Graphene Field effect transistors

Transcription

1 GDR Meso 2008 Aussois 8-11 December 2008 Graphene Field effect transistors Jérôme Cayssol CPMOH, UMR Université de Bordeaux-CNRS 1) Role of the contacts in graphene field effect transistors motivated by the experimental work of Benjamin Huard, Nimrod Stander and David Goldhaber-Gordon from Stanford (See next talk) J.C., B. Huard, and D. Goldhaber-Gordon, arxiv ) Crossed Andreev reflection in graphene bipolar transistors J.C., Phys. Rev. Lett. 100, (2008)

2 Introduction to graphene 2D allotropic form of carbon: electrons 3 electrons in covalent bonds 1 electron/carbon will form a two dimensional gas by jumping from A (red) to B (purple) sites E(k x, k y ) 2 Dirac points E(k x, k y ) Semiconductor with zero gap Metal with zero DOS at Fermi level Two dimensional material with Fermi points when undoped 2DEG or 2DHG when doped (first transistor 2004)

3 Graphene: a novel 2D electron (or hole) gas 1- Exotic nodal quasiparticles: massless Dirac-Weyl fermions in a simple 2D monolayer of carbon atoms. 2- Besides technological issues, Graphene based Field effet Transistors (gfets) are also very convenient labs to study the physics of those massless quasiparticles: Klein tunneling, localization, Quantum Hall effect, proximity effect Suspended gfet Sour ce Drai n From Xu Du et al., Nature Nanotechnology, 3, 491 (2008) (Eva Andrei s group, Rutgers) The source/drain contacts are used to inject/collect Dirac-Weyl fermions within a 2D-electron/hole gas whose density can be tuned by applying a gate voltage.

4 Band structure: electron/hole symmetry Graphene near Dirac points Wallace (1947) Silicium from Chelikowsky and Cohen (1974). Charge conjugaison symmetry at low energy No symmetry Hole vs electron liquids

5 Conductance: theory (ballistic transport) J. Twördzydlo, B. Trauzettel, M. Titov, A. Rycerz, and C.W. J. Beenakker, PRL 96, (2006). W p-doped n-doped Model: Infinite doping below the contacts sharp barriers (d=0) Same conductance for electron (n) and hole (p) gases Fermi energy in the central graphene region Monotonic increase since the doping below the leads is infinite

6 Experiments on suspended gfets (2008) (quasi-?) ballistic transport Rutgers (Eva Andrei) Xu. Du et al. ) Columbia (Philip Kim): K. Bolotin et al. L=0.5 microns n (10 12 cm -2 ) Large assymetry in spite of electron/hole symmetric band structure: Intrinsic band structure Conductance measurements probe (altogether) Impurity scattering Injection at the contacts

7 Importance of the contacts Experiments: Stanford: see Benjamin Huard s talk Stuttgart: photocurrent E.J.H Lee et al. Nanotech Nature Nano. 3, 486 (2008). B. Huard et al., Phys. Rev. B 78, (R) (2008). Theory G. Giovannetti et al., PRL 101, (2008): a metal may dope graphene while preserving its band structure Extended inhomogeneities of the carrier density near the contacts

8 Single metal-graphene contact The density of carriers in graphene underneath the metal is fixed The density far from the lead can be tuned by varying the gate voltage V(x) Questions: 1) How the transport is affected by the shape of the potential step V(x)? 2) UNIPOLAR (nn) vs BIPOLAR (np) steps?

9 Scattering problem for the MDF (single valley) Incident Massless Dirac Fermion d x filled states (here a nn junction) x

10 Solution: reflection probability normal incidence (*) x Absence of backscattering for any d (*)

11 Contrasted behavior: Uni(nn) vs. Bipolar (np) smooth Transmission T The smoother the step : sharp UNIPOLAR (nn junction) kd<0.01 (sharp) kd=0.1 kd=1 (smooth) the higher the transmission T k y /k F (m) =sin a (a = incidence angle) sharp BIPOLAR (np junction) the higher the reflection smooth Origin of the electron/hole asymmetry in transport

12 A single contact Conductance Noise: Fano factor abrupt Bipolar smooth abrupt smooth smooth Unipolar abrupt Bipolar Unipolar abrupt smooth The Fano factor vanishes on the potential step is removed by tuning the density to the density underneath the contact.

13 Two metallic contacts: gfet (Fabry-Pérot) Our model: Fabry-Pérot cavity defined by density steps (induced by the leads) Finite (low) doping underneath the contacts and smooth barriers Source (lead 1) Gate Drain (lead 2) filled states d d X J.C., B. Huard, and D. Goldhaber-Gordon, arxiv Transmission resonances (no true confinement at normal incidence)

14 Our results: conductance G/G Finite doping: Saturation Oscillations 0.5 d/l = 0.2 smooth 0.5 Asymmetry d/l = 0.1 Role of d d/l 0.01 abrupt k (g) F L Interference Contrast Effective length Suggestion: suspended graphene gfet with metal contacts that dopes lightly the graphene (10 11 and cm -2 )

15 ballistic Noise in gfets: theory From Twördzydlo PRL 96, (2006). diffusive Bulk F=1/3 At low density F=1/3 at any density Ballistic + realistic contacts (our result)

16 Fano factor: experiments (ballistic or diffusive) F Helsinki (Perti Hakkonen) From Danneau et al. PRL 100, (2008). F Harvard (C. Marcus) From DiCarlo et al. PRL 100, (2008). Coherent or not?

17 Conclusion: Here, we have provided a theoretical model of extented contacts which shows that conductance and the noise of gfet are dramatically modified especially when the metal dopes lightly the graphene under: J.C., B. Huard, and D. Goldhaber-Gordon, arxiv (2008)

18 GDR MESO 2008 Aussois 8-11 December nd Part Crossed Andreev reflection in graphene bipolar transistors J.C., Phys. Rev. Lett. 100, (2008). (ANR ELEC-EPR 07-NANO-011) Outline: Andreev reflection: retroreflection or specular? CAR in graphene

19 Andreev reflection Retroreflection Specular Classical Andreev reflection Quantum Andreev reflection at a NS interface JETP Lett 1964 In both cases, direction of the rays (photons, electrons, holes ) are obtained from the Snell-Descartes law

20 Andreev reflection in graphene (below the gap) C. Beenakker, PRL 97, (2006) Superconductor Intraband Transverse momentum is always conserved along the interface Energy<gap (Snell-Descartes law) Interband

21 CAR experiments in metals (not graphene) Russo et al. (Morpurgo), PRL 95, (2005)

22 Behind the Andreev mirror? (thin superconducting barrier) e h Andreev transmission e h e Electron (co)tunneling Negative refraction (Veselago lensing) n-graphene Superconductor p-graphene x

23 Quench AR and EC e h e n-graphene Superconductor p-graphene x

24 Oscillations of the Andreev transmission Probability for hole transmission

25 Regime dominated by electron-hole conversion I 1 I 2

26 Conclusion and aknowledgements Regime where Andreev transmission (Crossed Andreev reflection) dominates: Convertion of an electron beam into a hole beam Discussions: Nimrod Stander, Björn Trauzettel, Takis Kontos, F. Pistolesi and A. Buzdin Funding: J. Cayssol: Institut Universitaire de France (A. Buzdin chair) Agence Nationale de la Recherche under grant 07-NANO-011 (ELEC-EPR) B. Huard and D. Goldhaber-Gordon: MARCO/FENA and Air Force Office of Scientific Research.