1. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr M. Bescond, J-L. Autran, M. Lannoo 4 th European Workshop on Ultimate Integration of Silicon, March 20 and 21, 2003 Quantum Transport Simulation in DG MOSFETs using a Tight Binding Green’s function Formalism

2. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr OutlineOutline  Overview of the problem  Device considered  Theory: Tight Binding Green’s function formalism  Results and discussion  Conclusion  Overview of the problem  Device considered  Theory: Tight Binding Green’s function formalism  Results and discussion  Conclusion

3. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr Overview of the problem  Device dimensions scale into the nanometer regime.  The Green’s function formalism represents a basic method to describe the quantum behavior of the transistors : capacity to describe interactions and semi-infinite contact (source, drain).  However, most of the studies consider this formalism in the EMA, whose validity in the nanometer scale is debatable :  Device dimensions scale into the nanometer regime.  The Green’s function formalism represents a basic method to describe the quantum behavior of the transistors : capacity to describe interactions and semi-infinite contact (source, drain).  However, most of the studies consider this formalism in the EMA, whose validity in the nanometer scale is debatable : E(k) k 0 Parabolic approximation of an homogeneous medium Parabolic approximation of a finished system of atoms

4. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr Device considered  Single Atomic conduction channel DG MOSFET.  Mixed-mode approach :  The axis source-channel-drain is represented by an atomic linear chain treated in tight binding (1).  The other parts of the system are classically treated from a dielectric point of view. (1) M. Bescond, M. Lannoo, D. Goguenheim, J-L. Autran, Journal of Non-Cristalline Solids (2003) in press.  Single Atomic conduction channel DG MOSFET.  Mixed-mode approach :  The axis source-channel-drain is represented by an atomic linear chain treated in tight binding (1).  The other parts of the system are classically treated from a dielectric point of view. (1) M. Bescond, M. Lannoo, D. Goguenheim, J-L. Autran, Journal of Non-Cristalline Solids (2003) in press.

5. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr Device considered  Band profile versus position :  Hypothesis :  Source and drain are considered as metallic reservoirs.  We consider a negative Schottky barrier of –0.11 eV.  Band profile versus position :  Hypothesis :  Source and drain are considered as metallic reservoirs.  We consider a negative Schottky barrier of –0.11 eV. E FD E E EFSEFS V DS (eV) SourceChannelDrain E (eV) X 0.11 eV

6. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr Tight binding Green’s function formalism  Retarded Green’s function : (2) S. Datta, Superlatt. Microstruct., 28, 253 (2000). One defines and Electron density can be computed as : f : Fermi-Dirac distribution  Retarded Green’s function : (2) S. Datta, Superlatt. Microstruct., 28, 253 (2000). One defines and Electron density can be computed as : f : Fermi-Dirac distribution Self energies (2) HamiltonianEnergy A D = G  D G + A S = G  S G +  Spectral functions

7. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr Tight binding Green’s function formalism  The current :  The device is virtually cleaved into two regions :  The transmitted current I through the plane separating the two parts is :  The current :  The device is virtually cleaved into two regions :  The transmitted current I through the plane separating the two parts is :, where Q is the charge density of the system.

8. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr Tight binding Green’s function formalism  In the tight binding set, hamiltonian operator has the following form :  The associated retarded Green’s function of the uncoupled system is :  The final expression of the current is :  In the tight binding set, hamiltonian operator has the following form :  The associated retarded Green’s function of the uncoupled system is :  The final expression of the current is : Include the self energies of the semi-infinite source and drain.  Coupling matrix -2  in = g-g*  =(I-gVgV)) -1 Tr 1 trace restricted to part 1

9. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr Results and Discussion Self- consistent coupling New Electrostatic Potential New Electron density Poisson 2D Green Electrostatic Potential + Electron density CURRENT Simulation code : Electron density profiles :

10. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr Results and discussion  I DS versus V G at two different temperatures :  Tunneling current affects : - the magnitude of the current in the subthreshold region, - the quantitative shape of the curve.  I DS versus V G at two different temperatures :  Tunneling current affects : - the magnitude of the current in the subthreshold region, - the quantitative shape of the curve.

11. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr Results and discussion  I DS (V G ) for several values of the channel length :  For a 20 nm device, the curve has a nearly perfect slope of 60 mV/decade.  In smaller devices, the increase of the subthreshold current is due to electron tunneling through the ‘bump’ of the electric potential profile. V DS = 0.4 V

12. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr Results and discussion  I DS vs V DS. Dashed line represents the current obtained with a quantum of conductance G 0 = 2e²/h (3).  In thin channels, the conductance is quantified in units of G 0.  Saturation shows up only when the electron potential energy maximum in the channel is suppressed by positive gate voltage, and is due to the exhaustion of source electrons.  I DS vs V DS. Dashed line represents the current obtained with a quantum of conductance G 0 = 2e²/h (3).  In thin channels, the conductance is quantified in units of G 0.  Saturation shows up only when the electron potential energy maximum in the channel is suppressed by positive gate voltage, and is due to the exhaustion of source electrons. Reflections due to the drop voltage (3) R. Landauer, J. Phys. :Condens. Matter, 1, 8099 (1989).

13. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr Results and discussion  Transmission coefficient for V G = 0.7 V :  Even if injected ballistic particles transmit freely from source to drain without channel potential barrier, reflections due to the drop voltage V DS still exist.  Transparency attenuation is all the more pronounced as the applied voltage V DS increases.  Transmission coefficient for V G = 0.7 V :  Even if injected ballistic particles transmit freely from source to drain without channel potential barrier, reflections due to the drop voltage V DS still exist.  Transparency attenuation is all the more pronounced as the applied voltage V DS increases.

14. Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr ConclusionConclusion  Single conduction channel MOSFET device using tight binding Green’s function formalism has been simulated.  « Tunneling transistor » : tunneling effect changes the overall shape of the current characteristics : the subthreshold curve is no longer exponential.  Even in the strong-tunneling regime the transistor is still responsive to gate voltage.  Because of the decrease of the transverse number, the resonant level energies of the channel have to be determined with a high precision.  Single conduction channel MOSFET device using tight binding Green’s function formalism has been simulated.  « Tunneling transistor » : tunneling effect changes the overall shape of the current characteristics : the subthreshold curve is no longer exponential.  Even in the strong-tunneling regime the transistor is still responsive to gate voltage.  Because of the decrease of the transverse number, the resonant level energies of the channel have to be determined with a high precision. Next step : include the 3D silicon atomic structure.