Semiconductor Physics (BBS00015) Study Material - 2024-2025 PDF

B.Tech CSE-DS, B.Tech CSE-AIML, B.Tech CSE-CYS 2024 and Semester-I Semiconductor Physics (BBS00015) Class 2024-25 ODD Study Material (Semiconductor Physics, BBS00015) Module I part 1 Contents 1...

B.Tech CSE-DS, B.Tech CSE-AIML, B.Tech CSE-CYS 2024 and Semester-I Semiconductor Physics (BBS00015) Class 2024-25 ODD Study Material (Semiconductor Physics, BBS00015) Module I part 1 Contents 1 Introduction………………………………………………………………………………………………………………………………… 1 2 Limitation of Classical Physics……………………………………………………………………………………………………… 1 2.1 Blackbody Radiation and the Ultraviolet Catastrophe………………………………………………………….. 1 2.2 Photoelectric effect………………………………………………………………………………………………………………….. 2 3 Quantum vs. Classical Mechanics………………………………………………………………………………………………… 3 4 Planck’s theory of blackbody radiation ……………………………………………………………………………………… 3 4.1 Deduction from Planck’s law..………………………………………………………………………………………………… 5 4.1.1 Wien’s Displacement law………………….…………………………………………………………………………….. 5 4.1.2 Stefan-Boltzmann law……………………………………………………………………………………………………… 5 5 Photoelectric effect……………………………………………………………………………………………………………………… 6 5.1 Characteristics of photoemission………………………………………………………………………………………………. 6 5.2 Explanation of Photoelectric effect using Einstein’s postulate:…………………………………………………. 7 5.3 Numerical example:………………………………………………………………………………………………………………….. 7 1 Introduction This section deals with a “qualitative” overview of quantum physics and how it compares to classical physics. We shall learn about a few fundamental experiments that illustrate the limitation of classical mechanics and the need for a more fundamental theory. Next, we shall compare the basic properties of Quantum and Classical mechanics using non-technical terms (minimal usage of mathematical derivation) as far as possible. 2 Limitation of Classical Physics 2.1 Blackbody Radiation and the Ultraviolet Catastrophe Fig. 1 shows the energy distribution of blackbody radiation as a function of wavelength. Classical physics tells us that the amount of the radiation is inversely proportional to the wavelength, or, more precisely, the power emitted 1 per unit area per unit solid angle per unit wavelength is proportional to 𝜆4 where λ is the wavelength. This means that as the wavelength approaches zero, the amount of radiation becomes infinite! The black curve in Fig. 1 illustrates the same. This result is known as the ultraviolet catastrophe since ultraviolet light has shorter wavelengths than the visible range of em waves. Obviously, this does not match well with experimental data, since when we measure the total radiation emitted from a black body, we certainly do not measure it to be infinity! Hence Classical theory has a serious limitation in explaining the experimentally observed distribution of electromagnetic radiation emitted by a blackbody. Therefore, we need a more fundamental concept, Quantum theory. Prepared by the faculty members of the Physics department 1 Brainware University, Kolkata B.Tech CSE-DS, B.Tech CSE-AIML, B.Tech CSE-CYS 2024 and Semester-I Semiconductor Physics (BBS00015) Class 2024-25 ODD UVVISIBLE INFRARED 