Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 5 where: H S = 1 2M ∑ α ∑ n x ,n y   p α n x ,n y ,0  2 +  p α n x ,n y ,N z  2  + + 1 4 ∑ α C α ∑ n x ,n y  2  u α n x ,n y ,0  2 + 2  u α n x ,n y ,N z  2 + +  u α n x ,n y ,N z −1 −u α n x ,n y ,N z  2 +  u α n x ,n y ,1 −u α n x ,n y ,0  2 + +  u α n x ,n y ,0 −u α n x +1,n y ,0  2 +  u α n x ,n y ,0 −u α n x −1,n y ,0  2 + (4) +  u α n x ,n y ,0 −u α n x ,n y +1,0  2 +  u α n x ,n y ,0 −u α n x ,n y −1,0  2 + +  u α n x ,n y ,N z −u α n x +1,n y ,N z  2 +  u α n x ,n y ,N z −u α n x −1,n y ,N z  2 + +  u α n x ,n y ,N z −u α n x ,n y +1,N z  2 +  u α n x ,n y ,N z −u α n x ,n y −1,N z  2  ; H B = 1 2M ∑ α ∑ n x ,n y  p α n x ,n y ,n z  2 + 1 4 ∑ α C α × × ∑ n x ,n y  N z −1 ∑ n z =1   u α n x +1,n y ,n z −u α n x ,n y ,n z  2 +  u α n x −1,n y ,n z −u α n x ,n y ,n z  2 + +  u α n x ,n y +1,n z −u α n x ,n y ,n z  2 +  u α n x ,n y −1,n z −u α n x ,n y ,n z  2  + (5) + N z −2 ∑ n z =2   u α n x ,n y ,n z +1 −u α n x ,n y ,n z  2 +  u α n x ,n y ,n z −1 −u α n x ,n y ,n z  2  + +  u α n x ,n y ,N z −1 −u α n x ,n y ,N z −2  2 +  u α n x ,n y ,1 −u α n x ,n y ,2  2  . We have decided to use the approach of Heisenberg’s equations of motion (Toˇsi´c et al., 1992; ˇ Setrajˇci´c et al., 1992; ˇ Setrajˇci´c&Panti´c, 1994) for the determination of possible frequencies (energy spectrum) and the states of phonons. We start from the following system of the equations of motion for the phonon displacements: –forn z = 0 ¨ u α n x ,n y ,0 − Ω 2 α  u α n x +1,n y ,0 + u α n x −1,n y ,0 + u α n x ,n y +1,0 + + u α n x ,n y −1,0 + u α n x ,n y ,1 −6u α n x ,n y ,0  = 0; (6) –for1 ≤ n z ≤ N z −1 ¨ u α n x ,n y ,n z − Ω 2 α  u α n x +1,n y ,n z + u α n x −1,n y ,n z + u α n x ,n y +1,n z + + u α n x ,n y −1,n z + u α n x ,n y ,n z +1 + u α n x ,n y ,n z −1 −6u α n x ,n y ,n z  = 0; (7) 321 Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 6 Thermodynamics –forn z = N z ¨ u α n x ,n y ,N z − Ω 2 α  u α n x +1,n y ,N z + u α n x −1,n y ,N z + u α n x ,n y +1,N z + + u α n x ,n y −1,N z + u α n x ,n y ,N z −1 −6u α n x ,n y ,N z  = 0. (8) where Ω α = √ C α /M. The solution of this system of N z+1 homogeneous differential-difference equations for phonon displacements can be looked for in the form of the product of an unknown function (along z-axis) and harmonic function of the position (within XY-plane) known from the bulk solutions, i.e. I u α n x ,n y ,n z (t)= ∑ k x ,k y ,k z +∞  −∞ dω e ia(k x n x +k y n y )−itω Φ α n z ; Φ α n z ≡ Φ α n z ( k z , ω ) .(9) Substituting this expression into the equations 6–8 we obtain: R Φ α 0 + Φ α 1 = 0 Φ α 0 + R Φ α 1 + Φ α 2 = 0 ··· · Φ α n z −1 + R Φ α n z + Φ α n z +1 = 0 (10) ··· · Φ α N z −2 + R Φ α N z −1 + Φ α N z = 0 Φ α N z −1 + R Φ α N z = 0 where: R ≡ W 2 α −4 F k x k y −2; W α ≡ ω Ω α ; F k x k y ≡ sin 2 ak x 2 + sin 2 ak y 2 . (11) In this way the system of N z + 1 differential-difference equations 6–8 turns into a system of N z + 1 homogeneous algebraic difference equations 10. In order that this system possesses nontrivial solutions, its determinant: D N z +1 (R)=                R 100 ··· 000 0 1 R 10 ··· 000 0 01R 1 ··· 000 0 ···· . . . ···· 0000··· 1 R 10 0000 ··· 01R 1 0000 ··· 001R                (12) must vanish. The roots (poles) of this determinant represent one of the forms of Chebyshev’s polynomials of the second order (Cottam & Tilley, 1989) and can be written in the form: D N z +1 (R) ≡D N I z −1 (R)= sin ( N I z ζ ) sinζ ; N I z = N z + 2; ζ = 0, where: R = 2cosζ. Above mentioned condition (D N z +1 (R ν )=0) is satisfied for: ζ ν = πν N I z ; ν = 1,2,3, , N I z −1 , (13) 322 Thermodynamics Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 7 whose substitution into expressions 11 leads to the expression for demanded (possible) unknown phonon frequencies: I ω α k x k y (μ)=2 Ω α  G I μ + F k x k y , (14) where: G I μ ≡ sin 2 ak z (μ) 2 ; k z (μ)= π a μ N I z ; μ ≡ N I z −ν = 1,2, 3, , N I z −1 . (15) Onemustnoticethatcontrarytok x and k y which range from 0 to π/a, one has: k min z ≡ k z (1)= π a 1 N I z > 0; k max z ≡ k z (N I z −1)= π a N I z −1 N I z < π a , (16) because N I z   N x , N y  . If one divides the system of equations 10 by Φ α 0 ≡ Φ α 0 (k z ) and rejects the last equation, this system is obtained in the new form: R ν +  1 = 0, for n z = 0 1 + R ν  1 +  2 = 0, for n z = 1  n z −1 + R ν  n z +  n z +1 = 0, for 2≤n z ≤ N z −1 (17) where R ν ≡ 2cosζ ν and:  n z ≡  α n z = ( Φ α 0 ) −1 Φ α n z =⇒ Φ α n z = Φ α 0  α n z . (18) The last of the equations 18 is satisfied for:  n z =(−1) n z { P sin ( n z ζ ν ) + Qsin [( n z −1 ) ζ ν ]} , (19) and using this and expression 13 it follows:  1 = −P sin(ζ ν );  2 = P sin(2 ζ ν )+Q sin(ζ ν ). Substituting these expressions into the first and second equation in the system od difference equations 18 we arrive to the unknown coefficients P ≡ P ν = R ν sin −1 ζ ν and Q ≡ Q ν = − sin −1 ζ ν , while returning them into expressions 19 and 18, it follows: Φ α n z (k z )=(−1) n z sin [( n z + 1 ) ζ ν ] sinζ ν Φ α 0 . (20) According to above calculations – combining 9, 20 and standard normalization (Callavay, 1974), one can easily obtain the final expression for phonon displacements in the form: I u α n x ,n y ,n z (t)= ∑ k x k y N I z −1 ∑ μ=1 I N α n z  k x k y , μ  × × e ia(k x n x +k y n y )−itω α k x k y (μ) sin [( n z + 1 ) ak z (μ) ] ; (21) I N α n z  k x k y , μ  =(−1) n z  ¯h MN x N y N I z ω α k x k y (μ) . 323 Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 8 Thermodynamics Comparing the result obtained here with the corresponding one for ideal infinite structures, one can conclude that mechanical vibrations in the ideal unbounded structure are plane waves in all spatial directions, while in the thin film they represent the superposition of the standing waves in z-direction and plane waves in XY-planes. It is also evident that the displacement amplitude in the films is ∼ 10 4 √ 2/N I z times larger 2 than the amplitude in corresponding unbounded structures. Using expression 14 one can determine the dispersion law for phonons in thin undeformed ideal film 3 : I E α k x k y (μ) ≡ ¯h I ω α k x k y (μ)=E α I  G I μ + F k x k y , (22) where E α I = 2¯h Ω α and which is valid together with expressions 11 and 15. We represent graphically this energy spectrum at the Fig.2 vs XY-plane vector k 2 = k 2 x + k 2 y : I E z μ ≡  I E z k x k y (μ)/E z I  2 = I E z k z (μ)  F k x k y  . Fig.2 represent energy spectrum of phonons in the ideal (ultrathin N z = 4) crystalline films vs. two-dimensional (XY planar) wave vector. Within the band of bulk energies with continual spectrum (bulk limits are denoted by solid dashed lines) one can notice five allowed discrete phonon energies in the film studied (thin solid lines). One can notice the narrowing of the energy band and the existence of the energy gap. 05,10,15,20, 10, 05, 15, R xy E u D i Fig. 2. Phonon spectra in the ideal ultrathin crystalline films 2 For very thin films N I z ∼ 10, so the factor of the amplitude increase can achieve even 2000. 