Quantum transport simulation of exciton condensate transport ... - arXiv

Quantum transport simulation of exciton condensate transport physics in a double layer graphene system Xuehao Mou,1 Leonard F. Register,1 Allan H. MacDonald,2 and Sanjay K. Banerjee1 1

Department of Electrical and Computer Engineering and Microelectronics Research Center, The University of Texas at Austin, Austin, Texas 78758, United States

2

Department of Physics, The University of Texas at Austin, Austin, Texas 78712, United States

Abstract—Spatially indirect electron-hole exciton condensates stabilized by interlayer Fock exchange interactions have been predicted in systems containing a pair of two-dimensional semiconductor or semimetal layers separated by a thin tunnel dielectric. The layer degree of freedom in these systems can be described as a pseudospin. Condensation is then analogous to ferromagnetism, and the interplay between collective and quasiparticle contributions to transport is analogous to phenomena that are heavily studied in spintronics. These phenomena are the basis for pseudospintronic device proposals based on possible low-voltage switching between high (nearly shorted) and low interlayer conductance states and on near perfect Coulomb drag-counterflow current along the layers. In this work, a quantum transport simulator incorporating a non-local Fock exchange interaction is presented, and used to model the essential transport physics in the bilayer graphene system. Finite size effects, Coulomb drag-counterflow current, critical interlayer currents beyond which interlayer DC conductance collapses at sub-thermal voltages, non-local coupling between interlayer critical currents in multiple lead devices, and an Andreev-like reflection process are illustrated.

I. INTRODUCTION Coherent pairing of electrons and holes localized in separate III-V semiconductor layers has been observed at high magnetic fields and cryogenic temperatures in systems containing a pair of quantum wells separated by tunnel barriers (where holes are defined within as empty states in partially empty Landau levels).1-4 In these spatially indirect exciton condensates, interlayer coherence is mediated by the interlayer Fock exchange interactions.3 The consequences of condensate formation that have been studied experimentally include novel transport effects such as near perfect Coulomb drag-counterflow currents, and greatly enhanced interlayer conductance up to a critical current Ic and a corresponding critical interlayer voltage Vc beyond which the DC current collapses,5 the latter effect being partially analogous to transport phenomena in Josephson junctions. In this paper, we present a theoretical quantum transport simulation study of closely related phenomena in the absence of a magnetic field. Although spatially indirect exciton condensation has not yet been observed in this regime, it is expected to appear when nesting is established between electron- and hole-like Fermi surfaces of two-dimensional semiconductors, gapless semiconductors, or semimetals. The prospects for observing spatially indirect exciton condensation are therefore being enhanced by progress in fabricating and studying twodimensional materials. In this study we choose for the sake of definiteness to study graphene bilayers, but our conclusions apply equally well to other systems.

First successfully isolated about a decade ago, graphene already has exhibited a broad set of novel phenomena of interest to researchers from multiple disciplines, such as the half-integer quantum Hall effect and the related Berry’s phase that had been predicted theoretically by physicists.6,7 Its unique symmetric and conical band structure has allowed research on the properties of massless Dirac Fermions, including novel transport properties in semiconductor device physics. When separated by dielectric tunnel barriers such has hexagonal boron nitride, it is possible to take advantage of the symmetry between graphene’s electron and hole band structures to establish the Fermi surface nesting conditions that favor interlayer exciton condensates.8,9 Systems based on monolayer or few-layer transition metal dichalcogenide (TMD) semiconductors have extremely large exciton binding energies and may provide an even better platform for such condensates. The prospect of transport effects analogous to those in already studied in III-V quantum well systems, but absent magnetic fields and at higher temperatures9,10 perhaps even room temperature, have led to novel “beyond CMOS” device proposals with switching energies potentially of only a few tens of zepto-joules, including the Bilayer pseudoSpin Field-Effect Transistor (BiSFET)11-14 and, more recently, the Bilayer pseudoSpin Junction Transistor (BiSJT).15 Studies of the transport properties of systems containing a spatially indirect exciton condensate require simulators that incorporate the non-local Fock exchange interaction. We describe such a simulator in detail, and use it to model essential transport physics including finite size effects, critical currents at sub-thermal voltages, Coulomb dragcounterflow current, non-local coupling between critical currents in multi-terminal devices, and a reflection process for this system akin to the Andreev reflection at the boundary of a conventional superfluid. Although the simulation techniques described here should be transferable to TMD based system, or to other two-dimensional semiconductors or semimetals, simulations might in some cases have significantly greater computational costs, depending on the complexity of the electronic structure. The purpose of this work is to model essential transport properties in the presence of the exciton condensates, including but not limited to those serving as the basis for BiSFET and BiSJT. The challenges which must be met to achieve such condensates have been and are being addressed elsewhere.8-10,16 This work is intended to motivate such work, and to help with the interpretation and design of experiments as well as potential devices. Portions of this work were reported previously in conference publications.13,17,18 Here, more details of the method, more complete sets of data, a more comprehensive and synergistic analysis thereof, and a more thorough discussion of the underlying essential physics including interpretations in terms of electrons, holes, and excitons are provided. The quantum transport simulator and the system model used here are discussed in Section II; various transport properties and discussions of underlying essential physics are addressed in Section III. II. TIGHT-BINDING HARTREE-FOCK MODEL Although we discuss electron-hole interactions, transport simulations are performed entirely in terms of electron states within the conduction and valence bands of opposite layers. The Hartree-Fock approximation is used to model the

exchange interaction between electrons (due to their indistinguishability and odd parity) within a single-particle framework. The resulting time-independent Schrödinger’s equation for single-electron energy eigenstates {φβ(r), Eβ}, neglecting for the moment the external potential profile, takes the form:



2 2  φ  r   φ  r  dr VC r, r  r , r    dr VC r, r  r , r φ  r   E  φ  r  , 2m

(1a)

where ρ(r,rꞌ) is the density matrix,

 r, r   f     r  *  r .

(1b)



The second term on the left of Eq. (1a) is the local Hartree potential term describing the classical electrostatic potential at r due to a charge distribution −qρ(r) = −qρ(r,r), to which we can readily add the contributions of externally applied electrostatic fields. The third term is the non-local and purely quantum mechanical Fock exchange interaction, which lowers the electron-electron interaction energy due to tendency of the odd-parity electrons to be further away from each other than expected classically.19 The indices β (βꞌ) label the single electron states and the fβ (fβꞌ) are their occupation probabilities. VC(r,rꞌ) is the Coulomb interaction energy between electrons at positions r and rꞌ. (Notably, the contributions to the Hartree and Fock terms for β = βꞌ cancel, demonstrating that Eq. (1) avoids interaction between any electron and itself.) For transport simulations, we switch to an atomistic tight-binding approximation considering in this work just one 2pz (π) orbital per carbon atom site located at discrete positions R. (More general models are also described by the equations that follow if the indices R are considered to also represent the various orbitals on each atom.) Within this tight-binding framework, Eq. (1) can be written in the following form20:

H TB φ R T   VH R T φ R T    VF R T , R B φ R B   E φ R T  ,

(2a)

RB

VF R T , R B   VC R T , R B  R T , R B  ,

(2b)

 R T , R B    f  '  ' R T  * ' R B  ,

(2c)

'

for the top (T) layer. The form for the bottom (B) layer is obtained by switching the “T” and “B” indices. We have explicitly considered only the interlayer Fock exchange interaction, labeled VF, which is the basis for exciton condensation. VH is the nominal Hartree potential energy, which in principle incorporates contributions due to charges on either layer, any external charge (externally applied fields), and, here, any uniform correction to the energy due to intralayer Fock exchange interactions. HTB includes the intra-layer and (much weaker) interlayer “bare” tight-binding hopping energies. Note that VF(RT,RB) can become complex under non-equilibrium conditions through the density matrix ρ(RT,RB).

