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Spin-valley Hall transport induced by spontaneous symmetry breaking in half-filled zero Landau level of bilayer graphene

Intrinsic Hall conductivity, emerging when chiral symmetry is broken, is at the heart of future low energy consumption devices because it can generate non-dissipative charge neutral current. A symmetry breaking state is also induced by electronic correlation even for the centro-symmetric crystalline materials. However, generation of non-dissipative charge neutral current by intrinsic Hall conductivity induced by such spontaneous symmetry breaking is experimentally elusive. Here we report intrinsic Hall conductivity and generation of a non-dissipative charge neutral current in a spontaneous antiferromagnetic state of zero Landau level of bilayer graphene, where spin and valley contrasting Hall conductivity has been theoretically predicted. We performed nonlocal transport experiment and found cubic scaling relationship between the local and nonlocal resistance, as a striking evidence of the intrinsic Hall effect. Observation of such spontaneous Hall transport is a milestone toward understanding the electronic correlation effect on the non-dissipative transport. Our result also paves a way toward electrical generation of a spin current in non-magnetic graphene via coupling of spin and valley in this symmetry breaking state combined with the valley Hall effect.

Theory has predicted gap opening of clean graphene at the charge neutrality point (CNP) induced by electron-electron interaction [1,3,4]. Such gap opening was also experimentally confirmed in bilayer and trilayer graphene [5][6][7]. The gap is even enhanced in presence of a perpendicular magnetic field due to enhancement of effective Coulomb interaction, where the CNP correspond to half-filled zero-th Landau level (ν = 0 state). This gapped state is distinct from conventional quantum Hall states observed in semiconductor two-dimensional electron gas. It shows unique properties such as the absence of edge channels [8][9][10][11][12][13][14][15], phase transition driven by a tilted magnetic field or a perpendicular electric field [12,17], and the recently observed long-range spin transport [18,19]. While there has been a theoretical debate about the nature of ν = 0 state [2,[20][21][22][23][24][25], consensus from these experiments is the layer antiferromagnetic (LAF) state, where spins tend to align ferromagnetically due to the exchange interaction within each layer and antiferromagetically between the layers (Fig.1a) [2,20,21]. Neel vector is in-plane and spins are slightly canted out of plane due to the Zeeman energy.
It is known that breaking of crystal chiral symmetry opens a band gap in graphene and that gapped graphene has valley contrasting intrinsic Hall conductivity, which generates transverse valley current to the injected electronic current. This phenomenon is called the valley Hall effect and has also been observed in graphene [26][27][28]. In the LAF state, intrinsic Hall conductivity also arises with the gap opening, and similar Hall effect is predicted. Here, chiral symmetry is broken due to the spontaneous symmetry breaking driven by the electron-electron interaction. Because opposite spins occupy opposite layers, sign of effective mass term depends on the spin [1,2]. Instead of the conventional valley current, spin-valley current, which is defined as the difference of valley current between the spins, is expected to flow ( fig.1b). This theoretically predicted spin-valley Hall effect [1] has been experimentally elusive. Note that the spin-valley Hall effect allows for coupling of spin and valley degrees of freedom. If we inject opposite current for each spin, in other words inject a spin current, a valley current is expected to flow in its transverse direction (Fig.1c). While spin-orbit interaction is weak in graphene, this spin current / valley current conversion may open a new possibility of electric generation of a spin current in graphene with high efficiency, being combined with the conventional valley Hall effect.
In this study, we observed nonlocal transport in the LAF state, which is assigned to the spin-valley Hall effect as an only possible origin. There were a few previous reports on nonlocal transport in graphene under magnetic field [16,29,30], however, no discussion on the spontaneous symmetry breaking state was made. Using bilayer graphene with dual-gated structure, we found gate dependence of nonlocal transport specific for LAF state and cubic scaling relationship between the local and nonlocal resistance similarly to the previous experiments on the valley Hall effect [26,27], consistent with the scenario of the spin-valley Hall effect. We performed nonlocal transport measurement to demonstrate the spin-valley Hall effect (Fig.1e). It is the scheme widely used to detect the spin Hall effect or the valley Hall effect [26][27][28]. When the charge current is injected between terminals 2 and 6, the We also measured the local resistance ( L ) (4-terminal resistance 6−5 / 1−4 ) and bias dependence of the two-terminal differential conductance to evaluate the gap size. All data except for that of the temperature dependence was taken at 1.7 K.
