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![Page 1: The g DFTB method applied to transport in Si nanowires and carbon nanotubes 1 Dip. di Ingegneria Elettronica, Universita` di Roma Tor Vergata 2 Computational.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649ecf5503460f94bdcf15/html5/thumbnails/1.jpg)
The gDFTB method applied to transport in Si nanowires and
carbon nanotubes
1 Dip. di Ingegneria Elettronica, Universita` di Roma Tor Vergata2 Computational Material Science, Universitat Bremen
Alessandro Pecchia1 L. Latessa1, Th. Frauenheim2, A. Di Carlo1
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RRD
DRDDR
LDL
H
H
H
0
0
H
,r aE i S H G I
Retarded (r) and advanced (a) Green functions are defined as follow
Let us write H and G in a block form
RRDRL
DRDDL
LRLDL
GGG
GGG
GGG
G
NEGF + DFTB = gDFTB
1 RLDD HEG
LDLDLL g
RDRDRR g Self-energies
Device region
Lead Lead
HDHL HR
LD RD
- +
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Non equilibrium density
In order to compute V(r) we need the local (r)
1
( )2
dE G Ei
Density Matrix
( ) ( ) ( ) ( ) ( )L RL RG E i E f E i E f E
We can build the n.e. density matrix
L R
( ) ( ) ( ) ( )r aG E G E E G E
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Self-consistent loop0μνH
<G
Density matrix
ρ
Multigrid Poisson solver
Self-consistent
solutions iq eR RG Σ
Evaluation of<G
Green’s function
1μνH
External potential
Hartree term
Exchange-correlation (LDA)+ +
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Implementation details
<G Is performed by countour integration and has been parallelized (MPI)
1 RLDD HEG
• All matrices stored in dense format
• Green’s functions computed by direct inversions
• Sparse storage
• Implementation of a block-iterative construction
Old gDFTB (2003-2005)New gDFTB (2006-)
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Block-iterative algorithm
The device G.F. are computed with an iterative algorithm
1, , , , 1 1, 1 1,[ ]L L L L L L L L L L L Lg ES H T g T
1) Computation of partial Green’s
2) Computation of equilibrium Green’s
, , , , 1 1, 1 1, ,L L L L L L L L L L L L L LG g g T G T g
3) Additional blocks needed for non-equilibrium
, , , 1 1,L n L L L L L LG g T G
HPLHPLHPLHPLHPL
†
†
†
†
L
L
L
L
H
H
H
H
H
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Profiling
SWCNT(20,0): 2880 atoms, 36 Principal Layers
2,5 nm
14,7 nm
60,2
5,40
20
40
60
80
Tempo (s)
OldgDFTB
NewgDFTB
1743
1860
500
1000
1500
2000
Peak Memory(MB)
OldgDFTB
NewgDFTB
Single node (P4 - 3.2 GHz), Single energy point
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Self-consistent potential
eV
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Poisson Equation
The Poisson’s equation is solved with a 3D Multi-grid algorithm.
( ) ( )n r q n r
2 4V n
Discretize in real space
This allows to solve complex boundary conditions (bias, gate)
2-terminals gated
coaxially-gated
4-terminals
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projection back in AO
3 ( ) ( )V d rV r n r
0 1( )
2H H S V V
Now we need to project the solution into the local basis set
Can be viewed as an approximation of the rigorous matrix elements of V(r).
This is consistent with standard DFTB
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Summary gDFTB implementation
Construction of H and S directly in sparse format
Solution of Green’s functions via block-iterative methods
Current Bottle-neck: Poisson equation
- Very efficient for 1D type systems- Memory scales linearly (depends on PL size)- Can be used for O(N) calculations even in equilibrium- Considerable speed-up and memory save
- Dense matrices never allocated
- Multigrid with uniform grid, dense storage!- Need to implement more efficient methods (finite elements with adaptive grids)
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Applications of gDFTB
symmetric moleculesymmetric molecule
Molecular Electronics.
Incoherent Transport andInelastic Tunneling Spectroscopy
A. Pecchia et al., Nano Lett. 4, 2109 (2004)
G. Solomon et al., J. Chem Phys 124, 094704 (2006)
A. Pecchia, A. Di Carlo, Introducing molecular electronics, Springer Series, (2005)
A. Pecchia, A. Di Carlo, Molecular Electronics: Analysis design and simulations, Elsevier (2006)
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Applications to CNT and SiNW
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Coaxially gated CNT
VD VS
=0
VG
Semiconducting CNT
Insulator (εr=3.9)
10 nm
1.5 nm
x
yz
CNT contact
(INFINEON - Düsberg)
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Atomic Forces VGATE = 5 V
Ang.
An
g.
