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)
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