The effect of an intense laser field on magneto donors in semiconductors

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The effect of an intense laser field on magneto donors

in semiconductors

H. Sari

a,∗

_{, E. Kasapoglu}

a

_{, I. Sokmen}

b a_{Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey}

b_{Department of Physics, Dokuz Eylül University, ˙Izmir, Turkey} Received 11 February 2003; accepted 13 March 2003

Communicated by J. Flouquet

Abstract

The laser-field dependence of the binding energy of shallow-donor impurities in graded, and square quantum wells under the external magnetic field is calculated by a variational method and in the effective mass approximation. We have shown that, not only the ‘dressed’ potential, but also the shape of the confinement potential, the strength of the external magnetic field parallel to the growth direction, and the impurity position play very important roles in the determining the binding energy of a hydrogenic impurity.

PACS: 71.55.Eq; 73.20.Hb

Keywords: Quantum wells; Impurities; Intense field

1. Introduction

The development of very intense laser fields has provided an opportunity to explore the properties of matter in strong electromagnetic fields that greatly exceed the Coulomb binding fields in an atom. The effect of an intense high-frequency laser field on the physical properties of bulk semiconductors has been discussed and analyzed in the literature [1–6].

To date, there exists a significant literature de-lay with the effect of external fields on the physi-cal properties of the semiconductor heterostructures

* _{Corresponding author.}

E-mail address: sari@cumhuriyet.edu.tr (H. Sari).

has attracted the attention of many researches in the last decades [7–16]. More recently, these studies have been extended low-dimensional semiconductor het-erostructures under intense electric fields, created by an applied AC voltage or a high-intensity THz laser [17–20]. As is well known, the external electric field has important consequences for the electronic and op-tical properties of the structure. Fanyao et al. have re-ported calculation of the binding energy of an axial donor hydrogenic impurity in ideal, infinite, cylindri-cal quantum wire placed in an intense, high-frequency laser field [17]. They have also presented calculations of the binding energy of an on-center donor hydro-genic impurity in a quantum dot in an intense laser field [18]. It was found that the binding energy of an impurity in the low-dimensional systems decreases

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with increasing the laser-field amplitude [21,23]. Also it is shown that, although the individual external elec-tric field themselves act in a way to lower the binding energy of the impurity, one might accomplish a reduc-tion of the weakening effects by considering the joint action of the two external fields [18]. Oliveira et al. have used the dressed-band approach to treat the in-teraction of a laser field with semiconductor GaAs– (Ga,Al)As quantum wells and dots [20]. They suggest that the strong localization of the electronic and impu-rity states due to the quantum well and quantum dot and enhanced by laser confinement may result use-ful for manipulation of electronic and donor states in some proposed solid-state based quantum computers. A electron in a semiconductor nanostructure under the action of a magnetic field perpendicular to the growth direction of and monochromatic electromagnetic radi-ation, linearly polarized in a direction perpendicular to the magnetic field is considered by Perez-Maldonado et al. [21], and they have observed a symmetry break-ing in the quasi-energy spectra for intensities greater than a critical value. It was also found that, when an in-plane magnetic field is applied the nonlinear effects appear in superlattices for lower radiation intensities and for radiation polarized in any direction perpendic-ular to the magnetic field [24].

In this Letter we consider the magnetic field effect on the binding energy of a hydrogenic impurity placed in a graded quantum well (GQW) in the additional presence of a intense, high-frequency laser field, taking into account the laser ‘dressing’ effects on the both impurity Coulomb potential and the graded confinement potential [17]. It was shown that, not only the ‘dressed’ Coulomb potential, but also ‘dressed’ confinement potential, and impurity positions plays very important roles in the determining the binding energy of a hydrogenic impurity [17–19]. Nevertheless to the best of our knowledge, there is no study on the effect of an intense laser-field on the polarizability of a donor in a graded quantum well.

