ECE 209 Notes
Table of Contents

Materials Structure

Atomic Structure

Band Structure

Metals

Optics

Dielectrics

Thermal Properties

Magnetic Properties
Materials Structures
Crystalline, polycrystalline, and amorphous

materials can be classified into the 3 types: crystalline, polycrystalline, and amorphous
Crystalline

crystalline solid:
a solid in which atoms bond in a regular pattern to form a periodic array of atoms

long range order:
happens in a crystalline solid b/c periodicity;

means that each atom is in the same position relative to its relative

perfect order yay
Polycrystalline

longrange order exists over small distances only

has small crystal “grains” that are randomly oriented
Amorphous

no longrange order! It’s completely disordered
Crystals in microelectronics

different types of crystals should be used for different things in microelectronics

Polycrystalline silicon used for:

gate material in MOS transistors

interconnect lines

Amorphous silicon used for:

switching transistors for AMLCD displays

solar cells

crystalline used for all kinds of things

attractive because of perfect order, which:

simplifies theories

repeatable, predictable and uniform properties for material processing
Silicon structure; covalent bonding

covalent bonding: shares atoms to make a full valence shell (8 atoms for Si)

Si ends up in a tetrahedral shape due to the repulsion interactions

silicon has a diamond unit cell
Unit cell

lattice:
infinitely repeating array of geometric points in space

lattice crystal structure:
a lattice, with atoms on the lattice points

unit cell:
smallest repeating structure in the lattice crystal structure

lattice constant:
the length of the cubic unit cell  a

interatomic distance:
the distance between atoms in a unit cell (not the same as a!!)
Bragg’s Law

to measure the lattice constant of an atom, use xray diffraction

for a wave incident on a plane of atoms, reflective pattern will have bright and dark spots from constructive and destructive interference

Bragg’s Law for where bright spots appear:
$$ n\lambda = 2dsin\theta $$
 crystal characteristics (and xray diffraction) depend on the direction you are looking
Miller indices

since direction matters, we need a way of classifying it

miller indices: sets of 3 numbers that are used to identify groups of crystal planes and directions
Miller Indices for Planes

set up 3 axes along 3 adjacent edges of unit cell

choose unit cell length as unit distance along respective axis (a = 1)

chose a plane that passes through the centre of particular atoms. The plan intersects the axes at distances x1,y1, za (in example below, 1,2, 2/3)

take reciprocals of interception coordinates, change to set of smallest ints, write as (hkl)
Miller Indices for Directions

take a parallel line which passes through the origin

not the length of the projections of this line on x,y,z axes

change to smallest ints

write as [hkl]
Family of Planes and Directions

family of planes: {hkl)

family of directions:

represents all equivalent planes/directions

{110} represents all planes (110), (011), (101), etc
Transmission Electron Microscopy

TEM samples thinned and illuminated with accelerated electrons

electrons are absorbed in the sample depending on thickness and material composition

intensity variation of the transmitted electron beam is observed using a viewing screen
Scanning Tunneling Microscopy

scans across the surface of sample with a very sharp needle

needle kept 1nm from surface, voltage applied between needle and sample

current used as feedback signal to determine gap size (can only give information about surface of sample)
Silicon bulk crystal growth

to make ICs, we have to grow perfect crystals on a commercial scale

for crystal growth, a saturated solution or a molten liquid is usually used.

the material is then grown on a
seed
crystal which acts as a
template
for the new growth

for silicon:

raw material: silicon dioxide

reduction => metallurgical grade polycrystalline Si

purification => electronic grade polycrystalline Si

melting & growth => crystalline bulk Si

during melting and growth, a seed crystal is pulled slowly out of a bath of molten and rotated slowly

this is Czochralski (Cz) crystal pulling

after growth, ingot is trimmed and sliced into wafers
Epitaxial growth

electrical properties of Si are controlled roughly when the basic wafers are produced, but more precise cotrol is needed for reliable devices

the top few microns of the wafer are where devices are made

this region must be well controlled

achieved by growing a even more perfect crystal layer on top of the wafer

called the
epitaxial layer

during epitaxial growth, the surface of the wafer acts as the template

decomposes Sicontaining gases in chemical vapor dposition

monolayer
: a layer one atom thick
Molecular beam epitaxy

a technique for growing thin layers

a steam of neutral atoms or molecules are evaporated from a heated cell

then incorporated into the growing film onto a heated target

is
$$$
$ b/c:

only one wafer can be used at a time

wafer must be small to get uniform layer

very good vacuum required
Epitaxy principles

major feature of epitaxy is that the newly deposited film precisely follows the crystalline form of the substrate template

adsorption: proces of atoms attaching themselves to the surface

adsorption can occur anywhere on the surface with equal probability, so layer is unlikely to be crystalline

for crystalline growth, absorbed atoms must be able to find the
minimum energy positions

the adatoms must be able to move along the surface

heat up the substrate during epitaxy to allow movement

if not hot enough, the adatoms stick where they land and the film is amorphous
Modifying Crystals

to make an electronic device, crystals need to be modified and shaped according to the needs of the device

examples: introducing impurities, etching/shaping
Ion implantation

ionized impurities are accelerated into an electric field and “smashed into” Si surface

depth of penetration determined by:

accelerating voltage

masses of ions and target atoms

crystal direction (density of atoms)
Etching and micromachining

sometimes you need to etch crystals to get certain structures
e.g. for making DRAMs, you need to etch deep tranches to make trenchcapacitors

wetetching: uses liquid chemicals to remove materials from a wafer

isotropic etching: chemicals etch at the same rate in all directions

anisotropic etching: chemicals selectively etch one crystal plane more

see example below. KOH etches (100) faster than (111) planes
Atomic Structure
Nature of light

classic physics: light is an electromagnetic wave w/ perpendicular field Bx and Ey

electric field is given by the following equation:
$$ E_y = E_o sin(kx  \omega t) $$

where:

k = 2π/λ  the wavenumber (λ is the wavelength of light)

ω = 2πf  the angular frequency (f is the freq of light)

c= ω/k = fλ  speed of light / wave velocity

light intensity is given by:
$$ I = \frac{1}{2} c \epsilon_o E_o^2 $$
Experimental Evidence of Light as EM Wave

interference and diffraction

Young’s doubleslit experiment
Photoemission (photoelectric effect)

when a metal electrode is illuminated with light, it emits electron (can create a current with this!)

light must possess the energy needed to “free” the electron from the metal (W)

any excess energy it possesses will become the kinetic energy of the electron

according to classical theory of light, the energy balance should be: E
_{
L
}
= W + E
_{
K
}

if we reduce E
_{
L
}
by reducing the intensity of the light, E
_{
K
}
should also decrease

if light intensity is increased > saturation current increases

more electrons emiited

same voltage is required to stop the current, thus the kinetic energy of the electrons is the same

classical theory of light can’t explain this!
Photons

light contains particle of fixed energy called
photons

light frequency increases > energy of light increases

E
_{
L
}
= hf
WaveParticle Duality

light has properties of both a wave and a particle

light waves consist of a stream of photon particles, each with energy hf

energy carried by the wave consists of
discrete lumps
or
quanta
De Broglie relationship

electrons also have a waveparticle duality

particlelike properties confirm with cathode ray tube (1897)

deBroglie predicted that electrons would have a wavelength:
$$ \lambda = \frac{h}{p} $$

where p = mv is the electron momentum

confirmed with electron diffraction experiment
Electron Diffraction Experiment

voltage accelerates electron, strikes a thin carbon layer, hits the screen

produce a glow of light proportional to their number and energy

using this, get the deBroglie relationship
Wavefunction, wave vector, and Schrödinger equation

