling process.As a next step, we want to find the leading-order wave function Ψ ( x ). Although Eq. (21) appears as an eigenvalueproblem, it is actually much more complicated since S ( ξ ) in Eq. (23) and π x ( ξ ) in Eq. (24) also depend on particleenergy ε . In fact, Eq. (21) could be utilized to find various components of the following zero-order wave functionΨ ( ξ, π y | φ, τ ) = ϕ (0) A ( ξ | φ, τ ) ϕ (0) H ( ξ ) ϕ (0) B ( ξ | φ, τ ) , (25)such that κ ( ξ ) ϕ (0) A ( ξ ) + cos φ [ τ π x ( ξ ) − iπ y ] ϕ (0) H ( ξ ) = 0 , (26)sin φ [ τ π x ( ξ ) + iπ y ] ϕ (0) H ( ξ ) + κ ( ξ ) ϕ (0) B ( ξ ) = 0 . (27)Consequently, the wave function in Eq. (25) can be rewritten asΨ ( ξ | φ, τ ) = cos φ Θ( ξ | τ ) − φ Θ (cid:63) ( ξ | τ ) ϕ (0) H ( ξ ) , (28)where Θ (cid:63) ( ξ | τ ) represents the complex conjugate of Θ( ξ | τ ), which is given byΘ( ξ | τ ) = 1 κ ( ξ ) [ τ π x ( ξ ) − iπ y ] = − τ exp [ − iτ θ k ( ξ )] . (29)Here, θ k ( ξ ) = tan − [ k x ( ξ ) /k y ] is the angle of wave vector k = { k x ( ξ ) , k y } = 1 / (cid:126) { π x ( ξ ) , π y } with respect to the x -axis.It is clear from Eq. (29) that Θ( ξ | τ ) does depend on the valley index τ and position ξ but not on φ . Moreover, thespatial dependence of ϕ (0) H ( ξ ) in Eq. (28) still needs to be determined.Moreover, by taking λ = 0, we get from the recurrence equation in Eq. (19) thatˆ O T ( ξ, π y | φ, τ ) Ψ ( ξ, π y | φ, τ ) = √ (3) x ( φ ) ∂∂ξ Ψ ( ξ, π y | φ, τ ) , (30)where Ψ ( ξ, π y | φ, τ ) has already been obtained expect for the spatial dependence on ϕ (0) H ( ξ ). However, we still do notknow the exact form of Ψ ( ξ, π y | φ, τ ). Mathematically, Ψ ( ξ, π y | φ, τ ) can be constructed from a linear combinationof three arbitrary orthogonal state vectors (cid:12)(cid:12) v (cid:11) , (cid:12)(cid:12) v (cid:11) and (cid:12)(cid:12) v (cid:11) in a three-dimensional spinor space. Let us first choose (cid:12)(cid:12) v (cid:11) to be the spinor part of Ψ ( ξ | φ, τ ) in Eq. (28), yielding (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11) = cos φ Θ( ξ | τ ) − φ Θ (cid:63) ( ξ | τ ) . (31)With the given spinor state (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11) in Eq. (31), we can choose freely the remaining (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11) and (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11) as long as all of them are mutually orthogonal to each other. By referencing wave functions in Eqs. (10) and (11) forincident particles, we take accordingly (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11) = cos φ Θ( ξ | τ )+1sin φ Θ (cid:63) ( ξ | τ ) , (32) (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11) = sin φ Θ( ξ | τ )0 − cos φ Θ (cid:63) ( ξ | τ ) . (33)Consequently, Ψ ( ξ, π y | φ, τ ) can be formally written asΨ ( ξ, π y | φ, τ ) = ϕ (0) H ( ξ ) (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11) + ϕ (1 , H ( ξ ) (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11) + ϕ (1 , H ( ξ ) (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11) . (34)Now, substituting Eq. (34) into Eq. (30), we find (cid:10) v ( ξ | φ, τ ) (cid:12)(cid:12) ˆ O T ( ξ, π y | φ, τ ) Ψ ( ξ, π y | φ, τ ) (cid:11) = (cid:10) v ( ξ | φ, τ ) (cid:12)(cid:12) ˆ O T ( ξ, π y | φ, τ ) (cid:12)(cid:12) (cid:110)(cid:12)(cid:12) ϕ (0) H ( ξ ) (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11) + ϕ (1 , H ( ξ ) (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11) + ϕ (1 , H ( ξ ) (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11)(cid:111)(cid:11) = 0 (35)since (cid:12)(cid:12) v j ( ξ | φ, τ ) (cid:11) for j = 1 , , O T ( ξ, π y | φ, τ ) (cid:12)(cid:12) v ( ξ | φ, τ ) (cid:11) = 0 from Eq. (22).Consequently, using Eq. (30) we get ( ) a ( ) b ( ) c ( ) d ( ) e ( ) f FIG. 1: (Color online) Calculated energy dispersion relations (shown shaded) ε ( k, ∆ | φ ) in α − T based on Eq. (44). Panels( a )–( c ) correspond to phase φ = π/
