Applied Physics Lecture Notes 2021-2022 PDF

MRCET CAMPUS MALLA REDDY COLLEGE OF ENGINEERING & TECHNOLOGY (AUTONOMOUS INSTITUTION - UGC, GOVT.OF INDI A) Affiliated to JNTUH; Approved by AICTE, NBA-Tier 1 & NAAC with A-GRADE I ISO 9001:2015 Maisammaguda, Dhulapally, Komapll...

MRCET CAMPUS MALLA REDDY COLLEGE OF ENGINEERING & TECHNOLOGY (AUTONOMOUS INSTITUTION - UGC, GOVT.OF INDI A) Affiliated to JNTUH; Approved by AICTE, NBA-Tier 1 & NAAC with A-GRADE I ISO 9001:2015 Maisammaguda, Dhulapally, Komaplly, Secunderabad - 500100, Telangana State, India LABORATORY MA LECTURE NOTES. APPLIED PHYSICS 2021-22 (R20)  PREFACE Physics is a fundamental aspect of science on which all engineering sciences are made. The fundamental concepts of physics have given the way for the development of engineering branch and technologies. All modern technological advances from laser micro surgery to television, from computers to mobile phones, from remote controlled toys to space vehicles, directly work on the principles of physics. Accordingly the syllabus of engineering courses includes physics as an essential subject. The physics syllabus in engineering course is primarily divided into two parts i.e. applied physics & Engineering physics as per curriculum requirement in universities & engineering colleges in India. The scope of applied physics course is widely extended to various areas of engineering disciplines and emerging new technologies. Applied physics is very vast subject and hence important topics have been shortlisted and included in the hand- book /material. The present hand-book/material of Applied physics is divided into five units i.e. Unit-1 deals with Laser & fiber-optics , Unit-2 deals with Quantum Mechanics, Unit-3 deals with Electronic materials, Unit-4 deals with Semiconductor physics, Unit-5 deals with Dielectrics and Magnetic properties of materials. The first chapter of unit-1 deals with lasers where students were able to study the laser principles and features, absorption and emission(spontaneous & stimulated mechanisms), Population inversion and various pumping schemes of laser systems, laser mechanism and types of laser systems such as Ruby, He-Ne & Semiconductor lasers & its applications. The second chapter of unit-1 deals with fiber-optics where students were able to know about principle & construction of optical fibers, Acceptance angle & N.A. relations, modes of propagation of light through step-index and graded index fibers(Single & Multi-mode fibers),optical communication system & applications of fibers. Students were able to know the importance of optical fiber & lasers in photonics and fiber-optics industry. Optical fibers find application in communication and broadcasting the information from place to place. Unit-2 covers quantum mechanics which deals with origin of quantum mechanics, de- Broglie concepts of matter waves, experimental procedures adopted to verify the wave nature of matter waves and discuss about Schrödinger matter wave equation & its applications. Students able to know dual nature of light (wave & particle) & its interaction with micro-scopic particles at the atomic scale. Unit-3 covers Electronic materials deals with free electronic theory(Classical & Quantum), Fermi level, density of energy states, periodic potential, Bloch’s theorem, Kronig-Penny model , E-K diagram & effective mass of an electron, Origin of energy bands & classification of materials into metals, insulators & semiconductors. Students will be to think and judge the electric response of materials/solids largely stems from the dynamics of electrons, and their interplay with atoms and molecules. Students will be familiar with Fermi level and classification of materials based on band –gap of solids. Unit-4 covers semiconductor physics which deals with various types of semiconducting materials, carrier concentration in both intrinsic and extrinsic semiconductor, carrier transport, formation and V-I characteristics of PN-diode , Hall-effect, semiconductor devices-LED and solar cell. Students try to know the scope & usage of semiconductors in a wide range of components and devices such as diodes, transistors, photo sensor, and microcontroller, microprocessor and integrated chips. Unit-5 covers Dielectric & Magnetic properties of materials. The dielectric materials deals with introduction & basic definitions of dielectric materials, types of polarizations(Electronic & Ionic) and calculation of their polarizabilities, internal fields in a solid , Clausius-Mossotti relation. Magnetic materials with introduction & basic definitions, Origin of magnetic moment(Bohr magneton), classification of dia, para & ferro magnetic materials, properties of anti-ferro and ferri magnetic materials, Hysteresis curve based on domain theory, soft and hard magnetic materials. Dielectric materials finds useful in capacitor, power transformer, cables, spark generators, transducers. A magnetic material finds application in making of permanent magnets, core materials for inductance coils & transformers, relays & heavy current engineering. The concepts in this material are explained in very lucid manner & the contents are optimised so that student will follow & digest the content. Language is simple & self explanatory. Authors express their happiness on the encouraging welcome given to hand book/material made by physics faculty all over. We are thankful to all physics faculty in taking pain in preparation of the digital content/hand-book. Comments and feedback for the improvement of hand book is welcome & appreciated. I hope this material will be beneficial to both students for preparation of internal & final semester exams. Authors  CONTENTS UNIT NAME OF THE UNIT 1 LASERS AND FIBEROPTICS 2 QUANTUM MECHANICS 3 ELECTRONIC MATERIALS 4 SEMICONDUCTOR PHYSICS 5 DIELECTRICS AND MAGNETIC PROPERTIES OF MATERIALS  MALLA REDDY COLLEGE OF ENGINEERING & TECHNOLOGY B. TECH- I- YEAR- II-SEM L T P C 3 - - 3 (R20A0011) APPLIED PHYSICS COURSE OBJECTIVES: 1 To analyze the ordinary light with a laser light and realize the transfer of light through optical fibers. 2 To identify dual nature of the matter and behavior of a particle quantum mechanically. 3 To explore band structure of the solids and classification of materials. 4 To acquire the basic knowledge of various types of semiconductor devices and find the applications in science and technology. 5 To Compare dielectric and magnetic properties of the materials and enable them to design and apply in different fields. UNIT – I LASERS & FIBER OPTICS (9Hours) Lasers: Characteristics of lasers, Absorption, Spontaneous and Stimulated emissions, population inversion, meta stable state, types of pumping, lasing action, construction and working of Ruby Laser, Helium-Neon Laser, Semiconductor diode Laser, Applications of lasers. Fiber Optics: Introduction to optical fiber, Construction and working principle of an Optical Fiber, Acceptance angle and Numerical aperture, Types of Optical fibers - Mode and Propagation through step and graded index fibers ,Losses in optical fiber, Optical Fiber in Communication System, Applications of optical fibers. UNIT – II QUANTUM MECHANICS (7 Hours) Wave nature of particles, de Broglie’s hypothesis, matter waves, Heisenberg’s uncertainty principle, Davisson and Germer’s experiment, G.P Thomson experiment, Schrodinger time- independent wave equation-significance of wave function, particle in one dimensional square well potential. UNIT – III ELECTRONIC MATERIALS (7 Hours) Free electron theory(Classical & Quantum)- Assumptions, Merits and drawbacks, Fermi level, Density of states, Periodic potential, Bloch’s theorem, Kronig – Penny model (qualitative) , E – K diagram, Effective mass, Origin of energy bands in solids, Classification of materials : Metals, semiconductors and insulators. UNIT-IV SEMICONDUCTOR PHYSICS (10Hours) Intrinsic and extrinsic semiconductors, Direct and indirect band gap semiconductors, Carrier concentration in intrinsic and extrinsic semiconductors. Dependence of Fermi level on carrier concentration and temperature, carrier transport: mechanism of diffusion and drift, Formation of PN junction, V-I characteristics of PN diode, energy diagram of PN diode, Hall experiment, semiconductor materials for optoelectronic devices - LED, Solar cell. UNIT – V: DIELECTRICS AND MAGNETIC PROPERTIES OF MATERIALS (10 Hours) Dielectrics: Introduction, Types of polarizations (Electronic and Ionic) and calculation of their polarizabilities, internal fields in a solid, Clausius-Mossotti relation. Magnetism: Introduction, origin of magnetism, Bohr magneton, classification of dia, para and ferro magnetic materials on the basis of magnetic moment, Properties of anti-ferro and ferri magnetic materials, Hysteresis curve based on domain theory, Soft and hard magnetic materials. COURSE OUTCOMES:After completion of studying Applied Physics the student is able to 1 Observe the properties of light and its engineering applications of laser in fiber optic communication systems. 2 Apply the basic principles of quantum mechanics and the importance of behavior of a particle. 3 Find the importance of band structure of solids and their applications in various electronic devices. 4 Evaluate concentration & estimation of charge carriers in semiconductors and working principles of PN diode. 5 Examine dielectric, magnetic properties of the materials and apply them in material technology. TEXT BOOKS: 1. Engineering Physics by Kshirsagar&Avadhanulu, S Chand publications. 2. Engineering Physics- B.K.Pandey, S.Chaturvedi, Cengage Learning. REFERENCES: 1. Engineering Physics – R.K. Gaur and S.L. Gupta, DhanpatRai Publishers. 2. Engineering Physics, S Mani Naidu- Pearson Publishers. 3. Engineering physics 2nd edition –H.K. Malik and A.K. Singh. 