High mobility transport in isotopically-enriched 12 C and 13 C exfoliated graphene

Graphene quantum dots are promising candidates for qubits due to weak spin-orbit and hyperfine interactions. The hyperfine interaction, controllable via isotopic purification, could be the key to further improving the coherence. Here, we use isotopically enriched graphite crystals of both 12 C and 13 C grown by high-pressure-high-temperature method to exfoliate graphene layers. We fabricated Hall bar devices and performed quantum transport measurements, revealing mobilities exceeding 10 5 cm 2 /V s and a long mean free path of microns, which are as high as natural graphene. Shubnikov-de Haas oscillations, quantum Hall effect up to the filling factor of one, and Brown-Zak oscillations due to the alignment of hBN and graphene are observed thanks to the high mobility. These results constitute a material platform for physics and engineering of isotopically-enriched graphene qubits.


I. INTRODUCTION
Graphene quantum dots are among the most promising candidates as platforms for spin qubits [1][2][3] thanks to the weak spin-orbit coupling and hyperfine interactions.Understanding the hyperfine interaction is a strategy to improve the qubit coherence in several systems such as GaAs [4][5][6][7] and Si-based systems [8][9][10].The hyperfine interaction has also been investigated in 13 Cenriched carbon nanotube quantum dots [11] and it was theoretically proposed that graphene quantum dots could benefit from isototpe purificiaction.[1,[12][13][14], modulating the ratio of 12 C (nuclear spin 0) and 13 C (nuclear spin 1/2) (see Appendix A).Comparing the results of 12 C and 13 C-enrichment would enable the direct investigation of the effect of hyperfine interaction.While isotopicallyenriched graphene has been realized, they have been made by the chemical vapor deposition (CVD) method [15,16], whose electronic quality is usually not as high as for exfoliated samples and thus challenging for qubit fabrication.
In this work, we investigate high-quality exfoliated graphene devices out of 12 C-enriched (99.7%) and 13 Cenriched (91.4%) graphites.We synthesize enriched graphite by the high-pressure-high-temperature method [17,18].We then establish a recipe to exfoliate the layered crystals to obtain monolayer and bilayer graphene.The isotope effect is characterized by mass spectroscopy and Raman spectroscopy, showing a clear difference between 12 C and 13 C graphene.After confirming the enrichment, we fabricate a Hall bar device from each kind of graphite and perform quantum transport measurements.* siwakiri@phys.ethz.chFor both graphene samples, high mobility (µ ≥ 1 × 10 5 cm 2 /Vs) and long mean free path (l mfp ≥ 1 µm) are demonstrated.These values demonstrate that the electronic quality of graphene remains as high as natural graphene after isotopic enrichment.We also observe the Shubnikov-de Haas oscillations, quantum Hall effect, and Brown-Zak oscillations, which confirm the quality of the sample.This result forms the basis for building isotopically-enriched graphene qubits and investigating the role of hyperfine interactions.

A. Enriched graphite and exfoliation
Figure 1(a) shows the optical picture of the crystal of 12 C-enriched and 13 C-enriched graphite.Hereafter, we call them 12 C graphite/graphene and 13 C graphite/graphene for convenience.Isotopically enriched graphite crystals are obtained using Co-Ti solvents by a temperature difference method under high pressure.This method modifies the high-pressure synthesis conditions of diamond single crystals in the graphite stability region of 3 GPa and 1600 • C for 20 hrs [19].The carbon source material is dissolved in the molten solvent under high pressure and high temperature and precipitated as a graphite crystal.The source materials are 12 C-enriched diamond crystals prepared by CVD using 12 C-enriched methane gas (Tomei-diamond Co Ltd) and commercially available 13 C-enriched graphite.After high-pressure synthesis, the metallic solvents are removed with hot aqua regia, and the graphite crystals are purified with pure water.
Note that the crystal size (∼5 mm) is relatively small compared to the natural graphite crystals, which can be up to of the order of cm.The relative isotope ratio found in the crystals is determined using inductively coupled plasma time-of-flight mass spectrometry (ICP-TOFMS).
The ICP-TOFMS (icp-TOF2R, TOFWERK AG, Thun, Switzerland) is operated in low-mass mode and was coupled to an ArF excimer laser (193 nm, GeoLas C, Lambda Physik, Göttingen), equipped with a low dispersion ablation cell for minimally invasive sampling of the small crystals.The ratios of isotopes are determined for natural graphite, 12 C graphite, and 13 C graphite.The results of the natural graphite are used to calibrate the ratio for the other samples, assuming the natural abundance of 12 C: 13 C = 98.9 : 1.1.The result of the mass spectroscopy yields 12 C: 13 C = 99.7 : 0.3 for the 12 C sample and 8.6 : 91.4 for the 13 C sample, confirming the effect of enrichment.The relative concentrations of the isotopes determined by mass spectroscopy have a relatively large error because of the small amount of the sample available.
In spite of the relatively small size of the graphite crystals, exfoliation is possible for both 12 C and 13 C.We perform exfoliation by using the standard scotch tape method, where we deposit graphite onto the scotch tape and fold and peel apart the tape a certain number of times before the flakes are transferred onto a Si/SiO2 substrate.We compare exfoliation for folding and peeling apart the tape 4-6 times with 10-15 times.We analyze the chips under a microscope and record the size of the flakes.The statistics of the flake size (mono-, bi-, and trilayer 12 C graphene) is shown in Figure 1(b), together with an image of an example of a large bilayer flake.It turns out that after 4-6 times of exfoliation, it is possible to obtain 50 µm scale graphene flakes of mono, bi, and tri layers.The same result is obtained for the 13 C graphene.These results demonstrate that the enriched 12 C and 13 C exfoliation is possible and that the flakes are available for fabricating devices.

