HYDROGENIC IMPURITY STATES IN QUANTUM DOTS AND QUANTUM WIRES

Hydrogenic

impurity

states

in quantum

dots

and quantum

wires

D.

S.

Chuu,

C.

M. Hsiao, and W.

N.

Mei*

Department

of

Electrophysics, National Chiao Tung University, Hsinchu, Taiwan, Republic

of

China

(Received 14 January 1992;revised manuscript received 24March 1992)

The energies ofhydrogenic impurity states with an impurity atom located at the center ofaquantum

dot and on the axis ofaquantum-well wire are studied. These two systems are all assumed to have an infinite confining potential. In the caseofthe quantum dot, the impurity eigenfunctions are expressed in

terms ofWhittaker functions and Coulomb scattering functions. The calculated ground-state energy of

the impurity approaches the correct limit ofthree-dimensional hydrogen atom as the radius ofthe quan-tum dot becomes very large. In the case ofthe quantum-well wire, analytical solutions can be obtained if

we divide the space into atwo-dimensional subspace (perpendicular to the axis ofthe quantum-well wire) and a one-dimensional subspace (parallel tothe axis ofthe quantum-well wire). The calculated ground-state energy ofthe quantum-well wire approaches the ground-state energy ofthe shallow-impurity atom located on the surface as the radius ofthe wire becomes infinite. Variations ofthe state energies with the radius ofthe quantum dot and the quantum-well wire are obtained.

I.

INTRODUCTION

In the past ten years, impurity states in various confined systems, such as quantum wells, quantum-well wires, and quantum dots, have been a subject

of

extensive investigations in basic and applied research. ' Quasi-two-dimensional (quasi-2D) quantum wells have been widely studied and applied to various semiconductor de-vices, such as high-electron-mobility transistors. Quasi-one-dimensional systems, such as quantum-well wires, are known to have the advantage

of

high mobility and suppression

of

carrier scattering. The emission line for quantum well wires was observed to be two to three times broader than that

of

the two-dimensional quantum wells and occurred at 6

—

10-meV higher binding energy.

Studies

of

quantum dots or quantum-we11 wires are very interesting problems because specific properties

of

these lower-dimensional structures can be easily achieved by varying the radius

of

the quantum dot or the quantum-well wire. An electron bound to an impurity atom located at the center

of

the quantum dot or on the axis

of

the quantum-well wire may appear to be unaffected by the boundary when the radius is very large and behaves very much like an impurity atom in the three-dimensional case. However, as the radius is re-duced, spatial confinement begins tocause the kinetic en-ergy

of

the electron to increase due to the uncertainty principle and eventually it may overcome the attractive potential between the electron and the impurity atom; thus the total energy may change from negative to posi-tive at a certain radius

of

the confining system and finally diverges to infinity as the radius approaches zero. More-over, the effective strength

of

the Coulomb interaction between the electron and the impurity atom depends on the geometric dimension

of

the system and isenhanced as the size

of

the system is reduced. Thus, in quantum-well wires or quantum dots the effective strength

of

the Coulomb interaction can be changed by varying the ra-dius

of

the quantum-well wire or the quantum dot.

Con-versely, dramatic changes in the binding energies may serve as a clear signal for changes in the effective dimen-sion

of

quantum-well wires orquantum dots.

Upon reduction

of

the geometric dimension, the confinement

of

electron motion becomes a pronounced effect. The physical properties

of

electrons in quantum-well wires or quantum dots are thus very different from those in the bulk. The impurity state in aquantum-well wire (or a quantum dot) has a spectrum composed

of

sub-bands

of

one- (or zero-) dimensional single-particle states. Each subband is continuous as a function

of

the wave vector and the energy gap between subbands is deter-mined by the splitting

of

the levels in the confining sys-tern. The density

of

state (DOS)

of

electrons in each sub-band is proportional to

E'

for the bulk and is a con-stant for agiven quantum well. The DOS isproportional to

E

' for a quantum-well wire, while it behaves like a 5function for a quantum dot.

Many theoretical works have been devoted to the study

of

the properties

of

impurity states in various confining systems. Lee and Spector have calculated the bind-ing energies for the bound states

of

a hydrogenic impuri-ty placed on the axis

of

a cylindrical quantum-well wire

of

infinite confining potential. Later, Bryant ' im-proved the model calculation by assuming afinite barrier for the confining potential with the impurity on and off the axis

of

the cylinder wire. The binding energies forthe bound states

of

hydrogenic impurity in a quantum-well wire

of

GaAs surrounded by

Ga,

Al As have been found to be 2—3 times larger than those in _comparable two-dimensional wells.' In the calculation

of

the hydro-genic impurity states, a variational principle with a trial wave function which takes into account the confinement

of

the carriers in the quantum-well wire or the quantum dot was usually employed. ' ' Zhu, Xiong, and Gu' used hydrogenic-effective-mass theory to study donor states in a spherical GaAs-Ga& Al As quantum dot.

