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 = 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).
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
k T1VS
+
SOURCE
CHANNEL
DRAIN
Gates
EOT
VG
L
W
VD
Effective Mass
Tensor (EMT)
X
 XX
 M D1   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
 2YZ
 2 XZ

2 
x
y
z
xy
yz
xz 
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
 2YZ
  V  y , z   E'   y , z   0
2
2
2
y
z
yz 
Coupling
• Where E’ is given by:
 2 k x2
t2l
 2 k x2
E  E' 
 E' 
2
2 YY ZZ  YZ
2mtrans
• mtrans is the mass along the transport direction:
mtrans
YY ZZ  YZ2

t2l
•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): mlmt  non diagonal EMT
  (ellipsoidal): mlmt  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: 44 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: TT=55 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: 44 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 r2(m0)
Conduction band edge (eV)
Conduction band edge and effective masses
1
2
3
4
5
W ire width (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
(m )
r(m
a t
2 ) 0
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
44
5
irewwidth
WWire
id th (n
(nmm) )
6
77
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 (0.04m0):
From 0.071m0 at T=5.65nm to 0.29m0 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 0.7m0
(effective mass: mtrans=0.6m0 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 -0.18m0 to -0.56m0 (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
-
e
h
L

∞

EFL
 Total current density:
I  I  I
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
N be2
 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
R1
0.4
Source
0.2
0.0
0.8 V
-0.2
-0.4
0
1
T0
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
R0
1
T’0
VDS>0
T1
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