Pdf) 2015, 517, 64

Letter

Cite This: Nano Lett. 2019, 19, 5373−5379 pubs.acs.org/NanoLett

Exciton Gas Transport through Nanoconstrictions † † † † † Chao Xu,*, Jason R. Leonard, Chelsey J. Dorow, Leonid V. Butov, Michael M. Fogler, ‡ ‡ Dmitri E. Nikonov, and Ian A. Young † Department of Physics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093, United States ‡ Components Research, Intel Corporation, Hillsboro, Oregon 97124, United States

*S Supporting Information

ABSTRACT: An indirect exciton is a bound state of an electron and a hole in spatially separated layers. Two-dimensional indirect excitons can be created optically in heterostructures containing double quantum wells or atomically thin semiconductors. We study theoretically the transmission of such bosonic quasiparticles through nanoconstrictions. We show that the quantum transport phenomena, for example, conductance quantization, single-slit diffraction, two-slit interference, and the Talbot effect, are experimentally realizable in systems of indirect excitons. We discuss similarities and differences between these phenomena and their counter- parts in electronic devices.

KEYWORDS: Indirect excitons, quantum point contact, ballistic transport, split gate

ζ ζ ndirect excitons (IXs) in coupled quantum wells have electrochemical potentials, s and d [Figure 2(a-c)]. The ff ζ − ζ − I emerged as a new platform for investigating quantum di erence s d > 0 is analogous to the source drain voltage transport. The IXs have a long lifetime,1 long propagation in electronic devices. Whereas electrons are Fermions, IXs − distance,2 9 and long coherence length at temperatures T behave as bosons. This makes our transport problem unlike the 10 ff below the temperature T0 of quantum degeneracy (eq 7). electronic one. The problem is also di erent from the slit Although an IX is overall charge neutral, it can couple to an diffraction of photons or other bosons, such as phonons, electric field via its static dipole moment ed, where d is the considered so far. Indeed, one cannot apply a source−drain voltage to photons or phonons in any usual sense. (However, distance between the electron and hole layers (Figure 1a). − These properties enable experimentalists to study the transport quantized heat transport of phonons17 19 has been studied.) of quasi-equilibrium IX systems subject to artificial potentials To highlight the qualitative features, we do our numerical ζ controlled by external electrodes (Figure 1b). calculations for the case where the drain side is empty, d = In this Letter, we examine theoretically the transport of IXs −∞. The conductance of the QPC can be described by the through nanoconstrictions (Figure 1c−e).11 Historically, bosonic variant of the standard Landauer−Büttiker theory.20 It studies of transmission of particles through narrow con- predicts that the contribution of a given subband to the total strictions have led to many important discoveries. In particular, conductance can exceed N/h if its Bose−Einstein occupation investigations of electron transport through so-called quantum- factor is larger than unity.21 To the best of our knowledge, point contacts (QPCs) have revealed that at low T, the there have been no direct experimental probes of this conductance of smooth, electrostatically defined QPCs exhibits prediction in bosonic systems. The closest related experiment a step-like behavior as a function of their width.12,13 The steps is probably the study of 6Li atoms passing through a QPC in 22 appear in integer multiples of Ne2/h (except for the anomalous the regime of enhanced attractive interaction. That experi- first one)14 because the electron spectrum in the constriction ment has demonstrated the conductance exceeding N/h, and 23−25 region is quantized into one-dimensional (1D) subbands the theory has attributed this excess to the virtual pairing (Figure 2e), where N is the spin-valley degeneracy. Associated of Fermionic atoms into bosonic molecules by quantum with such subbands are current density profiles analogous to fluctuations. the diffraction patterns of light passing through a narrow slit. Below we present our theoretical results for the IX transport These patterns have been observed by nanoimaging of the through single, double, and multiple QPCs. We ignore electron flow.15 The conductance quantization has also been exciton−exciton interaction but comment on possible observed in another tunable Fermionic system, a cold gas of interaction effects at the end of the Letter. 6Li atoms.16 For the transport of bosonic IXs, we have in mind a Received: May 7, 2019 conventional in solid-state physics setup where the QPC is Revised: June 25, 2019 connected to the source and drain reservoirs of unequal Published: July 2, 2019

