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Figure 1: Three types of the integer quantum Hall effect.
a,b, Schematic illustration of the conventional integer QHE found in 2D semiconductor systems (a), incorporated from refs12, and the QHE in bilayer graphene described in the present paper (b). Plateaus in Hall conductivity σx y occur at values (g e2/h)N, where N is an integer, e2/h the conductance quantum and g the system degeneracy. The distance between steps along the concentration axis is defined by the density of states g B/φ0 on each Landau level, which is independent of a 2D spectrum1,2,3,4,5,6,7,8,9. Here, B is the magnetic field and φ0=h/e the flux quantum. The corresponding sequences of Landau levels as a function of carrier concentrations n are shown in blue and orange for electrons and holes, respectively. For completeness, c also shows the QHE behaviour for massless Dirac fermions in single-layer graphene.
图1：整数量子Hall效应的三种类型。
a，b，2D半导体系统中发现的常规整数QHE的示意图（a），结合自参考文献12，以及本文描述的双层石墨烯中的QHE（b）。Hall电导率σx y的平台出现在值(g e2/h)N处，其中N是整数，e2/h是电导量子，g是系统简并。沿着浓度轴的台阶之间的距离由每个朗道能级上的态密度g B/φ0定义，它与2D谱无关。这里，B是磁场， φ0=h/e是通量量子。对于电子和空穴，作为载流子浓度n的函数的相应朗道能级序列分别以蓝色和橙色显示。为了完整起见，c还显示了单层石墨烯中无质量Dirac费米子的QHE行为。
Figure 2: Quantum Hall effect in bilayer graphene.
a, Hall resistivitiesρx y and ρx x measured as a function of B for fixed concentrations of electrons n≈2.5×1012 cm−2 induced by the electric field effect. Inset: Scanning electron micrograph of one of more than ten bilayer devices studied in our work. The width of the Hall bar (dark central area) is approximately 1 μm. The known geometry of our devices allowed us to convert the measured resistance into ρx x with an accuracy of better than 10%. b,c, σx y and ρx x are plotted as functions of n at a fixed B and temperature T=4 K. Positive and negative n correspond to field-induced electrons and holes, respectively. The Hall conductivity σx y=ρx y/(ρx y2+ρx x2) was calculated directly from experimental curves for ρx y and ρx x. σx y allows the underlying sequences of QHE plateaus to be seen more clearly. σx y crosses zero without any sign of the zero-level plateau that would be expected for a conventional 2D system. The inset shows the calculated energy spectrum for bilayer graphene, which is parabolic at low ɛ. Carrier mobilities μ in our bilayer devices were typically around 3,000 cm2 V−1 s−1, which is lower than for devices made from single-layer graphene3,4. This is surprising because one generally expects more damage and exposure in the case of single-layer graphene that is unprotected from the immediate environment from both sides.
图2：双层石墨烯中的量子霍尔效应。
a，对于由电场效应诱导的固定浓度的电子n ≈ 2.5×1012 cm−2，测量的霍尔电阻率ρx y和ρx x作为B的函数。插图：他们工作中研究的十多种双层器件之一的扫描电子显微照片。大厅条（暗中心区域）的宽度约为1 μm。设备的已知几何形状允许将测量的电阻转换成ρx x，精度优于10%。在固定的B和温度T=4 K下，b，c，σx y和ρx x作为n的函数绘制。正n和负n分别对应于场致电子和空穴。Hall电导率σx y=ρx y/(ρx y2+ρx x2)直接从ρx y和ρx x的实验曲线计算。σx y允许更清楚地看到QHE平台的潜在序列。σx y穿过零，没有任何常规2D系统预期的零能级平台的迹象。插图显示了计算出的双层石墨烯的能谱，在低温度下呈抛物线形。在双层器件中，载流子迁移率μ通常约为3,000 cm2 V−1 s−1，低于单层石墨烯器件。这是令人惊讶的，因为人们通常预计在单层石墨烯的情况下会有更多的损坏和暴露，因为单层石墨烯从两侧都不受直接环境的保护。
Figure 3: Resistivity of bilayer graphene near zero concentrations as a function of magnetic field and temperature.
a–d, The peak in ρx x remains of the order of h/4e2, independent of B (a,b) and T (c,d). This yields no gap in the Landau spectrum at zero energy. b, For a fixed n≈0 and varying B, we observed only small magnetoresistance. The latter varied for different devices and contact configurations (probably indicating the edge-state transport) and could be non-monotonic and of random sign. However, the observed magnetoresistance (for bilayer devices without chemical doping10) never exceeded a factor of two in any of our experiments in fields up to 20 T.
图3：接近零浓度的双层石墨烯的电阻率是磁场和温度的函数。
a-d，ρx x的峰值保持在h/4e2的数量级，与B (a,b)和T (c,d)无关。这在零能量的Landau谱中不会产生空隙。对于固定的n ≈ 0和变化的B，作者只观察到小的磁阻。后者因不同的器件和接触配置而异（可能表明边缘态输运），并且可以是非单调的和随机符号的。然而，在他们的任何实验中，在高达20 T的磁场中，观察到的磁阻（对于没有化学掺杂的双层器件）从未超过2倍。
在双组分布洛赫函数（进一步称为赝自旋）的空间中起作用，描述属于两个不等价碳亚晶格A和B以及标记为1和2的两个石墨烯层的弱耦合最近位置A1和B2上的电子波振幅。对于给定方向的准粒子动量p=（Pcosá, Psiná），一般形式的哈密顿量
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