Introduction

Ever since the realization that the charge and energy are carried by the edge states in a quantum Hall (QH) system, the interest of edge dynamics has surged both theoretically and experimentally. The understanding of edge dynamics make an electron interferometer suitable for exploring exotic phenomena like fractional statistics, quantum entanglement, and non-abelian excitations1,2,3,4,5. A graphene p–n junction (PNJ) naturally harboring co-propagating electron and hole-like edge states offers an ideal platform6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22 to study the edge or equilibration dynamics. The equilibration of such edge states is predicted to be facilitated by inter-channel tunneling via either incoherent or coherent scattering mechanism23,24,25,26,27,28,29,30,31 depending on the microscopic details of the interface. As suggested by Abanin and Levitov24, for a graphene PNJ interface with random disorders, the edge mixing is expected to be dominated by the incoherent process. In the opposite limit, a cleaner PNJ interface23,29,32,33 is supposed to exhibit coherent scattering. A cleaner PNJ interface is also very intriguing for studying the equilibration dynamics as it has spin and valley symmetry broken polarized QH edges34,35,36,37. Although there are several conductance measurements33,38,39 showing spin-selective partial equilibration of the edges, but the equilibration dynamics for symmetry broken QH edges at a PNJ is still unknown.

Shot noise is a quintessential tool to unravel the equilibration dynamics of a junction and it is usually characterized by Fano factor (F), which is the ratio of the actual noise to its Poissonian counterpart. For coherent and incoherent scattering, F = (1 âˆ’ t) and t(1 âˆ’ t), respectively24,40,41,42,43, with t being the average transmittance of the PNJ. So far, shot noise studies44,45 at graphene PNJ in the QH regime have been performed on Si/SiO2 substrate-based devices, where the spin–valley symmetry broken conductance plateaus are not observed and the measured Fano44,45 fairly agrees with the incoherent model24 due to disorder-limited interface. Besides, the shot noise measurements44,45 are focused around only the filling factors ν = Â±2 and ν = Â±6 with no clear dependence of F on filling factors (ν). More importantly, there are no shot noise studies for spin–valley symmetry broken QH edges at graphene PNJ.

With this motivation, we have carried out the conductance together with shot noise measurements at a PNJ realized in a dual graphite-gated hexagonal boron nitride (hBN) encapsulated high-mobility graphene device. From the conductance measurement, we show that the spin and valley degeneracies of the edge states are completely lifted and at the PNJ the edge states undergo spin-selective partial equilibration. Our shot noise data as a function of filling factors shows the following important results: (1) The Fano strongly depends on the filling factors. It monotonically increases with p side filling factors, whereas it slowly varies with n side filling factors. (2) For lower values of p side filling factors (νp ≤ 2), the variation of Fano matches well with the calculated Fano based on incoherent scattering model, whereas for higher values of p side filling factors (νp ≥ 4) Fano follows the coherent scattering model. These results reveal a crossover of scattering process from incoherent to coherent regime in the equilibration of QH edges, which has not been observed in the previous shot noise studies44,45.

Results

Measurement setup

The schematics of our device with the measurement setup are shown in Fig. 1a. The PNJ device is fabricated by placing an hBN-encapsulated graphene on top of two graphite gates BG1 and BG2, each of which can independently control the carrier density of one-half of the graphene (details in Supplementary Note 1 and Supplementary Fig. 1). The PNJ (width  ~10 μm) is obtained at the interface of BG1 and BG2 by applying opposite voltages to the gates. During our entire measurement, the BG1 (BG2) side is maintained as n (p) doped, by setting gate voltage VBG1 > 0 (VBG2 < 0). When a perpendicular magnetic field is applied to the graphene, chirally opposite QH edge states co-propagating along the PNJ are created as shown by the colored arrow lines in Fig. 1a. As shown in the figure, the current (Iin) injected at the p-doped region is carried by clockwise edge states towards the PNJ. After partitioning at the PNJ, the transmitted current (It) at the n-doped region and reflected current (Ir) at the p-doped region is carried by the outgoing anti-clockwise and clockwise edge states, respectively. The shot noise generated due to partitioning at the PNJ is carried by both the transmitted and reflected paths. To measure It and the shot noise, the measurement setup consists of two parts: (1) A low frequency (~13 Hz) part, which determines It by measuring the voltage drop Vm at the n-doped region, with a lock-in amplifier (LA) as shown in Fig. 1(a) (also see Supplementary Note 2 and Supplementary Fig. 2a). (2) A high-frequency shot noise measurement part, where a direct current (DC) current (Iin) is injected at p-doped region and the generated noise is measured at reflected side using LCR resonant circuit at  ~765 kHz as shown in Fig. 1a (described in detail in Supplementary Fig. 2b). All the measurements were performed at 8 T magnetic field inside a cryo-free dilution fridge (with base temperature  ~10 mK), whose mixing chamber plate serves as the cold ground (Fig. 1a).

