Stark effect of D− ion in a quantum well

~)

Solid State Communications, Vol. 89, No. 7, pp. 601-603, 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/9456.00+.00 0038-1098(93)E0127-J

Pergamon

S T A R K E F F E C T OF D - I O N I N A Q U A N T U M W E L L Masamichi Fujito, Akiko Natori and Hitoshi Yasunaga The University of Electro-Communications, Chofu, Tokyo 182, Japan

(Received 17 September 1993 by H.Kamimura) The binding energies of D - ion in GaAs/Gao.rA10.3As quantum well are calculated by means of the variational method, as flmctions of the well width and the electric field perpendicular to the well interface. The well width dependence shows a peak at about 75/~ width, and it is explained by the competition between the confinement effect and penetration effect of a quantum well. The binding energy decreases monotonically with increasing an electric field, since the electronic polarization of D - ion is suppressed, compared to its one electron excited state. The bound state of D - ion vanishes in strong electric fields larger than 50 kV/cm.

1. I n t r o d u c t i o n

V~(z) -A negative-donor centre (D-) in a semiconductor is formed by a neutral donor (D °) trapping an extra electron. It is analogous to a negative hydrogen ion H - in vacuum, which binding energy is calculated to be 0.0555 Ry. 1 Thus the binding energy of D - ion in a bulk semiconductor is very small? -4 However in a quantum well, the binding energy of D - ion becomes much larger than a bulk value, due to its confinement effect? '6 Really, the binding energy of D - ion is calculated to be 0.48 Ry* in the 2-dimensional limit of strong confinement,r In the case of an infinite barrier height of a quantum well, the well width dependence of a D - ion binding energy has been reported, s However in the real case of a finite barrier height, the systematic study of the well width dependence has not been performed yet. With respect to the Stark effect, it has been studied extensively for a neutral donor in a quantum well,9'1° but there have been no studies for D - ion as far as authors know. The Stark effect is a promising method to control the binding energy of D ion, for its application to an infrared detector. In the present paper, we consider D - ion in a quantum well constituted of GaAs/Gao.rA10.3As. We study both the well width dependence and Stark effect of the binding energy of D - ion, by means of the variational method.

kO(r,,r2) ----~b'(rl)~b2(r2) % ~b2(r,)g',(r2)

The following linear combination of anisotropic Ganssian orbitals are taken for single electron orbitals ¢i(r) (i = 1,2). 1

u

e;,(r) = - ~ , E CJxj(r), v

~,

(5)

j=l

XJ(f)'Z):I2aj(2bj)n'+l-~znjexp(-ajp2-bjz2)'Tt'i~(nj -[" 2

(1)

(6)

where p is the distance fi'om the z axis. Xi are normalized Gaussian orbitals and Si are numerical factors which are determined by the normalization condition of ~bi(r). The

7"12

+ C.- ,

(4)

~/2(1 + (¢1t¢~> ~)

We consider a D - ion located at the center of a single quantum well in the electric field perpendicular to the well interface. The Hamiltonian is written in the effective mass approximation as

h(r) = -~V ~_ _1 + U~(z) ?-

(3)

Here V~(z) is the quantum well potential with the well width L and the barrier height Vb, and c is the strength of the electric field applied in the z direction. In eq. (2) we neglected the following effects, band nonparabolicity and both differences of effective mass and dielectric constant between the well and barrier regions.6 In eqs. (1),(2) and (3) the effective atomic units are used, i.e. an effective Bohr radius a~ = h2e,/m*e 2 as a unit of length and twice the energy of effective Rydberg 2Ry* = m*e4/h2~,2 as a unit of energy. From the values of static dielectric constant ~, of 12.6~0 and of the electron effective mass m* of 0.0655m0 for GaAs, the effective atomic units are calculated as a~ = 99.5/~ and 2Ry* = 11.44meV. The barrier height Vb is evaluated to be 0.6AEg, from the energy gap mismatch of A E 9 = 1.247x eV between GaAs and Gal_xAlxAs.TM Our numerical calculation was carried out in a case of x = 0.3, i.e. Vb = 224meV. We adopted the variational method for the calculation of the ground state of a D - ion, using the following Chandra.sekhar type wave function.4

2. F o r m u l a t i o n

1 H(r,,r2) = h(rl) + h(r2) + - - ,

{ Vb (Izl > L/2) 0 (Izl _< L/2)

(2)

Z

601

Vol. 89, No. 7

D- ION IN A QUANTUM WELL

602

20

300

,

,

'1L~/

>

I

"s

lO ~

; ~ 0,5 ¢0

~

~

lOO

°~ 0

100

200

300

~ 400

500

0

10'0

Well w i d t h L ( ~ ) Fig. 1. Binding energies of D - ion (thick solid line) and a neutral donor D o (thin solid line) in the absence of the electric field, as a fnnction of the well width L. The arrow heads on the left and right vertical axes represent the bulk binding energies of D - ion and a neutral donor, respectively.

ground state energy of a D - ion is obtained by minimizing (~[nl~) E D - - - [~------~ (~ , (7) with variational parameters aj, bj and Cj for each orbital ~bi(r) (i = 1,2). On the other band, the binding energy Eb of a D ion is defined as Eb ----EDo + EQW -- ED- ,