direction (Figure 1a), an IX experiences the energy shift U = − eEzd. This property makes it possible to create desired external potentials U(x,y) acting on the IXs. To engineer a QPC, U(x,y) needs to have a saddle-point shape. Such a potential can be created using a configuration of electrodes: a global bottom gate plus a few local gates on top of the device. As depicted in Figure 1b, two of such top electrodes (gray) can provide the lateral confinement, and another two (blue and orange) can control the potential at the source and the drain. More electrodes can be added if needed. Following previous work,11 in our numerical simulations we use a simple model for U(x,y): − −−−xa22/2 ( yya )22 /2 yy1 U(,xy )=− A (1 Cexy ) e0 + Uf sd F s (1) x −1 where f F(x)=(e +1) is the Fermi function.ji The widthzy ffi j fiz parameters ax and ay and the coe cients A and C kon the {rst line of eq 1 are tunable by the gate voltage Vg (Figure 1b). The Figure 1. (a) Schematic energy band diagrams of an atomically thin second line in eq 1 represents a gradual potential drop of electron−hole bilayer subject to an external electric field. The direct magnitude Usd, central coordinate y1, and a characteristic width exciton (DX) is indicated by the red oval, the indirect exciton (IX) by s in the y-direction. These parameters are controlled by the black one. The bars represent atomic planes. (b) Same as (a) for a voltages Vs and Vd. Examples of U(x,y) are shown in Figure spacer-free type II heterostructure. Various combinations of 2D 1c,d. materials can be used, examples include: For GaAs structures, GaAs For qualitative discussions, we also consider the model of a (green), AlAs (yellow), and AlGaAs (blue); for TMD type I fi quasi-1D channel of a long length Lc and a parabolic con ning structures, MoS2 (yellow) and hBN (blue); for TMD type II potential. The corresponding U(x,y) is obtained by replacing structures, WSe2 (green), MoSe2 (yellow), and hBN (blue). (c) the top line of eq 1 with Schematic of an IX QPC device. Voltage Vs (Vd) controls the electrochemical potential of the source (drain); Vg creates a potential yLy−−c yy − 1 constraining IXs from the sides. (d) Variation of the IX potential f 0 f 0 UUx+ 2 F s F s 0 2 xx along the line y = y0 across the QPC for ax = 180 and 400 nm (dashed (2) and solid trace, respectively) with other parameters as follows: A = 2 μ where Uxx = AC/ax. The energy subbands (Figure 2b) in such 4.30 meV, C =1,ay = 200 nm, Usd = 1.0 meV, y0 = 4.0 m, and y1 = ji zy i yi y μ a channelj are z j zj z 1.0 m; see eq 1. (e) False color map of U(x,y) for the larger ax in (d) j z j zj z . k { k {k 22 { 1 ℏ ky Ek()=+ℏ+ Uω j + ,j = 0, 1 , ... jy0 x 22m (3) The associated eigenstates are the products of the plane waves ikyy ji zy e and the harmonicj oscillatorz wave functions k { χ =−2 2 j ()xHj ( 2 xw /0 )exp( xw /0 ) (4)

where Hj(x) is the Hermite polynomial of degree j. The length ω w0 and frequency x are given by

2ℏ Uxx w0 = , ωx = mωx m (5) ℏω The subband energy spacing x can be estimated as 1/2 1/2 m0 AC nm ℏωx ≃ 8.8meV m meV ax (6)

which is approximately 0.1 meV for devicesi y made of transition- ji zy ji zy j z ff metal dichalcogenidesj z (TMDs),j z wherej excitonz e ective mass Figure 2. Energy spectra at the source (a, d), the QPC (b, e), and the j z m = 1.0m0 and ax k= 180{ nm.k Hence,{ k the onset{ of quantization drain (c, f). The electrochemical potential of the source is indicated occurs at T ∼ 1 K. The same characteristic energy scales in by the horizontal dashed line. The shading represents the occupation GaAs quantum well devices11 can be obtained in a wider QPC, factor. Panels (a−c) are for bosonic particles, such as IXs. Panels (d− f) are for Fermions, for example, electrons. ax = 400 nm, taking advantage of the lighter mass, m = 0.2m0, compare solid and dashed lines in Figure 1d. In both examples ℏω ∼ x is much smaller than the IX binding energy ( 300 meV in Model of a QPC. In the absence of external fields, the IXs TMDs26 and ∼4 meV in GaAs27,28). Therefore, we consider are free to move in a two-dimensional (2D) x−y plane. When the approximation ignoring internal dynamics of the IXs as fi 29 an external electric eld Ez = Ez(x,y) is applied in the z- they pass through the QPC and treat them as point-like

