Electronic properties of monolayer d bil h and bilayer graphene Vladimir Falko Vladimir Falko Lancaster University, UK I & II. Electrons i l h in monolayer graphene. III. Electrons III. Electrons in bilayer graphene and the quantum Hall effect. A.Geim and K.Novoselov Nature Mat. 6, 183 (2007) the quantum Hall effect.
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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)
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
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
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.