Dirac fermion heating, current scaling, and direct insulator-quantum Hall transition in multilayer epitaxial graphene
© Liu et al.; licensee Springer. 2013
Received: 2 July 2013
Accepted: 12 August 2013
Published: 22 August 2013
We have performed magnetotransport measurements on multilayer epitaxial graphene. By increasing the driving current I through our graphene devices while keeping the bath temperature fixed, we are able to study Dirac fermion heating and current scaling in such devices. Using zero-field resistivity as a self thermometer, we are able to determine the effective Dirac fermion temperature (TDF) at various driving currents. At zero field, it is found that TDF ∝ I≈1/2. Such results are consistent with electron heating in conventional two-dimensional systems in the plateau-plateau transition regime. With increasing magnetic field B, we observe an I-independent point in the measured longitudinal resistivity ρxx which is equivalent to the direct insulator-quantum Hall (I-QH) transition characterized by a temperature-independent point in ρxx. Together with recent experimental evidence for direct I-QH transition, our new data suggest that such a transition is a universal effect in graphene, albeit further studies are required to obtain a thorough understanding of such an effect.
Graphene, which is an ideal two-dimensional (2D) system, has been attracting worldwide interest since its discovery in 2004 . While the sizes of mechanically exfoliated graphene are limited, its ultrahigh quality allows one to observe fascinating physical phenomena such as ambipolar characteristics , anomalous integer quantum Hall steps , Berry's phase [2, 3], and fractional quantum Hall effect [4–6]. On the other hand, graphene prepared by chemical vapor deposition (CVD) and epitaxial graphene can be used for potential device applications because the sizes of these systems should allow realization of wafer-scale integrated circuits based on graphene .
When a charge system is appreciably heated by a driving current, the equilibrium between the phonons and the charges collapses. In this situation, effective charge temperature (T c ) can be substantially higher than lattice temperature (T L ) . This interesting physical phenomenon is normally called the charge heating effect. In some cases, there exists a simple effective charge temperature-current relation T c ∝ I α , where α is an exponent that depends on charge-phonon scattering . It is now well established that the two-bath model can be used to describe charge heating and charge energy loss rate by charge-phonon scattering . The charge heating effect has become increasingly important as device dimensions are reduced and charge mobility is increased . In particular, Dirac fermion heating in graphene is an important physical phenomenon since it affects thermal dissipation and heat management in modern electronics  and low-temperature applications such as quantum resistance metrology .
Insulator-quantum Hall (I-QH) transition [12–15] is an interesting physical phenomenon in the field of 2D physics. Especially, a direct transition from an insulator to a high Landau level filling factor ν ≥ 3 QH state which is normally described as the direct I-QH transition continues to attract interest [16–18]. Very recently, experimental evidence for direct I-QH transition in epitaxial monolayer graphene  and in mechanically exfoliated multilayer graphene  has been reported. In order to further study direct I-QH transition in the graphene-based system, one may wish to investigate Dirac fermion heating in graphene. Moreover, it is a fundamental issue to see if a current-independent point in the longitudinal resistivity when the bath temperature is fixed exists since such a point should be equivalent to the direct I-QH transition. Furthermore, one could probe current scaling on both sides of the direct I-QH transition to further study Dirac fermion-phonon scattering as well as Dirac fermion-Dirac fermion scattering, both of which are very fundamental physical phenomena.
In this paper, we report magnetotransport measurements on multilayer epitaxial graphene of few layers obtained under conditions which favor controlled growth at high temperatures . Dirac fermion heating in the high current limit is studied. It is found that in the low magnetic field regime, the effective Dirac fermion temperature obeys a simple power law TDF ∝ I≈0.5. Such results suggest that the Dirac fermion-phonon scattering rate 1/τDFP ~ T2, consistent with those in conventional 2D electron systems. With increasing magnetic field, interestingly, a current-independent point in the longitudinal resistivity is observed. It was demonstrated that such a point corresponds to the direct I-QH transition characterized by a T-independent point in ρxx. This result is further supported by the vastly different I dependences for both sides of the I-QH transition. Our new experimental results, together with recent experimental results [19, 20], indicate that direct I-QH transition is a universal effect in graphene. We suggest that further experimental and theoretical studies are required to obtain a complete picture for direct I-QH transition in graphene-based devices.
