Nanotechnology-Enabled Sensors

6.2 Density and Number of States 295
Obviously, the Fig. 6.5 graphs can be utilized to visualize how the properties
of nanomaterials can be quantized when the dimensions of a material
are reduced to quantum sizes.
For instance, the conductance of a 1D structure can be derived from the
DOS. In a 1D structure (such as a nanowire) with a length of L, if a potential
equal to V is applied, for a mobile electron the energy difference (ΔE)
between two sides of this 1D structure is equal to:
ΔE = eV/L. (6.27)
where e is an electron charge. Now, if q is charge (equal to –e for an electron),
and v is the velocity, the net current, I, flowing through the channel
will be:
I = N(E)qv = D(E)ΔE qv, (6.28)
where D(E) and N(E) are the density of states and the number of states in a
1D case, respectively, for which D(E) was presented by Eq. (6.24). Replacing
vi which is the velocity of an electron from Eq. (6.1) in sub-band i
the DOS can be obtained as: 3
D(
E)
1 D
L ⎛ m ⎞
dE = ⎜ ⎟ 2 π ⎝ � ⎠
1/
2
⎛
⎜
1
⎜
⎝ ( E − Ei
)
1/
2
⎞
⎟
4L
⎟
= , (6.29)
⎠ hvi
where Ei is the energy in each sub-band and m is the mass for an electron.
From Eqs. (6.28) and (6.29) we can obtain the equation defining current, I,
as:
I = 2e 2 V/h. (6.30)
Interestingly, Eq. (6.30) shows that in a 1D structure, the current only
depends on the voltage across it via fundamental constants. Consequently,
the two-terminal resistance is calculated to be 12.906 kΩ for such a 1D
structure.
As can be observed, an ideal 1D structure has a finite resistance. This is
called a resistance quantum (or the inverse conductance quantum). Such a
resistance can also be experimentally measured. At the end of the 1980s,
the University of Cambridge and Delft groups reported the measurements
of steps in the conductance of a quasi 1D configuration in a field effect
transistor structure by means of the voltage applied at the gate as shown in
Fig. 6.6. 4,5
All the above assumptions are for near zero degree Kelvin temperatures.
At higher temperatures a thermal effect also appears which smoothes out
the step shape.