2. Theory

The method used in the present calculation is based upon a nonperturbative theory that has been devel-oped to describe the atomic behavior in intense high-frequency laser fields [22,23]. We assume that the

ra-diation field can be represented by a monochromatic plane wave of frequency ω. For linear polarization, the vector potential of the field in the laboratory frame is given by A(t)=eA0cosωt, where e is the real unit vector of the polarization. By applying the time-dependent translation r→r+α(t)the semi-classical Schrödinger equation in the momentum gauge, de-scribing the interaction dynamics in the laboratory frame of reference, was transformed by Kramers as follows [25], (1) − ¯h2 2m∗∇ 2_ϕ(_r_{, t)}₊_V_r₊_α_(t)_{ϕ(r, t)}₌_i_h_¯∂ϕ(r, t) ∂t .

HereV (r)is the atomic binding potential and (2) α(t)=eα0sinωt, α0=

eA0 m∗cω,

represents the quiver motion of a classical electron in the laser field andV (r+α(t))is the ‘dressed’ potential energy. In terms of the average intensity of the laser,I,

α0can be written as [17],

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α0=

I1/2/ω2e/m∗(8π/c)1/2,

whereeandm∗are the charge and effective mass of an electron, respectively,cis the velocity of the light andA0is the amplitude of the vector potential.

Following the Floquet approach [22,23], the space-translated version of the Schrödinger equation, Eq. (1), can be cast in the equivalent form of a system of coupled time independent differential equations for the Floquet components of the wave function ϕ, containing the (in general complex) quasi-energyE. An iteration scheme was developed to solve this, for the zeroth Floquet componentϕ0 the system reduces

to the time-independent Schrödinger equation [17,22, 23]. (4) − ¯h2 2m∗∇ 2₊_{V (}_α_0,_r₎ ϕ0=Eϕ0,

where V (α0,r) is the ‘dressed’ potential which

de-pends on ω and I only through α0 [22]. For the

Coulomb potential case V (r)= −_εe_|2_r_|, the ‘dressed’ potential has the form [26]

(5) Vc(α0,r)= − e2/2ε 1 |r+α0|+ 1 |r−α0| ,

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3. Dressed binding energy

By changing the Al concentrationx in the G1−x

-AlxAs one obtains a linearly changing conduction

band profile as shown in Fig. 1. The functional form of the confinement potentialV (z)is given as,

V (z)=V0Θ|z−L/2| (6) +V0 4 z L/2+1 Θ|L/2−z|,

whereV0is the conduction band offset at the interface, Lis the well width, andΘis the step function.

Before preceding further and applying the above described dressed potential theory to our particular graded quantum well system, we write down the Hamiltonian of a system consisting of an electron bound to a donor impurity inside a graded quantum well in the presence of an intense high-frequency laser field (the laser-field polarization is along thez -direction) and an external magnetic field applied along the growth direction, is given by

H= − ¯h 2 2m∗ ∂2 ∂ρ2+ 1 ρ ∂ ∂ρ+ 1 ρ2 ∂2 ∂Φ2 − ¯h2 2m∗ ∂2 ∂z2+ ehB_¯ 2m∗cLz+ e2 8m∗c2B 2_ρ2 (7) +Vb(α0, z)+Vc(α0, r),

whereB is the magnetic field strength,Vc(α0, r) is

the ‘dressed’ Coulomb potential, andVb(α0, z)is the

‘dressed’ confinement potential which is given by the

Fig. 1. The schematic representation of the conduction band in the graded Ga1−xAlxAs structure. following expression: Vb(α0, z)= V0 2 Θz−(L/2+α0) +Θz−(L/2−α0) (8) +V0 8 z L/2−α0+ 1 ×Θ(L/2−α0)−z + z L/2+α0 + 1 ×Θ(L/2+α0)−z .

Using the variational method, it is possible to as-sociate a trial wave function, which is an approxi-mated eigenfunction of the Hamiltonian described in Eq. (7). The ground-state wave function of the impu-rity is given by

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ψ(r)=ψ(z)Φ(ρ, λ),

where the wave function in the (x–y) plane is chosen to be the wave function of the ground state of a two-dimensional hydrogen-like atom [27,28]:

(10) Φ(ρ, λ)=1 λ 2 π 1/2 exp(−ρ/λ),

in which λ is a variational parameter. The ground-state impurity energy is evaluated by minimizing the expectation value of the Hamiltonian,ψ(r)|H|ψ(r)

with respect toλ.

