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The question here is whether the Debye model is suited to evaluate the specific heat of nanocrystals. For this, the simplest possible nanocrystal is considered: a basic cubic structure made of atoms that interact through a harmonic potential. This simple model can be solved exactly. This allows the dispersion relation of the mechanical waves to be determined, so that calculating the exact specific heat turns out to be quite straightforward. Then, the same problem is solved using the Debye approximation. Our findings show that the specific heat of a nanocrystal evaluated exactly is higher than the value found in the thermodynamic limit, that is to say, the specific heat decreases as the nanocrystal size increases. In addition, it becomes clear that the Debye model is a poor approximation for calculating the specific heat of a nanocrystal. Naturally, the Einstein model yields an even worse result. The cause of the discrepancy is found in the role of the nanocrystal surface.

The boom of physics at a nanoscopic scale justifies questioning the validity of some of the usual claims when working with systems made up of a few hundred or a few thousand particles. Previous papers [

In addition, the question arises whether the usual models for solids are applicable at the nanoscale. A recent article [

The body of this article is structured in three main sections. Section 2 presents the model of a nanocrystal formed by N_{A} atoms and shows its exact solution. Section 3 introduces the approximations made to obtain the specific heat of the Debye solid and, for comparison purposes, the Einstein model is also included. By comparing both models with the exact results, it will become clear that they are not suited for systems with thousands of atoms. Finally, Section 4 summarizes the conclusions.

_{0}. There are a total of N_{A} atoms, which means that the side of the nanocrystal has n L = N A 1 / 3 atoms and the total volume is given by V = ( n L − 1 ) 3 a 0 3 . The atoms with mass M are connected to each other through springs characterized by a constant K.

It should be noticed that we deal with a toy model since there are no simple cubic crystals in nature. We use this crystal structure because our concern is to check the validity of the Debye model at the nanoscale and do not pretend to reproduce any actual experimental result.

In terms of classical mechanics, the behavior of this system is well-known: the system has 3 n L 3 different oscillation modes whose frequencies can be calculated easily. The case of a one-dimensional lattice with pinned boundary conditions is dealt with in textbooks [

ω 2 ( k ) = ( 4 K M ) sin 2 ( k a 0 2 ) (1)

Generalizing to the three-dimensional case [

k = π n L a 0 ( m x , m y , m z ) (2)

ω 2 = 4 K M ( sin 2 [ π 2 n L m x ] + sin 2 [ π 2 n L m y ] + sin 2 [ π 2 n L m z ] ) (3)

where m x , m y , m z = 0 , 1 , 2 , ⋯ , n L − 1 . The characteristic frequency of the system is:

ω 0 = 2 K / M . (4)

It is necessary to consider all possible combinations of m_{x}, m_{y} and m_{z}, except (0, 0, 0) that obviously corresponds to the body at rest.

So far for the classical treatment, in quantum terms, these results are interpreted as the system having phonons that are characterized by a wave vector k and a frequency ω(k), and that obey Bose-Einstein’s statistics. It is a usual exercise in any statistical mechanics course [

U = ∑ k ( 1 2 ℏ ω ( k ) + ℏ ω ( k ) e ℏ ω ( k ) k B T − 1 ) (5)

It follows immediately that the heat capacity C = d U / d T is:

C = ∑ k ( ℏ 2 ω 2 ( k ) e ℏ ω ( k ) k B T k B T 2 ( e ℏ ω ( k ) k B T − 1 ) 2 ) (6)

The sum of all the k vectors can be broken down into a triple sum: ∑ k ( ) → ∑ k x ∑ k y ∑ k z ( ) , and one should remember that there are three possible polarization states (two transverse and one longitudinal) of each phonon.

Also, some abbreviations are included to simplify the notation. We make use of an auxiliary function defined as follows:

R ( m x , m y , m z ) = ( sin 2 [ π 2 n L m x ] + sin 2 [ π 2 n L m y ] + sin 2 [ π 2 n L m z ] ) 1 / 2 (7)

The characteristic frequency of the system is ω 0 = 2 ( K / M ) 1 / 2 . The temperature can be expressed in terms of a dimensionless variable t as T = tT_{c} with T_{c} = ħω_{0}/k_{B}. Keeping in mind all these definitions and remembering that there are n L 3 atoms, the specific heat per atom c is written:

c e x = 3 k B n L 3 ∑ m x = 0 n L − 1 ∑ m y = 0 n L − 1 ∑ m z = 0 n L − 1 R 2 ( m x , m y , m z ) e R / t t 2 ( e R / t − 1 ) 2 ; m x + m y + m z ≠ 0 (8)

In the rest of this article, the specific heat is expressed in terms of units of k_{B}. In this way, c becomes an adimensional variable as well as the temperature t.