5374 DOI: 10.1021/acs.nanolett.9b01877 Nano Lett. 2019, 19, 5373−5379 Nano Letters Letter particles. Incidentally, we do not expect any significant 0 N EUj −−sd μ reduction of the exciton binding energy due to many-body G =−∑ θ()EUf0 s h j sd B T screening in the considered low-carrier-density regime in j (12) semiconductors with a sizable gap and no extrinsic doping. (At i y high carrier densities, the screening effect can be substantial.30) The dependence of G on Usdj is illustrated inz Figure 3a. The j z Conductance of a QPC. As with usual electronic devices, conductance exhibits asymmetricj peaks. Eachz peak signals the we imagine that our QPC is connected to a semi-infinite k { source and drain leads (labeled by l = s,d). Inside the leads the IX potential energy U(x,y) tends to asymptotic values Ul. Without loss of generality, we can take Ud =0,asineq 1. The ff − α − di erence Usd = Us Ud = (eVs eVd) is a linear function of ffi the control voltages Vs, Vd (Figure 1b). The coe cient of α ∼ proportionality can be estimated as d/dg, where dg is the vertical distance between the top and bottom electrodes. The IX energy dispersions at the source and the drain are parabolic and are shifted by Usd with respect to one another, see Figure 2a,c. In the experiment, the IX density of the reservoirs can be controlled by photoexcitation power. For example, the IX 11 density ns can be generated on the source side only, while the ≪ drain side can be left practically empty, nd ns. This is the μ case we focus on below. The chemical potentials l are related 31,32 to the densities nl via 2 − 2πℏ n μ =−Teln(1TT0/ ), T ≡l l 0 mN (7)

Figure 3. (a) Conductance of a bosonic QPC vs Usd from the Note that we count the chemical potentials from the minima of ℏω μ − ℏω the appropriate energy spectra and that we use the units system adiabatic approximation (eq 12) for T = 0.8 x, s = 0.05 x. ≡ Inset: Total current as a function of Usd. The notation used on the y- kB 1. In turn, the electrochemical potentials are given by ν ≡ ω π axis is x x/2 . (b) Conductance of a Fermionic QPC for T = ℏω μ ℏω 0.05 x, s = 2.7 x. Note that the number of conductance steps ζl =+μl Ul (8) occurring at U <=ℏE 0 1 ω is equal to μ /ℏω rounded up to the sd 0 2 x s x which implies nearest integer.

Usd=−()()ζζ s d −− μμsd (9) activation of a new conduction channel whenever U In the terminology of electron devices, the first term in eq 9 sd − ζ − ζ ≡ approaches the bottom of a particular 1D subband. All the is related to the source drain voltage Vsd, namely, s d fl − peaks have the same shape, which is the mirror-re ected Bose eVsd. The second term in eq 9 is referred to as the built-in function with a sharp cutoff.AsT decreases, the width of the potential, which is said to originate from charge redistribution Δ ∼ −μ peaks min(T, s) decreases. The magnitude of the peaks in the leads. −μ fi ζ − ζ Gmax =(N/h)f B( s/T) can be presented in the form: If the particle densities nl are xed, then variations of s d ff N rigidly track those of Usd. The di erential conductance can be G = f computed by taking the derivative of the source−drain particle max h max (13) current I with respect to Usd: where f max is the occupation of the lowest energy state at the source. For Fermions, f is limited by 1. For 2D bosons, f GIU=∂(/ ∂sd )nn , (10) max max s d T0/T − = e 1 exceeds 1, and, in turn, Gmax exceeds N/h at T < G has a dimension of 1/h and its natural quantum unit is N/h. T0/ln 2. 0 (The spin-valley degeneracy is N = 4 in both TMDs and The sudden drops of G at Usd = Ej (Figure 3a) occur GaAs.) because the current carried by each subband saturates as soon For the long-channel model (eq 2), G can be calculated as it becomes accessible to all the IXs injected from the source, analytically because the single-particle transmission coefficients down to the lowest energy Usd. These constant terms do not θ − 0 affect the differential conductance G. The total current as a through the QPC have the form tj(E)= (E Ej ), that is, they 0 ≡ θ function of U is plotted in the inset of of Figure 3a. take values of either 0 or 1. Here Ej Ej(0), and (x) is the sd Heaviside step-function. Adopting the standard Landauer− It is again instructive to compare these results with the more Büttiker theory for Fermions20 to the present case of bosons familiar ones for Fermions, which are obtained replacing the fi ∑ − Bose function f with the Fermi function f in eq 12. In the (see also ref 21), we nd the total current to be I = j(Is,j B F Id,j), where the partial currents are regime where Usd is large and negative, the Heaviside functions in eq 12 play no role, so that G traces the expected quantized N ∞ E − ζ staircase shown in Figure 3b. The conductance steps occur I = f a tEdE() aj, ∫ B j whenever U = E0 − μ . Once U approaches E0 = ℏω /2, a h Ua T (11) sd j s sd 0 x different behavior is found: The conductance displays and f (x)=(ex − 1)−1 is the Bose function. In turn, the additional sudden drops at U = E0, which causes it to B ji zy sd j conductance is j z oscillate between two quantized values. These drops appear for k { 5375 DOI: 10.1021/acs.nanolett.9b01877 Nano Lett. 2019, 19, 5373−5379 Nano Letters Letter

