transport in nanowire mosfets: influence of the band-structure m. bescond imep – cnrs – inpg...
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Transport in nanowire MOSFETs: Transport in nanowire MOSFETs: influence of the band-structureinfluence of the band-structure
M. BescondIMEP – 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
• Motivation: improve the device performances
• Gate-all-around MOSFET: materials and orientations
• Ballistic transport within the Green’s functions
• Tight-binding description of nanowires
• Conclusion
Outline
2
Towards the nanoscale MOSFET’s Scaling of the transistors:
New device architectures
New materials and orientations
Improve carrier mobility
Gate-all-around MOSFET1: Increasing the number of gates offers a better control of the potential
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.
Improve potential control
1M. Bescond et al., IEDM Tech. Digest, p. 617 (2004). 3
3D Emerging architectures3D Emerging architectures
3D simulations: The gate-all-around MOSFET
Gate-All-Around (GAA) MOSFETs
TSi=WSi=4nm
TOX=1nm
Source and drain regions: N-doping of 1020 cm-3.
Dimensions: L=9 nm, WSi=4 nm, and TSi=4 nm, TOX=1 nm.
Intrinsic channel.
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
XZ
Y
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
a) b)
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
XZ
Y
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
a) b)
WSiVG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
XZ
Y
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
a) b)
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
XZ
Y
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
a) b)
WSiVG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
XZ
Y
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
a) b)
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
XZ
Y
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
a) b)
WSi
5
3D Mode-Space Approach*
3D Problem = N1D 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).
The 3D Schrödinger = 2D (confinement) + 1D ( transport)
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesEOT
W
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesEOT
W
2D (confinement)
1D (transport)
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
i,nψ
ith eigenstate of the nth atomic plan
Y
X
Z
Y
X
Z
6
Different Materials and Different Materials and Crystallographic OrientationsCrystallographic Orientations
Different Materials and Orientations
Ellipsoid coordinate system (kL, kT1, kT2)
+
Device coordinate system (X, Y, Z)
+
Rotation Matrices
Z
X
Y
Z
X
Y
ZZZYZX
YZYYYX
XZXYXX
1
DM
Effective Mass Tensor (EMT)
t
t
l
1
00
00
00
MD
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesEOT
W
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesEOT
W
Y
X
Z
Y
X
Z
Y
X
Z
Y
X
ZVG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesEOT
W
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesEOT
W
kT2
kL
kT1
kT2
kL
kT1
k T2 k L
k T1
k T2 k L
k T1
8
Theoretical Aspects*
• 3D Schrödinger equation:
Potential energyZ
X
Y
Z
X
Y
H3D: 3D device Hamiltonian
Coupling
z,y,xEz,y,xz,y,xVTz,y,xH D3D3
zx
2zy
2yx
2zyx2
T2
XZ
2
YZ
2
XY2
2
ZZ2
2
YY2
2
XX
2
D3
* F. Stern et al., Phys. Rev. 163, 816 (1967).
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesEOT
W
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesEOT
W
9
Theoretical Aspects*
• The transport direction X is decoupled from the cross-section in the 3D Schrödinger equation:
0z,y'Ez,yVzy
z,y2
z
z,y
y
z,y
2
2
YZ2
2
ZZ2
2
YY
2
• Where E’ is given by:
trans
2
x
2
2
YZZZYY
l
2
t
2
x
2
m2
k'E
2
k'EE
• mtrans is the mass along the transport direction:
l
2
t
2
YZZZYYtransm
Coupling
•M. Bescond et al., Proc. ULIS Workshop, Grenoble, p.73, April 20th-21st 2006.•M. Bescond et al. JAP, submitted, 2006. 10
3D Mode-Space Approach
Resolution of the 2D Schrödinger equation in the cross-section: mYY, mZZ, mYZ. Resolution of the 1D Schrödinger equation along the transport axis: mtrans.
