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Drexel University ECE Department ECEE 302 Electronic Devices 30 September 2002 ECEE 302: Electronic Devices Lecture 2. Physical Foundations of Solid State Physics 30 September 2002 BMF-Lecture 3-093002-Page -1 Copyright © 2002 Barry Fell Drexel University ECE Department Outline ECEE 302 Electronic Devices • Characteristics of Quantum Mechanics – – – – – Black Body Radiation Photoelectric Effect Bohr’s Atom and Spectral Lines de Broglie relations Wave Mechanics • • • Probability Schrodinger Equation Uncertainty Relations • Applications of Wave Mechanics – Infinite Well – Finite Barrier (Tunneling) – Hydrogen Atom • Periodic Table – Pauli Exclusion Principle – Minimum Energy – Bohr’s Aufbauprinzip (Building-Up Principle) • Atomic Structure 30 September 2002 BMF-Lecture 3-093002-Page -2 Copyright © 2002 Barry Fell Drexel University ECE Department What is Quantum Mechanics? ECEE 302 Electronic Devices • Classical Mechanics – – – – Newton’s Three Laws applies to particles (localized masses) or mass distributions Well defined deterministic trajectories Initial Conditions and Equations of Motion determine particle behavior for all time • Classical Optics – Light is wave (non-localized, spread over region) – Interference and Diffraction effects are seen • Quantum Mechanics – Laws based on Geometrical Optics (short wave-length region) – Describes system behavior in terms of “Wave Function” • “square of wave function” provides probabilistic information about state of system – Evolution of Wave Function in Time is deterministic – An observation based on the wave function • • possible outcomes of observation probability of each outcome 30 September 2002 BMF-Lecture 3-093002-Page -3 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Black Body Radiation • Black Body is perfect absorber (and re-radiator) • Based on classical mechanics, the black body should have infinite energy • Planck found an empirical formula to fit Black Body experimental curve • To derive this formula from first principles he had to assume light energy (electromagnetic energy) was not spread out in space but came in small packages or bundles (quanta - german word for “dose”). E=hn Classical Equi-partition of Energy Model Planck' s Law for Black Body Radiation 8 n2 Un 3 c Thermal (Black Body) Energy hn hn kT e 1 Un Black Body spectral Radiant Energy per unit volume per Hertz Frequency 30 September 2002 BMF-Lecture 3-093002-Page -4 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Planck’s Black Body Radiation Law Consider Planck' s Law for Black Body Radiation 8 n2 Un 3 c hn hn kT e 1 Un Black Body spectral Radiant Energy per unit volume per Hertz n frequency of radiation c velocity of light h Planck' s constant 6.62 10- 27 erg sec (unit of action) k Boltzmann' s constant 1.38 10-16 erg / K energy per degree Kelvin T Temperatur e of the Black Body Radiator in degrees Kelvin Consider hn kT 8 n2 Un 3 c hn e hn kT 8 n2 hn 8 n2 3 3 c 1 hn 1 c 1 kT hn 8 n2 3 kT (classical limit) c hn kT Consider hn kT 8 n2 Un 3 c 30 September 2002 hn e hn kT hn 8 n2 hn 8 n2 kT 3 h n e (Boltzmann Factor - Wien' s Law) hn 3 c c 1 e kT BMF-Lecture 3-093002-Page -5 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Photo-Electric Effect • When light of frequency n is incident on a metal, electrons are emitted from the metal – Emission is instantaneous – kinetic energy of the emitted electrons are dependent on the frequency of the light (n) • Einstein (1905) used Planck’s idea of bundle of light to explain this effect – (1/2) mv2=hn-F – Einstein called this particle of light a “photon” • Einstein had shown that Planck’s hypothesis could be interpreted as showing that waves can exhibit particle like properties hn Ekinetic F=Work Function 30 September 2002 Ekinetic 1 mv 2 2 1 mv 2 hn F 2 F Work Function BMF-Lecture 3-093002-Page -6 of the metal Copyright © 2002 