Page 95 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 83

(2DEG) regions, Figure 3-3 (a), (b). A rendition of the first experimental
demonstration of the effect is shown in Figure 3-3(c). It is observed that the
conductance decreases approximately linearly as the gate voltage is
increased negatively, i.e., as the constriction or channel width narrows. In
particular, at V G=-2.2V, the channel is pinched-off and the conductance is
zero. Notice also, that the conductance decreases in discrete steps of e2 2 h .
An explanation of the observed quantized conductance was attributed to
the resistance of the constriction upon comparison with the semi-classical
formula for the conductance of a constriction in a 2DEG, denoted G S, after
Sharvin who derived it [68]. G S is given by,

G = e 2 dN 2 D v W , (8)
S π dE F

where dN 2D dE = m * = π is the quantum mechanical density of states,
including a factor of two for spin, v = k = m is the Fermi velocity, with
*
F F
k = 2 π λ = 2 n π being the Fermi vector and n S the 2DEG electron
F F S
density, and W is the width of the constriction. Rewriting (65) so that the
quantized conductance becomes explicit, one obtains,

2
G = 2 e 2 k F W = 2 e 2 W . (9)
S π λ
h h
F
The fact that this equation includes the ratio W λ suggested that,
F
experimentally, there should be deviations due to the manifestation of the
wave nature of electrons whenever λ ~ W . In particular, it was determined
F
that the plateau values of conductance are obtained whenever W is an
integral multiple of λ / 2 . Therefore, the quantized conductance is a
F
manifestation of the wave nature of electrons in that as the voltage is
increased from pinch-off, a new mode (band) for transport becomes
available every time the constriction widens by λ / 2 . The transmission
F
coefficient of the constriction captures this [115]. The deviations from
flatness of the conductance plateaus were attributed to scattering or to the
abruptness of the constriction. Finally, as the temperature increases, the
conductance steps smear out until at high temperature they disappear. This is
due to the non-monoenergetic, wider, distribution of electrons launched by
the reservoirs into the constriction [68] and exposes one of the practical
limitations of QPCs, namely, that their utilization requires extremely low
temperatures.