SEMICONDUCTORS

GENERAL

Draw the temperature dependence of the concentration of charge carriers in a p-type semiconductor. For each regime that you identify, draw a schematic of the band structure to indicate the origin of charge carriers. Give an expression for p(T) where appropriate.

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</p>

Freeze out, intrinsic, extrinsic

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</p>

<center>

<br>

<math>\frac

E_g + \frac

k_B T \ln \left ( \frac

\right )</math>

<br>

</center>

DEVICES

Describe the operating principle of a solar cell. How would you find the optimal load to be used with the solar cell? Support your answer with equations.

We want to fabricate a p-n junction in a Si wafer. What could you use as a p and n type dopant?

Draw the space charge distribution versus position for Na >> Nd assuming an abrupt junction at xd after equilibrium is achieved, identify the depletion region.

Express the chemical potential as a function of dopant concentration prior to equilibrium. What is the magnitude of the built-in voltage once equilibrium is established? Draw the band structure after equilibrium. Clearly indicate the position of the chemical potential curvature of the bands. Assuming an ideal diode behavior, give an expression for the I-V characteristic of the p-n junction. Explain how and why the different currents are affected under applied bias.

In what direction do the charge carriers flow right after the p-n junction is formed? What is the driving force for their flow and what ultimately stops the transfer of charges? Plot the I-V curve for a p-n junction.

Outlines the different methods for creating a built-in voltage in a material. Describe how the build-in voltage is generated and how we may control its magnitude. Support your answer with equations.

Describe the operating principle of a solar cell. How would you find the optimal load to be used with the solar cell? Support your answer with equations.

Draw the band diagrams reflecting the joining of the following pairs of materials, provide a brief-explanation on the properties and characteristics of each junction, suggest candidate materials for each junction. P-n junction with different Ec and Ev levels, p-n junction with save Ec and Ev levels, metal and n-doped.

AMORPHOUS

Plot on separate figures the DOS versus energy for an amorphous and crystalline semiconductor. Relate the features in the DOS to the specific crystal imperfections.

Plot qualitatively the electronic eigenfunctions versus position for the states in the mobility gap and out of the mobility gap of an amorphous semiconductor.

Compare and contrast the electronic properties of amorphous and crystalline semiconductors. Be sure to relate the electronic structure to the underlying physical structure. How are the electronic states in the mobility gap of an amorphous semiconductor different from the rest of the states. Compare the properties of amorphous Si and crystalline Si. What types of applications are these two forms of Si used in and why?

Give two examples of amorphous semiconductors that have a corresponding crystalline form. Compare and contrast the electronic properties of the amorphous and crystalline forms on one of the materials.

How would you describe the structure of an amorphous semiconductor? Provide a mathematical function that can be used to desribe the structure of an amorphous material, plot this function for a hypothetical 2D amorphous material.

Define the DoS and explain why it is used to desribe the electronic properties of amorphous semiconductors. Draw a typical DoS function for an amorphous structure and identify the different types of states associated with the different regimes. What characteristics of the amorphous structure give rise to the different states and the corresponding features in the DoS?

ORGANIC

Give an example and provide a bonding schematic of an organic material that would display semiconducting properties?

NUMERICAL METHODS

Using the variational approach demonstrate how you would proceed to calculate the molecular eigenfunctions in a linear conjugated polymer with 10 repeat units? Provide as much detail as you can about the trial function.

MAGNETIC MATERIALS

Explain briefly what the following forces are and in what way do they each influence the magnetic ordering, comment on their relative strength and dependence on distance: exchange, dipolar, anisotropic.

Explain why magnetic domains form.

OPTICAL PROPERTIES

What is the most general relation between the polarization vector and the electric field?

<center>

<br>

<math>P = \chi^

\vec E + \chi^

\vec E^2 + \chi^

\vec E^3 +...</math>

<br>

</center>

A material���s optical properties are described by its susceptibility. Outline the derivation steps for an analytical expression for the linear susceptibility (damped harmonic oscillator model).

<center>

<br>

<math>m \frac

+ m \gamma \frac

+ m \omega_o^2 X + ax^2= q E(t)</math>

<br>

</center>

There is a second-order process exhibited by KTP. The process is frequency doubling.

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</p>

A third order process is self-focusing. CdxHg1���xGa2S4

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</p>

Plot the materials dispersion (index of refraction versus wavelength) for Si, Ge, and SiO2 for wavelengths 1-10 microns.

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</p>

See notes

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</p>

What is the form of the harmonic oscillator potential? What are the energy eigenvalues? Plot the first 3 eigenfunctions. Why is the harmonic oscillator potential so important and general? Consider an experiment where a gas of diatomic molecules is being illuminated by light and is undergoing harmonic vibrations���the intensity of the illumination is increase by 3 orders of magnitude���what form of the potential would you use to analyze the vibrations if the quadratic form is no longer adequate?

<center>

<br>

<math>\frac

\frac

</math>

<br>

</center>

Large difference: <math>SiO_2</math> and <math>Si</math>

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</p>

Small difference: <math>GaSb</math> and <math>GaAs</math>

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</p>

Suppose you have a linearly polarized TE plane wave incident from left to right on a planar dielectric interface (between media 1 and 2) at an angle theta. Write explicit mathematical expression for all the waves that exist once the original wave encounters the interface. Clearly label the wavevectors, frequencies, and directions of the electric and magnetic fields for each wave on a schematic.

