+1-415-670-9189
info@expertsmind.com
Characterization technology for nanomaterials
Course:- Other Engineering
Reference No.:- EM13215




Assignment Help
Assignment Help >> Other Engineering

Question 1

(a) For the reciprocal lattice, we know that

Exp(jG.R) = 1

where G.R = 2Πt, t= an integer;

R= la+mb+nc = real-space/direct translational vector; l, m and n= integers;

G = hp+kq+lr = reciprocal lattice vector normal to the plane in the direct lattice having Miller indices (hkl);

and

1990_characterization technology for nanomaterials.png


(i) Calculate the reciprocal lattice of the body-centred cubic (bcc) structure.

(ii) Show that the reciprocal of the face-centred cubic (fcc) structure is itself a bcc structure.

(b) Draw (101) and (111) planes in a cubic unit cell. Determine the Miller indices of the directions which are common to both the planes.

(c) A plane makes intercepts of 1, 2 and 0.5 Å on the crystallographic axes of an orthorhombic crystal with a:b:c = 3:2:1. Determine the miller indices of the plane

 

(a) Given that 622_characterization technology for nanomaterials1.pngand a one-dimensional crystal of lattice constant a, which obeys

483_characterization technology for nanomaterials2.png ), calculate the effective mass of electrons for this energy dispersion relationship.

(b) The following is the E-k dispersion diagram for GaAs

106_characterization technology for nanomaterials3.png

Show that the motion of electrons near the conduction band minimum is isotropic.

(c) For the diagram in part (b) above, explain why two of the allowed energy valence bands (V1 and V2) are referred to as "heavy hole" and "light hole" bands, respectively.

(d) With the aid of simple diagrams show how the presence of a deep acceptor state can reduce the conductivity of n-type semiconductor.


Question

Consider a one-dimensional semiconductor with the following expressions for the energy as function of k in the conduction (Ec(k)) and valence (Ev(k)) band:

538_characterization technology for nanomaterials4.png

where a is the lattice constant, and EG and E0 are positive constants.

In the first Brillouin zone we know that -Π/a ≤ k ≤ Π/a .

(a) What is the minimum energy in the conduction band, and where can it be found in the Brillouin zone?

(b) What is the minimum energy in the valence band, and where can it be found in the Brillouin zone?

(c) What is the band gap energy of this material?

(d) Sketch the band structure in the first Brillouin zone, explicitly showing the energies in each band at the zone centre and the zone edge.

(e) Is this material a direct or indirect band semiconductor?

(f) What is the conduction band effective mass at the conduction band minimum?

(g) What is the conduction band effective mass at k=0?

Suppose an electron in the conduction band can be described by a wavepacket made up from k-states in the small range of k round about k= Π/a. what is the velocity of motion of this

Question

Assume that the E(k) relationship for electrons in the conduction band of a hypothetical tetravalent n-type semiconductor can be approximated by

E= ak2 + constant.

Given 2177_characterization technology for nanomaterials6.png (in 1D) and cyclotron resonance frequency

The cyclotron resonance for electrons in a field of B=0.1T occurs at an angular rotation frequency of ωc =1.8 x 1011 rad.S-1.

(a) Find the value of a

(b) Assume that the semiconductor is doped with pentavalent donors. Estimate the number of donors per cubic metre, given that the Hall coefficient at room temperature RH= -6.25x10-6 m3C-1 and the relative dielectric constant .

Optical excitation of intrinsic Ge creates an average density of 1012 conduction electrons per cubic centimetres in the material at liquid

nitrogen temperature. At this temperature, the electron and hole mobilities are equal, = 5000 cm2/Vs. The Ge relative dielectric constant is 20. We also know that conductivity , and the Einstein diffusion coefficient (D) relationship 1291_characterization technology for nanomaterials8.png where all terms have their usual
meaning.

(c) If 100 volts is applied across a 1 cm cube of Ge crystal under these conditions, approximately how much current is observed?

(d) What is the approximate value of the diffusion coefficient for the electrons and holes?

Question 5

(a) The electron and hole mobilities in a Si sample are 0.135 and 0.048 m2/V-s respectively. Determine the conductivity of intrinsic Si at 300 K if the intrinsic carrier concentration is 1.5×1016 atoms/m3. The sample is then doped with 1023 phosphorous atoms/m3.

Determine the equilibrium hole concentration, conductivity and position of the Fermi level relative to the intrinsic level.

(b) The resistivity of an intrinsic semiconductor is 4.5 Ωm at 20°C and 2 Ωm at 32°C. Calculate the energy band gap.

