PHYSICAL REVIEW B VOLUME 36, NUMBER 14 15 NOVEMBER 1987-I

Anomalous displacement of excitons in a quantum well

T. OdagakiDepartment of Physics, Brandeis Uniuersity, }vValtham, Massachusetts 02254

(Received 15 April 1987; revised manuscript received 19 May 1987)

1s excitons confined in a quantum well under a static electric field perpendicular to the well lay-

ers are studied. The exciton in the excited levels of the quantum well is shown to be displaced to-ward the direction opposite to the field for weak fields and as the field is increased further it moves

toward the direction of the field, in striking contrast with the exciton in the ground level of thewell, which is always pushed toward the direction of the field for these strengths of the field. Thepolarization of the exciton in the excited levels changes its sign during the course of this anoma-

lous displacement.

The recent progress of crystal-growth techniques hasmade it possible to fabricate high-quality heterostruc-tures with controlled configurations. ' The best knownheterojunction is GaAs/Ga& „Al As where GaAs has anarrower band gap than Ga

&Al As. In the

Ga, Al As/GaAs/Ga& Al„As heterostructure, elec-trons and holes are confined within the GaAs well. It isknown that a photoexited electron-hole pair forms an ex-citon which is also confined in the quantum well. Theproperties of excitons in the heterostructure have beenstudied by various authors. The Stark effect of theexciton has also been studied by Miller et al. '

In this Brief Report I study the displacement of exci-tons in the quantum well under a static electric field per-pendicular to the quantum-well layers. It will be shownthat, when the electric field is increased, 1s excitons inthe excited levels of the quantum well are first pulled to-ward the direction opposite to the field and then towardthe same side of the layers as the field points. The dis-placement of excitons is a striking contrast to the dis-placement of a 1s exciton in the ground level of thequantum well which is always pulled toward the sameside of the layers, and to the common-sense idea that acharged particle is displaced toward the direction deter-mined by its charge and mass. For much higher fields,these excitons begin to be polarized as usual.

I assume a sufficiently isolated quantum well and aperfect confinement of electrons and holes in the well.To facilitate the analysis within the effective-mass ap-proximation, I consider the Hamiltonian

2 2 2 2Ex+ByH= +

2m, 2m~ 2p

2

~[x'+y'+(z, —z„}']

+eF(z, —zt, )+ V;,„t(z, )+ V," f(zt, )

Here, m„z, and m&, z& are the effective mass and the zcoordinate (perpendicular to the layers) of the electron

This type of wave function is known to be better than se-parable wave functions for all thicknesses. The functionp'(z, ) [p (zh )] is the exact solution to the Schrodingerequation for a single electron (hole) confined in the wellunder a static field F and given by the linear combina-tion of Airy functions, Ai and Bi:

P'(z, ) ~ Ai(g, )+C, Bi(g, ),Q"(zt, ) ~ Ai(gh ) +Ch Bi(gt, ),

where

g, =a,' z, /L —e'a,

with a, =2m,~

e~

FL /fi and

Ai( &ea —2/3) /Bi( &ea —2/3

)

(3)

and similar formulas for gt, . L is the width of the well.Because of the quantum-size effect, the electron energyand hole energy, c' and c", take discrete values classifiedby quantum numbers n and m (=1,2, 3, . . . ). An exci-

and the hole in the well, respectively; x and y are the rel-ative x and y coordinates of the electron and hole; ~ isthe dielectric constant of the well; p is the reducedelectron-hole mass in the plane of the layers; F is theelectric field perpendicular to the layers; and V,',„f andV,",„f are the confinement potentials for electrons andholes due to the band-edge discontinuity. The center-of-mass kinetic energy of electrons and holes in the layerplane is omitted from the Hamiltonian since it contrib-utes only a trivial factor with unit amplitude to the wavefunctions. I assume a perfect confinement of electronsand holes in the quantum well, which is known to be agood assumption for the electron —heavy-hole exciton foralmost all thicknesses of the well.

I determine the exciton energy and wave functionvariationally taking a nonseparable trial wave function

4(z„zh, x,y ) ~ P'(z, )P"(zt, )

Xexp[ —[x +y +(z, —z„) ]' /A, I .

(2)

36 7653 1987 The American Physical Society

7654 BRIEF REPORTS 36

ton characterized by the nth electron level and the mthhole level will be called the (n, m) exciton. The photoex-citation creates excitons which satisfy the conditionn =m . In the following, I focus on excitons withn =m =1 and n =m =2. I determined the parameter A,

of these excitons for a given field strength by minimizingthe energy expectation value. For the parameterm, =0.067 (in units of the electron mass), mz /m, =6.7,Ir=13 (these values are selected so as to represent theheavy hole in GaAs well), and L =100 A, the effectiveBohr radius A, is found to be approximately 7.1 —7.4 Afor the field strength a, =0—50 (i.e., F=0 7. 5 &(—10V/m).

