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AlxGa1−xAs quantum well wire under tilted laser field
Journal of Luminescence 132 (2012) 1420–1426
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Density of states in a cylindrical GaAs/AlxGa1 xAs quantum well wire under tilted laser field Adrian Radu, Ecaterina Cornelia Niculescu n Department of Physics, ‘‘Politehnica’’ University of Bucharest, 313 Splaiul Independent¸ei, Bucharest, RO 060042, Romania
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 November 2011 Received in revised form 4 January 2012 Accepted 6 January 2012 Available online 21 January 2012
Within the effective-mass approximation the subband electronic levels and density of states in a semiconductor quantum well wire under tilted laser field are investigated. The energies and wave functions are obtained using a finite element method, which accurately takes into account the laserdressed confinement potential. The density of states obtained in a Green’s function formalism is uniformly blueshifted under the laser’s axial field whereas the transverse component induces an additional non-uniform increase of the subband levels. Our results confirm that the tilted laser field destroys the cylindrical symmetry of the quantum confinement potential and breaks down the electronic states’ degeneracy. Axial and transversal effects of the non-resonant laser field on the density of states compete, bringing the attention to a supplementary degree of freedom for controlling the optoelectronic properties: the angle between the polarization direction of the laser and the quantum well wire axis. & 2012 Elsevier B.V. All rights reserved.
Keywords: Quantum well wire Tilted laser field Finite element method Density of states.
1. Introduction By the advent of artificial semiconductor structures with sizes comparable to the de Broglie wavelength of electron, new opportunities arise for investigating the intense laser field (ILF) effects on quantum nanostructures. It is proved that the electronic transport characteristics and optical properties of low-dimensional systems irradiated by ILFs are different from those of a bulk semiconductor, more pronouncedly as the carriers’ confinement is increased by dimensionality reduction [1–14]. Since 1D semiconductor structures are usable for designing ultrafast electronic devices, there is a strong motivation to study their response to intense external fields such as lasers. Some theoretical studies have focused on shallow donor states [13,15–19] and intersubband transitions [20,21] in QWWs dressed by laser fields polarized perpendicularly to the wire axis. By making use of a nonperturbative method [13,21], a significant laser-induced shift of the electronic levels was found, more pronounced for thinner QWWs. For an ILF linearly polarized along the wire axis recent works [22–24] have predicted an axial localization effect, which leads to a transition from one-dimensional to zero-dimensional behavior of the density of states (DoS). However, the restriction on the polarization direction such as being along or perpendicular to the wire axis may not be always experimentally feasible. Moreover, due to the strong optical and magnetic anisotropies
n
Corresponding author. E-mail address: [email protected] (E.C. Niculescu).
0022-2313/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2012.01.020
induced by ILFs [19,21] the question of an optimal interaction setup (i.e. the relative orientation between a QWW and an optical field, which leads to a maximum interaction) becomes technologically important. The main objective of our study is to settle on the dependence of the laser-driven electronic DoS in a cylindrical QWW on the laser’s tilt angle between two limit cases, the first occurring for a laser radiation linearly polarized along the wire, and the last for a polarization in the QWW’s transverse plane. A proper understanding of the DoS is important for controlling the optical properties related to electrons in semiconductor QWWs. Therefore, besides providing a theoretical framework with which to interpret polarization-dependent absorption-emission spectra of QWWs, our results can be used for designing novel electronic devices in which the DoS tuning plays a significant role. The paper is organized as follows. Section 2 describes the theoretical framework. Section 3 presents the influence of the tilted laser field on the energy levels and DoS for electrons confined in a GaAs/AlGaAs cylindrical QWW. The possibility of tuning such quantities by varying the laser polarization direction and frequency was pointed out. Finally, our conclusions are summarized in Section 4.
2. Theory 2.1. Electron states under tilted laser fields The physical system we consider is the nanostructure formed by surrounding a long cylindrical wire made up of direct bandgap
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semiconductor material (GaAs) by a coaxial layer material with a wider bandgap (AlxGa1 xAs) and practically the same lattice constant (Fig. 1). The QWW is considered to interact with a polarized single-mode laser beam, non-resonant with the energy levels of the electrons in the wire. The z-axis is chosen along the wire direction. The quasimonochromatic laser beam is linearly polarized along an s direction, which makes an angle y with the z-axis. The orthogonal transverse axis x and y are chosen so that xC(z,s). The hard-wall confinement potential has the cylindrically symmetric form ( VðrÞ ¼
0, V 0,
r A ½0,RÞ , r A ½R,1Þ
ð1Þ