(8)

where EDO is the ground state energy of a neutral donor in a q u a n t u m well and EQW is the ground state energy of an electron in the quantum well. They are obtained by minimizing (~lhl~b) E D o - (¢lg'------~' (9) and EQw =

(¢1_1

2

(~b[¢)

20'0

30'0

40'0

5oo

Well w i d t h L ( ~ )

(10)

As for single electron orbitals in eqs. (9) and (10), the variational wave functions of eq. (5) are assumed. All the integrals in eqs. (7), (9) and (10) are calculated analytically. In numerical calculation, we used the conjugate gradient method for minimizing energies with variational parameters aj, bj and Cj. The Gaussian orbitals up to 12th order with z are necessary for representing the ground state in a q u a n t u m well. In the electric field the odd order terms become also necessary to express the polarization. In the absence of the electric field, 10 Gaussian orbitals were used for D - ion, 11 orbitals for a neutral donor and 7 orbitals for the ground state in a q u a n t u m well. In the presence of the electric field, 16 Gaussian orbitals for D - ion, 20 orbitals for a neutral donor and 13 orbitals for the ground state in a q u a n t u m well were used.

Fig. 2.

The root-mean-square radii ( q ~

(thick solid

line) and x( / ~ - ) / 2 (thick broken line) of D - ion with those of a neutral donor (corresponding thin lines), as a function of well width L.

3. N u m e r i c a l R e s u l t s Fig. 1 shows the calculated binding energies of a D ion and a neutral donor, as a function of the well width L in the absence of the electric field. Both binding energies approach each bulk value indicated by arrow heads at each vertical axis, in the limit of a large well width. As L decreases, the binding energies increase at first due to the confinement effect of a well, and subsequently they decrease due to penetration of the wave fnnctions outside the well. The competition behavior between confinement effect and penetration effect is clearly seen in the root mean square radii ( w / ~ of both a D - ion and a neutral donor in Fig. 2. The well width at the maximum of a D - binding energy is wider than that of a neutral donor, because the penetration effect becomes prominent at a wider well width for D - ion. ~/(p~)/2 are also plotted in Fig. 2, and it is seen that the anisotropy of radii show maxima near the peak width of binding energies. The electric field dependence of the binding energies of a D - ion and a neutral donor are shown in Fig. 3, for L=200A quantum well. The dependence in Fig. 3 can be explained by their polarization behavior. In a weak field ~', the polarizations are proportional to g', and the binding energies decrease proportional to £2. Fig. 4 shows the polarizations of a D - ion and a neutral donor with polarizations of their one electron excited states to the ground state of a q u a n t u m well, as a function of an electric field. For both a D - ion and a neutral donor, their excited states are more polarizable than the ground states. Then the effects of polarization lowers the binding energies. In a strong field, the polarization becomes suppressed by the barrier obstacle effect of a q u a n t u m well. Thus the binding energy of a neutral donor tends to saturate at large ~', as seen in Fig. 3. On the other hand for D - ion, the bound state of D - ion vanishes in these strong electric fields.

Vol. 89, No. 7 1.2

D- ION IN A QUANTUM WELL ~

,

,

12

603 0

>

-

-20 0.8 -

~

8~ -40

0.6

i~

0.4

4~

0.2

2

v

-80

o.~

°~

~1

-60

~

-

-02 o

26

46

60

86

'

-2 100

-100 0

Electric field E(kV/cm)

20

40

60

'~" 8 0'

100

Electric field E(kV/cm)

Fig. 3. The electric field dependence of binding energies of D - ion (thick solid line) and a neutral donor D O (thin solid line) in a quantum well of width 200/~. The negative value for D - ion means vanishing of the bound state. For a neutral donor, the previous result 1° is also shown by a thin broken line.

4. Discussion a n d C o n c l u s i o n We have calculated the ground state energies of D ion, a neutral donor and single electron in a quantum well under the electric field. The single electron wave functions are represented quite well by the linear combination of anisotropic Gaussian orbitals. It gives better results for the binding energy of a donor in the electric field, compared with the previous calculation 1° as seen in Fig. 3. However for D - ion, the assumed variational wave function takes into account only the radial correlation between two electrons, and so the approximation is not so good as for single electron states. Accordingly the calculated D - binding energies become smaller than the exact values. They give about halves of exact val-

Fig. 4. The polarizations of D - ion (thick solid line) and a neutral donor (thin solid line) with those of their one electron excited states (corresponding broken lines), as a function of the electric field g.

ues obtained in the bulk limit I and at the well width of 100/~. 6 This means that the degree of accuracy dose not depend on the well width. So, the calculated well width dependence gives sufficiently reliable result. To improve our calculation, consideration of the angular correlation between two electrons is required. The binding energy of D - ion depends on a quantum well width and has its maximum at about 75/~. With respect to the application of D - ion to a far infrared detector, a quantum well structure with the width of about 75/~ is favorable, since the absorption threshold energy is insensitive to deviation of widths around this value. The binding energy of D - ion decreases monotonically with increasing an electric field, and the bound state disappears in strong electric fields beyond 50 kV/cm. This variation of a binding energy with an electric field will achieve a tunable far infrared detector.

References

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