Supercurrent Generation in Unipolar Electro-optic Devices

Ming Xie,1 ,2 and A. H. MacDonald1 1Department of , The University of Texas at Austin, Austin, TX 78712, USA 2Department of Physics and Astronomy, , Pittsburgh, PA 15260, USA (Dated: July 26, 2018) We describe a mechanism by which an electrical bias voltage applied across a unipolar semicon- ductor quantum well can drive an or polariton supercurrent. The mechanism depends on the properties of electronic quasiparticles in quantum wells or two-dimensional materials that are dressed by interactions with the coherent exciton field of an exciton condensate or the coherent exciton and photon fields of a polariton condensate, and on approximate conservation laws. We propose experiments that can be performed to realize this new light-matter coupling effect, and discuss possible applications.

PACS numbers: 71.35.-y, 73.21.-b, 71.36.+c

I. INTRODUCTION or where the particle current exits the coherent cavity region and absorbs them when it enters, it does not yield a global exciton or exciton-polariton This work is motivated by remarkable progress over generation rate and thus it cannot on its own support a the past decade in realizing and exploring the physics finite bosonic particle population. It can however gener- of dynamical steady states of coupled light and mat- ate a supercurrent in the exciton or polariton condensate ter in which two-dimensional exciton-polaritons, bosonic that flows between source and drain regions. states that are coherent mixtures of vertical cavity pho- Our article is organized as follows. In Section II we ex- tons and quantum-well excitons, condense into states plain the basic mechanism, which applies to both exciton with macroscopic coherence.1–8 The achievement of and exciton-polariton condensate cases. Throughout this exciton-polariton condensation can be viewed as an end article we focus on the exciton-polariton case because it point of the long quest for exciton condensation in can be reliably realized experimentally. The basic idea is semiconductors10–12 containing steady state populations that when electronic quasiparticles are dressed by their of conduction band electrons and valence band holes. interaction with an exciton-polariton condensate, they From this point of view, the role of the resonant cou- do not separately conserve conduction and valence band pling to vertical cavity photons, which turns excitons currents. Conservation is recovered by compensating bo- into exciton-polaritons, is to provide the bosonic parti- son generation and absorption contributions to the con- cles with phase stiffness via the small polariton mass that densate equation of motion that drive the condensate is sufficient to combat disorder that would otherwise lo- into a state that carries a super-current. This effect is calize the excitons. In typical systems that steady-state partly analogous to supercurrent generation at normal- polariton population is provided by optically exciting a superdonducting interfaces, and to spin-transfer torques population of non-resonant excitons that can be scattered in ferromagnets, but has some surprising wrinkles. In into the condensate at a rate that balances the rate at particular the direction of the exciton-polariton super- which polariton population is lost by cavity-photon leak- current is opposite to the direction of charged particle age. For many purposes the polariton system environ- flow. The transfer process between the quasiparticles and ment can be approximated as a thermal reservoir. The the condensate in general leads to source terms for dif- polariton condensate can also be supported electrically ferent components of the exciton-polariton condensate, by applying a bias voltage across a bipolar system so in a ratio that depends on the strength of the coupling of that the current is carried through the system by in- the quasiparticles to that component of the condensate. jecting conduction-band electrons at an n-contact and (In practice the quasiparticle-condensate transfer process valence-band holes at a p-contact.13–15 mainly produces a source term for the exciton portion of In this article we describe a mechanism for electrical the condensate. ) In Section III we explain how sep-

arXiv:1710.05826v1 [cond-mat.mes-hall] 16 Oct 2017 manipulation of exciton-polariton condensates that can arate transfer processes, proportional to exciton-photon act even in unipolar systems. It is based on the proper- Rabi coupling strengths, rearrange the composition of the ties of electronic quasiparticles in quantum wells or two- current on a longer length scale so that it is ultimately dimensional materials that are dressed by interactions carried by the lower-polariton condensate. In Section with the coherent exciton field of an exciton condensate, IV we examine the possibility of identifying these trans- or the coherent exciton and photon fields of a polariton fer effects experimentally by measuring the finite trans- condensate. It relies on approximate conservation of total verse momentum of the lower-polariton condensate when exciton number in an exciton condensate or on approxi- it carries a supercurrent. Finally in Section V we sum- mate conservation of the sum of total photon and exciton marize our results, present conclusions, and speculate on numbers in polariton condensates. Because it generates possible applications. 2