where V0 is the conduction-band offset and r ¼(x2 þy2)1/2 is the electron’s radial position in the wire. The electrons have a free movement along the wire but in its transverse plane (x,y) the quantum confinement effect becomes important if radius R is comparable to the electron’s Bohr radius in the bulk semiconductor. We will demonstrate that the laser’s transverse component dresses the energy levels associated with the lateral confinement, whereas the axial laser field modifies the electronic energies related to the movement along the wire. If there is no laser irradiation on the QWW, the electron’s energy levels can be calculated by solving the atemporal ¨ Schrodinger equation in cylindrical coordinates " # _2 1 @ @ 1 @2 @2 þ CðrÞ þ VðrÞCðrÞ ¼ ECðrÞ þ 2 r @r 2mn r @r r @j2 @z2
ð2Þ
The wave functions of the electronic subbands may be set into the form
Cp,n, kz ðr, j,zÞ ¼ Fp,n ðr, jÞexpðikz zÞ ¼ Rp,n ðrÞexpðipjÞexpðikz zÞ, ð3Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where p is an integer and kz ¼ 2mn Ez =_ is the continuous wave number in the z direction. Ez is the axial, not quantized, electron’s energy. The radial eigenfunctions Rp,n ðrÞ are analytical solutions of the Bessel equation
r2
n 2 @2 R @R 2 2m þ r þ r EVð r Þ p R¼0 @r @r2 _2
ð4Þ
and have the forms 8 < C ip,n J p ðkip,n rÞ, Rp,n ðrÞ ¼ : C op,n K p ðkop,n rÞ,
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r A ½0,RÞ , r A ½R,1Þ
ð5Þ
where Jp is the p-order Bessel function,p Kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p is the modified Bessel pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi function, kip,n ¼ 2mnEp,n =_ and kop,n ¼ 2mnðV 0 Ep,n Þ=_. Ep,n are the discrete bound-state energy eigenvalues corresponding to the bottoms of 1D subbands. Coefficients C ip,n , C op,n and energies Ep,n may be obtained by matching the wave functions given by Eq. (5) and their derivates at the wire boundary and by wave functions’ normalization. It should be noticed from Eq. (4) that Ep,n ¼ E p,n, which means the states described by eigenfunctions Fp a 0,n(r,j) are double-degenerated. Total subband energies corresponding to the eigenfunctions Cp,n, kz ðr, j,zÞ are given by Ep,n ðkz Þ ¼ Ep,n þEz
ð6Þ
Under the tilted laser field described by the vector potential AðtÞ ¼ ðx^ sin y þ z^ cos yÞA0 sinðotÞ the quantum states can be ¨ obtained from the time-dependent Schrodinger equation in rectangular coordinates " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ^ x ðtÞÞ2 ðpz þ z^ eAz ðtÞÞ2 ðp? þ xeA @Cðr,tÞ , þ þVð x2 þ y2 Þ Cðr,tÞ ¼ i_ @t 2mn 2mn ð7Þ where p? (pz) is the electron momentum perpendicular to (along) the wire axis. When the vector potential has only an x-component (y ¼ p/2) Eq. (7) implies that " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ^ x ðtÞÞ2 ðp? þ xeA @Fðx,y,tÞ ð8Þ þ Vð x2 þ y2 Þ Fðx,y,tÞ ¼ i_ @t 2mn By applying the time-dependent translation x-x þððeA0 sin yÞ= ðmn oÞÞsinðotÞ, one has [25] 2
~ p? ~ ðx,y,tÞ ¼ i_ @F ðx,y,tÞ , ~ ðx,y,tÞ F ð9Þ þ V @t 2mn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where V~ ðx,y,tÞ ¼ Vð ðx þ a0x sinðotÞÞ2 þy2 Þ is the laser-dressed confinement potential with a0x ¼ ððeA0x Þ=ðmn oÞÞ ¼ ððeA0 sin yÞ= ðmn oÞÞ denoting the laser parameter in the x-direction. In the high-frequency limit [26] the laser-dressed bound states for the transversal motion are solutions of the time-independent ¨ Schrodinger equation " # ! _2 @2 @2 ~ ðx,yÞ ¼ E~ F ~ ðx,yÞ, ~ a ðx,yÞ F þ ð10Þ þ V 2mn @x2 @y2 where V~ a ðx,yÞ is the zero-order Fourier component of the dressed potential energy, i.e. its time average over one period Z o 2p=o ~ V~ a ðx:yÞ ¼ V ðx,y,tÞdt ð11Þ 2p 0 For the particular potential function given by Eq. (1), Eq. (11) may be analytically integrated to the closed form qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi < Ref R2 y2 g þ x V 0 2 A Re Yða0x xRe R y2 Þarc cos@ V~ a ðx,yÞ ¼ p : a0x qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 19 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ref R2 y2 gx = 2 2 @ A , R y Þarccos þ Yða0x þ xRe ; a0x
ð12Þ Fig. 1. Schematic view of a cylindrical GaAs/AlGaAs QWW irradiated by an ILF. R is the wire’s radius and y is the angle between s, the polarization direction of the laser and z, the longitudinal axis of the wire.
where Y is the Heaviside unit-step function. This is the first analytical form proposed in the literature for the laser-dressed
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confinement potential of a cylindrical QWW, which allows an accurate calculation of the dressed subband states. By solving Eq. (10) with a finite element method [27–29], one may obtain the laser dressed transverse energies E~ p,n and the ~ p,n ðx,yÞ. The orthogonal states, related transverse eigenfunctions F which degenerate in Fp a 0,n(r,j) for a0x ¼0 still have x and y as axis of symmetry, thus their energies and wave functions will be x ~ x ðx,yÞ and E~ y , F ~ y ðx,yÞ, respectively. denoted by E~ p,n , F p,n p,n p,n ¨ Note that Schrodinger Eq.(8) for the transversal motion is not affected by the presence of an additional axial component Az ðtÞ ¼ A0 cos y sinðotÞ of the vecftor potential; thus, the eigenva~ p,n ðx,yÞ obtained lues E~ p,n and the corresponding eigenfunctions F by solving Eq.(10) remain the same. By writing Cðr,tÞ ¼ ~ p,n ðx,yÞULðz,tÞ [24] in Eq. (7) we obtain two Schrodinger ¨ F equations: one related to a motion of electrons perpendicular to the wire axis, which is identical to Eq. (10), and the other for the parallel motion, that is " # ðpz þ eAz ðtÞÞ2 ~ @Lðz,tÞ ð13Þ þ E p,n Lðz,tÞ ¼ i_ @t 2mn Here L(z,t) depends only parametrically on the vector potential component perpendicular to the wire axis, via the energy E~ p,n . By temporal integration [23,24] of Eq. (13) one finds (
Lðz,tÞ ¼ Lðz,0Þexp i
"
2
_ kz E~ p,n þ þ DEz 2mn 2
!
