Thermodynamical properties of ultrathin layered structures

Journal of Physics and Chemistry of Solids 61 (2000) 931–936 www.elsevier.nl/locate/jpcs

Thermodynamical properties of ultrathin layered structures S.B. Lazarev a,*, M.R. Pantic´ b, S.M. Stojkovic´ c, B.S. Tosˇic c, J.P. Sˇetrajcˇic´ c a

Technological Faculty, Zvornik, S.R. Bosnia and Hercegovina and Higher School of Chemistry and Technology, Sˇabac, Serbia, Minor Yugoslavia b Institute for Nuclear Sciences Vincˇa, Belgrade, Serbia, Minor Yugoslavia c Institute of Physics, University of Novi Sad, Serbia, Minor Yugoslavia Received 3 June 1999; accepted 27 September 1999

Abstract Thermodynamical properties of ultrathin films were analysed using Green’s function method. Mean-square of the molecular displacements, mean-square of the velocities and the phonon contribution to the specific heat of the thin films were calculated. A comparison with crystal bulk has shown that thermodynamical properties of thin films are strongly influenced both by the sample dimensions and boundary conditions. The results of the specific heat analyses make the validity of free surface model doubtful, because the specific heat calculated in this model becomes negative at very low temperature. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Thin films; D. Phonons; Mean-square displacement; D. Specific heat

1. Introduction Harmonic vibrations of the ultrathin films were analysed in paper [1] by the method of two-time, temperature dependent Green’s functions. We have studied the phonon spectra of thin films and established the conditions for the existence of the surface states. We have also determined the local densities of phonon states and the Debye frequencies for thin films. The knowledge of the properties of such films is of crucial importance because the very-large-scale-integration technology [2] requires smaller devices with thinner insulating films. Amorphous SiO2 films in field-effect transistors, can be used as an example of ultrathin film whose thickness is only few nanometers. In that case, dramatic device failures due to dielectric breakdown are possible. The understanding of the mechanisms, which lead to the degradation and breakdown of the devices is dependent on the understanding of the mechanism of electron transport [3] and thermodynamical properties [4] of the ultrathin films.

* Corresponding author. Tel.: ⫹381-15-323-877; fax: ⫹381-15324-715. E-mail address: [email protected] (S.B. Lazarev).

Green’s function method, due to built-in statistics, gives opportunities for calculating the mean value of the series of the physical characteristics of ultrathin films [5–8]. In this paper we have analysed the dynamical properties of the atoms in various layers and thermodynamical properties of thin films. The dynamical properties of the atoms are described by the mean-square displacement and meansquare velocity in Section 3. Section 4 will be devoted to the thermodynamics of thin films. We shall evaluate internal energy and specific heat of the phonon subsystem of thin films.

2. Lattice dynamics In order to evaluate mean-square displacements and mean-square velocities at each film layer, it is necessary to determine the Green’s function of displacement– displacement type: Ganx ny nz ;mx my mz …t† ˆ Q…t†具‰u a;nx ;ny ;nz …t†; ua;mx ;my ;mz …0†Š典

…1†

where Q…t† is Heaviside’s function, and u a;nx ;ny ;nz …t† is the projection of molecular displacement at the site n~ onto reference system axis of the index a ˆ x; y; z: Fourier transform of the function (1) was calculated in the paper [1] and

0022-3697/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(99)00392-3

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has the form: a …kx ; ky ; v† ˆ Gn;n



NX z ⫹1 iប gan;n …@n † 2pCa n ˆ1 vna …kx ; ky †

1 1 ⫺ v ⫺ vna …kx ; ky † v ⫹ vna …kx ; ky †

! (2)

