Transport in nanowire MOSFETs: influence of the bandstructure

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Transport in nanowire MOSFETs influence of the
band-structure
M. Bescond IMEP CNRS INPG (MINATEC),
Grenoble, France Collaborations N. Cavassilas,
K. Nehari, M. Lannoo L2MP CNRS, Marseille,
France A. Martinez, A. Asenov University of
Glasgow, United Kingdom SINANO Workshop,
Montreux 22nd of September
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Outline

• Motivation improve the device performances
• Gate-all-around MOSFET materials and
  orientations
• Ballistic transport within the Greens functions
• Tight-binding description of nanowires
• Conclusion

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Towards the nanoscale MOSFETs

• Scaling of the transistors

? New device architectures
? Improve potential control
? Gate-all-around MOSFET1 Increasing the number
of gates offers a better control of the potential
? New materials and orientations
? Improve carrier mobility

• Ge, GaAs can have a higher mobility than silicon
  (depends on channel orientation).
• Effective masses in the confined directions
  determine the lowest band.
• Effective mass along the transport determines
  the tunnelling current.

1M. Bescond et al., IEDM Tech. Digest, p. 617
(2004).
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• 3D Emerging architectures
• 3D simulations The gate-all-around MOSFET

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Gate-All-Around (GAA) MOSFETs
TSiWSi4nm TOX1nm

• Source and drain regions N-doping of 1020 cm-3.
• Dimensions L9 nm, WSi4 nm, and TSi4 nm,
  TOX1 nm.
• Intrinsic channel.

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3D Mode-Space Approach
? The 3D Schrödinger 2D (confinement) 1D
( transport)
1D (transport)
2D (confinement)

• 3D Problem N?1D Problems ? Saving of the
  computational cost!!!!
• Hypothesis ?n,i is constant along the
  transport axis.

J. Wang et al., J. Appl. Phys. 96, 2192 (2004).
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Different Materials and Crystallographic
Orientations
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Different Materials and Orientations
?

Effective Mass Tensor (EMT)
Ellipsoid coordinate system (kL, kT1, kT2)
Device coordinate system (X, Y, Z)
?

? Rotation Matrices
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Theoretical Aspects

• 3D Schrödinger equation

Potential energy
H3D 3D device Hamiltonian
Coupling
F. Stern et al., Phys. Rev. 163, 816 (1967).
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Theoretical Aspects

• The transport direction X is decoupled from the
  cross-section in the 3D Schrödinger equation

Coupling

• Where E is given by

• mtrans is the mass along the transport direction

• M. Bescond et al., Proc. ULIS Workshop, Grenoble,
  p.73, April 20th-21st 2006.
• M. Bescond et al. JAP, submitted, 2006.

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3D Mode-Space Approach
? The 3D Schrödinger 2D (confinement) 1D
( transport)
1D (transport)
2D (confinement)

• Resolution of the 2D Schrödinger equation in the
  cross-section mYY, mZZ, mYZ.
• Resolution of the 1D Schrödinger equation along
  the transport axis mtrans.

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Semiconductor conduction band

• Three types of conduction band minima

Electron Energy

• ? (ellipsoidal) ml?mt ? non diagonal EMT

• ? (ellipsoidal) ml?mt ? non diagonal EMT

E? E? E?

• ? (spherical) mlmt ? diagonal EMT

? ? ?
?-valleys
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Results effective masses

• Wafer orientation lt010gt

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Material Ge

• Square cross-section 4?4 nm, lt100gt oriented wire

mYY0.2m0 mZZ0.95m0 mtrans0.2m0
Z
?4-valleys
1st
2nd
mYY0.117m0 mZZ0.117m0 mYZ-11/(0.25m0)
mtrans0.6m0
6 nm
?-valleys
Free electron mass
? Non-diagonal terms in the effective mass tensor
couple the transverse directions in the ?-valleys
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Material Ge

• Square cross-section T?T5?5 nm, lt100gt oriented
  wire

• Total current is mainly defined by the
  electronic transport through the ?-valleys
  (?bulk)
• Tunneling component negligible due to the value
  of mtrans in the ?-valleys (0.6m0)

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Material Ge

• Square cross-section 4?4 nm, lt100gt oriented wire

?4-valleys mYY0.2m0, mZZ0.95m0
?-valleys mYY0.117m0, mZZ0.117m0
? The ?4 become the energetically lowest valleys
due to the transverse confinement
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Material Ge
?4-valleys mtrans0.2m0 versus
?-valleys mtrans0.6m0
? The total current increases by decreasing the
cross-section!
M. Bescond et al., IEDM Tech. Digest, p. 533
(2005).
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• 3D Emerging architectures
• Influence of the Band structure Silicon

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Why?

