Advanced Solid State Physics Syllabus of the Phys 672 Electron-electron interactions, Electron-phonon interactions, Dielectric and Optical Properties Magnetism in solids: Magnetic Properties, Diamagnetism and Para magnetism, Magnetic
Resonance, Superconductivity and Ginzburg-Landau equations. Books 1. Introduction to solid state physics: by Charles Kittle
2. by Review of Free Electron Model 1. Classical Theory Drude Model
2. Quantum theory Drude model History Hundred years ago (around 1900) there was no understanding of physics of solids. There was no explanation of Photo electric effect.
Rutherford has not determined the size of the nucleus. Bohr had not given the shell (discrete energy level ) picture of atom. Formulation of quantum mechanics was decade behind. Thomson had discovered the electrons in 1897
Drude model continue Drude in 1900 formulate a model to explain the two of the most striking properties of the metallic state: namely conduction of electricity and heat. Physical model: In metallic state most loosely bound electrons some how become mobiles.
These electrons behaves like classical particles generally we call it electron gas. Energy of the electron is determined by the temperature of metal. The electrons dont interact with each other rather with the much larger atomic cores.
Drude model continue Geometrical consideration imply mean free path between collision of 1/(R2n) Where R is core radius and n # of atoms per unit volume Estimate the number of electrons per unit volume.
continue n =( NAZc m)/A = (6:02 x 10 23
atoms/mol x1e/atomx1x106g/m3)/ 29g/mol = 2 x 1028 e/m3 where NA is Avogadro's number, m is the density of the metal, A is the atomic number of the element and the numbers are for
Na (sodium). He assumed that after collision a conduction electron has random direction and speed and does not depend on velocity before. continue By analogy with ideal gas, Maxwell Boltzman
distribution was applied for electrons. Thermal velocity of electron is order of 105 m/s. But average velocity is zero. If the electron is subjected to external force, then the electrons accelerate. However in Drude model the collision of electrons with atomic cores prevents to
accelerate indefinitely rather results in establishment of drift velocity in the direction of the force. A statistical equation for describing the "mean drift velocity" in the direction of the field may be written:
where is the "mean free time" or "relaxation time" (the "mean free path" is given by t times the "mean thermal velocity", vtherm. When the field is zero, this equation, obviously, describes the decay of any gross charge carrier motion in any particular direction.
This equation has the simple solution: where is the mobility. Finally, if we have n charge carriers per unit volume, the net flux of charge carriers per unit time is
given by continue and the current density by Where
is electrical conductivity. If we have both negative and positive charge carriers then continue For metals the number carriers is fixed and the essentially linear
temperature dependence of metallic conductivity is attributed to the temperature dependence of the mobility. Electrical conductivity of sodium metal There are a couple of implications of this. For a metal like Na with a resistivity, Na = 1/= 50 n-m the relaxation time is m the relaxation time is
about 10-m the relaxation time is 14 s. QUANTUM MECHANICAL TREATMENT OF ELECTRON GAS We find the ground state properties of N electrons confined in Volume V. Ground state of N independent electrons can
be found by finding the energy levels of single electron in the volume V and filling the N electrons according to Pauli exclusion principle. A single electron will satisfy the time independent Schrdinger equation
continue Confinement of electrons is represented by boundary condition of The choose of boundary condition Whenever one is dealing with problems that are
not explicitly concerned with the effects of metallic surface, then choice of boundary condition is on ones disposal and can be determined with mathematical convenience. If metal is sufficiently large , we expect that bulk properties will be unaffected by the detailed
configuration of surface. Our choice will be cube of length L=V1/3 Boundary condition The boundary condition Every where outside the box
In particular at boundaries. Boundary condition Taking wave function 0 at boundary gives standing wave solution inside the box. Which is
unsatisfactory. Transport of charge and energy of electrons are more conveniently discussed in terms of running wave. This can be achieved by assuming the periodic boundary condition.
The analytical form (x,y,z+L)=(x,y,z))=(x,y,z) (x,y+L)=(x,y,z),z)=(x,y,z) (x+L)=(x,y,z),y,z)=(x,y,z) This boundary condition is known as Born-m the relaxation time is von Karman boundary condition. The solution of Schrodinger eqn is
(r)= Aeik.r = (1/V) eik.r where E = 2k2/2m continue Energy /2m
Energy Spectrum for many electrons Density of states K -Space
continue continue In terms of energy Solve it at home
Fermi energy How to calculate Ground state of N non interacting electrons gas We define the density of states g(k)
The number of states in a given energy range (E,dE) is equal to the volume enclosed between the spherical shells E +dE = constant and E = constant, multiplied by the density of states in kspace. From this we obtain the density of states
on the energy scale We now consider an electron gas with N electrons. The state of lowest energy (ground state) will then correspond to the N/2 points in k-space having the lowest energy, each being occupied by two
electrons (Fig. 2.1). In k-space these points fill a sphere of radius (Fermi sphere). is defined by the condition Here n= N/Vg is the electron concentration. We then have the following expression for the Fermi radius KF and for the energy of
electrons at the surface Excited states The excited states of the system consist of states where one or more fermions is excited to higher energy states. The average occupations of the states at a given temperature T is given by the
Fermi-Dirac distribution Fermi Dirac distribution Fermi Dirac distribution Excited state
Excited state Excited State Excited state
Excited State Average Energy and Specific heat Calculate ground state energy Specific heat
Please take the time to go through each aspect presented and think about how you will balance your field placement with other life responsibilities. These are just suggestions and ideas to get you thinking how to creatively rearrange your time.
A hospice patient, an evangelical Christian, was refusing the chaplain, but the pt would regularly ask the Hospice Aide, while receiving a bath, for a prayer. They would speak of spiritual ideas of heaven or fears of death as isolation/separation...
Sub/Site/106 4 fold deg pseudo CUG CUA CUC CUU Four fold degenerate site No matter what nucleotide is substituted, the codon specifies the same amino acid Leucine Frequency of codon usage for highly expressed genes Frequency of tRNA species Frequency...
Dr. H. Amar Ahmad, M.Si Plt. Sekretaris Deputi Pembudayaan Olahraga Hj. Suryati, S.Sos, M.Si Sekretaris Deputi Peningkatan Prestasi Olahraga Dra. Marheni Dyah Kusumawati, M.Pd. Kondisi 11 Januari 2019 Asisten Deputi Peningkatan Tenaga dan Sumber Daya Pemuda Dr. Deswan, M.Si Asisten...
Team Leadership Log Name(s) Date(s) Topic/Description Goals Audience Impact Statement • "Sampling" of activities • Each person listed at least once Identify district, school, or grade-level needs Form work groups (or pairs or individuals) Carry out work March-April-May Submit: (1)...
Christ demonstrated His unparalleled love to us by laying down His life to reconcile us to God. ... Come and feast with us around God's pure Word: we read it, study it, believe it and apply it. For your time...
Accommodations should not advantage the ELL unfairly and thus compromise the validity of the test results. Accommodations used during testing should be similar to those used by the ELL to complete classroom activities and assessments. Accommodations must be necessary for...
Ready to download the document? Go ahead and hit continue!