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Interlayer electron and hole can be paired up through coulomb interaction to form an exciton insulator when their kinetic energy is substantially smaller than the interaction energy. The traditional platform to realize such an interlayer interaction is the double quantum well with dielectric material between electron and hole, for which an external magnetic field is required to generate Landau level flat bands that can reduce the kinetic energy of charged carriers. When both quantum wells are at the half filling of the lowest landau level, the electron-electron repulsive interaction, by the particle-hole transformation in one well, will be equivalent to electron-hole attractive interaction, from which interlayer exciton and its condensation can emerge. In a two-dimensional twisted homostructure or an angle aligned heterostructure, there exists a moiré superlattice, in which bands are folded into the mini-Brillouin zone by the large moiré period. Gap opening at the boundary of mini-Brillouin zone can form the well-known moiré flat band. This review will discuss how to use the moiré flat bands to generate exciton insulator in the absence of external magnetic field in transitional metal dichalcogenide (TMD) moiré heterostructure. Unlike the double quantum well where symmetric well geometry is used, the moiré related sample can have multiple different geometries, including monolayer TMD-hexagonal boron nitride-moiré structure, moiré-moiré structure, and monolayer TMD-bilayer TMD structure. The carriers in those structures can be well tuned to locate equally in different layers, and particle-hole transformation in the moiré first Hubbard band can transform the interlayer repulsive coulomb interaction into attractive interaction, which is the same as that in quantum well under magnetic field. We will show that by using differential contrast reflection spectrum, interlayer photoluminescence, 2s exciton sensing, quantum capacitance and microwave impedance microscopy, the signature of exciton fluid can be identified. The excitonic coherence features in those structures will promise by using the coulomb drag technique and counter flow technique in future. In general, exciton in moiré lattice is a promising candidate for studying the Bose-Hubbard model in solids and can well realize exciton superfluidity, excitonic mott insulator as well as the crossover between them.
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Keywords:
- exciton insulator /
- moiré lattice /
- flat band /
- two-dimensional semiconductor
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图 1 激子绝缘体产生示意图 (a)左, 半导体带隙Eg, 激子束缚能Eb, 当Eb大于Eg时, 单粒子能带结构在电子-空穴吸引作用下表现得不稳定; 右, 半金属有着负的带隙, 黑色虚线表示费米能, 同样在电子-空穴的吸引下能带结构不稳定, 二者都可能自发形成激子并打开一个关联能隙; (b)—(d)一个电子-电子的双层结构在纵向磁场中等价于一个电子-空穴双层结构[18]; (b)电子处于平行的上下两层中的示意图; (c)在磁场中电子的动能量子化到一系列分立的朗道能级, 每个朗道能级包含若干简并的电子圆形轨道, 这里用方格子代表轨道; 当磁场足够强时, 所有电子都集中在最低朗道能级并且是部分填充(图中为1/3填充); (d)在下面一层中进行粒子-空穴转换, 此时原先空的格点等价为空穴占据(图中绿色格点), 原先两层的电子-电子库仑排斥在此变换后等价为两层的电子-空穴库仑吸引, 当两层电子空穴数目一致的时候(各自都半填充最低朗道能级), 最有可能发生激子的玻色爱因斯坦凝聚(图中展示的是电子-空穴不相等的情况)
Figure 1. Schematic for exciton insulator formation: (a) Left, a semiconductor with bandgap Eg and exciton binding energy Eb, when Eb is larger than Eg, the single particle band structure is not stable under electron-hole attraction; right, a semimetal with a negative bandgap, black dashed line indicates the Fermi level. Similarly, under electron-hole attraction, the band structure is not stable. Both will spontaneously form exciton and open a correlation gap. (b)–(d) An electron-electron bilayer system in a strong magnetic field is equivalent to an electron-hole bilayer[18]. (b) Cartoon depiction of two parallel layers of electrons. (c) In a magnetic field the kinetic energy of 2D electrons is quantized into discrete Landau energy levels. Each such Landau level contains a huge number of degenerate orbitals, here depicted schematically as a checkerboard of sites. If the field is strong enough, all electrons reside in the lowest Landau level, and only occupy a fraction (here one-third) of the available sites. (d) A particle-hole transformation applied to the lower electron layer places the emphasis on the unoccupied sites—that is, the holes (colored green) in that layer. This transformation changes the sign of the Coulomb interactions between layers from repulsive to attractive. Exciton BEC is most likely to occur when the number of electrons and holes are equal, that is, when each layer is half-filled (this is not the case in this figure).
