416
14 Nanostructures
(a)
(b)
Fig. 14.25 a Schematic drawing of a quantum dot (QD) with tunnel contacts and gate electrode. The inset depicts an
equivalent circuit with capacitances. b Realization with an in-plane gate structure. The distance between ‘F’ and ‘C’
(gate electrode) is 1 µm. Electron transport occurs from a 2DEG between 3/F to 4/F through the quantum points contacts
1/3 and 2/4. Part (b) from [1386], reprinted with permission, ©1991, Springer Nature
14.4.2 Electrical and Transport Properties
The classical electrostatic energy of a quantum dot with capacitance C G that is capacitively coupled
to a gate (Fig. 14.25) at a bias voltage V G is given by
E =
Q
2
2C G
− Q α V G ,
(14.17)
where α is a dimensionless factor relating the gate voltage to the potential of the island and Q is the
charge of the island.
Mathematically, minimum energy is reached for a charge Q min = α C G V G . However, the charge
has to be an integer multiple of e, i.e. Q = N e. If V g has a value, such that Q min /e = N min is an
integer, the charge cannot fluctuate as long as the temperature is low enough, i.e.
kT
e
2
2 C G
.
(14.18)
Tunneling into or out of the dot is suppressed by the Coulomb barrier e
2
/2C G , and the conductance is
very low. Analogously, the differential capacitance is small. This effect is called Coulomb blockade.
Peaks in the tunneling current (Fig. 14.26b), conductivity (Fig. 14.26a) and the capacitance occur, when
the gate voltage is such that the energies for N and N + 1 electrons are degenerate, i.e. N min = N +
1
2
.
The expected level spacing is
e α αV G =
e
2
C G
+ N ,
(14.19)
where N denotes the change in lateral (kinetic) quantization energy for the added electron. e
2
/C
will be called the charging energy in the following.
A variation of the source–drain voltage (for a given gate voltage) leads to a so-called Coulomb
staircase since more and more channels of conductivity contribute to the current through the device
(Fig. 14.27). In Fig. 14.28 the tunneling current is shown as a function of the potential of a dot, formed
by gates on a WSe 2 flake [1388], and the source-drain voltage. The iso-current lines form so-called
‘Coulomb blockade diamonds’.
The charge Q on the QD is determined by the charges on the gate, source and drain capacities,
Q = Q G − Q S + Q D . Together with Q S = C S V S , Q D = C D V D and V SD = V S + V D and Q G =
14 Nanostructures
(a)
(b)
Fig. 14.25 a Schematic drawing of a quantum dot (QD) with tunnel contacts and gate electrode. The inset depicts an
equivalent circuit with capacitances. b Realization with an in-plane gate structure. The distance between ‘F’ and ‘C’
(gate electrode) is 1 µm. Electron transport occurs from a 2DEG between 3/F to 4/F through the quantum points contacts
1/3 and 2/4. Part (b) from [1386], reprinted with permission, ©1991, Springer Nature
14.4.2 Electrical and Transport Properties
The classical electrostatic energy of a quantum dot with capacitance C G that is capacitively coupled
to a gate (Fig. 14.25) at a bias voltage V G is given by
E =
Q
2
2C G
− Q α V G ,
(14.17)
where α is a dimensionless factor relating the gate voltage to the potential of the island and Q is the
charge of the island.
Mathematically, minimum energy is reached for a charge Q min = α C G V G . However, the charge
has to be an integer multiple of e, i.e. Q = N e. If V g has a value, such that Q min /e = N min is an
integer, the charge cannot fluctuate as long as the temperature is low enough, i.e.
kT
e
2
2 C G
.
(14.18)
Tunneling into or out of the dot is suppressed by the Coulomb barrier e
2
/2C G , and the conductance is
very low. Analogously, the differential capacitance is small. This effect is called Coulomb blockade.
Peaks in the tunneling current (Fig. 14.26b), conductivity (Fig. 14.26a) and the capacitance occur, when
the gate voltage is such that the energies for N and N + 1 electrons are degenerate, i.e. N min = N +
1
2
.
The expected level spacing is
e α αV G =
e
2
C G
+ N ,
(14.19)
where N denotes the change in lateral (kinetic) quantization energy for the added electron. e
2
/C
will be called the charging energy in the following.
A variation of the source–drain voltage (for a given gate voltage) leads to a so-called Coulomb
staircase since more and more channels of conductivity contribute to the current through the device
(Fig. 14.27). In Fig. 14.28 the tunneling current is shown as a function of the potential of a dot, formed
by gates on a WSe 2 flake [1388], and the source-drain voltage. The iso-current lines form so-called
‘Coulomb blockade diamonds’.
The charge Q on the QD is determined by the charges on the gate, source and drain capacities,
Q = Q G − Q S + Q D . Together with Q S = C S V S , Q D = C D V D and V SD = V S + V D and Q G =