Gold nanoparticle single-ele ctron transistor simulation

Gold nanoparticle single-electron transistor simulation

Y. S. Gerasimov*ab, V. V. Shorokhova, E. S. Soldatova, O. V. Snigirevab a Department of Physics, Moscow State University, Moscow, Russia, 119899

b IMP, NRC “Kurchatov Institute”, Moscow, Russia, 123182

ABSTRACT

We suggest an approach for determining nanoobjects’ energy spectra depending on its total electric charge. The approach was tested studying golden nanoparticles of different sizes surrounded by ligands of dodecanethiols. Quantum methods of Hartree-Fock and DFT were used to calculate energy spectra and capacitances for such gold nanoparticles consisting of up to 33 atoms of gold. Presence of “extra” levels in energy gap of nanobject’s spectra was revealed. Also dodecanethiol SCH2(CH2)10CH3 ligands influence on the total capacitance and energy spectrum of molecular nanocluster was shown. Finally transport characteristics for gold nanocluster based molecular single-electron transistor (MSET) were calculated using the obtained energy spectra.The simple energy spectra structure model used in the present work may allow single-electron transistor (SET) simulation for larger molecular systems.

Keywords: electronic nanodevices, molecular electronics, single-electron transistor, discrete energy spectra, stability diagram, gold nanoparticles, dodecanethiols

1. INTRODUCTION

The problem of selecting appropriate molecular objects for creating nanoscale molecular electronic devices [1] is of significant practical importance today. It is well known that discrete structure of nanoobjects’ energy spectra influences transport characteristics (IV-curves, stability diagram) of MSET [2] (figure 1). The study of this influence will give opportunity to make the purposeful object selection for obtaining desired current-voltage characteristics.

Investigation of the molecular spectra influence on voltage-current characteristics features meets high computational complexity in the study of the energy spectra of large molecules (more than 200 atoms). The main computational challenge is the number of direct quantum calculations required for full MSET simulation. Today one of the largest currently known molecular objects which were directly calculated by Hartree-Fock method is a protein molecule consisting of 642 atoms [3]. Our estimations show that approximate number of atoms in a real ligand free gold nanocluster of 3 nm size is about 570 and for the same cluster covered by dodecanethiol ligand shell it is more than 104. That’s why it becomes clear that full-scale direct computations are inappropriate for our study and one should find a different approach.

To simplify the problem’s solution we suggest parameterization of the energy spectra of molecular objects (molecules, nanoparticles, nanograins). Analysing sets of spectra for single atoms, small and medium molecules (e.g. [4]) we observed that position of the most important for transport energy levels in a spectrum can be represented by a simple linear function of the additional electric charge Q of a molecular object. Therefore, for example, if we have calculated molecule spectra structure for some charges Q we can determine the spectra for the rest significant values of Q just by approximation. Such an approach may allow to reduce significantly the amount of direct quantum calculations of such spectra.

Figure 1 shows the schematic view (a) of the single-electron system under consideration and its equivalent electrical circuit scheme (b). A molecule or nanoparticle plays a role of a SET’s central island. It lies in the nanogap between two electrodes and thus forms 2 molecule-electrode tunnel junctions. The electric potential on the molecular island can be manipulated by the gate voltage which induces an additional charge on the island QG= CGVG. Gate is coupled to the molecule through non tunnel junction. The resulting capacitance СΣ =Cl+Cr+CG and typical resistance Rl and Rr values should satisfy the usual single-electron conditions [1].

*[email protected]

International Conference Micro- and Nano-Electronics 2012, edited by Alexander A. Orlikovsky, Vladimir F. Lukichev,Proc. of SPIE Vol. 8700, 870015 · © 2012 SPIE · CCC code: 0277-786/12/$18 · doi: 10.1117/12.2017078

Proc. of SPIE Vol. 8700 870015-1

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central island (a molecule)r

r R ITIiTr R-P"

Tr.

T cv)T

Thus in this work we study practically important gold nanoparticles to know whether their energy spectra have the peculariaties just mentioned above. At first we calculate gold nanoparticles energy spectra. Then we should determine electron transport energy levels for charge states of a taken molecular object. It can be done by applying the energy spectra model to the calcualted gold nanoparticles spectra. After we have the model energy spectra a SET based on the gold nanoparticles can be simulated. Finally we’ll be able to say if the proposed approximation model of energy spectra can be used for modeling the bigger nanoparticles of this kind for SET.

a b

Figure 1. (a) Principal view of the single-electron system under consideration and (b) common circuit scheme of a

single-electron device. Cl(r), Rl(r) – capacitance and resitance of the left (right) tunnel gap, VT – applied tunnel voltage, IT – tunnel current, VG, CG – gate volatge and non tunnel gate capacitance. The dash line indicates the central Coulomb island separated with tunnel gaps from the left and right electrodes.

