NANO LETTERS
Specific Heat of Nanocrystals
2002 Vol. 2, No. 1 21-24
Keshav N. Shrivastava† School of Physics, UniVersity of Hyderabad, Hyderabad 500046, India Received September 12, 2001
ABSTRACT We find that the phonons of certain frequencies are scattered by the boundary of the nanocrystal so that the specific heat depends on the size of the particles. At low temperatures, the Debye’s T3 law, which is known for a large number of particles, is completely broken and exponential behavior may be observed. In the case of a magnetically ordered ferromagnet, we find that the specific heat depends on the size of the nanocrystals because of the scattering of spin waves from the surface of the nanoferromagnets.
1. Introduction. Recently, there has been considerable interest in the physical properties of nanocrystals,1 which are being prepared by new methods.2 The specific heat of nanoparticles will be interesting but has not been calculated in the literature. The specific heat of a bulk solid was calculated by Einstein in 1910 by assuming that all of the atoms have one and the same frequency. Later in 1912 Debye found that there is a frequency spectrum from zero to a cutoff value that gives the T3 law for the specific heat. If the material is sufficiently simple in crystal structure, the Debye T3 law is well obeyed. In the case of a phase transition, there is a critical behavior and the specific heat near the critical temperature shows a peak.3 If there are two-level impurities in the system, the specific heat shows a peak, at a temperature corresponding to the energy level separation. In 1940, it was found that there is a soft-phonon mode, the frequency of which varies with temperature, moving toward the Rayleigh line and reaching zero at the transition temperature so that there is a contribution to the specific heat that vanishes at the transition temperature. The specific heat of magnetic materials such as MnO is very different from the value in the nonmagnetic materials, such as MgO, which shows that there is a contribution due to the spin waves. In this letter, we report that there is a phonon that is scattered at the boundary of a nanocrystal. This scattering generates new phonon frequencies in a nanocrystal, which contributes to the specific heat. The Debye T3 law will be completely suppressed by this kind of boundary scattering if there are a large number of particles of the size of a few nanometers for which the specific heat exhibits exponential dependence as a function of temperature. If the material is a ferromagnet, such as Fe2O3, the spin waves are scattered by the surface of the nanocrystal so that there is a spin-wave contribution to the specific heat of a nanocrystal. † Tel: +91-40-3010811. Fax: +91+40-3010145. E-mail: knssp@uohyd. ernet.in Webpage: www.htcs.org.
10.1021/nl010064n CCC: $22.00 Published on Web 11/16/2001
© 2002 American Chemical Society
2. Theory of Phonons. We consider that the material consists of an ensemble of nanocrystals. The phonons are scattered at the boundary of the nanocrystals of size, d. The X-rays are diffracted according to the Bragg’s formula, 2d sin θ ) nλ, where d is the distance between planes, θ is the angle which the incident ray makes from the normal to the planes, λ is the wavelength, and n is any positive integer, usually equal to 1 or 2 for the first and the second-order diffraction, respectively. Instead of the X-ray, we consider phonons and determine which frequency will be scattered by the nanocrystals. We consider that the wavelength of the phonon is λ ) V/ν, where ν is the phonon frequency and V is the velocity of sound. The phonon frequency scattered by the boundary of the nanocrystal is ω ) 2πnV/(2d sin θ)
(1)
Taking sin θ )1, we obtain the lower phonon frequency as ω1 ) πV/d
(2)
for n ) 1. When θ ) 0 and sin θ ) 0, there is a divergence in the phonon frequency. Therefore, we replace sin θ by its average value, 〈sin θ〉 ) 1/2, to obtain ω2 ) 2πV/d
(3)
The average value of 1/2 may be interpreted as if all scattering from 0 to 30° has been ignored. Actually, the Bose-Einstein factor becomes very small as the scattered frequency becomes large at small angles. Therefore, it is reasonable to ignore this range of angles. Thus we have phonons of frequencies between ω1 and ω2 that are scattered by the nanocrystals. For, ω2 ) 1012 Hz and V ) 105 cm/s = 103 m/s, d = 6.28 nm. Therefore, it is reasonable to expect phonon scattering from the boundaries of the nanocrystals.
