Solid State Communications, Printed in Great Britain.
BINDING
Vol.
83,
No.
ENERGY
8, pp. 555558,
OF EXCITONS
S. Le Goff
00381098/92$5.00+.00 Pergamon Press
1992.
IN CYLINDRICAL
QUANTUM
Ltd
DOIS
and B. Stebe
Universite de Metz et Ecole Superieure d’ElectricitC Laboratoire d’Optoelectronique et de Microelectronique 1 Bd Arago, 57078 Metz Cedex 3, France 9 Jlrrw 1992 by G. B~.s~~IIY/)
(Rcccilrrl
The binding energy of an exciton in a cylindrical semiconductor quantum dot is investigated by the variational method using electron and hole finite potential wells corresponding to the GaAs / Gu,_,Al,As nanostructure. It appears that the coulombic correlation energy reaches a minimum for sizes comparable to the bulk exciton effective Bohr radius.
For semiconductor heterostructures with sizes smaller than the bulk exciton radius, the effect of the quantum confinement becomes appreciable and restricts the electron and hole motions. Recent advances’ in submicron technology has made it possible to obtain such sizes in different geometries. In the ultimate step of the quantum confinement, the motions of the electrons and holes are restricted in all the three space dimensions. There are two kinds of zerodimensional ( OD ) nanostructures : the microcrystals and the quantum boxes. Until recently, progress in making ODnanostructures has been almost entirely limited to colloidal solutions of semiconductor microcrystallites embedded in a glass or alkali halide matrix.2J Quite recently, advances in material preparation have made it possible to obtain quasiOD quantum dots derived from GaAs / Ga,_,Al,As 2&heterostructures.*1s Whereas the microcrystallites are roughly spherical, the quantum dots are better described by disk.+ or cylinderslr+ Most theoretical studies have been devoted to excitonic states in spherical microcrystals, while less paper+I* concern excitons in quantum boxes. Up to now, two box shapes have been investigated within the infinite barriers approximation : square flat platesl**lr and cylindricalls boxes. In this paper, we present the results of the first variational calculation of the ground state energy of an exciton in a cylindrical dot with finite electron and hole barriers. We consider an isolated cylindrical GUAX box of radius R and height H, embedded in an infinitely large Gn,_,AI,As crystal. The conduction and valence band offsets are assumed to be weak enough so that the envelope function approximation may be used in a two bands model. These two bands are also assumed to be isotropic, parabolic and non degenerated. The effective Hamiltonian reads then : Hz_
&92_ 2m,
l
.+9 2mh
+ c ( re ) + where
K is
the
vt( rh dielectric
) constant,
l
m,
and
rni
are
respectively the electron and hole effective masses, supposed to be quite the same in both the dot and the outside materials. Pw and Vb are respectively the electron and the hole well potential
arising
V;=Vit’(piR)B(l+H/Z)
from the band offsets (i=e,h)
: (2)
where V, and Vh are the electron and hole barrier heights. 0 (x) is the step function ( 0 (x) = I if x > 0; B(x) = 0 if x < 0 ). z, and zh are the electron and hole coordinates along the cylinder axis chosen as zaxis, and pe , ph are the electron and hole coordinates in the plane perpendicular to We get the energy E and the envelope the cylinder axis. wave function 8 of the ground state by resolving the effectivemass SchrBdinger equation by means of the variation method :
H’P=(cc,)‘I’=El
(3)
to the dotmaterial energygap where ts corresponds discontinuity between conduction and valence bands. In order to choose the trial wave function, we study first some limiting cases. The bulk 3Dsituation may be achieved or in three other limiting when V, and V, vanish, situations : il R + 0 and H + 0 : ii) R + 00 and H + 0 : iii) R t 0 and H  00. In this limit,~ the ground state energy Ex so and wave function Vn are well known :
ExSD=_ where
l/p
_!d_ 2ns P’ = l/m:
VD = exp(r/aXsu) + l/m;
and r denotes
(4) the electronhole
distance, while oXsn = t#/@ stands for the 30 exciton Bohr radius. If H remains finite, we get a 1D quantum well ( QW ) whose confinement direction is parallel to the zaxis in the two cases : i) R + 0 ; ii) R + 00. On the other hand, if R remains finite, we get a quantum well wire ( QWW ) directed along the zaxis in the two cases : i) H + 0 ; ii) H + 00. We may define the exciton wave function using the six independant coordinates ( pi , di , zi ), where #i is the angular coordinate of the particle i ( i = e,h ). In the ground state, the invariance under every rotation about the zaxis make it possible to reduce the angular dependance of the system to the only variable + = 0.  & varying between z and 1. 0 may be replaced by the inplane electronhole distance peh although two opposite values of @ correspond to a given value of peh. We choose the following trial wave function :
*X = Fe (P, . ‘=I Fh (P,,. z,,) F, b&. I&  ,,I)
(5)
where the product function F, Fh describes the confinement of the uncorrelated electronhole pair and F, describes the internal motion of the exciton. For large R and H. Feh
556
E X C I T O N S IN C Y L I N D R I C A L Q U A N T U M DOTS
reduces to itS 3Dlimit ( 4 ). In the other cases, it be may approximated by a screened variational function. To avoid the occurence of fourfold integrals, which leads to heavy numerical computations, we choose a simpler wave function Feh (Peh,
Iz.