ff − − 9 Figure 4. IX density distribution in a QPC for three di erent Usd. Panels (a c) depict the density distribution n in the x y plane in units of 10 −2 − fi μ cm . Panels (d f) show the pro les of n along the midline, y =1 m, of the region plotted directly above. Values of Usd are adjusted to highlight ℏω the contributions of particular subbands: (a, d) j = 0 for Usd = 0.52, (b, e) j = 1 for Usd = 1.52, (c, f) j = 2 for Usd = 2.52, all in units of x = 0.10 meV; T = 1.0 K everywhere. ′ the same reason as in the bosonic case: current saturation for while the x momentum kx remains the same. Subsequently, fi 0 each subband that satis es the condition Ej < Usd. the incident wave impinges on the QPC at y = y0. Typically, The adiabatic QPC model considered above is often a good this causes a strong reflection back to the source. However, approximation12,13 for more realistic models, such as eq 1. The certain eigenstates have a non-negligible transmission. After latter cannot be treated analytically but we were able to passing the QPC, their wave functions expand laterally with the 33 ≪ compute G numerically, using the transfer matrix method characteristic angular divergence of 1/(kyw0) 1. Such wave (see below). The results are presented in the Supporting functions can be factorized ψ(,xy )= eiky yχ (, xy ), where the Information. The main difference from the adiabatic case is slowly varying amplitude χ(x,y) obeys the eikonal (or paraxial) that the conductance peaks are reduced in magnitude and equation: broadened. 2 ℏk Density Distribution in a QPC. In analogy to experiments ℏ 2 y 15 −ℏivyy ∂ − ∂+ xUvχ =0, y ≡ with electronic QPCs, it would be interesting to study 2m m (15) quantized conduction channels by the optical imaging of IX fl If the model of the long constriction (eq 2) is a good ow. Unlike electrons, IXs can recombine and emit light, so ji zy approximation,j the solutionz is as follows. Inside the QPC, χ(r) that the IX photoluminescence can be used for measuring their j z χ current paths. This motivates us to model the density is proportionalk to a particular{ oscillator wave function j(x)(eq distribution n(x,y) of IXs in the QPC. We begin with analytical 4). Outside the QPC, it behaves as a Hermite-Gaussian beam considerations and then present our numerical results (Figure whose probability density can be written in the scaling form 4). 221 x The calculation of the n(x,y) involves two steps. First, we |χχ(,xy )|= by()j by () solve the Schrödinger equation for the single-particle energies (16) E = ℏ2k′2/(2m) and the wave functions ψ(x,y) of IXs subject ()yy− ji 2 zy to the potential U(x,y). The boundary conditions for the =+ j0 z ψ by() 1 2j z (x,y) is to approach linear combinations of plane waves of w k { ′ ′ ± ′ → −∞ y (17) momenta k =(kx , ky )aty but contain no waves of momenta k″ =(k ″, k ″) with Re k ″ <0aty→ +∞. Such 1 x y y w = kw2 states correspond to waves incident from the source. Second, yy2 0 (18) − ζ |ψ |2 we sum the products f B(E s) (x,y) to obtain the n(x,y). − We choose not to multiply the result by N, so that our n(x,y)is This representation has been used in the study of a Bose Einstein condensate (BEC) expansion from a harmonic the IX density per spin per valley. 34,35 For a qualitative discussion, let us concentrate on the trap. Our problem of the steady-state 2D transport ′ through the QPC maps to the non-interacting limit of this simplest case of large ky , that is, fast particles. As the incident − wave propagates from the source lead, it first meets the smooth problem in 1 + 1D, with t =(y y0)/vy playing the role of potential drop U at y = y . Assuming k ′s ≫ 1, the over-the- time. In other words, the IX current emerging from the QPC is sd 1 y mathematically similar to a freely expanding BEC. This barrier reflection at y = y can be neglected. Therefore, the y 1 mapping is accurate if the characteristic width of the energy momentum increases to the value k dictated by the energy y distribution of the IX injected into the QPC from the source is conservation: small enough, Δ ≪ ℏω. Dimensionless function b(y) has the meaning of the expansion factor. A salient feature of the kk22= ′ +ℏ(2 mU / 2 ) yy sd (14) eigenfunctions is the nodal lines χ(x,y) = 0. The lowest

