Excitonic structure in a quantum well under the tilted magnetic fieldE. Kasapoglu, H. Sari, and I. Sökmen Citation: Journal of Applied Physics 88, 2671 (2000); doi: 10.1063/1.1287520 View online: http://dx.doi.org/10.1063/1.1287520 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/88/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Exciton bound to a neutral donor in a parabolic quantum-well wire J. Appl. Phys. 106, 053716 (2009); 10.1063/1.3213333 Hydrogenic-donor impurity states in coupled quantum disks in the presence of a magnetic field J. Appl. Phys. 102, 033709 (2007); 10.1063/1.2764232 Electron-confined longitudinal optical phonon interaction and strong magnetic field effects on the binding energyin GaAs quantum wells J. Appl. Phys. 91, 2093 (2002); 10.1063/1.1430532 Polaronic and magnetic field effects on the binding energy of an exciton in a quantum well wire J. Appl. Phys. 91, 232 (2002); 10.1063/1.1419261 Exciton binding energy in spherical quantum dots in a magnetic field J. Appl. Phys. 86, 4509 (1999); 10.1063/1.371394

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Excitonic structure in a quantum well under the tilted magnetic fieldE. Kasapoglu and H. SariCumhuriyet University, Physics Department, 58140 Sivas, Turkey

I. SokmenDokuz Eylu¨l University, Physics Department, I˙zmir, Turkey

~Received 1 May 2000; accepted for publication 30 May 2000!

By using an appropriate coordinate transform we have calculated variationally the ground stateexciton binding energy in the single square well under the tilted magnetic field. The dependence ofthe binding energy to the magnetic field strength, well width, and the direction of the field isdiscussed. We conclude that for small well widths the exciton binding energy is very sensitive to thetilt angle. © 2000 American Institute of Physics.@S0021-8979~00!05517-1#

I. INTRODUCTION

Effects of the tilted magnetic field on quasi-two-dimensional semiconductor heterostructures have been stud-ied in recent years. Application of a strong magnetic field toa confined electron or hole system has a pronounced effecton spatial quantization. The most of the theoretical and ex-perimental work regarding these systems is concerned withthe configuration in which the magnetic field is perpendicu-lar or parallel to interfaces in which electrons or holes areconfined.1–3 In this case, the Schro¨dinger equation is sepa-rable, and analytic solutions may be found. However, if themagnetic field is tilted in respect to the interface, the vari-ables in the Schro¨dinger equation can not be seperated andvariational4,5 or perturbation6,7 methods have been used. Sofar only the eigenenergies of two-dimensional electrons sub-jected to a tilted magnetic field have been solved analyticallyusing a parabolic potential well.8 Sokmen et al., however,have completely solved the Schro¨dinger equation using asquare well potential as the confining potential and obtainanalytical solutions without making any approximations fortwo-dimensional semiconductor heterostructures under anexternally applied tilted magnetic field.9 In this study, wehave calculated with the use of a variational approximationthe exciton binding energy under the tilted magnetic field asa function of quantum well width, magnetic field, and tiltangle.

II. THEORY

We considered a quantum well withL0 width consistingof a GaAs layer sandwiched between two semi-infiniteGa12xAl xAs slabs. The conduction band of this system isgiven schematically in Fig. 1. In the effective mass approxi-mation, the Hamiltonian of the electron-hole system underthe tilted magnetic field, the value of which in thex-z planeB5(B Cosu,0,B Sinu) is

H51

2meFpe1

e

cA~re!G2

11

2mhFph2

e

cA~rh!G2

2e2

«0ure2rhu1Ve~ze!1Vh~zh!, ~1!

whereVe(ze) andVh(zh) are the confinement potential pro-files for the electrons and holes in thez direction, respec-tively. We have neglected image force effects, which shouldbe rather small because of the nearly equal the electric con-stants of GaAs and GaAlAs we have found that the neglectof the mass discontinuity at the interfaces did not have anyappreciable effect on the excitonic binding energy. Since thecalculations are more complex effects, like hole subbandcoupling, conduction band nonparabolicity is ignored. Thefunctional form of the confinement potentials for the elec-trons and holes are given as

Ve~ze!5V0e~S~zL2ze!1S~ze2zR!!,

~2!

