PHYSICAL REVIEW B VOLUME 48, NUMBER 19
15 NOVEMBER 1993-I
Brief Reports are accounts of completed research which, while meeting the usual Physical Review standards of scientific quality, donot warrant regular articles. A Brief Report may be no longer than four printed pages and must be accompanied by an abstract. Thesame publication schedule as for regular articles is followed, and page proofs are sent to authors.
Delocalization of excitons in a magnetic field
P. SchmelcherTheoretische Chemic, Physikalisch Chemisches Institut, Universitat Heidelberg, Im Xeuenheimer Feld 253,
W-6900 Heidelberg, Federal Republic of Germany(Received 22 February 1993; revised manuscript received 3 August 1993)
The two-body effects for a Mott-Wannier exciton in a magnetic field are investigated. For excitonswith nonvanishing pseudomomentum there exists an outer potential well which contains a class of weak-ly bound and delocalized quantum states. The spectrum and eigenfunctions in this potential we11 are cal-culated in a harmonic approximation.
The behavior of matter in strong magnetic fields be-came in the past two decades a subject of great interest.The numerous investigations in different branches ofphysics like, for example, atomic, molecular, and solid-state physics, have shown that the properties of matter instrong external fields are very different from those of thefield-free case and we, therefore, encounter a variety ofnew phenomena due to the presence of the strong field.The term "strong field" has no absolute meaning butrather indicates that the forces due to the field are com-parable to or even larger than the interaction forces ofthe system. This can happen for different states of thesame physical system on different scales of the absolutefield strength. In particular, it is possible to investigatethe strong field regime of, for example, the hydrogenatom by studying its highly excited Rydberg states at lab-oratory magnetic-field strengths. ' In solid-state physics,the analog of such a "simple" system like the hydrogenatom would be an elementary excitation like, for exam-ple, an exciton. Let us assume that the distance of theparticle and hole is much larger than the lattice spacing,i.e., we consider a Mott-Wannier exciton. For this kindof exciton all effects due to the presence of the lattice canbe included in a simple way in the excitonic Hamiltonianmodel. ' This Hamiltonian model is a phenomenologicalapproach which neglects many residual interactions.However, it represents under certain circumstances agood zeroth-order approximation to the exact excitonicHamiltonian.
Excitons, excitonic molecules, as well as excitonicmatter, have been studied extensively in the literature.They are common phenomena occuring in insulator andsemiconductor physics. Since the dielectricity constantcan become large and since the effective masses of theparticle and hole are very often small compared to theproton and electron mass, the corresponding excitonicRydberg (binding energy of the exciton) can be much
H= (KeBXr) +1 2 12M 2p p
where @=[m mh/(mh m )] (we assume mh )m ). mhand m are the mass of the hole and particle, respective-ly. p = (m~m& /M), M, 8, and e are the reduced and totalmass, magnetic field vector, and dielectricity constant, re-
smaller than the hydrogenic one. This has drastic conse-quences if we turn on a magnetic field: the strong-Geldregime occurs for the ground or first few excited states ofcertain excitons already at laboratory magnetic-fieldstrengths. They are, therefore, ideal objects in order tostudy strong-field effects in the laboratory (see, for exam-ple, Refs. 2, 3, or 68). For an interacting two-body sys-tem, like an exciton, in a homogeneous magnetic field it isnot possible to perform a complete separation of thecenter-of-mass (CM) motion in the corresponding Hamil-tonian. In other words, it is not possible to completelydecouple the center-of-mass and internal motion. Sincethe masses of the particle and hole are, in general, ofcomparable order of magnitude, all two-body effects areof particular importance and should be treated exactly inthe underlying excitonic Hamiltonian. A study of thesetwo-body effects is precisely the subject of the present pa-per. (We remark that both the coupling of the collectiveto the relative motion as well as the two-body effects forthe internal relative motion have been investigated veryrecently ' for the case of the hydrogen atom in a mag-netic field. )
Our starting point is the nonrelativistic Hamiltonianfor the neutral exciton in a uniform magnetic field. Weassume that the pseudoseparation of the center-of-massmotion has been performed (for the details of this well-known transformation we refer the reader to the litera-ture' ' ). The resulting Hamiltonian takes on the fol-lowing appearance:
0163-1829/93/48(19)/14642(4)/$06. 00 48 14 642 Qc1993 The American Physical Society