6, while panels ( d )–( f ) to a dice lattice with φ = π/
4. The left column relates to the casewith ∆ / E (0) = 0; the middle column to ∆ / E (0) = 0 .
5; and the right column to ∆ / E (0) = 1 .
0. Here, the Fermi wave number k F is taken as an unit for k , and E (0) = (cid:126) v F k F as an unit for energy with Fermi velocity v F . (cid:10) v ( ξ | φ, τ ) (cid:12)(cid:12) ˆΣ (3) x ( φ ) ∂∂ξ Ψ ( ξ, π y | φ, τ ) (cid:11) = (cid:10) v ( ξ | φ, τ ) (cid:12)(cid:12) ˆΣ (3) x ( φ ) ∂∂ξ (cid:12)(cid:12) (cid:10) v ( ξ | φ, τ ) (cid:11) = 0 . (36)Here, Eq. (36) can be written as[Θ( ξ | τ ) + Θ (cid:63) ( ξ | τ )] ∂ ϕ (0) H ( ξ ) ∂ξ (37)+ 12 (cid:26) (cid:20) ∂ Θ( ξ | τ ) ∂ξ + ∂ Θ (cid:63) ( ξ | τ ) ∂ξ (cid:21) + (cid:20) ∂ Θ( ξ | τ ) ∂ξ − ∂ Θ (cid:63) ( ξ | τ ) ∂ξ (cid:21) cos 2 φ (cid:27) ϕ (0) H ( ξ ) = 0 . Writing Γ ( ± ) ( ξ | τ ) = Θ( ξ | τ ) ± Θ (cid:63) ( ξ | τ ), we finally arrive at the equation:Γ (+) ( ξ | τ ) ∂ ϕ (0) H ( ξ ) ∂ξ + 12 (cid:20) ∂∂ξ Γ (+) ( ξ | τ ) + cos 2 φ ∂∂ξ Γ ( − ) ( ξ | τ ) (cid:21) ϕ (0) H ( ξ ) = 0 , (38)in which we have F ( ξ | φ, τ ) ≡ ∂∂ξ (cid:2) Γ (+) ( ξ | τ ) + cos 2 φ Γ ( − ) ( ξ | τ ) (cid:3) = 2 i π y πξ / [cos 2 φ π x ( ξ ) − iπ y ] dπ x ( ξ ) dξ , (39)which is obtained directly from Eq. (29).The solution to Eq. (38) is easily found to be ϕ (0) H ( ξ ) = c exp − ξ (cid:90) ξ F ( ζ | φ, τ )Γ (+) ( ζ | τ ) dζ . (40)Specifically, for a dice lattice with φ = π/
4, Eqs. (38) and (40) are reduced to ( ) a ( ) b ( ) c ( ) d ( ) e ( ) f ( ) i ( ) ( ) j k FIG. 2: (Color online) Calculated energy dispersion relations ε ( k, ∆ | φ ) in α − T based on Eq. (50) in the presence of a finiteenergy gap ∆. Panels ( a )–( c ) correspond to phase φ = 0, panels ( d )–( f ) to φ = π/
