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Schottky Defects in Nanoparticles G. Guisbiers* UCL, Institute of Mechanics, Materials and Civil Engineering, 2 Place Sainte Barbe, B-1348 Louvain-La-Neuve (Belgium) ABSTRACT: Vacancies play a major role in the electrical and thermal transport as well as the mechanical behavior of materials. To understand the processes occurring in nanomaterials during heat treatment and mechanical deformation, the size effects on the vacancy formation energy and entropy have to be considered. As these material properties are hardly measurable experimentally, particularly in nanoparticles, a theoretical model calculating the size-dependent vacancy formation energy and entropy has been developed in this study. It has been found that size reduction makes the vacancy much easier to form; then the vacancy concentration increases when size reduces and temperature increases.
1. INTRODUCTION The materials properties of nanostructures are still investigated extensively today both theoretically and experimentally due to their scientific and industrial importance.1,2 In particular, nanoporous materials are more used compared to other nanomaterials due to the fact that they are good host materials in nanotechnology.3 At the nanoscale, the materials properties are modified from their corresponding bulk behavior due to the high surface area to volume ratio of nanostructures and to the possible quantum effects appearing at these scales.4 Defects as vacancies (also named Schottky defects) play an important role in the electrical and thermal transport as well as the mechanical behavior of materials. The size effects on the vacancy formation energy and entropy are the keys to understand the processes occurring in nanomaterials during heat treatment and mechanical deformation. However, it is difficult to measure experimentally nanomaterials properties2 like the vacancy formation energy5 and the vacancy formation entropy.6 Therefore, to overcome experimental difficulties, modeling appears to be an appropriate alternative.7 To model this size effect, thermodynamics is used; however, its validity at the nanoscale has to be discussed.8 From the theory of thermodynamics fluctuations, the size limit where thermodynamics applies can be estimated. Indeed, the thermal fluctuation is proportional to the reciprocal square root of the total number of particles present in the system, δT/T = ∼1/(N)1/2. In the case of a spherical nanoparticle with a diameter equal to 4 nm, δT/T = ∼2% (knowing that the density of solid and liquids, n = N/V, is ∼1029 m-3). One generally considers that thermodynamics is valid when the number of atoms is large, which occurs when the thermal fluctuations are small; that is it is valid until sizes around ∼4 nm. r 2011 American Chemical Society
Therefore, in this article, we develop a theoretical model to estimate the size effect on the vacancy formation energy, the vacancy formation entropy and the vacancy concentration. In section 2, a universal equation is used to describe size and shape effects on different materials properties. In section 3, the size effect on the lattice vibration frequency is discussed. In section 4, the size effect on the vacancy concentration, the influence of the vacancy concentration on the vibration frequency and the melting temperature are studied. Finally, section 5 deals with the conclusions.
2. UNIVERSAL RELATION In previous articles,9,10 we have developed a universal relation (eq 1), which can describe the size and shape effects on many materials properties (melting temperature, Debye temperature, superconducting temperature, Curie temperature, cohesive energy, activation energy of diffusion, and vacancy formation energy), based on the spin of the particles involved in the considered material property. The size/shape-dependent material property, ξ, to bulk material property, ξ¥, ratio is given by:9,10 ξðD, SÞ=ξ¥ ¼ ð1 - Rshape =DÞ1=ð2SÞ
ð1Þ
where Rshape is the parameter quantifying the size effect on the material property and depending on the shape of the nanostructure. This parameter is defined by Rshape = [D(γs - γl)/ΔHm, ¥](A/V) where A/V is the surface area to volume ratio, ΔHm,¥ is the bulk melting enthalpy and γs(l) the surface energy in the solid Received: August 25, 2010 Revised: December 24, 2010 Published: January 20, 2011 2616
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Figure 1. Size-dependent Debye temperature (Debye frequency) to the bulk Debye temperature (Debye frequency) ratio versus the diameter of the nanoparticle. Insert: Size-dependent Einstein temperature (Einstein frequency) to the bulk Einstein temperature (Einstein frequency) ratio versus the diameter of the nanoparticle. Experimental points, from ref 19 are indicated for selenium nanoparticles.