Gold nanoclusters’ peculiarities are described in section 2. The calculation model and the main principles of defining a molecular object are described in section 3. Further in this section ligand’s influence and energy spectra structure features are discussed. After all the necessary data and energy spectra model for the gold nanoclusters is obtained we performed Monte-Carlo SET simulation which is the point of section 4. Here stability diagram, IV-curves and signal characteristics for a taken gold nanocluster are presented. Finally we make conclusions on the proposed electron energy spectra model, discuss the results and the future prospects of researching molecular objects applicability.

2. GOLD NANOPARTICLES

Today gold nanoparticles are prospective candidates for the role of MSET’s central Coulomb island (see, for example, [5]). The size and shape of such nanoparticles can be reliably controlled with no more than 10% size deviation. For instance there are commercially available golden nanoparticles with 3-5 nm diameter [6] which are suitable for deposition into a single-electron device’s nanogap.

Usually metal nanoparticles have a protective organic shell which prevents nanoobjects from mutual adhesion (as pure metal clusters are unstable [7]). This method of chemical stabilization of metal nanoparticles is very reliable and it is widely used for golden nanoparticles in order to prevent their interreaction. The molecules forming such an organic shell are called ligands. They also provide nanoparticles a mechanism of hitching the substrate in a molecular device (fig.3).

In the present theoretical work dodecanethiol SCH2(CH2)10CH3 (Fig.2) ligands were considered to stabilize golden nanoparticles, see Figure 3. These ligands are used in gold 3-5 nm nanoparticles solution [6] in our group. So we consider thus formed metal nanoclusters AuNLM: here N – number of Au atoms, L – a dodecanethiol ligand, M – number of ligands. Nanoclusters and nanoparticles of sizes we study, molecules and atoms – all of them can be considered molecular objects for us in this work. So we use these kinds of nanoobjects in context and applied to MSET equally.

Figure 2. Dodecanethiol ligand molecule SCH2(CH2)10CH3.

~0.9 nm



u. 3.

CH3(CH2)1oCH2A

LM - ligand shell (dodecanethiols)

Figure 3. Gold nanoparticle AuN coated with ligand shell LM (N – number of gold atoms, M – number of ligands).

Estimated total diameter of the gold nanocluster Au27LM: 2.3 nm. Sulfur atom attaches a ligand to the gold core.

3. QUANTUM CALCULATIONS

3.1 The model

To simulate a taken molecule based MSET we first had to implement quantum calculations for every molecule charge in a considered charge range. This allows one to get stable molecule’s geometry configuration for a taken charge and molecule’s electron energy spectrum. During the process of electron transport in a MSET the total electric charge of the molecule it is based on typically can take values in a reasonable range from -2e to 2e (e – module of electron charge). The negative charges for a molecule are less realizable because in practice putting an electron on a molecule is much more difficult than removing one. In those cases where it was possible we computed the range -3e…3e.

Geometry optimization of gold atoms based molecular systems (without ligands) was performed using DFT B3LYP5 method along with LANL2DZ basis. The choice is determined by the optimization of time needed for large systems calculations. The combination of the DFT quantum-chemical method and basis set LANL2DZ is suffcient enough [8] for determining the stable geometry of a nanoobject in our case.

Further calculations of the energy spectra of metal based nanoclusters AuNLM (L – is a ligand - dodecanethiol) were performed by unrestricted Hartree-Fock (UHF) method in basis SBKJC. This basis set choice was made to satisfy the requirements for a uniform basis for two kinds of molecular objects:

• involving heavy and light atoms at the same time;

• involving heavy or light atoms separately.

We needed a uniform basis set for energy spectra of molecular objects to be comparable.

Ab initio Hartree-Fock method is the most appropriate for our problem as it provides the most adequate electron energy spectrum which is the basis of our MSET simulation.

All quantum calculations were made by means of Firefly package [9]. Initial parameters for a single state of a nanoobject calculation are as follows:

~ 2.8 nm (N=27)

~ 0.5 nm

(N=27)



a) Molecule’s atoms initial coordinates

b) Full electrical charge Q of a molecular object

c) A parameter m defining a molecule’s exitation state: ∑ +=i

ism 12 , where 21

±=is - a spin of an i-th

electron (summation is over all the electrons of an object). It’s the only parameter contemporary quantum calculations programs have which allows us to distinguish different molecule exitation states.

