Influence of Band-Structure on Electron Ballistic
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Transcript Influence of Band-Structure on Electron Ballistic
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
Outline
• 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
2
Towards the nanoscale MOSFET’s
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).
3
3D Emerging architectures
3D simulations: The gate-all-around MOSFET
Gate-All-Around (GAA) MOSFETs
Z
WSi
VG
Gates
X
SiO
SiO22
Y
VS
SOURCE
CHANNEL
DRAIN
VD
Si
TSi=WSi=4nm
TOX=1nm
TSi
Gates
Oxide
VG
W
TOX
L
a)
b)
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.
5
3D Mode-Space Approach*
1D( (transport)
The 3D Schrödinger = 2D
2D (confinement)
(confinement) + 1D
transport)
VG
Z
Gates
X
SiO
SiO22
Y
VS
SOURCE
CHANNEL
DRAIN
VD
Si
TSi
Gates
EOT
VG
TOX
W
ψ n ,i
L
ith eigenstate of the nth atomic plan
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).
6
Different Materials and
Crystallographic Orientations
Different Materials and Orientations
kT2
Z
VG
kL
Gates
X
VS
+
l
M 1 0
0
Y
kT1
SOURCE
VD
DRAIN
CHANNEL
D
Gates
EOT
VG
0
t
0
0
0
t
W
L
Ellipsoid coordinate Device coordinate
system (kL, kT1, kT2) system (X, Y, Z)
V
k T2
Z
G
Z X
Gates
kL
Y
V
k T1 S
+
SOURCE
CHANNEL
DRAIN
Gates
EOT
VG
L
W
VD
Effective Mass
Tensor (EMT)
X
XX
M D1 YX
ZX
Y
XY
YY
ZY
XZ
YZ
ZZ
Rotation Matrices
8
Theoretical Aspects*
• 3D Schrödinger equation:
H 3 D x , y , z T3 D V x , y , z x , y , z E x , y , z
VG
Potential energy
H3D: 3D device Hamiltonian
Z
Gates
VS
X
Y
SOURCE
CHANNEL
DRAIN
VD
Gates
EOT
VG
W
L
2
2
2
2
2
2
2
T3 D
XX 2 YY 2 ZZ 2 2 XY
2YZ
2 XZ
2
x
y
z
xy
yz
xz
Coupling
* F. Stern et al., Phys. Rev. 163, 816 (1967).
9
Theoretical Aspects*
• The transport direction X is decoupled from the crosssection in the 3D Schrödinger equation:
2
2 y , z
2 y , z
2 y , z
YY
ZZ
2YZ
V y , z E' y , z 0
2
y 2
z 2
yz
Coupling
• Where E’ is given by:
2 k x2
t2l
2 k x2
E E'
E'
2
2 YY ZZ YZ
2mtrans
• mtrans is the mass along the transport direction:
mtrans
YY ZZ YZ2
t2l
•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
The 3D Schrödinger = 2D
2D (confinement)
(confinement)
1D( (transport)
+ 1D
transport)
VG
Z
Gates
X
SiO
SiO22
Y
VS
SOURCE
CHANNEL
DRAIN
VD
Si
TSi
Gates
EOT
VG
L
W
TOX
σ n ,i
ith eigenstate of the nth atomic plane
Resolution of the 2D Schrödinger equation in the cross-section: mYY, mZZ, mYZ.
Resolution of the 1D Schrödinger equation along the transport axis: mtrans.
11
Semiconductor conduction band
(ellipsoidal): mlmt non diagonal EMT
(ellipsoidal): mlmt non diagonal EMT
(spherical): ml=mt diagonal EMT
Electron Energy
• Three types of conduction band minima:
E E
EΔ
kZ
kX
-valleys
kY
-valleys
12
Results: effective masses
• Wafer orientation: <010>
13
Material: Ge
• Square cross-section: 44 nm, <100> oriented wire
Z
X
Y
mYY=0.2*m0
mZZ=0.95*m0
mtrans=0.2*m0
4-valleys
Z
1st
2nd
mYY=0.117*m0
Z
X m
ZZ=0.117*m0
X
6 nm
-1=±1/(0.25*m )
m
YZ
0
YY
mtrans=0.6*m0
-valleys
Free electron mass
Z
Non-diagonal terms in the effective mass tensor
couple the transverse directions in the -valleys
14
Material: Ge
• Square cross-section: TT=55 nm, <100> oriented wire
10
10
10
ID (A) 10
10
10
10
-5
-6
-7
L=9nm
10
VDS=0.4V
10
I D (A)
10
T=5 nm
-8
-9
-10
-11
0.0
10
10
Total
10
Tunneling
10
-5
-6
-7
-8
-9
-10
-valleys
-11
4-valleys
Thermionic
0.2
0.4
0.6
VG (V)
0.8
0.0
0.2
0.4
0.6
VG (V)
0.8
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
Material: Ge
• Square cross-section: 44 nm, <100> oriented wire
10
I D (A)
10
10
10
10
10
-5
-6
-7
T=4 nm
VDS=0.4V
-8
-9
-10
-valleys
-11
0.0
4-valleys
0.2
0.4
0.6
VG (V)
0.8
electron sub-bands (eV)
10
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0
VG=0.8V
VDS=0.4V
L=9nm
st
LAMBDA (1 )
nd
LAMBDA (2 )
st
DELTA4 (1 )
nd
DELTA4 (2 )
4
8
12
X (nm)
16
4-valleys: mYY=0.2*m0, mZZ=0.95*m0
-valleys: mYY=0.117*m0, mZZ=0.117*m0
The 4 become the energetically lowest
valleys due to the transverse confinement
16
Material: Ge*
10
10
I D (A)
10
10
10
10
10
-5
-6
Total current
L=9nm
-7
-8
VDS=0.4V
-9
-10
T=4 nm
-11
T=5 nm
0.0
0.2
0.4
VG (V)
0.6
0.8
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 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.
