Transcript Vertical Devices Group

Full-band approaches to the
electronic properties of nanometerscale 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
Full-band methods
state-of-the-art MOSFETs :
gate lengths < 20nm , thin gate oxides < 1nm
required theoretical approaches that include
This
Work
• quantum description beyond limitations of EMA
• atomic structure modeling
Full-band atomistic MOS calculations
gate oxide tunneling
quantization of states
in MOS inversion layer
Methods
• semiempirical
tight binding
• empirical
pseudopotential
• transfer matrix
• bulk Bloch function
expansion
Tunnelling through thin oxide layers
Tight-binding
 s,s 1
L
C-1
C0
C s-2
C s -1
Cs
C s +1
R
C N+1 C N+2
  H 1s , s 1 ( H s , s  E )  H 1s , s 1 H

1
0

EFL
ECB = 3.1 eV
s , s 1
 (k|| , E)  Cs (E) | s, k|| 
Transfer
Matrix
  Cs   Cs 1 
 Cs 


T

 
 s

C
C
C
 s 1 
  s 1   s 
Self consistently calculated
potential profile
MOS
Vox
Transmission
Coefficient T(E,k||)
DT
Tunneling
current J(Vox)
EFR
n+-Si
p-Si
SiO2
J
e
 2 
2
 dk  T  E, k   f  E, E   f  E, E  dE
//
BZ //
//
R
FR
L
FL
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
• SiO2
sp3s*d
sp3
Si / -cristobalite / Si
Transmission Coefficients
-cristobalite model
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
• Interface / 3D
microscopic effects
Increases T
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
-cristobalite
model
tox = 3.05 nm
n+-Si
SiO2
p-Si
• 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
-cristobalite
n+-Si
SiO2
p-Si
• Good agreement with experimental
results
[Khairurrjial et al., JAP 87, 3000 (2000)]
• Microscopic calculation,no fitting parameters (contrary to EMA)
Tunneling current : SiO2 polymorphs
Norm. current (tox~1.6nm)
• 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
-quartz fails to reproduce
correct I/V slope
Tunneling current components
-cristobalite
All components
CBE
VBE
VBH
]
2
CBE
10
3
n+-Si
SiO2
p-Si
• CBE: Electron tunneling
from Gate Conduction band
(dominant for Vox < ~1.3 V)
• VBE: Electron tunneling
from Gate Valence band :
dominant for Vox > ~1.3 V
(interband tunneling)
Current Density [A/cm
VBE
10
10
10
1
-1
-3
0.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
V ox
• VBH: Holes tunneling from p-Si
Valence band (negligible)
FULL-BAND CALCULATION OF QUANTIZED
STATES
Method:
Self-consistent bulk Bloch Function Expansion
[ 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.
FULL-BAND CALCULATION OF QUANTIZED
STATES
Method:
Self-consistent bulk Bloch function expansion
Hcristal (r,r)  W  (R)V (r  R  d , r  R  d )

material

R
1 if r point belongs to the  material
W  (r)  
0 otherwise
atom in a cell
matrix element
nk   H cristal  kn  W   (k  k ) 
 

  Bnk  (G) Bnk (G)V  (  G  k , G  k)eid (k G Gk )


G,G 
structure independent
FULL-BAND CALCULATION OF QUANTIZED
STATES
2


Si states in MOS
inversion channel
2
50
SiO2
60
70
8030 9040
50
60
Si
2.5
2.5
80
90
2.0
2.0
1.5
1.5
Electron density
1.0
1.0
0.5
0.5
0.0
0.0
0
20
40
60
80
-3
F = 200kV/cm
70
Conduction band edge
19
n+
Si
40
3.0
Electron Density [10 cm ]
30
Conduction band edge [eV]
3.0
100 120 140 160 180 200
Depth [A]
Self consistently calculated band profile
FULL-BAND CALCULATION OF QUANTIZED
STATES
Si states in MOS
inversion channel
• Quantization energies :
good agreement with EMA
k||=kmin
pseudopotential
meff (non-parabolic)
meff (parabolic)
30
25
DOS, a.u.
20
• Parallel dispersion and DOS: good
15
10
5
0
0.0
in
Full band
EM
Non p EM
0.2
agreement only for E < ~0.3 eV.
• Large discrepancies for higher energies,
Large k contribution
when a greater part of Brillouin zone is
0.4
0.6
0.8
involved.
Energy, eV
• Higher scattering rates (lower
mobilities) are expected.
FULL-BAND CALCULATION OF QUANTIZED
STATES
Si states in Double
Gate MOSFET
SiO2
Si
SiO2
2.2nm
Full band
EM
Non p EM
2.5
DOS, a.u.
2.0
pseudopotential
meff (parabolic)
meff (non parabolic)
1.5
1.0
0.5
0.0
0.0
0.2
0.4
0.6
Energy, eV
• Sizable deviations from EMA for thin (2-3 nm) rectangular
wells and for energy E > ~ 0.3 eV.
• Only the 1st state energy is calculated correctly in the EMA.
0.8
CONCLUSIONS
Two examples of full-band quantum MOS simulations
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.
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.