FOREWORD Reasons for choosing topic On the investigations of the heat capacity of the free electron gas in metals, most of theoretical calculations are not consistent with experimental results The reasons that may be explained are the impurities and defects of crystals or approximate theory calculations The approximate methods have their own limitations Such as, perturbation theory can not easily found several physical phenomena as spontaneous symmetry breaking, phase-state transition … That requires new non-disturbance methods such as density-functional methods, Green-function method, ab initio method, algebraic deformation theory, statistical moment method,…which include all orders in perturbation theory and maintain nonlinear elements In recent years, algebraic deformation theory has attracted the attention of many theoretical physicists because these new mathematical structures are suitable for many theoretical physics problems such as quantum statistics, nonlinear optics, condensed matter physics, Algebraic deformation theory has been applied in field theory and elementary particle, especially, in nuclear physics It succeeded in researching and explaining problems related to the bosons In this thesis, we choose the algebraic deformation theory to investigate the fermion system Specifically, we use this theory to study the heat capacity and paramagnetic susceptibility of the free electron gas in metal at low temperatures Thin film is fascinating, exciting material which attracts the interest of many scientists both in theory and in experiment due to its wide applications Nanomaterials have different properties comparing to with bulk materials Nowaday, thin film material is widely used in many fields such as cutting tools, medical implants, optical elements, integrated circuits, electronic devices,… There are many different methods have been used to study the thermodynamic properties of metallic thin films Although these methods have achieved some certain results but they still don’t include the effect of anharmonic lattice vibrations In recent years, statistical moment method have been used successful in studying the thermodynamic properties and elasticity of crystals including the anharmonicity of lattice vibrations In this thesis, we apply the statistical moment method to study the thermodynamic properties of metallic thin films for the first time However, statistical moment method is not suitable to study the thermodynamic properties and magnetic properties of the free electron gas in metal With all of the reasons described above, we apply the q-deformed algebraic theory to study heat capacity and paramagnetic susceptibility of the free electron gas in metal at low temperatures and the statistical moment method to investigate the thermodynamic properties of metallic thin films The thesis title is "Applications of q-deformed Fermi-Dirac statistics and statistical moment method to study thermodynamic properties, magnetic properties of metals and metallic thin films" Purpose, object and scope of studying Apply of q-deformed Fermi-Dirac statistics to research heat capacity and paramagnetic susceptibility of free electron gas in metals at low temperatures, formula of heat capacity and magnetic susceptibility from the free electron gas in a metal depends on q deformation Use statistical moment method to study thermodynamic properties of metal thin films, build free energy calculation formula, build thermodynamic quantities of metal thin films determination theories, apply to metal thin films which have the face-centered cubic structure and body-centered cubic structure Influence of surface, size effect, the dependence on temperature, pressure on thermodynamic properties of metal thin films have also been considered From the obtained analytical results, we performed the numerical calculation for alkali metals, transition metals, metallic thin films We also make the comparing between the theoretical and experimental result to verify the reliability of the chosen method Research methodology In this thesis, we applied two methods: Algebraic deformation method: Based on this method, the q-deformed Fermi-Dirac statistics has been built We applied this statistics to investigate heat capacity and paramagnetic susceptibility of the free electron gas in metals Statistical moment method: This method is used to build the theory for calculating the thermodynamic properties of metallic thin films which have face-centered cubic structure and body-centered cubic structure We expanded approximately the interaction potential to the third and fourth orders of particle displacement from equilibrium position Based on these results, we determine the Helmholtz free energy of particles in metallic thin films Then we build the analytical expressions of thermodynamic quantities of metallic thin films such as thermal expansion coefficient, isothermal compression ratio, adiabatic compression ratio, isobaric heat capacity, isometric heat capacity, isothermal elastic modulus including the anharmonic effects, surface effects, size effects in different temperature and pressures Scientific and practical significance of the thesis • Making the investigation of fermion particles system with Fermi–Dirac deformation statistics we found heat capacity and paramagnetic susceptibility of free electron gas in metals at very low temperature From the shared values of the deformation parameter q of each metal group, we calculated the heat capacities for a series of alkali metals and transition metals • Initially constructing the torque statistical theory to calculate the thermodynamic quantities of metallic thin film; thermal expansion coefficient, isothermal compressibility, coefficient of adiabatic compression, the heats capacity, isothermal and adiabatic moduli • Investigating the dependence of the thermodynamic quantities on thickness, temperature and pressure of metallic thin films: Al, Cu, Au, Ag, Fe, W, Nb, Ta • Allowing the prediction of more information of thermodynamic properties of metallic thin films at various pressures, as well as other thin film materials such as Ni, Si, CeO2, • The success of the thesis has contributed to the perfection and development of the statistical moment theory in researching thermodynamic properties of metallic thin film Moreover, the theory can also be applied to study the elastic properties of metallic thin film New contributions of the thesis Successfully building the analytical expressions of heat capacity and paramagnetic susceptibility of free electron gas in metal based on deformation theory By developing statistical moment theory we study the thermodynamic properties of metallic thin films Constructing the analytical formulas of thermodynamic quantities for metallic thin films which have face-centered cubic (Al, Au, Ag, Cu) and body-centered cubic (Fe, W, Nb, Ta) structures depending on the temperature, thickness and pressure Numerical calculations have been performed and compared with other theoretical results and available experimental data to verify the correctness and effect of the theory The thesis also suggests us to develop the statistical moment method for studying the elastic properties of thin films Moreover, this theory can be developed to investigate the thermodynamic properties and elastic properties of other materials such as thin films mounted on the substrate, oxide thin films, semiconductors Thesis outline Beside the introduction, conclusion, references and appendices, the thesis