2K.
Wang et al. IEDM Tech. Dig., p. 537 (2005).
Nehari et al. Solid-State Electron. 50, 716 (2006).
19
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)
Diamond structure:
2nd (12)
1st (4)
Reference
20
Tight-Binding method
Band structure calculation
20 different coupling terms for Ge:*
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
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).
*Y.M. Niquet et al., Appl. Phys. Lett. 77, 1182 (2000).
21
Simulated device
Si Nanowire Gate-All-Around transistor
z
x
1.36nm
y
1.3
6n
m
9nm
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(W=TSi)xLG].
22
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
-valleys
23
Energy dispersion relations
T=2.72 nm
Energy [eV]
T=1.36 nm
T=5.15 nm
2.5
2.5
2.5
2.0
2.0
2.0
1.5
1.5
1.5
-1.0
-0.5
0.0
0.5
Wavevector kx [/a0]
1.0
-1.0
-0.5
0.0
0.5
Wavevector kx [/a0]
1.0
-1.0
-0.5
0.0
0.5
1.0
Wavevector kx [/a0]
Energy dispersion relations for the Silicon conduction band calculated with sp 3 tight-binding model.
The wires are infinite in the [100] x-direction.
Direct bandgap semiconductor
The minimum of 2 valleys are zone folded, and their positions are in k0=+/- 0.336
Splitting between 4 subbands
24
3.0
0 .6
Using Bulk m*
mx* a t (m0)
2.5
2.0
From TB E(k)
1.5
Bulk CB
Edge
F ro m T B E (k)
0 .4
S i B u lk
0 .2
1.0
1
1 .1
2
3
4
5
Wire width (nm)
6
7
Bandgap increases when the
dimensions of cross section decrease
m* increases when the dimensions of
cross section decrease :
2E
m 2
k
*
x
2
k 0, 0.336
1
mx* fo r2(m0)
Conduction band edge (eV)
Conduction band edge and effective masses
1
2
3
4
5
W ire w idth (n m )
6
7
6
7
F rom T B E (k )
1 .0
S i B ulk
0 .9
1
2
3
4
5
W ire w id th (n m )
25
Results
Current-Voltage Caracteristics
-4
10
15
-5
*
10
Bulk m
*
TB E(k) m
-6
10
-6
10
*
*
Bulk m
*
TB E(k) m
20
-5
10
-6
10
10
15
10
5
-10
10
10
5
0.0
0.2
0.4
VG (V)
0.6
0
0.0
10
5
-11
-13
10
10
10
-12
10
-12
-9
-10
-11
10
15
-8
10
10
-10
10
10
-11
10
ID (A)
-9
10
ID (µA)
10
10
ID (A)
ID (µA)
-9
20
-7
-8
-8
10
25
10
-7
-7
10
Bulk m
*
TB E(k) m
ID (µA)
-5
10
ID (A)
2.98 nm
1.9 nm
1.36 nm
0.2
0.4
VG (V)
0.6
0
-12
10
0.0
0.2
0.4
0.6
0
VG (V)
ID(VG) characteristics in linear/logarithmic scales for three nanowire MOSFET’s
(LG=9nm, VD=0.7V) 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)
26
Results
Overestimation on ON-Current
0 .6
1 .1
mmx** fo
)
r(2m(m
) 0
a t
0
60
50
F rom T B E (k )
F ro m T B E (k)
0 .4
1 .0
x
ION overestimation (%)
70
40
30
0 .2
0 .9
20
10
0
1
S i B ulk
S i B u lk
2
2
3 3 4 4
Wire width (nm)
5
5
6
1
2
3
4
5
ire wwid
idth
WWire
th (n
(nmm) )
6
7
When the transverse dimensions decrease,
the effective masses increase and the carrier
velocity decreases.
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.
