Electronic properties of graphene - I

Vladimir Falko

helped byV.Cheianov, E.McCannK.Kechedzhi, D.Abergel

T.Ando, B.Altshuler, I.Aleiner

Ultra-thin graphitic films: from flakes to micro-devices

for references, see the review articleGeim and Novoselov - Nature Mat. 6, 183 (2007)

Novoselov et al -Science 306, 666 (2004)

L1: band structure of monolayer graphene, ‘chiral’ electrons, Berry’s phase π in monolayer graphene, unusual properties of the PN junction in graphene.

L2: bilayer graphene and QHE in graphene.

L3: disorder and transport in graphene.

Monolayer graphene

Lattice, symmetry and band structure of monolayer graphene.

Intricate details: trigonal warping in the band structure.

‘Chiral’ electrons and Berry’s phase π in monolayer graphene, suppressed backscattering of chiral electrons.

Unusual properties of the PN junction in graphene focusing & caustics, Veselago lens for electrons.

σ - bonds

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

2sp

C

Carbon has 4 electrons in the outer s-p shell

)(πzp orbitals determine conduction properties of graphite

0γ

eV10~

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)

0ie 3/2πiepr

3/2πie−

0ie 3/2πie

3/2πie−

0)()( == −+ KK εε

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−++

1000

;

0100

;

0010

;

0001

ABBA

valley index

In the corners of the Brillouin zone, electron states on the A and B sub-lattices decouple and have exactly the same energy:

and also for both corners of the BZ,

0)1( 3/23/20 =++−= − ππγ ii

AB eeH

four-fold degeneracy

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−++

1000

;

0100

;

0010

;

0001

ABBA

Four degenerate states in the corners of the Brillouin zone realize a 4-dimensional

irreducible representation of the symmetry group of the honeycomb lattice

Translation

AAT →⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−

34

34

34

34

π

π

π

π

i

i

i

i

ee

ee

Generating elements: xAA SCT ,,3π→

3πCRotation

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−

32

32

32

32

π

π

π

π

i

i

i

i

ee

ee

−+ ⎯→←⎯→←

KKBA

Mirror reflection

xS

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

11

11

BA ⎯→←

}{ 6 TCG v ⊗

Gr

'Gr

Valley ‘+’

pr

(valleys)

pK rr++pK rr

+−

⎥⎦⎤

⎢⎣⎡ ++−=

−+−−

+

)()(0,

32232

332232 y

ax

ay

ay

ax

a ppiipippiiKAB eeeeeH

ππ

γ

sec8

023 10~ cmav γ=

πγγ aippaH yxKBA 023

023

, )( −=+−≈+

+−=−−≈ πγγ aippa yx 023

023 )(

yx

yx

ipp

ipp

+=

−=+

π

π

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−

=

−

−

+

+

+

+

A

B

B

A

H

ϕϕϕϕ

ππ

ππ ,

00

00

ˆ

valley index‘pseudospin’

sublattice index‘isospin’

yx

yx

ipp

ipp

−=

+=+π

π

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

prpr

π+π

ARPES: heavily doped graphene synthesized on silicon carbideA. Bostwick et al – Nature Physics, 3, 36 (2007)

Monolayer graphene

Lattice, symmetry and band structure of monolayer graphene.

Intricate details: trigonal warping in the band structure.

‘Chiral’ electrons and Berry’s phase π in monolayer graphene, suppressed backscattering of chiral electrons.

Unusual properties of the PN junction in graphene focusing & caustics, Veselago lens for electrons.

To write down the monolayer Hamiltonian describing electrons near the K-points, one has to construct all

possible invariants using 4x4 matrices (with sublattice and valley indices) acting within the 4-dimensional

representation and the momentum operator, . (phenomenology)

Alternatively, one can apply the tight-binding model including the dominant next-neighbour (AB) hop and also

longer-distance (AA) hops and to expand to higher order in (or ).

(microscopy)

pr

1<<pa +ππ ,

higher order invariants (expansion terms)

valley a weak electron-hole asymmetry due to

non-orthogonality oforbital basis

and AA, BB hopping⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−

++

,,,,

ABBA

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

+

+

2

2

2

2

00

0)(0

00ˆ

pp

vH δπ

πμ

ππ

ς

1±=ς+K−K

⎥⎦⎤

⎢⎣⎡ ++−=

−+−−

+

)()(0,

32232

332232 y

ax

ay

ay

ax

a ppiipippiiKAB eeeeeH

ππ

γ

2802

3 )()(2

0yx

ayx ippippa +−−≈ γγ

valley

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

+

+

0)(0

00ˆ

2

2

ππ

μπ

πςvH

1±=ς

Fp εε =)( rϕ

ϕ

π

πi

yx

iyx

peipp

peipp−=−=

=+=

)sin,(cos ϕϕpp =r

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

weak ‘trigonal warping’: which has interesting consequences for

the weak localisation effect (Lecture 3).

m

rr KKpp

tt

→−→

−→

±;

time-inversionsymmetry

Monolayer graphene

Lattice, symmetry and band structure of monolayer graphene.

Intricate details: trigonal warping in the band structure.

