Research ArticleQUANTUM PHYSICS

Mach-Zehnder interferometry using spin- and valley-polarized quantum Hall edge states in graphene

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Science Advances  18 Aug 2017:
Vol. 3, no. 8, e1700600
DOI: 10.1126/sciadv.1700600
  • Fig. 1 Creating an MZI using spin- and valley-polarized quantum Hall edge channels.

    (A) Schematic illustration of the formation of MZIs at a graphene pn junction. Green and purple denote quantum Hall edge channels of opposite spin. Top panel: At (νB, νT) = (1, − 1), where νB (νT) is the filling factor in the n-region (p-region); two edge channels run along the interface. Their opposite spin suppresses interchannel scattering. Middle panel: At (νB, νT) = (2, − 1), a pair of spin-up edge channels forms an MZI. Interchannel scattering occurs at the ends of the junction, as indicated by dotted lines. Bottom panel: At (νB, νT) = (2, − 2), two pairs of same-spin edge channels form two MZIs. (B) Device 1: An edge-contacted monolayer graphene flake encapsulated in hBN. The top gate (Au) and bottom gate (graphite) define the pn junction (p, red color; n, blue color). The top hBN dielectric is 20-nm thick and the bottom is 30-nm thick. The top gate is contacted by a lead that runs over a bridge fabricated from hard-baked PMMA to avoid shorting to the graphene flake. The back gate (Si) is used to strongly increase the p-doping of the graphene leading up to the right lead and to reduce the contact resistance. The SiO2 back-gate dielectric is 285-nm thick. (C) Two-terminal conductance of device 1 in the pn regime at B = 4 T. We distinguish four regions (dashed boxes). Region I corresponds to νB = 1 and νT = −1. Region II corresponds to νT = − 1 and νB ≥ 2. Region III corresponds to νB = 1 νT ≤ − 2. Region IV corresponds to νB ≥ 2 and νT ≤ − 2.

  • Fig. 2 Characterization of a single MZI.

    (A) Two-terminal conductance of device 1 at B = 9 T, over a range of filling factors corresponding to a single interferometer at the pn junction. (B) Modeling the charge-density dependence of the distance between the edge channels that form an MZI. The red and blue shading illustrates the spatial variation of the charge density close to the pn junction. The green line illustrates the spatial variation of the energies of the exchange-split ν = 1 and ν = −2 Landau sublevels. The edge channels are located at the positions where these sublevels intersect the Fermi energy. The distance between the edge channels determines the flux through the interferometer. Far from the pn junction, where the lowest LL is completely empty (ν = −2) or completely full (ν = 2), the exchange-splitting Uex vanishes. However, near the pn junction, the electronic ground state can develop an imbalance in the valley occupation, leading self-consistently to a nonzero Uex. (C) Increasing the electron and hole densities decreases the distance between the edge channels. (D) A strong imbalance between the electron and hole densities. (E) Simulation of the two-terminal conductance as a function of filling factors based on the model sketched in (B) to (D). (F) Local visibility of the conductance oscillations observed in (A). The gray dashed box indicates where the visibility was not extracted because of nonresolved oscillations. (G) Blue circles: Experimentally determined probability of finding a visibility, gmax, or gmin greater than x. Visibility is extracted from the color plot in (F); gmin and gmax are extracted from the color plots in fig. S3 (A and B). Purple dashed line: The theoretical prediction based on MZIs with beamsplitters described by random scattering matrices that correspond to beamsplitter transmission probabilities uniformly distributed between 0 and 1. Green solid line: The theoretical prediction based on MZIs with beamsplitters described by a skewed distribution of transmission probabilities (fig. S3C and note S2).

  • Fig. 3 Mach-Zehnder oscillations as a function of magnetic field and dc voltage bias.

    (A) Two-terminal differential conductance as a function of magnetic field B and dc voltage bias Vdc at (νB, νT) = (1, − 2), for which only one interferometer is formed at the pn interface. (B) Visibility of the conductance oscillations shown in (A) as a function of dc bias. (C) Conductance oscillations with B at zero dc bias corresponding to the red dotted line in (A). From the period ΔB = 66 mT, we calculate the distance between edge states to be 52 nm, assuming that the distance between the beamsplitters is given by the 1.2-μm width of the device. (D) Line trace corresponding to the purple dotted line in (A) showing oscillations with respect to Vdc.

  • Fig. 4 Gate-length dependence of the Mach-Zehnder oscillations.