The specific heat of a nanocrystal may be obtained by calculating the lattice energy as E)
∫4πVpωk[(2π)3{exp(pωk/kBT) - 1}]-13k2 dk
(4)
for a three-dimensional solid. Here V is the volume of the crystal, ωk is the phonon frequency, and we use the BoseEinstein distribution. A factor of 3 has been introduced to take into account two transverse and one longitudinal branch of the phonon spectrum. The above integral gives
∫
E ) 12πVp[(2π)3V3]-1(kBT/p)4 x3(e-x - 1)-1 dx (5) from which the specific heat, ∂E/∂T, is immediately found to depend on T3, which is called the Debye’s T3 law. In the case of a nanocrystal, we obtain the scattering contribution by considering the limit x(1) ) pπV/(kBTd) ) 1/t
(6)
and x(2) ) 2pπV/(kBTd) ) 2/t
(7)
t ) kBTd/(πpV)
(8)
with
Figure 1. Schematic drawing of the dispersion relation showing kink at π/d where d is the size of the nanocrystal. Here ω is the phonon frequency and k is the wave vector with ao as the unit cell constant in the bulk crystal.
in the dispersion relation in Figure 1. Accordingly, a peak is predicted in the specific heat at a temperature of pπV/d ) kBTpeak. For V ) 105 cm/s and d ) 100 × 10-8 cm, we predict Tpeak ) 2.4 K so that such a peak is within the observable range of temperatures. At this temperature the usual Debye waves are suppressed. In other words, there is a lot of nanofog so that the clear Debye solid is not visible. We integrate (9), under the low-temperature condition, x . 1, to find
The scattering contribution to the lattice energy of the nanocrystal is then found to become
∫x(1)x(2)x3(ex - 1)-1 dx
E ) 12πVp[(2π)3V3]-1(kBT/p)4
(9)
It is of interest to estimate the magnitude of this energy relative to the lattice energy (4). The frequency dependence of the integrand in (4) is the same as in (9). In the case of a Debye solid, the frequency is ωD ) πV/ao, where ao is the unit cell constant in the bulk crystal. The distance between two surfaces in a nanocrystal is d = 10ao, so that the frequency of the nanocrystal is ω1= πV/10ao= ωD/10. Therefore the frequency of the nanocrystal is one-tenth that of the Debye solid. There is a factor of ω3 in the integrand so that this factor is 1/1000 in a nanocrystal compared with that in the bulk crystal. However, the Bose-Einstein factor has an exponential of the frequency in the denominator so that when the frequency is reduced by a factor of 10, the Bose-Einstein number increases by 22 000. Therefore, the integrand in the nanocrystal is larger than in the bulk crystal by a factor of 22. However, the value in the limit is again smaller in the nanocrystal than in the bulk crystal, so that the value of the full integral is of the same order of magnitude in the nanocrystal as in the bulk crystal. In the dispersion relation of the bulk crystal, the frequency varies with the wave vector from 0 to π/ao, whereas in the nanocrystal, we predict a kink at π/d. An effort is made to show such a kink 22
E ) 9pV(π2V3)-1(kBT/p)4 exp[-(kB/pπ)(Td/V)] (10) which shows that the T4 law for energy is considerably modified by the exponential term, which depends on the grain size, d, as well as on the temperature. We write the above expression as E ) b1T4 exp[-b0T]
(11)
b1 ) 9pV(π2V3)-1(kB/p)4
(12)
b0 ) (kBd/pπV)
(13)
where
The specific heat at low temperatures is thus Cp ) ∂E(T)/∂T ≈ 4b1T3 exp(-b0T)
(14)
which describes the modification of the Debye’s T3 law in going from crystals to nanocrystals. The size dependence is described as Cp = 4b1T3 exp[-kBTd/(pπV)] ) c1 exp(-Rd)
(15)
Nano Lett., Vol. 2, No. 1, 2002
where R ) kBT/(pπV) and c1 is a constant. Thus, there is a large change in the specific heat for a small change in the size, d, of the nanocrystal due to phonon scattering at the surface. 3. Spin Wave Scattering. The dispersion relation of a ferromagnet is well-known to describe the variation of the frequency as a function of wave vector, pωk ) pDk
2
Un ) 2VpD(2π)-2
∫π/d2π/d[exp(pωk/kBT) - 1]-1k4 dk
(23)
The integral in the above may be written as
∫[exp(pω/kBT) - 1]-1k4 dk )