t ot DEz sinð2otÞ þ 2a0z kz sin2 2_o _ 2
#) ,
ð14Þ where DEz ¼ ((eA0z)2/(4mn))¼mn((a0zo)/(2))2 is the energy blueshift induced by the axial component of the laser field and a0z ¼ ððeA0z Þ=ðmn oÞÞ ¼ ððeA0 cos yÞ=ðmn oÞÞ denotes the laser parameter in the z direction. 2.2. Laser-driven density of states Following the Green’s function approach introduced in [22–24] for quasi-1D electronic systems under ILFs and taking into account Eq. (14), the retarded propagator (or Green’s function) for non-interacting electrons is written as
dp
,p
dn
,n
dk
k
G þ ðp1 ,n1 , kz1 ; p2 ,n2 , kz2 Þ ¼ 1 2 1 2 z1 z2 Yðt 2 t 1 Þ i_
t t ot 2 ot 1 2 1 2 þ 2a0z kz sin sin2 exp i E~ p,n ðkz Þ _ 2 2
DEz ½sin ð2ot 2 Þsin ð2ot 1 Þ 2_o
3. Results and discussion The numerical calculations were performed for a GaAs/Alx˚ The size of Ga1 xAs cylindrical QWW with x¼0.3 and R¼75 A. considered GaAs nanostructure is chosen to be slightly smaller ˚ A number than the electronic Bohr radius in bulk GaAs ( ffi100 A). of characteristic features of energy levels are revealed in this intermediate quantum confinement regime, when the strength of the electron-laser field interaction and quantum confinement effects are comparable. An isotropic effective mass mn ¼(0.0665þ(0.08 þ1/300)x)m0 of the electron throughout the nanostructure was assumed (m0 denoting the free electron mass). The conduction-band offset V0 ¼228 meV was introduced by taking into account the Miller’s rule (V0 ¼60%DEgap E0.7x þ0.2x2 in eV units). The laser parameter ˚ so that the QWW’s is considered to be a constant a0 ¼R¼ 75 A, confinement potential will be significantly modified as the laser polarization angle y varies from 0 to p/2. In the absence of the laser filed, there are four bound states in the QWW (the second and the third being double degenerated): E0,1 ¼37.22 meV, E1,1 ¼93.65 meV, E2,1 ¼164.67 meV and E0,2 ¼187.96 meV. 3.1. Laser-dressed electronic states Fig. 2 shows the time-averaged laser dressed confinement potential given by Eq. (11) for a0x ¼75 A˚ (y ¼901), when the radiation provokes maximum changes on the potential energy. At high energies the effective potential profile seems to be enlarged along the laser polarization direction, while at the bottom of the well the potential has a sharp profile. The mean width of the QWW confinement potential within the y¼0 section, RV 1 defined as ð2=V 0 Þ 0 0 V~ a ðx þ ,0ÞdV, remains equal with 2R. On the other hand, the mean width within the x¼0 section has a diminished value with respect to the no-dressing case, RV 1 ð2=V 0 Þ 0 0 V~ a ð0,y þ ÞdV o2R, because there is no enlargement in the y direction. These observations clearly suggest a laserinduced strong enhancement of the quantum confinement for low-energy states. Fig. 3 illustrates the changes of the transverse probability density for the electron ground state when rotating the laser from axial to transversal polarization. For a0x ¼0 (Fig. 3(a)) one may observe the angular independence characteristic to
ð15Þ
where E~ p,n ðkz Þ ¼ E~ p,n þ DEz þ ðð_2 k2z Þ=ð2mn ÞÞ. The laser-driven DoS per unit length in each 1D subband as a function of electron’s energy can be derived from the imaginary part of the Green’s function Fourier transform [23,24], so that Z Dp,n ðEÞ g ¼ s dp,n ðEÞ ¼ dðEE~ p,n ðkz ÞÞf 0 2 ðkz Þdkz , ð16Þ L 2p P where gs ¼2 is the spin degeneracy factor and f 0 ðkz Þ ¼ i A 2Z J i ððDEz Þ=ð2_oÞÞJ2i ða0z kz Þ, Ji denoting the order i Bessel function. By making use of the Dirac-d function properties, one can obtain [24] the following expression for the overall DoS per unit length dðEÞ ¼
pffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DðEÞ 2mn X YðEE~ p,n DEz Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 0 ð 2mn ðEE~ p,n DEz Þ=_Þ ¼ gs L 2p_ p,n EE~ p,n DEz
ð17Þ Due to the term DEz in the denominator and the Y function, an uniform blueshift of the DoS profile as a whole is expected under an axial laser field.
˚ Fig. 2. Laser-dressed confinement potential of the cylindrical QWW with R¼ 75 A.
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Fig. 3. Cross section of the electron probability density for the ground state, at (a) y ¼0; (b) y ¼ 901.
Fig. 5. Cross section of the electron probability density for the second excited state, at y ¼ 0 (a) and (c) and y ¼901 (b) and (d).
Fig. 4. Cross section of the electron probability density for the first excited state, at y ¼0 (a) and(c) and y ¼901 (b) and (d).
non-degenerate states and a maximum probability density on the wire axis. This on-axis maximum is enhanced for a0x ¼75 A˚ and a significant blueshift of the ground state energy is observed (Fig. 3(b)). The probability density of the electron in the first excited state is presented in Fig. 4. As expected, in the absence of the laser component on the x-axis, there are two orthogonal eigenstates with the same energy E1,1 (Fig. 4(a) and (c)). For both states the probability density has an on-axis minimum and two identical maxima symmetrically positioned with respect to the wire axis. The ILF breaks down the radial symmetry so that two laser dressed orthogonal states with different energies are obtained (Fig. 4(b) and (d)). Fig. 5(a) and (c) shows the density of probability for the second excited state of the electron, also double-degenerated. For both orthogonal states the localization probability is zero on the wire axis and has four equal maximums with equidistant angular separation. Fig. 5(b) and (d) illustrate the laser-induced degeneracy breaking with an obvious difference between the orthogo~ x ðx,yÞ and F ~ y ðx,yÞ. It’s also apparent from nal states F 2,1 2,1 Fig. 5(b) and (d), that the electron wave functions spread in the barrier regions of the dressed QWW. This later feature is a general
Fig. 6. Cross section of the electron probability density for the third excited state, at (a) y ¼ 0; (b) y ¼901.
characteristic of low-dimensional structures subjected to nonresonant laser fields [19]. The upper subband state is not degenerated because its wave function is also angularly independent (Fig. 6(a)). Two radial maxima can be observed on the density of probability image, one in the middle of the wire (on-axis maximum) and the other by the form of a circular ring (near-edge maximum). In high transverse laser fields (Fig. 6(b)), due to the leakage of the wave function into the barrier regions, the probability density is strongly affected, having two symmetrical large peaks outside the wire and another two weaker maxima inside. Fig. 7 offers a better image of the transverse subband levels dependence on the in-plane component of the laser parameter. One may observe that, although all the energy levels are blueshifted, the ground level is the most affected by the field, increasing with more than 60 meV. The effect of the radiation is less pronounced for higher energy levels due to a weaker confinement in the upper part of the dressed QWW potential. It is clear from Fig. 7 that large field intensity is needed to induce appreciable changes of the upper-lying states. Indeed, E2,1 level remains degenerated under relatively high transverse laser
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3.2. Laser-driven density of states Accordingly to previous discussion, we calculated the electron DoS for two different values of the laser frequency. Fig. 9(a) presents the results obtained for o/2p ¼10 THz, at three different orientations of the laser field. For y ¼0 there is only a relatively small and uniform blueshift of all energy levels, induced by the axial laser field, as compared with the unperturbed DoS. In addition, the DoS profile changes from the characteristic shape for an unperturbed QWW to a set of sharp peaks, which is a sign of an increased confinement due to the axial field effect on the electron [23]. For y ¼ p/4 the energy blueshift is the combined effect of both radiation components. The level splitting, which is apparent from this figure results from degeneracy breaking effect produced by the transverse laser component. For y ¼ p/2, only the
Fig. 7. Transverse subband energy levels versus in-plane laser parameter.