The ground-state donor binding energy is given by [29,30]

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EB=Ez+γ−

ψ(r)Hψ(r),

where Ez is the ground sate energy of the elec-tron obtained from the Schrödinger equation in the

z-direction without the impurity, andγis the first Lan-dau level,γ=ehB/(_¯ 2m∗c).

4. Results and discussion

For numerical calculations, we take,m∗=0.0665×

m0 (wherem0 is the free electron mass), ε=12.58

and the aluminum concentrationx=0.39. The posi-tion of the donor impurity is given as(xi,0,0)the bar-rier heightV0is obtained from the 60% (40%) rule of

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Fig. 2. The calculated impurity binding energies of a graded and square GaAs/Ga1−xAlxAs quantum wells as a function of the

laser-field amplitudeα0, for different values of the applied magnetic field, for a particular value of well widthL=125 Å, and (a)zi=0, (b)zi= −L/2, (c)zi=L/2.

the band-gap discontinuity"Egfor donor (acceptor)

states for aluminum concentrationx0.45, such that the band-gap is direct at theΓ point and"Egis given by [31] as,"Eg=1247x(meV).

Fig. 2. Continued.

The calculated impurity binding energies of a graded and square GaAs/G1−xAlxAs QW material as

a function of the laser-field amplitude α0 are given

in Fig. 2(a)–(c) for different values of the applied magnetic field, for a particular value of well width

L=125 Å, and zi=0,−L/2, L/2, respectively. As

seen in Fig. 2(a) for both GQW and SQW structure the binding energy of an impurity first does not depend onα0, but after a critical value ofα0,α0>20 Å, the

binding energy rapidly decreases with laser field. The reason is that; initially the spatial confinement is pre-dominant but for further large value ofα0the

geomet-ric confinement of the donor electron increases thus it gets to be more energetic and it can penetrate into the potential barriers easily where the wave function of the electron reflects its three-dimensional charac-ter. This penetration modifies the subband dispersion relations and weakens the Coulomb interaction be-tween the donor electron and ionized impurity atom, thus the overlap function gets smaller and the bind-ing energy begins to decrease rapidly. At the same time Fig. 2(a) shows that, the binding energies of the graded and square well structure get to be same for large enoughα0values (α030 Å), since in this case

three-dimensional character of the donor electron is predominant and the effect of the shape of the con-finement potential on the Coulombic interaction is not

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important. As expected the binding energy increases with magnetic fieldB, since magnetic field gives an additional lateral electronic-confinement and the prob-ability of the finding the donor electron around the im-purity ion increases withB, resulting in higher bind-ing energy. It should be pointed out that, for the on-center impurity in the range ofα0<30 Å where the

shape of the confinement potential on the Coulombic interaction is important, the donor electron is mostly confined in the left side of the graded well whereas in the SQW the electron moves freely in the whole well region, as a consequence the binding energy in the GQW is smaller than that of SQW. As seen in Fig. 2(b) due to the asymmetric character of the elec-tron wave function in the graded well in contrast to the results of on-center impurity, the binding energy of GQW is higher than that of SQW for all values of applied laser fields, since there is an additional elec-tron confinement in the left side of the GQW. No-tice also that especially for smallα0 values the dif-ference in the binding energy between two models increases with magnetic field. Also Fig. 2(b) shows that, for the donor located atzi= −L/2 while in the

