Shallow donor impurities in a quantum box

Solid State Communications, Vol. 85, No. 7, pp. 651-655, 1993. Printed in Great Britain.

0038-1098/93 $6.00 + .00 Pergamon Press Ltd

SHALLOW DONOR IMPURITIES IN A QUANTUM BOX Ka-Di Zhu and Shi-Wei Gu Institute of Condensed Matter Physics and Department of Applied Physics, Shanghai Jiao Tong University, Shanghai 200030, People's Republic of China (Received 14 April 1992; in revised form 22 October 1992 by P. Burlet)

The ground impurity binding energy and the effective potential due to the interaction between electron and the confined LO-phonons in quantum boxes are investigated by means of Lee-Low-Pines variational treatment. The results show that the effective potential induced by the interaction of the electron with the confined LO-phonons is very small at the boundary of the quantum box, while the potential rises to the maximum value at the center of the quantum box. 1. INTRODUCTION WITH RECENT advances in the art of microfabrication, quantum microstructures can now be made to exhibit quantum carrier confinement in all three dimensions (quantum boxes or quantum dots) [1-13]. These structures provide a great deal of uncovering of new phenomena and the potential device application in future lasers [7] and optical modulation [13]. The hydrogenic impurity states in quantum boxes have been investigated extensively [14-16]. Using different series forms in different regions of the radial equation, Zhu [14] obtained exact solutions of donor states in a spherically rectangular quantum well by a numerical method, and showed that there are stronger confinement and larger binding energy for a hydrogenic donor in the well of Gat_xAIxAs than in quantum-well wires and two-dimensional quantum-well structures. Further, Einevoll and Chang [15] calculated the binding energies of acceptors by using the effective bond-orbital model (EBOM). However, their evaluations have not taken into account the interaction of the electron with the confined LO-phonons. In the experimental work, Sikorski and Merkt [17] have realized arrays of quantum dots on InSb and obvserved intraband transitions between their discrete (zero-dimensional) electronic states with far-infrared magnetospectroscopy. Recently, Dellow et al. [18] demonstrated experimentally the resonant tunnelling through the bound states of a single donor atom in a quantum dot (in the presence of a magnetic field). Roussignol et al. [19] have shown experimentally and explained theoretically that phonon broadening is quite important in very small semiconductor quantum dots. Klein et al. [20] studied the size dependence of electron-phonon coupling in

semiconductor nanospheres, and derived the expression of the vibrational LO and SO eigenfunctions for a sphere in the continuum approximation. SchmittRink et al. [21] have evaluated the phonon broadening of optical spectra which the phonon modes are modeled by the corresponding bulk modes. This is an oversimplication. In [22] we have derived the interaction Hamiltonian of an electron with the confined LO-phonons in a quantum box and utilized this Hamiltonian to study a free polaron in a quantum box. It is shown for small boxes that the electron self-energy enhances rapidly to a maximum value and then shrinks slowly to the limit of the wire value as the box tends to infinity in one direction while remaining fixed in the other two directions. But the research work is restricted to the weak-couping systems. In this paper, on the basis of the same assumpion as in [22], we further investigate the bound polaron to a shallow-doped impurity located at the center of a quantum box by means of the Lee-Low-Pines variational treatment. The ground impurity binding energy and the effective potential induced by the interaction between the electron and the confined LO-phonons are derived. In Section 2 we shall present the calculation methods. Section 3 demonstrates the details of the numerical calculations and the results are discussed. Finally, a summary is given in Section 4. 2. THEORY For the sake of convenience, we assume that a shallow-doped impurity is located at the center of a quantum box. All other assumptions are similar to [22]. As a result, the total Hamiltonian of the electronphonon system is given by

651

=

/-/~ +/-/~

+ ~_~,.

(1)

SHALLOW DONOR IMPURITIES IN A QUANTUM BOX

652

The first term He is the Hamiltonian of the shallow donor and is expressed by

h2 ( 3 2 He =

2m* ~

Ix[ ~< T Li'

L2' lY[ ~< T

02 + ~

02 ) + ~

e2

eoor'

(2)

L3' [z[ ~< T

(3)

where m* is the band mass of the electron, r = (x2 + y2 + z2)~n, eo~is the optical dielectric constant and - e Z / e ~ r is the Coulombic potential. The second term is the LO-phonon operator, Hph

