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

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

Tunnelling through thin oxide layers Transfer Matrix Transmission Coefficient T(E,k || ) C  s-2 LR C  s -1 CsCs C  s +1 C0C0 C -1 C  N+1 C  N+2 Tight-binding Self consistently calculated potential profile SiO 2 p-Si n + -Si V ox  E CB = 3.1 eV DT MOS E FL E FR Tunneling current J(V ox )

Tunnelling through thin oxide layers based on crystalline-SiO 2 polymorphs  -cristobalite, tridymite,  -quartz 3D Si/SiO 2 /Si model structures lattice matching : no dangling bonds, no defects non stoichiometric oxide at Si/SiO 2 interface : SiO, SiO 2, SiO 3 Silicon sp 3 s * d SiO 2 sp 3 Tight Binding parameterization Si /  -cristobalite / Si

Transmission Coefficients  -cristobalite model TB vs. EMA EMA underestimates (up to 2-3 orders of magnitude) TB transmission for thicker oxides (t ox > 1.6 nm) Overestimation for thinner oxides Better agreement with non-parabolic correction, but always higher T(E) T(E,k || ) for k || = 0 Increases T Non – parabolicity of complex bands 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)]

Tunneling Current : TB vs. EMA SiO 2 p-Si n + -Si  -cristobalite model Current mainly determined by transmission at E = 0.2 Ev t ox = 3.05 nm EMA underestimates TB current for thicker oxides (t ox > 1.6 nm) Overestimation of TB for thinner oxides (t ox < 1.6 nm) Non-parabolic correction to EMA overestimates always TB, max 20 times

Tunneling current SiO 2 p-Si n + -Si  -cristobalite Good agreement with experimental results [Khairurrjial et al., JAP 87, 3000 (2000)] Microscopic calculation,no fitting parameters (contrary to EMA)

Tunneling current : SiO 2 polymorphs Better agreement with experiments for  -cristobalite (m eff = 0.34 m 0 )  -quartz : higher mass (0.62) Exponential decay with t ox ( agreement with experiments) Oxide thickness dependence of tunneling current lower contribution to transmission  -quartz fails to reproduce correct I/V slope Norm. current (t ox ~ 1.6nm)

Tunneling current components CBE: Electron tunneling from Gate Conduction band (dominant for V ox < ~ 1.3 V) V ox VBE: Electron tunneling from Gate Valence band : dominant for V ox > ~ 1.3 V (interband tunneling) VBH: Holes tunneling from p-Si Valence band (negligible)  -cristobalite SiO 2 p-Si n + -Si VBE CBE

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

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

n + Si Si SiO 2 FULL-BAND CALCULATION OF QUANTIZED STATES Si states in MOS inversion channel Self consistently calculated band profile F = 200kV/cm

FULL-BAND CALCULATION OF QUANTIZED STATES Si states in MOS inversion channel Quantization energies : good agreement with EMA in k || =k min 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. Large contribution

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

CONCLUSIONS Two examples of full-band quantum MOS simulations Atomistic tight-binding approach to oxide tunneling Strong dependence of tunneling currents on local oxide structure. Qualitative/quantitative discrepancies from effective mass approx. Calculated currents in good agreement with experiment. 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. Future work Transmission from quantized states in the channel. Calculation of scattering rates and extension to 2D systems.