Semiconductor Physics Study Material 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)...

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 2 Contents 1 Wave-particle duality........................................................................................................................................1 1.1 Phase and Group velocity.................................................................................................................................2 1.1.1 Relation between phase velocity and group velocity of matter waves.......................................................2 1.2 Numerical examples:........................................................................................................................................3 2 Heisenberg’s uncertainty principle...................................................................................................................3 2.1 Numerical examples.........................................................................................................................................3 3 Wave function and its physical significance.....................................................................................................4 3.1 Probability density:...........................................................................................................................................5 3.2 Normalization of the wave function.................................................................................................................5 4 Operators in quantum mechanics....................................................................................................................5 5 Expectation value of an operator.....................................................................................................................6 6 Numerical examples.........................................................................................................................................6 1 Wave-particle duality The photoelectric effect and Compton scattering are evidence of the particle nature of light. On the other hand, interference, diffraction, polarisation, etc., reveal the wave nature of light. French physicist Louis de Broglie proposed that the dual character exhibited by light may also be the property of the subatomic particles. De Broglie tried to fit into the Bohr orbit an exact number of standing waves in analogy with the integral number of standing waves in a stretched string. If an integer number of waves of wavelength λ are to be fitted into a Bohr orbit of radius r, then 2πr = nλ, where n is an integer number. Comparing this with Bohr’s condition of quantization of the angular momentum, we have 𝑛𝜆 𝑚𝑣𝑟 = 𝑚𝑣 = 𝑛ℏ 2𝜋 2𝜋ℏ ℎ So, 𝜆= = 𝑚𝑣 𝑝 (1) where p = mv is the momentum of the electron. A few relations to remember: ℎ 𝐸 = ℎ𝜈 = 2𝜋𝜈 = ℏ𝜔 2𝜋 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 and ℎ ℎ 2𝜋 𝑝= = = ℏ𝑘. 