d \epsilon g_c (\epsilon) \frac

{e^

+ 1 \right )</math>

<br>

<math>p_v (T) = \int_{-\infty}^

d \epsilon g_v (\epsilon) \left ( \frac

\left ( \frac

\right )^{{3/2) (E - E_c)}

</math>

<br>

</center>

THe term <math>m_c^*</math> is the effective mass related to the parabolicity of the band-edge, and <math>E_c</math> is the energy of the bottom of the conduction band.

<p>
</p>

Electrons and holes inside the material (both fermions) obey to the Fermi-Dirac statistics representing the occupation of the particles over the energy levels of the system depending on the Fermi energy <math>E_F</math>.

<center>

<br>

<math>f(E) = \frac

{1+ e^{\frac

{kT}}</math>

<br>

</center>

For an intrinsic semiconductor at thermal equilibrium the Fermi energy lies in the middle of the band-gap and is the same for electron and holes. When a non-equilibrium situation is considered (for example under optical or electrical pumping, with carrier densities strongly exceeding the thermal values), the electron and hole distribution can be described separately through two different quasi-Fermi levels if intraband relaxation processes are much faster than interband recombination, so that quasi-thermal equilibrium within the bands can be assumed.

<p>
</p>

The product of the quasi-Fermi distribution and the density of states in the conduction band will represent the carrier distribution over the given energy levels. If we integrate this distribution over all the available energy levels we will obtain the density of electrons in the band.

<center>

<br>

<math>n = \int_

^

\left (\frac

\right )</math>

<br>

<math>N_c = \frac

\left ( \frac

\right )^

</math>

<br>

<math>F_

{\pi^{1/2}} \int_0^

\frac{x^{1/2}}{1+e^{(x-u)}}dx</math>

<br>

</center>

If the number of particles in the system is known, one can extract from the expression of <math>n</math> the value of the quasi-Fermi energy at a given temperature. The integral in the previous relation cannot be calculated analytically and requires a numerical calculation.

<p>
</p>

If the condition <math>E - E_F \gg kT</math> is valid, the Fermi distribution can be replaced by the Boltzmann distribution and there is an analytical solution of the density of particles.

<center>

<br>

<math>n = N_c e^{- \frac

</math>

<br>

<math>E_F^c = E_c - kT \ln \left ( \frac

\right )</math>

<br>

</center>

How is the chemical potential engineered?

The chemical potential is engineered through the addition of impurities.

What is the Fermi Energy?

In physics and Fermi-Dirac statistics, the Fermi energy of a system of non-interacting fermions is the smallest possible increase in the ground state energy when exactly one particle is added to the system. It is equivalent to the chemical potential of the system in its ground state at absolute zero. It can also be interpreted as the maximum energy of an individual fermion in this ground state. The Fermi energy is one of the central concepts of condensed matter physics.

<p>
</p>

The position of the chemical potential is obtained from the expressions of the charge carrier density. In intrinsic semiconductors the number of electrons in the conduction band is equal to the number of holes in the valence band.

<center>

<br>

<math>n_c(T) = N_c (T) e^{-(\epsilon_c - \mu )/k_B T}</math>

<br>

<math>n_c(T) = e^{-(\mu - \epsilon_v)/k_B T}</math>

<br>

<math>n_c(T) = p_v(T)</math>

<br>

<math>n_c(T) = n_i(T)</math>

<br>

<math>n_i(T) = \sqrt

e^{\frac

</math>

<br>

</center>

There is then an expression of the chemical potential.

<center>

<br>

<math>\mu = \epsilon_v + \frac

+ \frac

{1 + 2 e^{(\mu - \epsilon_d)/k_B T}} - N_c e^

{1 + 2 e^{(\mu - \epsilon_d)/k_B T}}</math>

<br>

</center>

Impurity states modelled as modified hydrogen atoms

• Consider the weakly bound 5th electron in phosphorous as a modified hydrogen atom
• For hydrogenic donors or acceptors, think of the electron or hole, respectively, as an orbiting electron around a net fixed charge
• Estimate the energy to free the carrier into the conduction band or valence band by using a modified expression of the energy of an electron in the hydrogen atom.

<center>

<br>

<math>E_n = \frac

</math>

<br>

<math>E_n = - \frac

\frac

\frac

</math>

<br>

</center>

• In the ground state <math>n=1</math>, <math>\epsilon</math> is on the order of <math>10</math>. The binding energy of the carrier to the impurity atom is <math><0.1 eV</math>.
• Expect that many carriers are ionized at room temperature.

References

• Fermi Energy http://en.wikipedia.org/wiki/Fermi_energy

P-N Junctions

The p-n junction possesses some interesting properties which have useful applications in modern electronics. P-doped semiconductor is relatively conductive. The same is true of N-doped semiconductor, but the junction between them is a nonconductor. This nonconducting layer, called the depletion zone, occurs because the electrical charge carriers in doped n-type and p-type silicon (electrons and holes, respectively) attract and eliminate each other in a process called recombination. By manipulating this nonconductive layer, p-n junctions are commonly used as diodes: electrical switches that allow a flow of electricity in one direction but not in the other (opposite) direction. This property is explained in terms of the forward-bias and reverse-bias effects, where the term bias refers to an application of electric voltage to the p-n junction. Consider lecture 10, 11.

Doping Profile

A plot of the net doping concentration as a function of position is referred to as the doping profile.

<center>

Unable to render embedded object: File (Doping_profile.PNG) not found.

</center>

Only the doping variation in the immediate vicinity of the metallurgical junction is of prime importance. Two common idealizations are the step junction and the linearly graded junction profiles.

Poisson's Equation

Poisson's equation is a relationship from electricity and magnetism. It can be a starting point in finding quantitative solutions to electrostatic variables.

<center>

<br>

<math>\nabla \cdot E = \frac

</math>

<br>

</center>

Below is the charge density inside a semiconductor assume dopants to be totally ionized.