 Zh l) = exp(aPeh) e x p (  f ( z ,  Zh)2)
that a product s u c h / i (Pi) gi (zi) is not an exact solution o f the single particle SchrOdinger equation. Let us first investigate the inplane motion. In the following, we choose the effective Bohr radius % = r,h:/me*e~ o f the 3Ddonor for the length unit and 2 R y = e:/r, a9 (twice the effective ground state Rydberg of the 3Ddonor) for the energy unit. In these unitS, the 2D effective mass Schrodinger equation reads :  r i V~ fi (Pi) + v i w (Pi) fi (Pi) = Ei fi (Pi)
1 r e = ~.,
o r h = ~.;
I
 cr ~ [ Oa__~2h+  1  4 a
( 7 )

2
aa~,]
a2 ,,. Peh2 + Pe2  Ph2 p.p.. ap. Op~
a
n= ~ o7.+ ~ ~ +
(6 )
than that recently used TM. Two different variational parameters a and "t are introduced in order to take into account the possible anisotropy of the dot. Even the oneparticle problem is difficult, because the well potential ( 2 ) cannot be written as a sum like Viwp (Pl) + Vi,,~ (zi) of an inplane potential and a zaxis potential, so
with
1 [ 02
Vol. 83, No. 8
Ph OPh
#eh 2 + Ph 2  Pe 2 02 + Ph Peh OPh aP~h
+ ~eh
~Peh 2 4 (Ze Zh)2
O'~yh)
(14)
+ I~w (pc , Z , )+ Vhw (ph ,Z h ) T h e mean energy is then given b y :
(E(a,7))
=
= ~I [ (O.IR) 2 + o (Oh~R)2 + (~'elH)2 + o(xhlH) ~" ] 1 +o,.~ 
l+a c~P~( ~,R )
2  +
2
PI(cqR)
etP4(c~,R)+oPs(m,R)  2
Px(cqR)
Z3(',/ , H)+aZ 4(7 , !t) z~("t, H) + .7(1+o')  2~2(1+a) ~ + 2"/ z1(7, H)
vi,,p(Pl) = Vi0(p i  R) ( 8 ) zs(1, H) ]
where o = m: / m h is the effectivemass ratio. Equation ( 7 ) can be solved analytically. The ground state wave functions read : (15) fi (Pi) = Jo( 01 Pi /R ) if Pi < R = Ai Ko( Hi Pi ) if Pi > R
( 9 )
The integrals Pi and Z i are defined by : with /~i = ( Vi /ri  (0i / R ) 2 )H2 . Jo and K o are respectively the zeroorder Bessel function and modified Bessel function 19. 0i and Ai are constantS determined by the boundary conditions at Pl = R : A i = Jo(0i) / Ko(/8i R)
i; Pi = 8 x
X
and
i oo dPe
P. + Ph 0 dph fJlPe  Oh[
Fi (Pc ,Ph ,Peh) Pe Ph Peh dPeh
( 16 )
'[(p,, + ph) 2  p.h 2] [P,,h ~  (p.  ph) 2]
0i Jl(0i) /Jo(Oi) = Hi R KI(/~i R)/Ko(Bi R)
( l0 )
Similarly the zaxis ground state solution of the SchrOdinger equation becomes :
gi (zi) = COS (x i Zi / H ) = B i exp ( k i Izi
l)
if
Izi I < H / 2
if
Izl I " H/2
and
(12) For the 3D confinement, the product fi (Pi) gi (zi) is no longer solution of the SchrSdinger effective mass equation, hut is expected to be a reasonable approximation of the wave function for values of R and H close to ax ~D . The ground state wave function and energy are determined by minimizing the expectation value of energy :
for all variations of the two parameters a and "7.
' Ph , Peh)
( II )
tan('xi /2) = k i H/Ir i
(s(,,,.~)) = <~'xlnl~'x> / <~'xl~'x>
FI(Pe , Ph , Peh) = Ue (Pc) /h (Ph) exp(c~P~)] 2 F3(Pe , Ph , Peh) = ~ F I ( P e Veh
with k i = ( V i / r i  (xi / H) 2 )l/z . xi and Bi are constantS determined by the boundary conditions at z i = H / 2 : Bi = cos(~ri /2) / exp( k i 14/2)
where :
( 13 )
The Hamiltonian operator reads, in our systems of "atomic units" 
F4(Pe , Ph , Peh)
peh2 + pe~  ph2 fe (Pc) f'e (Pc)
Pe Peh
X [fh (Ph) exp(otPeh)]2 F~(Pe , Ph , Peh)
Peh2 + ph2  pc2 fh (Ph) f'h (Ph) Ph Peh
x [/, @.) e x p (  ~ p ~ h ) P F~(Pe , Ph , Peh) = F~(Pe , Ph , P ~ ) e(R  Pc) Fs(Pe , Ph , Peh) = Fx(Pe , Ph • Peh) 0(R  Ph)
( 17 )
and : oo
o~
Zi= [ dze [ dZh Gi(ze , Z h ) o00 400
(18)
Vol. 83, No. 8
EXCITONS
IN CYLINDRICAL
QUANTUM
with
557
DOTS
0.5.
