The calculation of the band structure of Single-Walled Carbon nanotubes proceeds in two steps. First, the band structure of graphene is calculated in a tight-binding model . As Single-Walled Carbon nanotubes can be understood as rolled-up graphene sheets, a process called zone-folding is used to take into account the chiral structure of the nanotube.
Graphene consists of a hexagonal lattice (see figure (2.1)). The lattice is defined by two primitive lattice vectors a1 and a2. The chirality vector Ch is defined by a linear combination of the primitive lattice vectors. Ch = na1 + ma2 (2.1).
The index numbers n and m are the chiral indices which contain all the informa-tion about diameter and the roll-up direction. A second vector is important: The translational vector a is perpendicular to the circumferential vector. Together with Ch it defines the unit cell of the Carbon Nanotube. When going to reciprocal space, the lattice vectors Ch and a are replaced by k⊥ and kz. Thereby k⊥ is associated with the circumference of the nanotube whereas kz refers to the axis along the tube. Compared to its diameter, the nanotube can be considered as infinitely long. Hence kz = 2π/a is continuous. Around the circumference, however, the situation is diﬀer-ent. A wave travelling around the nanotube must have the same value at its point of departure. Therefore we find: eik⊥r = eik⊥(Ch+r) (2.2).
Electron transport in Single-Walled Nanotubes
At the macroscopic scale, electron transport within a conductor is diﬀusive and the conductance is given by the ohmic relation G = σW/L. In this expression W and L are width and length, respectively, and σ is a material parameter. When dimensions become much smaller than the mean free path of an electron, however, another phenomenon called ballistic transport occurs. In this regime, the conductance is given by the Landauer-B¨uttiker formalism , , . The result for the cur-rent through a mesoscopic conductor connected to two electron reservoirs, i.e. the contacts, is given by
I = e d (fL( ) − fR( ))T( ) (2.4).
where T( ) is the transmission probability through the conductor and f is the Fermi- Dirac distribution: fL,R(E) = 1 (2.5) 1 + e(E−μL,R)/kBT. The conductance is defined by the following equation G( e2 ) = d ( )FT ( − ) (2.6) h where FT ( − ) is the thermal broadening function FT ( − ) = − d ( 1 )) d exp(( − )/kBT)+1 which can be approximated as a Dirac δ-function in the zero temperature limit. Calculating the integral we obtain: G( )= e2 T( ) (2.7).
Quantum dots and Coulomb blockade
A quantum dot develops if an artificial atom is weakly connected to the electron reservoirs (see section 1.1). This weak connection is described by quantum me-chanical tunneling matrix elements. If temperature and coupling between leads and quantum dot are suﬃciently low, a phenomenon called Coulomb blockade , , ,  becomes important. In order to understand Coulomb-blockade, electron-electron interaction must be taken into account, which was not considered yet in the derivation given in subsection 2.1.2.
In our experiment, we use Single-Walled Carbon Nanotubes to implement quantum dots. If two contacts are evaporated on a Single-Walled Carbon Nanotube, tunnel barriers form between the nanotube and the metallic leads. In the following we describe how electronic transport is possible through such a structure.
Coulomb blockade at zero bias
If a quantum dot is in the Coulomb blockade regime and if we assume that the quantum mechanical level separation is negligible, an energy EC is needed to add a supplementary charge to the dot. kBTel << EC = e2/CΣ (2.10)
The energy EC is called charging energy and CΣ is the sum of all capacitances of the dot to leads, gates and ground. It is important that the energy to add an electron to the dot is bigger than the thermal energy kBTel. Therefore many experiments are carried out at low cryogenic temperatures, though room temperature devices are described in the literature  .
In a simplified picture, the situation is as illustrated in figure (2.3 (a)). The chemical potential of the leads is μL and μR for left and right lead, respectively. In between the leads, the quantum dot has quantized energy levels. In the general case, this quantization is a sum of the electrostatic part EC and a contribution coming from single particle quantization. The classical part is due to the Coulomb repulsion between single electrons, the quantum mechanical part is due to the small length of the nanotube between the two leads which results in an energy quantization ΔE = hv2LF . Here, h is Planck’s constant and vF is the Fermi velocity in the nanotube. Additionally, a gate electrode acts on the dot which shifts the energy levels as a function of gate voltage. That is to say that the gate voltage alters the total charge on the dot. As E = Q2/2C, the energy levels shift as a function of the gate voltage. The gate voltage can thus be used as a switch which moves the dot either in the Coulomb blockade regime (see figure 2.3(a)) or drives the dot into resonance (see figure 2.3(b)) by aligning the dot energy level with the chemical potential of the leads. By sweeping the gate voltage, one can therefore drive the quantum dot through a sequence of resonance peaks and blockaded regions. This leads to so-called Coulomb-peaks if the conductance is plotted as a function of gate voltage (see figure 2.3(c)).
In the next step the gate voltage diﬀerence between two adjacent levels on the quantum dot will be calculated. We start with the expression for the electrostatic energy diﬀerence E = E(N + 1) − E(N), where N is the number of electrons on the dot.