3 Most common treatment is that using classical procedure, for example, second quantization method (Callavay, 1974), on the basis of expressions 3–??, 14 and 20, the Hamiltonian H IF is diagonalized, and then the energy spectrum in the form 22 is readily obtained. 324 Thermodynamics Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 9 One can clearly see from the plot explicate discreteness of the allowed energy levels of phonons in the ideal film with respect to the continuum of these values for the corresponding bulk-structures. All three acoustic frequencies in bulk-structures vanish when three-dimensional (spatial) vector k =     k    vanishes, while the minimal frequencies of phonons in the thin ideal film-structure are: Δ i ≡ ( ω I α ) min ≡ ω I α (k x = k y = 0, k z = k min z ) ≈ Ω α π N I z > 0. (23) On the other hand, maximal values of the frequencies of acoustic branches in the ideal infinite crystal tend to the value ( ω B α ) max = 2Ω α √ 3whenk α → π/a, α = x, y,z, while in the studied ideal film they are: ( ω I α ) max ≡ ω I α  k x = k y = π a , k z = k max z  ≈ 2Ω α √ 3  1 − π 2 /12 ( N I z ) 2  < ( ω B α ) max . (24) It can be also seen from the same figure that the width of the energy band in the film is narrower. From expressions 23 and 24 we can determine the total narrowing of the band of allowed energies of the phonons in the film-structures with respect to the bulk band: W I α ≡ ¯h {( ω B α ) max − [( ω I α ) max − ( ω I α ) min ]} ≈ ¯h Ω α π ( N I z + 1 ) ( N I z ) 2 > 0. (25) The functional behavior and the physical explanation, as well as the effects that might be caused by the existence of the frequency threshold 23 and the band narrowing 25 will be exposed in the next Section after the analysis of the phonon behavior in the deformed structures. 2.2 Deformed film-structures During sputtering perpendicular to the boundary surfaces (Cava et al., 1987; Chu et al., 1987; Politis et al., 1987; Segre et al., 1987; Dietrich et al, 1987; Kuwahara, 1992; Notzel et al., 1992), the atoms that are introduced locate themselves along z-axis since this is energetically most convenient: a z = 3a and a x = a y = a. For that reason the effective mass of the atom (as well as the elongation constant), at some site in the crystalline lattice, will depend on its relative position (with respect to the origin of z-axis), i.e. on the lattice index in the z-direction, but not in x and y directions: M n ≡ M n x ,n y ,n z = M n z . Using the Heisenberg equations of motion for u and p with the Hamiltonian (given by expression 2), taking into account above mentioned conditions, one arrives to the system of N z + 1 homogeneous differential-difference equations for the phonon displacements: –forn z = 0 ¨ u α n x ,n y ,0 = C α 0 M 0  u α n x +1,n y ,0 + u α n x −1,n y ,0 + u α n x ,n y +1,0 + u α n x ,n y −1,0 + u α n x ,n y ,1 −5u α n x ,n y ,0  + (26) + C α 1 2M 0  u α n x ,n y ,1 −u α n x ,n y ,0  − C α −1 2M 0 u α n x ,n y ,0 , 325 Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 10 Thermodynamics –for1≤ n z ≤ N z −1 ¨ u α n x ,n y ,n z = C α n z 2M n z  2  u α n x +1,n y ,n z + u α n x −1,n y ,n z + u α n x ,n y +1,n z + u α n x ,n y −1,n z  + + u α n x ,n y ,n z +1 + u α n x ,n y ,n z −1 − 10u α n x ,n y ,n z  + (27) + C α n z +1 2M n z  u α n x ,n y ,n z +1 −u α n x ,n y ,n z  + C α n z −1 2M n z  u α n x ,n y ,n z −1 −u α n x ,n y ,n z  , –forn z = N z ¨ u α n x ,n y ,N z = C α N z M N z  u α n x +1,n y ,N z + u α n x −1,n y ,N z + u α n x ,n y +1,N z + + u α n x ,n y −1,N z + u α n x ,n y ,N z −1 −5u α n x ,n y ,N