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Understanding Excitons Using Spherical Geometry Author's personal copy Physics Letters A 376 (2012) 1997–2000 Contents lists available at SciVerse ScienceDirect Physics Letters A www.elsevier.com/locate/pla Understanding excitons using spherical geometry Pierre-François Loos Research School of Chemistry, Australian National University, Canberra ACT 0200, Australia article info abstract Article history: Using spherical geometry, we introduce a novel model to study excitons confined in a three-dimensional Received 24 March 2012 space, which offers unparalleled mathematical simplicity while retaining much of the key physics. This Received in revised form 1 May 2012 new model consists of an exciton trapped on the 3-sphere (i.e. the surface of a four-dimensional ball), Accepted 5 May 2012 and provides a unified treatment of Frenkel and Wannier–Mott excitons. Moreover, we show that one Available online 8 May 2012 can determine, for particular values of the dielectric constant !,theclosed-formexpressionoftheexact Communicated by R. Wu wave function. We use the exact wave function of the lowest bound state for ! 2tointroducean = Keywords: intermediate regime which gives satisfactory agreement with the exact results for a wide range of ! Exciton values. Frenkel exciton 2012 Elsevier B.V. All rights reserved. Wannier–Mott exciton Exact solution Spherical geometry 1. Excitons An exciton (X) is a quasiparticle created by the association of an electron (e) and an electron hole (h) attracted to each other by the Coulomb force [1,2].Theelectronandholemayhaveei- ther parallel or anti-parallel spins, and form an electrically neutral quasiparticle able to transport energy without carrying net electric charge. The concept of an exciton was first proposed by Frenkel [3] in 1931 to described excitations in insulators. In semiconductors, aholeisusuallycreatedwhenaphotonisabsorbed,andexcites an electron from the valence to the conduction band, yielding a positively-charged hole. In such materials, excitons play a key role in optical properties [4]. Fig. 1. Four-level excitonic system as a prototype of a quantum gate. These systems are of particular interest in quantum information and computation to construct coherent combinations of quantum ensure quantum coherence between 00 and 11 leading to errors | ! | ! states [5].FollowingDiVincenzo’stheory[6],quantumgatesop- in the quantum logic device [7]. erating on just two qubits at a time are sufficient to construct There are two main kinds of excitons: Frenkel excitons [3] ageneralquantumcircuit.Thebasicquantumoperationscanbe (sometimes called molecular excitons [2])arefoundinmaterials performed on a sequence of pairs of physically distinguishable where the dielectric constant is generally small, and are character- quantum bits and, therefore, can be illustrated by a simple four- ized by compact, localized wave functions. Wannier–Mott excitons level system, as shown in Fig. 1.Inopticallydrivensystems,direct [8,9] are found in semiconductors with a large dielectric constant, excitation to the upper 11 level (lowest biexciton1 state) from and have large, delocalized wave functions. | ! the ground state (GS) 00 is usually forbidden and the most effi- | ! cient alternative is to use two distinguishable excitonic states with 2. The model orthogonal polarizations ( 01 and 10 )asintermediatestates. | ! | ! However, in atomic systems, excitation to the upper level does not In 1983, Laughlin [10] proposed an accurate trial wave function in order to explain and predict the fractional quantum Hall effect (FQHE), and eventually received the Nobel prize in physics in 1998 E-mail address: [email protected]. (jointly with Horst L. Störmer and Daniel C. Tsui) for the discov- 1 Abiexciton(B)correspondstotheassociationoftwoexcitons. ery of this new form of quantum fluid with fractionally charged 0375-9601/$ – see front matter 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.05.010 Author's personal copy 1998 P.-F. Loos / Physics Letters A 376 (2012) 1997–2000 excitations. A few months after the publication of Laughlin’s pa- Here, we consider exciton states with zero angular momentum per, Haldane [11] introduced spherical geometry for the study of and in which the two particles have opposite spin (singlet states). the FQHE, wherein the two-dimensional sheet containing electrons The present study can be easily generalized to higher angular mo- is wrapped around the surface of a 2-sphere, and a perpendicu- mentum states for both the singlet and triplet manifolds [20]. lar (radial) magnetic field is generated by placing a Dirac magnetic For zero angular momentum states, Ψ depends only on the rel- monopole at the centre of the 2-sphere. This geometry has played ative coordinate ω or u.Ananalysisof(1) reveals that Ψ ,likethe an important role in testing various theoretical conjectures. The para-positronium