Fig. 8. Energy levels blueshift induced by the axial laser filed, as a function of the laser frequency.
parameter, a0x ffiR/2, and for the highest excited state only a slow variation with growth of a0x is observed. The coalescence of the subband energy levels with the increase of the transverse laser parameter is a remarkable aspect, which has been also reported for square QWs under ILFs [11]. Fig. 8 depicts the energy blueshift induced by the axial component of the laser field as a function of the laser frequency. All levels present a parabolic increase while keeping the same energy separation. For relatively small frequency values (o/2p o20 THz), the blueshift DEz brought by an axial orientation of the laser field is observed to be less important than the typical energy levels increase induced by the same laser in the transverse orientation case (Fig. 7). By maintaining the same laser parameter at higher values of o, one may notice the opposite situation.
Fig. 9. Electron DoS versus energy for (a) o/2p ¼10 THz; (b) o/2p ¼25 THz and three different orientations of the laser field. Dotted line is for the DoS in the absence of laser radiation. a,b,c,d represent the ground and the three excited subbands, respectively. Notations x, y stand for different levels originating from the same electronic state through laser degeneracy breaking.
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transverse effect remains, the energy spectrum being non-uniformly blueshifted. Fig. 9(b) presents the electron DoS for o/2p ¼25 THz, at the same three orientations of the laser field. For y ¼0 (axial laser polarization) there is a large uniform blueshift of the energy levels and a strong decrease of the DoS profile height. For y ¼ p/4 the energy blueshift and the level splitting are the combined effects of the axial and transversal laser components. Unlike in Fig. 9(a), the DoS profile is redshifted as compared with y ¼0 case. For y ¼ p/2 (transverse laser polarization), there is an even larger redshift of the DoS, energies being still blueshifted as compared with the unperturbed DoS profile.
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As a more comprehensive analysis, in Fig. 10(a,b) we present the electronic DoS as a function of the energy and the laser polarization direction. High DoS areas are delimited by singularity lines, where d-N. For a relatively small laser frequency (Fig. 10(a)) the singularity lines are blueshifted by the increase of the angle y, starting with a DoS configuration characteristic to the axial laser (y ¼0) and ending up with a configuration typical to the transverse laser effect (y ¼ p/2). For higher values of the laser frequency (Fig. 10(a)) the DoS exhibits a different behavior, the singularity lines being redshifted by the increase of the angle y. One may now observe that the question of optimal geometry for the laser–QWW interaction depends on the electron’s states involved in optical transitions and on the wire size. Since a laser field polarized along the axis of the wire uniformly shifts the subband levels (Fig. 9), the intersubband transitions are not affected. Consequently, geometries with small y values are more likely to yield efficient tuning of the interband transitions, for which there is an uniform and predictable energy blueshift of ððe2 A20z Þ=ð4ÞÞðð1=mn Þ þ ð1=mnhh ÞÞ, where mn (mnhh ) is the electron (heavy-hole) effective masse. Note that this shift in the joint density of states is independent of the quantum wire’s size or shape. One the other hand, at large y values the conduction subband states are strongly and not uniformly affected by the transverse laser field (Fig. 10). This effect depends on the size and shape of the wire being more pronounced at stronger carrier confinement, i.e. for lowest-lying levels and small R values. Therefore, the desired energy range for the intersubband transitions may be obtained by changing the size and/or composition of the heterostructure as well as the polarization direction of the laser field. For GaAs/AlGaAs low-dimensional structures the hole states are less sensitive to the transversal laser action, for two reasons: (i) the mass of the heavy-hole inside the QWW (mnhh ¼ 0:35m0 ) is five times larger than the effective mass of the electron, so that the laser parameter a0x is five times smaller for the valence band; (ii) the offset for the heavy-hole is roughly 1.5 times smaller than the conduction band offset, so that the THz radiation effect is weakened. Consequently, for thin QWWs in which the electron ground-state subband energy is very sensitive to the radiation strength, the angle between the polarization direction of the laser and the wire axis could tune the blueshift of the lowest-energy interband transition (i.e. the bandgap energy).
4. Conclusions
Fig. 10. DoS as a function of the electron energy and the laser field orientation for (a) o/2p ¼10 THz; (b) o/2p ¼25 THz.
A semiconductor QWW under tilted ILF was investigated in the effective mass-approximation using a non-perturbative approach and a finite element method. The originality of this work consists in proposing a new degree of freedom for controlling the electronic DoS in 1D nanostructures and the emergent optical properties: the angle between the polarization direction of the laser and the wire axis. Moreover, an original closed-form expression for the laser dressed potential of a cylindrical semiconductor QWW was proposed. Our results revealed that the effects of the non-resonant laser field on the electron probability density, reflected in the behavior of the energy levels, are very dependent on the laser polarization direction. An axial laser field induces an uniform blueshift of the energy levels, which has a parabolic increase with the laser frequency and is more suitable for interband optical tuning in QWWs. A transverse laser field mostly affects the first conduction subband level being adequate for tuning the intersubband transitions and also the band-gap energy. Any intermediary laser orientation has a combined effect on the energy spectrum and the DoS, depending on the tilt angle and the frequency.
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Acknowledgment One of the authors (A.R.) recognizes financial support from the European Social Fund through POSDRU/89/1.5/S/54785 project: ‘‘Postdoctoral Program for Advanced Research in the field of nanomaterials’’. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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[13] A. Radu, E.C. Niculescu, Phys. Lett. A 374 (15–16) (2010) 1755. [14] C.A. Duque, E. Kasapoglu, S. Sakiroglu, H. Sari, I. Sokmen, Appl. Surf. Sci. 257 (2011) 2313. [15] Qu Fanyao, A.L.A. Fonseca, O.A.C. Nunes, Phys. Rev. B 54 (23) (1996) 16405. [16] H.D. Karki, S. Elagoz, P. Baser, R. Amca, I. Sokmen, Superlattices Microstruct. 41 (2007) 227. [17] A. John Peter, M. Santhi, J. Optoelectron. Adv. M. 11 (5) (2009) 565. [18] M. Santhi, A. John Peter, Eur. Phys. J. B 71 (2009) 225. [19] E.C. Niculescu, A. Radu, Curr. Appl. Phys. 10 (5) (2010) 1354. [20] L.M. Burileanu, A. Radu, Opt. Commun. 284 (7) (2011) 2050. [21] E.C. Niculescu, L.M. Burileanu, A. Radu, A. Lupas-cu, J. Lumin. 131 (6) (2011) 1113. [22] F.M.S. Lima, O.A.C. Nunes, A.L.A. Fonseca, M.A. Amato, E.F. da Silva Jr., Semicond. Sci. Technol. 23 (2008) 125038. [23] F.M.S. Lima, O.A.C. Nunes, A.L.A. Fonseca, M.A. Amato, C.P. Lima, E.F. da Silva Jr., Solid State Commun. 149 (2009) 678. [24] C.P. Lima, F.M.S. Lima, A.L.A. Fonseca, O.A.C. Nunes, New J. Phys. 13 (2011) 073005. [25] H. Kramers, Collected Scientific Papers, North-Holland, Amsterdam, 1956., p. 866. [26] M. Gavrila, J.Z. Kaminski, Phys. Rev. Lett. 52 (1984) 613. [27] G. Strang, G. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, USA, 1973. [28] P.L. George, Automatic Mesh Generation – Application to Finite Element Methods, Wiley, 1991. [29] Y. Saad, Linear Algebra Appl. 34 (1980) 269.