Fig. 1: Schematics and junction transmittance.
figure 1

a Schematics of the device and measurement setup. The encapsulated graphene flake is positioned on top of two bottom graphite gates BG1 and BG2, which are connected to the gate voltages VBG1 and VBG2. The chirality of the edge states during measurement, for p- and n-doped region, is shown by the red and purple arrowed lines. For both conductance and shot noise measurements, excitation current Iin is injected at the p side. The transmitted current (It) at the n side is determined by measuring the voltage Vm with a lock-in amplifier (LA). The shot noise generated at the graphene p–n junction (PNJ) is measured at the p side using a resonant tank circuit followed by a cryogenic amplifier (CA). The extreme left and right contacts were grounded to the dilution mixing chamber plate serving as cold ground (cg). b The trans-resistance Rt = Vm/Iin, as a function of VBG1 and VBG2. Rt shows a checkerboard-like pattern corresponding to the different combinations of p and n side filling factors νp and νn, shown in the white vertical and horizontal axis, respectively. The white dotted points on (νp, Î½n) = (−2, 2) plateau are to show how noise data were taken at several different (VBG1, VBG2) points of the same plateau. c Spin configuration of the edge states for two different ways of Landau level (LL) degeneracy lifting with increasing magnetic field (B): spin and valley polarized ground states. Red and black color indicates valley degrees of freedom. d Measured transmittance t (open circles) with error bar (standard deviation of t from average value) of the PNJ as a function of filling factor νn for νp = âˆ’1. The calculated t for full equilibration (tfull) and spin-selective equilibration (tspin) for spin-polarized ground state is shown by blue and red dashed lines, respectively.

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Conductance measurement

 Figure 1b shows trans-resistance, Rt = Vm/Iin as a function of gate voltages VBG1 and VBG2. The plot shows plateau-like features creating a checkerboard pattern for different combinations of p and n side filling factors, νp and νn, respectively (details in Supplementary Note 3 and Supplementary Fig. 3a). The transmittance t = It/Iin of each plateau is determined from the Rt as \(t=| {\nu }_{{\rm{n}}}| {R}_{{\rm{t}}}/\frac{h}{{e}^{2}}\), where Vm = ItRh and Rh\(=\frac{h}{{e}^{2}}/| {\nu }_{{\rm{n}}}|\) is the QH resistance of the n-doped region. To understand the filling factor dependence of t, we have shown the two possible sequences of spin polarization of the edge states (valley or spin-polarized ground state) in Fig. 1c39. The Fig. 1d shows the measured values of t (open circles with error bar) as function νn for νp = âˆ’1 with the corresponding theoretical values considering full equilibration24; tfull = ∣νn∣/(∣νp∣ + ∣νn∣) (blue dashed line), and spin-selective partial equilibration38,39,46; \({t}_{{\rm{spin}}}=\frac{1}{| {\nu }_{{\rm{p}}}| }\left[\frac{| {\nu }_{{\rm{p}}\uparrow }{\nu }_{{\rm{n}}\uparrow }| }{| {\nu }_{{\rm{p}}\uparrow }| +| {\nu }_{{\rm{n}}\uparrow }| }+\frac{| {\nu }_{{\rm{p}}\downarrow }{\nu }_{{\rm{n}}\downarrow }| }{| {\nu }_{{\rm{p}}\downarrow }| +| {\nu }_{{\rm{n}}\downarrow }| }\right]\) (red dashed line), where ∣νp↑∣(∣νp↓∣) and ∣νn↑∣(∣νn↓∣) are the total number of up (down) spin edge channels of the p- and n-doped region, respectively (Supplementary Note 4 and Supplementary Fig. 4). The error bars (standard deviations) in Fig. 1d and Supplementary Fig. 4 show the conductance fluctuation of different plateaus. It can be seen that, although the conductance fluctuation increases with higher filling factor, the magnitude of the fluctuation remains negligible compared to the average transmittance values. We note that for clarity we have removed the error bars that are smaller than the width of the symbol used. The red dashed line in Fig. 1d is based on the spin structure for the spin-polarized ground state and it is in very good agreement with the measured t. Note that the other spin sequence also gives good agreement with the experimental data. For simplicity, we will be presenting only one of them (spin-polarized ground state) throughout the manuscript. The measured t and the calculated values based on spin-selective equilibration for other plateaus are also in very good agreement and are shown in Supplementary Fig. 4.