‘Chiral’ electrons and Berry’s phase π in monolayer graphene, suppressed backscattering of chiral electrons.

Unusual properties of the PN junction in graphene focusing & caustics, Veselago lens for electrons.

Monolayer

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

DoS

gatecarriers Vn ∝

holes electrons vp=|| ε

xpyp2

4

0

ρ(kΩ

)

6

4-4 0

n (1012 cm-2)

280K

4K

140K

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

vp=ε

xpypWallace, Phys. Rev. 71, 622 (1947)

nvppvvH rrrr⋅=⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+

σσπ

π0

01

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

components of a relativistic Dirac fermion.

⎟⎟⎠

⎞⎜⎜⎝

⎛=

B

A

ϕϕ

ψ

vp=ε

xpyp

Chiral electronsisospin direction is linked to the axis

determined by the electron momentum.

for conduction band electrons,

valence band (‘holes’)

1=⋅nrrσ

1−=⋅nrrσ

pr

pr

yx

yx

ipp

ipp

+=

−=+

π

π

nvppvvH rrrr⋅=⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+

σσπ

π0

0ˆ1

)( pn rr

Berry phase πψψψ πσπ i

i

ee =→32

2

Chiral electrons:‘isospin’ direction is linked to the axis determined by the electron momentum,

Bloch function amplitudes on the AB sites - ‘isospin’ ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

B

A

ϕϕ

ψ1=⋅nrrσ

vp=ε

xpyp

Fε

nvppvvH rrrr⋅=⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+

σσπ

π0

01

vp=ε

xpyp

Chiral electrons: isospin direction is linked to the electron momentum.

for conduction band electrons,

valence band

1=⋅nrrσ

1−=⋅nrrσ

pr

pr Due to the isospinconservation, A-B symmetric

potential cannot backward scatter chiral fermions,

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

1=⋅nrrσ

|)'(|cos~)( 22 ppfw rr

−θθ

θ

)(θw⎟⎟⎠

⎞⎜⎜⎝

⎛=

− 2/

2/

21

θ

θ

ψ i

i

p ee

r

Monolayer graphene

Lattice, symmetry and band structure of monolayer graphene.

Intricate details: trigonal warping in the band structure.

‘Chiral’ electrons and Berry’s phase π in monolayer graphene, suppressed backscattering of chiral electrons.

Unusual properties of the PN junction in graphene focusing & caustics, Veselago lens for electrons.

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

Transmission of chiral electrons through the PN junction in graphene

vpc =ε

vpv −=εFp

Fp

π/2Fe

F

pN

vpeU

=

−=

π/

'2Fh

F

pN

vpeU

=

=

conduction band electrons

1=⋅nrrσ pr

Due to the isospin conservation, A-B symmetric potential cannot backward scatter chiral electrons

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

⎟⎟⎠

⎞⎜⎜⎝

⎛=

− 2/

2/

21

θ

θ

ψ i

i

p ee

r

pvH rr⋅= σˆ

Transmission of chiral electrons through the PN junction in graphene

0=U

⎟⎟⎠

⎞⎜⎜⎝

⎛=

− 2/

2/

21

θ

θ

ψi

i

p ee

r

Due to the ‘isospin’ conservation, electrostatic potential U(x) which

smooth on atomic distances cannot scatter chiral fermions in the exactly

backward direction.

θθ θπ 2sin cos)(2dpFew −=

θ

1−

μθ =++ )(sin 222 xUpp Fx

dkF/1

d

dp

he

Lg Fnp

π

22=

⊥

eIII )1( 21−=⋅

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)

A characteristic Fano factor in the shot noise:

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

Transmission of chiral electrons through the PN junction in graphene

⊥L

d

Heersche et al - Nature Physics (2007)

PN junctions should be taken into consideration in

two-terminal devices, since contacts dope

graphene.

ppv

pv

vppcr

rr

r

=∂∂

=

=ε

ε )(

ppv

pv

vppvr

rr

r

−=∂∂

=

−=ε

ε )(

vp

Fermi momentum

cp

Fermi momentum

π/2ce

c

pN

vpeU

=

−=

π/

'2vh

v

pN

vpeU

=

=

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

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

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

ppvVvp cc

rr== ;ε

pr

cθ

cVr

n-type

vvccyy pppp θθ sinsin' −=⇒=

nkk

c

v

v

c =−=θθ

sinsin

Snell’s law with negative

refraction index

''

'

ppvV

vp

v

vrr

−=

−=ε

'prvθ

vVrp-type

PN junction

vc pp =

cv pp 8.0=

Veselago Lens for photons

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

Graphene bipolar transistor: Veselago lens for electrons

w2

w

Focusing prism (beam-splitter) for electrons

Layered poorly conducting semimetal used in pencils and

nuclear fusion moderatorsM. Dresselhaus, G. Dresselhaus

Adv. Phys. 51, 1 (2002)

Graphitestudied from 1930th

‘Theoretical graphene’

construction block in the theory of

graphite and nanotubes

Physical Properties of Carbon NanotubesSaitoh, Dresselhaus, Dresselhaus, Imperial College Press 1998

BuckyballsCurl, Kroto

Smalley 1985Nanotubes

Iijima 1991Smalley 1993