in the coherent region in the bare conduction-valence ba- sis is ����� ��������  H gψ(r) + ∆(r) H = 0 (1) � gψ(r) + ∆(r) −H0 � � � � where H = − 2∇2/2m + δ/2 and ∇ is the 2D gradi- � � 0 ~ ent operator. For simplicity, we have taken the effective masses m of the two bands to be identical and isotropic. �� We further simplify the problem by assuming that the system is translationally invariant in the transverse (y) FIG. 1: (Color online) Schematic representation of the planar direction and neglecting electrostatic band bending ef- semiconductor microcavity with contacts made to the quan- fects. tum well. The Hamiltonian (1) is written in a rotating-wave frame in which the condensate energy is effectively ab- sorbed into the energy of electrons, so that the actual II. ELECTRICALLY DRIVEN EXCITON bandgap Eg is replaced by a reduced gap δ ≡ Eg − ~ω SUPERCURRENT where ~ω is the chemical potential of the condensate. For typical polariton condensates δ is of the same order17–19 We illustrate the quasiparticle-condensate transfer ef- as the two-dimensional exciton binding energy. In the fect by considering a planar semiconductor microcavity two band model rotating and fixed (f) representation with electrical contact established to an embedded quan- wavefunctions are related by tum well, as illustrated schematically in Fig. 1. For def- ˆ −1 initeness we assume that the contacts are n-type and Ψ(r, t) = U (t)Ψf(r, t) (2) that the quantum well is intrinsic, so that the electri- ˆ i cal system is a lateral n-i-n homojunction. As we ex- where U(t) = exp(− 2 ωtσz) and σz is the band Pauli- plain below, this geometry does not16 support steady- matrix. Here Ψ(r, t) = (u(r, t), v(r, t))T is a two- state net electrically-controlled exciton or polariton gen- component vector whose components are the conduction eration. We therefore assume that a non-resonant optical and valence components of the quasiparticle wavefunc- pump is also present to maintain a finite steady-state po- tion. lariton population in the cavity region. We are interested The two terms in the off-diagonal component of the in the regime in which the pumping power exceeds the Hamiltonian originate from electron-photon coupling and threshold for polariton condensate formation, and a bias from Coulomb interactions17–19 respectively, and are voltage is applied between the electrical contacts. time-independent in the rotating representation. In the The influence of electrical transport on the polari- electron-photon coupling term g is the electric dipole cou- ton condensate can be understood in a two-step model. pling constant and ψ = haˆi is the coherent photon field, First, electric charge is injected from the left (L) contact, i.e. the expectation value of the photon annihilation op- drained to the right (R) contact and transported through eratora ˆ in the coherent photon state. ∆ is a mean field the cavity region by fermionic quasiparticles that carry describing coherence induced by interband exchange in- an electrical charge and are dressed by the condensate. teractions, which are also the origin of the attractive During this process, a neutral exciton supercurrent is in- force that binds electrons and holes in an isolated ex- duced in response to the charge current via a mecha- citon. Both gψ and ∆ are chosen to be real and constant nism explained below. Second, the exciton supercurrent across the cavity region(0 < x < L). In the left (x < 0) is converted to a lower polariton supercurrent. During and the right (x > L) contact gψ = 0 and ∆ = 0. The this latter process, part of the exciton condensate mo- matter-photon transfer effect we describe below occurs mentum is transferred to the photonic component. This because of spatial variation in the inter-band coupling two step model is justified when the exciton-photon con- terms in (1). version length scale is much longer than the length scale In the rotating representation the quasiparticle Hamil- over which the exciton supercurrent is generated. tonian is time independent and its wavefunction Ψ(r, t) We describe both the quantum well and the contacts satisfies a time-independent Schr¨odingerequation. The with a two-band semiconductor model. The contacts, quasiparticle-condensate transfer effect is most simply il- which are located outside of the cavity, are not dressed lustrated by assuming ballistic transport. We therefore by the photon field, but the bare conduction and valence postpone a discussion of scattering effects to the discus- band states of the quantum well in the cavity region are sion section. Due to translational symmetry in the y coupled by both the coherent photon and the coherent direction, the transverse momentum ky is a good quan- exciton field. For the illustrative calculations we perform tum number. For each ky the x-dependent factor in below, we assume that the coherence amplitude rises the wavefunction satisfies a one-dimensional Schr¨odinger equation with H → Heff(k ) = − 2∇2 /2m + δeff/2, steeply at the n-i interface. The mean field Hamiltonian 0 0 y ~ x ky 3

