To characterize the scattering mechanisms in our sample system, we can locally evaluate the sheet resistance/conductivity and the defect resistance given in Eq.
(1.14) and (1.19), respectively. Since graphene is a 2D conductor, the definitions and dimensions of these quantities deviate from the better known 3D case.
The macroscopic sheet resistance is given by
(1.20)
Figure 1.6: Landauer residual resistivity dipole at a local scatterer. (a) Ballistic con-ductor with leads µ1 and µ2 including a scatterer (X) with a finite transmission T. (b) Elec-trochemical potential , (c) conduction band edge and (d) electron density as a function of position around the scatterer. Reprinted (adapted) with permission from Ref. [31]. Copyright (1997) Cambridge University Press.
with the width and length as well as the total resistance of the sample (sam-ple geometry shown in Figure 1.7a). The latter is defined as with the volt-age difference between the two contact and the total electric current . One of the major goals of this thesis is now to evaluate the electronic
1.2 ElectronicTransport in mesoscopic systems
11 transport locally. In this case, we ask for the fraction that drops across the dis-tance in the sample. Here, can be in the order of nanometers. Then we can write Eq. (1.20) as
(1.21)
where we introduced the local electric field and the current density . For a perfectly homogeneous system as shown in Figure 1.7b, this again
Figure 1.7: Electronic transport in 2D. (a) Diffusive 2D conductor of length and width with leads µ1 and µ2. (b) Voltage drop across the sample for a completely homogeneous system indicated by a homogeneous electric field . (c) Voltage drop for different local electric fields as well as additional localized voltage drops .
yields the macroscopic value . However, in the presence of local variations in sheet resistance or of local defects (Figure 1.7c) the local electric field deviates from the macroscopic one. This can be nicely seen by again writing Eq. (1.21) as
(1.22)
where we defined . Thus, the local resistivity is expressed as its mac-roscopic counterpart multiplied by the ratio of electric fields.
Analogously, localized defects can be characterized by:
1 Theoreticalbackground
12
(1.23)
The change in dimension arises here from their definition to be a 1D interface in a 2D conductor.
For the simple case of a system translational invariant in y-direction9, the macro-scopic sheet resistance can be written as the sum of the local parts
rule
(1.24)
With the fraction of each different local sheet resistance as well as the defect concentration with the total number of defects of each type.10 One can undoubtedly argue that is not representing a sheet resistance, since it now contains different contributions from localized defects as well as locally varying sheet resistances. Nevertheless, this is the situation in (most) large scale transport experiments. It motivates the use of local transport studies by scanning tunneling potentiometry and other techniques to dissect the different contributions in Eq. (1.24).
Magnetotransport
With the resistivity of a 2D system, we define the magnetoresistance (MR) , the change in resistance due to a magnetic field B.
(1.25) For a transverse magnetic field B, perpendicular to the current flow, charge carriers get deflected by the Lorentz-force. Generalizing the Drude conductivity introduced in Eq. (1.10) with an additional magnetic field leads to
(1.26) in the limit of 2D transport yields [31]
9 Additionally, the current density is also not necessarily constant, if the system is not transla-tional invariant in y-direction as it is the case in Figure 1.7. This is discussed in detail in 3.3.3.1 of chapter 3 for the case of inhomogeneities caused by graphene wrinkles in graphene on SiO2 and additionally in Ref. [42] for steps and interfaces in SiC-graphene.
10 We here neglect in this purely classical treatment phase-coherent effects such as weak localization treated in subsection 1.2.5.
1.2 ElectronicTransport in mesoscopic systems
13 (1.27) where we made use of the relations in Eq. (1.12). Moreover, the hall resistance
was introduced. The subscript for indicates the zero-field conductivity/resistivity. Inverting the resistivity matrix yields [77]
(1.28) The magnetic field increases the off-diagonal elements.
Figure 1.8: Corbino disk contact geometry. Contacts are given by an inner and an outer circle while a transverse (perpendicular) magnetic field B is applied. For increasing B the current I0 changes by the additional component IB. Thus, the path an electron has to travel in the medium with resistivity increases leading to a positive MR.
Figure 1.9: Potential distribution and Magnetoresistance for different contact geom-etries. (a) Sample in short-channel geometry with low ratio . (b) Long sample (Hall-geometry) with high ratio . (c) for the
1 Theoreticalbackground
14
sample geometry in a. (d) for the sample geometry in c. (Simulation parameters:
/ )
The influence on the MR can best be understood in the geometry of a Corbino disk [32] (Figure 1.8). For no magnetic field, the electrons can directly flow from the inner to the outer contact, while for finite field they get deflected and have to spend a longer time in the sample. Therefore, the increases. Since for this geometry the electric field does not change under the influence of a magnetic field, the change in MR can directly be seen from the conductivity tensor in Eq. (1.28) to scale with . This holds in general for other geometries. However, the effect can be more or less pronounced depending on the sample geometry.
Figure 1.9 shows finite element method simulations using COMSOL Multiphysics for different sample geometries in case of no magnetic field as well as for . As can be seen in spatially resolved potential images in Figure 1.9a, the contacts in-duce fixed boundary conditions, since they are on one potential and thus the equi-potential lines in their vicinity are heavily bent for applied magnetic field . These regions are responsible for the MR. Therefore, short-channel samples with
(Figure 1.9a) are dominated by the contact regions and show a strong MR (Figure 1.9c, relative increase ). In contrast, for long thin geometries with
shown in Figure 1.9b (Hall-geometry) this contribution vanishes, since a constant electric field gradient Ey in y-direction has been established here. The respective MR-curve is shown in Figure 1.9d with almost no dependence on magnetic field (obey different y-scale, relative increase ). Figure 1.10 quantifies this obser-vation demonstrating that the magnitude of the observed MR is especially present for short samples and is vanishing in the limit of perfect Hall geometries.
Figure 1.10: Magnetoresistance for different aspect ratios. Color-coded plots are sim-ulated for different length of the sample while the width is held constant. Thus, the aspect ratio changes. Accordingly, for increasing aspect ratio the MR decreases.
(Simulation parameters as in Figure 1.9)
1.2 ElectronicTransport in mesoscopic systems
15 In section 3.4 of chapter 3 the geometry was on purpose chosen to be short-chan-neled (Figure 1.9a) instead of using a Hall-geometry (Figure 1.9b). In this way de-viations from the quadratic behavior are easier to detect due to the larger absolute change in MR.