is divided into chapters and 11 subsections Contents of the thesis is presented in 132 pages with 37 tables, 60 figures and charts, 121 references CHAPTER OVERVIEW OF STUDY SUBJECTS AND RESEARCH METHODS 1.1 Algebraic deformation method Symmetry is a common feature in many physical systems, the mathematical language of symmetry theory is group theory Quantum symmetry theory based on quantum group is one of the topical subjects in physics, attracting the attention of many theoretical physicists Lie group theory is a mathematical tool of symmetry theory which plays an important role in unifying and predicting physical phenomena In particular, Lie groups became key tools in field theory and elementary particle theory In order to apply Lie group for studying many problems of theoretical physics, Drinfeld V G quantized Lie group and then derived algebra deformation structure known as quantum algebra Algebraic structure of quantum group is described formally as a deformation q of algebra U(G) of the Lie algebra G, so that in the limit case of deformation parameter q → 1, the algebra U(G) returns to Lie algebra G Thus quantum algebra can be seen as a distortion of classical Lie algebra In recent decades the investigation of quantum algebra has been developed strongly and obtained many good results, it is attracted the attention of many theoretical physicists These new mathematical structures are suitable with many problems in theoretical physics such as the theory of quantum inverse scattering, exactly solvable model in quantum statistics, rational Conformal field theory, two sided field theory with fractional statistics This theory has gained many successes in researching and explaining the issues related to the Higgs particle In the early of twentieth century, after successfully buiding Bose – Einstein statistics, based on the characteristics of Bose system which is that particles in a state can be arbitrary like photons, πmesons, K-mesons , Einstein predicted that there exists a special state, so-called Bose – Einstein condensation state From experiments, physicists have found the transition temperatures of some superconducting materials In 2001 three American physicists have experimentally generated condensate with alkali metals, all three physicists were awarded the Nobel Prize, this discovery opens up new technologies for science In 1927, using the concepts of quantum mechanics to the micro system, Sommerfeld was the first one proposing the model of free electron gas in metal which uses Fermi - Dirac statistics instead of classical Maxwell – Boltzmann statistics In the case of particles with half-integer spin (so-called Fermion particles) such as electrons, protons, Neuton, positron there is only or particle on an energy level (in other words, all Fermion must have different energies), this restriction is so-called the Pauli exclusion principle, Fermion particles obey Fermi–Dirac statistics Quantum groups and quantum algebra are surveyed conveniently in forms of deformed harmonic oscillator Representation theory of quantum algebra with a deformation parameter leading to the development of q deformation algebra in formalism of deformed harmonic oscillator Quantum algebra SU(2)q depends on the first parameter proposed by the research N Y Reshetikhiu when he used the quantum equation Yang-Baxter to investigate other quantum systems The investigation of deformed harmonic oscillator is fueled by more and more attention to the particles complying with statistical theories which are different from Bose-Einstein statistics and Fermi-Dirac statistics, especially para Bose statistics and para Fermi statistics as expanded statistics Para statistical particles are called para particle Since the appearance of para statistical theory many efforts have been done to expand the canonical commutation relations However, up to now the most notable expansion is in the scope of inventing quantum algebra There is an interesting thing that the studying of the deformed oscillators has shown that para boson oscillator can be seen as the deformation of the boson oscillator Para Bose algebra can also be seen as the deformation of the Heisenberg algebra On the other hand it's natural that the investigation of these above special statistics within the framework of quantum groups leads to the quantum para statistical theories Making the calculation of their statistical distribution, the results will become familiar statistics: Bose-Einstein statistics or Fermi-Dirac statistics in special cases The object is to study specific heat and paramagnetic susceptibility of the free electron gas in alkali metals and transition metals Numerical calculation have been performed for Fermion particles with the hope that quantum group will help us bring up the physical model more generally, and have more precise supplement with experiments; and the investigation of elementary particles by using this method will be more effective than using the concept of normal group 1.2 The statistical moment method Statistical moment method (SMM) is one of the modern methods of statistical physics In principle one can apply this method to research the structural properties, thermodynamics, elasticity, diffusion, phase transitions, of various different types of crystals such as metals, alloys, crystal and compound semiconductor, nano-size semiconductor, ionic crystals, molecular crystals, inert gas crystal, superlattices, quantum crystals, thin films,…with the cubic structure and hexagonal structure in the wide range of temperature from K to melting temperatures and under the effects of pressure SMM is simple and clear in terms of physics A series of thermomechanical properties of crystals are represented in the form of analytical expressions that take into account the effects of anharmonicity and correlation of lattice vibrations It can easily to numerically calculate the thermo-mechanical quantities And we don’t need to use the fitting technique and take the average as least squares method In many cases, SMM calculations can give better results comparing to experiments than other methods We also can combine the SMM with other methods such as first principles (FP), anharmonic correlated Einstein model (ACEM), the self-consistent method (SCF), The research object of this thesis are thermodynamic properties of metallic thin films which have face-centered cubic (FCC) and body-centered cubic (BCC) structures at different temperatures and pressures, in particular for metallic thin films: Al, Cu, Au, Ag, Fe, W, Nb, Ta The obtained results will be compared with other method calculations and experiments The pressure effects on thermodynamic quantities with no experiment data can be used to orientate and predict for future experiments 1.2.1 General formula of moments Considering a quantum system under the unchanged forces in the direction of generalized coordinate Qi Hamiltonian Hˆ of this system has form as follows: Hˆ = Hˆ − ∑ Qˆ i , (1.1) i where Hˆ is the Hamiltonian of the system with no external forces By some transformations, the authors derived two important equations: ) The relational expression between average value of generalized coordinate Qk and free energy ψ of quantum system under of external force a: ∂ψ < Qˆ k > a = − ∂ak (1.2) ) The relational expression between operator Fˆ and coordinator Qk of the system with Hamiltonian Hˆ : ˆ ˆ  F , Qk  +  − Fˆ a a Qˆ k a =θ ∂ Fˆ ∂ak a ∞ B  ih  − θ ∑ 2m   m =0 (2 m)!  