Klein Tunneling. PHYS 503 Physics Colloquium Fall /11

Transcription

1 Klein Tunneling PHYS 503 Physics Colloquium Fall /11 Deeak Rajut Graduate Research Assistant Center for Laser Alications University of Tennessee Sace Institute Web: htt://drajut.com 1of xx

2 Outline Classical icture Tunneling Klein Tunneling Biolar junctions with grahene Alications 2 2of xx

3 Classical Picture H Kinetic Energy = E Mass of the ball = m E < mgh E = mgh E > mgh 3 3of xx

4 Tunneling Transmission of a article through a otential barrier higher than its kinetic energy (V>E). It violates the rinciles of classical mechanics. It is a quantum effect. 4 4of xx

5 Quantum tunneling effect On the quantum scale, objects exhibit wave-like characteristics. Quanta moving against a otential hill can be described by their wave function. The wave function reresents the robability amlitude of finding the object in a articular location. 5 5of xx

6 Quantum tunneling effect If this wave-function describes the object as being on the other side of the otential hill, then there is a robability that the object has moved through the otential hill. This transmission of the object through the otential hill is termed as tunneling. E < V Ψ(x) V Ψʹ(x) Tunneling = Transmission through the otential barrier 6 6of xx

7 Tunneling Reflection Interference fringes Transmission Tunneling Source: htt://en.wikiedia.org/wiki/quantum_tunneling 7 7of xx

8 Klein Tunneling In quantum mechanics, an electron can tunnel from the conduction into the valence band. Such tunneling from an electron-like to hole-like state is called as interband tunneling or Klein tunneling. Here, electron avoids backscattering 8 8of xx

9 Tunneling in Grahene In grahene, the massless carriers behave differently than ordinary massive carriers in the resence of an electric field. Here, electrons avoid backscattering because the carrier velocity is indeendent of the energy. The absence of backscattering is resonsible for the high conductivity in carbon nanotubes (Ando et al, 1998). 9 9of xx

10 Absence of backscattering Let s consider a linear electrostatic otential U ( x) = Fx Electron trajectories will be like: y 0 x y = 0 Conduction band d min Valence band y r r r r > 0; v > 0; v y of xx

11 Absence of backscattering For y =0,no backscattering. The electron is able to roagate through an infinitely high otential barrier because it makes a transition from the conduction band to the valence band. e - Conduction band Valence band Potential barrier of xx

12 Band structure U Conduction band U 0 E F 0 d Valence band x conduction U > E F = E 0 valence F of xx

13 Absence of backscattering In this transition from conduction band to valence band, its dynamics changes from electron-like to hole-like. The equation of motion is thus, at energy E with v 2 dr dt r 2 E r r 2 = ( E U ) E U r r It shows that v in the conduction band (U < E) and r in the valence band (U > E). r v = 2 v r of xx

14 Klein tunneling States with r v r are called electron-like. States with r v r are called hole-like. Pairs of electron-like and hole-like trajectories at the same E and y have turning oints at: d min = 2 v F y Conduction band Valence band Electron-like (E, y ) d min Hole-like (E, y ) of xx

15 Klein tunneling d min = 2v F y The tunneling robability: exonential deendence on d min. T ( y ) = π ex y 2h d min = πv ex hf 2 y Condition: in x at x and out x at x is sufficiently large : in x out x, >> y, hf v of xx

16 Transmission resonance It occurs when a -n junction and an n- junction form a -n- or n--n junction. At y =0, T( y )=1 (unit transmission): No transmission resonance at normal incidence. P y =0 e - Conduction band Valence band Potential barrier No transmission resonance of xx

17 Biolar junctions Electrical conductance through the interface between -doed and n-doed grahene: Klein tunneling. grahene Ti/Au to gate Ti/Au Ti/Au Lead PMMA SiO 2 n++ Si (back gate) Lead of xx

18 Biolar junctions To gate: Electrostatic otential barrier Fermi level lies In the Valence band inside the barrier (-doed region) In the Conduction band outside the barrier (n-doed region) U Conduction band U 0 E F 0 d Valence band x of xx

19 Biolar junctions Carrier density n carrier is the same in the n and regions when the Fermi energy is half the barrier height U 0. U d U 0 E F n n x Fermi momenta in both the n and regions are given by: F hk F d 80 nm = U 2v 0 = Fd 2v Measured by Huard et al (2007) for their device. 19 of 19 xx

20 Biolar junctions The Fermi wave vector (kf = πncarrier ) for tyical carrier 12 densities of 10 cm 2 is > 10-1 nm -1. n carrier Under these conditions k F d >1, -n and n- junctions are smooth on the Fermi wavelength. The tunneling robability exression can be used. T ( y ) = π ex y 2h d min = πv ex hf 2 y of xx

21 Biolar junctions The conductance G -n of a -n interface can be solved by integration of tunneling robability over the transverse momenta The result of integration : G 2 4e n = = h W 4e W d yt ( y ) 2πh h 2π 2 F hv where W is the transverse dimension of the interface. Cheianov and Fal ko, of xx

22 Alications of tunneling Atomic clock Scanning Tunneling Microscoe Tunneling diode Tunneling transistor of xx

23 Questions? Who got the Nobel rize (1973) in Physics for his ioneering work on electron tunneling in solids? of xx

24 Dr. Leo Esaki (b. 1925, Osaka, Jaan) 24 of xx

25 Thanks!! 25 of xx