NANO LETTERS
Melting Temperature, Brillouin Shift, and Density of States of Nanocrystals
2002 Vol. 2, No. 5 519-523
Keshav N. Shrivastava* School of Physics, UniVersity of Hyderabad, Hyderabad 500046, India Received January 20, 2002; Revised Manuscript Received February 15, 2002
ABSTRACT The amplitude of displacement of atoms due to scattering at the surface is calculated. It is found that the zero-point vibrational correction to the mean-square displacement varies as the inverse square of the nanocrystal size and the temperature-dependent part also varies with the nanocrystal size. The melting temperature is found to depend on the nanocrystal size so that smaller melting temperature is obtained for smaller nanocrystals. The phonon−phonon scattering is found to shift the Brillouin line such that as the crystal size reduces, it moves toward the Rayleigh line. The phonon density of states is found to be larger for nanocrystals than for the bulk crystals as it depends on the inverse square of the nanocrystal size.
I. Introduction. Recently, it has been realized that modern electronic devices may work much better if made from nanometer size crystals. The nanocrystals can be free from defects and may have a single domain. However, physical properties have to be determined before they can be used in devices. It is reported by us1 that the Debye T3 law of specific heat is changed in going from the bulk crystal to a nanocrystal because of the scattering at the surface. It has been known for some time that the melting temperatures of nanocrystals are much smaller than those of the bulk crystals. For example, the melting temperature of CdS has been measured2 for particles of different sizes. The crystals of size ∼3 nm melt at about 1200 K while those of size 1.5 nm melt at 600 K. The relaxation in 4.5 nm size CdSe is faster than in the bulk crystal so that nanocrystals respond to the external pressure faster than the bulk crystals. Therefore, pressure induced symmetry changes and hence phase transitions can be measured more accurately with nanocrystals than with bulk crystals.3 In some cases, one starts with the known nanocrystal size and calculates the properties. In other cases, the physical properties are first measured and from that the nanocrystal size is determined. The unit cell dimensions in the microcrystals are different from those in the bulk crystal. In solid state theory, the particle size is usually infinite and hence calculations for finite-particle size have not been performed. In this letter, we calculate the mean-square displacement of an atom in a nanocrystal due to scattering of phonons of wavelength of the order of the size of the nanocrystal. This displacement shows that the melting temperature is linearly * To whom correspondence should be addressed. Tel: +91-40-3010811. Fax: +91-40-3010145. E-mail:
[email protected]. Webpage: www.htcs.org. 10.1021/nl020287t CCC: $22.00 Published on Web 03/12/2002
© 2002 American Chemical Society
proportional to the size of the nanocrystal. We calculate the shift of the Brilloiun line and find that it shifts toward the Rayleigh line in going from the bulk to the nanocrystal. We also find that the phonon density of states increases as the inverse square of the nanocrystal size. The calculations of the melting temperature are compared with the experimental data of Si nanocrystals, the calculated Brillouin shift is compared with the data for Sn and the phonon density of states are compared with the neutron scattering data. In all of the three cases, the calculated dependence on the particle size is in accord with that measured. II. Melting. We consider that the material consists of an ensemble of nanocrystals. The size of the nanograin is d. The sound waves are scattered by the nanocrystal if the distance between two sides of a nanocrystal is an integer multiple of the wavelength. In the case of X-rays, the diffraction is determined by the Bragg’s law. The distance between atoms is of the order of the wavelength, λ, so that the scattering occurs at, 2d sin θ ) nλ, where θ is the angle which the incident ray makes from the normal to the surface, parallel to which the next atomic layer is located at a distance, d, from the first surface and n is an integer. Usually, the first-order diffraction spots are seen for which n ) 1. The second-order spots also become visible in some cases with n ) 2. We consider the sound waves or phonons in the solid which may be scattered by the surface boundaries of the nanocrystals. The wavelength of the phonons or that of sound waves is λ ) V/ν where V is the sound velocity and ν is the frequency of the lattice wave. The phonon frequency which is scattered by the nanocrystals is ω ) 2πnV/(2d sin θ)