GI(ze , Zh ) = [ge (Ze) gh (Zh) exp(3(ze  Zh)Z)]2
I t~
G , ( z , , z h ) = (ze  z h )2 ax(z . ,z h )
1.0
C a ( z , . z h ) = (z,  zh ) 8, (z,) g', ( z , ) X [gh (Zh) exp(q'(ze  Zh)=)]=
G,I(z e , z h ) = (Zh  Ze ) gh (Zh) g'h ( Z h )
~
x [g. (z.) exo('y(z.  Zh)=)]2
t5
Gs(z o , zh) = Ga(zo , Zh) e ( n / 2  Iz,J) Ge(z , , Zh) = Gl(z " , zh) 0(H/2 
IZhl)
(19)
The Pi and Z i integrals are computed numerically. expectation value of the coulombic energy reads :
The
(V~o~a) = oo
 [
2.5
oO
dz® [
dzhGl(ze,Zh)Pc(ot,
(20)
R,zezh)
0
1
2
4
3
R ( ^TONC 0~S
5
)
where Pc is analog to the Pi integrals ( 16 ). Here : F¢ (p, , p h , Pea) = FI(Pe ,Ph ,Pea)
jPeh 2 + (=e  Zh)~:
(21) In order to simplify the numerical computations, we determine (Vcoul) using the following a p p r o x i m a t i o n for the Pc integrals :
F I G . I. Correlation energy W of an exciton confined in a q u a n t u m cylinder plotted against R for H = I. The curves labelled "LH" are devoted to a light hole exciton ( a = 0.196 ), while the curves labelled "HH" are devoted to a heavy hole exciton ( a = 0.707 ). Both are d r a w n for two values o f x ( 0.15 and 0.30 ).
a
¢o (z.  z h ) = b + [ze  z h [
( 22 ) 0.0
where the constants a and b are chosen so that exact values o f (Vcoul) are both obtained in the limits I z ,  z h [ , 0 and [z,  z h I ~ oo. These t w o conditions lead to a = PI(a,R) and a/b = P=(~z,R). So, the s i m p l i f i e d expression o f (Vcoul) is given b y :
(23)
(Voo=~) =  z= (a, R,~,H)/Z 1 (% H)
BJLKLlilr 0.5
where Z c is analog to the Z i integrals ( 18 ), with : Gc ( H , R , a , 7 , z ' , z h )
=
Cl(z. , Zh )
(24)
Px(a, R) e3(a,R) + Iz,

Zh I ~ " tO
R
In the numerica! calculations, we use the following material data zo : m e / m o = 0.0665 for the electron mass,
m h h / m o = 0.34 and mth/ m o = 0.094 for the heavy and light holes masses respectively. T h e band offsets are given by V e = Q e e g and V h = Qheg, where Q. = 0 . 5 7 = l  Q h • F u r t h e r we assume that the band gap eI and the a l u m i n i u m percentage x are related by 2x : eg = 1.155 x + 0.37 x 2 eV. Using the value r= ~ = 12.5 for the dielectric constant, we get a D = 99.5 Angstr6m for the effective Bohr radius and 2 Ry = e2/¢aD = 11.58 meV for twice the effective ground state R y d b e r g of the 3D donor. To study the influence of the c o n f i n e m e n t on the coulombic interaction, we define the correlation energy W :
w =
( 25 )
1.5
I
I
1
i
I
2
=
i
I
I
3
~
i
I
I
4
8 ( AmNC~ S )
F I G . 2. Correlation energy W of a heavy hole exciton confined in a q u a n t u m cylinder ( x ffi 0.15 ) plotted against H for R = 1 and R = 5 .
558
E X C I T O N S IN C Y L I N D R I C A L Q U A N T U M DOTS
where E e and E h are roughly evaluated by :
El = (fl gilHil/i gi) / (/i gil/i gl )
(26)
/i being the wave functions ( 9 ) and gi the wave functions ( 11 ). Fig. 1 shows the variations of W against R for a fixed value of H = 1, for heavy (a = 0.196) and light (a = 0.707) holes, with x ~ 0.15 and x = 0.30. We get a pronounced minimum near R = 40 Angstr6ms. Fig. 2 shows the variations of W against H for two fixed values of R for heavy holes and x = 0.15. Here, W becomes minimum for H = 60 AngstrOms. Highest oscillator strengths are expected near these minima. It appears that the influence of the lateral quantum confinement is more important than that arising from the axial confinement. ACKNOWLEDGEMENTS  This work has been performed at the Centre de Calcul de Strasbourg Cronenbourg ( C.N.R.S. ). It was supported by the Centre National de la Recherche Scientifique and the Minist~re de I'Education Nationale ( DRED ).
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Vol. 83, No. 8
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