Analytical and Numerical Investigation of Noise in nanoscale Ballistic Field Effect Transistors

1
Analytical and Numerical Investigation of Noise
in nanoscale Ballistic Field Effect Transistors
IWCE 2004
G. Iannaccone Dipartimento di Ingegneria
dellInformazione, Università degli Studi di
Pisa Via Diotisalvi 2, I-56122, Pisa,
Italy g.iannaccone_at_iet.unipi.it

• Acknowledgments
• Support from EU (SINANO), MIUR (FIRB), Fondazione
  CRP

2
Motivation

• Effects typical of mesoscopic devices can be
  observed in mundane MOSFETs at room temperature
  as the ballistic component of the drain current
  increases
• Suitable noise models are required by circuit
  designers, especially for analog and mixed signal
  applications
• Here we focus on the limit of ballistic transport.

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Noise in nanotransistors

• Noise of the drain current
• Transition from Thermal to shot in as the
  ballistic limit is approached
• Shot noise of the gate current
• Plus contributions due to defects (not considered
  here)

Shot noise of the gate tunnel current

• INTEL - Prototype 20 nm MOSFET
• NMOS Gate delay 0.6 ps

Noise of the drain current
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Shot Noise

• Noise is an extremely sensitive probe of
  electron-electron interaction.
• No interactions ? Poissonian process
• Interaction introduce coordination in the
  collective motion of electrons, making the
  process non Poissonian ( S ? Sfull).
• Interaction Pauli Exclusion and Coulomb Repulsion

Full shot noise
S power spectral density of the noise current
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Ballistic transport in MOSFETs (I)

• Density of states in the first subband in the
  channel
• Electron density at the subband peak in the
  channel

Source
Drain
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Charge fluctuations in MOSFETs

• Fluctuations of n2D as function of ?fS and ?fD
• fluctuations electrostatics
• in the contacts
• Subband maximum EM depends on n2D via
  electrostatics
• Electrostatic effects are included in a single
  capacitance per unit area CC

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Equivalent circuit

• Add quantum capacitance towards the source and
  the drain.
• Equivalent circuit

C
y
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Barrier modulation

• Fluctuation of channel barrier
• Current density is modulated by barrier height !

Total capacitance
Longitudinal velocity
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Current Fluctuations

• Current fluctuations depend on fluctuations of
  the occupation factors and of the channel
  barrier
• fluctuations electrostatics
• in the contacts
• Current fluctuations expressed as a function of
  contact fluctuations

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Shot noise power spectral density

• Power spectral density
• Far from equilibrium, if fD 0 , we have CQD
  0, and

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Noise suppression factor (Fano factor)
Effect of Pauli Exclusion
Effect of Coulomb Interaction

• Fano factor is always lt 1 and
• If CC is very large (e.g., large gate
  capacitance) then Coulomb interaction is
  completely screened
• For Maxwell-Boltzmann statistics (e.g. below
  threshold), fS ltlt 1

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25 nm Well tempered MOSFET

• Doping Profile of the 25 nm well tempered
  MOSFET (D. Antoniadis)
• Effective channel length 25 nm
• Super-halo doping in the channel minimizes charge
  sharing effects

• Lowest subband profile from 2D PS solver (G.
  Fiori et al., APL 81, 3672 (2002))
• With Vg1V, Vds0.1 V, 96.5 of current is
  carried by the 1st subband

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Subband Maximum and Source Quantum Capacitance CQS
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Shot noise suppression in well tempered MOSFETs

• VDS 0.5 V

• VDS 1 V

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Noise in the partially ballistic MOSFETs
(I)(with G. Mugnaini)

• The channel of an arbitrary MOSFET is
  decomposed in a chain of ballistic MOSFETs of
  length the mean free path.

• the first N-1 MOSFETs can be aggregated in an
  equivalent drift-diffusion MOSFET. (G. Mugnaini
  et al., submitted to IEEE-TED).

