Shot noise generated by graphene p–n junctions in the quantum Hall effect regime

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 electron-like and hole-like 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 mode-mixing 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 beam-splitter 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.


Su pplementary note 1 | Theoretical background
In this section, we present some of the theoretical elements used to analyze our data.
These elements are mainly based on (1), and describe the equilibration of edge channels in the PNJ in terms of inelastic and quasi-elastic scattering occurring between the edge channels in the PNJ. We then present various models of energy relaxation to account for the effect of the length of the junction, including energy losses towards phonons. Finally, for the sake of completeness, we mention a model of fully coherent scattering between the edge channels propagating in the PNJ, which does not agree with the experimental data.
In elastic and quasi-elastic scattering In this part, we discuss the effects of inelastic and quasi-elastic scattering between copropagating edge states on transport and noise. We consider the mixing of electron and hole edge modes with the chemical potential μ 0 + sd and μ 0 , respectively, at (ν ug , g ) = (2, −2). To model inelastic and quasi-elastic scattering, we add a fictitious contact (noted 4 in Supplementary Figure 1a and two channels stem from contact 2 (mimicking the incoming channels in the p-region). A second QPC (QPC2) is then used to divert two of the channels leaving contact 4 towards the contact 3 (mimicking the outgoing channels in the n-region) and the other two towards the contact 2 (mimicking the outgoing channels in the p-region).
Note that to achieve this configuration, one has to use four edge channels (yielding a total "virtual" filling factor of 4) and set the transmissions of the two innermost channels through both QPCs to zero. Doing so fully satisfies the important condition stated above, and gives a completely equivalent description of the system shown in The noise spectrum is defined as is the Fourier transform of the current operator at the contact α and ⟨̂( )⟩ is its average value. The zero frequency limit will be noted , = , ( = 0). Following the scattering approach, we obtain the noise power, with the notation, with , ( ) the elements of the scattering matrix of the entire system, where and denote the contact number and k and n the edge channel.
Since the current is conserved through the fictitious contact 4 (i.e., the current 4 flowing out of the sample from the contact is zero at any time), its chemical potential μ 4 can fluctuate. This allows to define an average energy distribution 4 ̅ ( ) for electrons in the contact, the shape of which depends on how current conservation through the contact is enforced. This is directly linked to the scattering processes occurring in the contact.
Now, we discuss the current and the noise in the presence of the contact 4. We where δ is the intrinsic part of the fluctuations. We also have = ⟨ ⟩ + Δ . Therefore Since We finally obtain the redistributed noise measured at contact 3, 33 in/qe = ⟨Δ 3 2 ⟩: For a PNJ at (ν ug , g ) = (2, −2), this gives In the inelastic case, 4 ̅ ( ) is the Fermi distribution with 4 ̅̅̅ = 0 + ( sd )/2 Figure 1c). Then we obtain 3 = 2 sd /ℎ and 33 in = 2 2 B eff /ℎ. In the absence of energy losses towards additional degrees of freedom, the effective temperature eff of the Fermi distribution is given by the balance between the input power from the contacts and the heat flow carried by the electronic channels leaving the junction (2): B eff = | sd |√4/3/ . The Fano factor thus becomes in = √4/3/ . In presence of additional energy losses (for instance due to electron-phonon coupling, see below), eff and thus the noise gradually decrease to zero with increasing interaction length.

(Supplementary
In the quasi-elastic case, not only the total current is zero on average ( 4 = ∫ 4 ( ) = 0) but also the contribution to the current of the states of energy E is zero  (3) and (4), we obtain the current 3 = 2 sd /ℎ and the noise 33 qe = 2( 2 /ℎ)| sd | 1/4. The Fano factor becomes qe = 1/4.

Length dependence
For the length dependence of the quasi-elastic and inelastic scattering models, we first recall some experimental results obtained at ν = 2 in a two-dimensional electron gas formed in a GaAs/AlGaAs heterojunction (5). In this study, one of the two edge channels is driven out of equilibrium (the resulting energy distribution is a double-step function).
After several propagation lengths, the electronic energy distribution is measured. In  Figure 3b). Moreover, the presence of a non-negligible heat transfer to phonons is generally indicated by a sublinear behavior of the noise for large enough sd , which we do not observe at our experimental accuracy.
One can explain this discrepancy with the following arguments: first, treating the PNJ as an homogeneous is obviously incorrect, since by definition the electronic density drastically changes in such a region. Second, a 2D model (even assuming an average density or electron-phonon coupling constant) ignores the particular physics of QH edge channels, especially the suppression of backscattering. As the electron-phonon coupling is generally assumed to increase with impurity scattering, one expects it to be strongly suppressed in a ballistic edge channel. As such, our results tend to suggest that even though they interact strongly in the PNJ, the "quantum Hall" nature of the copropagating edge channels is preserved in the PNJ. Nonetheless, the presence of relaxation mediated by electron-phonon coupling cannot be completely ruled out in our system, and, provided a more realistic model, perhaps explain the fluctuations in the noise shown in Fig. 3 of the main paper.
Coh erent scattering in copropagating edge channels In this part, we discuss the effects of coherent scattering between copropagating edge states on transport and noise in a system represented in Supplementary Figure 4. We where a state entering the PNJ on a given channel tunnels into the other channels with the same amplitude √ but with different phases: Note that to ensure normalization, ϵ cannot be larger than 1/3. This matrix is then encompassed into the elements αβ,kn ( ) of the scattering matrix of the entire system.
We now discuss the effect of the PNJ length on the coherent scattering. Increasing L will induce decoherence that we must take into account in the coherent scattering approach. To model decoherence, we introduce a fluctuating phase ( ) each scattering process being now described by an amplitude √ ( ) . Decoherence can be described by a succession of coherent scatterer in series. After N scatterers, the element of the scattering amplitude becomes ( ) = √ ∑ ( ) . If the source of decoherence is a Gaussian stochastic process of zero mean, ( ) is a Gaussian random variable with ⟨ ( )⟩ = 0 and ⟨ ( )⟩ can be expressed in terms of variance of the phase (3), Therefore, the expected noise becomes, where is the coherence length of the system.

Su pplementary Methods
We prepared a graphene wafer by thermal decomposition of a 6H-SiC(0001) substrate (6). As a result of doping from the SiC substrate and the HSQ layer, graphene has n-type carriers with the density of about 5 10 11 cm −2 .
In our devices, the top gate does not overlap with ohmic contacts to avoid a gate leakage. As a result, ungated regions present between the gated region and ohmic contacts on the gated side. To obtain a good contact to hole edge channels in the presence of the ungated regions, we etched graphene in a comb shape around in h , where four PNJs are formed in series (Supplementary Figure 5). When the chemical potential of incoming hole edge channels and in h are 0 and 0 + sd , respectively, the mode mixing and partitioning in a PNJ lead to the chemical potential 0 + sd /2 .
Undergoing the same process four times, the chemical potential of the outgoing hole edge channel becomes, Small difference from 0 + sd causes an error of several percent for the estimation of the bias between the electron and hole edge channels. This error does not affect our discussions.