GATE
GATE
Forces
[Ang]
Application of VG changesCNT diameter
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Screening problem
QinsG CCC
111
insC
QC
GV
CNTVQuantum correction to the induced charge
Vext
Distance
V
G
CNT electron gas
CNT is not able to accumulate the electronic charge to completely screen the gate bias (λ > electron gas extension)
λ CNTgate
rins RR
LC
/log
2 0
CNT completely screens the external field.
Classical electrostatics: charge induced on the CNT is
λ
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HOWEVER: In a CNT the DOS is not the only contribution. Many body correction should be considered (XC)
Screening in CNT: DOS limit
Why charge induction is limited?
DO
S
[Latessa et al., Phys. Rev. B 72, 035455 (2005)]Pauli exclusion principle limits the induced electrons to the number allowed by filling the DOS
FCNTQ EDOSeVQC 2/
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Many-body corrections
0
22
K
KEDOSe
neC FQ
Compressibilityof an interacting
electron gas
Compressibilityof an non-interact
electron gas
1 2K nn
1
QCQ
Compressibility Capacity
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Evidence of Negative K
Eisenstein, Pfeiffer and West, PRL 68, 674 (1992)Eisenstein, Pfeiffer and West, PRB 50, 1760 (1994)
N (1011 cm-2)
0.0
0.1
-0.1
-0.20.0 0.2 0.4 0.6 0.8 1.0 1.2
Nc
Compressibility of 2D electron gas
1/ 220 1 ce d N
N N
Thomas-Fermi screening
In 1D systems things can be more complicated because of D(E)
Including exchange
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Negative CQ in CNTs
CNT ox
G Q ox
V C
V C C
ox QG
g Q ox
C CdQC
dV C C
Overscreening, CQ<0
[Latessa et al., Phys. Rev. B 72, 035455 (2005)]
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Negative CQ in CNTs
Critical charge density nTS
TOTAL SCREENING
High density limit:PARTIAL SCREENING
CQ approaches e2DOS(EF)
gDFTB calculation
CQ proportional to DOS
Low density limit: OVER-SCREENING
Fit to analytic modelChalmers, PRB 52, 10841 (‘95)
Fogler, PRL 94, 056405 (2005)
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XC in DFTB
Hubbard and e-e repulsion integrals
0
( )[ ] [ ] ( ) ( ') '
( ')
xcxc xc i
k k
k
v rv n v n n r n r drdr
n r
T.A. Niehaus, PRA 71, 022508 (2004)
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Diameter Dependence
CNT (13,0) CNT (10,0) CNT (7,0)
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Output characteristics
dEEEfEEfETh
eI SFDF ,,
2h
VDS < 0
p pi
EF,SEF,D
Drain Source
“Electrostatic saturation”
L. Latessaet al.: IEEE Trans. Nanotechnol., in press
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• Small sub-threshold swing
(theoretical limit for silicon MOSFET: 60 mV/dec)• Ion/Ioff ~ 108
• Unipolar behavior
Trans-characteristics
DS
G
Id
dVS
log
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Band-to-well tunneling
Generation of confined states in a quantum well
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SiNW FET
SiO2 shells has been removed and silicon is terminated with H
D. D. D. Ma. et al., Science, vol. 299, pp. 1874-1877, 2003
L. J. Lauhon, et al., Nature, vol. 420, pp. 57-61, 2002.
Coaxially gated Si nanowire FET
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Geometry relaxations
d<10 nm
10<d<20 nm
d>30 nm
1.22 nm
0.87 nm
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Device geometry
P doped region P doped regionIntrinsic regionoxide
oxide
1.2 nm (2.4 nm)
7.7 nm
3.6 nm
6 nm
Drain Source
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CNT vs SiNW
CNT-FET
FQ EDOSeC 2
SiNW-FET
FQ EDOSeC 2
6 nm6 nm
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Differences in S
Coax. gated (7,0) CNTFET SiNW – FET
|VGS| (V)
C
urre
nt,
I DS (
A)
S = 180 mV/decS = 75 mV/dec
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Conclusions of part II
Atomistic Density Functional approach can be extended to account for current transport in molecular devicesby using self-consistent non-equilibrium Green function.
We use gDFTB is a good compromise between simplicity and reliability but there is room for improvement.
The use of a Multigrid Poisson solver allows for study very complicated device geometries
CNT and Silicon Nanowire FET has been studied with gDFTB
Quantum capacitance in CNT is governed by XC
Gate control in SiNW FETs is more delicate
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Surface Green’s function
The surface G.F. is computed by iteration (decimation technique)
HPLHPLHPLHPLHPL
g
†
†
†
†
L
L
L
L
H
H
H
H
H
12345 -> Converged surface Green’s function
Lopez Sancho et al., J. Phys. F: Met. Phys. 14 1205 (1984); ibid., 15 851 (1985)