<center>

<br>

<math>\rho = q(p - n +N_D - N_A)</math>

<br>

</center>

Qualitative Solution

There should be band bending and an internal electric field with a nonuniform doping of a pn junction diode. Assume a one-dimensional step junction. Expect regions far from the metallurgical junction behave identically to an isolated semiconductor. The fermi level is constant under equilibrium.

<center>

Unable to render embedded object: File (Step_junction_I.PNG) not found.

Unable to render embedded object: File (Step_junction_II.PNG) not found.

Unable to render embedded object: File (Step_junction_III.PNG) not found.

</center>

Deduce the functional form of the electrostatic variables. The relationship of V versus x is of the same functional form as the curve of <math>E_c</math>, <math>E_i</math>, or <math>E_v</math> upside-down. Find the <math>E</math> versus <math>x</math> relationship from the derivative of <math>E_c</math> with respect to position. Find <math>\rho</math> versus <math>x</math> from the slope of the plot of <math>E</math> versus <math>x</math>.

<center>

<br>

Unable to render embedded object: File (Step_junction_--_qVbi.PNG) not found.

<br>

Unable to render embedded object: File (Step_junction_--_electrostatic_potential.PNG) not found.

<br>

Unable to render embedded object: File (Step_junction_--_electric_field.PNG) not found.

<br>

Unable to render embedded object: File (Step_junction_--_charge_density.PNG) not found.

<br>

<math>\vec E = \int \frac

dx</math>

<br>

<math>V = \int \vec E dx</math>

<br>

</center>

There is a voltage drop across the junction under equilibrium conditions and there is charge near the metallurgical boundary. Consider the source of charge. In a p-material, the positive hole charges balance immobile acceptor-site charges. In an n-material, the electronic charge balances the immobile charge associated with the ionized donors. After the two materials are joined, the holes begin to diffuse from the p-side to the n-side. Electrons diffuse from the n-side to the p-side of the junction. The donors and acceptors are fixed. Unbalanced dopant site charge is left behind. There is a significant non-zero charge in the space charge region or depletion region. The built-up of charge and the associated electric field continues until the diffusion of carriers across the junction is balanced by carrier drift.

The Built-in Potential

Consider a nondegenerately-doped pn junction under equilibrium conditions. Ends of the equilibrium depletion region are at <math>-x_p</math> and <math>x_n</math>.

<center>

<br>

<math>E = - \frac

</math>

<br>

<math>0 = q \mu_n n E + qD_n \frac

</math>

<br>

<math>V_

= \frac

\frac

</math>

<br>

<math>W = x_n + x_p</math>

<br>

</center>

Applied Bias

Depletion widths decrease under forward biasing and increase under reverse biasing. A decreased depletion width when there is a forward bias means there is less charge around the junction and a correspondingly smaller electronic field. Reverse biasing creates a large space charge region and a bigger electric field. Potential decreases at all points with a forward bias and increases at all points with a reverse bias. The potential hill shrinks in both size and extent under forward biasing, whereas reverse baising gives rise to a wider and higher potential hill. One may conceive of the diode terminals as providing direct access to the p- and n-ends of the equilibrium Fermi level. Progress from the equilibrium diagram to the forward bias diagram by moving the n-side upward by <math>qV_a</math> while holding the p-side fixed. The reverse bias is obtained from the equilibrium diagram by pulling the n-side Fermi level downward.

Forward-bias

<p>
</p>

Forward-bias occurs when the P-type block is connected to the positive terminal of a battery and the N-type block is connected to the negative terminal, as shown below.

<p>
</p>

With this set-up, the 'holes' in the P-type region and the electrons in the N-type region are pushed towards the junction. This reduces the width of the depletion zone. The positive charge applied to the P-type block repels the holes, while the negative charge applied to the N-type block repels the electrons. As electrons and holes are pushed towards the junction, the distance between them decreases. This lowers the barrier in potential. With increasing bias voltage, eventually the nonconducting depletion zone becomes so thin that the charge carriers can tunnel across the barrier, and the electrical resistance falls to a low value. The electrons which pass the junction barrier enter the P-type region (moving leftwards from one hole to the next, with reference to the above diagram).

<p>
</p>

This makes an electric current possible. An electron starts flowing around from the negative terminal to the positive terminal of the battery. It starts at the negative terminal, moving towards the N-type block. Having reached the N-type region it enters the block and makes its way towards the p-n junction. The junction barrier can no longer keep the electron in the N-type region due to the forward-bias effect (in other words, the thin depletion zone produces very little electrical resistance against the flow of electrons). The electron will therefore cross the junction and move ahead into the P-type block. Once inside the P-type region, the electron, being thermally free (from bonding)���or mobile���will move through the rest of the crystal, making its way to the positive terminal of the power supply. Please note that the electron does not jump from one hole to the next in the p-region. This actually qualifies as electron-hole recombination which immobilises both hole and electron. The electron can move freely through the crystal without needing to jump into holes which is what happens when electrons do cross the depletion layer. This process will be repeated over and over again, producing a complete circuit path through the junction.

<p>
</p>

The Shockley diode equation models the operation of a p-n junction outside the avalanche region.

<p>
</p>

Reverse Bias

<p>
</p>

Connecting the P-type region to the negative terminal of the battery and the N-type region to the positive terminal, produces the reverse-bias effect. The connections are illustrated in the following diagram:

<p>
</p>

Because the P-type region is now connected to the negative terminal of the power supply, the 'holes' in the P-type region are pulled away from the junction, causing the width of the nonconducting depletion zone to increase. Similarly, because the N-type region is connected to the positive terminal, the electrons will also be pulled away from the junction.

<p>
</p>

This effectively increases the potential barrier and greatly increases the electrical resistance against the flow of charge carriers. For this reason there will be no (or minimal) electric current across the junction.

<p>
</p>

At the middle of the junction of the p-n material, a depletion region is created to stand-off the reverse voltage. The width of the depletion region grows larger with higher voltage. The electric field grows as the reverse voltage increases. When the electric field increases beyond a critical level, the junction breaks down and current begins to flow by avalanche breakdown.