(liquid) phase. D is the size of the nanostructure. S is equal to 0.5 or 1 if the particles involved in the considered phenomena follow a statistic of Fermi-Dirac or Bose-Einstein, respectively. Vibration is described by phonons following a Bose-Einstein statistic (S = 1); then eq 1 becomes eq 1a9-11 for the Debye temperature (temperature corresponding to the maximal energy which can excite lattice vibrations),: TDebye ðDÞ=TDebye, ¥ ¼ ð1 - Rshape =DÞ1=2
ð1aÞ
Where TDebye and TDebye,¥ are the size-dependent Debye temperature and the bulk Debye temperature, respectively. The melting temperature and the vacancy formation energy are both related to electrons following a Fermi-Dirac statistic (S = 1/ 2) and then eq 1 becomes eq 1b and 1c, respectively:9-11 Tm ðDÞ=Tm, ¥ ¼ 1 - Rshape =D
ð1bÞ
Ev ðDÞ=Ev, ¥ ¼ 1 - Rshape =D
ð1cÞ
where Tm and Tm,¥ are the size-dependent melting temperature and the bulk melting temperature, respectively. Ev and Ev,¥ are the size-dependent vacancy formation energy and the bulk vacancy formation energy, respectively. As pointed out by Gladkikh and Kryshtal,12 there exists experimentally a linear relation between the vacancy formation energy and the melting temperature, which validates our theoretical approach. Furthermore, the creation of vacancies implies breaking chemical bonds. Therefore, there is a relation between the bulk cohesive energy of a material and its bulk vacancy formation energy.13
3. VIBRATION FREQUENCY IN NANOPARTICLES Atomic vibration is of high interest because the behavior of phonons influences the thermal, electrical, and optical properties in nanomaterials. The large surface area to volume ratio of a nanomaterial strongly influences the electrical and thermal
properties due to the scattering of phonons. It also influences the optical properties because of the introduction of surface polarization and surface states.14 As the Debye temperature is defined as ωDebye = kTDebye/p (where k and p have their usual meaning)15 and as the Debye frequency is directly proportional to the Einstein frequency,16 we have the same size effect on these both properties (eq 1), therefore we can forget the subscripts Debye and Einstein and simply write the vibration frequency by ω. ωðDÞ ¼ ω¥ ð1 - Rshape =DÞ1=2
ð2Þ
Therefore, the vibration frequency decreases with size as also predicted by other theoretical models14,16-18 and in agreement with available experimental data.16,19 Indeed, in Figure 1, we have represented the Debye temperature and Debye frequency of selenium nanoparticles whereas in the insert of Figure 1 we have plotted the Einstein temperature and Einstein frequency of these selenium nanoparticles. It is clear that there is a fairly well agreement between the theoretical curves obtained with the universal equation (eq 1) and the available experimental data.
4. VACANCY CONCENTRATION IN NANOPARTICLES 4.1. Size Effect on the Vacancy Concentration. The vacancy, also named Schottky defect, is the simplest defect that can be created in a material; it corresponds to a lack of atom in the lattice.15 The formation of a vacancy can be considered as the removal of one interior atom from the crystal and the replacement of the atom on the crystal surface. Vacancies in materials are often taken to be equivalent to internal surfaces; therefore, the role of the surface may be taken similar to that of defects.20 Therefore, it is natural to expect that decreasing the size of a material will increase the vacancy concentration due to an increasing surface area to volume ratio. The two parameters governing the temperature dependence of the vacancy concentration are the vacancy formation energy and the vacancy formation entropy. Therefore, we will consider the size effect 2617