Notations used in expression (2) are the following: C a are Hooke’s constants, gan;n …@n † are spectral weights of Green’s functions, while vna …kx ; ky † gives phonons frequencies. The knowledge of phonon’s Green’s functions allows us to calculate the statistical average value of the molecular displacements and velocities in various film layers. Our previous study [1] shows that it is plausible to ignore the effects of the surface reconstruction (change of lattice constants at the surface and neighbouring layers) when one wishes to concentrate on the effects of energy parameter changes at the boundaries. In order to evaluate the mean-square of the molecular displacement in each film layer n ˆ 0; 1; …; Nz ; 1 XZ 具u2a;n 典 ˆ …3† dv 具u2a;n …kx ; ky †典v ; Nx Ny k k x y

it is necessary to determine first the Fourier transform of the correlation function: 具u2a;n …kx ; ky †典v ˆ

2ReGan;n …kx ; ky ; v ⫹ i0⫹ † : ebបv ⫺ 1

Using the expression (2), Fourier transform of the correlation function can be written in the form: 具u2a;n …kx ; ky †典v ˆ

ប 2M 

NX z ⫹1

gan;n …@v † vna …kx ; ky † n ˆ1

d‰v ⫺ vna …kx ; ky †Š ⫺ d‰v ⫹ vna …kx ; ky †Š ebបv ⫺ 1 …4†

nv Nx Ny X gn;n …@n †Q‰v ⫺ v na …0; 0†Šv; vnav …0; 0† 2 2pV a n ˆ1

ⱕ v ⱕ vnav ⫹1 …0; 0†;

(5)

where V a2 ˆ Ca =M; one obtains: 具u2a;n 典 ˆ 2 典 J具u a;n

ˆ

⫹1 2 u NX 典 gan;n …@n †J具u a;n …u†; a 2p C n ˆ1

Zxa2;n xa1;n

coth x dx;

coth x ˆ 1 ⫹ 2

∞ X

e ⫺2jx

jˆ1 典 the integral in expression (6) becomes J具u a;n …t† ˆ a a a a x2;n ⫺ x1;n ⫺ Z1 …2x2;n † ⫹ Z1 …2x1;n †; where 2

Zr …x† ˆ

∞ X e⫺nx ; nr nˆ1

…r ˆ 1; 2; …†

are Dyson’s Z¯-functions. Substituting the value of the integral obtained in this way into expression (6) we obtain the mean-square of the amplitude of molecular displacement in the units of ‰u0 =CŠ: 具u2a;n 典 ˆ

Nz ⫹ 1 p 1 X ga …@ †{D ⫺ Dn ⫺ 2t‰Z1 …D=t† 4p nˆ1 n;n n

p ⫺ Z1 … Dn =t†Š}:

…7†

D ˆ vD =V 0 is dimensionless Debye frequency. Using this expression we have performed a detailed numerical analysis of the mean-squares of the molecular displacements for various models of boundary conditions. Above models concern the behaviour of surface Hooke’s constants.Their values at the boundaries are given as: a C⫺1 ˆ C a …1 ⫹ s0 †;

CNa ˆ C a …1 ⫹ sN †;

…8†

where the parameters s0 i sN have different values for different models:

(M is the mass of the molecules.) Substituting this expression into Eq. (3), after the common transition from the sum over kx i ky to the integral over v using the local density of phonon state [1]: dna …v† ˆ

p where xa1;n ˆ Dna =…2t†; Dna are the minimal values of the phonon’s frequencies in the dimensional sub-bands, and xa2;n ˆ vaD =…2tV 0a †; vaD is Debye frequency for thin films. For the sake of comparison with the bulk, we have introduced here the dimensionless “temperature” p in the form t ˆ u =u0 ; where u0 ˆ បV 0 ; and V 0 ˆ C=M for simple cubic lattice. Expanding the function

