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Polariton Supercurrent Generation in Unipolar Electro-Optic Devices

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Polariton Supercurrent Generation in Unipolar Electro-Optic Devices Polariton Supercurrent Generation in Unipolar Electro-optic Devices Ming Xie,1 David Snoke,2 and A. H. MacDonald1 1Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA 2Department of Physics and Astronomy, University of Pittsburgh, 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 exciton 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 excitons or polaritons 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 g (r) + ∆(r) H = 0 (1) � g (r) + ∆(r) −H0 � � � � where H = − 2r2=2m + δ=2 and r 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 − ~! SUPERCURRENT where ~! 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- ^ −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- ^ i cal system is a lateral n-i-n homojunction. As we ex- where U(t) = exp(− 2 !tσz) 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 = ha^i 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 ^ 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 g 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 g = 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.
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