(1)
where n is an integer, n ) 1 or 2. The lower value of the
phonon frequency is obtained by taking sin θ ) 1 as ω(1)) πV/d
(2)
where V is the sound velocity, V = 5 × 105 cm/s. For the frequency ν ) 1012 Hz, we get d ) 5 π nm for the particle size. The high frequency can be cutoff by replacing sin θ by an average value. This is an important aspect of the theory because ω diverges at θ ) 0. Therefore, we replace sin θ by its average value, 〈sin θ〉 ) 1/2. For n ) 1, we obtain a limiting value of the phonons which are scattered by the boundary of the nanocrystal as
δR ) (p/2Mωk)1/2(ak + a-k†)
(4)
where M is the mass of the crystal, ωk the phonon angular frequency and k is the wave vector. The operators ak†(ak) create (annihilate) a phonon of wave vector k. The correlation function is given in terms of the Bose-Einstein distribution (ak + a-k†)(ak′ + a-k′†) ) (2nk + 1) δk,k′
(5)
with nk ) [exp(pω/kBT) - 1]-1. For a three-dimensional solid, the square of the above displacement is determined from 〈u2〉 ) Σk(p/2Mωk)(2nk +1)
(6)
We multiply the above by [4πV/(2π)3]3k2dk and integrate, instead of the sum to find
∫ (p/2Fω)(2n + 1)[4π/(2π) ]3k dk 3
2
∫ ω-1(2n +1)k2dk
〈u2〉 ) {6πp/(2π)3F}
[3p/2π2FV3]
∫oΩ [exp(pω/kBT) - 1]-1ωdω
(11)
∫ (2n + 1)ωdω
pω/kBT ) x, ωdω ) (kBT/p)2 xdx so that 〈u2〉 ) 3pΩ2/(16π2FV3) + [3p/2π2FV3](kBT/p)2
∫oX (ex-1)-1xdx
(12)
where X ) pΩ/kBT. Therefore, 〈u2〉 varies as T2 at low temperatures in a three-dimensional solid. We assume that ω(1) and ω(2) are both outside the range of the phonon spectrum so that there is scattering of phonons from the nanocrystal. The contribution to the square of the displacement in the nanocrystal may be written as 〈u2〉 ) [6πp/{(2π)3FnVn3}]
ω(2) (2n + 1)ωdω ∫ω(1)
(13)
where Fn is the mass density per unit volume of the nanocrystal, and Vn is the velocity of sound. The eq 13 has two terms, one depending on temperature and the other a zero-point contribution. The zero-point contribution is easily integrable and its value is (14)
(7)
(8)
We use the dispersion relation, k ) ω/V, where V is the sound velocity to obtain (9)
which can be written into two terms, one the zero-point vibrational contribution and the other, a temperature-dependent part 520
〈u2〉 ) [3pΩ2/{16π2FV3}]+
〈u2〉nano(0) ) [3p/(8π2FnVn3]{ω(2)2 - ω(1)2}
where F ) M/V is the mass density. A factor of 3 has been introduced to add the contributions of two transverse and one longitudinal branches of the phonon spectrum so that
〈u2〉 ) [6πp/{(2π)3FV3}]
We assume a continuous phonon spectrum of frequencies from zero to the Debye cutoff frequency, Ω, so that the above can be written as
(3)
Thus, we have determined the range of frequencies introduced by the nanocrystal in addition to the usual phonon frequency spectrum in the bulk crystal. In a harmonic oscillator, the displacement of an atom is given by
〈u 〉 )
(10)
We make the change of variables
ω(2) ) 2πV/d
2
∫ ∫2{exp(pω/kBT) - 1}-1ωdω
〈u2〉 ) [6πp/{(2π)3FV3}][ ωdω +
We substitute the values of the two limiting frequencies from eqs 2 and 3 into eq 14 to obtain 〈u2〉nano(0) ) 9p/(8FnVnd2)
(15)
Therefore, for V ) Vn, the above expression shows that the mean square displacement of atoms is proportional to the inVerse square of the nanoparticle size, d. The temperaturedependent part of eq 13 is given by 〈u2〉nano(T) ) [3p/(4π2FnVn3)](kBT/p)2
x(2) x (e -1)-1xdx ∫x(1)
(16)
where x(1) ) pω(1)/kBT and x(2) ) pω(2)/kBT. We use eqs 2 and 3 to find that x(1)) pπV/(kBTd) and x(2) )2pπV/ (kBTd). At very low temperatures, x . 1, we can ignore 1 Nano Lett., Vol. 2, No. 5, 2002
in the denominator, i.e., ex . 1 so that I1 )
which means that larger nanocrystals have larger melting temperature. Let us consider the zero-point contribution also. Then by using the Lindemann criterion we obtain
x(2) x(2) (ex-1)-1xdx = ∫x(1) xe-xdx ) ∫x(1)
-exp[-x(2)][1 + x(2)] + exp[-x(1)][1+x(1)] (17)
cao2 - (a1/d2) ) (a4/d) Tnm
(28)
Therefore eq 16 becomes 〈u 〉nano(T) ) 2
[3p/{2π2FnVn3}](kBT/p)2(pπV/kBTd)exp(-pπV/kBTd) (18) where we have written only the largest of the four terms. Using eqs 15 and 18, the displacement of an atom in a nanocrystal is found to become 〈u2〉nano(0) + 〈u2〉nano(T) ) a1/d2 + a2(T/d)exp[-a3/(Td)] (19) where the constants are given by a1 ) 9p/8FnVn a2 ) 3kBV/(2πFnVn3)
