4
PHYSICAL REVIEW B VOLUME 51, NUMBER 3 15 JANUARY 1995-I Ground state of a quantum disk by the efFective-index method Guy Lamouche and Yves Lepine Departement de Physique et Groupe de Recherche en Physique et Technologie des Couches Minces, Universite de Montreal, Case Postale 6128, Succursate Centre vill-e, Montreal, Quebec, Canada HBCBJ7 (Received 8 July 1994) A simple method for obtaining a good approximation of the ground state of a quantum disk is presented in the case where the electron radial con6nement is substantial. Developed in the envelope-function formalism, the method is derived from the efFective-index technique well known to dielectric wave-guide theory. Its validity is discussed by comparing its results with those obtained using a numerical relaxation technique. I. INTRODUCTION Disk-shaped quantum structures are suitable to model the quantum dots fabricated by the patterning of quan- tum wells. i They also represent an approximate way of describing the dots obtained by the coherent islanding effect and by the epitaxial growth on a tilted substrate. Theoretical study of Wannier excitons in such structures has been performed by Kayanuma for infinite potential barriers. 4 Le Goff and Stebe looked at the same prob- lem for cylindrical quantum dots with finite potential barriers. 5 They relied on a variational approach, start- ing with approximate eigenfunctions of the electron and hole confined by the quantum disk. For the disk case, those approximate functions are only valid for quantum dots with a high radial confinement. In this paper, we present a technique leading to a bet- ter approximation of the ground-state energy and wave function of an electron or a hole confined by a quantum disk. Our approach is developed within the envelope- function formalism and is derived &om the effective in- dex (EI) method well known to the dielectric waveguide theory. The precision of the method is evaluated by com- paring its results with those obtained by directly solving the Schrodinger equation using a numerical relaxation technique. In Sec. II, we present the EI approach applied to the quantum-disk structure. In Sec. III, we describe the numerical relaxation technique used for comparison. Effective-index and numerical results are then compared in Sec. IV for InAs/InP and GaAs/Gai Al As quan- tum dots, showing that our approximate method indeed gives a very good description of the ground state. II. APPROXIMATE GROUND STATE We develop an approximate description of the ground state of a quantum disk in the spirit of the EI method. s We consider a disk of radius a and of thickness b. The inner potential is V (V ) 0) and the outer potential is null. We treat the problem in the one-band envelope- function formalism (Ben Daniel-Duke model), imposing the function continuity and the current conservation at the disk boundaries. In the following, we use the term wave function in a loose sense to refer to the envelope function. For simplicity, we assume isotropic effective masses for the interior (m, *) and for the exterior (m, ') regions. But the approach could easily be generalized to anisotropic effective mass tensors. Our treatment is performed in cylindrical coordinates (p, P, z), the origin of the system lying at the center of the disk and the z axis being chosen along the rotation axis. For the ground state, the Schrodinger equation is independent of the P coordinate and reduces to a two-dimensional form: interior, al s 2m, *pap & ap)l 2m, *. az2 p V @(p z) =E 0(C» ); exterior, a /' a) 2 . *pap ~, 'al, )~ hz az (~ z) '///(p, z) = R(p) Z(z), (2) by artificially decoupling the radial and z-dependent parts of Eq. (1) as follows. For a quantum disk with a good radial confinement, the major part of the wave function is found in the region p ( a. The diameter being greater than the thickness, we expect the variation of B(p) to be smaller than the variation of Z(z) in that region. We can thus approxi- mate Z(z) by neglecting the contribution of R(p) to the kinetic energy term of Eq. (1). This leads to the problem of a one-dimensional square well of inner potential V. The well-known result is~ ~z~ ( 6/2: Z(z) = A cos 2m, *. (E, + V)/5 z ~z~ ) b/2: Z(z) = B ezp ( /2m', ~E, ~/b ~z~), z(b) where A and B are related by = Eg @()o, z), (1) where Eg is the ground-state energy of the system. For such a Gnite well, the spatial dependence of the potential makes the preceding equation not separable. We nevertheless look for an approximate solution of the form 0163-1829/95/51(3)/1950(4)/$06. 00 51 1950 1995 The American Physical Society