FIG. 1. Simulaated structure. Two T graphene layers l (infinite in i the y directioon) are coupled via bare and Foock exchange innteractions in a channel of lenggth L and with interlayer i separration d across a tunnel dielectrric. The effectivve dielectric constant εr represents the dielectric envirronment betweeen and, still morre importantly, above and beloow the graphenee layers as a whhole. The perioddicity in the y direction is reppresented by thee shaded stripes along the transsport direction encompassing e one o armchair chhain of graphenne atom per layer. Atoms w within the armchhair chain are laabeled by conseecutive integerss, starting from 1 on the left off the channel and ending with N on the right. Two consecutiv ve atomic sites within one arm mchair chain of atoms and withhin the same lay yer correspond to t a primitive m and laabeled TL (top left), BL unit cell of monnolayer graphenne. Four semi-infinite, perfect absorbing/injeccting leads are modeled, (bottom left), T TR (top right) an nd BR (bottom right), which conceptually c (thhey do appear only o implicitly in the boundaryy conditions) incorporate atooms 0, −1, −2, … on the left annd N+1, N+2, N+3, N … on the right r within a laayer.

Fig. 1 illustrates the fo our-terminal structure s usedd in the simulaator developed in Ref. 17, where two grraphene sheets are separatedd by a distannce d throughh an interlayeer/tunnel dieleectric and co oupled via thee quasi-singlee-particle baree coupling and the many-body Fock exchhange interaction, both withhin a channel of length L. The dielectricc properties of o the environm ment, including g the interlayyer dielectric,, the more im mportant dieleectrics above and below thhe condensatee region, and frree-carrier scrreening, are alll representedd very roughlyy by a single effective e dieleectric constannt εr, given ou ur focus on essential transportt physics in thhe presence off the exciton condensate c and not on the conditions c required to createe the condensatte. Beyond thee channel, thee interlayer Foock exchange interaction is taken to be zero, z qualitativvely consistennt with a designned increase inn the interlayyer separation or in the perrmittivity of th he dielectric environment e t localize thee to condensate. The T width off the graphenne sheets perppendicular to the transporrt direction arre taken to be b infinite fo or simulation puurposes. For specificity, s thee graphene is oriented suchh that there iss an armchair pattern of atooms along thee transport (x) direction. Thee pattern of bare b interlayerr tunnel-coupling through the dielectric could be quiite complex inn principle (we do not speciffy a particular dielectric herre), and will not in general correspond c to the “A-B” cooupling pattern n between A annd B sublattiice atoms of opposite grapphene layers as in a Bernaal stack. Morreover, given a mixed baree interlayer couupling pattern n, bulk calculations have shown s that the coherent innterlayer excitton condensatte would mosst readily couplee to the A-A (or, ( equivalenttly, B-B) com mponent of thee bare interlayer coupling paattern.21 Thereefore, we havee taken the bare interlayer coupling to bee of the latter A-A form. Thhe carriers in both layers are a conceptuallly created viaa external gatess that are om mitted in Fig. 1. For simulaation purposes, the requireed carrier conncentrations arre adjusted byy applying fixed electrostaticc potential eneergies V(RT) = VT = −V(RB) = −VB = −V Vdiff/2, with thee energy refereence being thee equilibrium ((no voltages applied) cheemical potential (Fermi leevel). Conseq quently, the nominal n electtron and holee concentrationns are equal given g the sym mmetric band structure s of graphene. g For all simulation ns, the variatiions in carrieer

concentration with applied voltages are negligible compared to the nominal concentrations, and hence only the Fock interaction, VF(RT,RB), is considered self-consistently. For this transport system, the single-electron state label β = {γ,σ,kx,i,ky,s} identifies the lead of injection γ, the energy “valley” σ, the incident crystal momentum ћkx,i of the injected wave, the transverse mode of well-defined crystal momentum ћky, and (real) spin s. The occupation probability fβ is determined by its energy and the voltage applied to the lead in which it originates, Vγ = −μγ/q, where μγ is the chemical potential (Fermi level) of the lead γ. As shown by the stripes in the top and bottom layer oriented parallel to the transport direction in Fig. 1, the simulated structure is periodic in the transverse (y) direction with a lattice constant of ay encompassing a single armchair pattern of carbon atoms with atomic locations within each layer ΓT(B), a subset of RT(B). (Note while we assumed A-A coupling here, in general there would be no requirement that the atoms on the top and bottom layers be vertically aligned for this purpose.) Therefore, the Fock exchange interaction term can be rewritten for, e.g., any ΓT as

V Γ F

T

RB

  ΓB

, R B φ R B    ΓB

 V Γ  

C



  VC,k ΓB

'

y

T

 , Γ B  a y y

V Γ  

 

 f '

C

T

 ' φ '



 

   , Γ B  a y y  Γ T , Γ B  a y y φ Γ B  a y y

Γ T φ* ' Γ B e ika φ Γ B eik a y

y

y



y

* k  Γ T , Γ B  f  ' φ ' Γ T φ ' Γ B φ Γ B 

,

(3a)

y

  VF,k Γ T , Γ B φ Γ B  ΓB

y

where VC,k

Γ T , Γ B     VC ΓT , Γ B  a y y ei ( k k )a 

y k y

y

y

y

(3b)

 

and

VF,k Γ T , Γ B   VC,k y

'

y  k y

Γ T , Γ B  f  'φ ' ΓT φ* ' Γ B  .

(3c)

Here, ŷ is a unit vector in the y direction and η is an integer. Moreover, absent explicit free-carrier screening as modeled here, the Coulomb interaction potential energy is just





 VC Γ T , Γ B  a y y 

q2

 , 4 r Γ T  (Γ B  a y y )

(4)

which allows the VC,ky−k′y(ΓT,ΓB) to be pre-calculated and stored. Using Eq. (3), Eq. (2) can be rewritten in the quasi-onedimensional (quasi-1D) form

H TBφ Γ T   VH Γ T φ Γ T   VF,k Γ T , Γ B φ Γ B   E φ Γ T  ΓB

y

(5)

for the top layer. A similar expression can be obtained for the bottom layer. For each β—suppressing the indices for notational convenience now—Eq. (5) can be written in matrix form including both top and bottom layers as

E m   H m,n  n  0 .

(6a)

n

Here, m(n) label individual atomic slices along the transport direction, i.e., pairs of atoms, one in the top layer and one in the bottom, along the shaded armchair atomic chains in the transport direction in Fig. 1. The φm(n) are the corresponding tight-binding 2×1 matrices/column vectors for each slice m(n) with components φT,m(n) and φB,m(n). (Here we have assumed that there is nothing to induce coupling between orthogonal spin states in this graphene-based system, so that we may treat each spin state separately, unlike what would be the case for TMDs.) The Hm,n are 2×2 tight-binding potential matrices coupling slices m and n via both the bare and Fock interactions of Eq. (5). Note that there are only nonzero onsite and nearest-neighbor intra-layer and interlayer bare coupling interactions, while the interlayer Fock interactions extend among all sites within the channel but do not, in this model, couple to points beyond the channel. The boundary conditions at the ends of the channel are obtained by assuming that the four leads, TL, BL, TR and BR, are semi-infinite and perfectly absorbing. Labeling the atomic sites within the channel from m = 1 to N, the incident (i), if any, and “outgoing” (o) (including outwardly evanescing or reflected into the lead of incidence) components of the wavefunctions are related across the simulation region boundaries—between slices 0 and 1 on the left boundary for example—by

 0   0;i   0;o   0;i  P1,0;o 1;o   0;i  P1,0;o 1  1;i   P1,0;o 1  I  P1,0;o P0,1;i  0;i .

