Module 1: Fundamentals of Semiconducting Materials PDF

OPTOELECTRONICS & DEVICE PHYSICS Module 1 “Fundamentals of Semiconducting Materials” 1  CONTENT Concept of energy bands Charge carriers Carrier concentration Concept of Fermi level Hall effect...

OPTOELECTRONICS & DEVICE PHYSICS Module 1 “Fundamentals of Semiconducting Materials” 1  CONTENT Concept of energy bands Charge carriers Carrier concentration Concept of Fermi level Hall effect Magnetic materials Superconductors Numericals  Band Theory- Concept of energy bands The band theory of solids is a fundamental concept in condensed matter physics and materials science that helps explain the electronic structure and electrical conductivity of solid materials. It provides a framework for understanding how electrons are arranged and behave in a crystalline solid. It helps explain why some materials are insulators, some are semiconductors, and others are conductors, based on the energy band structure and the presence or absence of an energy band. The key ideas and principles of band theory are: 1.Energy Bands: In a crystalline solid, the atoms are arranged in a periodic lattice structure. When electrons from individual atoms come together in a solid, they form energy bands. Energy bands represent the allowed energy levels that electrons can occupy in the crystal lattice. 2.Valence and Conduction Bands: The two most important energy bands are the valence band and the conduction band. The valence band is the lower- energy band, and it is typically filled with electrons. Electrons in the valence band are tightly bound to their respective atoms. The conduction band is the higher-energy band, and it is usually partially or completely empty. Electrons in the conduction band are free to move throughout the crystal. 3.Energy Band Gap: The region of energy levels between the valence band and the conduction band is called the energy band gap (or bandgap). It represents the energy range where electrons are forbidden to exist. The size of the band gap is a crucial factor in determining a material's electrical properties.  Classification of Solids on the Basis of Band Theory Based on the energy band gap, materials can be classified into three main categories: 1. Insulators: Materials with a large band gap that have very few electrons in the conduction band, making them poor conductors of electricity are called insulators e.g., diamond, glass. 2. Semiconductors: Materials with a moderate band gap that have a few electrons in the conduction band at room temperature, and their electrical conductivity can be controlled by adding small amounts of impurities (doping) or by applying external factors like temperature are called semiconductors e.g., silicon, germanium. 3.Conductors: Materials with overlapping valence and conduction bands that have a large number of free electrons in the conduction band are called conductors and are excellent conductors of electricity e.g., metals.  Classification of Semiconductors  Semiconductors are materials that have electrical conductivity between that of conductors (such as metals) and insulators (such as ceramics or plastics).  They are essential components in electronic devices and can be classified in several ways Doping: The process of adding controlled impurities to a semiconductor is known as doping  Differentiate N-type and P-type Semiconductors Sl N- type semiconductor P-type Semiconductor No 1 Extrinsic semiconductor Extrinsic semiconductor obtained by doping with obtained by doping with pentavalent impurity such trivalent impurity such as as Phosphorous, Antimony, Boron, Gallium, Indium etc Arsenic etc 2 The doped atoms give The impurity atoms added extra electrons in the create vacancies of structure, and are called electrons (i.e., holes) in donor atoms. the structure, and are called acceptor atoms. 3 Majority Charge carriers Majority Charge carriers are electrons and minority are holes and minority Energy band diagram of intrinsic, n-type and p-type semiconductors  Difference Between Intrinsic and Extrinsic Semiconductors Intrinsic Semiconductor Extrinsic Semiconductor It is in pure form It is formed when trivalent or pentavalent elements are added to pure semiconductor Holes and electrons are equal Number of free holes are very large in p- type and number of electrons are very large in n-type Fermi level lies exactly at the centre of Fermi level is near to valence band in p-type energy gap and near conduction band in n-type Carrier concentration is low Carrier concentration is very high Conductivity is low Conductivity is high Carrier concentration can be increased Carrier concentration can be increased by by increasing temperature increasing dopant concentration or by increasing temperature or both.  Fermi Energy Fermi energy, named after the Italian physicist Enrico Fermi. It is a fundamental concept in quantum mechanics and solid-state physics. It represents the highest energy state of an electron at absolute zero temperature (0 Kelvin or -273.15 degrees Celsius) in a system. Definition The energy corresponding to the highest occupied level of an electrons at 0 K is called Fermi energy. The corresponding energy level is called the Fermi level. At 0 K all the energy levels above Fermi level are empty and all those below are completely filled. Key points about Fermi energy: 1.Quantum Mechanical Basis: In a solid or any other system of fermions (particles with half-integer spin, such as electrons), the Pauli exclusion principle is defined that no two electrons can occupy the same quantum state. This principle results in a distribution of electron energies called the Fermi-Dirac distribution. 