(a) 1.5

1.0 1.0 decay length ξ j j n c s i

j 0.5 / jv j ) j 0 0.8 = 0.0 y k , E

( 0.5 T 0.6 R I B F T (b)

n E 0.10 o

i j j y x c x s s g s

i δ δ r −∇ −∇ 0 δ00 )

e 0.05 j j n x v x i

m 0.4 n s j −∇ −∇ E / n j a (

r 0.00 x T

0.2 ∇

Left contact QW Right contact − 0.05 kx 0.10 0.0 0 1 2 4 6 8 10 0 L Position(x) E/(δ00 /2)

FIG. 2: (color online) Energy dependence of the transmis- FIG. 3: (color online) Spatial dependence of (a) normalized sion probability for normal incidence ky = 0. The tran- conduction and valence band partial probability currents and sition probability decays exponentially with channel length (b) divergence of the probability currents inside the QW for 0 0 0 for |E| < δ0/2, where δ0 is the renormalized quasiparticle normal incidence ky = 0 with E = 0.55δ0. jin is the current gap. Inset: Model energy dispersions in different regions for incident from the left contact. For the chosen parameters, the δ = 10meV and gψ + ∆ = 10meV. decay length ξ ≈ 1.8 nm is much smaller than the channel length L.

obtain the following coupled continuity relations where we defined an effective bandgap δeff = δ + 2k2/m. ky ~ y 2 ∂|u| 2(gψ + ∆) ∗ In the cavity region, the bulk quasiparticle energies + ∇xjc = Im{u v} (3) ± p 2 2 ∂t are E = ±Ek ≡ ± ξ + (gψ + ∆) , where ξk = ~ k k 2 2k2/2m + δeff/2, and the quasiparticle wavefunction ∂|v| 2(gψ + ∆) ∗ ~ x ky + ∇xjv = − Im{u v} (4) has mixed conduction and valence band character. It ∂t ~ is useful to define a ky-dependent quasiparticle bandgap ∗ q where jc = ~Im{u ∇xu}/m is the conduction band cur- δ0 ≡ (δeff )2 + 4(gψ + ∆)2. Fig. 2 plots the transmis- rent and j = − Im{v∗∇ v}/m is the valence band cur- ky ky v ~ x rent. (We have suppressed the (E, ky, x) dependence of sion probability for a ky = 0 electron incident (I) from 0 the wavefunctions and the probability current densities the left contact as a function of energy E. For E > δ0/2 propagating modes are present in the cavity region and for notational convenience.) These relations lead directly the transmission probabilities between source and drain to the conservation law of particle current ∂ρ/∂t+∇xj = 2 2 contacts are large. Because the channel length exceeds 0, where ρ = |u| + |v| is the particle density and microscopic length scales, the transmission probability j = jc + jv the corresponding probability current den- jumps sharply from near 0 to near 1 at the band edge. sity. In a steady state, j is spatially constant as shown (The oscillatory behavior in Fig. 2 is a consequence of in Fig. 3(a). multiple reflection between source/bulk and bulk/drain The key observation from Fig. 3 is that because the interfaces.) quasiparticles in the coherent cavity region are coupled to the condensate, the individual currents jc and jv are Because the condensate is charge neutral, electric cur- not separately conserved. In analogy with the spin den- sity in a spin- 1 system, we can define a difference density rent in the QW is carried by quasiparticle excitations 2 2 2 only. For energy E > δ0 /2 the cavity region has two ρ− ≡ |u| − |v| which can be interpreted as the exciton ky probability distribution. The associated continuity rela- conduction-band dominated propagating Schr¨odinger ∗ + tion has the form ∂ρ−/∂t+∇xj− = 4(˜gψ+∆)Im{u v}/~, equation solutions with real wavevectors ±kx , and two valence-band dominated evanescent Schr¨odingerequa- where j− ≡ jc − jv is the corresponding electron-hole (or tion solutions with imaginary wavevectors ±k−. Here counter-flow) current. The source(or drain) term on the q x right-hand side is proportional to the coherence ampli- k± = 2m −δeff/2 ± (E2 − (gψ + ∆)2)1/2/ 2. We solve x ky ~ tude and large in the transition region between contact the elastic scattering problem by matching the wavefunc- and cavity (Fig. 3(b)). Since the microscopic model ne- tion and its derivative at the two interfaces and con- glects processes, other than coupling to the cavity pho- serving transverse momentum20,21. Quasiparticle flow is ton, which violate separate particle-number conservation, characterized by the counter flow of its conduction and a compensating source or drain term appears in the equa- valence component. From the mean field equation, we tion of motion for the condensate and restores the sym- 4 metry. (See Appendix A for a formal analysis.) These 3.5 conservation terms are precisely analogous to those that produce Cooper-pair source and drain terms when cur- 3 7 22 6 Itot Is )

