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Symposium on Quantum Mechanical Models of Materials
A Density Functional Theory Study
of Schottky Barriers
at Metal-nanotube Contacts
School of Electrical and Computer Engineering
Tuo-Hung Hou
Outlines
 Introduction
 What’s carbon nanotube (CNT)?
 What’s Schottky barrier ?
 What’s DFT?
 DFT simulation on (8,0) carbon nanotube
 DFT simulation on CNT/Pd and CNT/Au contacts
 Summary
Carbon Nanotube
Discovered by Dr. Sumio Iijima in NEC in 1991
Extraordinary properties:
1. Self-assemble nanostructure. (diameter 1 nm / aspect ratio as high as 107)
2. 1-D carrier transport. Reduced scattering. Very high mobility (100x faster
than silicon)
3. Sustain current density as high as 109 A/cm2. (100x higher than copper)
4. Stiffer and stronger than steel.
Proc. IEEE, vol.91, p.1772, 2003
Carbon Nanotube
Graphene: a layer of graphite
Chirality vector
C= na1 + ma2
metallic
semiconducting
Graphere: 2DEG (3,3) CNT: 1DEG
(10,10) CNT
(4,2) CNT: 1DEG
(20, 0) CNT
Proc. IEEE, vol.91, p.1772, 2003
Work Function & Schottky Barrier
Work Function:
Schottky Barrier:
Minimal energy necessary to extract an
electron from the metal
The barrier formed at the metalsemiconductor interface
W = Ve - EF
BP = EF – Ev ; BN = Ec - EF
vacuum
M
M
M
S
M
Ec
e
e
EF
e
BP
S
BN
Ec
EF
EF
Ev
Ev
Carbon Nanotube Field Effect Transistor
S
D
CNT
Bottleneck step of charge transport
Back Gate
The Schottky barrier at metal/ CNT interface determines the performance
of CNTFET.
Proc. IEEE, vol.91, p.1772, 2003
A Closer Look of Schottky Barrier
M
S
Evac
Evac
Ec
No charge transfer
EF
M
S
Evac
Evac
W
Ec
EF
Ev
Ev
M
S
Evac
Evac
charge transfer
Ec
EF
Ev
-+
M
charge transfer &
finite separation
S
Evac
Evac
Ec
EF
-
+
Ev
Ab-Initio Calculation of Schottky Barrier
DFT takes into account the interfacial
interactions (dipole formation, geometry
relaxation etc. ) in first-principle. Thus, it
is very accurate in calculating Schottky
barriers heights.
Schottky barrier height BP
w/o interfacial interaction
= [EF]metal – [ EV]CNT
w/i interfacial interaction
= [EF – <V>]metal – [ EV -<V>]CNT + [<V>metal - <V>CNT]IF
EF
EF
[EF – <V>]metal
EV
[EV – <V>]CNT
_ +
EV
[<V>metal - <V>CNT]IF
J. Vc. Sci. Tech. A, vol.11, p.848, 1993
Brief Review of DFT
Density Functional Theory (DFT):
Describing electrons in a many-body system using the density instead of the
many-body wave function. It dramatically reduces the dimension of freedom
from 3N for N electrons to just 3.
1st Hohenberg-Kohn Theorem:
A one-to-one mapping exists between the ground-state charge density and
the ground-state many-body wavefunction.
2nd Hohenberg-Kohn Theorem:
There is a variational principle so that



 
E[  (r )]  T [  (r )]  Ee e [  (r )]   vext (r ) (r )dr  E0
So we can continuously refine the charge density to find the ground-state
energy.
Brief Review of DFT



 
E[  (r )]  T [  (r )]  Ee e [  (r )]   vext (r ) (r )dr  E0
Kohn-Sham Equation & Energy Minimization:


1
1

(
r
)

(
r
)  