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Journal of Luminescence journal homepage: www.elsevier.com/locate/jlumin
Density of states in a cylindrical GaAs/AlxGa1 xAs quantum well wire under tilted laser field Adrian Radu, Ecaterina Cornelia Niculescu n Department of Physics, ‘‘Politehnica’’ University of Bucharest, 313 Splaiul Independent¸ei, Bucharest, RO 060042, Romania
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 November 2011 Received in revised form 4 January 2012 Accepted 6 January 2012 Available online 21 January 2012
Within the effective-mass approximation the subband electronic levels and density of states in a semiconductor quantum well wire under tilted laser field are investigated. The energies and wave functions are obtained using a finite element method, which accurately takes into account the laserdressed confinement potential. The density of states obtained in a Green’s function formalism is uniformly blueshifted under the laser’s axial field whereas the transverse component induces an additional non-uniform increase of the subband levels. Our results confirm that the tilted laser field destroys the cylindrical symmetry of the quantum confinement potential and breaks down the electronic states’ degeneracy. Axial and transversal effects of the non-resonant laser field on the density of states compete, bringing the attention to a supplementary degree of freedom for controlling the optoelectronic properties: the angle between the polarization direction of the laser and the quantum well wire axis. & 2012 Elsevier B.V. All rights reserved.
Keywords: Quantum well wire Tilted laser field Finite element method Density of states.
1. Introduction By the advent of artificial semiconductor structures with sizes comparable to the de Broglie wavelength of electron, new opportunities arise for investigating the intense laser field (ILF) effects on quantum nanostructures. It is proved that the electronic transport characteristics and optical properties of low-dimensional systems irradiated by ILFs are different from those of a bulk semiconductor, more pronouncedly as the carriers’ confinement is increased by dimensionality reduction [1–14]. Since 1D semiconductor structures are usable for designing ultrafast electronic devices, there is a strong motivation to study their response to intense external fields such as lasers. Some theoretical studies have focused on shallow donor states [13,15–19] and intersubband transitions [20,21] in QWWs dressed by laser fields polarized perpendicularly to the wire axis. By making use of a nonperturbative method [13,21], a significant laser-induced shift of the electronic levels was found, more pronounced for thinner QWWs. For an ILF linearly polarized along the wire axis recent works [22–24] have predicted an axial localization effect, which leads to a transition from one-dimensional to zero-dimensional behavior of the density of states (DoS). However, the restriction on the polarization direction such as being along or perpendicular to the wire axis may not be always experimentally feasible. Moreover, due to the strong optical and magnetic anisotropies
n
Corresponding author. E-mail address: [email protected] (E.C. Niculescu).
0022-2313/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2012.01.020
induced by ILFs [19,21] the question of an optimal interaction setup (i.e. the relative orientation between a QWW and an optical field, which leads to a maximum interaction) becomes technologically important. The main objective of our study is to settle on the dependence of the laser-driven electronic DoS in a cylindrical QWW on the laser’s tilt angle between two limit cases, the first occurring for a laser radiation linearly polarized along the wire, and the last for a polarization in the QWW’s transverse plane. A proper understanding of the DoS is important for controlling the optical properties related to electrons in semiconductor QWWs. Therefore, besides providing a theoretical framework with which to interpret polarization-dependent absorption-emission spectra of QWWs, our results can be used for designing novel electronic devices in which the DoS tuning plays a significant role. The paper is organized as follows. Section 2 describes the theoretical framework. Section 3 presents the influence of the tilted laser field on the energy levels and DoS for electrons confined in a GaAs/AlGaAs cylindrical QWW. The possibility of tuning such quantities by varying the laser polarization direction and frequency was pointed out. Finally, our conclusions are summarized in Section 4.