(21)
a3 ) pπV/kB
(22)
The displacement thus varies as 1/d2 at low temperatures and then there is a term of the form of (T/d)exp(-a3/Td). Thus, the particle size plays an important role at low temperatures. At high temperatures, x , 1, so that it is sufficient to write ex ) 1 + x, in the denominator of the integrand of eq 16 so that x(2) (ex - 1)-1xdx = pπV/(kBTd) ∫x(1)
(23)
and the displacement becomes 〈u2〉nano(0) + 〈u2〉nano(T) ) (a1/d2) + a4(T/d)
(24)
a4 = 3kB/(2πFnVn2)
(25)
where
We use the Lindemann criterion to determine the melting temperature of a nanocrystal. It is assumed that the lattice melts when 〈u2〉 ) c ao2
(26)
where ao is the usual lattice constant and c = 0.1. We apply eqs 26 to 24, so that by ignoring the quantum mechanical zero-point term, we obtain the melting point as Tnm ) 0.1 ao2d/a4 Nano Lett., Vol. 2, No. 5, 2002
(27)
for the melting temperature of the nanocrystal. Now, the value of Tnm is slightly reduced compared with the value given by eq 27, and whereas Tnm versus d is linear in eq 27, has now ceased to be linear and has acquired a negative d-1 term. The Tnm(d) as a function of the size of the nanocrystal slightly bends Tnm ) (d/a4)[0.1 ao2 - (a1/d2)]
(29)
For small d, the correction term is large. We compare eq 27 with the experimental data. The melting temperature, Tnm, of Si nanocrystals as a function of size of the particles has been measured by Goldstein.4 The data are d ) 1 nm, Tnm ) 400 K; d ) 3 nm, Tnm ) 800 K, and for d ) 4 nm, Tnm ) 1000 K, etc. Therefore, the predicted linearity of the melting temperature, Tnm as a function of particle size, d, is in reasonable agreement with the experimental data. In the case of CdS, for d ) 1.5 nm, Tnm ) 600 K and for d ) 3.5 nm, Tnm = 1200 K. This measured linearity2 of the melting temperature is also in agreement with the calculated expression 27 for small particle size. Johari5 has developed the thermodynamic theory of melting of submicron size crystals in terms of the surface and volume energy and found that smaller crystals melt at lower temperatures. In our theory, scattering occurs at the surface so that there is complete consistency with the surface energy and hence both the theories give smaller melting temperatures for smaller crystals. The ensemble of nanocrystals is a very complex system which differs considerably from the infinite perfect crystal. However, in a real sense, a nanocrystal is more likely to be perfect than a large crystal. Therefore, plane waves can propagate through the nanocrystal with scattering at the boundary. The total energy obtained by calculating the lattice energy of one nanocrystal and multiplying it by the number of crystals, can give quite a good value of the lattice heat. However, it does not include the lattice of the substrate or that of the gel materials which may be present. Therefore, one must find the lattice energy of the nanocrystals and that of the gel medium separately. The energy of the combined system is just the sum of the two values. In the case of experiments, the specific heat of the gel has to be subtracted from that of the total system, to obtain the value for the nanocrystals. Our theory is limited by two frequencies given by eqs 2 and 3, one of these is the result of the finite size of the nanocrystal and the other is required to have finite lattice energy. It is quite reasonable to believe that large energies are not present in the system. The mean square displacement varies as the inverse square of the size of the nanocrystals so that it is large, for smaller crystals. However, this effect has a tendency to saturate once the size becomes larger than about 20 unit cells. Therefore, 521
in about 10 cells, we expect that the amplitude in the nanocrystals is about twice that in the bulk crystals. In the case of large distribution in the sizes, smaller crystals melt at lower temperatures so that there is a liquid and the larger crystals though have a larger melting temperature, may get desolved and not wait for higher temperatures to melt. Therefore, a liquid may be seen at lower temperatures than is predicted by the theory for larger crystals. There is a problem of contact area between two nanocrystals so that the actual size is larger than that of single crystals. The theory assumes the size of the nanocrystals so that large contact area generates larger sizes and hence smaller