Dirac equation & Strain induced

graphene

Kyung-Joong KIM

Department of Physics, Chungnam National University

Outline

• Relativistic quantum mechanical wave equations • Hole theory• Prediction about antiparticle and the discovery of the

positron• Chiral tunneling and Klein paradox • Graphene and band structure• The Dirac equation and graphene • Strain induced graphene quatum dot

Relativistic quantum mechanical wave equation

결국 무엇이 다른가 ?????

이것이 바로 오늘의 메인 !!

Klein-Golden e.q.

Pauli theory!!

Dirac had formulated in 1928

Hole theory

• Dirac sea : the Sea of Negative Energy

디락 방정식은 Negative Energy 의 해를 하나 더 갖는다 . 그것은 당시 디락에겐 골칫거리였다 .

디락은 이 골칫거리를 해결하기 위해 가설을 세웠고 , 그 가설의 주인공은 처음엔 positron 이 아닌 proton 으로 생각했다 .

Prediction about antiparticle and the discovery of the positron

CPT symmetry …

"Pure cowardice!" Anyway, he shared the Nobel Prize, in 1933.

Carl Anderson

Chiral tunneling and Klein paradox

Massless Dirac particles cannot be confined by an electrostatic potential : Klein paradox but confinement can be achieved with a magnetic field

Graphene and band structure

The Dirac equation and graphene

0ie

ie0

0e

e0

i

i

i

i

r

rErpF

,,

,,

,,

2211222 rr

rrie

rrie

AArrrrF

ABri

rFi

BAri

rFi

Decoupled schrodinger-like eq. in cylindrical problem

Graphene disk with gaussian bump

2222Rh0 ))y(x-)(R( y)(x, 4 h

External field [E or G]

Landau & Lifshitz ‘Theory of elasticity’

Structure of graphene disk under a central point load

)))ln(y(x - ))y(x-(R)(( y)(x,222

yxR22222

21

Rh0

h

Landau & Lifshitz ‘Theory of elasticity’

The Dirac equation (contain the gauge potential)

0ie

ie0

0e

e0

i

i

i

i

r

rErF

,,)(,

,,)(,

rrrhetrie

rrrhetrie

ABri2

Bri

rFi

BAri2

Ari

rFi

2

2

Assumption : In-plane displacement Ux = Uy = 0

Effective magnetic fields

Assumption : In-plane displacement Ux = Uy = 0

Effective magnetic fields (point central load)

Assumption : In-plane displacement Ux = Uy = 0

Black hole of the electron