Vh~zh!5V0h~S~zL2zh!1S~zh2zR!!,

whereS is the step function, and the left and right boundariesof the well are located atz5zL52L0/2 andz5zR5L0/2,respectively. The magnetic field can be described by thevector potential A5(0,xB Sinu2zBCosu,0) using the“.A50 gauge, whereu is the angle between the direction ofthe magnetic field andx-axis. So Eq.~1! reduces to

H51

2meFpxe

2 1pye2 1pze

2 12e

cAyepye1

e2

c2 Aye2 G

11

2mhFpxh

2 1pyh2 1pzh

2 22e

cAyhpyh1

e2

c2 Ayh2 G

2e2

«0ure2rhu1Ve~ze!1Vh~zh!. ~3!

By using the following transformation,

S z8x8 D5S Cosu 2Sinu

Sinu Cosu D S zxD . ~4!

The Hamiltonian can be separable and it becomes as

JOURNAL OF APPLIED PHYSICS VOLUME 88, NUMBER 5 1 SEPTEMBER 2000

26710021-8979/2000/88(5)/2671/6/$17.00 © 2000 American Institute of Physics

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H51

2me~px8e

21pz8e

2!1

1

2mepye

2 1eB

mecze8 pye

1e2B2

2mec2 ze8

211

2mh~px8h

21pz8h

2!

11

2mhpyh

2 2eB

mhczh8pyh1

e2B2

2mhc2 zh82

2e2

«0ure2rhu1Ve~xe8 ,ze8!1Vh~xh8 ,zh8!. ~5!

For convenience, if we replacere , rh-electron andhole coordinates, respectively, by the center of massR5(mere1mhrh)/M , (M5me1mh) and relative coordi-nater5re2rh , the Hamiltonian in Eq.~5! becomes

H5PY

2

2M1

py2

2m1

1

2me~px8e

21pz8e

2!

11

2mh~px8h

21pz8h

2!2

eB

Mc~ze82zh8!PY

2eB

c S ze8

me1

zh8

mhD py1

e2B2

2c2 S ze82

me1

zh82

mhD

2e2

«0A~xe82xh8!21~ye2yh!21~ze82zh8!2

1Ve~xe8 ,ze8!1Vh~xh8 ,zh8!, ~6!

wherem5memh /M is the reduced mass. We can drop theterm associated withPY ; since the Hamiltonian does notdepend onY, PY is a good quantum number. Also, the ex-pectation value of the coupling terms linear inB,

eB

c S ze8

me1

zh8

mhD py

is identically zero for the chosen trial wave function.10

With this coordinate transformation the left and rightboundaries of the wells on thez8 andx8 axes are

zL,R8 5zL,R Cosu2x Sinu,~7!

xL,R8 5zL,R Sinu1x Cosu,

respectively. The solution of the corresponding Schro¨dingerequation is not straightforward since, after the coordinatetransformation, the potential energy of the electron and holein the wellVe,h(xe,h8 ,ze,h8 ) couples thex8 andz8 variables. In

order to decompose the potential energy of the electron andhole Ve,h(xe,h8 ,ze,h8 ) we rewrite the step functions in Eq.~2!as follows:

S~zL2ze,h!5Cos2 uS~zL82ze,h8 !1Sin2 uS~xL82xe,h8 !,~8!

S~ze,h2zR!5Cos2 uS~ze,h8 2zR8 !1Sin2 uS~xe,h8 2xR8 !.

By considering the above equations we can separate the po-tential as~see Ref. 9!

Ve,h~xe,h8 ,ze,h8 !5Ve,h~xe,h8 !1Ve,h~ze,h8 !, ~9!

where

Ve,h~xe,h8 !5V0e,h Sin2 u~S~xL82xe,h8 !1S~xe,h8 2xR8 !!,

~10!Ve,h~ze,h8 !5V0

e,h Cos2 u~S~zL82ze,h8 !1S~ze,h8 2zR8 !!.

