Download Electron Transport Through Thiolized Gold Nanoparticles in Single

J Supercond Nov Magn
DOI 10.1007/s10948-014-2661-6
ORIGINAL PAPER
Electron Transport Through Thiolized Gold Nanoparticles
in Single-Electron Transistor
Y. S. Gerasimov · V. V. Shorokhov · O. V. Snigirev
Received: 9 June 2014 / Accepted: 30 July 2014
© Springer Science+Business Media New York 2014
Abstract We propose an analytical parametric model for
defining energy spectra of nanoparticles with a number of
atoms of up to 3,300. This allows us to perform MonteCarlo simulations for single-electron transistor (SET) based
on gold nanoparticles with a size of up to 5.2 nm at temperatures from 0.1 to 300 K. At the first step, energy
spectra were calculated for isomers of gold nanoparticles, consisting of up to 33 gold atoms using methods of
quantum mechanics: density functional theory (DFT) with
LANL2DZ basis set for “geometry” optimization; unrestricted Hartree–Fock method (UHF)x with SBKJC basis
set to evaluate energy parameters of nanoobjects, which
include gold atoms with many electrons. It was found that
the general structure of the energy spectra changes unsignificantly if the number of atoms is greater than 27. Moreover,
the size of the energy gap and the position of energy levels
in it are linear functions of one parameter—the total electric
charge of the nanoparticle. These features of energy spectra
allowed us to perform calculations of the transport characteristics for a real SET using gold nanoparticle as a central
conducting island.
Keywords Single-electron transistor · Molecular
electronics · Gold nanoparticles · Electronic nanodevices ·
Discrete energy spectra · Stability diagram
O. V. Snigirev · V. V. Shorokhov
Department of Physics, Moscow State University, Moscow,
119991, Russia
e-mail: [email protected]
Y. S. Gerasimov ()
NRC Kurchatov Institute, Moscow, 123182, Russia
e-mail: [email protected]
V.V. Shorokhov
e-mail: [email protected]
1 Introduction
The development of single-electron transistors (SET) [1–
3] using molecular-scale nanostructures [4] as a central
conducting island seems very promising for the future of
electronics. The size of the conducting island is the main
factor that determines the maximum possible operating temperature of the SET. Because of their chemical-mechanical
stability and only small deviations in their size and shape,
gold nanoparticles are very often [5–8] used as the central
island in the SET structure.
Nanoparticles are covered by a layer of organic molecules, so called ligands, which protect them from mutual
contact and conglomeration and simultaneously defines
the tunneling resistance between the nanoparticle and the
source-drain electrodes, and the dielectric constant of the
ligand layer. These parameters can be controlled by selecting the appropriate type and length of the ligand molecule.
To calculate the main characteristics of the SET—the
current-to-voltage curves and a gate induced charge-tovoltage control characteristics—it is necessary to find
energy spectra of electrons confined inside the nanoparticle. We estimate that a gold particle with size 2 nm contains
about 215 atoms, and a particle covered by ligands with total
diameter 5 nm contains more than 3,300 atoms. At the same
time, one of the largest known molecular objects, calculated by the Hartree–Fock method is the crambin molecule
[9]—a protein consisting of 642 atoms. In our case, the
presence of the ligand shell and many-electron gold atoms
in the nanoparticles core dramatically increases the computational complexity for quantum chemistry methods. Thus,
the direct quantum calculation of all necessary energy characteristics for single-electron modeling of these particles is
not feasible even at the present level of supercomputer’s
development technology.
J Supercond Nov Magn
The most logical simplification to solve this problem is
to reduce dramatically the amount of computations required
for the description of the process of the electron transport
through a gold nanoparticle. Such a simplified description may be performed in case when it would be possible
to allocate a set of energy characteristics of the considered nanoparticles which can be parameterized. Recently, it
was shown for the molecules of carborane C2 B10 H12 and
fullerene C60 [10, 11] that a number of parameters (the
boundaries of an energy gap, the position of an “extra”
energy levels in the gap) may be described by a linear function of the additional number of electrons at the molecule.
