Density of impurity states and optical absorption spectra associated with shallow impurities in quantum - well wires

Density of impurity states and optical absorption spectra associated with shallow impurities in quantum - well wires

Superlattices and Microstructures, Vol. 9, No. I, 1991 5 DENSITY OF IMPURITY STATES AND OPTICAL ABSORPTION SPECTRA ASSOCIATED WITH SHALLOW IMPURITIE...

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Superlattices and Microstructures, Vol. 9, No. I, 1991

5

DENSITY OF IMPURITY STATES AND OPTICAL ABSORPTION SPECTRA ASSOCIATED WITH SHALLOW IMPURITIES IN QUANTUM WELL WIRES -

N. Porras Montenegro', J. L6pez-Gondar + and L. E. Oliveira t "Depto. de Fisica, Universidad del Valle, Call, Colombia ~Dept. of Theor. Physics, U. Havana, 10400, Havana, Cuba TInst. de Fisica, Unicamp, C.P. 6165, Campinas-SP, Brazil (Received 13 August 1990)

The binding energies of shallow donor impurities in cylindrical quantum-well w i r e s are calculated as functions of the well radii and of the donor location in the well for different radii of the wires using a variational procedure within the effective ma s s a p p r o x i m a t i o n . Assuming an homogeneous distribution of impurities inside the well, we have also calculated the d e n s i t y of impurity states as a function of the impurity b i n d i n g e n e r g y and the optical absorption spectra associated with transitions between the n=l valence subband and the donor impurity band. A comparison with previous t h e o r e t i c a l results obtained in the case of quantum wells - in contrast with the present results for cylindrical quantum-well wires - is performed.

Due to the interesting quantum size effects present in low dimensional structures such as semiconducting superlattices and heterostructures, a great deal of work has been devoted to these systems in the last decade. In the last few years, s ev~_~ 1 experimental I-4 and theoretical studies on electronic structure, transport properties, excitonic and impurity levels in quantum-well wires (QwWs) h a ~ been performed. Bryant-'- studied the effect of changing the cross-sectional form of the QWW on the impurity 's binding energy and found that in the case of wires with the same cross-sectional area the binding energies are nearly equal for the cylindrical and the rectangular QWWs provided that the rectangular form does not deviate too m u ~ from the square shape. Weber et al calculated the impurity binding energies as functions of the impurity position, and the density of impurity states in GaAs (Ga,AI)As Qwws with different rectangular cross-sections and for infinite well depths. Brown and 78 Spector ' calculated the impurity binding energies using infinite and finite cylindrical confining potentials for both axial and off axis impurities. 0 7 4 9 - 6 0 3 6 / 9 1 / 0 1 0 0 0 5 + 0 5 $02.00/0

In this work we assume an homogeneous distribution of shallow-donor impurities in a cylindrical GaAs-(Ga,A1)As QWW and calculate the density of impurity states as a function of the impurity binding energy and the optical absorption spectra associated with transitions between the n=l valence subband and the donor impurity band. Although until now there are no conclusive experimental reports on the impurity-related optical spectra of GaAs-(Ga,A1)As QWs and QWWs, it is certainly of interest to study the optical absorption associated with transitions between the valence and donor-impurity bands in GaAs-(Ga,AI)As cylindrical QWWs. The Hamiltonian of a single hydrogenic impurity in a GaAs-(Ga,AI)As QWW system is given by ~2

e 2

H=

+ V(p)

2m"

c

[ ( P -

Po )2 + z2]*/2

(1)

where m is the electronic effective mass, c the dielectric constant of the wire material , and V(e ) the confining

© 1991 Academic Press Limited

6

Superlatt~ces and Microstructures, Vol. 9, No. 1, 1991

potential. The r e l a t i v e s e p a r a t i o n of the carrier from the impurity along the axis of the wire is given by z and ~o

where

is

the

impurity

location

along

the

d i r e c t i o n which is p e r p e n d i c u l a r to the QWW axis. For the finite well the p o t e n t i a l V(p) in the above h a m i l t o n i a n will be taken as zero for pd w h e r e d is the QWW o

radius. According to Brown and Spector 7'e, the impurity b i n d i n g e n e r g y can be w r i t t e n in e f f e c t i v e R y d b e r g s as

E

=Eb(y,to)

is

the

impurity

b i n d i n g energy, L(E) is the p o r t i o n of the line E=E lying w i t h i n the c i r c u l a r i cross section and V means the g r a d i e n t L

with respect to the impurity position• Our results a r e p r e s e n t e d in r e d u c e d atomic units ( a . u . ) , w h i c h correspond to a length u n i ~ of an effective Bohr • 2 2 radlus ao= h e/m e , and an energy unit of

an

effective

Rydberg,

R" Q~Ws R

o

Eb (y, to) = _(la)2

4a(e+P) (d/dX) (Q+P)

(2)

with 1

Q=IodttJ~ (xt) Io (2 Aayt<) Ko (2~ayt>)

(3)

and

X

K~(cx)

