Análisis e interpretación de datos ficha de observación

The easiest way to understand some basic experimental results of transport measurements through single quantum dots is the constant-interaction (CI) model. Two important as- sumptions have to be made to apply this model: (i) the Coulomb interactions between the electrons in the dot are given by a single constant capacitance C, where C is defined as the total capacitance to rest of the world C=CS+CD+Cg. CS is the capacitance of the source, CD is that of the drain, andCg that of the gate. (ii) the discrete energy spectrum is independent of the number of electrons on the dot. Then, one can write the total energy

U(N) of an N-electron quantum dot as U(N) = [−|e|(N −N0) +CSVSD+CgVg] 2 2C + N X n=1 En(B), (2.1)

where −|e| is the electron charge and N0 the number of electrons in the dot at zero gate voltage. By tuning CSVSD and CgVg, one can adjust the the charge on the dot that is by the gate voltage (via the capacitance Cg) or by the bias voltage (via the capacitance CS). The last sum in Eq. (2.1) runs over the occupied single-particle energy levelsEn(B), which are separated by a difference ∆En = En−En−1. The characteristics of the confinement potential and of an eventually applied magnetic field are responsible for these energy levels. Instead of using the total energyU(N) as given by Eq. (2.1), it is often convenient to use the energy of the local dot level with N electrons, which is also called the electrochemical potential µ(N) of the N-electron quantum dot. This is defined as the energy that is required to add an electron to the dot, here from N −1 to N electrons:

µ(N) _≡ U(N)₋U(N ₋1) = µ N −N0− 1 2 ¶ EC− EC |e| (CSVSD+CgVg) +EN, (2.2) whereEC =e2/C is the charging energy of the dot. The so-called addition energyEadd(N) that separates the discrete levels in the dot is defined as

Eadd(N) =µ(N + 1)−µ(N) =EC + ∆E. (2.3)

∆E is the level spacing between two discrete quantum states. ∆E can be zero, if two electrons are added to the same spin-degenerate level. The other term in the addition energy is just a purely electrostatic part.

The necessary condition for transport through a quantum dot is, of course, energy conservation, i.e. it must be favorable for an electron to leave the source, tunnel through the dot and enter the drain. This can be achieved by tuning the local dot level (usually via the gate voltageVg, as long as the bias voltageVSDis fixed) of a specific number of electrons, e.g. µ(N+1) in Figure 2.4 (a), into the bias window between the electrochemical potentials of the source and the drain, i.e. µS ≥µ(N + 1) ≥ µD. The bias voltage is then given by the difference of the electrochemical potentials of source and drain: eVSD = µS −µD. In such a situation, electrons can tunnel sequentially (one by one and incoherently) from the source into the dot state with a local dot level µ(N + 1) and tunnel out into the drain.

Figure 2.4 (b) shows a different situation: here no local dot level µ(N) for a specific number N of electrons is situated in the bias window. Therefore the tunneling through

both depicted states with N + 1 or N electrons is blocked. This phenomenon is called

Coulomb blockade, since the charging energy EC would have to be overcome, which is energetically not possible in this case. The charging energy is the energy scale representing the repulsion of electrons in a quantum dot.

(a) eV_SD µS µD (N) µ µ(N+1) Eadd ΓL ΓR (b) eV_SD µS µD ΓL ΓR (N) µ µ(N+1) (c) eV_SD µS µD ΓL ΓR ∆E (N) µ (N+1) µ (N−1) µ

Figure 2.4: Similar to Ref. [46]: Schematic diagrams for tunneling processes through a quantum dot (initially occupied byN electrons): (a) the state withN+ 1 electrons and an electrochemical potentialµ(N+1) is in the so-calledbias windowbetween the electrochem- ical potentials of sourceµS and drainµD and therefore sequential (one electron) tunneling through this state is allowed, and the dot is occupied by either N + 1 or N electrons. (b) neither µ(N + 1) nor µ(N) is in the bias window, and therefore sequential tunneling through one of the dot states is not possible. The dot now is in Coulomb blockade. (c) the ground and excited states (separated by the level spacing ∆E) of a quantum dot filled with N electrons are in the bias window and can contribute to sequential tunneling through the quantum dot.

of the N-electron quantum dot are in the bias window defined by source and drain and therefore both can contribute to sequential tunneling through the quantum dot. The energy difference between the ground and the first excited state for N electrons is just the level spacing ∆E as shown in the Figure.

As already mentioned above, the local dot levels of the states in the quantum dot can be tuned via the gate voltage Vg and the bias voltage VSD. If one measures the Coulomb blockade and the resulting oscillations in the current through the quantum dot, one usually fixes the bias voltageVSD and only tunes the gate voltageVg. By makingVgmore negative, more electrons are pushed out of the quantum dot and the number of electrons on the dot decreases. This is illustrated in Figure 2.5 (a). The shown behavior is usually called Coulomb oscillations, since the number of electrons on the dot changes from peak to peak. In between the peaks, in the so-called Coulomb valley, the number of electrons on the dot is fixed and sequential transport through the quantum dot is not possible. Sequential transport is a first order process, consisting of uncorrelated one-electron tunnel events. Even in the Coulomb blockade regime or a Coulomb valley, higher order processes still can occur. We will discuss this issue later in more detail.

Figure 2.5 (b) depicts theCoulomb diamondsthat show up, if one measures the current or the differential conductance dI/dVSD dependent on the gate voltage Vg and the bias voltage VSD. If one crosses the diamonds for a very small bias voltage (dotted line), one would just find the Coulomb oscillation as in Figure 2.5 (a). For larger bias voltages, as already explained for Figure 2.4 (c), a charge ground state and the next excited state can be used for sequential transport. The onset of these excited state tunneling processes can

Figure 2.5: From Refs. [46, 49]: (a) Coulomb peaks as a function of the applied gate voltage Vg on the quantum dot. (b) Coulomb diamonds, i.e. the measured differential conductance dI/dVSD through the quantum dot as a function of the gate voltage Vg and the bias voltage VSD. The differential conductance is then usually color-coded, here one can only see the most important lines. The black diamond-shaped lines represent the onset of current through charge ground states, whereas the additional gray diagonal lines show the beginning of transport through excited charge states (with a larger bias voltage).

also be found in Figure 2.5 (b).

To summarize again, the Coulomb blockade can be lifted by adjusting the gate voltage in an appropriate way or by increasing the applied bias voltage over the quantum dot. By means of a Coulomb diamond measurement, a single quantum dot can be well characterized,

because the charging energy EC and the level spacing ∆E can be determined by the

distances in the plot (see Figure 2.5 (b)). The constant interaction model explains these effects successfully, but it is too simplified, if one wants to consider higher order processes or spin effects as well.

The amplitude and line shape of the Coulomb oscillations (and the appearance of the peaks at all) depend strongly on the interplay of the relevant energy scales in the system, one can distinguish between three temperature regimes [65]:

1. e2_{/C ¿k}

BT, where one cannot resolve the discreteness of charge.

2. ∆E _¿ kBT ¿ e2/C, the classical or metallic Coulomb blockade regime, where due to the small level spacing ∆E many excited states can contribute.

3. kBT ¿∆E ¿e2/C, thequantum Coulomb blockade regime, where just a few states are available for transport processes.

The shape of the Coulomb peaks as a function of the temperature has been calculated in the case of classical [70, 71] and quantum Coulomb blockade [71].

For the quantum dot charge qubit that we consider in this thesis, only the last regime of a quantum Coulomb blockade is relevant, since the charge levels in the two dots that

we would like to use should be well defined. Therefore, the level spacing should be quite large and the dot itself rather small.