Weak localization and anti-localization are phase-coherent transport phenomena that are directly linked to the presence of scattering centers in the conductor. Since both the phase coherence time and length are decreasing with increasing temperature, it is a low-temperature effect. Figure 1.11 depicts a random distribu-tion of scatterers.
Figure 1.11: Schematic for weak localization. For a closed trajectory of scattering cen-ters an incident electron wave (black) can possibly scatter clockwise (blue) and counter-clockwise (red). Adapted from Ref. [180].
An incident electron wave can now scatter (amongst many possible trajectories) clockwise and counter-clockwise along the blue and red trajectory, respectively. If time-reversal symmetry is not violated, then the same phase is collected in both loops leading to constructive interference. The electron is located in the loop and does therefore not contribute to the current flow. This leads subsequently to an increase (decrease) in resistance (conductance).
Using a semi-classical 2-dimensional approach [50], the correction to the conduct-ance is given by
(1.29)
1 Theoreticalbackground
16
With the momentum relaxation time already introduced in subsection 1.2.1 and the phase coherence time . Since drastically scales with temperature , the effect becomes mostly apparent only at low temperatures.
Under the influence of a magnetic field both paths collect different phases and the expression generalizes to (and assuming for ) [22]
(1.30)
with the digamma function . Figure 1.12 shows the changes in conductance for different momentum relaxation times . As a consequence of decreasing both the absolute value of [see Eq. (1.29)] and the change in magnetic field in-creases, since the electrons can better localize, if their free range of movement is limited.
Figure 1.12: Magnetic field dependence of weak localization. Absolute change in con-ductance as a function of magnetic field . Color-coded are different values for the momentum relaxation time (Additional parameters: , ).
Due to the peculiarities of graphene arising from its band structure and the pseudo-spin (see section 1.1) the correction in Eq. (1.30) changes and is described in the following. McCann et al. derived a theory of weak localization in graphene [123]
(1.31)
1.2 ElectronicTransport in mesoscopic systems
17 With
Additionally, the magnetic fields are related to the scattering times / scat-tering lengths
(1.32) Here, is the flux quantum and the diffusion constant. The relaxation time has been replaced by the two time scales and , the intervalley and the combined scattering time, respectively. They contain the scattering contributions . Intervalley scattering describes scattering between Dirac cones of different sublattices (see section 1.1). The combined scattering time includes the intervalley scattering time, the intravalley scattering time (scattering within one sublattice ) and a correction due to trigonal warp-ing . The latter is induced by a distortion of the energy dispersion in Eq. (1.15) at higher energies with three fold symmetry enabling additional scattering. The combined scattering time is defined as . The sign for the terms involving is different and thus describes anti-localization of the electrons.
This weak anti-localization in graphene is a consequence of .[123]
In section 4.3 we will study the influence of single dopant atoms, characterized by scanning tunneling microscopy and spectroscopy, on magnetotransport. As it turns out the presence of this atomic scale scattering centers is triggering a transition from a classical quadratic MR (subsection 1.2.4) to a pronounced weak localization behavior as discussed in above.
19 This chapter gives an overview of the sample preparation techniques and experi-mental methods used in this thesis and thus paves the way for the following chap-ters.
Section 2.1 introduces the sample preparation of graphene. Since its discovery, different preparation methods of graphene have been reported.[13, 103, 135, 164]
- -grown
gra-phene on silicon carbide (SiC) and gragra-phene grown by chemical vapor deposition (CVD) belong to the most prominent methods. In most studies in the framework of this thesis we investigated graphene on SiC-graphene (in chapter 3, section 3.2 and 3.4 as well as in chapter 4) which has already been studied in our group in recent years [42, 43]. An introduction to the growth method and its history is given in section 2.1.1. In the study presented in section 3.3 in chapter 3 we used com-mercial graphene on SiO2 which is treated here in section 2.1.2. We used this sam-ple system on the one hand, because the substrate is insulating already at room temperature necessary in this experiment conducted at ambient conditions. On the other hand, different kinds of defects are found in this system emerging from the differences in the growth process. Additionally, the samples used in both studies in chapter 4 have been treated with low energy ion bombardment in the group of Prof. Hans Hofsäss. In section 2.1.3 we introduce this method.
Moreover, different experimental techniques have been employed to study the lo-cal transport properties in graphene and are discussed in section 2.2. In general, scanning tunneling microscopy (STM), introduced in section 2.2.1 has been used in most studies (In chapter 3, section 3.2 and 3.4 as well as chapter 4). Addition-ally, scanning tunneling spectroscopy (STS) treated in section 2.2.2 was used in particular to investigate the local electronic structure of the graphene sheet as well as doping atoms in chapter 4, section 4.3. An introduction to scanning tunneling potentiometry is given in 2.2.3. This technique is later used in section 3.2 and 3.4 in chapter 3 to study the local voltage drop in SiC-graphene. The effect of thermo-voltage is discussed in 2.2.3.1 and the experimental implantation of a magnetic field potentiometry setup in 2.2.3.3. In section 3.3 of chapter 3 we use the atomic force microscopy (AFM)-based technique of Kelvin probe force microscopy (KPFM) to investigate the voltage drop in CVD-graphene on SiO2. It is introduced in 2.2.4. For ion-implanted graphene samples we performed macroscopic transport measurements in the study in section 4.3 of chapter 4. Therefore, an introduction to them is given in section 2.2.5.
20