Minimum conductivity of graphene and its bilayer
Minimum conductivity of graphene graphene and its bilayer
Minimum conductivity of graphene graphene and its bilayer
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<strong>Minimum</strong> <strong>conductivity</strong> <strong>of</strong><br />
<strong>graphene</strong> <strong>and</strong> <strong>its</strong> <strong>bilayer</strong><br />
Ali G. Moghaddam & Malek Zareyan<br />
IPM 15 th Spring Conference-May 2008<br />
١
Conductivity <strong>of</strong> <strong>graphene</strong><br />
Conductivity without<br />
charge carriers<br />
Conductivity, not<br />
conductance quantization<br />
Graphene remains metal<br />
at Dirac point<br />
Novoselov et al.,<br />
Zhang et al.,<br />
Nature 438, 197 (2005)<br />
Nature 438, 201 (2005)<br />
٢
Minimal <strong>conductivity</strong><br />
Mott’s argument<br />
Anderson localization<br />
2D system + Disorder<br />
Metal-insulator transition<br />
Ziman, Models <strong>of</strong> Disorder, (Cambridge 1979)<br />
Mott, N. F., Davis, E. A., (Oxford, 1979)<br />
Electron Processes in Non-Crystalline Materials<br />
The best or the worst metal<br />
Dirac fermions<br />
do not localize<br />
Morozov et al., PRL (2006)<br />
٣
Theoretical studies<br />
Eduardo Fradkin (1986)<br />
Boltzamann Eq. is not<br />
proper for Dirac Fermions<br />
Other theories<br />
Lee 1993<br />
Ludwig 1994<br />
Morita 1997<br />
Ziegler 1998<br />
Peres 2005<br />
Gusynin 2005<br />
Katsnelson 2006<br />
Tworzydlo 2006<br />
Cserti 2006<br />
Ostrovsky 2006<br />
Kubo formula<br />
Main reasons behind<br />
minimum <strong>conductivity</strong><br />
Intrinsic Impurities<br />
Self-doping<br />
Electron-hole puddles<br />
Macroscopic inhomogeneity<br />
Interactions<br />
Masslessness <strong>of</strong> carriers<br />
Chirality<br />
…<br />
٤
Ballistic transport<br />
Pseudo-diffusive<br />
transport<br />
Tworzydlo et al., PRL 2006<br />
٥
Missing pi<br />
Approximation in theory<br />
Limited range <strong>of</strong> parameters in exp.<br />
A. Geim P. Kim<br />
٦
Limited range <strong>of</strong> parameters in exp.<br />
Exp. with wider junctions<br />
Miao et al. Science 2007<br />
Danneau et al 2008<br />
٧
٨
Very recent experiment<br />
٩
Bilayer <strong>graphene</strong><br />
١٠
A1-A2 dimerization<br />
0<br />
Massive chiral fermions<br />
١١
Chirality or masslessness<br />
Chirality (Klein tunneling) is responsible for MC<br />
Scattering Theory again<br />
2 times larger than monolayer<br />
Snyman & Beenakker, PRB (2007) Morozov et al., PRL (2008) ١٢
Trigonal Warping <strong>of</strong> spectrum<br />
Dresselhaus (1974), Cheianov (2006)<br />
١٣
Anisotropy & Length dependence<br />
30<br />
20<br />
10<br />
( 0 )<br />
0<br />
5 10<br />
L/l<br />
15<br />
3<br />
2<br />
1<br />
20<br />
σ min /σ min<br />
⊥<br />
AGM & MZ, submitted to PRL<br />
١٤
Conclusion<br />
Possible mechanism for Minimal <strong>conductivity</strong><br />
Essential Differences between bi & monolayer<br />
TW leads to non-universal minimal <strong>conductivity</strong>.<br />
TW can be more important in clean ballistic<br />
experimental samples<br />
١٥
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