6, and panels ( i )–( k ) to φ = π/
4. The leftcolumn relates to the case with ∆ / E (0) = 0; the middle column to ∆ / E (0) = 0 .
5; and the right column to ∆ / E (0) = 1 . ( ) a ( ) d ( ) e ( ) b ( ) c ( ) f FIG. 3: (Color online) Plots for classically inaccessible regions, i.e., Im[ π x ( ξ, ∆ | φ )] (cid:54) = 0, of an incident electron in gappedgraphene under a potential barrier ν ( ξ ) = ν + a ξ . Here, ε is the kinetic energy of incoming electron, and ξ = 0 is selectedas the crossing point at which the particle switches between electron and hole states in the barrier region. The boundaries ofshaded region correspond to the turning points satisfying π x ( ξ, ∆ ) = 0, and each panel shows the shaded region for specific ∆ values as indicated. The upper panels ( a )–( c ) demonstrate how the area and shape of shaded regions depend on π y , while thelower panels ( d )–( f ) on the barrier slope a . All quantities and parameters used in this graphs and others later are dimensionlessand scaled by Eq. (13). ( ) a ( ) d ( ) e ( ) b ( ) c ( ) f ( ) i ( ) j ( ) k FIG. 4: (Color online) Plots for classically inaccessible regions (shown shaded) of an incident electron in α − T materials with ν ( ξ ) = ν + a ξ . The boundaries of shaded region correspond to the turning points satisfying π x ( ξ, ∆ | φ ) = 0, and each panelshows the shaded region for specific ∆ and π y values as indicated. Three upper panels show the results as functions o π y fora dice lattice with φ = π/ a = 1, while the remaining six plots for α − T lattices as functions of φ for various values of∆ and a = 2. ( ) a ( ) b FIG. 5: (Color online) Calculated transmissions T ( π y | a, ∆ ) from Eq. (54) for gapped graphene. Panel ( a ) presents T ( π y | a, ∆ ) as a function of transverse momentum π y for slope a = 0 . = 0 , . , . , . b ) shows the a dependence of T ( π y | a, ∆ ) with ∆ = 0 and various vales of π y = 0 , . , . , ( ) a ( ) b ( ) c FIG. 6: (Color online) Calculated transmissions T ( π y | a, ∆ , φ ) from Eq. (54) for gapped α −T lattices. Panel ( a ) demonstrates T ( π y | a, ∆ , φ ) as a function of transverse momentum π y for φ = π/ a = 0 . = 0 , . , . , . b ) and ( c ) display the φ dependence of T ( π y | a, ∆ , φ ) with π y = 0, a = 1 and various vales of ∆ = 0 , . , . , . b ) and with π y = 0, ∆ = 0 . a = 0 . , , . , c ). dϕ (0) H ( ξ ) ϕ (0) H ( ξ ) = −
12 Γ (+) ( ξ | τ ) ∂ Γ (+) ( ξ | τ ) ∂ξ dξ = − π y (cid:113) π x ( ξ ) − π y dπ x ( ξ ) , (41)while for gapless graphene with φ = 0 we acquire the same results as in Ref. [4]. At last, Eq. (41) leads to the solution ϕ (0) H ( ξ ) = (cid:115) π x ( ξ ) − π y π x ( ξ ) = (cid:115) − (cid:20) π y π x ( ξ ) (cid:21) , (42)and then Ψ ( ξ | φ, τ ) in Eq. (28) can be completely determined. All other higher-order wave function Ψ λ ( ξ | φ, τ ) for λ = 1 , , · · · can be found by using Eq. (19) repeatedly. Moreover, Eq. (42) becomes divergent as π x ( ξ ) = 0, whichis similar to the resulst of Schr¨odinger particles and graphene with a zero or finite bandgap. Such a unique featureindicates that the WKB approximation cannot be used in the vicinity of so-called turning points with π x ( ξ ) = 0. III. ENERGY BANDGAP IN A PSEUDOSPIN-1 LATTICE
In this section, we will concentrate on calculating electron transmission over a barrier with a linear potentialprofile ν ( ξ ) = ν + a ξ for an α − T material, where the constant a quantifies the strength of an applied electricfield. In fact, we are able to find the transmission solely based on the semi-classical action S ( ξ ) and the longitudinalcomponent of electron momentum π x ( ξ ), and we do not need compute the wave function, its phase factors or thespatial dependence. This is obviously an advantage of employing the WKB approximation, i.e., a possibility to acquirevery precise transmission of electrons with limited