Exact

solution and quantum-level structures were ob-tained. In the investigation

of

the behavior

of

the

46 HYDROGENIC IMPURITY STATES IN QUANTUM DOTS

AND.

.

.

3899 genlike impurity states in a small quantum-well wire, the

effects

of

disorder were also taken into account. ' Zhao et

al.

studied the collective excitations

of

edge-state electrons in a quasi-one-dimensional quantum-well wire under a large transverse magnetic field. The plasmon modes for both intrasubband and intersubband excita-tions were found. Briggs and Leburton

'

employed a Monte Carlo simulation

of

multisubband quasi-one-dimensional quantum-well wire. In their calculation, effects

of

polar-optical-phonon and inelastic acoustic-phonon scattering have been included. Although many works have investigated various aspects

of

the electronic properties in the lower-dimensional structures as men-tioned above, a full theoretical understanding

of

these bound states and their binding energies isstill lacking.

In this paper we present a detailed formulation for the state energies

of

a hydrogenic impurity located at the center

of

a quantum dot and or a quantum-well wire. Variations

of

the state energies with the radius

of

quantum-well wires and quantum dots are presented.

BR+2

BR+

1 A,

L(L+1)

4 (4)

where A,

=2pZe

/equi

a

and R

(aa)=0.

If

we further write R (g)

=

g

'F(g),

then

Eq.

(3) can be rewritten as

r

-'

—(L+

l)'

F"(g)+

—

—

+

—

+

4

_g g2

F(g)=0

. (5)

Equation (5) is the Whittaker equation with the solu-tions as

states is possible. Therefore, one can study solutions

of

the Schrodinger equation in two regions. (1)

For

E

&0,

the solutions can be expressed in terms

of

%hittaker functions. (2}

For

E

)

0, the solutions can be expressed in terms

of

the Coulomb-scattering wave functions, as can be seen in the following:

(1)Negative-energy region,

E

&0.

We first set

)=a,

r, with

a

= —

8pE/A &

0.

Then

Eq.

(3) becomes

II.

FORMULATION A. Quantum dot

Consider a hydrogenic impurity located at the center

of

a spherical dot which is confined by an infinite spheri-calpotential well with radius

a.

The Hamiltonian

of

this system can be written as

V

—

+

V(r)

2p H' where 0,

r(a

V(

)='

„)

F2

L(g)=e

~ g

+'4(L

+1

—

A,_,2L

+2,

g),

or

F

(g)

—

e—g/2(

L+1—

X4(

L+1

—

—

A,,

2L+2—

,

g),

where

4

isthe conAuent hypergeometric function

4(a,

b,

x)

=1+

—

a

—

x

+

a(a+1)1

x

2

+

b 1 b(b

+

1)2!

a(a+1)(a+2)

(a+k)1

b(b

+

1)(b

+2)

(b

+k)(k

+1)!

xx

k+1+

Hence, R (g)

of

Eq. (4) can be expressed as

(6a}

(6b)

(6c)

and

_p,

e,and

Z

are the effective mass, dielectric constant, and core charge.

V(r)

is the confining potential. The Schrodinger equation for

H

in spherical coordinates

(r,8, p) for

r

&acan be written as or

R(g)=g

'F&L(g)=e

~~ g

4(L+1—

A,,

2L+2,

g),

(7a)

a'

2a'

1 a . a sinO 2p Br

rs„r

sin8 ~8

_=e

&~ g 4&(

L+1

—

A,_, 2L

—

+2,

g) .

—

(7b)

+

2p Qy.2

r

Br

L(L+1)

Ze

=

_ER

₍

_r),

₍₃₎

with R

(a)

=0.

Since the motion

of

the electron is confined inside the dot, the existence

of

positive bound

B2

Z2

+

+

—

qI=EV

(2)

r

sin OBy

and

0'(r

=a,

8,

y)

=0

because the potential is infinite for

r

&

a.

%'(r,8,

g)

may be separated into the product R (r)8(8)g(q&),as in the case

of

the hydrogen atom,

8(8)

is the associated Legendre polynomial, and

_g(y)

=e'

m

=0, +1,

+2

... .