…6†

1. ideal surfaces …s0 ˆ sN ˆ 0†; 2. free surfaces …s0 ˆ sN ˆ ⫺1† and 3. asymmetric boundaries …s0 ˆ 0; sN ˆ ⫺1†: Besides this models of boundary conditions we shall study the more general case with “varying” values of surface parameters. Our case study will be the ten-layer film …Nz ˆ 9†: The results are presented in Fig. 1(a). In all three models of the boundary conditions, the mean-square of vibration amplitudes of atoms at the surfaces and in the interior of the film differ substantially, which agrees with the well-known theoretical results [9] and the recent measurements of the Debye temperature by RHEED technique [10]. For the free-surface film, these differences amount to 15–30% within the temperature range 0–u0 : Bulk values reach up to fifth layer from the film surface. Particularly interesting is the case of the asymmetric film which allows the conclusion that the increase of the elastic constant on the film surface leads to the decrease of the mean-square displacement, and vice versa.

S.B. Lazarev et al. / Journal of Physics and Chemistry of Solids 61 (2000) 931–936

Using the expression (7), we have studied the temperature dependence of the mean-square displacements of the atoms in various film layers. These results are presented in Fig. 1(b) for several surface layers …n ˆ 0; 1; 2† of the freesurface film (dotted lines) and ideal surface film (thin lines). Solid lines correspond to bulk. Temperature dependence is linear in both cases for high temperatures, similar to the bulk behaviour. The plot shows that the average square displacements of surface atoms in the free surface model increase faster with the temperature rise than it is the case in the bulk. In the ideal surface model, mean-square displacements of the surface atoms increase slower as compared to the bulk. It is interesting to note that at absolute zero, vibration amplitudes differ negligibly from the bulk values. A visible difference can be noticed only in the surface layer of free surface film. Our previous study [1] shows that the ratio D has the value close to 5 so, our “dimensionless” temperature t reaching 5 implies that the temperature of the system is simply of the order of Debye temperature, or less. So, we are far away from any melting effect. The calculation of the mean-square velocities of atoms in film layers is based on the expression [11]: 具v2a;n 典 ˆ

1 X Z∞ v2 具u2a;n …kx ; ky †典v dv; Nx Ny k ;k ⫺ ∞ x

…9†

y

which can be written in the following form, in the manner similar to the one presented above: 具v2a;n 典 ˆ 2 † J…v a;n …t†

Nz ⫹ 1 2 2u0 t3 X ga …@ †J…v † …t†; pM nˆ1 n;n n a;n

ˆ

Zxa2;n xa1;n

…10†

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The expression (12) allows the analysis of the meansquare velocities, quite similar to the one for mean-square displacements. Mean-square velocities differ from the bulk values only in the surface layers n ˆ 0 and n ˆ Nz ; Fig. 2(a). Bulk values are reached practically after the second layer from the film surface. In the model of ideal boundaries, the difference of the mean-square velocities over the film layers is negligible. Temperature dependence of the meansquare velocities of atoms is linear at high temperatures, Fig. 2(b), similar to the bulk. At the absolute zero, mean-square velocities of zero-vibrations in both models of boundary conditions are slightly lower from the bulk values.

3. Internal energy and specific heat In previous studies of the thin films, we have established the influence of the boundary conditions and film width on the phonon spectrum, the density of phonon states, Debye frequency [1] and dynamical properties of the molecules. Comparison with corresponding properties of the bulk indicated to significant differences, which are the consequence of the dimensional quantization and localization of the phonon states. One can expect on the basis of these results that thermodynamic properties will also differ. We are going to analyse here the internal energy and the specific heat of the phonon subsystem of thin films. Internal energy of the phonon subsystem, can be evaluated as quantum-statistical mean value of the Hamiltonian, U ˆ 具H典: After lengthy calculations, one obtains   Nz ⫹ 1 X 1 3ប X bប a U…t† ˆ vma …kx ; ky † coth vm …kx ; ky † Nx Ny 2 mˆ1 k ;k 2

