Calculate the largest diameter a spherical silver nanoparticle

4. (a) Show that the kinetic energy (T) of a 3D Fermi electron gas at 0 K is

T 53 NE F

where N is the number of free electrons and E F 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 V T ]

(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 .

V

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, n i, at 200 and 400 K for Ge and GaAs knowing that n i 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, n 0, and holes, p 0, is related to the intrinsic carrier concentration via the mass action law n 0 p 0 = n i 2 . 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 N d

= 5 x 1016 cm‐3 and N a = 5 x 1016 cm‐3, respectively. When calculating n i for silicon use a total effective number of available states per unit volume, N s [N s = (N v N c)1/2 where N v and N c are the effective density of

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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

N

1

(N d N a )2

4n i2

when N d > N a (n‐type)

n o

d

a

2

2

(c) Plot (using MatLab or another plotting progr am) the Ferm i energy E F w ith respect t o the intrinsic

Fermi level E Fi for n‐type silicon at 300 K for a donor concentration varying between 1 015 cm‐3 and 1 020 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.]

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