The differential equation for the radi-al part R

(r)

can be obtained as follows:

Since we require the wave function to be finite every-where, the wave function must bein the form

of

Eq. (7a) for

L

)

0

or Eq.(7b)for

L

&0,

i.e.

,

R

_(g)

=e

~ g'I _@(

_ILI+

₁

—

_A.,

2IL+2,

_g')

.

_(7c)

4(L

+1

—

A._,2L

+2,

aa)

=0

and the total energy

of

the system may begiven as pZ2e4

262/2 g2

(8) Because

I.

represents the total angular momentum, which is always positive, ~L~ can be replaced by

L

in the above solution. The value

of

A, can be determined from the

boundary condition R

(g=aa)

=0,

which isequivalent to setting

It

is easy to prove that

4

is an even function

of

A._; thus,

we need only consider the case where A,

)

0.

(2} Positive-energy region,

E

&

0.

First set

g=ar,

where

a

=

2p—

E/fi

&

0;

then

Eq.

(3) becomes

B'

_2B

2p Br

r

Br

L(L+1)

_R(

) Ze

R(

)=0

r2 E'r (16) 8 R 2 BR 2i}

L(L+1)

where n

=

pZ—e /equi

a

and R

(aa

)

=0.

If

we further set R (g)

=g

'F(g),

then

Eq.

(10)becomes

Comparing with the modified Bessel equation

2

By

By

x

+(1

—

2s)x

+[(s

—

y

a

)+a

y

x

jy

=0,

Bx

F"(g)+

1

— —

F(g)

=0

.

Equation

(11}

is a Couloinb equation. ' The solutions

of

Eq. (11)

are

_F„L(g)

and

_{G„I (():}

F„z(g)=e

'&g

+'4(L

+1

i',

_2L

—

_+2,

_2i

_g),

_(12a)

Ro(r)

=r

'

_J

2

1/2

8pZe _r

eR (18)

we obtain s

=

—,

', y=

—,

',

a

=(8mZe

)/(eiii ), and

a =(2L

+

1) . Thus, the radial function forL

=0,

which isfinite as r

~0,

can be written as

G„L(g)

=FI

(ri,g)

In(2()+

+8L(i},

g),

PL r} (12b)

To

satisfy the boundary condition

Ro(a)=0,

we require that

where

4

is the confluent hypergeometric function as shown in

Eq.

(6c).

_F„r

(g) can be expressed in another form as

(SpZe /eh' _)a

_=x;

where

x;

are roots

of

the Bessel function

J1.

The radial function for

L

=

1can be written as

F,

,

i(k}

=k'

"c'I.

(n

0»

where

41

(ri,g)is defined as

(12c)

Ri(r)=r

'

_J

2 1/2 8pZe _r equi (19) R

(g)=g 4L(i},

g) . (13)

The value

of

_{g can} be determined from the boundary con-dition

CLg,

",

pZe

a

—

—

0,

eA _g

and the total energy

of

the system may begiven as

A L( )gk L—

i-k=L+1

with

AL+1=1,

AL+i(i})

=ri/(L

+1)

and

(k

+L

(k

L

—

1)

Ak

=—

2riAk,

(i})

—

Ak

i(i})

for

k)L+2.

Since

_G„L(g)

is singular at

(=0,

only

F„L(g)

is used as the solution

of

the system. Hence, R

(g} of Eq.

(10)may be written as

B.

Quantum wire

Consider now an impurity located on the axis

of

an infinitely high cylindrical quantum well with radius d. The Hamiltonian for this system can be expressed as

+

Bx By AB

2I

IBz

Z2

+

V(p),

er (20a) The boundary condition

_R,

(a)

=0

y~~ld~

SpZe /equi )a

=y;

where _y; are the roots

of

the Bessel function

J3.

After substituting the values

of x;

and y; we can obtain the turning points.

For

example, 1s-state energy becomes positive when the radius is less than

1.

8325ao and 2p-state energy becomes positive when the radius is less than

5.

088 31ao,where ao isthe Bohr radius. From the above discussion, one can also note that the core charge and effective mass will effect the value

of

the turning point and a larger effective mass or core charge will yield a smaller turning point

of

the energy.

pZ2e4 1

2E.2g2 ~2 (15) where

Since 4&L

(i},

_g) is an even function

of

il, we only need to consider

g)

0.

(3) The turning point in energy from

E

&0

to

E

&0.

