Device fabrication<\/p>\n
The heterostructures were assembled by the layer-by-layer dry transfer technique45<\/a>. Bulk crystals were first exfoliated onto Si wafers with 285\u2009nm of oxide layer, and flakes of suitable thickness and shape were selected based on optical micrographs. Using a stamp built from a polycarbonate film supported on a polypropylene-carbonate-coated polydimethylsiloxane block, we picked up the constituent layers one at a time at 70\u2009\u00b0C. The assembled stack (Extended Data Fig. 1c<\/a>) was released onto a SiO2\/Si chip carrying pre-patterned Pt electrodes, after which the polycarbonate was dissolved away in chloroform and isopropanol. To create the conduction channel in the W layer, the top gate was patterned by electron-beam lithography and removed by oxygen-plasma reactive-ion etching (Oxford Plasmalab80Plus). Finally, a second electron-beam lithography step followed by the thermal evaporation of Bi was carried out to create Bi contacts to the Mo layer.<\/p>\n Optical micrographs of devices 1 and 3 are presented in Extended Data Fig. 1<\/a>. The top and bottom gates are outlined by dashed black and red lines, respectively; the channel (shaded orange) is set by the region in which the gates overlap. Extended Data Fig. 1c<\/a> shows a cross-sectional schematic. Inside the channel, a thin hBN barrier (1.5\u20132\u2009nm) separates the W and Mo layers, whereas a thicker hBN spacer (about 10\u2009nm) is placed outside the channel to facilitate exciton injection. The W and Mo layers are contacted with Pt and Bi electrodes46<\/a>,47<\/a>, respectively, to achieve ohmic contacts at low temperatures. Furthermore, the gates are maintained at a high value of \\(\\Delta\\) that heavily dopes the metal\u2013semiconductor contact regions. In these devices, bilayer MoSe2 was used for better mechanical stability but functions as monolayer MoSe2 because under high \\(\\Delta\\), electrons are confined to the layer that is closer to the W layer.<\/p>\n Electrical measurements<\/p>\n Two types of transport measurement (Extended Data Fig. 2<\/a>) were carried out, both in an Oxford TeslatronPT closed-cycle 4He cryostat operating down to 1.5\u2009K and magnetic field up to 12\u2009T. Details are described in refs. 32<\/a>,34<\/a>,38<\/a>. The first configuration probed the W layer alone in a conventional Hall-bar layout, yielding \\({R}_{{{xx}}}\\) and \\({R}_{{{xy}}}\\) with the Mo layer kept floating. The second configuration was a Coulomb drag measurement: current was sourced through the Mo layer, whereas the longitudinal and transverse voltages developed across the W layer gave the drag resistances \\({R}_{{{xx}},{\\rm{drag}}}\\) and \\({R}_{{{xy}},{\\rm{drag}}}\\), respectively. For both configurations, a 1-mV (r.m.s.) a.c. excitation at 7.33\u2009Hz was used, currents and voltages were recorded with Stanford Research SR860 and SR830 lock-in amplifiers, and the voltage signals first passed through an Ithaco DL1201 preamplifier with a 100-M\u03a9 input impedance. To reduce a.c. coupling between the Mo and W layers in the Coulomb drag measurements, we used a voltage transformer. The a.c. bias voltage was applied to the primary of a 1:1 voltage transformer. The secondary end of the transformer was connected across a 10-k\u03a9 potentiometer whose two ends were connected to electrode Mo-10 and Mo-7 on the