Barry Fell Drexel University ECE Department The Crisis in Atomic Theory ECEE 302 Electronic Devices • Ernst Rutherford determined the atom has a small positvely charged, solid nucleus which is surrounded by a swarm of negatively charged electrons • Classical Electrodynamics predicted that an accelerating particle (such as an electron moving around the nucleus of an atom) should radiate continuously, reduce its radius until it spirals into the nucleus • Observation shows us that – Atoms are stable – Radiation from atoms is with discrete spectral lines that follow a geometric series • Niels Bohr resolved these issues with a new atomic model based on the work of Einstein and Planck 30 September 2002 BMF-Lecture 3-093002-Page -7 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices The Atom of Neils Bohr • Postulates – Atoms are stable • • Electrons can exist in well defined orbits around a nucleus (Atomic State) Electrons do not radiate when in the orbit (Contrary to Classical Electromagnetism) – Discrete Spectral lines • • • Atoms radiate (emit a photon) or absorb energy (absorb a photon) when an electron makes a transition from one fixed orbit (initial state) to another orbit (final state) The frequency of the light emitted or absorbed is given by Planck’s formula n=DE/h The discrete orbits are determined by “quantization” of the orbital angular momentum. This is determined by the relation mvrn=nh • This theory successfully reproduced the spectral line pattern seen in H, He+, and Li++, single electron atoms 30 September 2002 BMF-Lecture 3-093002-Page -8 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Bohr’s Atom (1 of 3) For circular motion, Newton' s law for circular motion is mv 2 Ze 2 r 4 0r 2 The orbital angular momentum, L, is nh n 2 Where we have invoked the quantum Momentum of the Atom L mvr h 2 condition on the Orbital Angular e and mv 2 F r nh mv 2r and mv nh 2r 2 Ze 2 FElectrostatic 30 September 2002 Ze 2 4 0r 2 BMF-Lecture 3-093002-Page -9 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Bohr’s Atom (2 of 3) So m 2 v 2 mZe 2 r 4 0r 2 and 2 mZe 2 nh 1 3 4 0r 2 2 r and 4 0 nh rn mZe 2 2 2 Hence v nh 2r and 2 nh nh mZe 2 2 mZe 2 2 vn 2rn 2 4 0 nh 4 0 nh 30 September 2002 BMF-Lecture 3-093002-Page -10 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Bohr’s Atom (3 of 3) Kinetic Energy (KE n ) of Electon in the Orbit 1 mZ 2e 4 h 2 KE n mv n 2 2 2 24 0 n 2 2 The Potential Energy PE n of the electon in the Orbit is Ze 2 mZ 2e 4 PE n 2 rn 4 0 2 n 2 2 so mZ 2e 4 En KE n PE n 2 24 0 n 2 2 and mZ 2e 4 1 1 En Em hn n ,m 2 2 2 2 24 0 m n and 30 September 2002 n n ,m mZ 2e 4 1 1 2 2 2 2 m n 24 0 h BMF-Lecture 3-093002-Page -11 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices The atom of Bohr Kneels (“The Strange Story of the Quantum” - Banesh Hoffman) • Bohr’s Theory failed to predict the spectral behavior of more complex atoms with two or more electrons • Spectral line splitings (fine structure) was not predicted by Bohr’s Theory • The anomalous Zeeman effect (splitting of spectral lines in Magnetic Fields) was also not predicted successfully • This required a more fundamental theory 30 September 2002 BMF-Lecture 3-093002-Page -12 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Davision-Germer Experiment and the de Broglie Hypothesis • In 19XX Davision and Germer of the Bell Telephone Laboratories showed that high energy electrons could be diffracted by crystals • Particles had shown characteristics of waves • Louis de Broglie characterized the wave nature of particles by the expression l=h/p • localized particle properties: Energy & Momentum • non-local wave properties: frequency & wavelength Characteristics mass energy momentum 0 E=hn p=hk=h/l=E/c Particles (~electron) m E=hn p=hk=h/l=(2mE)1/2 Wave (light) 30 September 2002 BMF-Lecture 3-093002-Page -13 Copyright © 2002 Barry Fell Drexel University ECE Department Wave