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on these two materials properties to write the vacancy concentration at the nanoscale. Let us remember first that at thermodynamic equilibrium, the vacancy concentration in bulk materials is given by:5,15 Sv , ¥ - Ev , ¥ cv, ¥ ðTÞ ¼ exp exp ð3Þ k kT where Sv,¥ and Ev,¥ are the bulk vacancy formation entropy and the bulk vacancy formation energy, respectively. k is the Boltzmann’s constant and T is the temperature. For bulk materials, the vacancy formation energy and the vacancy formation entropy are always constant, which is not the case with nanomaterials.11,21 Therefore, by replacing Ev,¥ and Sv,¥ with Ev and Sv respectively into eq 3, we obtain the expression of the vacancy concentration into nanomaterials: Sv ðDÞ - Ev ðDÞ cv ðT, DÞ ¼ exp exp ð4Þ k kT As there seems to exist a linear relation between the vacancy formation energy and the entropy22 (Figure 2), it is natural to assume a similar size effect on the vacancy formation entropy as for the vacancy formation energy (eq 1c), Sv(D)/Sv,¥ = 1 - Rshape/D. By considering the size effect on the vacancy formation energy and entropy and combining it with eq 4, we can express the sizedependent vacancy concentration by the following relation: - Sv, ¥ Rshape Ev, ¥ Rshape cv ðT, DÞ ¼ cv, ¥ ðTÞ exp exp k D kT D ð5Þ Unfortunately, we cannot compare eq 5 with any experimental data because they are inexistent. Theoretically, a similar equation has been obtained previously by Qi et al.,21 who considered the size effect on the vacancy formation energy but neglected the size effect on the vacancy formation entropy. Nevertheless, these authors could confirm the correctness of eq 1c to describe the size effect on the vacancy formation energy. Moreover, eq 5 can be linked to the Gibbs-Thomson equation23 expressing the equilibrium of a two-phase system at a curved interface. Indeed, in a pure system, the only possible kind of solute atoms is point defects, that is interstitial atoms or vacancies. Therefore, the corresponding enhancement of vacancies in the particle can be expressed by the Gibbs-Thomson equation,23 Δcv(T,D) = 4Ωγcv,¥(T)/kTD, where Ω is the atomic volume and γ is the surface energy. By expressing the polynomial form of the exponential in eq 5, we get cv(T,D) cv,¥(T) ≈ cv,¥(T)[-Sv,¥TRshape þ Ev,¥Rshape]/kTD. Thus, we have -Sv,¥TRshape þ Ev,¥Rshape ≈ 4Ωγ, which is valid for most of the materials because generally Ω ≈ 10 cm3/mole,15 γ ≈ 2J/ m2,24 Sv,¥/k ≈ 5,6 Ev,¥ ≈ 1 eV6,20, and Rshape ≈ 1 nm.25 4.2. Experimental Evidence. Point defects like vacancies are thermodynamically stable because they increase the entropy of the particle.5 When a vacancy concentration gradient appears, it activates diffusion (1st Fick’s law15), which proves that vacancies and diffusion are closely related. At the nanoscale, diffusion in nanomaterials is considerably enhanced compared to bulk materials,26-28 therefore it suggests that the vacancy concentration increases when the size of structures is reduced because vacancies facilitate the displacement of atoms.
Figure 2. (a) Bulk vacancy formation entropy, Sv,¥/k, is plotted versus the bulk vacancy formation energy, Ev,¥. (b) The product Sv,¥Rsphere/k is plotted versus the product Ev,¥Rsphere. (c) The product Ev,¥R is plotted versus the bulk melting temperature, Tm,¥, of all the elemental materials indicated in Table 1.
Other mechanical properties are strongly affected by point defects. Indeed, the presence of vacancies in the crystal lattice will modify the lattice structure around the vacancies and then result in a decrease of the lattice parameter.23,29-31 The size effect on the 2618
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Figure 3. (a) Size-dependent vacancy concentration to the bulk vacancy concentration ratio of some elemental materials indicated in Table 1, versus the size, at T = 293 K. The sphere is the considered shape. (b) Vacancy concentration versus kT for a 10 nm spherical nanoparticle of some elemental materials indicated in Table 1. (c) Vacancy concentration of a spherical gold nanoparticle versus its size and temperature. The temperature varies between T = 293 and 1000 K.
