# 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

• long-range order exists over small distances only
• has small crystal “grains” that are randomly oriented

#### Amorphous

• no long-range 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 x-ray 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 x-ray 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 co-ordinates, 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 Si-containing 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  $$b/c: • only one wafer can be used at a time • wafer must be small to get uniform layer • very good vacuum required ### Epitaxy principles • major feature of epitaxy is that the newly deposited film precisely follows the crystalline form of the substrate template • adsorption: proces of atoms attaching themselves to the surface • adsorption can occur anywhere on the surface with equal probability, so layer is unlikely to be crystalline • for crystalline growth, absorbed atoms must be able to find the minimum energy positions • the ad-atoms must be able to move along the surface • heat up the substrate during epitaxy to allow movement • if not hot enough, the ad-atoms stick where they land and the film is amorphous ### Modifying Crystals • to make an electronic device, crystals need to be modified and shaped according to the needs of the device • examples: introducing impurities, etching/shaping #### Ion implantation • ionized impurities are accelerated into an electric field and “smashed into” Si surface • depth of penetration determined by: • accelerating voltage • masses of ions and target atoms • crystal direction (density of atoms) #### Etching and micromachining • sometimes you need to etch crystals to get certain structures e.g. for making DRAMs, you need to etch deep tranches to make trench-capacitors • wet-etching: uses liquid chemicals to remove materials from a wafer • isotropic etching: chemicals etch at the same rate in all directions • anisotropic etching: chemicals selectively etch one crystal plane more • see example below. KOH etches (100) faster than (111) planes ## Atomic Structure ### Nature of light • classic physics: light is an electromagnetic wave w/ perpendicular field Bx and Ey • electric field is given by the following equation: $$ 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 $$#### Experimental Evidence of Light as EM Wave • interference and diffraction • Young’s double-slit experiment • expand ### Photo-emission (photoelectric effect) • when a metal electrode is illuminated with light, it emits electron (can create a current with this!) • light must possess the energy needed to “free” the electron from the metal (W) • any excess energy it possesses will become the kinetic energy of the electron • according to classical theory of light, the energy balance should be: E L = W + E K • if we reduce E L by reducing the intensity of the light, E K should also decrease • if light intensity is increased -> saturation current increases • more electrons emiited • same voltage is required to stop the current, thus the kinetic energy of the electrons is the same • classical theory of light can’t explain this! ##### Photons • light contains particle of fixed energy called photons • light frequency increases -> energy of light increases • E L = hf • h = Planck’s constant ### Wave-Particle Duality • light has properties of both a wave and a particle • light waves consist of a stream of photon particles, each with energy hf • energy carried by the wave consists of discrete lumps or quanta ### De Broglie relationship • electrons also have a wave-particle duality • particle-like properties confirm with cathode ray tube (1897) • deBroglie predicted that electrons would have a wavelength: $$ \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 time-dependent and space-dependent 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 time-independent • wave function is continuous, smooth, and single- • Boundary conditions: 1. For x<0 and x>L, the term Vψ dominates $$ -V\Psi = 0 \\ \Psi = 0 |\Psi|^2 = 0 $$electron cannot be outside the well 2. 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 co-ordinate #### 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 (n-1) #### 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, … (z-1) m l = 0, ±1, ±2, ±3, … ±(z-1) 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 ### Multi-electron atoms, Pauli principle, and the periodic table • building atoms requires an organization of Z electrons around the nucleus • two principles for doing so: • electrons will occupy the lowest possible available energy state • no more than one electron can have the same set of quantum numbers (Pauli’s Exclusion Principle) • start filling the atomic states (quantum number states) with electrons • e.g.: For Lithium, there are 3 electrons: • n = 1, l = 0, m l = 0, m s = 1/2 • n = 1, l = 0, m l = 0, m s = -1/2 • n = 2, l = 0, m l = 0, m s = 1/2 • notation: ## 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 ### Energy band formation; metals, semiconductors and insulators • 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 2s-band, 6N states in 2p-band • in diamond crystal, bonding and anti-bonding 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 semi-conductors, 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 (electron-hole pairs)

### Carrier Generation

• electron-hole 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 electron-hole 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
• n-type semiconductor: predominant electron concentration
• p-type semiconductor: predominant hole concentration

#### n-type doping

• n-type 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

#### p-type doping

• p-type 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

 $$f(E) = \frac{1}{1 + e^{\frac{E - E_F}{kT}}}$$
- Fermi-Dirac 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 n-type material: closer to E C because the concentration of electrons in CB is higher than concentration of holes in VB
- E F in p-type 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 x-direction: 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

#### Haynes-Shockley 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 non-uniformity (gradient) of carrier concentration

# need more

### Direction and indirect bandgap semiconductors

• dielectrics and semi-conductors 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 electron-hole 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 non-radiative 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