Chiral tunneling in single and bilayer graphene Mikhail Katsnelson, - - PowerPoint PPT Presentation

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May 01, 2023 50 likes •293 views

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

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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

Radboud University Nijmegen

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

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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

Radboud University Nijmegen

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

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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

Radboud University Nijmegen

Klein tunneling and “magic angles”

[σxˆ px + σypy + V (x) − E]ψ(x) = 0 *Fermi velocity is 1

Stepwise barrier

V (x) = 0, |x| > a V0, |x| < a

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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

Radboud University Nijmegen

Velocity conservation

Normal incidence

py = 0 : [σxˆ px + V (x) − E]ψ(x) = 0 [. . .] commutes with σx = ⇒ velocity conservation! σ = +1 : [+ˆ px + V (x) − E]ψ(x) = 0 σ = −1 : [−ˆ px + V (x) − E]ψ(x) = 0 Classical mechanics: ± |px| + V (x) − E = 0, ± = ⇒ electron/hole Velocity conservation leads to a change of particle type Is the diagonalization possible for non-perpendicular incidence?

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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

Radboud University Nijmegen

Schr¨

dinger equations with COMPLEX potentials

[σp + U(x)]ψ(x) = 0, U(x) = V (x) − E

Simple transformation...

[σp − U(x)][σp + U(x)]ψ(x) = [ˆ p2

x + p2 y − U(x)2 − i σxU′(x)]ψ(x) = 0

σx is the constant matrix = ⇒ diagonalization!

Effective complex potential

[−2∆ + p2

y − (V (x) − E)2 ∓ i V ′(x)]η1,2(x) = 0

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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

Radboud University Nijmegen

Generic potential landscape: n-p junction

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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

Radboud University Nijmegen

Exact solution [Cheianov, Fal’ko 2006]

Linear potential: T = e−p2

y/α

V (x) = αx ψ1,2 = Diν(z) ± i√νeiπ/4Diν−1(z), ν = p2

y/2α

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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

Radboud University Nijmegen

Exact solution [Cheianov, Fal’ko 2006]

Linear potential: T = e−p2

y/α

V (x) = αx ψ1,2 = Diν(z) ± i√νeiπ/4Diν−1(z), ν = p2

y/2α

Incoming and reflected: x → −∞ : ψ1,2 → (−z)iνe−iαx2/2 ∓

i √ 2πν Γ(1−iν)(−z)−iνeiαx2/2−iπν+iπ/4

k = dS/dx = α|x| > 0 electron Transmitted: x → ∞ : ψ1,2 → ziνe−iαx2/2, z = √ 2αeiπ/4|x| k = dS/dx = −α|x| < 0 hole

TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 7 / 18

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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

Radboud University Nijmegen

General WKB treatment

T = Exp

−2

x1

y − [E − V (x)]2dx

Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 8 / 18

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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

Radboud University Nijmegen

Exact solution and the general WKB treatment

Potential profile, ux

Violet ⇒ numerics Red ⇒ linear potential T = e−p2 sin2(φ)/α Blue = ⇒ WKB

12 0. 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.1 0.2 0.3 0.4

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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

Radboud University Nijmegen

Generic potential landscape: n-p-n junction

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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

Radboud University Nijmegen

Magic angles as resonant scattering

1 π x1(py)

x0(py)

[E − V (x)]2 − p2

ydx = n + ν

4

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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

Radboud University Nijmegen

Magic angles as resonant scattering

1 π x1(py)

x0(py)

[E − V (x)]2 − p2

ydx = n + ν

4

TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 11 / 18

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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

Radboud University Nijmegen

Magic angles as resonant scattering

1 π x1(py)

x0(py)

[E − V (x)]2 − p2

ydx = n + ν

4

TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 11 / 18

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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

Radboud University Nijmegen

Maslov index and Berry phase

x p

x electrons

p

y

p

x

ν = 1 φB = ±π/2

x p

x holes

p

x

p

y ν = 2 φB = 0

TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 12 / 18

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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

Radboud University Nijmegen

Fabry-P´ erot resonances [Shytov, Rudner, Levitov 2008]

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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

Radboud University Nijmegen

Analytics vs. numerics

Π 12 Π 6 Π 4 Π 3 5 Π 12 2 0. 0.2 0.4 0.6 0.8 1. T = |t1|2|t2|2 |1 − |r1||r2|e−2iS+i∆Θ|2 ∆Θ = πν/2 + δ(py) SRL: ∆Θ is undefined! Blue: numerics Gray: SRL, ∆Θ = 0

Maslov index ν

= 2

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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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

Radboud University Nijmegen

Analytics vs. numerics

Π 12 Π 6 Π 4 Π 3 5 Π 12 2 0. 0.2 0.4 0.6 0.8 1. T = |t1|2|t2|2 |1 − |r1||r2|e−2iS+i∆Θ|2 ∆Θ = πν/2 + δ(py) SRL: ∆Θ is undefined! Blue: numerics Orange: WKB (δ = 0)

Maslov index ν

= 2

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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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

Radboud University Nijmegen

Analytics vs. numerics

Π 12 Π 6 Π 4 Π 3 5 Π 12 2 0. 0.2 0.4 0.6 0.8 1. T = |t1|2|t2|2 |1 − |r1||r2|e−2iS+i∆Θ|2 ∆Θ = πν/2 + δ(py) SRL: ∆Θ is undefined! Blue: numerics Red: RTK analytics

Maslov index ν

= 2

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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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

Radboud University Nijmegen

Analytics vs. numerics

Π 12 Π 6 Π 4 Π 3 5 Π 12 2 0. 0.2 0.4 0.6 0.8 1. T = |t1|2|t2|2 |1 − |r1||r2|e−2iS+i∆Θ|2 ∆Θ = πν/2 + δ(py) SRL: ∆Θ is undefined! Blue: numerics Orange: WKB (δ = 0) Red: RTK analytics Gray: SRL, ∆Θ = 0

Maslov index ν

= 2

RTK: δ is computed

TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 14 / 18

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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

Radboud University Nijmegen

Symmetric and asymmetric barriers

Π 12 Π 6 Π 4 Π 3 5 Π 12 2 0. 0.2 0.4 0.6 0.8 1. Orange = ⇒ symmetric Blue = ⇒ asymmetric For an asymmetric bar- rier transmission ampli- tudes are suppressed!

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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

Radboud University Nijmegen

Phase space for single and bilayer

Single layer Bilayer For normal incidence (separatrices) the smoothest classical trajectory corresponds to total transmission in single layer graphene and to total reflection in bilayer graphene

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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

Radboud University Nijmegen

Protected resonances in bilayer

15 ° 30 ° 45 ° 60 ° 75 ° 90 ° 0. 0.2 0.4 0.6 0.8 1. Blue = ⇒ symmetric Red = ⇒ asymmetric

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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

Radboud University Nijmegen

Conclusion

Semiclassical theory of Klein tunneling is presented
Magic angles are treated in terms of resonant scattering
Positions of magic angles have been correlated with the

quantization condition

The influence of the potential shape on the transmission

coefficient has been investigated Thank you for your attention! For a detailed discussion you are welcome to visit Koen’s poster

TCM Nanoelectronics beyond roadmap Chiral tunneling in single and bilayer graphene 18 / 18