(7)

Here, P0,1;i and P1,0;o are 2×2 diagonal matrices that relate the complex amplitudes of the incident (i) and outgoing (o) wave-function components between adjacent slices. The P0,1;i and P1,0;o are defined consistent with the required Bloch function form of the propagating modes in the leads, or the counterparts of Bloch functions with complex wave-vectors for “outgoing” evanescent modes into the leads. For example, for incident (outgoing) propagating wave-functions φγ″,m,i(o) in a given lead γ″ (with m ≤ 1 or m ≥ N for the left and right leads, respectively), φγ″,m,i(o) = eikγ″,i(o)·Γγ″,m ukγ″,i(o)(Γγ″,m), where ukγ″,i(o)(Γγ″,m) has the periodicity of the unit cell (four slices in the transport direction). The kγ″,i(o) are readily determined based on the energy E, potential energy in the given lead γ″, transverse wave-vector ky and required direction of propagation. Applying Eq. (7) to the Schrodinger’s equation, Eq. (6a), yields for the left boundary (slices 0 and 1),

EI1  H1,2 2  H1,11   H1,0 P1,0;o1  H1,0 I  P1,0;o P0,1;i  0;i .

(6b)

A similar expression is obtained for the right boundary (slices N and N+1),

EI N  H N , N 1 N 1  H N , N  N   H N , N 1PN , N 1;o N  H N , N 1 I  PN , N 1;o PN 1, N ;i  N 1;i .

(6c)

The injected wave-functions, 0;i and N+1;i, are chosen to be localized to either the left end or the right end of the channel and to the top or bottom layer consistent with the definition of the index γ, and are normalized to carry the appropriate amount of incident current per transverse mode ky per unit energy consistent with Landauer-Büttiker theory.22 Starting

with an initial guess of the modified Hartree-Fock potentials VF,ky(ΓT,ΓB), if just uniformly zero, we solve Eqs. (6a-c) for each value of ky independently. The Fock exchange interaction for each value of ky then must be recalculated as a function of all values of ky. We repeat this process iteratively to obtain a self-consistent solution (analogous to selfconsistent Schrödinger-Poisson’s calculations, except with non-local potential or off-diagonal density matrix elements for the exchange interactions). For reference, Eqs. (6a-c) are equivalent to the NEGF problem GS, where the Green’s function is G = (EI – H –

− with self-energy matrix  and source vector S.22 Therefore, GS can be rewritten as (EI – H – S. Comparing this latter form to Eqs. (6a-c) allows us to identify H1,0P1,0;o, HN,N+1PN,(N+1);o, and all other m,n = 0. Similarly, S1 = H1,0(I – P1,0;oP0,1;i)0;i, SN = HN,N+1(I – PN,N+1;oPN+1,N;i) N+1;i, and all other Sl = 0. However, in these ballistic transport equations, we have no need for the full G matrix, which includes terms representing transport between internal points. In the tight-binding model, the charge current flow from a top layer atomic site RT to a bottom layer site RB associated with any particular state β, Iβ(RT,RB), is





I  R T , R B    2q/h  Im  f   * R T H b R T , R B   VF R T , R B   R B  ,

(8)

where the coupling between sites RB and RT is decomposed in terms of the bare coupling Hb and the Fock exchange interaction VF. Spin-degeneracy is included with non-interacting spins. The total site-to-site current I(RT,RB), therefore can be written as I R T , R B    I  R T , R B  

 .   2q/h  Im  f   * R T H b R B , R T   VF R B , R T   R B    

(9)

Upon inspection, it can be seen that for the total current, but only for the total current, the component associated with the Fock exchange interaction includes the imaginary part of the product of the interlayer density matrix (Eqs. (2b-c)) with its Hermitian adjoint, which intrinsically vanishes. The expression for the total interlayer site-to-site current I(RT,RB) thus takes the simplified form,   I R T , R B    2q/h H b R T , R B  Im f    R T  * R B   ,   2q/h H b R T , R B   R T , R B  sinarg R T , R B 

(10)

a result that is independent of condensate formation.14,23 Therefore, bare coupling between sites is required for a nonzero net current flow between those sites, and, therefore, a so-called “spontaneous” condensate formed in the absence of any bare interlayer coupling—with a large interlayer density matrix ρ(RT,RB) self-consistently obtained via Fock interlayer exchange interaction that is proportional to ρ(RT,RB)—is incapable of carrying a net interlayer current. However, the

enhanced inteerlayer densitty matrix ρ(R RT,RB) in the presence p of a condensate can c enhance the interlayerr current flow w. Note that, forr a particular state β, non-zzero current Iβ(RT,RB) still can be carrieed between siites with no associated a baree coupling. III. SIMULAT TION RESULTS In this seection, we beggin showing tthe formation of the condennsate in nanosscale structurees, and with calculations c of o intra-layer annd interlayer transmission probabilities p u under equilibrrium condition ns, VTL = VBLL = VTR = VBRR = 0 V, in thee presence of thhe condensatee, and for refference also in n its absence. However, ass will becomee apparent, sim mply knowingg these transmiission probabiilities is not ennough to preddict all essenttial features off the non-equuilibrium transsport. We thenn simulate non--equilibrium transport t undeer three biasinng conditions: a “drag-counnterflow” biasiing condition with only onee layer biased, VBL = −VBR = Val/2 and VTLL = VTR = 0, annalogous to thhat previously explored in Ref. R 24 within a simpler 1-D D g condition mu uch like that which w was thee effective masss model withh a local exchaange interaction; an “interllayer” biasing basis for prevviously propossed BiSFET with w only one end e biased,11,112 VTL = −VBL = Vil/2 and VTR ” T = VBR = 0; and a “mixed” biasing schem me that has beecome the basis for the sepparately propoosed BiSJT,155 VTL = −VBL = Vctrl/2 and VTR = −VBR = Vdrv/2, also toouched upon inn Ref. 24.

FIG. 2. Local density d of state (LDOS) plotteed in the center of the channel. The top-layer LDOS L is plotteed with solid linnes and bottom-layer LDOS wiith dashed liness, respectively with w solid and open o circles reppresenting the coondensate-free LDOS for com mparison. A band gap startss to form and saaturates to its buulk value with increasing channel length. For stronger condeensates (low εr),, the (incomplete) baand gap saturates earlier, and is i larger for the same channel llength.

A. Formatiion of a nan noscale cond densate In this suubsection, wee illustrate how the condennsate forms inn nanoscale channels of in ncreasing lenggth. (Howeverr, scaling issuess per se will be b addressed subsequently.) s ) The calculatted local denssity of states (LDOS) ( in thee center of thee

channel for thhe structure in Fig. 1 undeer equilibrium m conditions is i plotted in Fig. F 2 for various channel lengths L andd effective dieleectric constannts εr, with carrrier concentraations n (top layer) l = p (botttom layer) ≈ 6×1012 cm-2 (V ( diff = 0.5 eV V) and interlayerr separation d = 1 nm. For these t simulatiions—and all others to folllow in this work—room tem mperature (3000 K) was assum med in calcullating the therrmal distributtions fβ of injected carrierss from the leaads. For increeasing channeel lengths, a bannd gap, if inco omplete, beginns to form abo out the chemiccal potential μ = 0 eV, whicch is also the point p at whichh the top and boottom layer Dirac D cones woould otherwisee cross, as inddicated by the reduction in the t LDOS. Thhe edges of thee band gap reggion saturate toward bulk results20,21 (aalthough less obviously soo for εr = 3.0 and the chhannel lengths considered heere). The deggree to whichh the gap form ms can be ussed as one measure of thee condensate strength.20 An n example of thhe LDOS vs.. energy alongg the channell is shown inn Fig. 3, whicch exhibits a gradual crosssover betweenn regions with and without thhe condensatee. The crossovver results froom both the ev vanescent deccay of lead staates in the gapp region, muchh as would occcur at a conveentional heterrojunction, and the associatted non-local self-consistenntly-calculatedd exchange inteeraction that iss inherently weakened w nearr the simulatioon region bounndaries.