2. Theory 2.1. Electron states under tilted laser fields The physical system we consider is the nanostructure formed by surrounding a long cylindrical wire made up of direct bandgap
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semiconductor material (GaAs) by a coaxial layer material with a wider bandgap (AlxGa1 xAs) and practically the same lattice constant (Fig. 1). The QWW is considered to interact with a polarized single-mode laser beam, non-resonant with the energy levels of the electrons in the wire. The z-axis is chosen along the wire direction. The quasimonochromatic laser beam is linearly polarized along an s direction, which makes an angle y with the z-axis. The orthogonal transverse axis x and y are chosen so that xC(z,s). The hard-wall confinement potential has the cylindrically symmetric form ( VðrÞ ¼
0, V 0,
r A ½0,RÞ , r A ½R,1Þ
ð1Þ
where V0 is the conduction-band offset and r ¼(x2 þy2)1/2 is the electron’s radial position in the wire. The electrons have a free movement along the wire but in its transverse plane (x,y) the quantum confinement effect becomes important if radius R is comparable to the electron’s Bohr radius in the bulk semiconductor. We will demonstrate that the laser’s transverse component dresses the energy levels associated with the lateral confinement, whereas the axial laser field modifies the electronic energies related to the movement along the wire. If there is no laser irradiation on the QWW, the electron’s energy levels can be calculated by solving the atemporal ¨ Schrodinger equation in cylindrical coordinates " # _2 1 @ @ 1 @2 @2 þ CðrÞ þ VðrÞCðrÞ ¼ ECðrÞ þ 2 r @r 2mn r @r r @j2 @z2
ð2Þ
The wave functions of the electronic subbands may be set into the form
Cp,n, kz ðr, j,zÞ ¼ Fp,n ðr, jÞexpðikz zÞ ¼ Rp,n ðrÞexpðipjÞexpðikz zÞ, ð3Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where p is an integer and kz ¼ 2mn Ez =_ is the continuous wave number in the z direction. Ez is the axial, not quantized, electron’s energy. The radial eigenfunctions Rp,n ðrÞ are analytical solutions of the Bessel equation
r2
n 2 @2 R @R 2 2m þ r þ r EVð r Þ p R¼0 @r @r2 _2
ð4Þ
and have the forms 8 < C ip,n J p ðkip,n rÞ, Rp,n ðrÞ ¼ : C op,n K p ðkop,n rÞ,
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r A ½0,RÞ , r A ½R,1Þ
ð5Þ
where Jp is the p-order Bessel function,p Kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p is the modified Bessel pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi function, kip,n ¼ 2mnEp,n =_ and kop,n ¼ 2mnðV 0 Ep,n Þ=_. Ep,n are the discrete bound-state energy eigenvalues corresponding to the bottoms of 1D subbands. Coefficients C ip,n , C op,n and energies Ep,n may be obtained by matching the wave functions given by Eq. (5) and their derivates at the wire boundary and by wave functions’ normalization. It should be noticed from Eq. (4) that Ep,n ¼ E p,n, which means the states described by eigenfunctions Fp a 0,n(r,j) are double-degenerated. Total subband energies corresponding to the eigenfunctions Cp,n, kz ðr, j,zÞ are given by Ep,n ðkz Þ ¼ Ep,n þEz
ð6Þ
Under the tilted laser field described by the vector potential AðtÞ ¼ ðx^ sin y þ z^ cos yÞA0 sinðotÞ the quantum states can be ¨ obtained from the time-dependent Schrodinger equation in rectangular coordinates " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ^ x ðtÞÞ2 ðpz þ z^ eAz ðtÞÞ2 ðp? þ xeA @Cðr,tÞ , þ þVð x2 þ y2 Þ Cðr,tÞ ¼ i_ @t 2mn 2mn ð7Þ where p? (pz) is the electron momentum perpendicular to (along) the wire axis. When the vector potential has only an x-component (y ¼ p/2) Eq. (7) implies that " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ^ x ðtÞÞ2 ðp? þ xeA @Fðx,y,tÞ ð8Þ þ Vð x2 þ y2 Þ Fðx,y,tÞ ¼ i_ @t 2mn By applying the time-dependent translation x-x þððeA0 sin yÞ= ðmn oÞÞsinðotÞ, one has [25] 2
~ p? ~ ðx,y,tÞ ¼ i_ @F ðx,y,tÞ , ~ ðx,y,tÞ F ð9Þ þ V @t 2mn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where V~ ðx,y,tÞ ¼ Vð ðx þ a0x sinðotÞÞ2 þy2 Þ is the laser-dressed confinement potential with a0x ¼ ððeA0x Þ=ðmn oÞÞ ¼ ððeA0 sin yÞ= ðmn oÞÞ denoting the laser parameter in the x-direction. In the high-frequency limit [26] the laser-dressed bound states for the transversal motion are solutions of the time-independent ¨ Schrodinger equation " # ! _2 @2 @2 ~ ðx,yÞ ¼ E~ F ~ ðx,yÞ, ~ a ðx,yÞ F þ ð10Þ þ V 2mn @x2 @y2 where V~ a ðx,yÞ is the zero-order Fourier component of the dressed potential energy, i.e. its time average over one period Z o 2p=o ~ V~ a ðx:yÞ ¼ V ðx,y,tÞdt ð11Þ 2p 0 For the particular potential function given by Eq. (1), Eq. (11) may be analytically integrated to the closed form qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi < Ref R2 y2 g þ x V 0 2 A Re Yða0x xRe R y2 Þarc cos@ V~ a ðx,yÞ ¼ p : a0x qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 19 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ref R2 y2 gx = 2 2 @ A , R y Þarccos þ Yða0x þ xRe ; a0x
ð12Þ Fig. 1. Schematic view of a cylindrical GaAs/AlGaAs QWW irradiated by an ILF. R is the wire’s radius and y is the angle between s, the polarization direction of the laser and z, the longitudinal axis of the wire.
where Y is the Heaviside unit-step function. This is the first analytical form proposed in the literature for the laser-dressed
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confinement potential of a cylindrical QWW, which allows an accurate calculation of the dressed subband states. By solving Eq. (10) with a finite element method [27–29], one may obtain the laser dressed transverse energies E~ p,n and the ~ p,n ðx,yÞ. The orthogonal states, related transverse eigenfunctions F which degenerate in Fp a 0,n(r,j) for a0x ¼0 still have x and y as axis of symmetry, thus their energies and wave functions will be x ~ x ðx,yÞ and E~ y , F ~ y ðx,yÞ, respectively. denoted by E~ p,n , F p,n p,n p,n ¨ Note that Schrodinger Eq.(8) for the transversal motion is not affected by the presence of an additional axial component Az ðtÞ ¼ A0 cos y sinðotÞ of the vecftor potential; thus, the eigenva~ p,n ðx,yÞ obtained lues E~ p,n and the corresponding eigenfunctions F by solving Eq.(10) remain the same. By writing Cðr,tÞ ¼ ~ p,n ðx,yÞULðz,tÞ [24] in Eq. (7) we obtain two Schrodinger ¨ F equations: one related to a motion of electrons perpendicular to the wire axis, which is identical to Eq. (10), and the other for the parallel motion, that is " # ðpz þ eAz ðtÞÞ2 ~ @Lðz,tÞ ð13Þ þ E p,n Lðz,tÞ ¼ i_ @t 2mn Here L(z,t) depends only parametrically on the vector potential component perpendicular to the wire axis, via the energy E~ p,n . By temporal integration [23,24] of Eq. (13) one finds (
Lðz,tÞ ¼ Lðz,0Þexp i
"
2
_ kz E~ p,n þ þ DEz 2mn 2
!
