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Intense laser ﬁeld effects on the electron Raman scattering in a strained InGaN/GaN quantum well M.J. Karimi n Department of Physics, College of Sciences, Shiraz University of Technology, Shiraz 71555-313, Iran

H I G H L I G H T S

G R A P H I C A L

Electron Raman scattering (ERS) in a strained InGaN/GaN quantum well is investigated. Impacts of spontaneous and piezoelectric polarization ﬁelds are taken into account. Effects of intense laser ﬁeld, indium composition and well width on the ERS are studied. The peak positions and their values change non-monotonically with laser ﬁeld strength. Peak values increase with the increasing indium composition as well as well width.

The emission spectra in a strained In yGa1 ℬyN/GaN quantum well for different values of the laser dressing Ũ. parameter ƀ 0 . The dotted (solid) curves correspond to ƀ 0 = 0 (ƀ 0 = 10 A)

art ic l e i nf o

a b s t r a c t

Article history: Received 25 June 2014 Received in revised form 22 September 2014 Accepted 23 September 2014 Available online 2 October 2014

The differential cross-section for an intersubband electron Raman scattering process in a strained InGaN/ GaN quantum well in the presence of an intense laser ﬁeld is studied. In the effective-mass approximation, the electronic structure is calculated by taking into account the effects of spontaneous and piezoelectric polarization ﬁelds on the conﬁnement potential. Effects of laser ﬁeld strength, indium composition and the well width on the differential cross-section of the strained quantum well are investigated. Results show that the position and the magnitude of the peaks of emission spectra considerably depend on the laser ﬁeld strength as well as structural parameters. & Elsevier B.V. All rights reserved.

Keywords: Strained quantum well Electron Raman scattering Intense laser ﬁeld

A B S T R A C T

1. Introduction Raman scattering is a highly efﬁcient experimental tools that enables us to investigate the electronic and optical properties of the low-dimensional semiconductor structures. In the past decade, several researchers have studied the differential cross-section (DCS) for the electron Raman scattering (ERS) to explicate the experimental results [1–18]. For example, ERS in asymmetrical

n

Corresponding author. Tel.: þ 98 711 726 1392; fax: þ 98 711 735 4523. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.physe.2014.09.017 1386-9477/& Elsevier B.V. All rights reserved.

multiple quantum wells (QWs) without [9] and with an external electric ﬁeld [10], effect of built-in electric ﬁeld on ERS in InGaN/ GaN coupled QWs [11], ERS in semiconductor quantum wire in an external magnetic ﬁeld [12], ERS in cylindrical quantum wires [13], magnetic ﬁeld effects on the ERS in coaxial cylindrical quantum well wires [14], one phonon-assisted electron Raman scattering in a wurtzite cylindrical quantum well wire [15], magnetic ﬁeld effect on the ERS in a cylindrical quantum dot (QD) [16], effects of hydrogenic impurity and geometrical size on the ERS in single and multilayered spherical QDs [17], combined effects of hydrostatic pressure, temperature and external electric ﬁeld on ERS in a parabolic disc-shaped QD [18] were presented by various authors.