2. Absolute Zero Temperature: Fermi energy is defined at absolute zero temperature because at this temperature, all available electron states up to the Fermi energy level are occupied, while all states above it are unoccupied. 3. Role in Conductivity: Fermi energy determines the electrical conductivity of a material. Materials with a high Fermi energy have a greater number of electrons near the top of their energy bands, making them good conductors. Conversely, materials with a low Fermi energy have fewer electrons near the top of their bands and are typically insulators. Key points about Fermi energy Continues: 4. Temperature Effects: At temperatures above absolute zero, electrons can gain thermal energy and occupy states above the Fermi energy, contributing to electrical conductivity. The probability distribution of electron energies broadens with increasing temperature. 5. Application in Semiconductor Physics: Fermi energy determines the electrical and thermal characteristics of solid. It helps to explain the behavior of charge carriers (electrons and holes) and their conductivity properties. It is a key parameter in the analysis of semiconductors, metals and insulators.  Carrier concentration in an intrinsic semiconductor  The number of electrons in the conduction band per unit volume and the number of holes in the valence band per unit volume of the material is called carrier concentration or density of charge carriers.  The number of electrons in the conduction band per unit volume of the material is called the electron concentration.  The number of holes in the valence band per unit volume of the material is called the hole concentration.  Expression for electron concentration in conduction band: where EC is the energy of conduction band and EF – fermi energy  Expression for hole concentration in valence band:  Expression for intrinsic carrier concentration: In the case of intrinsic semiconductors n = p Therefore  Expression for fraction of electron in conduction band in an intrinsic semiconductor: where n is the number of electrons excited to conduction band N is the total number of electrons available in the valence band initially. 1.Compute the concentration of intrinsic charge carriers in a germanium crystal at 300 K. Given that Eg = 0.72 eV and assume that me=mh. Ans Answer is 2.27 /m3 2. Estimate the fraction of electrons in the conduction band at 300K of (a) Germanium (Eg= 0.72 eV) (b) Silicon (Eg=1.1 eV) and (c) Diamond (Eg=5.6 eV). (d) What is the significance of these results? Ans a) Germanium: fraction = b) Silicon: fraction = 5.86 c) Diamond: fraction = d) The results shows that, larger the bandgap lesser the electrons that can go into the conduction band, at a given temperature. 3. Estimate the fraction of electrons in the conduction band at room temperature in Ge with Eg=0.72 eV. Ans a) Germanium: fraction = 4. Assuming that the number of electrons near the top of the valence band available for conduction is and the number of electrons exited to conduction band is /m3, calculate the energy gap of germanium at room temperature. = 2kT(14.508) Ans is 0.75 eV  Hall Effect  Hall Effect was discovered by Edwin Herbert Hall in 1879. Definition: When a current-carrying conductor or a semiconductor is introduced to a perpendicular magnetic field, a voltage can be measured at the right angle to the current path (a transverse electric field is developed). This effect of obtaining a measurable voltage is known as the Hall Effect and the voltage is called Hall voltage. Theory When a conductive plate is connected to a circuit with a battery, then a current starts flowing. The charge carriers will follow a linear path from one end of the plate to the other end. The motion of charge carriers results in the production of magnetic fields. When a magnet is placed near the plate, the magnetic field of the charge carriers is distorted. This distorts the straight flow of the charge carriers. The force which disturbs the direction of flow of charge carriers is known as Lorentz force. Due to the distortion in the magnetic field of the charge carriers, the negatively charged electrons will be deflected to one side of the plate and positively charged holes to the other side. A potential difference, known as the Hall voltage will be generated between both sides of the plate which can be measured using a meter.  Hall Coefficient The Hall coefficient RH is mathematically expressed as The Hall voltage represented as VH is given by the formula: Where, J is the current density of the carrier electron, E is the induced electric field and B is the magnetic strength. Here, I is the current flowing through the sensor  The hall coefficient is positive if the number of positive B is the magnetic Field Strength charges is more than the negative charges.  