# ECE 209 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 ### Energy-band 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 n-type ##### MOSFET Band Diagrams • to build energy band diagrams: 1. choose zero points on co-ordinates (for energy axis, 0 is vacuum level) 2. in equilibrium, E F = const everywhere on band diagram xo ## Metals ### Critical issues with metallization • 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 ### Metallic bonding • 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 ### Electrons in metals • 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 $$• only solution is at n=0, θ=0 • transmission occurs in direction of travel, magnitude unchanged • source of resistivity is either the temperature or non-crystallinity of a metal #### Temperature dependence of resistivity • At T>0K, atoms move away from ideal lattice position b/c vibrations • electrons become scattered • for atoms in a gas: • mean free path: average electron path length is defined • mean free time: time between collisions #### In the Presence of an Electric Field • when a potential difference is applied across metal, electrons drift towards larger positive potential • current density: $$ 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 non-magnetic 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 light-material 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 co-efficient (m -1 ) #### Bouger-Lambert-Beer’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 #### Absorption in Metals • 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 #### Absorption in Non-Metals • 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 electron-hole pairs • may or may not re-emit 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 • R = I R /I 0 • 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 0.5-2.0% 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 #### Emission in Non-Metals - Luminescence • luminescence: general term which describes the re-emission 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 n-layer 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
• Q = CV
• 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 5-10nm, 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 E-field 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, polymer-based materials tend to charge negatively:
• consider a material like paraffin. The side-arms 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 non-uniform
• 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
• μ= Qd
• 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 23-24

### 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 re-alignment 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:
• non-polar 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 ’‘
• 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 non-uniform 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
 $$Flux = 4\pi r^2 \frac{q}{4\pi r^2} = q$$
• 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
• 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
• currents of gas flow

### 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
• dQ = CdT
• 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
 $$C_V = c_vM = \frac{C_V^\prime} {n}$$

### Dulong-Petit Law

• C p = C v = 25 J*mol -1 K -1
• Dulong-Petit 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
• ability to conduct heat
• 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 x-direction - 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
• E = kT
• in a three-dimensional 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)
• Bose-Einstein 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:
• electrons
• phonons
• depending on the material, either one or the other tends to dominate
• good electrical conductors tends also to be good thermal conductor
• Wiedemann-Franz 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

#### Wiedemann-Franz 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 junction-to-air 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 chip-to-package
• 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

• in non-uniform magnetic field, diamagnetic materials are ejected while others are attracted to different degree
• A force: F = VH(dH/dx)

• where H = magnetic field strength
• V = sample volume
• two constants are used to characterize magnetic properties:

• susceptibility: χ, “responsiveness” of material
• permeability: μ = (1 + χ)μ 0
• μ 0 = permeability of free space
• magnetic field strength, H is related to magnetic field density, B
• outside the material: B 0 = μ 0 H
• inside the material: B = μ 0 (H + M)
• where M, magnetization of material is proportional to H: M = χH

### Para- and Dia-magnetics

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 non-zero 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 sub-shells 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 anti-parallel 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 anti-parallel to each other in adjacent atoms
• in the whole material, the number of atoms with anti-parallel 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 non-zero 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 high-frequency 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 sub-units 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 B-H 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 B-H 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 B-H 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 non-parallel 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 B-H curve
• heat above T c or strong reverse field -H use to reverse the effect