Electronic properties of monolayer d bil hand bilayer graphene

Vladimir FalkoVladimir FalkoLancaster University, UK

I & II. Electronsi l hin monolayer graphene.

III. ElectronsIII. Electronsin bilayer graphene and the quantum Hall effect.

A.Geim and K.Novoselov Nature Mat. 6, 183 (2007)

the quantum Hall effect.

Graphite GrapheneGeim & Novoselov 2004

How to make graphene.Tight-binding model for electrons in graphene.

Layered semimetal used in pencils, fuses and nuclear

f i d t

Dirac-type electrons, Berry phase π, and trigonal warping correction.

fusion moderators. Absorption spectrum and ARPES.Unusual properties of PN junctions in

h d V l l f l tgraphene and Veselago lense for electrons.

‘Do It Yourself’graphene

Geim & Novoselov(Manchester) 2004

Mechanical exfoliation: the favourite method of

h i d iwww.grapheneindustries.comto make individual10-100 µm flakes.

Graphene from a ‘nanopensil’Kim (Columbia Univ) 2005 10 µm10 µm

High-tech graphene: grown epitaxially by solid state graphitization of Si-face of SiC (Si sublimation from the surface)

Lauffer, Emtsev, Graupner, au e , sev, G aup e ,Seyller (Erlangen), LeyPRB 77, 155426 (2008)

10 nm

5 mm

Tzalenchuk, Lara-Avila, Kalaboukhov, Paolillo, Syväjärvi,

kiYakimova (Linkoping),Kazakova, Janssen, VF, Kubatkin, arXiv:0909.1220 Nature Nanotechnology - in press

Gaskill et al, (HRL Malibu) ECS Trans. 19, 117 (2009) 10 µm

How to make graphene.

Tight-binding model for electrons in graphene.

Dirac-type electrons, Berry phase π, and trigonal warping correction.

Graphene optical absorption and ARPES.

Absence of back-scattering of electrons from electrostatic potentialfrom electrostatic potential.

Unusual properties of PN junctions in h d V l l f graphene and Veselago lense for

electrons.

hybridisation forms strong directed bonds2sp

4 electrons in the outer s-p shell of carbon

hybridisation forms strong directed bonds which determine a honeycomb lattice structure.p

C

- bonds

)(zp orbitals determine conduction properties of graphite

0)(p orbitals determine conduction properties of graphite

AB

Wallace, Phys. Rev. 71, 622 (1947)Slonczewski, Weiss, Phys. Rev. 109, 272 (1958)

Wallace, Phys. Rev. 71, 622 (1947)Slonczewski, Weiss, Phys. Rev. 109, 272 (1958)

221121ananR nn

221121nn

A4.121 aaa

121 nnRGie

2G

GNGNG

Reciprocal latticea

221121GNGNG NN

aGaG 1111

2||;

G

unitcell

aGaG

S

2

111

2||;

||;

1G

unitcellSGaG 222 ||;

Reciprocal lattice

GNGNG

)()( kGk

Hexagonal

221121GNGNG NN )()(

21kGk NN

Hexagonal reciprocal lattice corresponding to

the hexagonalG the hexagonal

Bravais lattice2G

1st Brilloun zone

1G

1G

Fermi ‘point’in graphenein graphene

FEk )(

First BrillouinFirst Brillouinzone 0ie 3/2ie

3/2ie

G Valley

1

e

G

V ll

1

'G1

Valley

)()(0,

32232

332232 y

ax

ay

ay

ax

a ppiipippiiKAB eeeeeH

vippaH KBA )(023

)(023

yx ippa vippaH yxKBA )(02,

Bloch function amplitudes (e.g., in the valley K) on the AB sites (‘isospin’) mimic spinK) on the AB sites ( isospin ) mimic spin

components of a massless relativistic particle.