z  + (28) + C α N z −1 2M N z  u α n x ,n y ,N z −1 −u α n x ,n y ,N z  − C α N z +1 2M N z u α n x ,n y ,N z . The solution of this system of equations can be looked for in the form of the product of an unknown function in z-direction and plane harmonic waves in XY-planes: D u α n x ,n y ,n z (t)= ∑ k x ,ky,k z +∞  −∞ dw e ia ( k x n x +k y n y ) −it ω Ψ α n z ; Ψ α n z ≡ Ψ α n z ( k z , ω ) . (29) Calculating corresponding derivatives and introducing them into equation 27 we obtain the difference equation for the unknown functions Ψ α n z : 2  M n z ω 2 −4 C α n z F k x k y  Ψ α n z + C α n z  Ψ α n z +1 + Ψ α n z −1 −2 Ψ α n z  + (30) +C α n z +1  Ψ α n z +1 −Ψ α n z  + C α n z −1  Ψ α n z −1 −Ψ α n z  = 0 valid for n z = 1,2,3, , N z − 1. For n z = 0, using the same procedure, one obtains from equation 26 the first (upper) boundary equation: 2  M 0 ω 2 −4 C α 0 F k x k y  Ψ α 0 + C α 0 ( Ψ α 1 −2 Ψ α 0 ) + (31) C α 1 ( Ψ α 1 −Ψ α 0 ) −C α −1 Ψ α 0 = 0 and for n z = N z , using equation 28, the second (lower) boundary equation: 2  M N z ω 2 −4 C α N z F k x k y  Ψ α N z + C α N z  Ψ α N z −1 −2 Ψ α N z  + (32) +C α N z −1  Ψ α N z −1 −Ψ α N z  −C α N z +1 Ψ α N z = 0. The procedure for the determination of the allowed phonon states using the above equations is extremely complicated 4 . For that reason we are going to perform the transition to continuum 4 In fact, it is not known or elaborated in the literature for this, completely general case. 326 Thermodynamics Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 11 (continuum approximation), i.e. transition from the discrete to continual variables, and expand the corresponding quantities into the Taylor’s series: n z −→ z; Ψ α n z −→ Ψ α (z); M n z −→ M(z); a z −→ a(z); C α n z −→ C α (z). Besides that, as a consequence of sputtering, i.e. clustering of foreign atoms around the atoms of the basic matrix (Toˇsi´c et al., 1987; ˇ Setrajˇci´c et al., 1990; Toˇsi´c et al., 1992; Ristovski et al., 1989), the mass of the basic matrix must be substituted by the corresponding reduced mass: M −1 (z)=M −1 m + n(z) M −1 d , (33) where: M m – is the mass of the basic matrix, M d – the mass of doping atoms and n(z) –their number at the site z (measured from the lower boundary surface of the crystalline film). After these transformations and introduction into the difference equation 30, it becomes a second order differential equation: d 2 Ψ α (z) dz 2 + C −1 α (z) d C α (z) dz d Ψ α (z) dz −  4 F k x k y a 2 (z) − M(z) ω 2 C α (z) a 2 (z)  Ψ α (z)=0. (34) Further solving of this differential equation demands the specification of the functional dependence of the quantities M(z), C α (z) and a(z), and they depend not only on the procedure of the sputtering of the basic matrix – ideal crystalline film-structure, but also on the number, type and distribution of the sputtered atoms. 2.2.1 Asymmetrical deformation Taking into account that the production of oxide superconductive ceramics includes the sputtering with foreign atoms (Cava et al., 1987; Chu et al., 1987; Politis et al., 1987; Segre et al., 1987; Dietrich et al, 1987; Kuwahara, 1992; Notzel et al., 1992; Johnson, 1995), we shall assume that it is performed perpendicularly to one (upper) of the boundary surfaces of the model film-structure. For this reason, doping atoms cluster along z-direction, from this upper surface towards lower boundary surface and let us assume the (approximate) parabolic distribution of such ”weighted” atoms, i.e. their reduced masses: M (z) −→ M A (z)=A A M + B A M (z − L) 