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