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PHYSICAL REVIEW B VOLUME 51, NUMBER 3 15 JANUARY 1995-I

Ground state of a quantum disk by the efFective-index methodGuy Lamouche and Yves Lepine

Departement de Physique et Groupe de Recherche en Physique et Technologie des Couches Minces,Universite de Montreal, Case Postale 6128, Succursate Centre vill-e, Montreal, Quebec, Canada HBCBJ7

(Received 8 July 1994)

A simple method for obtaining a good approximation of the ground state of a quantum diskis presented in the case where the electron radial con6nement is substantial. Developed in theenvelope-function formalism, the method is derived from the efFective-index technique well knownto dielectric wave-guide theory. Its validity is discussed by comparing its results with those obtainedusing a numerical relaxation technique.

I. INTRODUCTION

Disk-shaped quantum structures are suitable to modelthe quantum dots fabricated by the patterning of quan-tum wells. i They also represent an approximate way ofdescribing the dots obtained by the coherent islandingeffect and by the epitaxial growth on a tilted substrate.Theoretical study of Wannier excitons in such structureshas been performed by Kayanuma for infinite potentialbarriers. 4 Le Goff and Stebe looked at the same prob-lem for cylindrical quantum dots with finite potentialbarriers. 5 They relied on a variational approach, start-ing with approximate eigenfunctions of the electron andhole confined by the quantum disk. For the disk case,those approximate functions are only valid for quantumdots with a high radial confinement.

In this paper, we present a technique leading to a bet-ter approximation of the ground-state energy and wavefunction of an electron or a hole confined by a quantumdisk. Our approach is developed within the envelope-function formalism and is derived &om the effective in-dex (EI) method well known to the dielectric waveguidetheory. The precision of the method is evaluated by com-paring its results with those obtained by directly solvingthe Schrodinger equation using a numerical relaxationtechnique.

In Sec. II, we present the EI approach applied tothe quantum-disk structure. In Sec. III, we describethe numerical relaxation technique used for comparison.Effective-index and numerical results are then comparedin Sec. IV for InAs/InP and GaAs/Gai Al As quan-tum dots, showing that our approximate method indeedgives a very good description of the ground state.

II. APPROXIMATE GROUND STATEWe develop an approximate description of the ground

state of a quantum disk in the spirit of the EI method. s

We consider a disk of radius a and of thickness b. Theinner potential is —V (V ) 0) and the outer potentialis null. We treat the problem in the one-band envelope-function formalism (Ben Daniel-Duke model), imposingthe function continuity and the current conservation atthe disk boundaries. In the following, we use the termwave function in a loose sense to refer to the envelopefunction. For simplicity, we assume isotropic effectivemasses for the interior (m,*) and for the exterior (m,')

regions. But the approach could easily be generalizedto anisotropic effective mass tensors. Our treatment isperformed in cylindrical coordinates (p, P, z), the originof the system lying at the center of the disk and the zaxis being chosen along the rotation axis. For the groundstate, the Schrodinger equation is independent of the Pcoordinate and reduces to a two-dimensional form:

interior,al s

2m,*pap&

ap)l 2m,*. az2p —V @(p z)

=E 0(C» );

exterior,a /' a)

2 .*pap ~,'al, )~

hz az(~ z)

'///(p, z) = R(p) Z(z), (2)

by artificially decoupling the radial and z-dependentparts of Eq. (1) as follows.

