9.5. SINGLE-ELECTRON TUNNELING
247
where E / E O is the dimensionless dielectric constant of the semiconducting material
that forms the dot, and c0 = 8.8542 x 10-'2F/m is the dielectric constant of free
space. For the typical quantum dot material GaAs we have & / E O = 13.2, which gives
the very small value C = 1.47 x lo-'' r farad for a spherical shape, where the
radius r is in nanometers. The electrostatic energy E of a spherical capacitor of
charge Q is changed by the amount AE - eQ/C when a single electron is added or
subtracted, corresponding to the change in potential AV = A E / Q
AV = e/C EX 0.109/r volts
(9.13)
where r is in nanometers. For a nanostructure of radius r = lOnm, this gives a
change in potential of 11 m\! which is easily measurable. It is large enough to
impede the tunneling of the next electron.
Two quantum conditions must be satisfied for observation of the discrete nature of
the single-electron charge transfer to a quantum dot. One is that the capacitor singleelectron charging energy $/2C must exceed the thermal energy k,T arising from
the random vibrations of the atoms in the solid, and the other is that the Heisenberg
uncertainty principle be satisfied by the product of the capacitor energy e2/2C and
the time z = RTC required for charging the capacitor
(9.14)
where RT is the tunneling resistance of the potential barrier. These two tunneling
conditions correspond to
e2 >> kBT
h
e2
RT >> -
(9.15a)
(9.15b)
where h/e2 = 25.8 13 kS1 is the quantum of resistance. When these conditions are
met and the voltage across the quantum dot is scanned, then the current jumps in
increments every time the voltage changes by the value of Eq. (9.13), as shown by
the I-versus- V characteristic of Fig. 9.18. This is called a Coulomb blockade because
the electrons are blocked from tunneling except at the discrete voltage change
positions. The step structure observed on the I-V characteristic of Fig. 9.18 is
called a Coulomb staircase because it involves the Coulomb charging energy e2/2C
of Eq. (9.15a).
An example of single-electron tunneling is provided by a line of ligand-stabilized
Au,, nanoparticles. These gold particles have what is referred to as a structural
magic number of atoms arranged in a FCC close-packed cluster that approximates
the shape of a sphere of diameter 1.4 nm, as was discussed in Section 2.1.3. The
cluster of 55 gold atoms is encased in an insulating coating called a ligand shell that
is adjustable in thickness, and has a typical value of 0.7nm. Single-electron
247
where E / E O is the dimensionless dielectric constant of the semiconducting material
that forms the dot, and c0 = 8.8542 x 10-'2F/m is the dielectric constant of free
space. For the typical quantum dot material GaAs we have & / E O = 13.2, which gives
the very small value C = 1.47 x lo-'' r farad for a spherical shape, where the
radius r is in nanometers. The electrostatic energy E of a spherical capacitor of
charge Q is changed by the amount AE - eQ/C when a single electron is added or
subtracted, corresponding to the change in potential AV = A E / Q
AV = e/C EX 0.109/r volts
(9.13)
where r is in nanometers. For a nanostructure of radius r = lOnm, this gives a
change in potential of 11 m\! which is easily measurable. It is large enough to
impede the tunneling of the next electron.
Two quantum conditions must be satisfied for observation of the discrete nature of
the single-electron charge transfer to a quantum dot. One is that the capacitor singleelectron charging energy $/2C must exceed the thermal energy k,T arising from
the random vibrations of the atoms in the solid, and the other is that the Heisenberg
uncertainty principle be satisfied by the product of the capacitor energy e2/2C and
the time z = RTC required for charging the capacitor
(9.14)
where RT is the tunneling resistance of the potential barrier. These two tunneling
conditions correspond to
e2 >> kBT
h
e2
RT >> -
(9.15a)
(9.15b)
where h/e2 = 25.8 13 kS1 is the quantum of resistance. When these conditions are
met and the voltage across the quantum dot is scanned, then the current jumps in
increments every time the voltage changes by the value of Eq. (9.13), as shown by
the I-versus- V characteristic of Fig. 9.18. This is called a Coulomb blockade because
the electrons are blocked from tunneling except at the discrete voltage change
positions. The step structure observed on the I-V characteristic of Fig. 9.18 is
called a Coulomb staircase because it involves the Coulomb charging energy e2/2C
of Eq. (9.15a).
An example of single-electron tunneling is provided by a line of ligand-stabilized
Au,, nanoparticles. These gold particles have what is referred to as a structural
magic number of atoms arranged in a FCC close-packed cluster that approximates
the shape of a sphere of diameter 1.4 nm, as was discussed in Section 2.1.3. The
cluster of 55 gold atoms is encased in an insulating coating called a ligand shell that
is adjustable in thickness, and has a typical value of 0.7nm. Single-electron