Quantum transport in GFET for a graphene monolayer Alfonso Alarcón

2.- Quantum transport model Outline 1.- Introduction 2.- Quantum transport model 3.- Results 4.- Conclusions 5.- More results A. Alarcón 15 December 2011 3

1.1.- Introduction: From microelectronic to nanoelectronic Technology Roadmap [International Technology Roadmap for semiconductor ; http://www.itrs.net] Electron devices have to be described by quantum mechanics ITRS considers GFET (Graphene Field Effect Transistors) among the candidate for post-CMOS electronics Theoretical approaches provide a necessary tool to guide the continuous progress of the electronics industry A. Alarcón 15 December 2011 1

1.2.- Introduction: Graphene Carbon can exist in several different forms. The most common form of carbon is graphite, which consists of stacked sheets of carbon with a hexagonal structure with only one atom in thickness. [Nobel prize of Physics (2010)] If we fabricate a isolated a sheet (one atomic layer) of graphite... [K. S. Novoselov et al. Science, 306, 666-669 (2004)] Graphene is a 2-D crystalline material (semi-metal or a semiconductor) It is representative of a whole class of 2D materials including for example single layers of Boron-Nitride (BN). The carbon-carbon horizontal distance of 0.142nm and 0.123nm in the vertical distance. Advantages of a GFET (Graphene Field Effect Transistors) respect to the conventional FET: for example in high-speed applications, solid state FETs suffer from short-channel effects. The most popular [Graphene transistors, nature nanotechnology , 5 (2010) Scaling theory predicts that a FET with a thin barrier and a thin gate-controlled region as GFET will be robust against short-channel. The possibility of a channel with one atomic layer thinness is perhaps the most attractive feature of graphene for use in transistors. A. Alarcón 15 December 2011 2 2

2. – Treatment of the tight binding Hamiltonian 1.3.- Introduction: Electronic structure of graphene Fermi surface: is characterized by six double cones. Conduction band Electrons and holes (quasi-particles) in graphene are relativistic massless particles that obey the Dirac equation Fermi level (Undoped graphene) Valence band Dispersion relation Milky way galaxy: Dirac points (or k points) Klein Tunneling If particles are describes by Schrödinger equation these have a finite possibility to across the barrier with a E< V (resonance energies) For perfectly normal incidence on the potential barrier step (kx = 0 and ky > 0), the quasi-particle is always fully transmitted, no matter if its energy is higher or lower than the barrier height. This occurs because, even if the electron energy is lower than the potential height, it across the barrier region as a hole state. [M.I Katsnelson, K. S. Novoselov and A. K. Geim, Nat Phys. 2, 620 (2006)] A. Alarcón 15 December 2011 20 3

2.- Quantum transport model Outline 1.- Introduction 2.- Quantum transport model 2.1.- Non-equilibrium Green’s functions approach 2.2.- General scheme of simulation for a GFET 3.- Results 4.- Conclusions 5.- Extra results A. Alarcón 15 December 2011 3

Non-equilibrium Green’s functions 2.1. Quantum transport model : Non-equilibrium Green’s functions approach Non-equilibrium Green’s functions (NEGF) It is a computational tool based on many-body quantum theory to modeling nanoelectronics devices. Solve exactly Dirac and Schrödinger equation under non-equilibrium conditions [Modeling of nanoscale devices, Proceedings of the IEEE, 96,9 (2008)] Some assumptions are needed: The method use a single-particle approach. Use a mean field approximations [Viet-Hung Nguyen, PhD. Thesis, Université de Paris sud 11 (2010)] [Alfonso Alarcon Pardo, PhD. Thesis, Universitat Autonoma de Barcelona (2010). ISBN: 978-846-9464-07-6] This is a method to study quantum electron transport taking into account the boundary conditions between the contacts and the device active region The most popular Limitation of approach: Computation of time-dependent current Treatment of noise A. Alarcón 15 December 2011 2 4

2.2. – Quantum transport model : General scheme of simulation for a GFET Input: Vgs, Vds,Eg,T… Nearest neighbor interaction One site energy Hopping energy t=2.7eV Retarded Green function Newton-Rhapson method Convergence criteria Charge density Computation of current [A. H. Castro Neto and al. Rev. Mod. Phys. 81, 109 (2009)] [Viet-Hung Nguyen, PhD. Thesis, Université de Paris sud 11 (2010)] A. Alarcón 15 December 2011 5 20

2.- Quantum transport model Outline 1.- Introduction 2.- Quantum transport model 3.- Results 3.1- Double gate GFET 3.2- Comparison between SiO2 and BN 3.3- Oscillations in Vgs<0 3.4- Dirac points 4.- Conclusions 5.- Extra results A. Alarcón 15 December 2011 3

3.1.- Results: Double gate GFET [Graphene transistors, nature nanotechnology , 5 (2010) Strong saturation Asymmetric Lineal region Electron conduction Hole conduction General parameters of simulation: T=77K Vds=0,2V Eg=0eV ; Range of energies: [0,-eVds] eV Wt=Wb=2nm N + =1013 cm-2 Ls,d=20nm SiO2 =3.9 ; BN =3.5 Dirac points Strong saturation Jmin A. Alarcón 15 December 2011 7

3.2.- Results: Comparison between SiO2 and BN small difference General parameters of simulation: T=77K Vds=0,2V Eg=0eV ; Efermi Wt=Wb=2nm N + =1013 cm-2 Ls,d=20nm SiO2 =3.9 ; BN =3.5 The 2D planar Hexagonal boron nitride(hBN) provides an alternative in the fabrication of isolator in GFET It has the same atomic structure as graphene and it shares many of its properties. It provides the same operation that SiO2 [Wang, IEEE Electron Device Letters, 32, 9 (2011)] A. Alarcón 15 December 2011 8