The turning point for the bound-state energy changing from positive tonegative can be obtained by setting

E

=0

in Eq. (3):

0,

p~d

I'(p)=

'

)d

and

r

=(x

+y

+z

)',

p=(x

+y )'~;

m, and mi are the transverse mass and longitudinal mass

of

the electron inside the well; eis the dielectric constant

of

the wire ma-terial; and Ze is the core charge. The axis

of

the wire is

HYDROGENIC IMPURITY STATES IN QUANTUM DOTS

AND.

.

. 3901 along the zdirection.

To

solve the Schrodinger equation

for the Hamiltonian defined in

Eq.

(20a),we first perform a coordinate transformation:

x

=x',

y

=y',

z

=(m,

/m()'/ z'

. The Hamiltonian in

Eq.

(20a) then becomes

represents aquantum-circle system in which a

2D

hydro-genic impurity islocated atthe center

of

an infinite circu-lar well. The solutions for this quantum-circle problem can be obtained in a similar way as discussed in the preceding section. The eigenfunctions for the quantum-circle system may be divided into two cases.

(1)

For

E

&0,

Z2

,

+I'(p')

.

(20b) ip(02)( ~ )

—

)/24(lmI+1/2 g X

@(

lml+I

—

~,

21m

1+2,

g);

(23a)

H

=H()1(p)+H()2(a)+H

(a

p)

where (20c) Here,

r'=(x'

+y'

+z'

)'/

and

p'=p=(x'

+y'

)'

For

convenience, we shall drop the prime forthe coordi-nate variables from now on. Now we rewrite the Hamil-tonian in the following form:

(2)for

E

)

0,

'(x,

_y;a)=g

4

_)/2(g,

_g),

(23b) where

4

_{)/2(g, g)} is defined in

Eq.

(12d}. The turning point for the energy from

E

)

0

to

E

&

0

in the quantum-circle system may be determined, as in the case

of

the _quantum dot, by setting

(}'

pe'

H()1(P)

=

2m Qz ez

f2

Q2 Q2 H()2(a)

=

+

2mt ~z ~y 2 2 2 pe

ae

ze 6z 6p 6r

ae

₊

v(p),

6'p (20d) (20e) (20f) and d '

J

0 2 '_1/2 8p

Z

_d'

₌₀

_{for m}

_=0,

eA ' 1/2 8

p

Z

_d'

₌₀

_{for m}

_=1.

eA

0,

p~d

I'(p)=

'

)d

and

a

and

_p

are the unknown parameters which are to be treated as variational parameters and can be determined later. Decomposing

H

into Hpi and H02 in

Eq.

(20c)is equivalent to dividing the space into two-dimensional (perpendicular to the axis) and one-dimensional (parallel to the axis) subspaces. Letus take

Hp(a, p)

Hpi(p)+H02(a)

as the unperturbed Hamiltonian and regard

H'(a,

p)

as the perturbation term (which can be adjusted as a small term by varying the parameters

a

and

p).

The unper-turbed ground-state wave function and the unperturbed ground-state energy for the Hamiltonian

_{Hp(a, P)}

may be written as

(I)(lml+1

—

A,_,

2lml+2,

ad)=0

.

(2}

For

E

&0,

(23c)

@m

—

1/2(9~ad }

The eigenvalues may then be given as E(p2)( ₎ )MZ2e4 1

2/2 g2

(23d)

(23e)

The first-order energy correction can now beobtained as

EEg"(a,

p)=()pg

'(r;a,

p)lH'(a,

p)l%'

'(r;a,

p) )

The requirement that )Il'

_'(x,

_y;a)

=0

_{at boundary}

_of

_the

circle implies the following: (1)

For

E

&0,

)p'

_'(r;a,

p)=%'

"(z'p))p'

'(x

y'a)

E(0)(

_p)

—

E(01)(p)+E(02)(

(21a) (21b) Ae

=

()P(

'(x,

_y;a)

)Ii( 2)(x,

_y;a)

₎

respectively, where (11'

"(z;p)

_{is the ground-state wave}

function

of

the 1D hydrogen atom (Hp, ), and

'(x,

y;a)

is the ground-state wave function

of

the 2D hydrogen atom in a circular well (H02).

For

the 1D hy-drogen atom, the eigenfunction and eigenvalue

of

the ground state may be given as

Ze

—

()Il'

_'(r;a,

_P)

)Ii'

_)(r;a,

_P))

_.

_(23f)

EP

The second term on the right-hand side (rhs) in the above equation can be integrated analytically togive

'Pg

"(z;P)=&2co

lzl exp(

—

colzl),

4 mte E(01)(p) (22a) (22b)

(

2 _mte4

'"'('p

=

'

_p'

ez (23g)

where

co=(2m,

e /eA }P The Hami.ltonian in

Eq.

