Acta Materialia 52 (2004) 501–505
www.actamat-journals.com
Size-effect on the electronic structure and the thermal stability
of a gold nanosolid
Chang Q. Sun
a
a,*
, H.L. Bai b, S. Li a, B.K. Tay a, E.Y. Jiang
b
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
b
Institute of Advanced Materials Physics and Faculty of Science, Tianjin University, Tianjin 30072, PR China
Received 16 July 2003; received in revised form 17 September 2003; accepted 19 September 2003
Abstract
Consistent insight into the size-enhanced E4f -level shift and the size-suppressed melting point of Au nanosolids has been obtained
based on the bond order-length-strength (BOLS) correlation mechanism [Sun et al., Acta Mater. 51 (2003) 4631]. Consistency
between theory predictions and observations reveals that the atomic-coordination number (CN)-imperfection induced bond contraction and the associated bond energy increase dictate these changes. The increase of binding energy density per unit volume in the
relaxed surface region perturbs the Hamiltonian that determines the core-level shift; the decrease of atomic cohesive energy (a
product of atomic CN and the single bond energy) determines the thermal energy required for melting. Extending the knowledge to
the lower end of the size limit suggests that the metallic bond in the gold monatomic-chain contracts by 30% associated with 43%
magnitude rise of the bond energy. The crystal binding intensity contributing from all the atoms in the solid to the Au-E4f electrons
in the bulk is determined to be )2.87 eV.
Ó 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Nanostructure; Electronic structure; Thermal stability; Mechanical strength
Metallic nanosolids, or nanoclusters, nanocrystals or
nanoparticles, have attracted tremendous interest, since
they show intriguing properties from a basic scientific
viewpoint, as well as great potential in upcoming technological applications such as nanoelectronics. The intriguing phenomena include quantum conductance,
enhanced mechanical strength and binding energy, and
the lowered chemical reactivity and thermal stability.
Quantifying the intensity of crystal binding energy to a
core electron of an isolated atom and the core-level
energy shift upon bulk and nanosolid formation are of
great challenge. Although several models have been
developed for the size dependence of melting-point (Tm )
suppression [1,2] and core-level shift, [3] consistent understanding of the electronic structure, mechanical
strength, and thermal stability of a metallic nanosolid is
highly desirable. In this presentation, we describe an
*
Corresponding author. Tel.: +65-6790-4517; fax: +65-6792-0415.
E-mail address: [email protected] (C.Q. Sun).
URL: http://www.ntu.edu.sg/home/ecqsun.
effective approach to determine the binding energy of a
core electron and the thermal stability of gold nanosolids by decoding the measured size dependence of both
the Au 4f-level energy [4] and the melting-point suppression based on the recent bond order-length-strength
(BOLS) correlation [5–7]. Decoding the size dependence
of the 4f core-level shift gives information about the 4flevel energy of an Au atom isolated from the solid and
the crystal binding intensity of the entire solid to the 4f
electron upon bulk formation. Extrapolation of the size
dependent Tm suppression suggests that the gold monatomic chain (MC) melt at a temperature of 320 K.
The BOLS correlation that is derived from the correlation of atomic-coordination number (CN)–atomicradius noted by Goldschmidt [8] and Febelman [9]
indicates that the CN-imperfection of atoms at the
surface or at sites surrounding defects causes the remaining bonds of the lower-coordinated atom to contract to di ¼ ci d, spontaneously. The spontaneous bond
contraction is associated with magnitude increase of the
bond energy, or potential well deepening, ei ¼ cm
i eb .
1359-6454/$30.00 Ó 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2003.09.033
502
C.Q. Sun et al. / Acta Materialia 52 (2004) 501–505
The m is an adjustable parameter depending on the
nature of the bond of the system. For elemental metals,
m ¼ 1. According to Goldschmidt, if an atom reduces its
CN from 12 to 8, 6, and 4, the radius of the specific atom
will contract by 3%, 4%, and 12%, respectively. The
dimmer bond of metals such as Ti, Zr and V has been
reported to contract by 30–40% [9]. The BOLS correlation is represented by
(
ci ðzi Þ ¼ di =d0 ¼ 2=½1 þ expðð12 zi Þ=8zi Þ;
ð1Þ
ei ¼ cm
i eb :
The i denotes the atom in the ith atomic layer. The ei and
eb are the zero-temperature binding energies per bond
which is independent of the type of the pair-wise interatomic potential. d0 is the corresponding bulk value of
bond length. For a flat or a slightly curved surface,
z1 ¼ 4 [1,5,10,11]; for a spherical dot, z1 ¼ 4ð1 0:75=Kj Þ
[12], z2 ¼ 6 and z3 ¼ 12. Kj ¼ Rj =d0 is the number of
atoms lined along the radius of a spherical dot.
