Project Topics
Get Complete Project Material File(s) Now! »
Band Structure Calculations
We now apply the above-presented model to the case of the graphene honeycomb lattice. As it was already mentioned, lattice symmetry is reflected by Bravais lattice, while honeycomb lattice can only be generated by a Bravais lattice with two atoms per unit cell, or in other words by two sublattices each having the same translational symmetry as the initial lattice. Thus a trial wavefunction, that exhibits the lattice symmetry should in general be described by a linear combination of Bloch waves of two sublattices: ψk(r) = akϕAk (r) + bkϕBk (r − δAB) (1.10) .Here, we have chosen the A sublattice sites to coincide with the sites of Bravais lattice, whereas the B sublattice sites are shifted by δAB with respect to them, hence this component appearance in the second function’s argument. Both functions ϕAk (r) and ϕBk (r) are given by (1.9), as the atoms of two sublattices are identical, the A and B indices are put here for pedagogical reason, we will omit them from now on. We may now search the solutions of the Schr¨odinger equation Hψk = kψk.
Zitterbewegung, Chirality and Klein Tunneling
Another intriguing consequence of Dirac equation, named by Erwin Schr¨odinger with German term Zitterbewegung[133], in other words jittery or trembling motion, states that any attempt to localize a relativistic quantum particle will fail, since it is inevitably accompanied by creation of particle-antiparticle pairs at the position of localization. Indeed, the momentum uncertainty of a confined relativistic particle impacts its energy uncertainty, in contrast to the non-relativistic case, where position-momentum and energy-time uncertainties are unrelated. As a consequence, such confined particle will be described by a wave-packet containing both particle (positive energy) and antiparticle (negative energy) components, which moreover will interfere, causing rapid oscillations of the particle. It is hence matter system, this phenomena is manifested by the means of special kind of inter-band transitions with creation of virtual electron-hole pairs. Before considering the phenomenon of Klein Tunneling, it is expedient to first introduce one important property of Dirac fermions, namely Chirality or (helicity), defined as the projection of its spin onto the direction of propagation. The corresponding operator reads hp = p · σ.
Quantum Phase Coherent Systems
In certain conditions a physical system manifests properties, that can not be explained within (semi-)classical framework, introduced in the previous section, since one has to account for wave-like effects, e.g. interferences, so the quantum mechanical description should be used. For such regime to be reached, the system has to be quantum coherent. By that we understand that the quantum phase of electronic wave-functions is preserved at least on the scale of the system. If we denote coherence length and time by tφ and lφ, this means that system size L lφ. The coherence is lost when the system interacts with the environment (phonons, other electrons, magnetic impurities, electromagnetic fields, etc.), since this will result in projection of system eigenstates into corresponding interaction operators eigenstates, that are not necessarily the formers. Moreover, these are usually inelastic processes, except in the cases where the environment is modified without energy transfer (e.g. magnetic impurity’s spin flip). On the other hand, during an elastic scattering process the phase is modified in a well-defined way, thus quantum coherence is preserved.
As it was discussed in the previous section, the inelastic scattering rate decreases with temperature so, in order to attain such regime experimentally, in addition to a small size and reduced dimension, a very low temperature is required.
As mentioned before, the main inelastic scattering mechanism at low temperature is electron electron interaction. Hence lφ le-e. Elastic scattering length lel le-d is also important since it allows to distinguish between two sub-regimes: diffusive in which electron propagation through the system is accompanied by many elastic collisions and ballistic in which electrons propagate without scattering (except the system boundaries). Typical length-scales in different systems are given in the table: Multiple are the devices in which at low temperature quantum coherence is attained. These are usually based on two-dimensional electron gases obtained at the interface of two semi-conductors or in graphene, one-dimensional conductors represented by various nano-wires or carbon nanotubes, quasi-0-dimensional nanoparticles or point defects. In the next section we will give examples of such devices and related mesoscopic phenomena.
Effects of the Coherence on Transport Properties
In the coherent conductors all transport processes can be described in terms of electronic waves transmission taking into account eventual interference effects.
Here we will give several examples of the most prominent transport phenomena that arise in quantum coherent systems. Maybe the best known manifestation of the quantum effects in transport properties of a coherent conductor is the conductance quantization. This phenomenon can be observed in Quantum Point Contacts (QPC) — ingenious devices in which the number of electronic wave propagation modes can be controlled externally by applying depleting gate voltage. In such systems, conductance, defined at mesoscopic scale as the sum of electron waves transmission probabilities over all modes, behaves as a step function of gate voltage as it can be seen in fig. 2.2.a, each new plateau corresponding to addition of a new mode to the total transmission. The plateaus are equally spaced by 2g0, where g0 is a value called conductance quantum, defined by fundamental physical quantities only: g0 = e2 h (the factor of 2 comes from the spin degeneracy). Another situation in which conductance quantization emerges is Quantum Hall Effect. Both effects will be detailed in the following sections.