The 3D Schrödinger = 2D (confinement) + 1D ( transport)
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesEOT
W
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesEOT
W
2D (confinement)
1D (transport)
Y
X
Z
Y
X
Z
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
i,nσ
ith eigenstate of the nth atomic plane
11
Semiconductor conduction band
(spherical): ml=mt diagonal EMT
• Three types of conduction band minima:
(ellipsoidal): mlmt non diagonal EMT
(ellipsoidal): mlmt non diagonal EMT
E E EΔ
Electro
n E
nerg
y
-valleys-valleys
kZ
kX
kY
kZ
kX
kY
12
Results: effective masses
• Wafer orientation: <010>
13
Material: Ge
mYY=0.2*m0 mZZ=0.95*m0 mtrans=0.2*m0
mYY=0.117*m0 mZZ=0.117*m0 mYZ
-
1=±1/(0.25*m0) mtrans=0.6*m0
4-valleys1st 2nd
-valleys
Z
6 nm
Non-diagonal terms in the effective mass tensor couple the transverse directions in the -valleys
Free electron mass
Z
X
Y
Z
X
Y
• Square cross-section: 44 nm, <100> oriented wire
Y
X
Z
Y
X
Z
Y
X
Z
Y
X
Z
14
Material: Ge• Square cross-section: TT=55 nm, <100> oriented wire
0.0 0.2 0.4 0.6 0.810-1110-1010-910-810-710-610-5
ID (A)
VG (V)
L=9nmV
DS=0.4V
T=5 nm
Total Tunneling Thermionic
0.0 0.2 0.4 0.6 0.8
10-1110-1010-910-810-710-610-5
I D (A)
VG (V)
-valleys 4-valleys
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 (0.6*m0) 15
• Square cross-section: 44 nm, <100> oriented wire
Material: Ge
0.0 0.2 0.4 0.6 0.8
10-1110-1010-910-810-710-610-5
T=4 nmV
DS=0.4V
I D (A)
VG (V)
-valleys 4-valleys
0 4 8 12 16-0.3-0.2-0.10.00.10.20.30.40.5
VG=0.8V
VDS
=0.4VL=9nm
elec
tron
sub
-ban
ds (eV
)
X (nm)
LAMBDA (1st)
LAMBDA (2nd)
DELTA4 (1st)
DELTA4 (2nd)
The 4 become the energetically lowest valleys due to the transverse confinement
4-valleys: mYY=0.2*m0, mZZ=0.95*m0
-valleys: mYY=0.117*m0, mZZ=0.117*m0
16
Material: Ge*
0.0 0.2 0.4 0.6 0.810-1110-1010-910-810-710-610-5
Total currentL=9nmV
DS=0.4V
I D (A)
VG (V)
T=4 nm T=5 nm
4-valleys: mtrans=0.2*m0 versus -valleys: mtrans=0.6*m0
The total current increases by decreasing the cross-section!
* M. Bescond et al., IEDM Tech. Digest, p. 533 (2005). 17
3D Emerging architectures3D Emerging architectures
Influence of the Band structure: Silicon
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).
2K. Nehari et al. Solid-State Electron. 50, 716 (2006). 19
Tight-Binding methodBand 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.
1st (4)
2nd (12)
3rd (12)
Diamond structure:
Reference
20
Tight-Binding methodBand structure calculation
ESS(000) -7.16671 eV ESS(111) -1.39517 eV
Exx(000) 2.03572 eV Esx(111) 1.02034 eV
Exx(111) 0.42762 eV Exy(111) 1.36301 eV
Ess(220) 0.09658 eV Ess(311) -0.11125 eV
Esx(220) -0.13095 eV Esx(311) 0.13246 eV
Esx(022) -0.15080 eV Esx(113) -0.05651 eV
Exx(220) 0.07865 eV Exx(311) 0.08700 eV
Exx(022) -0.30392 eV Exx(113) -0.06365 eV
Exy(220) -0.07263 eV Exy(311) -0.07238 eV
Exy(022) -0.16933 eV Exy(113) 0.04266 eV
20 different coupling terms for Ge:*
*Y.M. Niquet et al., Appl. Phys. Lett. 77, 1182 (2000).
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).
21
The dimensions of the Si atomic cluster under the gate electrode is [TSix(W=TSi)xLG].
1.36
nm
x
y
z
1.36
nm
x
y
z
SiliconHydrogen
Schematic view of a Si nanowire MOSFET with a surrounding gate electrode.Electron transport is assumed to be one-dimensional in the x-direction.
Simulated deviceSi Nanowire Gate-All-Around transistor
22
Energy dispersion relations In the bulk:
The minimum of the conduction band is the DELTA valleys defined by six degenerated anisotropic bands.