Qualitative Effect of Bias

• Applying a potential to the ends of a diode does not increase current through drift
• The applied voltage upsets the steady-state balance between drift and diffusion, which can unleash the flow of diffusion current
• "Minority carrier device"

<center>

<br>

<math>n_p = n_n e^{\frac{-q(V_

- V_a)}{k_B T}}</math>

</center>

• Forward bias, which is a positive potential applied to p and a negative potential applied to n, decreases the depletion region and increases the diffusion current exponentially
• Reverse bias, which is a negative potential applied to p and a positive potential applied to n, increases the depletion region, and no current flows ideally
• Solve minority carrier diffusion equations on each side and determine <math>J</math> at the depletion edge.

<center>

<br>

<math>J = q \left (\frac

\frac

{k_B T}} - 1 \right )</math>

<br>

<math>J = J_o \left (e^{\frac

= V</math>

<br>

<math>\frac

<br>

</center>

The variational principle states that

• <math>\varepsilon \geq E_0</math>, where <math>E_0</math> is the lowest energy eigenstate (ground state) of the hamiltonian
• <math>\varepsilon = E_0</math> if and only if <math>\Psi</math> is exactly equal to the wave function of the ground state of the studied system.

<p>
</p>

The variational principle formulated above is the basis of the variational method used in quantum mechanics and quantum chemistry to find approximations to the ground state.

<p>
</p>

Your guessed wavefunction, <math>\psi</math>, can be expanded as a linear combination of the actual eigenfunctions of the Hamiltonian (which we assume to be normalized and orthogonal)

<center>

<br>

<math>\phi = \sum_

\psi_

c_

|H|\sum_

\psi_

\left\langle c_

|E_

\psi_

c_

|\psi_

\sqrt{ \frac

{ \hbar^2}}</math>

<br>

<math>\mbox

</math>

<br>

<math>g_n(\epsilon) = \frac

\int_{S_{n(\epsilon)}} \frac

</math>

<br>

</center>

Band Diagram

<center>

Unable to render embedded object: File (Description_of_states_in_amorphous_semiconductors.PNG) not found.

</center>

Van Hove

Van Hove singularities occurr where there is a non smooth density of state. It seems there won't be Van Hove singularities in an amorphous semiconductor.

Anderson

Milestones in a-Si development

• There is poor electronic properties of pure a-Si. A dramatic increase in photoconductivity is a result of hydrogenation (10 atomic %). This was discovered in the late '60s.
• a-Si is typically made by glow discharge plasma decomposition of silane (<math>SiH_4</math>) gas.
• Doping of a-Si is achieved by adding <math>PH_3</math> (n-type) or <math>B_2H_6</math> (p-type).
• Typical carrier mobilities are ~<math>1 \frac
  Unknown macro: {cm^2}
  Unknown macro: {V-s}
  </math> versus <math>1000</math> in crystalline silicon, while minority carrier diffusion lengths are about <math>100 nm</math>.
• By the mid 80's, <math>10%</math> conversion efficiency was achieved in a-Si cells. Depending on technology <math>5-13%</math> is possible.
• The typical solar cell design is a <math>p-i-n</math> junction. The individual layers are about <math>50 nm</math>. They serve the purpose of establishing the <math>V_
  Unknown macro: {bi}
  </math> but do not provide photocarriers.
• Amorphous silicon is a more efficient absorber of light compared to crystalline silicon and therefore can be significantly thinner (<math>x100</math>).
• The first large scale a-Si commercial installation was in Davis, CA.

Cause and Effect Relations Between Atomic Structure and Electronic Properties

• The existence of short range order results in a similar overall electronic structure of an amorphous material compared to a crystal with a similar stoichiometry.
• Silicon is semiconducting while <math>SiO_2</math> is an insulator both in the crystalline and amorphous forms
• The abrupt band edges are replaced by broadened tails of states extending into the forbidden gap. These originate from the deviations in bond length and angle. The band tails are very important in determining the electronic properties as most of the electronic transport in semiconductors occurs near the band edges and is thus due to these states.
• Electronic states deep in the gap occur because of coordination defects. These defects dominate the electronic properties by acting as recombination centers.
• The presence of localized states due to the disorder- Anderson Localization

Notes

• There is no k-conservation in amorphous materials due to the lack of long range order and lack of discrete translational symmetry.
• short range order and correlation exists
• The basic description of the electronic states is in terms of the DOS function.
• Since k is not a "good quantum number" the distinction between direct bandgap and indirect bandgap semiconductors is lost. Consequently optical translations are allowed in materials such as silicon without the participation of a phonon
• Disorder reduces carrier mobility due to scattering
• Conductivity is due to extended as well as localized states

The nature of eigenfunctions of amorphous semiconductors

<p>
</p>

References

• Van Hove singularityhttp://en.wikipedia.org/wiki/Van_Hove_singularity
• Shimakawa, Koichi and Singh, Jai, Advances in Amorphous Semiconductors

Calculate Eigenfunctions

Use the variational approach and a trial wavefunction that consists of a linear combination of atomic orbitals.

<center>

<br>

<math>\overline E = <\hat H></math>

<br>

<math>\overline E = \frac{\int_{-\infty}^

\chi^* \chi dV} \ge E_

</math>

<br>

<math>\chi = \sum_

^N c_n \phi_n</math>

<br>

</center>

Search for coefficients that minimize <math>< \hat H ></math>.

<center>

<br>

<math>\frac

< \hat H> = 0</math>

<br>

</center>

Conductive Polymers

The Nobel Prize in Chemistry was awarded in 2000 to Heeger MacDiarmid Shirakawa for the discovery of electrically conductive polymers.

Steps leading to the 2000 Nobel Prize in Chemistry

• The original discovery focused on the role of conductivity in a simple conjugated polymer called polyacetylene
• Made in 1974 by Shirakawa and co-workers using the Zieglar-Natta catalyst. The material was initially non-conducting but appeared metallic.
• In 1977 Shirakawa, Heeger and MacDiarmid discovered that oxidation with chlorine, bromine or iodine vapor made polyacetylen 9-11 orders of magnitude more conductive
• The "doped" polyacetylene had a conductivity of <math>10^5 S/m</math>. Teflon is <math>10^{-16} S/m</math>, and silver is <math>10^8S/m</math>.