(20e)

The third term

of

Eq.

(23f}can be reduced to a one-variable integral and readily integrated over ztogive

f

(Q2

3/2)2 2 —2coz R (p)pdp2 dz

[x

+y

+(m

/m )z

]'

=(2''3~~)2

f

_R2(p)

_dp ~_—,

'[Nr

r(2''p)

N—

„+,(2''p)]

0

—

_—,

_'[H„,

_(2''p)

—

_H

_+,

_(2''p)+m.

'

_(p/2)'[1

_(v+

—,

'

)]

']

',

(23h)

where

co'=(ml/m,

)co;H„(p) and

N„(p)

are the Struve functions and Neumann functions, respectively. Using Eqs. (23g) and (23h), and after performing some numeri-cal integrations, the total-state energy, up to the first-order correction, may in principle be obtained as

E (a,

P)=E'

"(P)+E'

'(a)+QE'"(a,

_P),

(23i) which contains parameters

of

a

and

P.

The optimum values

of

cc and

_P

can be determined by noting that the total Hamiltonian

H

in Eq. (20b) or (20c)does not con-tain the parameters

a

or

_P

so that the exact eigenvalues

of H

should be independent

of

a

and

P.

However, in most cases occurring in practice, one cannot obtain the exact eigenvalues

of

the Hamiltonian H, and only an ap-proximate solution is attainable. Thus, in the perturba-tion treatment, the values

of

a

and

P

can be chosen so that the approximate total-energy eigenvalue in

Eq.

(23i) should be least sensitive to the parameters

a

and

P.

This is equivalent to the following conditions:

BE„BE„

₌₀

_and

₌₀

_.

Ba (24)

By using Eq. (24), the values

of

a

and

_P

can be deter-mined. Thus, the total-state energy

of

the quantum wire can beobtained.

III.

RESULTSAND DISCUSSIONS

Figure 1 shows the variation

of

the calculated 1s

(n

=1,

L

=0),

2s (n

=2

L

=0),

and 2p (n

=2

L

=1)

CC 10— O lD CJI Q)

c

5—

);:I

15 2s 2p l ''~._W I I 1Q 15

Radius (uni ts of ao)

FIG.

1. 1s-, 2s-,and 2p-state energies ofa hydrogenic impuri-ty located atthe center ofaconfining sphere as a function ofthe radius ofthe confining sphere. %'*and ao are the effective

ryd-berg energy and the effective Bohr radius.

Eb E&o Eo (25)

where

_E,

_o isthe ground-state energy

of

the spherical well without the impurity and Eo is the ground-state energy with the impurity inside the we11. Figure 2 presents the

Eb Eo and

E

&o as a function

of

the dot radius r. Since

E

&o is proportiona1 to the reciprocal

of

the square

of

the

dot radius, one can see that

E,

o is large except when the

well width isvery large. The corresponding binding ener-gy

of

the Coulomb potential is proportional to the re-ciprocal

of

the radius

of

dot. One can note that Eb ap-proaches a large value as rbecomes very small, since the -state energies

of

a hydrogenic impurity located at the center

of

a quantum dot for the given dot radius. The en-ergy is expressed in terms

of

the effective Rydberg

%*=e

_{/(2eao ), where}

_@=K'

_{Ep is} the dielectric con-stant

of

free space, and the radius is expressed in terms

of

the effective Bohr radius (ao

=e

/(pe

), where

@=a,

mo, with

a,

the ratio

of

the effective mass

of

electrons in different material to the bare electron mass. In the calcu-lation, the mo and E'p are assumed to be the free-electron

mass and dielectric constant

of

free space. One can see from the figure that the energy

of

the 1s state becomes negative when the dot radius is larger than

1.

833ao and approaches

1%',

which is the energy

of

the n

=1

state for the free-space hydrogen atom. The energy

of

the 2s state becomes negative as the dot radius becomes larger than

6.125ao.

As the radius

r

becomes larger than

15ao,

the energy

of

the 2s state in the quantum dot approaches the value

0.25%'

of

the 2sstate in the free space. A simi-lar situation can be observed in the case

of

the 2p state. The 2p state

of

the spherical dot has positive energy as the radius becomes smaller than

5.088ao.

These two states are degenerate in the free hydrogen atom and split from each other as the radius

of

the dot becomes smaller than

8ao.