A

(valleys) 0

B

p

(valleys)

pvipp

ippvH

yx

yx

00ˆ

sec8

023 10~ cmav

pp yx McClure, PR 104, 666 (1956)

valley indexvalley index‘pseudospin’

1

B

A

vH

,

00

ˆ

A

B

vH

0

0pp

A

sublattice indexyx ipp ‘isospin’

yx ipp

Also, one may need to take into account an additional real spin degeneracy of all states

Bloch function amplitudes (e.g., in the valley K) on the AB sites (‘isospin’) mimic spin

ii

ppp )sin,cos(

components of a relativistic Dirac fermion.

A

i

yx

iyx

peipp

peipp

0

B

isospin direction is

nvppvvH

0

0 isospin direction is linked to the axis

determined by the electron momentum.for conduction band

electrons,pvp

1n

1

p

ip e1

21 xpyp

valence band (‘holes’)1np

F pvH

B

A

F

conduction band )sincos( ppp conduction band

1n p)sin,cos( ppp

sublattice ‘isospin’ is linked to the direction of the electron

1

1

valence band 1n p

of the electron momentum

ip e21

)( pn Berry phase

3

322 ii

ee d2

32 ee

dddi

0

Electronic states in graphene observed using ARPES

B

A

vp2|~| BAARPESI

pKGk ||

22

sin~ 2 BARk

Mucha-Kruczynski, Tsyplyatyev, Grishin, McCann, VF, Boswick, Rotenberg - PRB 77, 195403 (2008)

ARPES of heavily doped grapheney p g psynthesized on silicon carbide

Bostwick et al - Nature Physics, 3, 36 (2007)

0)(0

00ˆ

2

2

vH

weak ‘trigonal warping

0)(0

)()(

32232

332232 y

ax

ay

ay

ax

a ppiipippiiH

valley 1

0,322333223 yyy

KAB eeeeeH 2

8023 )()(

20

yxa

yx ippippa

)(

i

iyx

peipp

peipp

Fp )( yx peipp

)sin,cos( ppp

KKpp

tt

;

time-inversionsymmetry

A. Bostwick et al – Nature Physics 3, 36 (2007)

pp ;

How to make graphene.

Tight-binding model for electrons in graphene.

Dirac-type electrons, Berry phase π, and trigonal warping correction.

Graphene optical absorption and ARPES.

Absence of back-scattering of electrons from electrostatic potentialfrom electrostatic potential.

Unusual properties of PN junctions in h d V l l f graphene and Veselago lense for

electrons.

Absorption of light by graphene membrane

pvH

H v

ce

pH

ce

e2ceg

/

1

1

%3.22

ce

Gusynin, Sharapov, Carbotte - PRL 96, 256802 (2006) Gusynin Sharapov - PRB B 73 245411 (2006)

Nair, Blake, Grigorenko, Novoselov, Booth, Stauber, Peres, Geim - Science (2008)

Gusynin, Sharapov - PRB B 73, 245411 (2006) Nilsson, Castro Neto, Guinea, Peres, - PRL 97, 266801 (2006)

Abergel, VF - PRB B 75, 155430 (2007)

Visibility of graphene flakes on SiO /Si substrateon SiO2/Si substrate

ssph2sin

s

N 212

o

nms10

300

o10

1o

ms10

1

Abergel, Russell, VF - Appl. Phys. Lett. 91, 063125 (2007) Blake, Hill, Castro Neto, Novoselov, Jiang, Yang, Booth, Geim - Appl. Phys. Lett. 91, 063124 (2007)

How to make graphene.

Tight-binding model for electrons in graphene.

Dirac-type electrons, Berry phase π, and trigonal warping correction.

Graphene optical absorption and ARPES.

Absence of back-scattering of electrons from electrostatic potentialfrom electrostatic potential.

Unusual properties of PN junctions in h d V l l f graphene and Veselago lense for

electrons.