• Scaling the transistor size
• devices nanostructures
• Electrical properties depend on
• Band-bap.
• Curvature of the bandstructure effective
  masses.
• Atomistic simulations are needed1,2.
• Aim of this work describe the bandstructure
  properties of Si and Ge nanowires.

1J. Wang et al. IEDM Tech. Dig., p. 537 (2005).
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2K. Nehari et al. Solid-State Electron. 50, 716
(2006).
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Tight-Binding method Band structure calculation

• Concept Develop the wave function of the system
  into a set of atomic orbitals.
• sp3 tight-binding model 4 orbitals/atom 1 s 3
  p
• Interactions with the third neighbors.
• Three center integrals.
• Spin-orbit coupling.

3rd (12)
2nd (12)
Diamond structure
1st (4)
Reference
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Tight-Binding method Band structure calculation

• 20 different coupling terms for Ge

• Coupling terms between atomic orbitals are
  adjusted to give the correct band structure
  semi-empirical method.

Y.M. Niquet et al. Phys. Rev. B, 62
(8)5109-5116, (2000).
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Y.M. Niquet et al., Appl. Phys. Lett. 77, 1182
(2000).
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Simulated device Si Nanowire Gate-All-Around
transistor
Silicon
Hydrogen
Schematic view of a Si nanowire MOSFET with a
surrounding gate electrode. Electron transport is
assumed to be one-dimensional in the x-direction.
The dimensions of the Si atomic cluster under the
gate electrode is TSix(WTSi)xLG.
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Energy dispersion relations

• In the bulk
• The minimum of the conduction band is the DELTA
  valleys defined by six degenerated anisotropic
  bands.

? Constant energy surfaces are six ellipsoids
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Energy dispersion relations
T2.72 nm
T5.15 nm
T1.36 nm

• Energy dispersion relations for the Silicon
  conduction band calculated with sp3 tight-binding
  model.
• The wires are infinite in the 100 x-direction.
• Direct bandgap semiconductor
• The minimum of D2 valleys are zone folded, and
  their positions are in k0/- 0.336
• Splitting between D4 subbands

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Conduction band edge and effective masses
Bandgap increases when the dimensions of cross
section decrease m increases when the dimensions
of cross section decrease
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Results Current-Voltage Caracteristics
1.9 nm
2.98 nm
1.36 nm
ID(VG) characteristics in linear/logarithmic
scales for three nanowire MOSFETs (LG9nm,
VD0.7V) with different square sections.

• No influence on Ioff, due to the reduction of
  cross section dimension which induces a better
  electrostatic control
• Overestimation of Ion (detailled on next slide)

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Results Overestimation on ON-Current
When the transverse dimensions decrease, the
effective masses increase and the carrier
velocity decreases.
Overestimation of the Ion current delivered by a
LG9nm nanowire MOSFET as a function of the wire
width when using the bulk effective-masses
instead of the TB E(k)-based values.
K. Nehari et al., Solid-State Electronics, 50,
716 (2006). K. Nehari et al., APL, submitted,
2006.
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• 3D Emerging architectures
• Influence of the Band structure Germanium

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Conduction band minima

• Three types of conduction band minima

• L point four degenerated valleys (ellipsoidal).
• ? point single valley (spherical).
• ? directions six equivalent minima
  (ellipsoidal).

?-valleys
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Dispersion relations
T5.65 nm
Ge lt100gt

• Indirect band-gap.
• The minimum of CB obtained in kX??/a
  corresponding to the 4 ? bulk valleys.
• Second minimum of CB in kX0, corresponding to
  the single ? bulk valley (75 of s orbitals).

