F. Sacconi, M. Povolotskyi, A. Di Carlo, P. Lugli University of Rome “Tor Vergata”, Rome, Italy

1. Full-band approaches to the electronic properties of nanometer-scale MOS structures F. Sacconi, M. Povolotskyi, A. Di Carlo, P. Lugli University of Rome “Tor Vergata”, Rome, Italy M. Städele Infineon Technologies AG, Munich, Germany

2. semiempirical • tight binding • empirical pseudopotential • transfer matrix • bulk Bloch function expansion Full-band methods state-of-the-art MOSFETs : gate lengths < 20nm , thin gate oxides < 1nm required theoretical approaches that include • quantum description beyond limitations of EMA • atomic structure modeling This Work Full-band atomistic MOS calculations quantization of states in MOS inversion layer gate oxide tunneling Methods

3. Tunnelling through thin oxide layers Tight-binding Transfer Matrix L R C-1 C0 Cs-2 Cs-1 Cs Cs +1 CN+1 CN+2 Transmission Coefficient T(E,k||) MOS Self consistently calculated potential profile Vox ECB =3.1 eV EFL DT Tunneling current J(Vox) EFR n+-Si p-Si SiO2

4. Tunnelling through thin oxide layers 3D Si/SiO2/Si model structures • based on crystalline-SiO2 polymorphs -cristobalite, tridymite, -quartz • lattice matching : no dangling bonds, no defects • non stoichiometric oxide at Si/SiO2 interface : SiO, SiO2, SiO3 Tight Binding parameterization • Silicon sp3s*d • SiO2 sp3 Si / -cristobalite / Si

5. Transmission Coefficients • -cristobalitemodel • TB vs. EMA T(E,k||) for k|| = 0 • EMA underestimates (up to 2-3 orders of magnitude) TB transmission for thicker oxides (tox > 1.6 nm) • Overestimation for thinner oxides • Better agreement with non-parabolic correction , but always higher T(E) • Non – parabolicity of complex bands • Increases T • Interface / 3D microscopic effects • Decreas T for thin oxides [see M. Städele, F. Sacconi, A. Di Carlo, and P. Lugli, J. Appl. Phys. 93, 2681 (2003)]

6. tox = 3.05 nm • -cristobalitemodel n+-Si p-Si SiO2 • Current mainly determined by transmission at E = 0.2 Ev • EMA underestimates TB current for thicker oxides (tox > 1.6 nm) • Overestimation of TB for thinner oxides (tox < 1.6 nm) • Non-parabolic correction to EMA overestimates always TB, max 20 times Tunneling Current : TB vs. EMA

7. Tunneling current • -cristobalite n+-Si p-Si SiO2 • Good agreement with experimental results[Khairurrjial et al., JAP 87, 3000 (2000)] • Microscopic calculation,no fitting parameters (contrary to EMA)

8. Norm. current (tox~1.6nm) -quartz fails to reproduce correct I/V slope Tunneling current : SiO2polymorphs • Oxide thickness dependence of tunneling current • Exponential decay with tox(agreement with experiments) • Better agreement with experiments for -cristobalite (meff = 0.34 m0) • -quartz : higher mass (0.62) lower contribution to transmission

9. -cristobalite CBE VBE n+-Si p-Si SiO2 ] All components CBE 2 3 10 VBE VBH 1 10 Current Density [A/cm -1 10 -3 10 0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 Tunneling current components • CBE: Electron tunneling from Gate Conduction band • (dominant for Vox < ~1.3 V) V ox • VBE: Electron tunneling from Gate Valence band : dominant for Vox > ~1.3 V • (interband tunneling) • VBH: Holes tunneling from p-Si Valence band (negligible)

10. FULL-BAND CALCULATION OF QUANTIZED STATES Self-consistent bulk Bloch Function Expansion Method: [ F. Chirico, A. Di Carlo, P. Lugli Phys. Rev B 64, 45314 (2001)] Diagonalize Hamiltonian in basis of Bloch functions  H  =  mq | Hcrystal + V | nk  Empirical pseudopotential band structure Hartree potential of free charges calculate charge density iteration calculate V from Poisson’s eq.

11. FULL-BAND CALCULATION OF QUANTIZED STATES structure independent Self-consistent bulk Bloch function expansion Method: atom in a cell material matrix element

12. FULL-BAND CALCULATION OF QUANTIZED STATES n+ Si Si SiO2 Self consistently calculated band profile Si states inMOS inversion channel F = 200kV/cm

13. FULL-BAND CALCULATION OF QUANTIZED STATES • Quantization energies : • good agreement with EMA in k||=kmin Full band EM Non p EM • Parallel dispersion and DOS:good agreement only for E < ~0.3 eV. • Large discrepancies for higher energies, when a greater part of Brillouin zone is involved. • Higher scattering rates (lower mobilities) are expected. Largecontribution Si states inMOS inversion channel

14. FULL-BAND CALCULATION OF QUANTIZED STATES SiO2 Si SiO2 2.2nm Full band EM Non p EM • Only the 1st state energy is calculated correctly in the EMA. Si states in Double Gate MOSFET • Sizable deviations from EMA for thin (2-3 nm) rectangular wells and for energy E > ~ 0.3 eV.

15. Pseudopotential approach to inversion layer quantization • Effective mass approximation is reliable (up to 2 nm) for quantization energy calculations for several lowest levels, but fails completely to reproduce the density of states for E > 0.3 eV. Atomistic tight-binding approach to oxide tunneling • Calculated currents in good agreement with experiment. • Qualitative/quantitative discrepancies from effective mass approx. • Strong dependence of tunneling currents on local oxide structure. CONCLUSIONS Two examples of full-band quantum MOS simulations • Future work • Transmission from quantized states in the channel. • Calculation of scattering rates and extension to 2D systems.