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F. Sacconi, M. Povolotskyi, A. Di Carlo, P. LugliUniversity of Rome “Tor Vergata”, Rome, Italy
M. StädeleInfineon Technologies AG, Munich, Germany
Full-band approaches to the electronic Full-band approaches to the electronic properties of nanometer-scale MOS properties of nanometer-scale MOS
structuresstructures
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 tunnelingquantization 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
1 1
1, 1 , , 1 , 1
1 1
( )
1 0s s ss s s s s s s s
ss s s
C C CH H E H HT
C C C
(,)()|, |||| s kECEsk Transfer Matrix
Transmission Coefficient T(E,k||)
Cs-2
L R
Cs -1 Cs Cs +1C0C-1 CN+1 CN+2
, 1s s Tight-
binding
//
// //2 , , ,2
k k R FR L FL
BZ
eJ d T E f E E f E E dE
Self consistently calculated potential profile
SiO2p-Si
n+-Si
VoxECB = 3.1 eVDT
MOSMOSEFL
EFR
Tunneling current J(Vox)
Tunnelling through thin oxide layers
• based on crystalline-SiO2 polymorphs -cristobalite, tridymite, -quartz
3D Si/SiO2/Si model structures
• lattice matching : no dangling bonds, no defects
• non stoichiometric oxide at Si/SiO2 interface : SiO, SiO2, SiO3
• Silicon sp3s*d • SiO2 sp3
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 (tox > 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
SiO2p-Si
n+-Si
-cristobalite model
• Current mainly determined by transmission at E = 0.2 Ev
tox = 3.05 nm
• 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
SiO2p-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 : SiO2 polymorphs
• Better agreement with experiments for -cristobalite (meff = 0.34 m0)
• -quartz : higher mass (0.62)
• Exponential decay with tox (agreement with experiments)
• Oxide thickness dependence of tunneling current
lower contribution to transmission
-quartz fails to reproduce correct I/V slope
Norm. current (tox~1.6nm)
Tunneling current components
• CBE: Electron tunneling from Gate Conduction band(dominant for Vox < ~1.3 V)
Vox
0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4
10-3
10-1
101
103
Cu
rre
nt
De
nsi
ty [
A/c
m2
] All components CBE VBE VBH
• VBE: Electron tunneling from Gate Valence band : dominant for Vox > ~1.3 V(interband tunneling)
• VBH: Holes tunneling from p-Si Valence band (negligible)
-cristobalite
SiO2p-Sin+-Si
VBE
CBE
FULL-BAND CALCULATION OF QUANTIZED STATES
Self-consistent bulk Bloch Function ExpansionMethod:
Diagonalize Hamiltonian in basis of Bloch functions
H = mq | Hcrystal + 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, 45314 (2001)]
FULL-BAND CALCULATION OF QUANTIZED STATES
Self-consistent bulk Bloch function expansion Method:
,cristal (r ,r) (R)V (r R d r R d )R
H W
,
cristal
d (k G G k )k k
G,G
k k (k k )
(G ) (G)V ( G k G k) in n
n H n W
B B e
structure independent
matrix element
1 if r point belongs to the material(r)
0 otherwiseW
material atom in a cell
n+ Si
SiSiO2
FULL-BAND CALCULATION OF QUANTIZED STATES
Si states in MOS inversion channel
Si states in MOS inversion channel
Self consistently calculated band profile
22
F = 200kV/cm
FULL-BAND CALCULATION OF QUANTIZED STATES
Si states in MOS inversion channel
Si states in MOS inversion channel
• Quantization energies :good agreement with EMA in k||=kmin
Full bandEMNon 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
k
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
SiSiO2 SiO2
Si states in Double Gate MOSFET
Si states in Double Gate MOSFET
Full bandEMNon p EM
• Only the 1st state energy is calculated correctly in the EMA.
CONCLUSIONSTwo 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.