𝜆 2𝜋 𝜆 1.1 Phase and Group velocity The phase velocity 𝑣⃗𝑝 of a wave is the velocity with which a definite phase of the wave, such as its crest (the highest point) or trough (the lowest point), propagates in a medium. The energy associated with a wave propagates with a velocity, 𝑣⃗𝑔 , known as the group velocity. De Broglie proposed that the particle’s velocity 𝑣⃗ should be equal to the group velocity of the corresponding matter-wave. 𝜔 𝑑𝜔 𝑣𝑝 = , 𝑣𝑔 =. 𝑘 𝑑𝑘 1.1.1 Relation between phase velocity and group velocity of matter waves 𝑚𝑜 𝛽𝑐 𝑝 = ℏ𝑘 = 𝑚𝑣 = (2) √1−𝛽2 𝑚𝑜 𝑐 2 𝐸 = ℏ𝜔 = (3) √1−𝛽2 𝐸 𝑐 𝑐2 ⇒𝑝= 𝛽 = 𝑣 (4) 𝐸 ℏ𝜔 𝜔 again 𝑝 = ℏ𝑘 = 𝑘 𝑐2 𝜔 So, = 𝑣 𝑘 According to de Broglie, 𝑣 = 𝑣𝑔 , therefore 𝑣𝑝 𝑣𝑔 = 𝑐 2. The relativistic expression of the total energy of E of a particle is given by 𝐸 2 = 𝑝2 𝑐 2 + 𝑚𝑜2 𝑐 4 ℏ2 𝜔2 = ℏ2 𝑘 2 𝑐 2 + 𝑚𝑜2 𝑐 4 ℏ2 𝑣𝑝2 𝑘 2 = ℏ2 𝑐 2 𝑘 2 + 𝑚𝑜2 𝑐 4 2 2 𝑚𝑜2 𝑐 4 𝑣𝑝 = 𝑐 + 2 2 ℏ 𝑘 𝑚𝑜2 𝑐 2 𝑣𝑝 = 𝑐√1 + ℏ2 𝑘 2 Therefore we have 𝒗𝒑 > 𝒄. Also, phase velocity of de Broglie wave depends on the wavelength λ even in the vacuum. This is different from light waves or any electromagnetic wave. 1.2 Numerical examples: 1. Calculate the wavelength of the matter wave of a thermal neutron at 400K. The kinetic energy of a particle at equilibrium temperature T is given by 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 3 𝐸 = 𝐾𝐵 𝑇 2 Now, de Broglie wavelength of a neutron with energy E is ℎ ℎ ℎ 𝜆= = = √2𝑚𝐸 3𝑚𝐾𝐵 𝑇 √2𝑚 3 𝐾𝐵 𝑇 √ 2 mass of neutron = 1.675 × 10−27 kg, Boltzmann’s constant (KB)= 1.38 × 10−23JK−1, T = 400 K, and h = 6.626 × 10−34Js. Therefore substituting the values we get λ = 1.25 Å. 2 Heisenberg’s uncertainty principle The fact that a particle exhibits wave nature requires us to develop a suitable wave equation that describes the motion of the particle. The solution of this equation gives a wave function ψ(x,t) which describes the state of motion of the particle everywhere in space at any instant of time. The square of the modulus of the amplitude of the wave function gives the probability density. Also, its integral over a finite region of space represents the probability of finding the particle over that region. The representation of a particle by a wave and its probability interpretation rules out the possibility of finding a particle precisely at a point in space and determining its definite momentum at the same time. Thus the wave representation of the particle implies some uncertainty ∆x of the particle’s position 𝑥⃗ and a corresponding uncertainty ∆p in specifying the momentum 𝑝⃗ of the particle simultaneously. Heisenberg in 1927 pointed out this limitation in determining the position and momentum of a particle as, ∆x ∆px ≥ ℏ (5) A more accurate calculation considering the definition of quantum mechanical definition of expectation values gives; ∆x ∆px ≥ ℏ/2 (6) Frequently, while solving numerical problems we use the Eqn. 6. A generalized statement of Heisenberg’s uncertainty principle is as follows. If p and q denote two canonically conjugate variables, then the uncertainty relation, using the above definition, becomes: ∆q ∆p ≥ ℏ/2 A few common pairs of [p, q] are [x, px], [E, t], etc. 2.1 Numerical examples Non-existence of an electron inside the nucleus of an atom: The radius of the nucleus of a typical atom is in the order of 10−14m. If an electron is considered to be inside the nucleus, the uncertainty in its position is ∆x = diameter of the nucleus ≃ 2r ≃ 2 × 10−14m From the uncertainty principle, the corresponding uncertainty in momentum of the electron is ℎ ∆𝑝𝑥 ≥ 2𝜋∆𝑥 6.632 × 10−34 ∆𝑝𝑥 = 𝑘𝑔𝑚𝑠 −1 2 × 3.14 × (2 × 10−14 ) 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 ≥ 5.27 × 10−21 kg m s−1 It means that if the elementary particle is inside the nucleus, its minimum momentum must be 𝑝min = 5.27 × 10−21 kg m s −1 The minimum energy of an electron of mass m is obtained from relativistic consideration, 𝐸 2 = 𝑝2 𝑐 2 + 𝑚𝑜2 𝑐 4 ≃ 𝑝2 𝑐 2 Rest mass energy (𝑚0 𝑐 2) of the electron ≃ 0.511 MeV is negligible in comparison to pc. 𝐸 = 𝑝𝑐 = 5.27 × 10−21 × 3 × 108 𝐽 5.27 × 10−21 × 3 × 108 = 1.6 × 10−19 = 10 MeV. So, if an electron remains inside the nucleus, its energy must be of the order of 10 MeV. But from the experimental measurement the energy of an electron is found to be approximately 3 to 4 MeV. Therefore, electrons cannot be present within the nucleus. For protons and neutrons, their rest mass is 𝑚0 ≃ 1.67 × 10−27kg. Therefore, the corresponding value of the kinetic energy of the elementary particles is given by 𝑝2 (5.27 × 10−21 )2 𝐸𝑘 = = J ≃ 52 keV. 2𝑚 2 × (1.67 × 10−27 ) Since this kinetic energy 𝐸𝑘 is smaller than the experimentally measured energies carried by the particles emitted from a nucleus, both these particles can exist inside the nucleus. 3 Wave function and its physical significance The space-time behaviour of a quantum mechanical particle can be described by a function, known as a wave function ψ(𝑟⃗,t). Usually, ψ is a function of both position 𝑟⃗ and time t. However, it may also depend on position only. 3.1 Probability density: |ψ(x,t)|2dx gives the probability of finding the corresponding particle between x and x+dx at time t. |ψ(x,t)|2 can also be written as ψ∗(x,t) ψ(x,t), where ψ∗(x,t) is the complex conjugate of ψ(x,t). The probability of finding a 𝑥 particle in space between xi and xf is therefore given by , 𝑃 = ∫𝑥 𝑓|Ѱ(𝑥, 𝑡)|2 𝑑𝑥 𝑖 The total probability of finding the particle anywhere in space is then, ∞ ∞ 𝑃 = ∫−∞|Ѱ(𝑥, 𝑡)|2 𝑑𝑥 = ∫−∞ Ѱ∗ (𝑥, 𝑡)Ѱ(𝑥, 𝑡) 𝑑𝑥 = 1. (7) 3.2 Normalization of the wave function In general, the wave function may be of some form that does not satisfy the Eq. 7. In such a case, we multiply the wave function by a constant number to satisfy the Eq. 7, i.e., ψN(x,t) = Cψ(x,t). ∞ ∞ ∫ |Ѱ(𝑥, 𝑡)|2 𝑑𝑥 = |𝐶|2 ∫ |Ѱ(𝑥, 𝑡)|2 𝑑𝑥 = 1 −∞ −∞ ∞ ⇒ |𝐶|2 = 1/ ∫ |Ѱ(𝑥, 𝑡)|2 𝑑𝑥. −∞ C is known as the normalization constant. 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 Usually, in this course, we consider that the probability density |ψ(x,t)|2 for a wave function (also called a state) is independent of time. Such states are known as stationary states. 