Layer Polarizability and Easy-Axis Quantum Hall ... - ACS Publications

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Layer Polarizability and Easy-Axis Quantum Hall Ferromagnetism in Bilayer Graphene C. Pan,† Y. Wu,† B. Cheng,† S. Che,† T. Taniguchi,‡ K. Watanabe,‡ C. N. Lau,† and M. Bockrath*,† †

Department of Physics and Astronomy, University of California, Riverside, California 92521, United States Advanced Materials Laboratory, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan

‡

S Supporting Information *

ABSTRACT: We report magnetotransport measurements of graphene bilayers at large perpendicular electric displacement fields, up to ∼1.5 V/nm, where we observe crossings between Landau levels with different orbital quantum numbers. The displacement fields at the studied crossings are primarily determined by energy shifts originating from the Landau level layer polarizability or polarization. Despite decreasing Landau level spacing with energy, successive crossings occur at larger displacement fields, resulting from decreasing polarizability with orbital quantum number. For particular crossings we observe resistivity hysteresis in displacement field, indicating the presence of a first-order transition between states exhibiting easy-axis quantum Hall ferromagnetism. KEYWORDS: Graphene, quantum Hall effect, Landau level crossings, quantum Hall ferromagnetism

B

Our devices are made by using Elvacite layers to transfer mechanically exfoliated bilayer graphene flakes onto atomically smooth h-BN flakes on SiO2/Si wafers.32 Clean regions are chosen and the rest removed by inductively coupled plasma (ICP) etching. The SiO2 is etched by buffered HF, and the graphene/h-BN stack is transferred32 onto a few-layer graphite flake acting as a back gate (e.g., ref 33). A Hall bar is patterned using electron-beam lithography and ICP etching. Another hBN flake is transferred onto the Hall bar to encapsulate the graphene. After an etch process, the device is contacted by Cr/ Au electrodes to make one-dimensional edge contacts.34 The Figure 1a left inset shows a device image, while the Figure 1a right inset shows a layer stack diagram. The top gate position is shown by the dotted line in the Figure 1a left inset, but the image was taken before its evaporation step. The completed devices are loaded into a flowing 4He gas cryostat, and Rxx is measured vs the top and bottom gates voltages (Vt and Vb, respectively) and a perpendicular magnetic field B. The measurements are performed at a temperature T ≈ 1.4 K unless otherwise noted. The Figure 1a main panel shows Rxx vs Vt with Vb = 0, showing a maximum corresponding to the charge neutrality point. Analyzing this data yields a mobility of 34 000 cm2 V−1 s−1. B is then swept. Figure 1b shows Rxx in a color plot vs n and B, showing a Landau fan pattern, similar to that obtained previously on bilayer graphene.1 From this data, we determine the back and top gate capacitances Cb and Ct as Cb = 134 nF/cm2 and Ct = 87 nF/cm2. We then determine the charge density qn = CbVb + CtVt, where q is the electric charge,

ilayer graphene’s electronic spectrum consists of chiral massive Fermions1 and has exhibited a number of spin- or valley-ordered many body phases and fractional quantum Hall states.2−13 Recent work has shown that Landau level (LL) crossings can be induced in bilayer graphene and used to study these states by applying a perpendicular electric displacement field D, which tunes the interlayer potential difference2−7,14 and the LL energies.2−6 These crossing LLs originate from symmetry-broken states belonging to the same quasi-degenerate multiplets.2−5 Here we study LL crossings under much larger displacement fields than reported previously, where we observe and investigate crossings between levels having different orbital quantum numbers split by an energy ∼ℏωc, where ℏ is Planck’s reduced constant and ωc is the cyclotron frequency. The Dvalues at the crossing points are mainly determined by energy shifts arising from the differing interlayer polarizability and polarization of the crossing levels. By measuring D at the crossing points, we find that the layer polarizability of states decreases with increasing orbital quantum number. This results because of increased effective layer coupling at higher energies, favoring eigenstates with more equal probability density on the two layers.1,15−17 From our data, we determine the nearest interlayer neighbor hopping parameter γ1 = 480 meV, in agreement with previous studies.1,16,18−20 Moreover, we demonstrate that near these interorbital crossing points the longitudinal resistivity Rxx is hysteretic in D at low temperature (T ≈ 1.4 K), with the hysteresis vanishing by T ∼ 10 K, indicating quantum Hall ferromagnetism (QHF),2,4,5,13,21−29 with easy-axis anisotropy in the space of the two states.30,31 These states are strongly layer polarized and exhibit an interaction energy scale ∼1 meV. © 2017 American Chemical Society