2 . Using boundary conditions: M A (0)=M m ; M A (L)= M m M d M d −nM m , we determine the unknown coefficients A A M and B A M , so that we obtain: M A (z)= M m O A M  1 −  1 −O A M   1 − z L  2  ; O A M = 1 − n M m M d . (35) The sputtering of the basic matrix causes also (parabolic) change of the lattice constant: a (z) −→ a A (z)=A A a + B A a (z − L) 2 , 327 Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 12 Thermodynamics with boundary conditions: a A (0)=a z ; a A (L)= a z n , from which it follows: a A (z)=a z  1 − n −1 n z L  2 − z L   . (36) Since Hooke’s constants may be expressed as C α (z)=const(α) a −γ (z), using expression 36 one can write: C A α (z)=C α  1 + γ n −1 n z L  2 − z L   . (37) Furthermore, instead of a A (z) and C A α (z) we shall use their values averaged over the total film width (L ): a A z ≡ a A (z)= 1 L L  0 a A (z) dz = f A n a z ; f A n ≡ n + 2 3n ; C α A ≡ C A α (z)= 1 L L  0 C A α (z) dz = g A n C α z ; g A n ≡ 1 + 2γ n −1 3n . (38) To simplify the solution of the last differential equation, besides 35 and 38, it is convenient to change variable z → η:1−z/L = Λ η, so that it becomes: d 2 Ψ A α dη 2 + K A α   1 −  Λ A α  2  1 −O A M  η 2  ω 2 −4 O A M M m C α A F k x k y  Ψ A α = 0. (39) Introducing new notations: K A α = L 2  Λ A α  2 M m a A z 2 C α A O A M ; Λ A α =  a A z Ω A α L ω ; Ω A α = Ω α τ A α (n) ; τ A α (n)=     g A n O A M 1 −O A M (40) and A Q α k x k y (ω)= L a A z  ω Ω A α  1 −O A M  −1 −4 Ω A α ω F k x k y  ≡ ≡ 2s + 1; s = 0,1,2, , (41) the above differential equation can be turned into Hermit-Weber one (Callavay, 1974): d 2 Ψ A α dη 2 +  A Q α k x k y (ω) −η 2  Ψ A α = 0, (42) with the solution: Ψ A α ≡ Ψ A α (η) −→ A Ψ α s (η)= A N α s H s (η)e −η 2 /2 , (43) where H s (η) – is Hermitian’s polynomial of the order s. In order that atomic displacements remain finite, it is necessary that A Q α k x k y (ω) satisfies the identity condition (expressed by 41) which, in fact, insures the physics-chemical (crystallographic) stability of the model 328 Thermodynamics Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 13 film-structure. This identity allows the determination of the allowed vibrational frequencies of the system: A ω α k x k y (s)= Ω A α 2  G A s +   G A s  2 + F k x k y  , (44) where F k x k y is defined in 11, and G A s ≡ 2s + 1 N A z ; N A z = L a A z  1 −O A M  −1 . It is clear from this expression that none of the possible frequencies A ω α k x k y (s) vanishes, neither for s = 0, nor for (dimensionless) twodimensional vector q = a −1  k 2 x + k 2 y → 0. Since we have solved Hermite-Weber’s equation 42 without taking into account the boundary conditions, it must be supplied by two boundary equations 31 and 32, for z = 0andz = L,i.e. its solution 43 must satisfy these supplementary conditions. The substitution of 43 into 31 for z = 0 ⇒ η =  Λ A α  −1 and q = 0gives:  2 −O A M  A ω α 0 (s) Ω A α  2  H s  1 Λ A α  = exp  1 − a A z 2L  A ω α 0 (s) Ω A α  H s  L − a A z Λ A α L  . (45) By analogous procedure, solution expressed by 43 with boundary equation 32, for z = L ⇒ η = 0andq = 0, gives:  2 +  1 −O A M   A ω α 0 (s) Ω A α  2  H s (0)=exp  a A z 2L A ω α 0 (s) Ω A α  H s  a A z Λ A α L  . (46) Using the relation H s (x + c)=H s (x)+(2c) s one can write: H s  L − a A z Λ A α L  =(−1) s H s  a A z Λ A α L  +  2 Λ A α  s , after which the equations 45 and 46 turn into a single one:  2 −O A M  A ω α 0 (s) Ω A α  2   exp  A ω α 0 (s) Ω A α  H s  1 Λ A α  +(−1) s+1 H s (0)  = =  (−1) s  A ω α 0 (s) Ω A α  2 H s (0)+  Λ A