M. Bescond et al. J. Comp. Electron., accepted
(2006).
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Dispersion relations
Ge lt100gt
T1.13 nm

• The four bands at kX??/a are strongly shifted.
• The minimum of the CB moves to kX0.
• The associated state is 50 s (? character) and
  50 p (? and ? character) ? Quantum
  confinement induces a mix between all the bulk
  valleys.
• ? These effects can not be reproduced by the
  effective mass approximation (EMA).

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Effective masses ? point
Ge lt100gt
(1/m)(4? ²/h²)?(? ²E/? k²)

• Significant increase compared to bulk value
  (0.04?m0)
• From 0.071?m0 at T5.65nm to 0.29?m0 at T1.13nm
  ? increase of 70 and 600 respectively.
• ? Other illustration of the mixed valleys
  discussed earlier in very small nanowires.

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Effective masses kX?/a
Ge lt100gt

• Small thickness the four subbands are clearly
  separated and gives very different effective
  masses.
• Larger cross-sections (Dgt4nm) the effective
  masses of the four subbands are closer, and an
  unique effective mass can be calculated around
  0.7?m0 (effective mass mtrans0.6?m0
  for T5nm)
• The minimum is not obtained exactly at kX?/a

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Band-gap Ge vs Si
Ge lt100gt

• For both materials the band gap increases by
  decreasing the thickness T (EMA).
• EG of Ge increases more rapidly than the one of
  Si Si and Ge nanowires have very close band
  gaps.
• ? Beneficial impact for Ge nano-devices on the
  leakage current (reduction of band-to-band
  tunneling).

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Effective masses Valence Band

• Strong variations with the cross-section from
  -0.18?m0 to -0.56?m0 (?70 higher than the mass
  for the bulk heavy hole).

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Conclusion

• Study of transport in MOSFET nanowire using the
  NEGF.
• Effective Mass Approximation different materials
  and orientations (Tgt4-5nm).
• Thinner wire bandstructure calculations using a
  sp3 tight-binding model.
• Evolution of the band-gap and effective masses.
• Direct band-gap for Si and indirect for Ge except
  for very small thicknesses ( mixed  state
  appears at kX0).
• Bang-gap of Ge nanowire very rapidly increases
  with the confinement band-to-band tunneling
  should be attenuated.
• Ge is much more sensitive then Si to the quantum
  confinement
• ? necessity to use an atomistic description
  Full 3D

A. Martinez, J.R. Barker, A. Asenov, A.
Svizhenko, M.P. Anantram, M. Bescond, J. Comp.
Electron., accepted (2006)

A. Martinez, J.R.
Barker, A. Svizenkho, M.P. Anantram, M. Bescond,
A. Asenov, SISPAD, to be published (2006)
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Description of ballisticity the Landauers
approach
? 1D case Concept of conduction channel and
quantum of conductance

• Current density from Left to right

• Total current density

• Quantum of conductance

extra
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Resistance of the reservoirs
? Resistance of the reservoirs the Fermi-Dirac
distribution limit the electron quantity injected
in a subband (D02e2/h).
extra
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Towards the nanoscale MOSFETs
1971
1989
1991
2003
2001
410M
42M
1.2M
transistors /chip
134 000
2300
Channel length of ultimate RD MOSFETs in 2006
10 µm 1 µm 0.1 µm 10 nm
De Broglie length in semiconductors
? quantum effects
Mean free path in perfect semiconductors
? ballistic transport
extra
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Semi-empirical methods
? Effective Mass Approximation (EMA)

• Near a band extremum the band structure is
  approximated by an parabolic function

?
(Infinite system at the equilibrium)
extra
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Numerical Aspects

• Simulation Code

• Potential energy profile (valley (010))

Electrostatic potential
2D Schrödinger Resolution
1D density (Green)
Self-consistent coupling
3D density (Green)
New electron density
Poisson
(Neumann)
New electrostatic potential
? The transverse confinement involves a
discretisation of the energies which are
distributed in subbands
Current
Extra