Conductivity versus temperature for conjugated polymer and metal

Conductivity of polyacetylene increases with temperature, while conductivity of silver decreases.

The bonding in conjugated polymers

• Application of a simple Huckel theory gives a pretty good approximation to the electronic eigenfunctions of the molecule.
  • Ground state properties are adequately described by Huckel like models
  • Excited states involve lattice interactions
• There are two types of bonds in the system characterized by an <math>sp^2</math> hybridization
• Angular momentum 0 orbitals called <math>\sigma</math> bonds in the plane of the molecule and <math>\pi</math> bonds comprising of superposition of atomic <math>p_z</math> orbitals out of the x-y planes.
• Electronic transport properties determined by the <math>\pi</math> bonds.
  • Given the delocalized nature of the <math>\pi</math> molecular orbitals, shouldn't polyacetylene be a metal?
  • In fact pure (undoped) polyacetylene is a semiconductor with a bandgap around <math>1.7 eV</math>.
  • Electronic distribution is predicted to be across the entire chain implying equal bond lengths
• Focus on the HOMO(<math>\pi</math>) and LUMO(<math>\pi^*</math> ) which correspond to the valence and conduction bands discussed previously in inorganic SC.

Trans and cis polyacetylene

• Shirakawa discovered how to produce all trans polyacetylene using the Ziegler Natta organo-metallic <math>Ti(OBu)_4</math> catalyst
• trans polyacetylen first isolated in early 70's demonstrated a conductivity <math>10^{-2}S/m</math> in all <math>cis</math> polyacetylene

Optical absorption in doped and undoped polyacetylene

• Like a classical semiconductor the sample is transparent to light with a photon energy smaller than the band gap (<math>1.7 eV</math>).
• There is an emergence of a midgap state in the doped material around <math>0.7 eV</math>

Pierls instability in one dimesional chains

• Pierls showed that a hypothetical chain of sodium atoms would undergo a metal to insulator transition because the equi-spaced lattice is unstable with respect to an alternating bond configuration (1930s).
• This small conformation change leads to the emergence of a small bandgap

The effect of doping conjugated polymers (first published in 1977)

• Conductivity increases with increased doping
• The role of the dopant (halogen <math>I_2/Br_2</math>) is to remove an electron from the polymer
  • The resulting cation is not completely delocalized.
• Radical cation ("polaron" formed by the removal of one electron. The polaron migrates
• The radical cation is localized partly because of the interaction with a counter ion.
• A high concentration of dopants is needed in order to facilitate polaron mobility because the anion is typically of very low mobility.

===SSH model of soliton formation in polyacetylene

• A soliton separates A and B phases

===Properties of PPV(poly (p-phenylene))

• Electroluminescence from conjugated polymers was first reported in 1990 by R. Friend and coworkers
• In PPV the polymer backbone is held together by <math>\sigma</math> bonds.
  • The remaining electron in each carbon is in a <math>p_z</math> or <math>p_y</math> atomic orbital which overlaps to produce a <math>\pi</math> MO.
• The HOMO LUMO separation in PPV is aboout 2.5 eV which corresponds to a green-yellow emission.
• The magnitude of the gap depends on the conjugation length
  • Longer lengths correspond to smaller gaps with an asymptote reached at about ten repeat units.
• In crystalline semiconductors the electron and hole pairs are delocalized in three-dimensions which leads to a small overlap. In organic semiconductors the electron and holes are confined to the same chain.
  • This leads to bound neutral excited states called excitons.
  • Excitons play an important role in determining the PL properties of PPV and other conjugated polymers.

Simple PLED structure (polymer light emitting diode)

• semiconducting polymer is sandwiched between a hole injecting material (ITO) and a low work function metal such as <math>Ca</math>, <math>Al</math>, or <math>Mg</math>
• Upon external bias, electrons are injected from the <math>Ca</math> electrodes. Holes come from the ITO and recombine in the semiconductor layer and emit light corresponding to the HOMO-LUMO separation of the polymer
• Devices that are <math>80%</math> efficient have been produced.

Applications

• Doped polyaniline is used as a conductor and for electromagnetic shielding of electronic circuits. Polyaniline is also manufactured as a corrosion inhibitor.
• Poly(ethylenedioxythiophene) (PEDOT) doped with polystyrenesulfonic acid is manufactured as an antistatic coating material to prevent electrical discharge exposure on photographic emulsions and also serves as a hole injecting electrode material in polymer light-emitting devices.
• Poly(phenylene vinyliden) derivatives have been major candidates for the active layer in pilot production of electroluminescent displays (mobile telephone displays).
• Poly(dialkylfluorene) derivatives are used as the emissive layer in full-color video matrix displays
• Poly(thiophene) derivatives are promising in use in field-effect transistors: they may possibly find use in a supermarket checkouts
• Poly(pyrrole) has been tested as microwave-absorbing "stealth" (radar-invisible) screen coatings and also as the active thin layer of various sensing devices.

Magnetic Materials

Energy Terms

Exchange Energy

Minimize exchange energy when all spins are parallel

<center>

<br>

<math>E = -2J \sum_

S_i S_j</math>

<br>

</center>

Domain walls are the interfaces between domains. There is a significant angle among the magnetic moments of the atoms at the border. The higher the exchange constant <math>A_

</math> and <math>K</math>. The higher the <math>A_

/ K</math> ratio, the wider the domain walls.

Magnetostatic Energy

<center>

<br>

<math>E = -M \cdot H</math>

<br>

</center>

Demagnetization factor

Anisotropy Energy

Easy and hard direction. The angle, <math>\theta</math>, is the angle from the easy angle. Extra energy to magnetize along a particular angle. Squared due to symmetry. Energy required to rotate spins from favored direction.

