Polarizability of a polaron in spherical quantum dots

Physica E 14 (2002) 184 – 189

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Polarizability of a polaron in spherical quantum dots H. Satoria; b; ∗ , A. Salib; c , K. Satorid a UFR

Optique et Dynamique des Materiaux, Faculte des Sciences d’Oujda, B.P. 524, Oujda, Morocco Normale Superieure, Department de Physique, B.P. 5206, 30000, Bensouda F)es, Morocco c D epartement de Physique, Faculte des Sciences, B.P. 4010, Beni M’Hamed, Mekn)es, Morocco d D epartement de Maths et Informatique, Faculte des Sciences, B.P. 1796, Dhar Mehraz F)es, Morocco b Ecole

Abstract Using a variational calculation within the e+ective mass approximation, we have calculated the binding energy and the polarizability of a polaron bound to a shallow hydrogenic impurity located at the center of spherical quantum dots embedded in a glass matrix, considering an in-nite con-ning potential. The binding energy of the ground state is calculated as a function of the radius of the dots for di+erent values of weak electric -eld applied in the z-direction. The polaronic e+ect has been considered by using the Lee–Low–Pines variational method. Our results show that the polarizability and the binding energy are very sensitive to the polaronic and the con-nement e+ects. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 71.38.+i; 71.55.−i; 73.20.Dx Keywords: Quantum dots; Impurity; Binding energy; Polarizability; Electric -eld; Polaronic e+ect

1. Introduction At present, much attention is being paid to physics of low-dimensional semiconductor heterostructures such as quantum wells (QWs), quantum well wires (QWWs), and quantum dots (QDs). The con-nement of the carriers in these structures is responsible for the appearance of quantum phenomena, which cannot be observed in bulk semiconductors. The e+ect of an applied electric -eld on the physical properties of low-dimensional systems has become a subject of considerable interest both from the theoretical and technological point of view, due to the importance of ∗

Corresponding author. Ecole Normale Superieure, DAepartment de Physique, 7ES 30000, Bensouda FCes, Morocco. Tel.: +212-672-501-92. E-mail address: [email protected] (H. Satori).

these systems in the development of new semiconductor devices. Generally, these microstructures are based on polar and semi-polar semiconductors for which the polar interactions between the -eld induced by vibrational motion and the electronic charge density are imposed. The description of longitudinal optical (LO), surface optical (SO) and interface optical (IO) phonon modes has recently been performed for a variety of shapes such as rectangle, cylinder and sphere. The results show that the phonon modes and their eigenfrequencies depend on the geometry of the material considered [1–15]. The application of an electric -eld along the growth axis of heterostructure gives rise to a polarization of the carrier distribution and to an energy shift of the quantum states. Such e+ects introduce considerable changes in the energy spectrum of the carriers, which

1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 2 ) 0 0 3 8 1 - 8

H. Satori et al. / Physica E 14 (2002) 184 – 189

may be used to control and modulate the intensity output of optoelectronic devices [16 –20]. In this paper, we report a calculation of the ground state binding energy and the polarizability of a polaron bound to a hydrogenic impurity in a spherical QD embedded in a glass matrix in the presence of an electric -eld. We use a variational method in which the trial wave function takes into account the con-nement of the carrier in the dot, the inLuence of the Coulomb interaction between the impurity ion and the electron (while the impurity is present), and the inLuence of the electric -eld. The polaronic e+ects due to the con-ned LO phonon and SO phonon are considered by means of the modi-ed Lee–Low–Pines (LLP) [21] transformation.

+ where b+ ‘m (k) (b‘m (k)) and a‘m (a‘m ) are the creation (annihilation) operators, respectively, for the con-ned LO and SO phonon modes. k is the wave number for the LO phonons mode. The third term and the last term in Eq. (1) represent, respectively, the electron– phonon interaction Hamiltonian (He−ph ) and the ion– phonon interaction Hamiltonian (Hion−ph ) and can be written as follows [1,2,9]:
LO f‘ (k)[j‘ (kr)Y‘m (r)b‘m (k) + h:c:] He−ph = ‘mk




+ Hion−ph =

2. Hamiltonian

H = He + Hph + He−ph + Hion−ph :

(1)

The -rst term is the Hamiltonian for a hydrogenic impurity con-ned in the well potential V (r), i.e. 20 He = −∇2 − + V (r) + r cos ( ): (2) ∞ |r − ri | Here, we use the reduced atomic units (a.u.), which correspond to a length unit of e+ective Bohr radius, a∗ = ˜2 0 =m∗ e2 , and an energy unit of e+ective Rydberg, R∗ = m∗ e4 =2˜2 02 . In Eq. (2), r denotes the vector position of the electron, m∗ the electronic e+ective-mass, 0 the static dielectric constant, ∞ the high-frequency optical dielectric constant of the quantum dot material, and ri is the impurity position, which we take at the center of the dot (ri = 0). In the electric term r cos ( ), =eFa∗ =R, is the measure of the electric -eld strength. V (r) is zero for r 6 R and in-nite for r ¿ R. The second term in Eq. (1) represents the free phonon Hamiltonian,
Hph = ˜!LO b+ ‘m (k)b‘m (k) ‘mk

