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.