FIG.3. LDOS vs. energy and position withinn the channel fo or εr = 2.2, Vb = 0.5 meV, d = 1 nm, L = 15 nm m and n = p ≈ 6×1012 cm-2. Thee d by the dark reegions. lack of LDOS iis demonstrated

B. Intra-lay yer and interlayer trans smission pro obabilities In the preesence of the condensate-innduced band gap, g the intra-layer transmisssion throughh the channel within w a singlee layer with thhe energy rannge of the gap ap will be attenuated substtantially, as is i perhaps to be expected. Perhaps less intuitively, thhe majority off the incidentt electrons aree, however, trransmitted to the other lay yer on the sam me end of thee channel in thee presence off the condensaate. The averaage transmissiion probabilitiies as a functiion of energyy for injections from leads BL L and TL withh and without the condensaate are contrassted in Fig. 4 for f εr = 2.2, d = 1 nm, L = 15 1 nm, n = p ≈ 6×1012 cm-2, aand interlayerr A-A bare cooupling, Hb(RT, en atoms on the t A sublatticce of the samee T RB) = Vb = 1 meV betwee bilayer unit ccell, and zero otherwise. (N Note that diffferent densitiees of transverrse modes in the top and bottom b layerss, which come ffrom the interllayer electrosttatic potentiall difference, cause the non-equality betw ween averagedd transmissions over propagatting modes froom BL to TL and those from TL to BL. However, dettailed balance is guaranteedd for any givenn single ky modde.) In the abssence of the condensate, c almost all injected current is i transmitted to the oppossite side of thee same layer (F Fig. 4(c)) with h the maximuum interlayer transmission over three orrders of magn nitude lower (F Figs. 4(b) andd

(d)). In the prresence of the condensate, intra-layer i trannsmission fallls to a few perrcent in the ceenter of the ennergy gap (Fig g. 4(g)), while tthe majority of o the injectedd current (~75 5%) tunnels thhrough the intterlayer dielecctric to the saame end of thee channel (Fig. 4(f)). The reeason for enhaanced interlayyer transmissioon probabilities is the conddensate-enhannced interlayeer coupling, com mposed of the bare couplingg as well as thhe exchange innteraction.

FIG. 4. The vaalue of the transsmission coefficcients T(E), including that for reflection to thee lead of incideence and averagged over all incident modess, for injection from f leads BL and a TL. Transm mission coefficiients without thhe condensate arre plotted in (a))-(d), and those with the condennsate in (e)-(h).. Simulation paarameters are dielectric constannt εr = 2.2, layerr separation d = 1 nm, channell length L = 15 nm, interlayer bbare coupling Vb = 1 meV andd carrier concenntrations n = p ≈ 6×1012 cm-2.

C. Drag-co ounterflow biasing b and exciton flow w As illustrrated by the example e of inn Fig. 5(b) and d (d), under the t drag-coun nterflow biasin ng condition, VBL = −VBR = Val/2 and VTL = VTR = 0, cuurrents of neaarly identical magnitude m flow along the upper u and low wer layers in thhe channel buut in opposite diirections. These currents floow despite thee lack of any bias b on the leads to the top p layer, and deespite the bandd gap within thhe channel in presence of thhe condensatee exhibited inn Fig. 3 and thhe associated low intra-layer equilibrium m transmission pprobabilities of o Fig. 4 nearr the chemicall potential. In addition, the interlayer currrent is relativvely limited byy comparison ddespite the laarge interlayeer transmissioon probabilitiees of Fig. 4 in the presennce of the coondensate. Too understand thhis behavior we w must delvve deeper, liteerally here in terms of the energy. Figs. 6(d)-(f) andd (h) show thee current flow subdivided s ab bout a cutoff energy e Eco = −65 − meV that iis both well below the chem mical potentiaal(s) and abovee the nominal llower edge off the condenssate-induced band b gap at approximately a −100 meV (Figs. ( 2 and 3). 3 The chargee current at thee energy of thee injected carrriers is indeed d substantiallyy attenuated in n the channel,, as shown in Figs. 6(d), (ee)

and (h), althoough not comp pletely for thiss short 15 nm m channel, andd flows primarrily between thhe layers, as shown s in Figss. 6(f) and (h). H However, an opposing o current loop along g and betweenn layers withinn the channel is excited bellow Eco—in an n entirely elastiic process—th hat essentiallyy vanishes at the t ends of the channel, as also shown in n Figs. 6(f) annd (h). (In thaat this sub-Eco ccurrent essentiially vanishess at the channnel ends and thhat its peak value v almost matches m that of o the incidennt and extractedd currents, it becomes b clearr that this sub--Eco current iss not just the thermal tail of o the injectedd and extractedd current distribbutions.) It is this nominallyy sub-band-gaap current thatt extends the total t current flow f all the waay through thee channel. Unliike for bare co oupling alone, the interlayeer phase relatiionship for diffferent states β becomes self-consistentlyy tied together through the complex exchaange potentiall that comes thhrough the gloobal interlayeer phase relatioonship presennt in the interlaayer density matrix m ρ(RT,R RB) (Eqs. (1))-(3)). It is inn this way thhat injected/exxtracted curreent within thee condensate-innduced band gap g can inducee current flow w below the baand gap.

FIG. 5. Illustraative samples foor distributions of intra-layer current c per widtth (Iintra) and intterlayer currentt per area (Jil). (a) ( and (c): BiSFET-like biiasing with Vil = 5 mV and Vb = 0.5 meV, andd (b) and (d): drrag-counterflow w current biasinng with Val = 400 mV and Vb = 0.5 meV. Otherr parameters are εr = 2.2, d = 1 nm, n = p ≈ 6× ×1012 cm-2 and L = 15 nm. Thee positive directions for intra- and interlayer currents are as shown in the pllots.

While the calculationss are performeed in terms off propagating electron statees only, althou ugh within thee valence bandd of the p-type layer, turning g to the languuage of electroons and holes can be helpfu ul in understaanding the essential physicss. At one end of the channel, the current flow f can be innterpreted as electrons and d holes being injected i from m (extracted to o) the n-type andd p-type leadss into (from) thhe end of the channel to creeate/emit/exciite/form (annihhilate/absorb/destroy/break kup) coherent excitons and associated exciton flow wiithin the channnel from (to) that end. For Coulomb draag-counterflow w w above Eco inn Figs. 6(f) annd (h) is assocciated with thee injection andd current biasinng, specificallyy, the interlayyer current flow extraction off electron-holee pairs at the opposite endds of the channnel. The opp positely direccted interlayerr current flow w below Eco in tthe same figurres is associatted with the coorresponding creation and annihilation a off the coherentt excitons. Thee intra-layer cuurrent flow below Eco in Figgs 6(d),(e) and (h) is assocciated with thee flow of the excitons alonng the channell,

with correspoonding oppossite charge current c flowss in the two layers, prod ducing the neear-perfect Coulomb drag gcounterflow ccurrent. Moreoover, while thhe electron poortion of the exxciton pairs flows fl at the lo ower edge of the t condensatee induced bandd gap within the t channel, the t hole portioon flows at thhe upper edgee of the band gap under thhe energy signn change requirred to define holes, h which makes m the tottal energy connservation cleaar even as thee energies of the t constituennt electrons andd holes changee as they are injected i to (exxtracted from)) the channel to create (ann nihilate) the excitons withinn the channel. F Finally, we noote that this prrocess is someewhat analogoous to Andreev reflection in n which an injjected electronn and reflected hole at the eddge of a conveentional superrfluid region creates c a coheerently bound Cooper pair of o electrons inn an energy connserving mannner, but here an a injected eleectron in the n-type n layer and a a reflected d electron (an injected holee) in the p-type layer l create a bound electroon-hole pair, or o an exciton.