4. Engineering Physics – P.K. Palaniswamy, Scitech publications. 5. Physics by Resnick and Haliday.  APPLIED PHYSICS UNIT – I LASER& FIBER-OPTICS INTRODUCTION: LASER stands for Light Amplification by Stimulated Emission of Radiation. Laser technology started with Albert Einstein in 1917, he has given theoretical basis for the development of Laser. The technology further evolved in 1960 when the very first laser called Ruby Laser was built at Hughes Research Laboratoriesby T.H. Mainmann. CHARACTERISTIC OF LASER RADIATION: The laser beam has the properties given below which distinguish it from an ordinary beam of light. Those are 1. Highly directional 2. Highly monochromatic 3. Highly intense 4. Highly coherence 1. Highly directional: A conventional light source emits light in all directions. On the other hand, Laser emits light only in one direction. The width of Laser beam is extremely narrow and hence a laser beam can travel to long distances without spreading. The directionality of laser beam is expressedin terms of divergence 𝑟2 − 𝑟1 ∆𝜃 = 𝑑2 − 𝑑1 Where r1and r2are the radii of laser beam spots at distances of d 1 and d2 respectively from laser source. 2. Highly monochromatic: A monochromatic source is a single frequency or single wavelength source of light. The laser light is more monochromatic than that of a convectional light source. This may be due to the stimulated characteristic of laser light. The band width of convectional monochromatic light source is 1000A0. But the band width of ordinary light source is 10 A0. For high sensitive laser source is 10-8 A0. 3. Highly intense: Laser light is highly intense than the conventional light. A one milli-Watt He-Ne laser is highly intense than the sun intensity. This is because of coherence and directionality of laser. Suppose when two photons each of amplitude ‘A’ are in phase with other, then young’s principle of superposition, the resultant amplitude of two photons is 2A and the intensity is 4a2. Since in laser many numbers of photons are in phase with each other, the amplitude of the resulting wave becomes ‘nA’ and hence the intensity of laser is proportional to n2A2. So 1mw He-Ne laser is highly intense than the sun. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 1  APPLIED PHYSICS 4. Highly coherence A predictable correlation of the amplitude and phase at any one point with other point is called coherence.In case of conventional light, the property of coherence exhibits between a source and its virtual source where as in case of laser the property coherence exists between any two sources of same phase. There are two types of coherence i) Temporal coherence ii) Spatial coherence. Temporal coherence (or longitudinal coherence): The predictable correlation of amplitude and phase at one point on the wave train w.r. t another point on the same wave train, then the wave is said to be temporal coherence. Spatial coherence (or transverse coherence): The predictable correlation of amplitude and phase at one point on the wave train w. r.t another point on a second wave, then the waves are said to be spatial coherence (or transverse coherence).Two waves are said to be coherent when the waves must have same phase & amplitude. INTERACTION OF LIGHT WITH MATTER AND THE THREE QUANTUM PROCESSES: When the radiation interacts with matter, results in the following three important phenomena. They are (i)Induced or Stimulated Absorption (ii)Spontaneous Emission (iii)Stimulated Emission STIMULATED ABSORPTION (OR) INDUCED ABSORPTION (OR) ABSORPTION: An atom in the lower energy level or ground state energy level (E 1) absorbs the incident photon and goes to excited state (E2) as shown in figure below. This process is called induced or stimulated absorption. Let E1 and E2 be the energies of ground and excited states of an atom. Suppose, if a photon of energy E2−E1 = hν interacts with an atom present in the ground state, the atom gets DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 2  APPLIED PHYSICS excitation form ground state E1 to excited state E2. This process is called stimulated absorption. Stimulated absorption rate depends upon the number of atoms available in the lowest energy state as well as the energy density of photons. SPONTANEOUS EMISSION: The atom in the excited state returns to ground state emitting a photon of energy (E) = E2 – E1 = hv, without applying an external energy spontaneously is known as spontaneous emission. Let E1 and E2 be the energies of ground and excited states of an atom. Suppose, if photon of energy E2− E1 = hν interacts with an atom present in the ground state, the atom gets excitation form ground state E1 to excited state E2 The excited atom does not stay for a long time in the excited state. The excited atom gets de-excitation after its life time by emitting a photon of energy E 2− E1 = hν. This process is called spontaneous emission. The spontaneous emission rate depends up on the number of atoms present in the excited state. The probability of spontaneous emission (P21) is independent of u(𝜗). 𝑃21 = 𝐴21 STIMULATED-EMISSION: The atom in the excited state can also returns to the ground state by applying external energy or inducement of photon thereby emitting two photons which are having same energy as that of incident photon. This process is called as stimulated emission. Stimulated emission was postulated by Einstein. Let E1 and E2 be the energies of ground and excited states of an atom. Let a Photon of energy E2-E1=hυ interacts with the excited atom with in their life time The atom gets de-excitation to ground state by emitting of another photon. These photons have same phase and it follows coherence. This phenomenon is DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 3  APPLIED PHYSICS called stimulated emission Stimulated emission rate depends upon the number of atoms available in the excitedstate as well as the energy density of photons. Comparison between Spontaneous and Stimulated emission: Spontaneous emission Stimulated emission 1. The spontaneous emission Was 1. The stimulated emission was Postulated by Bohr. Postulated by Einstein. 2. Additional photons are not required 2. Additional photons are required in in spontaneous emission. Stimulated emission. 3. One Photon is emitted in 3. Two photons are emitted in spontaneous emission. stimulated emission. 4. The emitted radiation is incoherent. 5. The emitted radiation is coherent. 5. The emitted radiation is less intense. 6. The emitted radiation is high intense. Light Amplification: Light amplification requires stimulated emission exclusively. In practice, absorption and spontaneous emission always occur together with stimulated emission. The laser operation is achieved when stimulated emission exceeds the other two processes due to its higher transitions rates of atomic energy levels. POPULATION INVERSION: The number of atoms present in the excited (or higher) state is greater than the number of atoms present in the ground energy state (or lower state) is called population inversion. Let us consider two level energy systems of energies E1and E2as shown in figure. Let N1 and N2 be the population (means number of atoms per unit volume) of E1and E2 respectively. According to Boltzmann’s distribution the population of an energy level E, at temperature T is given by 𝐸 (− 𝑖 ) 𝑁𝑖 = 𝑁𝑜 𝑒 𝑘𝐵𝑇 𝑤ℎ𝑒𝑟𝑒𝑖 = 1,2,3, …. 𝑁𝑖 where ‘N0’ is the number of atoms in ground or lower energy states & k is the Boltzmann constant. From the above equation the population of energy levels E 1& E2 are given by E (− 1 ) 𝑁1 = 𝑁𝑜 𝑒 𝑘𝐵 𝑇 DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 4  APPLIED PHYSICS 𝐸 (− 2 ) 𝑁2 = 𝑁𝑜 𝑒 𝑘𝐵 𝑇 At ordinary conditions N1>N2 i.e., the population in the ground or lower state is always greater than the population in the excited or higher states. The stage of making, population of higher energy level greater than the population of lower energy level is called population inversion i.e., N2>N1. METASTABLE STATE: In general the number of excited particles in a system is smaller than the non excited particles. The time during which a particle can exist in the ground state is unlimited. On the other hand, the particle can remain in the excited state for a limited time known as life time. The life time of the excited hydrogen atom is of the order of 10 -8 sec. However there exist such excited states in which the life time is greater than 10 -8sec. These states are called as Meta stable states. PUMPING MECHANISMS (OR TECHNIQUES): Pumping: The process of rising more no of atoms to the excited state by artificial means is called pumping. A system in which population inversion is achieved is called as an active system. The method of raising the particles from lower energy state to higher energy state is called pumping. (or the process of achieving of population inversion is called pumping). This can be done by number of ways. The most commonly used pumping methods are  Optical pumping  Electrical discharge pumping  Chemical pumping  Injection current pumping Optical pumping: Optical pumping is used in solid laser. Xenon flash tubes are used for optical pumping. Since these materials have very broad band absorption, sufficient amount of energy DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 5  APPLIED PHYSICS is absorbed from the emission band of flash lamp and population inversion is created. Examples of optically pumped lasers are ruby, Nd: YAG Laser ( Y 3 AL5G12 ) (Neodymium: Yttrium Aluminum Garnet), Nd: Glass Laser Electrical discharge pumping: Electrical discharge pumping is used in gas lasers. Since gas lasers have very narrow absorption band pumping them any flash lamp is not possible. Examples of Electrical discharge pumped