While the radius

of

the dot is larger than 10a o, they become almost degenerate and approach

0.25%",

which is the energy

of

the n

=2

state for the free-space hydrogen atom. One can also note that as the radius

of

the quantum dot decreases, the state energy in-creases. Furthermore, the energy increment forthe excit-ed state is much more pronounced than that

of

the ground state. As the radius r approaches zero, the state energy increases infinitely.

The binding energy Eb

of

the hydrogenic impurity is defined as the energy difference between the ground-state energy

of

the spherica1-well system without the impurity and the ground-state energy

of

the spherical-well system with the impurity,

i.

e.,

46 HYDROGENIC IMPURITY STATES IN QUANTUM DOTS

AND.

. .

3903 l 16— O 12 CXl L

c

LU 10 Radius(units of ci,*)

FIG.

2. The 1s-state energy (solid line), binding energy (dot-ted line), and kinetic energy (i.e., without the impurity, broken

line) of the hydrogenic impurity located at the center of a

confining sphere as the functions ofthe radius ofa confining

sphere.

electron is pushed toward the center

of

the spherical well by the confining potential as

r

approaches zero. Chu, Xiong, and

Gu'

performed a hydrogenic-effective-mass theory to calculate donor-state energies in a spherical quantum dot.

To

compare their result with ours, com-pare

Fig.

1

of

Ref.

17with the broken curves

of

our

Fig.

2. For

an infinite-potential well, the ground-state energy without the impurity obtained by Chu, Xiong, and Gu is around

2.

5%'

1.

5%',

1.

0%',

and

0.

SJ7'

for adot radius

of 2ao,

2.

5ao,

3ao,

and

4ao,

respectively. These results agree completely with our corresponding values shown in

Fig. 2. To

compare the ground-state binding energy for both works, one notes that the results for

Vo=

Do do not

differ much from

V0=80%"

or

40%*

for a dot radius larger than 2ao in the work

of

Chu et

al.

Therefore, we may compare the results shown in

Fig.

2

of Ref.

17with the dotted curve

of

our

Fig. 2. For

example, one can see our calculated binding energy is around

2.

6A'

and

2.

0%'

for a dot radius

of 2.

0ao and

3.0ao,

which are in very good agreement with those obtained _by Chu, Xiong, and

Gu.

' The same degree

of

agreement exists between our first excited state and that

of

Chu, Xiong, and Gu. From Figs. 1 and 2

of Ref.

17,one may easily find the excited-state energy without the impurity and the excited-state binding energy

of

the quantum dot. The difference between these two values yields the excited-state energy

of

the impurity located inside the dot. The excited-state energy

of

Chu, Xiong, and Gu can be ob-tained easily as

-3.

5%*,

1.

9%*,

and

1.

3% for adot ra-dius

_{of 2ao, 2.5ao,}

and

3.

0ao,

respectively. These values are in good agreement with ours shown in the dotted curve

of

Fig. 1.

Consider now a donor located at the center

of

a GaAs quantum dot. From our

Fig.

2,one ob-tains the ground-state binding energy

Eb-4.

6%

=24.

4 meV for the dot radius R

=

_lao (Eb

=

1%*=5.

3 meV for R

=

~

).

Ferreira da Silva' calculated the impurity states in a quantum-well wire

of

GaAs-Ga& Al As. They ob-tained the ground-state binding energy EI,

-24

meV for a wire radius

of

lao.

The fact that the

ground-state-O

8-E Ch

c

4 GaAs

---—

InAs 6 8 10 4 ~ ~ 0 % 4 4 p&dius(units of ao)

FIG.

3. The 1s-state (L =O,n

=1)

energy (in meV) ofa

hy-drogenic impurity located at the center ofsemiconductor GaAs (dashed line) dot and InAs (solid line) dot as a function ofthe radius ofthe dot. ao isthe effective Bohrradius.

impurity binding energy

of

a quantum dot isalmost equal to that

of

a quantum-well wire reflects the fact that the geometrical difference becomes less important as the ra-dius

of

the dot or wire becomes equal

to

the radius

of

the impurity atom.

Figure 3presents the ground-state energies

of

an im-purity atom located in GaAs and InAs quantum dots. The values

of a,

and a are

a,

=0.

067 and

~=13.

13 for GaAs, and

_a,

=0.

023 and

~=14.

6 for InAs. From

Fig.

3,one can see that our calculated ground-state energies

of

an impurity located in GaAs and InAs quantum dots for a very large radius

of

the confining system approach the correct limits

5.

3 and

1.

47 meV for the bulk GaAs and InAs semiconductors.