Shot noise measurement

In this section, we present the results of our shot noise measurement. The shot noise generated at the PNJ is measured at the reflected side (p side) as a function of Iin, as shown in Fig. 1a. In general, measured excess current noise (SI) with finite temperature broadening follows the expression:

$${S}_{{\rm{I}}}=2e{I}_{{\rm{in}}}{F}^{* }\left[{\mathrm{coth}}\left(\frac{e{V}_{{\rm{sd}}}}{2{k}_{{\rm{B}}}T}\right)-\frac{2{k}_{{\rm{B}}}T}{e{V}_{{\rm{sd}}}}\right],$$
(1)

where Vsd is the applied bias voltage across the PNJ, T is the temperature, and kB is Boltzmann constant. For eVsd > kBT shot noise dominates over thermal broadening and SI becomes linear with Iin. This can be seen in Fig. 2a, showing one representative SI versus Iin noise data for (νp, Î½n) = (−2, 2), (−3, 3), and (−4, 4) filling factor plateaus. It should be noted that SI is the excess current noise without the thermal equilibrium noise (Vsd = 0). The red lines in Fig. 2a are the fit using Eq. (1). The slopes of the fit have been used to determine the normalized noise magnitude (\({F}^{* }=\frac{{S}_{{\rm{I}}}}{2e{I}_{{\rm{in}}}}\)). For obtaining Fano (\(F=\frac{{S}_{{\rm{I}}}}{2e{I}_{{\rm{t}}}}\)), we just follow F = F*/t, which is conventionally used to characterize the noise and used in the previous shot noise studies on graphene PNJ44,45. Figure 2b shows the histogram of F values, obtained from the noise data taken at several (VBG1, VBG2) points (~50) (Supplementary Note 5) on each checkerboard (plateau) as shown by the white dotted points in Fig. 1b for (νp, Î½n) = (−2, 2). The histograms are fitted with the Gaussian function as shown by the solid red lines in Fig. 2b for (νp, Î½n) = (−2, 2), (−3, 3), and (−4, 4) plateaus. It can be seen that the histograms have a maximum at a certain value of F (mean value), which depends on the filling factors (νp, Î½n). The noise data and the corresponding histograms for some other plateaus are shown in Supplementary Figs. 5,  6, and 7. The variation in Fano is well captured by the Gaussian function for most of the plateaus, except for some plateaus with large Fano variation. To quantify how well the Gaussian fitting is, we have quoted the “R-squared” value of the fitting (also see Supplementary Figs. 6 and 7). We should note that the accuracy of the extracted Fano is very essential to pinpoint the exact scattering mechanism. The accuracy also depends on the amplifier gain and noise from the contacts, as well as on enough statistics. In Supplementary Note 6 and Supplementary Fig. 8, the precise gain calibration and in Supplementary Note 7 and Supplementary Fig. 9, the measured contact noise as a function filling factors are described. The contact noise has been subtracted in the histogram plots shown in Fig. 2b as well as in Supplementary Figs. 6 and  7.

Fig. 2: Shot noise and Fano factor.
figure 2

a Measured current noise (SI) generated by the graphene p–n junction (PNJ), as a function of injected current Iin, for filling factor (νp,  Î½n) = (2, −2), (3, −3) and (4, −4) plateaus. The gate voltage values (VBG1, VBG2), at which the data have been taken are shown in the figures. The solid red lines are the fit with Eq. (1) to extract the Fano factor. b Histogram of all Fano values obtained from noise data taken at several (VBG1, VBG2) points (~50) for the (νp, Î½n) = (2, −2), (3, −3) and (4, −4) plateaus. The solid lines are the Gaussian fit to extract the mean value of F and error is its standard deviation. The R-squared value shows the quality of the fitting.