knowledge on electronic states, and an easily evaluated S ( ξ ) evenfor a very complicated model Hamiltonian. In this paper, we focus on two special cases with a bandgap induced eitherby adding an insulating substrate to an α − T layer or by imposing an external off-resonance dressing field.We start with adding an α -independent energy gap ∆ to our previous Hamiltonian in Eq. (1) by using ˆΣ (3) z inEq. (7), namely ˆ H α ( k | τ, φ ) = (cid:126) v F k τ − cos φ k τ + cos φ k τ − sin φ k τ + sin φ + ∆ − , (43)which gives rise to an eigenvalue equation ε (cid:2) ε − ∆ − ( (cid:126) v F k ) (cid:3) + ( (cid:126) v F k ) ∆ cos 2 φ = 0 . (44)A similar gap model was adopted in Ref. [66] for studying effects of an ionized impurity atom on electronic states of α − T .Figure 1 displays the calculated energy dispersion from Eq. (44), from which we find that symmetry between thevalence and conduction bands under ∆ > φ except for φ = π/ F V and that between the flat and conduction bands ∆ CF areopened up, and satisfies ∆ CF (cid:54) = ∆ F V away from k = 0 for all considered cases. Therefore, the triple connection forthe valence, flat and conduction bands in each corner of the Brillouin zone is fully broken.In the presence of the energy gap ∆ , the operator ˆ O T ( ξ, π y | φ, τ ) initially introduced in Eq. (20) is modified intoˆ O T ( ξ, π y , ∆ | φ, τ ) = κ ( ξ ) + ∆ cos φ ( ∂ S ∆ ( ξ ) /∂ξ − iπ y ) 0cos φ ( ∂ S ∆ ( ξ ) /∂ξ + iπ y ) κ ( ξ ) sin φ ( ∂ S ∆ ( ξ ) /∂ξ − iπ y )0 sin φ ( ∂ S ∆ ( ξ ) /∂ξ + iπ y ) κ ( ξ ) − ∆ , (45)where the bandgap ∆ should be rescaled to ∆ /V , corresponding to the scaled particle’s kinetic energy ε andexternal potential ν ( ξ ) in Eq. (13). For simplicity, however, we will still adopt the same notation ∆ . Here, we wouldlike to emphasize that the Hamiltonian in Eq. (43) provides a good description for all electronic properties of α − T φ → (3) z in Eq. (43) cannot be properly transformed to a 2 × = 0.To overcome these limitations, we introduce an alternative model Hamiltonian, i.e., including a φ -dependent termˆ H ∆ ( φ ) = ∆2 ˆ S z ( φ ) = ∆ cos φ − cos 2 φ
00 0 − sin φ , (46)where the bandgap ∆ is included through ˆ S z ( φ ) = − i (cid:104) ˆ S x ( φ ) , ˆ S y ( φ ) (cid:105) , (47)as employed in Ref. [33]. Here, the φ -dependent gap term in Eq. (46) can also be viewed as a part of the Floquet-Magnus Hamiltonian for electron dressed state under a circularly-polarized dressing field, which also depends onthe valley index τ = ±
1. It is easy to show that Eq. (46), in the graphene limit φ →
0, reduces toˆ H ∆ ( φ →
0) = ∆ − , (48)and meanwhile, for a dice lattice with φ = π/
4, toˆ H ∆ ( φ → π/
4) = ∆2 − = ∆2 ˆΣ (3) z , (49)which implies that the effect from laser irradiation on a dice lattice is only half of that on graphene.By combining Eqs. (1) and (46), the energy dispersion for gapped α − T lattices is found to satisfy the followingequation, i.e., ε − (cid:18) k (cid:19) ε − ∆ (cid:20) ε cos 4 φ + ∆3 sin 2 φ sin 4 φ (cid:21) = 0 , (50)which gives rise to three solutions, given by ε λ ( k, ∆ | φ ) = 2 √ (cid:114) k + ∆ φ ) cos (cid:34) πλ − (cid:32) √ sin 2 φ sin 4 φ [8 k + ∆ (5 + 3 cos 4 φ )] / (cid:33)(cid:35) , (51)where λ = 0 , , φ = 0)and dice lattice ( φ = π/
4) are displayed in the top and bottom rows, respectively, along with the case with φ = π/ α − T material in the middle row. From both top and bottom rows, we find symmetric dispersions withrespect to k = 0, and the graphene gap at k = 0 is exactly twice of that for a dice lattice. As φ = π/ a )–( c ) of Fig. 1, i.e., switchingbetween peak and valley at k = 0. Moreover, the mirror symmetry between the valence and conduction band is alsobroken for a finite value of ∆ and all values of 0 < φ < π/ O T ( ξ, π y | φ, τ ) in Eq. (45) has been changed toˆ O T ( ξ, π y , ∆ | φ, τ ) = κ ( ξ ) + ∆ cos φ cos φ ( ∂ S ∆ ( ξ ) /∂ξ − iπ y ) 0cos φ ( ∂ S ∆ ( ξ ) /∂ξ + iπ y ) κ ( ξ ) − ∆ cos 2 φ sin φ ( ∂ S ∆ ( ξ ) /∂ξ − iπ y )0 sin φ ( ∂ S ∆ ( ξ ) /∂ξ + iπ y ) κ ( ξ ) − ∆ sin φ , (52)2where ∆ = ∆ /V . Therefore, from Eqs. (52) and (23) we are able to find explicitly the spatially-dependent longitu-dinal momentum π x ( ξ, ∆ | φ ) as[ π x ( ξ, ∆ | φ )] = κ ( ξ ) − π y − ∆ φ ) + ∆ κ ( ξ ) sin 2 φ sin 4 φ . (53)As φ = 0, w get from Eq. (53) that [ π x ( ξ, ∆ | φ = 0)] = κ ( ξ ) − ∆ − π y , which is the same as that in Ref. [4]. For adice lattice with φ = π/
