Low power Project 2,3 and Nanotechnology Hw 3, project2
Omar Abughori
nanotech_sp18_hw3.pdf
1
EE-451: Introduction to Nanotechnology, Spring’18, Homework #3
1.(10') In quantum dot, due to quantum confinement, energy of photon emitted can be calculated as
2
2
8 r
h E
c hhE g
where ν: photon frequency, λ: photon wavelength, c: speed of light (c=3×108m/s), h: Planck’s constant (h=6.63×10-34m2kg/s), Eg is the gap energy in bulk semiconductor, r is the quantum dot radius, µ is reduced mass of the exciton,
**
**
he
he
mm
mm
, where me* and mh* are effective masses of electron and hole.
For semiconductor CdSe, me*=0.13m0, and mh*=0.45m0, where m0 is the rest mass of electron, m0=9.1×10-31kg. Energy gap of bulk CdSe material: Eg=1.74eV (1eV=1.6×10-19J). Quantum dots of CdSe passivated with ZnS are used to produce fluorescence of different QD colors. Determine the size of the CdSe core required (i.e. radius r=?) to produce fluorescence at a wavelength of λ=400nm (violet). 2.(20’) A particle of mass m in one-dimensional space is in the state
])/[( 2),( itmxaAetx where ψ(x,t) is the wave function of particle, A and a are positive real constants, x is location, t is time, i is imaginary unit, )2/( h , where h is Planck’s constant. (1). Find A=? (Hint: Wave function ψ(x,t) should satisfy normalization condition. Thus at
t=0, we should have: 1)0,( 2
dxx . Furthermore, based on Gaussian integrals, we
have:
20 / 22 adxe ax , where a is any constant.)
(2). Wave function should satisfy Schrodinger’s equation:
),()( ),(
2
),( 2
22
txxV x
tx
mt
tx i
Based on Schrodinger’s equation, find out the potential field V(x) applied on the particle. (3). Calculate the expectation values of x, x2, and momentum p. That is, find <x>=? <x2>=? <p>=? Hint: The expectation value (or average value) of a function f(x) is:
dxxxfxf
2 )()()( , where ψ(x)=ψ(x,0) (i.e. t=0). Thus:
dxxxx
2 )( ; dxxxx
222 )( , dt
xd mp .
Gaussian integrals equation:
3.(20') 1). What is quantum computing? What is quantum computer? What is a “qubit”? What’s the difference between qubit and a classical digital bit? If we have 6-bit classical digital memory, how many patterns can we store at any moment? If we replace them with 6-bit qubits, how many patterns can we store at any moment?
2
2). In quantum computation, assume input state as 1| 2
3 0|
2
1 in
. After passing through
X, Z and H gates in turn, find out the quantum state of the output out using matrix calculation. Clearly show your calculation step by step, including the states of ΨX and ΨZ after passing through X and Z gates.
Hint: The input wave function can be expressed as state matrix:
2/3
2/1 1|
2
3 0|
2
1 in
.
For quantum gate: (Output state matrix) = (Quantum gate matrix) × (Input state matrix). 3). A quantum XOR gate is represented as a unitary operator:
In quantum XOR gate, the first particle acts as a conditional gate to flip the state of the second particle. The state of the second particle corresponds to the action of XOR gate. The quantum circuit diagram of the XOR gate is shown as below.
Assume the input state of 2-bit qubits is:
0
5/4
0
5/3
10| 5
4 00|
5
3 AB
. Use matrix operation to find the output state of the 2-bit
qubits ( ?'AB ). 4. (15') 1). What is break-junction technique for molecular electronics? Briefly explain the procedures of the technique. How can we control the electrode gap in it? Can we use steel as substrate for it? Why? When we measure the conductance of benzene-1,4- dithiolate molecule using break-junction technique, how can we judge the point when a single molecule is connecting two gold electrodes? 2). What is sequential tunneling and what is coherent tunneling in molecular electronics? 3). Briefly explain the working principle of a Catenane-based molecular switch. Use a graph to help explain the concept. Why does the I-V characteristics of catenane molecular switch show hysteresis? Please briefly explain. 4). Does a single molecule have the same conductivity as its bulk material? When we use STM tip to probe the conductance of a single molecule, list a method which can help the molecule to be probed to be perpendicularly aligned to the substrate surface. Please use a figure to show the concept. Can we use conductive molecules to prop up the molecule to be probed? Why? What is the requirement on the length of the molecules used to prop up the molecule to be probed? 5.(20') 1). What is electron pump and what is it used for? Assume one electron is pumped through the electron pump in every 1ms (1ms=10-3sec), what is the resulted current in
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unit of A? (Note: 1e=1.6×10-19C). If we use it as a current standard to measure other current, can we achieve very high resolution? 2). The requirement for a particle with diameter d to work as a single-electron transistor (SET) using Coulomb blockade is that the energy U required to place a single electron charge on it must be much greater than its thermal energy kBT:
Tk d
e U B
0
2
4
Where dielectric constant of vacuum ε0=8.85×10-12F/m, electron charge e=1.6×10-19C, Boltzmann’s constant kB=1.38×10-23 m2·kg·s-2·K-1·C, T is absolute temperature. Based on above information, determine the maximum particle diameter dmax of a gold nanoparticle to work as a single-electron transistor using Coulomb blockade at room temperature (T=300K)? 6.(15’) Nanowire crossbar architecture is a most promising implementation for future nanoelectronic VLSI circuits. At each crosspoint of horizontal and vertical nanowires, a p-n junction can be programmed to be either connected or disconnected. In this way, the crossbar structure can be programmed to implement different logic functions. As shown in Figure 1, a full adder circuit is implemented with nanowire crossbar structure. For input pattern ABCin=011, clearly mark the state (“ON” or “OFF”) of each p-n junction in the figure, and mark the voltage level at each output line of the AND plane. Also mark the voltage level at each output line of the OR plane (i.e. Sum, Sum’, Cout, Cout’). What is the value of each output (logic “0” or “1) in the circuit for this given input pattern? Is it the correct function of a full-adder?
Figure 1. A full-adder implemented with nanowire crossbar circuit
Due on 04/18/2018, Wednesday in class.
