For the heterostructures described in this work1,ns is typically of order2 × 1015 m−2, and µ ∼ 40 m2/V s.
1.2 Quantum dots
1.2.1 Creation
Having introduced some of the relevant characteristics of 2DEGs, we now turn to the interesting quantum systems that can be defined and measured within these environ-ments, including single and double quantum dots, and quantum point contacts. 2DEGs are typically controlled via electrostatic fields set by electrode gates. These metallic structures generally comprise Ti/Au stacks (∼ 100 nm thickness) evaporated on the surface of a heterostructure. At sufficiently cold temperatures, the application of ne-gative voltages reverse-biases the gates, allowing the metal to be held at a constant voltage with respect to the 2DEG. The resulting electric fields locally change the elec-trochemical potential, creating depletion zones and residual puddles of electrons. By making the gate voltages increasingly negative until the size of these charge islands are on the order of the Fermi wavelength, zero-dimensional quantum dots can be formed.
Quantum dots are weakly coupled to their environment and can be filled with one or more electrons (see Fig. 1.3). The quasi-metallic charge reservoirs on either side of these dots (known asleads) provide a continuum of states for the shuttling of electrons on and off the central island. In order to access the leads, metallic regions (known as ohmics) are annealed at high temperature into the layer stack to make a low-resistance connection with the 2DEG (ideally around∼ 50 Ω)2. Setting up a potential difference across two ohmics on opposite sides of a quantum dot allows a current to flow and for the conductance of the dot to be directly measured.
1See Chapters 3 - 5.
2These ohmic contacts are made from a eutectic alloy stack of AuGe, with an atomic weight ratio of 88:12, or 2:1 layer height when evaporating.
1. TRANSPORT IN GAAS-ALGAAS TWO-DIMENSIONAL
Figure 1.3: Single quantum dot. (a,b) Electron micrograph of a quantum dot gate structure measured in the Quantum Nanoscience Lab at the University of Sydney. The device in (a) can be configured as a single (white dotted region) or double (blue circles) quantum dot via the application of negative voltages on the gate electrodes (light grey patterns). Squares with crosses represent ohmic contacts, which are annealed into the heterostructure to make contact with the 2DEG. In (b), conductance, G, through a single quantum dot is measured (colour axis) while sweeping the voltages on the left and right wall gates (VL and VR). Bright yellow stripes indicate charge transitions. (c) shows a cartoon illustrating transport in a single quantum dot for two different regimes. On the left, an energy level of the dot (denoted µN) lies within the bias window of the source and drain reservoirs (µS and µD respectively), allowing an electron to tunnel on and off the central island. On the right, tunnelling is not energetically allowed and the occupation of the dot remains constant. (c) adapted from Ref. [14].
1.2 Quantum dots
1.2.2 Artificial atoms
The electrostatic confining potential that houses a quantum dot can support a number of spin-degenerate energy levels. It is for this reason that quantum dots are often referred to as artificial atoms, as their well-defined energy states can be compared to atomic orbitals [17]. The allowable energy states can be understood within the constant interaction model, where the relationship between electrons in a dot and its electrostatic environment is characterised by a total capacitance term, C [14]. The energy of the dotU (N ) in its ground state with N electrons is given by
U (N ) = (−|e|(N − N0) +PiCiVi)2
2C +
N
X
j
Ej (1.12)
whereEj is a summation over the single particle energy levels up toN , and Ci and Vi
refer to the capacitance and voltage contributions of the nearby structures (including the source and drain leads, and the electrode gates). The electrochemical potential,µ, on the island is then defined as
µ ≡ U (N ) − U (N − 1). (1.13)
The additional energy associated with bringing an extra electron onto the dot is
Eadd(N ) = µ(N + 1) − µ(N ) = EC+ ∆E (1.14)
whereEC = e2/C is the charging energy (similar to the ionisation energy for an atom) and ∆E gives the level spacing [14].
For small sized dots the self capacitance term,C, is small and so the energy required to increase the electron occupation of the dot becomes large. Considering the dot and its environment as the plates of a capacitor, this value ofC also sets the time taken to charge up an island with an extra electron. This process occurs with a time constant of
∆t = RtC where Rt is the tunnelling resistance. Invoking the Heisenberg uncertainty
1. TRANSPORT IN GAAS-ALGAAS TWO-DIMENSIONAL ELECTRON GASES
relation in the form∆Ec∆t > h, it is therefore necessary that
Rt > h
2e2 (1.15)
in order to resolveEC (where the factor of1/2 accounts for the degenerate spin up and spin down states). This rather simple, intuitive derivation gives rise to the conductance quantum
G0= 2e2
h , (1.16)
and also sets the condition for observing single electron tunnelling events through the energy levels of a quantum dot [18]. A second key requirement is that the thermal energy of the electrons must be sufficiently low such that they cannot be easily excited on or off the dot, i.e.
kBT < e2
C (1.17)
wherekB is Boltzmann’s constant.
1.2.3 Transport
When the electrochemical potential of a quantum dot lies within the energy states or bias window set by the reservoirs, µs ≥ µ(N ) ≥ µD, a flow of current proportional to the source-drain voltage is observed,µs− µd = −|e|VSD (blue square in Figure 1.3 (c)) [14]. This is visible as peaks in the conductance. In this way, varying the source-drain bias can be used to effect transport through the dot. Equivalently, the potential level of the island itself can be tuned via the capacitive coupling of aplunger gate (VP 1 orVP 2
in Fig. 1.3 (a)). Both scenarios can be used to induce charge tunnelling events, until the energy needed to fill the dot with one extra electron is greater than the chemical potential at the leads. In this situation the energy level of the dot lies outside the bias window, and tunnelling is barred. This is known as Coulomb blockade, and is indicated by the orange star in Fig. 1.3 (b) and (c).