The possibility to use the similar approach for gold nanoparticles was shown in present work. Transport characteristics
(stability diagrams, CVC, and control curves) of a SET
based on gold nanoparticles were simulated using the calculated spectra and the parameters of the nanoparticles,
defined by the proposed analytical parametric approach.
external finished atomic layer. The nanoparticle Au33 at
Fig. 1b is the largest one we managed to calculate directly
by means of so-called self-consistent field quantum methods. The second atomic layer of the Au33 particle is not
completely filled (in case the second layer of atoms is fully
filled the metal nanoparticle is expected to consist of 55
atoms of gold).
2.1 Energy Spectra
The calculated energy spectra for Au27 are shown in Fig. 2
and Au27 L2 was given in [14]. It was revealed that the
addition of two ligands results in a shift of the electron
spectrum in its structure by two charge states respectively.
This means that the addition of each ligand charges the gold
nanoparticle core with one +e and, consequently, the entire
spectrum of structural shifts by -1 on charge state n. In this
case, the spectrum of a nanoparticle gold core itself changes
insignificantly if the number of gold atoms N is greater
than 27.
2 Quantum Calculation of Golden Particles’ Properties
2.2 Energy Parameters of Gold Nanoparticles
The estimation of the properties of gold nanoparticles was
done by means of quantum-chemical calculations from the
basic principles. The Firefly standard package [12] was
used. The following combinations of quantum calculation
method and wave function basis sets were used to investigate nanoparticles containing N from 1 to 33 atoms:
density functional theory (DFT) with LANL2DZ basis
set for “geometry” optimization; unrestricted Hartree–Fock
method (UHF) with SBKJC basis set [13] to evaluate energy
parameters of nanoobjects, which include “heavy” atoms.
Methodologically, the calculation was started from a single atom and then the particle size was increased by adding
symmetric gold atoms to the previously achieved optimized
nanoobjects’ structure. The particle Au13 , shown in Fig. 1a,
is the first structurally stable polyhedron in a series of threedimensional metal particles. It has just one inner atom,
while all the rest atoms are external, thus forming the first
Fig. 1 Optimized spatial structure of gold nanoparticles a Au13 b
Au33 , which is the result of quantum calculation
The full energy of a nanoparticle in the ground (not exited)
energy state can be presented as a quadratic function of
charge state n:
e 2 n2
(1)
+ μn + Efull,0 (0) ,
2C
here μ—chemical potential of the nanoparticle, C—its self
electrical capacitance [15], Efull,0 (0) = const—full energy
of electrically neutral nanoparticle. The values of full energy
were obtained simultaneously with the spectra from quantum calculations [12]. In turn, this allowed us to calculate
Efull,0 (n) =
Fig. 2 Diagram of calculated energy spectra over the energy gap for a
gold nanoparticle Au27
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the ionization energy I1 and the affinity to electron A1 , that
gives the values of coefficients in (1):
I1 + A1
=μ,
2
(2)
I1 − A1
e2
=
.
(3)
2
2C
The values of (I1 + A1 )/2 and (I1 − A1 )/2 are shown at
Fig. 3.
The coefficient (I1 + A1 )/2 can be interpolated by the
function y(x) = a + b/x. Parameter a is the asymptote of μ
when the number of atoms N → ∞. The value (I1 + A1 )/2
is also a Mulliken electronegativity χ by definition. For
crystalline gold Mulliken electronegativity in units of Pauling scale equals 4.55 eV which is close to the work function
of gold Wex = 4.58 eV. The asymptote at Fig. 3 gives
χN→∞ = 2.3 ± 0.7 [eV],
(4)
which is approximately two times smaller than the tabulated
values of the electronegativity and work function of the bulk
gold. The reason for this may be that in classical physics
the vertical ionization potential is measured, but in our case
two things cause significant effects: the rearrangement of
the atoms in the particle when its charge will change and
the fact that most of the atoms in the particle are superficial
(“tearing” off an electron from such particles is easier than
from the surface of the gold crystal). Anomalous behavior
of the physical properties of metal nanoparticles is emphasized for sizes of 2 to 10 nm and is still little studied [16]. In
presence of ligands (I1 +A1 )/2 increases, as the alkanethiol
bonding increases electronegativity.