Figure 1 shows the probability of finding the electronat z, [Fig. 1(a)] and the hole at zi, [Fig. 1(b)] in the (2,2)exciton for various strengths of the field which aredefined by

dx dy dzp(e) + dx dy dzadze

for the electron (hole). Note the direction of the field is

in the positive z direction (I shall call this direction rightand the other left). One can clearly see the probabilitiesare skewed toward the negative z direction for weak

fields and then pulled toward the positive direction as thefield is increased further. The average position of theelectron (solid line) and the hole (dashed line) are shown

in Fig. 2 as functions of the field strength for the (1,1)

and (2,2) excitons. Compared to the displacement of the

(1,1) exciton under the weak field which is to the right,

0.7

0.6

v 05,

50

FIG. 2. Average z coordinates of electron (solid line) andhole (dashed line) in a 1s exciton in quantum levels (1,1) and(2,2) are plotted against the dimensionless field

a, =2m,~

e~

FL /A' . A crossing of these two lines for the(2,2) exciton indicates the negative polarization for a =35—80.

the (2,2) exciton is pulled to the left for field up toa, =20. When the field a, is between 20 and 100 the ex-citon moves toward the positive direction. A crossing ofthe average positions indicates the change in sign of thepolarization of the exciton. I show in Fig. 3 the perspec-tive view of the joint probability ( ~ j ~

N~

dx dy ) forthe (2,2) exciton when a, =10, 30, 50, and 100. Forlower fields the (2,2) exciton has two well-defined peaks

5 I I

a~ =10

0-

ae = 30

0

C3O

CL

ae- 50I

c 50-(DC3

ae -100

050-

0ze/L

(a)

1 0 zh/L

(b)

50-

FICx. 1. Probability density of finding (a) an electron and (b)a hole in a 1s exciton in the (2,2) quantum level under a staticuniform electric field directed along the positive z direction.The electron and hole in the exciton are displaced to the nega-tive z direction for weak fields and for stronger fields theymove toward the positive z direction. For much stronger field,the electron is pulled to the negative direction.a, =2m,

~

e~

FL '/iri2 is the dimensionless field.

0-

FICx. 3. Perspective view of the joint probability density offinding an electron at z, and a hole at zz in a 1s exciton in the

(2,2) quantum level for various field strengths. The electricfield points in the direction from [0,0] to [1,1].

36 BRIEF REPORTS 7655

with more weight in the left side of the well. As the fieldis increased the third peak appears and at the same timeit moves toward the right side of the well and other twopeaks diminish. For fields stronger than a, —100 theelectron tends to move back to the negative directionwhile the hole tends to keep moving towards the positivedirection. An exciton in the ground level shows adifferent displacement: the electron and hole move tothe right for weak fields. For stronger fields it is polar-ized as usual. The displacement of excitons can be ex-plained from the behavior of a charged particle in aquantum well which shows a negative polarization if itis in excited quantum levels of the well. One can under-stand this negative polarization from the classical pointof view. First, the classical electron corresponding to aquantum one in the excited quantum levels carries non-vanishing velocity. When a weak external field is ap-plied the electron decelerates in the direction of the fieldand stays a longer time in the right half of the well thanin the left half. When the field is increased further, theelectron executes a periodic motion between the left walland the classical turning point, and the spatial distribu-tion is skewed to the opposite direction of the field.Similarly a hole is displaced to the left for weak fieldsand then moves towards right.

For the exciton, first note that the reduced field ez forthe hole is mq /m, times larger than a, . Also note thatthe wave functions for electron and hole in the secondexcited level have one node and thus the spatial proba-

bility distribution has two peaks. For fields up toe, —10, the hole is strongly pulled to the left while theelectron stays in more or less symmetric distribution.Thus the exciton is likely to be found in the left half ofthe well. The left peak of the electron wave function isat the left of the peak of the hole wave function and thepolarization of the excition is positive. When the hole ispulled back to the right, so is the exciton. At abouta, =50 the electron is displaced maximally in the posi-tive direction and the exciton shows a negative polariza-tion. For much higher fields the electron and holesbehave normally and exciton shows a positive polariza-tion again.

In conclusion I have demonstrated that 1s excitons inexcited quantum levels in a quantum well show anoma-lous displacement under electric field. Similar effects inexcitons will be observed in real quantum heterostruc-tures, though one has to take account of the finite bar-rier height. It should be noted that the field energyacross the well for a, =100 and L =100 A is still muchsmaller than the band gap in GaAs. Electrons in quan-tum wells show similar anomalous behavior which willaffect various transport properties under electric fieldsuch as the resonant tunneling current.

I would like to thank L. Friedman and P. Norris foruseful discussion. This work was supported in part by agrant from Research Corporation.

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