t ot DEz sinð2otÞ þ 2a0z kz sin2 2_o _ 2
#) ,
ð14Þ where DEz ¼ ((eA0z)2/(4mn))¼mn((a0zo)/(2))2 is the energy blueshift induced by the axial component of the laser field and a0z ¼ ððeA0z Þ=ðmn oÞÞ ¼ ððeA0 cos yÞ=ðmn oÞÞ denotes the laser parameter in the z direction. 2.2. Laser-driven density of states Following the Green’s function approach introduced in [22–24] for quasi-1D electronic systems under ILFs and taking into account Eq. (14), the retarded propagator (or Green’s function) for non-interacting electrons is written as
dp
,p
dn
,n
dk
k
G þ ðp1 ,n1 , kz1 ; p2 ,n2 , kz2 Þ ¼ 1 2 1 2 z1 z2 Yðt 2 t 1 Þ i_
t t ot 2 ot 1 2 1 2 þ 2a0z kz sin sin2 exp i E~ p,n ðkz Þ _ 2 2
DEz ½sin ð2ot 2 Þsin ð2ot 1 Þ 2_o
3. Results and discussion The numerical calculations were performed for a GaAs/Alx˚ The size of Ga1 xAs cylindrical QWW with x¼0.3 and R¼75 A. considered GaAs nanostructure is chosen to be slightly smaller ˚ A number than the electronic Bohr radius in bulk GaAs ( ffi100 A). of characteristic features of energy levels are revealed in this intermediate quantum confinement regime, when the strength of the electron-laser field interaction and quantum confinement effects are comparable. An isotropic effective mass mn ¼(0.0665þ(0.08 þ1/300)x)m0 of the electron throughout the nanostructure was assumed (m0 denoting the free electron mass). The conduction-band offset V0 ¼228 meV was introduced by taking into account the Miller’s rule (V0 ¼60%DEgap E0.7x þ0.2x2 in eV units). The laser parameter ˚ so that the QWW’s is considered to be a constant a0 ¼R¼ 75 A, confinement potential will be significantly modified as the laser polarization angle y varies from 0 to p/2. In the absence of the laser filed, there are four bound states in the QWW (the second and the third being double degenerated): E0,1 ¼37.22 meV, E1,1 ¼93.65 meV, E2,1 ¼164.67 meV and E0,2 ¼187.96 meV. 3.1. Laser-dressed electronic states Fig. 2 shows the time-averaged laser dressed confinement potential given by Eq. (11) for a0x ¼75 A˚ (y ¼901), when the radiation provokes maximum changes on the potential energy. At high energies the effective potential profile seems to be enlarged along the laser polarization direction, while at the bottom of the well the potential has a sharp profile. The mean width of the QWW confinement potential within the y¼0 section, RV 1 defined as ð2=V 0 Þ 0 0 V~ a ðx þ ,0ÞdV, remains equal with 2R. On the other hand, the mean width within the x¼0 section has a diminished value with respect to the no-dressing case, RV 1 ð2=V 0 Þ 0 0 V~ a ð0,y þ ÞdV o2R, because there is no enlargement in the y direction. These observations clearly suggest a laserinduced strong enhancement of the quantum confinement for low-energy states. Fig. 3 illustrates the changes of the transverse probability density for the electron ground state when rotating the laser from axial to transversal polarization. For a0x ¼0 (Fig. 3(a)) one may observe the angular independence characteristic to
ð15Þ
where E~ p,n ðkz Þ ¼ E~ p,n þ DEz þ ðð_2 k2z Þ=ð2mn ÞÞ. The laser-driven DoS per unit length in each 1D subband as a function of electron’s energy can be derived from the imaginary part of the Green’s function Fourier transform [23,24], so that Z Dp,n ðEÞ g ¼ s dp,n ðEÞ ¼ dðEE~ p,n ðkz ÞÞf 0 2 ðkz Þdkz , ð16Þ L 2p P where gs ¼2 is the spin degeneracy factor and f 0 ðkz Þ ¼ i A 2Z J i ððDEz Þ=ð2_oÞÞJ2i ða0z kz Þ, Ji denoting the order i Bessel function. By making use of the Dirac-d function properties, one can obtain [24] the following expression for the overall DoS per unit length dðEÞ ¼
pffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DðEÞ 2mn X YðEE~ p,n DEz Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 0 ð 2mn ðEE~ p,n DEz Þ=_Þ ¼ gs L 2p_ p,n EE~ p,n DEz
ð17Þ Due to the term DEz in the denominator and the Y function, an uniform blueshift of the DoS profile as a whole is expected under an axial laser field.
˚ Fig. 2. Laser-dressed confinement potential of the cylindrical QWW with R¼ 75 A.
A. Radu, E.C. Niculescu / Journal of Luminescence 132 (2012) 1420–1426
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Fig. 3. Cross section of the electron probability density for the ground state, at (a) y ¼0; (b) y ¼ 901.
Fig. 5. Cross section of the electron probability density for the second excited state, at y ¼ 0 (a) and (c) and y ¼901 (b) and (d).
Fig. 4. Cross section of the electron probability density for the first excited state, at y ¼0 (a) and(c) and y ¼901 (b) and (d).
non-degenerate states and a maximum probability density on the wire axis. This on-axis maximum is enhanced for a0x ¼75 A˚ and a significant blueshift of the ground state energy is observed (Fig. 3(b)). The probability density of the electron in the first excited state is presented in Fig. 4. As expected, in the absence of the laser component on the x-axis, there are two orthogonal eigenstates with the same energy E1,1 (Fig. 4(a) and (c)). For both states the probability density has an on-axis minimum and two identical maxima symmetrically positioned with respect to the wire axis. The ILF breaks down the radial symmetry so that two laser dressed orthogonal states with different energies are obtained (Fig. 4(b) and (d)). Fig. 5(a) and (c) shows the density of probability for the second excited state of the electron, also double-degenerated. For both orthogonal states the localization probability is zero on the wire axis and has four equal maximums with equidistant angular separation. Fig. 5(b) and (d) illustrate the laser-induced degeneracy breaking with an obvious difference between the orthogo~ x ðx,yÞ and F ~ y ðx,yÞ. It’s also apparent from nal states F 2,1 2,1 Fig. 5(b) and (d), that the electron wave functions spread in the barrier regions of the dressed QWW. This later feature is a general
Fig. 6. Cross section of the electron probability density for the third excited state, at (a) y ¼ 0; (b) y ¼901.