For a quantum disk with a good radial confinement,the major part of the wave function is found in the regionp ( a. The diameter being greater than the thickness,we expect the variation of B(p) to be smaller than thevariation of Z(z) in that region. We can thus approxi-mate Z(z) by neglecting the contribution of R(p) to thekinetic energy term of Eq. (1). This leads to the problemof a one-dimensional square well of inner potential —V.The well-known result is~

~z~ ( 6/2: Z(z) = A cos 2m,*. (E, + V)/5 z

~z~ ) b/2: Z(z) = B ezp (—/2m', ~E, ~/b ~z~), z(b)

where A and B are related by

= Eg @()o,z), (1)

where Eg is the ground-state energy of the system.For such a Gnite well, the spatial dependence of the

potential makes the preceding equation not separable.We nevertheless look for an approximate solution of theform

0163-1829/95/51(3)/1950(4)/$06. 00 51 1950 1995 The American Physical Society

51 BRIEF REPORTS 1951

cos g2 m,' (E, + V)/h2 b/2B=Aexp — 2m' E, h2 b 2

and the ground-state energy E, of this quantum well isobtained by solving7

tan 2 m,' (E, + V) /h2 b/2m,'. lE,

l

:(E,+V)

p&a: Z(z)h' 0 ( (9 )

2 .PKliPKP IR(.)

= Eg Z(z)R(p), (6)with m*(z) = m,*. for lzl & b/2 and m'(z) = m,* forlzl ) b/2. If the interior and exterior efFective massesare different, one cannot find an analytic function R(p)which satisfies Eq. (6) for all values of z. We thus definean effective radial Hamiltonian by replacing m (z) by itsaverage value m,* given by

Z .() Z

m, ,=

m,*(z)=

(ZlZ)

Our efFective radial Hamiltonian corresponds to theproblem of a circular well of inner potential E, and withinterior and exterior effective masses given, respectively,by m,' and by m,*. This Hamiltonian is solved to obtainthe function R(p) and its ground-state energy Eg is usedas an approximation for the ground-state energy of thequantum disk. We get

p&a: R(p)= CJ() 2m,' (Eg —E,)/h2 p

p ) a: R(p) = DKl

o2m,*lE l/h' p l, (8))

where J„and K are, respectively, a Bessel function anda modified Bessel function. The constants C and D arerelated by

Jp 2m,* (Eg —E,)/h2a

Kp 2 *Eg 5 c

and. E~ is solution of

Jp

2 m,'. (Ea —E )/Ka a ICa (/2 m,'~Ea~/ha a)

2m,'. (Ea —E,)/2'a IC, (/2m, '[Ea[/2a a)

m,' lElm.* (Eg —E,)

. (1o)

To define an effective radial equation, we introduceZ(z) in Eq. (1) and we neglect the kinetic energy contri-bution of this z-dependent part for p & a. We obtain

h' 0 ( 8)p&a: Z(z) — —l

p —l

+ E, R(p)(') p'« 'pr= Eg Z(z)R(p)

With this approximate treatment, the ground-state en-ergy of the quantum disk is obtained by solving the twotranscendental equations [Eqs. (5) and (10)]. For the cal-culations of Sec. IV, this task is easily performed withthe Maple Computer Algebraic System. s

The envelope function [Eqs. (2), (3), and (8)] deservessome continents.

(i) The z-dependent part is evaluated only by consid-ering the &action of the wave function contained withinp & a. The precision of this approximation will thus in-crease with the radial confinement of the wave function.

(ii) Z(z) verifies the good boundary conditions acrossthe disk limits for p ( a. It presents a spurious discon-tinuity in its derivative across the planes lzl = b/2 forp) a.

(iii) R(p) does not verify the good current continuitycondition at the disk boundary. It also presents a spu-rious current discontinuity at p = a for ]zl ) b/2. Ourdefinition of m, is an attempt to minimize both misbe-haviors and it enforces the continuity of the total currentcrossing the surface defined by p = a.

In Sec. IV, we will nevertheless see that the EI wavefunction gives a good approximation to the true solutionfor a wide range of parameters.