+


‘m

˜!SO a+ ‘m a‘m ;

‘m


‘m

+

Under the e+ective mass approximation, the Hamiltonian of a hydrogenic impurity system con-ned in a polar spherical QD semiconductor in the presence of a uniform and weak electric -eld, F, applied in the z-direction can be written as

(3)

185

f‘SO [(r=R)‘ Y‘m (r)a ˆ ‘m + h:c:];

(4)

f‘SO [(0=R)‘ Y‘m (r)a ˆ ‘m + h:c:]




‘mk

f‘LO (k)[j‘ (0)Y‘m (0)b‘m (k) + h:c:]; (5)

where h.c. denotes the Hermitian conjugate, j‘ (x) is the spherical Bessel function of the ‘th order and Y‘m (r) ˆ is the spherical harmonic function. The coupling coePcients are [1,9]  1=2 4˜!LO e2 f‘LO (k) = ± 3 2 2 R k j‘+1 (kR) ×(1=∞ − 1=0 )1=2

(6)

for LO modes, and √  1=2 2˜e2 ‘∞ !LO SO f‘ = ± ‘∞ + (‘ + 1)d R!‘ ×(1=∞ − 1=0 )1=2

(7)

for SO modes. The sign + is related to the ion–phonon coupling and to the electron–phonon coupling. d is the matrix dielectric constant.

3. Model In the adiabatic approximation, we therefore choose the trial wave function for the ground state as |

tot 

= U | e |0;

(8)

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H. Satori et al. / Physica E 14 (2002) 184 – 189

where |0 is the vacuum state of the polaron and U is a unitary displacement transformation 
[g‘m a+ U = exp ‘m − h:c:] ‘m

+


‘mk

[h‘m (k)b+ ‘m (k)

 − h:c:]

(9)

The binding energy Eb of the hydrogenic impurity is de-ned as the ground state energy of the system E( = 0; )min without the coulomb interaction, minus the impurity ground state energy E(; )min , taking into account the phonons e+ect and the electric -eld in both situations [23]. Eb = E( = 0; )min − E(; )min :

(16)

in which g‘m and h‘m (k) are the variational functions to be determined by minimizing the energy of the system. Following the HassAe variational method [22], the wave function | e  describing the electron ground state in the presence of the weak applied electric -eld is given by

The polarizability of the polaron is de-ned [16] as E(0) − E( ) p = 2 × lim : (17) →0 2

| e  = (1 + r cos ( ))| 0 :

The numerical results are applied to the material CuCl whose electron–phonon coupling is intermediate (CuCl = 2:45) and to the weakly polar material GaAs (GaAs = 0:06). The parameters used to calculate the binding energy of a bound polaron in spherical QDs are 0 =7:9; ∞ =3:61 and m∗ =m0 =0:5 for CuCl, where m0 is the free electron mass and 0 = 12:5; ∞ = 10:9 and m∗ =m0 = 0:06 for GaAs. The results are displayed Q and R∗ =109:0 meV in atomic units (a.u.) a∗ =8:36 A ∗ Q for CuCl and a = 100:2 A and R∗ = 5:75 meV for GaAs. The dielectric constant of the glass matrix is taken to be d = 2:25. In Fig. 1a, the binding energy of hydrogenic impurity in a spherical CuCl QD embedded in a glass matrix with (broken curves) and without (full curves) phonons correction, is plotted versus the dot radius R for di+erent values of electric -eld. From this -gure, for a given value of the electric -eld and for both cases, as the dot radius decreases, the binding energy increases. This originates from the shrinkage of the electronic wave function with the geometric con-nement. For dots with R ¡ 3a∗ , the binding energy is relatively insensitive to the electric -eld e+ect, since for small radii of the dot, the dominant contribution to the binding energy comes from the Coulomb and the con-nement potential. For larger dots, the electric -eld governs the behavior of the binding energy because it overcomes the spatial location. Besides, the increasing of its strength tends to extend the electronic ground state wave function and to ionize the electron. Consequently, the binding energy decreases with the increase of the electric -eld for large dot radius [19].