FIG. 6. Bottom m-layer (IB), Toop-layer (IT) currrent per width and interlayer (J ( il) current per area distributio ons for (a)-(c) B BiSFET biasingg, and (d)~(f) drag-counterflow biasing, b using the t same system m parameters, biasing condition ns and rules forr signs of currennt flow as for a that origiinate at the positions they reprresent for (g) BiiSFET-like and d Fig. 5. These cuurrent distributiions are also visualized with arrows (h) drag-counteerflow biasing, with vertical annd horizontal coomponents propportional to the interlayer and intra-layer currrents, respectively. However, H with thhe interlayer annd intra-layer cuurrent having diifferent units, th he shown relativ ve lengths of thhe horizontal and vertical com mponents are chosen for illusttration clarity on nly. The currennts above Eco coorrespond to eleectron and hole injection from the leads into oor extraction to the leads from the channel reggion, while thosse below Eco corrrespond to cohherent exciton flow f within the channel.

Finally, w we note that as a the bias Vala is increasedd, the near perrfect Coulomb b drag-counteerflow currentt (Fig. 7(a)) is maintained unntil ultimately y the condenssate itself colllapses, as show wn by the collapse of the band b gap in Fig. F 7(b). This eventual collapse appears to be with increasing i freee carrier conncentrations abbove and bellow the band gap with thee voltage-defined splitting of o the chemiccal potentials,, which collappses the conddensate in muuch the samee way as highh temperatures or large chargge imbalancess would do.20

FIG. 7. (a) Colllapse of the neear-perfect Coullomb drag-counnterflow currentt in the presencce of the condennsate in a nanosscale channel with parameterrs provided in th he figure. The bottom b layer is biased, and thuus the drag-efficciency η is defin ned as the top-tto-bottom ratio between corressponding currennt amplitudes avveraged across the channel. (b)) LDOS in the center c of the chhannel for repreesentative voltages that doo and do not su upport the near-pperfect Coulom mb drag-counterrflow current, as indicated in (aa). The collapsee of the nearperfect Coulom mb drag-counterrflow current is accompanied by b the collapse of the condensate.

D. Critical current and d voltage within the BiS SFET-like bia asing schem me

FIG. 8. (a) Am mplitude of the Fock F exchange interactions (prroportional to thhe pseudospin amplitude) a betw ween bare-couppled pairs of top p and bottom atooms as a functioon of position inn a 15 nm channnel. Other param meters are Vdiff = 0.5 eV, Vb = 0.5 meV, d = 1.2 nm and εr = 2.2. (b) Total ppseudospin phasse between the same s pairs of atoms a as a functtion of position with the same system parameters. Apparently, thee pseudospin am mplitude is posiition-dependentt yet voltage-independent, whiile the pseudosppin phase is volltage-dependentt yet largely posiition-independeent.

FIG. 9. Linearr sin(θ) vs. Vil reelation for BiSF FET biasing connditions, with εr = 2.2, d = 1 nm, n Vb = 0.75 meV, m and n = p ≈ 6×1012 cm−2, for varying chaannel lengths L..

For the BiSFET-like B b biasing condittion, VTL = −VBL = Vil/2, VTR = VBR = 0, 0 we find thaat condensate formation cann greatly increaase the interllayer conductance, reachin ng ~75% of the Landaueer-Büttiker baallistic limit for the leadss, consistent witth expectationns based on thhe interlayer trransmission probabilities. p Remember R thaat the equilibrrium interlayeer

transmission probabilities are orders of magnitude lower (Fig. 4) without the condensate. (The BiSFET nominally has contacts to only one end, while here a negligible current flows to the grounded contacts for the “BiSFET-like” biasing, as seen in Fig. 5(a), so that the systems are the same in terms of this aspect of current flow.) However, this high, at least, DC conductance can only be maintained up to a critical current Ic, again as determined by self-consistent effects of the current flow on the entire interlayer density matrix ρ(RT,RB). The interlayer density matrix ρ(RT,RB) may be defined as a collective “pseudospin” ρ(RT,RB) with pseudospin magnitude |ρ(RT,RB)| and phase θ(RT,RB) ≡ arg[ρ(RT,RB)], which is analogous to a collective real spin state—magnetic moment—with strength and orientation. (This terminology underlies the BiSFET and BiSJT names.) As illustrated by Fig. 8, under BiSFET-like biasing, the pseudospin magnitude exhibits a position dependence consistent with the localization of the condensate, but is essentially independent of the interlayer bias, while the pseudospin phase is weakly dependent of position but varies with Vil, θ(RB,RT) ≈ θ.13 With θ evaluated at mid-channel for specificity, sin(θ) is proportional to Vil as shown in Fig. 9. The inter-atom current flow between layers of Eq. (10) can be written as I R T , R B   I max R T , R B sin( ) ,

(11a)

where I max R T , R B    2e/h H b R T , R B   R T , R B  .

(11b)

Similarly, the total interlayer current I can be written as I  I c sin( ) ,

(12a)

I c   I max R T , R B  ,

(12b)

where, in general,

RT RB

although, as previously noted, in this work we only include bare coupling, and thus total inter-atomic-site current flow, between top- and bottom-layer A atoms in the same unit cell. With sin(θ) having a maximum magnitude of unity when |θ| = π/2, the maximum or “critical” current magnitude is reached, as indicated by the “max” and “c” subscripts in Eqs. (11) and (12). In practice, as |θ| approaches π/2 in our simulations, the rate of convergence slows asymptotically.17 Therefore, we use the observed linear dependence of sin(θ) on Vil to extract the critical currents Ic and corresponding values of associated “critical voltage” Vc from somewhat smaller interlayer currents and voltages.13 (Such linearity is observed also at |θ| close to π/2 in Ref. 17.) Thus-extracted Vc are shown in Fig. 10 as a function of the bare interlayer coupling, and for L = 15 and 20 nm. It was not possible to converge any solution in our steady-state calculations beyond the critical voltage. Instead, the solutions are not numerically stable in a conventional sense. Instead, the pseudospin magnitude remains essentially constant with iteration while the phase smoothly rotates through all angles periodically, the larger Vil beyond Vc, the faster the rotation with iteration.14,17 The nearly constant pseudospin magnitude suggests that the condensate does not

collapse beyoond Vc, in conntrast to cases for the collappse of the Coulomb drag-ccounterflow cuurrent (Fig. 7)). The rotationn with iterationn suggests insstead time-deependent curreent oscillationns analogous to those in superconductiing Josephsonn junctions, j whhich have voltage-dependennt DC currentss below a crittical value folllowed by colllapse of the DC D current andd an onset of ann AC current of fixed amplitude after thhe critical currrent is reachedd.11,25 Similarrly, we expectt an oscillationn rate f of f/(Vili – Vc) ≈ 2q/h h ≈ 0.5 THz/m mV, which would w be well into the THzz regime for the t simulated systems heree. Consistent wiith the “pseud dospin” terminnology, this trransition from m steady-state to oscillatory y behavior unnder increasingg drive voltagee also is analoogous to that expected for in-plane easyy axis nanom magnets (magn nets with in-pplane magneticc anisotropy) driven d by spin n transfer torrque (STT) viia interlayer ccharge curren nt flow throug gh perpendicuular easy axis nanomagnets.26 However, to directly model m the behaavior beyond Vc would reqquire time-deppendent simullations beyond d our current caapabilities.22

FIG. 10. Lineaar dependence of o critical voltagge Vc on interlaayer bare coupliing energy Vb. System S parametters are n = p ≈ 6×1012 cm−2, εr = 2.2, and d = 1 nm.