M.J. Karimi / Physica E 66 (2015) 18–23

On the other hand, the wurtzite GaN-based heterostructures have attracted much attention recently due to their potential applications. They are widely used in the fabrication of (I) optoelectronic devices such as high-brightness blue light emitting diodes (LEDs) and laser diodes [19], (II) high power electronic devices such as high-electron mobility transistors (HEMTs) [20,21], hetero-structure bipolar transistors (HBTs) [22,23] and ﬁeld effect transistors (FETs) [24,25]. The III-Nitride semiconductors with wurtzite crystal structure have spontaneous polarization and exhibit much large piezoelectric polarization than the zinc-blende family [26,27]. These polarization ﬁelds can produce a strong internal built-in electric ﬁeld (BEF) and lead to the quantum conﬁned Stark effect in wurtzite GaN-based QWs [19]. Therefore, the spontaneous and strain-induced piezoelectric polarizations can play an important role in electronic and optical properties of wurtzite GaN-based QWs. In recent years, several works have been done both experimentally and theoretically in order to investigate the inﬂuence of BEF on the electronic and optical properties of strained nitride QWs [11,28–39]. For example Ha et al. investigated the intersubband absorption in strained AlGaN/GaN double QWs [28]. Their results indicate that the coupling between two wells can be enhanced by strain modulation. Yarar made a comparative study on the electron mobility in modulation doped AlGaN/GaN and InGaN/GaN QWs [29]. He found that InGaN/GaN QWs have superior mobility characteristics compared to AlGaN/GaN QWs with regard to temperature, doping concentration and spacer length. BEF effect on the linear and nonlinear intersubband optical absorptions in InGaN/AlGaN strained single QWs has been studied by Chi and Shi [30]. They found that the increase of the compositions of In and Al reduces the peak absorption wavelength. Minimala et al. investigated the effect of internal electric and external magnetic ﬁelds on the nonlinear optical properties in a strained wurtzite GaN/AlGaN QD [31]. They show that the optical absorption coefﬁcients and the refractive index changes are strongly dependent on the incident optical intensity, the internal electric ﬁeld and the magnetic ﬁeld. In addition to the BEF, the high-power tunable laser sources, such as free electron lasers, can signiﬁcantly change the potential proﬁle of the QW structures. Recently, the effects of intense laser ﬁeld on the electronic and optical properties of conventional unstrained GaAs QWs have been investigated in several works [8,40–54]. Overall, these studies concluded that the energy levels and intersubband optical transitions in QWs with different shapes can be signiﬁcantly controlled and modiﬁed by an intense laser ﬁeld. To our knowledge, the effect of non-resonant intense laser ﬁeld on the ERS process in the strained wurtzite QWs has not been studied yet. In this work, we employ the numerical methods to obtain the electronic structure and the corresponding wave functions of the system. Then, we will investigate the effects of the intense laser ﬁeld, indium (In) composition and the well width on the ERS of the strained InGaN/GaN QWs, by considering the BEF. The paper is organized as follows: we describe the theoretical framework in Section 2. Then, the results are discussed in Section 3, and ﬁnally, the conclusions are given in Section 4.

2. Theory 2.1. Electronic structure We consider a strained single QW consisting of an In yGa1 ℬyN well with thickness Lw, and two GaN barriers with thickness Lb. The layers have grown on GaN substrate and along the [0 0 0 1]

19

direction (c-axis) denoted as z direction. The origin is taken at the left corner of the QW. In the absence of the laser ﬁeld, the Hamiltonian of the electron in the effective-mass approximation is

H=ℬ

?2 d ⎡ 1 d ⎤ ⎥ + V (z) ⎢ 2 dz ⎣ mύ(z) dz ⎦

(1)

where

⎧ m wύ , 0 ↿ z ↿ L w mύ = ⎨ ύ ⎩ mb , z < 0, z > L w ⎪

⎪

(2)

is the position dependent effective mass of the electron and the conﬁnement potential V(z) is given by

⎧V + eF z, z<0 b ⎪ 0 V (z) = ⎨ eFwz, 0 ↿ z ↿ Lw ⎪ ⎩ eFwL w + eFb(z ℬ L w) + V0, z > L w

(3)

Here, e is the absolute value of the electron charge and V0 is the potential height between In yGa1 ℬyN and GaN, which can be determined by the band-gap discontinuity. Fw and Fb denote the built-in electric ﬁeld in the well and barrier regions, respectively, which are obtained from the following [55,56]:

Fw =

2(Pb ℬ Pw)Lb ƙ0(ƙbL w + 2ƙwLb)

(4)

Fb =

(Pw ℬ Pb)L w ƙ0(ƙbL w + 2ƙwLb)

(5)

where ε0 is the vacuum permittivity, ƙƫ and Pƫ are the dielectric constant and the total polarization of layer ƫ ( = w, b). The total polarization in each layer is written as

Pƫ = Pƫsp + Pƫpz

(6)

where Pƫsp and Pƫpz denote the spontaneous and piezoelectric polarizations, respectively. The piezoelectric polarization is given by [36,55]

