PHYSICAL REVIEW B VOLUME 48, NUMBER 19

Brief Reports

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= (K—eBXr) +1 2 1

2M 2p p —e~Ap

2 2

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 6—8). 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

(2)

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'

I—

5 x 105

C3

LLI 00

T I M E T (no i

FIG. 1. The mean-square distance (X2+ Y2) of an ensembleof 500 center-of-mass trajectories in the fully chaotic regime asa function of time. The parameter values are m~=0. 2m„mz =5m„@=5, B=2.35 T, and a total energy of E= —0. '75

meV.

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 interactions and one may ask howthese interactions perturb and modify the excitonicmotion in the magnetic field. Answers to these questionswould help to clarify the role of the excitonic CM motionfor the properties of solids. However, they go beyond thescope of this paper and are left to future investigations.

Next let us investigate the inAuence of the two-bodycharacter on the behavior and properties of the excitonwith nonvanishing pseudomomentum. Our starting pointis again the Hamiltonian (1) for KAO. The first quadra-tic term of this Hamiltonian represents the kinetic energy

14 644 BRIEF REPORTS 48

of the CM motion (M/2)R [see Eq. (2)] whereas thesecond quadratic term is the internal kinetic energy(p/2)r . The important observation is now that eachterm of the quadratic kinetic energy of the center of mass(1/2M)(K —eBXr) is gauge inuariant and acts as a po-tential energy for the internal relative motion (see Ref. 12for an explanation). From the point of view of the inter-nal motion the Hamiltonian (1) can, therefore, be dividedin a unique way in kinetic (T) and potential ( V) energyterms, i.e., H =T+ Vwhere

2

T= p —— BXr1 e Ll

2p 2 p(3a)

and

V= (K—eBXr)—1

2M

2

(3b)

M

In the following, we discuss the properties of the po-tential V and the resulting consequences for the excitonicdynamics. Apart from the constant K /2M, the poten-tial Vis a combination of a diamagnetic (e /2M )(BXr),a Stark (e/M)(BXK)r, and a Coulomb potential term(—e /e~r~). With the choices B=(O,O, B) and

K=(O, K, O) (K )0) the electric field points along the neg-ative x direction. Figure 2 shows an intersection of thepotential V along the direction of the electric field

(y =z =0). Close to the origin, i.e., for small absolutevalues of the x coordinate, the Coulomb potential dom-inates. With increasing absolute values of the x coordi-nate, the Stark term gains in significance and eventuallybecomes comparable with the strength of the Coulombpotential. The diamagnetic term provides in this coordi-nate region only a small correction to the Coulomb andStark terms. As a consequence of the competition of thelatter two terms, a saddle point arises which is, in Fig. 2,located at approximately x = —807 A. For even largerabsolute values of the x coordinate the Coulomb potentialbecomes smaller and the shape of the potential is more

and more determined by the quadratic potential term(e /2M)B x . Due to the competition of the Stark anddiamagnetic terms the potential V now develops aminimum which is, in Fig. 2, located at approximatelyx = —3004 A. Not only the quantitative appearance ofthe potential V depends, of course, on the values ofM, e,K,B but also the existence of both the saddle point aswell as the minimum. Quantitative conditions for theirexistence will be given below.

The discussed properties of our potential V have im-portant implications on the dynamical behavior of the ex-citon. One observation is that the ionization of the exci-ton, i.e., the infinite separation of the particle and holecan take place only in the direction parallel to the mag-netic field: in the direction perpendicular to the magneticfield, the diamagnetic potential term is dominating forlarge distances and causes a confining behavior of the po-tential V. Another important observation is the fact thatthe minimum of the potential V leads to a potential welland, as we shall see below, to a new class of weakly boundquantum states inside this well. Excitons in these statesare very extended delocalized objects for which the parti-cle and hole are separated far from each other.

Let us turn now to a quantitative investigation of thepotential V. The outer potential well described above andillustrated in the intersection of Fig. 2 exists only if thepotential V possesses both a saddle point and a minimum.In order to obtain a condition for their existence we haveto determine the extrema of the potential V. A simplecalculation shows that the components of the coordinatevectors Iro] of the extrema obey the equations yo =zo=0and

xo+(K/B)xo (M/eB —)=0 . (4)

The existence of both the saddle point and the minimumyield the following condition on the parameter values:

K & "(BM/—e) . (5)

This is a necessary and sufhcient condition which saysthat the pseudomomentum must exceed some criticalvalue in order to form the outer potential well. In the fol-lowing, we assume that the condition (5) is fulfilled. Thequestion arises then: does there exist bound states in theouter potential well and how do they look?

The simplest way to investigate this question is to ex-pand the potential V around its minimum position ro upto second order and to solve the resulting equations ofmotion analytically. The Taylor expansion of V aroundro yields the following approximate potential:

V, =C+(p/2)co x +(p/2)co y +(p/2)co, z, (6)

where

—3--4000 -2000 0

X COORD INATE (A)FIG. 2. The potential V along the direction (x axis) of the

motional electric field. The parameter values are the same as inFig. 1 and in addition we have K=1.3X10 rn, A/ns.