4, on the other hand, we find [ π x ( ξ, ∆ | φ = π/ = κ ( ξ ) − (∆ / − π y .Equation (53) becomes quadratic if its last term equals zero, which can be satisfied for either graphene with φ = 0or a dice lattice with φ = π/
4. In these two case, the classically inaccessible regions are simply connected, as seenin Figs. 3 and 4. For all other φ values, the turning points, or the boundaries of classically forbidden regions, aredetermined by a cubic κ ( ξ ) equation and these regions consist of several parts with non-trivial shapes and connections,as demonstrated in Fig. 4( d )–( k ).In WKB theory, the transmission amplitude T ( π y | a, ∆ , φ ), or the probability for electron tunneling, can beestimated by the integral of | π x ( ξ, ∆ | φ ) | presented in Eq. (53), which is equivalent to Im[ π x ( ξ, ∆ | φ )], over theclassically forbidden regions (CFR). This leads to T ( π y | a, ∆ , φ ) = exp − (cid:126) (cid:90) CFR | π x ( ξ ) | dξ (54)under the condition of (cid:90) CFR | π x ( ξ ) | dξ (cid:29) (cid:126) . (55)Here, CFR are defined by π x ( ξ, ∆ ) <
0, i.e., the particle acquires an imaginary longitudinal momentum as thena strongly decayed transmission. The calculated location and size of CFR from Eq. (53) for gapped graphene arepresented as shaded regions in Fig. 3. The boundaries of CFR, determined by π x ( ξ, ∆ ) = 0, are not linear for κ ( ξ ) inthe presence of a finite energy gap, as seen from Eq. (53). The obtained CFR is always symmetric with the selectedelectron-to-hole crossing point ξ = 0, given by κ ( ξ ) = [ ν ( ξ ) − ε ] / (cid:126) = 0. For the case of ∆ = 0, we find from Eq. (53)that κ ( ξ ) = − (cid:113) π x ( ξ ) + π y ≡ − π tot ( ξ ) which is opposite to the total momentum π tot ( ξ ).Even though the result in Eq. (54) appears only as an estimation, for some trivial cases, e.g., gapless graphene,this result becomes accurate. In fact, for ∆ = 0 we know that both Eq. (53) for π x ( ξ ) and the conditions fordetermining the turning points π x ( ξ ) = 0 coincide with gapless graphene. Therefore, the obtained transmission T ( π y | a ) = exp [ − ( π/a ) π y ] is expected also true for an arbitrary α − T material with 0 < α < = 0.Furthermore, we expect that the width of CFR will increase with π y , as verified from Eq. (53), since it leads to decreaseof π x ( ξ ) for a given energy ε of incoming particle. However, the CFR width reduces for increasing slope a becausethe electron/hole crossover and the new classically allowable state is now achieved within a shorter displacement ofa charger particle along its trajectory under a non-uniform potential. These predicted effects are indeed observedfor electron transmission in gapped graphene, as presented in Fig. 5. Apart from the energy gap ∆ , which alwaysresults in reduced transmission, we see a similar effect from π y kept as a constant in our model, as seen in Fig. 5( a ).In contrast, a larger potential slope a leads to an enhanced transmission for fixed ∆ , as found from Fig. 5( b ).Results for transmission of electrons in gapped α − T are presented in Fig. 6. Similar to gapped graphene, theKlein paradox occurs only for a head-on collision, or π y = 0, and ∆ = 0, as found from Figs. 6( a ) and 6( b ).Moreover,we also find that the Klein paradox remains true for all values of a