The criticcal currents and, a as illustraated in Fig. 100, the associaated critical vooltages Vc alsoo are linearly dependent onn the bare interrlayer hopping g energy Vb, consistent c witth Eq. (11b). Physically, thhis dependencce can be undderstood in thee following waay. Above Eco most of the current c flow is between layyers, but againn a current looop is excited in the channeel below Eco (F Figs. 6(a)-(c) and (g)) that is associatedd with exciton flow. Withh incident (ou utgoing) electrron and holes creating (annnihilating) exccitons at one end of the ch hannel but noot extracted (iinjected) at thhe other end, a steady-statee exciton popullation can onlyy be maintained through baare-coupling-aassisted recom mbination (genneration) withiin the channell. Therefore, thhe greater thee bare couplinng, the greateer current of injected (outtgoing) electrrons and holees that can bee maintained inn the opposite layers. Conveersely, with exxcitons injected at one end of the channeel and extracted at the otheer under drag-coounterflow biaasing, much laarger current injection i and extraction cann be supportedd with the sam me or lower Vb as already seeen (Fig. 5(b) vs. v Fig. 5(a), e.g.). e Critical ffor applicationn to the propoosed BiSFET and BiSJT, thhe calculated critical voltagges can be scaaled below thee thermal voltagge kBT/q (wheere kB is Boltzzmann’s consttant), which iss approximately 26 mV in these t 300 K siimulations, viaa the linear deppendence on the t bare couplling, as in thee examples shhown in Fig. 10. 1 The basis for these subb-kBT/q criticaal

voltages is thee collective efffect, represennted here throuugh the Fock exchange inteeraction that couples c many electron states β together.

FIG. 11. Effecctive interlayer critical c voltage on the “drive” (drv) end of thee channel at wh hich the critical interlayer curreent is reached as a function off the interlayer voltage appliedd to the “control” (ctrl) end of the channel. Thhe dashed line with w slope of −11 corresponds to perfect criticcal voltage consservation such that t sum of the two voltages att which the critiical current is reached would be b fixed. System parameeter are εr = 2.2,, Vb = 0.5 meV, L = 15 nm, d = 1.0 nm and n = p ≈ 6×1012 cm c -2.

E. Mixed B BiSJT biasin ng While biaasing both endds of the channnel can allow w for more intrra-layer currennt flow, it doees not increasee the ability too drive interlayyer current, at least not a neet current. Insstead, when we w mix the BiiSFET-like an nd drag-counteerflow biasingg schemes to obbtain the BiSJJT regime, VTL w observed behavior b that is, indeed, thee T = −VBL = Vctrl c /2, VTR = −VBR = Vdrv/2, we basis for the proposed BiiSJT15 from w which we hav ve borrowed the “control”” (ctrl) and “drive” “ (drv) nomenclaturee. Specifically, we w find that thhe total interlayer critical current c and associated total voltage appliied to both channel ends aree essentially coonserved, as shown in Figg. 11. Equivaalently, appliccation of a sub-critical vooltage on onee end with ann associated subb-critical interrlayer currentt can change the t effective crritical currentt and voltage seen s at the othher end. In thee 1D-effective--mass based siimulations off Ref. 24, it was w found thatt bare interlay yer coupling was w necessary to allow lesssthan-perfectlyy matched couunterflow intraa-layer currennts, which is a different persspective on thee same basic cconclusion. F. Channe el scaling, sy ystem param meters, bulk k critical tem mperature an nd normaliz zed wavelen ngth Interlayer critical curreent, voltage annd the conducctance have beeen calculatedd with variouss system param meters such as channel lengtth L, interlayeer bare couplinng energy Vb, interlayer seeparation d, efffective dielecctric constant εr, and carrieer concentrationns n and p. All the simulations are basedd on the BiSFE ET-like biasinng scheme, VTL T = −VBL = Vil/2 and VTR = VBR = 0, withhout losing gennerality givenn the critical cuurrent and volltage conservaations. For refference we alsso use the bulkk value of the ccritical tempeerature at whicch the conden nsate collapses, Tc, which is i a measure of o the latter’s strength, as a unifying paraameter. (As weell as being caalculated direcctly) Tc can bee estimated froom the 0K indduced bulk baand gaps Eg0 as Tc ≈ Eg0/(4kB) essentially inndependent off dielectric en nvironment, innterlayer separration or carriier concentrattion.20 We alsoo use λ, the Feermi wavelength normalizeed to the trannsition lengthh required to form the con ndensate in thhe channel, too

characterize tthe abruptnesss of the channnel in a way that t is relevannt to quantum m mechanical reflection r proobabilities. For this purpose, the transition length is defi fined as the disstance across which the LD DOS at the cheemical potenttial drops from m 90% to 10% of its value in the absencce of the conddensate. These two parameeters are interrdependent inn that the bulkk condensate sttrength affectss how abruptlyy the condensaate can form with w position in i the channell, as will be seeen below.

FIG. 12. Interllayer conductannce Gil normalizzed to Laudauerr-Büttiker limitt for the leads att the chemical potential p (0 eV)), interlayer current densityy Ic (per unit wid dth) and criticall voltage Vc witth varying effecctive dielectric constant c εr = 2.2 2, 2.5 and 3.0 and a corresponding bulk critical tem mperatures Tc ≈ 655K, 524K and a 377 K, respectively. (For εr = 3.0 (Tc ≈ 37 77 K) the condeensate is only partially formed within the 255 nm channel, annd therefore thee transition lenggth cannot be determined by thhe definition in the text. mply less than thhe Fermi wavellength divided by b half the maxximum channel length (12.5 nm m), which is 1.22.) Therefore, λ is taken to be sim meters are Vb = 0.5 meV, d = 1 nm and n = p ≈ 6×1012 cm-2. The other param

We first consider whaat is a voltagee-independent interlayer coonductance Gill below the crritical current Ic and criticaal voltage Vc as a function off channel lenggth for differinng effective dielectric d consstants, εr = 2.22, 2.5 and 3.00, as in Fig. 12 2 with the correesponding Tc and a λ listed inn the collectivee figure inset. Other parameeters are d = 1 nm, Vb = 0.55 meV and n = p ≈ 6×1012 cm m−2. As seen in i Fig. 12(a), with increasiing channel leength, Gil incrreases and satturates to aboout 75% of thee Landauer-Bütttiker limit GLB ds, to which it is normalizedd here. (GLB ≅ (Vdiff/t0)(1.15 5 kS/cm), wheere t0 = 2.7 eV V L for the lead is the magnituude of the neaarest-neighborr intra-layer baare coupling potential p in thee tight-bindingg model). The rate at which Gil saturates s increeases with thee strength of the t condensatee as measure by Tc. Consisstent with bulkk estimates,20,211 Fig. 12(b) exxhibits a lineaar dependencee of interlayerr critical curreent Ic (actually a current deensity per uniit width for thiss figure and those t to follow w) on channeel length L, suuggesting a co onstant criticaal current perr unit area if a shorter effectiive channel leength Leff—thee actual L minnus its L axis intercept for each linear Ic vs. L curve— —is consideredd to allow for the t non-abrupt condensate formation f illu ustrated in Figg. 3.20 Leff incrreases with Tc. The differennce between L and Leff lies inn the non-locaal nature of thhe exchange innteraction. With a half-widtth (radius) R = |RT – RB| off roughly 2 nm m at half maxim mum of bulk density matrrix elements ρ(R ρ T,RB) for this graphenee system, deffined by both the Coulombb interaction annd coherence length of the interlayer density d matrixx,20 a graduall build-up off the condensaate within thee channel is inhherent. Moreo over, in determ mining the nett strength of the t exchange interaction inn, e.g., the top layer at somee point RT withhin the transittion region, a reduced ampplitude in ρ(R RT,RB) for couupling to poin nts RB closer to t the channeel edge must be compensatedd for by an inccreased amplittude in ρ(RT,RB) for coupliing to points closer c to the channel c centerr. The latter beecomes increaasingly difficuult to obtain as the criticcal temperaturre Tc approacches the 300 K simulationn