3.3.- Results: Oscillations in Vgs<0 Vgs=-1.1V ; Lg=10nm Resonance Vgs=-1.1V ; Lg=25nm Reduction of the amplitude of oscillations Resonance peaks The oscillations for Vgs < 0 decrease with Lg because of the confinement states (resonant peaks) in the barrier. The oscillations decreases with the T. A. Alarcón 15 December 2011 9

3.4.- Results: Dirac points Gap Gap Jmin decrease with Lg But tends to a constant value for large Lg. VDirac shifts to positives values when we increase the Lg. VDirac tends to a constant value for large Lg. A. Alarcón 15 December 2011 10

4. - Summary We present a quantum transport model to simulate GFET using NEGF. The Hamiltonian of the system is studied by means of a tight binding approach. We present an scheme with the different steps to simulate the density of current solving the Poisson equation of a self-consistent form. We study the transfer characteristic a double gate system. Comparison between SiO2 and BN are showed. Also we study the oscillation in Vgs<0 and Dirac points. The oscillations decreases when we increase the Lg. The Dirac point ( and the minimum value of the current) tends to a constant value when we increase Lg. A. Alarcón 15 December 2011 11 19

References [International Technology Roadmap for semiconductor ; http://www.itrs.net] [K. S. Novoselov et al. Science, 306, 666-669 (2004)] [A. H. Castro Neto and al. Rev. Mod. Phys. 81, 109 (2009)] [M.I Katsnelson, K. S. Novoselov and A. K. Geim, Nat Phys. 2, 620 (2006)] [Viet-Hung Nguyen, PhD. Thesis, Université de Paris sud 11 (2010)] [Modeling of nanoscale devices, Proceedings of the IEEE, 96, 9 (2008)] [Alfonso Alarcon Pardo, PhD. Thesis, Universitat Autonoma de Barcelona (2010). ISBN: 978-846-9464-07-6] [Graphene transistors, nature nanotechnology , 5 (2010)] The most popular [Wang, IEEE Electron Device Letters, 32, 9 (2011)] A. Alarcón 15 December 2011 2

Questions A. Alarcón 15 December 2011

2.- Quantum transport model Outline 1.- Introduction 2.- Quantum transport model 3.- Results 4.- Conclusions 5.- More results 5.1.- Single gate vs double gate comparison 5.2- Single gate vs double gate comparison: Oscillations in Vgs < 0 5.3.- Single gate vs double gate comparison: Dirac point 5.4- Single gate vs double gate: Different oxide thickness comparison 5.5- Single gate vs double gate: Negative differential conductance 5.6- Double gate SiO2: Negative differential conductance for different gap energy A. Alarcón 15 December 2011 3

5.1.- Results: Single gate vs double gate comparison Oscillations Significant difference Oscillations Dirac points Dirac points small difference General parameters of simulation: T=77K Vds=0,2V Eg=0eV; Range of energies: [0,-eVds] eV Wt=2nm ; Ws=100nm N + =1013 cm-2 Ls,d=20nm SiO2 =3.9 ; BN =3.5 A. Alarcón 15 December 2011 15

5.2.- Results: Single gate vs double gate comparison: Oscillations in Vgs < 0 Resonance Resonance peaks Reduction in the number of oscillations and of the amplitude of theses oscillations respect to the double gate system A. Alarcón 15 December 2011 16

5.3.- Results: Single gate vs double gate comparison: Dirac point Gap Gap VDirac Gap VDirac Decreasing of the Jmin respect to the double gate. VDirac shifts to negatives values respect to the double gate. VDirac tends to a constant value more slowly respect to the double gate. A. Alarcón 15 December 2011 17

5.4.- Results: Double and single gate GFET: Different thickness of oxide Shift of the Dirac points to negative values Shift of the Dirac points to negative values more marked for single gate A. Alarcón 15 December 2011 18

5.5.- Results: Single gate vs double gate: Negative differential conductance General parameters of simulation: Vgs = -0.5V; T=77K ; Eg=0eV ; Wt=Wb=2nm ; Ws = 100nm ; N + =1013 cm-2 Ls,d=20nm ; Lg =50nm ; SiO2 =3.9 ; BN =3.5 Vds= 0.3V Gap c Vds= 0.35V Resonance peaks Resonant states NDC Gap Resonance peaks Resonant states Very small NDC c Appreciable NDC to double gate system (BN) GFET A. Alarcón 15 December 2011 19

5.6.- Results: Double gate SiO2: Negative differential conductance for different gap energy General parameters of simulation: T=300K ; Wt=Wb=2nm ; N + =1013 cm-2 ; Ls,d=20nm ; Lg =15nm ; SiO2 =3.9 NDC depend on the Lg. For long Lg the NDC is more important (compare with previous slide) When we increase the energy of the gap the NDC is more important A. Alarcón 15 December 2011 20

Extra Summary We have presented difference between single and double gate GFET. Comparison between SiO2 and BN ar also showed. We have studied the oscillation in Vgs<0 and Dirac points. In single gate GFET the oscillations decreases. We have studied the oscillation in Vgs<0 and Dirac points. The Dirac point is shifted to negatives values in single gate GFET. The shift of the Dirac point is strong when we increase de thickness of the oxide in both GFET system. Negative differential conductance is observed in double gate GFET for non-zero gap. A. Alarcón 15 December 2011 20 19

Questions A. Alarcón 15 December 2011