Magnetotransport behavior

The magnetotransport properties of graphene on GaAs and InGaAs are demonstrated on examples of a mono- and bilayer graphene sample on GaAs and a bilayer sample on InGaAs. Both field sweeps of a perpendicularly applied magnetic field at constant gate voltage and gate voltage sweeps at a finite applied magnetic field have been carried out.

Temperature dependent magnetotransport measurements, accurate analysis of the data and possible explanations will be presented in the following.

From measurements of the longitudinal Rxx and Hall resistance Rxy, information about quality, mobility and carrier density of the investigated sample can be achieved. In Fig-ure 6.4 the two-terminal resistance Rxx as a function of the backgate voltage measured at T =4.2 K and with an applied magnetic field of B =10 T for the graphene bilayer (InGaAs-BLG(a)) on InGaAs is displayed. The number of layers in this case has been confirmed by Raman spectroscopy [112]. As demonstrated in inset of Figure 6.4, for both sweep directions starting from -1 V to -2.5 V (black trace) and back from -2.5 V to -1 V (red

6.2. MAGNETOTRANSPORT BEHAVIOR 69

Figure 6.4:Gate dependent resistance for bilayer graphene on InGaAs substrate (InGaAs-BLG(a)).

At B = 10 T SdHs are observable. The inset shows up and down sweep of the gate voltage in the negative region. Besides a small offset the signals proof themselves.

trace) Shubnikov de-Haas (SdH) oscillations are clearly visible. This substantiates the two-dimensional nature of the charge carriers and the high quality with rather high charge carrier mobility of the two-dimensional graphitic crystal. The small offset between these two traces may be caused by changes of the electrical field at a certain voltage between down and up sweep due to (de-)charging of deep traps and misfit dislocations in the meta-morphic grown buffer for strain relaxation [88], which is also part of the dielectric layer.

Since the distinct SdH oscillations are monitored for low negative bias voltages in the hole transport region, the type of intrinsic charge carriers for this BLG on InGaAs seems again to be holes, as found for all other graphitic flakes on GaAs or InGaAs substrates. The rea-son for the resistance behavior around zero gate voltage and for positive gate voltages is still unclear. The signal becomes noisy and no further (distinct) oscillations are detecable as visible in Figure 6.4. One may speculate, that measurements in two terminal geometry are influenced by polarity of the charge carriers relative to the current path and polarity of the magnetic field [98], that the lower Landau levels (first and second) are broadened by disorder and can not be resolved or that the dielectric material or the gate contact is damaged. One or a combination of these possibilities could explain the observed behav-ior. After these sweeps, the gate-gate contacts became diode-like, even though they were almost ohmic before these measurements. Due to that, no further measurements on this BLG were feasible.

-10 -8 -6 -4 -2 0 2 4 6 8 10

Figure 6.5: Magnetic field and gate voltage dependence of a graphene monolayer on GaAs (GaAs-MLG(b)) atT=4.2K. a) Four-terminal longitudinal magnetoresistanceR_xx(black) and Hall-resistanceR_xy forU_Gate= 0 V, an optical image of the device is inserted. b) Gate sweeps at B = 0 T (black) and B = -10 T. The resistance increases with increasing the voltage, but neither clear Shubnikov-de Haas oscillation nor clear quantum Hall plateaus are observable.

The magnetotransport properties with both, magnetic field a) and gate voltage sweep b) at T =+4.2 K measured in a multi-terminal geometry on a MLG sample (GaAs-MLG(b)) on GaAs are shown in Figure 6.5. The intrinsic longitudinal resistance Rxx, without

ap-6.2. MAGNETOTRANSPORT BEHAVIOR 71

Figure 6.6: Gate voltage dependence of a graphene monolayer on GaAs (GaAs-MLG(b)) atT = 4.2 K. a) Gate-voltage dependence ofR_xy for applied voltages between 3.5 V and -1.5 V (Same sample as in Figure 6.5). Inset: Carrier density extracted from Hall measurements in a) as a function of the effective gate voltage.

plied gate voltage measured as a function of a perpendicular applied magnetic field swept from -10 T to +10 T shows signatures of quantized transport (Figure 6.5). Small mag-netoconductance oscillations interpreted as universal conductance fluctuations are visible superimposed to a parabolic background in the whole magnetic field range ofB=±10T.