This is expression (8) is exact, noted by subscript “ex”, and it gives the specific heat per atom for a nanocrystal with a simple cubic structure and n_{L} atoms per side. The first question that arises is how this parameter changes according to the size of the system. And the answer is that it changes very slowly, as seen in

In the next section, the specific heat will be calculated again but introducing step-by-step the approximations of the Debye model.

Actually, several approximations are made in the Debye model for solids [

Linearizing the dispersion relation of the phonons implies approximating the sine that appears in (1) by its argument. In one dimension:

ω ( k ) = 2 ( K m ) 1 / 2 | sin ( k a 0 2 ) | ≅ ω 0 k a 0 2 = ω 0 π m 2 n L (9)

with m = 0 , 1 , 2 , ⋯ , n L − 1 .

In three dimensions, the auxiliary function R ( m x , m y , m z ) defined previously needs to be modified as follows:

R ′ ( m x , m y , m z ) = ( ( π 2 n L m x ) 2 + ( π 2 n L m y ) 2 + ( π 2 n L m z ) 2 ) 1 2 (10)

To calculate the specific heat, the same expression (8) is used, but with function R ′ given by (10).

Then, the frequencies are assumed to be continuous and the sum in (6) is replaced by an integral that includes the density of states for phonons. And finally, a maximum frequency, known as the Debye frequency, is introduced. Physically, the existence of a maximum frequency is determined by the existence of a minimum wavelength associated with the interatomic distances. In the case of a finite sample, there will also be a minimum frequency because there is a maximum wavelength associated with the linear size of the sample.

The minimum frequency ω_{min} is obtained taking the wavelength along the edge of the cube as maximum half-wavelength:

λ max 2 = ( n L − 1 ) a 0 = V 1 / 3 (11a)

ω min = π c V 1 / 3 (11b)

where c is the speed of sound in the lattice, given by [

To establish the maximum frequency, or Debye frequency ω_{D}, the procedure is similar to the usual treatment: the integral for the density of states between ω_{min} and ω_{D} has to be equal to the number of degrees of freedom for the system. Here, attention should be paid to the atoms in the surface that, consequently, have fewer degrees of freedom than those inside the cube. A careful count [

∫ ω min ω D D ( ω ) d ω = ∫ ω min ω D 3 V 2 π 2 c 3 ω 2 d ω = 3 ( N A − N A 2 / 3 ) (12)

This gives:

ω D 3 = 6 c 3 N A π 2 V + c 3 π 3 V − 6 c 3 N A 2 / 3 π 2 V (13)

It is interesting to analyze each term in the previous expression. The first term is the one usually present in the treatment of the Debye solid. The second is associated with the existence of a minimum frequency due to the size of the sample. And the third accounts for the existence of a surface. It becomes clear that the effect caused by the finite size of the sample is negligible in the presence of the other two terms that explicitly depend on N_{A}. Now, the problem is reduced to calculating the specific heat of this nanoparticle keeping in mind the linearized dispersion relation and the existence of a minimum and a maximum frequency.

Taking into account the density of states for phonons D(ω) and expression (6), the heat capacity in the Debye approximation is:

C Debye = 9 N A ℏ 2 ω D 3 k B T 2 ∫ ω min ω D ω 4 e ℏ ω k B T ( e ℏ ω k B T − 1 ) 2 d ω (14)

To enable comparisons with the previous results, it is convenient to rewrite the previous expression in terms of the dimensionless temperature t defined earlier, and to divide by the number of atoms. The specific heat per atom c_{Debye} in the Debye approximation is obtained as follows:

c Debye = 9 k B u D 3 t 2 ∫ u min u D u 4 e u / t ( e u / t + 1 ) 2 d u (15)

The integration limits are u min = ω min / ω 0 y u D = ω D / ω 0 . It is useful to write these magnitudes in terms of our model parameters, as seen in

u D = ( 3 n L 3 π 2 4 ( n L − 1 ) − 3 n L 2 π 2 4 ( n L − 1 ) + π 3 8 ( n L − 1 ) ) 1 / 3 (16a)

and:

u min = π 2 ( n L − 1 ) (16b)

It is clear that expression (15) depends only on the number of atoms n_{L} and the reduced temperature t.