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Noise in the partially ballistic MOSFETs
(II)(with G. Mugnaini - preliminary)
NL/l

• Thermal noise source shot noise source
• As the ratio between the device length and the
  mean free path is reduced, Noise has a transition
  THERMAL ? SHOT
• Presently including the effect of electrostatics
  on noise

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Gate currents Fresh and stressed
oxidesExperimental results by F. Crupi From
G. Iannaccone et al. IEEE-TED 50, 1363 (2003)
Noise properties of the current through fresh
oxides full shot noise at large currents
I-V characteristics 6 nm oxide

• Stress voltage 7.8 V(8 V is the breakdown
  voltage)
• SILCs should introduce alter also the noise
  properties

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Current through fresh oxides(tunneling native
TAT)theory and exp.
Trap distribution is a gaussian centered at 1.8
eV below the oxide CB, with 0.1 eV standard
deviation

• Exp. By F. Crupi

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TAT modelG. Iannaccone et al. IEEE-TED 50, 1363
(2003

• Generation and Recombination rates

• Trap occupation factor

FANO Factor
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Extraction of SILC trap distributionComparison
with experiments

• 6 nm oxide
• Gaussian distribution centered -0.5 eV below Si
  CB, standard deviation 82 meV

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Extraction of SILC trap distribution (V)
For thicker oxides shot noise suppression is due
to transitions through native traps
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Conclusion

• We have derived an analytical expression of noise
  in ballistic MOSFETs with two well defined
  contributions from Pauli exclusion and Coulomb
  repulsion.
• Noise properties can be computed from a numerical
  simulation of DC electrical properties.
• Numerical results for well tempered MOSFETs
  operating in the ballistic regime have been
  shown, exhibiting room temperature suppression of
  shot noise, in typical operating conditions, down
  to 0.25.
• Shot Noise of the gate current contribution of
  native traps may be important also for noise
  properties (experiments here are still missing)
• For thicker oxides a distribution of traps can be
  extracted that reproduces both DC and noise
  characteristcs

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Current Fluctuations

• Current fluctuations depend on fluctuations of
  the occupation factors and of the channel
  barrier
• fluctuations electrostatics
• in the contacts
• We can introduce two average velocities vS and vD

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Equilibrium and far from equilibrium

• If fS fD, S reduces to 4KTG, as it must be,
  where
• Far from equilibrium, if fD 0 , we have CQD
  0, and
• The noise suppression factor is a weighted average

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Extraction of native and SILC trap distribution
(I)

• Simulations with a distribution uniform in
  energy
• do not provide satisfactory results

• Integral equation with

as the unknown
Hp
Trap distribution indipendent of position
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Extraction of SILC trap distribution (II)

• Model A Riccò, Gozzi, Lanzoni, IEEE TED 45,
  1998.
• mean quantities fluxes and capture cross
  section

• Model B Ielmini, Spinelli, Lacaita, IEEE TED
  47, 2000.
• Transient SILC components

Electrons from cathode VB
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Extraction of SILC trap distribution (III)
J-V Curves

• 5.9 nm oxide
• Comparison with exp. performed in Pisa
• Other thicknesses
• Comparison with experiments drawn from the
  literature (Ricco et al.)
• Effect of surface traps for very low voltages

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Extraction of SILC trap distribution (IV)
Fano Factor
Stronger suppression for Em approaching the
silicon gap center
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Understanding the nature of SILCs

• Stress-induced leakage currents (SILCs) are the
  single most important limit to downscaling of
  non-volatile memories
• read disturb, retention degradation
• SILCs are due to tunneling assisted by traps
  generated by electric field stress.
• The energy distribution of traps is not known
• We show that detailed modeling, coupled with DC
  and noise characterization, can provide enough
  information to extract information about the
  energy distribution of traps

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Tunneling Current fresh oxides

• Electron effective mass in the oxide conduction
  band
• Determination of the oxide thickness
• Native traps are required for fitting the current
  at low fields

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TAT model (I)

• Two-Step tunneling

• Anelastic TAT

G. Iannaccone et al., IEEE-TED 2003
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Tunneling Current Model

• The electron density n(x) at the Si-SiO2
  interface is computed by solving the Schrödinger
  equation for the two-fold and four-fold
  degenerate conduction band minima.
• 1D Poisson and Schrödinger equations are solved
  iteratively.
• Once the band profiles and charge densities are
  obtained, we can compute the tunneling current

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25 nm Well tempered MOSFET

• Doping Profile of the 25 nm well tempered
  MOSFET (D. Antoniadis)
• Effective channel length 25 nm
• Super-halo doping in the channel minimizes charge
  sharing effects

• Quantum confinement in the middle of the channel
  (z 45 nm)
• With Vg1V, Vds0.1 V, 96.5 of current is
  carried by the 1st subband

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Subband profile and characteristics

• 2D simulation
• First subband profile in the longitudinal
  direction for increasing Vds.