Mo layer. The potentiometer was adjusted to equally distribute the a.c. bias voltage at the two electrodes. The d.c. bias voltage \\({V}_{{\\rm{b}}}\\) was applied to the midpoint of the potentiometer. This sets the potentials at the two Mo electrodes to be \\({V}_{\\mathrm{Mo} \\mbox{-} 10}=\\,{V}_{{\\rm{b}}}+\\,\\frac{{V}_{\\mathrm{AC}}}{2}\\) and \\({V}_{{\\rm{Mo}}-7}=\\,{V}_{{\\rm{b}}}-\\,\\frac{{V}_{{\\rm{AC}}}}{2}\\) so that the time-averaged potential of the Mo layer remains at \\({V}_{{\\rm{b}}}.\\,\\)The method reduces a.c. coupling between the layers by minimizing the out-of-plane a.c. voltage drop. For the drive current measurement, a 100-k\\(\\Omega\\) resistor was connected in series with the Mo layer.<\/p>\n Optical measurements<\/p>\n Optical spectroscopy was carried out in an attocube attoDRY2100 closed-cycle 4He cryostat (1.5-K base temperature, fields to 9\u2009T). Broadband illumination from a Thorlabs tungsten\u2013halogen lamp was delivered through a single-mode fibre and brought to normal incidence on the sample by a cryogenic objective with a 0.8 numerical aperture, with the optical power held below 50\u2009nW\u2009\u00b5m\u22122 to avoid heating. Light reflected from the sample passed through the same objective and was resolved in a spectrometer and measured by a liquid-nitrogen-cooled silicon charge-coupled device (Princeton Instruments). We defined the RC spectrum as RC\u2009=\u2009\\(\\frac{I-{I}_{0}}{{I}_{0}}\\), normalizing the spectrum of interest I against a featureless reference I0 recorded where the sample was heavily doped by electrons or holes. Circularly polarized \u03c3+ (left-handed) and \u03c3\u2212 (right-handed) light for the helicity-resolved measurements were generated by pairing a linear polarizer with an achromatic quarter-wave plate.<\/p>\n Identification and characterization of trions<\/p>\n The (X\u2009+\u2009X\u00b1) regions host both excitons and trions. When the Mo layer is in open circuit, the current in the W layer is carried by free holes. This allows us to identify the (X\u2009+\u2009X\u00b1) regions as they appear as insulating states. On increasing charge imbalance, excitons and trions dissociate because of screening of the interlayer Coulomb interactions, and the transport becomes metallic. We outlined the (X\u2009+\u2009X\u00b1) regions in the \\({V}_{{\\rm{b}}}\\)\u2013\\({V}_{{\\rm{g}}}\\) phase space by tracing the metal\u2013insulator transition. In the absence of a magnetic field, trions are confined near the \\(p=2n\\) and \\(n=2p\\) lines (Fig. 1d<\/a>). At a finite magnetic field, the trion regions expand presumably due to a field-induced enhancement in their binding energies. The trion regions also move up in \\({V}_{{\\rm{b}}}\\) due to the valley Zeeman splitting. With a further increase in the B field, the trions, as a spin singlet, are no longer stable in regions where the valley Zeeman splitting exceeds the trion binding energy; the trion regions shrink. The evolution of the trion phase diagram with magnetic field is illustrated in Extended Data Figs. 4<\/a> and 8<\/a>.