The calculations were carried out for different combinations of initial parameters of Q and m which define current state of an object. We call the states with minimal permitted m at a given Q the ground states. The states with higher m are excited states.

Thus we calculated electron spectra of gold nanoparticles AuNLM for the cases of 1≤N≤33 and 0≤M≤12. At first ligand free gold nanoparticles optimization calculations were performed (increasing N by adding 1 atom to the stable N-1 atoms molecule). After that the quantum-chemical energy calculation process was repeated by linking ligands one by one to AuN metal core. The outputs of calculations are electron spectra Ei(n) and full energies EFULL(n), here n is a charge state which is defined as:

Σ−=∑ NZn

αα

, (1)

where e αZ - α -th nucleus charge, ΣN - total number of electrons in a molecule with total electric charge Q.

The results are presented and discussed below.

3.2 Self capacitance of gold nanoparticles

The capacitance of a molecular object is one of the most necessary parameters for simulating a MSET. To obtain it the approach developed in our previous works [10-11] was used. To get an objects capacitance one have to calculate the first ionization energy I1 and electron affinity A1 and substitute them to the formula [10]:

11

22AI

eC−

= , (2)

where

)0()1(1 FULLFULL EEI −= , (3)

)1()0(1 −−= FULLFULL EEA . (4)

Since EFULL(n) values are computed directly the capacitance CAuN of studied gold nanoclusters AuN can be easily determined.

The main physical properties of metal nanoparticles may noticeably differ from the properties of a bulk metal [7]. The estimated number of atoms for a golden nanoparticles of size > 2 nm without ligands is more than 155. As the gold nanoparticle Au155 is very hard to calculate as mentioned above we started calculating capacitance from a single Au atom increasing the number of atoms N gradually. There is no prefered most optimal structure for small clusters AuN at N<13 and the gold atoms can form 1D, 2D and 3D isomers: so called lines, chains and polyhedra [7]. Starting from N=13 a dense packing of atoms becomes more preferable, for instance cluster Au13 is an icosahedron with 1 inner atom and 12 surface atoms. As the number of atoms in a nanoparticle grows the ratio of surface atoms number to internal atoms number decreases. For 2 nm metal nanoparticle about 60% of atoms are surface.

It can be shown that capacitance CAuN(N;d) is a linear function of N1/d where d – the object’s dimensionality (d=1 for 1D, d=2 for 2D etc.) Linear dependence approximation of values CAuN calculated by UHF and DFT (for comparison) is shown in figure 4. Most of the values fit linear approximation quite well. There are also some deviating values which can be observed mostly for N<13. We suppose it should be explained by not enough optimization quality (and thereby



22

20

18

16

14

12

10

8

6

CHF, polyhedra (d =3)

° CDFT, polyhedra (d =3)

CHF, chains (d =2)C(N1/d) =B N1ia +B

1 2

; -----------;- --

r._._._._..

4u155' D=2 nm

=155; N^(1/d)=5,37; -

>21,5 10-20 Fi

i

--------; ---------

i i i - i i i i i i/ t--; ; --; ; - ; ---- -; ----------- ; ------------; -

i

i

i i i i i i

-----i------------- ---- -i------------------------------------i =------------i------------------------%---------------------i Ï!/i i

--_._._._.J,_._._._._.L._._._._._.J._._._._._._1_._._._._._.1._._._._._._1_._._._._._.l-__.. ..__-_._._.L._._._._._.1_._._._._._.L._._._._.i i i i i i i i i

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5 6,0

N1/d

>dJW73

0,014

0,012

0,010

0,008

0,006

0,004

0,002

0,000

-0,002

-0,004

Au27 , a=2,3e-3, b=3,2e-3

Au27L2 , a=1,9e-3, b=5,1 e-3

- Au27L4 , a=1,6e-3, b=6,9e-3

dEFu,/dQ=aQ+b -

iMMW7-=== .M=r ....

-3 -2 0

Q, e

1 2 3

2,5

2,0

V 1,0

0,5

0

M

unaccurate full energy values) or rather by ambiguous geometry with transitional (fractional) dimensionality [7] (e.g. d~2.5). Thus the Au155 capacitance can be estimated and according to the linear approximation C(N;d) by the value of about 21.5·10-20 F (215 zF) (see figure 4).

Figure 4. Calculated capacitance CAuN dependence on the number of atoms N powered dimension d factor (1/d).