Similarly, it is negative when electrons are more than holes. q is the charge n is the number of charge carriers per unit Other formulae volume  or d is the thickness of the sensor. 1. Calculate the Hall voltage when a conductor carrying a current of 100 A, is placed in a magnetic field of 1.5 T. The conductor has a thickness of 1 cm, and the number density of charges inside the conductor is 5.9 ×1028 /m3. 𝐼𝐵 𝑉 𝐻= 𝑞𝑛𝑑 Ans = 15.89×10-7 V 2. The Hall coefficient of certain silicon specimen was found to be –7.35 × 10 –5 m3 C–1 from 100 to 400 K. Calculate the number of charge carriers and determine the nature of the semiconductor. Nature is n type semiconductor. 3. A semiconducting crystal with 12 mm long, 5 mm wide and 1 mm thick has a magnetic density of 0.5 Wbm–2 applied perpendicular to largest faces. When a current of 20 mA flows lengthwise through the specimen, the voltage measured across its width is found to be 37μV. What is the Hall coefficient of this semiconductor? 𝑉𝐻𝑑 𝑅𝐻 = 𝐵𝐼 Applications of Hall Effect  Find whether the semiconductor is N-type or P-type  Find carrier concentration  Magnetic field sensing equipment  Proximity detectors  Hall effect Sensors and Probes  MAGNETIC MATERIALS - TERMS Magnetic Susceptibility: Ratio of intensity of magnetisation produced in the sample to the magnetic field intensity which produces magnetization. It has no units. χ 𝐻 𝑀 Magnetization: The process = of converting a non magnetic material to a magnetic material. Intensity of magnetization: It is magnetic moment per unit volume. Relative permeability: The ratio of flux density produced in a material to the flux density produced in vacuum by the same magnetising force.  MAGNETIC MATERIALS - TERMS Magnetic flux (Φ): The total no: of magnetic lines of force in a magnetic field (unit- Weber) Magnetic flux density (B): Magnetic flux per unit area at right angles to the direction of flux. (Wb/𝑚2) Magnetic field intensity (H): Magneto motive force per unit length of the magnetic circuit. It is also called magnetic field strength or magnetizing force. (A-turns/m) Permeability (µ): The ability of a material to conduct magnetic flux through it. (H/m) Magnetic dipole moment is a vector quantity that measures the strength and orientation of a magnet or other object that creates a magnetic field. It's also a measure of the strength of a small current loop that acts like a tiny magnet. The magnetic dipole moment of an object determines how much torque the object experiences in a given magnetic field. The stronger the magnetic moment, the stronger the magnetic field and the torque. The magnetic dipole moment is estimated using the equation m=NIA where 𝑚 is the magnetic dipole moment, 𝑁 is the number of turns 𝐼 is the current, and 𝐴 is the area of the coil. The units of magnetic dipole moment are ampere meters squared (Am 2). The direction of the magnetic dipole moment is determined by the right-hand thumb rule. For a magnet, the direction is from the south pole to the north pole ORIGIN OF PERMENANT MAGNETIC DIPOLES  A moving electric charge is responsible for Magnetism. ORIGIN OF PERMENANT MAGNETIC DIPOLES ORBITAL MOMENTUM ORBITAL SPIN CLASSIFICATION OF MAGNETIC MATERIALS ▶ Diamagnetic – materials which lack permanent dipoles are called diamagnetic ▶ Paramagnetic – if the permanent dipoles do not interact among themselves, the material is paramagnetic ▶ Ferromagnetic – if the interaction among permanent dipoles is strong such that all the dipoles line up in parallel, the material is ferromagnetic ▶ Antiferromagnetic – if the permanent dipoles line up in antiparallel direction, the material is antiferromagnetic ▶ Ferrimagnetic – antiparallel with unequal magnitude  CLASSIFICATION OF MAGNETIC MATERIALS DIAMAGNETIC MATERIALS No permanent dipoles are present so net magnetic moment is zero. Dipoles are induced in the material in presence of external magnetic field. The magnetization becomes zero on removal of the external field. Magnetic dipoles in these substances tend to align in opposition to the applied field. Hence, they produce an internal magnetic field that opposes the applied field and the substance tends to repel the external field around it. This reduces the magnetic induction in the specimen.  CLASSIFICATION OF MAGNETIC MATERIALS DIAMAGNETIC MATERIALS Magnetic susceptibility is small and negative. Relative permeability is less than one. It is present in all materials, but since it is so weak it can be observed only when other types of magnetism are totally absent. Ex: Gold, water, mercury, B, Si, P, S, ions like Na+, Cl- and their salts, diatoms like H2, N2,..  CLASSIFICATION OF MAGNETIC MATERIALS DIAMAGNETIC MATERIALS They repel the magnetic lines of force. The existence of this behavior in a diamagnetic material is shown  CLASSIFICATION OF MAGNETIC MATERIALS PARAMAGNETIC MATERIALS If the orbital's are not completely filled or spins are not balanced, an overall small magnetic moment may exist The magnetic dipoles tend to align along the applied magnetic field and thus reinforce the applied magnetic field. Such materials get feebly magnetized in the presence of a magnetic field i.e. the material allows few magnetic lines of force to pass through it. The magnetization disappears as soon as the external field is removed.  