-valleys
Constant energy surfaces are six ellipsoids
23
Energy dispersion relations
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 semiconductorThe minimum of 2 valleys are zone folded, and their positions are in k0=+/- 0.336Splitting between 4 subbands
T=1.36 nm T=2.72 nm T=5.15 nm
24
Conduction band edge and effective masses
Bandgap increases when the dimensions of cross section decreasem* increases when the dimensions of cross section decrease :
1 2 3 4 5 6 71.0
1.5
2.0
2.5
3.0
Bulk CBEdge
Using Bulk m*
From TB E(k)
Con
duct
ion
band
edg
e (e
V)
Wire width (nm)
1 2 3 4 5 6 70.2
0.4
0.6
Si Bulk
From TB E(k)
mx*
at (m
0)
W ire width (nm)
1 2 3 4 5 6 70.9
1.0
1.1
S i Bulk
From TB E (k)
mx*
fo
r 2(m
0)
W ire w id th (nm )
1
336.0,0
2
22*
k
x k
Em
25
ResultsCurrent-Voltage Caracteristics
0.0 0.2 0.4 0.6
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
0
5
10
15
I D (
A)
I D (
µA
)
VG (V)
Bulk m*
TB E(k) m*
0.0 0.2 0.4 0.610-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
0
5
10
15
20
I D (
A)
VG (V)
Bulk m*
TB E(k) m*
I D (
µA
)
0.0 0.2 0.4 0.610-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
0
5
10
15
20
25
I D (
A)
VG (V)
Bulk m*
TB E(k) m*
I D (
µA
)
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)
ID(VG) characteristics in linear/logarithmic scales for three nanowire MOSFET’s (LG=9nm, VD=0.7V) with different square sections.
1.36 nm 1.9 nm 2.98 nm
26
K. Nehari et al., Solid-State Electronics, 50, 716 (2006).
K. Nehari et al., APL, submitted, 2006.
ResultsOverestimation on ON-Current
Overestimation of the Ion current delivered by a LG=9nm nanowire MOSFET as a function of the wire width when using the bulk effective-masses instead of the TB E(k)-based values.
1 2 2 3 3 4 4 5 5 60
10
20
30
40
50
60
70
I ON o
vere
stim
atio
n (%
)
Wire width (nm)
1 2 3 4 5 6 70.2
0.4
0.6
Si Bulk
From TB E(k)
mx*
at (m
0)
W ire width (nm)
1 2 3 4 5 6 7
0.9
1.0
1.1
S i Bulk
From TB E (k)
mx*
fo
r 2(m
0)
W ire w id th (nm )
When the transverse dimensions decrease, the effective masses increase and the carrier velocity decreases.
27
3D Emerging architectures3D Emerging architectures
Influence of the Band structure: Germanium
• Three types of conduction band minima:
-valleys-valleys
Conduction band minima
• L point: four degenerated valleys (ellipsoidal).• point: single valley (spherical).• directions: six equivalent minima (ellipsoidal).
29
• Indirect band-gap.• The minimum of CB obtained in kX=/a corresponding to the 4 bulk valleys.• Second minimum of CB in kX=0, corresponding to the single bulk valley (75% of s orbitals).
T=5.65 nm
Dispersion relations*
4 bulk valleys
2 bulk valleys
Single bulk valley4 bulk valleys
Y
X
Z
Y
X
Z
4 bulk valleys
Ge <100>
*M. Bescond et al. J. Comp. Electron., accepted (2006). 30
• The four bands at kX=/a are strongly shifted.• The minimum of the CB moves to kX=0.• 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).
T=1.13 nm
Dispersion relationsGe <100>
31
Effective masses: pointGe <100>
• Significant increase compared to bulk value (0.04m0):From 0.071m0 at T=5.65nm to 0.29m0 at T=1.13nm increase of 70% and 600% respectively.
Other illustration of the mixed valleys discussed earlier in very small nanowires.
(1/m*)=(4 ²/h²)( ²E/ k²)
32
Effective masses: kX=/a
• Small thickness: the four subbands are clearly separated and gives very different effective masses.
• Larger cross-sections (D>4nm): the effective masses of the four subbands are closer, and an unique effective mass can be calculated: around 0.7m0 (effective mass: mtrans=0.6m0 for T=5nm)
• The minimum is not obtained exactly at kX=/a:
Ge <100>
33
Band-gap: Ge vs Si
• 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).
Ge <100>
34
Effective masses: Valence Band
• Strong variations with the cross-section: from -0.18m0 to -0.56m0 (70% higher than the mass for the bulk heavy hole).
35
Conclusion• Study of transport in MOSFET nanowire using the NEGF.• Effective Mass Approximation: different materials and orientations
(T>4-5nm).
