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These PowerPoint color diagrams can only be used by instructors if the 3rd Edition has been adopted for his/her course. Permission is given to individuals who have purchased a copy of the third edition with CD-ROM Electronic Materials and Devices to use these slides in seminar, symposium and conference presentations provided that the book title, author and © McGraw-Hill are displayed under each diagram. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Elementary Quantum Physics From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Photons: Light as a wave The classical view of light as an electromagnetic (EM) wave. Phenomena such as interference, diffraction, refraction and reflection can be explained by theory of waves. An electromagnetic wave is a traveling wave with time-varying electric and magnetic Fields that are perpendicular to each other and to the direction of propagation. Fig 3.1 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Light as a wave Traveling wave description E y ( x, t ) Eo sin(kx t ) k is the wavenumber (propagation const) = 2p/l is the angular frequency of the wave (or 2pf or 2pn where n is the frequency A similar equation describes the variation of magnetic field Bz with x at any time t Represents a traveling wave in the x direction, which, in the present example, is a sinusoidally varying function. The phase velocity is c = /k = nl From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Intensity of light wave 1 I c oE o2 2 The energy flowing per unit area per second, of the wave. o is the permittivity From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) S1P – S2P = nl S1P – S2P = (n+1/2)l Schematic illustration of Young’s double-slit experiment. Fig 3.2 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Diffraction patterns obtained by passing X-rays through crystals can only be explained by using ideas based on the interference of waves. (a) Diffraction of Xrays from a single crystal gives a diffraction pattern of bright spots on a photographic film. (b) Diffraction of X-rays from a powdered crystalline material or a polycrystalline material gives a diffraction pattern of bright rings on a photographic film. Fig 3.3 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (c) X-ray diffraction involves constructive interference of waves being "reflected" by various atomic planes in the crystal. Fig 3.3 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Bragg’s Law Bragg diffraction condition 2d sinθ nλ n 1, 2, 3, ... The equation is referred to as Bragg’s law, and arises from the constructive interference of scattered waves. Aside from exhibiting wave-like properties, light can behave like a stream of particles of zero rest-mass. It can be viewed as a stream of discrete entities or energy packets called photons, each carrying a quantum energy hn, and momentum h/l, where h is a universal const = Planck’s const = 6.626 x 10-34 Js = 4.136 x 10-15 eVs From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The photoelectric effect. Fig 3.4 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) Photoelectric current vs. voltage when the cathode is illuminated with light of identical wavelength but different intensities (I). The saturation current is proportional to the light intensity (b) The stopping voltage and therefore the maximum kinetic energy of the emitted electron increases with the frequency of light u. (Note: The light intensity is not the same) Results from the photoelectric experiment. Fig 3.5 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The effect of varying the frequency of light and the cathode material in the photoelectric Experiment. The lines for the different materials have the same slope h but different intercepts Fig 3.6 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Photoelectric Effect Photoemitted electron’s maximum KE is KEm KEm hu hu0 Work function, F0 The constant h is called Planck’s constant. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The PE of an electron inside the metal is lower than outside by an energy called the workfunction of the metal. Work must be done to remove the electron from the metal. Fig 3.7 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Intuitive visualization of light consisting of a stream of photons (not to be taken too literally). SOURCE: R. Serway, C. J. Moses, and C. A. Moyer, Modern Physics, Saunders College Publishing, 1989, p. 56, figure 2.16 (b). Fig 3.8 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Light Intensity (Irradiance) Classical light intensity 1 2 I c oE o 2 Light Intensity I ph hu Photon flux ph N ph At From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Light consists of photons From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) X-rays are photons X-ray image of an American one-cent coin captured using an x-ray a-Se HARP camera. The first image at the top left is obtained under extremely low exposure and the subsequent images are obtained with increasing exposure of approximately one order of magnitude between each image. The slight attenuation of the X-ray photons by Lincoln provides the image. The image sequence clearly shows the discrete nature of x-rays, and hence their description in terms of photons. SOURCE: Courtesy of Dylan Hunt and John Rowlands, Sunnybrook Hospital, University of Toronto. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Scattering of an X-ray photon by a “free” electron in a conductor. Fig 3.9 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The Compton experiment and its results Fig 3.10 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Schematic illustration of black body radiation and its characteristics. Spectral irradiance vs. wavelength at two temperatures (3000K is about the temperature of The incandescent tungsten filament in a light bulb.) Fig 3.11 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Black Body Radiation Planck’s radiation law 2phc Il hc 5 l exp 1 lkT 2 Stefan’s black body radiation law PS ST 4 Stefan’s constant 2p 5 k 4 S 2 3 5.670108 W m 2 K 4 15c h From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Stefan’s law for real surfaces Electromagnetic radiation emitted from a hot surface Pradiation = total radiation power emitted (W = J s-1) Pradiation S S [T T ] 4 4 0 S = Stefan’s constant, W m-2 K-4 = emissivity of the surface = 1 for a perfect black body < 1 for other surfaces S = surface area of emitter (m2) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electron as a wave From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Young’s double-slit experiment with electrons involves an electron gun and two slits in a Cathode ray tube (CRT) (hence, in vacuum). Electrons from the filament are accelerated by a 50 kV anode voltage to produce a beam that Is made to pass through the slits. The electrons then produce a visible pattern when they strike A fluorescent screen (e.g., a TV screen), and the resulting visual pattern is photographed. SOURCE: Pattern from C. Jonsson, D. Brandt, and S. Hirschi, Am. J. Physics, 42, 1974, p.9, figure 8. Used with permission. Fig 3.12 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Fig 3.13 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Fig 3.13 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The diffraction of electrons by crystals gives typical diffraction patterns that would be Expected if waves being diffracted as in x-ray diffraction with crystals [(c) and (d) from A. P. French and F. Taylor, An Introduction to Quantum Mechanics (Norton, New York, 1978), p. 75; (e) from R. B. Leighton, Principles of Modern Physics, McGraw-Hill, 1959), p. 84. Fig 3.13 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) De Broglie Relationship Wavelength l of the electron depends on its momentum p h l p De Broglie relations h l p OR p h l From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Time-Independent Schrödinger Equation There is a general equation that describes this wave-like behavior and, with the appropriate potential energy and boundary conditions, will predict the results of experiments. The equation is called “Schrödinger equation” and it forms the foundations of quantum theory. In EM theory, a traveling EM wave resulting from a sinusoidal current oscillations, or The traveling voltage wave on a long transmission line, can generally be described by a traveling wave equation of the form E( x, t ) Eo exp j(kx t ) E( x) exp( jt ) where E(x) = Eoexp(jkx) represents the spatial dependence, which is separate from the time variation. We note that the time dependence is harmonic and therefore predictable. For this reason, we put aside the exp(-jt) term until we need the instantaneous magnitude of the voltage. Born suggested there may be a similar wave function for the electron, which is Represented by Y(x,t). The amplitude squared represent the probability of finding Electron per unit distance. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Y ( x, y , z , t ) 2 is the probability of finding the electron per unit volume at x,y,z at time t Y ( x, y, z , t ) dxdydz 2 is the probability of finding the electron in a small elemental volume dxdydz at x, y z at time t Only |Y|2 has meaning, not Y, the latter function need not be real; it can be complex function with real and imaginary parts. For this reason, we tend to use Y*Y where Y* is the complex conjugate of Y, instead of |Y|2 to represent the probability per unit volume. To obtain the wavefunction Y(x,t) for the electron, we need to know how the electron interacts with its environment. This is embodied in its potential energy function V(x,t) because the net force the electron experiences is given by V (r ) e2 4p o r r x 2 y 2 z 2 is the distance between the electron and the proton. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) If the PE of electron is time independent, then the spatial and temporal dependences Of Y(x,t) can be separated, and the total wavefunction Y(x,t) of e- can be written as jEt Y ( x, t ) Y ( x) exp( ) h Where Y(x) is the electron wavefunction that describes only the spatial behavior, and E is the energy of e-. E/h The fundamental equation that describes the electron’s behavior by determining Y(x) is called the time-independent Schrodinger equation. It is given by d 2 2m 2 ( E V ) 0 2 dx h This is a second-order differential equation. In 3-D, it becomes 2 2 2 2m 2 2 2 ( E V ) 0 2 x y z The notation (∂ψ/∂x) differentiates ψ(x,y,z) with respect to x but keeping y and z const From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Steady-state probability distribution of electron is given by Y ( x, y , z , t ) Y ( x, y , z ) 2 2 Two important boundary conditions are often used: 1) Y must be continuous 2) ∂ψ/∂x must be continuous Y must be single-valued and smooth, without any discontinuities From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The enforcement of these boundary conditions results in strict requirements on the wavefunction (x), as a result of which only certain wavefunctions are acceptable. These wavefunctions are called the eigenfunctions (characteristic functions) of the system., and they determine the behavior and energy of the electron under steadystate conditions. The eigenfunctions (x) are also called stationary states, in as much as we are only considering steady-state behavior. Example 3.5 solve the schrodinger eq for a free electron whose energy is E. Solution Since the e- is free, its potential energy is zero, V = 0. This leads to d 2 2m 2 E 0 2 dx h d 2 2 k 0 2 dx Solving the differential equation, we get (x) = Aejkx or Be-jkx Traveling waves to the x and –x directions. Thus, free e- has a traveling wave solution with a wavenumber k = 2p/l, that can have any value. The energy of electron is Simply KE, so From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (h k ) 2 p 2 KE E 2m 2m p = h/l Its momentum is given by p h k or This is the de Brogile relationship. The probability distribution for the e- is ( x) A exp j (kx ) A2 2 2 which is constant over the entire space. Thus, the electron can be anywhere between x = -∞ and x = +∞. The uncertainty ∆x in its position is infinite. Since the electron has a well-defined wavenumber k, its momentum p is also well-defined by virtue of p = hk/2p. The uncertainty ∆p in its momentum is thus zero. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Time-Independent Schrodinger Equation Steady-state total wave function jEt Y( x,t ) ( x)exp Schrodinger’s equation for one dimension d 2 2m 2 E V ) 0 2 dx Schrondinger’s equation for three dimensions 2m 2 2 2 ( E V ) 0 2 x y z 2 2 2 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Infinite potential well: a confined electron From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Its PE is zero inside that region and infinite outside. The electron cannot escape because it would need an infinite PE. The probability ||2 of finding an electron per unit volume is zero outside 0 < x < a. Thus, = 0 when x <= 0 and x >= a, and is determined by the Schrodinger eq in 0 < x < a with V = 0. Therefore, in region 0 < x < a d 2 2m 2 E 0 2 dx This is a second-order linear differential equation. As a general solution, we take ( x) A exp(jkx) B exp( jkx) where k is some constant. We note that y(0) = 0; therefore B = -A so that ( x) A[exp(jkx) exp( jkx)] 2 Aj sin kx Substitute to relate the energy E to k. Thus 2m 2 Ajk 2 (sin kx) 2 E (2 Aj sin kx) 0 h px2 h 2k 2 E KE 2m 2m where px= ±hk/2p From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) k is a wavenumber and is determined by the boundary condition at x = a where = 0, or (a) = 2Ajsin ka = 0 ka = np, where n = 1, 2, 3 … called a quantum number For each n, there’s a special wavefunction n ( x) 2 Aj sin( npx ) a called eigenfunction. For each n, there is a special k value, kn = np/a, and special energy value En h 2 k n2 En 2m h 2 (pn) 2 h 2 n 2 called eigenenergies of the system En 2m a2 8m a2 A is determined from the normalization condition. The total probability of finding the