B. Raman spectroscopy
The exfoliated graphene is further characterized by Raman spectroscopy (Horiba LabRAM HR Evolution UV-VIS-NIR).The laser energy and wavelength are 1 mW and 532 nm, respectively.We choose a flake larger than 10 µm and perform a Raman spectrum measurement by changing the laser position.The data shown in Fig 1(c) is from the part of the graphene that is a few µm inside from the edge of the flake.The measurements are done on mono and bilayer flakes.As shown in Fig. 1(c), both 12 C and 13 C graphene show prominent G (∼1520-1580 cm −1 ), D+D ′′ (∼2360-2490 cm −1 ), 2D (∼2540-2710 cm −1 ), and 2D ′ (∼3120-3250 cm −1 ) peaks.The numbers in the parentheses are typical ranges of the observed peak wavenumbers.We see a clear peak wavenumber shift between 12 C and 13 C.These differences are one of the most prominent signatures of the isotope effect because the nuclear mass difference results in a phonon frequency and Raman shift difference.We do not observe a shift between 12 C and natural graphene within the experimental resolution.After subtracting a linear background from the data, we fit all the peaks with Lorentzian functions.The fit parameters are in the table in Appendix B.
The obtained parameters are consistent with the ones reported in exfoliated natural graphene and isotopically enriched CVD graphene [20][21][22][23][24].The measured wavenumber of the G peak, ω G , for 12 C or natural graphene reported in Refs.[20][21][22][23][24] are between −2 cm −1 and 10 cm −1 of our measured values for 12 C graphene (for mono-and bilayer).The ω G for 13 C graphene reported in [22,24] is in the range of −5 cm −1 to 8 cm −1 of the values we measure for mono-or bilayer.Comparing the wavenumber of the 2D peak, ω 2D , to literature values, it turns out that the values reported are generally slightly higher (up to 1.5 %) than the values we measure.For the monolayer 12 C graphene ω 2D , we find that our measurement aligns very well (within 1 cm −1 ) with the natural graphene measurement in [20].However, the measurements in Refs.[22][23][24] find values 15-35 cm −1 higher than what we measure for 12 C monolayer graphene as well as for 13 C monolayer graphene.Looking at the bilayer measurements for ω 2D peaks [20], we find two peaks that are 2 cm −1 and 3 cm −1 apart.For ω 2D ′ [25] reports 3250 cm −1 for 12 C graphene and 3130 cm −1 for 13 C graphene.These values are consistent with what we measure.
We also estimate the isotopic concentration of 13 C graphene from the Raman shift using the relation ω = ω 12C m m+x∆m [26].Here, ω 12C is the Raman shift of pure 12 C graphene, m is atomic mass of 12 C, x is the concentration of 13 C, and ∆m the atomic mass differ-ence of 12 C and 13 C.For 13 C graphene, we obtain a 13 C enrichment x = 92.9%using the 2D and 2D' peaks and x = 86.2%using the G peak.Since the G peak is sensitive to the distortion and carrier density [27], it can be affected by unintentional doping due to the charged impurity in the silicon substrate, making the peak shift.The estimation from 2D and 2D' peaks is closer to the value obtained by the mass spectroscopy with a slight overestimation (±1.5%).At around 93% of 13 C, the deviation of Raman shifts by one cm −1 modulates the concentration estimation by ∼ 1.7%.Therefore, considering the resolution of the measurement, the estimation from the 2D and 2D' peaks is consistent with the one from the mass spectroscopy.

