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Klein TunnelingKlein TunnelingPHYS 503 Physics Colloquium Fall 2008PHYS 503 Physics Colloquium Fall 2008

9/119/11

Deepak RajputGraduate Research AssistantCenter for Laser ApplicationsUniversity of Tennessee Space InstituteEmail: drajput@utsi.eduWeb: http://drajput.com

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Outline

Classical picture

Tunneling

Klein Tunneling

Bipolar junctions with graphene

Applications

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Classical Picture

Kinetic Energy = EMass of the ball = m

H

E < mgH E = mgH E > mgH

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Tunneling

Transmission of a particle through a potential barrier higher than its kinetic energy (V>E).

It violates the principles of classical mechanics.

It is a quantum effect.

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Quantum tunneling effect

On the quantum scale, objects exhibit wave-like characteristics.

Quanta moving against a potential hill can be described by their wave function.

The wave function represents the probability amplitude of finding the object in a particular location.

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Quantum tunneling effect

If this wave-function describes the object as being on the other side of the potential hill, then there is a probability that the object has moved through the potential hill.

This transmission of the object through the potential hill is termed as tunneling.

Tunneling = Transmission through the potential barrier

V

Ψ(x)

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Ψʹ(x)E < V

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Tunneling

Source: http://en.wikipedia.org/wiki/Quantum_tunneling

Reflection Interference fringesTransmission Tunneling

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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

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Tunneling in Graphene

In graphene, the massless carriers behave differently than ordinary massive carriers in the presence of an electric field.

Here, electrons avoid backscattering because the carrier velocity is independent of the energy.

The absence of backscattering is responsible for the high conductivity in carbon nanotubes (Ando et al, 1998).

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Absence of backscattering

Let’s consider a linear electrostatic potential

Electron trajectories will be like:

FxxU )(

10

dmin

0 x

y

Conduction band Valence band

pvpy

;0 pvpy

;0

0yp

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Absence of backscattering

For py = 0, no backscattering.

The electron is able to propagate through an infinitely high potential barrier because it makes a transition from the conduction band to the valence band.

Conduction band Valence band

e-

Potential barrier

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Band structure

d

U

EF

Conduction band

Valence band

U0

0x

valenceF

conductionF EEU 0

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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

It shows that in the conduction band (U < E) and in the valence band (U > E).

UE

pv

p

E

dt

rd

2

222 )(|| UEpv

pv

pv

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Klein tunneling

pv

States with are called electron-like.

States with are called hole-like.

Pairs of electron-like and hole-like trajectories at the same E and py have turning points at:

pv

F

pvd y ||2

min

dmin

Electron-like(E,py)

Hole-like(E,py)

Conduction band Valence band

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Klein tunneling

F

vpdppT

yyy

2min

exp2

||exp)(

The tunneling probability: exponential dependence on dmin.

F

pvd y ||2

min

v

Fppp

xpxp

youtx

inx

outx

inx

|

:largely sufficient isat and at

|,||||,

Condition:

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Transmission resonance

It occurs when a p-n junction and an n-p junction form a p-n-p or n-p-n junction.

At py=0, T(py)=1 (unit transmission): No transmission resonance at normal incidence.

Conduction band Valence band

e-

Potential barrier

Py=0

No transmission resonance16

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Bipolar junctions

Electrical conductance through the interface between p-doped and n-doped graphene: Klein tunneling.

n++ Si (back gate)

graphene

Ti/Au Ti/Au

Lead Lead

top gate

SiO2

PMMA

Ti/Au

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Bipolar junctions

Top gate: Electrostatic potential barrierFermi level lies

In the Valence band inside the barrier(p-doped region)

In the Conduction band outside the barrier(n-doped region)

d

U

EF

Conduction band

Valence band

U0

0 x

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Bipolar junctions

Carrier density ncarrier is the same in the n and p regions when the Fermi energy is half the barrier height U0.

U0

EF

xn np

dU

Fermi momenta in both the n and p regions are given by:

v

Fd

v

Ukp FF 22

0

d ≈ 80 nm†

† Measured by Huard et al (2007) for their device. 19

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Bipolar junctions

The Fermi wave vector for typical carrier densities of is > 10-1 nm-1.

Under these conditions kFd >1, p-n and n-p

junctions are smooth on the Fermi wavelength.

The tunneling probability expression can be used.

)n(k carrierF 2cm 1210carriern

F

vpdppT

yyy

2min

exp2

||exp)(

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Bipolar junctions

The conductance Gp-n of a p-n interface can be solved by integration of tunneling probability over the transverse momenta

The result of integration †:

where W is the transverse dimension of the interface.

v

FW

h

epTdp

W

h

eG yynp 2

4)(

2

4 22

† Cheianov and Fal’ko, 2006

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Applications of tunneling

Atomic clock

Scanning Tunneling Microscope

Tunneling diode

Tunneling transistor

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Questions ?

Who got the Nobel prize (1973) in Physics for his pioneering work on electron tunneling in solids?

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Dr. Leo Esaki (b. 1925, Osaka, Japan)

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Thanks !!