<center>

<br>

<math>K = K_1 \sin^2 \theta + K_2 \sin^4 \theta + K_3 \sin^6 \theta</math>

<br>

</center>

Magnetostrictive Energy

The magnetization of a ferromagnetic material is in nearly all cases accompanied by changes in dimensions. The resulting strain is called the magnetostriction <math>\lambda</math>. From a phenomenological viewpoint there are really two main types of magnetostriction to consider: spontaneous magnetostriction arising from the ordering of magnetic moments into domains at the Curie temperature, and field-induced magnetostriction.

<p>
</p>

Spontaneous magnetostriction within domains arises from the creation of domains as the temperature of the ferromagnet passes through the Currie (or ordering) temperature. Field-induced magnetostriction arises when domains that show spontaneous magnetostriction are reoriented under action of a magnetic field.

<p>
</p>

The field-induced bulk-magnetostriction is the variation of <math>\lambda</math> with <math>\vec H</math> or <math>\vec B</math> and is often the most interesting feature of the magnetostrictive properties to the materials scientist. However, the variations <math>\lambda (\vec H)</math> or <math>\lambda (\vec B)</math> are very structure sensitive so that it is not possible to give any general formula for the relation of magnetostrictino to field.

Demagnetizing fields

In view of the fact that the magnetization <math>\vec M</math> and the magnetic field <math>\vec H</math> point in opposite directions inside a magnetized material of finite dimensions, due to the presence of the magnetic dipole moment, it is possible to define a demagnetizing field <math>\vec H_d</math> which is present whenever magnetic poles are created in a material

<p>
</p>

The demagnetizing field depends on two factors only. These are the magnetization in the material (i.e. the pole strength) and the shape of the specimen (i.e the pole separation which is determined by sample geometry). The demagnetizing field is proportional to the magnetization.

<p>
</p>

When dealing with samples of finite dimensions in an applied magnetic field <math>\vec H_

- N_d \vec M</math>

<br>

</center>

Examples involving energy terms

• Spins in sphere
• Small, long rods to store energy

References:

• David Jiles Introduction to Magnetism and Magnetic Materials

Spintronics

In order to make a spintronic device, the primary requirement is to have a system that can generate a current of spin polarised electrons, and a system that is sensitive to the spin polarization of the electrons. Most devices also have a unit in between that changes the current of electrons depending on the spin states.

<p>
</p>

The simplest method of generating a spin polarised current is to inject the current through a ferromagnetic material. The most common application of this effect is a giant magnetoresistance (GMR) device. A typical GMR device consists of at least two layers of ferromagnetic materials separated by a spacer layer. When the two magnetization vectors of the ferromagnetic layers are aligned, then an electrical current will flow freely, whereas if the magnetization vectors are antiparrallel then the resistance of the system is higher. Two variants of GMR have been applied in devices, current-in-plane where the electric current flows parallel to the layers and current-perpendicular-to-the-plane where the electric current flows in a direction perpendicular to the layers.

<p>
</p>

The most successful spintronic device to date is the spin valve. This device utilizes a layered structure of thin films of magnetic materials, which changes electrical resistance depending on applied magnetic field direction. In a spin valve, one of the ferromagnetic layers is "pinned" so its magnetization direction remains fixed and the other ferromagnetic layer is "free" to rotate with the application of a magnetic field. When the magnetic field aligns the free layer and the pinned layer magnetization vectors, the electrical resistance of the device is at its minimum. When the magnetic field causes the free layer magnetization vector to rotate in a direction antiparallel to the pinned layer magnetization vector, the electrical resistance of the device increases due to spin dependent scattering. The magnitude of the change, (Antiparallel Resistance - Parallel Resistance) / Parallel Resistance x 100% is called the GMR ratio. Devices have been demostrated with GMR ratios as high as 200% with typical values greater than 10%. This is a vast improvement over the anisotropic magnetoresistance effect in single layer materials which is usually less than 3%. Spin valves can be designed with magnetically soft free layers which have a sensitive response to very weak fields (such as those originating from tiny magnetic bits on a computer disk), and have replaced anisotropic magnetoresistance sensors in computer hard disk drive heads since the late 1990s.

<p>
</p>

Future applications may include a spin-based transistor which requires the development of magnetic semiconductors exhibiting room temperature ferromagnetism. The operation of MRAM or magnetic random access memory is also based on spintronic principles.

<p>
</p>

Reference

• Spintronics

GMR

The development of devices that sense and store information is driven by the demand for more capable computers. The utilization of the phenomenon of giant magneto-resistance (GMR) has led to the creation of more sensitive sensors that read bits coded on magnetized regions of disks. An understanding of GMR has led to a large increase in information storage capacity. The phenomenon of GMR has also been of interest to physicists.

<p>
</p>

The magneto-resistance effect (MR effect) describes the change in the electrical resistance of a material due to the application of a magnetic field. Research advances related to electron spin, tunneling effects, and production of ultra-thin layers have made possible the design of electrical devices based on quantum mechanical effects of the electron. The giant magneto-resistance effect is dependent on spin properties of electrons. Several other magneto-resistive effects have been discovered, and they are classified under XMR effects or XMR technology.

<p>
</p>

An example of a structure where the giant magneto-resistive effect is observed consists of a layer of copper, a normal metal, between layers of cobalt, a ferromagnetic material. When the magnetic moments of the ferromagnetic layers are parallel, there is less resistance to the flow of current. In current devices, the direction of magnetization of one layer is typically fixed in one direction, while the other is determined by an external field. The difference of resistance between parallel and anti-parallel configurations is 10-15% in current devices.

<p>
</p>

The effect of GMR is similar to phenomenon of polarization. In a polarization experiment, light cannot pass through if two polarizers are perpendicular to one another. Similarly, one magnetic layer may allow electrons of one type of spin to pass, and, if the second structure is aligned in the same direction, current can easily pass through. If the second layer is misaligned, neither spin channel can pass through the structure easily, and the electrical resistance is high.