Monolayer graphene: two-dimensional gapless semiconductor with the Dirac-type spectrum of electrons

DoS6

4K

140K

bandvalence

bandconduction

2

4

(k

)

280K

140K

gatecarriers Vn band band

04-4 0

n (1012 cm-2)

K. Novoselov et al., Science 306, 666 (2004)

vp

xpyp

nvppvvH

0

0

0

isospin direction is linked to the electron momentum.

ip e

12

1

vp for conduction band electrons

1n p

1

xpyp

1n

1n p Due to the isospin conservation, A-B symmetric

1n

valence band conservation, A B symmetric

potential cannot backward scatter chiral fermions,

Ando, Nakanishi, Saito J Ph S J 67 2857 (1998)

)(wJ. Phys. Soc. Jpn 67, 2857 (1998)

)(

|)'(|cos~)( 22 ppfw

PN junctions in the usual gap-full semiconductors are non-transparent for incident electrons therefore they are highly resistivetransparent for incident electrons, therefore, they are highly resistive.

PN junctions in in graphene are different.

Transmission of chiral electrons through the PN junction in graphene

vp vpc

Fp/

'2Fh

F

pN

vpeU

vpv Fp

/2Fe

F

pN

vpeU

conduction band electrons

1n p1nDue to the isospin conservation, A-B symmetric potential

cannot backward scatter chiral electrons cannot backward scatter chiral electrons

For graphene PN junctions: Cheianov, VF - PR B 74, 041403 (2006)pvH ˆ For graphene PN junctions: Cheianov, VF PR B 74, 041403 (2006)‘Klein paradox’: Katsnelson, Novoselov, Geim, Nature Physics 2, 620 (2006)pvH

Transmission of chiral electrons through the PN junction in graphene

0U

ip

12

1

)(sin 222 xUpp Fx

ip e2

d

Due to the ‘isospin’ conservation, 2sin cos)(

2dpFew 1p ,

electrostatic potential U(x) which smooth on atomic distances cannot scatter chiral fermions in the exactly

1

backward direction. dkF/1

Transmission of chiral electrons through the PN junction in graphene

L

d

dp

he

Lg Fnp

22

Due to transmission of electrons with a small incidence angle, θ<1/pFd , a PN junction in graphene should display a finite conductance (no pinch-off) dhL

eIII )1( 21

should display a finite conductance (no pinch off)

A characteristic Fano factor in the shot noise: )( 2

Cheianov, VF - PR B 74, 041403 (2006)

PN junctions should be taken into consideration in

t t i l d i two-terminal devices, since metallic contacts

dope graphene, due to the p g pwork function difference.

Heersche et al - Nature Physics (2007)

w

PNP junction with a suspended gate: an almost ballistic regime: w~l.

A Young and P Kim - Nature Physics 5, 222 (2009)

Wishful thinking about graphene microstructuresFocusing and Veselago lens for electrons in ballistic graphene

Cheianov, VF, Altshuler - Science 315, 1252 (2007)

pvppc

)( Fermi momentum

/

'2v

pN

vpeU

ppv

pv

vppv

)(

cp/vh pN

ppv

pv

vp

Fermi momentum/2

ce

c

pN

vpeU

vp

The effect we’ll discuss would be the strongest in sharp PN junction, ith d λwith d~λF .

vvccyy pppp sinsin' PN junction

cV

nppvc

sinsin

ppcvsin

Snell’s law with negative

cnegative

refraction index

v'p

pvVvp cc

;

n-type

''vpv

vVp-type

pVp cc ;

''ppvVv

vc pp vc pp

cv pp 8.0 cv pp

Veselago Lens for photonsg p

Veselago, Sov. Phys.-Usp., 10, 509 (1968) Pendry, Phys. Rev. Lett., 85, 3966 (2000)

Graphene bipolar transistor: Veselago lens for electrons

ww

w2

How to make graphene.

Tight-binding model for electrons in graphene, Dirac-type electrons, Berry phase π, and trigonal warping correction, graphene optics and ARPES.

Absence of back-scattering of electrons from Absence of back scattering of electrons from electrostatic potential and unusual properties of PN junctions in graphene and Veselago lense for electronselectrons.

Lecture III:Electronic properties of bilayer graphene, Landau levels in monolayers and bilayers, and the Quantum Hall effect.and the Quantum Hall effect.