Pƫpz = 2

a ℬ aƫ ⎛ c ⎞ ⎜e31 ℬ e33 13 ⎟ aƫ ⎝ c33 ⎠

(7)

where a is the lattice constant of the substrate, aƫ is the lattice constant of the well or barrier, c13, c33 and e31, e33 are the elastic and piezoelectric constants, respectively [26,27]. The theory of interaction of non-resonant laser ﬁeld with an electron conﬁned in a QW has been given in detail in Ref. [40–42] and we will not enter into details here. The time-independent Schrödinger equation in one dimension for an electron inside a strained QW in the presence of non-resonant intense laser ﬁeld is given by

?2 d ⎡ 1 dǰ (z) ⎤ ⎢ ⎥ + Vd(ƀ 0, z)ǰ (z) = Eǰ (z) ℬ 2 dz ⎣ mύ(z) dz ⎦

(8)

where ƀ 0 = eI /mύŵ 2 is the laser-dressing parameter, I and ŵ are the strength and frequency of the laser ﬁeld, respectively. Also, Vd(ƀ 0, z) is the “dressed” conﬁnement potential which depends on I and ŵ only through α0. After some calculations similar to those performed in Ref. [42], the following expression for Vd(ƀ 0, z) is

20

M.J. Karimi / Physica E 66 (2015) 18–23

obtained:

where ez is the unit vector in the z direction: ℸ

Vd(ƀ 0, z) = (V0 + eFbz)ℬ(V0 + eFbzℬeFwz)ƞ[(z ℬ ƀ 0)(ℬz + L w ℬ ƀ 0)] ⎡ (V + eF z ℬ eF z) ⎛ z⎞ b w ℬ⎢ 0 cosℬ1⎜ℬ ⎟ ư ⎝ ƀ0 ⎠ ⎣

Tl(n, k) =

e(Fb ℬ Fw) ⎛ 2 L ℬ z ⎞⎤ ¬ ⎜ ƀ 0 ℬ (z ℬ L w)2 + L wcosℬ1 w ⎟⎥ ư ƀ 0 ⎠⎦ ⎝ (9)

where θ is the step function. We solve Eq. (8) numerically using the fourth-order Runge–Kutta method to ﬁnd energy eigenvalues and wave functions of the system.

The DCS of electron Raman scattering in a volume V per unit solid angle dŵ for the incident light with the frequency ω l and the scattered light with the frequency ω s is given by [10,12]

V 2Ǻ s2ƛ(Ǻ s) d2ƾ W (Ǻ s , es) = dǺ s dŵ 8ư 3c 4ƛ(Ǻ l)

Å Å f

a

ⅈ ℬℸ ǰ n(z) mύ1(z)

Ei = ?Ǻ l + Eni

dǰ k(z)

dz

dz

(14)

and

Ef = ?Ǻ s + Enf

(15)

For the intermediate states |a⊂ and |b⊂ we have two possibilities [10], the electron absorbs the incident photon or the electron emits the secondary photon of energy ℏωs, so we can write

Ea = Ena

and

Eb = ?Ǻ l + ?Ǻ s + Enb

(16)

Å ni, nf

Ũf Ş f2|M0|2 [?Ǻ l ℬ ?Ǻ s + Eni ℬ Enf ]2 + Ş f2

(17)

where

4e4 ƛ(Ǻ s)? ưm02c 4ƛ(Ǻ l)Ş f

|(el ”ez )(es ”ez )|2 ,

(18)

(10)

M0 =

?2 2m 0

Å n

⎡ T (n , n ) T (n , n ) Tl(nf , n)Ts(n, ni ) ⎤ l i ⎥ ⎢ s f ℬ ⎢⎣ ?Ǻ s + En ℬ En + iŞa ?Ǻ l ℬ Enf + En ℬ iŞb ⎥⎦ f

(19)

and [10]

⎧1, ?Ǻ l ℬ ?Ǻ s + En ℬ En ⇀ 0 ⎪ i f Ũf = ⎨ ⎪ ⎩ 0, ?Ǻ l ℬ ?Ǻ s + Eni ℬ Enf < 0

2 ^ ^ ^ ^ ⊁f |Hs|a⊂⊁a|Hl|i⊂ ⊁f |Hl|b⊂⊁b|Hs|i⊂ +Å Ei ℬ Ea + iŞa Ei ℬ Eb + iŞb b

ƕ(Ef ℬ Ei)