14 5000 K 12 Classical theory (5000 K) 10 8 6 4000 K 4 2 3000 K 0 0 0.5 1 1.5 2 2.5 3 Wavelength (μm) Fig. 1: The electromagnetic spectrum of energy radiated by a black body. Source:Wikipedia. According to quantum theory, radiation can only be emitted in a discrete “packet” of energy called quanta. If we assume this idea of emission of quanta of energy, we can theoretically calculate the distribution curve of blackbody radiation that agrees well with the experimentally measured spectrum. The law describing the amount of radiation at each wavelength is called Planck’s law of blackbody radiation, after the name of Max Planck who proposed this theory. 2.2 Photoelectric effect When light falls on a metal surface, the metal surface emits electrons if the frequency of the incident light is greater than the minimum frequency, known as the threshold frequency of the metal. Such a phenomenon is known as the photoelectric effect. Using classical physics and the assumption that light is a wave, one can make the following predictions: Brighter light should have more energy, so it should cause the emitted electrons to have more kinetic energy and thus move faster. Light with higher frequency should hit the material more often, so it should cause a higher rate of electron emission, resulting in a larger electric current. However, what happens is the exact opposite: The kinetic energy of the emitted electrons increases with frequency, not brightness. The electric current increases with brightness, not frequency. Prepared by the faculty members of the Physics department 2 Brainware University, Kolkata B.Tech CSE-DS, B.Tech CSE-AIML, B.Tech CSE-CYS 2024 and Semester-I Semiconductor Physics (BBS00015) Class 2024-25 ODD Since the classical theory appears inadequate in explaining this effect we need Quantum theory. Similar to Plank’s theory of blackbody radiation, Einstein proposed that light consists of discrete quanta called photons. Each photon has energy proportional to the frequency of the light. Brighter light of the same frequency has more photons; however, each photon has the same amount of energy. This proposed model agrees well with the experimentally observed phenomenon. 3 Quantum vs. Classical Mechanics Quantum mechanics is, as far as we know, the exact and fundamental theory of reality. Quantum mechanics is necessary to describe small objects, like, elementary particles, atoms, and molecules. All big objects are effectively made of microscopic particles; therefore, in principle, quantum mechanics can describe humans, planets, galaxies, and even the whole universe. This is where Classical mechanics comes into the picture; when many small quantum systems make up one large system, classical mechanics generally offers an adequate description for all practical purposes. This is similar to how relativity is always the correct way to describe physics, but at low velocities, much smaller than the speed of light, Newtonian physics is a good enough approximation. 