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The Journal of Physical Chemistry C lattice parameter is attributed to the lattice strain induced by the formation of vacancies.32 The creation of vacancies also reduces the force constants seen by the atoms neighboring the vacant site and thereby decreases the local vibration frequencies5,33 in agreement with our eq 2. Furthermore, the hardness and the yield strength of the material typically increase with decreasing size, a phenomenon known as the Hall-Petch effect.34,35 The size-dependent hardness enhancement at a temperature well below the melting temperature is due to the bond length contraction and the associated bond strength gain.36 Recently, it has been shown by nanoindentation, microcompression and molecular dynamics simulations that nanoporous gold can be as strong as bulk gold.37 Furthermore, it has been shown that increasing vacancy concentration results in an increase of the electrical resistivity.5,38 Indeed, electrons are scattered by vacancies, impurities, phonons, grain boundaries, surfaces, and interfaces; resulting in a less efficient electrical transport. This also has an impact on the thermal transport by decreasing the thermal conductivity.5,39 Indeed, thermal transport is carried out by electrons and phonons that undergo electrons scattering and lattice softening. Many couples of immiscible elements have been formed into solid solutions through ball-milling when the grains are milled down to the nanometer scale.23 This observation strongly supports the theory presented in the section 4.1 that the equilibrium solubility between two components (for a pure system, vacancies and atoms; for an alloy, atoms A and B) increases at the nanoscale. Furthermore, it has also been confirmed by Li40 that miscibility should increase with increasing porosity due to diminishing internal stress. 4.3. Discussion. In parts a and b of Figure 2, we have plotted Sv,¥ versus Ev,¥ and Sv,¥Rsphere versus Ev,¥Rsphere, respectively. Globally, for the materials considered, there exists a linear relation between these two material properties; however, it is clear from part b of Figure 2 that the vacancy formation entropy of indium seems too high and has to be remeasured experimentally. In part c of Figure 2, the graph, Ev,¥Rsphere versus Tm,¥, shows a linear relation between these two parameters. Therefore, according to parts b and c of Figure 2, a linear relation between Sv,¥Rsphere and Tm,¥ is also expected. From eq 5, it is clear that we cannot exhibit any particular trend on the vacancy concentration based on the distinction between low and high melting point materials. This is illustrated in part a of Figure 3, where the size-dependent vacancy concentration to bulk vacancy concentration ratio of some low and high melting point materials is plotted versus the diameter, at T = 293 K. At room temperature, in a bulk silicon material, the bulk vacancy concentration is around 10-42; therefore, according eq 5 in a 10 nm spherical silicon nanoparticle, the vacancy concentration is around 10-37. The nanoscaled melting temperature of this nanoparticle is ∼1491 K. At this temperature, the bulk vacancy concentration is around 10-7 and the vacancy concentration in the nanoparticle is around 10-6. In part b of Figure 3, the vacancy concentration of a 10 nm spherical nanoparticle of some elemental materials indicated in Table 1 is plotted versus the thermal energy, kT. As expected, the vacancy concentration increases with temperature. In part c of Figure 3, the vacancy concentration of a spherical gold nanoparticle is plotted versus its size and temperature. It is clear with this figure that the vacancy concentration reaches high values (∼10-3) for small sizes (∼5 nm) and high temperatures (T = ∼1000 K). 4.4. Influence of the Vacancy Concentration on the Vibration Frequency. In the case of vacancy defects, one will expect a softening of the vibrations of their neighboring atoms due to the missing coupling to the vacant site. Indeed, atoms near a vacancy tend to explore a larger free volume in their vibrations and tend to