⎡ 2 ⎤ ⎢ d ƾ ⎥ = ƾ Ǻs 0 ⎢ dǺ dŵ ⎥ Ǻl ⎦ ⎣ s

ƾ0 =

here, ƛ(Ǻ ) is the frequency dependent refractive index, c is the light velocity in vacuum, es is the polarization vector for the emitted secondary radiation ﬁeld, and W (Ǻs , es) is the transition rate calculated according to Fermi's golden rule [10,12]:

2ư ?

dzTs(n, k)

Using the above-mentioned equations, the following expression for the DCS is obtained:

2.2. Raman cross-section

W (Ǻ s , es) =

dz

In the initial state there is an electron in a conduction subband and an incident radiation photon with energy ℏωl; the ﬁnal state of the process involves an electron in the other conduction subband and a secondary radiation photon with energy ℏωs. Thus, we have the following expressions for the energies of initial and ﬁnal states:

⎡ (V + eF z ℬ eF z) ⎛ z ℬL ⎞ b w w ℬ⎢ 0 cosℬ1⎜ ⎟ ư ⎝ ƀ0 ⎠ ⎣

¬ ƞ(ℬ|z ℬ L w| + ƀ 0) ℬ eL w(Fw ℬ Fb)ƞ(z ℬ L w ℬ ƀ 0).

dǰ k(z)

ℸ

= m0

⎤ e(Fb ℬ Fw) ƀ 02 ℬ z 2 ⎥ƞ(ℬ|z| + ƀ 0) + ư ⎦

ℬ

ⅈ ℬℸ ǰ n(z)

(20)

The emission spectra, (1/ƾ0)(d ƾ /dǺsdŵ ) versus ℏωs, have two types of peaks: (1) Resonant peaks: ?Ǻs = En ℬ Enf . These peaks are shown with 2

(11)

where Ei and Ef are the energies of the initial |i⊂ and ﬁnal |f ⊂ states of the system, respectively, |a⊂ and |b⊂ denote the intermediate states with energies Ea and Eb, respectively, and their correspond^ ^ ing lifetime widths Γa and Γb. Hl and Hs are, respectively, the Hamiltonian operators for interaction with the incident and secondary radiation ﬁelds. Based on the dipole approximation these operators are given by [10,12]

(n, nf ) in the following ﬁgures. (2) Step-like peaks: ?Ǻs = ?Ǻ l + Eni ℬ Enf . These peaks are shown with SL(ni , nf ) in ﬁgures. Also, a singularity can be observed when ?Ǻs = ?Ǻ l . This singularity is shown with SL in ﬁgures.

3. Results and discussion

^ Hl(s) =

e 2ư? (el(s)”p) ml(s) VǺ l(s)

(p = ℬ i?ℛ). (12)

where ml = m0(ms = mύ(z)) is the free electron mass (position dependent effective mass). For the calculation of the DCS, we should evaluate the matrix elements appeared in Eq. (11). Using the Hamiltonian operator for the radiation ﬁeld, Eq. (12), and the wave functions corresponding to the strained QW, the following matrix elements are obtained:

i?e 2ư? ^ ⊁ǰ n|Hl(s)|ǰ k⊂= ℬ (el(s)”ez )Tl(s)(n, k) m0 VǺ l(s)

(13)

The material parameters used in our computation are listed in Table 1. All the parameters of the In yGa1 ℬyN are calculated with the linear interpolation method except energy gap, Eg (In yGa1 ℬyN)= [57]. The conduction yEg (InN) + (1 ℬ y)Eg (GaN) ℬ 1.4y(1 ℬ y) band discontinuity is assumed to be in the form V0= ʁ Ec = 0.75|Eg (GaN) ℬ Eg (In yGa1 ℬyN)| [29]. In this section, the time-independent Schrödinger equation is solved numerically for a strained QW in the presence of laser ﬁeld. Then, we have investigated the effects of the laser ﬁeld, In composition y and the well width on the DCS of the system. We perform our calculations for the case Lb = 2L w , Şa = Şb= Ş f = 0.005 eV (corresponds to a lifetime of about 140 fs) [58] and ?Ǻ l = 1.8 eV .