Question 6

(a) The energy near the valence band edge of a crystal is given by

E= -Ak2 

where A= 10-39 Jm2 . An electron with wave vector k= 1010 kx m-1 is removed from an orbital in the completely filled valence band. Determine the effective mass, velocity, momentum and energy of the hole.

(b) In intrinsic GaAs, the electron and hole mobilities are 0.85 and 0.04 m2 /V-s, respectively and the corresponding effective masses are 0.068 m0 and 0.5 m0, where m0 is the rest mass of an electron. Given the energy band gap at 300 K as 1.43 eV, determine the intrinsic carrier concentration and conductivity.

Question 7

(a) An electron is presumed to be represented by a wavepacket centred initially around k=0 in the otherwise empty conduction band of a hypothetical one-dimensional crystal. The conduction band of the crystal has the dispersion relation between energy E and wavevector k E = A[1- cos(ka)], where A is a constant and a is the lattice constant. An electric field E is applied along the axis of the crystal (the x-axis) in the positive direction. Assuming that the crystal is perfect and that there is no scattering of the electron, write an expression for the velocity v of the electron as a function of  time. Remember group velocity,1060_characterization technology for nanomaterials9.png


(b) Suppose we have a crystal that has orthorhombic symmetry (three mutually perpendicular axes in the x, y and z directions, but all of different lengths). As a consequence of the crystal symmetry, there is now no requirement that the properties of the crystal should look the same in the three crystal directions. The crystal has one conduction band minimum centred at k=0 (i.e. in the middle of the Brillouin zone), and the electron energy in that minimum obeys the equation

2230_characterization technology for nanomaterials10.png

This corresponds to a constant energy surface that is ellipsoidal, but is not an ellipsoid of revolution; the cross-sections in each of the three coordinate directions are ellipses.

What is the density-of-states effective mass mDS in this material? (In other words, what would be the effective mass mDS for a hypothetical spherically symmetric band that would give the same total number of states with energy less than some particular energy E0?)

You may wish to consider the number of k states contained within an ellipsoidal volume of k space, and compare that to the number within a spherical volume. The volume of an ellipsoid described by the equation

2426_characterization technology for nanomaterials11.png

Question 8

(a) The electron is represented by a wave function y for which E = hω = hν where ω=2πν= corresponding radian frequency of the wave. Show that the group velocity (vg) and effective mass (m*) of the electron in one dimension are given by the following expressions:


(b) A hypothetical semiconductor has a band structure in one dimension of the form E = Eo[1-cos(ka)],

where Eo= 1.5 eV and a = 5Å. What are the group velocity and effective mass at the following points in this band structure?

(i) k=0;

(ii) k=π/4a;

(iii) k= π/2a;

(iv) k= π/a.

(c) An electron is presumed to be represented by a wavepacket centered initially round k = 0 in the otherwise-empty conduction band of a hypothetical one-dimensional crystal. The conduction band of the crystal has the dispersion relation between energy E and wavevector k

E = A[1-cos(ka)],

where A is a constant, and a is the lattice constant. An electric field F is applied along the axis of the crystal (the x axis) in the positive direction. Assuming that the crystal is perfect and there is no scattering of the electron, write an expression for the velocity, v, of the electron as a function of time.




Put your comment
 
Minimize


Ask Question & Get Answers from Experts
Browse some more (Other Engineering) Materials
Construct the circuit with MultiSIM using a 5% tolerance for the resistor and the capacitor. Adjust the frequency according to the table below and obtain measurements from M
What are the processes for developing a new mine in Western Australia -  Schedule production for the mine detailed below to optimize copper concentrate at minimum mining cost
Digital Electronics Assignment Combinational/Sequential Logic design with Integrated Circuits (Dual in line package) Car wash concept with the following steps in a Combination
A 100-kW, three-phase, 60-Hz, 460-V, six-pole wound-rotor induction motor develops its rated full-load output at a speed of 1158 r/min when operated at rated voltage and fre
Design a digital clock according to the following specifications: The clock will display time in the standard 12 hour format HH:MM with an indicator for AM/PM. Your clock wi
Temperature of outlet water, 25.3SoC Partial pressure of water in inlet air, 6.27 ton; and in outlet air, 20.1 torr The value for the diffusivity of water vapor in air is 0.
You are the Lead Controls Engineer for Gniewkowski Engineering Company (GEC), Attached is a Sequence of Operations that gives you a basic idea of how the systems/units operat
objectives of Vibration Analysis of Gear Fault - The aim of this experiment is to conduct vibration analysis for a motor-gearbox system with some intended damaged on the gear