ECE 209 Midterm Notes
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