M.J. Karimi / Physica E 66 (2015) 18–23

21

Table 1 Material parameters used in the calculations. Parameter

GaN

InN

Reference

Eg (eV)

3.43

0.7

[57]

Ũ a (A)

3.189

3.545

[26]

mύ(m0)

0.20

0.11

[26]

0.029

0.032

[27]

0.49 0.73 103 398

0.57 0.97 92 224

[27] [27] [26] [26]

P sp (C/m2) e31 (C/m2) e33 (C/m2) c13 (Gpa) c33 (Gpa)

In Fig. 1, the variation of the conﬁnement potential proﬁle is plotted as a function of the position z for different values of the laser dressing parameter α0 and indium composition y. In this ﬁgure, the minimum point of the potential energy (Vmin) is also shown. From Fig. 1(a), it is seen that the effective width of the well at the bottom (top) decreases (increases) with the laser ﬁeld strength. Also, Vmin shifts upward and hence the conﬁning potential height reduces. For these reasons, the lowest subband is affected strongly by the conﬁnement and therefore the ground state energy E1 increases. The enhancement of the width at the top of the well and the reduction of the potential height eliminate the effects of each other and hence the higher subbands have small variations. The calculations indicate that the ﬁrst excited energy E2 reduces and the second excited energy E3 increases slightly. Fig. 1(b) shows that with the increase of In composition y, the bottom of the QW as well as Vmin shift downward and the potential of the barrier region enhances. The reason for the above result is as follows: as In composition increases, the difference between material parameters of the well and barrier layers increases and leads to the enhancement in the magnitudes of the potential height V0 and built-in electric ﬁelds Fw and Fb. Since Fw is negative, the bottom of the QW shifts to the lower values of the energies. As a result, the ground state energy E1 is shifted down. The calculations reveal that E2 reduces slightly and E3 remains almost constant with the increasing y. The reason for the above result is as follows: the enhancement of the potential height V0 increases the energy eigenvalues. But, the lowering of the value of Vmin leads to decrease in the energy eigenvalues. The effects of the variations of V0 and Vmin on the energy levels E2 and E3 are comparable, and thus they eliminate the effects of each other. Therefore, E2 and E3 have small variations with the increasing y. In Fig. 2, the energy difference between subbands (ʁ E = Eij = Ei ℬ Ej ) is plotted as a function of α0 for y¼0.5 and

α =0

Ũ Fig. 2. The energy difference between subbands of the strained QW with L w = 20 A versus α0. The dashed curves correspond to y¼ 0.5 and the solid curves correspond to y¼ 0.7.

y¼0.7. This ﬁgure shows that E21 and E31 decrease, and E32 increases with the enhancement of α0. It is due to the behaviors of E1, E2 and E3, which are mentioned in the results of Fig. 1(a). From Fig. 2, we can also see that the energy differences between subbands of y ¼0.7 case are larger than those of y¼0.5 case, especially for E21 and E31. This is because the increase of y augments the quantum conﬁnement of the electron and hence the energy difference between subbands will increase. The variations of E31 and E21 with the increase of y are greater than that of E32. This is due to the ground state energy E1 which is shifted down considerably, while E2 and E3 have negligible changes with the increase of y. Note that Eij and Eji have opposite behavior, i.e., E12 and E13 (E32) enhance (reduces) with the laser ﬁeld. Also, E12, E13 and E23 are negative and decrease in the magnitude with the increasing y. Fig. 3 displays the emission spectra in a strained QW Ũ and y¼ 0.5 for two different cases; ƀ = 0 (without with L w = 20 A 0 laser ﬁeld) and ƀ = 10 AŨ. From this ﬁgure, we can see that the 0

position and the value of the peaks depend on the absence or presence of the laser ﬁeld. This is because the laser ﬁeld changes the energy eigenvalues and the corresponding wave functions of the system. The displacement in the peaks position with the variation of the laser ﬁeld can be discussed as follows: with regard

α =0

y = 0.5

0

V(z)

V(z)

α =5A

α = 10 A

y= 0.3 y= 0.5

V

V

y= 0.7

z

z

Fig. 1. The conﬁnement potential proﬁle as a function of the position z for different values of laser dressing parameter α0 (a) and In composition y (b).