wave function, possesses an “anti-Kato” behav- main reason for the popularity of this compact geometry is that ior [24] it does not have edges, which makes it suitable for an investiga- Ψ %(u) 1 tion of bulk properties. Spherical geometry has been instrumental , (5) Ψ(u) =−2 in establishing the validity of the FQHE theory, and provides the 'u 0 cleanest proof for many properties [12]. ' = which' shows that Ψ must behave as Ψ(u) 1 u/2 O (u2) for Following Haldane’s footsteps, we introduce a simple model ' = − + using spherical geometry to study excitons confined in a three- small interparticle distance. After a suitable scaling of energy (E µ R2), the Schrödinger dimensional space for any value of the dielectric constant. It yields ← E aunifiedtreatmentofFrenkelandWannier–Mottexcitons[13], equation (1) is and we will show that one can determine, for a particular value 1 of the dielectric constant, the closed-form expression of the exact Ψ %%(ω) 2cotωΨ %(ω) E Ψ(ω) 0, (6) + + !√2 2cosω + = wave function associated with the lowest bound state (i.e. associ- ! − " ated with a negative energy). where ! 1/(µR) can be regarded as the relative dielectric con- = Our model consists of an exciton trapped on the 3-sphere (i.e. stant of the system, and µ 2m m /(m m ) is the reduced = e h e + h the surface of a four-dimensional ball). Excitons on the surface mass of the exciton [25]. of a 2-sphere have been previously studied theoretically [14,15] and experimentally [16,17].However,ourmodelhastheadvan- 4. Frenkel regime tage of having the same dimensionality as real three-dimensional solid-state or molecular systems. Moreover, previous studies on In the Frenkel (small-!)regime(Ψ Ψ and E E ), the ≡ F ≡ F two-electron [18–21] and many-electron systems [22] have shown Coulomb interaction is dominant and the electron and hole form many similarities between real and spherically-confined systems. atightlyboundpair(ω 0) [14,26].Assumingcotω (2 1/2 1 ≈ ≈ − In Ref. [15],Pedersenreportsexactsolutionsfortheunbound 2cosω)− ω− ,wefind states (i.e. associated with a positive energy) of an exciton on the ≈ surface of a 2-sphere based on the recursive approach developed in 2 1 Ψ %%(ω) Ψ %(ω) EF ΨF(ω) 0. (7) Ref. [18].However,excitonsonathree-dimensionalsphericalsur- F + ω F + !ω + = face have not been considered before, and this Letter presents the ! " The above equation is a hydrogenic-like Schrödinger equation, and first study of exact solutions and asymptotic regimes of excitons in yields, for the nth bound state, the following zeroth-order eigen- asphericalthree-dimensionalspace.Toourbestknowledge,thisis function and eigenvalue: also the first study reporting an exact closed-form solution associ- ated with the lowest bound state of an exciton. (n, ) ω ω Ψ 0 (ω) L1 exp , (8) F ∝ n n! − 2n! 3. Schrödinger equation ! " ! " (n, ) 1 E 0 , (9) Let us consider an exciton created on the surface of a 3-sphere F =−4!2n2 of radius R.Thecoordinatesofaparticleona3-spherearegiven m where n N∗ and L (x) is a generalized Laguerre polynomial [27]. by the set of hyperspherical angles Ω (θ,φ, χ).Inatomicunits ∈ n = The lowest state (n 1) is a bound state, and the zeroth-order (h e 1), the Schrödinger equation of the system is (1,0)= ¯ = = wave function ΨF is an exponential function strongly peaked at 2 2 1 (1,0) 2 e h ω 0associatedwiththezeroth-orderenergyEF 1/(4! ). ∇ ∇ Ψ(Ωe,Ωh) Ψ(Ωe,Ωh), (1) = =− 2m R2 + 2m R2 − u = E Taking into account the first-order correction (cot ω ω 1 ω/3 e h ≈ − − ! " and (2 2cosω) 1/2 ω 1 ω/24) yields where − − ≈ − + (1,1) 1 9 2 1 ∂ 2 ∂ 1 ∂ ∂ E , (10) sin θ sin φ F 2 ∇ = 2 ∂θ ∂θ + sin φ ∂φ ∂φ =−4! − 8 sin θ # ! " ! " 1 ∂2 which is plotted in Fig. 2,andhastheeffectofstabilizingthe (2) lowest bound state. A similar calculation for an exciton on a 2- 2 2 (n, ) + sin φ ∂χ sphere [14] yields E 0 1/(!2(2n 1)2).Itshowsthat,similar $ F =− − is the Laplace operator in hyperspherical coordinates [23], me and to anisotropic semiconductors [28],thebindingenergyofthelow- mh are the masses of the electron and the hole, and est state of the two-dimensional exciton is four times larger than in three dimensions. The similarity between flat and spherical ge- 1 1 1 u− r r − R√2 2cosω − (3) ometries is actually not surprising because, in the small-! (large =| 1 − 2| = − radius) limit, the surface of a 2- or 3-sphere is locally flat, and the is the Coulomb interaction between the two particles, where the % & tightly bound pair behaves as on a flat space. cosine of the interparticle angle is r2 r2 u2 5. Wannier–Mott regime cos ω 1 + 2 − = 2r1r2 In the Wannier–Mott (large-!)regime(Ψ ΨWM and E cos θe cos θh sin θe sin θh cos φe cos φh ≡ ≡ = + EWM), the kinetic energy is dominant, and the exciton is uniformly sin θ sin θ sin φ sin φ cos(χ χ ). (4) delocalized over the 3-sphere [29].Thisregimecanbestudied + e h e h e − h Author's personal copy P.-F.
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