III. NUMERICAL SOLUTION

To obtain a numerical solution for the ground state ofthe quantum disk, we rely on a relaxation approach. Thetime-independent Schrodinger equation [Eq. (1)] can ar-tificially be transformed in a time-dependent problem,

—P(p, z, t) = HP(p, z, t)—

At time t = 0, any function satisfying the good spa-tial boundary conditions can be expressed as a linearcombination of the eigenfunctions of the Hamiltonian.All the coefFicients of this expansion evolve exponentiallyin time, according to the negative of the energy corre-sponding to the related eigenfunction. Thus, the time-evolution process favors the low energy components andafter a sufIiciently long time, the ground-state componentdominates. iP

Our trial wave function is the approximate wave func-tion of the previous section. The time evolution isperformed with finite difFerences using an alternating-direction implicit scheme. The good boundary condi-tions at the disk limits are forced. The wave functionis normalized at each step and the energy is evaluatedusing Eq. (1), so that the evolution to the true groundstate can be followed. Although the exact wave func-tion should vanish only at infinity, we artificially imposea null value at some finite distance of the disk, for nu-merical purpose. This artificial boundary condition isapplied far enough so that it does not significantly afFectthe results. We estimate the precision of the numericalcalculations presented in Sec. IV to be better than 1%.

BRIEF REPORTS1952

'cular well in theof a circular-dimensional well wi

inner potential-one- im

ls having an into

y par calcu a ion,

li th tod db o tho [ E.pg a sctures consi ere

ia c'1 confinement is ig

are a mos int indentical to osdth

n gives rroximation and with the EI approxtaine wi r

r the confinemenh those given by the

7

ue shows that the EI mete results wit os

ethoderica re ah t}1 bt 'more precise t an

EIves va ue

th t t}1s im ortant o noitht e pulations. In ig.

d~Thn o

n a radius in1t h 1e see a roach gives resu s me see that the appro

the numerica'

al results.

CONCLUS IONV.

d the effective indexwe have applied t eI this work, wenn to the gui ed d wave theory omethod well known

0.12

0.10

0.08

o 0.06

0.04

0.0

0.0

IV. APPLICATION

ddd Istructures of InAs ru As embe ethe parameter rs of Re.

usinn electrons using

= 0.71 eV .b d fF

n electronu- tate energy of an eradunction of

Figure 1 presents e

ses. Theax ach are also s own

func ion

th the r gls ex ected since e

d k'""' "' w'ttion in the isSec. II. Forf hprecision o e r

we macin the worst casere has the lowest

p, . Notet a

ction can eb Wlfof th fradial confinement: only a o( a region.and b) the

found in the ph EI wave function ag s,a, the w

ed above

mainlye solution in the in

erformed eeresti-

cacua i pmates the wave fu nction or p e

urious discontinu-i ies in the gradient of t e wd' k s discussed above.

d- tate energies givendb hthose obtaine y

ed''( ~

/ O.ss o. i5g

d"k 'fmass in o

a roxima ion,h o d t t

function by multiplying t e so u-0.08

-0.10—

-0.12—

-0.14—)Q)

cn -0.16—PQ)

-0.18—CD

GJ

-0.20—

-0.22—

-0.24—

Effective Index

Numerical Relaxation0.12

0.02

O.OO-

0.10-

0.04

-0.26—

-0.289 105 6 7 82 3 4

radius (nm)

b an1 rou — ner y of an electron confined y anFIG.InAs-InP quantum disk othe radius.

~(o/Pyy

'n l tron con6ned-state wave functionnofaneec r

76 h' kh EI o Il d

uantum disk ob the nu-dius evaluated by (a) the apnm radius eva

merica re1 laxation technique.

51 BRIEF REPORTS 1953

-10—CO

-12—CD

-14—0)

-16—Q)

05

c -18—O

-20—

Effective Index

Numerical Relaxation

0.035

0.030

0.025—

E0.020—

OOc

0.015—

65

0.010—

Effective Index

Numerical Relaxation

LG

-22—

-24—0.005—

-26I I I I I I

10 12 14 16 18 20 22 240.000

I

10 15 20 25radius (nm)

FIG. 3. Ground-state energy of an electron con6ned by anGaAs/Gap. ssAlp. ysAs quantum disk of 2.98 nm thickness asa function of the radius.

calculation of the ground state of a quantum disk struc-ture. By comparing its results with those obtained by anumerical relaxation approach, we have seen that it leadsto a good approximation of the ground-state energy andwave function as long as the radial confinement is large.We estimate that this approximation will remain good forstructures for which half of the wave function is radiallyconfined within the radius of the disk. The generalizationof this approach to a cylindrical structure with a thick-ness greater than the diameter is straightforward sinceone only has to inverse %he procedure.