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Papers on the subject

• Y. Naveh, A. N. Korotkov, K. K. Likharev, Shot
  noise suppression in multimode ballistic Fermi
  conductors, Phys. Rev. B, 60 (1998), R2169-2172.
• O. M. Bulashenko and J. M. Rubì, Shot-noise
  suppression by Fermi and Coulomb correlations in
  ballistic conductors, Phys. Rev. B, 65 (2001)
  045307.
• O. M. Bulashenko and J. M. Rubì, Self-consistent
  theory of current and voltage noise in multimode
  ballistic conductors, Phys. Rev. B, 66 (2002),
  045310.
• G. Gomila, I. R. Cantalapiedra, T. Gonzalez, L.
  Reggiani, Semiclassical Theory of shot noise in
  ballistic n-i-n semiconductor structures
  Relevance of Pauli and long-range Coulomb
  correlations, Phys. Rev. B, 66 (2002) 075302.

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Noise in the partially ballistic MOSFETs
(II)(with G. Mugnaini)

• Far-from equilibrium transport in each ballistic
  MOSFET, gives a local shot noise van der
  Ziel,1986
• If the chain is long enough, local equilibrium
  holds in the whole sructure and then local shot
  noise reduces to conventional thermal noise

• Similarly to the aboveseen current macromodel, a
  noise macromodel for a device operating in
  intermediate tranposrt regime, is given by the
  series of a thermal noise generator with a shot
  noise generator.
• We expect that when the ratio between the device
  length and the mean free path reduces, a more
  pronounced far-from equilibrium behavior emerges
  both in the static current and in the noise.

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Pauli and Coulomb interactions
Limits density in real space
Limits density in phase space

• In most cases interactions make the collective
  motion more regular

Reduced fluctuations
Sub-poissonian process
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Fully ballistic transport regime

• Electrons with sufficient energy to overcome the
  barrier near the source reach the drain
  conserving energy and transversal momentum

• Electron states originating from the source obey
  the Fermi-Dirac statistics with source Fermi
  Energy Efs
• Electron states originating from the drain obey
  the Fermi-Dirac statistics with drain Fermi
  Energy Efd
• This ensures continuity of current density per
  unit energy in each subband

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Model

• Poisson equation in 2D
• The electron density n(f) in the quantum region
  is obtained from the solution of the Schrödinger
  equation with density functional theory
• p(f), ND(f), NA-(f) and of n(f) out of the
  quantum region are given by the corresponding
  semiclassical expressions

• Discretization with the box integration method
• Newton-Raphson method with predictor-corrector
  iteration scheme

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Mass anisotropy and electron density

• The Schrödinger equation must be solved twice
• For the 2 minima along the vertical (x) direction
• For the other 4 minima
• The quantum electron density becomes

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Model out of equilibrium

• When the Poisson-Schrödinger equation is solved,
  and charge density and potential profiles are
  known, we compute the current density in the i-th
  subband
• The total current density is

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Examples of nanotransistors

• INTEL, in production now
• oxide thickness 2 nm

• INTEL test device
• In production by 2005 (ITRS 2002 update
• oxide thickness 0.8 nm

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Summary

• Motivation
• VLSI devices are already nanoelectronic devices
  !!
• Mesoscopic Noise in MOSFETs
• Shot Noise of the drain current in ballistic
  MOSFETs
• Shot Noise of the gate current in fresh oxides
  and in the case of tunneling assisted by traps
• Conclusion
• Acknowlegments
• F. Crupi, A. Nannipieri, G. Curatola, G. Fiori