frequencies. There is a problem of compactness of the particles without touching, the effect of which is to change the number of nanocrystals per unit volume. The larger the number of nanocrystals, the larger is the lattice energy. The Lindemann’s criterion for melting makes use of the amplitude of oscillations of the atoms which are calculated in the harmonic approximation. The phonon-phonon scattering gives rise to anharmonicity which leads to the thermal expansion. To get an idea of these anharmonic processes we shall study the shift of the Brillouin line in the next section. III. Brillouin Scattering. In ref 1, we have shown that there is a kink in the frequency versus wave vector dispersion relation when the wave vector becomes equal to π/d, where d is the particle size. The frequency goes to zero when the wave vector, k ) 0. Actually, in crystals, there is an anisotropy, so that ω does not become equal to zero at k ) 0. Instead of zero frequency at zero wave vector, a finite value is found, i.e., ω ) ωB at k ) 0. The velocity of light, c, is very large, compared with the velocity of sound, V. Therefore, ω ) ck has very large slope and cuts the phonon dispersion, ω ) ωB + Vk, very near ωB. Therefore, there is a peak when the light frequency is ωR ( ωB where ωR is the Rayleigh frequency, and ωB is called the Brillouin frequency. We calculate the shift of the Brillouin line as a function of particle size, d. We consider the phonon-phonon scattering, for which good calculations of the self-energy are obtainable from the thermodynamic Green functions.6,7 The Hamiltonian for such a system can be written as H ) Σq pωq aq†aq + Σkqφk,q(ak-q†aq†ak + h.c.) (30) where pωq is the single-particle phonon energy, and φk,q determines the interaction, with h.c. as the hermitian conjugate of the previous term. For this type of interaction, the single-phonon energy is pωq′ ) pωq- Σq |φk,q|2(pωq)-1(2nq+1)
(31)
where nq)[exp(pωq/kBT) -1]-1 is the Bose-Einstein factor for phonons. The shift of the Brillouin line is therefore determined by δνB )
∫ |φk|2(2nk+1)[4πV/(2π)3]3k2dk
(32)
where we have changed the summation into integration with 522
V as the volume of the sample with limits of integration as appropriate to a crystal of infinite dimensions with the usual Debye type cutoff in the frequency to prevent the divergence in the energy of the solid. We introduce the size of the crystal, d, by limiting the wave vector from π/d to 2π/d as given in ref 1 δνB )
∫π/d2π/d |φk|2[12πV/{pkV(2π)3}](2n + 1)k2dk
(33)
The calculation depends on the wave-vector dependence of the phonon-phonon scattering, which we take as, φk ) φok, so that the above becomes δνB ) [12πV|φo|2/pV(2π)3]
∫ (2n + 1)k3 dk
(34)
where the limits are from π/d to 2π/d. We can separate the above into two terms, one depending on the temperature and the other a zero-point contribution δνB ) δνB(T) + δνB(0)
(35)
where δνB(0) ) 45V|φo|2π2/(8pVd4) and δνB(T) ) [3|φo|2V/(π2pV)]
∫π/d2π/d [exp(pω/kBT) - 1]-1k3 dk
(36)
Therefore, the Brillouin shift is determined by ν ) νo - (c1/d4)
(37)
as far as the particle size is concerned. For small particles, the Brillouin frequency also becomes small. In other words, the Brillouin line moves toward the Rayleigh line as the particle size reduces, as shown schematically in Figure 1. Bottani et al.8 have successfully prepared Sn nanocrystals on Si(100) substrate and have measured the Brillouin scattering using 514.5 nm Rayleigh light from argon ion laser. It is found that the shift of the Brillouin line for 20 nm particle size is larger than for 2.5 nm in agreement with our calculations. Assuming that the Rayleigh light has a wavelength of 514.5 nm, equivalent to the frequency of ν ) 582507 GHz, the Brillouin wavelength is 356 × 105 nm and the frequency is 8.4 GHz. Therefore, Brillouin frequency is smaller than Rayleigh frequency by 5 orders of magnitude. Therefore, although Brillouin9 predicted the scattering as early as 1922, it could not be observed untill 1960 when sophisticated equipment became available. IV. Density of States. In three dimensions, the wave vector frequency relation is, k ) ω/V. Most of the scattering occurs when k varies from π/d to 2π/d so that the frequency Nano Lett., Vol. 2, No. 5, 2002
Figure 1. Schematic display of shift of the Brillouin line in going from the bulk crystal to the nanocrystal. The dashed line shows a broad Rayleigh line with a sharp Brillouin line shown by continuous curve.