The value of (I1 − A1 )/2 is defined by the capacitance.
In experiments, the model of a metallic sphere coated with
a dielectric shell is often used when one needs to take into
account the ligand shell for the nanoparticle’s capacitance
(e.g., [17]):
CAuN LM = 4π ε0 εL (r/d)(r + d),
(5)
where ε0 —electric constant, εL —permittivity of the ligand
shell (for dodecanethiol εL = 2.6 from experimental data
[17, 18]), d—the thickness of the dielectric shell (for dodecanethiol d = 1.74 nm). Hereinafter, AuN LK denotes a gold
nanoparticle with the number N of gold atoms and K ligands
L. Our calculations showed that formula (5) is applicable if
the core size of the nanoparticle is more than 1 nm.
3 Parametric Analytical Model
The full energy change of the single-electron transistor
under consideration is entirely determined by the change in
the full energy of the central island (the nanoparticle). The
difference of the full energies before and after electron tunneling allows one to take into account the change in the
electron-electron interaction in the nanoobject. Thus, our
parametrization of the single-electron system is reduced to
the approximation of the total energy of the nanoparticles
Efull (n) as a function on n.
In general, the full energy depends on the degree of
electronic excitation (ex) which is defined by electron distribution in the spectrum. For the ground states of molecularscale objects, it is known that Efull,0 (n) is a quadratic
function (1) of the charge state. Based on our calculations,
we concluded that this approximation could be used to
parameterize both ground and excited energy states of gold
nanoparticles. If a nanoparticle jumps/transits from charge
state n1 with the degree of excitation ex1 to the charge state
n2 with the degree of excitation ex2 , then the change in the
full energy will have the form:
E ex1 ,ex2 (n2 , n1 ) = EC (n22 − n21 ) + χ(n2 − n1 ) +
+E ex1 ,ex2 (0),
(6)
where E ex2 ,ex1 (0) stands for full energy change due to
reorganization of energy levels in spectrum for the states
with the degree of excitation ex2 and ex1 (the values EC ≈
const, χ ≈ const).
It is important to emphasize that these transitions are not
optical, but they are transitions of a molecular object from
one energy state to a neighboring one. If one deals with spin
excitation and it is given by the nanoobject’s multiplicity1 ,
Fig. 3 The dependance
of calculated magnitudes (I1 + A1 )/2 and
√
(I1 − A1 )/2 on 3 N of gold nanoparticles; the effective nanoparticle
radius was evaluated by the classical formula reff = C/4πε0
1 Spin
multiplicity M equals
the sum of all electrons spins in the sys
tem: M = 2S + 1 = 2 i ms + 1, where S – angular spin moment, ms
– spin value of s-th electron, ms = ± 12 [19].
J Supercond Nov Magn
Fig. 4 Examples of the nanoobject energy transition between states
(I) and (II) when charging are shown for the case of a two-level system where: a both states are ground; b state (II) is spin-excited; c the
upper electron in state (II) is excited to a higher electron shell with a
larger principal quantum number; d state (I) have an opposite spin in
comparison to the condition (I) in the case a)
then that is a transition from a state characterized by a pair
of quantum numbers (n1 ;M1 ), to the energy state (n2 ;M2 )
due to the electron tunneling through one of two tunnel
junctions.