α  −s exp  a A z 2L A ω α 0 (s) Ω A α  . (47) It is obvious from here that the parameters M m , M d , n, L and quantum number s are not mutually independent. In fact, for given values, from expression 44 they define the conditions for the existence of phonon states with the energies ¯h A ω α k x k y (s). From this equation, one can determine for which value of quantum number s the function A ω α 0 (s) attains minimal value. A graphical-numerical solving method gives s min = 2. Numerical calculations and estimates were performed for the compound La ( Ba 2−ε La ε ) Cu 3 O 7+δ , where it was taken n = 3, γ = 12, O A M = 1/0,83, ε = 0,125, δ = 0, 11, v z ≈ 3, a z ≈ 1, 2; all based on data from 329 Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 14 Thermodynamics ( ˇ Setrajˇci´c et al., 1990; Cava et al., 1987; Chu et al., 1987; Politis et al., 1987; Segre et al., 1987; Dietrich et al, 1987; Kuwahara, 1992; Notzel et al., 1992; Johnson, 1995) and (Ristovski et al., 1989; Djaji´c et al., 1991; ˇ Setrajˇci´c et al., 1994). Due to the discreteness of the solutions (2.43) and the initial model, their total number must be equal to N z + 1. It follows from here that the quantum number s is bounded also from above: s max = N z + 2, i.e. s ∈ [ 2, N z + 2 ] . Substituting of the solution expressed by 43 into difference equation 30, and normalizing it, the expression for the phonon displacements becomes: A u α n x ,n y (z, t)= ∑ k x k y N A z −2 ∑ s=2 A N α n z  k x k y , s  H s (z)e −2 (1−z/L) 2 /(2Λ A α ) 2 × × e i  a ( k x n x +k y n y ) −t A ω α k x k y (s)  ; (48) A N α n z  k x k y , s  ≡ (−1) n z  ¯h M z N x N y N A z A ω α k x k y (s) ; N A z = N z + 4. The analysis of this expression shows that, contrary to phonon displacements in ideal unbounded structures (plane waves in all three spatial directions), and similar to the ideal films (superposition of standing wave and plane waves), here they represent the superposition of the plane waves in XY-planes and collective vibrational harmonic motion along z-direction. The amplitude of the phonon displacements is here ∼ 10 4 √ 2/N A z times larger than the corresponding one in the bulk structures, and approximately equal (in fact slightly smaller) than in the ideal films 5 . According to all above mentioned, it follows from expression 44 that the dispersion law for phonons in the asymmetrically deformed crystalline films has the following form: A E α k x k y (s) ≡ ¯h A ω α k x k y (s)=E α A  G A s +   G A s  2 + F k x k y  , (49) where E α A ≡ ¯h Ω A α /2, and s = 2,3,4, , N A z −2. Graphical presentation of this dispersion law in the form A E z s ≡  A E z k x k y (s)/E z A  2 = A E z s  F k x k y  is given in the Fig.3. Fig.3 represent the energy spectrum of phonons in the asymmetrically deformed (ultrathin N z = 4) crystalline films vs. two-dimensional (XY planar) wave vector. Besides the narrowing of the energy band with five discrete levels and the presence of the energy gap (with respect to the bulk band denoted by solid dashed lines) a shift of this band outside bulk limits can be noticed, corresponding to the appearance of the localized phonon modes. One can see from this plot that non of the allowed energies, i.e. possible frequencies A ω α q (s) does not vanish for q → 0, implying that the presence of boundaries together with the deformation of the atom distribution of the parabolic type (expressed by 35–38) leads to the appearance of the energy gap in the phonon spectrum, i.e. to the possible creation of the phonons of only the optical type. Contrary to the dispersion law for phonons in unbounded and nondeformed structures, where minimal and maximal frequency of the acoustic phonon 5 See the comment bellow the expression 22. 