Energy levels and far-infrared optical absorption of impurity doped semiconductor nanorings: Intense laser and electric fields effects

Chemical Physics 479 (2016) 1–4

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Energy levels and far-infrared optical absorption of impurity doped semiconductor nanorings: Intense laser and electric fields effects M.G. Barseghyan ⇑ Department of Solid State Physics, Yerevan State University, Alex Manoogian 1, 0025 Yerevan, Armenia National University of Architecture and Construction of Armenia, Teryan 105, 0009 Yerevan, Armenia

a r t i c l e

i n f o

Article history: Received 30 June 2016 In final form 2 September 2016 Available online 3 September 2016

a b s t r a c t The effects of electron-impurity interaction on energy levels and far-infrared absorption in semiconductor nanoring under the action of intense laser and lateral electric fields have been investigated. Numerical calculations are performed using exact diagonalization technique. It is found that the electron-impurity interaction and external fields change the energy spectrum dramatically, and also have significant influence on the absorption spectrum. Strong dependence on laser field intensity and electric field of lowest energy levels, also supported by the Coulomb interaction with impurity, is clearly revealed. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction Quantum rings (QRs) have attracted a lot of attention due to their unique properties and have been under extensive theoretical and experimental investigations. Aharonov–Bohm effect has been observed in QRs [1] which shows potential to use these structures for quantum computational devices. In addition, QRs have found application in nanoelectronics and spintronics [2,3]. It is well known that the characteristic wavelengths determined by the position of the energy levels are very important in many applications. One way to change the position of the levels is to tailor the size of the nanostructures. However, for a given structure the transition energy between two levels is almost fixed (neglecting the fluctuations of temperature and external hydrostatic pressure) in the absence of external fields. Other way to change and control the position of the levels is the additional interaction of charge carriers, such as electron–electron [4–8] and electron-impurity interaction [9–15]. Recently a few number of investigations were devoted to the study of electronic and impurity states and intraband optical properties in zero-dimensional semiconductor nanostructures under the simultaneous influence of intense laser field (ILF) and external electric field. Using the effective mass and parabolic band approximations and a variational procedure Duque et al. calculated the combined effects of intense laser radiation, hydrostatic pressure, and applied electric field on a shallow-donor impurity states confined in cylindrical-shaped single and double GaAs=Ga1
x Alx As ⇑ Address: Department of Solid State Physics, Yerevan State University, Alex Manoogian 1, 0025 Yerevan, Armenia. E-mail address: [email protected] http://dx.doi.org/10.1016/j.chemphys.2016.09.001 0301-0104/Ó 2016 Elsevier B.V. All rights reserved.

QD [16]. Using a perturbative method Burileanu has investigated the behaviour of the binding energy and photoionization cross-section of a donor impurity in spherical QD under the influence of electric and laser fields [17]. The ILF effect on the impurity states in a CdS=SiO2 QD under applied electric field was studied within the effective mass approximation by using a finite difference method [18]. In our previous works effects of an ILF and static electric field on one-electron states and intraband optical absorption coefficient are investigated in two-dimensional GaAs=Ga0;7 Al0:3 As QR [19,20]. We have observed the splitting and increase of energy levels induced by the ILF. Meanwhile, the energy splitting, decrease and increasing of energy levels induced by the lateral electric field were obtained. Our results show that the incident light polarization can induce redshifts and blueshifts in the far-infrared absorption spectrum of the QR. In present work the combined influences of the ILF and lateral electric field on one-electron states and intraband optical absorption coefficient in GaAs=Ga0;7 Al0:3 As two-dimensional QR have been investigated taking into consideration the electron-impurity Coulomb interaction. Our study indicates that, strong electronimpurity interaction brings a very profound influence on the electronic states and on intraband optical absorption coefficient of QR. The paper is organized as follows: in Section 2 we describe the theoretical framework. Section 3 is dedicated to the results and discussion, and our conclusions are given in Section 4. 2. Theoretical framework We consider a cylindrical GaAs QR with the electron confined in the z ¼ 0 plane [21]. In this work we have used the method which is based on the non-perturbative theory that was developed

2

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originally to describe the atomic behaviour under intense, highfrequency laser field conditions [22,23]. We assume the system to be under the action of laser radiation and x-axis oriented lateral electric field. The laser field is represented by a monochromatic plane wave of frequency x0 . The laser beam is non-resonant with the semiconductor structure, and linearly polarized along a radial direction of the structure (chosen along the x-axis). In the high-frequency regime the particle is subjected to the time-averaged potential [24,25]

V d ðx; yÞ ¼

x0 2p

Z 2p=x0 0

Vððx þ a0 sinðx0 tÞÞi þ yjÞdt;

ð1Þ

where a0 ¼ eA0 =ðmx0 Þ denotes the laser field parameter, m is the electron effective mass, A0 ¼ A0 i is the vector potential, and i and j are the unit vectors along the laser polarization (x-axis) and the y-axis respectively. In the case of finite square lateral confining potential well, from Eq. (1) one may obtain a closed analytical form of V d ðx; yÞ, as in [19]. For the time-averaged laser-dressed hydrogenic donor impurity potential we use the Ehlotzky approximation [24,26,27]:

2

V i ðx; yÞ ¼


3

2

e 6 1 1 7 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5; 2e 2 2 2 2 Dþ þ y D
þ y

ð2Þ

where e is the dielectric constant of the material, which, for simplicity, is taken the same inside and outside the QR. Here

D2 ¼ ðx
x0  a0 Þ2 and x0 ¼ ðR1 þ R2 Þ=2, R1 and R2 are the inner and outer radii of QR respectively. The laser-dressed energies in the presence of hydrogenic donor impurity are obtained from the time-independent Schrödinger equation " #
h2 2
r þ V T ðx; yÞ þ eFx Ud ðx; yÞ ¼ Ed Ud ðx; yÞ; ð3Þ 2m ? where r2? ¼ @ 2 =@x2 þ @ 2 =@y2 ; V T ðx; yÞ ¼ V d ðx; yÞ þ V i ðx; yÞ and F is the electric field strength. The laser-dressed energy eigenvalues and eigenfunctions may be calculated using 2D diagonalization technique. The eigenfunctions are presented as a linear expansion of the eigenfunctions of the rectangle [19,20]. The light absorption process can be described as an optical transition that takes place from an initial state to a final one assisted by a photon. The optical absorption calculations for the intraband transitions are based on Fermi’s golden rule derived from timedependent perturbation theory [19,28–30]:

að
h xÞ ¼

16 p2 bFS
h x Nif jMfi j2 dðEf
Ei

h xÞ; nr V

In Fig. 1(a)–(d) the dependencies of first three dressed energy levels of on-centre (x0 ¼ ðR1 þ R2 Þ=2) impurity on the laser field parameter a0 are presented for different values of electric field strength F. As it can be seen the strengthening of the laser field (the increment of intense laser field intensity) produces a raising of the energy levels of impurity state, the reason of which is the following: with the strengthening of the laser field the dressing effect decreases the effective length of the confinement potential (for low lying states) along the laser field polarization direction. A similar phenomena has been found in the absence of hydrogenic impurity in our previous work [20]. In the presence of on-centre impurity due to the electron-impurity interaction the energy levels (especially the ground state energy) are lower, which makes the impurity states more localized and more sensitive to the laser field. It is important to compare with results without impurity in [20]: in the absence of impurity, increase in the laser field parameter from 0 to 5 nm produces 3.3 meV increase in the ground state energy (see the black curve in Fig. 4(a) in [20]), while in the presence of it the variation of ground state for the same interval of the laser field is about 18 meV, which shows the black curve in Fig. 1(a) in the present work. The energy levels of impurity states’ for higher values of electric field are more stable on dependencies on the laser field. This effect is caused by competition of both fields on the electron probability density in QR. The electron wave function is depicted in Fig. 2. The results are presented for different values of ILF parameter a0 and electric field strength F. It is observed that the electron cloud is no longer ringshaped as it must be in the absence of the impurity, but is centred at the impurity site. The laser dressing visibly reduces the ground state electron cloud localization around the impurity. On the other hand x-axis directed electric field forces the peak of the electron wave function to shift to the left part of QR, that is clearly visible in Fig. 2(b) where the localization is almost zero in the right side of QR, for j = 1, 2, 3 states. For the investigation of intraband optical absorption spectra of investigated nanosystem it is necessary to study also the dipole matrix elements induced by incident light. In our previous work the selection rules of allowed optical transitions have been

ð4Þ

where nr is the refractive index of the material, V is the volume of
x the sample per structure [1], bFS is the fine structure constant, h is the incident photon energy and Ef and Ei are the energies of the final and initial states, respectively. N if ¼ N i
Nf is the difference between number of electrons in the initial and final states. Since we consider a one-particle problem, we assume N i ¼ 1 for the ground state and N f ¼ 0 for all upper states. M fi is the matrix element of coordinate. We calculate the dipole matrix elements for x and y-polarizations of the incident light and replace the d-function by a Lorentzian profile with a full width at half maximum of 0.8 meV [31]. 3. Results and discussion The calculations are performed for a GaAs=Ga0:7 Al0:3 As QR with parameter values V 0 ¼ 228 meV, nr ¼ 3:6; m ¼ 0:067m0 , where m0 is the free-electron mass [32], and the radii of the ring are fixed to R1 ¼ 5 nm and R2 ¼ 25 nm.

Fig. 1. The dependences of the low-lying energy levels on the laser field parameter a0 . Several values of the electric field strength F are considered.