θ  2m ∂Fˆ (2 m ) ∂ak , (1.3) , (1.4) a where θ = k BT , B2m is the Bernoulli factor From equation (1.3), one can derive inductive formula of moment: Kˆ n+1 a = Kˆ n a Qˆ n+1 2m ∞ ∂ < Kˆ n > a B2 m  ih  ∂Kˆ n(2 m ) +θ −θ ∑   a ∂an+1 ∂an+1 m =0 (2m)!  θ  a where Kˆ n is the n-order correlative operator: Kˆ n = n−1 [ [Qˆ1 , Qˆ ]+ Qˆ ]+ Qˆ n ]+ 42 43 n −1 1.2.2 General formula of free energy Considering a quanum system specified by Hamiltonian Hˆ in the form of: Hˆ = Hˆ − αVˆ ∂ψ (α ) We can write: < Vˆ >α = − ∂α This equation is equivalent to the following formula: (1.5) (1.6) α ψ (α ) = ψ − ∫ < Vˆ >α d α (1.7) CHAPTER THE q-DEFORMED FERMI-DIRAC STATISTICS AND APPLICATION 2.1 The Fermi-Dirac statistics and q-deformed Fermi-Dirac Statistics 2.1.1 The Fermi-Dirac statistics In order to build the Fermi-Dirac statistics, we can use the quantum field theory We start from the average expression of physical quantity F (corresponding to the operator Fˆ ) based on the grand canonical distribution Fˆ = { { } } Tr exp  − β ( Hˆ − µ Nˆ )  Fˆ , Tr exp  − β ( Hˆ − µ Nˆ )  (2.1) where µ is chemical potential, Hˆ is the Hamiltonian of the system, β = with k B is the kBT Boltzmann constant and T is the absolute temperature of the system If we choose the origin of hω potential energy is E = then Hˆ n = hω n or Hˆ = ε Nˆ with ε is a quantum energy Note that TrFˆ = ∑ n Fˆ n , f ( Nˆ ) n = f ( n ) n (2.2) n The average number of particles on an energy level is given by Nˆ = { T r e xp  − β ( Hˆ − µ Nˆ )  Nˆ T r e xp  − β ( Hˆ − µ Nˆ )  { } } (2.3) Making the calculation of expression (2.3), we obtain the average particle number in a quantum state Nˆ = n (ε ) = f ( ε ) = ε −µ (2.4) e kBT + (2.4) is the Fermi-Dirac distribution function It represents the probability of finding an electron on energy level ε at temperature T 2.1.2 The q-deformed Fermi-Dirac statistics The q-deformed Fermion oscillator q number corresponding to the normal number x is defined by [ x ]q = qx − q−x , q − q −1 (2.5) where q is a parameter If x is an operator, we can also define similarly (2.5) Note that q number is invariant under the inverse transformation q → q-1 In the limit q → ( τ → ), q returns to the normal number (operator) lim [ x ]q = x q →1 (2.6) q-deformed Fermion oscillator is characterized by creation and annihilation operators, bˆ + , bˆ and particle number operator Nˆ = bˆ + bˆ In q-deformed Fermion oscillator these operators satisfy the anti-commutative relation ˆ bˆ bˆ + + q bˆ + bˆ = q − N (2.7) When q → ( τ → ), (2.7) returns to the normal anti-commutative relation and then { } bˆ + bˆ = Nˆ q { } ˆ ˆ + = Nˆ + , bb (2.8) q For q-deformed Fermion q − n − ( − 1) n q n = q + q −1 {n}q (2.9) The q-deformed Fermi-Dirac statistics In order to build the Fermi-Dirac statistics for q-deformed Fermion oscillators, we also derived from the average expression of a physical quantity F as (2.1) The average particles on an { } energy level are determined based on (2.3), but here we replace Nˆ by Nˆ We obtained the qq deformed Fermi-Dirac statistics distribution function as Nˆ q = n (ε ) = f q (ε , T ) = e β (ε − µ ) e β (ε − µ ) − + (q − q −1 )e β (ε − µ ) − (2.10) 2.2 Heat capacity and paramagnetic susceptibility of the free electron gas in metal 2.2.1 Heat capacity of free electrons gas The temperature-dependent heat capacity of metal is described in the form as CV = γ T + β T , (2.11) in which the linear part γ T is the heat capacity of the free electron gas and the nonlinear part β T is the heat capacity of the cations in the network node Total number and total energy of the free electron gas at temperature T are determined by ∞ N = ∫ ρ (ε ) n (ε ) d ε , (2.12) ∞ E = ∫ ερ (ε ) n (ε ) d ε (2.13) g (ε )V In which n (ε ) is the average particle number with energy ε , ρ (ε ) = (2 m ) / ε / is 4π h the density state, g( ε ) is multiple degeneracy of each energy level ε Because each energy level ε corresponds to states s = ± h so g( ε ) = 2s + = 2 Applying q-deformed Fermi-Dirac statistics, the average particle number with energy ε is n (ε ) that can be determined as in (2.10) V (2 m ) / If we put α = , we have 2π h ε −µ ∞ e k BT − N = α ∫ ε 1/ 2 e ε −µ d ε = α I1 / , ε −µ (2.14) + ( q − q −1 )e kBT − kBT ε −µ ∞ e kBT − E = α ∫ ε 3/2 e ε −µ d ε = α I3 / ε −µ −1 + ( q − q )e kBT (2.15) −1 k BT µ is the chemical potential at the temperature T = 0K and µ = lim µ ( T ) T →0 Notice that ε −µ e kBT − lim n (ε ) = lim T →0 T →0 e ε −µ k BT ε −µ −1 + (q − q )e −1 kBT 1 = 0 ( ε < µ0 ) , ( ε > µ0 ) (2.16) We can say that at temperature T = 0K, free electrons in turn "fill" the quantum states with energies < ε < µ and the limited energy level µ is called the Fermi energy level We can identify µ according to this relation µ0 N = α ∫ ε 1/ d ε = αµ 03/ (2.17) From (2.17), we can derive 3 N  ε F = µ0 =   2α  /3 h2  N  =  3π  2m  V  /3 (2.18) Total energy of the free electron gas at T = K is E0 = α ε −µ µ0 e kBT − 3/ ∫ε ε −µ e k BT dε = α ε −µ −1 + ( q − q )e kBT −1 µ0 ∫ε 3/ dε = N µ0 (2.19) µ This means that that at the ground state (T = 0K), the energy of free electron gas is not equal to zero Thus, the average energy of a free electron is At very low temperature is greater than zero, in pursuance of identifying E and µ we need to calculate this integral ε −µ I = ∞ e kBT − ∫ g (ε ) e ε −µ kBT ε −µ −1 + ( q − q )e kBT dε = −1 ∞ ∫ g (ε ) f (ε )d ε , (2.20) in which g (ε ) = ε 1/ or g (ε ) = ε / If ε − µ ≈ k B T and k B T are very small, we obtained 2  N = α I1/2 = α  µ 3/2 + µ −1/2 F (q)(k BT )2 +  , 3  (2.21) 2  E = α I3/2 = α  µ 5/2 + 3µ1/2 F (q )(k BT )2 +  5  (2.22) with F (q ) = ∞ ∞ ∞ −1  (q)k (− q )k (q )k q ( q − 1) + (1 + q ) − q +  ∑ ∑ ∑ q2 +1  k2 k =1 k k =1 k =1 k ∞ ∑ k =1 (− q )k k3   (2.23)  From (2.21), (2.22), (2.17) and (2.19) we derived the approximation results  µ ≈ µ 1 − F ( q )( k B T ) µ   +  ,  (2.24)  F ( q )( k B T ) 75 F ( q )( k B T )  E ≈ E 1 + −  µ 02 µ 04   (2.25) Thus, the total energy of the free electron gas at very low temperature T is   F ( q )( k B T )  F ( q )( k B T )  E ≈ E 1 +  = N µ 1 +  µ 02 µ 02     (2.26) Heat capacity at constant volume of free electron gas in the q-deformed is then formulated N k B2 F ( q )T  ∂E  C Ve =  = =γ  µ0  ∂ T V LT (2.27) T 2.2.2 Paramagnetic susceptibility of the free electron gas According to the quantum theory, paramagnetic susceptibility of the free electron gas obtained by Pauli in the form χP = I N µΒ2 = H k BTF (2.28) Here, I is the magnetization, H is the magnetic field strength, N is the total number of free electrons, µ B is the manheton Bohr and TF is the Fermi temperature According to (2.28), paramagnetic susceptibility of the free electron gas in metal does not depend on the temperature and the results calculated by Pauli were in very good agreement with experimental data Moreover, measurements point out that the paramagnetic susceptibility of nonferromagnetic metal depends very weakly on the temperature When applying the q-deformed theory, we can identify paramagnetic susceptibilities of free electron gas in metal from q-deformed Fermi-Dirac statistics distribution function According to the principles of quantum mechanics, the dependence of density state on energy at temperature T is f ( ε , T ) D ( ε ) in which f ( ε , T ) is q-deformed Fermi-Dirac statistics q q distribution (2.10) and D ( ε ) = V  m2  ε Therefore, 2π  h  β (ε − µ ) −1 V  2m  12 f q ( ε , T ) D ( ε ) = β (ε − µ ) ε   β ε −µ e + ( q − q −1 ) e ( ) − 2π  h  e (2.29) If there is no magnetic field, the total magnetic moment of free electron gas is equal to zero Because