Abstract
Graphene offers a unique system to investigate transport of Dirac Fermions at p–n junctions. In a magnetic field, combination of quantum Hall physics and the characteristic transport across p–n junctions leads to a fractionally quantized conductance associated with the mixing of electronlike and holelike modes and their subsequent partitioning. The mixing and partitioning suggest that a p–n junction could be used as an electronic beam splitter. Here we report the shot noise study of the modemixing process and demonstrate the crucial role of the p–n junction length. For short p–n junctions, the amplitude of the noise is consistent with an electronic beamsplitter behaviour, whereas, for longer p–n junctions, it is reduced by the energy relaxation. Remarkably, the relaxation length is much larger than typical size of mesoscopic devices, encouraging using graphene for electron quantum optics and quantum information processing.
Introduction
In graphene, owing to the linear and gapless band structure, ntype (electronlike) and ptype (holelike) regions can adjoin without a gap in between. Investigations of charge carrier transport across such a p–n junction (PNJ) have revealed unique phenomena reflecting the Dirac Fermion character in graphene, such as Klein tunneling^{1,2,3}, Veselago lensing^{4} and snake state^{5,6}. In the quantum Hall (QH) effect regime under high magnetic field B, the conductance across a PNJ shows plateaus at G_{PNJ}=G_{0}ν_{1}ν_{2}/(ν_{1}+ν_{2}), where G_{0}=e^{2}/h is the conductance quantum (h is Planck’s constant), ν_{1}=2, 6, 10, … and ν_{2}=−2, −6, −10, … are the Landau level filling factor in the n and p regions, respectively^{7,8,9}. This conductance quantization in bipolar QH states has been explained by the mixing of countercirculating electron and hole edge modes^{10,11}: the current injected to the PNJ is distributed to electron and hole modes in the PNJ by the mode mixing with the ratio depending on the number of each modes, thus on ν_{1} and ν_{2}, and then partitioned at the exit of the PNJ (Fig. 1b). This process gives rise to the conductance quantization at the values depending on ν_{1} and ν_{2}. However, experimental study of the modemixing mechanism is lacking. If the mode mixing is caused by quasielastic scattering as suggested in ref. 10, 11, a graphene PNJ acts as a beam splitter of electrons and holes. A better understanding of these properties is a crucial step towards the development of electron quantum optics experiments in graphene; beam splitters together with edge states are key components for electronic interferometry^{12,13,14}.
Shot noise measurements can provide insight into the modemixing mechanism (Supplementary Figs 1 and 4): when the electron and hole modes biased by V_{sd} are mixed, the energy distribution in the PNJ f_{PNJ}(E) becomes outofequilibrium and the subsequent partitioning of the modes gives rise to the shot noise. If the mode mixing is quasielastic, f_{PNJ}(E) is a doublestep function. At zero temperature, the shot noise generated by the partitioning of the modes with doublestep f_{PNJ}(E) is expected to be (^{10}, Supplementary Note 1),
characterized by the Fano factor F=S_{I}/2eI, yielding F=0.25 for (ν_{1}, ν_{2})=(2, −2). Energy losses towards external degrees of freedom can drive f_{PNJ}(E) towards a Fermi distribution with a chemical potential eV_{sd}/2 (Fig. 1c)^{15,16,17}, causing the noise (and thus the Fano factor) to vanish as the carrier dwell time in the PNJ becomes larger than the energy relaxation time (^{18}). Inelastic processes between modes in the PNJ may occur, causing f_{PNJ}(E) to relax towards a Fermi distribution with a finite temperature T_{eff}(V_{sd}) given by the balance between the Joule power dissipated in the PNJ and the heat flowing along the outgoing electronic channels^{10,19} (Supplementary Note 1). In this case, the Fano factor becomes [ for (ν_{1}, ν_{2})=(2, −2)]. Note that standard transport measurements yield the same value of G_{PNJ} for all cases, and thus cannot distinguish them.
In the following, we investigate the evolution of f_{PNJ}(E) with carrier dwell time in the PNJ, which can be tuned by changing the length of the PNJ. We demonstrate that the amplitude of the noise is consistent with an electronic beamsplitter behaviour when PNJ is short and the energy losses are negligible.
