Page 81 - Nanotechnology an introduction

possible, and also guarantees that single-electron devices are not quantum devices in the sense of Section 7.3. Tunneling of a single electron is a
random event with a rate (probability per unit time) Γ depending solely on the reduction ΔW of the free (electrostatic) energy of the system as a
result of the event and expressed as

(7.8)
where I is the d.c. current (often satisfactorily given by Ohm's law) in the absence of single-electron charging effects. Useful general expressions for
ΔW are [108]

(7.9)
where V and V are the voltage drops across the barrier before and after the tunneling event, or
i
f
(7.10)
where

(7.11)
where C is the capacitance matrix of the system, k and l being the numbers of islands separated by the barrier. Equation (7.8) implies that at low
temperatures (i.e., k T ≪ ΔW) only tunneling events increasing the electrostatic energy and dissipating the difference are possible; their rate is
B
proportional to ΔW, since an increase in the applied voltage increases the number of electron states in the source electrode, which may provide an
electron capable of tunneling into an empty state of the drain electrode (Figure 7.10, upper diagram). Note that the simplicity of the above
equations does not imply that it is necessarily simple to calculate the properties of single-electron systems—several tunneling events may be
possible simultaneously, but we can only get the probability of a particular outcome, implying the need for a statistical treatment to yield average
values and fluctuations of variables of interest. For large systems with very many possible charge states a Monte Carlo approach to simulate the
random dynamics may be the only practical method.

Figure 7.10 Top: energy diagram of a tunnel junction with quasicontinuous spectra; bottom: energy diagram showing discrete levels of the source electrode.
In a nanosized device, electron confinement engenders significant energy quantization (cf. Section 2.5). Experimental work has revealed features
not accounted for by the orthodox theory, in particular cotunneling (the simultaneous tunneling of more than one electron through different barriers at
the same time as a single coherent quantum-mechanical process) and discrete energy levels (Figure 7.10, lower diagram), resulting in a different
energy dependence of the tunneling rate (cf. (7.8)):

(7.12)
where Γ is a constant; the orthodox rate (7.8) is the sum of the rates (7.12) over all the levels of a continuous spectrum of the island. Hence if ΔW
0
≫ k , in the miniature device the tunneling rate is a constant.
B
The “single-electron box” (Figure 7.9) is the simplest single-electron device. The Gibbs free energy of the system is

(7.13)
where n is the number of uncompensated electrons, C is the total islet capacitance and the parameter Q (the “external charge”) is given by
e
Σ
(7.14)
where C is the islet–gate capacitance.
0
The average charge Q = ne of the single-electron box increases stepwise with the gate voltage (i.e., external charge Q ); this phenomenon is called
e
the Coulomb staircase, with a fixed distance ΔQ = e between steps (for E ≫ k T). The reliable addition of single electrons despite the presence
c
e
B
of thousands of background electrons is possible due to the great strength of the unscreened Coulomb attraction.