The striking difference between a bulk solid and a
spherical nanosolid is that the portion of the lowercoordinated atoms is considerably large for a nanosolid.
Generally, the mean relative change of a detectable
quantity Q of a nanosolid containing Nj atoms can be
expressed as QðRj Þ ¼ Nj q, and as Qð1Þ ¼ Nj q, for the
same solid without considering the effect of surface
CN-imperfection. Using a shell structure, the QðRj Þ is
expressed as
X
Nj q ¼ Nj q þ
Ni ðqi qÞ;
ð2Þ
i63
the relative change of the QðRj Þ follows a scaling law [5]:
8
P
DQðRj Þ
>
< Qð1Þ ¼ i 6 3 cij ðqi =q 1Þ ¼ Dqj ;
ð3Þ
1 ðKj 6 3Þ;
Rsout;i Rsin;i
Ni
>
c
¼
¼
: ij Nj
i
Rsj
/ sc
ðK
>
3Þ;
j
Kj
where q and qi correspond to the density of the Q inside
the bulk and in the ith surface layer, respectively. s ¼ 1; 2
and 3 correspond to dimensionality of a thin plate, a rod,
and a spherical dot. The i is counted up to three from the
outermost atomic layer to the center of the solid as no CNimperfection is expected when i > 3. Rin;i and Rout;i correspond to the inner and outer radius of the ith atomic
layer (Rout;i Rin;i ¼ di ). Eq. (3) represents that the sizeand-shape dependence of a detectable quantity of a
nanosolid originates from the Dqi =q that determines the
sigh and magnitude of change. Such a change follows the
portion of surface atoms, cij , of the entire solid.
For a spherical dot at the lower end of the size limit,
Kj ¼ 1:5 (Rj ¼ Kj d0 ¼ 0:43 nm for an Au spherical dot),
c1j ¼ 1, c2j ¼ c3j ¼ 0, and z1 ¼ 2, which is identical in
situation to an atom in a monatomic chain despite the
geometrical orientation of the two interatomic bonds.
Actually, the geometrical orientation of the bond configuration contributes not to the modeling exercise.
Therefore, the performance of an atom in the smallest
nanosolid is a mimic of an atom in a monatomic chain
of the same element without presence of external stimulus such as stretching or heating.
As a scaling law, the measured size dependence of the
QðRj Þ is always proportional to the inverse Rj (converges
at Kj < 3) with a slope b. Combining the scaling law
based on measurement and theory of Eq. (3), we have
1
bRj
ðmeasurementÞ;
QðRj Þ Qð1Þ ¼
ð4Þ
Qð1Þ Dqj ðtheoryÞ
with Qð1Þ b=ðDqj Rj Þ. The Dqj / R1
j (Eq. (3)) varies simply with the parameter m and the known dimensionality and known size of the solid. There are only
two independent variables, m and Qð1Þ. If a certain
known Qð1Þ value such as the Tm ð1Þ of the considered
system is given, the m can be readily obtained. With the
obtained m, any other unknown quantities Qð1Þ such
as the crystal binding intensity, E4f ð1Þ, of the same
system can be determined uniquely with the above relations. Not surprisingly [5], the size-and-shape dependence of a detectable quantity of the same system can
also be predicted for materials design purpose once the
corresponding qðz; eÞ function is established.
The CN-imperfection and the associated bond energy
rise contributes not only to the cohesive energy (Ecoh;i ¼
zi ei ) per atom but also to the binding energy density
(EB;i ¼ ni ei ) per unit volume in the relaxed region. The
atomic Ecoh;i determines the thermodynamic behavior of
a nanosolid such as melting, or phase transition
(Tm / Ecoh ) [1]. The compressibility (under pressing
force) or extensibility (under stretching force) at constant temperature is given by
1 1 oV o2 u
ds
b¼
;
¼
V
/
V oP T
oV 2
e
T
ou P¼ ;
oV T
ð5Þ
b is the inverse YoungÕs modulus that increases at surfaces due to the CN-imperfection enhanced bond energy
[5,13]. The EB;i contributes to the overall potential in the
Hamiltonian of an extended solid which leads to the
perturbation of the core-level energy of the solid [10]
DEm ðRj Þ ¼ DEm ð1Þb1 þ DHj c;
ð6Þ
where DEm ð1Þ ¼ Em ð1Þ Em ð1Þ is the energy shift upon
bulk formation. The DHj , being independent of the
particular form of the interatomic potential, is the
contribution from bond relaxation of the lower-coordinated atoms at the curved surface of the nanosolid.
