1. Quantum transport in GFET for a graphene monolayer
Alfonso Alarcón

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

3. 1.1.- Introduction: From microelectronic to nanoelectronic
Technology
Roadmap
[International Technology Roadmap for semiconductor ;
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 December 2011 1

4. 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, (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 December 2011 2 2

5. 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 December 2011 20 3

6. 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 December 2011 3

7. 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: ]
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 December 2011 2 4

8. 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 December 2011 5 20

9. 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 December 2011 3

10. 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 December 2011 7

11. 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 December 2011 8

12. 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 December 2011 9

13. 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 December 2011 10

14. 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 December 2011 11 19

15. References
[International Technology Roadmap for semiconductor ;
[K. S. Novoselov et al. Science, 306, (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: ]
[Graphene transistors, nature nanotechnology , 5 (2010)]
The most popular
[Wang, IEEE Electron Device Letters, 32, 9 (2011)]
A. Alarcón December 2011 2

16. Questions
A. Alarcón December 2011

17. 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 December 2011 3

18. 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 December 2011 15

19. 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 December 2011 16

20. 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 December 2011 17

21. 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 December 2011 18

22. 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 December 2011 19

23. 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 December 2011 20

24. 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 December 2011 20 19

25. Questions
A. Alarcón December 2011