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The measured values of F (mean value) as a function of filling factor are shown in Fig. 3 as open circles with the error bars (standard deviations of Gaussian fit in Fig. 2b). In Fig. 3a, b, F is plotted as a function of νp, while the n side filling factor is kept fixed at νn = 2 and νn = 5, respectively. It can be seen that F increases monotonically from  ~0.05 to 0.6 with increasing νp. Similarly, in Fig. 3c, d, F is plotted as a function of νn, while the p side filling factor is kept fixed at νp = âˆ’2 and νp = âˆ’4, respectively. However, in this case, the F does not increase monotonically with νn, rather slowly varies around  ~0.2 and 0.6 for νp = âˆ’2 and νp = âˆ’4, respectively. A similar dependence of F on νn or νp for other fixed values of νp or νn (Supplementary Note 8) are shown in Supplementary Figs. 10 and  11.

Fig. 3: Fano versus filling factor.
figure 3

Fano as a function of p side filling factor νp, for n side filling factor νn = 2 (a) and νn = 5 (b). Open circles with error bars represent the experimentally measured Fano. The error bars correspond to the standard deviation from the Gaussian fits of the Fano histograms. Red and blue dashed lines correspond to the theoretically calculated Fano for coherent and incoherent scattering (quasi-elastic), respectively. c, d shows Fano as a function of filling factor νn for νp = âˆ’2 and νn = âˆ’4, respectively. Red and blue dashed lines correspond to the calculated Fano for coherent and incoherent scattering (quasi-elastic), respectively.

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Discussion

To understand the above results, we theoretically calculate F for coherent and incoherent processes. In coherent scattering, the injected hot carriers from p side (Fig. 1a) coherently scatter to the n side and the inter-channel scattering can be described by scattering matrix approach40,41. In this case, F follows as (1 âˆ’ t) similar to that of a quantum point contact (details in Supplementary Note 9). Furthermore, for our symmetry broken PNJ, we also impose the constraints that the two opposite spin channels do not interact with each other39,46. Thus, the Fano can be written as Fcoherent = (∣νp↑∣t↑(1 âˆ’ t↑) + ∣νp↓∣t↓(1 âˆ’ t↓))/(∣νp↑∣t↑ + ∣νp↓∣t↓), where t↑ = ∣νn↑∣/(∣νn↑∣ + ∣νp↑∣) and t↓ = ∣νn↓∣/(∣νn↓∣ + ∣νp↓∣) are the transmittance of up and down spin channels, respectively (Supplementary Note 4). The calculated Fcoherent is shown as red dashed lines in Fig. 3. Fcoherent increases with νp but decreases with νn, which can be qualitatively understood as the transmittance of the PNJ decreases and increases with νp and νn, respectively. For incoherent scattering, we consider the quasi-elastic24,41 scattering. In quasi-elastic case, known as chaotic cavity model, the injected hot carriers from p side scatters to the n side and subsequently scatters back and forth due to the presence of disorders along the PNJ giving rise to double-step distribution24,41,43,44,47,48. Following Abanin and Levitov24 the expression for F is t(1 âˆ’ t) and for our symmetry broken PNJ Fano can be written as \({F}_{{\rm{incoherent}}}=(| {\nu }_{{\rm{p}}\uparrow }| {t}_{\uparrow }^{2}(1-{t}_{\uparrow })+| {\nu }_{{\rm{p}}\downarrow }| {t}_{\downarrow }^{2}(1-{t}_{\downarrow }))/(| {\nu }_{{\rm{p}}\uparrow }| {t}_{\uparrow }+| {\nu }_{{\rm{p}}\downarrow }| {t}_{\downarrow })\). The calculated Fincoherent is shown as blue dashed lines in Fig. 3 (Supplementary Note 9). Fincoherent remain almost constant around F ~ 0.2 and much smaller in magnitude compared to Fcoherent. For the completeness, we also mention the Fano values for inelastic scattering as described by Abanin and Levitov24, which is very similar in magnitude with the quasi-elastic case as shown in Supplementary Fig. 16e.