M.G. Barseghyan / Chemical Physics 479 (2016) 1–4

nm x

j=1

nm

3

j=3

j=2

F

(a)

j=1

y

j=3

kV/cm

j=2

F

(b)

Fig. 2. The wave functions of the first three dressed states of the electron (j = 1, 2, 3). Three values of the laser field parameter (0, 3, 5 nm) and two values of the electric field strength (0, 30 kV/cm) are considered.

obtained in the cases of absence of impurity for different types of light polarization. It was shown that in the case of y-polarization of the incident light the transitions from the ground state to the first excited state are allowed. While in the case of x-polarization the 1 ! 3 transitions are allowed. The presence of the two-dimensional hydrogenic impurity doesn’t destroy the symmetry of the corresponding electron wave functions and for that reason the selection rules of intraband optical absorption remain the same. It can be justified considering the symmetry of the wave functions of corresponding states, which is presented in Fig. 2. In the presence of hydrogenic impurity the symmetry of the wave functions, even in the absence of external fields, corresponds to the symmetry of the electron wave function without hydrogenic impurity with applied electric field. Particularly, as it can be seen from Fig. 2, there is no symmetry in respect to the y-axis, so there will be no defined parity of the wave functions in x variable; first and third states have even wave functions in y variable while second state

Fig. 3. The absolute value of the matrix elements jMfi =ej as functions of the laser field parameter a0 . Different directions of the light polarization and different values of electric field strength F are considered.

The intraband absorption coefficient dependence on the incident photon energy for different values of ILF parameter a0 , electric field strength F and different light polarization direction are shown in Figs. 3 and 4. It is worth to note that in the case of zero electric field the strengthening of ILF brings the red shift and increment of

will have an odd wave function in y : U1d ðx;
yÞ ¼ U1d ðx; yÞ;

U2d ðx;
yÞ ¼
U2d ðx; yÞ; U3d ðx;
yÞ ¼ U3d ðx; yÞ. Fig. 3 shows the dependencies of absolute values of the matrix elements jMfi =ej on ILF parameter a0 for different values of electric field. Different directions of the light polarization are considered. As it can be seen from Fig. 3(a) and (b) the matrix elements for allowed transitions increase with strengthening ILF, which is explained by the overlapping of the wave functions of corresponding states (see Fig. 2. On the other hand in the cases of relatively strong electric fields (see Fig. 3(c) and (d)) the matrix elements are not so sensitive to the ILF parameter, also showing increasing and decreasing behaviours. This effect happens for both directions of light polarization. It can be explained by the corresponding behaviour of wave functions, which is visible in Fig. 2. Comparing the wave functions of j ¼ 1; 2; 3 ground and excited states we can observe that the wave functions in the case of F ¼ 30 kV/cm (see Fig. 2(b)) are almost insensitive to the laser field. Which brings to the mentioned above reduced influence of ILF.

Fig. 4. Dependence of the intraband optical absorption coefficient on incident photon energy in a single nanoring. The results are presented for y-polarization of the incident light and laser field parameter a0 is considered from 0 (bottom) to 5 nm (top) with a 1 nm step.

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M.G. Barseghyan / Chemical Physics 479 (2016) 1–4

shown that in the presence of on-centre impurity an increase in the laser field parameter from 0 to 5 nm augments the ground state energy in 18 meV, while in the absence of electron-impurity interaction the variation of ground state in the same region of the laser field is about 3 meV. The energy levels of impurity states showed more stable dependencies on the laser field for higher values of electric field. Also, the presence of hydrogenic impurity creates red as well as blue shifts of the intraband absorption spectrum with the variation of both electric and laser fields. The obtained results show the importance of intense laser fields as well as homogeneous electric fields and hydrogenic impurity as potential tools to manipulate with the energy spectrum and in turn with optical properties of ring-like structures, which can form the material basis for opto-electronic devices operating in far-infrared region. Acknowledgments The author thank A.A. Kirakosyan and H.M. Baghramyan for useful discussions. The work was supported by the Armenian State Committee of Science (Project No. 15T-1C331) and Armenian National Science and Education Fund (ANSEF Grant No. nano4199). References

Fig. 5. Dependence of the intraband optical absorption coefficient on incident photon energy in a single nanoring. The results are presented for x-polarization of the incident light and laser field parameter a0 is considered from 0 (bottom) to 5 nm (top) with a 1 nm step.

the absorption spectra maximum, while in the absence of impurity red (y-polarization) and blue (x-polarization) shifts have been observed (see black curves in Fig. 7(a) and (d) in [20]). This is because of the corresponding behaviour of the matrix elements and energy distances between investigated states. In the presence of hydrogenic impurity and under the action of both fields the red as well as blue shifts are observed. The interesting one is in the case of F ¼ 15 kV/cm (see Figs. 4(b) and 5(b)). By changing the polarization direction of the incident light for the same range of ILF parameter in the infrared spectrum significant change has been observed. In the case of y-polarization only red shift has been observed, but in the case of x-polarization both shifts have been observed. If we compare the threshold energies with and without electron-impurity interaction the following result will be observed: without the external fields due to the electron-impurity interaction the maximum of the absorption coefficient moves to high energy region, which means that the blue shift have been observed (the threshold energy with impurity seven times bigger then threshold energy without impurity). 4. Conclusions We have studied the combined influences of the intense laser and lateral electric field on one-electron states in GaAs=Ga0:7 Al0:3 As single QRs taking into consideration the electron-impurity interaction. The mentioned above influences of external fields and incident light polarization direction are investigated on the intraband absorption coefficient as well. The laser dressed effect is considered both on electron confinement and electron-impurity Coulomb interaction potentials. It has been

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