4 Operators in quantum mechanics An operator is a mathematical rule that when acting on a function results in a new function. If 𝛼̂ be an operator, 𝑑 then; 𝛼̂𝑓 = g, where, f and g are two functions. If 𝛼̂ = 𝑑𝑥 and f(x) = x2e−x then, 𝛼̂𝑓 = (2𝑥 − 𝑥 2 )𝑒 −𝑥. In quantum mechanics, each dynamical variable is represented by an operator. Table 1 shows a list of a few common operators, Dynamical variable Symbol Operator representation Position vector 𝑟⃗ 𝑟̂ = 𝑟⃗ Cartesian components of 𝑟̂ x,y,z 𝑥̂ = 𝑥⃗, 𝑦̂ = 𝑦⃗, ̂𝑧 = 𝑧⃗ Linear momentum 𝑝⃗ 𝜕 𝜕 𝜕 𝑝̂ = −𝑖ℏ (𝑖̂ + 𝑗̂ + 𝑘̂ ) = −𝑖ℏ𝛻⃗⃗ 𝜕𝑥 𝜕𝑦 𝜕𝑧 Angular momentum ⃗𝐿⃗ = 𝑟⃗ × 𝑝⃗ ℏ 𝐿̂ = (𝑟⃗ × 𝛻⃗⃗) 𝑖 Cartesian components of 𝐿̂ 𝜕 𝜕 𝐿̂𝑥 = −𝑖ℏ (𝑦 − 𝑧 ) 𝜕𝑧 𝜕𝑦 𝜕 𝜕 𝐿̂𝑦 = −𝑖ℏ (𝑧 −𝑥 ) 𝜕𝑥 𝜕𝑧 𝜕 𝜕 𝐿̂𝑧 = −𝑖ℏ (𝑦 −𝑥 ) 𝜕𝑥 𝜕𝑦 Tab. 1: Example of a few operators Eigenfunction and Eigenvalue An eigenfunction and eigenvalue of an operator 𝛼̂ are a function ϕ and a constant λ such that 𝛼̂ϕ = λϕ 𝑑 For example, The operator 𝑑𝑥 has an eigenfunction 𝑒 𝑘𝑥 with eigenvalue k 𝑑 𝑘𝑥 𝑒 = 𝑘𝑒 𝑘𝑥 , 𝑑𝑥 𝑑2 where k can have any constant value. Similarly, the operator 𝑑𝑥2 has an eigenfunction of the form cos ωx, where ω is a constant. The corresponding eigenvalue is −ω2. 5 Expectation value of an operator The expectation value of an observable or in other words, of the corresponding operator, can be defined such that if a large number of independent measurements of the concerned observable are done then the most probable value would be the expectation value. Mathematically, the expectation value of an operator 𝛼̂ can be defined as; ∞ 〈𝛼〉 = ∫−∞ Ѱ∗ (𝑥)𝛼̂Ѱ(𝑥) 𝑑𝑥 (8) 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 6 Numerical examples 1. The wavefunction for a quantum particle of mass m confined to move in the region of space 0 ≤ x ≤ L is given by 2 4𝜋𝑥 Ѱ(𝑥) = √ sin ( ) 𝐿 𝐿 Calculate the expectation value of x and kinetic energy Ek for the particle. 2 𝐿 4𝜋𝑥 4𝜋𝑥 〈𝑥̂〉 = ∫ sin ( ) 𝑥 sin ( ) 𝑑𝑥 𝐿 0 𝐿 𝐿 4𝜋𝑥 𝐿 Let us consider, 𝑦 = 𝐿 ⇒ 𝑑𝑥 = 4𝜋 𝑑𝑦 4𝜋 2 𝐿 2 4𝜋 2𝐿 𝑦 2 𝑦 sin 2𝑦 cos 2𝑦 2𝐿 (4𝜋)2 𝐿 〈𝑥̂〉 = ( ) ∫ 𝑦 sin2 𝑦 𝑑𝑦 = [ − − ] = =. 𝐿 4𝜋 𝑜 (4𝜋)2 4 4 8 0 (4𝜋)2 4 2 2 𝐿 4𝜋𝑥 ℏ2 𝑑 2 4𝜋𝑥 〈𝐸̂𝑘 〉 = ∫ sin ( ) (− 2 ) sin ( ) 𝑑𝑥 𝐿 0 𝐿 2𝑚 𝑑𝑥 𝐿 𝑑2 4𝜋𝑥 4𝜋 2 4𝜋𝑥 2 sin ( ) = − ( ) sin ( ) 𝑑𝑥 𝐿 𝐿 𝐿 2 ℏ2 4𝜋 2 𝐿 2 4𝜋𝑥 〈𝐸̂𝑘 〉 = ( ) ∫ sin ( ) 𝑑𝑥 𝐿 2𝑚 𝐿 0 𝐿 2 16ℏ2 𝜋 2 𝐿 16ℏ2 𝜋 2 = =. 𝐿 2𝑚𝐿2 2 2𝑚𝐿2 2. The state of a one-dimensional quantum system is represented by the wavefunction ψ(x) = N sin(3πx) for 0 ≤ x ≤ 1 with N being the normalization constant. Determine the normalization constant and calculate the probability of finding the particle within the range 2/3 ≤ x ≤ 1. 1 𝑁 2 ∫ sin2 3𝜋𝑥 𝑑𝑥 = 1 0 1 𝑦 = 3𝜋𝑥 ⇒ 𝑑𝑥 = 𝑑𝑦 3𝜋 3𝜋 𝑁 2 3𝜋 2 𝑁2 𝑦 1 𝑁2 ∫ sin 𝑦 𝑑𝑦 = [ − sin 2𝑦] = =1 3𝜋 0 3𝜋 2 4 0 2 𝑁 = √2 The probability of finding the particle in the range 2/3 ≤ x ≤ 1 is given by; 1 1 𝑃 = ∫ |Ѱ(𝑥)|2 𝑑𝑥 = 2 ∫ sin2 3𝜋𝑥 𝑑𝑥 2/3 2/3 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 3𝜋 2 3𝜋 2 2 𝑦 1 = ∫ sin 𝑦 𝑑𝑦 = [ − sin 2𝑦] 3𝜋 2𝜋 3𝜋 2 4 2𝜋 2 𝜋 1 = =. 3𝜋 2 3 Prepared by the faculty members of the Physics department 7 Brainware University, Kolkata