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.

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ds, 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 0 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 ai in the direction of generalized coordinate Qi. Hamiltonian of this system has form as follows:
(1.1)
where 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 and free energy of quantum system under of external force a:
(1.2)
The relational expression between operator and coordinator of the system with Hamiltonian
(1.3)
where B2m is the Bernoulli factor.
From equation (1.3), one can derive inductive formula of moment:
(1.4)
where is the n-order correlative operator:
1.2.2. General formula of free energy
Considering a quanum system specified by Hamiltonian in the form of:
(1.5)
We can write: (1.6)
This equation is equivalent to the following formula:
(1.7)
CHAPTER 2
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 based on the grand canonical distribution
(2.1)
where is chemical potential, is the Hamiltonian of the system, with is the Boltzmann constant and is the absolute temperature of the system. If we choose the origin of potential energy is then or with is a quantum energy. Note that
(2.2)
The average number of particles on an energy level is given by
(2.3)
Making the calculation of expression (2.3), we obtain the average particle number in a quantum state
(2.4)
(2.4) is the Fermi-Dirac distribution function. It represents the probability of finding an electron on energy level 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
(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 returns to the normal number (operator)
(2.6)
q-deformed Fermion oscillator is characterized by creation and annihilation operators, and particle number operator . In q-deformed Fermion oscillator these operators satisfy the anti-commutative relation
. (2.7)
When (), (2.7) returns to the normal anti-commutative relation and then
, . (2.8)
For q-deformed Fermion
. (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 by We obtained the q- deformed Fermi-Dirac statistics distribution function as
(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
(2.11)
in which the linear part is the heat capacity of the free electron gas and the nonlinear part 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
(2.12)
(2.13)
In which is the average particle number with energy is the density state, g() is multiple degeneracy of each energy level . Because each energy level corresponds to 2 states so g() = 2s + 1 = 2.
Applying q-deformed Fermi-Dirac statistics, the average particle number with energy is that can be determined as in (2.10).
If we put , we have
(2.14)
(2.15)
is the chemical potential at the temperature T = 0K and Notice that
(2.16)
We can say that at temperature T = 0K, free electrons in turn "fill" the quantum states with energies and the limited energy level is called the Fermi energy level. We can identify according to this relation
(2.17)
From (2.17), we can derive
(2.18)
Total energy of the free electron gas at T = 0 K is
(2.19)
Thus, the average energy of a free electron is This means that that at the ground state (T = 0K), the energy of free electron gas is not equal to zero.
At very low temperature is greater than zero, in pursuance of identifying E and we need to calculate this integral
(2.20)
in which or . If and are very small, we obtained
(2.21)
(2.22)
with (2.23)
From (2.21), (2.22), (2.17) and (2.19) we derived the approximation results
(2.24)
(2.25)
Thus, the total energy of the free electron gas at very low temperature T is
(2.26)
Heat capacity at constant volume of free electron gas in the q-deformed is then formulated
(2.27)
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
(2.28)
Here, I is the magnetization, H is the magnetic field strength, N is the total number of free electrons, µB 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 non-ferromagnetic 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 in which is q-deformed Fermi-Dirac statistics distribution (2.10) and . Therefore,
(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 and vice versa. The electron distribution curve is shifted as shown in Figure 2.1.
(b)
Figure 2.1. Electron distribution in magnetic field at 0 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
. (2.30)
In which are respectively the electron concentration with spin in the same direction and opposite direction of magnetic field and are defined by
(2.31)
(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
(2.33)
Substituting into (2.33), we obtained the paramagnetic susceptibility of free electrons gas in metal as
. (2.34)
CHAPTER 3
APPLICATIONS OF STATISTICAL MOMENT METHOD TO INVESTIGATE THERMODYNAMIC PROPERTIES OF METALLIC THIN FILMS WITH THE FACE-CENTER 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
Let us consider a metallic free standing thin film with layers and thickness d. It is supposed that the thin film has two atomic surface layers, two next surface layers and () 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.
