Exercice session:Klein tunnel effect in graphene= petit effet tunnel dans le graphène…

Pierre Allain et Jean-Noël Fuchs LPS Orsay

OutlineI) IntroductionII) Potential step (np junction)III) Potential barrier (npn structure)IV) Conclusion

Cargèse, October 2008

Oskar Klein a.k.a. « Oskar-le-petit »Sweden's quantum king (1894-1977)‏

Klein paradox 1929

II) Potential step (np junction)

ZONE 1 ZONE 2

II) Potential step

II) Potential step

e=0.25

e=0.5: T= cos²Φ

e=0.75

e=1: T(Φ)=0 except T(0)=1

e=0: T=2cosΦ/[1+cosΦ]

II) Potential step

Critical angle

Evanescent waves

Propagative waves

III) Potential barrier (npn structure)

ZONE 1 ZONE 2 ZONE 3

III) Potential barrier

III) Potential barrier

e≡E/V0=0.42, l≡d/(h*v

F/V

0)=4.85

e=0.29, l=6.91

T=1

Katsnelson, Novoselov and Geim, Nat. Phys. 2006

III) Potential barrier

e≡E/V0

T

Φ

l≡d/(h*vF/V

0)=1