1. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -1 Copyright © 2002 Barry Fell 30 September 2002 ECEE 302: Electronic Devices Lecture 2. Physical Foundations of Solid State Physics 30 September 2002 In Preparation for Next Monday

2. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -2 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -3 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -4 Copyright © 2002 Barry Fell 30 September 2002 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=h Classical Equi-partition of Energy Model Thermal (Black Body) Energy Frequency

5. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -5 Copyright © 2002 Barry Fell 30 September 2002 Planck’s Black Body Radiation Law

6. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -6 Copyright © 2002 Barry Fell 30 September 2002 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 ( ) Einstein (1905) used Planck’s idea of bundle of light to explain this effect –(1/2) mv 2 =h -  –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  =Work Function

7. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -7 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -8 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -9 Copyright © 2002 Barry Fell 30 September 2002 Bohr’s Atom (1 of 3) e Ze

10. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -10 Copyright © 2002 Barry Fell 30 September 2002 Bohr’s Atom (2 of 3)

11. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -11 Copyright © 2002 Barry Fell 30 September 2002 Bohr’s Atom (3 of 3)

12. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -12 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -13 Copyright © 2002 Barry Fell 30 September 2002 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  =h/p localized particle properties: Energy & Momentum non-local wave properties: frequency & wavelength Characteristicsmass energymomentum Wave (light) 0 E=h p=hk=h/ =E/c Particles (~electron) mE=h p=hk=h/ =(2mE) 1/2

14. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -14 Copyright © 2002 Barry Fell 30 September 2002 Wave Mechanics (1 of 2) Schrodinger used classical geometrical optics to formulate a new mechanics he called wave mechanics

15. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -15 Copyright © 2002 Barry Fell 30 September 2002 Wave Mechanics (2 of 2) The Schrodinger Wave Equation

16. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -16 Copyright © 2002 Barry Fell 30 September 2002 Physical Interpretation of  Schrodinger initially believed that  was a guiding wave Max Born introduced a probability interpretation for  The change in probability within a volume V is due to the “flow” of propability across the bounding surface A

17. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -17 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -18 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -19 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -20 Copyright © 2002 Barry Fell 30 September 2002 Applications Potential Well - (Stationary States) Potential Barrier - (Tunneling) Hydrogen Atom (quantum numbers) Electron Spin Hydrogen Molecule Bonds –Ionic Bond –Valence Bond

21. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -21 Copyright © 2002 Barry Fell 30 September 2002 Potential Well (in 1-dimension) and Bound States Schrodinger Equation Boundary Conditions Solutions Quantization of Energy Levels Uncertainty Principle

22. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -22 Copyright © 2002 Barry Fell 30 September 2002 Potential Barrier in 1 dimension (Tunneling) Schrodinger Equation Boundary Conditions Solution Uncertainty Principle

23. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -23 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -24 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -25 Copyright © 2002 Barry Fell 30 September 2002 Hydrogen Molecule Schrodinger Equation Solution Interpretation Electron Spin

26. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -26 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -27 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -28 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -29 Copyright © 2002 Barry Fell 30 September 2002 Periodic Table

30. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -30 Copyright © 2002 Barry Fell 30 September 2002 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. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -31 Copyright © 2002 Barry Fell 30 September 2002 Bohr’s Building up (Aufbauprinzip) Principle Determines the basic structure of atoms Based on the Pauli Exclusion Principle Minimum Energy Principal

32. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -32 Copyright © 2002 Barry Fell 30 September 2002 Minimum Energy Criteria Over-arching principle is minimum energy explains electronic structure of rare earth elements

33. ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -33 Copyright © 2002 Barry Fell 30 September 2002 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 = 1s1XPotassium1s2,1p6,2s1 –Helium= 1s2Carbon1s2, 1p41s2,1p6,2s2 –Lithium= 1s2,11p1Nitrogen –Boron= 1s2, 1p2 Neon1s2, 1p6