rent flows across a normal-metal/superconductor inter- 1

− 5 23 2.5 face, and to the spin-transfer torque contribution to the m

µ 4 ) · 1 − equation of motion for a magnetic condensate in inho- A 2 3 102 m m

( × µ mogeneous magnetic systems. In both of these cases the 2 · I

A 1 conservation law is restored by the action of the quasi- µ 1.5 (

I 0 particles on the collective degrees of freedom of the con- 0 0.1 0.2 0.3 densate. In our case, because the quasiparticles interact 1.0 eV˜b(eV) much more strongly19 with the excitonic part of the con- densate than with the cavity photons, the quasiparticles 0.5 act primarily as a source for the excitonic part of the con- 0.16 ∗ 0 densate. It follows that ∇xjs = −2(˜gψ + ∆)Im[u v]/~. 0 0.002 0.004 0.006 0.008 0.01 0.012 Including this superfluid component, the conservation is eV˜b(eV) recovered for the total electron-hole probability current je-h = j− +2js, where the factor 2 appears because we de- FIG. 4: (Color online) Current-voltage characteristics at tem- fine js as the pair probability current. Fig. 3(b) shows the peratures T = 2K, 10K and 20K (increasing from bottom to divergence of the probability currents along the transport top.) V˜ is the effective bias voltage in the rotating frame. direction which represents the rate of transfer between The grey line marks the threshold at the quasiparticle gap ˜ 0 min the quasiparticles and the condensate. eV = δ0/2. Black dot marks the minimum current Is re- As indicated in Fig. 3, the length scale over which the quired for an observable momentum shift at T = 20K. quasiparticle-condensate transfer occurs is short. From our calculation we can associate this length with the decay length of the evanescent valence-band dominated while µL = eVb > 0 varies. The Schr¨odingerequation Schr¨odingerequation solutions at conduction band ener- is time-independent in the rotating frame in which the gies. Away from the interfaces, only propagating modes time-dependence eiωt is gauged away. Provided that elec- sat survive so js saturates at its asymptotic value js . An trons are injected into and extracted from the conduction analytical result can be obtained for jsat as follows. Con- band the gauge transformation does not shift the effective s ˜ sider two cross sections one deep in the left contact bias voltage Vb. We calculate the saturated supercurrent x = −∞ and another in the asymptotic region of the from Eqs.(7) and Eq.(8) which yield quantum well, e.g. near x = L/2. For an electron inci- Z dent at an energy E > δ0 /2, conservation laws require e X  ˜  ky Is = T (E, ky)α(E) f(E − eVb) − f(E) dE hLy k sat y jc|x=−∞ = jc|x=L/2 + js , (5) (9) sat jv|x=−∞ = jv|x=L/2 − js . (6) where we have arbitrarily multiplied the exciton number For transport through conduction quasiparticle channel, current by e in order to place both currents in the same jc|x=−∞ = jin = j and jv|x=−∞ = 0. Using the relation units. Here α(E, ky) = α(E) is independent of ky and 2 2 sat (jc/jv)x=L/2 = −uk/vk, we have js = jv|x=L/2 and vanishes at energies far above the band gap. As a conse- quence, when the voltage is increased, the supercurrent sat 2 js vk will at first increase gradually, before saturating at higher = − ≡ αk, (7) 2 2 voltages. (See the inset of Fig. 4.) The supercurrent Is j uk − vk is plotted as a function of the total input charge current 2 2 where uk = (1 + ξk/Ek)/2 and vk = (1 − ξk/Ek)/2 are Itot in Fig. 5. At low bias voltages, Is increases linearly 2 2 the coherence factors . Because uk − vk > 0, the minus with Itot with a slope proportional to the exciton den- sign indicate the direction of supercurrent density js is sity nex. When the bias moves deep into the conduction opposite to the particle current density j. band, Is saturates, in agree with the saturation shown in We calculate the total charge current density in the the I − V curve. contact using the Landauer-B¨uttiker formula Z −e X   III. EXCITON-PHOTON TRANSFER Itot = T (E, ky) f(E − eV˜b) − f(E) dE (8) hLy ky In this section, we address the length scale over which where e > 0 is the unit of electric charge and f(E) = the exciton supercurrent is converted to a lower polariton 1/[e−(E−µR)/kB T ] the Fermi distribution function. We supercurrent. The large value of this length scale com- have assumed that the chemical potential of the right pared to the length scale on which the exciton supercur- contact is fixed at the middle of the bandgap µR = 0 rent is generated justifies our simplified two-step model. 5

