Weak localization

Weak localization arises from constructive interference between time reversed partial waves of the charge carriers in disordered samples. Due to the chirality of the charge carriers, quantum interference in graphene is different to conventional semiconductors [53]. Conse-quently the carriers in graphene have an additional Berry phase ofπ. As shown in Figure 6.12 a) for various gate voltages the quantum correction of the resistance is usually mea-sured by applying a perpendicular magnetic fieldsB. WL in graphene depends not only on the inelastic scattering of electrons, characterized by the phase coherence lengthLΦ, but also on the elastic scattering caused by impurities and lattice defects. The inelastic scat-tering is represented by two characteristic lengths, the intravalley scatscat-tering lengthL?and the intervalley scattering lengthLi. LargeL?suppresses the interference within one valley due to chirality breaking defects and random magnetic fields due to ripples and disloca-tions, whereas largeLi restores the interference by mixing the two valleys with opposite chirality [53]. Intervally scattering is caused by sharp defects, e.g. sample edges, whereby charge carriers can be scattered between valleys. Interestingly, in graphene both, weak localization and weak anti-localization was reported to appear in one sample depending on the ratioLΦ/L?andLΦ/Lithat can be tuned by changing carrier density and tempera-ture [53].

The data in Figure 6.12 were measured in a four-terminal geometry at T =4.2 K on the monolayer sample GaAs-MLG(b), which exhibits a charge carrier mobility of µ= 1660cm²/Vs as shown in section 6.2. The magnetoresistance traces were taken for dif-ferent two-dimensional hole densities ranging from8.5·10¹²cm⁻²to6.3·10¹²cm⁻². The hole density is varied by changing the backgate voltage from -1.5 V to 3.5 V. Please note that the traces are not vertically shifted for clarity. The offset is caused by the fact that decreasing the carrier density increases simultaneously the sheet resistivity. Remember, the high-field magnetotransport behavior of this monolayer sample was already shown in Figure 6.5, where also an optical micrograph of the device is inserted. For almost all curves, a positive magnetoresistance due to weak localization appears aroundB=0 T.

Additionally, UCFs are visible for the whole swept magnetic field range. For better

com--1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 450

500 550 600 650 700

B (T)

R(W)

T = 4 K

GaAs - MLG(b)

a)

b)

W 40

-27

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1 0 1 2 3

8.2 7.7 7.2 6.8 6.6

B (T)

p(10cm) 12-2

U(V)Gate

Figure 6.12: Weak localization measurements for different gate-voltages of a monolayer sample on GaAs (GaAs-MLG(b)). a) Gate-voltage dependence of the low-field region ofR_xx for applied voltages between 3.5 V and -1.5 V. b) Data from a) in a false color plot of the low-field region of R_xx after subtraction of a second order polynomial background. The WL signal is aperiodically suppressed with the hole density. The suppressed regions are marked with arrows.

6.4. PHASE COHERENT TRANSPORT 81

Figure 6.13:Vertical traces through the plots in Figure 6.12 along B=0 T. a) The data without any correction. b) The data of a) corrected by a linear regression (corresponding to the B=0 T profile of Figure 6.12 b)). The regions, where the WL signal is supressed are marked with arrows.

parison of the WL signal, the curves are displayed in Figure 6.12 b) in a false color plot, where the second order polynominal background is subtracted. Bright areas correspond to high resistance values and dark blue ones to low resistance values. Surprisingly, the WL signal aperiodically disappears for certain carrier densities. The regions, where the WL is suppressed are marked with arrows in Figure 6.12 b). Linecuts through this plots along B=0 T are shown in Figure 6.13 without a) and after b) correction with a linear regression taking into account the carrier density dependency of the sheet resistivity. In both graphs the suppression of the WL signal is clearly observed. From simple analysis it was not possible to extract a periodicity in the curves. The regions with vanishing WL sig-nals, and simultaneously regions with very distinct WL signals are spaced in gate voltage between∆U =0.9 V and∆U =1.1V corresponding to changes in the two dimensional charge carrier density of between∆p=0.37·10¹²cm⁻²and∆p=0.47·10¹²cm⁻². This is not a clear periodicity, but the change in the hole density needed to come from one to the next suppressed region are similar. The visibility of the suppression of the WL signals that could also be interpreted as a starting transition from WL to WAL behavior is improved by plotting the normalized magnetoconductance for all measured hole densities as revealed in Figure 6.14. The curves for vanishing WL effect are red colored and correspond to the

-0.2 0.0 0.2 -0.4

0.0 0.4

B (T)

T = 4 K

GaAs - MLG(b)

ds(e/h)2

Figure 6.14:Normalized magnetoconductance traces for different backgate voltages measured on sample GaAs-MLG(b) (extracted from Figure 6.12). The red traces correspond to gate voltages, where the WL signal vanishes.

density (backgate voltage) values marked by arrows in Figures 6.12 and 6.13. We found such an aperiodic suppression of the WL signal for graphene on GaAs independent of the number of layers.

Suppression of WL or transition from WL to weak anti-localization (WAL) was reported for graphene on SiO₂ dependent on the carrier density or the temperature [53, 55, 139], but to best of our knowledge no recurrent suppression by continuously tuning the carrier density is reported in literature so far. Only in reference [124] a similar plot was shown for graphene on SiO₂, however the authors did not discuss this issue.