图 2 莫尔平带 (a)在TMD异质结中形成的莫尔晶格[25], 其中A, B, C 三个位置是莫尔格点中的高对称点, am为莫尔周期, 其上下层原子堆叠情况参照右边的放大示意图(此图为转角0°附近); (b)在形成莫尔晶格后, 布里渊区从原来的
$ 1/a $ (1010 m–1)(a为原子间距)折叠为1/am(108 m–1), 形成迷你布里渊区, 在其边界通过布拉格反射打开能隙, 形成莫尔平带Figure 2. Moiré flat band: (a) Moiré lattice formed in TMD heterobilayer. A, B, C are three high symmetry points in the moiré unit cell[25], here am is the moiré lattice period. And the zoomed in lattice configuration was shown on the right (this is a zero-angle twist case). (b) After having moiré lattice, the Brillouin zone shrinks from the original 1/a (～1010 m–1) (a is the original distance between neighboring atoms) to 1/am (～108 m–1), forming mini Brillouin zone. This opens a gap at its boundary due to Bragg reflection followed by the emergence of moiré flat band.
图 3 单层-莫尔结构中的激子绝缘体[153,154] (a)该结构中的粒子-空穴转换示意图; (b)单层WSe2与莫尔异质结在不加电场时的能带相对位置示意图, 其中黑色虚线代表费米能; (c)—(e)莫尔异质结层间激子光致发光强度(c), 激子探测手段中的2s态强度(d), 以及穿透电容(e)随电场和电荷浓度的变化. 其中穿透电容可以清楚看到在区域III (单层与莫尔晶格都同时被掺杂了的区域), 在总的浓度为1的时候, 连续调节电场改变体系激子浓度(浓度范围
${v }_{x}$ 为0—0.7), 体系始终有带隙Figure 3. Exciton insulator in monolayer-hBN-moiré structure[153,154]: (a) Schematic for particle-hole transformation in this system; (b) band alignment between monolayer WSe2 and moiré heterobilayer without electric field, black dashed line indicates the Fermi level; (c)–(e) photoluminescence intensity from moiré interlayer exciton (c), exciton sensor’s 2s intensity (d) and penetration capacitance (e) as a function of electric field and dope. The penetration capacitance clearly shows in region III, where both monolayer and moiré are doped and when total filling is at 1, the system is always gaped when continuously tune the exciton density (
${v}_{x}\sim$ 0–0.7) by varying the electric field.
图 4 单层-双层结构中(a)—(c)[155]以及双莫尔结构中(d)—(g)[157]的激子绝缘体 (a)电场调控空穴在两层WSe2中的分布, 电场向上时(上图), 空穴全部分布于靠近莫尔晶格的那层WSe2, 当电场向下时(下图), 空穴可以被转移到最下层WSe2中, 并且与上层中的空位(电子)束缚在一起形成激子; (b)扫描背向栅极电压调控两层空穴比例, 在总的浓度为1的时候出现绝缘态的性质; (c)通过MIM测量到体系在
$ v=1 $ 的确处于绝缘态, 其带隙在120 K消失; (d)双莫尔结构样品示意图, 最上层WSe2用于2s激子探测; (e) WS2-2L WSe2-WS2导带能带排列; (f) 2s激子反射强度随电场E和电子浓度$ v $ 的变化关系; (g)激子密度波示意图, 在总的电子浓度$ v $ = 1/3时, 红(蓝)色点表示上(下)层莫尔晶格中的电子, 激子在晶格中移动时会被限制在1/3的格点中(如图中箭头所示)Figure 4. Exciton insulator in monolayer TMD-bilayer TMD structure (a)–(c) [155] and in double moiré structure (d)–(g) [157]: (a) Electric field tunes hole distribution in both WSe2 layers, when it points upward (upper picture), holes are all located in the moiré WSe2 layer, when it points downward (lower picture), holes can be transferred to another WSe2 layer. Those holes can bond with vacancies (electron) in the moiré to form excitons. (b) Back-gate dependent reflection spectrum. Insulator behavior emerges at total filling 1. (c) MIM showing the insulator state at
$ v=1 $ has a gap equivalent to 120 K. (d) Double moiré sample schematic, the top most WSe2 layer is used for 2s exciton sensing. (e) Conduction band alignment in WS2-2L WSe2-WS2 structure. (f) 2s reflection amplitude as a function of electric field E and electron density$ v $ . (g) Exciton density wave schematic. At total electron filling$ v= 1/3 $ , the red (blue) dots represent charges in top (bottom) moiré, exciton hops only under the 1/3 lattice (as shown by the black arrow).