supercurrent carried in the bulk of the coherent region 25 where the ratio between photon and exciton parts of the condensate is that of the LP mode. The length scale over 20 which the LP form of the condensate is recovered is the )

1 decay length λUP of the evanescent upper polariton (UP) − 15 m

µ wave which satisfies · A µ ( 10 s p I 0 2 2 −2 2mUP (~ω0 − ex) + 4Ω λUP = 2 (12) 5 ~ assuming that ~ω is close to the LP band bottom. 0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Given the typical detuning ω0 −  ∼ 10meV and 1 ~ ex I (mA µm− ) · Rabi coupling Ω ∼ 10meV values we estimate that 2 2 4 λ ∼ 200nm. Generally λ /ξ ≈ m0/mph ∼ 10 , pro- FIG. 5: (Color online) Supercurrent as a function of total 0 charge current at T = 2K. The parameters used here are the vided that the renormalized gap δ0 and the UP-LP split- p 0 2 2 same as in Fig.(4) ting (~ω0 − ex) + 4Ω are similar in magnitude. This large ratio validates our simplification of the electronic to polaritonic current conversion process into two steps oc- To identify this length scale, we turn to the bosonic de- curring sequentially in space. scription of coupled coherent exciton Φex and coherent photon Φph fields. These are described by the coupled linear Gross-Pitaevskii equations24,25 IV. CONDENSATE MOMENTUM IN THE SUPERFLUID STATE  2 2  ~ ∇ 0 ~ωΦex(r) = − + ex Φex(r) + Ω(x)Φph(r), (10) 2mex When it carries a supercurrent the condensate has a  2 2  non-zero phase gradient. As explained above, over the ~ ∇ ~ωΦph(r) = − + ~ω0 Φph(r) + Ω(x)Φex(r), bulk of the coherent region we expect a pure LP con- 2mph densate with uniform density and phase gradient, i.e. (11) i(ksx−ωt) ΦLP(r, t) = φLPe where ks is the condensate mo- mentum in the x direction. The condensate momentum where we have neglected the weak nonlinear term that and current density are related by accounts for exciton-exciton interactions and assumed that the condensate is in a quasi-equilibrium in which I = en v (13) compensating pump and decay terms can be ignored. s s s We focus on the right boundary of the cavity where 2 where ns = |φLP| is the density and vs = ~ks/mLP the exciton supercurrent is generated and flows towards the superfluid velocity of the LP condensate. Owing left boundary, and define the cavity region by impos- to the dynamic nature of polariton condensates, pho- ing a step-like profile on the Rabi coupling: Ω(x) = tons are emitted continuously out of the cavity, car- Θ(x)Θ(L − x)Ω0. The reverse process at the left bound- rying precise information about the momentum space ary region of the cavity can be considered in a similar distribution of the parent polariton particles3,26. The fashion. In the bulk, the resonant coupling between exci- condensate momentum ks can thus be measured us- ton and photon modes produces two new eigen-branches, ing angle-resolved emission spectroscopy.27 At zero bias, the lower polariton(LP) and the upper polariton(UP), the quasi-equilibrium condensate can either be formed which have quadratic dispersion around zero momen- −1 −1 spontaneously at zero momentum using a standard non- tum and effective masses mUP/LP ≈ mph 1 ± (~ω0 − resonant pumping scheme, or pumped directly with a 0 p 0 2 2  ex)/ (~ω0 − ex) + 4Ω /2. resonant laser pump at finite momentum ks 6= 0. When − At x = L the exciton component of the condensate an electrical bias is applied, ks is shifted with respect to carries a finite supercurrent. The condensate may also the unbiased ground state (e.g. ks = 0) and the emitted have a photon component which does not carry a cur- photons acquire an extra transverse momentum ∆k = ks. rent. When the condensate is decomposed into UP and Quasiparticle-condensate transfer can be observed if a LP components, and the condensate has formed with a detectable momentum shift ∆k is achievable. Because chemical potential ~ω near the bottom of the LP branch, of their finite size, even deep in the condensate regime, its UP component is evanescent. Because Rabi coupling polariton condensates have a finite width in their momen- conserves total boson number while allowing for conver- tum distribution, which places a fundamental limit on the sion between excitons and photons, the total current is resolution of the condensate momentum ks. It has been identical with the exciton supercurrent generated at the shown28,29 that the 1/e width (or the FWHM) of the mo- first step. It follows that the exciton supercurrent gener- mentum distribution above the threshold pumping den- ated by quasiparticle-condensate transfer at the bound- sity is typically 0.5 to 1µm−1. As the pumping density ary of the coherent region is identical to the polariton increases, the momentum distribution broadens slightly 6 possibly due to polariton-polariton interactions. Taking it is transformed by Rabi coupling into a purely lower- the minimum detectable ∆k interval as ∆kmin ∼ 1µm−1 polariton condensate current over a longer length scale. and using Eq.