Applications of q-deformed Fermi-Dirac statistics and statistical moment method to study thermodynamic properties, magnetic properties of metals and metallic thin films

Specifically, we use this theory to study the heat capacity and paramagnetic susceptibility of the free electron gas in metal at low temperatures.. In this thesis, we apply the statistic

FOREWORD

1 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 andmagnetic 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"

2 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

3 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

4 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

5 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

6 Thesis outline

Beside the introduction, conclusion, references and appendices, the thesis is divided into 4 chapters and 11 subsections Contents of the thesis is presented in 132 pages with 37 tables, 60 figures and charts, 121 references

CHAPTER 1 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, π-mesons, 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 0 or 1

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 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 a i in the direction of generalized

coordinate Q i Hamiltonian Hˆof this system has form as follows:

ˆ ˆ0 i ˆi,

i

H =H −∑a Q (1.1) where H is the Hamiltonian of the system with no external forces ˆ0

By some transformations, the authors derived two important equations:

The relational expression between average value of generalized coordinate Q)k and free energy ψ of quantum system under of external force a:

2 0

where θ =k T B , B 2m is the Bernoulli factor

From equation (1.3), one can derive inductive formula of moment:

1.2.2 General formula of free energy

Considering a quanum system specified by Hamiltonian Hˆin the form of:

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 Fˆ )based on the grand canonical distribution

T r exp ( H N ) F ˆ

(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

b+ b and particle number operator Nˆ = b bˆ ˆ+ In q-deformed Fermion oscillator these operators

satisfy the anti-commutative relation

ˆ ˆ ˆ ˆ N

b b+ + q b b+ = q− (2.7) When q → 1(τ → 0), (2.7) returns to the normal anti-commutative relation and then

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 { }ˆ .

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

β 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

1 / 2

1 / 2 2

ε µ

ε µ

<

> (2.16)

We can say that at temperature T = 0K, free electrons in turn "fill" the quantum states with

energies 0 < <ε µ0 and the limited energy level µ0 is called the Fermi energy level We can

identify µ0 according to this relation

0

0 0

2

.3

µ

= ∫ = (2.17) From (2.17), we can derive

2 0

Thus, the average energy of a free electron is 3 0

5 µ 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 µ we need

to calculate this integral

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 3 2

P

B F

N I

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

2 2

2 2

magnetic field H is reduced by an amount µΒH 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 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

CHAPTER 3 APPLICATIONS OF STATISTICAL MOMENT METHOD TO INVESTIGATE

THERMODYNAMIC PROPERTIES OF METALLIC THIN FILMS WITH THE

FACE-CENTER CUBIC AND BODY-FACE-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 *

n layers and thickness d It is

supposed that the thin film has two atomic surface layers, two next surface layers and ( *

n −4)

atomic internal layers (see Fig 3.1) N ng , N ng1 and N tr 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 :

2 2

1 1

1

22

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

which, a( )0 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 P, zezo

temperature and temperature T are determined as

ng

ng1tr

ng io

3.1.2 Free energy of thin film

Free energies of the surface, next surface and internal layers of thin film are determined as,

Let us consider the system consisting ofN 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