3.3 Ligand shell influence

Ligand shell coating a core golden nanoparticle changes object’s properties relative to the initial ligand free ones. First, electrical capacitance of a taken nanoobject is changed. In order to estimate it we performed our calculations for a number of energy states of 3 kinds of AuNLM nanoobjects: M=0;2;4 at N=27. The correspondig Coulomb quadratic dependences of EFULL on molecule charge Q fit the calculated EFULL values. It can be well demonstrated by dependence of derivative dEFULL/dQ on Q:

dEFULL/dQ=a·Q+b, (5)

where a ~ 1/C, b/e ~ χ=(I1+A1)/2, χ - molecule electronegativity,

shown at figure 5a.

a b

Figure 5. (a) Derivative of total energy EFULL of Au27LM on the additional electric charge Q and (b) the values of

Au27LM capacitance for selected M values.



The increase of ligands number M magnifies the total capacitance C of a gold nanocluster AuNLM. That can be seen from the change of a slope of dEFULL/dQ lines in figure 5a.

Figure 5b shows the values of Au27LM capacitance for selected M values. One can see the change in capacitance from CAu27=1.16·10-19 F to CA27L4=2.12·10-19 F is quite insignificant in contrast to the nanoparticle size change in 2.8 times (see figure 3). For specific number of ligands M/N=0.07 Coulomb energy EC (EC = e2/C) change in respect to the ligand free one is 3% and for M/N=0.15 it is 83%. We suppose that this nonlinear increase of capacitance may point on the overall nonlinear dependence of “core-shell” nanocluster capacitance on the ligands number M (although we have only 3 values of C here which is not enough). We also revealed that the molecular system’s AuNLM capacitance is not equal to the sum of its parts AuN and LM the way it is in the classical electrostatics. So every AuNLM molecular system should be quantum calculated to obtain its capacitance.

It have been found also that ligands influence energy spectrum of a gold nanoparticle as well. The electron energy spectrum of a gold nanocluster is shifted in the same way as the ground state energy minimum does – to the left on the Q axis. But the structurally it remains the same (its structure is shown below). The behavior of a nanocluster electronegativity χ is in strong connection with the electron energy spectrum: it increases proportionally to the number M of the ligands attached to the metal core (see b-factor at figure 5a).

Thus it has been shown that the increase of nanoparticle’s AuNLM capacitance with the increase of M is not significant in respect to the metal core’s AuN initial capacitance. But every additional dodecanethiol ligand L linked to the golden core AuN increases the wholse nanoparticle’s AuNLM electronegativity.

The view of electron energy spectra is discussed in the next subsection.

3.4 Energy spectra calculation

For MSET simulation one have to know exact values of electron energy levels which are being occupied or left by an electron transitioning through the single-electron system. We analysed a set of such spectra of gold nanoparticles AuNLM for different combinations of N and M (13≤N≤33 and 0≤M≤12).

Figure 6 shows UHF calculated electron energy levels Ei of the electronic ground states typical spectrum of Au27 nanoparticles on the charge state n (the values over the energy HOMO-LUMO gap are only depicted). Having analyzed such spectra for different nanoclusters we came to a conclusion that a spectrum of any nanoobject can be divided into two types of levels:

• Energy levels which shift as the whole along with the energy gap ΔEHL “up” and “down” by the value of Coulomb energy EC (EC = e2/C) when the charge of a nanoobject is varied by e (module of electron charge):

Ei(n-1) ≈ Ei(n-1)+EC

The energy gap ΔEHL value remains almost the same:

ΔEHL ≈ const

• The “extra” levels in the energy gap area (circled red at Fig.6). These levels appear and disappear one by one along with varying the nanoobject’s charge state n. These levels cause additional features on IV-curves of a MSET based on the nanoobject.

As far as we know the “extra” levels presence in the electronic spectrum was not previously mentioned or used in the simulation of electron transport. Though they are located in the energy gap, by definition in our case such a level is a HOMO (highest occupied molecular orbital), i.e. the bottom of commonly used HOMO-LUMO gap. The energy gap value ΔEHL itself equals to HOMO-LUMO for n=1 here. Another remarkable fact is that the size of the gap ΔEHL approximately does not change in wide range of N (up to N=33) and regardless to the ligands number M. As stated above ligands just shift electron energy-levels diagram to the left by charge state n axis. For comparison and demonstration similar diagram for Au27L2 is shown in figure 7. ΔEHL=2.4 eV, which is 0.3 eV more than, e.g., for thiolate-protected Au20 clusters in work [12].