As such, the bond contraction, Ddj , the melting-point
suppression, DTmj , the Hamiltonian perturbation, DHj ,
and the compressibility/extensibility modification, Dbj ,
of a nanosolid are in the form:
C.Q. Sun et al. / Acta Materialia 52 (2004) 501–505
8
P di d0 P
>
Ddj ¼
cij d0 ¼
cij ðci 1Þ < 0;
>
>
>
i63
i63
>
>
P ni ei nb e0 P
>
>
>
DHj ¼
cij nb e0 ¼
cij ðnib cm
1Þ > 0;
>
i
<
i63
i63
P zi ei zb e0 P
>
cij zb e0 ¼
cij ðzib cm
1Þ < 0;
DTmj ¼
i
>
>
i
6
3
i
6
3
>
>
>
>
P dis e1
P
>
d s e1
>
>
cij id s e10 0 ¼
cij ðcsþm
1Þ < 0:
: Dbj ¼
i
i63
0 0
ð7Þ
i63
One needs to note that the bond number density in the
relaxed region does not change upon relaxation and
hence ni =nb ¼ 1. Incorporating the above relations into
the measured size-and-shape dependence of the corresponding properties, we can obtain quantitative information that is beyond direct measurement. As discussed,
at the lower end of the size limit (z ¼ 2, c1j ¼ 1 and
c2j ¼ c2j ¼ 0), all the property changes are related to the
behavior of a single bond of the lower-coordinated
atoms.
For instance, with the measured size dependence of
the core-level shift, we can discriminate the crystal
binding (bulk shift) from atomic trapping (core-level
position of an isolated atom) of an isolated atom [10].
We may let Qð1Þ ¼ DE4f ð1Þ ¼ E4f ð1Þ E4f ð1Þ in Eq.
(4) for a Au solid. Least-mean-square linearization of
the measured E4f ð1=Rj Þ of Au nanosolids deposited on
various substrates such as octanedithiol [4], TiO2 [14], Pt
[15], and thiol-caped Au particles [16] gives intercepts a
that correspond to the bulk value of E4f ð1Þ ¼ 84:37
eV [17], and a slopes b that vary with the dimensionality
of the Au solids grown on different substrates. With the
known m ¼ 1 value for pure metals [5], the DE4f ð1Þ and
E4f ð1Þ can be determined as given in Table 1.
Comparing the two theoretical curves for a spherical
dot and a thin plate with the measured profiles in
Fig. 1(a) reveals that the Au solid on octanedithiol
substrate follows ideally the curve of a spherical dot.
The BOLS correlation in Eq. (1) suggests that at
R ¼ 1:5d ¼ 0:43 nm, the Au–Au distance contracts by
30% from 0.288 to 0.200 nm, which is closes to the
value, 0.23 0.04 nm, measured at 4.2 K [18] and
the calculated shorter distance of 0.232 nm [19] as well.
Table 1
The length and energy of the Au–Au bond in the monatomic chain
and the core-level energy of an isolated Au atom obtained from
decoding the E4f ðRj Þ and Tm ðRj Þ of nanosolid Au
b
m
s
E4f (eV)
DE4f ð1Þ (eV)
Dchain (nm)
echain =ebulk
Tm;chain =Tm ð1Þ
503
Au/
octanedithiol
Au/TiO2
3.7804
1
3
)81.504
)2.866
0.2001
1.43
1/4.2
1.5253
1
)81.506
)2.864
Au/
Pt
Au/
thiol
1
)81.504
)2.866
3
)81.505
)2.865
Fig. 1. Comparison of theory with the measured size dependence of (a)
½E4f ðKÞ E4f ð1Þ=½E4f ð1Þ E4f ð1Þ of Au (nanodot) on octanedithiol
[4] and Au (nanoplate) on TiO2 [14] and Pt [15] substrates and thiolcaped [16] with derived information as given in Table 1, and (b)
[Tm ðKÞ Tm ð1Þ=Tm ð1Þ of Au nanosolids on W [21] and C [22]
substrates and embedded in SiO2 matrix [23], showing strongly interfacial effects and dimensionality transition on the Au nanosolid melting. K ¼ R=d: Melting at the lower end of size limit (K ¼ 1:5; z ¼ 2)
corresponds to the situation of a gold monatomic chain.
The cohesive energy per bond increases by 43%, which
coincides with the measured E4f from the smallest Au
nanosolid as shown in Fig. 1(a). The deviation between
theory and measurement from the thinner Au/TiO2 film
is due to the initial island-like growth mode of metal on
oxide surface [14,20], as well as the Au/TiO2 interfacial
effect. Measurements in Fig. 1(a) show the interesting
trend that the core-level drops abruptly from that of an
isolated atom by a maximum (40%) upon the smallest
nanosolid being formed and then the shift recovers in a
R1 fashion to the bulk value when the solid grows from
atomic scale to macroscopic size.