<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>

What components of the wave vector are preserved across the dielectric interface? Show how Snell���s law is related to these conserved components.

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</p>

To calculate the behavior of the normal components of the field with respect to the interface unit vector Gauss' theorem

<center>

<br>

<math>\int_

\vec \nabla \cdot \vec D dv = \int_

\vec \nabla \times \vec E \cdot \hat n dS = \int_

\vec E \cdot d \vec r</math>

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</center>

• <math>\hat n \times (\vec E_2 - \vec E_1) = 0</math>
  • The tangential component of the electric field is continuous across the interface.
• <math>\vec n \times (\vec H_2 - \vec H_1 = \frac
  Unknown macro: {4 pi}
  Unknown macro: {c}

  \vec K</math>

  • • In the presence of a surface current the tangential component of the magnetic induction changes abruptly.

  Focus on the interface plane, <math>x=0</math>. Boundary conditions dictate relations between the field components on both sides of the interface. The phase of the fields must be equal.

  <center>

  <br>

  <math>( \vec k_1 \cdot \vec r )_

  Unknown macro: {x=0}

  = ( \vec k'1 \cdot \vec r )

  = ( \vec k_2 \cdot \vec r )_

  Unknown macro: {x=0}

  </math>

  <br>

  <math>( \vec k_

  Unknown macro: {1y}

  y + \vec k_

  Unknown macro: {1z}

  z) = ( \vec k'_

  y + \vec k'_

  Unknown macro: {1z}

  z) = ( \vec k_

  Unknown macro: {2y}

  y + \vec k_

  Unknown macro: {2z}

  z)</math>

  <br>

  <math>k_

  Unknown macro: {1y}

  = k'_

  = k_

  </math>

  <br>

  <math>k_

  = k'_

  Unknown macro: {1z}

  = k_

  Unknown macro: {2z}

  </math>

  <br>

  </center>

  Consider an arbitrary vector that lies in the <math>z-y</math> plane. in this case, the following is true.

  <center>

  <br>

  <math>\vec r = (x=0,y,z) = \vec r_

  Unknown macro: {z-y}

  </math>

  <br>

  <math>(\vec k_

  Unknown macro: {1t}

  \cdot \vec r_t ) = (\vec k'_

  \cdot \vec r_t ) = (\vec k_

  Unknown macro: {2t}

  \cdot \vec r_t )</math>

  <br>

  </center>

  The vectors <math>\vec k_1</math>, <math>\vec k'_1</math>, <math>\vec k_2</math> all lie in a place called the plane of incidence. Orient the coordinate system so that the plane of incidence coincides with the <math>x-z</math> plane.

  <center>

  <br>

  <math>\vec E = \vec E e^

  Unknown macro: {i(omega t - k_x x - k_z z)}

  </math>

  <br>

  </center>

  The tangential components of the wavevector are all identical regardless of the medium that they are in and regardless of whether a medium is lossless or absorbing.

  <center>

  <br>

  <math>k_

  = k_

  Unknown macro: {1'z}

  = k_

  Unknown macro: {2z}

  = \beta</math>

  <br>

  <math>|\vec k_1| = |\vec k'_1| = n_1 \vec

  Unknown macro: {omega}
  </math>

<br>

<math>|\vec k_2| = n_2 \vec

\sin \theta_1 = \frac

\sin \theta_2</math>

<br>

</center>

Consider the following dielectric constant of <math>SiO_2</math> glass and quartz: at <math>1kHz</math>, glass=7.2 and quartz=3.3, and the index of refraction for both materials is approximately n=1.5 in the visible range. Explain the observed difference between the two materials, and between the two frequencies.

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</p>

What form do Maxwell���s equations take in a homogenous anisotropic medium? What do the solutions look like in this medium? Draw the dispersion relations for this type of medium. How does this dispersion relation differ from that of a free particle? How is the reflection coefficient calculated for an interface between a medium with refractive index n1 and index n2?

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</p>

MISC.

Calculate the density of states function for a 2D free electron gas. Define the Fermi energy for a 2D electron gas containing N electrons contained in an area A.

<center>

<br>

<math>N = \frac

{\frac

^2}</math>

<br>

<math>N = \left ( \frac

</math>

<br>

</center>

What is the difference between localized and delocalized bloch states?

First of all, a Bloch state is the wavefunction of a particle (e.g., electron) placed in a periodic potential. It consists of the product of a plane wave and a periodic function (Bloch envelope) that has the same periodicity as the potential. This is a wavefunction of an electron which spans throughout the crystal. Thus, by defination Bloch waves are always delocalized. The concept of localized states were introduced in extrinsic semiconductors. On doping, the extra hole or electron (depending on the type of doping) are bound by its center nucleus and thus these electron or holes form localized states, which may become delocalized and hence a bloch electron (or hole) when excited thermally.

<math>\psi_{n\mathbf{k}}(\mathbf

)</math>

Why do the band edges "happen" to be horizontal?

When do we use Maxwell-Boltzmann versus Fermi-Dirac and Bose-Einstein statistics?

In statistical mechanics, Maxwell-Boltzmann statistics describes the statistical distribution of material particles over various energy states in thermal equilibrium, when the temperature is high enough and density is low enough to render quantum effects negligible.
Fermi���Dirac and Bose���Einstein statistics apply when quantum effects have to be taken into account and the particles are considered "indistinguishable".