In the case

of

the quantum-well wire, instead

of

spheri-cal symmetry, there is cylindrical symmetry. Figure

4

shows the ground-state energy

of

a hydrogenic impurity located on the axis

of

a cylindrical wire as a function

of

the wire radius. As the radius becomes very large, the impurity should behave like a free hydrogen atom with the ground-state energy

of

1 Ry. However, our result

shows that for larger radii, our calculated energy ap-proaches

0.

22 Ry, not 1 Ry. The inconsistency is due

to

the dividing

of

the space into two orthogonal subspaces. The 1D subspace (the one-dimensional hydrogen atom) requires that the electron wave function vanish at posi-tion z

=0,

while in the real case the electron wave func-tion should vanish at the wire surface. This means that the method

of

dividing the space into 1D and 2D sub-spaces forces the creation

of

an additional node at z

=0

plane. This is equivalent to saying that we considered a half-cylindrical wire. Thus, as the radius

of

the half-cylindrical wire becomes very large, our ground-state en-ergy should approach the ground-state energy

of

a surface-impurity system. In the surface-impurity case, the lowest state isthe 2PO state

of

the free hydrogen atom

due to the surface selection rule. This explains why our ground-state energy for the quantum wire is

0.

22 Ry

in-lA CFl L CLl 2— IJJ

)

0) E 4 C) C7) EV C LLI Radius(units ofao*) 1 T 4 6 8 Radius(units of a',)

FIG.

4. The ground-state energy (solid line) and the binding energy (broken line) ofa hydrogenic impurity located at the axis ofa cylindrical wire asa function ofthe radius ofthe wire.

%*

and ao are the effective rydberg and the effective Bohr radius.

FIG.

5. The ground-state energy (in meV, solid line) and the

binding energy (in meV, broken line) ofa hydrogenic impurity

located at the axis ofa silicon cylindrical wire as a function of the radius ofthe wire. ao isthe effective Bohr radius.

stead

of

1Ry. Figure 5 shows the ground-state energy

of

a hydrogenic impurity located at the axis

of

a Si quantum wire. When the radius is very large, our calculated ground-state energy approaches the ground-state energy

10.

7 meV

of

_{the surface impurity}

of

a Si semiconductor. On the contrary, the binding energies for the Si bulk semiconductor is around 28.6 meV. Bryant' used a vari-ational trial wave function to calculate the binding ener-gy

of

ahydrogenic impurity placed on the axis

of

a cylin-drical quantum-well wire. He found a set

of

even and odd z-parity states. The binding energy

of

the lowest odd z parity state is lower than that

of

the lowest even z parity state. When the wire radius becomes very small, Bryant found that the binding energy

of

the lowest even z-parity state approaches 1

_{Ry. To}

_compare our _quantum wire shown in

Fig.

4 with those

of

Bryant, ' one should note that our results are similar to the odd z-parity case

of

Bryant because our wave function vanishes at z

=0,

which is equivalent to the case

of

the odd z-parity wave functions

of

Bryant.' The binding energies

of

the lowest even z-parity state for GaAs wi.re in the work

of

Bryant (Fig. 2

of Ref.

10) are around 5.

0%',

3.

4R*,

2.

2%*,

and

1.

9%* (1%'

=

5.3 meV) while the binding energies

of

the lowest odd z-parity state are around

1.

0%*, 0.7%*,

0.

6%',

0.

55%*,

and

0.

4%'

for the wire radius equals to

0.

5ao,

1Oao,

1.5a0,

2.

0ao,

and

3.

0a0

_(lao

—

100 A

—

200ao).

Our corresponding results shown in Fig. 4 are around

1.

1%",0.

7537*,

0. 65%*,

0. 59%',

and

0. 5%'

for the wire radii equal to

0.

5ao, 1.0ao, 1.5ao, 2.0ao,

and

3.

0ao.

Therefore, our results agree very reasonably with those

of

the lowest odd z-parity state.

IV. CONCLUSION

We obtained the analytical solutions for the state ener-gies

of

an impurity located inside a quantum dot and a quantum wire. A method

of

dividing the space into a one-dimensional subspace and a two-dimensional sub-space has been employed

to

solve the impurity-state ener-gies for a quantum wire. Whittaker functions and the scattering Coulomb wave functions are used in the case

of

quantum dots.

It

is found that as the radius

of

the quantum dot or the quantum-well wire becomes very large, the eigenenergies approach the corresponding state energies

of

the free-space hydrogen atom and become positive when the radius

of

the dot or wire is small. Al-though the present method has only been applied to infinite-confining-potential systems, when the confining potential is finite, one may employ an approximate method for the finite well or use direct numerical calcu-lation.