The Hall resistanceRxy shows a linear slope in the low-field region and rudimentary Hall-plateaus demonstrating transport in a two-dimensional charge carrier system. The positive Hall coefficient suggests transport in a hole system. From the slope ofR_xythe intrinsic hole density was determined to p=7.7·10¹² cm⁻². In Figure 6.5 b) two gate sweeps of the longitudinal resistance Rxx are plotted. The black trace is measured without an applied magnetic field (B=0 T), whereas the red line corresponds to the behavior with an applied magnetic field ofB=-10 T. The trace without field shows an increase of the resistanceRxx

by increasing the gate voltage, further verifying holes as intrinsic charge carriers. The CNP point is expected to appear for much higher positive voltages, namely forU_Gate=18.8 V as will be demonstrated later. From the linear dependence of the conductivity taking into account the shape of the contact geometry and the charge injection parameter again the mobility can be determined. Due to the high intrinsic doping only the hole mobility for this monolayer sample (GaAs-MLG(b)) can be observed, which constitutesµ=1660cm²/Vs and is almost a factor three lower compared to the bilayer sample GaAs-BLG(b). The data

acquired with an applied magnetic field ofB=−10T also shows an increase inRxx. Some changes in the slope, but no clear signatures for SdH oscillations are detectable by varying the carrier density over the gate voltage. Reasons for the absence of clear signatures for quantized transport can be found in the low charge carrier mobility and the high intrinsic charge carrier density together with the low breakthrough voltage that precludes larger changes of the two-dimensional charge carrier density.

In Figure 6.6 a), the Hall resistanceRxyfor the same monolayer sample (GaAs-MLG(b)) is measured for different gate voltages fromUGate=3.5V toUGate=−1.5V. Variations in the Hall-slopes indicates gate-induced changes of the two-dimensional hole density, which can be determined from the Hall slope. The two-dimensional hole density pexhibits a lin-ear dependency on the applied gate voltageUGate as demonstrated in the inset of Figure 6.6, where a shift of the CNP of about 18.8 V has been corrected. From the dependency of the carrier density on the gate voltage,n,p=α·UGate,e f f, the charge injection parameter αfor the GaAs substrate was determined to beαGaAs=4.1·10¹¹cm⁻²/V. The measured value for the used GaAs/AlGaAs substrate is about four times higher than the theoretically estimated one. Nevertheless, both, theoretical and measured values are higher compared to the literature value for SiO₂withαSiO₂ ≈7.2·10¹⁰cm⁻²/V [22].

The magnetotransport measurements on the bilayer sample on GaAs (GaAs-BLG-(a)) shown in Figure 6.7 have been performed in two-terminal geometry with a perpendicu-lar applied magnetic field up to B=±10 T without changing the carrier density by ap-plying a backgate voltage. In the high-field region (|B>1.5|T) signatures of quantized transport, namely SdH oscillations are detectable that are superimposed on a positive parabolic background. In the low field region quantum interference phenomena such as weak localization (WL) and universal conductance fluctuations (UCFs) were found. In the low field region these effects are superimposed by a negative parabolic background. Both regions are depicted for T = 4.2 K in Figure 6.7 a). As depicted in Figure 6.7 b), SdH oscillations, WL signal, UCF and the negative parabolic background exhibit strong tem-perature dependent damping, whereas the positive parabolic background in the high field region is almost temperature independent. The temperature dependent damping of the SdH oscillations is demonstrated in Figure 6.8 a) after subtraction of the second order polynomial background at higher fields and plotted vs. the inverse magnetic field1/B. The Fast-FourierTransformations (FFT) of the data shown in Figure 6.8 a) reveal two distinct maxima corresponding to densities p₂andp₃in the whole investigated temperature range (1.7 K≤T ≤50 K) and a smaller maximum p1(see Figure 6.8 b)). The two-dimensional carrier density seems to be p3=1.29·10¹² cm⁻²= 2·p2=4·p1. The relation between p₁,p₂and p₃may be caused by spin- and valley degeneracy. The degeneracy of such a graphene or thin graphite layer is known to beg=4[45]. From gate dependent measure-ments the type of charge carriers has been determined to be holes and from the density and the charge injection parameterαthe shift of CNP has been ascertained to 3.15 V. This means that the intrinsic hole density is rather low for this bilayer sample on GaAs (GaAs-BLG(a)) compared to most of all other investigated mono-, bi- and few-layer sample on