5376 DOI: 10.1021/acs.nanolett.9b01877 Nano Lett. 2019, 19, 5373−5379 Nano Letters Letter subband j = 0 has no such lines, whereas the higher subbands have exactly j = 1, 2, ... of them. For a quantitative modeling, we carried out simulations using the transfer matrix method.33 This method gives a numerical solution of the Schrödinger equation discretized on a finite-size real-space grid for a given energy E and the boundary conditions described above. To obtain the total particle density, we summed the contributions of individual states, making sure to include enough E’stoachieve convergence. In all of our calculations, the chemical potential μ fi s was xed to produce the density (per spin per valley) ns = 1.0 × 1010 cm−2 at the source. Examples of such calculations for temperature T = 1.0 K are shown in Figure 4. Since the partial densities |ψ(x,y)|2 are weighted with the Bose−Einstein − ζ factor f B(E s), the lowest-energy j = 0 subband typically dominates the total density, making it look like a nodeless Gaussian beam (Figure 4a,d). However, if Usd is tuned slightly above the bottom of the j = 1 subband, the contribution of this subband is greatly enhanced by the van Hove singularity of the πℏ ∝ 1D density of states N/( vy) (inside the QPC. As a result, n(x,y) develops a valley line (local minimum) at x = 0, which is χ the nodal line of function 1(x/b,y)(Figure 4b,e). A similar phenomenon occurs when we tune Usd to slightly above the bottom of j = 2 subband. Here the density exhibits two valley lines, which approximately follow the nodal lines of function χ 2(x/b,y)(Figure 4c,f). These theoretical predictions may be tested by imaging IX emission with high enough optical resolution. Note that these van Hove singularities do not ff Figure 5. Double-QPC IX device. (a) The profile of the IX potential enhance the di erential conductivity G because of the across the two constrictions. (b) The IX density in units of 108 cm−2. cancellation between the density of states and the particle (c) The IX density along the line y =2μm in (b). The ripples at the velocity vy. As explained above, this leads to current saturation flanks of the fringes are finite-size artifacts. and thus a negligible contribution of jth subband to G at Usd > 0 Ej . Double and Multiple QPC Devices. We have next fi eld optical techniques. Note that for a grating with Ns slits, explored the exciton analog of the Young double-slit the crossover to the far-field diffraction occurs at the distance fi interference. A short distance downstream from the rst ∼N y , which is prohibitively large for the transfer matrix μ s T QPC (1.0 m along y), we added another potential barrier simulations. For a qualitative illustration of this crossover, we constructed from two copies of the single-QPC potential (eq computed the interference pattern simply by adding a number ’ fi 1) shifted laterally in x. As in Young s classical setup, the rst of Gaussian beams; see Figure 6b,c. QPC plays the role of a coherent source for the double-QPC. Discussion and Outlook. In this Letter, we considered a An example of the latter with the center-to-center separation of few prototypical examples of mesoscopic IX phenomena. We 800 nm is shown in Figure 5a. We computed the IX density analyzed the subband quantization of IX transport through a fl distribution in this system for Usd at which the IXs uxes single QPC, the double-slit interference from two QPCs, and through all of the QPCs that are dominated by the j =0 the Talbot effect in multiple QPCs. As for electrons, these subband. The IXs transmitted through the double-QPC create phenomena should be experimentally observable at low distinct interference fringes (Figure 5b,c). enough temperatures. The present theory may be straightfor- Finally, we considered a four-QPC array. To show the wardly expanded to more complicated potential landscapes results more clearly, we simulated the zero-temperature limit, and structures. where a single energy E contributes. As illustrated by Figure 6a, In the present work, we neglected IX interactions. This ff the interference pattern begins to resemble that of a di raction interaction is dipolar, which in 2D is classified as short-range, grating. Near the QPCs, it exhibits the periodic refocusing parametrized by a certain interaction constant g. As mentioned known as the Talbot effect. The repeat distance for the 36 above, the problem of the IX transport through a QPC is complete refocusing is closely related to the problem of a BEC expansion from a 34 λ harmonic trap. Following the studies of the latter, interaction y = fi − T 2 can be included at the mean- eld level by using the Gross 11−−λ Pitaevskii equation instead of the Schrödinger one. For the a2 (19) single-QPC case, we expect the GPE solution to show a faster λ π “ ” where =2 /ky is the de Broglie wavelength of the IXs and a lateral expansion of the exciton jet , that is, a more rapid is the distance between the QPCs. For λ = 100 nm and a = growth of function b(y). For the double- and multi-QPC cases, ≃ μ 1000 nm, this distance yT 20 m is beyond the range plotted we expect repulsive interaction to suppress the interference in Figure 6a. Therefore, only the so-called fractional Talbot fringes, similar to what is observed in experiments with cold effect is seen in that figure. Although too fine for conventional atoms.38 The line shape and amplitude (visibility) of the imaging, these features may in principle be resolved by near- fringes remain to be investigated.