<p>
</p>

Early studies in which GMR was first observed may have been published in 1987. The patent of the effect is owned by Peter Grunberg, who led a team at the Julich Research Centre and observed the effect in trilayers of Fe/Cr/Fe. The effect was simultaneously and independently observed in multilayers of Fe/Cr.

<p>
</p>

Three types of GMR are multilayer, granular, and spin-valve GMR. The effect was first observed in multilayer configurations and the effect in granular GMR is not as large as multilayer GMR. Spin-valve GMR is the most useful industrially, and the magnetic recording industry is researching structures that are very sensitive to magnetic fields. IBM and Seagate develop high density disk and tape playback heads. Additional applications include avionic compasses, swipe-card readers, wheel rotation sensors in ABS brakes, and current sensors for use in safety powerbreakers and electricity meters.

<p>
</p>

References:

• Giant Magneto-Resistive Tunnelling Effect http://en.wikiversity.org/wiki/Materials_Science_and_Engineering/magnetic_materials/giant_magneto-resistive_tunnelling_effect
• Giant Magnetoresistive Effect http://en.wikipedia.org/wiki/Giant_magnetoresistive_effect
• Giant Magnetoresistance http://www.stoner.leeds.ac.uk/research/gmr.htm
• Science Week Materials Science: On Magnetoresistive Tunnel Junctions http://scienceweek.com/2005/sc050107-2.htm
• New Magneto Coupler on magnetic GMR-technology http://www.yet2.com/app/list/techpak?id=26114&sid=350&abc=0&page=details
• Giant Magnetoresistance of (001)Fe/(001)Cr Magnetic Superlattices http://prola.aps.org/pdf/PRL/v61/i21/p2472_1

TMR

In physics, the tunnel magnetoresistance effect, commonly abbreviated as TMR, occurs when two ferromagnets are separated by a thin (about 1 nm) insulator. Then the resistance of the tunneling current changes with the relative orientation of the two magnetic layers. The resistance is normally higher in the anti-parallel case.

<p>
</p>

It was discovered in 1975 by M. Julliere, using iron as the ferromagnet and germanium as the insulator.

<p>
</p>

Room temperature TMR was discovered in 1995 by Moodera et. al. following renewed interest in this field fueled by the discovery of the giant magnetoresistive effect. It is now the base for the magnetic random access memory (MRAM) and read sensors in hard disk drives. For more technical information see [Moodera and Mathon 1999].

<p>
</p>

References:

• Tunnel Magnetoresistance http://en.wikipedia.org/wiki/TMR

Organic Materials

References

• Organic Semiconductorhttp://en.wikipedia.org/wiki/Organic_semiconductor
• Conductive Polymershttp://en.wikipedia.org/wiki/Conductive_polymers

Optical Properties

Maxwell's Equations

In electromagnetics, Maxwell's equations are a set of four equations, compiled by James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.

<p>
</p>

Maxwell's four equations express, respectively, how electric charges produce electric fields (Gauss' law), the experimental absence of magnetic monopoles, how currents and changing electric fields produce magnetic fields (the Ampere-Maxwell law), and how changing magnetic fields produce electric fields (Faraday's law of induction). Refer to article in wikipedia to see in depth discussion.

+ \frac{\partial \vec{D}}

</math> |

\vec

+ \int_S \frac{\partial\vec{D}}

\cdot \mathrm

</math> |

Index of Refraction

The refractive index (or index of refraction) of a material is the factor by which the phase velocity of electromagnetic radiation is slowed in that material, relative to its velocity in a vacuum. It is usually given the symbol n, and defined for a material by:

<center>

<br>

<math> n=\sqrt

</math>

<br>

</center>

where ��r is the material's relative permittivity, and ��r is its relative permeability. For a non-magnetic material, <math>\mu_r</math> is very close to 1, therefore n is approximately <math>\sqrt

</math>.

<p>
</p>

The phase velocity is defined as the rate at which the crests of the waveform propagate; that is, the rate at which the phase of the waveform is moving. The group velocity is the rate that the envelope of the waveform is propagating; that is, the rate of variation of the amplitude of the waveform. It is the group velocity that (almost always) represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an optical fiber.

Stoke's Theorem

Divergence Theorem

Wave Equation

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. The homogeneous form of the equation, written in terms of either the electric field <math>E</math> or the magnetic field <math>H</math>, takes the form:

<center>

<br>

<math>(\nabla^2-

) \mathbf

= 0</math>

<br>

<math> \nabla \times \mathbf

= -\mu_o \varepsilon_o \frac{\partial^2 \mathbf

= -\mu_o \varepsilon_o \frac{\partial^2 \mathbf

\over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf

\ \ = \ \ 0</math>

<br>

</center>

where <math>c</math> is the speed of light in free space.

<center>

<br>

<math>c = { 1 \over \sqrt

} = 2.998 \times 10^8 \frac

</math>

<br>

</center>

Solutions

See article in wikipedia.

EM Fields at Interfaces

Boundary Conditions

The tangential components of <math>\vec B / \mu</math> can be discontinuous at an interface if there is a layer of surface current confined there. In the equation below, <math>\mu_1</math> and <math>\mu_2</math> are the magnetic permeabilities of the two media and <math>\vec j_s</math> is the surface current density. If both materials behave as perfect insulators, no surface currents can exist, and the tangential components of \vec B / \mu are continuous across the interface.

<center>

<br>

<math>\frac

\vec B_2 \times \vec n = \vec j_s</math>

<br>

</center>

The tangential components of the electric field are continuous at an interface.

<center>

<br>

<math>\vec E_1 \times \vec n = \vec E_2 \times \vec n</math>

<br>

</center>

Transfer Matrix

If the electric and magnetic fields in the <math>x-y</math> plane at <math>z=0</math> are known, use Maxwell's equations at a fixed frequency to integrate the wavefield and find electric and magnetic fields in the <math>x-y</math> plane at <math>z=c</math>. Assume that there is zero divergence of <math>\vec B</math> and <math>D</math>. It is only necessary to know two components of each field.