in each state there are two electrons with their spins in opposite directions, when we put magnetic field into system, the energy of electron which its spin is in the same direction of the magnetic field H is reduced by an amount µΒ H and vice versa The electron distribution curve is shifted as shown in Figure 2.1 (a) (b) Figure 2.1 Electron distribution in magnetic field at K according to Pauli theory Figure 2.1 (a) points out the states occupied by electrons which their spins are in the same direction and the opposite direction to the magnetic field Figure 2.1 (b) shows spins which are in excess due to the effect of external magnetic fields If the redistribution of electrons does not occur, the energy of system will be adverse Therefore, some electrons which their spins are in opposite direction of magnetic field will move to states with contrary spin direction This leads to the contribution to the magnetization I = ( N + − N − ) µΒ (2.30) In which N ± are respectively the electron concentration with spin in the same direction and opposite direction of magnetic field and are defined by ε N+ = F d ε f q ( ε , T )D ( ε + µ Β H ) , − µ∫Β Η N− = F d ε f q ( ε , T )D ( ε − µ Β H ) + µ∫Β Η (2.31) ε (2.32) At very low temperature is greater than zero, the integral (2.31) and (2.32) can be calculated approximately From (2.30), we inferred the paramagnetic susceptibility of the free electron gas in metal as χ = Substituting α= I µ2 = α ε F1 / µ B2 − α ε F− / B/ F H εF V  2m    2π  h  3/2 ,N = V  2mε F    3π  h  3/2 ,ε F = ( q )( k B T ) h  3π N    2m  V  (2.33) 2/3 into (2.33), we obtained the paramagnetic susceptibility of free electrons gas in metal as χ = N µ B2 N µ B2 − F εF ε F3 10 (q )(k B T ) (2.34) CHAPTER APPLICATIONS OF STATISTICAL MOMENT METHOD TO INVESTIGATE THERMODYNAMIC PROPERTIES OF METALLIC THIN FILMS WITH THE FACECENTER CUBIC AND BODY-CENTERED CUBIC STRUCTURES 3.1 Thermodynamic properties of metallic thin films at zero pressure 3.1.1 The atomic displacemente and the average nearest-neighbor distance ng Let us consider a metallic free standing thin film with n* layers and thickness d It is supposed that the thin film has two atomic surface layers, two next surface layers and ( n* − ) atomic internal layers (see Fig 3.1) Nng, Nng1 and Ntr are respectively the atom numbers of the surface layers, next surface layers and internal layers of this thin film ng a a d Thickness (n*- 4) Layers tr a Fig 3.1 The metallic free standing thin film Using the general formula of statistical moment method, we derive the displacements of atoms in the surface, next surface and internal layers of thin film in the absence of external forces and at temperature T : y 0tr = γ tr θ Atr ; y 0ng = 3 k tr 2γ ng 1θ 3k ng Ang ; y 0ng = − γ ng θ k ng x ng coth x ng (3.1) Thus, by using SMM, we can determine the atom displacement from the equilibrium and then the nearest neighbour distance between two intermediate atoms at a temperature T as a tr (T ) = atr ( ) + y 0tr , a ng (T ) = a ng ( ) + y 0ng , a ng (T ) = a ng ( ) + y 0ng , (3.2) which, a ( ) is the nearest neighbor distance between two particles at K which can be determined from the minimum condition of potential interaction or obtained from the equation of state The average nearest neighbor distance between two atoms of thin film at pressure P , zezo temperature and temperature T are determined as a0 = a= ( ) ( ) 2ang ( ) + 2ang1 ( ) + n* − atr ( ) n −1 * , 2ang (T ) + 2ang1 (T ) + n* − atr (T ) n −1 * where 11 , (3.3) (3.4) uitr γ 1tr = a ≡ ytr , xtr =  ∂ 4ϕ iotr ∑ 48 i  ∂u i4α ,tr γ tr =   ∂ 4ϕ iotr   , γ tr = ∑ 48 i  ∂u i2β ,tr ∂ ui2γ ,tr  eq  ∂ 4ϕ tr ∑  io 12 i  ∂ui4α ,tr  y ng ≡< u ing > a , x ng = γ 1ng = γ ng hωtr  ∂ 2ϕiotr , θ = k BT , ktr = ∑  2θ i  ∂uiα ,tr h ω ng 2θ  ∂ 4ϕ iong 1  ∑ 48 i  ∂ u i4α , ng   ∂ 4ϕ ng 1   io = ∑ 12 i   ∂uiα , ng  y ng =< u ing > a , xng = γ ng =  ∂ 4ϕiotr   +  2  eq  ∂uiβ ,tr ∂uiγ ,tr 2θ   ( β ≠ γ ) ,  eq (3.5)     = ( γ 1tr + γ 2tr )  eq   ng  , θ = k BT ,k ng = ∑  ∂ ϕ2 io  = mω ng2 ,  ∂u   iα ,ng  eq i    ∂ 4ϕ iong  , γ ng =   , ∑ 48 i  ∂ u i β , ng 1∂ u i2γ , ng   eq eq   ∂ 4ϕ iong  +  2  eq  ∂uiβ , ng ∂uiγ , ng h ω ng   = mωtr , eq , θ = k B T , k ng =  ∂ 3ϕ ng io  ∑  i ,α , β ,γ  ∂uiα , ng α ≠β  (3.6)     = ( γ 1ng + γ ng ) ( β ≠ γ )  eq  ( ) ( ) ∑  ϕ ing0 aix2 + 0ϕ ing0  = mω ng2 , i    ∂ 3ϕiong +   ng eq  ∂uiα ,ng ∂uiγ     eq  (3.7) 3.1.2 Free energy of thin film Free energies of the surface, next surface and internal layers of thin film are determined as, respectively Ψtr = U0tr + 3Ntrθ [ xtr + ln( − e−2 xtr )] + + 3Ntrθ ktr2  2γ 1tr  xtr coth xtr  2 1+ γ 2tr xtr coth xtr −  +     6N θ  x coth xtr  x c oth xtr  + tr4  γ 22tr ( + tr )xtr coth xtr −2 ( γ 12tr + 2γ 1tr γ 2tr ) 1 + tr (1 + xtr coth xtr ) ,  2 ktr     (3.8) 3Nng1θ  2γ1ng1  xng1 cothxng1  )] + γ 2ng1xng2 coth2 xng2 − Ψng1 ≈ U + 3Nng1θ [ xng1 + ln(1− e 1+  +  kng1   (3.9) 6Nng1θ 4 xng1 cothxng1 x cothx  ng1 ng1 +  γ 2ng1(1+ )xng1 cothxng1 − 2( γ12ng1 + 2γ1ng1γ 2ng1 ) (1+ )(1+ xng1 cothxng1 ) , 2 kng1 3  −2xng1 ng1 Ψ ng ≈ U 0ng + N ng θ [ x ng + ln(1 − e where U 0tr = Ntr ∑ϕ i tr i0 − x ng (3.10) )], N ng1 N r r ( ri tr ), U 0ng1 = ϕing0 ( ri ng1 ),U 0ng = ng ∑ i ∑ϕ ng i0 r ( ri ng ), (3.11) i Let us consider the system consisting of N atoms with n* layers, the number of atoms on each layer are the same and equal to N L , then free energy of thin film is given by Ψ = Ntrψ tr + Nng1ψ ng1 + Nngψ ng − TSC = ( N − N L )ψ tr + N Lψ ng1 + N Lψ ng − TSC , 12 (3.12) where Sc is the configuration entropy, ψ tr ,ψ ng ,ψ ng ψ ng are the free energies of the atomic surface layers , next surface layers and internal layers of metallic thin film, respectively From (3.12), free energy of an atom is determined as Ψ   2 TS =  − * ψ tr + * ψ ng + * ψ ng − C N  n  n n N (3.13) Using a as the average nearest-neighbor distance and d is the thickness of the metallic thin film, then we have For the metallic thin film with the (FCC) structure: d = (n* − 1) a (3.14) , TS Ψ d − 3a 2a 2a = ψ tr + ψ ng + ψ ng − C N N d 2+a d 2+a d 2+a (3.15) For the metallic thin film the (BCC) structure: ( ) a d = n* − , (3.16) Ψ d − 3a 2a 2a TS = ψ tr + ψ ng + ψ ng − C N N d 3+a d 3+a d 3+a (3.17) 3.1.3 Thermodynamic quantities of the metallic thin film 3.1.3.1 The isothermal compressibility and isothermal elastic modulus The isothermal compressibility χT and isothermal elastic modulus BT are determined as χT = 1  ∂V  =−   , BT V0  ∂P T (3.18) where, V0 is the volume of the system at K By some transformations, we obtained respectively the isothermal compressibility of the metallic thin film with the (FCC) and (BCC) structures as χT = a2 2P + 3V  d − a ∂ Ψ tr   d + a ∂ a tr  a  3   a0  ∂ Ψ ng ∂ Ψ ng 2a 2a + + 2 d + a ∂ a ng d + a ∂ a ng   T (3.19) , χT = a  d − a ∂ Ψ tr 2P +  3V  d + a ∂ a tr2  a  3   a0  ∂ Ψ ng ∂ Ψ ng 2a 2a + + 2 d + a ∂ a ng d + a ∂ a ng 13   T , (3.20) where V = Nv ( v is the atomic volume at temperature T, v = face-centered cubic structure, v = ( ) a ( ) a for the thin film with for the thin film with body-centered cubic structure) In 3 there  ∂ Ψ  is determined by the following formula  ∂a  T  ∂ 2Ψ  = 3N    ∂a T  ∂ 2u hω 1 +  4k  ∂ aT  ∂ 2k  ∂ k     −     ∂ aT k  ∂ aT    (3.21) 3.1.3.2 Thermal expansion coefficient Thermal expansion coefficient of thin metal films can be calculated as follows α = where α tr = k B da d ng α ng + d ng 1α ng + ( d − d ng − d ng ) α tr = , a0 d θ d tr ng ng k B ∂ y (T ) k B ∂ y (T ) k B ∂ y (T ) ;α ng = ;α ng1 = , a , tr ∂θ a ,ng ∂θ a ,ng ∂θ (3.22) (3.23) with