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Table 1. Material Properties of Some Metals and Semiconductors; the Bulk Melting Temperature, Bulk Vacancy Formation Energy, Bulk Vacancy Formation Entropy and Rsphere Parameter Are Indicated material
Tm,¥ (K) a
Ev,¥ (eV)b
Sv,¥/kc
Rsphere (nm) d,e
Al
933
0.76
2.4
1.56
Sc
1814
1.31
Ti
1941
1.55
V
2183
1.77
4.8
Cr
2180
1.85
6.34
Fe
1811
1.61
Co Ni
1768 1728
1.49 1.55
2.40 1.78 1.56 1.50 2.46 1.25
2.49 1.56
Cu
1358
1.12
1.5
1.63
Zn
693
0.56
0.5
1.32
Y
1795
1.45
Zr
2128
1.75
Nb
2750
2.65
4.2
1.51
Mo
2893
2.24
5.7
1.06
Ru Rh
2607 2237
2.20 1.92
2.34 1.91
1.03 1.33
Pd
1828
1.31
Ag
1235
1.13
1.5
1.68 1.88
Cd
594
0.46
0.5
2.36
In
430
0.49
5.0
3.79
Sn
505
0.55
2.52
Hf
2506
2.12
Ta W
3290 3693
2.86 3.10
Re
3458
2.70
1.50
Os
3306
2.70
0.83
Ir
2719
2.31
Pt
2042
1.45
4.5
1.52
Au
1337
0.95
0.7
1.83
Hg
234
0.12
Tl Pb
577 601
0.39 0.38
2.04 1.63
6.5
1.56 1.26
0.97
4.41 1.6
3.55 2.90
Si
1687
2.50
4.4
1.16
Ge
1211
3.1
4.9
1.72
Se
494
0.80
1.02
Ref 43. b Ref 20. c Ref 6. d Ref 25. e Rsphere varies for a sphere, a cylinder, and a film in the following ratio 3:2:1.44 a
vibrate at lower frequencies.33 To write explicitly the vibration frequency as a function of the vacancy concentration, let us combine eq 2 with eq 5 and get: " !#1=2 kT cv ðT, DÞ ln ð6Þ ωðT, DÞ ¼ ω¥ 1 Ev, ¥ - Sv, ¥ T cv, ¥ ðTÞ From this equation, it is clear that the lattice vibration frequency decreases with the vacancy concentration. The physical interpretation of this behavior is that the creation of a vacancy leads to a decreasing number of bonding states and then to a softening of the vibration lattice frequency because the phonon frequency is modified by a change in the number of these states. In Figure 4, the vibration frequency of a silicon nanoparticle is plotted versus the vacancy concentration ratio. 2620
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’ ACKNOWLEDGMENT The author thanks the Belgian Federal Science Policy Office (BELSPO) for financial support through the “Mandats de retour” action. ’ REFERENCES
Figure 4. Size-dependent vibration lattice frequency of a spherical silicon nanoparticle versus its vacancy concentration ratio. Insert: Sizedependent melting temperature of a spherical silicon nanoparticle versus its vacancy concentration ratio.
4.5. Influence of the Vacancy Concentration on the Melting Temperature. Since Lindemann’s work in 1910,41 many
authors42 have linked melting with thermal vibrations, as Wautelet with his surface phonon instability model.20 In this model, Wautelet assumed a linear relation between the mean phonon frequency and the vacancy concentration. From eq 6, we can see that this relation is more complicated than just a linear one. eq 6 can be considered as an improvement of the Wautelet’s hypothesis. Therefore, by combining eq 5 with eq 1b, the nanoscaled melting temperature is obtained as function of the vacancy concentration: " # kT cv ðT, DÞ ln ð7Þ Tm ðDÞ=Tm, ¥ ¼ 1 Ev, ¥ - Sv, ¥ T cv, ¥ ðTÞ Applying eq 7 to a bulk material, that is cv = cv,¥, the bulk melting temperature is retrieved, Tm = Tm,¥. The insert of Figure 4 represents the size-dependent melting temperature versus the vacancy concentration ratio.
5. CONCLUSIONS In this article, we have calculated the size effect on the vacancy formation energy, the vacancy formation entropy and the vacancy concentration into nanomaterials through a top-down approach by using classical thermodynamics. From this study, it has been found that the vacancy concentration increases when size reduces and temperature increases. Furthermore, the size/temperature-dependent vacancy concentration expression could be linked to the Gibbs-Thomson equation. Reduction in the vibration frequency has also been predicted with size reduction in agreement with available experimental data. Moreover, a relation between the melting temperature and the vacancy concentration has been found. Finally, this model helps to understand the apparition of porosity in nanomaterials. ’ AUTHOR INFORMATION Corresponding Author
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*To whom correspondence should be addressed. E-mail:
[email protected]. 2621
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