C. Device fabrication
To characterize the transport quality of the isotopically enriched graphene, we fabricate Hall bars of 12 C and 13 C bilayer graphene as shown in Figure 1(d).A stack of hexagonal boron-nitride (top hBN)/bilayer graphene/hBN (bottom hBN)/graphite (back gate) is made by the poly-dimethylsiloxane/poly-carbonates dry transfer method.Electric contacts to the edge of the bilayer graphene are fabricated by etching the top hBN and depositing Cr/Au.After etching, a Hall bar is shaped by reactive ion etching (CHF 3 and O 2 ).We measure the Hall bars in a dilution refrigerator with a base temperature of 55 mK.The bottom panel of Fig. 1(d) shows the schematic of the device.The two contacts at each sample end are used as source and drain electrodes, where we inject current from the source to the grounded drain.The longitudinal and transverse voltages (V x and V y ) are measured between the two contacts along and across the sides.The spacing between source and drain, and between the voltage contacts is 4 µm and 1 µm, respectively.We use a lock-in amplifier (Stanford Research Systems SR830) connected to the source in series with a 100 MΩ resistance and apply an AC voltage of 1 V, generating an AC current of 10 nA.We synchronize the lock-in amplifier driving the current I with the other two lock-in amplifiers.These lock-in amplifiers are then used to measure the resistances (R xx = dVx dI and R xy = dVy dI at zero bias current).We apply a DC voltage (Yokogawa 7651 Programmable DC Source) to the back gate (not shown in the schematic).n|e| .We also obtain consistent values from a parallel plate capacitor model (n = C bg V bg , where C bg is the capacitance between the graphene and the back gate and V bg is the back gate voltage) and Shubnikov-de Haas measurements (R xx ∝ cos( 2πnh 4eB )).For both samples, clear fan-like structures departing from the charge neutrality point n = 0 and expanding with magnetic field are seen, which is attributed to the Shubnikov-de Haas (SdH) oscillations.The oscillations appear already at around 1 T, testifying to our samples' high mobility.White dotted tilted lines show the fitting to the filling factor of 1, 2, and 3 at around n = 0 (for 12 C and 13 C) and filling factor for 2, 6, and 10 at around n = ±2.3× 10 12 cm −2 (for 12 C only).

III. TRANSPORT MEASUREMENT
Additionally, in the 12 C sample, we see multiple Landau fans appearing at densities of around 4.09, 3.29, 2.30, 1.78, -2.06, and -2.31 ×10 12 cm −2 .This is attributed to the additional Dirac points due to the unintended alignment between the graphene and one of the hBNs, forming a moiré superlattice [28].This results in an energy spectrum for the charge carriers known as the Hofstadter butterfly and causes satellite Dirac peaks.Furthermore, at the intersections of the Landau fans, it predicts horizontal lines of peaks in R xx called Brown-Zak oscillations [29,30].The horizontal lines in Fig. 2(a) indicate the position of ϕ/ϕ 0 = 1/p with p an integer and ϕ 0 = h/e the flux quanta.As we discuss in Appendix C, the moiré unit cell size estimated from the Brown-Zak oscillations agrees with the one formed by graphene/hBN alignment.This observation is another piece of evidence for having a high-quality sample.
At a high magnetic field, we observe the quantum Hall effect, as shown in Figs.2(b) and (e).For both samples, the integer quantum Hall effect is observed up to filling factor ν = 1 with quantized plateaus of R xy and dips of R xx .The significant drop of R xy in 12 C beyond filling factor 1 (B ≥ 7 T) is due to the Brown-Zak oscillations.We observe a similar drop whenever B crosses the horizontal lines indicated in Fig. 2(a).
We also estimate the mobility µ and mean free path |e| µ.As seen in Figure 2(c) and (f), the mobility mostly ranges from 1 × 10 4 to 3 × 10 5 cm 2 /Vs.The mobility becomes zero around charge neutrality and forms a peak with increasing n.This behavior can be attributed to the difference in dominant scattering mechanisms (long-range Coulomb scattering at low density and short-range impurity scattering at high density) [31].
The mean free path l mfp reaches up to 1 − 5 µm for both 12 C and 13 C.This value is comparable to the one reported in natural graphene.Note that the spacing between the contacts is of the same order of magnitude as the estimated mean free path, meaning that the transport in the sample is in the ballistic regime.In this regime, the Drude model has limited validity, and the mobility and mean free path are lower bound estimates.
We further perform a quantum Hall effect breakdown measurement, applying a DC current up to 10 µA so that the quantum Hall effect (plateau in ρ xy and zero in ρ xx ) is no longer observed.In GaAs 2DEG systems, a large hysteresis in the current sweep direction before and after the breakdown is observed typically for odd filling factors due to dynamic nuclear polarization (spin transfer from electrons to nuclei) [32,33].We do not observe any clear hysteresis within our measurement precision (data is not shown here).To observe the effect of hyperfine interaction, more sophisticated measurements such as resistivity-detected NMR [34][35][36] or actually building a quantum dot and performing a T 1 and T 2 coherence time measurement would be useful.