The monotonic increase of F with νp in Fig. 3a, b is in contradiction with the incoherent scattering model and is consistent with the coherent case, except for lower values of νp. However, the measured F with νn for νp = âˆ’2 matches very well with the incoherent scattering, but for νp = âˆ’4 it perfectly matches the coherent model. This suggests there is a crossover from incoherent to the coherent regime with the increasing number of edge channels at the p side. This is also seen in Fig. 4, where the measured F plotted as a function of ∣νp∣ = ∣νn∣ and increases monotonically from ~0.2 to 0.55 (open circles), whereas Fincoherent (blue dashed line) and Fcoherent (red dashed line) remain constant around ~0.25 and 0.5, respectively. The coherent nature is further manifested as an increase in Fano error bar, showing a wider spread of Fano values at higher filling factors. This can be qualitatively understood from the increased mesoscopic conductance fluctuations of the junction, observed at higher filling factors plateaus. As the coherence increases, conductance fluctuation increases23,31, as can be seen in Fig. 1b, d.

Fig. 4: Crossover from incoherent to coherent regime.
figure 4

Fano (open circle with error bars) as a function of n or p side filling factors νn or νp, when ∣νn∣ = ∣−νp∣. The error bars correspond to the standard deviation of the Gaussian fits to the Fano histograms. Red and blue dashed lines correspond to the theoretically calculated Fano for coherent and incoherent scattering (quasi-elastic), respectively. The crossover from incoherent to coherent regime can be seen with increasing filling factors.

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One plausible reason for the observed crossover with increasing filling factor is the velocity-dependent phase coherence of the edge states49. The velocity, hence the phase coherence of the edge states, increases with a higher filling factor, as the confining potential becomes steeper. It qualitatively explains the observed crossover from incoherent to coherent scattering regime with increasing filling factor. Also, the asymmetric dependence of the Fano factor on p and n side filling factors (Fig. 3) can be qualitatively understood, by considering different steepness of the confining potential on either side of the PNJ14. In our device, the graphite gate BG2 is closer (~21 nm) to the graphene flake, than the graphite gate BG1 (~50 nm). Hence, at the p-doped side of the junction the confinement potential is steeper compared to that at n-doped side. As a result, the phase coherence of the edge states at the p side will change more rapidly than that for edge states at the n side. One more possibility could be that the screening might be playing a big role in dynamics as observed in GaAs-based two-dimensional electron gas47,50,51,52,53. The coherent scattering dominates as the screening increases with more number of participating edges at the PNJ. However, it does not explain the asymmetry observed in Fig. 3.

In summary, we have carried out conductance together with shot noise measurement on a high-quality graphene PNJ, for the first time, with spin and valley symmetry broken QH edges. We have shown that the conductance data follows the spin-selective partial equilibration, and most importantly, our shot noise data reveal the intricate dependence of Fano on filling factors with a crossover in dynamics from incoherent to the coherent regime, which cannot be obtained from the conductance measurements. These results will help to design future electron optics experiments using the polarized QH edges of graphene.

Methods

Device fabrication

To make the encapsulated device, the hBN and graphene, as well as the graphite flakes for bottom gates, were exfoliated from bulk crystals on Si/SiO2 substrates. Natural graphite crystals were used for exfoliating graphene and the graphite flakes. The suitable flakes for the device were first identified under an optical microscope and then sequentially assembled with the residue-free polycarbonate-polydimethylsiloxane stamp technique54,55,56. We have used 15- and 21-nm-thick hBN flakes for encapsulating the graphene flake and 10–15-nm-thick graphite flakes for the bottom gates. To make the metal edge contacts on the device, first, the contacts were defined with e-beam lithography technique. Then, along the defined region, only the top hBN flake was etched out using CHF3−O2 plasma. After that Cr(2 nm)/Pd(10 nm)/Au(70 nm) was deposited using thermal evaporation.

Shot noise setup

To measure the shot noise, first, the voltage noise generated from the device is filtered by a superconducting resonant LC tank circuit, with resonance frequency at 765 kHz and bandwidth 30 kHz57,58. The filtered signal is then further amplified by the HEMT cryo amplifier followed by a room temperature amplifier. The amplified signal is then fed to a spectrum analyzer, to measures the root mean square of the signal. The gain of the amplifier chain is determined from the temperature dependence of the thermal noise of νp = âˆ’2 filling factor plateau, while the n  side is in the insulating state. The thermal noise measurement is carried out using the same noise circuit.