=

Bpl,p2,p3(ml,m2, m3), (4) where B~,p2,p3(ml,m2, m3) [Bp,.p2,,3(m,,m2, m3)] is the X

creation (annihilation) operator for the LO phonon of frequency ~OLO.m~, m2 and m3 are the x, y and z components of the total wave vector K. When rn~, m2 and rn3 are odd or even, p~, Pz and P3 are positive ( + ) or negative ( - ) . The last term in equation (1) is the Hamiltonian of an electron interacting with the confined LO-phonons and is directly taken from [22]. In the following we shall adopt the variational treatment developed by Lee et al. [23] to deal with the Hamiltonian (1). To faciliate the deductions, we use the following unitary transformation 2ZZ

k ml ,plm2,p2m3,P3

X

YEE

mI,plm2,p2m3,P3

(5)

h°)LoBm.p2.p3(m,,m2,m3)

Li,p2.p3(ml, + m2, m3) }

z))

=

I¢(x, y, z))10q),

(9)

We take the variational minimum of F as the effective Hamiltonian of the bound polaron in a quantum box, that is

h2 ( 0 2 //~ = min{F} =

632

2m* ~

+ ~

02 ) +

e2 -- eo~(x2 + y2 + z2)112 "[- VLo(X, Y, z), where VLo(X,y, z) is the effective potential induced by the interaction between the electron and the confined LO-phonons and expressed by VLo(X, y,

{[~

z) = -aho) 32NIN2N3

(m, nx) (m2ny'~ E.2Eo3cos k--U-, cos \ L~ /

+

[~

Era2Era3cos

(8)

(rn, nx'] (m2rcy'] L, / cos \-ZT-~ /

Fn3/~'Z]~I(m I, m2, m3)12

x sin \3 ( L

(mllrx~

(m2nY]

m3)] 2 x cos\(m3rcz~I(ml,mz, L~ / [~m~

(6)

(7)

(11)

kLo ~/~

x cos\(rn3~zZ L~ /~ l[(m,, m2, m3) ] 2

+

where ~b(x, y, z) is the wavefunction of the electron moving in the box and 10q) is the vacuum state of the phonon. The expectation value of the Hamiltonian H in such a state is given by

E = (dP(,,,°)p(X,y, z)IU-~HUI~m%(X, y, z)) = (dp(x, y, z)lFl(a(x, y, z)),

F[x, y, z, fp+,p2.p3(ml,m2, m3),

fpl,p2,p3(ml, m2, m3) ].

[~m~

wherefp~p2,p3(rn,, m2, m3) andfpl.p2.p3(mt, rn2, m3) are variational parameters which will subsequently be selected by minimizing the energy of the system. The wavefunction of the system can be written as

~(o) ,,,,,ex, y,

F = (01n*10) =

h~°LoB;,p2,p3(m,,m2,m3)

x L,,,2,,3(m,, m2, m3) -

where

(10)

E E E h(OLoB;,p2,P3(ml, m2, m 3 ) ml ,Plm2,p2m3,P3

U = expf

Vol. 85, No. 7

(mlrcX'~

E~sink-'L---~l J c o s \

(m2rcY'~

m2 m3

L2 ]

× cos\ L3 ] t(m,, ,~, m~)|~l

+ x sin

cos (rn, lrX ~ sin (m2rrY ~ L3 ,I I(ml' m2, m3)

+ [ ~ ~ ~ sin (m'Trx~

(m2~rY~

\ Ll /c°s\--ZT/

× sin f m3m'~

I(m,, m2, m3)-]2

+ [~'~

\ Lt ,]

\--~2 ,]

Vol. 85, No. 7

SHALLOW DONOR IMPURITIES IN A QUANTUM B O X

m3)] 2 x c o s (m37cz~I(m,,m2, \ L3 ) +I~sin\--'~-i

(m, ltx~

653

10

(a)

q

(m2rcy~

$

]sin\-'-~-'2 ]

~ I(m,, m2, m3) \L3 }

x sin (m3nz

(12)

where

> [(mln~ 2

(m2~ 2

(m37t'~2 ] I/2

41-

I(m"m2'm3) = l_\-ZT-i/ + \ L2 / + \ - 2 7 / _ 1 " (13) In order to calculate the impurity binding energy for the ground state electron subband we only choose the simplest approximation for the trial wavefunction, that is q~(x, y, z) = Ll L2, ~3 cos

cos

cos

.