图 5 单层-单层结构中的激子绝缘体[158] (a)样品结构示意图,
${V}_{\rm b}$ 和$ {V}_{\rm g} $ 分别代表施加的偏压和栅压,$ \varDelta $ 为额外施加的面外电场(以产生高掺杂的接触区域); (b)能带和化学势相对位置示意图, 其中$ {\mu }_{\rm e} $ ,$ {\mu }_{\rm h} $ ,$ {\mu }_{\rm X} $ 分别代表电子、空穴、激子的化学势,${E}_{\rm G}$ 是系统的带隙; (c)静电相位图(忽略层间激子耦合), 其中p, n, i 分别代表空穴、电子掺杂和本征状态; (d), (e) 穿透电容(d)和层间电容(e)随偏压$ {V}_{\rm b} $ 和栅压$ {V}_{\rm g} $ 的变化关系; 穿透电容(d)观测到的绝缘区域(红色)与层间电容(e)观测到的电子-空穴对可注入区域(红色)重叠的部分(白色虚线与红色虚线围出的三角区域)指示了激子绝缘体的存在; (f)激子绝缘体相图Figure 5. Exciton insulator in monolayer-monolayer structure[158]: (a) Sample schematic in which
$ {V}_{\rm b} $ and$ {V}_{\rm g} $ represent bias voltage and gate voltage, respectively; (b) band diagram of the device in which$ {\mu }_{\rm e} $ ,$ {\mu }_{\rm h} $ and$ {\mu }_{\rm X} $ represent chemical potential of electron, hole and exciton, respectively;$ {E}_{\rm G} $ is the band gap; (c) electrostatic phase diagram in which p, n, i represent hole, electron doped and intrinsic layers, respectively; (d), (e) penetration capacitance (d) and interlayer capacitance (e) as a function of bias$ {V}_{\rm b} $ and gate$ {V}_{\rm g} $ ; the charge insulating region indicated by penetration capacitance (red in (d)) and electron-hole pair injectable region measured by interlayer capacitance (red in (e)) has an overlap, which is the triangle region surrounded by red and white dashes; that indicates the existing of excitonic insulator; (f) thermal-dynamic phase diagram of interlayer excitons in the system.
图 6 激子电输运实验示意图 (a)库仑拖拽实验; 单层TMD 与莫尔异质结晶格之间有2 nm 的氮化硼中间层, 单层TMD 和莫尔异质结分别处于不同回路, 在单层TMD 中施加电流I1, 测量在莫尔异质结中的拖拽电流I2, 这里莫尔异质结回路没有外加电流源, 激子形成宏观相干态后, I1 = I2, 并且回流实验(b)观测到没有耗散的层内电流 (
$ \Delta V=0 $ )Figure 6. Exciton transport experiment: (a) Coulomb drag measurement. Monolayer TMD and moiré structure are separated by a 2 nm hBN and they are in different circuits. A driving current I1 is in the monolayer TMD circuit and the drag current in moiré circuit can be measured, here moiré circuit is not connected with any external source. When exciton superfluidity forms, I1 = I2 and longitudinal voltage drop of counter-flow configuration shown in (b) should be ΔV = 0.
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