(10)

The trial wave function | 0  for the ground state in the absence of electric -eld, is taken as  sin(k10 r) exp(−r); r 6 R; (11) 0 (r) = r 0; r ¿ R; where  and  are the variational parameters, and k10 = =R. The expectation value of the energy of the polaron is given by E=

tot (r)|H | tot (r)

 

tot (r) tot (r)

=

 e (r)|H  | e (r) ;  e (r) e (r)

(12)

where H  = 0|U −1 HU |0: The variational conditions 9E 9E 9E 9E = ∗ = 0 and = ∗ =0 9h‘m 9h‘m 9g‘m 9g‘m

(13)

are used to determine the form of h‘m (k) and g‘m . The total energy of the system is found to be LO SO + Eph : E = Ee + Eph

(14)

LO SO and Eph are the correction due to the con-ned Eph LO and SO phonon modes; respectively; with

Ee =

 e (r)|He | e (r) :  e (r) e (r)

(15)

4. Results and discussions

H. Satori et al. / Physica E 14 (2002) 184 – 189

187

1.0

6

4

2

Polarisability (e a* /R*)

a

3

2

Binding energy (R*)

5

b

c

2

a 1

b

c 0.0

0 1 (a)

0.5

2

3

4

5

6

7

R (a*)

0 (b)

1

2

3

4

R (a*)

Fig. 1. (a) Binding energy of CuCl QD as function of the dot radius with phonon correction (broken curves) and without phonon correction (full curves) for di+erent electric -elds F = 0:1, 100 and 200 kV=cm curves a; b and c, respectively; (b) polarizability of CuCl QD as function of the dot radius for both cases with phonon correction (broken curve) and without phonon correction (full curve).

It is interesting to note that this e+ect is more pronounced in the presence of electron–phonon coupling. The comparison of results obtained with and without electron–phonons interaction shows that the e+ects of electron–phonons coupling on the binding energy are more signi-cant for small radii of the dot than for large radii. These results are in accordance with those obtained by Klimin et al. [14] who showed, by using a perturbation theory, that when the dot radius decreases, the interface contributions to a polaronic shift goes to zero, but the bulk-like phonon contribution increases in its module. Furthermore, we note that in the absence of the electric -eld, the correction given by the SO phonon vanishes. This result is in agreement with that obtained in Refs. [1,3,14]. Whereas, for F = 0 or for the impurity displaced on the center of the dot, the correction due to the SO phonon is weak but does not vanish. It is important to mention that the results of electron–phonon e+ect we have obtained in a QD system are more signi-cant than in QW [24] and QWW structures [25]. This is due to the increasing electronic con-nement achieved by the reduction of the dimensionality.

Fig. 1b illustrates the variation of the polarizability of the spherical polar CuCl QD, as a function of the dot radius. The results show that polarizability decreases as the radius of the dot decreases for both cases. This clearly shows the importance of the electronic con-nement on the polarizability. For large dots, the electronic con-nement becomes negligible and hence the results tend towards the 3D bulk case [16,18]. This means that when the electric -eld is applied along one direction (z), the electron accelerates along this direction. The interaction with phonons tends to weaken this acceleration specially for dot with R ¿ 1a∗ where the polaronic e+ect is more manifested in the polarizability. For R 6 1a∗ , the polarizability is quite similar for both cases with and without phonons correction. The general characteristic features and implications of the present calculations agree with those obtained in the recent paper of Feddi et al. [20]. In Figs. 2a and b, we have reproduced the case of the weakly polar material QD GaAs. The same behavior is exhibited as in Figs. 1a and b. The comparison between Fig. 2b and 1b shows that the correction due to the phonons is more important for the more polar QD CuCl than for the weakly polar QD GaAs. This

188

H. Satori et al. / Physica E 14 (2002) 184 – 189 5

1.5

Polarisability(e2a*2/R*)

Binding energy (R*)

4

3

2

a b

1

a

1.0

0.5

b 0

0.0 0

1

(a)

2

3

4 R (a*)

5

6

7

0

4

2 R (a*)

(b)

Fig. 2. (a) Binding energy of GaAs QD as function of the dot radius with phonon correction (broken curves) and without phonon correction (full curves) under di+erent electric -elds F = 0:1 and 3 kV=cm curves a and b, respectively; (b) polarizability of GaAs QD as function of the dot radius for both cases with phonon correction (broken curve) and without phonon correction (full curve).

result is in accordance with that obtained by Pokatilov et al. [15]. The same -gure shows that the binding energy is a+ected even for small values of electric -eld (F = 3 kV=cm). This results clearly explains that the electric -eld is more ePcient for the case of weakly polar QD GaAs than for the intermediate CuCl polar QD. In Fig. 2b we plot the polar polarizability of spherical GaAs QD versus the dot radius. The comparison with Fig. 1b shows that the polarizability is more weakened by the polaronic e+ect for CuCl QD than for the GaAs weakly polar material QD.

more polar QD CuCl than for the weakly polar QD GaAs. The electric -eld tends to decrease the binding energy for large dot radius. We have also shown that the polarizability decreases as the dot radius decreases and it is more weekend by the polaronic e+ect. We expect that our results may be helpful to simulate and explain new experimental works. Acknowledgements This work has been done with the -nancial support by CNR under “Programme d’Apui aC la recherche Scienti-que PARS” Physique 06.

5. Conclusion We have calculated the binding energy and polarizability for a polaron bound to a hydrogenic impurity located at the center of a spherical QD in both weakly and more polar materials in the in-nite potential model case. The interaction between the electron– phonons and the ion–phonons coupling has been taken into account in our study using a modi-ed LLP variational treatment. We have found that the correction in energy due to the phonons is more important for the

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