  
E[  (r )]     i* (r ) 2 i (r )dr   1  2 dr1dr2  Exc[  (r )]   vext (r ) (r )dr  E0
2
2
r1  r2
i 1
n
Plane Wave Basis
Errors:
1. LDA
2. Pseudopotential approximation
3. Energy cutoff of plane wave basis
4. K-point selection for BZ sampling
5. Finite unit cell size
LDA
Pseudopotential
Carbon Nanotube Unit Cell
Carbon nanotube 3-D coordinates: generated by the wrapping program.
Unit cell: A hexagonal close-pack lattice with larger enough separation
between tubes. Periodic in the z direction.
Y
X
a
(8, 0) CNT
Cross section the
hexagonal unit cell
Convergence
Plane wave energy cutoff
Unit cell size
relaxed structure
-365.186
Z0 = 4.26 Å
-4
8.0x10
Total Energy [ Ryd ]
Energy Convergence [ Ryd ]
** Energy difference between a unrelaxed
and a relaxed structure.
a = 22 Bohr
-4
6.0x10
-4
4.0x10
Below 1 meV
-4
2.0x10
20
25
30
35
Energy Cutoff [Ryd]
40
45
Ecutoff = 40 Ryd
a = 27 Bohr
-365.188
-365.190
a = 22 Bohr
-365.192
-365.194
-365.196
4.18
4.20
4.22
4.24
4.26
Z0 [Å]
Energy cutoff 40 Ryd, unit cell distance 22 Bohr and 11 special k-points along
the z direction are found to give good convergence.
4.28
Geometry Optimization of CNT
Y
Z
Z0 = 4.221 Å
Total Energy [ Ryd ]
(8, 0) CNT
-365.178
-365.180
-365.182
Ecutoff = 40 Ryd
a = 22 Bohr
-365.184
-365.186
-365.188
Min. 4.221 Å
-365.190
-365.192
-365.194
-365.196
4.18
4.20
4.22
4.24
4.26
4.28
4.30
Z0 [Å]
Geometry optimization in the XY plane was carried
out for each Z0 to find the most stable structure.
(Force < 20 meV/ Å in X,Y,Z directions)
Y
X
D = 6.29 Å
Unrelaxed structure from the graphene sheet:
Z0 = 4.26 Å , D = 6.26 Å
Band Structure of CNT
(8, 0) CNT
EG = 0.6 eV
Γ
X
(8,0) CNT is semiconducting with EG 0.6 eV, agreed with the value reported
by Blase etc. ( 0.62 eV by LDA, Phys. Rev. Lett. 72, 1878 (1994) )
Work Function of CNT
Potential
Charge Density
A’ A
A
A’
Y
High density
low potential
1x10
1
1x10
-1
1x10
-3
1x10
-5
1x10
-7
-3
Total Potential [ eV ]
4
2
0
-2
-4
-6
WF
4.7eV
EF
-8
-10
-10
A
Charge Density [ Bohr ]
X
The potential and charge density are averaged over the z direction.
C
C
-5
0
X [Å]
5
10
A’
VASP, Shan and Cho, Phys. Rev.
Lett. 94, 236602 (2005)
Total potential V = VIon + VH-F + Vxc
Important!! Unit is Ry not eV
Metal/Nanotube Contact Unit Cell
Cross section the tetragonal unit cell
b
Y
a
Y
X
Z
Y
X
The initial structure has a relaxed (8,0) CNT on top of (100) surface of a two
or three atomic-layer metal slab. The lattice parameters of metals are first
calculated from the bulk (Pd 3.88Å , Au 4.05Å). The tensile strain is applied
on metal at the z direction to match the lattice constant of CNT. The strains
at x y directions are calculated by the Poisson ratio.
Geometry Optimization of Contact
Rotational Angel
Translational Distance
-1122.88
d = 2.0Å
-1122.88