<center>

<br>

<math>\vec F(z = 0) = \left [E_x (z=0), E_y (z=0), H_x (z=0), E_y (z=0) \right]</math>

<br>

</center>

An equation below defines the transfer matrix, <math>\vec T</math>.

<center>

<br>

<math>\vec F (z = c) = \vec T (c, 0) \vec F(z=0)</math>

<br>

</center>

The transfer matrix defines how waves cross a slab of material. It is closely related to the transmission and reflection coefficients of the slab. If the slab is infinitely repeated thourout the material in a periodic array, it is possible to impose the Bloch condition and calculate the band structure, <math>K(\omega)</math>, from the eigenvalues of <math>T</math>.

<center>

<br>

<math>e^

\vec

(z=0) = \vec T (c,0) \vec F (z=0)</math>

<br>

</center>

• • Reflection
  • Multiplicity of interfaces
  • Periodic Medium
• Photonic Band Diagrams
• Damped Harmonic Oscillator

  References

• Transfer Matrix Techniques for Electromagnetic Waves

Index of Refraction, Band Gap, and Wavelength

Index of refraction measure of how much light interacts with material. Closer to one, more transparent.

<p>
</p>

Check graph in notes. <math>Ge</math> above <math>Si</math> above <math>SiO</math>. Index of refraction of oxide less than index of non-oxide. The insulator lies below semiconductor materials. <math>Ge</math> smaller band gap. Band gap goes down with increasing index of refraction.

Definition and Examples of Isotropic, Uniaxial and Biaxial Optical Materials

Define coordinate axes (principal axes) such that the dielectric tensor is diagonal.

<center>

<br>

<math>\epsilon = \begin

</math>

<br>

</center>

Isotropic

• Cubic or amorphous materials
• <math>\epsilon_1 = \epsilon_2 = \epsilon_3</math>
• <math>Y_3Al_5O_12</math> (Yttrium Aluminum Garnet)

Uniaxial

• Trigonal, tetragonal, hexagonal
• <math>\epsilon_1 \ne \epsilon_2 = \epsilon_3</math>
• <math>LiNbO_3</math>

Biaxial

• Triclinic, monoclinic, orthorhombic
• <math>\epsilon_1 \ne \epsilon_2 \ne \epsilon_3</math>

Maxwell's Equations and the Constitutive Equation

V</math> |

= 0</math> |

\over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf

\over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf

</math>

<br>

<math>\frac

{\sqrt{\mu \epsilon}} |\vec k| = \omega</math>

<br>

</center>

The Properties of Plane Wave Solutions, Wave Vector, Transversality

Phase velocity

The solution is a complex valued function. An expression of the phase is below. Consider the fronts where the phase is constant.

<center>

<br>

<math>\vec E (x,y,z) = \vec E_o e^

= \frac

\hat k</math>

<br>

<math>\vec v_

{\sqrt{\epsilon \mu}}</math>

<br>

</center>

The direction is defined by the wavevector's direction. The phase velocity is changed in a material compared with its value in vacuum.

Transversality of E-M Fields

There are three Cartesian terms of the solutions of the wave equations.

<center>

<br>

<math>\vec E(x,y,z) = E_x \hat x + E_y \hat y + E_z \hat z</math>

<br>

</center>

Consider a medium that is homogeneous and charge free. Two equations below imply the transversality of E-M fields.

<center>

<br>

<math>\vec \nabla \cdot \vec E = 0</math>

<br>

<math>\vec \nabla \cdot \vec H = 0</math>

<br>

<math>\vec E \cdot \vec k = 0</math>

<br>

<math>\vec H \cdot \vec k = 0</math>

<br>

</center>

Dispersion Relations

<center>

<br>

<math>\epsilon ( \omega ) \omega^2 = c^2 k^2</math>

<br>

</center>

The term <math>k</math> is the wavevector or propagation vector and it can be related to the wavelength, as shown below.

<center>

<br>

<math>|k| = \frac

</math>

<br>

<math>|k| = \frac

\vec \nabla \cdot \vec B dv = \int_

\vec B \cdot \hat n dS = 0</math>

<br>

<math>\int_

\vec D \cdot \hat n dS = 4 \pi \int_

\rho dv</math>

<br>

</center>

• <math>\hat n \cdot ( \vec B_2 - \vec B_1 ) = 0</math>
  • The normal component of the magnetic induction is continuous across the surface of discontinuity
• <math>\hat n \cdot ( \vec D_2 - \vec D_1 ) = 4 \pi \rho</math>
  • In the presence of a layer of surface charge, the normal component of the electric displacement changes abruptly by an amount equal to <math>4 \pi \sigma</math>. The tangential components are found through the application of Stokes Theorem.

<center>

<br>

<math>\int_

\vec E \cdot \frac

+ \int \left ( \vec E \times \vec H \right ) \cdot \hat n dS = 0</math>

<br>

</center>

Use materials equations

<center>

<br>

<math>\frac

</math>

<br>

<math>\frac

\vec E \cdot \frac

= \frac

\frac

\vec E \cdot \frac

= \frac

\frac

= \frac

</math>

<br>

</center>

Poynting Vector

The Poynting vector describes the energy flux <math>\left ( \frac

\right )</math> of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside independently co-discovered the Poynting vector.

<p>
</p>

It points in the direction of energy flow and its magnitude is the power per unit area crossing a surface which is normal to it. (The fact that it points perhaps contributes to the frequency with which its name is misspelled.) It is derived by considering the conservation of energy and taking into account that the magnetic field can do no work. It is given the symbol <math>S</math> (in bold because it is a vector) and, in SI units, is given by:

<center>

<br>

<math>\mathbf

= \vec

\times \vec

</math>

<br>

</center>

where <math>E</math> is the electric field, <math>H</math> and <math>B</math> are the magnetic field and magnetic flux density respectively, and <math>\mu</math> is the permeability of the surrounding medium.