By scaling all lengths in the exciton Bohr radius(aB5«0\2/me2), and energies in the exciton Rydberg(R5me4/2«0

2\2), and considering the above results, we candecompose the dimensionless Hamiltonian of the system intofive terms: the electron~hole! Hx8

e (Hx8h ) part in thex8 direc-

tion, the electron~hole! part Hz8e (Hz8

h ) in the z8 direction,

and the exciton partHexc:

H5Hx8e

1Hx8h

1Hz8e

1Hz8h

1Hexc, ~11!

where

Hx8e

52m

me

]2

] x8e2 1Ve~ xe8!,

Hx8h

52m

mh

]2

] x8h2 1Vh~ xh8!, ~12a!

Hz8e

52m

me

]2

] z8e2 1

e2B2\2

4mmec2R2 z8e

21Ve~ ze8!,

FIG. 2. Variation of the ground state exciton binding energy as a function ofthe well width foru515° and two different magnetic field values.

FIG. 1. ~a! Schematic representation of the potential well.~b! The directionof axes and externally applied magnetic fieldB.

2672 J. Appl. Phys., Vol. 88, No. 5, 1 September 2000 Kasapoglu, Sari, and Sokmen

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Hz8h

52m

mh

]2

] zh82 1

e2B2\2

4mmhc2R2 zh821Vh~ zh8!, ~12b!

Hexc5py

2

2m2

2

A~ xe82 xh8!21~ ye2 yh!21~ ze82 zh8!2. ~12c!

The Schro¨dinger equation of the excitonic structure is

HF~xe8 ,xh8 ,ze8 ,zh8 ,y,a!5EF~xe8 ,xh8 ,ze8 ,zh8 ,y,a!, ~13!

where E is the total energy andy5ye2yh . We take thefollowing trial wave function:

F~xe8 ,xh8 ,ze8 ,zh8 ,y,a!

5ce~xe8!ch~xh8!ce~ze8!ch~zh8!w~y,a!, ~14!

where the wave function in they directionw(y,a) is chosento be Gaussian-type orbital function:10–12

w~y,a!51

AaS 2

p D 1/4

e2y2/a2, ~15!

in which a is a variational parameter.To solve the Schro¨dinger equation in thez8 direction,

Hce,h~ze,h8 !5Ez8e,hce,h~ze,h8 !,

we take as base the eigenfunction of the infinite potentialwell with the Lb width. These bases are formed as

Cne,h~ze,h8 !5A 2

LbCosFnp

Lbze,h8 2dnG , ~16a!

where

dn5H 0 if n-odd

p

2if n-even

.

So, solutions in thez8 direction are described by

ce,h~ze,h8 !5 (n51

`

cnCne,h~ze,h8 !. ~16b!

We have also used this technique in our previous studies.13

In calculating the wave functionsce(ze8) andch(zh8) we haveensured that the eigenvalues are independent of the choiceinfinite potential well widthLb and that the wave functionsare localized in the well region. The wave functionsce(xe8)andch(xh8) represent the motion of the electron and hole inthe x8 direction which are exactly obtained from the Schro¨-dinger equation in thex8 direction

Hce,h~xe,h8 !5Ex8e,hce,h~xe,h8 !.

The total energy of the system is evaluated by minimiz-ing the expectation value of the Hamiltonian in Eq.~6! withrespect toa:

mina

^FuHuF&5E. ~17!

From the Eq.~17!, expectation value of the Hamiltonian isfound as follows:

^H&51

a2 1Ex8e

1Ex8h

1Ez8e

1Ez8h

2^Fu2

A~ xe82 xh8!21 y21~ ze82 zh8!2uF&. ~18!

Ground state exciton binding energy can be found using therelation

EB5Ex8e

1Ex8h

1Ez8e

1Ez8h

2^H&, ~19!

where Ex8e , Ex8

h and Ez8e , Ez8

h are the lowest electron andhole subband energies in thex8 and z8 directions, respec-tively. Substituting the expectation value of the Hamiltonianinto the Eq.~18!, we get the ground state exciton bindingenergy as follows:

FIG. 3. Variation of the ground state exciton binding energy as a function ofthe well width foru530° and two different magnetic field values. FIG. 4. Variation of the ground state exciton binding energy as a function of

the well width foru545° and two different magnetic field values.