<\/p>\n The assignment of trions is further supported by the handedness-resolved optical RC measurements. The attractive exciton polaron of the W layer, XW,a, is sensitive to hole doping and is used to probe the positive trions (Fig. 3<\/a>). Extended Data Fig. 5<\/a> summarizes the spectrally integrated RC of XW,a as a function of \\({V}_{{\\rm{b}}}\\) and \\({V}_{{\\rm{g}}}\\) (6\u2009T, device 2). The (X\u2009+\u2009X+) region (dashed lines) was identified from the transport measurement. Regions with finite RC in the \\({\\sigma }^{+}\\) and \\({\\sigma }^{-}\\) channels correspond to hole doping in the K and K\u2032 valleys, respectively. As expected, hole doping in the K\u2032 valley (majority valley) is observed in the entire pi and pn regions (Extended Data Fig. 5b<\/a>). The threshold for hole injection in the K valley (minority valley) is pushed up in \\({V}_{{\\rm{b}}}\\) because of the Zeeman splitting except in the (X\u2009+\u2009X+) region. RC is finite in the (X\u2009+\u2009X+) region at 1.5\u2009K although it is very weak in its lower part (Extended Data Fig. 5a<\/a>). On the thermal dissociation of trions at 10\u2009K, RC in the (X\u2009+\u2009X+) region vanishes (Extended Data Fig. 5c<\/a>). The results show that holes occupy both valleys in the (X\u2009+\u2009X+) region. Similarly, the attractive exciton polaron of the Mo layer, XMo,a, is sensitive to electron doping and is used to probe the negative trions (Extended Data Fig. 6<\/a>). Similar behaviours are observed for the (X\u2009+\u2009X\u2212) region.<\/p>\n We can estimate the trion binding energy, \\({E}_{{\\rm{T}},{\\rm{b}}}\\), from the difference between the charge injection threshold for the minority valley in the (X\u2009+\u2009X+) region at 1.5\u2009K and 10\u2009K (Extended Data Fig. 5a<\/a>, red line). At 10\u2009K, interlayer Coulomb interactions are negligible. At 1.5\u2009K, the system spontaneously lowers the injection threshold to gain the trion binding energy. The value is about 25\u2009mV in \\({V}_{{\\rm{b}}}\\). It corresponds to a trion binding energy of \\({E}_{{\\rm{T}},{\\rm{b}}}\\approx \\frac{{C}_{{\\rm{Q}}}^{-1}}{{C}_{{\\rm{Q}}}^{-1}+{C}_{2{\\rm{L}}}^{-1}}\\times\\)\u2009(25\u2009meV)\u2009\u2248\u20091.2\u2009meV. Here \\({C}_{{\\rm{Q}}}=\\,{e}^{2}\\frac{{g}_{{\\rm{s}}}{g}_{{\\rm{v}}}{m}_{{\\rm{h}}}}{2\\uppi {\\hslash }^{2}}\\approx\\)\u200930\u2009\u00b5F\u2009cm\u22122 is the quantum capacitance of the hole layer with spin\u2013valley degeneracy \\({g}_{{\\rm{s}}}{g}_{{\\rm{v}}}\\)\u2009=\u20092 and hole effective mass \\({m}_{{\\rm{h}}}\\approx 0.45{m}_{0}\\) in monolayer WSe2 (\\({m}_{0}\\) denotes the free electron mass); \\({C}_{2{\\rm{L}}}=\\frac{{\\epsilon }_{{\\rm{r}}}{\\epsilon }_{0}}{d}\\approx\\)\u20091.55\u2009\u00b5F\u2009cm\u22122 is the interlayer geometrical capacitance of the double layer with the out-of-plane dielectric constant of hBN \\({\\epsilon }_{{\\rm{r}}}\\approx\\)\u20093.5, vacuum permittivity \\({\\epsilon }_{0}\\) and interlayer separation d\u2009\u2248\u20092\u2009nm. The estimated trion binding energy is in good agreement with the temperature scale and the Zeeman energy scale for trion dissociation.