Coulomb blockade at finite bias
At finite bias, the chemical potential of the leads is changed. Single electron tun-neling occurs if either the chemical potential of the source contact or the chemical potential of the drain contact are aligned with the chemical potential of the dot. As the energy change needed for tunneling depends strongly on the position of the energy levels in the dot, single electron tunneling at finite bias is strongly dependent on the gate voltage. If one plots the diﬀerential conductance as a function of both gate voltage and source-drain voltage, the so-called Coulomb-diamonds become vis-ible in a 2D-plot. A typical Coulomb-diamond can be seen in figure (2.5(Left)). For each of the four corner points a sketch illustrating how tunneling works, is provided. Generally it can be said that tunneling occurs only if the condition μDrain ≤ μDot ≤ μSource (2.14).
Capacitive coupling: The electrostatic model
Throughout this subsection, we assume that the intermediate tunnel coupling is very small and the intermediate capacitive coupling dominates. At the end of the subsection, we treat the two boundary cases for very low and very large intermediate capacitive coupling.
In a double dot system, transport is only possible at certain gate voltages. Measuring the conductance at zero bias through the whole system as a function of the two gate voltages Vg1 and Vg2, one obtains the so-called honeycomb pattern (see figure 2.7(a)). Electron transport is only possible along the black lines. Within each hexagon, the number of charges on the double dot is fixed and transport is not possible, as illus-trated in figure (2.7(b)). Concerning the lines of conductance within the stability diagram, one can discriminate 3 diﬀerent situations, labeled (c) to (e). Each sit-uation is explained by the mutual position of the energy levels of dots and leads, as illustrated in the corresponding sketch at the up right side of figure (2.7). If cotunneling is negligible, transport in the double dot is only possible at the so-called triple points. An example of a triple point is labeled by (e). As can be seen in the sketch, all energy levels of dots and leads are aligned and hence transport is possible. In situations (c) and (d) only one dot level is aligned with the leads. At very small but finite interdot tunnel coupling, electrons can nevertheless tunnel from source to drain by means of cotunneling , as is explained above for a single quantum dot. In figure (2.7(f)) it is shown how the stability diagram changes in the case CM → 0. The degeneracy of the triple points, which is due to the capacitive coupling of the two dots, vanishes. In the other boundary case, if the capacitive coupling between the two dots becomes very high, the double dot eﬀectively behaves like one big dot. Within the stability diagram this would mean that the hexagonal structure vanishes and only straight lines remain (see figure 2.7(g)).
Applying a finite bias to the double dot, the triple points enlarge to triangular regions. This is visible in figure (2.8(a)). In black solid lines, the original position of two adjacent honeycomb-cells including two triple points is depicted. Due to the applied bias voltage Vsd both vertical and horizontal lines shift by δVg1 and δVg2. With the prefactors αg1 = Cg1 (2.15).
Tunneling processes involving superconductors
An important issue in our beamsplitter experiment is the tunneling from the super-conducting electrode into the normal lead. The situation can be understood visualiz-ing sketch (2.12(c)) which shows the so-called Semiconductor model of N-S junctions. The normal metal is represented by a continuous distribution of independent-particle energy states. The superconductor is represented by an ordinary semiconductor with an independent-particle density of states given by equation 2.31. If no external potential is applied, the Fermi-level of the normal metal lies in-between the super-conducting gap. Only if an external potential is applied, electrons in the normal state can populate levels which are suﬃciently high to overcome the gap and then tunnel into the superconductor.
Table of contents :
1.1 Artificial Atoms and Molecules
1.2 EPR-experiments in solid state
2.1 Single-Walled Carbon Nanotubes
2.1.1 Band structure
2.1.2 Electron transport in Single-Walled Nanotubes
2.2 Quantum dots and Coulomb blockade
2.2.1 Coulomb blockade at zero bias
2.2.2 Coulomb blockade at finite bias
2.3 Double Quantum dots
2.3.1 Capacitive coupling: The electrostatic model
2.3.2 High interdot tunnel coupling: The molecular state
2.4 BCS-theory and the spin singlet state
2.4.1 The BCS ground state
2.4.2 Quasiparticles and the density of states
2.4.3 Tunneling processes involving superconductors
2.5 Injecting superconducting correlations in a normal conductor: Andreev Reflection
2.5.1 Andreev Reflection at an NS interface
2.5.2 Crossed Andreev Reflection
2.6 Putting the puzzle together: Theoretical description of the beamsplitter
2.6.1 Qualitative Theory and Splitting argument
2.6.2 Methods to obtain the Γ’s
3 Sample preparation and Measurement environment
3.1 CVD growth
3.2 Lithographical patterning
3.3.1 Shadow Evaporation
3.3.2 Two-step Process
3.4 Measurement setup
4 Transport in a double quantum dot connected to a superconducting lead
4.1 Spectroscopy of the double dot
4.1.1 Stability diagram of the sample
4.1.2 Extraction of electrostatic parameters
4.1.3 Testing the NS-junction
4.2 Evidence for splitting Cooper pairs
4.2.1 Measurements along the axis of detuning at zero and finite magnetic field
4.2.2 Unbalanced Anticrossings
4.2.3 Quantitative comparison of theory and experiment
4.3 Nonlinear transport at triple points
5 Discussion and Outlook
A CVD growth of Single-Walled Carbon Nanotubes
A.1 Catalyst recipe for Single-Walled Carbon Nanotubes
A.2 Growth process – Paris
A.3 Growth process – Regensburg
C Finding the working point of the beamsplitter
D Determination of the current going to the superconductor in the side-injection setup