This method can be used to obtain a simple evalua-tion of the confinement energy of excitons in disk shapedquantum dots. A more complete treatment should in-clude the coulombic interaction as it is done by Le Goffand Stebe. s The EI method could then be used to ex-tend the treatment of Ref. 5 to less confining quantumdisks. For such structures, we have seen that the EI ap-proach gives a better evaluation of confinement energiesand wave functions than the approximation used in Ref.5.

The EI approach can also be used to approximate ex-cited states. Because of the symmetry of the problem,three types of excited are possible: excited state solutionsfor R(p) or for Z(z) and excited states for the angularpart of the wave function. However, the quality of these

z, p (nm)

FIG. 4. Ground-state wave function of an electron con6nedby an GaAs/Gap. ssAlp. ]sAs quantum disk of 2.98 nm thick-ness and 11.9 nm radius. The evaluation is done along thez-axis and along a radius in the x —y plane.

solutions is expected to be lower than that found for theground state because of the smaller confinement associ-ated with excited states. We have limited our study tothe ground state here since the numerical relaxation tech-nique used for reference is not well suited for the char-acterization of excited states. A detailed study of thevalidity of the EI approximation for the excited statescould only be performed by comparing its results with amore powerful numerical technique like the finite elementmethod.

Finally, we note that a similar decoupling approachhas been used to approximate the ground-state energy ofan electron in a quantum wire. iz Because of its symme-try, our problem reduces to two dimensions and can betreated using the same decoupling technique. To zerothorder, this approximation leads to equations similar tothe present EI approach with the exception of a nonzeropositive external potential for the radial equation. Con-sequently, this decoupling approximation should lead toan overestimation of the radial confinement while ourEI approach leads to an underestimation of this confine-ment. It should also give an underestimation of the bind-ing energy while the present EI approach overestimatesit. Combining EI results with those obtained from thisdecoupling method should help putting upper and lowerlimits for the characterization of the ground-state energy.

K. Kash, A. Scherer, J.M. Worlock, H.G. Craighead, andM.C. Tamargo, Appl. Phys. Lett. 49, 1043 (1986).D. Leonard, M. Krishnamurthy, C.M. Reaves, S.P. Den-baars, and P.M. Petrol, Appl. Phys. Lett. BS, 3203 (1993).R. Leonelli, C.A. Tran, J.L. Brebner, J.T. Graham, R.Tabti, R.A. Masut, and S. Charbonneau, Phys. Rev. B48, 11135 (1993).Y. Kayanuma, Solid State Commun. 59, 405 (1986)

s S. Le Goff and B. Stebe, Phys. Rev. B 47, 1383 (1993).G.B. Hocker and W.K. Burns, Appl. Opt. 16, 113 (1977).G. Bastard, Wane Mechanics Appli ed to SemiconductorHeterostructures (Les Editions de Physique, Paris, 1988).

Handbook of Mathematical Functions, edited by M.Abramowitz and I.A. Stegun (Dover, New York, 1965).Copyright Waterloo Maple Software and University of Wa-terloo, 1981—1994.S.E. Koonin, Computational Physics (Addison-Wesley,Reading, MA, 1990).W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vet-terling, Numerical Recipes (Cambridge University Press,Cambridge, 1986).G. Bastard. , J. A. Brum, and R. Ferreira, in Solid StatePhysics, edited by H. Ehrenreich and D. Turnbull (Aca-demic Press, New York, 1991), Vol. 44, p. 229.