IV. CONCLUSION
In conclusion, we have presented high-mobility transport of exfoliated 12 C and 13 C graphene synthesized by the high-pressure-high-temperature technique.We identified distinct differences between 12 C and 13 C graphene by the mass and Raman spectroscopy.We also fabricated Hall bar devices and performed quantum transport measurements, revealing high mobility and a long mean free path.Shubnikov-de Haas, quantum Hall, and Brown-Zak effects up to the filling factor of one were observed thanks to the high mobility.These results pave the way for developing isotopically-enriched graphene qubits and investigating the role of hyperfine interactions in graphene quantum dots.

FIG. 1 .
FIG. 1.(a) Optical picture of 12 C and 13 C graphite crystals.The minimum grid size in the background is 1 mm.(b) Histogram of 12 C graphene flake size.Inset shows a picture of the example bilayer flake.The scale bar is 50 µm.(c) Raman spectrum of 12 C and 13 C graphene.Top: Monolayer graphene.Bottom: Bilayer graphene.(d) Top panel: Optical picture of the fabricated Hall bar.The scale bar is 10 um.Bottom panel: Schematic of the fabricated Hall bar.Source, drain, and voltage terminals are shown together with the geometric measurements.

4 FIG. 2 .
FIG. 2. Quantum transport measurement of 12 C (a,b,c) and 13 C (d,e,f) bilayer graphene.(a,d) Rxx as a function of magnetic field B and carrier density n. White dotted tilted lines fit the Shubnikov-de Haas oscillations with filling factors shown in the Figure.White dotted horizontal lines show the position at which the Brown-Zak oscillations appear.(b,e) Example traces of the quantum Hall effect at n = 1.13 × 10 11 ( 12 C) and n = 1.9 × 10 11 ( 13 C).(c,f) Carrier density n dependence of the mobility µ and the mean free path l mfp estimated from the low magnetic field data.

Figures 2 (
Figures 2(a) and (d) show the magnetic field B and the carrier density n dependence of the longitudinal resistance R xx of 12 C and 13 C devices.The carrier density on the horizontal axis is estimated by measuring the classical Hall effect up to 100 mT and using the relation R xy = Bn|e| .We also obtain consistent values from a parallel plate capacitor model (n = C bg V bg , where C bg is the capacitance between the graphene and the back gate and V bg is the back gate voltage) and Shubnikov-de Haas measurements (R xx ∝ cos( 2πnh 4eB )).For both samples, clear fan-like structures departing from the charge neutrality point n = 0 and expanding with magnetic field are seen, which is attributed to the Shubnikov-de Haas (SdH) oscillations.The oscillations appear already at around 1 T, testifying to our samples' high mobility.White dotted tilted lines show the fitting to the filling factor of 1, 2, and 3 at around n = 0 (for 12 C and 13 C) and filling factor for 2, 6, and 10 at around n = ±2.3× 10 12 cm −2 (for 12 C only).Additionally, in the 12 C sample, we see multiple Landau fans appearing at densities of around 4.09, 3.29, 2.30, 1.78, -2.06, and -2.31 ×10 12 cm −2 .This is attributed to the additional Dirac points due to the unintended alignment between the graphene and one of the hBNs, forming a moiré superlattice[28].This results in an energy spectrum for the charge carriers known as the Hofstadter butterfly and causes satellite Dirac peaks.Furthermore, at the intersections of the Landau fans, it predicts horizontal lines of peaks in R xx called Brown-Zak oscillations[29,30].The horizontal lines in Fig.2(a) indicate the position of ϕ/ϕ 0 = 1/p with p an integer and ϕ 0 = h/e the flux quanta.As we discuss in Appendix C, the moiré unit cell size estimated from the Brown-Zak oscillations agrees with the one formed by graphene/hBN alignment.
l mfp by applying the classical Drude model to ρ xx = W L R xx (W and L are the width and the length of the sample) and ρ xy = R xy at a low magnetic field (≤ 100 mT).From the slope of the linear fit of R xy up to 100 mT together with R xx at 0 T, we determine µ and l mfp using the equations µ = 1 ρxx(B=0) dρxy dB and l mfp = ℏ √ πn