for an electromagnetic wave:
$$ E_y (x,t) = E_o sin(kx  \omega t) $$

for an electron wave:
$$ \psi = A sin(kx  \omega t) $$
$$ \psi = Ae^{j(kx  \omega t)} $$

where:

k = 2π/λ  the wave vector

ω = 2πf

A = constant

can separate the timedependent and spacedependent parts and write:
$$ \psi = Ae^{jkx}e^{j\omega t}$$

wave function related to the
probability
of finding the electron at a given point in space and time

represents the distribution of the electron wave in time

Probability = ΨΨ* = Ψ
^{
2
}

probability is a real value
Wave Vector and Potential Energy

electron wave momentum is related to the wavelength by this equation: p = h/v

momentum is a vector  therefore we need a vector form of the wavelength

wave vector:

direction: direction of wave travel

magnitude: k = 2π\λ

momentum now written as:
$$ p = \frac{h}{2π}k$$

kinetic energy is:
$$ E_k = \frac{p^2}{2m} = \frac{h^2}{8π^2}\frac{k^2}{m}$$

electron also has electrostatic potential energy

defined as the work done in pulling the negatively chargely electron from an infinite distance to a distance, r, from the positively charged nucleus:
$$ E_p = \frac{e^2}{4πε_0 r} $$
total energy is E = E
_{
k
}
+ E
_{
p
}
$$ k = \frac{2π}{h} \sqrt{2m(E  E_p)} $$
Schrodinger’s Equation

describes the electron wave function

if you know the electron potential energy and boundary conditions, you can calculate the parameters of electron orbital (wave function and energy)
$$ \frac{d^2 \Psi}{dx^2} + \frac{8\pi^2 m}{h^2}(E  E_p )\Psi = 0 $$
“Electron in a box” problem
Electron in a 1D Potential Well
setup:
 inside the box, potential energy is zero
 outside, is infinitely large
 we need to find the wave equation
$$ \frac{d^2 \Psi}{dx^2} + \frac{8\pi^2 m}{h^2}(E  V)\Psi = 0 $$

assumptions:

the case is timeindependent

wave function is continuous, smooth, and single

Boundary conditions:

For x<0 and x>L, the term Vψ dominates
$$ V\Psi = 0 \\
\Psi = 0
\Psi^2 = 0
$$
electron cannot be outside the well

Since d
^{
2
}
ψ/dx
^{
2
}
must be continuous, ψ = 0 at x={0,L}

Differential equation:

second order differential equation to solve for within the well

general solution equation is:
$$ \Psi (x) = 2Ajsin(\frac{n\pi x}{L}) $$

can solve for A because we know the probability of electron being in the box is 1 (integral of equation from 0 to L is 1)

final form of the equation is:
$$ \Psi (x) = j(\frac{2}{L})^{\frac{1}{2}}sin(\frac{n\pi x}{L}) $$
Electron Energy in Potential Well
$$ E = \frac{h^2 n^2}{8mL^2} $$
 energies E(n) are the eigenenergies of the electron
 energy is quantized
 n is the
quantum number
 min energy is at n=1, this is the
ground state
 energy of electron wave can only have discrete values
 energy of electron particle can take any value
Uncertainty Principle

free electron:

has single energy, momentum, wavelength  Δp = 0 (uncertainty 0)

electron wave is spread all over the space, so Δx = ∞

electron in a potential well:

Δx = L

Δp = hk/π

for n=1, k: = π, Δp = h/L
$$ \Delta x \Delta p = L \frac{h}{L} = h $$

Heinsenberg’s uncertainty principle: we cannot simultaneously and exactly know both the position and momentum of an electron along a given coordinate
Tunnelling

important application of the uncertainty principle

if an electron of energy E meets a potential energy barrier of height V_o_ > E, it might leak (“tunnel”) throug the barrier

probability of that depends on the energy and width of the barrier
Hydrogen atom

consider the H atom: an electron attached to a nucleus

electron is electrostatically bound to a single proton

since proton is so big, it behaves more like a particle

potential energy:
$$ V(r) = \frac{Ze^2}{4\pi \epsilon_o r} $$

where:

Z = number of electrons

r = (x
^{
2
}
+ y
^{
2
}
2 + z
^{
2
}
)
^{
1/2
}

considerng electron in H atom as confined in a potential well with PE V(r), electron’s wave function can be derived to be:
$$ E = \frac{Z^2 e^4 m}{8h^2 \epsilon_o^2}\frac{1}{n^2} $$

different energy values (different values on) are called
Energy Levels
Atomic Spectra

electrons can be excited into higher energy levels  requires energy

they can also return to a lower levels  releases energy in the form of a photon with appropriate energy E = hf = E
_{
higher
}
 E
_{
lower
}
Quantum Numbers
Principal Quantum Number, n

determines the radius of electron orbit and the energy level
Orbital Angular Quantum Number, l

determines the shape of the orbital

the electron wave at each orbit (at each r) may be standing or moving along the orbit

wave must be continuous and smoothly varying

must fit an integral number of wavelengths: lλ = 2πr
$$ L = pr = \ell \frac{h}{2\pi} $$

L is the angular momentum, which is quantized.

l can take any value from 0 to (n1)
Magnetic Quantum Number, m
_{
l
}

determines orientation of the orbital in space (the tilt of the electron cloud), and the energy of its electron in a magnetic field

angular momentum about the electron orbit is quantized as:
$$ L_z = \frac{m_\ell h}{2\pi} $$

l ≤ m
_{
l
}
≤ l
Electron Spin Quantum Number, m
_{
s
}

determines the rotation of electron about its own axis

has the values 1/2, 1/2 (spin up, spin down)
The Full Sert of Quantum Numbers and Values
n = 1, 2, 3, … z
l = 0, 1, 2, 3, … (z1)
m
_{
l
}
= 0, ±1, ±2, ±3, … ±(z1)
m
_{
s
}
= ±1/2
Summary of Quantum Numbers

Radius of orbit → n

Orbital angular momentum → l

Tilt of orbit’s plane → m
_{
l
}

Spin of electron → m
_{
s
}
Electron Clouds

we can define electron clouds corresponding to different combos of quantum numbers

probability density distribution
Multielectron atoms, Pauli principle, and the periodic table
Band Structure
Hydrogen molecule and molecular bonding

when atoms interact, they change behaviour

no two electrons in an interacting systems may occupy same quantum state

consider the case of two H atoms

when they are infinitely far apart, they have the same wave function

when they approach each other, their wave functions overlap and two new
molecular wave functions
emerge (see the image below for the two new functions

molecular wave functions are linear combinations of atomic orbitals; in this case, one is the sum and one is the difference

Ψ
_{
σ
}
is more confined to the nuclei, whereas Ψ
_{
σ*
}
is more spread

thus, Ψ
_{
σ*
}
has higher energy

Ψ
_{
σ
}
then is more energetically favourable, so both electrons occupy this state

bonding orbital:
the wave function Ψ
_{
σ
}
corresponding to the lowest energy level

antibonding orbital:
Ψ
_{
σ*
}

total energy of two electrons in H
_{
2
}
molecule is lower than in two single H atoms

one electron has to flip its electron spin but the energy gain due to dropping to bonding orbital is higher than the energy spent

consider 3 hydrogen atoms. They will also add their atomic wave functions, like so:

the more atoms in our function, the more molecular orbitals they’ll form. n atoms = n orbitals

if an energy band is not entirely filled, there are states available for electrons. Consider N Li atoms (2s half filled)

thermal energy is enough at room temp for electrons to jump between nearest energy levels

since the levels may belong to different atoms, electrons can easily travel from atom to atom
conducting current
Hybridization