Received: January 16, 2017 Revised: April 19, 2017 Published: April 21, 2017 3416

DOI: 10.1021/acs.nanolett.7b00197 Nano Lett. 2017, 17, 3416−3420

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the deep minima in the Figure 2a upper panel for LL filling factors ν that are multiples of 4.35 Typically symmetry breaking perturbations lift the LL degeneracies, producing additional Rxx minima such as the one at ν = −6 in the Figure 2a upper panel.36,37 As D varies in the Figure 2a lower panel, it creates an interlayer potential difference Δ/e, yielding a band gap Δ and tuning the LL energies. The Rxx minima produce vertical features that are mostly dark in the color plot but with a number of bright regions evident, such as that shown by the black arrow. These peaks occur when the LLs cross and become degenerate.38 For −0.5 < D (V/nm) < 0.5 these peaks arise from crossings between symmetry-broken LLs from the same nearly degenerate multiplet, i.e., between l = 0 and l = 1, and those with the same l.2−5,13,21,24 The ability to resolve these features attests to our device quality. Here we focus on larger D, where crossing features are visible between LLs with different l with at least one LL having l ≥ 2, such as indicated by the blue arrow at ν = −4. We note that a ν = −4 crossing was previously observed at lower D and B in ref 13, but we investigate here the behavior with respect to B and D at large D. We also observe and investigate additional crossings at ν = −5 and −8. Figure 2b shows line traces through these features at constant n for ν = −4, −5, and −8. Using this data, we plot D for these crossings in the Figure 2c inset. We measure the primary inter-LL energy gaps by measuring Rxx at the minima vs T and extracting the activation energies from Arrhenius plots. These energy gaps are plotted in Figure 2c. They decrease as |ν| increases. This may arise from increasing effective mass at larger energies.39 It may also arise from the smaller observed range of activated temperature behavior at high filling factors due to low temperature saturation, which has been previously attributed to variable range hopping (e.g., ref 40). Nevertheless, from this data we estimate ℏωc ≈ 30 meV at B = 9 T, yielding a cyclotron mass m ≈ 0.03 of the free electron mass at ν = −4. This is consistent with previous values.1,16,41 The D values corresponding to the interorbital LL degeneracies vs B are shown in Figure 3a. The measured |D| increases with B. Also, as shown in the Figure 2c inset and

Figure 1. (a) Left inset: Optical image of a bilayer graphene device. The dashed outline indicates the top gate location. Scale bar, 8 μm. Right inset: Schematic diagram of the multilevel stack. Main panel: Rxx vs Vt at T = 1.5 K with Vb = 0. (b) Rxx vs B at D = 0. Filling factors are labeled for quantum Hall states at the top.

and n is the carrier density, and displacement field 1 D = 2 (C bVb − C tVt) applied to the bilayer. The Figure 2a lower panel shows a color plot of Rxx vs n and D, taken at B = 9 T. Larger negative values in D are presented to improve the visibility of the crossings on the hole side. (The downwardsloping linear feature starting around 0.5 V/nm in D occurs at constant Vb, indicating that it originates outside the dual-gated region of interest, and did not occur in any other samples studied.) A D = 0 line trace is shown in the Figure 2a upper panel. A series of minima appear due to the LLs in the bilayer whenever the Fermi level is tuned to an inter-LL gap. At D = 0, the LL spectrum in the absence of symmetry breaking perturbations is given theoretically by El = ±ℏωc l(l − 1) , where l is the LL orbital quantum number index, and ωc = qB/m, with m the electronic effective mass.1 The LLs are 4-fold degenerate for each l, accounting for