A controlled sublimation method was used for graphene growth on a 6H-SiC (0001) surface . First, the SiC substrate was cleaned using a standard procedure for substrate cleaning . Second, the optically polished Si-face surface was placed face-to-face with a polished graphite disk (FTG) and arranged such that uniform Newton rings were observed in fluorescent light . The optically finished substrate surfaces resulted in a higher rate of SiC decomposition compared to chemical–mechanical processed (CMP) surfaces and created multiple graphene layers.
The epitaxial growth process was controlled by annealing in a sequence of temperature ramp and dwell stages in Ar background gas at a pressure slightly higher than 1 atm using a commercial furnace. The substrates were first dehydrated and cleaned in the furnace at 725°C for approximately 16 h. The temperature was ramped to 1,200°C for 30 min and then ramped at 100°C/min for graphene growth at a temperature (dwell time) of 1,850°C (45 min; samples 1 and 2) or 1,950°C (30 min; samples 3 and 4). The temperatures were measured and controlled using molybdenum-sheathed type ‘C’ thermocouples.
Results and discussion
If p = 2 [10, 25], then the exponent in the temperature-scaling relation is 0.5 [21, 26–28] which is consistent with our experimental results obtained on Dirac fermions. We note that our experimental results are equivalent to a T4 dependence of energy loss rate for Dirac fermions as calculated  and observed in epitaxial, CVD-grown and exfoliated graphene [10, 30]. It is worth pointing out that previous results are obtained in the plateau-plateau transition regime [26, 27, 31] and Shubnikov-de Haas region , which is in contrast with our case in the weak insulating regime where Landau quantization is not significant. Nevertheless, our data indeed indicate such a universal exponent at approximately 0.5 for heating in various 2D systems. Moreover, our results suggest that the Dirac fermion-phonon scattering rate 1/τDFP is proportional to T2. It is worth noting that enhanced mobility can be achieved in semiconductor quantum wires  and in semiconducting graphene nanoribbons  by a high dc electric field. Such interesting results are highly desirable for practical applications in narrow graphene devices in the high current limit.
n (1013 cm−2)
ρxx/ρxy at Bc
At the crossing fields, the corresponding Landau filling factors are much larger than 2. Therefore, we have observed direct I-QH transition in all our devices [17–20]. It was argued that for direct I-QH transition in conventional semiconductor-based 2D systems, near the crossing field, ρxx is approximately ρxy, and the product of μBc is close to 1 . However, in all our devices, ρxx/ρxy is much greater than 1, and μBc is always smaller than 1. Therefore, our data suggest that further studies are required to obtain a thorough understanding of the direct I-QH transition not only in conventional 2D systems but also in disordered graphene. The observation of a current-independent point in ρxx which corresponds to its temperature-independent counterpart suggests that applying a high current is equivalent to heating up the graphene lattice.
In conclusion, we have presented magnetoresistivity measurements on multilayer epitaxial graphene. It is found that a relation between the effective Dirac fermion temperature and the driving current can be given by TDF ∝ I≈0.5 in the low magnetic field regime. With increasing magnetic field, an I-independent point in ρxx is observed which is equivalent to its T-independent counterpart in the low current limit. Evidence for direct I-QH transition has been reported in four different graphene samples. Near the crossing field where the longitudinal resistivity is approximately T-independent, ρxx is at least two times larger than ρxy. Moreover, the product of Drude mobility and Bc is smaller than 1. We suggest that further studies are required to obtain a complete understanding of direct I-QH transition in disordered graphene.
This work was funded by the National Science Council (NSC), Taiwan and National Taiwan University (grant number 102R7552-2).
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