5377 DOI: 10.1021/acs.nanolett.9b01877 Nano Lett. 2019, 19, 5373−5379 Nano Letters Letter ■ AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. ORCID Chao Xu: 0000-0003-3834-1452 Notes The authors declare no competing ﬁnancial interest.

■ ACKNOWLEDGMENTS The work at UCSD is supported by the National Science Foundation under grant ECCS-1640173 and by the NERC, a subsidiary of Semiconductor Research Corporation through the SRC-NRI Center for Excitonic Devices.

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5378 DOI: 10.1021/acs.nanolett.9b01877 Nano Lett. 2019, 19, 5373−5379 Nano Letters Letter

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5379 DOI: 10.1021/acs.nanolett.9b01877 Nano Lett. 2019, 19, 5373−5379 Supporting Information for Exciton Gas Transport Through Nano-Constrictions

Chao Xu,˚,: Jason R. Leonard,: Chelsey J. Dorow,: Leonid V. Butov,: Michael M. Fogler,: Dmitri E. Nikonov,; and Ian A. Young;

:Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA ;Components Research, Intel Corporation, Hillsboro, Oregon 97124, USA

Electrostatic simulations Here H = 0.66 nm/d = 0.50 is the fraction of the distance d occupied by the hBN spacer. We found that To better assess the practicality of the proposed IX de- a reasonably close fit to U(x, y) plotted by the dashed vices, we numerically modeled several systems of the line in Fig. 1(d) of the main text can be achieved if the type shown in Fig. 1(a) of the main text. The first exam- width of the side gates is 100 nm, the width of the slits ple we considered is a TMD heterostructure with the is 100 nm as well, and the width of the neck of the following layer sequence. First, there is a thin crystal central gate is 500 nm. Such dimensions are attainable of hBN, which serves as both the encapsulating layer with standard nanofabrication methods. The applied and the gate dielectric. Next, there are MoS2 monolay- voltages need to be V = 4.1 V on the central gate and ers with two monolayers of hBN in between. Finally, d Vg = 3.8 V on the side gates. there is a thicker hBN capping layer. Assuming the ef- The second example we studied is an GaAs/AlGaAs d fective electron-hole separation is equal to the center- heterostructure composed of 100 nm of undoped Al- to-center distance of the two MoS2 layers and using GaAs, followed by two 8-nm-thick GaAs quantum 0.33 nm and 0.65 nm for the inter-layer spacing of, re- wells separated by a 4-nm-thick AlGaAs barrier, d = spectively, hBN and MoS2, we get 1.31 nm in this capped by another undoped AlGaAs layer to the total design. We took the thickness of the lower hBN layer to thickness of dg = 500 nm. The center-to-center quan- be 10 nm, corresponding to about 30 atomic layers, and tum well distance in this case is d = 12 nm. The usual d = the total thickness of the structure to be g 175 nm. choice of the bottom electrode is an n+-GaAs substrate. The structure is assumed to reside on a ground plane, We treated both GaAs and AlGaAs as isotropic mate- e.g., a graphite substrate, and to possess a set of top rials with identical permittivity of e = 12.5. We found gates depicted in Fig. 1(c) of the main text. Specifically, that the potential shown by the solid line in Fig. 1(d) two symmetrically placed slits shaped as square hair- can be realized for the following electrode