Figure 2. (a) Top panel: Rxx vs n at D = 0. Main panel: Color plot of Rxx vs n and D at B = 9 T. (b) Vertical line cuts passing LL crossing points along the dashed lines in panel a. (c) Inset: D at the LL crossings vs ν, for device 1 at 8 T and device 2 at 9 T. Main panel: Energy gap ΔE vs ν, for device 1 at 9 T. 3417

DOI: 10.1021/acs.nanolett.7b00197 Nano Lett. 2017, 17, 3416−3420

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Figure 3. (a) The D values of crossing points vs B for LL crossings at ν = −4, −8, −12. (b) Schematic LL energy diagram for B = 9 T representing the LL evolution with Δ, with crossings at ν = −4, −8, −12 shown. The crossing at Δ = 120 meV corresponds to D ≈ 0.71 V/nm and approximately agrees with the experimental crossing point at 0.8 V/nm. Inset: zoomed-in view of crossings within the shaded region in the main panel. (c) Main panel: activation plot at various D values at ν = −6. Inset: inferred gap vs D at ν = −6.

for l ≥ 2, while the l = 0 and l = 2 levels |0,+,↓⟩ and |2,−,↑⟩ cross at Δ = ℏωc 1/[η(1 − η)] .15 Figure 3b shows these predicted LL energies and crossings, with the inset showing a zoomed-in region. Although neglected above because it has only a small impact on the crossing point positions at large D, a small phenomenological energy splitting ∼1 meV is added to the curves to reproduce the low-D behavior. A zoomed-in plot near the crossing of the |2,−,↑⟩ and |0,+,↓⟩ states within the shaded rectangle of the main panel is shown in the inset. The first interorbital crossing occurring at ν = −4 results from the difference between the energy dependence on Δ between the l = 0 LL states and the l = 2 LL states. The l = 0 states’ energy dependence is much stronger than the l = 2 states because their wave functions are concentrated on one of the two layers depending on whether the state belongs to the K or K′ valley.15−17 This leads to a crossing near Δ ∼ 120 meV. Aside from a K−K′ splitting proportional to η and Δ, the energy dependence for l ≥ 2 with Δ is quadratic at small Δ, which reflects the energy from the induced layer dipole moment P = αE for an electronic state within a given LL, where α is the polarizability and E is the total electric field between the two layers. The polarizability α is given by α = ϵdP/dD = ϵ(dP/ dΔ)(dΔ/dD) = cϵ(dP/dΔ), where c ≡ dΔ/dD is taken to be a constant for small Δ, and ϵ is the interlayer dielectric constant. Using eq 1, we find that dP /dΔ = qd /(4ℏωc (l − 1)l ), where d = 3.35 Å is the interplanar spacing, and we have neglected terms of order η compared to 1. The induced dipole 1 adds an energy − 2 PE , so that the energy shift is ∝ αΔ2. For the LLs with l ≥ 2 shown in Figure 3b, this energy term dominates the LL energy shifts and their crossings for Δ > 20−50 meV. To determine c and γ1 from the data, we first measure Rxx vs T at various D values near a crossing point. The circles in Figure 2a show these D values. In this region, the inter-LL gap ΔLL is expected to be ηΔ = ηcD, so that Rxx = Ae−ηcD/(2kBT), where A is a constant. Figure 3c shows the Rxx data plotted against 1/T and fit to the above Rxx equation. The inset shows the inferred ΔLL vs D. A linear fit to this data yields cη = 10.7 meV nm/V. We then rewrite the above crossing Δ expression in terms of D and cη to find the lowest D crossing for ν = −4, Dν=−4