<\/p>\n Modelling transport in Coulomb-coupled double layers<\/p>\n We model transport in Coulomb-coupled electron\u2013hole double layers, phenomenologically following ref. 48<\/a>. We consider contributions of free electrons, free holes, excitons and trions, each of which is characterized by a 2\u2009\u00d7\u20092 in-plane conductivity tensors \\({\\sigma }_{{{i}}}\\) (i\u2009=\u2009e, h, X and T) when interlayer tunnelling is ignored. The current densities in the electron and hole layers (\\({J}_{{\\rm{e}}}\\) and \\({J}_{{\\rm{h}}}\\)) are related to the electric fields in the electron and hole layers (\\({E}_{{\\rm{e}}}\\) and \\({E}_{{\\rm{h}}}\\)) through a 4\u2009\u00d7\u20094 conductivity tensor \\({\\Sigma }\\):<\/p>\n $$\\left(\\begin{array}{c} {J}_{{\\rm{e}}x}\\\\ {J}_{{\\rm{e}}y}\\\\ {J}_{{\\rm{h}}x}\\\\ {J}_{{\\rm{h}}y}\\end{array}\\right)=\\Sigma \\left(\\begin{array}{c} {E}_{{\\rm{e}}x}\\\\ {E}_{{\\rm{e}}y}\\\\ {E}_{{\\rm{h}}x}\\\\ {E}_{{\\rm{h}}y}\\end{array}\\right)$$<\/p>\n \n (1)\n <\/p>\n In the (X\u2009+\u2009X\u2212) region, the 4\u2009\u00d7\u20094 conductivity tensor can be approximated as<\/p>\n $${{\\Sigma }}^{-}=\\left(\\begin{array}{cc}{\\sigma }_{{\\rm{X}}}+2{\\sigma }_{{\\rm{T}}}+{\\sigma }_{{\\rm{e}}} & -{\\sigma }_{{\\rm{X}}}-2{\\sigma }_{{\\rm{T}}}\\\\ -{\\sigma }_{{\\rm{X}}}-{\\sigma }_{{\\rm{T}}} & {\\sigma }_{{\\rm{X}}}+{\\sigma }_{{\\rm{T}}}\\end{array}\\right)$$<\/p>\n \n (2)\n <\/p>\n Here we assume perfect Coulomb drag for excitons and trions and negligible frictional drag for free electrons and holes at low temperature. The current density in the electron layer \\({J}_{{\\rm{e}}}\\) responds to the electric field in both electron and hole layers: the former carries contributions from excitons, trions and free electrons (\\({\\sigma }_{{\\rm{X}}}+2{\\sigma }_{{\\rm{T}}}+{\\sigma }_{{\\rm{e}}}\\)); the latter carries contributions only from excitons and trions from Coulomb drag (\\(-{\\sigma }_{{\\rm{X}}}-2{\\sigma }_{{\\rm{T}}}\\)); the factor of 2 for \\({\\sigma }_{{\\rm{T}}}\\) accounts for the presence of two electrons in a negative trion. The current density in the hole layer \\({J}_{\\rm{h}}\\) can be understood in a similar fashion, except that the free hole contribution is ignored. Free holes arise from dissociation of excitons in the (X\u2009+\u2009X\u2212) region, which is negligible due to the substantially larger exciton binding energy than the trion binding energy. Similarly, the conductivity tensor in the (X\u2009+\u2009X+) region can be expressed as<\/p>\n $${{\\Sigma }}^{+}=\\left(\\begin{array}{cc}{\\sigma }_{{\\rm{X}}}+{\\sigma }_{{\\rm{T}}} & -{\\sigma }_{{\\rm{X}}}-{\\sigma }_{{\\rm{T}}}\\\\ -{\\sigma }_{{\\rm{X}}}-2{\\sigma }_{{\\rm{T}}} & {\\sigma }_{{\\rm{X}}}+2{\\sigma }_{{\\rm{T}}}+{\\sigma }_{{\\rm{h}}}\\end{array}\\right).$$<\/p>\n \n (3)\n <\/p>\n In the standard Hall measurements, the hole layer was biased and probed, whereas the electron layer was in the open-circuit geometry. The only non-zero current density was \\({J}_{{\\rm{h}}x}\\), where x (y) denotes the longitudinal (transverse) direction. We can rewrite equation (1<\/a>) for square resistances:<\/p>\n $$\\left(\\begin{array}{c}0\\\\ 0\\\\ 1\\\\ 0\\end{array}\\right)=\\Sigma \\left(\\begin{array}{l}{R}_{{{xx}},{\\rm{drag}}}\\\\ {R}_{{{xy}},{\\rm{drag}}}\\\\ {R}_{{{xx}}}\\\\ {R}_{{{xy}}}\\end{array}\\right)$$<\/p>\n \n (4)\n <\/p>\n and solve it for \\({R}_{{{xx}}}\\) and \\({R}_{{{xy}}}\\). Similarly, for the Coulomb drag measurements, the hole layer was probed as a longitudinal current was driven in the Mo layer. The only non-zero current density was \\({J}_{{\\rm{e}}x}\\). We can rewrite equation (1<\/a>) for square resistances:<\/p>\n $$\\left(\\begin{array}{c}1\\\\ 0\\\\ 0\\\\ 0\\end{array}\\right)=\\Sigma \\left(\\begin{array}{l} {R}_{{{xx}}}\\\\ {R}_{{{xy}}}\\\\ {R}_{{{xx}},{\\rm{drag}}}\\\\ {R}_{{{xy}},{\\rm{drag}}}\\end{array}\\right)$$<\/p>\n \n (5)\n <\/p>\n and solve it for \\({R}_{{{xx}},{\\rm{drag}}}\\) and \\({R}_{{{xy}},{\\rm{drag}}}\\). Under a small perpendicular magnetic field B, we linearize the conductivity tensor in B:<\/p>\n $${\\sigma }_{{\\rm{X}}}={\\sigma }_{{\\rm{X}}0}\\left(\\begin{array}{cc}1 & 0\\\\ 0 & 1\\end{array}\\right),{\\sigma }_{i}={\\sigma }_{i0}\\left(\\begin{array}{cc}1 & -{\\omega }_{{\\rm{c}},i}{\\tau }_{i}\\\\ {\\omega }_{{\\rm{c}},i}{\\tau }_{i} & 1\\end{array}\\right),(i={\\rm{e}},{\\rm{h}},{\\rm{T}}).$$<\/p>\n \n (6)\n <\/p>\n Here \\({\\sigma }_{{\\rm{X}}0}=\\frac{{n}_{{\\rm{X}}}{e}^{2}{\\tau }_{{\\rm{X}}}}{{m}_{X}}\\) is the conductivity of excitons, and \\({\\omega }_{{\\rm{c}},{{i}}}=\\frac{{eB}}{{m}_{{{i}}}}\\) and \\({\\sigma }_{i0}=\\frac{{n}_{i}{e}^{2}{\\tau }_{i}}{{m}_{i}}\\) (i\u2009=\u2009e, h, T) are, respectively, the cyclotron frequency and d.c. conductivity of the charged particles (\\({m}_{{{i}}}\\), \\({n}_{{{i}}}\\) and \\({\\tau }_{{{i}}}\\) denote the particle mass, density and transport scattering time, respectively). In the (X\u2009+\u2009X\u2212) region, we solve equations (4<\/a>) and (5<\/a>) with \\({\\Sigma }={{\\Sigma }}^{-}\\) to obtain<\/p>\n $$\\begin{array}{l}{R}_{{{xy}}}\\approx -\\displaystyle\\frac{B}{e}\\left(\\displaystyle\\frac{1+f}{{n}_{{\\rm{e}}}}+\\displaystyle\\frac{{f}^{2}\\left(1-\\displaystyle\\frac{{\\sigma }_{{\\rm{X}}0}}{{\\sigma }_{{\\rm{e}}0}}\\right)}{{n}_{{\\rm{T}}}}\\right)\\,,\\,{R}_{{{xx}}}=\\displaystyle\\frac{\\left(1+f\\,\\right)}{{\\sigma }_{{\\rm{e}}0}}+\\displaystyle\\frac{f}{{\\sigma }_{{\\rm{T}}0}},\\\\\\qquad\\,\\,\\,\\,\\,{R}_{{{xy}},{\\rm{drag}}}\\approx -\\displaystyle\\frac{B}{{n}_{{\\rm{e}}}e}\\,,\\,{R}_{{{xx}},{\\rm{drag}}}\\approx \\displaystyle\\frac{1}{{\\sigma }_{{\\rm{e}}0}}\\end{array}$$<\/p>\n \n (7)\n <\/p>\n where \\(f=\\,{\\sigma }_{{\\rm{T}}0}\/({\\sigma }_{{\\rm{T}}0}+{\\sigma }_{{\\rm{X}}0})\\) is a dimensionless constant between 0 and 1. In the (X\u2009+\u2009X+) region, we solve equations (4<\/a>) and (5<\/a>) with \\({\\Sigma }={{\\Sigma }}^{+}\\) to obtain<\/p>\n $$\\begin{array}{l}{R}_{\\mathrm{xy}}\\approx \\displaystyle\\frac{B}{{n}_{{\\rm{h}}}e}\\,,\\,{R}_{\\mathrm{xx}}\\approx \\displaystyle\\frac{1}{{\\sigma }_{{\\rm{h}}0}},\\\\ {R}_{\\mathrm{xy},\\mathrm{drag}}\\approx \\displaystyle\\frac{B}{e}\\left(\\displaystyle\\frac{1+f}{{n}_{{\\rm{h}}}}-\\displaystyle\\frac{{f}^{2}\\left(\\displaystyle\\frac{{\\sigma }_{{\\rm{X}}0}}{{\\sigma }_{{\\rm{h}}0}}\\right)}{{n}_{{\\rm{T}}}}\\right)\\,,\\,{R}_{\\mathrm{xx},\\mathrm{drag}}\\approx \\displaystyle\\frac{(1+f\\,)}{{\\sigma }_{{\\rm{h}}0}},\\end{array}$$<\/p>\n \n (8)\n <\/p>\n On increasing the temperature, trions and excitons dissociate and \\({\\sigma }_{{\\rm{X}}}\\) and \\({\\sigma }_{{\\rm{T}}}\\) approach zero. Transport in the two layers is then decoupled and described by \\({\\Sigma }=\\left(\\begin{array}{cc}{\\sigma }_{{\\rm{e}}} & 0\\\\ 0 & {\\sigma }_{{\\rm{h}}}\\end{array}\\right)\\). \\({R}_{{{xy}}}\\) is positive and simply measures the Hall density of free holes in the hole layer.<\/p>\n","protected":false},"excerpt":{"rendered":"Device fabrication The heterostructures were assembled by the layer-by-layer dry transfer technique45. 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