ACKNOWLEDGMENT

This work was supported by National Science Council, Taiwan, Republic

of

China.

Onleave from University ofNebraska, Omaha, Nebraska. ~R. Dingle, H.L.Stormer, A.C.Goassard, and W. Wiegmann,

Appl. Phys. Lett. 33,665(1978).

2H. Sakaki, Jpn.

J.

Appl. Phys. 19, L735 (1980);P.M. Petroff, A. C. Gossard,

R.

A. Loan, and W. Wiemann, Appl. Phys. Lett. 41,645 (1982)~

P.

F.

Yuh and

K.

L.Wang, Appl. Phys. Lett. 49,1738(1986). 4J.Lee and H. N. Spector,

J.

Appl. Phys. 54, 3921 (1983);57,

366(1985).

5V.

K.

Arora, Phys. Rev.B23, 5611 (1981).

6M.A.Reed,

J.

N.Aggarwal,

R.

J.

Matyi,

T.

M.Moore, and A. E.Wetsel, Phys. Rev.Lett. 60,535(1988).

7Ch. Sikorski and U.Merkt, Phys. Rev. Lett.62,2164(1989). 8G.W.Bryant, Phys. Rev.B37,8763(1988).

G.W.Bryant, Phys. Rev.B 31,7812 (1985). oG. W.Bryant, Phys. Rev.B29,6632(1984).

J.

W. Brown and H. N. Spector,

J.

Appl. Phys. 59, 1179 (1986).

E.

A. de Andrada e Silva,

I.

C.da Cunha Lima, and A. Fer-reira da Silva, Phys. Rev.

8

37,8537(1988).

'

_F.

_{A.P.Osorio, M. H.Degani, and O.Hipolito, Phys. Rev.}

₈

37,1402(1988).

' _{MH.Degani and O.Hipolito, Phys. Rev. B35, 9345 (1987).}

46 HYDROGENIC IMPURITY STATES IN QUANTUM DOTS

AND.

.

.

3905

' _{T.Kodama, Y.Osaka, and M.Yamanishi, Jpn.}

_J.

_{Appl. Phys.}

24, 1370(1985).

J.

L.Zhu,

J. J.

Xiong, and

B.

L.Gu, Phys. Rev. B 41,6001 (1990).

' _{A.D'Andrea and}

_R.

_{Del Sole, Solid State Commum. 74, 1121} {1990).

' _{A. Ferreira da Silva, Phys. Rev.B 41,1684(1990).}

H. L.Zhao, Y.Zhu, L.%'ag, and S.Feng, Phys. Lett. A 142, 36(1989).

S.Briggs and

J.

P.Leburton, Phys. Rev.

B

38, 8163 (1988).

22A._{Gold and} A.Ghazali, Phys. Rev.B41,7626(1990). Y.Z.Hu, S.W. Koch, and D.

B.

TraThoai, Mod. Phys. Lett.

B4, 1009(1990)~

Y.Zhu and S.Zhou,

J.

Phys. A21,1361(1988).

25L.Banyai,

I.

Galbraith, C.Ell, and H.Haug, Phys. Rev.B36, 6099(1987).

T.

Kodama and

Y.

Osaka, Jpn.

J.

Appl. Phys. 25, 1875{1986).

W.

Y.

Lai and S.DasSarma, Surf. Sci. 170, 43 (1986). SK.

B.

Wong, M. Jaros and

J.

P.Hagon, Phys. Rev.B35,2463

{1987).

J.

Cibert,

P.

M.Petrol; G.

J.

Dolan, D.

J.

%'erder, S.

J.

Pear-ton, A.C.Gossard, and

J.

H.English, Superlatt. Microstruct. 3,35(1987).

3oA._{Jeffrey, Table}

_of

Integrals, Series, and Products (Academic,

New York, 1980),p. 1059.

3~Handbook

_of

Mathematical Function With Formulas, Graphs,

and Mathematical Tables, Natl. Bur. Stand. Appl. Math. Series No. 55, edited by A. Abramowitz and

I.

A. Stegun, (U.S.GPO, Washington, D.C., 1964), Chap. 14,p. 538.

32S.Wolfram, Mathematica, ASystem forDoing Mathematics by

Computer (Addison-Wesley, Reading, MA, 1988).

R.

J.

Bell,W.

T.

Housman, G.M.Goldman, and D. G. Rath-bun, Surf.Sci.7, 293 (1967).