electron in the whole region 0 < x < a is unity. xa xa npx ( x ) dx 2 Aj sin( ) dx 1 x0 x0 a 2 2 1/ 2 1 A 2a From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) 1/ 2 The resulting wavefunction for the electron is 2 ( x) j a npx sin a The ground state, lowest energy state corresponds to n = 1. Energy is not zero! The node of wavefunction is defined as the point where = 0 inside the well. The energy increases as the number of nodes increases in a wavefunction. The energy differences between the two consecutive energy levels, as follows: h 2 (2n 1) E En 1 En 8m a2 As a increases to macroscopic dimensions, a ∞, the electron is completely free and ∆E 0. Since ∆E = 0, the energy of a completely free electron is continuous. The energy of a confined electron, however, is quantized, and ∆E depends on the dimension (or size) of the potential well confining the electron Confined electron energy is quantized Free electron energy is continuous The symmetry of a wavefunction is called parity. Whenever the PE function V(x) exhibits symmetry about a certain point C, for example, about x = 0.5a, then the wavefunctions have either even parity (such as 1, 3 … that are symmetric) or have Odd parity (such as 2, 4 … that are antisymmetric). From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electron in a one-dimensional infinite PE well. The energy of the electron is quantized. Possible wavefunctions and the probability distributions for the electron are shown. Fig 3.15 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Infinite Potential Well Wavefunction in an infinite PE well npx n ( x) 2 Aj sin a Electron energy in an infinite PE well (pn) h n En 2 2 2m a 8m a 2 2 2 2 Energy separation in an infinite PE well h (2n 1) E En1 En 2 8m a 2 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Heisenberg’s Uncertainty Principle We cannot exactly and simultaneously know both the position and momentum of a particle along a given coordinate. There will be an uncertainty ∆x in the position and an uncertainty ∆px in the momentum of the particle and these uncertainties will be related by Heisenberg’s uncertainty relationship. Heisenberg uncertainty principle for position and momentum xpx These uncertainties are not in any way a consequence of the accuracy of a measurement or the precision of an instrument. Rather, they are the theoretical limits to what we can determine about a system. Heisenberg uncertainty principle for energy and time Et It is important to note that the uncertainty relationship applies only when the position and momentum are measured in the same direction (such as the x direction). Hence, for example, ∆x∆py can be zero. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Tunneling Phenomenon: Quantum Leak! From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) The roller coaster released from A can at most make it to C, but not to E. Its PE at A is less than the PE at D. When the car is at the bottom, its energy is totally KE. CD is the energy barrier that prevents the care from making it to E. In quantum theory, on the other hand, there is a chance that the care could tunnel (leak) through the potential energy barrier between C and E and emerge on the other side of hill at E. (b) The wavefunction for the electron incident on a potential energy barrier (V0). The incident And reflected waves interfere to give 1(x). There is no reflected wave in region III. In region II, the wavefunction decays with x because E < V0. Fig 3.16 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Classically, just like the roller coaster, the electron should bounce back and thus be confined in the region x < 0, because its total energy E is less than Vo. In the quantum world, however, there is a distinct possibility that the electron will “tunnel” thru the potential barrier and appear on the other side; it will leak thru. To show this, we solve the Schrodinger eq. We divide the electron’s space in to 3 regions, I, II, III. Solve for each region, to obtain I(x), II(x), III(x). In regions I and III, (x) must be traveling waves, as there is no PE. In region II, E – Vo is negative, so the general solution of the schrodinger eq is the sum of an exponentially decaying function and an exponentially increasing function. where I ( x) A1 exp(jkx) A2 exp( jkx) II ( x) B1 exp(x) B2 exp(x) III ( x) C1 exp(jkx) C2 exp( jkx) 2m(Vo E ) 2mE 2 2 k 2 h2 h Because the electron is traveling toward the right in region III, there is no reflected wave, so C2 = 0. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Apply the BCs and the normalization condition to determine the various constants A1, A2, B1, B2, C1. We must match the three waveforms at their boundaries (x = 0 and x = a) so that they form a continuous single-valued wavefunction. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Tunneling Phenomenon: Quantum Leak Probability of tunneling is the relative probability that the electron will tunnel from region I to III. T where III ( x) I ( x) 2 2 C12 1 2 A1 1 D sinh2 (a) Vo2 D 4 E (Vo E ) and is the rate of decay of II(x) . For a wide or high barrier, using a >>1, and sinh(a) 0.5exp(a), we can write Probability of tunneling through T To exp(2a) Reflection coefficient R where To 16E (Vo E ) Vo2 A22 R 2 1 T A1 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Scanning Tunneling Microscopy (STM) image of a graphite surface where contours represent electron concentrations within the surface, and carbon rings are clearly visible. Two Angstrom scan. |SOURCE: Courtesy of Veeco Instruments, Metrology Division, Santa Barbara, CA. Fig 3.18 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Fig 3.17 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) STM image of Ni (100) surface STM image of Pt (111) surface SOURCE: Courtesy of IBM SOURCE: Courtesy of IBM From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Potential Box: Three Quantum Numbers From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electron confined in three dimensions by a three-dimensional infinite PE box. Everywhere inside the box, V = 0, but outside, V = . The electron cannot escape from the box. Fig 3.19 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) We need to solve the 3-D version of the Schrodinger equation 2 2 2 2m 2 2 2 ( E V ) 0 2 x y z via the method of separation of variables, assuming that the wave function is “separable” into ( x, y, z) x ( x) y ( y) z ( z) Substitute this back into the previous eq, we can obtain 3 ordinary differential eqs similar to the one for the 1-D potential well. So the wavefunction is simply ( x, y, z) Asin(kx x) sin(k y y) sin(kz z) Apply the boundary conditions at x = a, y = b, and z = c to determine the const k’s. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Potential Box: Three Quantum Numbers Electron wavefunction in infinite PE well n1px n2py n3pz n1n2n3 ( x, y, z ) A sin sin sin a b c Electro energy in infinite PE box En1n2 n3 ) h 2 n12 n22 n32 h 2 N 2 2 8m a 8m a2 N n n n 2 2 1 2 2 2 3 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The electron in the hydrogenic atom is atom is attracted by a central force that is always directed toward the positive Nucleus. Spherical coordinates centered at the nucleus are used to describe the position of the electron. The PE of the electron depends only on r. Fig 3.20 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electron wavefunctions and the electron energy are obtained by solving the Schrödinger equation Electron’s PE V(r) in hydrogenic atom is used in the Schrödinger equation Ze V (r ) 4p o r 2 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) Radial wavefunctions of the electron in a hydrogenic atom for various n and values. (b) R2 |Rn,2| gives the radial probability density. Vertical axis scales are linear in arbitrary units. Fig 3.21 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electron energy is quantized Electron energy in the hydrogenic atom is quantized. n is a quantum number, 1,2,3,… 4 2 me Z En 2 2 2 8 o h n Ionization energy of hydrogen: energy required to remove the electron from the ground state in the H-atom 4 me 18 EI 2 2 2.1810 J 13.6eV 8 o h From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) K L M N O Sharp Principal diffuse fundamental From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) The polar plots of Yn,(, ) for 1s and 2p states. (b) The angular dependence of the probability distribution, which is proportional to | Yn,(, )|2. Fig 3.22 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The energy of the electron in the hydrogen atom (Z = 1). Fig 3.23 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The physical origin of spectra. (a) Emission (b) Absorption Fig 3.24 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) An atom can become excited by a collision with another atom. When it returns to its ground energy state, the atom emits a photon. Fig 3.25 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electron probability distribution in the hydrogen atom Maximum probability for = n 1 2 rmax n ao Z From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The Li atom has a nucleus with charge +3e, 2 electrons in the K shell , which is closed, and one electron in the 2s orbital. (b) A simple view of (a) would be one electron in the 2s orbital that sees a single positive charge, Z = 1 The simple view Z = 1 is not a