2 B 1

p 2M ex 03

1 B 1

p M Ex 03

1/21 1

CO

elxp I'

48 BRIEF REPORTS 14 645

C=(&~/2~)+(2/exo) —(B /2M)xo is a constant. V,is the potential for a three-dimensional anisotropic har-monic oscillator. The approximate Hamiltonian describ-ing the motion inside the potential we11 has the formH, = ( T+ V, ) where T is the kinetic energy for a particlewith charge e(p/p) and mass p in a magnetic field. Inthe following we investigate the classical trajectories aswell as the quantum-mechanical spectrum of the Hamil-tonian H, . To this end we solve the gauge-invariantNewtonian equations of motion belonging to the Hamil-tonian H, by the ansatz

x aexp(imt), z =c exp(ice, t), (7)

which yields the following solutions for the normalmodes in the (x,y) plane:

—I(co +co +co, )+[(co +co +co, )1

2

4 2 2 )1/2) 1/2CO~ CO&

where co, =(B/p, ) is the cyclotron frequency. We obtaintwo normal frequencies co+ for the oscillatory motionperpendicular to the magnetic field and one frequency co,for the harmonic motion parallel to the magnetic field.The quantum-mechanical spectrum belonging to theabove classical trajectories is a harmonic-oscillator spec-trum and the binding energies of the corresponding statesread as follows:

E =Co (n + + —,'

)co—+ (n + —,'

)co——( n, + —,

')co, ,

where Co =(B/2p) —C and (B/2p) is the zero point en-

ergy of the free particle and hole in a magnetic field, i.e.,the ionization threshold.

For our example in Fig. 2, the binding energy of theground state inside the potential well is approximatelyE~ =0.9 meV. The order of magnitude of the spatial ex-

tension of the ground state is roughly 300 A in both the xand y direction and about 1100 A in the z direction (forcomparison, the corresponding exciton in field-free spacehas a binding energy of 105 meV and a Bohr radius of 14A). The energies associated with the frequencies co+, co

and ~, take on the values E+ =1.36 meV, E =0.053meV, and E, =0.065 meV. Since the threshold energy isapproximately 0.71 meV we have a large number ofbound states in the potential well. However, only theground and first few excited states can, for our exampleof the parameter values, be described accurately withinthe harmonic approximation. For large amplitudes in-side the well, i.e., higher excitations, anharmonicitycorrections become important and our harmonic-oscillator expansion breaks down. The range of validityof the harmonic-oscillator approach depends sensitivelyon the parameter values, i.e., on the masses, dielectricityconstant, and field strength. For certain excitons, anhar-monicity corrections are important already for theground state in the potential well. We remark that thesize of the excitonic states varies strongly for di6'erent pa-rameter values.

For the delocalized quantum states discussed above,the particle and hole are always separated far from eachother and are prevented from coming close together by apotential barrier. In reality, these quantum states possessa finite lifetime due to the possibility of the tunneling pro-cess to the Coulomb singularity well. However, in thebulk the potential barrier might prevent the particle andhole from recombination and might increase the lifetimeof the exciton. It is now a challenging question to askwhether the two phenomena described in this papermight be observable in direct experiments or might con-tribute to certain properties of solids.

The author thanks D. L. Shepelyansky for fruitful andilluminating discussions and T. Pacher for a criticalreading of the manuscript.

iH. Friedrich and D. Wintgen, Phys. Rep. 183, 37 (1989), andreferences therein.

D. C. Reynolds and T. C. Collins, Excitons: Their Propertiesand Uses (Academic, London, 1981).

M. Ueta, H. Kanzaki, K. Kobayashi, Y. Toyozawa, and E.Hanamura, Excitonic Processes in Solids (Springer-Verlag,Berlin, 1986).

4R. R. Sharma, Phys. Rev. 170, 770 (1968).5W. F. Brinkmann, T. M. Rice, and B. Bell, Phys. Rev. B 8,

1570 (1973)~

M. Altarelli, and N. O. Lipari, Phys. Rev. B 9, 1733 (1974).7S. T. Chui, Phys. Rev. B 9, 3438 (1974).8K. J. Bajaj and C. H. Aldrich, Solid State Commun. 35, 163

(1980).P. Schmelcher and L. S. Cederbaum, Phys. Lett. A 164, 305

(1992); Z. Phys. D 24, 311 (1992); Phys. Rev. A 47, 2634

(1993).I. Dzyaloshinskii, Phys. Lett. A 165, 69 (1992).D. Baye, N. Clerbaux, and M. Vincke, Phys. Lett. A 166, 135(1992).P. Schmelcher and L. S. Cederbaum, Chem. Phys. Lett. 208,548 (1993).J. E. Avron, I. W. Her-bst, and B. Simon, Ann. Phys. (N.Y.)114, 431 (1978).

~4H. Herold, H. Ruder, and G. Wunner, J. Phys. B 14, 751(1981).

i58. R. Johnson, J. O. Hirschfelder, and K. H. Yang, Rev. Mod.Phys. 55, 109 (1983).

' M. Vincke and D. Baye, J. Phys. B 21, 2407 (1988).' P. Schmelcher, L. S. Cederbaum, and H. D. Meyer, Phys.

Rev. A 38, 6066 (1988).