4 Planck’s theory of blackbody radiation The failure of classical theory to explain the experimentally observed energy distribution of the radiation emitted by a blackbody made the path for a new theory. At the beginning of the twentieth century, Max Planck proposed a new concept to explain the blackboy radiation and absorption of radiation is not a continuous process but occurs discretely as an integral multiple of a basic unit, called the quantum of energy. Each quantum of energy carries a definite amount of energy E = hν, where h is a universal constant known as Planck’s constant (h = 6.626 × 10−34Js). Planck estimated the value of h by fitting the theory to the experimentally measured data (Fig. 1). The derivation of the expression of energy density radiated by a blackbody using Planck’s postulate is beyond the scope of this course and therefore we assume the following form of the energy density of a blackbody radiated for wavelengths between λ and λ + dλ 8𝜋ℎ𝑐 𝑑𝜆 𝑢𝜆 𝑑𝜆 = 5 … … … … … … … … (1) 𝜆 exp(ℎ𝑐/𝜆𝐾𝐵 𝑇) − 1 8𝜋ℎ 𝜈3 𝑑𝜈 ⇒ 𝑢𝜆 𝑑𝜆 = ………………………………...(2) 𝑐 3 exp(ℎ𝜈/𝐾𝐵 𝑇)−1 where KB is Boltzmann constant (1.38 × 10−23JK−1) and c is speed of light in vacuum. Planck’s formula for the energy distribution of the blackbody radiation agrees well with the experimentally measured data (Fig. 1) for any value of wavelength (λ). Previously, two other laws using Classical theory were known that can partially reproduce the experimentally measured energy distribution of the blackbody radiation (Fig. 1). Wien’s law of blackbody radiation Wilhelm Wien, from thermodynamical considerations and some arbitrary assumptions regarding the mechanism of emission and absorption of radiation, proposed a functional form of uλ for a given temperature T, 𝑎 −𝑏 𝑢𝜆 (𝑇)𝑑𝜆 = 𝜆5 exp ( 𝜆𝑇 ) 𝑑𝜆 (3) where a and b are constants. Prepared by the faculty members of the Physics department 3 Brainware University, Kolkata B.Tech CSE-DS, B.Tech CSE-AIML, B.Tech CSE-CYS 2024 and Semester-I Semiconductor Physics (BBS00015) Class 2024-25 ODD Rayleigh-Jeans Law The British physicist Lord Rayleigh and Sir James Jeans, based on classical theory arguments and empirical facts, proposed that a functional form of the energy density of blackbody radiation as, 𝑎𝑇 𝑢𝜆 𝑑𝜆 = 𝑑𝜆 (4) 𝜆4 where a is a constant and T is the temperature of the blackbody. This form agrees well with the experimentally measured spectra of radiation of the blackbody only at long wavelengths. This empirical law, however, severely fails at short wavelengths known as the ultraviolet catastrophe. Below we will show that for short and long wavelength limits functional form of Planck’s law of blackbody radiation reduces to Wien’s law (for λ → 0) and Rayleigh-Jean’s law (for λ →∞), respectively. ℎ𝑐 For very short wavelengths, i.e., λ → 0, we have, ≫ 1, and therefore 𝜆𝐾𝐵 𝑇 ℎ𝑐 ℎ𝑐 exp ( ) − 1 ≈ exp ( ) 𝜆𝐾𝐵 𝑇 𝜆𝐾𝐵 𝑇 if we consider 8πhc = a and hc/KB = b, for short wavelength limit Eq. 2 reduces to, 8𝜋ℎ𝑐 1 8𝜋ℎ𝑐 ℎ𝑐 𝑎 𝑏 lim 𝑢𝜆 𝑑𝜆 = 𝑑𝜆 = exp (− ) = exp (− ) 𝑑𝜆 (5) 𝜆→0 𝜆5 exp( ℎ𝑐 ) 𝜆5 𝜆𝐾𝐵 𝑇 𝜆5 𝜆𝑇 𝜆𝐾𝐵 𝑇 which is the same as Eq. 3. Therefore, Planck’s law for blackbody radiation reduces to Wien’s law at short wavelengths of the radiation. ℎ𝑐 𝑥2 𝑥3 Again for long wavelengths, i.e., 𝜆 → ∞, 𝜆𝐾 ≪ 1. we know, for 𝑥 ≪ 1, 𝑒 𝑥 = 1 + 𝑥 + + +⋯ 𝐵𝑇 2! 