Electronic Properties of GrapheneElectronic Properties of GrapheneElectronic Properties of GrapheneElectronic Properties of Graphene

Qinlong Luo Qinlong Luo Qinlong Luo Qinlong Luo qluo@utk.eduqluo@utk.eduqluo@utk.eduqluo@utk.edu

Instructor: Prof. DagottoInstructor: Prof. DagottoInstructor: Prof. DagottoInstructor: Prof. DagottoSolid State IISolid State IISolid State IISolid State IIMar 18, 2010Mar 18, 2010Mar 18, 2010Mar 18, 2010

Department of PhysicsDepartment of PhysicsDepartment of PhysicsDepartment of PhysicsUniversity of Tennessee, KnoxvilleUniversity of Tennessee, KnoxvilleUniversity of Tennessee, KnoxvilleUniversity of Tennessee, Knoxville

OUTLINESOUTLINESOUTLINESOUTLINES1. Introduction1. Introduction1. Introduction1. Introduction

2. Crystal Structure2. Crystal Structure2. Crystal Structure2. Crystal Structure

3. Dirac Cones3. Dirac Cones3. Dirac Cones3. Dirac Cones(1) theoretical results(1) theoretical results(1) theoretical results(1) theoretical results(2) experimental results(2) experimental results(2) experimental results(2) experimental results

4. Conclusions4. Conclusions4. Conclusions4. Conclusions

IntroductionIntroductionIntroductionIntroduction

� 1935 L.D. Landau and R.E. Peierls

� 1946 P. R. WallaceP.R. Wallace, Physical Review 71717171, 622-634 (1947)

� 2004 K.S. NovoselovK.S. Novoselov et al, Science 306306306306, 5696 (2004)

� 2005 K.S. NovoselovK.S. Novoselov et al, Nature 438438438438, 197-200 (2005)

IntroductionIntroductionIntroductionIntroduction

http://en.wikipedia.org/wiki/Graphene

� what is graphene?what is graphene?what is graphene?what is graphene?

� carbon� monolayer� honeycomb

Crystal StructureCrystal StructureCrystal StructureCrystal Structure

� Left: structure latticetwo triangular sublattices

� Right: correponding BZDirac cones are located at:

A.H. Castro Neto et al, Rev. Mod. Phys 81818181, 109-162 (2009)

Dirac conesDirac conesDirac conesDirac cones

� Tight-binding Hamiltonian:

NN NNN

A.H. Castro Neto et al, Rev. Mod. Phys 81818181, 109-162 (2009)

Dirac conesDirac conesDirac conesDirac cones

⎥⎦

⎤⎢⎣

⎡

32

13

TTTT

A.H. Castro Neto et al, Rev. Mod. Phys 81818181, 109-162 (2009)t1=2.7 eV t2=-0.2t1

Dirac conesDirac conesDirac conesDirac cones

expanding around point as with

12 tt <<

Dirac conesDirac conesDirac conesDirac cones

Electric field effect Shubnikov-de Hass oscillations (SdHOs)

SdHOs frequency vs. carrier concentration Cyclotron mass vs. carrier concentration

K.S. Novoselov et al, Nature 438438438438, 197-200 (2005)

Dirac conesDirac conesDirac conesDirac cones

gH

VR

∝1

neRH

1=

Electric field effect Electric field effect Electric field effect Electric field effect

nVg ∝

Dirac conesDirac conesDirac conesDirac cones

� An oscillation in the conductivity of a material that occurs at low temperatures in the presence of very intense, time varying magnetic fields

� often used to determine the effective mass of charge carriershttp://en.wikipedia.org/wiki/Shubnikov%E2%80%93de_Haas_effect

FBgV

Shubnikov-de Hass oscillations (SdHOs)

� plot the dependence of SdHO frequency on gate voltage by using standard fan diagrams

+

nVg ∝

SdHOs frequency vs. carrier concentration

nBF β=

Dirac conesDirac conesDirac conesDirac cones

square-root dependence

Cyclotron mass vs. carrier concentration

mc: cyclotron mass of carrier (effective mass) nVg ∝

nBF β=

Dirac conesDirac conesDirac conesDirac cones� Semi-classical expression (Ashcroft & Mermin):

where is the area in k-space of the orbits at the Fermi energy

( )EESmc ∂

∂=

π2

2ℏ

( ) 2kES π=

( )ESe

BF π2ℏ

=

Dirac conesDirac conesDirac conesDirac cones

( )ESe

BF π2ℏ

=

( )EESmc ∂

∂=

π2

2ℏ

nBF β=( ) nneES ∝= β

πℏ

2

2/1nmc ∝ ( ) 2/1nEES

∝∂

∂+

( ) )(ESEES

∝∂

∂

( ) 2kES π= ( ) 2EES ∝+

kE ∝ kc*ℏ=smc /106* ≈

ConclusionsConclusionsConclusionsConclusions

smvF /106≈

� A tight-binding model is investigated to calculate the band dispersion of graphene

� Linear dependence of energy on momentum is deduced from quantum oscillations and electric field effect experiments

� Both calculation and the experiments lead to a linear band dispersion with the same Fermi velocity

CommentsCommentsCommentsComments

� The detail information of relativistic theory about graphene can be found in: A.H. Castro Neto et al, Rev. Mod. Phys 81818181, 109-162 (2009)and also http://en.wikipedia.org/wiki/Graphene