Electronic properties of graphene, from from high …Electronic properties of graphene, from from...

Electronic properties of graphene, from ‘high’ to ‘low’ energiesfrom high to low energies.

Vladimir Falko, Lancaster University

Graphene for beginners: tight-binding model.Berry phase π electrons in monolayers. Trigonal warping. Stretched graphene.

PN junction in graphene. PN junction in graphene.

Berry phase 2π electrons in bilayer graphene.Landau levels & QHE. Interlayer asymmetry gap. y y y g p

Lifshitz transition and magnetic breakdown in BLG. Stretched BLG. Renormalisation group theory for interaction and spontaneous

symmetry breaking in BLGsymmetry breaking in BLG.

hybridisation forms strong directed bonds 2sp4 electrons in the outer s-p shell of carbon

which determine a honeycomb lattice structure.

C

SP yx,- bonds

)(zP orbitals determine conduction properties of graphite

0)(P orbitals determine conduction properties of graphite

AB

Graphene: gapless semiconductor

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)

)(0ˆ 0 kfH

0)(*

0 kfH

|| f || 0 f

Reciprocal lattice

GNGNG

)()( kGk

Hexagonal

221121GNGNG NN )()(

21kGk NN

Hexagonal reciprocal lattice corresponding to

the hexagonal G the hexagonal

Bravais lattice2G

1st Brilloun zone

1G

1G

Fermi ‘point’in undopedGraphene:

M-point, van Hove singularity

pK(K’) point

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

0 pp

A

sublattice indexyx ipp ‘isospin’

yx ipp

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

Electronic properties of graphene, from ‘high’ to ‘low’ energies.

Graphene for beginners: tight-binding model.Berry phase π electrons in monolayers. Trigonal warping Stretched graphene Trigonal warping. Stretched graphene.

PN junction in graphene.

Berry phase 2π electrons in bilayer grapheneBerry phase 2π electrons in bilayer graphene.Landau levels & QHE. Interlayer asymmetry gap.

Lifshitz transition and magnetic breakdown in BLG. Stretched BLG. R li ti th f i t ti d t Renormalisation group theory for interaction and spontaneous

symmetry breaking in BLG.

vp 0 ppp )sin,cos(

xpyp pvvH

0

0

i

yx

iyx

peipp

peippppp

)(

conduction band

vp

conduction band

vpn ,1p

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

ip e1

21

valence band vpn ,1

pof the electron momentum e

p

3

322 i

i

ee

Berry phase d2

32 ee

dddi

0

Electronic states in graphene observed using ARPES

ip e1

21

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

valley1

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 ;

Slightly stretched monolayer graphene

0

00 '

''0

'''000 0

032

32

000

ii ee 032

32 '''

0''

0'0

yxii iuuee

][ˆ u

vpvupvH

shift of the Dirac point in the momentum space, opposite in K/K’ valleys Ando - J. Phys. Soc. Jpn. 75, 124701 (2006)shift of the Dirac point in the momentum space, opposite in K/K’ valleys, like a vector potential

Foster, Ludwig - PRB 73, 155104 (2006)Morpurgo, Guinea - PRL 97, 196804 (2006)zeff ruB )]([

Monolayer graphene: two-dimensional gapless semiconductor with Berry phase π electrons

)(1̂ rUpvH

vp)(p

xpyp rpi

ip ee

12

1

1n

p e

2

0)(1̂ pp xU Due to the ‘isospin’ conservation, A-B symmetric perturbation does not backward scatter electrons, )(w

Ando, Nakanishi, Saito J. Phys. Soc. Jpn 67, 2857 (1998)

2'2

2 ||cos~)( ppUw

)(1̂ xUpvH

Potential which is smooth at the scale of lattice constant (A-B

symmetric) cannot scatter Berry h l t i tl

i

i

phase π electrons in exactly backward direction.

p

2

iw 00][222

abbaippw i

ippw

i zAe 2

][),(),(

baba

abbai

ipp

pi

ab

pba

zAe

Ae

2

2

abab

iba

ze

B h l tBerry phase π electrons

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 )(1̂ rUpvH

1n p

Due to the isospin conservation A-B symmetric potential

)(1 rUpvH

Due to the isospin conservation, A B symmetric potential cannot backward scatter electrons in monolayer graphene.

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


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 electrons in the exactly

1

backward direction. dkF/1


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

/

'2

v

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

Graphene bipolar transistor: Veselago lens for electrons

ww

w2

Electronic properties of graphene, from from high …Electronic properties of graphene, from from...

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