2673J. Appl. Phys., Vol. 88, No. 5, 1 September 2000 Kasapoglu, Sari, and Sokmen

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EB521

a2 12

aA2

p E2`

1`

uce~ xe8!u2dxe8

3E2`

1`

uch~ xh8!u2dxh8^ E2`

1`

uce~ ze8!u2dze8

~20!

3E2`

1`

uch~ zh8!u2dzh8

3E2`

1` e22y2/a2

A~ xe82 xh8!21 y21~ ze82 zh8!2dy.

III. RESULTS AND DISCUSSIONS

The values of the physical parameters used in ourcalculations areme50.067m0 , mh50.45m0 , m50.04m0 ,«0512.5 ~the static dielectric constant is assumed tobe the same everywhere!, aB5165.591 Å, Ryd53.473 69 meV,V0

e5228 meV,V0h5176 meV. Without los-

ing generality, and for simplicity in numerical calculations,we have chosen the boundaries of the well atzR85(L0/2)Cosu, zL852(L0/2)Cosu, xR85(L0/2)Sinu, andxL852(L0/2)Sinu, after the coordinate transformationwhich satisfies the following equation:

zR82zL85L0 Cosu, xR82xL85L0 Sinu ~21!

derived from Eq.~7!.In Fig. 2 for u515°, we present exciton binding energy

as a function of the well width for two different magneticfield values. In this figure, the exciton binding energy in-creases asL0 ~well width! increases, since the probability offinding the particles in the well increases and reaches a maxi-mum value. Where the binding energy is maximum the sys-tem has quasi-two-dimensional character. After the certainL0 value (L0>150 Å), the exciton binding energy decreasesas L0 increases, since confinement of the particles in thez8 direction decreases and the exciton binding energy ap-

proaches about 7 meV. In this case, the system may be con-sidered weakly confined in two dimensions, rather thanstrongly confined in one dimension. At small values ofL0 ,the magnetic field is not effective in squeezing the excitonwave function since geometric confinement predominates. Atlarge values ofL0 , the geometric confinement becomesweak, magnetic confinement becomes stronger, and the ex-citon binding energy increases due to the magnetic field.Consequently, as seen in Fig. 2 after the certainL0 value(L0>150 Å), the exciton binding energy is sensitive to themagnetic feild. The same physics also explain the magneticfield dependence of the exciton radius.

In Fig. 3, we display the same results, when the tilt angleis equal to 30°. The exciton binding energy increases withincreasing tilt angle, since the effective well width and thepotential height of the particles in thez8 direction decreasesand the localization of the particles in the well increases~seeFig. 2!.

In Fig. 4, we show the variation of the exciton bindingenergy versus the well width foru545° and two differentmagnetic field values. For this tilt angle value, the effectivewell widths and potential heights of the particles in both thex8 andz8 directions are equal, since the particles are underthe effect of the same geometric confinement~effective QWprofile! in both directions and the exciton binding energybecomes maximum. For this angle value, the well width inwhich the binding energy begins to be more sensitive to themagnetic field isL0>200 Å. As a consequence, as the tiltangle increases, both geometric confinement in thez8 direc-tion and the well width in which binding energy begins to besensitive to the magnetic field increases. For example, foru515°, while the well width in which the binding energybegins to be sensitive to the magnetic fieldL0>120 Å, foru545° this value becomesL0>200 Å.

For u560°, the variation of the exciton binding energyversus the well width is given in Fig. 5. In this value ofthe tilt angle, the probability of finding the particles inthe barrier increases, since the particles are confined in

FIG. 5. Variation of the ground state exciton binding energy as a function ofthe well width foru560° and two different magnetic field values.

FIG. 6. Variation of the ground state exciton binding energy as a function ofthe well width foru575° and two different magnetic field values.

2674 J. Appl. Phys., Vol. 88, No. 5, 1 September 2000 Kasapoglu, Sari, and Sokmen

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the narrow well in thez8 direction. That is, the particlesundergo delocalization, since the particles become more en-ergetic in the narrow wells. As seen in Fig. 5, foru560° thewell width in which the binding energy begins to be sensitiveto the magnetic field becomes smaller than the previous one(L0>50– 100 Å).