lasers are He-Ne laser, CO2 laser, argon-ion laser, etc Chemical pumping: Chemical reaction may also result in excitation and hence creation of population inversion in few systems. Examples of such systems are HF and DF lasers. Injection current pumping: In semiconductors, injection of current through the junction results in creates of population inversion among the minority charge carriers. Examples of such systems are InP and GaAs. PRINCIPLE OF LASER/LASING ACTION: Let us consider many no of atoms in the excited state. Now the stimulating photon interacts with any one of the atoms in the excited state, the stimulated emission will occur. It emits two photons, having same energy & same frequency move in the same direction. These two photons will interact with another two atoms in excited state & emit 8-photons. In a similar way chain reaction is produced this phenomenon is called “Principle of lasing – action”. We get a monochromatic, coherent, directional & intense beam is obtained. This is called laser beam. This is the principle of working of a laser. Components of a LASER: Any laser system consists of 3-important components. They are (i) Source of energy or pumping source (ii) Active-medium (Laser Material) (iii)Optical cavity or resonator (i) Energy Source : It supply energies & pumps the atoms or molecules in the active medium to excited states. As a result we get population inversion in the active medium which emits laser. Ex: Xenon flash lamp, electric field. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 6  APPLIED PHYSICS (ii) Active medium: The medium in which the population inversion takes place is called as active medium. Active-centre: The material in which the atoms are raised to excited state to achieve population inversion is called as active center. (iii) Optical-cavity or resonator: The active medium is enclosed between a fully reflected mirror & a partially reflective mirror. This arrangement is called as cavity or resonator. As a result, we get highly intense monochromatic, coherence laser light through the non-reflecting portion of the mirror. DIFFERENT TYPES OF LASERS On the basis of active medium used in the laser systems, lasers are classified into several types I. Solid lasers : Ruby laser, Nd;YAG laser, Nd;Glass II. Liquid lasers : Europium Chelate laser, SeOCl2 III. Gas lasers : CO2, He-Ne, Argon-Ion Laser IV. Dye lasers : Rhodamine 6G V. Semiconductor lasers : InP, GaAs. RUBY LASER Ruby laser is a three level solid state laser and was developed by Mainmann in 1960. Ruby (Al2O3+Cr2O3) is a crystal of Aluminium oxide, in which 0.05% of Al+3 ions are replaced by the Cr+3 ions. The colour of the ruby rod is pink. The active medium in the ruby rod is Cr+3 ions. Principle or Characteristics of a ruby laser: Due to optical pumping, the chromium atoms are raised to excited states then the atoms come to metastable state by non-radiative transition. Due to stimulated emission the transition of atoms takes place from metastable state to ground state and gives a laser beam. Construction:  In ruby laser 4cm length and 5mm diameter rod is generally used.  Both the ends of the rods are highly polished and made strictly parallel.  The ends are silvered in such a way, one becomes partially reflected and the other end fully reflected.  The ruby rod is surrounded by xenon flash tube, which provides the pumping light to excite the chromium ions in to upper energy levels. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 7  APPLIED PHYSICS  Xenon flash tube emits thousands joules of energy in few milli seconds, but only a part of that energy is utilized by the chromium ions while the rest energy heats up the apparatus.  A cooling arrangement is provided to keep the experimental set up at normal temperatures. Working:  The energy level diagram of chromium ions is shown in figure.  The chromium ions get excitation into higher energy levels by absorbing of 5600A0 of wave length radiation.  The excited chromium ions stay in the level H for short interval of time (10 -8 Sec).  After their life time most of the chromium ions are de-excited from H to G and a few chromium ions are de-excited from H to M.  The transition between H and M is non-radioactive transition i.e. the chromium ions gives their energy to the lattice in the form of heat.  In the Meta stable state the life time of chromium ions is 10 -3 sec.  Due to the continuous working of flash lamp, the chromium ions are excited to higher state H and returned to M level.  After few milli seconds the level M is more populated than the level G and hence the desired population inversion is achieved.  The state of population inversion is not a stable one.  The process of spontaneous transition is very high.  When the excited chromium ion passes spontaneously from H to M it emits one photon of wave length 6943A0.  The photon reflects back and forth by the silver ends and until it stimulates an excited chromium ion in M state and it to emit fresh photon in phase with the earlier photon.  The process is repeated again and again until the laser beam intensity is reached to a sufficient value.  When the photon beam becomes sufficient intense, it emerges through the partially silvered end of the rod.  The wave length 6943A0 is in the red region of the visible spectrum on returning to ground state (G). DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 8  APPLIED PHYSICS Uses of Ruby laser: - Used in distance measurement using ‘pulse echo technique’ - Used for measurement of plasma properties such as electron density and temperature. - Used to remove the melanin of the skin. - Used for recording pulsed holograms. - Used as target designators and range finders in military. Draw backs of Ruby laser: - It requires high pumping power. - The efficiency of ruby laser is very small. It is a pulse laser. - He-Ne LASER It was discovered by A. Javan & his co-workers in 1960. It is a continuous wave gas laser. It consists of mixture of He & Ne in 10:1 ratio as a active medium. Principle/Characteristics of He-Ne laser: This laser is based on the principle of stimulated emission, produced in the He & Ne. The population inversion is achieved due to the interaction between He & Ne gases.Using gas lasers, we can achieve highly coherent, directional and high monochromatic beam. Construction:  In He-Ne gas laser, the He and Ne gases are taken in the ratio 10:1 in the discharge tube.  Two reflecting mirrors are fixed on either ends of the discharge tube, in that, one is partially reflecting and the other is fully reflecting.  In He-Ne laser 80cm length and 1cm diameter discharge tube is generally used.  The output power of these lasers depends on the length of the discharge tube and pressure of the gas mixture.  Energy source of laser is provided by an electrical discharge of around 1000V through an anode and cathode at each end of the glass tube. Working:  When the electric discharge is passing through the gas mixture, the electrons accelerated towards the positive electrode.  During their passage, they collide with He atoms and excite them into higher levels.  F2 and F3 form F1. In higher levels F2 and F3, the life time of He atoms is more.  So there is a maximum possibility of energy transfer between He and Ne atoms through atomic collisions.  When He atoms present in the levels F2 and F3 collide with Ne atoms present ground DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 9  APPLIED PHYSICS  state E1, the Ne atoms gets excitation into higher levels E 4 andE6.  Due to the continuous excitation of Ne atoms, we can achieve the population inversion between the higher levels E4 (E6) and lower levels E3 (E5).  The various transitions E6→ E5, E4→ E3 and E6→ E3 leads to the emission of wavelengths 3.39A⁰, 1.15 A⁰ and 6328A0.  The first two corresponding to the infrared region while the last wavelength is corresponding to the visible region.  The Ne atoms present in the E3 level are de-excited into E2 level, by spontaneously emission of photon.  When a narrow discharge tube is used, the Ne atoms present in the level E 2 collide with the walls of the tube and get de-excited to ground level E1. Uses of He-Ne laser: - Used in laboratories foe all interferometric experiments. - Used widely in metrology in surveying, alignment etc. - Used to read barcodes and He-Ne laser scanners also used for optical character recognition. - Used in holography. SEMICONDUCTOR LASER A Semiconductor diode laser is specially fabricated p-n junction device that emits coherent light when it is forward biased. The wavelength of the emitted photon depends upon the activation energy of crystal. Principle: When a p-n junction diode is forward biased, the electrons from n – region and the holes from the p- region cross the junction and recombine with each other.During the recombination process, the light radiation (photons) is released from a certain specified direct band gap semiconductors like Ga-As. This light radiation is known as recombination radiation. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 10  APPLIED PHYSICS The photon emitted during recombination stimulates other electrons and holes to recombine. As a result, stimulated emission takes place which produces laser. Construction Figure shows the basic construction of semiconductor laser.  The active medium is a p-n junction diode made from the single crystal of gallium arsenide.  This crystal is cut in the form of a platter having thickness of 0.5μmm.  The platelet consists of two parts having an electron conductivity (n-type) and hole conductivity (p-type).  The photon emission is stimulated in a very thin layer of PN junction (in order of few microns).  The electrical voltage is applied to the crystal through the electrode fixed on the upper surface.  