ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 3-093002-Page -1 Copyright...

30 September 2002

ECEE 302 ElectronicDevices

Drexel UniversityECE Department

BMF-Lecture 3-093002-Page -1Copyright © 2002 Barry Fell

ECEE 302: Electronic Devices

Lecture 2. Physical Foundations of Solid State Physics

30 September 2002

In Prepara

tion fo

r

Next M

onday

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

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

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(BlackBody)Energy

Frequency

Hertzper volumeunit per

EnergyRadiant spectral Body Black U1e

h

c

8U

Radiation Body Blackfor Law sPlanck'

kT

h3

2

30 September 2002




Planck’s Black Body Radiation Law

Law) s Wien'-Factor (Boltzmann ehc

8

e

h

c

8

1e

h

c

8U

kThConsider

limit) (classical kTc

8

kThh

c

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1kTh

1

h

c

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1e

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kThConsider

Kelvin degrees inRadiator Body Black the of eTemperatur T

Kelvin degreeper energy K/erg101.38 constant sBoltzmann' k

action) of(unit secerg10 6.62constant sPlanck' h

light of velocity c

radiation of frequency

Hertzper volumeunit per EnergyRadiant spectral Body Black U1e

h

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Radiation Body Blackfor Law sPlanck'Consider

kT

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2

kT

h3

2

3

2

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2

3

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2

-16

27-

kT

h3

2

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) mv2=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

hmv2

1E 2

kinetic

2kinetic mv

2

1E

h

metal the of

Function Work

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

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

30 September 2002




Bohr’s Atom (1 of 3)

2

2

20

22

r2

nhvm

andr2

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and

Atomthe of Momentum

AngularOrbital the on condition quantum the invoked have we Where

2

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30 September 2002





nh

2

4

mZe

nh

2

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mZe

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nhv

andr2

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Hence

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nh

mZe

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and

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r

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and

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vm

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0

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n

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2

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222

30 September 2002





22220

42

m,n

m,n22220

42

mn

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1KE

Orbit the in Electon of )(KE Energy Kinetic

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

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

Characteristics mass energy momentum

Wave (light) 0 E=h p=hk=h/=E/c

Particles (~electron) m E=h p=hk=h/=(2mE)1/2

30 September 2002




Wave Mechanics (1 of 2)

• Schrodinger used classical geometrical optics to formulate a new mechanics he called wave mechanics

x

tx,jtx,p and ,

t

tx,jtx,Ethat Note

eetx, Transform

h

p22k ,k

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relations h/p Broglie de ),h(E Einstein the of use Make

etx, function, wave"" a withStart

xp

tE

jkxt2j

xt2j

30 September 2002




Wave Mechanics (2 of 2)The Schrodinger Wave Equation

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find Weequation. above the intox

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from determined is equation waver Schrodinge The

2

22

2

30 September 2002




Physical Interpretation of

• Schrodinger initially believed that was a guiding wave

• Max Born introduced a probability interpretation for

current" yprobabilit" the called is which

tx,tx,gradtx,gradtx,jm2

t,xS where

0t,xSdivt

t,xP

that show can We

dtt tot time theat dx x tox

position theat is E energy of particle a

that yprobabilittx,tx,tx, tx,P

amplitude yprobabilit a is tx,

EEEE

EE

2

E

E

t

t,xP

t,xS

The change in probability within a volume V is due to the “flow” of propability across the bounding surface A

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

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

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

30 September 2002




Applications

• Potential Well - (Stationary States)

• Potential Barrier - (Tunneling)

• Hydrogen Atom (quantum numbers)

• Electron Spin

• Hydrogen Molecule

• Bonds– Ionic Bond

– Valence Bond

30 September 2002




Potential Well (in 1-dimension) and Bound States


• Boundary Conditions

• Solutions

• Quantization of Energy Levels

• Uncertainty Principle

30 September 2002




Potential Barrier in 1 dimension (Tunneling)


• Boundary Conditions

• Solution

• Uncertainty Principle

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

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

30 September 2002




Hydrogen Molecule


• Solution

• Interpretation

• Electron Spin

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

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)

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

30 September 2002




Periodic Table

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

30 September 2002




Bohr’s Building up (Aufbauprinzip) Principle

• Determines the basic structure of atoms

• Based on the Pauli Exclusion Principle

• Minimum Energy Principal

30 September 2002




Minimum Energy Criteria

• Over-arching principle is minimum energy

• explains electronic structure of rare earth elements

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 = 1s1 X Potassium 1s2,1p6,2s1– Helium = 1s2 Carbon 1s2, 1p4 1s2,1p6,2s2– Lithium = 1s2,11p1 Nitrogen– Boron = 1s2, 1p2 Neon 1s2, 1p6

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