temperature. T Thus, L − Lefff increases wiith decreases in i Tc − T. (Thhe relationshipp of the non-lo ocal nature off the exchangee interaction to L − Leff also illustrates thee need for expplicit modelingg of the non-local interactioon to judge thhe potential fo or nano-scale coondensates.) Inn addition, thee effective criitical current density, d Jc,avg = Ic/Leff, increeases with Tc even allowingg for the increases in Leff withh Tc, consistennt with the dep pendence of Imax(RT,RB) onn |ρ(RT,RB)| (E Eq. 12(b)). At this pooint, we note that for computational expeediency we geenerally have considered veery strong conndensates, withh the lowest Tc of 377 K stilll not produciing Gil saturattion with a 255 nm channel length in thiss modeled sysstem at 300 K. K ry to achieve greatly g enhancced Gil still onn the scale of the LandauerrHowever, we also note thaat saturation iss not necessary o magnitude greater than the condensaate-free interllayer conductance expectedd Büttiker limitt (Fig. 12(a)) and orders of based on the transmission probabilities of o Fig. 4. Moreover, saturaation may not even be ideall from one peerspective. Thee critical voltagge, Vc = Ic/Gil, decreases with w L initially y due to the faster increasse in Gil than in Ic, and inccreases with L afterwards beecause of the saturation off Gil and conttinuous increaase of Ic. Thuus, the point at which Vc—an — importannt parameter forr BiSFET and d BiSJT circuuit applicationns—is the leasst sensitive too variations inn channel lenggth is reachedd prior to saturaation.

FIG. 13. Interllayer conductannce Gil normalizzed to Laudauerr-Büttiker limitt for the leads, interlayer i criticaal current densiity Ic per unit width, and (nonn-equilibrium) reflection r probaabilities, for inccident electronss in the bottom--left lead BL, RBL i B for varying interlayer 12 separation d. The other param meters are εr = 2..2, Vb = 0.5 meV V and n = p ≈ 6×10 6 cm-2. Thhe interlayer sep parations d are 0.8, 0 1.0 and 1.22 K 655 K and 5885 K, respectiveely. nm, with associated bulk criticcal temperaturees of Tc ≈ 747 K,

For the simulation resuults of Fig. 133, differing innterlayer separrations, d = 0.8, 1.0 and 1.22 nm, are conssidered with εr = 2.2, Vb = 00.5 meV, andd n = p ≈ 6×1012 cm-2. Thhe basic trendds with respecct to Tc are much m the sam me for both Gil saturation andd Ic with L as a seen in Figg. 12. Howeveer, the spreadd in Tc is less as expected given its gennerally weakeer 2 dependence on o d than to εr20 despite the slightly larger percentile vaariation in d considered c herre, and thus inn the results as

well. In addittion, we see thhat that the saaturated value of Gil can varry, although the t overall efffect is not largge in this casee. While the sm maller the layeer separation the more quiickly Gil saturrates as expeccted, the satuuration is slighhtly larger foor larger d (Fig. 13(a)). This increase i in satturated Gil is associated a witth less abrupt change in thee potential upoon entering thee channel—albeit the interlaayer exchangge mediated hopping h energgy—indicatedd by smaller λ. The averaaged reflectionn coefficients as a a function of o energy (Figg. 13(c)), in thhe vicinity off the chemicall potential (0 eV) where thhe incident andd outgoing currrents flow, exxhibit this effeect even moree strongly. (Thhe reflection probability p deecreases abovee the chemicaal

potential as the t energy off incidence appproaches thee condensate band edge alllowing more transmissionn, and initiallyy increases beloow because off a lack of avaailable final staates in the oppposite layer.) The T overall efffect, howeverr, is not large..

FIG. 14. Interllayer conductannce Gil normalizzed to Laudauerr-Büttiker limitt for the leads, interlayer i criticaal current densiity Ic per unit width, and assoociated critical voltage v Vc for vvarying interlay yer hopping eneergies Vb = 0.5, 0.75 and 1.0 meV. m The other parameters p are εr = 2.2, d = 1 nnm and n = p ≈ 6×1012 cm-2. Tc ~ 655K and λ ~ 2.25 for all thhree cases.

For the results r of Fig.. 14, differentt interlayer baare coupling strengths s weree considered, Vb = 0.5, 0.775 and 1 meV V, with εr = 2.2, d = 1 nm, annd n = p ≈ 6× ×1012 cm-2. Thhe strength of the condensaate and associaated Tc are noot significantly y affected by Vb at this scalle, and, thus, the interlayerr conductancee Gil also rem mains essentiaally unaffectedd (Fig. 14(a))). However, thee critical currrent (Fig. 144(b) and assoociated criticaal voltage (Fig. 14(c)) increase with increasing i Vb , consistent with Eq. 11(b). For strong bare coupling, past bulk woork has shown n that the streength and eveen the detailedd shape of the interlayer dennsity matrix ρ(R ρ T,RB) can be affected.21 However, such strong innterlayer bare coupling alsoo would likely produce p Vc su ubstantially greeater that kBT/q. T/

FIG. 15. Interlaayer conductancce Gil normalized to Laudauerr-Büttiker limit for the leads, an nd interlayer crritical current deensity, Ic per 12 unit width, withh varying carrieer concentrationns n = p ≈ 3.8×1012 cm-2, 6.0×1012 cm-2 and 8.6×10 8 cm-2 (V Vdiff = 0.3, 0.5 and a 0.7 eV respectively) annd correspondinng bulk critical temperatures Tc ≈ 598 K, 6555 K and 702 K, respectively. r

Finally, we w consider differing d nomiinal carrier cooncentrations, n = p ≈ 3.8×1012, 6.0×1012, and 8.6×10012 cm-2 (Vdiff = 0.3, 0.5 and 00.7 eV, respecctively) in Figg. 15, with εr = 2.2, Vb = 0..5 meV, and d = 1.0 nm. These T parametter sets exhibiit the largest ddifference in saturated Gil as normalizzed to the Laandauer-Büttikker limit for the correspoonding carrieer concentrationn. Here, the decrease d in thee Fermi waveelength relativve to the vallley center acccompanying thhe increase in n carrier conceentration exceeeds the decreease in transiition length caused c by stroonger conden nsates, produccing a smalleer wavelength-nnormalized po otential changge (smaller λ)). Moreover, this smootheer potential chhange is assoociated with a

stronger condensate. The combination produces a significant increase in the saturated normalized Gil (Fig. 15(a)). Of course, the Landauer-Büttiker limit for Gil also increases with carrier concentration. By comparison, it is likely that resulting smoother potential change and weaker condensate in the cases for Figs. 12 and 13 partially compensate each other producing smaller if any changes in the saturated normalized Gil. In addition, the stronger condensate leads to a larger critical current per channel length (Fig. 15(b)) or even effective channel length. IV. CONCLUSION