The end faces of the junction diode are well polished and parallel to each other. They act as an optical resonator through which the emitted light comes out Working: Figure shows the energy level diagram of semiconductor laser.  When the PN junction is forward biased with large applied voltage, the electrons and holes are injected into junction region in considerable concentration.  The region around the junction contains a large amount of electrons in the conduction band and a large amount of holes in the valence band.  If the population density is high, a condition of population inversion is achieved. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 11  APPLIED PHYSICS  The electrons and holes recombine with each other and this recombination’s produce radiation in the form of light.  When the forward – biased voltage is increased, more and more light photons are emitted and the light production instantly becomes stronger.  These photons will trigger a chain of stimulated recombination resulting in the release of photons in phase.  The photons moving at the plane of the junction travels back and forth by reflection between two sides placed parallel and opposite to each other and grow in strength.  After gaining enough strength, it gives out the laser beam of wavelength 8400o A. The wavelength of laser light is given by Where Eg. is the band gap energy in Joule.The power output from this laser is 1mW. The nature of output is continuous wave or pulsed output Advantages: 1. It is very small in dimension and the arrangement is simple and compact. 2. It exhibits high efficiency.It can have a continuous wave output or pulsed output. 3. The laser output can be easily increased by controlling the junction current 4. It is operated with lesser power than ruby and CO2 laser. Disadvantages: 1. The output is usually from 5 degree to 15 degree i.e., laser beam has large divergence. 2. Threshold current density is very large (400A/mm2). 3. It has poor coherence and poor stability. Applications: 1. It is well suited for interface with fiber optic cables used in communication. 2. It is used to heal the wounds by infrared radiation APPLICATIONS OF LASERS Due to high intensity, high mono-chromaticity and high directionality of lasers, they are widely used in various fields like 1. communication 2. computers 3. chemistry DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 12  APPLIED PHYSICS 4. photography 5. industry 6. medicine 7. military 8. scientific research 1. Communication:  In case of optical communication, semiconductors laser diodes are used as optical sources.  More channels can be sent simultaneously Signal cannot be tapped as the band width is large, more data can be sent.  A laser is highly directional and less divergence, hence it has greater potential use in space crafts and submarines. 2. Computers :  In LAN (local area network), data can be transferred from memory storage of one computer to other computer using laser for short time.  Lasers are used in CD-ROMS during recording and reading the data. 3. Chemistry :  Lasers are used in molecular structure identification.  Lasers are also used to accelerate some chemical reactions.  Using lasers, new chemical compound can be created by breaking bonds between atoms or molecules. 4. Photography :  Lasers can be used to get 3-D lens lessphotography.  Lasers are also used in the construction of holograms. 5. Industry :  Lasers can be used to blast holes in diamonds and hard steel.  Lasers are also used as a source of intense heat.  Carbon dioxide laser is used for cutting drilling of metals and nonmetals, such as ceramics plastics, glass etc.  High power lasers are used to weld or melt any material.  Lasers are also used to cut teeth in saws and test the quality of fabric. 6. Medicine :  Pulsed neodymium laser is employed in the treatment of liver cancer.  Argon and carbon dioxide lasers are used in the treat men of liver and lungs.  Lasers used in the treatment of Glaucoma. 7. Military :  Lasers can be used as a war weapon.  High energy lasers are used to destroy the enemy air-crafts and missiles.  Lasers can be used in the detection and ranging likes RADAR. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 13  APPLIED PHYSICS 8. Scientific research:  Lasers are used in the field of 3D-photography.  Lasers used in Recording and reconstruction of hologram.  Lasers are employed to create plasma.  Lasers used to produce certain chemical reactions.  Lasers are used in Raman spectroscopy to identify the structure of the molecule.  Lasers are used in the Michelson- Morley experiment.  A laser beam is used to confirm Doppler shifts in frequency for moving objects. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 14  APPLIED PHYSICS FIBER-OPTICS INTRODUCTION TO OPTICAL FIBER: Fiber optics is a branch of physics which deals with the transmission & reception of light waves using optical fibers which acts as a guiding media. The transmission of light waves by fiber optics was first demonstrated by John Tyndall in 1870. Optical Fiber: Optical fiber is a thin & transparent guiding medium or material which guides the information carrying light waves. It is a cylindrical wave-guide system which propagates the data & speech signals in the optical frequency range. A light beam acting as a carrier wave is capable of carrying more information than radio waves & microwaves because of its high frequency as shown below. Radio waves - 104 Hz , Micro waves - 1010 Hz, Light waves - 1015 Hz Construction: An optical fiber is a very thin, flexible transparent made with plastic or glass. It has cylindrical shape consisting of three layers or sections 1 ) The Core 2 ) The Cladding 3 ) The Outer jacket or Buffer jacket 1)The Core:It is the central layer surrounded by another layer called cladding. Light is transmitted within the core which has refractive index (n1). It is a denser medium. Core is made of silica (SiO2). 2)The Cladding:It is the second layer, surrounded by a third layer called the outer jacket. It has refractive index n2 which is less than the refractive index of core i.e (n1 >n2 ). It acts as a rarer medium. It keeps the light within the core because n1>n2. To lower the refractive index of cladding the silica is doped with phosphorous or bismuth material. 3)The Outer or Buffer Jacket:It is the third layer it protects the fiber from moisture & abrasion. To provide necessary toughness & tensile strength, a layer of strength member is arranged surrounding buffer jacket. It is made of polyurethane material. Working Principle of Optical Fiber: Total Internal Reflection: The principle of optical fiber is total internal reflection. Condition for Total Internal Reflection: 1) The light ray should move from denser to rarer medium. 2) The refractive index of core must be greater than cladding i.e. n1>n2 3) The angle of incidence (i) must be greater than the critical angle(θc) i.e. i>θc. 𝑛 4) The critical angle𝜃𝑐 = 𝑠𝑖𝑛−1 𝑛2. 1 DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 15  APPLIED PHYSICS Explanation: Let us consider a denser medium & rarer medium of refractive indices n1 & n2 respectively and n1>n2. Let a light ray move from denser to rare medium with ‘i’ as the angle of incidence & ‘r’ as angle of refraction. The refracted ray bends away from the normal as it travels from denser to rarer medium with increase of angle of incidence ‘i’. In this we get three cases Case-1: When in2 For air, n2 =1, 1 𝜃𝑐 = 𝑠𝑖𝑛−1 𝑛1 Case-3:When i> θc, then the light ray reflected back into the medium as shown in figure ACCEPTANCE ANGLE & ACCEPTANCE CONE: Def: Acceptance angle is the maximum angle of incidence at the core of an optical fiber so that the light can be guided though the fiber by total internal reflection. This angle is called as acceptance angle. It is denoted by ‘αi’.  Consider a cross-sectional view of an optical fiber having core & cladding of refractive indices n1 and n2. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 16  APPLIED PHYSICS  Let the fiber be in air medium (no). The incident light while entering into the core at ‘A’ makes an incident angle of ‘αi’ with the fiber-axis.  In core it travels along AB & is incident at part B on cladding interface.  Let αr be the angle of refraction at part ‘A’ & ‘θ’ be the angle of incidence at ‘B’.  When ‘θ’ is greater than the critical angle ‘θc’, then total internal reflection takes place into the core & light takes the path BD.  Due to multiple total internal reflections the propagation of light ray takes place through the fiber.  Applying Snell’s law at AC core-air interface :- 𝑠𝑖𝑛α𝑖 𝑛1 = 𝑠𝑖𝑛α𝑟 𝑛𝑜 𝑛𝑜 𝑠𝑖𝑛α𝑖 = 𝑛1 𝑠𝑖𝑛α𝑟 → (1)  Let a normal ‘BC’ be drawn from the point ‘B’ to the fiber axis. Then from ∆ABC, we get α𝑟 = 90° − 𝜃 → (2) Substitute eq - (2) in eq - (1) 𝑛𝑜 𝑠𝑖𝑛α𝑖 = 𝑛1 𝑠𝑖𝑛(90° − 𝜃) 𝑛𝑜 𝑠𝑖𝑛α𝑖 = 𝑛1 𝑐𝑜𝑠𝜃 → (3)  To get total internal reflection at point B (Core-Classing Interface) i.e.  Let the maximum angle of incidence at point A be α𝑖 (max)for which. From eqn (3), we get 𝑛𝑜 𝑠𝑖𝑛α𝑖 = 𝑛1 𝑐𝑜𝑠𝜃 → (4) α𝑖 (max) = α𝑖 , when 𝜃 = 𝜃𝑐 𝑛1 𝑠𝑖𝑛 α𝑖 (max) = 𝑐𝑜𝑠𝜃𝑐 → (5) 𝑛𝑜 𝑛2 We know that 𝑠𝑖𝑛𝜃𝑐 = 𝑛1 𝑛22 𝑛12 − 𝑛22 √(𝑛12 − 𝑛22 ) √(𝑛12 − 𝑛22 ) 𝐶𝑜𝑠𝜃𝑐 = √(1 − 𝑠𝑖𝑛 𝜃𝑐2 ) = √(1 − ) => √ = ∴ 𝑐𝑜𝑠𝜃𝑐 = 𝑛12 𝑛12 𝑛1 𝑛1 → (6) Substitute the eq(6) in eq(5) , we get DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 17  APPLIED PHYSICS 𝑛1 𝑛12 − 𝑛22 𝑠𝑖𝑛 α𝑖 (max) =√ 𝑛0 𝑛1 𝑛1 − 𝑛22 2 𝑠𝑖𝑛 α𝑖 (max) = √ 𝑛0 (OR) 𝑛2 − 𝑛22 α𝑖 (max) = 𝑠𝑖𝑛−1 √ 1 → (7) 𝑛0 For air medium, n0=1 α𝑖 (max) = 𝑠𝑖𝑛−1 √ 𝑛21 − 𝑛22 → (8) Fractional Index Change (∆): It is the ratio of refractive index difference in core & cladding to the refractive index of core. n1 -n2 ∆= n1 𝑛1 − 𝑛2 = ∆𝑛1 → (1) NUMERICAL APERTURE (N.A.): Def:It is defined as light accepting efficiency of the fiber and is equal to sine of the acceptance angle of the fiber i.e. 