(13), we estimate that the corresponding This simplification allows us to ignore the finite momen- min LP supercurrent density for ks = ∆k is, tum of the condensate when we calculate the conduction and valence band contributions to the quasiparticle cur- min −1 −1 Is ≈ 1.6 × 10 µA · µm (14) rent, and is justified by the large difference between the length scales of the two processes. −4 −2 where we have taken mLP = 10 m0 and nLP = 1µm , The process by which a quasiparticle current driven which is slightly larger than the threshold density. We through a polariton condensate generates a polariton min have marked the bias voltage at which Is is reached on supercurrent is partly analogous to the process by the Is − V curve in Fig. 4. The required voltage is well which a quasiparticle current driven through a normal- in the sub-gap regime in which the transport is mainly superconductor-normal metallic circuit generates a su- due to the tail of the Fermi distribution. percurrent in the superconducting metal. Andreev scat- tering is a process which involves change of band char- acter of the outgoing particle. In the case of a metal- V. SUMMARY AND DISCUSSION superconductor junction, broken particle-number conser- vation in the mean-filed Hamiltonian allows both An- In this article we have described a mechanism by which dreev reflection (AR), a process in which an incident elec- an electrical bias voltage applied across a unipolar semi- tron is reflected as a hole on the normal-metal side of the conductor quantum well can drive an exciton or polari- junction, and Andreev transmission (AT), a process in ton supercurrent, and illustrated using a simplified model which an incident electron is transmitted into the super- calculation that assumes ballistic transport and abrupt conductor with partial hole-like character. Because the boundaries between a cavity region in which a polari- contacts are n-type, the model we have studied for polari- ton condensate is established. Our mechanism is based ton condensation does not allow the analog of AR or AT, on spatially inhomogeneous mixing between conduction but does allow a normal transmission channel involving and valence bands in quantum wells or two-dimensional the dressed conduction quasiparticle band. The violation materials when they are dressed by interactions with the of particle-number conservation is weaker in the polari- coherent exciton and photon fields of a polariton conden- ton case because the dressed conduction bands are still sate. Because of this dressing, the mean-field Hamilto- dominantly conduction band in character, but the effect nian of the quasiparticles, unlike the microscopic Hamil- that launches a Cooper-pair supercurrent in a supercon- tonian, does not conserve particle number in conduction ductor is still similar to the effect discussed here. and valence bands separately. The conservation law is re- The simplified model we have used to illustrate our stored by the action of the quasiparticles on the conden- quasiparticle-condensate effect assumes ballistic scatter- sate, and this action is the source which generates an ex- ing. Even in samples that are completely free of disorder, citon supercurrent. A similar mechanism applies to pure this assumption is not realistic. For example, the high- excitonic condensates, and indeed is thought to produce energy electrons emitted from the source can scatter by an excitonic supercurrent when a charge current flows emitting phonons. Even more important, the scenario we through the strong-field quantum-Hall excitonic conden- are imagine here requires some separate process, whether sate state of bilayer electron systems30. In the polariton optical or electronic, that creates a population of conduc- case, unlike the quantum Hall case, the supercurrent can tion band electrons and valence band holes that supports in principle be directly measured by examining the pho- the polariton condensate steady state. Electronic quasi- tons that leak out of the cavity. According to a qualita- particles will scatter off these electrons or holes, or off the tive estimate based on the simplified model calculation, population of non-resonant excitons that appears when the effect is strong enough to produce a measurable con- they form bound states. Scattering is important in lim- densate momentum shift. iting the amount of charge current that can flow through The polariton steady state in the presence of an elec- the system at a given bias voltage. Because the supercur- tronic bias voltage should in principle be determined rent is generated immediately upon entry of the quasi- by solving coupled mean-field equations that differ from particles into the coherent region, however, we do not those described in Ref. 19 only because the fermionic expect that the amount of supercurrent generated for a quasiparticles are in contact with source and drain reser- given charge current will differ greatly from the estimate voirs that have different chemical potentials. In such a we have obtained. calculation, the finite momentum of the polariton would Quasiparticle-condensate transfer can be used to de- emerge naturally from the self-consistent calculation. A flect polariton supercurrents providing, for example the similar calculation has been performed previously31 for a possibility of electrical control of polariton flow at junc- model excitonic condensate. In the present case we have tions. Because Rabi coupling converts the condensate simplified the calculation by assuming a two step pro- current to a lower polariton current, and the lower po- cess in which the condensate current is generated very lariton state is in general spread across a number of quan- close to the boundary of the coherent region, and that tum wells or two-dimensional material layers, the dressed 7

