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Molecular electronics: a new challenge for O(N) methods. Roi Baer and Daniel Neuhauser (UCLA) Institute of Chemistry and Lise Meitner Center for Quantum Chemistry The Hebrew University of Jerusalem, Israel. IPAM, April 2, 2002. Collaboration. Derek Walter , PhD. Student (UCLA) - PowerPoint PPT Presentation
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1
Molecular electronics: a new challenge for O(N) methods
Roi Baer and Daniel Neuhauser (UCLA)
Institute of Chemistry and Lise Meitner Center for Quantum ChemistryThe Hebrew University of Jerusalem, Israel
IPAM, April 2, 2002
2
Collaboration
Derek Walter, PhD. Student (UCLA) Prof. Eran Rabani, Tel Aviv University Oded Hod, PhD. student (Tel Aviv U) Acknowledgments:
Israel Science Foundation Fritz Haber center for reaction dynamics
3
Overview
Molecular electronics is interesting Formalism O(N3) algorithm: non-interacting electrons Possible O(N) algorithm Electron correlation: O(N2) algorithm
4
Introduction
Why are coherent molecular wires interesting?
5
Conductance of C60
(a)
I
QDV
R1,C1
R2,C2
-1.0 -0.5 0.0 0.5 1.0
-0.8
-0.4
0.0
0.4
I (nA)
Tip Voltage (V)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
(c)
(b)
(a)
dI/dV (a
.u.)
0.0
0.2
0.4
0.6
0.8
1.0
T = 4.2 K
T = 4.2 KdI/d
V (a.u.)
0.0 0.3 0.6 0.9 1.20.0
0.3
0.6
0.9
dI/dV
Tip Voltage (V)
-1.0 -0.5 0.0 0.5 1.0
0.0
0.2
0.4
0.6
0.8
1.0T = 300 K
dI/dV (a
.u.)
Tip Voltage (V)
-1.0 -0.5 0.0 0.5 1.0
-0.8
-0.4
0.0
0.4
I (nA)
Tip Voltage (V)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
(c)
(b)
(a)
dI/dV (a
.u.)
0.0
0.2
0.4
0.6
0.8
1.0
T = 4.2 K
T = 4.2 K
dI/dV (a
.u.)
0.0 0.3 0.6 0.9 1.20.0
0.3
0.6
0.9
dI/dV
Tip Voltage (V)
-1.0 -0.5 0.0 0.5 1.0
0.0
0.2
0.4
0.6
0.8
1.0T = 300 K
dI/dV (a
.u.)Tip Voltage (V)
-1.0 -0.5 0.0 0.5 1.0
-0.8
-0.4
0.0
0.4
I (nA)
Tip Voltage (V)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
(c)
(b)
(a)
dI/dV (a
.u.)
0.0
0.2
0.4
0.6
0.8
1.0
T = 4.2 K
T = 4.2 K
dI/dV (a
.u.)
0.0 0.3 0.6 0.9 1.20.0
0.3
0.6
0.9
dI/dV
Tip Voltage (V)
-1.0 -0.5 0.0 0.5 1.0
0.0
0.2
0.4
0.6
0.8
1.0T = 300 K
dI/dV (a
.u.)
Tip Voltage (V)
Voltage [V]
dI/dV [a.u]
1.0
0.5
0.0
-1.0 0.0 1.0
T=4.2 K
STM
tip
Tunnel
Junction 1
Tunnel
Junction 2
(b)
D. Porath and O. Millo, J. Appl. Phys. 81, 2241 (1997).
6
Conductance of a nanotube
S. Frank and W. A. de Heer et al, Science 280, 1744 (1998).
7
Conductance of C6H4S2
Reed et al,
Science 278,252 (1997)
Chen et al,
Science 286,1550 (1998)
8
Coherent electronics
Size: ~ 1013 logic gates/cm2 (108) Response times: 10-15 sec (10-9)
Quantum effects: Interference Uncertainty Entanglement Inclonability
9
Interference effects
de-Broglie: electrons are waves Interference Nonlocal particle nature
Electrons are not photons! Fermions: cannot scatter into “any
energetically open state” Correlated: inelastic collisions, Coulomb
blockade… Tunneling: reducing/killing interference
effects, sensitive
10
A simple wire
W: Huckel parameters S D M: chain of 20 “gold” atoms, G G MW coupling = b Expect: current should grow with b
Units: eV
30 Carbons long
ML MR
V
bb
11
Sometimes more is less
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
b=1b=1.5b=2b=2.5b=3
I (e
2 /h V
olt)
V (Volt)
Inversion
12
Current from transmittance
R LI I I
Landauer current formula
R LI n E T E dE
30 Carbons long
ML MR
V
2eV 2eV
1
1 LL E
n Ee
13
Just because of the coupling…
0
0.2
0.4
0.6
0.8
1
-9 -8 -7 -6 -5 -4
b=1b=2
T(E
)
E (eV)
14
A switch based on interference
Simplest model of interference effects
2 4
6 8
10
15
Current-Voltage
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
0 0.5 1 1.5 2
0246810
V (Volt)
I (e
2 /h V
olt)
Destructive
Constructive
2 4
6 8
10
16
Fermi Wave length
=L
a=CC
Band bottom
Totally bonding
=2a Band top:
Totally non bonding
F=4a Band middle
Half filling
17
XOR gate based on interference
V1
V2
Current I
V1V2I
000
110
011
101
18
SensitivityDFT electronic structure. Molecule connected to gold wire, acting as a lead
Cu
rre
nt (
nA
)
Bias (Volt)