6.2. MAGNETOTRANSPORT BEHAVIOR 73

Figure 6.7: Temperature dependent two-terminal magnetoresistance of a bilayer graphene on GaAs (GaAs-BLG(a)). a) Magnetoresistance at 4.2K. The low-field region is marked. b) Tem-perature dependent magnetoresistance.

GaAs.

In Figure 6.9 the temperature dependent low fieldmagnetorsistance (MR) traces are

de-0 1x10¹⁶ 2x10¹⁶ 3x10¹⁶ 4x10¹⁶ 5x10¹⁶

Figure 6.8: Analysis of the temperature dependent two-terminal magnetoresistance from Figure 6.7 a) high-field region (B≥2T) after subtraction of a second order polynomial. b) Fast-Fourier transformation of the reciprocal traces of a).

picted for temperatures between 1.7 K and 10 K after subtraction of a temperature de-pendent negatively curved parabolic background which is shown in Figure 6.10 a). Both, WL and UCFs are reproducible, symmetric regarding polarity of the B-field and strongly

6.2. MAGNETOTRANSPORT BEHAVIOR 75

Figure 6.9: Temperature dependent two-terminal magnetoresistance of the bilayer graphene on GaAs (GaAs-BLG(a)- same sample as in Figure 6.7 and Figure 6.8). The temperature dependent magnetoresistance for low magnetic fields (B≤2T) is plotted after subtraction of a second order polynomial background.

polynominalbackgroundLF()W

b) a)

Figure 6.10: Analysis of the second order polynomial background of the electron-electron-interaction of Figure 6.9. a) Temperature dependent polynomial background in the low field region.

b) Slopeγof the polynomial background as a function ofB² plotted vsln(1/T)corresponding to τ_ee. Black traces correspond to the data taken for positiveB-fields and red from negative fields.

temperature dependent. Surprisingly, WL signatures seems to be suppressed for lower

temperatures or superimposed by increased UCF-signal. As outlined at the end of this chapter, from this quantum correction to the classical Drude conductivity, the phase coher-ence lengthL_Φcaused by inelastic scattering and the temperature dependence ofL_Φwill be determined as demonstrated in references [124,139,140]. The temperature dependent negatively curved parabolic background could also be induced by a quantum correction to the classical Drude conductivity due to impurity (electron-electron) interaction. Assuming transport in a diffusive channel in the metallic regime (EFτee/~>>1), the impurity inter-action time τee can be deduced from the parabolic background following the procedure demonstrated by Choi and coworkers [141]. The parabolic background can be written as

∆ρ_xx(B)∝ ρ µ eτee

m^∗σ₀

¶₂

·B²·δσ_ee (6.2)

withm^∗ effective mass,σ0 classical Drude conductivity andδσee correction to the Drude conductivity due to electron-electron interaction. ForkBTτee/~<1theory predicts for the two-dimensional case length. The slopes γ of R(B) plotted as a function of B² for different temperatures are collected in Figure 6.10 b). The slope γ is linearly dependent on ln(1/T). From this correlation τee can be extracted using equations (6.2) and (6.3). The resulting electron-electron interaction time is τ_ee= (0.157·^m_m^∗

0) ps. Assuming m^∗=0.1 m₀, the mass of charge carrier in BLG [142] or the mass of the heavy holes in FLG [1], one obtainsτee= 15.7fs. This value seems to be reasonable compared to values found for two-dimensional charge carrier systems in conventional heterostructures [141].