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Physica B 393 (2007) 213–216 www.elsevier.com/locate/physb

Temperature effect on impurity-bound polaronic energy levels in a GaAs parabolic quantum dot Shi-Hua Chena,, Jing-Lin Xiaob a

College of Sciences, Huzhou Vocational Technology College, Huzhou 313000, China College of Physics and Electro-Engineering, Inner Mongolia National University, Tongliao 028043, China

b

Received 21 November 2006; received in revised form 26 December 2006; accepted 4 January 2007

Abstract Energy levels of an impurity atom and its binding energy in a GaAs parabolic quantum dot (QD) are obtained by the second-order Rayleigh–Schrodinger perturbation theory, taking into account of the electron-bulk LO–phonon interaction. The binding energy of the ground state and the low-lying excited state is expressed as a function of the temperature and the effective conﬁnement length of the QD. It is found that the binding energy is a decreasing function of temperature, and the temperature effect becomes obvious in small quantum dots (QDs). r 2007 Elsevier B.V. All rights reserved. PACS: 73.21.La; 71.38.k Keywords: Quantum dot; Polaron; Binding energy

1. Introduction The rapid advances of nanofabrication technology have made it possible to work with quasi-zero-dimensional quantum dots (QDs) in laboratories. Such systems are of great interest in fundamental studies because of the completely discrete electronic states, as well as in practical applications for microelectronic devices because of their design ﬂexibility. Consequently there has been a large amount of work, both experimental [1–7] and theoretical [8–13], on QDs of materials such as GaAs/GaxAl1xAs compounds. One of the major concerns in such systems is the impurity states, which have attracted extensive attention in recent years [1–3,8,9]. The ground state and the ﬁrst excited state of an electron in a QD may be employed as a two-level quantum system (qubit). An electromagnetic pulse can be applied to drive an electron from the ground state to the ﬁrst excited state or to a superposition of the ground state and the ﬁrst excited state. To perform a quantum-controlled NOT Corresponding author. Tel.: +86 572 2364302; fax: +86 572 2363000.

E-mail address: [email protected] (S.-H. Chen). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.01.004

manipulation, one may simply apply a static electric ﬁeld by placing a gate near the QD [14,15]. The same scheme can be implemented using the ground state and the ﬁrst excited state of an impurity electron in a QD. Since the electron–phonon interaction is essential to understand the optical absorption spectra in semiconductors, [16] research on the polaron effect has become a main subject in the physics of low-dimensional quantum systems. Especially in the QD system, the electron–phonon interactions are enhanced by the geometric conﬁnement. Therefore, a number of studies have focused on the inﬂuence of the electron–phonon interactions on impurity properties in a QD [17–24]. Chen et al. [23] studied the thickness dependence of the binding energy of an impurity bound polaron in a parabolic QD in magnetic ﬁelds by using the second-order perturbation theory. Au-Yeung et al. [25] investigated the combined effects of a parabolic potential and a Coulomb impurity on the cyclotron resonance of a three-dimensional bound magnetopolaron by using Larsen’s perturbation method, under the condition of strong parabolic potential. Wang et al. [26] recently studied the binding energy of hydrogenic impurities in a GaAs cylindrical QD by using a two-parameter variational wave

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function. By introducing a trial wave function constructed as a direct product form of an electronic part and a part of coherent phonons. Kandemir et al. [19] investigated the polaronic effect on the low-lying energy levels of an electron bound to a hydrogenic impurity in a threedimensional anisotropic harmonic potential subjected to a uniform magnetic ﬁeld. In these works, however, temperature dependence of bound polarons in the system is ignored. In the present paper, we investigate the temperature dependence of the binding energy of an impurity bound polaron in a GaAs parabolic QD by using the second-order Rayleigh–Schrodinger perturbation theory. We will also analyze the effect of the effective conﬁnement length of the QD on the binding energy. This paper is comprised of the following. First, we derive the expressions of the binding energy of an impurity bound polaron by using the secondorder RSPT method. Then, our numerical results are presented and discussed. Finally, a brief conclusion is drawn in our investigation. 2. Theoretical model The electrons are much more strongly conﬁned in one direction (taken as z direction) than in the other two directions. Therefore, we shall conﬁne ourselves to consider only the motion of the electrons in the xy plane. We assume that the conﬁning potential in a single QD is isotropic and harmonic: V ðrÞ ¼ 1=2mn o20 r2 , where mn stands for the electron band mass, q is the coordinate vector of a two-dimensional quantity and o0 is a parameter characterizing the conﬁnement strength in the xy plane. The impurity atom is situated at the origin. The Hamiltonian of electron–phonon systems is given by H ¼ H 0 þ H 1, H0 ¼

p2 1 n 2 2 X e2 þ m , þ o r þ _o b b LO q q 0 2mn 2 1 r q

X H1 ¼ ðV q eiqr bq þ H:c:Þ.