3! ℎ𝑐 Hence neglecting higher-order terms of 𝜆𝐾 𝑇 , we can write 𝐵 ℎ𝑐 ℎ𝑐 exp (𝜆𝐾 𝑇 ) ≈ 1 + 𝜆𝐾 𝐵 𝐵𝑇 ℎ𝑐 ℎ𝑐 ℎ𝑐 exp ( )−1 = 1+ −1= 𝜆𝐾𝐵 𝑇 𝜆𝐾𝐵 𝑇 𝜆𝐾𝐵 𝑇 Thus we have, 8𝜋ℎ𝑐 𝑑𝜆 8𝜋ℎ𝑐 𝑑𝜆 lim 𝑢𝜆 𝑑𝜆 = lim = ℎ𝑐 𝜆→∞ 𝜆→∞ 𝜆5 exp( ℎ𝑐 )−1 𝜆5 𝜆𝐾𝐵 𝑇 𝜆𝐾𝐵 𝑇 8𝜋ℎ𝑐 𝜆𝐾𝐵 𝑇 𝑇 𝑎𝑇 = 𝑑𝜆 = 8𝜋𝐾𝐵 4 𝑑𝜆 = 4 𝜆5 ℎ𝑐 𝜆 𝜆 which is the same as Rayleigh-Jeans law (Eq. 4). Prepared by the faculty members of the Physics department 4 Brainware University, Kolkata B.Tech CSE-DS, B.Tech CSE-AIML, B.Tech CSE-CYS 2024 and Semester-I Semiconductor Physics (BBS00015) Class 2024-25 ODD 4.1 Deduction from Planck’s law 4.1.1 Wien’s Displacement law When the temperature of a blackbody increases, the overall radiated energy increases and the peak of the radiation curve moves to shorter wavelengths. According to the Wien’s displacement law, the product of the wavelength at which the energy density of radiation of a blackbody becomes maximum and the corresponding temperature of the blackbody is constant. From Planck’s law (Eq. 2) one can easily find that uλ is maximum when the denominator is minimum, since the numerator is a constant. The denominator can be written as, ℎ𝑐 𝑧 = 𝜆5 (exp ( ) − 1) 𝜆𝐾𝐵 𝑇 𝑑𝑧 Since 𝑑𝜆 = 0 at 𝜆 = 𝜆𝑚 , we have, 𝑑𝑧 ℎ𝑐 ℎ𝑐 ℎ𝑐 𝑑𝜆 = 5𝜆4 (exp (𝜆𝐾 𝑇 ) − 1) + 𝜆5 exp (𝜆𝐾 𝑇) (− 𝜆2 𝐾 𝑇 ) = 0 for λ = λm 𝐵 𝐵 𝐵 ℎ𝑐 ℎ𝑐 ⇒ 𝑒 𝜆𝑚𝐾𝐵 𝑇 (5𝜆4𝑚 − 𝜆3𝑚 ) − 5𝜆4𝑚 = 0 𝐾𝐵 𝑇 ℎ𝑐 ℎ𝑐 ⇒ 𝜆4𝑚 𝑒 𝜆𝑚𝐾𝐵 𝑇 (5 − ) − 5𝜆4𝑚 = 0 𝜆𝑚 𝐾𝐵 𝑇 ℎ𝑐 ⇒ 𝜆4𝑚 (𝑒 𝑥 (5 − 𝑥) − 5) = 0 (𝑥 = ) 𝜆𝑚 𝐾𝐵 𝑇 ⇒ 𝑒 𝑥 (5 − 𝑥) − 5 = 0 𝑎𝑠 𝜆𝑚 ≠ 0 ⇒ 𝑒 𝑥 (5 − 𝑥) = 5 ⇒ (5 − 𝑥) = 5𝑒 −𝑥 ⇒ 5(1 − 𝑒 −𝑥 ) = 𝑥 𝑥 ⇒ (1 − 𝑒 −𝑥 ) = 5 This is a transcendental equation that cannot be solved analytically. Therefore, we solve it graphically by plotting y = 1 − exp(−x) and y = x/5 on the same graph paper. The point of intersection of two curves gives the solution for x which is x = 4.96. We thus get ℎ𝑐 ℎ𝑐 𝜆𝑚 𝑇 = = = 0.0029𝐾𝑚 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐾𝐵 𝑥 4.96𝐾𝐵 This is Wien’s displacement law. 4.1.2 Stefan-Boltzmann law The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a blackbody across all wavelengths per unit time is proportional to the fourth power of its absolute temperature. The corresponding proportionality constant is known as Stefan constant, σ = 5.67 × 10−8J/(s m2 K4). Prepared by the faculty members of the Physics department 5 Brainware University, Kolkata B.Tech CSE-DS, B.Tech CSE-AIML, B.Tech CSE-CYS 2024 and Semester-I Semiconductor Physics (BBS00015) Class 2024-25 ODD If Eλ is the intensity of the emitted radiation between wavelength λ and λ + dλ, the total energy across all wavelengths emitted by the blackbody is given by ∞ 𝐸 = ∫ 𝐸𝜆 𝑑𝜆 0 The energy density uλ is related to the intensity of the emitted radiation as 𝑐 𝐸𝜆 = 𝑢𝜆 4 ∞ 𝑐 ∞ 𝑐 ∞ 1 𝑑𝜆 𝐸 = ∫ 𝐸𝜆 𝑑𝜆 = ∫ 𝑢𝜆 𝑑𝜆 = 8𝜋ℎ𝑐 ∫ 5 0 4 0 4 0 𝜆 exp ( ℎ𝑐 )−1 𝜆𝐾𝐵 𝑇 ℎ𝑐 1 𝐾𝐵 𝑇 Let’s assume 𝜆𝐾𝐵 𝑇 = 𝑥. So, − 𝜆2 𝑑𝜆 = ℎ𝑐 𝑑𝑥 ∞ 𝐾𝐵 𝑇 4 𝑥 3 Therefore we can write,𝐸 = 2𝜋ℎ𝑐 2 ∫0 ( ℎ𝑐 ) 𝑒 𝑥 −1 𝑑𝑥 2𝜋𝐾𝐵4 𝑇 4 ∞ 3 𝑥 𝑑𝑥 = ∫ ℎ3 𝑐 2 0 𝑒 𝑥 − 1 2𝜋𝐾𝐵4 𝑇 4 𝜋 4 = ℎ3 𝑐 2 15 2𝜋 4 𝐾𝐵4 𝑇 4 = 15ℎ3 𝑐 2 = 𝜎𝑇 4 where σ is Known as Stefan constant. Putting all the values of the constants we get, 2𝜋 4 𝐾𝐵4 = 5.67 × 10−8 𝐽/(𝑠𝑚2 𝐾 4 ) 15ℎ3 𝑐 2 Thus Stefan-Boltzmann’s law can be derived from Planck’s law of blackbody radiation. 