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Shubnikov-de Haas (SdH) oscillations in the longitudinal magnetoresistivity are caused by the B-dependent variation in the chemical potential due to Landau quantization as introduced in Sec. 1.4. Resistivity minima (maxima) appear when the Fermi energy F resides in between Landau levels (in the center of

a Landau level). In order for SdH oscillations to be observed, Landau level (LL) broadening must be overcome. The latter is set by the quantum lifetime τ_q, which is a measure of how long a charge carrier remains in a momentum eigenstate in the presence of scattering.70,71,¶ _{From the onset of SdH oscillati-}

ons in B one may derive the so-called quantum mobility µq as the uncertainty

principle requires LL broadening in energy due to charge carrier scattering, ∆ = ~/τq, to be smaller than the cyclotron energy ~ωc (gap between neighbo-

ring LLs) where ωc = eB/m∗, i.e. (using Eq. 1.11),

µqB  1. (1.17)

¶_{In contrast to the transport time τtrit takes into account the total scattering probability,}

Figure 1.11: Shubnikov-de Haas (SdH) oscillations measured at dierent backgate voltages |UG| ≤ 50 V. The data is recorded in section 1 of the bilayer graphene

device shown in Fig. 1.6a. (a) ∆ρxx values at T = 1.4 K plotted with a constant

oset. Minima (open symbols) and maxima (lled symbols) of SdH oscillations with associated lling factors ν. (b) Filling factor fan chart of the extracted minima (open symbols) and maxima (lled symbols) of the SdH oscillations (∆ρxx). (c) Frequency

f_SdHof SdH oscillations as a function of charge carrier density n.

Fig. 1.11 shows SdH oscillations in the longitudinal magnetoresistivity me- asured in section i = 1 of the bilayer graphene device shown in Fig. 1.6a at dierent values of the applied backgate voltage UG. In Fig. 1.11a the variation

in the magnetoresistivity ∆ρxx is plotted as function of inverse magnetic eld

strength B−1_{for U}

G= 15 V, . . . , 50 V. The gate voltage is varied in steps of 5 V

and the data is recorded at T = 1.4 K. SdH oscillations with a single frequency component fSdH can clearly be seen in each data set starting from B ≈ 4 T

(compare with Fig. 1.10) yielding µq≈ 2500 cm2/Vson the same order of mag-

nitude as values of µ in Fig. 1.8a. Minima (open symbols) and maxima (lled symbols) can directly be extracted as illustrated for the trace at UG = 50 V.

Using the charge carrier density n extracted from a simultaneous measurement of the Hall voltage, one can determine the lling factor ν = nh/Be (number of lled LLs) for each value of B as indicated in the gure. The step size of ∆ν = 4between adjacent maxima (or minima) of the SdH oscillations eviden- ces the fourfold degeneracy δ = 4 of the LLs in bilayer graphene. This allows us to calculate the charge carrier density n directly from each SdH oscillation

Figure 1.12: Temperature dependence of SdH oscillation amplitudes at an electron density nSdH= 3.3 · 1012cm−2. Data was recorded between T = 1.4 K and T = 110 K

in steps of 5 K from T = 5 K. (a) Amplitude of SdH oscillations ∆ρxx (light grey

lines), positions of extrema (dots), and ts to Eq. 1.19 (solid colored lines) using the given values for m∗ _{and τ}

q. (b) Dingle plot: The SdH oscillation amplitude ∆ρxx

normalized by ρ0(T ) and corrected for thermal damping by X(T ) plotted against

reciprocal eld. The solid line is a t to the data constrained to go through the theoretical value of 4 at B−1 = 0. as n = δ Φ0 · f_SdH=4 · e h · fSdH. (1.18)

According to Eq. 1.18, a complementary approach to identify δ is to plot fSdH

as a function of n, see Fig. 1.11c. For both electrons and holes, we can extract a linear dependence fSdH = β|n| with β = (1.03 ± 0.01) · 10−15Tm2. Since

β = Φ0/δ with Φ0= h/e ≈ 4.14 · 10−15Tm2, we get δ = 4.

We can further prove the trivial Berry phase ΦB in bilayer graphene as

follows. In the fan chart shown in Fig. 1.11b open (lled) symbols represent minima (maxima) of the SdH oscillations in ρxx(B). At each value of UG,

the points fall on straight lines extrapolating to ν = 0 at B−1 _{= 0. Minima}

of SdH oscillations in bilayer graphene are found at integer multiples of δ, corresponding to the conventional case of SdH oscillations in a 2DES with a parabolic band dispersion.72 _{Despite the same LL degeneracy δ, in single}

layer graphene the oscillations would be shifted featuring maxima instead of minima in ρxx(B)at integer multiples of δ, due to the non-trivial Berry phase

SdH oscillations also provide a measure of the eective mass m∗_{, which}

we may extract together with the quantum lifetime τq from the temperature

dependent amplitude ∆ρxx. As an example, we measured SdH oscillations at

an electron density of nSdH= 3.3·1012cm−2(determined according to Eq. 1.18)

at T = 1.4 K as well as between T = 5 K and T = 110 K in steps of 5 K. After subtracting the non-oscillating component of each trace ρxx(B), we plot the

absolute value of SdH oscillations as grey lines in Fig. 1.12a. The dependence of the oscillation amplitude ∆ρxx on B and T can be described as73

∆ρxx= 4ρ0X(T ) exp  − π ωcτq  , with X(T ) = 2π 2_k BT /~ωc sinh (2π2_k BT /~ωc) . (1.19) Here, X(T ) is a thermal damping factor and kB is the Boltzmann constant.

We identify well-resolved extrema of the oscillations (dots in Fig. 1.12a) in order to nd a least-square t of ∆ρxx(B)for all temperatures simultaneously,

using Eq. 1.19 with m∗ _{and τ}

q as the only free parameters. The best t (solid

colored lines in the gure) is obtained using m∗ _{= 0.043 · m}

e and τq = 31.5 fs.

The Dingle plot of ∆ρxx normalized by ρ0(T ) and corrected for temperature

by X(T ) in Fig. 1.12b demonstrates the accurate description of SdH oscillation amplitudes by Eq. 1.19 using these parameters.

From this analysis we learn that at the given electron density, m∗_{is in fact ≈}

10 %smaller than what we expect according to our description in Sec. 1.3. This nding is in close agreement with an analysis from Zou et al.33 _{and could be}

accounted for when using dierent values of the bilayer graphene band structure parameters γ0 and γ4 (stated in Table 1.1), the latter of which we neglect

altogether in our simplied band description (Eq. 1.3). The extracted value of the quantum lifetime τq = 31.5 fs is in good agreement with values reported for both single and bilayer graphene samples on SiO2.33,71 When compared to

the transport time, which we determine to be τtr = 84 fs using Eqs. 1.16 and

1.11 with both m∗ _{and n}

SdH as extracted in the preceding analysis, we obtain

the ratio τtr/τq = 2.67. This is a representative value for graphene samples

prepared on SiO2 and identies long-ranged disorder potentials to dominate

charge carrier scattering.71 _{In addition to charged impurities, random strain}

uctuations are likely their major source.55,74_{Further support for this assertion}