varies as given in eqs 2 and 3. Therefore, the density of states is given by F(d) ) [4πV/(2π)3]{3(2π)2/V d2] ) c2/d2
(38)
with c2 ) 6V/V. Here, we have used 1/2 for the averaged value of sin θ for the scattering of phonons from the boundaries of the nanocrystals. Therefore, smaller crystals have larger density of states. In nanometer size crystals, the phonon density of states is larger than in the bulk crystals. This theoretical result is in agreement with that measured10,11 from the neutron scattering at low frequencies where enhanced density of states have been found for 12 nm size nanocrystals. The enhancement of density of states in going from the bulk crystals to nanocrystals has been found by Derlet et al.12 from the numerical integration of Schroedinger equation for Ni and Cu nanometer size samples. Thus, our analytical calculation of the density of states is in agreement with both the computed values as well as with the neutron data. V. Historical Note. Lord Rayleigh’s work13 was done during the year 1871. Later in 1908, Mie found14,15 that more light is emitted in the forward direction than in the backward direction when light transmits through spherical particles of
Nano Lett., Vol. 2, No. 5, 2002
finite size, such as water droplets which are not subject to the crystal structure. The explanation of the “rainbow” was found at about that time.16 In those days, the concept of the Brillouin zone was not known. Therefore, they could not discover the light emitted when dispersion relation of vibrations crosses that of light. The concept of Brillouin zone was discovered in 1931 but in those days Brillouin scattering remained undetected. Thus, our calculations for the nanocrystals are new and will be of interest in the study of science of nanometer size material. We have already reported the specific heat17 of nanocrystals.1 VI. Conclusions. We have calculated the fluctuations in the distance between atoms and from that we have deduced the melting temperature. It is found that the melting temperature reduces in going from the bulk crystal to a nanocrystal. We have found the shift in a phonon frequency as a function of particle size. It is found that the Brillouin scattered light shifts toward the Rayleigh line in nanocrystals compared with that in the bulk. The density of states of phonons is larger in nanocrystals than in the bulk. In all of the three cases, melting temperature, Brillouin shift and density of states, the calculated values are in accord with the experimental data. References (1) Shrivastava K. N. Nano Lett. 2002, 2, 21. (2) Goldstein, A. N.; Ether, C. M.; Alivisatos, A. P. Science 1992, 256, 1425. (3) Wickham, J. N.; Herhold, A. B.; Alivisatos, A. P. Phys. ReV. Lett. 2000, 84, 923. (4) Goldstein, A. N. Appl. Phys. A 1996, 62, 331. (5) Johari, G. P. Philos. Mag. A 1998, 77, 1367. (6) Suguna, A.; Shrivastava, K. N. Phys. Stat. Solidi B 1980, 99, 305. (7) Suguna, A.; Shrivastava, K. N. Phys. Lett. A 1979, 71, 121. (8) Bottani, C. E.; Bassi, A. Li.; Tanner, B. K.; Stella, A.; Tognini, P.; Cheyssac, P.; Kofman, R. Phys. ReV. B 1999, 59, 15601. (9) Brillouin, L. Ann. Physique (Paris) 1922, 17, 88. (10) Fultz, B.; Robertson, J. L.; Stephens, T. A.; Nagel, L. J.; Spooner S. J. Appl. Phys. 1996, 79, 8318 (11) Fultz, B.; Anthony, L.; Nagel, L. J.; Nicklow, R. M.; Spooner, S. Phys. ReV. B 1995, 52, 3315. (12) Derlet, P. M.; Meyer R.; Lewis, L. J.; Stuhr, U.; van Swygenhoven, H. Phys. ReV. Lett. 2001, 87, 205 501. (13) Rayleigh, Lord Philos. Mag. 1871, XLI, 274, 447. (14) Mie, G. Ann Physik 1908, 25, 429. (15) Blumer, H. Zeit. Phys. 1926, 38, 304. (16) Born, M.; Wolf, E. Principles of Optics; Pergamon Press: Oxford. (17) Shrivastava, K. N. SuperconductiVity: Elementary Topics; World Scientific: New Jersey, London, 2000.
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