CLASSIFICATION OF MAGNETIC MATERIALS PARAMAGNETIC MATERIALS The magnetization (M) of such materials was discovered by Madam Curie and is dependent on the external magnetic field (B) and temperature T as: χ= 𝐶 where, C= Curie Constant The orientation of magnetic dipoles depends on temperature and applied field. Relative permeability µr >1 Susceptibility is independent of applied magnetic field and depends on temperature  CLASSIFICATION OF MAGNETIC MATERIALS PARAMAGNETIC MATERIALS Susceptibility is small and positive These materials are used in lasers. Ex: Liquid oxygen, sodium, platinum, salts of iron and nickel, rare earth oxides  CLASSIFICATION OF MAGNETIC MATERIALS FERROMAGNETIC MATERIALS They exhibit strongest magnetic behavior. Permanent dipoles are present which contributes a net magnetic moment. Possess spontaneous magnetization because of interaction between dipoles Origin for magnetism in Ferro magnetic materials are due to Spin magnetic moment. All spins are aligned parallel & in same direction When placed in external magnetic field it strongly attracts magnetic lines of force. The domains reorient themselves to reinforce the external field and producea strong internal magnetic field that is along the external field.  CLASSIFICATION OF MAGNETIC MATERIALS FERROMAGNETIC MATERIALS Most of the domains continues to be aligned in the direction of the magnetic field even after removal of external field. Thus, the magnetic field of these magnetic materials persists even when the external field disappears. This property is used to produce Permanent magnets. Transition metals, iron, cobalt, nickel, neodymium and their alloys are usually highly ferromagnetic and are used to make permanent magnets.  CLASSIFICATION OF MAGNETIC MATERIALS FERROMAGNETIC MATERIALS Susceptibility is large and positive, it is given by Curie Weiss Law; χ= 𝑇− where, C is Curie constant & θ is Curie temperature. θ When temperature is greater than Curie temperature then the material gets converted in to paramagnetic. They possess the property of Hysteresis. Ferromagnetic Material in magnetic field The total magnetic induction (magnetic flux density) B can be written as B = (H+M) H and M would have the same units, amperes/meter. The quantity M in this relationship is the magnetization of the material in A/m. H is the magnetising force (A/m) is the permeability of free space  Superconductivity  Superconductivity is a state of materials in which their electrical resistance is zero.  The resistivity of metals increases linearly with increase in temperature.  For certain metals, when they are cooled, the resistivity abruptly falls to zero at a temperature characteristic of the metal. This phenomenon is known as superconductivity.  The temperature at which transition takes place from normal conducting to superconducting state is called transition temperature or critical temperature TC.  The phenomenon was discovered by Kamerlingh Onnes during his studies on the properties of mercury at very low temperatures.  He found that the resistivity of mercury suddenly dropped to zero at 4.2 K.  Temperature Dependence of Resistivity of Superconductors  The variation of resistivity ‘ρ’ of a superconductor with temperature is shown in the graph.  As temperature is decreased the resistivity decreases and suddenly falls to zero at TC, which is called the critical temperature.  Below TC it continues to remain a superconductor.  Critical temperature is different for different materials.  Mercury loses resistance completely at 4.2 K, whereas above this temperature mercury behaves as a normal conductor.  2. Meissner effect When a specimen of a conducting material is placed in a magnetic field and cooled below its critical temperature, the magnetic field is completely expelled from the specimen. “The phenomenon of expulsion of magnetic flux completely from the specimen during the transition from normal state to the superconducting state is called as ‘Meissner effect.” Normal state Superconducting state The magnetic induction inside the specimen is given by B = o (H + M) Where, B is magnetic induction, H is external applied field and M is magnetization produced within the specimen. At T < Tc, B = 0 inside the super conductor. Hence we can write, B = o (H + M ) = 0, or H = - M M/H= -1 χ (Negative) , which is the characteristic of diamagnetism This shows that the superconductors are perfectly diamagnetic in nature.  General features (characteristics) of superconductors 1. Critical field HC  Superconductivity vanishes when a sufficiently strong magnetic field is applied.  “The minimum magnetic field required to destroy the superconductivity in the material is called critical field HC”.  