2

1

Au27

- Occupied levels

- Vacant levels

-Bo(n), Bp)

á) -"ca) _5

7)(D> -6

AEHLle2/C

IFJ-7

-81e2/C

9-1 0

Charge state n

1

6

4

Au27L2

- Occupied levels

-go(n), Bu(n)

. 1

iAEHL

III II

IIIIII

IIII

I111

1111

11

IIIIII

IIIIII

III II

IIIIII

IIIIII

III II

IIIII

III

I1

IIIII

IIII

I

y IIIIII

III

III

III11

1111111111

III11

1111111111

I1u11=II

II IIIIIIIN

1l

III11I1111111111II

III1I1111111II

11111111111111

111

111=111111

I

111111II 111

1111

IIII11111111111

-14mi =i -r

-3 -2 -1 0 1 2 3

Charge state n

Figure 6. Diagram of the Ei energy levels of the electronic ground states spectrum of Au27 nanoparticle at the charge

state n. Two columns of energies for each n depict “up” and “down” electron spin states. Bo(n) and Bu(n) – “lower” and “upper” boundaries of the energy gap ΔEHL. One can see that for Au27 and n=1 energy gap size equals by defenition the usual HOMO-LUMO gap. “Extra” levels are circled in the energy gap. ΔEHL=5 eV. EC=e2/C=2.4 eV.

Thus we consider that the charging energy EC, the size of the energy gap ΔEHL, its bottom value (Bo(n)) and “extra” levels position in the energy gap can define the main features of a taken molecule’s electron spectra for the most important charge states n (Q). It is also should be taken into account that in electron transport simulation one needs to know only the energy of the highest occupied level (HOMO) an electron go on/from. We’ll call such energy levels transport levels.

Investigations of energy spectra for a number of gold nanoparticles allowed us to assume that the occupied “extra” levels values can be linear approximated as well. Hereby the position of all the energy levels taking part in the electron transport can be easily parametrized in form of a linear function Ei(n).

Figure 7. Diagram of the energy levels of the electronic ground states spectrum Ei on the charge state n for nanoparticle

Au27L2. ΔEHL=4.8 eV. EC=e2/C=2.4 eV.



Summarizing all the above said the following approach of parametric determining of a simulated molecule’s energy spectra can be formulated:

1) At first one should calculate the electron energy spectra for the molecule charge states n=-1;0;1 to determine the molecule capacitance. If the capacitance value can be estimated as it was shown for Au155 in section 3.2 then at least one electron energy spectra should be calculated that contains an “extra” level.

2) The parameters EC and ΔEHL are evaluated from the obtained spectra. The position of an “extra” level in the calculated specra energy gap is determined.

3) The energy of transport levels of the first type (the energy gap’s bottom value Bo(n)) can be calculated as:

Bo(nx)= Bo(n0)+(n0-nx)·EC,

where nx – the charge state of a desired spectrum, n0 – the known quantum calculated charge state.

The “extra” levels position Eex(n) is similarly assumed to be:

Eex(nx)≈ Eex(n0)+(n0-nx)·EC+Δε(nx),

where Δε(nx) – is the distance between extra energy levels which approximately equals average energy-level density of energy spectrum below Bo(nx).

We assume the possibility of using such a simplified approach of calculating a nanoobject’s energy spectra depending on its electric charge (and spin multiplicity) for even much larger molecules not having to calculate directly the energy spectra for all the necessary states.

Thus we have established that energy spectra stucture for the investigated “core-shell” gold nanoparticles AuNLM is identical. But the most remarkable fact is that such a nanoparticle’s electron energy spectrum shifts by charge energy EC as the Fermi level in a solid-state shifts by electrostatic energy. Indeed the energy gap size ΔEHL reduces when the size of a nanoparticle increases. In limit the discrete energy spectrum becomes continuous and the energy gap bottom level Bo (or HOMO) transitions into a bulk metal Fermi level.

4. SINGLE-ELECTRON TRANSISTOR

4.1. Simulation model of electron transport

We base on a common system of kinetic equations for modeling the transport process in single-electron molecular transistor from the first principles. We use an assumption of weak coupling of a central island to electrodes. Basic form of these equations is the following:

)()()( tRDDCCtt

ppppp +−−+=

∂∂ ∑ −+−+ρ , (6)

here )(tρ – probability distribution function of electron configurations, C and D account for probabilities of creation and

destruction of molecular electron states respectively, +pC and +

pD – probabilities of electron tunneling to molecule on p

level, −pC and −

pD – probabilities of electron tunneling from p level to an electrode, R(t) – summand describing electron relaxation in a molecule.