Figure 3a, although the LL spacing apparently becomes smaller at larger index, the |D| values to tune them to degeneracy becomes larger. To understand this behavior, we first use a single-particle picture of the LLs, keeping the dominant terms in the Hamiltonian. This model has been extensively discussed and developed in previous works.1,16,17 The low-energy Hamiltonian is given by ⎡ ⎛ ⎤ 0 ⎞⎥ 1 1 0⎞ v 2 ⎛ π †π ⎟⎟ ⎟ − 2 ⎜⎜ H = σ Δ⎢ ⎜ ⎢⎣ 2 ⎝ 0 −1⎠ γ1 ⎝ 0 −ππ † ⎠⎥⎦ † 2⎞ ⎛ 1 ⎜ 0 (π ) ⎟ − 2m ⎜⎝ π 2 0 ⎟⎠

(1)

where π = px + ipy, where px and py are the x and y momentum operators, respectively, γ1 is the interlayer hopping parameter, v ∼ 106 m/s is the Fermi velocity, and σ = 1 for the K valley and σ = −1 for the K′ valley. The basis for the states near the K point is the amplitude to be on the A sites in the upper layer and the B sites in the lower layer.17 The roles are reversed for states near the K′ point. Thus, in terms of the amplitude ψA to be on A sites and ψB to be on the B sites, the states can be described by a layer pseudospin vector (ψA, ψB) for the K valley and (ψB, ψA) for the K′ valley. Using LL states eikyϕj, where k is the wavevector and j is an integer, the π (π†) operators become lowering (raising) operators.17 With the basis states |2l − 2⟩ = (0, ϕl−2), |2l − 1⟩ = (ϕl, 0), for integer l ≥ 2 and |0⟩ = (ϕ0, 0), |1⟩ = (ϕ1, 0) the Hamiltonian becomes block diagonal. This yields the eigenvalue spectrum for l ≥ 2 1 Elσ = − ησ Δ ± 2

1 2 Δ [η(2l − 1) − 1]2 + ℏ2ωc2l(l − 1) 4 (2)

where η = ℏωc/γ1. Two 4-fold quasi-degenerate energy states 1 1 also exist with energies E0 = 2 σ Δ and E1 = 2 σ Δ − σηΔ,1 where the degeneracy is lifted by additional small spin and valley splittings not explicitly included in eq 1.36,37 We denote the LLs by |l,p,s⟩, where l is the LL index, p indicates the valley index, and s the spin. Using eq 2 we find that the crossing 15

between the l and l+1 levels occurs at Δ = ℏωc l /(η − η2l) 3418

DOI: 10.1021/acs.nanolett.7b00197 Nano Lett. 2017, 17, 3416−3420

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⎛ η ⎞1/2 1 = ℏωc⎜ ⎟ cη ⎝1 − η ⎠

neously broken symmetry2,4,5,13,21−29 and a macroscopic occupation of one state, yielding a magnetic-like response in the space of the two LL states. A single isospin giving the amplitude for each electron to be in each one of the two LLs characterizes the many electron state. The angle θ of the isospin can be determined in the same way as that for an SU(2) spinor. The magnetic response can be easy-axis or easy-plane depending on the whether the effective exchange interaction or the electrostatic Coulomb energy associated with different layer charge distribution in the two LLs dominates, respectively. In easy-axis QHF, hysteresis occurs because of the barrier to tunneling via a global rotation of the isospin between two directions. This traps the system in a metastable state until the barrier is eliminated by sufficiently strong detuning of the Landau level energies by D. Quantitatively this can be described by taking the energy per electron for N electrons as

(3)

Using the known values for cη and ℏωc at B = 9 T enables the data in Figure 3a to be fit to eq 3. From this we find γ1 = 480 meV. This value is in good agreement to that reported previously.18−20 Using these values the blue dashed line and green dashed line are plotted without free parameters, showing a good fit to the data. This indicates that the behavior is wellcharacterized by the Hamiltonian of eq 1. We also find c ∼ 170 meV nm/V, using ℏωc = 30 meV. This value is in reasonable agreement with previous values as well, for example c ≈ 100 meV nm/V in the work of ref 42. From this we find a dielectric constant of ∼2ϵ0, where ϵ0 is the vacuum permittivity, which yields a polarizability for the l = 2 state ∼1.1 × 10−37 C2 m2/J. The decreasing polarizability with l in conjunction with the slowly decreasing level spacing leads to successive crossing points at larger D, as observed. Finally, using eq 2 and η we can infer a value for the energy splitting of the LLs that cross at low D. For example, the ν = −6, D ∼ 0.15 V/nm crossing indicates a D = 0 splitting ∼1 meV. On the electron side, no similar crossings are observed. This is consistent with theoretical calculations including additional terms in the Hamiltonian breaking electron−hole symmetry.16 These lead to predicted crossings at D values outside of the range of n and D shown in Figure 2a.9,16 We now turn to the behavior near the LL crossing points. Figure 4 shows Rxx vs D at constant filling factor (ν = −8) from