3.1 Excited States
In the ground state, the total spin of the system and
the energy are minimum for any n. An example of
such a transition between the ground states is shown
in Fig. 4a. In the proposed model of energy transitions, we will also consider two types of excited energy
states:
1. Spin excitations, energy states, which have different
multiplicity at the same charge state n. An example of
an energy transfer to the one of such states is shown in
Fig. 4b.
2. Rydberg (shell) excitation within one energy state
(n;M) (i.e., occurring without changes in the total spin)
when an electron jumps to a higher molecular shell.
Schematically, such a state transition (II) is depicted
in Fig. 4c. In case of single atoms, optical transitions
between the states of this type are usually represented
by Grotrian diagrams [19, 20]. Similarly, the distance
Fig. 5 Temperature dependence of the differential conductance diagrams of single-electron transistor based on 5.2 nm gold nanoparticle
a) T = 9 K; b) T = 160 K
Fig. 6 Series of simulated current-voltage curves of a SET based on
5.2 nm gold nanoparticle for the temperature T = 9 K
J Supercond Nov Magn
between the molecular terms was defined in the model
as
1
1 I1 + A1
E ≈
,
(7)
− 2
2
n21
n2
is greater than 13 K then the features at the current curves
become undistinguishable.
where n1 , n2 – principal quantum numbers of the
electronic energy levels.
The analytical parametric model for defining energy spectra
of the nanoparticles with a number of atoms up to 3,300 was
proposed and enabled us to perform Monte-Carlo simulations for SETs based on gold nanoparticles with a size up to
5.2 nm at temperatures from 0.1 to 300 K. As a result, using
this model, the stability diagram of the tunneling current,
current-voltage, and control characteristics were obtained.
Comparison of the calculated stability diagram for the
SET with a nanoparticle size 5.2 nm (Au591 L182 ) with the
experimental one [21] showed a good qualitative agreement.
The proposed approach can be used in the future for the
analysis of SETs based on similar nanostructures including
single-dopant atoms [22]. The work was performed with
the support of Russian Ministry of Science and Education.
Moreover, the spin degeneracy is taken into account, as
the same multiplicity may correspond to the states with different spin directions. Figure 4d shows an example of the
transition between the ground states, equiprobable to one at
Fig. 4a. Thus, the order of degeneracy for each state equals
2M−1 .
4 Transport Characteristics Calculation
We used a standard Monte-Carlo method of simulation
to calculate the transport properties of a SET based on
gold nanoparticles—it is described in details in our previous work [14]. The preparation of single-electron transistor
based on gold particles of a controlled size was reported
in [21]. The average size of the nanoparticles including
the ligand shell was 5.2 ± 0.5 nm. The ligands used are
decanethiols with the length of about 1.2 nm. Hence, the
size of the gold nucleus is about 2.8 nm. The nanoparticle of this size according to our estimations consists of
N = 591 gold atoms approximately and about K = 182
ligand molecules – Au591 L182 .
Figure 5 shows the calculated differential conductance
diagrams of a SET based on 5.2 nm gold nanoparticle
according to our model at temperatures 9 and 160 K. At a
temperature T = 9 K in Fig. 5a, there are distinguishable
features, associated with the first and the second excited
states. At higher temperatures, the information about the
features in the tunneling current is completely smoothed by
thermal fluctuations that one can see in Fig. 5b. In spite of
this, the on/off ratio of the tunneling current in Coulomb
oscillation characteristics κon/off = (Imax − Imin )/Imax at
temperature T = 160 decreases just to ∼ 55%.
The estimation of acceptable width of distinct lines on the
stability diagram let us also estimate the critical temperature
of the SET based on the 5.2 nm gold particle.
T ≈ 0.025 · EC /kB = 12.9 K,
(8)
here Boltzmann constant kB = 8.62 · 10−5 [eV/K]. Consequently, at temperatures greater than 12.9 K features on
stability diagrams obviously will not be seen due to the
presence of thermal noise.
The CVC for three gate voltages VG are shown in Fig. 6a
for T = 9 K. Our simulations showed that if the temperature
5 Conclusions
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