    (A) Optical microscopy image of device 2: An edge-contacted, hBN-encapsulated monolayer of graphene with five top gates of different lengths. The top-gate dielectric (hBN) is 17-nm thick. The bottom hBN layer is 16-nm thick. The back-gate dielectric (SiO2) is 285-nm thick. Leads (L1 to L6) are yellow. Top gates (TG1 to TG5) are orange. Using the top and back gate, we induce an npn charge configuration with two pn junctions and their associated MZIs connected in series. (B) Atomic force microscopy (AFM) image of device 2. The graphene is indicated by the white dashed line. Top gates are outlined in green, leads are in yellow, and etched regions are in blue. The lengths of both sides of each top gate are indicated in micrometers. (C) Two-terminal conductance measured across top gate 1 (TG1) using leads L1 and L2 at B = 8 T. Region I corresponds to νB = −1 and νT = 1. Region II corresponds to νT = −1 and νB ≥ 2. Region III corresponds to νT ≤ −2 and νB = 1. Region IV corresponds to νB ≥ 2 and νT ≤ −2. Inset: Close-up of (B) showing the top gate and the two leads used in this measurement. The edge channels are indicated by black lines. (D) Frequency spectrum (FFT, fast Fourier transform) of the conductance oscillations for all top gates. The x axis is normalized to the approximate length of TG5. The expected frequencies for each gate are indicated by the black dashed lines.

  • Fig. 5 Absence of equilibration between edge channels running along a gate-defined edge.

    (A to D) Schematic of device 3: An edge-contacted, hBN-encapsulated monolayer of graphene with two top gates and a side gate. An AFM image shows the top gates (TG1 and TG2, false-colored purple) on top of the hBN-encapsulated graphene flake. Yellow (green) indicates the leads (side gate). The progression of (A) to (D) illustrates the changing locations of the edge channels in the central region as the filling factor under the side gate is tuned from νS = 0 to νS = 3, whereas the regions under TG1 and TG2 are kept at νT = 1, and the rest of the device is kept at νB = 4. The circulating edge states near the contacts are omitted for clarity. (E) Voltage measured at the right contact as a function of the side-gate voltage VS that tunes the side-gate filling factor νS for νT = 1, 2, and 3, as indicated by the insets. The data (blue) is an average of a set of traces taken at different magnetic fields between 7.9 and 8 T (fig. S7). The red line indicates the expected values given by a model that assumes no equilibration along the gate-defined edge and full equilibration between same-spin channels along the physical edge, taking into account the independently measured contact resistances (note S5). The top trace corresponds to the sequence depicted in (A to D).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/8/e1700600/DC1

    fig. S1. Characterization of device 1 in the regime where νB > 0 and νT > 0 (which we call the nn′ regime).

    fig. S2. Two-terminal conductance of device 1 in the pn regime (in which νB > 0 and νT < 0) at B = 4 T and large filling factors.

    fig. S3. Analysis of transmission and reflection in Mach-Zehnder beamsplitters.

    fig. S4. The effect of a dc bias on the differential conductance of a pn junction.

    fig. S5. Analyzing the average conductance observed in npn measurements on device 2.

    fig. S6. Analyzing the gate-length dependence of the Mach-Zehnder oscillation frequencies observed in npn devices.

    fig. S7. Device 3: verifying the presence of broken symmetry quantum Hall states and measurements of edge channel equilibration as a function of magnetic field.

    note S1. Modeling the distance between the edge channels forming an MZI.

    note S2. Scattering model for an MZI at a graphene pn junction.

    note S3. Conductance of two MZIs in series.

    note S4. Analyzing the gate-length dependence of the Mach-Zehnder oscillation frequencies observed in the npn measurements on device 2.

    note S5. Gate-defined equilibration studies.

    note S6. Calculating charge densities and filling factors from gate voltages.

    References (34, 35)

  • Supplementary Materials

    This PDF file includes:

    • fig. S1. Characterization of device 1 in the regime where νB > 0 and νT > 0 (which we call the nn′ regime).
    • fig. S2. Two-terminal conductance of device 1 in the pn regime (in which νB > 0 and νT < 0) at B = 4 T and large filling factors.
    • fig. S3. Analysis of transmission and reflection in Mach-Zehnder beamsplitters.
    • fig. S4. The effect of a dc bias on the differential conductance of a pn junction.
    • fig. S5. Analyzing the average conductance observed in npn measurements on device 2.
    • fig. S6. Analyzing the gate-length dependence of the Mach-Zehnder oscillation frequencies observed in npn devices.
    • fig. S7. Device 3: verifying the presence of broken symmetry quantum Hall states and measurements of edge channel equilibration as a function of magnetic field.
    • note S1. Modeling the distance between the edge channels forming an MZI.
    • note S2. Scattering model for an MZI at a graphene pn junction.
    • note S3. Conductance of two MZIs in series.
    • note S4. Analyzing the gate-length dependence of the Mach-Zehnder oscillation frequencies observed in the npn measurements on device 2.
    • note S5. Gate-defined equilibration studies.
    • note S6. Calculating charge densities and filling factors from gate voltages.
    • References (34, 35)

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