Critical field is a function of temperature and varies with temperature T in accordance with the relation 2  T   H C H 0 1       TC     Variation of critical field as a function of temperature where Ho is the critical magnetic field at 0 K. 1. A superconducting tin has a critical temperature of 3.7 K at zero magnetic field and a critical field of 0.0306 Tesla at 0 K. Find the critical field at 2 K. 2. The critical temperature of Nb is 9.15K. At zero kelvin the critical field is 0.196 T. In the laboratory the temperature of Nb is increased i) Using a magnetic field, Nb can be changed to normal conductor below critical temperature. State True or False. ii) The magnetic field required to change superconducting Nb to normal conductor at 0K is a) 0 T b) 1 T c) 0.5 T d) 0.196 T iii) The magnetic field required to change superconducting Nb to normal conductor above 0K and below 9.15K is a) More than 0.196 T b) Less than 0.196 T c) Equal to 0.196T d) Zero  Types of Superconductors Based on the magnetic behavior, superconductors are classified into two categories, viz. Type I and Type II superconductors. Type I Superconductors (Soft Superconductors):  In type I superconductors, the transition from superconducting to normal state, in the presence of magnetic field, occurs sharply at the critical field HC.  Below the critical field the magnetization in the specimen is proportional to the applied field and it behaves like a diamagnet.  For H > HC, magnetization of the specimen is negligible and the specimen becomes a normal conductor.  This kind of behavior is exhibited by pure elements. The critical field is relatively low for these superconductors.  Examples are aluminium, lead and indium.  Type II Superconductors (Hard Superconductors):  Type II superconductors are characterized by two critical fields HC1 and HC2.  Up to HC1, below the critical field, the specimen is perfectly diamagnetic and hence is superconducting.  From HC1 to HC2, Meissner effect is incomplete.  This state is called the mixed or vortex state.  Above the critical field HC2, the specimen becomes an ordinary conductor.  HC2 can be as high as 30 T, and retention of superconductivity at such high magnetic fields makes them useful in creating very high magnetic fields.  Examples of type II superconductors are transition metals and alloys containing niobium, silicon and vanadium  Type-I and Type-II superconductors S.No Type-I superconductors Type-II superconductors 1 The material loses its The material loses its magnetization abruptly. magnetization gradually. 2 They exhibit complete They do not exhibit Meissner effect complete Meissner effect 3 They are characterized by They are characterized by one critical magnetic field two critical magnetic fields (Hc) lower critical field (Hc1) and upper critical field (Hc2) 4 No mixed state is present. Mixed state is present 5 They are called soft They are called Hard superconductors. superconductors (Hc2=30T) (Hc=0.1T) 6 Ex: Lead, Tin, Hg, Al etc. Ex: Nb-Sn, Nb-Zr, Nb-Ti etc.  BCS Theory Explanation to superconductivity was given by John Bardeen, Leon N Cooper and John Robert Schrieffer on the basis of quantum theory. The theory is named after the three scientists as BCS theory. When an electron approaches a positive ion, there is a Coulomb attraction between them, which produces a distortion in the lattice. Now if another electron passes through the distorted lattice, interaction between the distorted lattice and the electron occurs, which results in lowering of energy of the electron. This effect is an interaction between two electrons via the lattice distortion or the phonon (quantum of energy of lattice vibration), which results in the lowering of energy for the two electrons.  This lowering of energy implies that the force between the two electrons is attractive. This type of interaction is called the electron- phonon-electron interaction. This interaction is strongest if the two electrons have equal and opposite momentums and spin. Such pairs of electrons are called Cooper pairs. At very low temperatures the density of Cooper pairs is large and they collectively move through the lattice with small velocity. Low speed reduces collisions and thereby resistivity decreases, which explains superconductivity. The attraction between the electrons in the Cooper pair is very weak. Therefore they can be separated by small increase in temperature, which results in a transition back to the normal state. Applications  Sample Questions: Remembering 1. Phosphorous can be used as a) Pentavalent Impurity b) Trivalent Impurity c) Intrinsic semiconductor d) Extrinsic Semiconductor 2. The temperature at which conductivity of a material becomes infinite is called a) Critical Temperature b) Absolute Temperature c) Mean Temperature d) Crystallization temperature 3. Superconductors are paramagnetic in nature. State True/ False  Sample Questions :Thought Provoking 1. Fermi energy of a Copper wire is found to be 8.5 eV. If the length of the wire is doubled, the Fermi energy will be a) 8.5 eV b) 4.25 eV c) 17 eV d) 65 eV 2. For calculating the Fermi energy of a material, the graph of Resistance vs. Temperature is plotted. The slope is found to be exactly same as the slope obtained for Copper. The density of Copper is 8.96 g/cm³ and the density of the material is 10 g/cm³. We know that Fermi Energy is proportional to the square root of the product of density and the slope of the above graph, (. If everything else remains same, we can say for sure that the Fermi energy of the material is a) lower than that of copper b) same as that of copper c) higher than that of copper d) inverse of the Fermi energy of copper Thank You