Here we shows the top gate and back gate dependence of L in Figure 2a-c. At zero magnetic field, L at the CNP monotonically increases with the displacement field due to the opening of a single particle band gap. We identified the = 0, = 0 point as the minimum along the CNP line (red circle). When the magnetic field of 2 T is applied, a local maximum appears around the = 0, = 0 point similarly to the previous report [12], indicating the development of LAF insulating state (Fig. 2d). This state becomes more robust for the higher magnetic field.
We performed nonlocal transport measurement for this state.   Second, Ettingshausen and Nernst effect can cause nonlocal voltage [29,30] but it is difficult to explain the L dependence of NL discussed later (see Supplementary   Information). Third, current leakage through two measurement terminals into the ground also can cause nonlocal voltage, but this is not the origin of NL neither because input impedance of the voltage amplifiers is high enough ( Supplementary Information   Fig.S1b, Table S1).
Another possible origin of the nonlocal transport could be the edge transport (Fig.1f).
Although it has been established both experimentally [8][9][10][11][12][13][14][15] and theoretically [2,20] that the LAF state does not have an edge channel, gate inhomogeneity at the sample edge can cause carrier doping and produces less-resistive regions along the edge. In order to eliminate the edge contribution, we employed a long-edge (Fig.1d) sample, which has a same dimension with normal edge sample except for a 10 times longer edge than the neighboring standard Hall bar. If we assume the simplest situation where the transport is only caused by the edge transport and neglect the bulk transport, NL should become 10 times smaller for the long-edge sample. Figure 3 shows We now turn to the relationship between the L and NL obtained by changing the magnetic field and temperature (see Figure 3). We find that all data points lie on a single curve of the cubic scaling. In analogy with the spin Hall effect and valley Hall effect [26,27] Fig.3, however, we observed cubic scaling in the entire measurement region for the long-edge sample. In the other sample of the normal-edge length with higher resistance at the CNP, we found linear scaling above L~9 0 kΩ (~0.7 e 2 /h) . The reason for the cubic scaling up to higher resistivity than h/4e 2 is not yet clearly understood but it suggests that the effective conductivity of Eq.1 is higher than the measured conductivity, or that the effective spin-valley Hall conductivity is smaller than 4e 2 /h. Actually the cubic scaling up to the higher resistance regime was also observed in previous experiments on the valley Hall effect in bilayer graphene [26,27]. It was shown in ref. 26 that the cubic scaling holds for thermally activated component of the transport even for high L . Swirly behavior seen in fig. 3 under low magnetic field and high temperature is related to non-monotonic temperature dependence of L and NL (see Supplementary   Information), whose mechanism is not yet clearly understood. Local and nonlocal transport experiments were performed using lock-in amplifiers (around 13 Hz) and preamplifier with input impedance of 100 MΩ . This input impedance is large enough for nonlocal measurement (see Supplementary   Information). Bias dependence of the two-terminal differential conductance was measured using a voltage source, preamplifiers, voltmeters, and ammeters.

Supplemental information
Ⅰ Elimination of other possible origins of the nonlocal transport Here we discuss and eliminate the possibility of the normal Hall voltage, Ettingshausen and Nernst effect, and current flow into measurement terminals as an origin of nonlocal transport.

[Hall voltage]
If measurement terminals are shifted from the vertical direction to the longitudinal direction of the Hall bar, Hall voltage appears. But this normal Hall voltage is not the origin of NL in our device because NL becomes maximal at zero carrier density. In addition, we confirmed that sign of NL is not altered by reversing the magnetic field ( Fig. S1a,b). This result is inconsistent with the normal Hall voltage while it is consistent with the scenario of the spin-valley Hall effect.

