418
14 Nanostructures
(a)
(b)
Fig. 14.28 a Top: Sketch of WSe 2 flake with gate structure, defining a lateral quantum dot. Bottom: SEM image of the
structure. The outline of the WSe 2 flake (thickness 4.5 nm) is indicated as dotted line. BG: back gate, PG: plunger gate.
b Current (in false colors) as a function of the back gate voltage V BG and the source-drain voltage V SD (T = 240 mK).
On the top is a current trace as a function of V G (for V SD = 0, along the red dashed line in the diagram). SEM image in
panel (a) and panel (b) adapted from [1388] with permission from RSC
Fig. 14.29 Schematic
Coulomb diamond stability
diagram. The slopes γ 1 and
γ 2 are discussed in the text.
The periodicity in the gate
voltage is given by e/C G .
The charge on the dot is
Q = (−e) n
A lot of research so far has been done on lithographically defined systems where the lateral quantization energies are small and smaller than the Coulomb charging energy. In this case, periodic oscillations
are observed, especially for large N . A deviation from periodic oscillations for small N and a characteristic shell structure (at N = 2, 6, 12) consistent with a harmonic oscillator model ( 0 ≈ 3 meV)
has been reported for ≈500-nm diameter mesas (Fig. 14.30b,c). In this structure, a small mesa has been
etched and contacted (top contact, substrate back contact and side gate). The quantum dot consists of
a 12-nm In 0.05 Ga 0.95 As quantum well that is laterally constricted by the 500-nm mesa and vertically
confined due to 9- and 7.5-nm thick Al 0.22 Ga 0.68 As barriers (Fig. 14.30a). By tuning the gate voltage,
the number of electrons can be varied within 0 and 40. Measurements are typically carried out at a
sample temperature of 50 mK.
In the sample shown in Fig. 14.31, self-assembled QDs are positioned in the channel under a splitgate structure. In a suitable structure, tunneling through a single QD is resolved.
In small self-assembled quantum dots single-particle level separations can be larger than or similar
to the Coulomb charging energy. Classically, the capacitance for a metal sphere of radius R 0 is given as
C 0 = 4π π 0 r R 0 ,
(14.22)
e.g.. C 0 ≈ 6 aF for R 0 = 4 nm in GaAs, resulting in a charging energy of 26 meV. Quantum mechanically, the charging energy is given in first-order perturbation theory by
14 Nanostructures
(a)
(b)
Fig. 14.28 a Top: Sketch of WSe 2 flake with gate structure, defining a lateral quantum dot. Bottom: SEM image of the
structure. The outline of the WSe 2 flake (thickness 4.5 nm) is indicated as dotted line. BG: back gate, PG: plunger gate.
b Current (in false colors) as a function of the back gate voltage V BG and the source-drain voltage V SD (T = 240 mK).
On the top is a current trace as a function of V G (for V SD = 0, along the red dashed line in the diagram). SEM image in
panel (a) and panel (b) adapted from [1388] with permission from RSC
Fig. 14.29 Schematic
Coulomb diamond stability
diagram. The slopes γ 1 and
γ 2 are discussed in the text.
The periodicity in the gate
voltage is given by e/C G .
The charge on the dot is
Q = (−e) n
A lot of research so far has been done on lithographically defined systems where the lateral quantization energies are small and smaller than the Coulomb charging energy. In this case, periodic oscillations
are observed, especially for large N . A deviation from periodic oscillations for small N and a characteristic shell structure (at N = 2, 6, 12) consistent with a harmonic oscillator model ( 0 ≈ 3 meV)
has been reported for ≈500-nm diameter mesas (Fig. 14.30b,c). In this structure, a small mesa has been
etched and contacted (top contact, substrate back contact and side gate). The quantum dot consists of
a 12-nm In 0.05 Ga 0.95 As quantum well that is laterally constricted by the 500-nm mesa and vertically
confined due to 9- and 7.5-nm thick Al 0.22 Ga 0.68 As barriers (Fig. 14.30a). By tuning the gate voltage,
the number of electrons can be varied within 0 and 40. Measurements are typically carried out at a
sample temperature of 50 mK.
In the sample shown in Fig. 14.31, self-assembled QDs are positioned in the channel under a splitgate structure. In a suitable structure, tunneling through a single QD is resolved.
In small self-assembled quantum dots single-particle level separations can be larger than or similar
to the Coulomb charging energy. Classically, the capacitance for a metal sphere of radius R 0 is given as
C 0 = 4π π 0 r R 0 ,
(14.22)
e.g.. C 0 ≈ 6 aF for R 0 = 4 nm in GaAs, resulting in a charging energy of 26 meV. Quantum mechanically, the charging energy is given in first-order perturbation theory by