We used a standard Monte-Carlo method of simulation [13] to calculate the transport properties of an MSET shown in figure 1. The estimated lifetime of an exited state in a molecule is τ ~ 10-10 s. Thus extremely slow electron energy relaxation [14] was assumed which makes R(t)=0. The method is a probabilistic way of solving the kinetic equations (6) and it allows to study electron tunneling transport directly in dynamics. In particular, one of its advantages for us is that it gives the opportunity to see what concrete state of molecules and their energy levels are involved in tunneling transport through the molecule in the SET.



W" (n) 15143110

PW'

I aft Right

electrodeL1 r electrode

f(n)-

rar t1 Q. W''+(n),, '"/

' '

CG, VG

Figure 8. Scheme of electron transport in a single-electron transistor with central Coulomb island with discrete energy

spectrum Ep(n).

Figure 8 schematically shows the possible two types of tunneling events in monomolecular single-electron transistor:

a) The electron leaves the molecule through the left or right junction. The rate of this process is )()()( ,, nWnWnW r

plpp

−−− += ;

b) An electron comes to the molecule through the left or right junction. The rate of this process is )()()( ,, nWnWnW r

plpp

+++ += .

Each of these processes is the result of the multiplication of 3 independent (because of the condition of weak coupling) probabilities:

))(1)((),()(

))(1)((),()(

)()()),(1()(

)()()),(1()(

,,

,,

,,

,,

nfnnpPnW

nfnnpPnW

nfnnpPnW

nfnnpPnW

rfp

rp

rp

lfp

lp

lp

rip

rp

rp

lip

lp

lp

−Γ=

−Γ=

Γ−=

Γ−=

−

−

+

+

, (7)

where P(p,n) – probability of finding p level occupied at charge state n, )()( nrlpΓ

– tunneling probability through the left

(right) barrier,

)()(),( nf rlfip

– Fermi-Dirac function of electron distribution in electrodes, which for substitueted tunneling electron energy values (defined by energy balance) is given by:

],,)1()([)(

],,)([)(

],,)1()1([)(

],,)1([)(

,

,

,

,

TeVeVnEfnf

TeVeVnEfnf

TeVeVnEfnf

TeVeVnEfnf

GrFTp

rip

GlFTp

lfp

GrFTp

rip

GlFTp

lip

+−−−=

+−+=

+−−−+=

+−++=

εη

εη

εη

εη

(8)

here ɳ - applied tunnel voltage division ratio on the molecule, )(rlFε - Fermi energy of the left (right) electrode (for gold in

our work), Ep(n) – energy of an electron p level in a molecule’s spectrum at charge state n, which was quantum calculated in the section 3.3. Fermi-Dirac function of electron distribution is defined as:

)/exp(1

1],[TkE

TEfB+

= . (9)

The index i is used for an electron in the electrode before tunneling and index f is used for an empty state in the electrode after tunneling.

The total tunnel current finally was calculated as:



∑+

=i i

rlT

nneIτ2

, (10)

where nl and nr – the calculated (for all Monte-Carlo iterations) number of electron transitions through left and right tunnel barriers at a given VT, considering the direction of a transition in a sign, τi – lifetime of the current i-th molecule state (≈10-10 s).

We perforemed calculation for the temperature T = 0 K in order to better define and examine Coulomb diamonds on simulated contour diagrams. Temperature does not result in additional features on IV-curves, but only blurs the features which are visible at null temperature. For instance, as the calculated charging energy of a Au27L4 nanoparticle turned out to be 0.95 eV, the Coulomb dimonds features at room temperature T = 300 K ≈ 0.026 eV won’t be very much blured.

The used model electric current unit is )/( rlrle Γ+ΓΓΓ . The ratio of tunnel barriers resistances is given by the parameter γ:

rl

l

Γ+ΓΓ

=γ , (11)

We considered the simulated MSET to be fully symmetric and assumed ɳ=0.5 and γ=0.5.In practice these parameters selected to fit an experiment conditions.

In the next section we present and discuss the results of simulation: IV-curves and stability contour diagrams of MSET based on AuNLM nanoparticles.

4.2. Simulation results

Finally the transport characteristics of single-electron transistor based on gold nanoparticles were simulated at N = 13 (particle size 0.8 nm) and N = 27 and for the number of ligands M={0;2;4;12} using all the calculated data on electron energy spectra Ep (thus having got the transport levels) and capacitances.