E /N = ( −b + Uρ− I)cos(θ) +

1 UI − I cos2(θ ) 2

(4)

where UI−I is the isospin−isospin interaction energy, Uρ−I is the density-isospin interaction energy, and b is the single-particle splitting between the LLs.30 If UI−I > 0 the QHF is easy-plane, while if UI−I < 0 the QHF is easy-axis. The bare splitting b is taken to follow D−D0 = βb, where β is a constant and D0 is the D-value of the crossing point. Using a similar activation energy fitting procedure as that used to obtain c, we find that β ≈ 0.3 V meV−1 nm−1 for ν = −8. From eq 4, we see that the barrier between the isospin states will be removed when (−b + Uρ−I) = ±UI−I. Thus, we can estimate |UI−I| from the range of hysteresis, yielding |UI−I| ∼ 1 meV. The hysteresis vanishes by ∼10 K, which is consistent with this energy scale. To understand the occurrence of easy-axis QHF in this atomic bilayer system, we note that at the scale of D corresponding to the crossing points, eq 1 suggests that the wave functions of the two degenerate LLs are similarly layer polarized, which favors easy-axis ferromagnetism.30 The value of UI−I is comparable to that found typically in previous work on semiconductor heterostructures.31 This may result from the confinement of the electrons to an atomically thin layer, which enhances the interelectronic Coulomb interactions. In conclusion, we have investigated the properties of graphene bilayers at large displacement fields. The D values of the LL crossings can be understood in terms of LL layer polarizability based on a model that incorporates interlayer tunneling and screening. Finally, we observe hysteresis as the applied displacement field is swept up and down, indicating the presence of first-order transitions between states of a quantum Hall ferromagnet. The interaction energy scale is found to be ∼1 meV.

Figure 4. Rxx hysteresis in D due to quantum Hall ferromagnetism. Rxx vs D, at B = 9 T showing up (blue) and down (red) sweeps. Data taken from the region in Figure 2 are indicated by the red dashed line. The gate sweep rate in both directions was 0.01 V/s, yielding a D sweep rate ∼0.0003 V/nm·s. The data were very similar for sweep rates of 0.005, 0.01, and 0.02 V/s, although the hysteresis is slightly more pronounced at faster sweep rates than in the main panel.

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ASSOCIATED CONTENT

S Supporting Information *

the color plot data in Figure 2a, showing both up and down sweeps for T = 1.6, 5, and 9 K. Interestingly, the two lowest T traces show clear hysteresis. Such hysteresis is not observed while sweeping over the same range of D but while varying n to cross the ν = −11 to −12 transition (to provide a feature with which to detect any hysteresis), nor for the lower-D crossings, indicating that the hysteresis does not result from effects such as gate charging (please see Supporting Information for details). Such hysteresis has been previously observed while sweeping B and is a signature of a first-order QHF transition.31 In this phenomenon, interlayer interactions produce sponta-

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.7b00197. Additional figure concerning the hysteretic behavior (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 3419

DOI: 10.1021/acs.nanolett.7b00197 Nano Lett. 2017, 17, 3416−3420

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C. Pan: 0000-0001-6318-7878 Present Address

S.C., C.N.L., and M.B.: Department of Physics, The Ohio State University, Columbus, Ohio 43210, United States. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by DOE ER 46940-DE-SC0010597. Additional support for device fabrication was from the UCR CONSEPT Center.

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REFERENCES

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DOI: 10.1021/acs.nanolett.7b00197 Nano Lett. 2017, 17, 3416−3420