2s and 2p energy levels are close, so when they approach each other, 2 2s and 2 2p orbitals can mix to form hybrid orbitals

hybrid orbitals directed in tetrahedral directions and have the same energy

process called
sp
^{
3
}
hybridization
Energy Bands

when interatomic distance decreases so that electrons interact, their energy levels broadens (splits) into bands

there are 2N states in the 2sband, 6N states in 2pband

in diamond crystal, bonding and antibonding orbitals split and form
valence band
and
conduction band
, respectively

band gap  E
_{
G
}
 the difference in energy between the conduction and valence bands
Fermi Energy

at T=0K, all electrons will occupy states with lowest energy (valence band), so conduction band empty

fermi energy (E
_{
F
}
) = energy level corresponding to highest filled electron state at 0K.

as T increases, bands above E
_{
F
}
start to get filled

to conduct electric current, there must be vacant states in the band

no states available in energy levels within each band, no conduction
need more but confusing tho
Effective mass

acceleration of an free electron in vacuum is a = F
_{
ext
}
/ m
_{
e
}
, m
_{
e
}
= electron mass in vacuum

in a solid, electron interacts with crystal lattice atoms and experiences internal forces F
_{
int
}

thus, acceleration is: a = (F
_{
ext
}
+ F
_{
int
}
) / m
_{
e
}

since atoms in a crystalline solid are periodically positioned, variation of F
_{
int
}
is also periodic, we can simplify our acceleration equation:
$$ a_crystal = \frac{F_ext}{m_e^*}

where m
_{
e
}
^{
2
}
is the
effective mass
of the electron

effective mass depends on the material
Electrons and holes

in semiconductors, in order to get excited to empty states, electrons jump across the band gap

when excited to the conduction band, a vacant state is left in the valence band

this is called a
hole
 the absence of an electron

electrical conduction in a semiconductor involves movement of electrons in the conduction band and holes in valence band

electron and hole currents
Intrinsic Semiconductor

a pure semiconductor (no foreign atoms present) is an intrinsic semiconductor

electrons and holes can only be created in pairs (electronhole pairs)
Carrier Generation

electronhole pair generation: the act of exciting an electron from the valence band to the conduction band

electrons can be excited even though E
_{
T
}
is much smaller than E
_{
G
}
because atoms in the crystal are constantly vibrating (due to thermal energy) and
deforming interatomic bonds

thus, some bonds may be overstretched, and the bond energy can be smaller than thermal energy

electron concentration in the conduction band, n, (electrons per cm
^{
3
}
) is always equal to hole concentration in the valence band, p, (holes per cm
^{
3
}
)

n = p = n
_{
i
}

n
_{
i
}
= intrinsic carrier concentration

g
_{
i
}
: rate of generation
Recombination

opposite of carrier generation: the act of an electron falling back to VB

excess energy is released in the form of heat or light

rate of recombination, r
_{
i
}
, is proportional to equilibrium concentration of electrons/holes

r
_{
i
}
= αn
_{
0
}
p
_{
0
}
= αn
_{
i
}
^{
2
}

α = constant

n
_{
0
}
= equilibrium electron concentration

p
_{
0
}
= equilibrium hole concentration

in steady state, r
_{
i
}
= g
_{
i
}
Conduction

takes place only when electronhole pairs are created

conduction not great in intrinsic semiconductors at room temperature
Doping, extrinsic semiconductors

doping:
creation of carriers in semiconductors by introducing impurities

we get extra carriers, and better conductivity

doped semiconductor = extrinsic semiconductor

ntype semiconductor:
predominant electron concentration

ptype semiconductor:
predominant hole concentration
ntype doping

ntype Si obtained by adding small amounts of group V elements (P, As, Sb)

these elements have 5 valence electrons, but the atoms bond to Si (4 e
^{

}
), so one of the electrons is
weakly
bonded to the impurity atom

very tiny amount of energy needed to excite electrons, so at most temperatures most of the donor electrons will be ionized
ptype doping

ptype Si obtained by adding small amount of group III elements (B, Al, Ga, In)

these elements have 3 valence electrons, atoms bond to Si (4 e
^{

}
), one of the bonds will miss an electron

impurity atoms = acceptors (accept an extra electron)
Carrier concentration

how to calculate the number of electrons and holes available for conduction? need to know:

number of states available at a particular energy to be occupied

fraction of these states that are in fact occupied at a particular temperature
$$ n_o = \int_{E_c}^\infty \! f(E)N(E) \, \mathrm{d}E. $$

where:

N(E)  density of states

f(E)  Fermi function
Density of States

DOS: number of available states per unit volume

expressions for valence and conduction band are:
$$ N_V(E) = \frac{8\sqrt{2}\pi}{h^3}(m_p^*)^{\frac{3}{2}}(E_V  E)^{\frac{1}{2}} for E < E_V \\
N_V(E) = \frac{8\sqrt{2}\pi}{h^3}(m_n^*)^{\frac{3}{2}}(E  E_C)^{\frac{1}{2}} for E > E_C
$$
Fermi function
$$ f(E) = \frac{1}{1 + e^{\frac{E  E_F}{kT}}} $$
 FermiDirac distribution function gives us the probability that an available energy state at E will be occupied by an electron at temperature T

probability that an available energy state will be occupied by a hole is 1  f(E)
 at E=E
_{
F
}
, f(E) = 1/2
 E
_{
F
}
in intrinsic material: middle of band gap b/c concentration of holes in VB = concentration of electrons in CB
 E
_{
F
}
in ntype material: closer to E
_{
C
}
because the concentration of electrons in CB is higher than concentration of holes in VB
 E
_{
F
}
in ptype material: close to E
_{
V
}
because concentration of holes in VB is higher than concentration of electrons in CB
Equilibrium Carrier Concentration

for equilibrium conditions, can use the
effective density of states
* N
_{
C
}
at energy E_C. Thus:
$$ n_0 = N_C f(E_C) $$

Then, f(E
_{
C
}
) can be expressed as:
$$ f(E_C) = \frac{1}{1 + e^{\frac{E_C  E_F}{kT}}} = e^{\frac{E_C  E_F}{kT}} $$
$$n_0 = N_C e^{\frac{E_C  E_F}{kT}} $$

where N
_{
C
}
is a constant

similarly, concentration of holes is:
$$ p_0 = N_V [ 1  f(E_V) ] $$

where N
_{
V
}
is the effective density of states in the valence band

$$ p_0 = N_V e^{\frac{E_F  E_V}{kT}} $$

where N
_{
V
}
is a constant
Mass Action Law
$$ n_0 p_0 = n_i^2 \\
n_0 = n_i e^{\frac{E_F  E_i}{kT}} \\
p_0 = n_i e^{\frac{E_i  E_F}{kT}} $$
Conductivity and mobility

current of electrons and holes depends on:

carrier concentration (n, p)

carrier speed (v
_{
n
}
, v
_{
p
}
)

carrier charge (q or e)

current density can be written as:
$$ J_n = nev_n \\
J_p = pqv_p $$

at low electric field, the carrier velocity is proportional to the field: υ = με

the proportionality constant μ is called the
mobility

total current density is: J = σε

ε is called the
conductivity
Hall Effect

mobility in semiconductors can be estimated using the Hall effect

if we apply electric field E
_{
x
}
in direction x across a semiconductor and submit it to magnetic field B
_{
z
}
in direction z, then another electric field E
_{
y
}
(Hall field) occurs perpendicular to both E
_{
x
}
and B
_{
z
}