characteristic of low-dimensional structures subjected to nonresonant laser fields [19]. The upper subband state is not degenerated because its wave function is also angularly independent (Fig. 6(a)). Two radial maxima can be observed on the density of probability image, one in the middle of the wire (on-axis maximum) and the other by the form of a circular ring (near-edge maximum). In high transverse laser fields (Fig. 6(b)), due to the leakage of the wave function into the barrier regions, the probability density is strongly affected, having two symmetrical large peaks outside the wire and another two weaker maxima inside. Fig. 7 offers a better image of the transverse subband levels dependence on the in-plane component of the laser parameter. One may observe that, although all the energy levels are blueshifted, the ground level is the most affected by the field, increasing with more than 60 meV. The effect of the radiation is less pronounced for higher energy levels due to a weaker confinement in the upper part of the dressed QWW potential. It is clear from Fig. 7 that large field intensity is needed to induce appreciable changes of the upper-lying states. Indeed, E2,1 level remains degenerated under relatively high transverse laser
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3.2. Laser-driven density of states Accordingly to previous discussion, we calculated the electron DoS for two different values of the laser frequency. Fig. 9(a) presents the results obtained for o/2p ¼10 THz, at three different orientations of the laser field. For y ¼0 there is only a relatively small and uniform blueshift of all energy levels, induced by the axial laser field, as compared with the unperturbed DoS. In addition, the DoS profile changes from the characteristic shape for an unperturbed QWW to a set of sharp peaks, which is a sign of an increased confinement due to the axial field effect on the electron [23]. For y ¼ p/4 the energy blueshift is the combined effect of both radiation components. The level splitting, which is apparent from this figure results from degeneracy breaking effect produced by the transverse laser component. For y ¼ p/2, only the
Fig. 7. Transverse subband energy levels versus in-plane laser parameter.
Fig. 8. Energy levels blueshift induced by the axial laser filed, as a function of the laser frequency.
parameter, a0x ffiR/2, and for the highest excited state only a slow variation with growth of a0x is observed. The coalescence of the subband energy levels with the increase of the transverse laser parameter is a remarkable aspect, which has been also reported for square QWs under ILFs [11]. Fig. 8 depicts the energy blueshift induced by the axial component of the laser field as a function of the laser frequency. All levels present a parabolic increase while keeping the same energy separation. For relatively small frequency values (o/2p o20 THz), the blueshift DEz brought by an axial orientation of the laser field is observed to be less important than the typical energy levels increase induced by the same laser in the transverse orientation case (Fig. 7). By maintaining the same laser parameter at higher values of o, one may notice the opposite situation.
Fig. 9. Electron DoS versus energy for (a) o/2p ¼10 THz; (b) o/2p ¼25 THz and three different orientations of the laser field. Dotted line is for the DoS in the absence of laser radiation. a,b,c,d represent the ground and the three excited subbands, respectively. Notations x, y stand for different levels originating from the same electronic state through laser degeneracy breaking.
A. Radu, E.C. Niculescu / Journal of Luminescence 132 (2012) 1420–1426
transverse effect remains, the energy spectrum being non-uniformly blueshifted. Fig. 9(b) presents the electron DoS for o/2p ¼25 THz, at the same three orientations of the laser field. For y ¼0 (axial laser polarization) there is a large uniform blueshift of the energy levels and a strong decrease of the DoS profile height. For y ¼ p/4 the energy blueshift and the level splitting are the combined effects of the axial and transversal laser components. Unlike in Fig. 9(a), the DoS profile is redshifted as compared with y ¼0 case. For y ¼ p/2 (transverse laser polarization), there is an even larger redshift of the DoS, energies being still blueshifted as compared with the unperturbed DoS profile.
1425
As a more comprehensive analysis, in Fig. 10(a,b) we present the electronic DoS as a function of the energy and the laser polarization direction. High DoS areas are delimited by singularity lines, where d-N. For a relatively small laser frequency (Fig. 10(a)) the singularity lines are blueshifted by the increase of the angle y, starting with a DoS configuration characteristic to the axial laser (y ¼0) and ending up with a configuration typical to the transverse laser effect (y ¼ p/2). For higher values of the laser frequency (Fig. 10(a)) the DoS exhibits a different behavior, the singularity lines being redshifted by the increase of the angle y. One may now observe that the question of optimal geometry for the laser–QWW interaction depends on the electron’s states involved in optical transitions and on the wire size. Since a laser field polarized along the axis of the wire uniformly shifts the subband levels (Fig. 9), the intersubband transitions are not affected. Consequently, geometries with small y values are more likely to yield efficient tuning of the interband transitions, for which there is an uniform and predictable energy blueshift of ððe2 A20z Þ=ð4ÞÞðð1=mn Þ þ ð1=mnhh ÞÞ, where mn (mnhh ) is the electron (heavy-hole) effective masse. Note that this shift in the joint density of states is independent of the quantum wire’s size or shape. One the other hand, at large y values the conduction subband states are strongly and not uniformly affected by the transverse laser field (Fig. 10). This effect depends on the size and shape of the wire being more pronounced at stronger carrier confinement, i.e. for lowest-lying levels and small R values. Therefore, the desired energy range for the intersubband transitions may be obtained by changing the size and/or composition of the heterostructure as well as the polarization direction of the laser field. For GaAs/AlGaAs low-dimensional structures the hole states are less sensitive to the transversal laser action, for two reasons: (i) the mass of the heavy-hole inside the QWW (mnhh ¼ 0:35m0 ) is five times larger than the effective mass of the electron, so that the laser parameter a0x is five times smaller for the valence band; (ii) the offset for the heavy-hole is roughly 1.5 times smaller than the conduction band offset, so that the THz radiation effect is weakened. Consequently, for thin QWWs in which the electron ground-state subband energy is very sensitive to the radiation strength, the angle between the polarization direction of the laser and the wire axis could tune the blueshift of the lowest-energy interband transition (i.e. the bandgap energy).
4. Conclusions
Fig. 10. DoS as a function of the electron energy and the laser field orientation for (a) o/2p ¼10 THz; (b) o/2p ¼25 THz.
A semiconductor QWW under tilted ILF was investigated in the effective mass-approximation using a non-perturbative approach and a finite element method. The originality of this work consists in proposing a new degree of freedom for controlling the electronic DoS in 1D nanostructures and the emergent optical properties: the angle between the polarization direction of the laser and the wire axis. Moreover, an original closed-form expression for the laser dressed potential of a cylindrical semiconductor QWW was proposed. Our results revealed that the effects of the non-resonant laser field on the electron probability density, reflected in the behavior of the energy levels, are very dependent on the laser polarization direction. An axial laser field induces an uniform blueshift of the energy levels, which has a parabolic increase with the laser frequency and is more suitable for interband optical tuning in QWWs. A transverse laser field mostly affects the first conduction subband level being adequate for tuning the intersubband transitions and also the band-gap energy. Any intermediary laser orientation has a combined effect on the energy spectrum and the DoS, depending on the tilt angle and the frequency.
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Acknowledgment One of the authors (A.R.) recognizes financial support from the European Social Fund through POSDRU/89/1.5/S/54785 project: ‘‘Postdoctoral Program for Advanced Research in the field of nanomaterials’’. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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