5 Photoelectric effect The photoelectric effect is the phenomenon of the emission of electrons from the surface of a metal when a beam of electromagnetic radiation of appropriate frequency is incident on it. Electrons thus emitted are known as photoelectrons and the current produced by the emission of the photoelectrons is known as photocurrent. 5.1 Characteristics of photoemission 1. The number of photoelectrons emitted per second, that is, the photoelectric current is directly proportional to the intensity of the incident radiation but it is independent of the frequency (or wavelength) of incident light. 2. The maximum speed of the emitted photoelectron is independent of the intensity of the incident light, but depends on its frequency, increasing linearly with the increase of the frequency of the incident light. 3. For each material emitting photoelectrons, there is a minimum energy, φ0, that must be supplied to the material to have photoelectrons emitted from its surface. This minimum energy is called the work function of the material. In other words, there is a minimum frequency ν0 of the incident light below which no Prepared by the faculty members of the Physics department 6 Brainware University, Kolkata B.Tech CSE-DS, B.Tech CSE-AIML, B.Tech CSE-CYS 2024 and Semester-I Semiconductor Physics (BBS00015) Class 2024-25 ODD photoelectron would be ejected from the surface of the metal. This cut-off frequency is known as the threshold frequency. 5.2 Explanation of Photoelectric effect using Einstein’s postulate: In 1905, Albert Einstein proposed a new theory of electromagnetic radiation to explain the photoelectric effect. According to this theory, photoelectric emission does not take place by continuous absorption of energy from radiation. Light energy is built up of discrete units – the so-called quanta of energy of radiation. Each quantum of radiant energy has energy hν, where h is Planck’s constant and ν is the frequency of light. Such a quantum or packet of light energy is known as a photon. When light of energy hν falls on the surface of a metal having work function φ0, photoelectrons are ejected only if hν > φ0. The excess energy hν −φ0 is taken by the electrons as its kinetic energy. Thus the maximum kinetic energy of the emitted photoelectrons is 1 2 2 𝑚𝑣𝑚𝑎𝑥 = ℎ𝜈 − 𝜑0 (6) One can apply a negative potential to stop the ejected photoelectrons. The minimum negative potential must be applied to stop the fastest moving photoelectron is known as stopping potential. If Vs be the potential required to stop the fastest moving photoelectron having speed vmax then, 1 2 𝑒𝑉𝑠 = 𝑚𝑣𝑚𝑎𝑥 2 Therefore the Equ.6 can also be in an alternative form as follows, hν − φ0 = eVs hν = hν0 + eVs (7) 5.3 Numerical example: In an experiment using a tungsten cathode that has a threshold wavelength 2300Å is irradiated by the ultraviolet light of wavelength 1800 Å. Calculate the maximum kinetic energy of the emitted photoelectrons and the work function of tungsten. 1 2 𝑚𝑣𝑚𝑎𝑥 = ℎ𝜈 − ℎ𝜈𝑜 2 = ℎ𝜈 − ℎ𝜈0 1 1 = ℎ𝑐 ( − ) 𝜆 𝜆0 1 1 = 6.626 × 10−34 × 3 × 108 ( −10 − ) 1800 × 10 2300 × 10−10 −19 = 2.4 × 10 𝐽 2.4×10−19 = 1.6×10−19 𝑒𝑉 = 1.5 𝑒𝑉 Prepared by the faculty members of the Physics department 7 Brainware University, Kolkata