Other very vivid examples of wave interference effects in coherent systems are Weak localisation and Aharonov-Bohm effect. The first effect manifests itself in disordered systems and is commonly interpreted in terms of coherent backscat system depicted by the sketch of fig. 2.1.a. Then, to calculate the probability PAB for a particle to travel from point A to point B, one has to account for all possible trajectories, leading from A to B (shown on the sketch in grey). Moreover, coherence of the system requires that the calculation of the total probability be done in terms of probability amplitudes Ai of each classical trajectory, rather than classical probabilities |Ai|2.
Framework, Hypotheses, Formulation
For the sake of simplicity we consider a one-dimensional, two-terminal coherent conductor to which two reservoirs of fermions — the “left” (L) and the “right” (R) — are connected and in addition we will omit the spin degree of freedom (the extension to multi-terminal, multi-mode case with spin being more or less straightforward).
The system is considered to be in stationary regime and, furthermore, at thermal equilibrium. Moreover we neglect electron-electron interactions effects. The reservoirs are infinitely long and much larger than the conductor, they can be characterized by temperature T and chemical potentials μL(R) and obey Fermi- Dirac statistics: fL(R)(ε) = exp (ε − μL(R))/kBT + 1 −1 . In addition, they are supposed to be perfect sources and sinks of electrons such that entering particle looses immediately its phase. From this point of view the specified problem is irreversible. On the other hand, the coherent conductor can be seen as a kind of stochastic “frontier guard” transferring the particles from one reservoir to another with certain (generally energy dependent) probability or otherwise rejecting them back to their reservoir of origin (such transfer processes being of course energy conservative). Mathematically this is expressed by scattering matrix that we will introduce soon.
Table of contents :
Introduction
I Overview of Graphene Physics and Mesoscopic Transport
1 Physical Properties of Graphene
1.1 Crystal Structure of Graphene
1.2 Graphene Band Structure
1.2.1 Basic Principles
1.2.2 Band Structure Calculations
1.2.3 Low-Energy Excitations
1.3 Properties of Dirac Fermions
1.3.1 Probabilty Current Density
1.3.2 Zitterbewegung, Chirality and Klein Tunneling
2 Transport in Mesoscopic Systems
2.1 Mesoscopic Scale
2.1.1 Classical Discription of Transport
2.1.2 Quantum Phase Coherent Systems
2.1.3 Effects of the Coherence on Transport Properties
2.2 Scattering Approach
2.2.1 Framework, Hypotheses, Formulation
2.2.2 Scattering Approach at Work
2.3 Quantum Hall Effect
2.3.1 Landau Quantization
2.3.2 Integer Quantum Hall Effect
3 Mesoscopic Transport in Graphene
3.1 Conducance and Shot Noise in Graphene
3.2 Quantum Hall Effect in Graphene
3.3 Experimental Results
II Experimental Setup and Device
4 Measurement System Principle
4.1 Experimental Requirements and Techniques
4.1.1 Typical Scales
4.1.2 Noise Measurement Techniques
4.2 Technical Realisation
4.2.1 Device Design
4.2.2 Measurement System
4.2.3 Cryogenic Amplification System
5 Device Fabrication
5.1 Methods
5.1.1 Obtaining Graphene
5.1.2 Making Graphene Visible
5.1.3 Raman Spectroscopy of Graphene
5.1.4 Graphene Oxygen Plasma Etch
5.2 Processes
5.2.1 Wafers Preparation
5.2.2 Graphite Deposition
5.2.3 Graphene Flakes Detection
5.2.4 Microcircuit Deposition
5.2.5 Nano-Constriction in Graphene
5.2.6 Device Test
5.2.7 Difficulties and Solutions
III Experimental Results
6 Measurement System Calibration
6.1 Low Frequency Calibration
6.1.1 Lines Tuning
6.1.2 Two-point Measurement
6.1.3 Calibration
6.2 High Frequency Calibration
6.2.1 Lines Tuning
6.2.2 Current Measurement
7 Conductance Measurements at Zero Magnetic Field
7.1 Conductance at Zero Bias
7.1.1 Ballistic Regime Hypothesis
7.1.2 Diffusive Regime Hypothesis
7.1.3 Discussion
7.1.4 Model for the “altered” Curves
7.2 Conductance Spectroscopy
8 Conductance in the Magnetic Field
8.1 Measurement Principle
8.2 Results and Discussion
9 Shot Noise in Graphene
9.1 Measurement Principle
9.2 Shot Noise in Presence of Joule Heating
9.2.1 Cooling by Electron Diffusion
9.2.2 Data Fit
9.2.3 Cooling by Phonon Emission
9.3 Noise Power Fluctuations
9.4 Results and Discussion
Conclusion
Appendices
A Device Fabrication
A.1 Recipes
A.2 Introduction to Raman scattering
A.3 Common Nano-fabrication Techniques
A.3.1 Microlithography Principle
A.3.2 Optical and E-Beam lithography
A.3.3 Thin Films Deposition
B Measurement System
B.1 Cryogenic Inset
B.2 Data Acquisition Module
B.3 RLC-Filter Pass-Band Calculation
Bibliography