Results
Measurement setup
We obtained bipolar graphene devices using a top gate covering half of the graphene (Fig. 1a); the carrier type in the gated region can be tuned by the gate voltage V_{G}, while that in the ungated region is fixed to electron by the doping (Methods). Therefore, the PNJ is formed at the interface between the gated and ungated regions when the carrier type in the gated region is hole for ΔV_{G}≡V_{G}−V_{CNP}<0 (V_{CNP} is the gate voltage at the charge neutrality point). We prepared five samples with different interface lengths L=5, 10, 20, 50, and 100 μm. The direction of B is chosen so that electron and hole modes from the ohmic contacts and , respectively, merge at the PNJ. For the noise measurement, V_{sd} is applied to either or and the noise is detected on C_{det} (Fig. 1a; Supplementary Methods and Supplementary Fig. 5). Magnetic fields up to B=16 T have been applied. The base temperature is T=4.2 K.
Detection of shot noise generated at PNJ
The inset of Fig. 2 shows the reflection of the averaged current from to C_{det} in the sample with L=50 μm. The magnetic field is B=10 T, at which the filling factor in the ungated region is fixed at ν_{ug}=2. When the bipolar QH state at (ν_{ug}, ν_{g})=(2, −2) is formed for ΔV_{G}<−10 V, the current injected from is partitioned equally to the electron and hole modes at the exit of the PNJ, yielding a reflection of 1/2. A current noise S_{I} appears in this regime. As V_{sd} applied to is increased, the excess noise ΔS_{I}≡S_{I}−S_{I}(V_{sd}=0) increases (solid cyan circles in Fig. 2). ΔS_{I} approaches linear behaviour for eV_{sd}>k_{B}T, characteristic of the shot noise. A similar signal appears when V_{sd} is applied to (open cyan circles). In the unipolar QH state at (ν_{ug}, ν_{g})=(2, 2) for 20<ΔV_{G}<50 V, on the other hand, the shot noise is zero (solid black circles), proving that the shot noise is indeed generated at the PNJ.
Quantitatively, we extracted F by fitting ΔS_{I} as a function of V_{sd} using the relation including temperature broadening^{20}:
where G_{PNJ} is obtained by average current measurements. The fit yields F=0.015, which is one order of magnitude smaller than F=0.25 expected for the noise from the doublestep energy distribution. This indicates that f_{PNJ}(E) evolves during the charge propagation for L=50 μm, reducing the shot noise.
Evolution of noise amplitude with PNJ length
The evolution of f_{PNJ}(E) can be investigated using samples with different L. Figure 3a shows the results of the noise measurement in the bipolar QH state at (ν_{ug}, ν_{g})=(2, −2) for the five samples with L between 5 and 100 μm. The data show that the shot noise decreases with increasing L and almost disappears at L=100 μm (Fig. 3b), indicating that f_{PNJ}(E) relaxes to the thermal equilibrium through interactions with external degrees of freedom. An exponential fit of the data yields a relaxation length L_{0}=15 μm. The extrapolation to L=0 gives F∼0.27, consistent with the limit of quasielastic scattering F=0.25. Furthermore, the decrease is well reproduced by a model gradually coupling the modes propagating in the PNJ to cold external states (Supplementary Note 1 and Supplementary Fig. 2). Note that we are not able to observe whether inelastic scattering occurs inside the PNJ, because of the large error bars explained below. An important implication of the results is that, within the typical scale of usual mesoscopic devices (<1 μm), the energy loss towards external degrees of freedom is negligible and the current channels in the PNJ can be regarded as an isolated system.
Fluctuation of noise
We further investigate the properties of the PNJ focusing on the energy relaxation mechanism by measuring the shot noise for a wide range of B and ΔV_{G}. We identify the electronic states in the gated and ungated regions as a function of B and ΔV_{G} by a lowfrequency current measurement from to C_{det} (Fig. 4a) and then investigate the relation between those states and the noise. The electronic state in the ungated region depends only on B and the ν_{ug}=2 QH state is formed for B>4 T. In the gated region, the nonQH states at ν_{g}=8, 4, 0 and −4 appear as a current peaks. The bipolar QH state at (ν_{ug}, ν_{g})=(2, −2) is formed for B>4 T and between ν_{g}=0 and −4 (the region indicated by dashed lines), in which the current is almost constant, consistent with the quantized conductance^{7,8,9}. The shot noise in the sample with L=10 μm becomes small (Fig. 4b) when either or both ungated and gated regions are in a nonQH state. This confirms that shot noise is generated by the PNJ in a welldeveloped bipolar QH state. Within the bipolar QH state, the shot noise fluctuates largely, depending on B and ΔV_{G}. This noise fluctuation cannot be ascribed to G_{PNJ}, which is almost constant in the bipolar QH state. Furthermore, since the noise is generated in the welldeveloped QH state, the existence of multiple noise sources is unlikely. These facts indicate that the noise fluctuation is due to the fluctuation of the energy relaxation rate, which induces the fluctuation of Fano factor. Figure 4c shows the histogram of the Fano factor in the bipolar QH state calculated using equation (2). The s.d. is about 50% of the mean value.