T quasiparticle Hamiltonian in one layer can be influenced where Ψˆ = (Ψˆ c, Ψˆ v) and σ is the Pauli matrix by bias voltages applied to another layer separated by in the two-band basis. Because of this spontaneous some number of photon wavelengths. We anticipate that symmetry breaking, the continuity equation associated † the effect we have described might therefore enable in- with the electron-hole densityρ ˆe-h = Ψˆ (r)σzΨ(ˆ r) ˆ  †  teresting possibilities for coherent electrical coupling be- and the counter flow current je-h = Ψˆ (r) pΨ(ˆ r) − tween two-dimensional electron systems separated by mi- pΨˆ †(r)Ψ(ˆ r)/(2m) is no longer satisfied. Because the cron length scales. full Hamiltonian of the system respects this symmetry, This work was supported by the Army Research Office we expect the continuity equation to hold. In the steady under Award No. W911NF-15-1-0466 and by the Welch state case considered in Sec. II, the conservation of the Foundation under Grant No. F1473. counter-flow current is restored by assuming that a fi- nite condensate supercurrent is induced which cancels the non-conserving part of j− exactly. As pointed out in Appendix A: Self-consistent microscopic theory of Sec. II, in the heavily studied analogous inhomogeneous quasiparticle-condensate transfer magnetic systems the source(drain) term acts as a torque and is referred to as the spin-transfer torque. In this Appendix, we provide a microscopic expla- In the following, we show that this conservation of nation of supercurrent generation due to quasiparticle- counter-flow current is naturally satisfied in a self- condensate transfer based on the time-dependent consistent manner in the TDHF approach. The Hartree-Fock (TDHF) formalism. We focus on a purely quasiparticle-condensate transfer effect can be illustrated excitonic case for simplicity. The polariton case can be by mapping to a metallic system with ferromagnetic or- derived similarly by including the photon field and its der. coupling to fermionic degree of freedom. Define The BCS-like mean field Hamiltonian employed in our calculation breaks the global U(1) symmetry under which 0 ˆ † ˆ 0 ραβ(r, r ; t) = Ψα(r, t)Ψβ(r , t) (A2) the fermionic field operator transforms as where α, β = c, v. The equation of motion of the density ˆ ˆ 0 iθσz ˆ Ψ → Ψ = e Ψ. (A1) matrix writes