E
_{
y
}
occurs due to deflection of electrons from direction z due to
Lorentz force
F
_{
y
}
= ev
_{
x
}
B

electron velocity in xdirection: v
_{
x
}
= μ
_{
x
}
E
_{
x
}

in steady state, deflection is steady and Hall field counterbalances Lorentz force:

eE
_{
H
}
= ev
_{
x
}
B
_{
z
}

eE
_{
H
}
= J
_{
x
}
B
_{
z
}
/n

E
_{
H
}
/J
_{
x
}
B
_{
z
}
= 1/en = R
_{
H
}
 Hall coefficient

μ =  σR
_{
H
}
  Hall mobility
HaynesShockley Experiment

direct way of measuring mobility
Temperature dependence of carrier concentration
$$ n_i (T) = 2{\frac{2\pi kT}{h^2}}^{\frac{3}{2}}{m_n^* m_p^*}^{\frac{3}{4}}e^{\frac{E_G}{2kT}} $$
Compensation doping

semiconductor could have both acceptors and donors in it: this is compensation doping

the concentrations of electrons, holes, donors and acceptors can be obtained from
space charge neutrality law

the material must remain electrical neutral overall

p
_{
0
}
+ N
_{
d
}
^{
+
}
= n
_{
0
}
+ N
_{
a
}
^{

}

a material doped equally with donors and acceptors becomes “intrinsic” again
Diffusion Current

diffusion: net motion of carriers from regions of high carrier concentration to low carrier concentration if there is nonuniformity (gradient) of carrier concentration
need more
Direction and indirect bandgap semiconductors

dielectrics and semiconductors behave essentially the same way  the only difference is the
bandgap width

photons with energy exceeding E
_{
g
}
are absorbed by giving their energy to electronhole pairs

may or may not reemit the light during recombination depending on whether the gap is
direct
or
indirect

direct bandgap
semiconductors: electron drops from bottom of CB to top of VB, excess energy emitted as a photon

also known as
radiative recombination

indirect bandgap
semiconductors: recombination occurs in two stages via recombination centres (usually defects) in the bandgap:

electron falls from bottom of CB to the defect level, then down to the top of VB

electron energy is therefore lost in two portions by the emission of
phonons
(lattice vibrations)

this process is also known as
nonradiative recombination
Photoconductivity

increase of conductivity under illumination
$$ \Delta \sigma = \sigma_photo  \sigma_dark = \frac{e\eta I\lamda \tau (\mu_e + \mu_h)}{hcD}

η is quantum efficiency, and τ is average excess carrier lifetime
Energyband diagrams and MOSFET

no current from source to drain b/c diodes

channel is conductive because gate electrode is used

there is an insulator between metal and semiconductor, electric field builds across oxide layer if V is applied

similar to parallel plate capacitor
MOSFET Operation Modes

accumulation mode

negative voltage at the gate increases number of holes at interface

depletion mode

small positive voltage repels holes

inversion mode

large positive voltage attracts electrons to the interface, making it locally ntype
MOSFET Band Diagrams

to build energy band diagrams:

choose zero points on coordinates (for energy axis, 0 is vacuum level)

in equilibrium, E
_{
F
}
= const everywhere on band diagram xo

more devices per unit area  more metal interconnection lines

issues:

longer interconnects

higher capacitance per unit area

increasing heat (more devices per chip and higher frequency => increased heat production)

interconnects: resistance is R =ρl/A, resistance increases as width/length decrease

capacitance: metal lines end at MOSFET gate, forming RC line  potential source of slowing down the circuit speed

heat production: at junctions between metal layers, metal is thin => resistance higher, higher heat production

metal atoms have 1 to 3 valence electrons and ionize easily

electrons are shared between all atoms, so metal ions are surrounded by electrons

electrostatic forces are equal in all directions

ions positions are fixed

electrons move around freely

in a perfect metal crystal at T=0, there is
no resistance
due to the wave nature of electrons

an electron moving at constant velocity behaves as a plane wave

after interactions with wave, ions become the “sources” of secondary wavelets
$$ n\lambda = 2dsin\theta $$

in case of small interatomic distance and low electron speed
$$ \lambda > 2d \\
\frac{\lambda}{2d} = \frac{sin\theta}{n} > 1 $$

only solution is at n=0, θ=0

transmission occurs in direction of travel, magnitude unchanged

source of resistivity is either the
temperature
or
noncrystallinity
of a metal
Temperature dependence of resistivity

At T>0K, atoms move away from ideal lattice position b/c vibrations

electrons become scattered

for atoms in a gas:

mean free path:
average electron path length is defined

mean free time:
time between collisions
In the Presence of an Electric Field

when a potential difference is applied across metal, electrons drift towards larger positive potential

current density:
$$ J = nqv^d$$
Drift Velocity

in electric field, electrons experience acceleration

electron collisions with lattice ions causes velocity loss

net acceleration of electrons between collisions
$$ \frac{dv_d}{dt)_ACC = frac{q\epsilon}{m} $$

velocity loss at each collision:
$$ \frac{dv_d}{dt}_LOSS = \frac{v_d}{\tau} $$
$$ \frac{dv_d}{dt}_TOTAL = \frac{v_d}{\tau} + frac{q\epsilon}{m} = 0 $$

mobility: μ = qτ/m
Phonons

due to bonding, atom motions are connected (behave like a wave)

types of wave motion:

these waves act like
phonons

can model interaction between lattice and electron wave as interaction between electron and phonon
Structural dependence of resistivity

structural disorder gives rise to resistivity

list of imperfections in a chip includes:

impurity atoms

dislocations

grain boundaries
Superconductivity

for many elemental metals and alloys, the resistivity falls to an immeasurably small value at some point before the critical temperature (T
_{
C
}
)
The Meissner Effect

when superconducting material at temperature above T
_{
C
}
is placed in a magnetic field and then cooled down, all magnetic field lines are ejected from the material at T=T
_{
C
}
Optical Properties
Light wave propagation, Refraction Index

the velocity of the wavefront of light depends on the material in which it is travelling (because waves, yo)

in dielectric nonmagnetic material, electric field part of the wave interacts with electrons etc, polarizes atoms and molecules at the frequency of the wave

since wave propagation is coupled with dipole formation, polarization slows down propagation

rate of propagation is characterized by dielectric permittivity and magnetic permeability
$$ v = \frac{1}{\epsilon_r \epsilon_0 \nu_r \nu_0} $$
Refraction Index

refraction:
the bending of light as it passes from one material to another (due to the change in velocity)

refraction index is
n = c/v
, v = speed of light in material

is a consequence of electric polarization

when a light wave passes through a material, energy is lost to the electrons of the material

energy transferred comes from the velocity change

since polarization is frequency dependent, refractive index also depends on the wavelength of light
Dispersion

dispersion:
a general name give to effects that vary with wavelength

wavelength dependence of the refractive index is the
dispersion of the refraction index
$$ n^2 = 1 + \frac{A_1 \lambda^2}{\lambda^2  \lambda_1^2} + \frac{A_2 \lambda^2}{\lambda^2  \lambda_2^2} + ... } $$