satisfactory description for the outer electron because it has a probability distribution that penetrates the inner shell. We can instead use an effective Z, Zeffective = 1.26, to calculate the energy of the outer electron in the Li atom. Fig 3.26 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Ionization energy from the n-level for an outer electron EI ,n Z 2 effective (13.6 eV) 2 n From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) The electron has an orbital angular momentum, which has a quantized component L along an external Magnetic field Bexternal. (b) The orbital angular momentum vector L rotates about the z axis. Its component Lz is quantized; Therefore, the L orientation, which is the angle , is also quantized. L traces out a cone. (c) According to quantum mechanics, only certain orientations ( ) for L are allowed, as determined by and m Fig 3.27 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Orbital Angular Momentum and Space Quantization Orbital angular momentum L 1) 1/ 2 where = 0, 1, 2, ….n1 Orbital angular momentum along Bz Lz m Selection rules for EM radiation absorption and emission 1 and m 0, 1 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Spin angular momentum exhibits space quantization. Its magnitude along z is quantized, so the angle of S to the z axis is also quantized. Fig 3.29 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electron Spin and Intrinsic Angular Momentum S Electron spin S ss 1) 1/ 2 1 s 2 Spin along magnetic field S z ms 1 ms 2 the quantum numbers s and ms, are called the spin and spin magnetic quantum numbers. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Orbital angular momentum and space quantization The electron is attracted to the nucleus by a central force, just like the Earth is attracted by the central gravitational force of the sun. So it has an orbital angular momentum L: where l = 0, 1, 2, … < n L 1) 1/ 2 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) In the presence of external magnetic field Bz, taken arbitrarily in the z direction, the component of the angular momentum along the z axis, Lz is also quantized and given by Lz m For any given l, quantum mechanics requires that ml must have values in the range -l, -(l-1), …, -1, 0, 1, …, (l -1), l. We see that |ml| < l ml can be negative since Lz can be negative or positive. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electron Spin and Intrinsic Angular Momentum: S The electron has a spin or intrinsic angular momentum, denoted by S. In classical mechanics, in the absence of external torques, spin angular momentum is conserved. In quantum mechanics, this spin angular momentum is quantized. Electron spin S ss 1) Spin along Magnetic field S z ms 1/ 2 1 2 1 ms 2 s Spin angular momentum is space quantized and has constant magnitude. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Selection Rules for EM radiation And Absorption : the allowed photon emission processes. Photon emission involves = 1, ml = 0, 1 Fig 3.28 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The Zeeman Effect In 1896, the Dutch Physicist, Pieter Zeeman showed that the spectral lines emitted by atom in a magnetic field split into multiple energy levels. This is called “Zeeman Effect”. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Magnetic Dipole Moment of the Electron Orbital magnetic moment Spin magnetic moment e μ orbital L 2me μ spin e S ms Energy of the electron due to its magnetic moment interacting with a magnetic field In the presence of an external magnetic field B, the electron has an additional energy term that arises from the interaction of these magnetic moments with B. Potential energy of a magnetic moment EBL orbitalB cos where is the angle between orbital and B. Energy is min when orbital (the magnet) and B are parallel, = 0. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The Zeeman Effect •A magnetic field splits the mℓ levels. The potential energy is quantized and now also depends on the magnetic quantum number mℓ. •When a magnetic field is applied, the 2p level of atomic hydrogen is split into three different energy states with energy difference of ΔE = BB Δmℓ. mℓ Energy 1 E0 + μBB 0 E0 −1 E0 − μBB From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The Zeeman Effect •The transition from 2p to 1s, split by a magnetic field. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Magnetic Dipole Moment of the Electron From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) The orbiting electron is equivalent to a current loop that behaves like a bar magnet. (b) The spinning electron can be imagined to be equivalent to a current loop as shown. This current loop behaves like a bar magnet, just as in the orbital case. Fig 3.30 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electron orbiting nucleus with radius r and angular frequency , the magnetic moment is 2 e e r IA ( )(pr 2 ) 2p 2 Consider the orbital angular momentum L, which is the linear momentum p multiplied by the radius r, or L = pr = mevr = mer2 Therefore μ orbital e L 2me Similarly, the spin angular momentum of the electron S leads to a spin magnetic moment spin, which is in the opposite direction to S and given by μ spin e S ms From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Magnetic Dipole Moment of the Electron Orbital magnetic moment e μ orbital L 2me Spin magnetic moment μ spin e S ms From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Energy of the electron due to its magnetic moment interacting with a magnetic field In the presence of an external magnetic field B, the electron has an additional energy term that arises from the interaction of these magnetic moments with B. Potential energy of a magnetic moment EBL orbitalB cos where is the angle between orbital and B. Energy is min when orbital (the magnet) and B are parallel, = 0. A magnetic moment in a magnetic field experiences a torque that tries to rotate the magnetic moment to align the moment with the field. A magnetic moment in a nonuniform magnetic field experiences force that depends on the orientation of the dipole. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) Schematic illustration of the Stern-Gerlach experiment. A stream of Ag atoms passing through a nonuniform magnetic field splits into two. Fig 3.31 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (b) Explanation of the Stern-Gerlach experiment. (c) Actual experimental result recorded on a photographic plate by Stern and Gerlach (O. Stern and W. Gerlach, Zeitschr. fur. Physik, 9, 349, 1922.) When the field is turned off, there is only a single line on the photographic plate. Their experiment is somewhat different than the simple sketches in (a) and (b) as shown in (d). Fig 3.31 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Stern-Gerlach memorial plaque at the University of Frankfurt. The drawing shows the original Stern-Gerlach experiment in which the Ag atom beam is passed along the long- length of the external magnet to increase the time spent in the nonuniform field, and hence increase the splitting. The photo on the lower right is Otto Stern (1888-1969), standing and enjoying a cigar while carrying out an experiment. Otto Stern won the Nobel prize in 1943 for development of the molecular beam technique. Plaque photo courtesy of Horst Schmidt-Böcking from B. Friedrich and D. Herschbach, "Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics", Physics Today, December 2003, p.53-59. Fig 3.31 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Orbital angular momentum vector L and spin angular momentum vector S can add either In parallel as in (a) or antiparallel, as in (b). The total angular momentum vector J = L + S, has a magnitude J = [j(j+1)], where in (a) j = + ½ and in (b) j = - ½ Fig 3.32 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) The angular momentum vectors L and S precess around their resultant total angular Momentum vector J. (b) The total angular momentum vector is space quantized. Vector J precesses about the z axis, along which its component must be m j Fig 3.33 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) A helium-like atom The nucleus has a charge +Ze, where Z = 2 for He. If one electron is removed, we have the He+ ion, which is equivalent to the hydrogenic atom with Z = 2. Fig 3.34 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The Helium Atom PE of one electron in the He atom 2 2 2e e V (r1 , r12 ) 4po r1 4po r12 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Energy of various one-electron states. The energy depends on both n and . The dependence on is weaker than on n. Denote En,.As n and increase, En, also increases. Fig 3.35 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Pauli exclusion principle: (based on experimental observations). No two electrons within a given system (e.g. an atom) may have all four identical Quantum numbers, n, l, ml and ms. A wavefunction denoted by n,l,ml,ms, as an electronic state. For example, an e- with quantum numbers given by 2, 1, 1, ½ will have a wave function n,l,ml,ms = 2,l,1,1/2 and Is said to be in the state 2p, ml = 1 and spin up. Its energy will be E2p. The orbital motion of an electron is determined by n, l, and ml, whereas ms determines the spin direction (up or down). Suppose two electrons are in the same orbital state with identical n, l, ml. By the Pauli exclusion principle, they would have to spin in the opposite directions. One would have to spin up, the other down. e- are spin paired. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Paired spins in an orbital. Fig 3.36 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electronic configurations for the first five elements. Each box represents an orbital (n, , m) Fig 3.37 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Hund’s rule: Electrons in the same n, l orbitals prefer their spins to be parallel (same ms). Reason: If electrons enter the same ml state by pairing their spins (different ms), their quantum numbers n, l ml will be the same and their will both occupy the same region of space (same n,l,ml orbital). They will then experience a large Coulombic repulsion and will have a large Coulombic potential energy. On the other hand, if they parallel their spins (same ms), they will each have a different ml and will therefore occupy different regions of space ( different n,l,ml), thereby reducing their Coulombic repulsion. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electronic configuration for C, N, O, F and Ne atoms. Notice that in C, N, and O, Hund’s rule forces electrons to align their spins. For the Ne atom, all the K and L orbitals are full. Fig 3.38 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Absorption, spontaneous emission and stimulated emission Absorption, spontaneous emission, and stimulated emission. Fig 3.39 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The principle of the LASER. (a) Atoms in the ground state are pumped up to the energy level E3 by incoming photons of energy hu13 = E3-E1. (b) Atoms at E3 rapidly decay to the metastable state at energy level E2 by emitting photons or emitting lettice vibrations. hu32 = E3-E2. Fig 3.40 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (c) As the states at E2 are metastable, they quickly become populated and there is a population inversion between E2 and E1. (d) A random photon of energy hu21 = E2-E1 can initiate stimulated emission. Photons from this stimulated emission can themselves further stimulate emissions leading to an avalanche of stimulated emissions and coherent photons being emtitted. Fig 3.40 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Schematic illustration of the HeNe laser. Fig 3.41 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The principle of operation of the HeNe laser. Important HeNe laser energy levels (for 632.8 nm emission). Fig 3.42 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) Doppler-broadened emission versus wavelength characteristics of the lasing medium. (b) Allowed oscillations and their wavelengths within the optical cavity. (c) The output spectrum is determined by satisfying (a) and (b) simultaneously. Fig 3.43 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Laser Output Spectrum Doppler effect: The observed photon frequency depends on whether the Ne atom is moving towards (+vx) or away ( vx) from the observer vx v1 v0 1 c vx v2 v0 1 c Frequency width of the output spectrum is approximately u2 – u1 2v0v x v c Laser cavity modes: Only certain wavelengths are allowed to exist within the optical cavity L. If n is an integer, the allowed wavelength l is l n L 2 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Energy diagram for the Er3+ ion in the glass fiber medium and light amplification by Stimulated emission from E2 to E1. Dashed arrows indicate radiationless transitions (energy emission by lattice vibrations). Fig 3.44 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) A simplified schematic illustration of an EDFA (optical amplifier). The erbiumion doped fiber is pumped by feeding the light from a laser pump diode, through a coupler, into the erbium ion doped fiber. Fig 3.45 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) The retina in the eye has photoreceptors that can sense the incident photons on them and hence provide necessary visual perception signals. It has been estimated that for minimum visual perception there must be roughly 90 photons falling on the cornea of the eye. (b) The wavelength dependence of the relative efficiency ηeye(λ) of the eye is different for daylight vision, or photopic vision (involves mainly cones), and for vision under dimmed light, (or scotopic vision represents the dark-adapted eye, and involves rods). (c) SEM photo of rods and cones in the retina. SOURCE: Dr. Frank Werblin, University of California, Berkeley. Fig 3.46 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Some possible states of the carbon atom, not in any particular order. Fig 3.47 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)