330 Thermodynamics [...]... finite thickness Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 27 343 It is known that the phonon part in thermal capacitance of the system is described with cubic temperature dependence By introducing nondimensional reduced temperature, Δf 3 3 12 4 x For comparison... cm, we find ΔE ∼ 1 meV, which is observable, experimentally accessible quantity Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 21 337 D Ds 0,4 Da (s) 0,3 (a) Di (i) 0,2 0,1 4 6 8 10 12 14 Nz Fig 5 Phonon energy gap vs ultrathin film thickness mean the elimination of the... 1992; Notzel et al., 1992; ˇ Johnson, 1995) and (Ristovski et al., 1989; Djaji´ et al., 1991; Setrajˇ i´ et al., 1994) for the compound c cc Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 19 335 Eu 1,5 1,0 0,5 Ds 0,5 1,0 1,5 2,0 R xy /vy Fig 4 Phonon spectra in the...Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 15 331 Eu 1,5 1,0 0,5 Da 0,5 1,0 1,5 2,0 R xy Fig 3 Phonon spectra in the asymmetrically... least some light in the shadows surrounding the mechanism of high-temperature superconductivity, so that is the reason for our choice of data Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 23 339 be expressed as: NF = NF 2 2 F a k ; NF = Nx Ny Nz 4π (71) The density... matrix elements (V ) of the effective electron-electron interaction within the framework of the BCS theory there occur Debye’s frequencies: V ∼ Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 25 341 − ω D 1/2 , which, taking into account the results obtained, undoubtedly... Fk x k y Ψα = 0 , (56) Mm dζ 2 2 S S Kα = L2 Λα 2 Mm az Cα O M S S S S S ; Λα = S aS Ωα z S S ; Ωα = Ωα τα (n ) ; τα (n ) = 2 Lω S gn O M S Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 17 333 Since the atoms in the studied film (along z-direction) represent the system... same material with the same crystalline structure This statement, which is an experimental fact is supported by the following of our results Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 29 345 1 In the films there appear standing phonon waves along z-directions, the... Sajfert V.D & Toˇ i´ B.S (1992) Perturbation Method in the cc Analysis of Thin Deformed Films and a Possible Application Physica A, Vol 190, Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 31 347 No 3-4, December 1992, 363–374, ISSN 0378-4371 ˇ Setrajˇ i´ J.P & Panti´... Sajfert V.D (2004) Thermodynamics of Mechanical c cc sc Oscillations in Superlattices Materials Science Forum, Vol 453–454, September 2004, 33–36, ISSN 0255-5476 ˇ Setrajˇ i´ J.P.; Zori´ V.M.; Vuˇ enovi´ S.M.; Mirjani´ D.Lj.; Sajfert V.D.; Ja´ imovski S.K & Ili´ D.I cc c c c c c c (2007) Phonon Thermodynamics in Crystalline Nanofilms Materials Science Forum, 32 348 Thermodynamics Thermodynamics Vol . γ = 12, O A M = 1/0,83, ε = 0 ,125 , δ = 0, 11, v z ≈ 3, a z ≈ 1, 2; all based on data from 329 Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 14 Thermodynamics ( ˇ Setrajˇci´c. constant: a (z) −→ a A (z)=A A a + B A a (z − L) 2 , 327 Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 12 Thermodynamics with boundary conditions: a A (0)=a z ;. experimentally accessible quantity. 336 Thermodynamics Phonon Participation in Thermodynamics and Superconductive Properties of Thin Ceramic Films 21 4681 0121 4 0,1 0,2 0,3 0,4 N z D (i) (a) (s) D D D s a i Fig.