48 BRIEF REPORTS 14 643
spectively. K is the pseudomomentum. A is the vectorpotential which is in the following chosen to be in thesymmetric gauge A(r)=(1/2)BXr. r is the relativecoordinate of the particle and hole and p its canonicalconjugated momentum. The model Hamiltonian (1) forthe excitation is based on two approximations: theeffective-mass approximation and the dielectric continu-um model for Coulomb screening. These approximationscan hold only if the radius of the exciton is much largerthan the lattice spacings of the solid. The canonical con-jugated coordinate belonging to the pseudomomentum Kis the center-of-mass coordinate R. Since K is a constantof motion, the center-of-mass coordinate is a cyclic coor-dinate and does not appear in the transformed Hamil-tonian (1). However, this does not mean that the center-of-mass motion decouples from the internal motion.Indeed the equation of motion for the center of mass re-sulting from the Hamiltonian (1) reads as follows:
R= K BXr,M M
i.e., the center-of-mass velocity is, apart from a constant,completely determined by the internal relative coordi-nate. From Eq. (2) it is clear that the center of mass is,even for the case of vanishing pseudomomentum, inti-mately coupled to the internal motion. Let us investigatethe classical CM motion of' the exciton for the specialcase of vanishing pseudomomentum K=O and for vary-ing total energy E. The Hamiltonian (1) possesses forK=O an additional internal rotational symmetry aroundthe magnetic-field axis (which is assumed to be orientedalong the z axis) and the projection i., of the internal an-gular momentum onto the magnetic-field axis is, there-fore, a conserved quantity. For simplicity we considerthe case I., =O. The resulting Hamiltonian Ho shows,with increasing energy, a transition from regularity tochaos for the internal motion [r(t), p(t)]. It is now in-teresting to ask the following: what does happen with theCM motion of the exciton if its internal motion passesfrom regularity to chaos?
For regular internal motion the CM trajectories per-form in the plane perpendicular to the magnetic field(note: the CM motion parallel to the field axis is free)quasiperiodic oscillations with small amplitude which areconfined to a bounded range of coordinate space. If weincrease the total energy, chaotic trajectories for theinternal motion appear and take over more and more ofphase space. Eventually, complete phase space becomeschaotic. For chaotic internal motion, the CM motion isno longer restricted to some bounded volume of phasespace but experiences with increasing time an increasingvolume of the two-dimensional coordinate space. Asecond important observation is that the CM motionclosely resembles a random motion which has its origin inthe intrinsic chaotic force e(BXr) [see Eq. (2)].characteristic feature of a random motion is its diffusionlaw, i.e., the time dependency of the traveled mean-square distance. In order to investigate this property forour case of the excitonic CM motion (for the case ofchaotic integral motion) we have calculated the mean-square distance (X + Y ) as a function of time for an
10 x 10'
5 x 105
T I M E T (no iFIG. 1. The mean-square distance (X2+ Y2) of an ensemble
of 500 center-of-mass trajectories in the fully chaotic regime asa function of time. The parameter values are m~=0. 2mmz =5m@=5, B=2.35 T, and a total energy of E= 0. '75meV.
ensemble of 500 CM trajectories. The result is presentedin Fig. 1. Within statistical accuracy, the plot shows alinear dependency (X + Y ) =Dt. D is the diff'usionconstant for the spreading of the ensemble of CM trajec-tories and has, in our case, an approximate value of
0D=5.75X10 A /ns. To summarize, we have shownthat the classical transition from regularity to chaos inthe internal relative degrees of freedom is accompaniedby a transition from localized bounded quasiperiodic todelocalized unbounded randomlike motion in the centerof mass of the exciton. The randomlike CM motion ofthe exciton has diffusive properties in the sense that itobeys a linear diffusion law for the spreading in a planeperpendicular to the magnetic field. This might have im-plications on those properties of solids which have a con-tribution from excitonic processes like, for example, theelectro(optical) properties of semiconductors in a magnet-ic field. However, there are two points which should bekept in mind and which deserve further investigation.The first point is the fact that we have studied the classi-cal CM motion of the exciton and it is not obviously whatwill happen if we quantize both the internal as well as theCM motion. Does the CM diffusion survive or doesquantum localization take place? The second point isthat we have chosen a zeroth-order model Hamiltonianfor the Mott-Wannier exciton in a magnetic field whichneglects many residual