1. Calculate the largest diameter a spherical silver nanoparticle can have for there to be no more than 109 electron energy states over a 1 eV interval at an energy of 2 eV.
2. Determine the temperature at which a perfect cube of gold (with a volume of 27 nm3) becomes quantum confined (via the metal-to-insulator transition).
3. The wave function, (x), of an electron trapped in a potential well with a finite depth and width, L, is given by:
(x) AeCx for x 0 for 0
(x) F sin(kx) Gcos(kx) xL
(x) BeCx for x L
(a) Given that F = G = 5, k = 1, and C= 2 plot (using MatLab or another plotting program) the wave function over the range -2 ≤ x ≤ L + 2.
(b) Given that (L) = (0), determine the values of L and B if the wave function corresponds to the third lowest energy state.
4. (a) Show that the kinetic energy (T) of a 3D Fermi electron gas at 0 K is
T 3/5 NEF
where N is the number of free electrons and EF is the Fermi energy.
(b) Using the result from (a) derive a relation connecting the pressure and volume of an electron gas at 0 K.
[Hint: At absolute zero all processes are at constant entropy (Third Law) meaning p T/V ]
(c) Derive an expression for the bulk modulus, B, of an electron gas at 0 K in terms of Fermi energy and volume knowing B V p .
5. (a) Show for a simple square lattice (in 2D) that the kinetic energy of a free electron at a corner of the first zone (i.e., first Brillouin zone) is higher by a factor of 2 than that of an electron at the midpoint of a side face.
(b) What is the corresponding factor for a simple cubic lattice in 3D?
6. (a) Calculate the intrinsic carrier concentration, ni, at 200 and 400 K for Ge and GaAs knowing that ni at 300 K for Ge and GaAs has a value of 2.4 x 1013 cm-3, and 1.79 x 106 cm-3, respectively.
(b) The thermal equilibrium concentration of electrons, n0, and holes, p0, is related to the intrinsic carrier concentration via the mass action law n0 p0 = ni2. Determine the thermal equilibrium concentration of electrons and holes in an n-type silicon semiconductor at T = 300K which has a donor and acceptor concentration of Nd = 5 x 1016 cm-3 and Na = 5 x 1016 cm-3, respectively. When calculating ni for silicon use a total effective number of available states per unit volume, Ns [Ns = (NvNc)1/2 where Nv and Nc are the effective density of states in t he valence band and con duction band, respectively ], of 2.76 x 1019 cm-3. The following equation will be helpf ul:
N/no( N 1 /d) a/2 1/2√(Nd Na)2 4ni2 when Nd > Na (n-type)
(c) Plot (using MatLab or another plotting progr am) the Ferm i energy EF w ith respect t o the intrinsic Fermi level EFi for n-type silicon at 300 K for a donor concentration varying between 1015 cm-3 and 1020 cm-3. Wh at is significant about the donor concen tration of 3 x 1019 cm-3?
7. Imagine you are a research scientist at a local start-up and a senior R&D scientist puts you in charge of a new pro ject focused on measuring and following local neuronal firing ev ents along the axon of sin gle neurons in the brain. A fter some thought, and knowing that neuron actio n potentials travel as stro ng positive potentials do wn the neuron (see figure below), you decide to fabricate ultrase nsitive nanowire field-effe ct transistors (FETs) that can be positioned on the s urface of the neuron. After testing various devices y ou find that silicon nanowire FETs fit the bill and yo u create both p-type and n-type nanow ire FET arrays.
(a) Assumi ng you have ohmic metal-semiconductor contacts, draw out a typical current- voltage (I-V) plot for a single p-type and n-type F ET when the voltage is sw ept from -2 V to +2V for three different gate voltages (+10 V, 0 V, -10 V). [you should have 2 plots each with three different traces correspo nding to the g ate voltages]
(b) Now you have interfa ced the FETs with living neuronal tissue. If the neuro ns have a firing rate of 100 Hz, draw out conductivity time-courses (i.e., electrical conductivity through n anowire vs time) for both the p-type and n-type FETs over a 100 ms recording time. [Plots for a and b are only qualitative since we don't know the actual resistance of th e devices.]