Most investigations of WL and WAL effects of graphene on SiO₂ are done by averaging over a certain range of carrier density to reduce the UCFs and hence pronounce the WL signal. In this way, the curves can also better fitted by equation (2.14) [53, 54]. But by averaging only over a voltage range of∆U=1V [53], what seems minor for SiO₂, where gate voltages of aboutUGate=±100V can be applied, the uncertainty in the carrier den-sity constitutes∆p≈7.4·10¹⁰ cm². Whereas the separation of the suppressed regions in our experiments are found to be three to five time larger. One might speculate, that the effect of recurrent vanishing WL signal by tuning the carrier density could also be visible for graphene on SiO₂. Moreover, the WL signal and hence its (possible) suppression is more pronounced for graphene on GaAs compared to graphene on SiO₂. The suppression of WL may be induced by mesoscopic corrugations of the graphene sheets or the underlying

6.4. PHASE COHERENT TRANSPORT 83

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -1.0

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4

B (T)

ds(e/h)2

T = 25 K GaAs - MLG(b)

Figure 6.15:Measured magnetoconductance atT=25K andU_Gate=0for sample GaAs-MLG(b) (black) and fitted magnetoconductance (red).

substrate or by interaction with the substrate. The origin of this effect is still unclear and also first discussions with theory have not yet given any result.

From careful analysis of magnetoconductance curves as depicted in 6.14, the phase co-herence lengthLΦ and the electric scattering lengths, intervalley scattering lengthLiand intravalley scattering lengthL_?can be determined regarding to equation (2.14). In Figure 6.15 the magnetoconductance curves (black) for the monolayer GaAs-MLG(b) taken for UGate=0V atT=25K is fitted (red) using the equation mentioned above. Obviously, the fit matches the measured data very well in the magnetic field range ofB=±0.8T. The WL signal was measured as function of different carrier densities changed by tuning the back-gate voltage and in dependence of the temperatures betweenT =4.2K andT =65K.

Since this relative high temperatures the quantum interference effect still survives. But it is generally known that the WL effect in graphene is visible up to much higher temperatures (T =200 K) [53] compared to conventional semiconductors and metals. In Figure 6.16, the three characteristic scattering lengthL_Φ(black) ,L_?(red) andL_i(green) achieved from fits of the magnetoconductance are collected. The dependence from the carrier density (backgate voltage) is displayed in Figure 6.16 a) and from the temperature in b). The solid black line denotes the value forL_Φ determined from the averaged magnetoconductance curve over all measured backgate voltages−1.5V≤U_Gate≤3.5V atT =4.2K and con-stitutes toLΦ=223nm. This value lies in between all resulting values found forLΦ lying

0 10 20 30 40 50 60 70

Figure 6.16:Phase coherence lengthL_Φ(black) intravelley scattering lengthL_?(red), and interval-ley scattering lengthL_i(green) a) atT =4.2K in dependence of the carrier density (gate voltage) and b) in dependence of the temperature atU_Gate=0.

between 122 nm ≤L_Φ ≤317 nm. The grey shaded lines denotes the two-dimensional hole density regions where the suppression of the WL signals has been found. The same regions are marked with arrows in 6.12 and 6.13. Here, not an unambiguous relation can be entitled, however a clear direction seems to emerge. For the suppressed regions the phase coherence lengthLΦis definitely lowered relative to neighboring regions. Moreover, LΦseems slightly to increase in average with decreasing carrier density, but this trend has to be confirmed. The values for intravalley scattering lengthL_? (red) and the intervalley scattering length Li (green) are both struggling between 70 nm ≤L?,i ≤212 nm. The differences in the scattering lengths between neighboring hole density regions are very large. The trend regarding the hole density forL_?andL_iseems to be the opposite toL_Φ. Decreasing the hole density slightly decreases both elastic scattering lengths in average.

AlsoL?andLiare generally lower in the grey mark regions.

The temperature dependency of the three coherence lengths are given in Figure 6.16 b).

The values for L? and Li constitute both to aboutL?,i=110 nm ±10 nm and are inde-pendent from the temperature between T =4.2 K and T = 35 K and decrease minor with increasing temperature forT >35K. In contrast to that, the phasecoherence length L_φ decreased linearly with increasing temperature from 212 nm atT =7 K to96 nm at T =65K. This behavior was expected from literature [45, 98], and references therein.

Generally, the relation L_φ>L_?,i obtained for graphene on SiO₂ [53, 98] is also valid for graphene on GaAs. The determined values for the characteristic length for graphene on GaAs, especially values and temperature dependency of the phase coherence lengthLΦ

caused by inelastic scattering are in good comparison of the values reported in litera-ture [53, 54], and references therein. Finally, we had to refer about the uncertainty and the error done by fitting the magnetoconductance data to equation (2.14). Due to three free

6.4. PHASE COHERENT TRANSPORT 85

Figure 6.17: Temperature dependent conductance fluctuations amplitude extracted from the low field region for a) the bilayer sample GaAs-BLG(a) and b) monolayer sample GaAs-MLG(a).

parameters, the fit routine was not able to adjust all values automatically. Therefore, the fit had to be optimized by hand, what was a little challenging, because all parameters affect each other.