• 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 kX=0).
• 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)
36
1D case: Concept of conduction channel and quantum of conductance
Current density from Left to right:
dE-fh
e-E-fv
L
e-nev-I ∑ ∫
i
∞
∞
FLFLii
Total current density:
dE-f-E-fh
e-III ∫
∞
∞
FRFL
Quantum of conductance:
=V
Ilim=D
RL0→V +
RL+
Rq: If bosonic particles: h
eN=D
2b
b
Due to the Fermi-Dirac distribution (1 e-/state) which limits the electron injection in the active region
Resistance of the reservoirs
0.0 0.5 1.0
-0.2
-0.1
0.0
0.1
0.2
E F L
E ( e
V )
f(-EFL)0.0 0.5 1.0
-0.2
-0.1
0.0
0.1
0.2
E F L
E ( e
V )
f(-EFL)
Left electrode Right electrodeBallistic conductor
EFL
EFR
L
eVRL
Left electrode Right electrodeBallistic conductor
EFLEFL
EFR
L
eVRL
h
e)2(=
2
Description of ballisticity: the Landauer’s approach
extra
Resistance of the reservoirs
0.0 0.5 1.0
-0.2
-0.1
0.0
0.1
0.2
E F D
( e V )
E ( e
V )
f
0.0 0.5 1.0
-0.2
-0.1
0.0
0.1
0.2
E F S
( e V )
E ( e
V )
f
E F SE F D
V D S> 0
0.0 0.5 1.0
-0.2
-0.1
0.0
0.1
0.2
E F D
( e V )
E ( e
V )
f
0.0 0.5 1.0
-0.2
-0.1
0.0
0.1
0.2
E F S
( e V )
E ( e
V )
f
E F SE F D
V D S> 0
Resistance of the reservoirs: the Fermi-Dirac distribution limit the electron quantity injected in a subband (D0=2e2/h).
0 5 10 15 20-0.4
-0.2
0.0
0.2
0.4
0.6
Drain
Source
0.8 V
VG=0 V
Fir
st s
ub
band
of
the
(0
10)
valle
y (e
V)
Channel axis (nm)
VDS
=0.4 V
L=9 nm
ΔE
<0.
4 eV
0 5 10 15 20-0.4
-0.2
0.0
0.2
0.4
0.6
Drain
Source
0.8 V
VG=0 V
Fir
st s
ub
band
of
the
(0
10)
valle
y (e
V)
Channel axis (nm)
VDS
=0.4 V
L=9 nm
ΔE
<0.
4 eV
Source Drain
VDS>0
Off regime
1
T’0
1T0
Source Drain
VDS>0
On regime
1
T’0
1T1
R’1
R0
R’1
R1
Source Drain
VDS>0
Off regime
1
T’0
1T0
Source Drain
VDS>0
On regime
1
T’0
1T1
R’1
R0
R’1
R1
extra
Towards the nanoscale MOSFET’s
2003
1971 20011989
2300134 000
410M
42M
1991
1.2M
tran
sist
ors
/ch
ip
10 µm 1 µm 0.1 µm 10 nm
Mean free path in perfect semiconductors
ballistic transport
De Broglie length in semiconductors
quantum effects
Channel length of ultimate R&D MOSFETs in 2006
extra
Semi-empirical methods Effective Mass Approximation (EMA):
E(k)
k0
Parabolic approximation of an
homogeneous materialParabolic
approximation of a finished system of
atoms
1
2
2*
k
E1m
*m2
kkE
22 (Infinite system at the equilibrium)
• Near a band extremum the band structure is approximated by an parabolic function:
extra
New electrostatic potential
New electron density
1D density (Green)
Poisson
Electrostatic potential
Current
Simulation Code Potential energy profile (valley (010))
Numerical Aspects
2D Schrödinger Resolution
3D density (Green)
Self-consistent coupling
The transverse confinement involves a discretisation of the energies which are distributed in subbands
1st
2nd
3rd
y
0 5 10 15 20-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
EFD
EFS
L=9 nmV
DS=0.4 V
VG=0 VP
oten
tial e
nerg
y (e
V)
X (nm)1st
2nd
3rd
y
0 5 10 15 20-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
EFD
EFS
L=9 nmV
DS=0.4 V
VG=0 VP
oten
tial e
nerg
y (e
V)
X (nm)
(Neumann)
Extra