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 :
(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
(3.2)
which, is the nearest neighbor distance between two particles at 0 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 , zezo temperature and temperature T are determined as
(3.3)
(3.4)
where
, ,
(3.5)
.
,,
(3.6)
(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
(3.8)
(3.9)
(3.10)
where (3.11)
Let us consider the system consisting of atoms with layers, the number of atoms on each layer are the same and equal to then free energy of thin film is given by
(3.12)
where is the configuration entropy, 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
(3.13)
Using 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:
(3.14)
(3.15)
For the metallic thin film the (BCC) structure:
(3.16)
(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 and isothermal elastic modulus are determined as
(3.18)
where, V0 is the volume of the system at 0 K.
By some transformations, we obtained respectively the isothermal compressibility of the metallic thin film with the (FCC) and (BCC) structures as
(3.19)
(3.20)
where ( is the atomic volume at temperature T, for the thin film with face-centered cubic structure, for the thin film with body-centered cubic structure). In there is determined by the following formula
(3.21)
3.1.3.2. Thermal expansion coefficient
Thermal expansion coefficient of thin metal films can be calculated as follows
(3.22)
where (3.23)
with and 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:
(3.24)
Energies of thin film with the (FCC) and (BCC) structures are determined as, respectively
(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
(3.27)
(3.28)
The heat capacities at constant pressure of thin film with the (FCC) and (BCC) structures as
(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
(3.30)
we obtain the equation of state of metallic thin film as
(3.31)
Where, the parameters are determined from the nearest neighbour distance between two atoms of thin film. The nearest neighbour distance between two atoms is determined at pressure and at temperature T.
At temperature T = 0 K, equation (3.31) is reduced to
(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 and at absolute zero temperature T = 0 K. Using the Maple software to solve equation (3.32), we find out approximately the values of , , . 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 as
(3.33)
where
(3.34)
The average nearest neighbor distance of metallic thin film with the (FCC) and (BCC) structures at zero temperature T = 0 K and at pressure P as
(3.35)
In expression (3.34),
(3.36)
with the parameters , and at pressure and T = 0K.
Thermal expansion coefficient of metallic thin film with the (FCC) and (BCC) structures at pressure is given by
(3.37)
where
(3.38)
Energy of thin film with the (FCC) structure has form
(3.39)
The heat capacity at constant volume of thin film with the (FCC) structure at pressure as
(3.40)
The isothermal compressibility and isothermal elastic modulus of thin film with the (FCC) structure at pressure are determined
(3.41)
Energy of thin film with the (BCC) structure has form
(3.42)
The heat capacity at constant volume of thin film with the (BCC) structure at pressure as
(3.43)
The isothermal compressibility and isothermal elastic modulus of thin film with the (BCC) structure at pressure are determined
(3.44)
The heat capacity at constant pressure of thin metal film with the (FCC) and (BCC) structures is determined from the thermodynamic relations
(3.45)
CHAPTER 4
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
(4.1)
Substituting the values of N by Avogadro’s number NA, Boltzmann constant , experimental Fermi energy level and the electron thermal constant for each metal into the right-hand side of (4.1), we obtain the value of . 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.
Fig. 4.1. Temperature dependence of heat capacity of free electron gas of Na
Fig. 4.2. Temperature dependence of heat 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 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 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
fermi energy level , 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
+29
+20,8
+16
+27
+30,67
+22,86
+15
+26,21
According to (2.34), at T = 0 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.
Fig. 4.3. Temperature-dependent paramagnetic susceptibility of free-electron gas for Na
Fig. 4.4. Temperature-dependent paramagnetic 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 Lennard-Jones 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.
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.
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.
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
Fig. 4.11. Temperature dependence of the thermal expansion coefficient for Al thin film at various thickness
Fig. 4.12. Temperature dependence of the thermal expansion coefficient for Al, Cu, Au and Ag thi

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