We assume . • • 14 dzscontlnuzty

15 '

given

A1 c o n c e n t r a t i o n

(4) t<(t>)

is the

t and to, t = p/d,

lesser

(greater)

t o = Po/d,

of

y = d/a~ e

x = r,od , c =(2mVo/r~oh2-1)Zl2-_ with as the e f f e c t i v e Bohr radius variational parameter. J

a =

ao and X is a is the

o

o r d i n a r y Bessel function of order zero, and I and K are the modified Bessel o

f u n c t i o n s of order zero and of the first and second kind, r e s p e c t i v e l y . It is clear that in the case of a cylindrical quantum-well wire with infinite confining potential, the i m p u r i t y b i n d i n g energy is also given by Eq. (2) w i t h P = 0. Assuming that the circular cross s e c t i o n of the Q W W is not too small we m a y treat the impurity position as a continuous random variable (we a s s u m e i m p u r i t i e s only inside the QWW when calculating the density of impurity states) and define a density of impurity states .I'13 per unit binding energy as

i IL(EI)

g(Ei)

Tl'd 2

--

band-gap A1 As

1-x

x

2p d---~ ~

dE i

(5)

IVLEII -I

as a f u n c t i o n x<0.45

of

as 16 AE (eV)

the =

g

Itdt tK~(cxt)Io(2Xayt<)Ko(2Aayt>)

o

that the in a G a A s - G a

g

Gaz-xAlxAS'

where

o

5.72 meV in the case of donor impurities. We have ignored variations in the e f f e c t i v e mass and dielectric c o n s t a n t and c o n s i d e r e d the v a l u e s for GaAs t h r o u g h o u t the h e t e r o s t r u c t u r e .

heterostructure is d i s t r i b u t e d about 40% on the v a l e n c e band and 60% on the c o n d u c t i o n band w i t h the total b a n d - g a p difference, AE , b e t w e e n GaAs and

J2(x) P

=

o

22

" 4

m e /2h e . For Ga~s (Ga.AI)As these units are a = i00 ~ and

1.247x. In Fig. 1 we d i s p l a y the d e n s i t y of d o n o r - i m p u r i t y states as a f u n c t i o n of the b i n d i n g e n e r g y for both infinite and finite Qwws and for various well radii. Results in the case of the infinite Q W W are in ~ood a g r e e m e n t w i t h those of W e b e r et al-- for an infinite QWW of c o m p a r a b l e square cross section. For d > 300 ~ and i n c r e a s i n g Qww radii, the s t r e n g t h of the E el" peak (which is ! a s s o c i a t e d to donors at the edge of the QWW) d i m i n i s h e s w h e r e a s the s t r e n g t h of the EJ~x s i n g u l a r i t y (associated to i

i m p u r i t i e s at the center of the QWW) is enhanced, leading - for d >> i00 ~ - to a center of gravity of the impurity band w h i c h c o n v e r g e s to the E "ax v a l u e i

(which becomes the impurity binding energy in bulk GaAs). This b e h a v i o u r is q u a l i t a t i v e l y the sa~e as obtained by O l i v e i r a and F a l i c o v in the case of GaAs-(Ga,AI)As quantum-wells. It is clear from Fig.l that experimental values for the binding energies of i m p u r i t i e s in n o m i n a l l y undoped QWWs (or in h o m o g e n e o u s l y d o p e d QWWs) s h o u l d not be compared with the on-center i m p u r i t y value. We stress this because some p r e v i o u s t h e o r e t i c a l w o r k s *e-m on impurities in a u a n t u m - w e l l s seem to have o v e r l o o k e d that aspect and have compared calculated results for o n - c e n t e r i m p u r i t i e s w i t h experiment.

Superlattices and Microstructures, VoL 9, No. 1, 1991

7

Ei [ meV) 15

1.5

20

2-,5

El (meV) 15

10

20

l

,

L~) i

(bid-loo~

(a) d-,5o~ 1.5" " 1.O-

|

a:

1.0-

w v

r

1

0.5" 0.5-

,

........i

3

1

: 2

4

E i / R:

Eli meV) 3

0

5

3 E i / R .~

10

EiimeV} 10

0

15

15

(did .10o0

(¢) d . 3oo ~ 8

;.~2-

6;1.

0

0

0

0

U Ei/R;

E i IR:

F i g u r e 1. D e n s i t y of donor impurity states as a f u n c tion of the binding e n e r g y in infinite (dashed lines) and finite (full curves) GaAs Ga A1 As QWWs w i t h well radii (a) d

0.7 0.3 = 50 ~ , (b)

d== i 0 0 ~ ,

(c)

d =

and (d) d i000 ~. Notice d i f f e r e n t scales are used.