𝑁. 𝐴. = 𝑠𝑖𝑛 α𝑖 (max) 𝑛12−𝑛22 N.A=𝑠𝑖𝑛 α𝑖 (max) = √ 𝑛0 𝑛1 −𝑛2 We know that ∆ = 𝑛1 𝑛1 − 𝑛2 = ∆𝑛1 → (2) 𝑊𝑒𝑘𝑛𝑜𝑤 , 𝑁. 𝐴. = 𝑆𝑖𝑛 α𝑖 (max) = √𝑛22 − 𝑛12 for air n0=1 𝑁. 𝐴. = √(𝑛1 + 𝑛2 )(𝑛1 − 𝑛2 ) 𝐼𝑓𝑛1 = 𝑛2 , 𝑡ℎ𝑒𝑛𝑁. 𝐴. = √2𝑛1 × ∆𝑛1 𝑁. 𝐴. = 𝑛1 √2∆→ (3) TYPES OF OPTICAL FIBERS: Optical fibers can be classified based on either the mode they support or the refractive index profile of the fiber. They can also be classified based on the material of the fiber. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 18  APPLIED PHYSICS Based on Mode: The rays travelling in the fiber by total internal reflection are called modes. 1) Single mode fibers:  If the thickness of the fiber is so small that it supports only one mode then the fiber is called single modefiberor mono mode fiber.  The core diameter of this fiber is about 8 to 10µm and the outer diameter of cladding is 60 to 70µm. 2)Multi mode fibers:  If the thickness of the fiber is very large that it supports more than one mode then the fiber is called multi mode fiber.  The core diameter of this fiber is about 50 to 200µm and the outer diameter of cladding is 100 to 250µm. Based on refractive index profile: 1) Step-Index Optical fiber:  In a step-index optical fiber, the entire core has uniform refractive index n1 slightly greater than the refractive index of the cladding n2.  Since the index profile is in the form of a step, these fibers are called step-index fibers.  The transmission of information will be in the form of signals or pulses.  These are extensively used because distortion and transmission losses are very less.  Step-index optical fibers are of two types. They are ( i) Single mode step-index fiber ( ii) Multi-mode step-index fiber DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 19  APPLIED PHYSICS Transmission / propagation of signal in Step-index fibers:  Generally the signal is sent through the fiber in digital form i.e. in the form of pulses.  The same pulsed signal travels in different paths.  Let us now consider a signal pulse travelling through step index fiber in two different paths (1) and (2).  The pulse (1) travelling along the axis of the fiber and pulse (2) travelling away from the axis.  At the receiving end only the pulse (1)which travels along the fiber axis reaches first while the pulse(2) reaches after some time delay.  Hence the pulsed signal received at the other end is broadened. This is called internal dispersion.  This reduces transmission rate capacity of the signal.  This difficulty is overcome by graded index fibers. 2) Graded index optical fiber:  In this fiber, the refractive index of the core varies radially.  It has maximum refractive index at its centre, which gradually falls with increase of radius and at the core-cladding interface matches with refractive index of cladding.  Variation of refractive index of the core with radius is given by x p n(x) = 𝑛1 [1 − 2Δ (a) ]2 Where 𝑛1 -> refractive index at the centre of the core a-> radius of the core 𝑛1 −𝑛2 Δ-> Fractional index change,∆ = 𝑛1 p-> grating profile index number  This fiber divided into two types. ( i ) Single-mode graded index fiber ( ii ) Multi-mode graded index fiber DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 20  APPLIED PHYSICS Transmission / propagation of signal in Step-index fibers:  Let us now consider a signal pulse travelling through graded index fiber in two different paths (1) and (2).  The pulse (1) travelling along the axis of the fiber though travels along shorter route it travels through higher refractive index.  The pulse (2) travelling away from the axis undergo refraction and bend as shown in fig. though it travels longer distance, it travels along lesser refractive index medium.  Hence both the pulses reach the other end simultaneously.  Thus the problem of inter model dispersion can be overcome by using graded index fibers. Based on types of materials: 1) glass-glass optical fiber 2) glass-plastic optical fiber 3) plastic-plastic optical fiber ATTENUATION (POWER-LOSS) IN OPTICAL FIBERS When light propagates through an optical fiber, then the power of the light at the output end is found to be always less than the power launched at the input end. The loss of power is called Attenuation. It is measured in terms of decibels per kilometer. Attenuation:It is defined as the ratio of the optical power output (P out) from a fiber of length ‘L’ to the power input (Pin). 10 𝑃𝑖𝑛 𝐴𝑡𝑡𝑒𝑛𝑢𝑎𝑡𝑖𝑜𝑛(𝛼 ) = − ( ) 𝑑𝐵/𝑘𝑚 𝐿 𝑃𝑜𝑢𝑡 Attenuation occurs because of the following reasons (1) Absorption (2) Scattering loss 3) Bending loss (1) Absorption:  It occurs in two ways, i)Absorption by impurity or impurity absorption ii) Intrinsic absorption or internal absorption  Impurity absorption: The impurities present in the fiber are transition metal ions, such as iron, chromium, cobalt & copper. During signal propagation when photons interact with these impurity atoms, then the photons are absorbed by atoms. Hence loss occurs in light power.  Intrinsic absorption or internal absorption: The fiber itself as a material has a tendency to absorb light energy however small it may be. The absorption that takes place in fiber material assuming that there are no impurities in it, is called intrinsic absorption. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 21  APPLIED PHYSICS (II) Scattering loss:  When the signals travels in the fiber, the photons may be scattered due to variations in the refractive index inside the fiber. This scattering is called as Rayleigh scattering. It is also a wavelength 1 dependent loss.𝑅𝑎𝑦𝑙𝑒𝑖𝑔ℎ𝑠𝑐𝑎𝑡𝑡𝑒𝑟𝑖𝑛𝑔𝑙𝑜𝑠𝑠 ∝ 𝜆4 (III)Bending losses:  These losses occur due to (a) Macroscopic bending (b) Microscopic bending  Macroscopic bending: If the radius of core is large compared to fiber diameter causes large curvature at the bends. At these bends, the light will not satisfy the condition for total internal reflection & light escapes out from the fiber. It is called as macroscopic bending.  Microscopic bending: These are caused due to non- uniform pressures created during the cabling of the fiber or during the manufacturing the fiber. It causes irregular reflections. This lead to loss of light by leakage through the fiber. OPTICAL FIBER IN COMMUNICATION SYSTEM The most important application of optical fibers occurs in the field of communication. Fiber optic communication systems comprise of the following units. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 22  APPLIED PHYSICS Information signal source:  The information signal to be transmitted may be voice, video or computer data (analog signals).  In order to communicate through optical fiber, the analog signals are converted into electrical signals.( by Analog to Digital converter)  The converted electrical signals are passed through the transmitter. Transmitter:  The transmitter is a modulator device used to receive electrical input signal, and then modulate it into digital pulses for propagation into an optical fiber.  The modulator consists of a driver and a light source as shown in fig.  The driver receives the electrical signals and then converts into the digital pulses.  These digital pulses are converted into optical signals after passing through a light source, generally either light emitting diodes (LED’s) or a semi conductor laser is used as light source.  The optical signals are then focused into the optical fiber as shown in fig. Optical Fiber (or) Transmission medium:  The optical fiber is used as transmission medium between the transmitter and the receiver.  The optical signals are then fed into an optical fiber cable where they are transmitted over long distances using the principle of total internal reflection. Receiver:  The receiver is a demodulator device used to receive the optical signals from the optical fiber and then convert into electrical signals.  The demodulator consists of a photodetector, an amplifier and a signal restorer.  The optical signals which are emerging from the optical fiber are received by photo detector.  The photodetector converts the optical signals into electrical signals.  The electrical signals are then amplified by the amplifier and the amplified electrical signals are converted into digital form.  The amplified electrical signals are fed to a signal restorer where the original voice is recovered. Advantages of Optical Fibers in communication system:  High data transmission rates and bandwidth.  Low losses.  Small cable size and weight.  Immunity to EM radiations.  Safety due to lack of sparks.  Data security. APPLICATIONS OF OPTICAL FIBERS: 1) Due to high band-width, light can transmit at a higher rate up to 1014 to 1015 Hz. Than radio or micro-frequencies. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 23  APPLIED PHYSICS 2) Long distance signal transmission. 3) They are used for exchange of information in cable television, space vehicles, sub-marines 4) Optical fibers are used in industry in security alarm systems, process control & industrial automation. 5) They are used in pressure sensors in biomedical & engine control applications. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 24  APPLIED PHYSICS Unit II QUANTUM MECHANICS Quantum Mechanics:- It is the branch of physics which explains about the motion of microscopic particles like electrons, protons etc. It was introduced by Max Planck in 1900. Waves & particles: Wave:- A wave is nothing but spreading of disturbance in a medium. The characteristics/properties of waves are 1) Amplitude 2) Time period 3) Frequency 4) Wave- length 5) Phase 6) Intensity. Particle:- A particle is a point in space which has mass & occupies space or region. The characteristics/properties of a particle are 1) Mass 2) velocity 3) Momentum 4) Energy etc. Matter Waves or de-broglie-waves: The waves associated with a material particle are called as matter waves. Difference between matter waves and electro-magnetic waves:- Matter Wave Electromagnetic waves(Light waves) 1. Matter wave is associated with moving 1. Oscillating charged particle gives rise to particle or material particle. EM wave. 2. Wavelength of an electromagnetic wave is 2. Wavelength of matter wave is given as given by h 𝜆= hc mv 𝜆= E 3. Wavelength of an electromagnetic wave 3. Wavelength of matter wave depends depends upon the energy of the photon. upon mass of the particle & velocity. 4. It can travel with a velocity greater than 4. It can travel with a velocity equal to the the velocity of light in vacuum. speed of light in vacuum i.e c = 3 × 108 m/s. 5. Electric field and Magnetic field oscillate 5. It is not a EM wave. perpendicular to each other & generate EM Waves. de-broglie concept of dual nature of matter waves:- In 1924 ,Louis de-broglie suggested that matter waves also exhibit dual nature like radiation(light). They are I. Wave nature II. Particle nature Wave nature of matter waves is verified by Davisson & Germer experiment, G.P.Thomson experiment etc. Particle nature of matter waves is verified by photo-electric effect, Compton effect etc. de-broglie hypothesis:- 1) The universe consists of matter and radiation(light) only 2) Matter waves also exhibit dual nature like radiation. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 25  APPLIED PHYSICS 3) The waves associated with the material particles are called as debroglie-waves or matter waves & the wave length associated with matter waves are called as de-broglie wave-length or matter wave-length (λ). ℎ ℎ 4) de-broglie wave-length is given by 𝜆 = 𝑝 = 𝑚𝑣 Expression for de-broglie wave-length(λ) in various form:- According to the planck’s theory of radiation, the energy of photon is given by ℎ𝑐 𝐸 = ℎ𝜈 = ----(1) 𝜆 h- planck’s constant, ν-frequency of photon According to Einstein mass energy relation E = mc2-----(2) m-mass of a photon c-velocity of light From equation of (1) & (2) ℎ𝑐 = mc2 𝜆 ℎ ℎ 𝜆= = − − − (3) 𝑚𝑣 𝑝 Where p-momentum of of photon = mc m-mass of photon, c-speed of light But according to de-broglie theory Momentum of electron particle(p) = mv m-mass of e’s , v-velocity of electron particle ℎ ℎ de-broglie wave-length(λ) =𝜆 = 𝑚𝑣 = 𝑝------(4) Eq.(4) gives the expression for de-broglie wave-length. Other forms of de-broglie wavelength(λ):- (i)In terms of Energy(E):- 1 we know that the kinetic energy of particle i.e. 𝐸 = 𝑚𝑣 2 − − − (5) 2 Multiply Eq-(5) by ‘m’ on both sides, we get 1 mE = 2 𝑚𝑣 2 2Em = m2v2 mv = √2mE ℎ 𝜆= ---(6) √2𝑚𝐸 (ii) de-broglie wavelength in terms of voltage(V):- If a charged particle is accelerated through a potential difference(V), then the kinetic energy of the particle is given as E = eV---(7) 1 But we have kinetic energy(E) of particle i.e. E = 𝑚𝑣 2 2 1 eV = 2 𝑚𝑣 2 2eV = mv2 DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 26  APPLIED PHYSICS Multiply by ‘m’ on both sides we get 2meV = 𝑚2 𝑣 2 => mv = √2𝑚𝑒𝑣 h = 6.6× 10−34 J-S ℎ ℎ 12.27 We have de-broglie wavelength 𝜆 = = or 𝜆 = 𝐴° mv √2𝑚𝑒𝑣 √𝑉 h = 6.625× 10−34 Js 𝑚𝑒 = 9.1 × 10−31 𝐾𝑔 e = 1.6 × 10−19 𝑐 (iii) de-broglie wavelength in terms of Temperature(T):- According to the kinetic theory of gases, the average kinetic energy of a particle at temperature ‘T’ is given by 3 E = 2 k𝐵 𝑇 ℎ ℎ ℎ 𝜆= = 3 = √2𝑚𝐸 √2𝑚 × 2 k 𝐵 𝑇 √3𝑚k 𝐵 𝑇 k 𝐵 is the Boltzmann constant. Properties or characteristics of matter waves or de-broglie waves:-  Lesser the mass of the particle, greater is the wavelength associated with it.  Smaller the velocity of the particle, longer is the wav-length associated with the particle.  When V = 0, 𝜆 = ∞ & 𝑉 = ∞, 𝜆 = 0.  Matter waves produced when the particles in motion are charged or uncharged.  Matter wave are not electro-magnetic waves.  Matter waves travel faster than the velocity of light.  Wave nature of matter gives an uncertainty in the position of the particle. Experimental Verification of matter waves:- Here, we have two methods to verify the dual natureof matter waves. They are 1. Davisson and Germer’s experiment 2. G.P. Thomson’s experiment 1. Davisson and Germer’s experiment:- First practical evidence for the wave nature of matter waves was given by C.J.Davisson and L.H. Germer in 1927. This was the first experimental support to debroglie’s hypothesis. Principle: The e’s which are coming from the source are incident on the target and the e’s get diffracted. These diffracted e’s produce a diffraction pattern. It shows(explains)the wave nature of matter waves. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 27  APPLIED PHYSICS Experimental Arrangement:- Construction: The experimental setup is shown in above figure. It consists of mainly 3-parts (i)Filament (ii)Target (iii)Circular scale arrangement. It also consists of a low tension battery(LTB), High tension battery(HTB) & a cylinder(A). Working: When tungsten filament ‘F’ is heated by a LTB then e’s are produced. Thesee’s are accelerated by High voltage(HTB). The accelerated e’s are collimated into a fine beam of pencil by passing them through a system of pin-holes in the cylinder’A’.This beam of electrons is allowed to incident on nickel crystal which acts as target. Then e’s are scattered in all the directions. The intensity of scattered e’s is measured by the circular scale arrangement. In this arrangement, an electron or movable collector(Double walled faraday cylinder) is fixed to circular scale which can collect the electrons and can move along the circular scale. The electron collector(Double walled faraday cylinder) is connected to a sensitive galvanometer to measure the intensity of electron beam entering the collector at different scattering angles(∅). A graph is plotted between the scattering angle(∅) and the number of scattered electron’s as shown in above figure. The intensity of scattered e’s is maximum at ∅ = 50° & accelerating voltage =54V. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 28  APPLIED PHYSICS Calculation of wave-length associated with e’s: 2dsin 𝜃 = 𝑛𝜆---(1) For nickel crystal, d = 0.909𝐴° = 0.909 × 10−10 m n = 1, First order (i)Angle of diffraction(𝜽)(𝐏𝐫𝐚𝐜𝐭𝐢𝐜𝐚𝐥 𝐯𝐚𝐥𝐮𝐞): From the the figure, 180° = 𝜃 + 𝜃 + 50° 180° = 2𝜃 + 50° ∴ Diffraction angle( (𝜃) = 65° ----(2) Substituting the above values in eq − 1 we get 2× 0.909 × 𝑆𝑖𝑛65° = 1×𝜆 𝜆 = 1.65𝐴° ---(3) (𝐢𝐢) From de-broglie wave length(𝝀): 12.27 λ= A° √V But V = 54v 12.27 λ= A° = 1.67A° √54 λ = 1.67A°----(4) From eq-(3) & eq-(4) it was been proved both the practical & theoretical wavelengths are almost equal. Hence the wave nature of particle is proved experimentally. (2) G.P.Thomson experiment: In 1928, G.P.Thomson, experimentally proved the dual nature of matter(particles). Principle: The electrons which are coming from the source are incident on the thin metal foil e’s are diffracted. These diffracted electrons produce a diffraction fringes on the photographic plate placed behind the foil. It explain the dual nature of matter. Experimental Arrangement : DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 29  APPLIED PHYSICS Construction: The experimental set up is shown in fig. It consists of (i) Filament(F) (ii) Anode(A) (iii) Cathode(C) (iv) Slit(S) (v) Gold foil(G) (vi) Photographic plate(P) (vii) Evacuated chamber. Working: Using a suitable battery the filament’F’ can be heated, so that electrons get emitted and pass through a high positive potential to the anode(A).Then ‘e’ beam passes through a slit incident on the gold foil(G) of thickness 10−8 cm. The e’s passing through the gold foil’G’ are recorded on a photographic plate(p). Since the gold foil consists of large no of micro crystals oriented at random ,the e’s striking the gold foil diffracts according to bragg’s law. 2dsin 𝜃 = 𝑛𝜆 After developing the plate, the diffraction pattern consists of concentric rings about a central spot is obtained. This pattern is similar as produced by x-rays. To make sure this ,the pattern or fringes is deflected by application of magnetic field. Since the e’s are deflected by magnetic field so that pattern is shifted. But x-rays are not deflected by electric & magnetic fields.So Thomson concluded that e’s(matter) behave like waves. Calculation’s of wavelength of e’s: From fig, let us consider ‘OA’ is the radius(r) of the ring, ‘O’ is the center of the ring. Let ‘L’ be the distance between the gold foil(G) and the photographic plate(p). Then from ∆𝑙𝑒 QOA , 𝑟 tan 2𝜃 = 𝐿 If’ ‘′𝜃′ is small, then r 𝜃= 2L According to Bragg’s law, 2dsin 𝜃 = 𝑛𝜆 If ′𝜃′ is small, then sin 𝜃 = 𝜃 => 2d.𝜃 = 𝑛𝜆 r 𝑛𝜆 = 2 × 𝑑 × 2L dr dr 𝑛𝜆 = L =>𝜆 = nL---(1) de-Broglie wavelength of e’s: 12.27 ∴𝜆= − − − (2) √𝑉 eq-(1) & eq-(2) are almost equal. This confirms the existence of matter waves. The wave nature of matter(particles) is verified experimentally. Heisenberg uncertainty principle:- The discovery of dual nature of material particle imposes a serious hurdle in locating the exact position and momentum of a particle simultaneously. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 30  APPLIED PHYSICS This wax was removed by “Werner. Heisenberg” in 1927 by proposing a significant principle. Later it was called Heisenberg’s uncertainty principle or Heisenberg’s principle of indeterminacy. “It is impossible to measure both the position and momentum of a particle simultaneously to any desired degree of accuracy.” If ′∆x′& ′∆p′ are uncertainties in the measurement of position & momentum of the particle then mathematically this uncertainties of this physical variables is written as h ∆x. ∆p ≥ 4π -----(1) Similarly the uncertainties in measuring energy and time interval we can write h (i) ∆E. ∆t ≥ 4π and the uncertainties in measuring angular momentum & angular displacement as h (ii) ∆J. ∆θ ≥ 4π Explanation:- (i)If ∆x = 0. i.e., the position of a particle is measured accurately, then from eq-(1). ℎ ∆p = ∆x. 4π ℎ ∆p = 0 = ∞ It means that, the momentum of the particle can’t be measured. (ii)If ∆p = 0. i. e. , the momentum of a particle is measured accurately, from eq. (1). h h ∆x = = =∞ ∆p. 4π 0 From the above said observations made by Heisenberg, he clearly states that it is impossible to design an experiment to prove the wave & particle nature of matter at any given instant of time. If one measures position or momentum accurately, then there will be an uncertainty in the other.Thus, the Heisenberg’s uncertainty principle gives the probability of determining the particle at any given instant of time in place of certainty. Applications:- (i).It explains the non-existence of e’s in the nucleus. (ii).It gives the binding energy of an e’s in atom. (iii).It calculates the radius of Bohr’s first orbit. Schrodinger Wave Equation:- Schrodinger describes the wave nature of a particle in mathematical form and is known as Schrodinger wave equation(SWE). There are two types of SWE (i). Schrodinger Time independent wave equation(STIWE) (ii). Schrodinger Time dependent wave equation(STDWE) (i) Schrodinger Time independent wave equation(STIWE):- To derive an expression for “(STIWE)” let us consider an electron(particle)moving along a positive direction along the axes. Let x,y,z be the coordinates of the particle &′ψ′ is the DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 31  APPLIED PHYSICS wave-displacement or wave-function of the matter wave at any time’t’. It is assumed that ′ψ′ is finite, single-valued, continuous and periodic function. We can express the classical differential wave equation of the material particle in three- dimension axes is given as ∂2 ψ 2 ∂2 ψ ∂2 ψ ∂2 ψ = v ( + + ) ∂t 2 ∂x 2 ∂y 2 ∂z 2 ∂2ψ In Three dimension we write = V 2 ∇2 ψ − − − (1) ∂t2 The solution of eq-(1) is given by ψ = ψ0 sin ωt − − − (2) ψ = ψ0 sin 2πυt − − − (3) Where 𝜔 = 2πυ is the angular frequency of the particle Differentiating eq-(3) w.r.t. ‘t’ twice we get ∂ψ = ψ0 × cos 2πυt × 2πυ ∂t 2 𝜕 𝜓 = −ψ0 × sin 2πυt × 2πυ × 2πυ 𝜕𝑡 2 ∂2ψ = −4π2 ϑ2 ψ---(3) ∂t2 c v ϑ= = λ λ ∂2ψ 4π2 v2 ψ = − λ2 ---(4) ∂t2 Substituting eq-(4) in eq-(1), we get, 4π2 v 2 ψ V 2 ∇2 ψ = − λ2 4𝜋 2 2 ∇ ψ + 2 ψ = 0 − − − (5) 𝜆 According to de-broglie wave-length h h 𝜆 = p = mv ---(6) Sub. eq(6) in eq(5), we get 4𝜋 2 𝑚2 𝑣 2 ∇2 ψ + ψ = 0 − − − (7) ℎ2 We have, the total energy (E) is given by E = P.E. + K.E. 1 E = V + 2 mv 2 1 mv 2 = E − V 2 mv 2 = 2(E − V) m2 v 2 = 2m(E − V) − − − (8) Sub. eq-(8) in eq-(7) we get, 4𝜋 2 ∇2 ψ + 2 2m(E − V)ψ = 0 ℎ DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 32  APPLIED PHYSICS 8𝜋 2 𝑚(E − V)ψ ∇2 ψ + = 0 − − − (9) ℎ2 2𝑚(𝐸−𝑉)ψ ℎ ∇2 ψ + = 0 − − − (10) Where ℏ = 2𝜋 ℏ2 Eq-(9) & Eq-(10) is called as Schrodinger time independent wave equation in three dimension For a free particle, V= 0 , 2𝑚𝐸ψ ∇2 ψ + = 0 − − − (11) ℏ2 (ii)Schrödinger Time dependent wave equation:- ℏ 2 ̂Hψ = ̂ Eψ where ̂H = − 2m ∇2 + V = Hamiltonian operator ∂ ̂ E = iℏ ∂t = Energy operator Physical significance of wave-function(𝛙)(𝐄𝐢𝐠𝐞𝐧 − 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧): − Wave-function(𝛙) or Eigen-function(𝛙):- It is a variable or complex quantity that is associated with a moving particle at any position (x,y,z) and at any time ‘t’. (i) ‘ψ′ of a particle is represented by ψ = ψ0 e−iωt (ii) ‘ψ′ explains the motion of microscopic particles. (iii) ‘ψ′ is a complex quantity & it does not have any meaning. (iv) |ψ|2 = ψψ∗ is real and positive, it has physical meaning. (v) |ψ|2 represents the probability of finding the particle per unit volume. (vi) For a given volume d τ, the probability of finding the particle is given by probability density(p) = ∭|ψ|2 dτ where dτ = dxdydz (vii) ‘ψ′ gives the information about the particle behavior. (viii) ‘p’ values are between 0 to 1. (xi) wave-function ‘ψ′ is a single valued, finite and periodic function. (x) If p = ∭|ψ|2 dτ = 1, then ‘ψ′ is called normalized wave function. Application of Schrödinger Time independent wave equation(STIWE):- (1) Particle or electron in a one dimensional box or particle in an infinite square well potential:- Consider a particle or electron of mass ’m’ moving along x-axis enclosed in a one dimensional potential box as shown in figure.Since the walls are of infinite potential the particle does not penetrate out from the box. i.e. potential energy of the particle V=∞ at the walls. The particle is free to move between the walls A & B at x=0 and x=L. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 33  APPLIED PHYSICS The potential energy of the particle between the walls is constant because no force is acting on the particle. ∴ The particle energy is taken as zero for simplicity 𝑖. 𝑒. V= 0 between x=0 & x=L. Boundary Conditions:- (i)The potential energy for particle is given as V(x) = 0, for 0≤ 𝑥 ≤ 𝐿 --------- (1) V(x) =∞, when 0≥ 𝑥 ≥ 𝐿 The Schrödinger time independent wave equation for the particle is given by, d2 ψ 8π2 m + (E − V)ψ = 0 dx 2 h2 But v=0, between walls, d2 ψ 8π2 m + (E)ψ = 0----( 2 ) dx2 h2 8π2 m Let (E) = k 2 h2 Then eq-(2), becomes 𝑑2𝜓 + k 2 ψ = 0 − − − (3) 𝑑𝑥 2 The general solution of eq-(3) is given by 𝜓(𝑥 ) = Asinkx + Bcoskx------(4) Where A, B are two constants, ‘k’ is the wave-vector Applying the boundary conditions eq-(4), we get (i) 𝜓 = 0 𝑎𝑡 𝑥 = 0 0 = Asink(0) + Bcosk(0) 0 = 0 +B B=0 Then eq-(4) is written as 𝜓(𝑥 ) = Asinkx or ψn (x) = Asin(kx)----(6) (ii) 𝜓 = 0 𝑎𝑡 𝑥 = 𝐿 Eq-(6) can be written as 0 =ASin(kL) => A≠ 0, and sin(kL) = 0 sin(kL) = sin(n𝜋) n𝜋 kL = n𝜋 => k = 𝐿 -----(7) 1) Energy of the particle(Electron):- n𝜋 n2 h 2 k= =>k 2 = 𝐿 L2 8π2 m But we have, (E) = k 2 h2 𝑛 2𝜋2 8π2 m = ( E) ℎ2 h2 𝑛 2ℎ 2 𝑛 2ℎ 2 E = 8𝑚𝐿2 =>𝐸𝑛 = 8𝑚𝐿2 − − − (8) DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 34  APPLIED PHYSICS (ii)Energy levels of an particle(Electron):- h2 when n=1, 𝐸1 = 8mL2 4ℎ 2 when n=2, 𝐸2 = = 4𝐸1 8𝑚𝐿2 2 In general, 𝐸𝑛 = 𝑛 𝐸1 ∴ Energy levels of an electron are discrete. (iii) Normalization of the wave-function of the particle to find ‘A’:- To find the ‘A’ value, by applying the normalization conditions. 𝐿 ∫0 |𝜓(𝑥)|2 𝑑𝑥 = 1 𝐿 𝑛𝜋𝑥 ∫0 𝐴2 𝑆𝑖𝑛2 ( 𝐿 ) 𝑑𝑥 = 1 𝐿 𝑛𝜋𝑥 𝐴2 ∫ 𝑆𝑖𝑛2 ( ) 𝑑𝑥 = 1 0 𝐿 By integrating and applying the appropriate limits , finally we obtain the value of ‘A” as 2 A =√L ----(9) (iv) Electron wave functions ,probability density functions and energy functions:- (i) Wave function of electron/particle 2 𝑛𝜋𝑥 ψn (x) = √L sin( )-----(10) 𝐿 (ii) Probability density function(P) 2 𝑛𝜋𝑥 P = |𝜓(𝑥)|2 = L 𝑆𝑖𝑛2 ( )-----(11) 𝐿 (iii)Energy function of the particle 𝑛2 ℎ2 𝐸𝑛 = 8𝑚𝐿2 ----(12) DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 35  APPLIED PHYSICS Unit III ELECTRONIC MATERIALS Free electron theory: In solids, electrons in outer most orbits of atoms determine its electrical properties. Electron theory is applicable to all solids, both metals and non-metals. In addition, it explains the electrical, thermal and magnetic properties of solids. The structure and properties of solids are explained employing their electronic structure by the electron theory of solids. It has been developed in three main stages: 1. Classical free electron theory 2. Quantum Free Electron Theory. 3. Zone Theory.  Classical free electron theory: The first theory was developed by Drude & Lorentz in 1900. According to this theory, metalcontains free electrons which are responsible for the electrical conductivity and metals obey the laws of classical mechanics.  Quantum Free Electron Theory: In 1928Sommerfield developed the quantum free electron theory. According toSommerfield, the free electrons move with a constant potential. This theory obeys quantum laws.  Zone Theory: Bloch introduced the band theory in 1928. According to this theory, free electrons move in a periodic potential provided by the lattice. This theory is also called “Band Theory of Solids”. It gives complete informational study of electrons. Classical free electron theory: Even though the classical free electron theory is the first theory developed to explain the electrical conduction of metals, it has many practical applications. The advantages and disadvantages of the classical free electron theory are as follows: Advantages: 1. It explains the electrical conductivity and thermal conductivity of metals. 2. It verifies ohm’s law. 3. The free electrons moves in random directions and collide with either positive ions or other free electrons. Collision is independent of charges and is elastic in nature 4. The movements of free electrons obey the laws of classical kinetic theory of gases 5. Potential field remains constant throughout the lattice. Drawbacks: 1. It fails to explain the electric specific heat and the specific heat capacity of metals. 2. It fails to explain Electrical conductivity (perfectly) of semiconductors or insulators. 3. The classical free electron model predicts the incorrect temperature dependence of 𝜎.According to the classical free electron theory ,𝜎 𝛼 𝑇 −1. 4. It fails to give a correct mathematical expression for thermal conductivity. 5. Ferromagnetism couldn’t be explained by this theory. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 36  APPLIED PHYSICS Quantum free electron theory of metals: Advantages: 1. All the electrons are not present in the ground state at 0 K, but the distribution obeys Pauli’s exclusion principle. At 0 K, the highest energy level filled is called Fermi- level. 2. The potential remains constant throughout the lattice. 3. Energy levels are discrete. 