GQW the binding energy decreases with α0 in the

SQW vice versa. Since in the GQW the donor elec-tron is mostly confined in the left side of the well region, and by increasing the laser field the dressed confinement increases and the donor electron can eas-ily penetrates into the barrier regions thus the overlap between the electron and impurity ion begins to de-crease withα0. On the other hand, in the SQW due to

the symmetric character of the electron wave function the lateral confinement increases with laser field and the electronic probability around the impurity ion in-creases up to the value ofα0∼=45 Å, therefore result-ing in higher bindresult-ing energy. This behavior is evident in Fig. 2(c), which shows the variation of the bind-ing energy as a function of the laser field for several magnetic field values in square and graded QW with

zi=L/2. As expected in this case, zi=L/2, in the

both structures the binding energy increases with laser field up to the value of laser-fieldα0∼=45 Å. Also it

is observed that, due to the asymmetric character of the GQW system binding energy is not degenerate for impurity ground state corresponding to the symmetri-cal positions of impurity. ForB =0, our results ob-tained for the SQW are in good agreement with other theoretical calculations [18]. As a general feature, the

Fig. 3. The variation of the donor binding energy of the ground state for a donor at the center of a square GaAs QW as a function of the well width for different values ofα0, and two different magnetic field values.

shape of the confinement potential has a significant effect on the impurity binding energy for large well widths.

The explicit dependence of the donor impurity binding energy on the quantum well widthLis given in Fig. 3, where we show the variation of the binding energy of a donor located at the center of the SQW as a function of the well width for different values ofα0 and two different magnetic field values. It can

be observed in Fig. 3 that, for α0=0, i.e., no laser

field, as the well width decrease the donor binding energy increases first, reflecting the two-dimensional confinement effect resulting from the squeezing of the QWs walls. But after a critical well width the donor binding energy starts decreasing, reflecting the three-dimensional character of the wave function after it penetrates into the classically forbidden region. Also it is seen that, for large enough well width values (L∼400 Å) the geometric confinement becomes very weak and the binding energy goes to the Rydberg value, and the bulk-like limit is approached. As the laser-field increases the binding energy is less affected by the well width. For example, for α0=40 Å the

variation of the binding energy versus the well width is very small up to the value ofL=350 Å. It should

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Fig. 4. The variation of the donor binding energy of the ground state for a donor at the center of a graded GaAs QW as a function of the well width for different values ofα0, and two different magnetic field values.

be pointed out that, for very large well width values (L >300 Å) the geometric confinement gets to be very weak, i.e., three-dimensional character of the donor electron is dominant, the binding energy is not sensitive to the laser-field. In this range of well width the binding energy curves corresponding to different laser-fields coincide. This feature is a characteristic of the QWs in the presence of the laser-field [17].

In Fig. 4 we present the binding energies of donor placed at the center of the GQW as a function of the well width for several values of laser-field and for magnetic field valuesB=0,10 T. It is seen that in contrast to the results of SQW the donor binding energy of GQW is sensitive to the laser-field even tough at large well widths. We should point out that; in the range of small well widths,L200 Å, the donor binding energy decreases with the laser-field whereas forL200 Å the binding energy increases with the laser-filed. As a general feature in the GQW structure the donor binding energy is sensitive to the laser-field in the whole range of quantum well widths, at least in the range we used.

As a summary, in this Letter we have studied the effect of the high-frequency laser field and impurity position on the binding energy of a hydrogenic donor

placed in a GQW, and SQW heterostructure in the presence of the external magnetic field. The calcu-lations were performed within the effective-mass ap-proximation and by using a variational method. We conclude that, for quantum well structures the laser-field amplitude provides an important effect on the electronic and optical properties and the changes in the binding being dependent on shape of the confinement potential and the position of the impurity. By consid-ering the additional presence of an intense laser field, the above predictions could be observed for instance, in photoluminescence experiments [32–34], and in the optical spectra of donors in QW materials [35]. By in-creasing the laser intensity, the effective well width (dressed width) decreases which in turn increases the spacement of the energy levels in the quantum well so that this mechanism can be used as a way to con-trol the carrier density in these quantum well materi-als. For α0=0, i.e., no laser field, our results agree with other theoretical calculations [7,9]. The method used in this Letter is capable of describing the correct behavior of shallow donor impurities in QWs with dif-ferent shapes in an intense laser-field and an external field applied parallel to the growth axis of the struc-ture.

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