and φ , as long as these two conditions are met.For a finite bandgap ∆ ≥ . α − T , we see a strong dependence of transmission on phase φ , which becomesconsiderable less than one as shown in Figs. 6( b ) and 6( c ). In this case, transmission increases with φ due to existenceof the flat band in energy dispersion. Furthermore, we also find increasing transmission with a as seen in Fig. 6( c ).Such a feature could be attributed to the fact that switching between electron and hole states will be faster for asteeper potential, and this explanation holds true for both gapped graphene and all types of α − T materials.3 IV. CONCLUDING REMARKS AND SUMMARY
In conclusion, we have generalized the WKB semi-classical approximation for pseudospin-1 α − T lattices byderiving a complete set of recurrence transport equations. The solutions of these coupled differential equations haveprovided the semi-classical wave functions for gapped and the phase dependent α − T Hamiltonian, and led to thecorrect description of quantum states for charge carriers in the ballistic regime. Additionally, we have obtained closedform analytic expression for WKB wave functions which could be applied to various analytical models as well as forstudying the tunneling properties of electrons in gapped α − T materials.Our derivation of the generalized WKB equations and a pseudospin-1 Dirac-Weyl Hamiltonian for the α − T modelare shown to be quite different from a Schr¨odinger particle considered in standard quantum mechanics and even aDirac electron in graphene. For this case, we are facing with 3 × α − T materials. However, our focus in this paper was on the phase dependent electrontunneling, band gap modification and the suppression of the Klein paradox with non-square potential barriers. Even alimited knowledge regarding the semi-classical action and time-dependent momentum can provide us with importantinformation on electron dynamics and help us evaluate the tunneling transmission of electrons through the integralof absolute electron momentum over the classically inaccessible regions along the tunneling-electron path.Based on our generalized WKB theory, we have investigated the electron tunneling through an electric field biasedpotential barrier and revealed unimpeded Klein tunneling for the head-on collisions in the absence of an energy gap.This event applies to all α − T materials independent of the geometry phase φ , i.e., the calculated transmissionin gapless α − T materials does not depend on φ . Moreover, we have found that the π y -dependent transmissionis greatly reduced in the presence of a band gap, and is decreased when π y is increased. On the other hand, theslope of the potential profile always enhances the transmission despite the energy gap, which is due to speeding upan electron-to-hole (or hole-to-electron) transition over a shortened distance in ξ position space. As a result, thegeneralized WKB theory in this paper could be utilized to discern the array of localized and trapped electronic statesthrough their barrier scattering effects.Our effort in generalizing the WKB approximation to deal with the Hamiltonian for α − T lattices providesadditional tools to explore additional important and unknown modifications to tunneling mechanism in such materials,as demonstrated by our obtained analytical expressions for electron transmission and Klein tunneling. We believe thatour current study has revealed the most remarkable and exclusive physics features of novel low-dimensional materials.Meanwhile, all these discoveries will definitely find their applications in Dirac cone based tunneling transistors inanalog RF devices, along with their tunneling current control by constructive barrier scattering across designed arrayof coherent scatters. Acknowledgments
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