Calibrated with Qð1Þ ¼ Tm ð1Þ ¼ 1337:33 K and
using the same mð¼ 1Þ value used in calculating the
E4f (Rj ), we obtained the theoretical Tm -suppression
curves for different shapes, which are compared in
Fig. 1(b) with the measured size-dependent Tm of Au on
W [21] and on C [22] substrates, and Au encapsulated in
silica matrix [23]. Differing from the core-level shift data,
the melting profiles show that at the smaller size, the
Au/W interface promotes more significantly the melting
of Au (super-cooling) than the Au/C interface. The silica
matrix causes slightly super-heating of the embedded Au
solid compared with the rest two substrates. Besides, the
Tm measurement is a thermodynamic process that may
504
C.Q. Sun et al. / Acta Materialia 52 (2004) 501–505
affect the results to some extent. Nevertheless, the theoretically predicted melting curves merge at the lower
end of the size limit, K ¼ 1:5 with 75% suppression.
Therefore, it is anticipated that thermal rupture of the
Au–Au chain occurs at 320 K (to Tmb zib c1
i ¼
2Tmb =ð12 0:7Þ ¼ Tmb =4:2), much lower than the gold
bulk melting point (1337.33 K).
According to Eq. (7), the compressibility/extensibility
of the monatomic chain is 0.5 ( ¼ 0:72 1) times the
bulk values (for a metallic monatomic chain, s ¼ 1,
m ¼ 1). The shortened bond is twice stronger
2
(strength ¼ ei =di / c2
2), agreeing with prei ¼ 0:7
dictions of [24]. This means that more force is required
for stretching or compressing by the same length a single
bond in the atomic chain compared to the force needed
to stretch the same single bond in the bulk by the same
amount, in general. Whereas, high-pressure X-ray diffraction revealed that the lattice constant of small alumina solid is easier to compress than larger ones [25],
which disagrees with the current prediction. However, it
should be noted that the temperature (Tsm ) of a solid in
the semi-solid state is always lower than the Tm that is
suppressed by reducing particle size. At Tsm , the interatomic bond can be stretched and unfolded more easily,
and hence, the plasticity or the plastic energy of a suspended atomic chain is much higher than the atomic
chain embedded in the solid. Meanwhile, the heat released from bond unfolding and stretching should raise
the actual temperature of the specimen as well. Therefore, relation (Eq. (4)) for the constant temperature
compressibility or extensibility no longer holds for
atomic chains or nanowires of which the Tsm < Tm is
suppressed by size reduction.
In reality, the breaking limit of a bond is determined
by
Z dp ðT Þ
Eplastic ðT Þ ¼
f ðxÞ dx;
ð8Þ
dk ðT Þ
where dk ðT Þ and dp ðT Þ are the corresponding elastic and
plastic limit at T. The Eplastic remains constant at a given
temperature, while dp ðT Þ depends on both f ðxÞ and T .
Any fluctuation of the T or the stretching force f ðxÞ
applied in measurement will affect the measured dp ðT Þ,
and therefore, it understandable now why the roomtemperature Au–Au breaking limit (vary from 0.29 nm
[26], 0.36 nm (30%) [27], 0.35–0.40 nm [28] to even a
single event of 0.48 nm [29]) is much longer than that at
4 K (0.23 0.4 nm).
In the current BOLS approach, we assumed that
contribution from particle–substrate interaction is negligible. Actually, the interfacial reaction modifies the m
value slightly [3]. Matching predictions to measurements
in Figs. 1(a) and (b) evidence the validity of the assumption. Taking the temperature effect on the total
energy of a single bond into account would be significant in understanding the high ductility and the exten-
sibility/compressibility of metallic nanowires. Further
investigation is in progress.
We have thus incorporate the effect of atomic CNimperfection to the core-level shift and the melting point
of gold nanosolids, which derives a new method calibrating the DE4f ð1Þ and the E4f ð1Þ of an Au atom
isolated from the solid with a prediction of the length,
the strength, the extensibility and the thermal stability of
the Au–Au single bond under the conditions with and
without external stimuli, by simultaneously decoding the
known size dependence of the E4f ðRj Þ and Tm ðRj Þ.
Findings provide consistent insight into the CN-imperfection-enhanced binding intensity, mechanical strength,
the suppressed thermal stability, and the compressibility/extensibility of gold nanosolids, which could be extended to the thermal and mechanical behavior of other
metallic nanowires. Practice should be a helpful exercise
for information of bonding identities and the single
electron energy of an isolated atom that determine the
performance of nanosolid materials.
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