Total Energy [ Ryd ]
Total Energy [ Ryd ]
-1122.87
z=0Å
-1122.89
-1122.90
-1122.91
-1122.92
-1122.93
-1122.94
0
5
10
15
20
[
25
o
]
30
35
40
45
-1122.89
-1122.90
-1122.91
-1122.92
z
-1122.93
-1122.94
-1122.95
d = 2.0Å
-1122.96
 = 0o
-1122.97
-0.5
0.0
0.5
1.0
1.5
2.0
Z [Å]
Major degrees of freedom are first sampled before full ab-initio optimization
to avoid trapping at local minima.
2.5
Binding Energy
Interfacial Distance
Total Energy [ Ryd ]
-1122.60
-1033.00
-1122.65
z=0Å
-1033.05
-1122.70
 = 0o
-1033.10
-1122.75
Au
Pd
-1033.15
-1122.80
-1033.20
-1122.85
-1033.25
-1122.90
-1033.30
d
Pd
Au
(8,0) CNT
-365.1943
-365.1943
Metal slab
-757.5965
-667.9404
CNT/Metal
-1122.9598
-1033.1867
Binding Energy (meV)
12.4285
3.8194
EBinding = ECNT + EMetal – ECNT/Metal
-1033.35
-1122.95
-1123.00
Total Energy (Ryd)
1.6
1.8
2.0
2.2
2.4
2.6
2.8
-1033.40
3.0
d [Å]
The equilibrium interfacial distance between CNT and Pd is smaller
than CNT and Au with stronger binding energy.
Full ab-initio Optimization
CNT/Pd
Energy = -1502.0944 rdy
Step1
fixed
Y
Y
X
Z
Energy = -1502.1555 rdy
Step25
All force < 0.02 eV/Å
Potential Energy
A
CNT / Pd
Total Potential [ eV ]
10
A’
A
EF
-10
C
Pd Pd Pd C
-20
-10
-5
0
5
10
Y [Å]
10
Total Potential [ eV ]
CNT / Au
A’
0
0
EF
-10
Au Au Au C
-20
-10
-5
C
0
Y [Å]
5
10
No physical tunneling
barrier existed
between CNT and
either Pd or Au.
Although Au with
larger interfacial
distance does show
an additional bump.
The carrier
transportation across
the interface is then
determined by the
band lineup, i.e.
Schottky barrier.
Charge Density
CNT / Pd
C
Pd
CNT / Au
C
Au
Less charge density between C / Au due to the additional bump in the potential
profile.
Charge Transfer
+
_
+
_
+
_
+ +
_
_
_
Charge difference = [e] CNT/Metal – [e] CNT – [e] Metal
Electron transfers from CNT to Pd and Au. More dipole formation at CNT/Pd is
due to its proximity, but the dipole moment is not necessarily larger (p=qd).
Schottky Barrier
CNT/Pd
10
Pd
-10
-20
-40
-50
5.2
5.0
4.8
-60
-70
Pd WF = -4.79 eV; CNT EV= -5.01 eV;
4.6
C
Pd Pd Pd C
-5
Total Potential [ eV ]
-30
-64.2
-64.4
w/o interaction Ep = 0.22 eV
-64.6
-64.8
0
5
10
Y [Å]
10
CNT/Au
0
CNT
Au WF = -5.27 eV; CNT EV= -5.01 eV;
-20
w/o interaction Ep = -0.26 eV
-3.2
-3.4
-3.8
-60
-70
C
Au Au Au C
-80
-10
w/i interaction Ep = 0.39 eV
-3.6
-5
0
Y [Å]
Total Potential [ eV ]
Total Potential [ eV ]
-30
-50
w/i interaction Ep = 0.185 eV
Au
-10
-40
BP = [EF]metal – [ EV]CNT w/o interaction
= [EF – <V>]metal – [ EV -<V>]CNT
+ [<V>metal - <V>CNT]IF w/i interaction
5.4
-80
-10
Total Potential [ eV ]
CNT
0
Total Potential [ eV ]
Total Potential [ eV ]
20
-64.6
-64.8
-65.0
-65.2
5
10
Summary
1. Review the theory of the ab initio Schottky barrier calculation based on DFT.
The method is applicable for many interfacial problems in nanoscale.
2. Detailed DFT calculations on (8,0) carbon nanotube are performed,
including the geometry optimization, band structure, and work function.
3. Geometry optimization at CNT/metal contacts are carefully examined
though a 2-step process. The major degrees of freedom are first optimized,
and followed by the ab initio relaxation.
4. CNT is more closely bounded to Pd than Au with larger binding energy and
shorter interfacial distance.
5. Although the number of dipoles is larger in CNT/Pd, the total dipole moment,
which is responsible for the potential shift across the interface is larger in
CNT/Au, which therefore shows larger Schottky barrier.
6. Very good quantitative agreement in this study in comparison with previous
works and experimental results.

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