<p>
</p>

For example, the Poynting vector near an ideally conducting wire is parallel to the wire axis - so electric energy is flowing in space outside of the wire. The Poynting vector becomes tilted toward wire for a resistive wire, indicating that energy flows from the e/m field into the wire, producing resistive Joule heating in the wire.

Snell's Law and its Derivation

See below

Conserved E-M Quantities at Interfaces (Parallel Component of Wave Vector, Frequency) and the Consequence of their Conservation

Consider a wave incident from the left onto an interface.

<center>

<br>

<math>\mbox

</math>

<br>

<math>\vec E_i e^{\omega t - i \vec k_i \cdot \vec r</math>

<br>

<math>\mbox

</math>

<br>

<math>\vec E_r e^{\omega t - i \vec k_r \cdot \vec r</math>

<br>

<math>\mbox

</math>

<br>

<math>\vec E_t e^{\omega t - i \vec k_t \cdot \vec r</math>

<br>

</center>

Consider a homogeneous medium and dispersion relations.

<center>

<br>

<math>|\vec k_i| = |\vec k_r| = \frac

\sin \theta_2</math>

<br>

</center>

This leads to two important consequences.

• Angle of reflection is equal to the angle of incidence

<center>

<br>

<math>\theta_i = \theta_r</math>

<br>

</center>

• Snell's law:

<center>

<br>

<math>k_

+ H'_

= H_

</math>

<br>

<math>\vec \nabla \times \vec E + \frac

\frac

E_y</math>

<br>

<math>H_z = \frac

E_1 e^{-i (k_x x + k_z z)}</math>

<br>

<math>k_

= |k| \cos \theta_1 = \frac

= -n_1 \cos \theta_1 E_1 e^{-i(k_x x + k_z z)}</math>

<br>

<math>H'_

• E'_

• E'_

1 & 1
n_1 \cos \theta_1 & -n_1 \cos \theta_1\end

E_

\end

1 & 1
n_2 \cos \theta_2 & -n_2 \cos \theta_2\end

E_

\end

</math>

<br>

<math>D_

E_

\end

E_

\end

\right ) = \frac

{n_1 \cos \theta_1 + n_2 \cos \theta_2</math>

<br>

<math>r_s \left ( \frac

</math>

<br>

</center>

Define amplitudes on both sides of each interface for an s polarized field.

<center>

<br>

<math>x<0</math>

<br>

<math>E_1 e^{-ik_

x</math>

<br>

<math>0<x<d</math>

<br>

<math>E_2 e^{-ik_

x</math>

<br>

<math>d<x</math>

<br>

<math>E_3 e^{-ik_

x</math>

<br>

</center>

The propagation matrix captures the effect of propagation through a medium of index <math>n_2</math> and thickness <math>d</math>.

<center>

<br>

<math>P_2 = \begin

</math>

<br>

<math>D_

P_2 D_

\begin

= \begin

</math>

<br>

</center>

Transfer Matrix Approach to Solving Periodic Systems of Dielectric Materials

<center>

<br>

<table cellpadding=10>
<tr>

<td>
<math>na<x<na+d_2</math>
</td>

<td>
<math>na-d_1<x<na</math>
</td>

</tr>

<tr>

<td>
<center>
<math>n_1</math>
</center>
</td>

<td>
<center>
<math>n_2</math>
</center>
</td>

</tr>

</table>

<br>

<math>n(x+na) = n</math>

<br>

</center>

The solution in each medium is of the form below.

<center>

<br>

<math>E = E e^

(x-na)} + B_n e^{ik_

(x-na+d_1)} + B_n e^{ik_

= \sqrt{ \left ( \frac

\right )^2 - k_

\right )^2 - \beta^2} = \frac

= \sqrt{ \left ( \frac

\cos \theta_2</math>

<br>

</center>

Dynamic matrices are below.

<center>

<br>

<math>D_l^s = \begin

</math>

<br>

<math>D_

= D_l^{-1}D_

</math>

<br>

</center>

Consider the solution form. Relate the fields across the interface.

<center>

<br>

<math>E|_{x=na^{(1)}} = A_n + B_n</math>

<br>

<math>E|{x=na-d_1^{(1)}} = A_ne^{-ik

(na-d_1-na)}</math>

<br>

<math>E|{x=na-d_1^{(1)}} = A_ne^{ik

d_1}</math>

<br>

<math>E|_{x=na-d_1^{(1)}} = C_n + D_n</math>

<br>

<math>P_1 = \begin

</math>

<br>

<math>D_2 \begin

= D_1 P_1 \begin

</math>

<br>

<math>\begin

= D_2^{-1} D_1 P_1 \begin

</math>

<br>

</center>

The solution brings one to the LHS of the <math>x=na-d_1</math> interface (i.e. in medium 2). Propagate the wave in medium 2 back to the RHS of interface <math>x = (n-1)a</math>.

<center>

<br>

<math>E|_{x=na-d_1^

= C_n +D_n</math>

<br>

<math>E|_{x=(n-1)d^

= C_ne^{ik_

d_2</math>

<br>

<math>P_2 = \begin

e^{ik_

d_2}\end

</math>

<br>

</center>

Match fields across the interface to extract <math>A_

</math>.

<center>

<br>

<math>E|_{x=na-d^

= C_ne^{ik_

+ B_

A_

\end

C_

\end

A_

\end

C_

\end

A_

\end

A_

\end

A_

\end

\begin

</math>

<br>

<math>D_1^{-1} D_2 P_2 D_2^{-1} D_1 P_1 \begin

= e^

A_

\end

M_

& M_

M_

& M_

\end

(M_

+ M_

) \pm \left ( \frac

and e^{-iKa} are two eigenvalues.

<center>

<br>

<math>\frac{e^

A_

\end

\begin

M_

e^

\end

</math>

<br>

</center>

Photonic Band Diagrams and Bandgaps

Consider the case where <math>\beta=0</math> which corresponds to normal incidence. The dispersion relation <math>\omega</math> vs. <math>K</math> can be expressed as below.

<center>

<br>

<math>\cos Ka = \frac{M_

</math>

<br>

<math>k_2 = \frac