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Problems
Q1. Buckyball FETs In 2000, Park et al1 reported measurements of a buckyball (C60)
FET. An approximate model of their device is shown in Fig. P5.1. The measured
conductance as a function of VDS and VGS is shown in Fig. P5.2.
+

VGS
gat
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Ohms law3
What happens when we increase the size of a conductor? Eventually, we quantum
phenomena should transform into the familiar model of classical conduction: we should
obtain Ohms law.
V = IR
(4.17)
A linear relationship between V and I is
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2 l
.
(6.98)
a0
As shown in Fig. 6.47, this set of allowed k values does not include the K points. Thus
(0,4) tubes are insulating/semiconducting.
2 3k x + 6k y =
Bandstructure of carbon nanotubes
Since the conduction properties of graphene are
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Bandstructure of bulk semiconductors
As stated above, most of the common semiconductors are constructed from sp3hybridized atoms assembled in the diamond crystal structure.
Unfortunately, sp3hybridization makes the bandstructure calculation muc
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Comparison of ballistic and nonballistic MOSFETs.
To determine whether we should use the ballistic or semiclassical models of charge
transport we need to know the likelihood of electron scattering in the channel. This
depends on the channel le
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The tight binding approximation
Each atom in a conductor typically possesses many electrons. We can simplify molecular
orbital calculations significantly by neglecting all but a few of the electrons. The basis for
discriminating between the elec
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Solving for the energy
Considering the tight binding matrix of Eq. (6.39), non trivial solutions for the weighting
factors, c1 and c2 are obtained from
E
12
(6.40)
det 1
=0
21 2 E
Lets assume that the hopping interactions are equal 12 = 21 = .
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Tetrahedral alignment with four neighbors (sp3 hybridization)
Consider a central atom with four equispaced neighbors. Repulsion between these atoms
will push them to the points of a tetrahedron; see Fig. 3.28.
z
(+1,+1,+1)
y
x
(1,1,+1)
(1,+1,
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Thus, the splitting increases with the similarity in energy of the participating atomic
orbitals, i.e. the bonding orbital becomes more stable. This is a general attribute of the
interaction between two quantum states. The more similar their int
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Polyacetylene
Next, lets consider a longer chain of carbon atoms. Very long molecules are known as
polymers, and a polymer equivalent of the idealized conductor in Fig. 6.28 is known as
polyacetylene.
Specifically, lets solve for a carbon chain
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Molecular orbitals
Unfortunately, even when we apply the BornOppenheimer approximation and hold the
nuclear coordinates fixed, the solution to the Schrdinger equation (Eq. (A.2.5) is
extremely complex in all but the simplest molecules. Usually
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Simple cubic lattice
Face centered cubic lattice
Diamond lattice
a3
a1
a3
a1
a2
z
x
a1
a2
a3
z
x
y
a2
z
x
y
y
Fig. 6.34. A diamond lattice is simply a facecentered cubic lattice with a two atom
unit cell (outlined in red).
Bloch functions: wave
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Problems
Q1. Consider the conventional MOSFET illustrated below:
V2
 +
+ V1

The transistor is off when V2 = 0V. Assume VT = 1V and the transistor turns on for V1 >
VT.
Sketch the expected IV characteristics (I vs. V1) and explain, with refere
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The complete hydrogen atom wavefunction
Finally, we allow the radius of the electron to vary. The full kinetic energy operator is
2
1 d2
1
(A.1.30)
T =
r + 2 2
2
r
2me r dr
and since the radius varies, we can no longer set the potential to zer
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Part 7. Fundamental Limits in Computation
This course has been concerned with the future of electronics, and especially digital
electronics. At present digital electronics is dominated by a single architecture,
Complementary Metal Oxide Semicond
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i.e. for hydrogen the ground state radial wavefunction R(r) ~ exp(r/a0).
The principal quantum number also specifies the range of the orbital angular momentum
quantum number
l = 0, 1, ., n 1
(A.1.38)
Labeling Atomic Orbitals
Atomic orbitals are
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Appendix 1
The hydrogen atom: electron on a circle model
Hydrogen is the simplest element. There are just two components: an electron, and a
positively charged nucleus comprised of a single proton.
In a primitive model of the hydrogen atom, cons
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of empty states in the channel accessible to electrons from the source. In the limit that
there are no available states, the channel is a perfect insulator.
Switching between ON and OFF states is achieved by using the gate to push empty
channel
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where Lx is the length of the channel, and vx is the velocity component of the electron
parallel to the sourcedrain current. It is important to note that in 1d, 2d and 3d
conductors the transit time is dependent on the energy of the electron
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Part 6. Atomic orbitals and molecular bonds
The particle in the box approximation completely ignores the internal structure of
conductors. For example, it treats an insulator such as diamond the same as a conductor
such as gold. Despite this it
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Part 5. Field Effect Transistors
Field Effect transistors (FETs) are the backbone of the electronics industry. The
remarkable progress of electronics over the last few decades is due in large part to
advances in FET technology, especially their
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Brief notes on information theory and the thermodynamics of computation
We now examine the thermodynamics of computation.
(i) Minimum energy dissipated per bit
Assume we have a system, perhaps a computer, with a number of possible states. The
un
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The future of electronics?
The immediate path is clear: we have not yet reached the limits of scaling, or the
fundamental limits of field effect transistors. The electronics industry will push to smaller
length scales to minimize the power delay
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Switching Speed
The dynamic model of Fig. 7.4 relates the switching speed to the charging and
discharging time of the gate capacitor.
I
f max =
(7.5)
CVDD
Thus, switching speed can be improved by
increasing the on current of the transistors
(i)
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The quantum limit of conductance
Weve seen that in a quantum wire, current flow requires a difference in the quasi Fermi
levels for electrons moving with and against the current. Furthermore, only electrons
between the quasi Fermi levels, i.e. F
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Part 4. Two Terminal Quantum Wire Devices
Lets consider a quantum wire between two contacts. As we saw in Part 2, a quantum
wire is a onedimensional conductor. Here, we will assume that the wire has the same
geometry as studied in Part 2: a rec
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Crystals and periodic molecules
The particle in a box approximation is too crude for most problems. Tight binding, on the
other hand, is often quite computationally intensive. But fortunately, we can make
simplifications when the material is per
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Conventional MOSFETs:
Finally, we turn our attention to the backbone of digital electronics, the nonballistic
metal oxide semiconductor field effect transistor (MOSFET).
The channel material is a bulk semiconductor typically silicon. Here, we w
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Problems
Q1. (i) Consider two identical balls each 1cm in diameter and of mass m = 1g. One is
kept fixed, and the second is dropped directly on it from a height of d = 10cm. From the
uncertainty principle alone, what is the expected number of ti
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Reversible computers
In the previous section, we defined computation as a process that increases information
and decreases uncertainty. But if uncertainty (i.e. entropy) decreases within the computer,
entropy must increase outside the computer.