22

M.J. Karimi / Physica E 66 (2015) 18–23

Fig. 3. The DCS versus the secondary photon energy (?Ǻ s = hƫ s) in a strained QW Ũ and y ¼0.5. The dotted curves correspond to ƀ = 0 and the solid with L w = 20 A 0 Ũ. curves correspond to ƀ 0 = 10 A

Ũ (dotted curves) and Fig. 5. The emission spectra in a strained QW for L w = 20 A Ũ (solid curves), with ƀ = 5 A Ũ, y¼ 0.5. For more clarity the results of L w = 25 A 0 Ũ are multiplied to 5. L w = 20 A

to the results of previous ﬁgure, for the allowed transition (3, 2), E32 increases with the increasing laser ﬁeld, i.e. the peak suffers a blue-shift. However, for the transition (2, 1), E21decreases with the increase of the laser ﬁeld, i.e. the peak experiences a red-shift. For the SL peaks such as SL(1, 2) and SL(1, 3), ʁ E is negative and decreases in magnitude with the increasing α0, thus these peaks have a blue-shift. But, for the SL(2, 3) peak, ʁ E is negative and increases in magnitude and hence this peak experiences a redshift. From Fig. 3, it is also seen that the peaks (2, 1) and SL(1, 3) of ƀ 0 = 0 case are identical. This is because for this particular case, E21 is equal to (E13 + ?Ǻ l). Ũ In Fig. 4, the emission spectra in a strained QW with L = 20 A

position of the peaks for transitions between ground state and excited states, such as (2, 1) and SL(1, 3), are greater than those of transitions between excited states, such as (3, 2) and SL(2, 3). The reason for this result is described in the results of Fig. 2. Fig. 5 indicates the emission spectra of the strained QW for two different values of the well width. It is seen that the position of the peaks has a small variation with the increase of the well width. However, the magnitude of the peaks increases considerably with the increasing well width. Moreover, due to the decrease of the conﬁnement effects, the increase of the well width leads to the appearance of new peaks in the emission spectra, such as (4, 3) and (4, 2).

w

and ƀ 0 = 5.0 AŨ are plotted for two different values of the In composition y. This ﬁgure shows that the magnitude of the peaks increases with the increasing y. The resonant peaks (2, 1) and (3, 2) experience a blue-shift, while the SL peaks such as SL(1, 2), SL(1, 3) and SL(2, 3) suffer a red-shift. The displacement in the

4. Conclusions We have investigated theoretically the DCS for the intersubband ERS process in a strained InGaN/GaN QW in the presence of the intense laser ﬁeld. The energy eigenvalues and the corresponding wave functions have been obtained numerically using the fourth-order Runge–Kutta method. Then, the results are employed to investigate the effects of the laser ﬁeld, indium composition and the well width on the ERS in a strained QW. The main results of the calculations are as follows: the peak positions and their values change non-monotonically with the application of laser ﬁeld. An increase in the indium composition leads to the blue (red) shift for the resonant (step-like) peaks of the emission spectra. The magnitude of the peaks of the emission spectra increases with the increasing indium composition. Moreover, the increase of the well width causes the appearance of some new levels and leads to the considerable changes in the emission spectra. Finally, we believe that the dependence of the DCS on some factors, such as laser ﬁeld and structural parameters, could be utilized for spectroscopic characterization of the strained quantum well structures.

Acknowledgments Ũ and ƀ = 5 A Ũ. The Fig. 4. The emission spectra of the strained QW with L w = 20 A 0 dotted (solid) curves correspond to y¼ 0.5 (y¼ 0.7).

Financial support from the Shiraz University of Technology research councils is gratefully acknowledged.

M.J. Karimi / Physica E 66 (2015) 18–23

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