Discussion
The random variation of the energy relaxation rate as a function of B and ΔV_{G} suggest that localized states in bulk graphene play a main role for the energy relaxation. Energy in the PNJ can escape to the bulk graphene through Coulomb interaction with localized states: high frequency potential fluctuations in the PNJ, which is the source of the shot noise, are dissipated in the localized states. Since the energy level and the profile of the localized states depend on B and ΔV_{G}, fluctuations of the relaxation rate can be induced. On the other hand, the average current through the PNJ, which merely reflects the transmission coefficient, is hardly affected by the localized states. Note that a simple model of interaction with twodimensional phonons in the PNJ fails to quantitatively reproduce our observations (Supplementary Note 1 and Supplementary Fig. 3). It is reported that the electron–phonon coupling is expected to be vanishingly small in usual unipolar edge channels^{21,22,23}. To understand the energy relaxation length quantitatively, detailed analysis including interactions with phonons and any other possible mechanisms for the energy relaxation is necessary.
In conclusion, we showed that the mode mixing at PNJ in graphene bipolar QH states leads to nonequilibrium f_{PNJ}(E), generating shot noise. For a short PNJ (L<<15 μm), the energy loss towards external states is negligible and the noise is consistent with a quasielastic mode mixing. This suggests that a graphene PNJ can act as a beam splitter. Since 15 μm is much larger than typical length scale of mesoscopic devices, our results encourage using graphene for electron quantum optics experiments and quantum information.
Methods
Device fabrication
We prepared a graphene wafer by thermal decomposition of a 6HSiC(0001) substrate. SiC substrates were annealed at around 1,800 °C in Ar at a pressure of <100 torr. For the fabrication of devices, graphene was etched in an O_{2} atmosphere. After the etching, the surface was covered with 100nmthick hydrogen silsesquioxane (HSQ) and 60nmthick SiO_{2} insulating layers. As a result of doping from the SiC substrate and the HSQ layer, graphene has ntype carriers with the density of about 5 × 10^{11} cm^{−2}. The width of the PNJ roughly corresponds to the thickness of the insulating layers and estimated to be 200 nm at most. In the QH effect regime, because of the Landau level quantization, the width becomes smaller with B. An important advantage of the SiC graphene is its size: it is single domain for 1 cm^{2}, allowing us to investigate the effect of PNJ length.
Noise measurement
For the noise measurement, the current noise is converted into voltage fluctuations across one 2.5 kΩ resistor in series with the sample. A 500kHz bandwidth 3MHz tank circuit combined with a homemade cryogenic amplifier is used. After further amplification and digitization, the autocorrelation voltage noise spectra is calculated in realtime by a computer. Accurate calibration of the noise is done using Johnson–Nyquist noise that relies on the quantification of the resistance at ν=2 and the temperature of the system.
Additional information
How to cite this article: Kumada, N. et al. Shot noise generated by graphene p–n junctions in the quantum Hall effect regime. Nat. Commun. 6:8068 doi: 10.1038/ncomms9068 (2015).
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Acknowledgements
We acknowledge funding from the ERC Advanced Grant 228273 MeQuaNo and the ANR MetroGraph grant. We are grateful to S. Tanabe, P. Jacques and M. Ueki for experimental support.
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N.K. and P.R. performed the experiments. N.K., F.D.P., P.R. and D.C.G. analysed the data and wrote the manuscript. H.H. grew the wafer.
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Supplementary Figures 15, Supplementary Note 1, Supplementary Methods and Supplementary References (PDF 776 kb)
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Kumada, N., Parmentier, F., Hibino, H. et al. Shot noise generated by graphene p–n junctions in the quantum Hall effect regime. Nat Commun 6, 8068 (2015). https://doi.org/10.1038/ncomms9068
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DOI: https://doi.org/10.1038/ncomms9068
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