[Ettingdhausen and Nernst effect]
Ettingshausen effect is the magnetic analogue of Peltier effect where thermal gradient appears perpendicular to the magnetic field and charge current. Heat flow generated by the Ettingshausen effect causes nonlocal voltage due to Nernst effect (Fig. S1c) [29,30].
The Ettingshausen and Nernst effects are described by the following formulas: Here, is the charge current, is the heat flow, is the electric field, is the temperature, is the spectrum function of conductivity matrix, and is Fermi distribution function. We assume , in y direction and , in x direction. Using the electrical conductivity = 0 , Nernst coefficient = 1 0 ⁄ , and the electron thermal conductivity = − 2 2 ⁄ , Eqs. S1 and S2 are written as The heat flow generated by the current injection and Ettingshausen effect is derived from Eq. S3 as follows, Nonlocal electric field between the detection terminals induced by the Nernst effect is derived from inverse matrix of Eq. S3 and becomes where is decay rate of the heat flow. Therefore nonlocal resistance is given as [Current flow into measurement terminals] Our measurement was performed in the highly resistive regime of the CNP. When the resistance of the sample is high at the CNP under a high magnetic field, an electrical current flows into the voltage amplifiers that are connected to the two measurement terminals (Fig. S1d). NL originated from this effect is given by NL,amp = + amp /2 ≲ 2 amp (S10) Where is the resistance between the current injection terminals, is difference of the contact resistance between the two measurement terminals obtained from the two-terminal measurement, and amp =100 MOhm is the input impedance of the voltage amplifiers. Table S1 shows measured NL and calculated NL,amp by eq.S10 for several data points.
NL,amp is much smaller than obtained NL . The current flow into the voltage amplifiers is therefore not the origin of the observed nonlocal resistance. Table.S1 | Estimated resistance due to current flow into measurement terminals.
L and NL extracted from Fig.3 of main text, estimated values of (= L ) and (< ), amp , and calculated values of NL,amp using eq. S9.

Ⅱ Sample dependence
In the main text, we compared NL of two samples with different edge lengths. Here, we show the data of other two samples (Fig. S2a, sample3, 4). They have the same order of NL is spite of large difference of the edge length, which is consistent with the scenario of the bulk spin-valley current.
However their peak positions of NL as functions of the gate voltages were slightly shifted from those of L (Fig. S2b) and therefore unsuitable for further analysis. This shift most probably originates from inhomogeneity of the carrier density. dependence of the L and NL at = 0 at 1.5 K for the magnetic field of 2 T in sample 3 and Ⅲ Temperature dependence in high temperature region We show temperature dependence of L and NL for = 1.7 ~ 280 K (Fig. S3a-d). L and NL do not depend monotonically on the temperature. For example, by increasing the temperature, the sample changes to be metallic around = 8 K and again becomes insulating around = 26 K for = 2 T (see slope of Fig. S3a). We define 1(2) as the temperature at which slope of the resistance as a function of changes from negative (positive) to positive (negative), and plot them in Fig. S3f. 1 of L ( c1,local ) and that of NL ( c1,nonlocal ) increase with the magnetic field. c1,local is smaller than c1,nonlocal for ≤ 4 T, but larger for ≥ 6 T.
c2 of L ( c2,local ) is almost linear with and the proportionality factor is similar to that of the gap size estimated from bias measurement (main text Fig.3 inset), although we cannot define c2,local for ≥ 5 T. c2 of RNL ( c2,nonlocal ) is larger than c2,local .
In addition, we found that cubic scaling relationship between L and NL does not hold for > c1,local .
We suppose that decrease of L above c2,local is related to the breaking of the spin order of LAF state, whose robustness is characterized by the gap size, because c2,local is comparable with the gap size. However, some features are not yet understood such as the metallic temperature dependence for c1 < < c2 , and difference between c,local . and c,nonlocal . Further theoretical and experimental investigation is necessary to account for the temperature dependence of L and NL . In particular, experiments using spin injection or NMR may be useful to reveal the breaking of the spin order of LAF state at high temperature.