2

x coth x dx:

x

This integral can be evaluated by the multiple partial integration

⫹

1 a † J…v …x ⫺ xa1;n † ⫺ …xa2;n †2 Z1 …2xa2;n † a;n …t† ˆ 3 2;n 2

⫹

…xa1;n †2 Z1 …2xa1;n †

⫺

1 1 Z …2xa † ⫹ Z3 …2xa1;n †: 2 3 2;n 2

⫺

xa2;n

Z 2 …2xa2;n †

⫹

xa1;n

3ប 4

NX z ⫹1

y

X

mˆ1 kx ;ky

⫹ sN gaNz ;Nz …@m †Š

Z 2 …2xa1;n †

p p p ⫺ D n Z1 … D n =t†Š ⫺ t2 ‰DZ 2 …D=t†⫺ Dn Z2 … D n =t†Š ) p ⫺ t3 ‰Z3 …D=t† ⫺ Z3 … Dn =t†Š : …12†

1

vma …kx ; ky †

 coth

 bប a vm …kx ; ky † : 2

…13†

(11)

Thus we arrive to the final expression for the mean-square velocities in the units of ‰u0 =MŠ: ( 3 Nz ⫹ 1 1 X D ⫺ …Dn †3=2 t 2 a ⫺ ‰D2 Z1 …D=t† gn;n …@n † 具va;n …t†典 ˆ 12 p n ˆ1 2

‰s0 ga0;0 …@m †

The first term of the above expression represents the internal energy of the ideal structure, while the second term represents the correction due to boundaries. One can see that the second term depends on the parameters s0 and sN which determine the type of boundary conditions. Performing the standard transition from the sum to integral in the expression (13), one can calculate the specific heat per unit cell CV ˆ

kB 2U : Nx Ny …Nz ⫹ 1†u0 2t

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Fig. 1. (a) The values of mean-square displacements over the film layers …Nz ˆ 9† at the temperature t ˆ 3; for different models of boundary conditions. Solid line corresponds to the bulk structure. (b) Temperature dependence of the mean-square displacements. …Nz ˆ 9; n ˆ 0; 1; 2†:

Fig. 2. (a) The values of mean-square velocities over the film layers …Nz ˆ 9† at the temperature t ˆ 3. (b) Temperature dependence of the mean-square velocities in the film boundary layer.

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Fig. 3. Specific heat of the crystal lattice of ten-layered film: (a) low temperatures; (b) intermediate temperatures (1) s0 ˆ sN ˆ ⫺0:5; (2) s0 ˆ sN ˆ 0; (3) s0 ˆ sN ˆ 1:

In this way, one obtains CV, in the units of [kB]: " 2 NX z ⫹1 3 D 2D ⫹ s0 g0;0 ⫹ sN gNz ;Nz CV …t† ˆ ⫺ p…Nz ⫹ 1† n ˆ1 4t eD=t ⫺ 1 ⫹

p 1 Dn 2Dn ⫹ s0 g0;0 ⫹ sN gNz ;Nz p ⫺ …6D2 ⫹ s0 g0;0 Dn =t 4t 4 e ⫺1

p 1 ⫹ sN gNz ;Nz †Z1 …D=t†⫹ …6Dn ⫹ s0 g0;0 ⫹ sN gNz ;Nz †Z1 … Dn =t† 4 p p ⫺ 3tDZ2 …D=t† ⫹ 3t Dn Z1 … Dn =t† ⫺ 3t2 Z3 …D=t† # p 2 ⫹ 3t Z3 … Dn =t† (14)

Ref. [4] remains valid: specific heats of the bulk and film intersect at two points. One should however stress that the model of free surfaces leads to the negative specific heat at extremely low temperatures, so its correctness is the subject of further discussion. It is our opinion that this might be an artefact due to the crudeness of our approximation (neglecting of surface reconstruction and other effects), but also it might be the general weakness of free-surface model. Fig. 3(a) is based on the values of parameters …s0 ˆ sN ˆ ⫺0:5†; which lead to the positive specific heat. Further lowering of the surface parameters leads to the negative values for CV close to absolute zero.