Figure 8 shows all the main electron transport characteristics of a simulated MSET based on Au27L4 nanocluster. One can see at figure 8a that we got usual Coulomb diamond stability diagram. Color gradation depicts the tunnel current value. For IT = 0 we see cycles of Coulomb blockade diamonds. These diamonds can be associated with particular charge states [15], e.g. the largest diamond accounts for the charge state n = -1. Other diamonds are associated with nonblockade currents steps on IV-curves. The numbered horizontal dashed lines in figure 8a indicate the corresponding to different gate charge value QG IV-profiles in figure 8b.

The 1 IV-curve (figure 8b) shows the largest Coulomb blockade; its size is 2.6 V. Such a relatively big value is due to the nanoparticle little size (golden core is about 1 nm). As the golden nanocluster size will grow the maximun Coulumb blockade size will decrease. For example, in work [5] a SET based on Au nanoparticles 5.2 nm was presented. It is about 10 times larger than Au27LM. The measured charging energy here was 48 meV. Taking into account that for relatively big Au nanoparticles capacitance is proportinal to nanoparticle’s size R, it is clear that E ~ 1/C ~ 1/R. So estimating from our calculated data the charging energy for 5.2 nm we get ~91 meV. That is just 1.9 times more than the experimental one which is quite satisfactory. Respectively the measured in [5] maximum Coulomb blockade size is approximately 0.16 V which is 16 times bigger than for Au27L4.

Proc. of SPIE Vol. 8700 870015-10


3

2

3-2 -1 o

Tunnel Voltage VT, V2

13:ia

0,34

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0,24

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0,6

-2 -1 0 1

Tunnel Voltage VT, V

2

Gate Charge QG, e

ON

C

0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16

Tunnel Current IT, eF

Ii

LH

L CHC

II

CklH

L. CH`7

CH

' Au ,- 4t CHA- Au Au

AuAu Au i

Ati u Au AuH, Au- ilt)H: C}I r

H

H. CH H

H CH

H'

H 'HC

H

Lkir CHH r CH

Ii c"H rCH

a b

c d

Figure 8. (a) Simulated stability diagram of Au27L4 based MSET, (b) corresponding IV-curves series (“2”-“5” curves

are shifted down for convenience) and (c) Control curves for specific VT (curves β and δ are shifted to the right on IT axis as well). (d) – schematic view of the MSET central island – gold nanocluster.

Figure 8c shows control curves: applied tunnel voltage VT fixed, the gate potential VG (or corresponding charge QG) is varied. There is a very useful way of determining transport energy spectra levels of a nanoobject in a SET that can be applied in practice. One should get a signal characteristic for VT = 0 and the achieved system of current peaks will give the transport levels of the nanoobjects as it does curve α in figure 8c (in our case charge should be converted to energy units).

5. SUMMARY

We have studied gold nanoparticles AuNLM stabilized by dodecanethiol ligands and for the range of number of gold atoms N=1…33. It was shown that the capacitance of a gold nanoparticle C~N1/d (d – dimentionality of a nanoparticle) and that ligands L influence it insignificantly.

The analysis of the quantum calculated energy spectra of AuNLM nanoclusters revealed the possibility of electron energy spectra parametrization. Energy spectrum of a molecular nanoobject shifts by charging energy EC as a whole when the

Coulomb blockade diamond

Nonblockade diamond

Proc. of SPIE Vol. 8700 870015-11


nanoobject’s charge state is changed. Also there were found “extra” energy levels, which lie in the energy gap and appear and disappear one by one with the charge state change. Thus the characteristic parameters for a considered molecular object are its charging energy EC, the size of the energy gap ΔEHL, its bottom value (Bo(n)) and “extra” levels position in the energy gap. They can define the main features of a taken molecule’s electron energy spectra for the most important charge states n.

The proposed approach for determining molecular objects energy spectra for different charge states can be used for minimizing the quantity of required quantum calculations. It is applicable for “core-shell” nanoobjects as it is for small molecules (e.g. carborane or fullerene [16]).

We used our molecular energy spectra model to define transport energy levels of gold nanoparticles and demonstrated simulation of a SET based on nanocluster Au27L4. The obtained IV-curves and contour diagrams with Coulomb diamonds showed good applicability of our energy spectra approach for SET-simulating this kind of nanoobjects. More over it is quite promising for modeling energy spectra of 3-5 nm gold nanoparticles AuN with N>155 which is to be fulfilled shortly.