A
_{
n
}
and λ
_{
n
}
are the
Sellmeier coefficients

since white light is a collection of multiple wavelengths, its velocity is a
group velocity
, and has a
group index
Snell’s Law, Total Internal Refraction

angles of incidence and refraction are related by Snell’s Law
$$ \frac{sin\theta_1}{sin\theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1} $$

in the case n
_{
2
}
< n
_{
1
}
, refraction angle θ
_{
2
}
exceeds 90º, the light does not exit material 1 but is
totally internally reflected

respective incidence angle is called critical angle θ
_{
c
}
$$ sin\theta_c = \frac{n_2}{n_1} $$
Optical Fibers

got that total internal reflection going on

fiber optic tech has super fast speed of data transmission

HELP
Photon interaction with materials. Absorption, Reflection, Transmission, Refraction
from conservation of light, I
_{
0
}
= I
_{
T
}
+ I
_{
A
}
+ I
_{
R
}
,
 I
_{
0
}
is the intensity of incident light, I
_{
T
}
, I
_{
A
}
, I
_{
R
}
are intensity of transmitted, absorbed, and reflected light
 three types of lightmaterial interactions:
 transmission
 absorption
 reflection
 materials divided into:
 transparent (little absorption and reflection)
 translucent (light scattered within material)
 opaque (relatively little transmission)
 if material not perfectly transparent, light intensity decreases exponentially with distance
 if the light intensity drop in δx is δI δI = α δx I
 α = absorption coefficient (m
^{
1
}
)
BougerLambertBeer’s Law
$$ ax = ln(frac{I}{I_0}) $$
 light could be absorbed by the nuclei (all materials) or by the electrons (metals and narrow E
_{
g
}
semiconductors)
Atomic Absorption

type of absorption strongly depends on the type of material that absorbs

ionically bonded solids show high absorption because oppositely charged ions move in opposite directions creating more interactions

phonons exist in bands but only one of the phonon energies is excited by the radiation, so there is only one absorption frequency

transmission spectrum shows just one dark and is called
line spectrum

absorption spectrum is dominated by the absorption due to the molecules themselves

air pollution monitoring: can fitting known spectra of various gases to the measured atmospheric spectra over the same frequency range

in metals, photons are absorbed by electrons

almost any frequency of light is absorbed

practically all light absorbed within about 100nm of metal surface; thinner metal films will partially transmit light

excited electrons in the surface layers of metal  recombine again, emitting the light

metals are both opaque and reflective

reflection can be explained in terms of electrostatics

EM field forces the free electrons to move, moving charge is source of EM waves. Therefore, wave is reflected

band structure of metals not as simple as we assumed  there can be absorption below E
_{
F
}

metals are more transparent to very high energy radiation, where the inertia of electrons is the limiting factor

dielectrics and semiconductors behave essentially the same way  only difference is the
bandgap width

photons with energy exceeding E
_{
g
}
are absorbed by giving their energy to electronhole pairs

may or may not reemit the light during the recombination, depending on whether the gap is direct or indirect

direct bandgap:
excess energy emitted as a photon

indirect bandgap:
energy is lost in two portions by the emission of
phonon
(lattice vibration)
Reflection

occurs at the interface between two materials and is therefore related to refraction index

reflectivity is the ratio of incident and reflected light intensities

assuming light is incident normally to the interface
$$ R = \frac{n_2  n_1}{n_2 + n_1}^2 $$
Transmission

reflection and absorption are wavelength dependent

transmission  a “leftover” after reflection and absorption

to get transmission spectrum, just subtract reflection and absorption spectra from the incident light spectrum
$$ I_T = I_0  I_A  I_R $$

small differences in composition may lead to large differences in appearance
Colours

Al
_{
2
}
O
_{
3
}
(sapphire) is colourless  adding 0.52.0% of Cr
_{
2
}
O
_{
3
}
turns the material red  ruby!

Cr atoms substitute Al in the crystalline lattice and introduce impurity levels in sapphire bandgap

these levels give strong absorption at 400nm (violet) and 600nm (orange), leaving only red light to go through

similar technique is used to colour glasses by adding impurities while in the molten state
Emission. Luminescene and Fluroscence

luminescence:
general term which describes the reemission of previously absorbed radiative energy

common types: photo, electro, cathodoluminescence

depends on source of incident radiation: light,(fluorescent light) electric field (LED), or electrons (CRT)

also chemoluminescence due to chemical reactions (which makes glow sticks)

luminescence is further divided into
phosphorescence
and
fluorescence

fluroescence: electron transitions that require no change of spin

phosphorescence: electron transitions that require a change of spin

hence, fluroescence is faster!
Luminescence

if the energy levels are actually a range of energies, after electron excitation we observe a series of transitions accompanied by phonon emission, and then fluorescent transition

since part of electron energy is released as phonons, then the light emitted by fluorescence is of longer wavelength than incident light
Lasers

LASER: Light Amplification by the Stimulated Emission of Radiation

before, we considered spontaneous light emission, which happened due to randomly occurring effects

stimulated emission refers to electron transmissions that are stimulated by the presence of other potons

an incident photon with E >= E
_{
g
}
is equally likely to cause stimulated emission of another photon as be absorbed

emitted photon has the same energy and phase as the incident photon (i.e. they are coherent)

normally we have less electrons in the excited state than ground state

if we somehow get more electrons in the excited state than ground state, than we get stimulated emission > much more photons in the output than in the input > we get
amplification

population inversion:
when we have more electrons in the excited state than in the ground state

since random spontaneous emission gives incoherent output, it should be minimized in LASERs

done by using transmissions from which spontaneous emission is less likely; transmission from
metastable
states

common material for solid state lasers is ruby, sapphire with Cr impurities

to make laser, we have to achieve

population inversion

enough photons to stimulate emission

first condition is met by filling the metastable states with electrons using a zenon flash lamp (in ruby laser) or by electron injection (in semiconductor laser)

second condition is achieved by making laser in the rod shape. By mirroring the ends of the rod, we let photons travel back and forth along the rod

in order to keep the coherent emission, we must ensure that the light completes the round trip between the mirrors and returns in phase with itself

in order to produce coherent output, the distance between the rod ends must obey the relationship:
nλ= 2L

in semiconductor lasers, thin films of direct semiconductors are epitaxially deposited on top of each other

central layer is degenerately doped (the doping is so heavy that E
_{
F
}
< E
_{
V
}
and there are lots of empty electron states in the valence band)

under bias, electrons are injected from nlayer into central later and get trapped there

thus, population inversion in central layer
Dielectric Materials

dielectrics increase the capacitance between parallel plates
$$ C + \frac{\epsilon_o \epsilon_r A}{d}

increased capacitance allows more charge storage

why do we care about studying dielectric materials?

electron devices become smaller! A is always decreasing as the devices shrink; d cannot decrease indefinitely as bellow 510nm, tunnelling occurs. Therefore you get less charged stored as devices shrink.

but you cannot increase V too much as breakdown will occur. So the only way to increase C is to the increase the relative permittivity, which depends on the dielectric

types of dielectric materials:

MOSFETs: charge has to be accumulated at the semiconductor/gate dielectric interface

DRAM: charge is injected and retained in a capacitor with a MOSFET switch

CCD cameras: charged generated by photons is retained in the capactiros and read out by charge transfer between them

LCD: each pixel is a capacitor; charge stored generates Efield across liquid crystal layer and causes LC molecules to rotate
Surface Charge and Surface Electricity

dielectrics are insulators; do no conduct electric current at room temp

electric charge deposited on the dielectric surface cannot move and stays on the surface (e.g. being attached to surface defects)

dielectrics are therefore efficient in storing electrostatic charge!
Surface Charging

electron interactions at surfaces are complex and still poorly understood

all materials have “surface states” caused by the incomplete bonds

strongly affect the behaviour of many electronic devices

all ICs are covered with protective layer (‘passivation’) to prevent contamination by water vapour, etc.