d ng and d ng are the thickness of surface layers and next surface layers, respectively 3.1.3.3 Energy of thin film Using the Gibbs – Helmholtz thermodynamic expression:  ∂Ψ E = Ψ −θ   ∂θ   ∂  ∂Ψ    = −T    ,   ∂ T  ∂ T  V (3.24) Energies of thin film with the (FCC) and (BCC) structures are determined as, respectively E= d − 3a 2a 2a Etr + Eng + Eng1 , d 2+a d +a d 2+a E= d − 3a 2a 2a Etr + Eng + Eng1 , d 3+a d 3+a d 3+a (3.25) (3.26) 3.1.3.4 The heat capacities at constant volume and at constant pressure The heat capacities at constant volume of thin film with the (FCC) and (BCC) structures are determined as, respectively CV = d − a tr 2a 2a CV + C Vng + C Vng , d 2+a d 2+a d 2+a (3.27) d − a tr 2a 2a CV + C Vn g + C Vng , d 3+a d 3+a d 3+a (3.28) CV = The heat capacities at constant pressure of thin film with the (FCC) and (BCC) structures as  ∂V   ∂P  C P = CV − T     = C V + 9T V α BT  ∂T  P  ∂ V T (3.29) 3.2 Thermodynamic properties of metallic thin films under the effect of pressure 3.2.1 Equation of state of metallic thin film Equation of state plays an important role determining the properties of thin film under the effect of pressure Since the hydrostatic pressure P is determined from the following formula 14 a  ∂Ψ   ∂Ψ  P = −  = − 3V  ∂a  , ∂ V  T  T (3.30) we obtain the equation of state of metallic thin film as  ∂u ∂k  Pv = − a  + θ x coth x  k ∂a   ∂a (3.31) Where, the parameters u0 , k , x, ω are determined from the nearest neighbour distance between two atoms of thin film The nearest neighbour distance between two atoms is determined at pressure P and at temperature T At temperature T = K, equation (3.31) is reduced to  ∂u0 h ω (0 ) ∂k  Pv = −a  + , 4k ∂a   ∂a (3.32) If we know the atomic interaction potential of thin film with the (FCC) and (BCC) structures, we can determine the nearest neighbour distance between two intermediate atoms at pressure P and at absolute zero temperature T = K Using the Maple software to solve equation (3.32), we find out approximately the values of atr ( P, ) , ang1 ( P,0) , ang ( P,0) After that we determine the thermodynamic quantities of the metallic thin film under the effect of pressure as well as at zero pressure 3.2.2 The average nearest neighbor distance and thermodynamic quantities under the effect of pressure The average nearest neighbor distance of metallic thin film with the (FCC) and (BCC) structures at temperature T and at pressure P as a ( P, T ) = ( ) 2ang ( P, T ) + 2ang1 ( P, T ) + n* − atr ( P, T ) n −1 * (3.33) , where ang ( P,T ) = ang ( P,0) + y0ng ( P,T ) ; ang1 ( P,T ) = ang1 ( P,0) + y0ng1 ( P,T ) ; atr ( P,T ) = atr ( P,0) + y0tr ( P,T ) (3.34) The average nearest neighbor distance of metallic thin film with the (FCC) and (BCC) structures at zero temperature T = K and at pressure P as a0 ( P, ) = ( ) 2ang ( P, ) + 2ang1 ( P, ) + n* − atr ( P, ) n −1 * (3.35) In expression (3.34), y0tr ( P,T ) = 2γ ng1θ γ ngθ 2γ trθ ng1 A P,T ; y P,T = Ang1 ( P,T ) ; y0ng ( P,T ) = − xng ( P,T ) coth( xng ( P,T ) ) , (3.36) ) ( ) tr ( 3 3ktr 3kng1 kng with the parameters kng ( P,0 ) , and γ ng ( P,0 ) at pressure P and T = 0K Thermal expansion coefficient of metallic thin film with the (FCC) and (BCC) structures at pressure P is given by α= da ( P, T ) d ng α ng ( P, T ) + d ng 1α ng ( P, T ) + ( d − d ng − d ng ) α tr ( P, T ) kB = , a0 ( P, ) dθ d where 15 (3.37) tr ng1 ng kB ∂y0 ( P,T ) kB ∂y0 ( P,T ) kB ∂y0 ( P,T ) ;αng1 ( P,T ) = ;αng ( P,T ) = (3.38) atr (P,0) ∂θ ang1 (P,0) ∂θ ang (P,0) ∂θ αtr ( P,T ) = Energy of thin film with the (FCC) structure has form E ( P, T ) = d − 3a ( P , T ) d + a ( P, T ) Etr ( P, T ) + 2a ( P, T ) d + a ( P, T ) E ng ( P , T ) + 2a ( P, T ) d + a ( P, T ) E ng ( P , T ) , (3.39) The heat capacity at constant volume of thin film with the (FCC) structure at pressure P as CV = d − 3a d +a C Vtr ( P , T ) + 2a +a d C Vn g ( P , T ) + 2a +a d C Vn g ( P , T ) , (3.40) The isothermal compressibility and isothermal elastic modulus of thin film with the (FCC) structure at pressure P are determined  a ( P,T )  3  a ( P)    χT ( P,T ) = (3.41) == 2 2 BT ( P,T )   ∂ Ψ ∂ Ψ a ( P,T ) d − 3a ( P,T ) ∂ Ψtr 2a ( P,T ) 2a ( P,T ) ng1 ng 2P + + +   3V  d + a ( P,T ) ∂atr2 d + a ( P,T ) ∂ang2 d + a ( P,T ) ∂ang2  T Energy of thin film with the (BCC) structure has form E ( P, T ) = d − 3a ( P, T ) d + a ( P, T ) Etr ( P, T ) + 2a ( P, T ) d + a ( P, T ) Eng ( P, T ) + 2a ( P, T ) d + a ( P, T ) Eng ( P, T ) , (3.42) The heat capacity at constant volume of thin film with the (BCC) structure at pressure P as CV = d − 3a d 3+a C Vtr ( P , T ) + 2a d 3+a C Vn g ( P , T ) + 2a d 3+a C Vn g ( P , T ) , (3.43) The isothermal compressibility and isothermal elastic modulus of thin film with the (BCC) structure at pressure P are determined  a ( P,T )  3  a ( P,0)    χT ( P,T ) = = (3.44) 2 BT ( P,T ) a ( P,T )  d − 3a ( P,T ) ∂ Ψtr 2a ( P,T ) ∂2 Ψng1 2a ( P,T ) ∂2 Ψng  2P + + +   3V  d + a ( P,T ) ∂atr2 d + a ( P,T ) ∂ang2 d + a ( P,T ) ∂ang2  T The heat capacity at constant pressure of thin metal film with the (FCC) and (BCC) structures is determined from the thermodynamic relations C P ( P , T ) = C V ( P , T ) + 9T V α ( P , T ) BT ( P , T ) 16 (3.45) CHAPTER RESULTS AND DISCUSSION 4.1 Heat capacity and paramagnetic susceptibility of the free electron gas 4.1.1 Heat capacity of the free electron gas From equation (2.27), we obtain the expression of F(q), F (q ) = µ 0γ LT N k B2 (4.1) Substituting the values of N by Avogadro’s number NA, Boltzmann constant k B , experimental Fermi energy level µ0 and the electron thermal constant γ LT = γ TN for each metal into the right-hand side of (4.1), we obtain the value of F(q) Then, from the obtained value of F(q), using Maple software, we find out the value of parameter q for each metal And from here, we can choose the value of q = 0,642 for alkali metal group and q = 0,546 for transition metal group We use the same parameter q for each metal group and plot the temperature dependence of the heat capacity of the free electron gas in metal based on the deformation theory, free-electrons model and experiment in Fig 4.1 and Fig 4.2 So at low temperature, from expression (2.27), we found that the heat capacity of free electron gas in metal based on the deformation theory increases linearly with absolute temperature T This result is in agreement with quantum Sommerfeld’s theory using Fermi-Dirac statistics 90 50 Ex.[92] M.e[92] Present 80 Ex.