dimensions: Ţ Ť pins ( and ), divide the top plane into two rectangu- width 300 nm, slit width 200 nm, central neck width lar side gates maintained at voltage Vg plus a narrow- 1100 nm. The applied voltages are Vd = 2.00 V and necked central gate kept at voltage Vd. Vg = 1.85 V. Realizing this design again appears to be To compute the electrostatic potential distribution a straightforward task. in this system we used a commercial finite-element 1 Finally, we simulated the potential produced in the solver. Note that both hBN and MoS2 are materi- same GaAs/AlGaAs heterostructure by a different top 2 e = als with anisotropic permittivities x-y, hBN 6.71, electrode pattern, in which the slits delineating the side e = 3.56 and e = 14.29, e = 6.87. To z, hBN x-y, MoS2 z, MoS2 electrodes are curved, Fig. S1(a). The computed U(x, y) simplify the simulations, we assigned MoS2 the same is plotted in Fig. S1(b). This design gives a better ap- permittivities as hBN. In doing so, neglecting the dif- proximation to an ideal adiabatic QPC, see below. ference of the in-plane permittivities is justified because the IX potential U(x, y) is determined by the Ez- field. However, it is necessary to account for the dif- Derivation of Eq. (12) ference of the z-axis permittivities, which we did by The quasiparticle velocity of jth subband is multiplying the calculated U(x, y) by the factor

1 dEj ez, hBN H + (1 ´ H) = 1.46 . (S1) vj(ky) = . (S2) h¯ dky ez, MoS2

1 Therefore, the current injected by the source into the subband j is (a) 4 8 dky Is,j = N tj(Ej) fB,F(Ej ´ ζs)θ(Ej ´ Usd)vj(ky) 2 2π ż0 8 N 0 = t (E) f (E ´ ζ )dE , (S3) h j B,F s Uż -2 sd which agrees with Eq. (11) of the main text. Here B, F -4 label the bosonic and fermionic cases, respectively. The -4 -2 0 2 4 current Id,j injected by the drain is computed similarly. For an adiabatic QPC with the transmission coefﬁcients (b) t (E) = θ(E ´ E0), the net current I = I ´ I of jth 4 j j j s,j d,j subband is 4 8 2 N 3 I = θ(E ´ E0) f (E ´ U ´ µ )dE j h j F,B sd s 0 Uż 2 sd (S4) 8 N -2 ´ θ(E ´ E0) f (E ´ µ )dE . 1 h j F,B d ż0 -4 0 -4 -2 0 2 4 Accordingly, the differential conductance deﬁned by Eq. (10) of the main text can be written as

Figure S1. An example of simulated QPC devices. (a) top gate ge- G = BI /BU = (N/h) g , (S5) ometry (b) IX potential U(x, y). In the proposed GaAs device (see j sd j j j text), this potential is realized for applied voltages Vd = 2.00 V and ÿ ÿ Vg = 1.85 V. where

0 0 gj = θ(Ej ´ Usd) fF,B(Ej ´ Usd ´ µs) (S6)

is the dimensionless conductance of subband j. This quantity depends on Usd the same way as the mirror- reﬂected quasiparticle occupation factor fB,F until Usd (a) � 0 reaches Ej , at which point gj drops to zero, see Fig. S3. As explained in the main text, the reason for this cut- �/ − 1 off behavior is that the current contributed by channel j remains constant if Usd increases past the subband 0 bottom Ej . Therefore, it does not affect the differential conductance. However, if the QPC is not perfectly adiabatic, the transmission coefﬁcient tj(E) has a grad- 0 � − Δ � � ual rather than the sharp onset at Ej , and so the cutoff 0 (b) of gj at Usd = Ej is also gradual (see below). Finally, � summing over all the subbands, we obtain the total dif- 1 ferential conductance shown in Fig. 3 of the main text.