Mechanics (1 of 2) ECEE 302 Electronic Devices • Schrodinger used classical geometrical optics to formulate a new mechanics he called wave mechanics x j 2 nt l Start with a " wave" function, x, t e Make use of the Einstein (E hn ), de Broglie p h/ l relations h 2E E hn , 2 n 2 h h hk 2p p k , k 2 l l 2 h E p j t x Transform x, t e j2 t kx e x, t x, t Note that E x, t j , and p x, t j t x 30 September 2002 BMF-Lecture 3-093002-Page -14 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Wave Mechanics (2 of 2) The Schrodinger Wave Equation The Schrodinge r wave equation is determined from the classical relationsh ip p2 Ep, x V( x) 2m Substitute the relations E x, t j x, t x, t , and p x, t j t x into the above equation. We find x, t 2 2 x, t j V( x) x, t 2 t 2m x What is the physical significan ce of x, t ? 30 September 2002 BMF-Lecture 3-093002-Page -15 Copyright © 2002 Barry Fell Physical Interpretation of y Drexel University ECE Department ECEE 302 Electronic Devices • Schrodinger initially believed that was a guiding wave • Max Born introduced a probability interpretation for E x, t is a probabilit y amplitude Px, t E x, t E x, t E x, t probabilit y that 2 a particle of energy E is at the position x to x dx at the time t to t dt We can show that Px, t div Sx, t 0 t Sx, t E x, t grad E x, t grad E x, t E x, t 2 jm where which is called the " probabilit y current" The change in probability within a volume V is due to the “flow” of propability across the bounding surface A 30 September 2002 P x, t t Sx, t BMF-Lecture 3-093002-Page -16 Copyright © 2002 Barry Fell Drexel University ECE Department Solutions of the Schrodinger Equation ECEE 302 Electronic Devices • The Schrodinger equation is a second order differential equation • It can be split into the product of a time solution and a spatial solution by the method of separation of variables – Q(r,t)=A( r) B(t) – Show equations for t and for r • The spatial equation is called a sturm-louisville equation. Solutions exist only for certain values of the separation constant. These are called eigen (proper) solutions and eigen (proper) values • These possible solutions are called quantum levels and the specific values are called quantum values (or quantum numbers) • The corresponding functions are called quantum state functions 30 September 2002 BMF-Lecture 3-093002-Page -17 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Boundary Conditions • Solutions must obey the following conditions called boundary conditions – Q must be bounded (finite) everywhere – At boundaries between multiple solutions the magnitude and the derivative of the solutions must be continuous 30 September 2002 BMF-Lecture 3-093002-Page -18 Copyright © 2002 Barry Fell Drexel University ECE Department Normalization of the Wave Function ECEE 302 Electronic Devices • Since Q is a probability amplitude and QQ* is a probability we must have – integral of QQ* over all space = 1 – This is called normalization of the wave function • Examples – Particle in a box – Hydrogen Equation 30 September 2002 BMF-Lecture 3-093002-Page -19 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Applications • • • • • • Potential Well - (Stationary States) Potential Barrier - (Tunneling) Hydrogen Atom (quantum numbers) Electron Spin Hydrogen Molecule Bonds – Ionic Bond – Valence Bond 30 September 2002 BMF-Lecture 3-093002-Page -20 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Potential Well (in 1-dimension) and Bound States • • • • • Schrodinger Equation Boundary Conditions Solutions Quantization of Energy Levels Uncertainty Principle 30 September 2002 BMF-Lecture 3-093002-Page -21 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Potential Barrier in 1 dimension (Tunneling) • • • • Schrodinger Equation Boundary Conditions Solution Uncertainty Principle 30 September 2002 BMF-Lecture 3-093002-Page -22 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Hydrogen