∫
(1/2)(kBT/pD)5/2 [ex - 1]-1x3/2 dx (24)
(16) so that the lattice energy of a ferromagnetic material becomes
where D is called the magnetic stiffness constant. The excitations are called the magnons and their velocity is determined by
U ) VpD(2π)-2(kBT/pD)5/2I
(25)
where -1
Vm ) h (∂/∂k)pω ) 2kD
(17) I)
The wavelength of the magnon is determined from the ratio of magnon velocity, Vm, to the magnon frequency, ν, λ ) Vm/ν ) 4π/k
U)
∑k pωk[exp(pωk/kBT) - 1]-1 ∫
) 4πV(2π)-3 [exp(pωk/kBT) - 1]-1pωkk2 dk
∫
U ) pVD(2π)-2(kBT/pD)5/2 [ex - 1]-1x3/2 dx
(20)
which shows that the lattice energy of a ferromagnet varies as T5/2 and hence the specific heat as T3/2. We consider that the size of the nanocrystal is d and spin waves are scattered from this dimension so that the scattered wave vector is k ) 2π/λ
x(o) ) pDπ2/(kBTd2)
(21)
with
(27)
where d is the size of the nanoferromagnet. At low temperatures, x . 1, so that e-x occurs in the numerator. For the upper limit e-x is very small, so that we can consider that the value of the integrand is zero at 4x(o). This is clearly a peaked function so that to get the value of the integral, we take the value of the integrand at x(o) and multiply it by x(o). Therefore, at low temperatures, U ) (a1/d5) exp[-a2/(Td2)]
(19)
where V is the volume of the whole crystal. We use the dispersion relation (16) and make the change of variables, x ) pωk/kBT, so that the lattice energy of a ferromagnet becomes
(26)
with
(18)
The specific heat is determined from the lattice energy, which is the energy of one magnon multiplied by the distribution function summed over all magnons. We assume a threedimensional system and change the summation into integration as
4x(o) x [e - 1]-1x3/2 dx ∫x(o)
(28)
where a1 ) VpDπ3/4
a2 ) pπ2D/kB
(29)
so that the specific heat of a nanoferromagnet at a constant pressure becomes Cpn ) ∂U/∂T ) [a1a2/(d7T2)] exp{-a2/(Td2)}
(30)
from which we obtain two results. First, the specific heat varies with temperature as Cpn ∝ T-2 exp(-a3/T)
(31)
and with nanocrystal size, d, as λ ) 2d sin θ
or
k ) π/(d sin θ)
(22)
When θ ) 0, sin θ ) 0 so that k goes to ∞. The largest value of sin θ is equal to 1 so that the minimum value of k is π/d and the upper limit is taken as 2π/d so that reasonable values of the Bose-Einstein factor are used and when it becomes very small, it is cut off. Therefore, in the case of a nanocrystal, we integrate the wave vector in (19) as Nano Lett., Vol. 2, No. 1, 2002
Cpn∝ d-7 exp(-a4/d2)
(32)
where a3 ) a2/d2 and a4 ) a2/T. It is clear that there is a considerable effect of size on the magnetic contribution to the specific heat. 4. Conclusions. The specific heat of a nanocrystal is predicted to be considerably different from that of the bulk 23
Debye solid. We find that the low-temperature T3 law is completely broken. The low-temperature specific heat is expected to vary as exp(-b0T) and as a function of particle size as exp(-Rd), where d is the particle size. In the case of ferromagnetic nanocrystals the spin waves are diffracted by the grains and hence the specific heat varies as exp(-a3/T) as a function of temperature and as exp(-a4/d2) with the particle size. Thus we predict a new type of temperature and particle-size dependence of the specific heat of nanocrystals and nanoferromagnets. An enhancement in the density of phonon states of Fe has been found by Fultz et al.,4 but detailed specific heat in nanoferromagnets has not yet been
24
reported. In our theory, enhancement arises from the frequencies in the Bose-Einstein factor. References (1) Goldstein, A. N.; Ether, C. M.; Alivisatos, A. P. Science 1992, 256, 1425. (2) Zanchet, D.; Micheel, C. M.; Parak, W. J.; Gerion, D.; Alivisatos, A. P. Nano Lett. 2001, 1, 32. (3) Shrivastava, K. N. SuperconductiVity: Elementary Topics; World Scientific: New Jersey, London, 2000. (4) Fultz, B.; Ahn, C. C.; Alp, E. E.; Sturhahn, W.; Toellner, T. S. Phys. ReV. Lett. 1997, 79, 937.
NL010064N
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