K. Nehari et al., Solid-State Electronics, 50, 716 (2006).
K. Nehari et al., APL, submitted, 2006.
27
3D Emerging architectures
Influence of the Band structure: Germanium
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
-valleys
29
Dispersion relations*
T=5.65 nm
Ge <100>
1.5
Energy (eV)
2 bulk valleys
1.0
4 bulk valleys
4 bulk valleys
Single bulk valley
Z
-0.5
X
-1.0
-1.0
Y
-0.5
0.0
0.5
Wavevector kX (/a)
1.0
4 bulk valleys
• 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).
*M.
Bescond et al. J. Comp. Electron., accepted (2006).
30
Dispersion relations
T=1.13 nm
Ge <100>
kX=0
1.8
2.5
L
Band minima (eV)
Energy (eV)
2.0
1.5
-1.0
-1.5
-2.0
-1.0
1.6
1.4
1.2
1.0
0.8
-0.5
0.0
0.5
Wavevector kX (/a)
1.0
1
2
3
T (nm)
4
5
• 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).
31
Effective masses: point
Ge <100>
0.30
(1/m*)=(4 ²/h²)( ²E/ k²)
m* (m 0 )
0.25
wire
0.20
0.15
0.10
Bulk
0.05
1
2
3
T (nm)
4
5
• 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.
32
Effective masses: kX=/a
Ge <100>
• 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:
1.00
2.5
0.99
Energy (eV)
2.0
X
| ( /a)
Average values
Wavevector |k
0.98
0.97
1.5
D=1.13nm
-1.0
-1.5
0.96
1
2
3
T (nm)
4
5
-2.0
-1.0
-0.5
0.0
0.5
Wavevector kX (/a)
1.0
33
Band-gap: Ge vs Si
Ge <100>
2.8
Si
Ge
E G (eV)
2.4
2.0
1.6
1.2
0.8
1
2
3
T (nm)
4
5
• 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-toband tunneling).
34
Effective masses: Valence Band
m* H (m 0 )
-0.2
-0.3
-0.4
-0.5
-0.6
1
2
3
T (nm)
4
5
• 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 tightbinding 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
Description of ballisticity: the Landauer’s approach
1D case: Concept of conduction channel and quantum of conductance
Current density from Left to right:
e
e
I -nev - ∑vi f i - E FL L i
h
Left electrode
∞
∫f - E d
FL
I I I
e
h
L
∞
EFL
Total current density:
Right electrode
Ballistic conductor
eVRL
EFR
∞
∫f - E - f - E d
FL
FR
∞
Quantum of conductance:
VRL →0
VRL
=
+ =
(2)e 2
h
Due to the Fermi-Dirac distribution (1 e-/state) which
limits the electron injection in the active
region
Nbe2
Rq: If bosonic particles: D b =
Resistance
0.2
h
of the reservoirs
0.1
E ( eV)
D = lim +
I
0.0
EFL
-0.1
-0.2
0.0
0.5
1.0
f(-EFL)
extra
Resistance of the reservoirs
VDS=0.4 V
VG=0 V
0.6
L=9 nm
R1
0.4
Source
0.2
0.0
0.8 V
-0.2
-0.4
0
1
T0
Off
regime
5
Drain
10
15
Channel axis (nm)
Drain
VDS>0
Source
20
1
R’1
T’0
Source
ΔE<0.4 eV
First subband of the (010) valley (eV)
VDS>0
R0
1
T’0
VDS>0
T1
On
regime
1
Drain
R’1
0.2
0.2
0.0
Resistance of the reservoirs:
the Fermi-Dirac distribution limit
the electron quantity injected in a
subband (D0=2e2/h).
0.1
EFS ( eV)
E ( eV)
E ( eV)
0.1
-0.1
EFS
-0.2
0.0
0.5
f
1.0
0.0
EFD ( eV)
-0.1
EFD
-0.2
0.0
0.5
1.0
f
extra
Towards the nanoscale MOSFET’s
1971
1989
1991
2001
2003
transistors /chip
410M
42M
1.2M
134 000
2300
Channel length of ultimate
R&D MOSFETs in 2006
10 µm
1 µm
Mean free path in perfect
semiconductors
ballistic transport
0.1 µm
10 nm
De Broglie length in
semiconductors
quantum effects
extra
Semi-empirical methods
Effective Mass Approximation (EMA):
• Near a band extremum the band structure is approximated
by an parabolic function:
1 E
*
m 2
k
2
1
2k 2
E k
2m *
E(k)
Parabolic
approximation of a
finished system of
atoms
0
(Infinite system at the equilibrium)
Parabolic
approximation of an
homogeneous material
k
extra
Numerical Aspects
Simulation Code
Potential energy profile
(valley (010))
Electrostatic potential
(Neumann)
Potential energy (eV)
1D density
(Green)
3D density
(Green)
Poisson
0.6
0.4
EFS
0.2
0.0
New electron
density
2nd
L=9 nm
VDS=0.4 V
-0.2
-0.4
0
New electrostatic
potential
Current
3rd
0.8
2D Schrödinger
Resolution
Selfconsistent
coupling
y
EFD
VG=0 V
5
10
X (nm)
15
20
1st
The transverse confinement involves
a discretisation of the energies which
are distributed in subbands
Extra