The t r a n s i t i o n p r o b a b i l i t y per u n i t time for v a l e n c e - t o - d o n o r t r a n s i t i o n s associated with a donor impurity l o c a t e d at ~o in a QWW of r a d i u s d is proportional to the square of the m a t r i x e l e m e n t of the e l e c t r o n - p h o t o n interaction Hin t between the wave functions (valence) i.e. , Wd(Po,~)

of the initial and final (impurity)

~, that

v e c t o r p o t e n t i a l 23 . The above element may be w r i t t e n as 22'z4

< f [ H i n t l i> = Ce. PfiSfi ,

matrix

(7)

with

r Pfi = ~ J fl dr u r ( r )

state states,

p u (r)

,

(s)

and

-- 2.h Z [JZS(Er-Ez -h~) i

(s)

Hint = Ce.p, where e is the p o l a r i z a t i o n v e c t o r in the d i r e c t i o n of the e l e c t r i c field of the radiation, p is the m o m e n t u m operator, and C is a prefactor which contains the p h o t o n

with

300

Sfi = ~ dr F ; ( r ) F l ( r

)

,

(9)

where fl is the volume of the unit cell, Ff (Fi) is the envelope function and uf (ul) is the periodic part of the Bloch state for the final (initial) state. Details of the calculation will be published elsewhere.

8

SuperlatticesandMicrostructures, Vol. 9, No. 1, 1991 E

E ;,

1

I

! i

I i

"o

d

50 d = I000

3

Oi

35-h~-

I

_.

143

-,

0 L

I 55

~.

v

i

0L

i

-I0

-5

[O

(TIC,.)-Eg)(meV)

(]5 GO- Eg){ meV)

(o)

(b)

F i g u r e 2. Total a b s o r p t i o n Wd(~ ) [in units of Wo; see as a f u n c t i o n

5

F)

["\ l,

-tO

K;

h (~-Eq)(meV) ( c/

probability Eq. (ii)],

of he - Eg, for a GaAs

-

(Ga,AI)As infinite QWW of radius d equal to (a) 50 ~, (b) 500 ~ and (c) i000 ~. E,(E2) i n d i c a t e s the onset of transitions from the first s u b b a n d to the lower (upper) the impurity band.

For a h o m o g e n e o u s distribution of i m p u r i t i e s n, and a s s u m i n g that the QWW r a d i u s is m u c h larger than the lattice parameter, one has for the total t r a n s i t i o n p r o b a b i l i t y per unit of time

wd(~) =

d2

Po Wd(Po'~)

(i0)

In our c a l c u l a t i o n s we have used c = 12.58, E = 1.424 eV for the bulk GaAs g band gap, and m = 0.0665 m where m c

o

is the f r e e - e l e c t r o n mass. R e s u l t s the optical absorption associated d o n o r - i m p u r i t i e s will be expressed units of W w h i c h is given by

o

for to in

o

4m w°

=

where

n aolCl21e.P 12 o

a

o

2

(11)

is the Bohr radius.

The total a b s o r p t i o n p r o b a b i l i t y for an infinite GaAs-(Ga,AI)As QWW with r a d i u s d=50 ~ is shown in Fig. 2(a). In c o m p a r i n g this result w i t h that of the QW of t h i c k n e s s 10p ~ [cf. F i ~ 3(b) of Oliveira and P e r e z - A l v a r e ~ ~ ], we observe that their magnitudes are s i m i l a r but the energy range of the Q W W a b s o r p t i o n is larger due to the wider i m p u r i t y band in the QWW. In Fig. 2(b) we show the total absorption probability for an infinite

valence edge of

G a A s - ( G a , A I ) A s QWW of radius d=500 ~. We observe that now there is a noticeable peak structure associated w i t h i m p u r i t i e s located at the center of the wire, but also a large peak r e l a t e d to i m p u r i t i e s located next to the edge of the wire is p r e s e n t with higher intensity than in the QW of c o m p a r a b l e well w ~ d t h [cf. Fi~. 4(d) of Oliveira and Perez-Alvare~-]. Our r e s u l t s for a d = i000 ~ Q W W are shown in Fig. 2 (c): the s t r u c t u r e a s s o c i a t e d w i t h i m p u r i t i e s located at the center of the w i r e is much larger than the s t r u c t u r e a s s o c i a t e d w i t h i m p u r i t i e s at the edge of the wire meaning that we have essentially reached the bulk limit. S u m m i n g up, we have calculated the density of impurity states and the optical - a b s o r p t i o n spectra a s s o c i a t e d with transitions between the n=l v a l e n c e s u b b a n d and the d o n o r impurity band for a GaAs - ( G a , A I ) A s i n f i n i t e and cylindrical QWW. We have found two special s t r u c t u r e s in the transition p r o b a b i l i t y per unit time Wd(~), i.e., an edge associated with transitions i n v o l v i n g i m p u r i t i e s at the c e n t e r of the well and a peak a s s o c i a t e d with transitions related to i m p u r i t i e s at the edge of the well. Although experimental results for the optical-absorption spectra associated with impurities in QWWs are not yet

Superlattices and Microstructures, Vol. 9, No. 1, 1991 available, we believe our results are of importance in the quantitative understanding of future experimental work in this field.

AcknowledgementsN.P.M. and J.L-G. would like to acknowledge the Brazilian Conselho Nacional [email protected] Desenvolvimento Cient~fico e Tecnologico (CNPq) for partial financial support during their stay in Brazil. This work was supported by Brazilian Agencies FAPERJ, CNPq and FINEP.

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