I

I

0.2

0.4

N

I

I

0.6 X (Rp)

0.8

1.0

10

(b) q 8 3

(14) According to [24], this trial function only applies to the complete confinement case (i.e. for small boxes). The expectation value of the effective Hamiltonian is expressed by

gg

<~blnefrl~> =

Econ +

Ec +

ELO,

>__

IN1 = 2

0.2 2m*~ h27~2~ ( 1 1 12) ~212 + ~222 + ~ 7

\

(lS)

where Econ =

I

0.4

0.6 x (Rp)

0.8

(16) 10

(c)

is the confined energy of the electron.

Ec =

64~

nl2 n/2 n/2

,2e®a ! ! l cOs2(x) cOs2(y) cOs2(z) x x/N?x 2 + N~y 2 + N~z2dxdydz

1.0

(17)

is the average of the Coulombic impurity. ELO is the electron self-energy due to the interaction between the electron and the confined LO-phonons. Since its expression is too complicated we omit it here. 3. NUMERICAL RESULTS AND DISCUSSIONS According to the formula obtained in Section 2, we choose GaAs as an example and evaluate the effective potential of the confined LO-phonons, the confined energy and the Coulombic energy. In the calculations we consider that the shallow doped impurity is located at the center of the quantum box and for simplification only focus on the cubic quantum box ( m I = N2 = N3). Figure l(a)'(c) illustrate the effective potential VLo(X, y, Z) induced by the interaction between the electron and the confined LO-

3

7 6

>__ 2

~; 4

0

I

0.2

0.4

0.6 x (Rp)

= i 0.8

1.0

Fig. 1. The effective potential VLo(x, y, z) induced by the interaction between the electron and the confined LO-phonons as a function of x, y and z for several values of Nl. Rp is the polaron radius. (a) is the effective potential along the x-axis (y = z = 0); (b) is the effective potential along the diagonal in x-y plane (x = y, z = 0); (c) is the effective potential along the diagonal of the quantum box (x = y = z).

Vol. 85, 'No. 7

SHALLOW DONOR IMPURITIES IN A QUANTUM BOX

654

*o

~

50O

50

400

40-

300

*o

30

20O

20

100

10

0

I

4

I

I

8

12

l

16

[

20

4

N1

I

I

8

12

I

t6

20

N1

Fig. 2. The confined energy scaled by the electron effective Ryderg Re* = 5.3 meV as a function of the size (N1) of the quantum box.

Fig. 3. The Coulombic energy of the impurity scaled by the electron effictive Ryderg R* = 5.3meV as a function of the size (N~) of the quantum box.

phonons as a function of x, y and z, which (a) is the effective potential along the x-axis ( y = z = 0); (b) is the effective potential along the diagonal in x-y plane (x = y, z = 0); (c) is the effective potential along the diagonal of the quantum box (x = y = z). For each figure, we give two cases: NI = 7 and Nt = 14. From the figures we can see that at the boundary of the quantum box, the effective potential is equal to zero. This result implies that the interaction between the electron and the confined LO-phonons at the boundary of the box is very weak. At the center of the quantum box, the effective potential rises to the maximum value, the larger the size of the box, the larger the effective potential. Figure 2 presents the confined energy of the electron as a function of the size Nj of the quantum box. For a very small box, the confined energy is very large and gives the dominant contribution to the total energy. With increasing of the size of the box, the confined energy rapidly decreases according to the inverse square of N1 and monotonically tends to zero as the size of the quantum box approaches infinity. Figure 3 shows the Coulombic energy of the impurity as a function of the size N~ of the quantum box. For small boxes, the Coulombic energy is also large but is less than the confined energy in an order of magnitude. With enhancing the size of the box, the Coulombic energy rapidly shrinks according to the inverse N~ and monotonically tends to the electron effective Rydberg Re* of the three-dimensional case as the size of the quantum box approaches infinity.

tion of the electron with the confined LO-phonons is very weak at the boundary of the quantum box. At the center of the quantum box, the effective potential rises to the maximum value. (2) The confined energy of the electron rapidly decreases with increasing the size of the quantum box and monotonically tends to zero as the size of the box approaches infinity. The Coulombic energy of impurity also rapidly shrinks with enhancing the size of the quantum box and tends to the electron effective Rydberg R* of the three-dimensional case as the size of the box approaches infinity. (3) All the results obtained in this paper are available for both weak-coupling and intermediate-coupling electron-LO-phonon systems. Magnetooptical measurements in quasi-zero-dimensional excitation systems have just begun [25]. It is hoped that the theoretical results presented here might provide useful insights on the experimental investigations of shallow donor impurities in quasi-zero-dimensional systems.

4. CONCLUSIONS From the discussions above, we obtain the following conclusions: ( I ) The effective potential induced by the interac-

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