19
Quantum conductance formalism
( )ˆ1 ˆlimTr H N
R RtI Z e e I tb bm- - ×
®¥
é ù= ê úë û
rr
ˆ ˆ ˆ ˆ,l e l e l
iI qN q H Né ù= - = - ê úë û
&h
L R
IR
Wire
hR=1hR=0
( )ˆ ˆiH t iHt
R RI t e I e-
=h h ˆ
Tr H NZ e eb bm- ×é ù= ê úë û
rr
R. Baer and D. Neuhauser, submitted (2002).
20
Weak Bias: Linear Response
( ) ( ) ( )ˆ1 ˆˆlimTrlH N
lR e l RtG qZ e I I t
b mm m b¢ - -é ù -=ë û
®¥
é ù= ê úë û
Conductivity is a current-current correlation formula
R. Kubo, J Phys. Soc. Japan 12, 570 (1957).
21
Non-interacting Electrons
( ) ( )2 e
R l lRl R
qI F E N E dE
h ¹
= å ò
( )( )
1
1 llF
eb e me
-=
+
NlR(E) = cumulative transmission probability (from l to R)
R. Landauer, IBM J. Res. Dev. 1, 223 (1957).
22
Calculating conductance
Non-interacting particle formalism 4 step O(N3) algorithm
23
Step #1: Structure under bias
Use SCF model like DFT/HF etc. Optimize structure and e-density
ss
+
+
+
+
+
+
+
-
-
-
-
-
-
-
Right slabLeft slab
24
Step #2: Add Absorbing boundaries
effL RH H i
D. Neuhauser and M. Baer, J. Chem. Phys 90, 4351 (1989)
ss
+
+
+
+
+
+
+
-
-
-
-
-
-
-
Left slab Right slab
LG RG
25
Step #3: Trace Formula
( ) ( ) ( )†4 L RN E Tr G E G Eé ù= G Gë û
( ) ( ) 1G E E H -= -
T. Seideman and W. H. Miller, J. Chem. Phys. 96, 4412 (1992).
26
Step #4: Current formula
L R LRI F E F E N E dE
1
1 llF
e
l leVm m= +
(Landauer formula)
0
0.2
0.4
0.6
0.8
1
1.2
-1.5 -1 -0.5 0 0.5 1 1.5
DF
0
0+eV/2
0-eV/2
27
Efficient O(N3) Implementation
*
16Re L Rn
nm mnnm n m
e BI W W
h ff=
-å
( ) ( )L Rn
n
F E F EB dE
E f-
=-ò
N(E) is spiky Integrate energy analytically
†L LW U U= G
n n nHU Uf=
28
O(N) Algorithm
N(E) is averaged over E → A sparse part of G needed
The trace can be computed by a Chebyshev series
All energies computed in single sweep: integration is trivial
( ) ( ) ( )†
†
4 L RN E Tr G E G E
Tr T T
é ù= G Gë ûé ù= ë û ( )2 L RT G E= G G
R. Baer, Y. Zeiri, and R. Kosloff, Phys. Rev. B 54 (8), R5287 (1996).
0
0.2
0.4
0.6
0.8
1
1.2
-1.5 -1 -0.5 0 0.5 1 1.5
DF
0
0+eV/2
0-eV/2
29
Including electron correlation
Time Dependent Density Functional Theory
30
Linear response
, ,t
I t L G t t E t dt
r r
, ,I LG E r r
Uniform, weak, time dependent electric field:
----
++++
2
220
t
E t E e
31
Building the model
Small jellium sandwich
Large jellium sandwich
Embed small in large
Frozen Jellium (leads)
Dynamic system (w+contacts)
ImaginaryImaginary
potentialpotential
ImaginaryImaginary
potentialpotential
32
The setup for C3
a) Dynamic density
b) Frozen density
c) Total Density
d) Kohn-Sham potential
33
Conductance of C3
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0 0.25 0.5 0.75 1
z2 = -8 a
0
z2 = -4 a
0
z2 = 0 a
0
z2 = 4 a
0
z2 = 8 a
0
G(z
2,z0;
) [g
o]
[au]
34
Are correlations important?
Conductance is smaller by a factor 10. Possible reason: the same reason that
causes DFT to underestimate HOMO-LUMO gaps
35
Summary
Molecular electronics Theory of conductance Linear scaling calculation of conductance Importance of electrson-electron correlations TDDFT is expensive and at least O(N2)