4. Conclusion We have calculated, using expression (14) the specific heat for the ten layer film at extremely low temperatures, Fig. 3(a), and for somewhat higher temperatures, Fig. 3(b). One can conclude that the behaviour of the specific heat substantially depends on the boundary conditions. For the case s0 ˆ sN ˆ 0; the results are compatible with the results of the paper [4]. At extremely low temperatures, the specific heat of the film is lower than the specific heat of the bulk, then the curves intersect, and at intermediate temperatures film has the higher value of the specific heat. No plot was made for extremely high temperatures, but it is easy to estimate that at these temperatures, the specific heat of the bulk is higher than the specific heat of the film. So, the conclusion from

Lattice dynamics and thermodynamical properties of ultrathin films were analysed by the Green’s functions method. This approach enables the consistent derivation of several statistical characteristics of the system, like meansquare of the molecular displacements and velocities, internal energy and specific heat of the phonon subsystem of thin films. The analysis of the thermodynamical characteristics of thin films has indicated a large influence of the boundary conditions. Most often used till now, the model of free surfaces, gives results opposite to all other models. This model has mean-square of the displacements and the velocities at boundaries higher than in the bulk. All other models lead to the values of these quantities lower than in

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the bulk. As for the internal energy and specific heat, free surfaces model leads also to unacceptable results, as negative values of the specific heat at extremely low temperatures. All other models lead to the expected results: at extremely low and extremely high temperatures, the film possesses lower specific heat than the bulk. In the range of intermediate temperatures, the specific heat of the film is somewhat higher than the specific heat of the bulk. These results agree with the results of the paper [4] where the calculation was performed by a different method. From the point of view of direct application, the most important conclusion is that one can influence the thermodynamical characteristic of the film by the production technology. Higher specific heat in the case of insulating films improves their dielectric properties. Detailed calculations of the dielectric properties of thin films of molecular crystals [12], and in particular crystalline SiO2 films, are in progress.

Acknowledgements We would like to thank Prof. D. Rakovic for critical reading of the manuscript.

References [1] S.B. Lazarev, D.Lj. Mirjanic´, M.R. Pantic´, B.S. Tosˇic´, J.P. Sˇetrajcˇic´, J. Phys. Chem. Solids 60 (1999) 849. [2] H.S. Wong, K.K. Chang, Y. Lee, P. Doper, Y. Taur, IEEE Trans. Electron Devices 44 (1997) 1131. [3] M.V. Fischetti, D.J. DiMaria, S.D. Brorsob, T.N. Theis, J.R. Kirtley, Phys. Rev. B 31 (1985) 8124. [4] B.S. Tosˇic´, J.P. Sˇetrajcˇic´, D.Lj. Mirjanic´, Z.V. Bundalo, Physica A 184 (1992) 354. [5] S.B. Lazarev, M. Pantic´, B.S. Tosˇic´, Physica A 246 (1997) 53. [6] B.S. Tosˇic´, M. Pantic´, S.B. Lazarev, J. Phys. Chem. Solids 58 (1997) 1995. [7] S.B. Lazarev, PhD thesis, University of Novi Sad, Serbia, Yugoslavia, 1997. [8] M. Pantic´, Lj.D. Masˇkovic´, B.S. Tosˇic´, Int. J. Mod. Phys. B 12 (1998) 177. [9] S.G. Davison, M. Steslika, Basic Theory of Surface States, Clarendon Press, Oxford, 1996. [10] R.F. Elsaeyd-Ali, J. Appl. Phys. 79 (1996) 6853. [11] A. Isihara, Statistical Physics, Academic Press, New York, 1971. [12] J.P. Sˇetrajcˇic´, S.M. Stojkovic´, S.B. Lazarev, I.D. Vragovic´, D.Lj. Mirjanic´, Dielectric properties of thin molecular films, Proceedings of DRP’98, Bielsko-Biala, Poland, in press.