Z ∂ 0 X 00 00 00 0 00 00  i ρ (r, r ; t) = dr h 0 (r, r ; t)ρ 0 (r , r ; t) − ρ 0 (r, r ; t)h 0 (r , r; t) (A3) ~∂t αβ αα α β αα α β α0 where the full TDHF Hamiltonian is32 Z 0 0  X 00 00 00 00  0 0 0 hαα0 (r, r ; t) = hr|Tˆα|r iδαα0 + dr V (r − r )ρββ(r , r ; t) δ(r − r )δαα0 − V (r − r )ραα0 (r, r ; t), (A4) β

ˆ 2 ˆ where Tc(v) = ±(p /2m + Eg/2) is the kinetic energy where ρe-h = hρˆe-hi, je-h = hje-hi and we define operator. We simplify our discussion by approximat- ing the Coulomb interaction by the contact interaction ∆(r, t) = V ρvc(r, t). (A7) V (r) = V δ(r) which is valid when the relevant length scale is much longer than the exciton size. We also ignore This definition can be viewed as a self-consistency equa- the Hartree term of the interacting Hamiltonian since it tion which, when satisfied, will lead to the cancellation of doesn’t contribute to the “torque” effect caused by inter- the two terms in the second line of Eq. (A6). In a steady band coherence. Define state case ∂ρ/∂t = 0, je-h is conserved. The condensate-quasiparticle effect can be understood ραβ(r, t) = ραβ(r, r; t). (A5) by considering the time evolution of a state with a bare conduction electron with finite velocity traveling toward We obtain an equilibrium exciton condensate at time t = 0. The ∂ density matrix can be decomposed into ρe-h(r, t) = − ∇ · je-h(r, t) ∂t ρ(t) = ρs(t) + ρqp(t) (A8) 2  ∗  − ∆(r, t)ρcv(r, t) − ∆ (r, t)ρvc(r, t) i~ where ρs and ρqp corresponds to the condensate part and (A6) the quasiparticle part, respectively. If we assume that ρs 8

qp is kept unchanged at its initial value, only ρ evolves where ∆ = Re[∆]ex + Im[∆]ey is the effective in-plane according to the equation of motion Eq.(A6). This is the field and s⊥ the in-plane component of s. jz ≡ je-h is approximation used in Eq. (3) and (4) where a super- the current with spin polarized in z direction. The sec- current is included ad hoc. When the evolution of ρs is ond term on the right-hand side of Eq. (A10) is the well included, ρs and ρqp undergo a mutual evolution in which known spin-transfer torque term. When the spin polar- quasiparticle-condensate transfer occurs. ization of the injected current differs from the exchange To show this transfer effect more explicitly, we map field direction ∆, a mutual torque between the spin of the equation of motion of the exciton condensate system the current and the exchange field is generated and ro- to that of a ferromagnetic metal by defining tates the direction of both of them around z axis. This rotation is equivalent to a change in phase of ∆(r, t). X s(r, t) = σαβραβ(r, t). (A9) The propagation of this phase perturbation corresponds 33 αβ to the propagation of spin-wave . In a steady state, we expect a finite phase gradient of ∆(r, t) is generated Under this mapping, Eq. (A6) becomes which corresponds to a steady exciton supercurrent.

∂sz 4 = −∇ · jz + ∆ × s⊥ (A10) ∂t ~

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