[92] M.e[92] Present 40 CV (mJ/mol.K), Au 60 50 40 30 20 e 30 e CV (mJ/mol.K), Na 70 20 10 10 0 10 20 30 40 50 60 T (K) 10 20 30 40 50 60 T (K) Fig 4.1 Temperature dependence of heat Fig 4.2 Temperature dependence of heat capacity of free electron gas of Na capacity of free electron gas of Au At the same temperature, the alkali metals with one electron in the outermost shell have the values of q as well as function F(q) which are larger than those of transition metal Therefore, the alkali metals contribute to heat capacity of free electron gas largely than transition metals In contrast, transition metals with the electron in the outermost shell belonging to the subclass d or f have the values of q as well as function F(q) which are smaller than those of alkali metals Thus, the transition metals contribute to heat capacity of free electron gas smaller than alkali metals 4.1.2 Paramagnetic susceptibility of the free electron gas Let us considered at low temperature and used the values k B = 1, 380622.10 − 16 erg.K − , µ B = 9, 274096.10 − 21 erg ( g aus s ) , −1 N = N A = 6, 022169.10 23 ( mol ) , eV = 1, 6021917.10 −12 erg , −1 17 fermi energy level ε F = µ0 , F(q) function for each group alkali or transition metals on the right hand side of (2.34), in the CGS system, we calculated the paramagnetic susceptibility values of free electron gas in a series of metals by deformation theory which were presented in Table 4.1 Our calculation results of paramagnetic susceptibility have been compared with those in the literatures at room temperature Table 4.1 Paramagnetic susceptibility of the free electron gas in metals in literatures and theoretical calculations Elements Cs K Na Rb χTN (×10-6cm mol-1 ) +29 +20,8 +16 +27 χ LT (×10 -6 cm m ol -1 ) +30,67 +22,86 +15 +26,21 According to (2.34), at T = K, the paramagnetic susceptibility of the free electron gas in metal with deformation theory returns to the Pauli’s paramagnetic susceptibility in Sommerfeld quantum theory The second term in the right-hand side of (2.34) is almost negligible with temperature It means that the paramagnetic susceptibility of the free electron gas in metal does not depend on temperature These results are in good agreement with those mearsured by experiments Therefore, the temperature-dependent paramagnetic susceptibility of free-electron gas in metals based on deformation theory are presented as horizontal lines in Fig 4.3 and Fig 4.4 33 16.5 Na Cs 32 χ (×10 cm /mol) 15.5 31 15.0 χ (×10 cm /mol) 16.0 14.5 30 29 14.0 13.5 28 10 20 30 40 50 60 70 10 T (K) 20 30 40 50 60 70 T (K) Fig 4.3 Temperature-dependent paramagnetic Fig 4.4 Temperature-dependent paramagnetic susceptibility of free-electron gas for Na susceptibility of free-electron gas for Cs 4.2 Thermodynamic quantities of metallic thin film at zero pressure and under the effect of pressure In order to numerically calculate these above theoretical results, we choose the LennardJones interaction potential with parameters which were proposed in First of all, using Maple software, we obtained the average nearest neighbor distance for thin films Al, Cu, Au, Ag, Fe, W, Nb and Ta at temperature T, zero pressure and under the effect of pressure From which we determined thermodynamic quantities including the isothermal compressibility, isothermal elastic modulus, thermal expansion coefficient, heat capacities at constant volume and constant pressure for metallic thin film These quantities which depend on temperature and the thickness at zero pressure and under the effect of pressure are presented in the following tables and figures 18 2.715 2.900 10 layers 20 layers 200 layers bulk [15] 2.895 2.890 2.705 2.880 2.700 2.875 2.695 2.870 a (A ) a (A ) 2.885 2.710 2.865 2.690 2.685 2.860 2.855 10 layers 20 layers 70 layers 200 layers 2.680 2.850 2.675 2.845 2.840 2.670 200 300 400 500 600 700 800 500 1000 T (K) 1500 2000 2500 3000 T (K) Fig 4.5 Temperature-dependent nearest neighbor distance of Ag thin film at various thickness Fig 4.6 Temperature-dependent nearest neighbor distance of W thin film at various thickness As it can be seen in Fig 4.5 and Fig 4.6, the average nearest neighbor distance of thin film depends strongly on temperature and the thickness With the same thickness, the average nearest neighbor distances increase with temperature With the same temperature, the average nearest neighbor distances increase with the thickness When the thickness increases from 10 to 30 layers, the average nearest neighbor distance of thin film increases strongly When the thickness is larger than 30 layers, the average nearest neighbor distance slightly increases and approaches the nearest neighbor distance of the bulk 10.5 11.0 Al-70 layers Cu-70 layers Au-70 layers Ag-70 layers 10.0 10.5 9.5 10.0 9.0 8.5 χΤ (10 Pa) 9.0 −12 −12 χΤ (10 Pa) 9.5 8.5 10 layers 20 layers 70 layers 200 layers bulk [15] 8.0 7.5 7.0 100 200 300 400 500 600 700 800 8.0 7.5 7.0 6.5 6.0 5.5 5.0 900 100 200 300 400 500 600 700 800 900 T (K) T (K) Fig 4.7 Temperature-dependent isothermal compressibility of Ag thin film at various thickness Fig 4.8 Temperature-dependent isothermal compressibility of Al, Cu, Au and Ag thin films at 70 layers thickness According to Fig 4.7 to Fig 4.9, at the same thickness when the temperature increases, isothermal compressibility of thin films increases non-linearly and strongly in high temperature At the same temperature, isothermal compressibility decreases with the increasing of the thickness When the thickness increases from 10 to 70 layers, the isothermal compressibility strongly decreases When the thickneses is larger than 70 layers, the isothermal compressibility slightly decreases and approaches the value of the bulk The temperature and thickness dependence of the thermal expansion coefficient are presented in figures from Fig 4.10 to Fig 4.12 According to these figures, at the same thickness, the thermal expansion coefficient increases with the absolute temperature T At the same temperature, when the thickness increases, the thermal expansion coefficient of thin film increases and approaches the value of the bulk This result is in consistent with the experimental study of Al thin film on the substrate When the thickness increases to about 50 nm, the thermal expansion coefficient of thin films approaches the value of the bulk At room temperature, the coefficient of thermal expansion of the Al and Pb thin films increase with the increasing of thickness These results are in good agreement with our calculations 19 2.00 9.0 1.98 8.5 1.96 1.94 8.0 1.92 1.90 -1 α (10 K ) 7.0 1.88 −12 χΤ (10 Pa) 7.5 Al Cu Au Ag 6.5 6.0 1.86 1.84 1.82 1.80 5.5 Al Ag 1.78 1.76 5.0 20 40 60 80 10 20 30 d (nm) 40 50 60 70 d (nm) Fig 4.9 Thickness dependence of the isothermal compressibility for Al, Cu, Au and Ag thin films at T=300K Fig 4.10 Thickness dependence of the thermal expansion coefficient for Al, Au and Ag thin films at T=300K 2.8 2.4 10 layers 70 layers 200 layers bulk [58] 2.6 2.2 2.1 2.0 -1 -5 -5 -1 α (10 K ) 2.4 α (10 K ) Al-70 layers Cu-70 layers Au-70 layers Ag-70 layers 2.3 2.2 1.9 1.8 1.7 2.0 1.6 1.5 1.8 1.4 200 300 400 500 600 100 200 300 T (K) 400 500 600 700 800 900 T (K) Fig 4.12 Temperature dependence of the thermal expansion coefficient for Al, Cu, Au and Ag thin film at 70 layers thickness Fig 4.11 Temperature dependence of the thermal expansion coefficient for Al thin film at various thickness The temperature and thickness dependence of heat capacity at constant volume of metallic thin films are described on the Fig 4.13 and Fig 4.14 According to these figures when the temperature increases, heat capacity at constant volume increases sharply at low temperature and slightly decreases at high temperature This result can be explained as the contribution of anharmonic effect increases with the increasing of temperature At the same temperature, when the thickness increases, heat capacity at constant volume reduces and approaches the value of the bulk It proposes that the contribution of anharmonic effects decrease when the thickness increases 6.0 5.8 5.5 5.6 5.0 Cv (cal/mol.K) Cv (cal/mol.K) 5.4 5.2 5.0 10 layers 20 layers 70 layers 200 layers bulk [15] 4.8 4.6 4.5 4.0 3.5 Al-70layers Cu-70layers Au-70layers Ag-70layers 3.0 2.5 4.4 100 200 300 400 500 600 700 800 900 T (K) 100 200 300 400 500 600 700 800 900 T (K) Fig 4.13 Temperature-dependent specific heats at constant volume for Ag thin film at various thickness 20 Fig 4.14 Temperature-dependent specific heats at constant volume for Al, Cu, Au and Ag thin films at 70 layers thickness 7.0 6.5 Cp (cal/mol.K) CP (cal/mol.K) 6.0 5.5 10 layers 20 layers 70 layers 200 layers bulk [7] bulk [67] 5.0 4.5 Al-70 layers Cu-70 layers Au-70 layers Ag-70 layers 100 200 300 400 500 600 700 800 900 100 200 300 T (K) 400 500 600 700 800 900 T (K) Fig 4.15 Temperature dependence of the specific heats at constant pressure for Ag thin film at various thickness Fig 4.16 Temperature dependence of the specific heats at constant pressure for Al, Cu, Au and Ag thin films at 70 layers thickness The temperature and thickness dependence of the heat capacity at constant pressure for metallic thin films are presented on the Fig 4.15 and Fig 4.16 According to these figures, when the temperature increases, the heat capacity at constant pressure increases sharply in the low temperature and slightly decrease in high temperature range This is due to the contribution of anharmonic effects which increases with the increasing of temperature, especially in high temperature range At the same temperature, when the thickness increases, the heat capacity at constant pressure increases slowly and approaches to the value of bulk material The appearance and law of heat capacity at constant pressure of the thin film are the same as the bulk material When thickness increases to about 35 nm, the heat capacity at constant pressure of the thin film approaches the value of bulk The temperature and the thickness dependence of isothermal elastic modulus for metallic thin film are described on the Fig 4.17 and Fig 4.18 In contrast to isothermal compressibility, at the same thickness when the temperature increases, the isothermal elastic modulus non-linearly decreases but significantly reduces in the high temperature This is in consistent with the law and posture of the bulk material At the same temperature, when the thickness increases, the isothermal elastic modulus increases nonlinearly When the layer number increases from 10 to 100 layers, the isothermal elastic modulus increases sharply And when the layer number is greater than 100 layers, the isothermal elastic modulus of thin films increases slightly 1.9 14.0 10 layers 20 layers 70 layers 200 layers 13.5 13.0 Al Cu Au Ag 1.7 11.5 11.0 -11 12.0 1.6 11 BT (10 Pa ) 10 -1 BT (10 Pa ) 12.5 1.8 1.5 1.4 10.5 1.3 10.0 9.5 1.2 9.0 1.1 100 200 300 400 500 600 700 800 900 T (K) Fig 4.17 Temperature dependence of the isothermal elastic modulus for Ag thin film at various thickness 20 40 60 80 d (nm) Fig 4.18 Thickness dependence of the isothermal elastic modulus for Al, Cu, Au and Ag thin films at T=300K The thickness dependence of isothermal elastic modulus of metallic thin film at room temperature is described in Fig 4.18 Here, the isothermal elastic modulus increases with the increasing of the thickness and it increases sharply if the thickness is smaller than 25 nm When the thickness is greater than 25 nm, the isothermal elastic modulus increases slightly This means that the anharmonic effects decrease when the thickness increases 21 2.831 2.3 2.2 2.830 2.1 2.0 -5 -1 α (10 K ) a (A ) 2.829 2.828 1.9 1.8 1.7 1.6 Al- 0GPa Al- 0.24GPa Al- 0.64GPa 2.827 Au-10 layers,0GPa Au-10 layers,0.24GPa Au-10 layers,0.94GPa 1.5 2.826 1.4 20 40 60 80 100 200 300 d (nm) 400 500 600 700 800 900 T (K) Fig 4.19 Thickness-dependent nearest neighbor distance of Al thin film at various pressure and at T=300K Fig 4.20 Temperature-dependent thermal expansion coefficient of Au thin film at various pressure and at T=300K In Fig 4.19, the average nearest neighbor distance of thin film strongly depends on the thickness and pressure at room temperature The nearest neighbor of thin film increases with the increasing of thickness and decreases with the increasing of pressure The dependence of the average nearest neighbor on pressure can be explained that when the pressure rises, the surface is compressed, the atoms are closer, and the influence of surface effects leads to the decreasing of the average nearest neighbor distance It can be seen in Fig 4.20 that the thermal expansion coefficient of thin film increases with the increasing of temperature and thickness and it reduces with the rising of pressure These results can be explained as in the case of the nearest neighbor distance 16 11.0 Ag-10 layers,0GPa Ag-10 layers,0.24GPa Ag-10 layers,0.94GPa bulk [15],0GPa 10.5 10.0 14 13 -1 9.0 10 −12 BT (10 Pa ) 9.5 λΤ (10 Pa) Au-10 leyers,0GPa Ag-10 leyers,0GPa Ag-10 leyers,0.24GPa Ag-10 leyers,0.94GPa Au-10 leyers,0.24GPa Au-10 leyers,0.94GPa 15 8.5 12 11 8.0 10 7.5 7.0 0 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 T (K) T (K) Fig 4.21 Temperature-dependent isothermal compressibility of Ag thin film at various pressure and at 10 layers thickness Fig 4.22 Temperature-dependent isothermal elastic modulus of Au and Ag thin films at various pressure and at 10 layers thickness thickness In Fig 4.21 and Fig 4.22, we display the temperature dependence of the isothermal compressibility and the isothermal elastic modulus of metallic thin films at various pressures It can be seen in Fig 4.21, the isothermal compressibility increases with the increasing of temperature and sharply increase in high temperature, and it reduces with the increasing of the thickness and pressure The variation of the isothermal compressibility of thin film is the same as the bulk material In contrast to Fig 4.22, at the same thickness, the isothermal elastic modulus reduces with the increasing of temperature, while at the same temperature, the isothermal elastic modulus increases with the increasing of thickness and pressure The dependence of the isothermal compressibility and the isothermal elastic modulus on temperature and thickness under pressure have the postures like those at zero pressure 22 6.0 6.8 6.6 5.8 6.2 5.4 6.0 Cp (cal/mol.K) Cv (cal/mol.K) 6.4 5.6 5.2 5.0 4.8 Ag-10 layers,0GPa Ag-10 layers,0.24GPa Ag-10 layers,0.94GPa bulk[15] 4.6 4.4 5.8 5.6 5.4 5.2 Au-10 Ag-10 Au-10 Ag-10 5.0 4.8 4.6 layers,0GPa layers,0GPa layers,0.24GPa layers,0.24GPa 4.4 100 200 300 400 500 600 700 800 900 100 200 300 400 T (K) 500 600 700 800 900 T (K) Fig 4.23 Temperature-dependent heat capacity Fig 4.24 Temperature-dependent heat capacity at at constant volume of Ag thin film at various constant pressure of Au and Ag thin films at various pressure and at the thickness of 10 layers pressure and at the thickness of 10 layers 1.01 1.01 Cu[TKMM] Cu[60] 1.00 Ag[TKMM] Ag[60] 1.00 0.99 0.98 0.98 0.97 V/ V0 V/ V0 0.99 0.97 0.96 0.96 0.95 0.94 0.95 0.93 0.92 0.94 0.91 10 P (GPa) 10 P (GPa) Fig 4.25 Pressure dependent V/V0 of Cu thin film at T= 300K and at 80nm thickness Fig 4.26 Pressure dependent V/V0 of Ag thin film at T= 300K and at 55nm thickness In Fig 4.23 and Fig 4.24, we show the temperature dependence of the heat capacities at constant volume and at constant pressure of thin film under pressure Heat capacity at constant volume sharply increase at low temperature and slightly decrease at high temperature The heat capacity at constant pressure increases with the increasing of the thickness and depends weakly on pressure Meanwhile the isobaric heat capacity increases sharply with temperature at low temperature and slightly increases at high temperature At the same temperature, the heat capacity at constant pressure increases with the increasing of pressure The temperature and thickness dependence of the heat capacity at constant pressure and constant volume for thin film formation under pressure are the same as those at zero pressure Pressure dependence of volume V  a( P,T )  = compression  of thin films at T = 300 K are described in Fig 4.25 and Fig 4.26 V0  a( ,T )  Our results at 80nm and 55nm thickness are in good agreement with those of nanomaterials Cu and Ag [60], respectively 23 CONCLUSION In this thesis, the q-deformed Fermi-Dirac statistics has been used to study the specific heats, paramagnetic susceptibility of free-electron gas in metal; and the statistical moment methods in quantum statistics has been used to study the thermodynamic properties of metallic thin film with the (FCC) and (BCC) structures The results of the thesis are as follows By using the q-deformed Fermi-Dirac statistics, we derived the analytical expressions of the specific heats and paramagnetic susceptibility of the free-electron gas in metal at low temperature These quantities depend on the q parameters Our results showed that, at low temperature, while the specific heat at constant volume of the free-electron gas in metal is in direct proportion to the absolute temperatures, the paramagnetic susceptibility depends very weakly on temperature For numerical calculations, we used the same empirical parameters q for each group of alkali metal and transition metal Our results of the specific heats and paramagnetic susceptibility of the free electron gas in metal are in good agreement with those of experiments Building the analytical expressions of the thermodynamic quantities such as the Helmholtz free energy, the average displacement of atom from equilibrium position, the average nearest neighbor distance between two atoms, isothermal compressibility, the thermal expansion coefficient, the heat capacity at constant volume and constant pressure, isothermal elastic modulus, for metallic thin film with the (FCC) and (BCC) structures These expressions have taken into account the contribution of anharmonic effect of lattice vibrations, surface effects, size effects in different temperatures and pressures We also used the Lennard–Jones interaction potential to numerically calculate the obtained thermodynamic quantities Our results showed that average nearest neighbor distance and the thermodynamic quantities depend on temperature, pressure and thickness of thin films The obtained results show the good agreement with experimental results and other theoretical results The nearest neighbor distance, isothermal compressibility, thermal expansion coefficient, the heat capacity at constant volume and constant pressure, isothermal elastic modulus of thin films have the same laws and posture change of bulk materials When the thickness of thin films increases from 20 nm to 70 nm, thermodynamic properties of metallic thin film return to bulk material properties The analytical formulas derived are not only applied to thin films with the (FCC) and (BCC) structures but also used as a theoretical basis to investigate the thin films with different structures such as transistor thin films with diamond structure and zinc sulfide, The project can be extended to the study elastic properties and thermodynamic properties of metallic thin films on the substrate with the (FCC) and (BCC) structures, The success of the thesis participates in perfecting and developing the statistical moment method application to study the properties of crystals materials We will continue to expand this theory to study the elastic properties, thermodynamic properties of thin films on the substrate and semiconductor thin films in the future 24 [...]...CHAPTER 3 APPLICATIONS OF STATISTICAL MOMENT METHOD TO INVESTIGATE THERMODYNAMIC PROPERTIES OF METALLIC THIN FILMS WITH THE FACECENTER CUBIC AND BODY-CENTERED CUBIC STRUCTURES 3.1 Thermodynamic properties of metallic thin films at zero pressure 3.1.1 The atomic displacemente and the average nearest-neighbor distance ng Let us consider a metallic free standing thin film with n* layers and thickness... structure and zinc sulfide, The project can be extended to the study elastic properties and thermodynamic properties of metallic thin films on the substrate with the (FCC) and (BCC) structures, The success of the thesis participates in perfecting and developing the statistical moment method application to study the properties of crystals materials We will continue to expand this theory to study the... posture change of bulk materials When the thickness of thin films increases from 20 nm to 70 nm, thermodynamic properties of metallic thin film return to bulk material properties The analytical formulas derived are not only applied to thin films with the (FCC) and (BCC) structures but also used as a theoretical basis to investigate the thin films with different structures such as transistor thin films with... free-electron gas in metal; and the statistical moment methods in quantum statistics has been used to study the thermodynamic properties of metallic thin film with the (FCC) and (BCC) structures The results of the thesis are as follows 1 By using the q-deformed Fermi-Dirac statistics, we derived the analytical expressions of the specific heats and paramagnetic susceptibility of the free-electron gas in... coefficient of thin film increases and approaches the value of the bulk This result is in consistent with the experimental study of Al thin film on the substrate When the thickness increases to about 50 nm, the thermal expansion coefficient of thin films approaches the value of the bulk At room temperature, the coefficient of thermal expansion of the Al and Pb thin films increase with the increasing of thickness... that the thin film has two atomic surface layers, two next surface layers and ( n* − 4 ) atomic internal layers (see Fig 3.1) Nng, Nng1 and Ntr are respectively the atom numbers of the surface layers, next surface layers and internal layers of this thin film ng 1 a a d Thickness (n*- 4) Layers tr a Fig 3.1 The metallic free standing thin film Using the general formula of statistical moment method, we... 3.2 Thermodynamic properties of metallic thin films under the effect of pressure 3.2.1 Equation of state of metallic thin film Equation of state plays an important role determining the properties of thin film under the effect of pressure Since the hydrostatic pressure P is determined from the following formula 14 a  ∂Ψ   ∂Ψ  P = −  = − 3V  ∂a  , ∂ V  T  T (3.30) we obtain the equation of. .. dependence of volume 3 V  a( P,T )  = compression  of thin films at T = 300 K are described in Fig 4.25 and Fig 4.26 V0  a( 0 ,T )  Our results at 80nm and 55nm thickness are in good agreement with those of nanomaterials Cu and Ag [60], respectively 23 CONCLUSION In this thesis, the q-deformed Fermi-Dirac statistics has been used to study the specific heats, paramagnetic susceptibility of free-electron... isothermal compressibility of Ag thin film at various pressure and at 10 layers thickness Fig 4.22 Temperature-dependent isothermal elastic modulus of Au and Ag thin films at various pressure and at 10 layers thickness thickness In Fig 4.21 and Fig 4.22, we display the temperature dependence of the isothermal compressibility and the isothermal elastic modulus of metallic thin films at various pressures... 4.12 Temperature dependence of the thermal expansion coefficient for Al, Cu, Au and Ag thin film at 70 layers thickness Fig 4.11 Temperature dependence of the thermal expansion coefficient for Al thin film at various thickness The temperature and thickness dependence of heat capacity at constant volume of metallic thin films are described on the Fig 4.13 and Fig 4.14 According to these figures when the