This case is understood better when observing the varia-tion of the exciton binding energy according to the wellwidth for u575° in Fig. 6. In this angle value, the wellwidth in which the particles are confined in thez8 directionbecame so small that the particles are delocalized in allL0

values and the excitonic structure is formed by the magneticfield confinement in allL0 values. As a result, we can saythat the exciton binding energy is sensitive to the magneticfield in the narrow and wider well limits.

In Fig. 7, we show the variation of the exciton bindingenergy versus the tilt angle for two different magneticfield values and the single well width value. In the range of15°<u<45°, the exciton binding energy increases as the tiltangle increases and reaches a maximum value atu545° andthen decreases rapidly for the larger angles (45°<u<75°).Here the tilt angle is the parameter providing a change from5 meV to> 16 meV on the excitonic binding energy. Twoobservations are that:~i! in the range 15°<u<45°, as thelocalization of the particles in thez8 direction and Coulom-bic interaction increases, the binding energy increases;~ii ! inthe range of 45°<u<75°, the well width becomes smallerin thez8 direction and interaction decreases between electronand hole as delocalization for the particles begins, and thusthe exciton binding energy also decreases. As seen again inthis figure, the range in which binding energy begins to besensitive to the magnetic field is the range in which there isa delocalization regime.

Figures 8~a! and 8~b! shows the exciton binding energyas a function of the well width in allu values forB510 kG andB5100 kG, respectively. For two magneticfield values, the maximum binding is obtained due to the

geometric confinement of the system atu545°. For smallmagnetic field values, at the angles completing each other to90° ~that is, 15°–75° and 30°–60°!, the variation of the bind-ing energy gives nearly the same value with respect to thewell width, because at these angle values the effective QWprofiles in both thex8 andz8 directions are the same and theparabolic contribution to the confinement in thez8 directionis very small. But for large magnetic field values confine-ment in thez8 direction is affected from the parabolic term,and the 15°–75° and 30°–60° curves are quite different@seeFig. 8~b!#. As seen in these figures, the excitonic binding isdue to the magnetic quantization since the geometric con-finement disappears at the largeL0 values and Coulombicinteraction between the electron and hole is independent ofthe magnetic field direction~u!. When there is no geometricconfinement, the particles move the circular motion in theperpendicular plane to the magnetic field and the same physi-cal properties are observed in all directions. At smallL0

FIG. 7. Variation of the ground state exciton binding energy versus the tiltangle for the single magnetic field value and well width value.

FIG. 8. ~a! Variation of the ground state exciton binding energy as a func-tion of the well width in allu values forB510 kG. ~b! Variation of theground state exciton binding energy as a function of the well width in alluvalues forB5100 kG.

2675J. Appl. Phys., Vol. 88, No. 5, 1 September 2000 Kasapoglu, Sari, and Sokmen

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values (L0,100 Å), the magnetic field predominates only atlarge u values since, in this case (L0,100 Å,u.45°) theparticles are nearly free in thez8 direction. Foru,45° andL0,120 Å, the binding energy is independent of magneticfield.

In this study, we calculated, using a variational approxi-mation, the ground state exciton binding energy in the singlesquare quantum well under the externally applied tilted mag-netic field as a function of the tilt angle, the magnetic field,and the well width. In moderate well widths, it is indepen-dent of the magnetic field. It is seen that the direction of themagnetic field causes important changes in the excitonicbinding. For example, forL0550 Å, the change of the bind-ing energy betweenu515° andu545° is approximately 10meV. This change provides important results in the deviceapplications.

These results are obtained by the change direction of themagnetic field, without the need for the growth of manydifferent samples. However, we can say that the system isreduced from quasi 3D to 1D by the change in direction ofthe magnetic field since, the binding energy due to the direc-

tion of magnetic field, or the tilt angle change from 5 meV to16 meV in the smallL0 values.

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2676 J. Appl. Phys., Vol. 88, No. 5, 1 September 2000 Kasapoglu, Sari, and Sokmen

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