Atom • • • • • Schrodinger Equation in 3 dimensions Schrodinger Equation in spherical coordinates Separation of Variable for r, q, h Solution for h = magnetic quantum number Legendre polynomials for q = angular momentum quantum number • Legarre polynomials for r = principle (orbital) quantum number • Electron spin s=+/- 1/2 • Designation of a quantum state n,l,m,s 30 September 2002 BMF-Lecture 3-093002-Page -23 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Uncertainty Principle • Introduced by Werner Heisenberg • Statement of Uncertainty Relations – Uncertainty in position x uncertainty in momentum > h – Uncertainty in energy x uncertainty in time > h 30 September 2002 BMF-Lecture 3-093002-Page -24 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Hydrogen Molecule • • • • Schrodinger Equation Solution Interpretation Electron Spin 30 September 2002 BMF-Lecture 3-093002-Page -25 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Ionic Bond • Ionic Bonding – exchange of an electron between two atoms so each acheives a closed shell – result is a positive (electron donor) and negative (electron acceptor) ion – ions attract forming a bond • Examples: NaCl, KCl, KFl, NaFl 30 September 2002 BMF-Lecture 3-093002-Page -26 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Valance Bond • Valance Bond: Bonding due to two atoms of complementary valance combining chemically – Valance Band 4 (and 4): C, Si, Ge, SiC – Valance Band 3 and 5: GaAs, InP, – Valance Band 2 and 6: CdS, CdTe • Examples – Face Centered Cubic: Diamond (C), Silicon (Si) 30 September 2002 BMF-Lecture 3-093002-Page -27 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Periodic Table of Elements • History – Developed by Medelaev in 1850 based on chemical properties of atoms – Understood initially in terms of chemical affinities (Valence) – Quantum Mechanics provides the Physical Theory of Valence • Atoms are arranged in 8 basic columns related to the valence of each atom • Transition elements build up their electronic structure • Periodic Table can be understood in terms of two principles – Pauli Exclusion Principal – Minimum Energy Principal 30 September 2002 BMF-Lecture 3-093002-Page -28 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Periodic Table 30 September 2002 BMF-Lecture 3-093002-Page -29 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Pauli Exclusion Principle • No two electrons can be in the same quantum state at the same time • Fundamental in understanding the structure of the periodic table 30 September 2002 BMF-Lecture 3-093002-Page -30 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Bohr’s Building up (Aufbauprinzip) Principle • Determines the basic structure of atoms • Based on the Pauli Exclusion Principle • Minimum Energy Principal 30 September 2002 BMF-Lecture 3-093002-Page -31 Copyright © 2002 Barry Fell Drexel University ECE Department ECEE 302 Electronic Devices Minimum Energy Criteria • Over-arching principle is minimum energy • explains electronic structure of rare earth elements 30 September 2002 BMF-Lecture 3-093002-Page -32 Copyright © 2002 Barry Fell Drexel University ECE Department Atomic Structure ECEE 302 Electronic Devices • Quantum Numbers – n-principal quantum number signifies the electron orbit – l-orbital quantum number signifies the angular momentum in orbit n (l=0,1,2,…,n-1) – m - magnetic quantum number signifies the projection of the angular momentum quantum number on a specific axis (z), (m=-l,-(l-1), -(l-2), …,1,0,1,…,(l-1),l) – s - electron spin signifies and internal state of the electron (s=+1/2, -1/2) • Atom is described by the set of quantum numbers that describe each electron state – – – – Hydrogen Helium Lithium Boron 30 September 2002 = 1s1 = 1s2 = 1s2,11p1 = 1s2, 1p2 X Carbon Nitrogen Neon 1s2, 1p4 Potassium 1s2,1p6,2s1 1s2,1p6,2s2 1s2, 1p6 BMF-Lecture 3-093002-Page -33 Copyright © 2002 Barry Fell