ECEE 302: Electronic Devices

1. 30 September 2002 ECEE 302: Electronic Devices In Preparation for Next Monday Lecture 2. Physical Foundations of Solid State Physics

2. Outline • 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

3. What is Quantum Mechanics? • 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

4. 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 Thermal (Black Body) Energy Frequency

5. Planck’s Black Body Radiation Law

6. 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 F=Work Function

7. The Crisis in Atomic Theory • 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

8. 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

9. Bohr’s Atom (1 of 3) e Ze

10. Bohr’s Atom (2 of 3)

11. Bohr’s Atom (3 of 3)

12. 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

13. 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 Characteristicsmassenergymomentum Wave (light) 0 E=hn p=hk=h/l=E/c Particles (~electron) m E=hn p=hk=h/l=(2mE)1/2

14. Wave Mechanics (1 of 2) • Schrodinger used classical geometrical optics to formulate a new mechanics he called wave mechanics

15. Wave Mechanics (2 of 2)The Schrodinger Wave Equation

16. Physical Interpretation of y • Schrodinger initially believed that Y was a guiding wave • Max Born introduced a probability interpretation for Y The change in probability within a volume V is due to the “flow” of propability across the bounding surface A

17. Solutions of the Schrodinger Equation • 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

18. 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

19. Normalization of the Wave Function • 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

20. Applications • Potential Well - (Stationary States) • Potential Barrier - (Tunneling) • Hydrogen Atom (quantum numbers) • Electron Spin • Hydrogen Molecule • Bonds • Ionic Bond • Valence Bond

21. Potential Well (in 1-dimension) and Bound States • Schrodinger Equation • Boundary Conditions • Solutions • Quantization of Energy Levels • Uncertainty Principle

22. Potential Barrier in 1 dimension (Tunneling) • Schrodinger Equation • Boundary Conditions • Solution • Uncertainty Principle

23. 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

24. 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

25. Hydrogen Molecule • Schrodinger Equation • Solution • Interpretation • Electron Spin

26. 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

27. 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)

28. 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

29. Periodic Table

30. 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

31. Bohr’s Building up (Aufbauprinzip) Principle • Determines the basic structure of atoms • Based on the Pauli Exclusion Principle • Minimum Energy Principal

32. Minimum Energy Criteria • Over-arching principle is minimum energy • explains electronic structure of rare earth elements

33. Atomic Structure • 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 = 1s1 X Potassium 1s2,1p6,2s1 • Helium = 1s2 Carbon 1s2, 1p4 1s2,1p6,2s2 • Lithium = 1s2,11p1 Nitrogen • Boron = 1s2, 1p2 Neon 1s2, 1p6