Transport simulations � − � � � To calculate the conductance of a bosonic QPC, we 3 Figure S2. Dimensionless differential conductance of a single sub- used the transfer matrix method. In this method the band j of a long (adiabatic) QPC: (a) bosonic case, ∆ = min(T, ´µs). 2D Schrödinger equation with energy E and potential (b) fermionic case. U(x, y) in the simulation region ´L/2 ă x, y ă L/2 is represented by a tight-binding model on a square lattice with a suitably small lattice constant l. We take L = 8 µm and l = 20 nm, giving us a simulation grid

2 with Nx = 400 points along the x-axis. The boundary is indeed transformed to Eq. (S3) in this limit. The conditions for the Schrödinger equation correspond to corresponding analytical results for the current and semi-infinite leads inside which the particles are sub- conductance as functions of Usd are plotted in Fig. S3 ject to y-independent potentials by the thick lines. Shown in the same Figure by the dashed lines are our numerical results for the poten- Us(x) = U(x, ´L/2) , Ud(x) = U(x, +L/2) , (S7) tial of Fig. (S1)(b). They prove to be fairly close to the adiabatic limit except for some rounding of the conduc- for the source and the drain, respectively. The energy tance peaks and the current steps, as expected. On the spectrum of the source lead consists of Nx subbands other hand, the results for the potential U(x, y) given s + s with dispersion Ei Ek. Here Ei are the quantized en- by Eq. (1) of the main text, which are indicated by the ergies of motion in the potential Us(x), thin lines, deviate more from the ideal case. h¯ 2 Ek = (1 ´ cos kl), (S8) ml2 15 8 is the kinetic energy of the y-motion, and k is the y-axis momentum. The y-axis velocity is 6 10 4 2 1 dEk 2Ek ml ui(Ek) = = 1 ´ Ek . (S9) 2 h¯ dk d m 2¯h2 5 0 Similar energy spectrum exists in the drain. The trans- 0 1 2 3 fer matrix algorithm computes the Nx ˆ Nx matrix of the transmission coefficients tij(E) between every sub- 0 band i of the source and every subband j of the drain. -3 -2 -1 0 1 2 3 To reduce the computation load, we did not explicitly include the potential drop Usd in U(x, y). As explained Figure S3. Conductance of a bosonic QPC as a function of electro- in the main text, this drop simply changes the particle static potential energy difference between the source and the QPC ˜ velocity. We accounted for it by including the ratio of center Usd ” Usd ´ U(0, y0). The three lines correspond to three velocities before and after the drop as a multiplicative models considered: an ideal adiabatic constriction (thick line), a QPC with the potential of Fig. S1 (dashed line), a QPC defined by Eq. (1) factor. Accordingly, our formula for the total current of the main text (thin line). Inset: the total current vs. U˜ sd in these injected into the QPC by the subband i of the source is three models. Parameters: T = 0.8¯hωx, µs = ´0.05¯hωx.

1 ui(Ek) s Ii = fB(Ek + Ei ´ µs) L ui(Ek + Usd) ÿk (S10) s ˆ tij(Ek + Ei + Usd)uj(Ek + Usd) . ÿj References The k-grid in the summation must be dense enough to (1) COMSOL Multiphysicsr v. 4.0. http://www. achieve accuracy. We found the following choice ade- comsol.com. quate: (2) Fogler, M. M.; Butov, L. V.; Novoselov, K. S. High- 2πn L temperature superfluidity with indirect excitons in k = , n = 0, 1, . . . , n = . (S11) L max 4l van der Waals heterostructures. Nature Communi- cations 2014, 5, 4555. The net current through the QPC for the case of empty drain is simply i Ii. (3) Usuki, T.; Saito, M.; Takatsu, M.; Kiehl, R. A.; For consistency check, consider first an ideal adi- ř Yokoyama, N. Numerical analysis of ballistic- abatic QPC, where the transmission coefficients are electron transport in magnetic fields by using a 0 tij(E) = δijθ(E ´ Ej ). The rule for changing variables quantum point contact and a quantum wire. Physi- from k to ky is found from the energy conservation cal Review B 1995, 52, 8244–8255.

s Ej(ky) = Ek + Ei + Usd , (S12) which entails

1 dky vj(E) ... Ñ . . . (S13) L 2π ui(Ek) ÿk ż Using these expressions, it is easy to see that Eq. (S10)