6. CONCLUSIONS

In summary, in this work we presented the new approach for simulating of electron transport on the basis of the energy spectra model for nanoobjects. It gives a unique opportunity to obtain appropriate transport spectra of molecules consisting of more than 100 atoms. Our approach significantly reduces quantity of necessary quantum calculations of molecule energy properties (up to two or three states of the molecule). The inroduced energy spectra determinig method was applied for quantitative simulation of molecular single-electron transistor using gold nanoparticle as the Coulomb central island. The calculated results were estimated to be in satisfactory agreement with available experimental data on gold nanoparticles based MSET.

In practice parametrization of the obtained gold nanoparticles energy spectra allowed us to define transport energy levels as for small particles (N <155) and further it may also be helpful for the gold nanoparticles size of 2, 3, and 5 nm as well.

The obtained peculiarities of energy molecular spectra in the future can significantly simplify the solution of problems associated with simulating SET and study of electron transport in molecular systems of different sizes.

ACKNOWLEDGMENTS

This work has been supported by RFBR (Pr. No. 12-07-00816-a, 10-07-00712-а), Federal Target Programme «Scientific and scientific-pedagogical personnel of innovative Russia» in 2009 - 2013 years (Pr. No. 14.740.11.0389, 16.740.11.0020, 14.740.11.0370) and Federal Target Programme «Research and Development in Priority Fields of S&T Complex of Russia for 2007-2013» (Pr. No. 16.513.11.3063).

REFERENCES

[1] Likharev, K. K., "Single-electron devices and their applications", Proc. IEEE, 87, 606-632 (1999). [2] Shorokhov, V. V., Johansson, P., Soldatov, E. S., "Simulation of characteristics of a molecular single-electron

tunneling transistor with a discrete energy spectrum of the central electrode", J. Appl. Phys. 91(5), 3049-3053 (2002).

[3] Van Alsenoy, C., Yu, C.-H., Peeters, A. et al., "Ab Initio Geometry Determinations of Proteins. 1. Crambin", The Journal of Physical Chemistry A 102(12), 2246-2251 (1998).

[4] Gerasimov, Y. S., Shorokhov, V. V., Soldatov, E. S., Snigirev, O. V., "Calculation of the characteristics of electron transport through molecular clusters", Proc. SPIE. 7521, 75210U (2009).

[5] Norio Okabayashi, Kosuke Maeda, Taro Muraki, Daisuke Tanaka, Masanori Sakamoto, Toshiharu Teranishi, and Yutaka Majima, "Uniform charging energy of single-electron transistors by using size-controlled Au nanoparticles", Appl. Phys. Lett. 100(3), 033101 (2012).

[6] Sigma-Aldrich Co. LLC. "Dodecanethiol functionalized gold nanoparticles". URL: http://www.sigmaaldrich.com/catalog/product/aldrich/660434

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[7] Gubin, S.P. , [Himiya klasterov: osnovy klassifikacii i stroenie], Nauka, Moskva, 11-29 (1987). [8] Goel, S., Velizhanin, K. A., Piryatinski, A. et al., "DFT Study of Ligand Binding to Small Gold Clusters",

J.Phys.Chem.Lett. 1(6), 927-931 (2010). [9] Granovsky, A. A. Firefly version 7.1.G. URL: http://classic.chem.msu.su/gran/firefly/index.html. [10] Shorokhov, V. V., Soldatov, E. S., Gubin, S. P., "Self-capacitance of nanosized objects", Journal of

Communications Technology and Electronics 56(3), 352–369 (2011). [11] Gerasimov, Ya. S., Shorokhov, V. V., Maresov, A, G., Soldatov, E. S., Snigirev, O. V., "Calculation of the mutual

capacitance of nanoobjects", Journal of Communications Technology and Electronics 56(12), 1483-1489 (2011). [12] Manzhou Zhu, Huifeng Qian, and Rongchao Jin, "Thiolate-Protected Au20 Clusters with a Large Energy Gap of 2.1

eV", 131 (21), 7220-7221 (2009). [13] Honerkamp, J., [Stohastic dynamical systems: Concepts, numerical methods, data analysis], Wiley-VCH, New

York, (1994). [14] Averin, D.V., Korotkov, A.N., "Influence of discrete energy spectrum on correlated single-electron tunneling via a

mesoscopically small metal granule", Sov. Phys. JETP 70(5), 937-943 (1990). [15] Malinin, V. A., Shorokhov, V. V., Soldatov, E. S., "Determination of electronic properties of molecular objects on

the basis of nanodevices’ transport characteristics", Proc. SPIE. 7521, 75210 (2009). [16] Gerasimov, Y. S., Shorokhov, V. V., Soldatov, E. S., Snigirev, O.V., "Calculation of the characteristics of electron

transport through molecular clusters", Proc. SPIE. 7521, 29 (2009).

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