V nmn0 m0 q==

(3)

ð7Þ

1=2 n0 !n! ¼2 0 ðn þ jm0 jÞ!ðn þ jmjÞ! Z 1 0

0 dxxjm jþjmjþ1 Lnjm0 j x2 0

Ljnmj x2 J mm0 q== l 0 x exp x2 , ð8Þ

where m ¼ 0,71,72,y is the angular quantum number

and n ¼ 0,1,2,y is the radial quantum number. l 0 ¼ _= mn o0 Þ1=2 and Jn are the Bassel functions of the ﬁrst kind. The corresponding wave functions are given by 1=2 n

2 n; m; nq ¼ p1ﬃﬃﬃﬃﬃﬃ eimy 2m o0 n! xjmj Ljnmj x2 ex =2 nq , _ðn þ jmjÞ! 2p (9)

n 1=2 jmj where x ¼ r m o0 =_ , Ln are the associated Laguerre polynomials. The electronic self-energy shift can be found by the second-order Rayleigh–Schrodinger perturbation theory. 1=2 _ dE nm ¼ að_oLO Þ2 2mn oLO X X

Qnmn0 m0 nq þ 1 n0

m0

2ðn0 nÞ þ m0 jmj _o0 þ _oLO 1 þ Qnmn0 m0 nq

1 o 2ðn0 nÞ þ m0 jmj _o0 _oLO . ð10Þ Here

Z

1

Qnmn0 m0 ¼

h i2 dq== V nmn0 m0 q== .

(11)

0

(4)

(5)

At ﬁnite temperatures, we choose jnq i for the wave function to describe the phonon state, in which nq represents the number of LO-phonons, 1 nq ¼ expð_oLO =kB TÞ 1 , (6) where kB is the Boltzmann constant.

(2)

Here bþ q ðbq Þ creates (annihilates) a longitudinal optical (LO) phonon of frequency oLO and wave vector q(q ¼ q//,qz) and r ¼ (q,z) is the coordinate of the electron.

Using the Fourier expansion 1 X 4p ¼ expðiq rÞ. r V q2 q

q

X 4pe2 0 m0 q V nmn == , V 1 q2 q

(1)

q

V q ¼ ið_oLO =qÞð_=2mn oLO Þ1=4 ð4pa=V Þ1=2 .

It is not difﬁcult to ﬁnd that the unperturbed energy levels are X nq _oLO E 0nm ¼ ð2n þ jmj þ 1Þ_o0 þ

In Eq. (10), the ﬁrst term corresponds to the emission of virtual phonons excited by the electron–phonon interaction and the second term represents the absorption of virtual phonons in the process of electron–phonon interaction. The numerical results show that both of these two processes must be taken into account at ﬁnite temperatures. Combining the above results, we ﬁnd the energy levels of a polaron bound to an impurity atom at the origin 0Þ E nm ¼ E ðnm þ dE nm .

(12)

If Ee denotes the electron energy levels in the QD without any impurity, then the binding energy is given by X 4pe2

E b ¼ E e E nm ¼ V nmn0 m0 qk dE nm (13) 2 V 1 q q

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18

Eq. (13) expresses the binding energy for every state (n,m) as a function of the temperature and the effective conﬁnement length of the QD.

16

34 32

12

10 l0=3.0r0 8

0

50

26 24

20 18 50

100

200 T(K)

250

300

350

400

Fig. 2. The excited state binding energy in GaAs parabolic quantum dots as a function of the temperature for four effective conﬁnement lengths of the quantum dots.

30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

Eb0

Eb1(Eb-1)

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

for T ¼ 200 K. It is shown that the binding energy of the ground state and the low-lying excited state increase with decreasing the effective conﬁnement length of the QD. The result indicates that the polaron effect is quite important in small QDs at ﬁnite temperature. Figs. 1–3 show that the ground-state binding energy is larger than the low-lying excited binding energy at ﬁnite temperatures.

l0=2.5r0

l0=3.0r0

0

150

Fig. 3. The variation of impurity binding energy in GaAs parabolic quantum dots as a function of the effective conﬁnement length of the quantum dot for T ¼ 200 K.

l0=2.2r0

22

100

l0(r0)

28 Eb0(meV)

l0=2.2r0

l0=2.0r0

30

16

14

l0=2.5r0

Eb(meV)

As an illustration, we calculate the binding energy of an impurity atom with the electron-bulk LO–phonon interaction in a realistic sample GaAs for which a ¼ 0.068, _oLO ¼ 36.25meV, and me ¼ 0.067m0 [27], where m0 is the electron bare mass. The ground-state binding energy Eb0 corresponding to n ¼ 0, m ¼ 0, the excited binding energy Eb1(Eb1) corresponding to n ¼ 0, m ¼ 1 (n ¼ 0, m ¼ 1) in Eq. (10). The numerical results of the temperature and the size dependences on the binding energy of the ground state and the low-lying excited state in GaAs parabolic QDs are presented in Figs. 1–3. Figs. 1 and 2 show the ground state binding energy Eb0 and the low-lying excited state binding energy Eb1(Eb1) as a function of temperature for four effective conﬁnement lengths of the QDs (l0 ¼ 2.0r0, 2.2r0, 2.5r0 and 3.0r0, where r0 is the polaron radius), respectively. From Figs. 1 and 2 we can see that both the ground state binding energy Eb0 and the low-lying excited state binding energy Eb1(Eb1) are slowly decreasing function of temperature. That is to say, the binding energy of the ground state and the low-lying excited state will be weakened with the enhancement of temperature. Figs. 1 and 2 also show that as the size of the QDs becomes small, the change in the binding energy of the ground state and the low-lying excited state with temperature will be obvious. This result indicates that the temperature effect is quite signiﬁcant in small QDs. Fig. 3 illustrates the binding energy of the ground state and the low-lying excited state as a function of the effective conﬁnement length of the QD

Eb1(Eb-1)(meV)

l0=2.0r0

3. Numerical results and discussions

150

200

250

300

350

400

T(K) Fig. 1. The ground-state binding energy in GaAs parabolic quantum dots as a function of the temperature for four effective conﬁnement lengths of the quantum dots.

4. Conclusion In conclusion, we investigated the impurity-bound polaronic energy levels in a GaAs parabolic QD at ﬁnite

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