We have developed an atomistic tight-binding NEGF quantum transport method (except that we need not solve for the full Green’s function matrix in these ballistic simulation) for modeling non-equilibrium transport through nanoscale exciton condensates realized via the non-local many-body Fock exchange interaction. Specifically, we have modeled the graphene-dielectric-graphene system here. However, the essential physics should be similar in TMD-based material systems also under consideration or other material systems. In addition, the method itself should also be extendable to these latter systems, although with a greater computational burden. We have exhibited the possibility for condensate formation within nanoscale regions, and its dependence on the critical temperature of the bulk condensate. We have exhibited essentially transport effects that serve as the basis for beyond-CMOS device proposals including: interlayer conductances approaching the Landauer-Büttiker limits imposed by the leads but limited by critical currents (beyond which collapse of DC conductance and onset of THz AC conductance are expected, which is at least consistent with the oscillation in the pseudospin phase with iteration seen here), with the latter reached at sub-kBT/q voltages (BiSFET); critical current conservation for currents injected into two—or likely more—regions of the condensate (BiSJT); and nearperfect Coulomb drag-counterflow current between layers. We also have exhibited the underlying transport physics including the process by which incident (outgoing) electron-hole pairs in opposite layers excite (absorb) coherent excitons, in a manner somewhat analogous to Andreev reflection at the edge of conventional superfluids. The work is not intended to address the likelihood of achieving such condensates, which requires much greater attention to the details of screening and ultimately must be resolved experimentally. It is intended to motivate such work through exhibiting the novel transports effect that could thus be achieved, and perhaps to help with the interpretation of such experimental results. REFERENCES 1

B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Resonantly enhanced tunneling in a double layer

quantum Hall ferromagnet,” Phys. Rev. Lett., vol. 84, no. 25, pp. 5808-5811, Jun.2000. 2

E. Tutuc, M. Shayegan, and D. A. Huse, “Counterflow measurements in strongly correlated GaAs hole bilayers:

Evidence for electron-hole pairing,” Phys. Rev. Lett., vol. 93, no. 3, Jul. 2004. 3

J. P. Eisenstein and A. H. MacDonald, “Bose-Einstein condensation of excitons in bilayer electron systems,” Nature,

vol. 432, pp. 691-694, Dec. 2004.

4

L. Tiemann, W. Dietsche, M. Hauser, and K. von Klitzing, “Critical tunneling currents in the regime of bilayer

excitons,” New J. Phys., vol. 10, 045018, Apr. 2008. 5

D. Nandi, A. D. K. Finck, J. P. Eisenstein, L. N. Pfeiffer and K. W. West, “Exciton condensation and perfect Coulomb

drag, ” Nature ,vol. 488, pp. 481-484, Aug. 2012. 6

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katnelson, I. V. Grigorieva, S. V. Dubonos and A. A.

Firsov, “Two-dimensional gas of massless Dirac Fermions in graphene,” Nature,. Vol. 438, pp. 197-200, Nov. 2005. 7

Y. Zhang, Y.-W. Tan, H. L. Stormer and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s

phase in graphene,” Nature, vol. 438, pp. 201-204, Nov. 2005. 8

H. Min, R. Bistritzer, J.-J. Su and A. H. MacDonald, “Room-temperature superfluidity in graphene bilayers,” Phy. Rev.

B, vol. 78, 121401(R), Sep. 2008. 9

I. Sodemann, D. A. Pesin and A. H. MacDonald, “Interaction-enhanced coherence between two-dimensional Dirac

layers,” Phy. Rev. B, vol. 85, 195136, May 2012. 10

F.-C. Wu, F. Xue, and A.H. MacDonald, “Theory of Spatially Indirect Equilibrium Exciton Condensates,”

arXiv:1506.01947, submitted for publication, 2015. 11

S. K. Banerjee, L. F. Register, E. Tutuc, D. Reddy and A. H. MacDonald, “Bilayer pseudoSpin Field-Effect Transistor

(BiSFET): A proposed new-logic device,” Elec. Dev. Lett. IEEE, vol. 30, pp. 158-160, Jan. 2009. 12

L. F. Register, X. Mou, D. Reddy, W. Jung, I. Sodemann, D. Pesin, A. Hassibi, A. H. MacDonald and S. K. Banerjee,

“Bilayer pseudo-Spin Field Effect Transistor (BiSFET): Concepts and Critical Issues for Realization,” ECS Trans., vol. 45, no. 4, pp. 3-14, 2012. 13

X. Mou, L. F. Register and S. K. Banerjee, “Quantum transport simulations on the Feasibility of the Bilayer

Pseudospin Field-Effect Transistor (BiSFET)”, in IEEE IEDM, 2013, 4.7.1-4.7.4. 14

X. Mou, L. F. Register and S. K. Banerjee, “Ultra-low-power pseudospintronic devices via exciton condensation in

coupled two-dimensional material systems, ” in Nanoscale Materials and Devices for Electronics, Photonics and Solar Energy, Eds. S. Goodnick, A. Korkin and R. Nemanich, Springer (in press). 15

X. Mou, L. F. Register and S. K. Banerjee, “Bilayer pseudoSpin Junction Transistor (BiSJT): A New ‘Beyond-CMOS’

Logic Device,” submitted for publication, 2015. 16

R. T. Weitz, M. T. Allen, B. E. Feldman, J. Martin and A. Yacoby, “Broken-symmetry states in doubly gated

suspended bilayer graphene,” Science, vol. 330, pp. 812-816, Nov. 2010. 17

X. Mou, L. F. Register and S. K. Banerjee, “Quantum Transport Simulation of Bilayer pseudoSpin Field-Effect

Transistor (BiSFET) with Tight-Binding Hartree-Fock Model,” in IEEE SISPAD, 2013, pp. 420-423. 18

X. Mou, L. F. Register and S. K. Banerjee, “Interplay among Bilayer pseudoSpin Field-Effect Transistor (BiSFET)

performance, BiSFET scale and condensate strength,” in IEEE SISPAD, 2014, pp. 309-312. 19

C. F. Fischer, “Hartree-Fock method for atoms: A numerical approach,” J. Wiley & Sons, Inc., 1977.

20

D. Basu, L. F. Register, D. Reddy, A. H. MacDonald and S. K. Banerjee, “Tight-binding study of electron-hole pair

condensation in graphene bilayers: Gate control and system-parameter dependence,” Phys. Rev. B, vol. 82, 075409, Aug. 2010. 21

D. Basu, L. F. Register, A. H. MacDonald and S. K. Banerjee, “Effect of interlayer bare tunneling on electron-hole

coherence in graphene bilayers,” Phys. Rev. B, vol. 84, 035449, Jul. 2011. 22

S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge Studies in Semiconductor Physics and

Microelectronic Engineering, 1997. 23

D. Reddy, L. F. Register and S. K. Banerjee, “Bilayer pseudoSpin Field Effect Transistor,” in CMOS and Beyond:

Logic Switches for Terascale Integrated Circuits, Cambridge University Press, 2014. 24

J.-J. Su and A. H. MacDonald, “How to make a bilayer exciton condensate flow,” Nat. Phys., vol. 4, pp. 799-802, Aug.

2008. 25

K. K. Ng, Complete Guide to Semiconductor Devices, pp. 569, J. Wiley & Sons, New York, 2002.

26

H. Chen, A. Kent, A. H. MacDonald and I. Sodemann, “Nonlocal transport mediated by spin supercurrents,” Phys. Rev.

B., vol. 90, 220401(R), Dec. 2014.