many materials that bond covalently or ionically charge positively when rubbed

friction removes unbonded electrons

however, polymerbased materials tend to charge negatively:

consider a material like paraffin. The sidearms of molecular structure tend to attract water. Friction removes the H+ ions, leaving the OH behind
Coulomb’s Law

summary of electrostatics:

like charges repel, opposites attract

the force between charges:

inversely proportional to distance
^{
2
}

dependent on surrounding medium

acts along a line joining charges

proportional to each charge

Coulomb’s Law:
$$ F = \frac{q_1 q_2}{4\pi \epsilon r^2} u $$

in dielectrics, ideally no electrons available in the conduction band

in reality, some electrons are present caused by random phenomenon (e.g. UV rays, cosmic rays)

will be a small leakage current
Capacitance Effects: Fringing Fields

simple expression for capacitance has ignored several factors  most important being
Edge effects

electric field at the edges of a capacitor is nonuniform

edge effects are important also when considering the RC time constants of the IC interconnects (higher RC values limit the speed of the ICs)

fringing fields increase the effective area of the capacitor, leading to significant error

in ICs, also have to consider
capacitive coupling
between metals in different layers causing
crosstalk
Polarization and Relative Permittivity
What Happens: Dielectric Between Plates of a Capacitor

more general definition of electric field: E
_{
x
}
= dV/dx = ∇V

when a dielectric is inserted into parallel plate capacitor, additional charge is being stored on the plates

relative permittivity increases as a result

increase in the stored charge is due to polarization of the dielectric in the electric field

atoms and molecules become
polarized
when they are subjected to an electric field, and form electric dipoles

polarization occurs when voltage is applied to capacitor  first electron’s worth of charged is induced on the plates

the charge causes the dielectric to polarize, and therefore does not contribute to building up the potential difference across the capacitor

this same thing happens to the next electron’s worth of charge > until all atoms in the dielectric are polarized

only then due charge induced on capacitor start to contribute to build up the potential difference across the capacitor

therefore more charge has to flow in before capacitor charged up to the supply voltage

charge storage capacity has increased

dielectric has increased its capacitance
Polarization

in the presence of electric field, centres of each charge become slightly misplace and the particles become polarized  electric dipoles

polarization is equal to the bound charge per unit area of the dielectric surface; measured in Coulombs/m
^{
2
}

dipole moment:
the (absolute) charge on each of the two dipoles separated by their distance

consider P bound charges per unit area on opposite sides of a cube with side l, A = l
^{
2
}

oppositely direct dipoles inside the cube cancel out each other; the only uncompensated charge is next to the surfaces of the cube

total dipole moment is therefore: μ= P
A
l

P is
electric dipole moment per unit volume

INSERT MORE HERE 2324
Polarization Mechanisms

dipole moment also depends on the electric field
within
the material

remember that zero field gives no polarization, so μ must be field dependent

μ= αE
_{
int
}

α =
polarizability
of the material

the average dipole moment per unit of internal field

Clausius Equation:

P = (ε
_{
r
}
 1)ε
_{
o
}
E = N αE
_{
int
}

several mechanisms that contribute to α: α = α
_{
e
}
+ α
_{
a
}
+ α
_{
d
}
+ α
_{
i
}

α
_{
e
}
: electrical polarizability

α
_{
a
}
, α
_{
d
}
: molecular polarizability

α
_{
i
}
: interfacial polarizability
Electrical Polarizability

also called optical polarizability because the polarization can keep up with even optical frequencies

with no electric field, electron clouds are symmetric around the nucleus

when field is applied, electron cloud is distorted

the centers of the negative and positive charges are now offset, leading to an electric dipole μ
_{
e
}
= α
_{
e
}
E
_{
int
}
Molecular Polarizability

arises when the molecules of the material naturally forms dipoles (e.g. H
_{
2
}
O)

two things can happen when an electric field is applied:

atomic polarizability α
_{
a
}

orientational polarizability α
_{
d
}

no α
_{
d
}
in ionically bonded solids because strong bonding forces prevent wholesale realignment of molecules

however, small change in “centre of mass” of the bonds so a small α
_{
a
}
can be present
Interfacial Polarizability

accounts for the presence of lattice imperfections, ionized contaminants, a few electrons etc.

in an electric field, some or all of these can move through the material until they come to an interface
Frequency Dependence of Polarization

how do we distinguish the effect from different types of polarization?

if you put a dipole in an alternating field, the dipole will attempt to follow the oscillating field

but the dipole has inertia, so it takes a finite time to respond to field

if we oscillate the field fast enough, the dipole will eventually cease to respond fast enough

relaxation frequency:
the frequency at which the dipoles cannot move at all before the field reverses direction (hence the polarization “goes away”)

relaxation occurs at different frequencies for different mechanisms of polarization

relaxation frequency higher when switching smaller masses of material (less inertia)

so if we plot the permittivity as a function of frequency, should find changes at each relaxation frequency

electronic: only electrons must be moved, they’re very light so relaxation occurs at high frequencies

atomic: relies on ions moving their positions so freq should be about the same as thermal oscillations of the atoms

orientational: requires reorganization of groups of dipoles  freq lower than atomic

interfacial: caused by charge that percolates slowly through entire thickness of material; very low freq
Classification of Dielectrics

classify dielectrics into three categories:

nonpolar materials:
show variations of permittivity in the optical range of frequencies only

polar materials:
display both atomic and electric polarizability

dipolar materials:
display atomic, electric, and orientational polarizability
AC Permittivity and Dielectric Loss

dielectrics may have different response time in case of AC voltage

current leads the voltage when voltage applied to capacitor

in a real capacitor, there will be a small leakage current through the dielectric, which will be
in phase with V

thus, real capacitor consists of a capacitive component connected in parallel with a resistive component

since the impedance now consists of real and imaginary components of 0
^{
o
}
and 90
^{
o
}
phase, total phase difference is slightly less than 90
^{
o
}
, by δ
^{
o
}

δ is the
loss angle

mathematically, this is accounted for by defining the permittivity to be a complex number:

ε
_{
r
}
= ε
^{
’
}
 jε
^{
’‘
}

loss angle is therefore tanδ = ε
^{
’‘
}
/ε
^{
’
}

since power dissipated in the capacitor is proportional to the value of ε
^{
’‘
}
, dielectrics want as small a δ as possible

capacitive component of the capacitance has the same frequency dependence as the polarizability

resistive component of the capacitance has several maxima at the frequencies at which capacitive components of the capacitance cease to respond

these frequencies (relaxation peaks) should be avoided due to power dissipation
Gauss’ Law and Boundary Conditions

what if we have nonuniform dielectric between the plates of the capacitor?

electric flux density, D, in the material is given by:
D = εE

from Coulomb’s law, we get:
$$ D = \frac{q}{4\pi r^2} a $$

this means D is independent of ε for fixed q

the total flux crossing the sphere area is given by D*Area

this leads to:
$$ Flux = 4\pi r^2 \frac{q}{4\pi r^2} = q $$

total flux out of the surface is equal to the enclosed charge; the generalization of this to all closed surfaces is
Gauss’ Law
Dielectric Breakdown

we cannot apply an infinitely large voltage across our capacitor without it breaking down

dielectric breakdown usually evidenced by a sudden increase in current to a very large value

breakdown voltage (V
_{
BD
}
):
voltage that causes dielectric breakdown

dielectric strength (E
_{
BR
}
):
maximum electric field that can be applied to a dielectric without a breakdown

in real insulators, breakdown voltage can be hard to predict since it depends on surface and ambient conditions
Breakdown Mechanisms

avalanche breakdown:
occurs if the electric field across the insulator is high enough

the few electrons present can achieve enough energy to ionize other atoms (impact ionization)

secondary electrons are also accelerated, causing further ionization and avalanche develops

thermal breakdown:
occurs if the leakage current (loss angle) is large enough to cause significant heating > more leakage > more heating

discharge breakdown:
occurs is small gas bubbles are present in the material
Capacitors and Memories