4. It was successful to explain not only conductivity, but also thermionic emission paramagnetism, specific heat. Drawbacks: 1. It fails to explain classification of solids as conductors, semiconductors and insulators. Fermi level and Fermi energy: The distribution of energy states in a semiconductor is explained by Fermi –Dirac statistics since it deals with the particles having half integral spin like electrons. Consider that the assembly of electrons as electron gas which behaves like a system of Fermi particles or fermions. The Fermions obeying Fermi –Dirac statistics i.e., Pouli, s exclusion principle. Fermi energy: Itis the energy of state at which the probability of electron occupation is ½ at any temperature above 0K. It separates filled energy states and unfilled energy states. The highest energy level that can be occupied by an electron at 0 K is called Fermi energy level Fermi level: It is a level at which the electron probability is ½ at any temp above 0K (or) always it is 1 or 0 at 0K. Therefore, the probability function F(E) of an electron occupying an energy level E is given by, 1 𝐹 (𝐸 ) = 𝐸−𝐸 … … … … … … (1) 1 + exp( 𝐾𝑇 𝐹 ) F Where 𝐸𝐹 known as Fermi energy and it is constant for a system, ( K is the Boltzmann constant and T is the absolute temperature. E Case I : Probability of occupation at T= 0K,and 𝐸 < 𝐸𝐹 ) Case II:Probability of occupation at T= 0K, and 𝐸 > 𝐸𝐹 Then DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 37  APPLIED PHYSICS 1 1 1 1 𝐹 (𝐸 ) = 1 = ∞ = = =0 1+𝑒 0 1+𝑒 1+∞ ∞ i.e., all levels below 𝐸𝐹 are completely filled and al levels above 𝐸𝐹 are completely empty. As the temperature rises F (E). Case III:Probability of occupation at T= 0K, and 𝐸 = 𝐸𝐹 1 1 1 𝐹 (𝐸 ) = 0 = = = 0.5 1+𝑒 1+1 2 The probability function F(E) lies between 0 and 1. Hence there are three possible probabilitities namely Density of States (DOS): The number of electrons per unit volume in an energy level at a given temperature is equal to the product of density of states (number of energy levels per unit volume) and Fermi Dirac distribution function (the probability to find an electron). 𝑛𝑐 = ∫ g(𝐸 ) × 𝑓(𝐸 )𝑑𝐸 … … …. (1) where𝑛𝑐 is the concentration of electrons, g (E) is the density of states & F(E) is the occupancy probability. The number of energy states with a particular energy value E is depending on how many combinations of quantum numbers resulting in the same value n. To calculate the number of energy states with all possible energies, we construct a sphere in 3D- space with ‘n’ as radius and every point (nx, ny and nz) in the sphere represents an energy state. As every integer represents one energy state, unit volume of this space contains exactly one state.Hence; the number of states in any volume is equal to the volume expressed in units of cubes of lattice parameters).Also 𝑛2 = 𝑛𝑥2 + 𝑛𝑦2 + 𝑛𝑧2 Consider a sphere of radius n and another sphere of radius n+dn with the energy values are E and (E+dE) respectively. Therefore, the number of energy states available in the sphere of radius ‘n’ is by considering one octant of the sphere (Here, the number of states in a shell of thickness dn at a distance ‘n’ in coordinate system formed by 𝑛𝑥 , 𝑛𝑦 𝑎𝑛𝑑 𝑛𝑧 and will take only positive values ,in that 1 sphere 8 of the volume will satisfy this condition). DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 38  APPLIED PHYSICS The number of energy states within a sphere of radius (n+dn) is 1 4𝜋 ( ) (𝑛 + 𝑑𝑛)3 8 3 Thus the number of energy states having energy values between E and E+dE is given by 1 4𝜋 1 4𝜋 g(𝐸 )𝑑𝐸 = ( ) (𝑛 + 𝑑𝑛)3 − ( ) 𝑛3 8 3 8 3 1 4𝜋 𝜋 𝜋 = ( ) ⌊(𝑛 + 𝑑𝑛)3 − 𝑛3 ⌋ = (3𝑛2 𝑑𝑛) = 𝑛2 𝑑𝑛 8 3 6 2 compared to ‘dn’,𝑑𝑛2 𝑎𝑛𝑑 𝑑𝑛3 𝑎𝑟𝑒 𝑣𝑒𝑟𝑦 𝑠𝑚𝑎𝑙𝑙. Neglecting higher powers of dn 𝜋 g(𝐸 )𝑑𝐸 = 𝑛2 𝑑𝑛 … ….. (2) 2 The expression for𝑛𝑡ℎ energy level can be written as , 𝑛2 ℎ2 2 8𝑚𝐿2 𝐸 𝐸= 𝑜𝑟, 𝑛 = … ….. (3) 8𝑚𝐿2 ℎ2 8𝑚𝐿2 𝐸 1 ⟹𝑛=( )2 … … …. (4) ℎ2 Differentiating eq. (3): 8𝑚𝐿2 1 8𝑚𝐿2 2𝑛𝑑𝑛 = 𝑑𝐸 => 𝑑𝑛 = ( ) 𝑑𝐸 ℎ2 2𝑛 ℎ2 ∵ by substituting 1/n value in dn, 1 8𝑚𝐿2 1 𝑑𝐸 𝑛 = ( 2 )2 1 … … … (5) 2 ℎ 𝐸2 Substitute n2 and dn from eq. (3) and (5), we get 𝜋 8𝑚𝐿2 1 8𝑚𝐿2 3 dE g(𝐸 )𝑑𝐸 = ( 2 ) 𝐸 𝑑𝐸 × ( 2 )2 1 2 ℎ 2 ℎ E ⁄2 𝜋 8𝑚𝐿2 3 1 g(𝐸 )𝑑𝐸 = ( 2 )2 𝐸 2 𝑑𝐸 … … … (6) 4 ℎ According to Pauli’s Exclusion Principle, two electrons of opposite spin can occupy each energy state Equation (6) should be multiplied by 2 𝜋 8𝑚𝐿2 3 1 g(𝐸 )𝑑𝐸 = 2 × ( 2 )2 𝐸 2 𝑑𝐸 4 ℎ After mathematical simplification, we get g 4𝜋 3 1 (𝐸 )𝑑𝐸 = 3 (2𝑚)2 E 2 L3 𝑑𝐸 ℎ The density of energy states g(E) dE per unit volume is given by, DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 39  APPLIED PHYSICS 4𝜋 3 1 g(𝐸 )𝑑𝐸 = (2𝑚) 2 E 2 𝑑𝐸 ∵ L3 = 1 ℎ3 Bloch Theorem: According to free electron model, a conduction electron in metal experiences constant potential. But in real crystal, there exists a periodic arrangement of positively charged ions through which the electrons move. As a consequence, the potential experienced by electrons is not constant but it varies with the periodicity of the lattice.In zone theory, as per Bloch, potential energy of electrons considered as varying potential with respect to lattice ‘a’. Fig: Variation of potential energy in a periodic lattice. Let us examine one dimensional lattice as shown in figure. It consists of array of ionic cores along X-axis. A plot of potential V as a function of its position is shown in figure. From graph: At nuclei or positive ion cores, the potential energy of electron is minimum and in-between nuclei, the P.E.is considered as maximum w.r.to. Lattice constant ‘a’. This periodic potential V(x) changes with the help of lattice constant a, 𝑉 (𝑥 ) = 𝑉 (𝑥 + 𝑎) (‘a’ is the periodicity of the lattice) To solve, by considering Schrodinger’s time independent wave equation in one dimension, 𝑑 2 𝜓 8𝜋 2 𝑚 + [𝐸 − 𝑉(𝑥 )]𝜓 = 0 … … (1) 𝑑𝑥 2 ℎ2 Bloch’s 1D solution for Schrodinger wave equation (1)𝜓𝑘 (𝑥 ) = 𝑢𝑘 (𝑥 )𝑒𝑥𝑝(𝑖𝑘𝑥 ) … … …. (2) where 𝑢𝑘 (𝑥 ) = 𝑢𝑘 (𝑥 + 𝑎) 2𝜋 Here 𝑢𝑘 (𝑥 ) -periodicity of crystal lattice, modulating function, k- propagation vector = 𝜆 𝑒 𝑖𝑘𝑥 is plane wave. By applying eq.n (2) to eq.n (1),it is not easy to solve Schrodinger wave equation and Bloch cannot explains complete physical information about an 𝑒 − in periodic potential field. Then Kronig Penny model was adopted to explains the electrical properties of an 𝑒 −. Kronig-Penney model: Kronig –penny approximated the potentials of an 𝑒 − 𝑠 inside the crystal in terms of the shapes of rectangular steps as shown, i.e. square wells is known as Kronig Penny model. DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 40  APPLIED PHYSICS i.e. The periodic potential is taken in the form of rectangular one dimensional array of square well potentials and it is the best suited to solve Schrodinger wave equation. Width w I re It is assumed that the potential energygis zero when x lies between 0 and a, and is considered as I region. Potential energy is 𝑉0 , when x lies between –b and 0. And considered as II region. Boundary conditions: 𝑉(𝑥) = 0 , 𝑤ℎ𝑒𝑟𝑒 𝑥 𝑙𝑖𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 0 < 𝑥 < 𝑎 – 𝐼 𝑟𝑒𝑔𝑖𝑜𝑛𝑉(𝑥 ) = 𝑉0 , 𝑤ℎ𝑒𝑟𝑒 𝑥 𝑙𝑖𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 − 𝑏 < 𝑥 < 0 – 𝐼𝐼 𝑟𝑒𝑔𝑖𝑜𝑛 This model explains many of the characteristic features of the behavior of electrons in a periodic lattice. The wave function related to this model may be obtained by solving Schrodinger equations for the two regions, 𝑑 2 𝜓 2𝑚 + 2 E𝜓 = 0, for 0 < 𝑥 < 𝑎 𝑤𝑖𝑡ℎ 𝑉 (x) = 0 … … … … (1) 𝑑𝑥 2 ℏ 𝑑 2 𝜓 2𝑚 + 2 (E − V0 )𝜓 = 0, for − b < 𝑥 < 0 with V(x) = V0 … … … (2) 𝑑𝑥 2 ℏ Again, 𝑑2𝜓 2mE 2Π + α2 𝜓 = 0 … ….. (3) where α2 = and α = √2𝑚𝐸 𝑑𝑥 2 ℏ2 h 𝑑2𝜓 2m − β2 𝜓 = 0 … …. (4) where β2 = (V − E) 𝑑𝑥 2 ℏ2 0 The solution of these equations from Bloch theorem, 𝜓𝑘 (𝑥 ) = 𝑢𝑘 (𝑥 )𝑒𝑥𝑝(𝑖𝑘𝑥).From figure, square well potentials, if 𝑉0 increases, the width of barrier ‘w’ decreases, if 𝑉0 decreases the width of barrier w increases. But the (product) barrier strength 𝑉0 𝑤 remains constant. To get this, differentiating above Schrodinger wave equations 3 & 4 w.r.to x, and by applying boundary conditions of x (w.r.to their correspondingΨ), to known the values of constants A, B of region -I, C,D-for reg-II,we get mathematical expression (by simplification) DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 41  APPLIED PHYSICS 𝑠𝑖𝑛𝛼𝑎 𝑐𝑜𝑠𝑘𝑎 = 𝑃 + 𝑐𝑜𝑠𝛼𝑎 where, 𝛼𝑎 4𝜋 2 𝑚𝑎 2π 𝑃= 𝑉0 w and α= √2mE ℎ2 h P-varying term, known as scattering power. And ‘𝑣₀ b’ is known as barrier strength. Forbidden energy gap Conclusions: 1. The L.H.S is a cosine term which varies between the limits -1 and +1, and hence the R.H.S also varies between these limits. It means energy is restricted within -1 to +1 only. 2. If the energy of 𝑒 − lies between -1 to +1, are called allowed energy bands and it is shown by shaded portion in energy spectrum. This means that ‘αa’ can take only certain range of values belonging to allowed energy band. 3. As the value of αa increases, the width of the allowed energy bands also increases. 4. If energy of𝑒 − s not lies between -1 to +1 are known as forbidden energy bands and it is decreases w.r.to increment of 𝛼𝑎. 5. Thus, motion of 𝑒 − s. in a periodic lattice is characterized by the bands of allowed & forbidden energy levels. Case 1: DEPARTMENT OF HUMANITIES & SCIENCES ©MRCET (EAMCET CODE: MLRD) 42  APPLIED PHYSICS 1. P→∞ If P→∞, the allowed band reduces to a single (line) energy level, gives us steeper lines. We have 𝑠𝑖𝑛𝛼𝑎 𝑐𝑜𝑠𝑘𝑎 = 𝑃 + 𝑐𝑜𝑠𝛼𝑎 𝛼𝑎 𝛼𝑎 𝛼𝑎 𝑐𝑜𝑠𝑘𝑎 ( ) = sin 𝛼𝑎 + 𝑐𝑜𝑠𝛼𝑎 ( ) 𝑝 𝑝 1 𝑃→∞ , =0 𝑡ℎ𝑒𝑛 𝑠𝑖𝑛𝛼𝑎 = 0 ∞ 𝑠𝑖𝑛𝛼𝑎 = sin 𝑛𝜋 𝛼𝑎 = 𝑛𝜋 𝛼 2 𝑎2 = 𝑛2 𝜋 2 𝑛2 𝜋 2 𝛼2 = 2 𝑎 2𝑚𝐸 𝑛2 𝜋 2 = 2 ℏ2 𝑎 𝑛2 𝜋 2 ℏ2 𝑛 2 𝜋 2 h2 𝐸= = 2𝑚𝑎2 2𝑚𝑎2 4𝜋 2 2 2 𝑛 h 𝐸= , ℎ𝑒𝑟𝑒 𝑎 𝑖𝑠 𝑙𝑎𝑡𝑡𝑖𝑐𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 8𝑚𝑎2 It means, it (zone theory) supports quantum free electron theory. Case 2: P→0, We have 𝑠𝑖𝑛𝛼𝑎 𝑐𝑜𝑠𝑘𝑎 = 𝑃 + 𝑐𝑜𝑠𝛼𝑎 𝛼𝑎 𝛼𝑎 = 𝑘𝑎 𝛼=𝑘 𝛼2 = 𝑘2 2𝑚𝐸