Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Chiral tunneling in single and bilayer graphene Mikhail Katsnelson, Koen Reijnders, Timur Tudorovskiy Institute for Molecules and Materials – Theory of Condensed Matter Radboud University Nijmegen Nanoelectronics beyond the roadmap Keszthely, Lake Balaton, Hungary 14th June 2011 TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 1 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Outline Klein tunneling and “magic angles” Generic potential landscape: n-p junction Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 2 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Klein tunneling and “magic angles” [ σ x ˆ p x + σ y p y + V ( x ) − E ] ψ ( x ) = 0 *Fermi velocity is 1 Stepwise barrier � 0 , | x | > a V ( x ) = V 0 , | x | < a TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 3 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Velocity conservation Normal incidence p y = 0 : [ σ x ˆ p x + V ( x ) − E ] ψ ( x ) = 0 [ . . . ] commutes with σ x = ⇒ velocity conservation! σ = +1 : [+ˆ p x + V ( x ) − E ] ψ ( x ) = 0 σ = − 1 : [ − ˆ p x + V ( x ) − E ] ψ ( x ) = 0 Classical mechanics: ± | p x | + V ( x ) − E = 0, ± = ⇒ electron/hole Velocity conservation leads to a change of particle type Is the diagonalization possible for non-perpendicular incidence? TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 4 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Schr¨ odinger equations with COMPLEX potentials [ σ p + U ( x )] ψ ( x ) = 0 , U ( x ) = V ( x ) − E Simple transformation... [ σ p − U ( x )][ σ p + U ( x )] ψ ( x ) = y − U ( x ) 2 − i � σ x U ′ ( x )] ψ ( x ) = 0 p 2 x + p 2 [ˆ σ x is the constant matrix = ⇒ diagonalization! Effective complex potential y − ( V ( x ) − E ) 2 ∓ i � V ′ ( x )] η 1 , 2 ( x ) = 0 [ − � 2 ∆ + p 2 TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 5 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Generic potential landscape: n-p junction TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 6 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Exact solution [Cheianov, Fal’ko 2006] Linear potential: T = e − p 2 y /α V ( x ) = α x ψ 1 , 2 = D i ν ( z ) ± i √ ν e i π/ 4 D i ν − 1 ( z ) , ν = p 2 y / 2 α TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 7 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Exact solution [Cheianov, Fal’ko 2006] Linear potential: T = e − p 2 y /α V ( x ) = α x ψ 1 , 2 = D i ν ( z ) ± i √ ν e i π/ 4 D i ν − 1 ( z ) , ν = p 2 y / 2 α Incoming and reflected: x → −∞ : √ ψ 1 , 2 → ( − z ) i ν e − i α x 2 / 2 ∓ Γ(1 − i ν ) ( − z ) − i ν e i α x 2 / 2 − i πν + i π/ 4 2 πν i k = dS / dx = α | x | > 0 electron √ ψ 1 , 2 → z i ν e − i α x 2 / 2 , 2 α e i π/ 4 | x | Transmitted: x → ∞ : z = k = dS / dx = − α | x | < 0 hole TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 7 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment General WKB treatment � x 1 � � � p 2 y − [ E − V ( x )] 2 dx T = Exp − 2 x 0 TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 8 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Exact solution and the general WKB treatment Potential profile, u � x � Violet ⇒ numerics Red ⇒ linear potential T = e − p 2 sin 2 ( φ ) /α Blue = ⇒ WKB 0.4 0.4 0.3 12 0.2 0.2 0.1 0. 0 0.2 0.4 0.6 0.8 1.0 TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 9 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Generic potential landscape: n-p-n junction TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 10 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Magic angles as resonant scattering � x 1 ( p y ) 1 y dx = n + ν � [ E − V ( x )] 2 − p 2 π 4 x 0 ( p y ) TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 11 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Magic angles as resonant scattering � x 1 ( p y ) 1 y dx = n + ν � [ E − V ( x )] 2 − p 2 π 4 x 0 ( p y ) TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 11 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Magic angles as resonant scattering � x 1 ( p y ) 1 y dx = n + ν � [ E − V ( x )] 2 − p 2 π 4 x 0 ( p y ) TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 11 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Maslov index and Berry phase p p electrons y x ν = 1 φ B = ± π/ 2 p x x p p y x ν = 2 holes φ B = 0 p x x TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 12 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Fabry-P´ erot resonances [Shytov, Rudner, Levitov 2008] TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 13 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Analytics vs. numerics 5 Π 2 | t 1 | 2 | t 2 | 2 12 1. T = Π | 1 − | r 1 || r 2 | e − 2 iS + i ∆Θ | 2 3 0.8 Π ∆Θ = πν/ 2 + δ ( p y ) 4 0.6 SRL: ∆Θ is undefined! Π 6 Blue : numerics 0.4 Gray : SRL, ∆Θ = 0 Π 12 0.2 • Maslov index ν = 2 0. 0 • RTK: δ is computed RTK resonances S − δ/ 2 = π ( n + ν/ 4) TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 14 / 18
Klein tunneling and “magic angles” Generic potential landscape: n-p junction Radboud University Nijmegen Generic potential landscape: n-p-n junction Magic angles as resonant scattering: WKB treatment Analytics vs. numerics 5 Π 2 | t 1 | 2 | t 2 | 2 12 1. T = Π | 1 − | r 1 || r 2 | e − 2 iS + i ∆Θ | 2 3 0.8 Π ∆Θ = πν/ 2 + δ ( p y ) 4 0.6 SRL: ∆Θ is undefined! Π 6 Blue : numerics 0.4 Orange : WKB ( δ = 0) Π 12 0.2 • Maslov index ν = 2 0. 0 • RTK: δ is computed RTK resonances S − δ/ 2 = π ( n + ν/ 4) TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 14 / 18
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