main applications of dielectrics in electronics: capacitors and memory cells

capacitors:
can be made as discrete elements or be integrated with other elements on the same wafer

discrete caps made from alternating layers of metal and dielectric, wrapped up in a package

dielectric material is chosen to get the right range of values with the minimum loss angle
RAM

same problem of fitting capacitor into a small space occurs with RAM cells

in RAM, SiO
_{
2
}
is used as the dielectric

data is stored as the charge in a capacitor cell with a MOSFET as a switch

as the devices shrink, the challenge is how to maximize the charge storage and minimize the cell area

reducing cell area is important for increased storage densities

chip manufacturers make the capacitors 3D, which requires complex fabbing and may be more susceptible to faults

could increase charge storage capacity by using another material but manufacturers are slow to change their fabbing process
Piezoelectric Materials

piezeoelectric materials:
a mechanical stress (tension or compression force/area) causes a dielectric polarization or an applied electric field will cause a mechanical strain

used for electromechanical sensors and actuators

mechanism of piezoelectricity involves an asymmetry in the arrangement of positive and negative ions in the material

have a hexagonal unit cell

with no pressure applied, the centers of positive and negative charges concide

when mechanical pressure is applied in the vertical direction, the centers split  a dipole forms

symmetric material would not change its dipole moment when stressed

pressure applied in the horizontal direction does not induce polarization  piezoresistivity is
anizotropic

polarization, P, is related to mechanical stress, T

electric stress, E, is related to the mechanical strain, S
$$ d = (\frac { \partial P}{\partial T})_E = (\frac { \partial S}{\partial E})_T = "polarization coefficient" $$

quartz is historically the first piezoelectric material

quartz crystal is cut into a disc and electrodes are plated onto opposite sides

now, the disc has a mechanical resonant frequency precisely determined by its size

we can excite mechanical oscillations by applying an AC voltage

the resonant frequency of that voltage is therefore the same as that of the mechanical oscillation
Thermal Properties of Materials
Heat Issues in ICs

electrons release their energy to vibrating atoms upon collisions, causing heating

if generate heat is not removed, temperature increases

consequences:

carrier mobility may change

carrier concentration may change

dielectrics may leak more or even break down

heat generation in IC due to electric power:

P = IV = JAV = V
^{
2
}
/R = I
^{
2
}
R

total generated heat increases with:

more devices or interconnection lines per unit area (higher J)

higher operating voltage

higher operating frequency
Mechansms of Heat Generation and Loss

heat loss mechanisms in IC:

heat conduction

heat radiation

convection

heat conduction:
flow of heat through a solid, analogous to electronic transport but with:

driving force being ΔT instead of ΔV

constant of proportionality is thermal conductivity instead of electrical conductivity

heat radiation
: loss of energy by the emission of EM radiation in the IR wavelengths

heat convection:
transfer of heat away from a hot object because the gas next to it heats up and becomes less dense and then rises
Overview of Heat

heat (Q) flows from hotter objects to cooler

First Law of Thermodynamics: ΔE = W + Q
Heat capacity

heat capacity (C): ability to absorb heat from the external surroundings; amount of energy needed to heat particular material

depends on conditions measured under (volume and pressure)

C
_{
P
}
^{
’
}
: under constant pressure

C
_{
V
}
^{
’
}
: under constant volume
$$ C_V^\prime = C_P^\prime  \frac{\alpha TV}{K} $$

specific heat capacity:
heat capacity by unit mass
$$ c_p = \frac{C_P^\prime}{m} $$
$$ c_v = \frac{C_V^\prime}{m} $$

molar heat capacity:
heat capacity per moles of atom
$$ C_V = c_vM = \frac{C_V^\prime} {n} $$
DulongPetit Law

C
_{
p
}
= C
_{
v
}
= 25 J*mol
^{
1
}
K
^{
1
}

DulongPetit Law: heat capacity of metals saturates at 25 J*mol
^{
1
}
K
^{
1
}

heat capacity and thermal conductivity can be calculated by:

classical theory based on thermodynamics

quantum mechanics based on interaction
Thermal Conductivity

thermal conductivity (K):
proportionality coefficient between heat flow density and temperature gradient

Fourier’s Law:
$$ J_Q = K\frac{dT}{dx} $$

slightly temperature dependent, usually decreases as temperature increases
Classical treatment of heat capacity

treat electrons in a metal as a gas

ideal gas law: PV = nRT

will use in calculation of kinetic energy of gas molecules (or electrons in a metal)

in a small volume of gas, about 1/3 on average will move in xdirection  1/6 will move in +x direction

number of particles per unit time that hit the end of the volume per unit area will be:
$$ Z = \frac{1}{6}n_v v $$

where v= velocity, n
_{
v
}
= particles per unit volume

momentum transferred per unit area is:
$$ p^* = Z2mv = \frac{1}{6}n_v mvm = \frac{1}{3}\frac{N}{V}mv^2 $$

we find that the average kinetic energy of a gas molecule:
$$ E_kin = \frac{3}{2} kT $$

atoms vibrate about their ideal lattice positions due to their thermal energy

such an atom can be thought of as being like a sphere supported by springs

the atom acts like a simple harmonic oscillator which “stores” an amount of thermal energy

in a threedimensional solid, oscillator has energy E = 3kT; energy per atom

total internal energy per mole is therefore:
E = 3N
_{
o
}
kT
$$ C_V = \frac{\partial E}{\partial T} = 3kN_o $$
 classical treatment works well at high temperatures
 implies that heat capacity is constant and independent of temperature
 however, heat capacity is dependent on temperature

Debye Temperature (θ
_{
D
}
)
: temperature at which C
_{
V
}
has reached 96% of its final value
Quantum Theory of Heat Capacity

key assumption: the energies of the “atomic oscillators” are quantized

such quantized lattice oscillations are called
phonons

number of phonons increase with temperature

energy of each phonon is constant (electrons  number is constant but energy changes)

BoseEinstein Distribution
: the average number of phonons at any temperature was found to obey a distribution

turns out, electrons play a small part in the heat capacity; only a small fraction of the total number of electrons can gain thermal energy

1% of C
_{
V
}
is contributed by electrons at room temperature
Quantum treatment of thermal conductivity

what is the mechanism of heat transfer?

in a solid, only two things can move:

depending on the material, either one or the other tends to dominate

good electrical conductors tends also to be good thermal conductor

WiedemannFranz Law:
relationship between electrical and thermal conductivities (for metals), suggesting that
electrons can carry thermal energy as well as electrical

because of electrical neutrality, equal number of electrons move from hot > cold and from cold > hot

but their thermal energies are different and so the heat transported is proportional to the difference between electrical and thermal energies

in electrical insulators, there are few free electrons so the heat must be conducted in some other way  phonons (lattice vibrations)

the major difference between conduction by electrons and by phonons:

electrons: number constant but energy varied

phonons: number is variable (more phonons at hot end) but energy is quantized
Classical theory of Thermal Conductivity

let us consider a bar of material with a thermal gradient dT\dx

we calculate the flow of energy through a volume due to the temperature gradient and use this to calculate the number of “hot” electrons

an equal number of “cold” electrons must flow the opposite way, so we can solve for K
 λ is the
mean free path
between collisions with the lattice
 the idea is that an electron must have undergone a collision in this space and hence will have the energy/temperature of this location
WiedemannFranz Law
$$ \frac{K}{\sigma T} = \frac{\pi
^{
2 k
}
2}{3q^2} = L $$
 L = Lorentz number
 works quite well for metals, but not for “phonon materials”
Thermal resistance

just as we can relate electrical conductivity to electrical resistance, we can obtain a thermal resistance

for the temperature of the device on the chip:

T
_{
j
}
= T
_{
a
}
+ θ
_{
ja
}
P

T
_{
j
}
junction temperature

T
_{
a
}
ambient temperature

P dissipated power

θ
_{
ja
}
junctiontoair thermal resistance

θ
_{
ja
}
can be composed of several resistances in series

θ
_{
ja
}
= θ
_{
D
}
+ θ
_{
j1
}
+ θ
_{
P
}
+ θ
_{
j2
}

θ
_{
D
}
chip

θ
_{
j1
}
chiptopackage

θ
_{
P
}
package

θ
_{
j2
}
package to air

would like to minimize θ
_{
ja
}
in order to get a smaller drop in temperature per unit power

recall we can lose heat due to radiation, conduction, and convection

all of these depend on surface area, so we can improve heat dissipation by using a high sink of high thermal conductivity
Magnetic Properties
Basic Concepts
Faraday’s Experiment
Para and Diamagnetics
Paramagnetic and Ferromagnetics:
$$ B = \nu H = \nu_0 (H + M) $$
Diamagnetics:
 M opposes H
$$ B = \nu H = \nu_0 (H  M) $$
Magnetic Moments

magnetic materials always act as dipoles (even as a small piece of magnetic material, carries double magnetic charge  N and S poles)

therefore, possible to define a
magnetic dipole moment

each atom of magnetic material acts as a magnetic dipole

electric orbits acts like a coil generating magnetic field.

the material can be considered as an assembly of blocks  current loops

the magnitude of magnetic dipole μ
_{
m
}
= Ai

A = area of current loop

i = value of the current in the loop

in macroscopic body, net current in interior blocks is 0 because neighbouring currents cancel each other

on the surface, there is nonzero net current which equals iAn, where n is the number of blocks

every atoms except noble gases should demonstrate magnetic properties (inequal number of + m
_{
l
}
and + m
_{
s
}
)

this is valid for gases only

all magnetic compound materials include at least one transition element

these elements have an
inner
incomplete electron shell

as we learned, when atoms bond they electrons fill their *
outer
electron shell

only materials with incomplete inner shells after bonding
show magnetic properties
Size of Atomic Magnetic Moment

when atomic subshells are being filled, the m
_{
l
}
are first filled with electrons have m
_{
s
}
= +1/2 and only then electrons having m
_{
s
}
= 1/2 start to occupy their states

the consequence of this is that electrons with two different functions and the same spin have a lower energy then electrons with different wave functions and different spins
Bohr Magneton

β = qh/4πm  fundamental unit of magnetic moment
Alignment of Magnetic Moments in a Solid

rules for magnetic solids:

closed shells give no magnetic moment

this is no μ
_{
orb
}
for 3d shells

third effect is the interaction between the electron spins of adjacent atoms

this interaction can vary in strength. When it is strongest, it caries magnetic moments of adjacent atoms to either be
parallel
or
antiparallel
to each other
Paramagnetics

spin interactions are negligible compared to thermal agitation of the atomic magnetic moments

directions of magnetic moments in atoms are randomly oriented

an external magnetic field can induce some degree of alignment but it disappears when field is removed
Ferro, Antiferro and Ferrimagnetism
Ferromagnetics

strong spin interactions cause the atomic magnetic moments to align parallel to each other

in these metals, 3d electrons from the incomplete shells can move around the materials

the ability of electrons to move reduces the interaction ebtween the electrons and hence the magnetic moments are also reduced
Antiferromagnetics

strong spin interactions lead to these atomic magnetic moments to be antiparallel to each other in adjacent atoms

in the whole material, the number of atoms with antiparallel magnetic moments are equal

net magnetic moment in the material is zero
Ferrimagnetics

the moments for adjacent atoms are in opposite directions HOWEVER the moments have different magnitudes, thus giving us a nonzero net magnetic moment

this feature stems from the nature of these materials: they are all compounds

ferrimagnetics are electrical insulators, unlike all useful ferromagnets

ferrimagnetics can be used for highfrequency applications to avoid high eddy currents and resultant losses

ferrimagnetic compounds are called
ferrites
Diamagnetics

in diamagnetics, intrinsic magnetic field is opposite to external field therefore it is expelled from the field

all materials are diamagnetic, but the effect is weak and only shows up when none of the other effects are present

diamagnetism arises due to Lenz’ Law: “when a magnetic is moved towards a loop of wire, it induces a current in the loop which in turn generates a magnetic field to oppose the magnetic’s motion”
Curie Temperature

beyond a certain temperature, a material ceases to be ferromagnetic at all, and M drops to 0

at T = 0K, all atomic moments are perfectly aligned as we assume in our calculation. At T > 0K, thermal excitation of atoms reduces the degree of alignment and M drops

the temperature at which M > 0 is known as the
Curie temperature
Domains and Hysteresis

if all the atomic moments are lined up, this represents the maximum value of the moment  the condition known as
saturation

why are ferromagnetic elements not always permanent magnets?

materials are divided into subunits known as
domains

each domain is magnetized to saturation (inside domain, all moments are aligned)

however, direction of each domain can be randomly oriented
The BH Curve

since M relates B and H and M is not constant, what is the relationship between B and H in practice?

H = external field, B = magentic flux density inside material
 as H increases, B follows the path O P Q R
 at R, B value saturates at B
_{
s
}
 B
_{
s
}
corresponds to complete alignment of atomic moments
 if the applied field is now reduced
after
increasing to R, the path follows a different direction
 at H =0, B = B
_{
r
}
, known as remanence
 material now has a “permanent” magnetic flux density, becoming a permanent magnet
 to turn B to zero, apply a reverse field
 H
_{
c
}
= coercive field or coercivity
 if we further increase B in the reverse direction, get saturation at B
_{
s
}
again

hysteresis:
the different behaviour of the BH curve for different directions of H, and this curve is called the
hysteresis loop
Hard and Soft Materials

magnetic materials are divided into two broad groups based on their BH curves:
magnetically hard
and
magnetically soft
materials

this refers to how easy it is to magnetize or demagnetize them

measure of magnetic hardness is the value of H
_{
c
}

soft materials have a high permeability
Domains

domain’s formation is caused by energy minimization reasons

if all domains were aligned, then material would be magnetic and much of the field would be outside the material

but the field is a “storage” of potential energy called
magnetostatic energy

the whole energy in the system could be reduced by forming more domains

following this logic, the best result can be achieved if we have atomized domains

however, we have to consider the boundaries between domains, the
domain walls

domain wall:
the boundary between the two domains where the magnetization changes from one direction to another and hence the atomic spins do too

not simply one atomic spacing wide

exchange forces between neighbouring atomic spins favour very little relative rotation

magnetic moments that are oriented away from the field direction possess excess potential energy call the
anisotropy energy

in domain wall region, there is competition between exchange forces and the anisotropy energy minimization

what happens when we apply external magnetic field to the ferromagnet?

domains aligned parallel to the field will expand at the expense of domains with other orientations

growth is called the
motion of domain walls

the stronger the magnetic field, the more nonparallel domains shrink. As their size is small enough, the magnetic moments in these domains can be rotated until the material is uniformly magnetic

in magnetically soft materials, domain size shrinkage and magnetic moments rotation occur simultaneously and at low field
Domains in Polycrystals

in polycrystalline materials, grain boundaries do not allow domain walls to expand

after external field is applied, domain wall motion will result in crystal grains with single domains

in many grains, magnetic moments will be pointing at some angle to H

after we remove the field, some of these domains will return to previous conditions but some remain resulting in
remanence
B
_{
r
}
on the BH curve

heat above T
_{
c
}
or strong reverse field H use to reverse the effect
Magnetic Recording