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Title: PRESENTACIN DEL CV, PROYECTO DOCENTE Y PROYECTO DE INVESTIGACIN


1
Analytical 2D Modeling of Sub-100 nm MOSFETs
Using Conformal Mapping Techniques
Benjamin Iñiguez Universitat Rovira i Virgili
(URV), Tarragona, E-43001, SPAIN,
E-mailbinyigue_at_etse.urv.es Jarle Østhaug, Tor
A. Fjeldly UniK- University Graduate Center,
N-2027 Kjeller, NORWAY, E-mail torfj_at_unik.no
2
  • Need of new compact MOSFET modeling concepts
  • The behavior of sub-100 nm MOSFETs is critically
    determined by physical mechanisms that are not
    observed in larger devices.
  • To allow circuit designers to use the potentials
    of sub-100 nm technology, these mechanisms must
    be formulated and implemented into CAD tools.

3
  • Need of a 2-D compact model
  • The present MOSFET standard models are based on a
    1D theory, initially developed for long-channel
    devices.
  • Short channel effects have been progressively
    included (as the feature size has been shrunk
    down) using additional equations (often
    empirical).
  • This has resulted in an enormous increase of the
    number of parameters.

4
  • Purpose of our compact modeling work
  • We present a new modeling approach for nanoscale
    MOSFETs, in order to derive a model based on a
    careful consideration of device physics.
  • The scalability property is therefore inherent to
    the model, therefore provoking a dramatic
    reduction of the number of parameters with
    respect to standard models.

5
  • New 2D approach
  • Our new approach is based on a self-consistent
    solution of the 2D distribution of the
    longitudinal electrical field in the device.
  • Using this approach, short-channel effects and
    scaling properties are intrinsic to the model.
  • As a consequence, only a minimum set of
    parameters with clear physical meaning is needed,
    and a close accord is established with the
    fabrication process.
  • This 2D strategy allows to obtain accurate
    scaling properties of key parameters, such as
    threshold voltage and subthreshold current.

6
2D Strategy
  • Our method is based on separating the 2D
    potential distribution in the depletion region
    under the gate into that corresponding of a 1D
    Poissons equation and that of a Laplacian with
    well defined boundary conditions.
  • The Laplacian equation is solved using conformal
    mapping.
  • Our method has been applied to classical and SSR
    bulk MOSFETs
  • The method can easily be adapted to SOI MOS
    structures (including DG MOSFETs and FinFETs)

7
2D Strategy
  • We consider a bulk MOSFET in which the contact
    regions are approximated by rectangular boxes and
    the potential distributions in the drain and
    source depletion regions are calculated using
    Poisson 1D.
  • The MOSFET structure is split into three regions.
    In region 1, under the gate the 2D potential
    distribution is separated into a component
    corresponding to 1D Poissons equation and a
    Laplacian.
  • The potential distribution has to be determined
    from the normal electric field En(x) which points
    to region 1 from the channel. It is split into a
    contribution E0 coming from a 1D analysis and a
    contribution E2D(x), coming from a 2D analysis.
  • Once En(x) is determined, we will be able to
    obtain the potential in the channel, which in
    turn, will allow us to derive the threshold
    voltage VT and the subthreshold current Isub

8
2D Strategy
Schematic MOSFET geometry
Boundary conditions for the Laplacian of Region
1.
9
Conformal mapping
  • To solve the Laplacian, we perform conformal
    mapping of region 1 into the upper half of the
    (u,v) complex plane (using Schwartz-Christoffel
    transformation)
  • It will be easier to find in that plane the
    potential distributions, because of the relative
    simplicity of the boundary conditions in it.

10
Conformal mapping
Along the u-axis
11
Conformal mapping
  • This mapping, together with some approximations,
    allows us to obtain an analytical expression of
    the component E2D(u) of the Laplacian, which
    results in an analytical expression of E2D(x).
  • Therefore, we obtain an expression of the
    potential distribution in the channel, which
    allows us to derive analytical expressions of the
    threshold voltage and the subthreshold current

12
Results
(b)
(a)
Comparison between experimental (symbols) and
modeled (solid lines) results. Bulk MOSFETs with
Ns 2x1017 cm-3 and tox 8.6 nm (a) Threshold
voltage variation with VDS (b) Threshold voltage
variation with channel length at VDS0.05 V.
13
Results
Model calculations of the channel potential in
70 nm SSR MOSFET relative to the substrate for
(a) VDS 1.6 V at VGS 0 V (lower curve), 0.1
V (middle curve), and 0.31 V (upper curve),
Model calculations of the channel potential
relative to the substrate for VDS 0.05 V and 3
V for gate lengths of 210 nm and 250 nm and tox
8.6 nm
14
Results
Experimental (symbols) and modeled (lines)
subthreshold transfer characteristics for a 70
nm SSR MOSFET with tox 3nm. VDS 0.1 V
(lower curve) and 1.6 V (upper curve).
Measured (symbols) and modeled (line) of the
subthreshold transfer characteristic for a 250
nm MOSFET with and tox 5.6 nm at VDS 0.05 V.
15
Conclusions
  • We have developed a closed-form 2D modeling
    technique for sub-100 nm MOSFETs
  • The technique is based on conformal mapping,
    where the 2D Poissons equation in the depletion
    regions is separated into a 1D long-channel case
    and a 2D Laplace equation.
  • With a minimal parameter set, the present
    modeling reproduces both qualitatively and
    quantitatively the experimental data of
    deep-submicron and sub-100 nm bulk MOSFETs
  • Our technique can be extended to SOI MOS
    structures
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