In

The most readily visible conclusion is that the Debye approximation is unsuited for systems with tens of particles

It is useful to compare the exact specific heat of a nanocrystal with the usual value of the Debye solid in the thermodynamic limit [

And how do these results compare with the Einstein solid? In this case, all the phonons have the same frequency ω_{0} or, more precisely, the density of states is delta function center in that frequency: D ( ω ) = δ ( ω − ω 0 ) . Therefore, expression (6) is reduced to [

C Einstein = 3 ( N A − N A 2 / 3 ) ℏ 2 ω 0 2 e ℏ ω 0 k B T k B T 2 ( e ℏ ω 0 k B T − 1 ) 2 (17)

Naturally, it is convenient to divide by the total number of atoms N A = n L 3 and to introduce the dimensionless temperature t. It is necessary to keep in mind that the characteristic frequency was defined as (K/M)^{1}^{/2} in [_{B} units as:

c Einstein = 3 ( 1 − 1 n L ) 4 e 2 / t t 2 ( e 2 / t − 1 ) 2 (18)

_{ex}, in the Debye approximation c_{Debye} and in the Einstein approximation c_{Einstein}.

It is reasonable to wonder why the Debye approximation is unsuited for systems with hundreds or thousands of atoms. A first conjecture is related to the finite

size of the sample, which is reflected in the existence of a lower limit different from zero in the integral; however, this is not the case. The existence of a lower limit in the frequency has a negligible contribution. For example, when in the integral in (15) the lower limit is replaced by 0, the specific heat changes from 1.4325 to 1.4324 for n_{L} = 10 and t = 1. So, clearly, the existence of a lower integration limit different from zero is not relevant.

Another possible cause is the linearization of the dispersion relation.

The cause of the discrepancy seems to be twofold. On the one hand, the assumption of a continuous frequency distribution for the phonons is appropriate in the thermodynamic limit, but it is a very strong hypothesis when the number of phonons is in the order of n L 3 . And on the other hand, the existence of a surface seems to have a crucial effect, as already demonstrated in the case of the Einstein nanocrystal [

Finally, it should be noted that this model predicts that, for nanocrystals, the specific heat goes exponentially to zero, as inferred from expression (8). And this behavior is different from that observed in macroscopic crystals. There are no experimental data of the nanocrystal specific heat behavior for very low temperatures to confirm or rebut this prediction.

In this article, we have explored the validity of the Debye model at the nanoscale. For this, we have considered a very straightforward nanocrystal model: a simple cubic structure whose atoms are assumed to interact through a harmonic potential, as portrayed in

Once the exact specific heat is calculated for our nanocrystal, we apply the Debye approximation to allow for comparisons. To do this, the usual steps are followed, but keeping in mind that the nanocrystal has a finite size and, therefore, there is a maximum wavelength, or rather a minimum frequency. So we obtain an analytical expression for the Debye frequency―Equation (13)―where three terms are easily identified: the usual term and the corrections on account of the finite size and the surface. The first correction has proven to be irrelevant, but not the second. This agrees with our findings for the Einstein model in a previous paper [^{6} atoms, the exact value and that obtained with the Debye approximation differ considerably. The surface vibrational modes might be responsible for this disagreement. The message is then: do not forget the surface when dealing with nanocrystals. A naïve application of bulk results to a nanoparticle would lead to wrong conclusions for two reasons: 1) the surface introduces a non negligible effect (the surface phonons in the case of specific heat); 2) as the size of the particle is decreased, the approximation of using a smoothed density of states becomes less and less accurate.

The final conclusion of this article is that the standard Debye model is not very suited to model the specific heat of nanocrystals. But we also think there is a path for doing this better. Although nanocrystals with a simple cubic structure do not exist in nature, the steps outlined here could be replicated for a realistic crystal structure. It would simply be a matter of finding the frequencies of the normal modes for that realistic structure assuming that the atoms interact through a harmonic potential (“springs”). And once the formula linking the wave vector with the frequency is known, evaluating the exact specific heat becomes an exercise of statistical mechanics.

The authors declare no conflicts of interest regarding the publication of this paper.

Miranda, E.N. and Mora-Barzaga, G. (2019) How Good Is the Debye Model for Nanocrystals? Journal of Modern Physics, 10, 601-612. https://doi.org/10.4236/jmp.2019.106042